diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 18ef7bd..4316b78 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.1","generation_timestamp":"2024-02-26T08:51:57","documenter_version":"1.2.1"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.2","generation_timestamp":"2024-03-28T14:12:51","documenter_version":"1.3.0"}} \ No newline at end of file diff --git a/dev/API/index.html b/dev/API/index.html index 7b43823..8460dcf 100644 --- a/dev/API/index.html +++ b/dev/API/index.html @@ -1,7 +1,7 @@ -API reference · Antique.jl
GitHub

API reference

Antique.DeltaPotential — Type

DeltaPotential(α=1.0, m=1.0, ℏ=1.0)

$\alpha$ is the potential strength, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.HarmonicOscillator — Type

HarmonicOscillator(α=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

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Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D__\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.PoschlTeller — Type

PoschlTeller(lambda=1.0)

$\lambda$ determines the potential strength. This model is defined dimensionless, i.e. $x$ is given in units of a characteristic length $x_0$, and $E$ in units of a characteristic energy, e.g. $E_\mathrm{char} = \frac{\hbar^2}{2 m x_0^2}$.

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Antique.E — Method

E(model::DeltaPotential)

\[E = - \frac{m\alpha^2}{2\hbar^2}\]

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Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

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Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n +API reference · Antique.jl

GitHub

API reference

Antique.DeltaPotential — Type

DeltaPotential(α=1.0, m=1.0, ℏ=1.0)

$\alpha$ is the potential strength, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.HarmonicOscillator — Type

HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D__\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source
Antique.PoschlTeller — Type

PoschlTeller(lambda=1.0)

$\lambda$ determines the potential strength. This model is defined dimensionless, i.e. $x$ is given in units of a characteristic length $x_0$, and $E$ in units of a characteristic energy, e.g. $E_\mathrm{char} = \frac{\hbar^2}{2 m x_0^2}$.

source
Antique.E — Method

E(model::DeltaPotential)

\[E = - \frac{m\alpha^2}{2\hbar^2}\]

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Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source
Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n = -\frac{m_\mathrm{e} e^4 Z^2}{2n^2(4\pi\varepsilon_0)^2\hbar^2} -= -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

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Antique.E — Method

E(model::InfinitePotentialWell; n=1)

\[E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]

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Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

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Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

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Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} += -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source
Antique.E — Method

E(model::InfinitePotentialWell; n=1)

\[E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]

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Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source
Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

source
Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. @@ -17,7 +17,7 @@ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots -\end{aligned}\]

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Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

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Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -40,7 +40,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

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Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source
Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -62,7 +62,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

source
Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source
Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -84,22 +84,22 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

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Antique.P — Method

P(model::PoschlTeller, x; n=0, m=0)

Associated Legendre polynomials are the associated Legendre functions for integer indices. Please note here, that for the Poschl-Teller potential we use a slightly different notation of the associated Legendre functions as compared to the model HydrogenAtom. Here we have an additional factor $(-1)^\m$.

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Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::DeltaPotential, x)

\[V(x) = -\alpha \delta(x).\]

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Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) +\end{aligned}\]

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Antique.P — Method

P(model::PoschlTeller, x; n=0, m=0)

Associated Legendre polynomials are the associated Legendre functions for integer indices. Please note here, that for the Poschl-Teller potential we use a slightly different notation of the associated Legendre functions as compared to the model HydrogenAtom. Here we have an additional factor $(-1)^\m$.

source
Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

source
Antique.V — Method

V(model::DeltaPotential, x)

\[V(x) = -\alpha \delta(x).\]

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Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 x^2 -= \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

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Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} += \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.

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Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = +\end{aligned}\]

The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = \left\{ \begin{array}{ll} \infty & x \lt 0, L \lt x \\ 0 & 0 \leq x \leq L \end{array} -\right.\]

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Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} +\right.\]

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Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

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Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} V(x) &= -\frac{\lambda(\lambda+1)}{2} \mathrm{sech}(x)^2 &= -\frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x)^2}. -\end{aligned}\]

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Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.nmax — Method

nmax(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

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Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

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Antique.ψ — Method

ψ(model::DeltaPotential, x)

\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]

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Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

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Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]

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Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

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Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

source
+\end{aligned}\]

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Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

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Antique.nmax — Method

nmax(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

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Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

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Antique.ψ — Method

ψ(model::DeltaPotential, x)

\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]

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Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

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Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

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Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]

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Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

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Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

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diff --git a/dev/DeltaPotential/index.html b/dev/DeltaPotential/index.html index 4f80dda..0f9691c 100644 --- a/dev/DeltaPotential/index.html +++ b/dev/DeltaPotential/index.html @@ -1,5 +1,5 @@ -Delta Potential · Antique.jl
GitHub

Delta Potential

The Delta potential is one of the simplest models for quantum mechanical system in 1D. It always has one bound state and its wave function has a cusp at the origin.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.DeltaPotential — Type

DeltaPotential(α=1.0, m=1.0, ℏ=1.0)

$\alpha$ is the potential strength, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::DeltaPotential, x)

\[V(x) = -\alpha \delta(x).\]

source

Eigen Values

Antique.E — Method

E(model::DeltaPotential)

\[E = - \frac{m\alpha^2}{2\hbar^2}\]

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Eigen Functions

Antique.ψ — Method

ψ(model::DeltaPotential, x)

\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by DeltaPotential and several parameters α, m and ℏ are set as optional arguments.

using Antique
+Delta Potential · Antique.jl
GitHub
Delta Potential
The Delta potential is one of the simplest models for quantum mechanical system in 1D. It always has one bound state and its wave function has a cusp at the origin.
Definitions
This model is described with the time-independent Schrödinger equation
\[  \hat{H} \psi(x) = E \psi(x),\]
and the Hamiltonian
\[  \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]
Parameters are specified with the following struct.
Parameters
Antique.DeltaPotential — Type
DeltaPotential(α=1.0, m=1.0, ℏ=1.0)
$\alpha$ is the potential strength, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).
source
Potential
Antique.V — Method
V(model::DeltaPotential, x)
\[V(x) = -\alpha \delta(x).\]
source
Eigen Values
Antique.E — Method
E(model::DeltaPotential)
\[E = - \frac{m\alpha^2}{2\hbar^2}\]
source
Eigen Functions
Antique.ψ — Method
ψ(model::DeltaPotential, x)
\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]
source
Usage & Examples
Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by DeltaPotential and several parameters α, m and ℏ are set as optional arguments.
using Antique
 DP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)
Parameters:
julia> DP.α
 1.0
 
@@ -43,4 +43,4 @@
 7.0 | 7.0 | 0.1 |    1.000000000000 |    1.000000000000 ✔
 7.0 | 7.0 | 1.0 |    1.000000000000 |    1.000000000000 ✔
 7.0 | 7.0 | 7.0 |    1.000000000000 |    1.000000000000 ✔
-

+
diff --git a/dev/HarmonicOscillator/index.html b/dev/HarmonicOscillator/index.html index 33ad4c7..7d66c18 100644 --- a/dev/HarmonicOscillator/index.html +++ b/dev/HarmonicOscillator/index.html @@ -1,8 +1,8 @@ -Harmonic Oscillator · Antique.jl
GitHub

Harmonic Oscillator

The harmonic oscillator is the most frequently used model in quantum physics.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.HarmonicOscillator — Type

HarmonicOscillator(α=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) +Harmonic Oscillator · Antique.jl

GitHub

Harmonic Oscillator

The harmonic oscillator is the most frequently used model in quantum physics.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.HarmonicOscillator — Type

HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

$k$ is the force constant, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HarmonicOscillator, x)

\[V(x) = \frac{1}{2} k x^2 = \frac{1}{2} m \omega^2 x^2 -= \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source

Eigen Values

Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

source

Hermite Polynomials

Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} += \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.

source

Eigen Values

Antique.E — Method

E(model::HarmonicOscillator; n=0)

\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]

where $\omega = \sqrt{k/m}$ is the angular frequency.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HarmonicOscillator, x; n=0)

\[\psi_n(x) = A_n H_n(\xi) \exp{\left( -\frac{\xi^2}{2} \right)},\]

where $\omega = \sqrt{k/m}$, $\xi = \sqrt{\frac{m\omega}{\hbar}}x$, $A_n = \sqrt{\frac{1}{n! 2^n} \sqrt{\frac{m\omega}{\pi\hbar}}}$, $H_n(x) = (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2}$ are defined.

source

Hermite Polynomials

Antique.H — Method

H(model::HarmonicOscillator, x; n=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. @@ -18,7 +18,7 @@ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots -\end{aligned}\]

source

Reference

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.

using Antique
+\end{aligned}\]
source

Reference

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.

using Antique
 HO = HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)

Parameters:

julia> HO.k
 1.0
 
@@ -430,4 +430,4 @@
 5.0 |  7 |   16.770509831248 |   16.770492175222 ✔
 5.0 |  8 |   19.006577808748 |   19.006555152416 ✔
 5.0 |  9 |   21.242645786248 |   21.242617504750 ✔
-
+ diff --git a/dev/HydrogenAtom/index.html b/dev/HydrogenAtom/index.html index f800b29..6f06431 100644 --- a/dev/HydrogenAtom/index.html +++ b/dev/HydrogenAtom/index.html @@ -1,12 +1,12 @@ -Hydrogen Atom · Antique.jl
GitHub

Hydrogen Atom

The hydrogen atom is the simplest 2-body Coulomb system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r),\]

where $\mu=\left(\frac{1}{m_\mathrm{e}}+\frac{1}{m_\mathrm{p}}\right)^{-1}$ is the reduced mass of electron $\mathrm{e}$ and proton $\mathrm{p}$. $\mu = m_\mathrm{e}$ holds in the limit $m_\mathrm{p}\rightarrow\infty$. Parameters are specified with the following struct.

Parameters

Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} +Hydrogen Atom · Antique.jl

GitHub

Hydrogen Atom

The hydrogen atom is the simplest 2-body Coulomb system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r),\]

where $\mu=\left(\frac{1}{m_\mathrm{e}}+\frac{1}{m_\mathrm{p}}\right)^{-1}$ is the reduced mass of electron $\mathrm{e}$ and proton $\mathrm{p}$. $\mu = m_\mathrm{e}$ holds in the limit $m_\mathrm{p}\rightarrow\infty$. Parameters are specified with the following struct.

Parameters

Antique.HydrogenAtom — Type

HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)

$Z$ is the atomic number, $m_\mathrm{e}$ is the electron mass, $a_0$is the Bohr radius, $E_\mathrm{h}$ is the Hartree energy and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::HydrogenAtom, r)

\[\begin{aligned} V(r) &= - \frac{Ze^2}{4\pi\varepsilon_0 r} &= - \frac{e^2}{4\pi\varepsilon_0 a_0} \frac{Z}{r/a_0} &= - \frac{Z}{r/a_0} E_\mathrm{h}, -\end{aligned}\]

The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n +\end{aligned}\]

The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::HydrogenAtom; n=1)

\[E_n = -\frac{m_\mathrm{e} e^4 Z^2}{2n^2(4\pi\varepsilon_0)^2\hbar^2} -= -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

source

Associated Laguerre Polynomials

Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} += -\frac{Z^2}{2n^2} E_\mathrm{h},\]

where $E_\mathrm{h}$ is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, $E_\mathrm{h} = 27.211~386~245~988(53)~\mathrm{eV}$ from here.

source

Eigen Functions

Antique.ψ — Method

ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::HydrogenAtom, r; n=1, l=0)

\[R_{nl}(r) = -\sqrt{\frac{(n-l-1)!}{2n(n+l)!} \left(\frac{2Z}{n a_0}\right)^3} \left(\frac{2Zr}{n a_0}\right)^l \exp \left(-\frac{Zr}{n a_0}\right) L_{n+l}^{2l+1} \left(\frac{2Zr}{n a_0}\right),\]

where Laguerre polynomials are defined as $L_n(x) = \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$, and associated Laguerre polynomials are defined as $L_n^{k}(x) = \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x)$. Note that replace $2n(n+l)!$ with $2n[(n+l)!]^3$ if Laguerre polynomials are defined as $L_n(x) = \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right)$. The domain is $0\leq r \lt \infty$.

source

Associated Laguerre Polynomials

Antique.L — Method

L(model::HydrogenAtom, x; n=0, k=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -29,7 +29,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +\end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::HydrogenAtom, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::HydrogenAtom, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -51,7 +51,7 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]

source

References

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.

using Antique
+\end{aligned}\]
source

References

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.

using Antique
 H = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)

Parameters:

julia> H.Z
 1
 
@@ -449,4 +449,4 @@
 0.5 |  1 |   39.478417274195 |   39.478417604357 ✔
 1.0 |  1 |    9.869604318963 |    9.869604401089 ✔
 7.0 |  1 |    0.201420496383 |    0.201420497981 ✔
-
+ diff --git a/dev/InfinitePotentialWell/index.html b/dev/InfinitePotentialWell/index.html index 5a9c2e5..23a1b80 100644 --- a/dev/InfinitePotentialWell/index.html +++ b/dev/InfinitePotentialWell/index.html @@ -1,11 +1,11 @@ -Infinite Potential Well · Antique.jl
GitHub

Infinite Potential Well (Particle in a Box)

The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = +Infinite Potential Well · Antique.jl

GitHub

Infinite Potential Well (Particle in a Box)

The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2m} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified with the following struct.

Parameters

Antique.InfinitePotentialWell — Type

InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

$L$ is the length of the box, $m$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::InfinitePotentialWell; x)

\[V(x) = \left\{ \begin{array}{ll} \infty & x \lt 0, L \lt x \\ 0 & 0 \leq x \leq L \end{array} -\right.\]

source

Eigen Values

Antique.E — Method

E(model::InfinitePotentialWell; n=1)

\[E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]

source

Eigen Functions

Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]

source

Proofs

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.

using Antique
+\right.\]
source

Eigen Values

Antique.E — Method

E(model::InfinitePotentialWell; n=1)

\[E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]

source

Eigen Functions

Antique.ψ — Method

ψ(model::InfinitePotentialWell, x; n=1)

\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]

source

Proofs

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.

using Antique
 IPW = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)

Parameters:

julia> IPW.L
 1.0
 
@@ -372,4 +372,4 @@
 0.5 |  1 |   39.478417274195 |   39.478417604357 ✔
 1.0 |  1 |    9.869604318963 |    9.869604401089 ✔
 7.0 |  1 |    0.201420496383 |    0.201420497981 ✔
-
+ diff --git a/dev/MorsePotential/index.html b/dev/MorsePotential/index.html index 6e8e429..5e14f08 100644 --- a/dev/MorsePotential/index.html +++ b/dev/MorsePotential/index.html @@ -1,5 +1,5 @@ -Morse Potential · Antique.jl
GitHub

Morse Potential

The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(r) = E \psi(r),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r)\]

Parameters are specified with the following struct.

Parameters

Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D__\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source

Number of Bound States

Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

source

Eigen Functions

Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} +Morse Potential · Antique.jl

GitHub

Morse Potential

The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(r) = E \psi(r),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \frac{\mathrm{d}^2}{\mathrm{d}r ^2} + V(r)\]

Parameters are specified with the following struct.

Parameters

Antique.MorsePotential — Type

MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)

$r_\mathrm{e}$ is the equilibrium bond distance, $D__\mathrm{e}$ is the the well depth , $k$ is the force constant, $\mu$ is the reduced mass and $\hbar$ is the reduced Planck constant (Dirac's constant).

source

Potential

Antique.V — Method

V(model::MorsePotential, r)

\[V(r) = D_\mathrm{e} \left( \mathrm{e}^{-2a(r-r_e)} - 2\mathrm{e}^{-a(r-r_e)} \right),\]

where $a = \sqrt{\frac{k}{2Dₑ}}$ is defined. The domain is $0\leq r \lt \infty$.

source

Eigen Values

Antique.E — Method

E(model::MorsePotential; n=0)

\[E_n = - D_\mathrm{e} + \hbar \omega \left( n + \frac{1}{2} \right) - \chi \hbar \omega \left( n + \frac{1}{2} \right)^2,\]

where $\omega = \sqrt{k/µ}$ and $\chi = \frac{\hbar\omega}{4D_\mathrm{e}}$ are defined.

source

Number of Bound States

Antique.nₘₐₓ — Method

nₘₐₓ(model::MorsePotential)

\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]

where $\omega = \sqrt{k/µ}$ is defined.

source

Eigen Functions

Antique.ψ — Method

ψ(model::MorsePotential, r; n=0)

\[\psi_n(r) = N_n z^{\lambda-n-1/2} \mathrm{e}^{-z/2} L_n^{(2\lambda-2n-1)}(\xi),\]

$N_n = \sqrt{\frac{n!(2\lambda-2n-1)a}{\Gamma(2\lambda-n)}}$, $\lambda = \frac{\sqrt{2\mu D_\mathrm{e}}}{a\hbar}$, $a = \sqrt{\frac{k}{2Dₑ}}$, $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$, $\xi := 2\lambda\mathrm{e}^{-a(r-r_e)}$ are defined. The domain is $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Antique.L — Method

L(model::MorsePotential, x; n=0, α=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -21,7 +21,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]

source

References

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.

# Parameters for H₂⁺
+\end{aligned}\]
source

References

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.

# Parameters for H₂⁺
 # https://doi.org/10.1002/slct.202102509
 # https://doi.org/10.5281/zenodo.5047817
 # https://physics.nist.gov/cgi-bin/cuu/Value?mpsme
@@ -837,4 +837,4 @@
 0.1 |  7 |   -0.038630997356 |   -0.038631017157 ✔
 0.1 |  8 |   -0.032413875662 |   -0.032413886246 ✔
 0.1 |  9 |   -0.026741858566 |   -0.026742018376 ✔
-
+ diff --git a/dev/PoschlTeller/index.html b/dev/PoschlTeller/index.html index fef245d..2580463 100644 --- a/dev/PoschlTeller/index.html +++ b/dev/PoschlTeller/index.html @@ -1,9 +1,9 @@ -Pöschl-Teller Potential · Antique.jl
GitHub

Pöschl-Teller Potential

The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter. It is defined for one-dimensional systems.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified within the following struct.

Parameters

Antique.PoschlTeller — Type

PoschlTeller(lambda=1.0)

$\lambda$ determines the potential strength. This model is defined dimensionless, i.e. $x$ is given in units of a characteristic length $x_0$, and $E$ in units of a characteristic energy, e.g. $E_\mathrm{char} = \frac{\hbar^2}{2 m x_0^2}$.

source

Potential

Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} +Pöschl-Teller Potential · Antique.jl

GitHub

Pöschl-Teller Potential

The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter. It is defined for one-dimensional systems.

Definitions

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(x) = E \psi(x),\]

and the Hamiltonian

\[ \hat{H} = - \frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{d}x ^2} + V(x).\]

Parameters are specified within the following struct.

Parameters

Antique.PoschlTeller — Type

PoschlTeller(lambda=1.0)

$\lambda$ determines the potential strength. This model is defined dimensionless, i.e. $x$ is given in units of a characteristic length $x_0$, and $E$ in units of a characteristic energy, e.g. $E_\mathrm{char} = \frac{\hbar^2}{2 m x_0^2}$.

source

Potential

Antique.V — Method

V(model::PoschlTeller, x)

\[\begin{aligned} V(x) &= -\frac{\lambda(\lambda+1)}{2} \mathrm{sech}(x)^2 &= -\frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x)^2}. -\end{aligned}\]

source

Number of Bound States

Antique.nmax — Method

nmax(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

source

Eigen Values

Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

source

Eigen Functions

Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by PoschlTeller and a single parameter lambda. It is assumed to be used within rescaled, dimensionless variables.

using Antique
+\end{aligned}\]
source

Number of Bound States

Antique.nmax — Method

nmax(model::PoschlTeller)

\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]

source

Eigen Values

Antique.E — Method

E(model::PoschlTeller; n=0)

\[E_n = -\frac{\mu^2}{2},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.

source

Eigen Functions

Antique.ψ — Method

ψ(model::PoschlTeller, x; n=0)

\[\psi_n(x) = P_\lambda^{\mu}(\mathrm{tanh}(x)) \sqrt{\mu\frac{\Gamma(\lambda-\mu+1)}{\Gamma(\lambda+\mu+1)}},\]

where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$ and $P_\lambda^{\mu}$ are the associated Legendre functions.

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by PoschlTeller and a single parameter lambda. It is assumed to be used within rescaled, dimensionless variables.

using Antique
 PT = PoschlTeller(lambda=6.0)

Parameters:

julia> PT.lambda
 6.0

Eigen values:

julia> E(PT,n=0)
 -18.0
@@ -153,4 +153,4 @@
  9 |  7 |    0.000000000000 |    0.000000000000 ✔
  9 |  8 |    0.000000000000 |    0.000000000000 ✔
  9 |  9 |    1.000000000000 |    1.000000000000 ✔
-
+ diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js index f531160..c6562b5 100644 --- a/dev/assets/documenter.js +++ b/dev/assets/documenter.js @@ -4,7 +4,6 @@ requirejs.config({ 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', - 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min', 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', @@ -103,9 +102,10 @@ $(document).on("click", ".docstring header", function () { }); }); -$(document).on("click", ".docs-article-toggle-button", function () { +$(document).on("click", ".docs-article-toggle-button", function (event) { let articleToggleTitle = "Expand docstring"; let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; debounce(() => { if (isExpanded) { @@ -116,7 +116,7 @@ $(document).on("click", ".docs-article-toggle-button", function () { isExpanded = false; - $(".docstring section").slideUp(); + $(".docstring section").slideUp(animationSpeed); } else { $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); $(".docstring-article-toggle-button") @@ -127,7 +127,7 @@ $(document).on("click", ".docs-article-toggle-button", function () { articleToggleTitle = "Collapse docstring"; navArticleToggleTitle = "Collapse all docstrings"; - $(".docstring section").slideDown(); + $(".docstring section").slideDown(animationSpeed); } $(this).prop("title", navArticleToggleTitle); @@ -224,224 +224,465 @@ $(document).ready(function () { }) //////////////////////////////////////////////////////////////////////////////// -require(['jquery', 'minisearch'], function($, minisearch) { - -// In general, most search related things will have "search" as a prefix. -// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc +require(['jquery'], function($) { -let results = []; -let timer = undefined; +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); -let data = documenterSearchIndex["docs"].map((x, key) => { - x["id"] = key; // minisearch requires a unique for each object - return x; + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } }); -// list below is the lunr 2.1.3 list minus the intersect with names(Base) -// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) -// ideally we'd just filter the original list but it's not available as a variable -const stopWords = new Set([ - "a", - "able", - "about", - "across", - "after", - "almost", - "also", - "am", - "among", - "an", - "and", - "are", - "as", - "at", - "be", - "because", - "been", - "but", - "by", - "can", - "cannot", - "could", - "dear", - "did", - "does", - "either", - "ever", - "every", - "from", - "got", - "had", - "has", - "have", - "he", - "her", - "hers", - "him", - "his", - "how", - "however", - "i", - "if", - "into", - "it", - "its", - "just", - "least", - "like", - "likely", - "may", - "me", - "might", - "most", - "must", - "my", - "neither", - "no", - "nor", - "not", - "of", - "off", - "often", - "on", - "or", - "other", - "our", - "own", - "rather", - "said", - "say", - "says", - "she", - "should", - "since", - "so", - "some", - "than", - "that", - "the", - "their", - "them", - "then", - "there", - "these", - "they", - "this", - "tis", - "to", - "too", - "twas", - "us", - "wants", - "was", - "we", - "were", - "what", - "when", - "who", - "whom", - "why", - "will", - "would", - "yet", - "you", - "your", -]); - -let index = new minisearch({ - fields: ["title", "text"], // fields to index for full-text search - storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results - processTerm: (term) => { - let word = stopWords.has(term) ? null : term; - if (word) { - // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names - word = word - .replace(/^[^a-zA-Z0-9@!]+/, "") - .replace(/[^a-zA-Z0-9@!]+$/, ""); - } +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { - return word ?? null; - }, - // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification - tokenize: (string) => string.split(/[\s\-\.]+/), - // options which will be applied during the search - searchOptions: { - boost: { title: 100 }, - fuzzy: 2, +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results processTerm: (term) => { let word = stopWords.has(term) ? null : term; if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names word = word .replace(/^[^a-zA-Z0-9@!]+/, "") .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); } return word ?? null; }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification tokenize: (string) => string.split(/[\s\-\.]+/), - }, -}); + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); -index.addAll(data); + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } -let filters = [...new Set(data.map((x) => x.category))]; -var modal_filters = make_modal_body_filters(filters); -var filter_results = []; + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } -$(document).on("keyup", ".documenter-search-input", function (event) { - // Adding a debounce to prevent disruptions from super-speed typing! - debounce(() => update_search(filter_results), 300); + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } }); +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + $(document).on("click", ".search-filter", function () { if ($(this).hasClass("search-filter-selected")) { - $(this).removeClass("search-filter-selected"); + selected_filter = ""; } else { - $(this).addClass("search-filter-selected"); + selected_filter = $(this).text().toLowerCase(); } - // Adding a debounce to prevent disruptions from crazy clicking! - debounce(() => get_filters(), 300); + // This updates search results and toggles classes for UI: + update_search(); }); -/** - * A debounce function, takes a function and an optional timeout in milliseconds - * - * @function callback - * @param {number} timeout - */ -function debounce(callback, timeout = 300) { - clearTimeout(timer); - timer = setTimeout(callback, timeout); -} - /** * Make/Update the search component - * - * @param {string[]} selected_filters */ -function update_search(selected_filters = []) { - let initial_search_body = ` -
Type something to get started!
- `; - +function update_search() { let querystring = $(".documenter-search-input").val(); if (querystring.trim()) { - results = index.search(querystring, { - filter: (result) => { - // Filtering results - if (selected_filters.length === 0) { - return result.score >= 1; - } else { - return ( - result.score >= 1 && selected_filters.includes(result.category) - ); - } - }, - }); + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } let search_result_container = ``; + let modal_filters = make_modal_body_filters(); let search_divider = `
`; if (results.length) { @@ -449,19 +690,23 @@ function update_search(selected_filters = []) { let count = 0; let search_results = ""; - results.forEach(function (result) { - if (result.location) { - // Checking for duplication of results for the same page - if (!links.includes(result.location)) { - search_results += make_search_result(result, querystring); - count++; - } - + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; links.push(result.location); } - }); + } - let result_count = `
${count} result(s)
`; + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; search_result_container = `
@@ -490,125 +735,37 @@ function update_search(selected_filters = []) { $(".search-modal-card-body").html(search_result_container); } else { - filter_results = []; - modal_filters = make_modal_body_filters(filters, filter_results); - if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { $(".search-modal-card-body").addClass("is-justify-content-center"); } - $(".search-modal-card-body").html(initial_search_body); + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); } } /** * Make the modal filter html * - * @param {string[]} filters - * @param {string[]} selected_filters * @returns string */ -function make_modal_body_filters(filters, selected_filters = []) { - let str = ``; - - filters.forEach((val) => { - if (selected_filters.includes(val)) { - str += `${val}`; - } else { - str += `${val}`; - } - }); +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); - let filter_html = ` + return `
Filters: ${str} -
- `; - - return filter_html; -} - -/** - * Make the result component given a minisearch result data object and the value of the search input as queryString. - * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult - * - * @param {object} result - * @param {string} querystring - * @returns string - */ -function make_search_result(result, querystring) { - let search_divider = `
`; - let display_link = - result.location.slice(Math.max(0), Math.min(50, result.location.length)) + - (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div - - if (result.page !== "") { - display_link += ` (${result.page})`; - } - - let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); - let text = - textindex !== null - ? result.text.slice( - Math.max(textindex.index - 100, 0), - Math.min( - textindex.index + querystring.length + 100, - result.text.length - ) - ) - : ""; // cut-off text before and after from the match - - let display_result = text.length - ? "..." + - text.replace( - new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence - '$&' - ) + - "..." - : ""; // highlights the match - - let in_code = false; - if (!["page", "section"].includes(result.category.toLowerCase())) { - in_code = true; - } - - // We encode the full url to escape some special characters which can lead to broken links - let result_div = ` - -
-
${result.title}
-
${result.category}
-
-

- ${display_result} -

-
- ${display_link} -
- - ${search_divider} - `; - - return result_div; -} - -/** - * Get selected filters, remake the filter html and lastly update the search modal - */ -function get_filters() { - let ele = $(".search-filters .search-filter-selected").get(); - filter_results = ele.map((x) => $(x).text().toLowerCase()); - modal_filters = make_modal_body_filters(filters, filter_results); - update_search(filter_results); +
`; } }) @@ -635,103 +792,107 @@ $(document).ready(function () { //////////////////////////////////////////////////////////////////////////////// require(['jquery'], function($) { -let search_modal_header = ` -
-
-

- - - - -

-
-
- -
-
-`; - -let initial_search_body = ` -
Type something to get started!
-`; - -let search_modal_footer = ` - -`; - -$(document.body).append( - ` - -

Future Works

List of quantum-mechanical systems with analytical solutions

Developer's Guide

This is the guideline for adding new models.

  1. First, please add a new issue here. We need to find a reference for the definition and analytical solutions (eigenvalues and eigenfunctions) before the development.
  2. Fork the repository on GitHub.
  3. Clone the forked repository to your local machine by Git.
  4. Add the new model name :ModelName to the models = [...] array in src/Antique.jl. : is required at the beginning.
  5. Add the file src/ModelName.jl with the same name as the model name. The most helpful code examples are harmonic oscillators for one-dimensional systems and hydrogen atoms for three-dimensional systems. We recommend that you copy these.
  6. Write the code in that file. First we need to create a structure struct ModelName with the same name as the model name (The best way is Find & Replace). Create V, E, ψ and other functions. Because the function names conflict, you must always give the structure as an argument. Multi-dispatch avoids conflict. We recommend using Revice.jl while coding. Run include("./developer/revice.jl") on the REPL or use dev.ipynb.
  7. Add test code test/ModelName.jl. At a minimum, it is recommended to check the normalization and the orthogonality of wavefunction using QuadGK.jl. All tests will be executed by executing include("./developer/test.jl"). It will take about 2 minutes to complete.
  8. Add documentation. Add either docs/ModelName.md or docs/jmd/ModelName.jmd (if you have a jmd file, the md file will be automatically generated). Include at least the definition of the Hamiltonian and the analytical solutions (eigenvalues and eigenfunctions).
  9. Add the new model into pages=[...] in docs/make.jl.
  10. Execute include("./developer/docs.jl") to compile. Please check docs/build/*.html in your browser.
  11. Push the code.
  12. Submit a pull request on GitHub.

Acknowledgment

This package was named by @KB-satou and @ultimatile.

+

Future Works

List of quantum-mechanical systems with analytical solutions

Developer's Guide

This is the guideline for adding new models.

  1. First, please add a new issue here. We need to find a reference for the definition and analytical solutions (eigenvalues and eigenfunctions) before the development.
  2. Fork the repository on GitHub.
  3. Clone the forked repository to your local machine by Git.
  4. Add the new model name :ModelName to the models = [...] array in src/Antique.jl. : is required at the beginning.
  5. Add the file src/ModelName.jl with the same name as the model name. The most helpful code examples are harmonic oscillators for one-dimensional systems and hydrogen atoms for three-dimensional systems. We recommend that you copy these.
  6. Write the code in that file. First we need to create a structure struct ModelName with the same name as the model name (The best way is Find & Replace). Create V, E, ψ and other functions. Because the function names conflict, you must always give the structure as an argument. Multi-dispatch avoids conflict. We recommend using Revice.jl while coding. Run include("./developer/revice.jl") on the REPL or use dev.ipynb.
  7. Add test code test/ModelName.jl. At a minimum, it is recommended to check the normalization and the orthogonality of wavefunction using QuadGK.jl. All tests will be executed by executing include("./developer/test.jl"). It will take about 2 minutes to complete.
  8. Add documentation. Add either docs/ModelName.md or docs/jmd/ModelName.jmd (if you have a jmd file, the md file will be automatically generated). Include at least the definition of the Hamiltonian and the analytical solutions (eigenvalues and eigenfunctions).
  9. Add the new model into pages=[...] in docs/make.jl.
  10. Execute include("./developer/docs.jl") to compile. Please check docs/build/*.html in your browser.
  11. Push the code.
  12. Submit a pull request on GitHub.

Acknowledgment

This package was named by @KB-satou and @ultimatile.

diff --git a/dev/objects.inv b/dev/objects.inv new file mode 100644 index 0000000..9086bc6 Binary files /dev/null and b/dev/objects.inv differ diff --git a/dev/search_index.js b/dev/search_index.js index 2f26e43..1cdb8b2 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"CurrentModule = Antique","category":"page"},{"location":"PoschlTeller/#Pöschl-Teller-Potential","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter. It is defined for one-dimensional systems.","category":"page"},{"location":"PoschlTeller/#Definitions","page":"Pöschl-Teller Potential","title":"Definitions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"and the Hamiltonian","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH = - frac12 fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters are specified within the following struct.","category":"page"},{"location":"PoschlTeller/#Parameters","page":"Pöschl-Teller Potential","title":"Parameters","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.PoschlTeller","category":"page"},{"location":"PoschlTeller/#Antique.PoschlTeller-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.PoschlTeller","text":"PoschlTeller(lambda=1.0)\n\nlambda determines the potential strength. This model is defined dimensionless, i.e. x is given in units of a characteristic length x_0, and E in units of a characteristic energy, e.g. E_mathrmchar = frachbar^22 m x_0^2.\n\n\n\n\n\n","category":"type"},{"location":"PoschlTeller/#Potential","page":"Pöschl-Teller Potential","title":"Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.V(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.V-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\n V(x)\n = -fraclambda(lambda+1)2 mathrmsech(x)^2\n = -fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Number-of-Bound-States","page":"Pöschl-Teller Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.nmax(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.nmax-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.nmax","text":"nmax(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Values","page":"Pöschl-Teller Potential","title":"Eigen Values","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.E(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.E-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Functions","page":"Pöschl-Teller Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.ψ(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.ψ-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(x)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Usage-and-Examples","page":"Pöschl-Teller Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by PoschlTeller and a single parameter lambda. It is assumed to be used within rescaled, dimensionless variables.","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using Antique\nPT = PoschlTeller(lambda=6.0)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"julia> PT.lambda\n6.0","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Eigen values:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"julia> E(PT,n=0)\n-18.0\n\njulia> E(PT,n=1)\n-12.5\n\njulia> E(PT,n=2)\n-8.0\n\njulia> E(PT,n=3)\n-4.5\n\njulia> E(PT,n=4)\n-2.0\n\njulia> E(PT,n=5)\n-0.5","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"lambda = 4.0\nPT = PoschlTeller(lambda)\n\nusing Plots\nplot(xlim=(-4,4), ylim=(-11.0,1.0), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:3\n # classical turning point:\n xE = acosh(sqrt(lambda*(lambda+1)/abs(E(PT,n=n))/2))\n # energy\n hline!([E(PT, n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-xE,xE], fill(E(PT, n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> E(PT, n=n) + ψ(PT, x,n=n), lc=n+1, lw=2, label=\"\\$n = $n\\$\")\nend\n# potential\nplot!(x -> V(PT, x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"(Image: )","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"lambda = 4.0\nPT = PoschlTeller(lambda)\n\nusing Plots\nplot(xlim=(-4,4), ylim=(-11.0,1.0), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:3\n # classical turning point:\n xE = acosh(sqrt(lambda*(lambda+1)/abs(E(PT,n=n))/2))\n # energy\n hline!([E(PT, n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-xE,xE], fill(E(PT, n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> E(PT, n=n) + ψ(PT, x,n=n), lc=n+1, lw=2, label=\"\\$n = $n\\$\")\nend\n# potential\nplot!(x -> V(PT, x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"(Image: )","category":"page"},{"location":"PoschlTeller/#Testing","page":"Pöschl-Teller Potential","title":"Testing","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"PoschlTeller/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Pöschl-Teller Potential","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"int psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 0.999999999999 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 6 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 8 | 0 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | -0.000000000000 ✔\n 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"CurrentModule = Antique","category":"page"},{"location":"InfinitePotentialWell/#Infinite-Potential-Well-(Particle-in-a-Box)","page":"Infinite Potential Well","title":"Infinite Potential Well (Particle in a Box)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell/#Definitions","page":"Infinite Potential Well","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"and the Hamiltonian","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters are specified with the following struct.","category":"page"},{"location":"InfinitePotentialWell/#Parameters","page":"Infinite Potential Well","title":"Parameters","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.InfinitePotentialWell","category":"page"},{"location":"InfinitePotentialWell/#Antique.InfinitePotentialWell-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.V(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.V-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell/#Eigen-Values","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.E(::InfinitePotentialWell)","category":"page"},{"location":"InfinitePotentialWell/#Antique.E-Tuple{InfinitePotentialWell}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell/#Eigen-Functions","page":"Infinite Potential Well","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.ψ(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.ψ-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell/#Proofs","page":"Infinite Potential Well","title":"Proofs","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen Functions & Eigen Values\nNormalization","category":"page"},{"location":"InfinitePotentialWell/#Usage-and-Examples","page":"Infinite Potential Well","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Antique\nIPW = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.L\n1.0\n\njulia> IPW.m\n1.0\n\njulia> IPW.ℏ\n1.0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> E(IPW, n=1)\n4.934802200544679\n\njulia> E(IPW, n=2)\n19.739208802178716","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Plots\nplot(xlim=(0,1), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> ψ(IPW, x, n=1), label=\"n=1\", lw=2)\nplot!(x -> ψ(IPW, x, n=2), label=\"n=2\", lw=2)\nplot!(x -> ψ(IPW, x, n=3), label=\"n=3\", lw=2)\nplot!(x -> ψ(IPW, x, n=4), label=\"n=4\", lw=2)\nplot!(x -> ψ(IPW, x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L = 1\nusing Plots\nplot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 1:5\n # energy\n plot!([0,L], fill(E(IPW,n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(0:0.01:L, x->E(IPW,n=n) + 5*ψ(IPW,x,n=n), lc=n, lw=2, label=\"\")\nend\n# potential\nplot!([0,0,L,L], [140,0,0,140], lc=:black, lw=2, label=\"\")","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/#Testing","page":"Infinite Potential Well","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Infinite Potential Well","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 10 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 10 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 10 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 4 | 10 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 10 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 10 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 10 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 8 | 10 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n 9 | 10 | 0.000000000000 | 0.000000000000 ✔\n10 | 1 | 0.000000000000 | 0.000000000000 ✔\n10 | 2 | 0.000000000000 | 0.000000000000 ✔\n10 | 3 | 0.000000000000 | 0.000000000000 ✔\n10 | 4 | 0.000000000000 | 0.000000000000 ✔\n10 | 5 | 0.000000000000 | 0.000000000000 ✔\n10 | 6 | 0.000000000000 | -0.000000000000 ✔\n10 | 7 | 0.000000000000 | -0.000000000000 ✔\n10 | 8 | 0.000000000000 | 0.000000000000 ✔\n10 | 9 | 0.000000000000 | 0.000000000000 ✔\n10 | 10 | 1.000000000000 | 1.000000000000 ✔","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values-2","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n E_n\n = int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | m | ℏ | n | analytical | numerical \n--- | --- | --- | -- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1 | 49.348021579139 | 49.348022005447 ✔\n0.1 | 0.1 | 0.1 | 2 | 197.392081461942 | 197.392088021787 ✔\n0.1 | 0.1 | 0.1 | 3 | 444.132165131018 | 444.132198049021 ✔\n0.1 | 0.1 | 0.1 | 4 | 789.568248175979 | 789.568352087149 ✔\n0.1 | 0.1 | 0.1 | 5 | 1233.700296336187 | 1233.700550136170 ✔\n0.1 | 0.1 | 0.1 | 6 | 1776.528266243334 | 1776.528792196084 ✔\n0.1 | 0.1 | 0.1 | 7 | 2418.052103857080 | 2418.053078266893 ✔\n0.1 | 0.1 | 0.1 | 8 | 3158.271745875927 | 3158.273408348594 ✔\n0.1 | 0.1 | 0.1 | 9 | 3997.187119264267 | 3997.189782441189 ✔\n0.1 | 0.1 | 0.1 | 10 | 4934.798141994514 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 1 | 4934.802157913905 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 2 | 19739.208146194214 | 19739.208802178713 ✔\n0.1 | 0.1 | 1.0 | 3 | 44413.216513101754 | 44413.219804902103 ✔\n0.1 | 0.1 | 1.0 | 4 | 78956.824817597895 | 78956.835208714852 ✔\n0.1 | 0.1 | 1.0 | 5 | 123370.029633618717 | 123370.055013616948 ✔\n0.1 | 0.1 | 1.0 | 6 | 177652.826624333364 | 177652.879219608410 ✔\n0.1 | 0.1 | 1.0 | 7 | 241805.210385707964 | 241805.307826689212 ✔\n0.1 | 0.1 | 1.0 | 8 | 315827.174587592541 | 315827.340834859409 ✔\n0.1 | 0.1 | 1.0 | 9 | 399718.711926426622 | 399718.978244118916 ✔\n0.1 | 0.1 | 1.0 | 10 | 493479.814199451241 | 493480.220054467791 ✔\n0.1 | 1.0 | 0.1 | 1 | 4.934802157914 | 4.934802200545 ✔\n0.1 | 1.0 | 0.1 | 2 | 19.739208146194 | 19.739208802179 ✔\n0.1 | 1.0 | 0.1 | 3 | 44.413216513102 | 44.413219804902 ✔\n0.1 | 1.0 | 0.1 | 4 | 78.956824817598 | 78.956835208715 ✔\n0.1 | 1.0 | 0.1 | 5 | 123.370029633619 | 123.370055013617 ✔\n0.1 | 1.0 | 0.1 | 6 | 177.652826624333 | 177.652879219608 ✔\n0.1 | 1.0 | 0.1 | 7 | 241.805210385708 | 241.805307826689 ✔\n0.1 | 1.0 | 0.1 | 8 | 315.827174587593 | 315.827340834859 ✔\n0.1 | 1.0 | 0.1 | 9 | 399.718711926427 | 399.718978244119 ✔\n0.1 | 1.0 | 0.1 | 10 | 493.479814199451 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 1 | 493.480215791390 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 2 | 1973.920814619422 | 1973.920880217871 ✔\n0.1 | 1.0 | 1.0 | 3 | 4441.321651310176 | 4441.321980490210 ✔\n0.1 | 1.0 | 1.0 | 4 | 7895.682481759791 | 7895.683520871485 ✔\n0.1 | 1.0 | 1.0 | 5 | 12337.002963361872 | 12337.005501361695 ✔\n0.1 | 1.0 | 1.0 | 6 | 17765.282662433339 | 17765.287921960840 ✔\n0.1 | 1.0 | 1.0 | 7 | 24180.521038570794 | 24180.530782668924 ✔\n0.1 | 1.0 | 1.0 | 8 | 31582.717458759253 | 31582.734083485939 ✔\n0.1 | 1.0 | 1.0 | 9 | 39971.871192642662 | 39971.897824411892 ✔\n0.1 | 1.0 | 1.0 | 10 | 49347.981419945128 | 49348.022005446779 ✔\n1.0 | 0.1 | 0.1 | 1 | 0.493480215948 | 0.493480220054 ✔\n1.0 | 0.1 | 0.1 | 2 | 1.973920815419 | 1.973920880218 ✔\n1.0 | 0.1 | 0.1 | 3 | 4.441321651944 | 4.441321980490 ✔\n1.0 | 0.1 | 0.1 | 4 | 7.895682481265 | 7.895683520871 ✔\n1.0 | 0.1 | 0.1 | 5 | 12.337002965030 | 12.337005501362 ✔\n1.0 | 0.1 | 0.1 | 6 | 17.765282661715 | 17.765287921961 ✔\n1.0 | 0.1 | 0.1 | 7 | 24.180521036064 | 24.180530782669 ✔\n1.0 | 0.1 | 0.1 | 8 | 31.582717460023 | 31.582734083486 ✔\n1.0 | 0.1 | 0.1 | 9 | 39.971871195191 | 39.971897824412 ✔\n1.0 | 0.1 | 0.1 | 10 | 49.347981417827 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 1 | 49.348021594816 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 2 | 197.392081541864 | 197.392088021787 ✔\n1.0 | 0.1 | 1.0 | 3 | 444.132165194438 | 444.132198049021 ✔\n1.0 | 0.1 | 1.0 | 4 | 789.568248126463 | 789.568352087149 ✔\n1.0 | 0.1 | 1.0 | 5 | 1233.700296503016 | 1233.700550136170 ✔\n1.0 | 0.1 | 1.0 | 6 | 1776.528266171473 | 1776.528792196084 ✔\n1.0 | 0.1 | 1.0 | 7 | 2418.052103606433 | 2418.053078266892 ✔\n1.0 | 0.1 | 1.0 | 8 | 3158.271746002275 | 3158.273408348594 ✔\n1.0 | 0.1 | 1.0 | 9 | 3997.187119519121 | 3997.189782441190 ✔\n1.0 | 0.1 | 1.0 | 10 | 4934.798141782662 | 4934.802200544679 ✔\n1.0 | 1.0 | 0.1 | 1 | 0.049348021595 | 0.049348022005 ✔\n1.0 | 1.0 | 0.1 | 2 | 0.197392081542 | 0.197392088022 ✔\n1.0 | 1.0 | 0.1 | 3 | 0.444132165194 | 0.444132198049 ✔\n1.0 | 1.0 | 0.1 | 4 | 0.789568248126 | 0.789568352087 ✔\n1.0 | 1.0 | 0.1 | 5 | 1.233700296503 | 1.233700550136 ✔\n1.0 | 1.0 | 0.1 | 6 | 1.776528266171 | 1.776528792196 ✔\n1.0 | 1.0 | 0.1 | 7 | 2.418052103606 | 2.418053078267 ✔\n1.0 | 1.0 | 0.1 | 8 | 3.158271746002 | 3.158273408349 ✔\n1.0 | 1.0 | 0.1 | 9 | 3.997187119519 | 3.997189782441 ✔\n1.0 | 1.0 | 0.1 | 10 | 4.934798141783 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 1 | 4.934802159482 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 2 | 19.739208154186 | 19.739208802179 ✔\n1.0 | 1.0 | 1.0 | 3 | 44.413216519444 | 44.413219804902 ✔\n1.0 | 1.0 | 1.0 | 4 | 78.956824812646 | 78.956835208715 ✔\n1.0 | 1.0 | 1.0 | 5 | 123.370029650302 | 123.370055013617 ✔\n1.0 | 1.0 | 1.0 | 6 | 177.652826617147 | 177.652879219608 ✔\n1.0 | 1.0 | 1.0 | 7 | 241.805210360643 | 241.805307826689 ✔\n1.0 | 1.0 | 1.0 | 8 | 315.827174600228 | 315.827340834859 ✔\n1.0 | 1.0 | 1.0 | 9 | 399.718711951912 | 399.718978244119 ✔\n1.0 | 1.0 | 1.0 | 10 | 493.479814178266 | 493.480220054468 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x","page":"Infinite Potential Well","title":"Expected Value of x","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"for only n=1.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.050000000000 | 0.050000000000 ✔\n0.5 | 1 | 0.250000000000 | 0.250000000000 ✔\n1.0 | 1 | 0.500000000000 | 0.500000000000 ✔\n7.0 | 1 | 3.500000000000 | 3.500000000000 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x2","page":"Infinite Potential Well","title":"Expected Value of x^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.002826727415 | 0.002826727415 ✔\n0.5 | 1 | 0.070668185378 | 0.070668185378 ✔\n1.0 | 1 | 0.282672741512 | 0.282672741512 ✔\n7.0 | 1 | 13.850964334096 | 13.850964334096 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p","page":"Infinite Potential Well","title":"Expected Value of p","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p rangle_n=1\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2fracmathrmd psi(x)mathrmdx Delta x\n + Oleft(Delta x^3right)\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n \n 2fracmathrmd psi(x)mathrmdx Delta x\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n - Oleft(Delta x^3right)\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n - fracOleft(Delta x^3right)2Delta x\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n psi(x+Delta x)\n =\n psi(x)\n + fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\n \n psi(x-Delta x)\n =\n psi(x)\n - fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.000000000003 | 0.000000000000 ✔\n0.5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.0 | 1 | 0.000000000000 | 0.000000000000 ✔\n7.0 | 1 | 0.000000000000 | 0.000000000000 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p2","page":"Infinite Potential Well","title":"Expected Value of p^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p^2 rangle\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 986.960431582781 | 986.960440108936 ✔\n0.5 | 1 | 39.478417274195 | 39.478417604357 ✔\n1.0 | 1 | 9.869604318963 | 9.869604401089 ✔\n7.0 | 1 | 0.201420496383 | 0.201420497981 ✔\n","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"CurrentModule = Antique","category":"page"},{"location":"MorsePotential/#Morse-Potential","page":"Morse Potential","title":"Morse Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.","category":"page"},{"location":"MorsePotential/#Definitions","page":"Morse Potential","title":"Definitions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH psi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"and the Hamiltonian","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"MorsePotential/#Parameters","page":"Morse Potential","title":"Parameters","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.MorsePotential","category":"page"},{"location":"MorsePotential/#Antique.MorsePotential-MorsePotential","page":"Morse Potential","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D__mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.V(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.V-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.E(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.E-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Number-of-Bound-States","page":"Morse Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.nₘₐₓ(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.nₘₐₓ-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.ψ(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.ψ-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.L(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.L-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k left(beginarrayl n+alpha n-k endarrayright) fracx^kk \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \nendaligned\n\nExamples:\n\nbeginaligned\n L_0^(0)(x) = 1 \n L_1^(0)(x) = 1 - x \n L_1^(1)(x) = 2 - x \n L_2^(0)(x) = 1 - 2 x + 12 x^2 \n L_2^(1)(x) = 3 - 3 x + 12 x^2 \n L_2^(2)(x) = 6 - 4 x + 12 x^2 \n L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#References","page":"Morse Potential","title":"References","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"P. M. Morse, Phys. Rev. 34, 57 (1929)\nJ. P. Dahl, M. Springborg, J. Chem. Phys. 88, 4535 (1988). (62), (63)\nW. K. Shao, Y. He, J. Pan, J. Nonlinear Sci. Appl., 9, 5, 3388 (2016). (1.6) \nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.12","category":"page"},{"location":"MorsePotential/#Usage-and-Examples","page":"Morse Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"# Parameters for H₂⁺\n# https://doi.org/10.1002/slct.202102509\n# https://doi.org/10.5281/zenodo.5047817\n# https://physics.nist.gov/cgi-bin/cuu/Value?mpsme\nrₑ = 1.997193319969992120068298141276\nDₑ = - 0.5 - (-0.602634619106539878727562156289)\nk = 2*((-1.1026342144949464615+1/2.00) - (-0.602634619106539878727562156289)) / (2.00 - rₑ)^2\nµ = 1/(1/1836.15267343 + 1/1836.15267343)\nℏ = 1.0\n\nusing Antique\nMP = MorsePotential(rₑ=rₑ, Dₑ=Dₑ, k=k, µ=µ, ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.rₑ\n1.997193319969992\n\njulia> MP.Dₑ\n0.10263461910653993\n\njulia> MP.k\n0.1027265041900817\n\njulia> MP.µ\n918.076336715\n\njulia> MP.ℏ\n1.0","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> E(MP, n=0)\n-0.09741377794418261\n\njulia> E(MP, n=1)\n-0.08738092406760907","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(0.1:0.01:15, r -> V(MP, r), lw=2, label=\"\", xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"r\", ylabel=\"V(r)\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Wave functions:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(xlim=(0,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> ψ(MP, x, n=0), label=\"n=0\", lw=2)\nplot!(x -> ψ(MP, x, n=1), label=\"n=1\", lw=2)\nplot!(x -> ψ(MP, x, n=2), label=\"n=2\", lw=2)\nplot!(x -> ψ(MP, x, n=3), label=\"n=3\", lw=2)\nplot!(x -> ψ(MP, x, n=4), label=\"n=4\", lw=2)\nplot!(x -> ψ(MP, x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve, Energy levels, Comparison with harmonic oscillator:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP = MorsePotential()\nHO = HarmonicOscillator(k=MP.k, m=MP.μ)\nusing Plots\nplot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"\\$r\\$\", ylabel=\"\\$V(r), E_n\\$\", legend=:bottomright, size=(480,400), dpi=300)\nfor n in 0:nₘₐₓ(MP)\n # energy\n EM = E(MP, n=n)\n EH = E(HO, n=n) - MP.Dₑ\n plot!(0.1:0.01:15, r -> EH > V(HO, r-MP.rₑ) - MP.Dₑ ? EH : NaN, lc=\"#BC1C5F\", lw=1, label=\"\")\n plot!(0.1:0.01:15, r -> EM > V(MP, r) ? EM : NaN, lc=\"#578FC7\", lw=1, label=\"\")\nend\n# potential\nplot!(0.1:0.01:15, r -> V(HO, r-MP.rₑ) - MP.Dₑ, lc=\"#BC1C5F\", lw=2, label=\"Harmonic Oscillator\")\nplot!(0.1:0.01:15, r -> V(MP, r), lc=\"#578FC7\", lw=2, label=\"Morse Potential\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"where, the potential of harmonic oscillator is defined as V(r) simeq frac12 k (r - r_mathrme)^2 + V_0.","category":"page"},{"location":"MorsePotential/#Testing","page":"Morse Potential","title":"Testing","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Generalized Laguerre Polynomials L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=0 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_0^(0)(x)\n = e^ - x e^x\n = 1 \n = 1\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(0)(x)\n = fracmathrmdmathrmdx x e^ - x e^x\n = 1 - x \n = 1 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(1)(x)\n = frace^x fracmathrmdmathrmdx x^2 e^ - xx\n = 2 - x \n = 2 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(0)(x)\n = frac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x e^x\n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(1)(x)\n = fracfrac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^xx\n = 3 - 3 x + frac12 x^2 \n = 3 - 3 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(2)(x)\n = fracfrac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^xx^2\n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(0)(x)\n = frac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^x\n = 1 - 3 x + frac32 x^2 - frac16 x^3 \n = 1 - 3 x + frac32 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(1)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx\n = 4 - 6 x + 2 x^2 - frac16 x^3 \n = 4 - 6 x + 2 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(2)(x)\n = fracfrac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - x e^xx^2\n = 10 - 10 x + frac52 x^2 - frac16 x^3 \n = 10 - 10 x + frac52 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(3)(x)\n = fracfrac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - x e^xx^3\n = 20 - 15 x + 3 x^2 - frac16 x^3 \n = 20 - 15 x + 3 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(0)(x)\n = frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4 \n = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(1)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx\n = 5 - 10 x + 5 x^2 - frac56 x^3 + frac124 x^4 \n = 5 - 10 x + 5 x^2 - frac56 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(2)(x)\n = fracfrac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - x e^xx^2\n = 15 - 20 x + frac152 x^2 - x^3 + frac124 x^4 \n = 15 - 20 x + frac152 x^2 - x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(3)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^7 e^ - xx^3\n = 35 - 35 x + frac212 x^2 - frac76 x^3 + frac124 x^4 \n = 35 - 35 x + frac212 x^2 - frac76 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=4 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(4)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^8 e^ - xx^4\n = 70 - 56 x + 14 x^2 - frac43 x^3 + frac124 x^4 \n = 70 - 56 x + 14 x^2 - frac43 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Normalization & Orthogonality of L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty L_i^(alpha)(x) L_j^(alpha)(x) x^alpha mathrme^-x mathrmdx = fracGamma(n+alpha+1)n delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" α | i | j | analytical | numerical \n---- | -- | -- | ----------------- | ----------------- \n0.01 | 0 | 0 | 0.994325851192 | 0.994325852936 ✔\n0.01 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 1 | 1.004269109703 | 1.004269111483 ✔\n0.01 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 2 | 1.009290455252 | 1.009290456144 ✔\n0.01 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 2 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 3 | 3 | 1.012654756769 | 1.012654758579 ✔\n0.01 | 3 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 7 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 3 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 4 | 4 | 1.015186393661 | 1.015186394564 ✔\n0.01 | 4 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 4 | 6 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 4 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 5 | 5 | 1.017216766449 | 1.017216768275 ✔\n0.01 | 5 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 6 | 4 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 6 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 6 | 1.018912127726 | 1.018912128636 ✔\n0.01 | 6 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 7 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 7 | 3 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 7 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 7 | 1.020367716480 | 1.020367717392 ✔\n0.01 | 7 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 8 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 8 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 8 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 8 | 1.021643176126 | 1.021643177967 ✔\n0.01 | 8 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 9 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 9 | 2 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 9 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 5 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 9 | 1.022778335210 | 1.022778336127 ✔\n0.05 | 0 | 0 | 0.973504265563 | 0.973504267703 ✔\n0.05 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 9 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 1 | 1.022179478841 | 1.022179479980 ✔\n0.05 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 1 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 8 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 1 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 2 | 1.047733965812 | 1.047733966390 ✔\n0.05 | 2 | 3 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 6 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 3 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 3 | 3 | 1.065196198575 | 1.065196199813 ✔\n0.05 | 3 | 4 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 8 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 3 | 9 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 4 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 4 | 4 | 1.078511151058 | 1.078511152326 ✔\n0.05 | 4 | 5 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 7 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 8 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 5 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 5 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 5 | 5 | 1.089296262568 | 1.089296263862 ✔\n0.05 | 5 | 6 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 5 | 7 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 5 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 5 | 9 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 6 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 6 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 5 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 6 | 6 | 1.098373731423 | 1.098373732739 ✔\n0.05 | 6 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 9 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 7 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 7 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 7 | 1.106219258076 | 1.106219258720 ✔\n0.05 | 7 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 8 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 8 | 1 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 8 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 8 | 3 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 8 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 8 | 5 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 8 | 1.113133128439 | 1.113133129790 ✔\n0.05 | 8 | 9 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 9 | 1 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 9 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 9 | 3 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 9 | 4 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 9 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 9 | 7 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 8 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 9 | 1.119317201375 | 1.119317202034 ✔\n0.10 | 0 | 0 | 0.951350769867 | 0.951350771636 ✔\n0.10 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 1 | 1 | 1.046485846854 | 1.046485847852 ✔\n0.10 | 1 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 2 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 2 | 1.098810139196 | 1.098810140297 ✔\n0.10 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 8 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 3 | 1.135437143836 | 1.135437145012 ✔\n0.10 | 3 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 3 | 6 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 8 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 3 | 9 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 4 | 1.163823072432 | 1.163823073667 ✔\n0.10 | 4 | 5 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 7 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 4 | 9 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 5 | 5 | 1.187099533881 | 1.187099535166 ✔\n0.10 | 5 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 5 | 7 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 5 | 9 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 6 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 6 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 5 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 6 | 1.206884526112 | 1.206884527440 ✔\n0.10 | 6 | 7 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 9 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 7 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 7 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 7 | 5 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 7 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 7 | 7 | 1.224125733628 | 1.224125734265 ✔\n0.10 | 7 | 8 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 7 | 9 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 8 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 8 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 8 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 8 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 7 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 8 | 8 | 1.239427305298 | 1.239427306699 ✔\n0.10 | 8 | 9 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 9 | 1 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 9 | 4 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 9 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 9 | 6 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 9 | 7 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 9 | 8 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 9 | 1.253198719802 | 1.253198721234 ✔\n0.50 | 0 | 0 | 0.886226925453 | 0.886226925863 ✔\n0.50 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 1 | 1.329340388179 | 1.329340389103 ✔\n0.50 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 2 | 1.661675485224 | 1.661675485734 ✔\n0.50 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 3 | 1.938621399428 | 1.938621400123 ✔\n0.50 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 4 | 2.180949074356 | 2.180949075236 ✔\n0.50 | 4 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 5 | 2.399043981792 | 2.399043982856 ✔\n0.50 | 5 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 6 | 2.598964313608 | 2.598964314050 ✔\n0.50 | 6 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 7 | 7 | 2.784604621723 | 2.784604622230 ✔\n0.50 | 7 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 8 | 8 | 2.958642410581 | 2.958642412199 ✔\n0.50 | 8 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 9 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 9 | 7 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 8 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 9 | 3.123011433391 | 3.123011435194 ✔\n1.00 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n1.00 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 1 | 2.000000000000 | 2.000000000000 ✔\n1.00 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 2 | 3.000000000000 | 3.000000000000 ✔\n1.00 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 3 | 4.000000000000 | 4.000000000000 ✔\n1.00 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 9 | 0.000000000000 | -0.000000000001 ✔\n1.00 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 4 | 5.000000000000 | 4.999999999999 ✔\n1.00 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 5 | 6.000000000000 | 6.000000000000 ✔\n1.00 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 6 | 7.000000000000 | 7.000000000000 ✔\n1.00 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 7 | 8.000000000000 | 8.000000000000 ✔\n1.00 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 8 | 9.000000000000 | 9.000000000000 ✔\n1.00 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 3 | 0.000000000000 | -0.000000000001 ✔\n1.00 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 9 | 10.000000000000 | 10.000000000002 ✔","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-\\psi_n(r)","page":"Morse Potential","title":"Normalization & Orthogonality of psi_n(r)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty psi_i^ast(r) psi_j(r) mathrmdr = delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n 0 | 8 | 0.000000000000 | -0.000000000026 ✔\n 0 | 9 | 0.000000000000 | -0.000000000104 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | 0.000000000001 ✔\n 1 | 8 | 0.000000000000 | -0.000000000022 ✔\n 1 | 9 | 0.000000000000 | -0.000000000067 ✔\n 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000009 ✔\n 2 | 9 | 0.000000000000 | -0.000000000030 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000001 ✔\n 3 | 8 | 0.000000000000 | -0.000000000002 ✔\n 3 | 9 | 0.000000000000 | -0.000000000006 ✔\n 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000001 ✔\n 4 | 9 | 0.000000000000 | 0.000000000001 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000001 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | -0.000000000001 ✔\n 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000002 ✔\n 6 | 9 | 0.000000000000 | -0.000000000003 ✔\n 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n 7 | 1 | 0.000000000000 | 0.000000000001 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000001 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000001 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000004 ✔\n 8 | 0 | 0.000000000000 | -0.000000000026 ✔\n 8 | 1 | 0.000000000000 | -0.000000000022 ✔\n 8 | 2 | 0.000000000000 | -0.000000000009 ✔\n 8 | 3 | 0.000000000000 | -0.000000000002 ✔\n 8 | 4 | 0.000000000000 | -0.000000000001 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000002 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 0.999999999995 ✔\n 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 9 | 0 | 0.000000000000 | -0.000000000104 ✔\n 9 | 1 | 0.000000000000 | -0.000000000067 ✔\n 9 | 2 | 0.000000000000 | -0.000000000030 ✔\n 9 | 3 | 0.000000000000 | -0.000000000006 ✔\n 9 | 4 | 0.000000000000 | 0.000000000001 ✔\n 9 | 5 | 0.000000000000 | -0.000000000001 ✔\n 9 | 6 | 0.000000000000 | -0.000000000003 ✔\n 9 | 7 | 0.000000000000 | 0.000000000004 ✔\n 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000015 ✔","category":"page"},{"location":"MorsePotential/#Eigen-Values-2","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n E_n\n = int psi^ast_n(r) hatH psi_n(r) mathrmdx \n = int psi^ast_n(r) left hatV + hatT right psi(r) mathrmdx \n = int psi^ast_n(r) left V(r) - frachbar^22m fracmathrmd^2mathrmd r^2 right psi(r) mathrmdx \n simeq int psi^ast_n(r) left V(r)psi(r) -frachbar^22m fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2 right mathrmdx\n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n 2psi(r)\n + fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n + Oleft(Delta r^4right)\n =\n psi(r+Delta r)\n + psi(r-Delta r)\n \n fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n =\n psi(r+Delta r)\n - 2psi(r)\n + psi(r-Delta r)\n - Oleft(Delta r^4right)\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n - fracOleft(Delta r^4right)Delta r^2\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n + Oleft(Delta r^2right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\npsi(r+Delta r)\n= psi(r)\n+ fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n+ frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\n\npsi(r-Delta r)\n= psi(r)\n- fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n- frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | -0.097482629904 | -0.097482629943 ✔\n0.1 | 1 | -0.087576629073 | -0.087576629208 ✔\n0.1 | 2 | -0.078201265005 | -0.078201265359 ✔\n0.1 | 3 | -0.069356537702 | -0.069356538266 ✔\n0.1 | 4 | -0.061042447162 | -0.061042448777 ✔\n0.1 | 5 | -0.053258993386 | -0.053258996131 ✔\n0.1 | 6 | -0.046006176374 | -0.046006177829 ✔\n0.1 | 7 | -0.039283996126 | -0.039283997743 ✔\n0.1 | 8 | -0.033092452642 | -0.033092467851 ✔\n0.1 | 9 | -0.027431545922 | -0.027431467792 ✔\n0.2 | 0 | -0.095387461081 | -0.095387461144 ✔\n0.2 | 1 | -0.081689100176 | -0.081689100427 ✔\n0.2 | 2 | -0.069052012799 | -0.069052013380 ✔\n0.2 | 3 | -0.057476198949 | -0.057476199867 ✔\n0.2 | 4 | -0.046961658628 | -0.046961660317 ✔\n0.2 | 5 | -0.037508391834 | -0.037508393202 ✔\n0.2 | 6 | -0.029116398568 | -0.029116400340 ✔\n0.2 | 7 | -0.021785678830 | -0.021785684062 ✔\n0.2 | 8 | -0.015516232619 | -0.015516237539 ✔\n0.2 | 9 | -0.010308059937 | -0.010308062755 ✔\n0.3 | 0 | -0.093795214605 | -0.093795214695 ✔\n0.3 | 1 | -0.077310338322 | -0.077310338694 ✔\n0.3 | 2 | -0.062417372330 | -0.062417373167 ✔\n0.3 | 3 | -0.049116316630 | -0.049116318029 ✔\n0.3 | 4 | -0.037407171221 | -0.037407173073 ✔\n0.3 | 5 | -0.027289936105 | -0.027289938027 ✔\n0.3 | 6 | -0.018764611280 | -0.018764613693 ✔\n0.3 | 7 | -0.011831196747 | -0.011831198102 ✔\n0.3 | 8 | -0.006489692505 | -0.006489694275 ✔\n0.3 | 9 | -0.002740098556 | -0.002740100893 ✔\n0.1 | 0 | -0.097413777944 | -0.097413777967 ✔\n0.1 | 1 | -0.087380924068 | -0.087380924205 ✔\n0.1 | 2 | -0.077893174789 | -0.077893175145 ✔\n0.1 | 3 | -0.068950530107 | -0.068950530660 ✔\n0.1 | 4 | -0.060552990023 | -0.060552989095 ✔\n0.1 | 5 | -0.052700554537 | -0.052700557255 ✔\n0.1 | 6 | -0.045393223648 | -0.045393222818 ✔\n0.1 | 7 | -0.038630997356 | -0.038631017157 ✔\n0.1 | 8 | -0.032413875662 | -0.032413886246 ✔\n0.1 | 9 | -0.026741858566 | -0.026742018376 ✔\n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"CurrentModule = Antique","category":"page"},{"location":"HydrogenAtom/#Hydrogen-Atom","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The hydrogen atom is the simplest 2-body Coulomb system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"and the Hamiltonian","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"where mu=left(frac1m_mathrme+frac1m_mathrmpright)^-1 is the reduced mass of electron mathrme and proton mathrmp. mu = m_mathrme holds in the limit m_mathrmprightarrowinfty. Parameters are specified with the following struct.","category":"page"},{"location":"HydrogenAtom/#Parameters","page":"Hydrogen Atom","title":"Parameters","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.HydrogenAtom","category":"page"},{"location":"HydrogenAtom/#Antique.HydrogenAtom-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.V(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.V-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.E(::HydrogenAtom)","category":"page"},{"location":"HydrogenAtom/#Antique.E-Tuple{HydrogenAtom}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.ψ(::HydrogenAtom, ::Any, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.R(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.R-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)\n\nwhere Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.L(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.L-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\n L_0^0(x) = 1 \n L_1^0(x) = 1 - x \n L_1^1(x) = 1 \n L_2^0(x) = 1 - 2 x + 12 x^2 \n L_2^1(x) = 2 - x \n L_2^2(x) = 1 \n L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^1(x) = 3 - 3 x + 12 x^2 \n L_3^2(x) = 3 - x \n L_3^3(x) = 1 \n L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_4^2(x) = 6 - 4 x + 12 x^2 \n L_4^3(x) = 4 - x \n L_4^4(x) = 1 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.Y(::HydrogenAtom, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.Y-Tuple{HydrogenAtom, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.P(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.P-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\nendaligned\n\nwhere Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.\n\nExamples:\n\nbeginaligned\n P_0^0(x) = 1 \n P_1^0(x) = x \n P_1^1(x) = left(+1right)sqrt1-x^2 \n P_2^0(x) = -12 + 32 x^2 \n P_2^1(x) = left(-3 xright)sqrt1-x^2 \n P_2^2(x) = 3 - 6 x \n P_3^0(x) = -32 x + 52 x^3 \n P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n P_3^2(x) = 15 x - 30 x^2 \n P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n P_4^4(x) = 105 - 420 x + 420 x^2 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#References","page":"Hydrogen Atom","title":"References","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre\nThe Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.3 Table1, 18.5 Table1, 18.5.17, 18.3 Table1, 18.5 Table1, 18.5.12\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13), p.603 (d.13)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), p.79 (14.12), p.93 (16.19)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999), p.493 (B.72), p.494 Table, p.493 (B.72), p.483 (B.12), p.483 (B.12)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), p.83 (4), p.83 (5), p.149 (21)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), p.126 (4.28), p.96 Table3.1, p.126 (4.27), p.139 (4.88), p.140 Table4.4, p.139 (4.87), p.140 Table4.5\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), p.195 Table6.1, p.196 (6.26), p.196 Table6.2, p.207 Table6.4\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), p.234\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021), p.245 Problem 3.30.b, ","category":"page"},{"location":"HydrogenAtom/#Usage-and-Examples","page":"Hydrogen Atom","title":"Usage & Examples","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antique\nH = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.Z\n1\n\njulia> H.Eₕ\n1.0\n\njulia> H.mₑ\n1.0\n\njulia> H.a₀\n1.0\n\njulia> H.ℏ\n1.0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> E(H, n=1)\n-0.5\n\njulia> E(H, n=2)\n-0.125","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave length (n=2rightarrow1, the first line of the Lyman series):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv\nprintln(\"ΔE = \", E(H,n=2) - E(H,n=1), \" Eₕ\")\nprintln(\"λ = \", ((E(H,n=2)-E(H,n=1))*Eₕ2nm⁻¹)^-1, \" nm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ΔE = 0.375 Eₕ\nλ = 121.50227341098497 nm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Hyperfine Splitting:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"# constants: https://doi.org/10.1103/RevModPhys.93.025010\ne = 1.602176634e-19 # C https://physics.nist.gov/cgi-bin/cuu/Value?e\nh = 6.62607015e-34 # J Hz-1 https://physics.nist.gov/cgi-bin/cuu/Value?h\nc = 299792458 # m s-1 https://physics.nist.gov/cgi-bin/cuu/Value?c\na0 = 5.29177210903e-11 # m https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nµ0 = 1.25663706212e-6 # N A-2 https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nµB = 9.2740100783e-24 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mub\nµN = 5.0507837461e-27 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mun\nge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngp = 5.5856946893 # https://physics.nist.gov/cgi-bin/cuu/Value?gp\n\n# calculation: https://doi.org/10.1119/1.12733\nδ = abs(ψ(H,0,0,0))^2\nΔE = 2 / 3 * µ0 * µN * µB * gp * ge * δ * a0^(-3)\nprintln(\"1/π = \", 1/π)\nprintln(\"<δ(r)> = \", δ, \" a₀⁻³\")\nprintln(\"<δ(r)> = \", δ * a0^(-3), \" m⁻³\")\nprintln(\"ΔE = \", ΔE, \" J\")\nprintln(\"ν = ΔE/h = \", ΔE / h * 1e-6, \" MHz\")\nprintln(\"λ = hc/ΔE = \", h*c/ΔE*100, \" cm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"1/π = 0.3183098861837907\n<δ(r)> = 0.3183098861837908 a₀⁻³\n<δ(r)> = 2.1480615849063944e30 m⁻³\nΔE = 9.427622831641132e-25 J\nν = ΔE/h = 1422.8075794882932 MHz\nλ = hc/ΔE = 21.070485027063118 cm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h},~E_n/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nplot!(0.1:0.01:15, r -> V(H,r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve, Energy levels:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nfor n in 0:10\n plot!(0.0:0.01:15, r -> E(H,n=n) > V(H,r) ? E(H,n=n) : NaN, lc=n, lw=1, label=\"\") # energy level\nend\nplot!(0.1:0.01:15, r -> V(H,r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Radial functions:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\\$\", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=300)\nfor n in 1:3\n for l in 0:n-1\n plot!(0:0.01:20, r->r^2*R(H,r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label=\"\\$n = $n, l=$l\\$\")\n end\nend\nplot!()","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/#Testing","page":"Hydrogen Atom","title":"Testing","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 10 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 10 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 10 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 4 | 10 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 10 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 10 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 10 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 8 | 10 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n 9 | 10 | 0.000000000000 | 0.000000000000 ✔\n10 | 1 | 0.000000000000 | 0.000000000000 ✔\n10 | 2 | 0.000000000000 | 0.000000000000 ✔\n10 | 3 | 0.000000000000 | 0.000000000000 ✔\n10 | 4 | 0.000000000000 | 0.000000000000 ✔\n10 | 5 | 0.000000000000 | 0.000000000000 ✔\n10 | 6 | 0.000000000000 | -0.000000000000 ✔\n10 | 7 | 0.000000000000 | -0.000000000000 ✔\n10 | 8 | 0.000000000000 | 0.000000000000 ✔\n10 | 9 | 0.000000000000 | 0.000000000000 ✔\n10 | 10 | 1.000000000000 | 1.000000000000 ✔","category":"page"},{"location":"HydrogenAtom/#Eigen-Values-2","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n E_n\n = int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | m | ℏ | n | analytical | numerical \n--- | --- | --- | -- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1 | 49.348021579139 | 49.348022005447 ✔\n0.1 | 0.1 | 0.1 | 2 | 197.392081461942 | 197.392088021787 ✔\n0.1 | 0.1 | 0.1 | 3 | 444.132165131018 | 444.132198049021 ✔\n0.1 | 0.1 | 0.1 | 4 | 789.568248175979 | 789.568352087149 ✔\n0.1 | 0.1 | 0.1 | 5 | 1233.700296336187 | 1233.700550136170 ✔\n0.1 | 0.1 | 0.1 | 6 | 1776.528266243334 | 1776.528792196084 ✔\n0.1 | 0.1 | 0.1 | 7 | 2418.052103857080 | 2418.053078266893 ✔\n0.1 | 0.1 | 0.1 | 8 | 3158.271745875927 | 3158.273408348594 ✔\n0.1 | 0.1 | 0.1 | 9 | 3997.187119264267 | 3997.189782441189 ✔\n0.1 | 0.1 | 0.1 | 10 | 4934.798141994514 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 1 | 4934.802157913905 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 2 | 19739.208146194214 | 19739.208802178713 ✔\n0.1 | 0.1 | 1.0 | 3 | 44413.216513101754 | 44413.219804902103 ✔\n0.1 | 0.1 | 1.0 | 4 | 78956.824817597895 | 78956.835208714852 ✔\n0.1 | 0.1 | 1.0 | 5 | 123370.029633618717 | 123370.055013616948 ✔\n0.1 | 0.1 | 1.0 | 6 | 177652.826624333364 | 177652.879219608410 ✔\n0.1 | 0.1 | 1.0 | 7 | 241805.210385707964 | 241805.307826689212 ✔\n0.1 | 0.1 | 1.0 | 8 | 315827.174587592541 | 315827.340834859409 ✔\n0.1 | 0.1 | 1.0 | 9 | 399718.711926426622 | 399718.978244118916 ✔\n0.1 | 0.1 | 1.0 | 10 | 493479.814199451241 | 493480.220054467791 ✔\n0.1 | 1.0 | 0.1 | 1 | 4.934802157914 | 4.934802200545 ✔\n0.1 | 1.0 | 0.1 | 2 | 19.739208146194 | 19.739208802179 ✔\n0.1 | 1.0 | 0.1 | 3 | 44.413216513102 | 44.413219804902 ✔\n0.1 | 1.0 | 0.1 | 4 | 78.956824817598 | 78.956835208715 ✔\n0.1 | 1.0 | 0.1 | 5 | 123.370029633619 | 123.370055013617 ✔\n0.1 | 1.0 | 0.1 | 6 | 177.652826624333 | 177.652879219608 ✔\n0.1 | 1.0 | 0.1 | 7 | 241.805210385708 | 241.805307826689 ✔\n0.1 | 1.0 | 0.1 | 8 | 315.827174587593 | 315.827340834859 ✔\n0.1 | 1.0 | 0.1 | 9 | 399.718711926427 | 399.718978244119 ✔\n0.1 | 1.0 | 0.1 | 10 | 493.479814199451 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 1 | 493.480215791390 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 2 | 1973.920814619422 | 1973.920880217871 ✔\n0.1 | 1.0 | 1.0 | 3 | 4441.321651310176 | 4441.321980490210 ✔\n0.1 | 1.0 | 1.0 | 4 | 7895.682481759791 | 7895.683520871485 ✔\n0.1 | 1.0 | 1.0 | 5 | 12337.002963361872 | 12337.005501361695 ✔\n0.1 | 1.0 | 1.0 | 6 | 17765.282662433339 | 17765.287921960840 ✔\n0.1 | 1.0 | 1.0 | 7 | 24180.521038570794 | 24180.530782668924 ✔\n0.1 | 1.0 | 1.0 | 8 | 31582.717458759253 | 31582.734083485939 ✔\n0.1 | 1.0 | 1.0 | 9 | 39971.871192642662 | 39971.897824411892 ✔\n0.1 | 1.0 | 1.0 | 10 | 49347.981419945128 | 49348.022005446779 ✔\n1.0 | 0.1 | 0.1 | 1 | 0.493480215948 | 0.493480220054 ✔\n1.0 | 0.1 | 0.1 | 2 | 1.973920815419 | 1.973920880218 ✔\n1.0 | 0.1 | 0.1 | 3 | 4.441321651944 | 4.441321980490 ✔\n1.0 | 0.1 | 0.1 | 4 | 7.895682481265 | 7.895683520871 ✔\n1.0 | 0.1 | 0.1 | 5 | 12.337002965030 | 12.337005501362 ✔\n1.0 | 0.1 | 0.1 | 6 | 17.765282661715 | 17.765287921961 ✔\n1.0 | 0.1 | 0.1 | 7 | 24.180521036064 | 24.180530782669 ✔\n1.0 | 0.1 | 0.1 | 8 | 31.582717460023 | 31.582734083486 ✔\n1.0 | 0.1 | 0.1 | 9 | 39.971871195191 | 39.971897824412 ✔\n1.0 | 0.1 | 0.1 | 10 | 49.347981417827 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 1 | 49.348021594816 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 2 | 197.392081541864 | 197.392088021787 ✔\n1.0 | 0.1 | 1.0 | 3 | 444.132165194438 | 444.132198049021 ✔\n1.0 | 0.1 | 1.0 | 4 | 789.568248126463 | 789.568352087149 ✔\n1.0 | 0.1 | 1.0 | 5 | 1233.700296503016 | 1233.700550136170 ✔\n1.0 | 0.1 | 1.0 | 6 | 1776.528266171473 | 1776.528792196084 ✔\n1.0 | 0.1 | 1.0 | 7 | 2418.052103606433 | 2418.053078266892 ✔\n1.0 | 0.1 | 1.0 | 8 | 3158.271746002275 | 3158.273408348594 ✔\n1.0 | 0.1 | 1.0 | 9 | 3997.187119519121 | 3997.189782441190 ✔\n1.0 | 0.1 | 1.0 | 10 | 4934.798141782662 | 4934.802200544679 ✔\n1.0 | 1.0 | 0.1 | 1 | 0.049348021595 | 0.049348022005 ✔\n1.0 | 1.0 | 0.1 | 2 | 0.197392081542 | 0.197392088022 ✔\n1.0 | 1.0 | 0.1 | 3 | 0.444132165194 | 0.444132198049 ✔\n1.0 | 1.0 | 0.1 | 4 | 0.789568248126 | 0.789568352087 ✔\n1.0 | 1.0 | 0.1 | 5 | 1.233700296503 | 1.233700550136 ✔\n1.0 | 1.0 | 0.1 | 6 | 1.776528266171 | 1.776528792196 ✔\n1.0 | 1.0 | 0.1 | 7 | 2.418052103606 | 2.418053078267 ✔\n1.0 | 1.0 | 0.1 | 8 | 3.158271746002 | 3.158273408349 ✔\n1.0 | 1.0 | 0.1 | 9 | 3.997187119519 | 3.997189782441 ✔\n1.0 | 1.0 | 0.1 | 10 | 4.934798141783 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 1 | 4.934802159482 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 2 | 19.739208154186 | 19.739208802179 ✔\n1.0 | 1.0 | 1.0 | 3 | 44.413216519444 | 44.413219804902 ✔\n1.0 | 1.0 | 1.0 | 4 | 78.956824812646 | 78.956835208715 ✔\n1.0 | 1.0 | 1.0 | 5 | 123.370029650302 | 123.370055013617 ✔\n1.0 | 1.0 | 1.0 | 6 | 177.652826617147 | 177.652879219608 ✔\n1.0 | 1.0 | 1.0 | 7 | 241.805210360643 | 241.805307826689 ✔\n1.0 | 1.0 | 1.0 | 8 | 315.827174600228 | 315.827340834859 ✔\n1.0 | 1.0 | 1.0 | 9 | 399.718711951912 | 399.718978244119 ✔\n1.0 | 1.0 | 1.0 | 10 | 493.479814178266 | 493.480220054468 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-x","page":"Hydrogen Atom","title":"Expected Value of x","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"for only n=1.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.050000000000 | 0.050000000000 ✔\n0.5 | 1 | 0.250000000000 | 0.250000000000 ✔\n1.0 | 1 | 0.500000000000 | 0.500000000000 ✔\n7.0 | 1 | 3.500000000000 | 3.500000000000 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-x2","page":"Hydrogen Atom","title":"Expected Value of x^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.002826727415 | 0.002826727415 ✔\n0.5 | 1 | 0.070668185378 | 0.070668185378 ✔\n1.0 | 1 | 0.282672741512 | 0.282672741512 ✔\n7.0 | 1 | 13.850964334096 | 13.850964334096 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-p","page":"Hydrogen Atom","title":"Expected Value of p","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n langle p rangle_n=1\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n 2fracmathrmd psi(x)mathrmdx Delta x\n + Oleft(Delta x^3right)\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n \n 2fracmathrmd psi(x)mathrmdx Delta x\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n - Oleft(Delta x^3right)\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n - fracOleft(Delta x^3right)2Delta x\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n psi(x+Delta x)\n =\n psi(x)\n + fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\n \n psi(x-Delta x)\n =\n psi(x)\n - fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.000000000003 | 0.000000000000 ✔\n0.5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.0 | 1 | 0.000000000000 | 0.000000000000 ✔\n7.0 | 1 | 0.000000000000 | 0.000000000000 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-p2","page":"Hydrogen Atom","title":"Expected Value of p^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n langle p^2 rangle\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 986.960431582781 | 986.960440108936 ✔\n0.5 | 1 | 39.478417274195 | 39.478417604357 ✔\n1.0 | 1 | 9.869604318963 | 9.869604401089 ✔\n7.0 | 1 | 0.201420496383 | 0.201420497981 ✔\n","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"CurrentModule = Antique","category":"page"},{"location":"DeltaPotential/#Delta-Potential","page":"Delta Potential","title":"Delta Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"The Delta potential is one of the simplest models for quantum mechanical system in 1D. It always has one bound state and its wave function has a cusp at the origin.","category":"page"},{"location":"DeltaPotential/#Definitions","page":"Delta Potential","title":"Definitions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"and the Hamiltonian","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"DeltaPotential/#Parameters","page":"Delta Potential","title":"Parameters","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.DeltaPotential","category":"page"},{"location":"DeltaPotential/#Antique.DeltaPotential-DeltaPotential","page":"Delta Potential","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"DeltaPotential/#Potential","page":"Delta Potential","title":"Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.V(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.V-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Values","page":"Delta Potential","title":"Eigen Values","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.E(::DeltaPotential)","category":"page"},{"location":"DeltaPotential/#Antique.E-Tuple{DeltaPotential}-DeltaPotential","page":"Delta Potential","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Functions","page":"Delta Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.ψ(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.ψ-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Usage-and-Examples","page":"Delta Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by DeltaPotential and several parameters α, m and ℏ are set as optional arguments.","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using Antique\nDP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"julia> DP.α\n1.0\n\njulia> DP.m\n1.0\n\njulia> DP.ℏ\n1.0","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Eigen values:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"julia> E(DP)\n-0.5","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Wave functions:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"DP = DeltaPotential(α=0.1, m=0.5, ℏ=0.1)\nx = LinRange(-2,2,500);\n\nusing Plots\nplot(x, x->ψ(DP,x), linewidth=3)\nplot!(xlim=[-2,2], ylim=[0,2.5], legend=false)\nplot!(xlabel=\"x\", ylabel=\"ψ(x)\", title=\"Delta Potential\")","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"(Image: )","category":"page"},{"location":"DeltaPotential/#Testing","page":"Delta Potential","title":"Testing","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"DeltaPotential/#Normalization-of-\\psi(x)","page":"Delta Potential","title":"Normalization of psi(x)","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"int_-infty^infty psi^ast(x) psi(x) mathrmdx = 1","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" α | m | ℏ | analytical | numerical \n--- | --- | --- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 0.1 | 7.0 | 1.000000000000 | 1.000004676239 ✔\n0.1 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 1.0 | 7.0 | 1.000000000000 | 0.999999999999 ✔\n0.1 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 0.1 | 7.0 | 1.000000000000 | 0.999999999999 ✔\n1.0 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 1.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 0.1 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 1.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"CurrentModule = Antique","category":"page"},{"location":"HarmonicOscillator/#Harmonic-Oscillator","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The harmonic oscillator is the most frequently used model in quantum physics.","category":"page"},{"location":"HarmonicOscillator/#Definitions","page":"Harmonic Oscillator","title":"Definitions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"and the Hamiltonian","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters are specified with the following struct.","category":"page"},{"location":"HarmonicOscillator/#Parameters","page":"Harmonic Oscillator","title":"Parameters","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.HarmonicOscillator","category":"page"},{"location":"HarmonicOscillator/#Antique.HarmonicOscillator-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(α=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.V(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.V-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.E(::HarmonicOscillator)","category":"page"},{"location":"HarmonicOscillator/#Antique.E-Tuple{HarmonicOscillator}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.ψ(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.ψ-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are defined.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.H(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.H-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\nendaligned\n\nExamples:\n\nbeginaligned\n H_0(x) = 1 \n H_1(x) = 2 x \n H_2(x) = -2 + 4 x^2 \n H_3(x) = -12 x + 8 x^3 \n H_4(x) = 12 - 48 x^2 + 16 x^4 \n H_5(x) = 120 x - 160 x^3 + 32 x^5 \n H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n H_7(x) = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n H_9(x) = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Reference","page":"Harmonic Oscillator","title":"Reference","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"DLMF 18.5.18\ncpprefjp\nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)","category":"page"},{"location":"HarmonicOscillator/#Usage-and-Examples","page":"Harmonic Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Antique\nHO = HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.k\n1.0\n\njulia> HO.m\n1.0\n\njulia> HO.ℏ\n1.0","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> E(HO, n=0)\n0.5\n\njulia> E(HO, n=1)\n1.5","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(-5:0.1:5, x -> V(HO, x), lw=2, label=\"\", xlabel=\"x\", ylabel=\"V(x)\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> ψ(HO, x, n=0), label=\"n=0\", lw=2)\nplot!(x -> ψ(HO, x, n=1), label=\"n=1\", lw=2)\nplot!(x -> ψ(HO, x, n=2), label=\"n=2\", lw=2)\nplot!(x -> ψ(HO, x, n=3), label=\"n=3\", lw=2)\nplot!(x -> ψ(HO, x, n=4), label=\"n=4\", lw=2)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times0.5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:4\n # energy\n hline!([E(HO, n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-sqrt(2*HO.k*E(HO, n=n)),sqrt(2*HO.k*E(HO, n=n))], fill(E(HO, n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> E(HO, n=n) + 0.5*ψ(HO, x,n=n), lc=n+1, lw=2, label=\"\")\nend\n# potential\nplot!(x -> V(HO, x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/#Testing","page":"Harmonic Oscillator","title":"Testing","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials-H_n(x)","page":"Harmonic Oscillator","title":"Hermite Polynomials H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=0 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_0(x)\n = e^ - x^2 e^x^2\n = 1 \n = 1\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=1 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_1(x)\n = - e^x^2 fracmathrmd e^ - x^2mathrmdx\n = 2 x \n = 2 x\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=2 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_2(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -2 + 4 x^2 \n = -2 + 4 x^2\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=3 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_3(x)\n = - fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx e^x^2\n = - 12 x + 8 x^3 \n = - 12 x + 8 x^3\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=4 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_4(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 12 - 48 x^2 + 16 x^4 \n = 12 - 48 x^2 + 16 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=5 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_5(x)\n = - fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx e^x^2\n = 120 x - 160 x^3 + 32 x^5 \n = 120 x - 160 x^3 + 32 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=6 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_6(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n = -120 + 720 x^2 - 480 x^4 + 64 x^6\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=7 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_7(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = - 1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n = - 1680 x + 3360 x^3 - 1344 x^5 + 128 x^7\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=8 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_8(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=9 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_9(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-H_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int_-infty^infty H_j(x) H_i(x) mathrme^-x^2 mathrmdx = sqrtpi 2^j j delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.772453850906 | 1.772453850906 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000001 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 3.544907701811 | 3.544907701811 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000014 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 14.179630807244 | 14.179630807244 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000011 ✔\n 2 | 9 | 0.000000000000 | -0.000000000002 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 85.077784843465 | 85.077784843465 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000139 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 680.622278747718 | 680.622278747718 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000002 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000063 ✔\n 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 6806.222787477181 | 6806.222787477180 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000009 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000001339 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000002 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 81674.673449726179 | 81674.673449726135 ✔\n 6 | 7 | 0.000000000000 | 0.000000000004 ✔\n 6 | 8 | 0.000000000000 | 0.000000000397 ✔\n 6 | 9 | 0.000000000000 | -0.000000000087 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000009 ✔\n 7 | 6 | 0.000000000000 | 0.000000000004 ✔\n 7 | 7 | 1143445.428296166472 | 1143445.428296166705 ✔\n 7 | 8 | 0.000000000000 | -0.000000000007 ✔\n 7 | 9 | 0.000000000000 | 0.000000011649 ✔\n 8 | 0 | 0.000000000000 | -0.000000000001 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000011 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000063 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000397 ✔\n 8 | 7 | 0.000000000000 | -0.000000000007 ✔\n 8 | 8 | 18295126.852738663554 | 18295126.852738667279 ✔\n 8 | 9 | 0.000000000000 | 0.000000001630 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000014 ✔\n 9 | 2 | 0.000000000000 | -0.000000000002 ✔\n 9 | 3 | 0.000000000000 | 0.000000000139 ✔\n 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000001339 ✔\n 9 | 6 | 0.000000000000 | -0.000000000087 ✔\n 9 | 7 | 0.000000000000 | 0.000000011649 ✔\n 9 | 8 | 0.000000000000 | 0.000000001630 ✔\n 9 | 9 | 329312283.349295914173 | 329312283.349295675755 ✔","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 8 | 0 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔","category":"page"},{"location":"HarmonicOscillator/#Virial-Theorem","page":"Harmonic Oscillator","title":"Virial Theorem","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The virial theorem langle T rangle = langle V rangle and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = 2 langle V rangle = 2 langle T rangle.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"2 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.1 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.1 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.1 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.1 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.1 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.1 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.1 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.1 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.1 | 9 | 9.500000000000 | 9.500000000000 ✔\n0.5 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.5 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.5 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.5 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.5 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.5 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.5 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.5 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.5 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.5 | 9 | 9.500000000000 | 9.500000000000 ✔\n1.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n1.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n1.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n1.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n1.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n1.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n1.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n1.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n1.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n1.0 | 9 | 9.500000000000 | 9.500000000000 ✔\n5.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n5.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n5.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n5.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n5.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n5.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n5.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n5.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n5.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n5.0 | 9 | 9.500000000000 | 9.500000000000 ✔","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values-2","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n E_n\n = int psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int psi^ast_n(x) left V(x) - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int psi^ast_n(x) left V(x)psi(x) -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.158113883008 | 0.158113879883 ✔\n0.1 | 1 | 0.474341649025 | 0.474341633410 ✔\n0.1 | 2 | 0.790569415042 | 0.790569374409 ✔\n0.1 | 3 | 1.106797181059 | 1.106797102928 ✔\n0.1 | 4 | 1.423024947076 | 1.423024818987 ✔\n0.1 | 5 | 1.739252713093 | 1.739252522506 ✔\n0.1 | 6 | 2.055480479109 | 2.055480213500 ✔\n0.1 | 7 | 2.371708245126 | 2.371707891950 ✔\n0.1 | 8 | 2.687936011143 | 2.687935558100 ✔\n0.1 | 9 | 3.004163777160 | 3.004163211450 ✔\n0.5 | 0 | 0.353553390593 | 0.353553374944 ✔\n0.5 | 1 | 1.060660171780 | 1.060660093649 ✔\n0.5 | 2 | 1.767766952966 | 1.767766749878 ✔\n0.5 | 3 | 2.474873734153 | 2.474873343556 ✔\n0.5 | 4 | 3.181980515339 | 3.181979874817 ✔\n0.5 | 5 | 3.889087296526 | 3.889086343463 ✔\n0.5 | 6 | 4.596194077713 | 4.596192749665 ✔\n0.5 | 7 | 5.303300858899 | 5.303299093519 ✔\n0.5 | 8 | 6.010407640086 | 6.010405374197 ✔\n0.5 | 9 | 6.717514421272 | 6.717511593266 ✔\n1.0 | 0 | 0.500000000000 | 0.499999968773 ✔\n1.0 | 1 | 1.500000000000 | 1.499999843774 ✔\n1.0 | 2 | 2.500000000000 | 2.499999593764 ✔\n1.0 | 3 | 3.500000000000 | 3.499999218732 ✔\n1.0 | 4 | 4.500000000000 | 4.499998718747 ✔\n1.0 | 5 | 5.500000000000 | 5.499998093755 ✔\n1.0 | 6 | 6.500000000000 | 6.499997343602 ✔\n1.0 | 7 | 7.500000000000 | 7.499996468887 ✔\n1.0 | 8 | 8.500000000000 | 8.499995468843 ✔\n1.0 | 9 | 9.500000000000 | 9.499994343445 ✔\n5.0 | 0 | 1.118033988750 | 1.118033832523 ✔\n5.0 | 1 | 3.354101966250 | 3.354101184969 ✔\n5.0 | 2 | 5.590169943749 | 5.590167912524 ✔\n5.0 | 3 | 7.826237921249 | 7.826234014984 ✔\n5.0 | 4 | 10.062305898749 | 10.062299492494 ✔\n5.0 | 5 | 12.298373876249 | 12.298364344997 ✔\n5.0 | 6 | 14.534441853749 | 14.534428572309 ✔\n5.0 | 7 | 16.770509831248 | 16.770492175222 ✔\n5.0 | 8 | 19.006577808748 | 19.006555152416 ✔\n5.0 | 9 | 21.242645786248 | 21.242617504750 ✔\n","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Antique","category":"page"},{"location":"#Antique.jl","page":"Home","title":"Antique.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Self-contained, Well-Tested, Well-Documented Analytical Solutions of Quantum Mechanical Equations.","category":"page"},{"location":"#Install","page":"Home","title":"Install","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install this package, run the following code in your Jupyter Notebook:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg; Pkg.add(\"Antique\")","category":"page"},{"location":"#Usage-and-Examples","page":"Home","title":"Usage & Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. Here are examples in hydrogen-like atom. The analytical notation of energy (eigen value of the Hamiltonian) is written as","category":"page"},{"location":"","page":"Home","title":"Home","text":"E_n = -fracZ^22n^2 E_mathrmh","category":"page"},{"location":"","page":"Home","title":"Home","text":"Hydrogen atom has symbol mathrmH and atomic number 1 (Z=1). Therefore the ground state (n=1) energy is -frac12 E_mathrmh.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antique\nH = HydrogenAtom(Z=1)\nE(H)\n# output> -0.5","category":"page"},{"location":"","page":"Home","title":"Home","text":"Helium cation has symbol mathrmHe^+ and atomic number 2 (Z=2). Therefore the ground state (n=1) energy is -2 E_mathrmh.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antique\nHe⁺ = HydrogenAtom(Z=2)\nE(He⁺)\n# output> -2.0","category":"page"},{"location":"","page":"Home","title":"Home","text":"There are more examples on each model page.","category":"page"},{"location":"#Supported-Models","page":"Home","title":"Supported Models","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
\n
\n \n \"InfinitePotentialWell\"/\n \n InfinitePotentialWell\n
\n
\n \n \"HarmonicOscillator\"/\n \n HarmonicOscillator\n
\n
\n \n \"MorsePotential\"/\n \n MorsePotential\n
\n
\n \n \"HydrogenAtom\"/\n \n HydrogenAtom\n
\n
\n \n \"DeltaPotential\"/\n \n DeltaPotential\n
\n
\n \n \"PoschlTeller\"/\n \n PoschlTeller\n
\n
","category":"page"},{"location":"#Future-Works","page":"Home","title":"Future Works","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"List of quantum-mechanical systems with analytical solutions","category":"page"},{"location":"#Developer's-Guide","page":"Home","title":"Developer's Guide","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This is the guideline for adding new models.","category":"page"},{"location":"","page":"Home","title":"Home","text":"First, please add a new issue here. We need to find a reference for the definition and analytical solutions (eigenvalues and eigenfunctions) before the development.\nFork the repository on GitHub.\nClone the forked repository to your local machine by Git.\nAdd the new model name :ModelName to the models = [...] array in src/Antique.jl. : is required at the beginning.\nAdd the file src/ModelName.jl with the same name as the model name. The most helpful code examples are harmonic oscillators for one-dimensional systems and hydrogen atoms for three-dimensional systems. We recommend that you copy these.\nWrite the code in that file. First we need to create a structure struct ModelName with the same name as the model name (The best way is Find & Replace). Create V, E, ψ and other functions. Because the function names conflict, you must always give the structure as an argument. Multi-dispatch avoids conflict. We recommend using Revice.jl while coding. Run include(\"./developer/revice.jl\") on the REPL or use dev.ipynb.\nAdd test code test/ModelName.jl. At a minimum, it is recommended to check the normalization and the orthogonality of wavefunction using QuadGK.jl. All tests will be executed by executing include(\"./developer/test.jl\"). It will take about 2 minutes to complete.\nAdd documentation. Add either docs/ModelName.md or docs/jmd/ModelName.jmd (if you have a jmd file, the md file will be automatically generated). Include at least the definition of the Hamiltonian and the analytical solutions (eigenvalues and eigenfunctions).\nAdd the new model into pages=[...] in docs/make.jl.\nExecute include(\"./developer/docs.jl\") to compile. Please check docs/build/*.html in your browser.\nPush the code.\nSubmit a pull request on GitHub.","category":"page"},{"location":"#Acknowledgment","page":"Home","title":"Acknowledgment","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package was named by @KB-satou and @ultimatile.","category":"page"},{"location":"API/","page":"API reference","title":"API reference","text":"CurrentModule = Antique","category":"page"},{"location":"API/#API-reference","page":"API reference","title":"API reference","text":"","category":"section"},{"location":"API/","page":"API reference","title":"API reference","text":"Order = [:type, :function]\nModules = [Antique]","category":"page"},{"location":"API/#Antique.DeltaPotential","page":"API reference","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HarmonicOscillator","page":"API reference","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(α=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HydrogenAtom","page":"API reference","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.InfinitePotentialWell","page":"API reference","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.MorsePotential","page":"API reference","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D__mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.PoschlTeller","page":"API reference","title":"Antique.PoschlTeller","text":"PoschlTeller(lambda=1.0)\n\nlambda determines the potential strength. This model is defined dimensionless, i.e. x is given in units of a characteristic length x_0, and E in units of a characteristic energy, e.g. E_mathrmchar = frachbar^22 m x_0^2.\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.E-Tuple{DeltaPotential}","page":"API reference","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HarmonicOscillator}","page":"API reference","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HydrogenAtom}","page":"API reference","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{InfinitePotentialWell}","page":"API reference","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{MorsePotential}","page":"API reference","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{PoschlTeller}","page":"API reference","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.H-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\nendaligned\n\nExamples:\n\nbeginaligned\n H_0(x) = 1 \n H_1(x) = 2 x \n H_2(x) = -2 + 4 x^2 \n H_3(x) = -12 x + 8 x^3 \n H_4(x) = 12 - 48 x^2 + 16 x^4 \n H_5(x) = 120 x - 160 x^3 + 32 x^5 \n H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n H_7(x) = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n H_9(x) = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\n L_0^0(x) = 1 \n L_1^0(x) = 1 - x \n L_1^1(x) = 1 \n L_2^0(x) = 1 - 2 x + 12 x^2 \n L_2^1(x) = 2 - x \n L_2^2(x) = 1 \n L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^1(x) = 3 - 3 x + 12 x^2 \n L_3^2(x) = 3 - x \n L_3^3(x) = 1 \n L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_4^2(x) = 6 - 4 x + 12 x^2 \n L_4^3(x) = 4 - x \n L_4^4(x) = 1 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k left(beginarrayl n+alpha n-k endarrayright) fracx^kk \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \nendaligned\n\nExamples:\n\nbeginaligned\n L_0^(0)(x) = 1 \n L_1^(0)(x) = 1 - x \n L_1^(1)(x) = 2 - x \n L_2^(0)(x) = 1 - 2 x + 12 x^2 \n L_2^(1)(x) = 3 - 3 x + 12 x^2 \n L_2^(2)(x) = 6 - 4 x + 12 x^2 \n L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\nendaligned\n\nwhere Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.\n\nExamples:\n\nbeginaligned\n P_0^0(x) = 1 \n P_1^0(x) = x \n P_1^1(x) = left(+1right)sqrt1-x^2 \n P_2^0(x) = -12 + 32 x^2 \n P_2^1(x) = left(-3 xright)sqrt1-x^2 \n P_2^2(x) = 3 - 6 x \n P_3^0(x) = -32 x + 52 x^3 \n P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n P_3^2(x) = 15 x - 30 x^2 \n P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n P_4^4(x) = 105 - 420 x + 420 x^2 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.P","text":"P(model::PoschlTeller, x; n=0, m=0)\n\nAssociated Legendre polynomials are the associated Legendre functions for integer indices. Please note here, that for the Poschl-Teller potential we use a slightly different notation of the associated Legendre functions as compared to the model HydrogenAtom. Here we have an additional factor (-1)^m.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)\n\nwhere Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\n V(x)\n = -fraclambda(lambda+1)2 mathrmsech(x)^2\n = -fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{HydrogenAtom, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nmax-Tuple{PoschlTeller}","page":"API reference","title":"Antique.nmax","text":"nmax(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nₘₐₓ-Tuple{MorsePotential}","page":"API reference","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(x)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"}] +[{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"CurrentModule = Antique","category":"page"},{"location":"PoschlTeller/#Pöschl-Teller-Potential","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"The Pöschl-Teller potential is one of the few potentials for which the quantum mechanical Schrödinger equation has an analytical solution. It has a finite number of bound states, which can be inferred easily from its potential strength parameter. It is defined for one-dimensional systems.","category":"page"},{"location":"PoschlTeller/#Definitions","page":"Pöschl-Teller Potential","title":"Definitions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"and the Hamiltonian","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" hatH = - frac12 fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters are specified within the following struct.","category":"page"},{"location":"PoschlTeller/#Parameters","page":"Pöschl-Teller Potential","title":"Parameters","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.PoschlTeller","category":"page"},{"location":"PoschlTeller/#Antique.PoschlTeller-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.PoschlTeller","text":"PoschlTeller(lambda=1.0)\n\nlambda determines the potential strength. This model is defined dimensionless, i.e. x is given in units of a characteristic length x_0, and E in units of a characteristic energy, e.g. E_mathrmchar = frachbar^22 m x_0^2.\n\n\n\n\n\n","category":"type"},{"location":"PoschlTeller/#Potential","page":"Pöschl-Teller Potential","title":"Potential","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.V(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.V-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\n V(x)\n = -fraclambda(lambda+1)2 mathrmsech(x)^2\n = -fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Number-of-Bound-States","page":"Pöschl-Teller Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.nmax(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.nmax-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.nmax","text":"nmax(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Values","page":"Pöschl-Teller Potential","title":"Eigen Values","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.E(::PoschlTeller)","category":"page"},{"location":"PoschlTeller/#Antique.E-Tuple{PoschlTeller}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Eigen-Functions","page":"Pöschl-Teller Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Antique.ψ(::PoschlTeller, ::Any)","category":"page"},{"location":"PoschlTeller/#Antique.ψ-Tuple{PoschlTeller, Any}-PoschlTeller","page":"Pöschl-Teller Potential","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(x)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"},{"location":"PoschlTeller/#Usage-and-Examples","page":"Pöschl-Teller Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by PoschlTeller and a single parameter lambda. It is assumed to be used within rescaled, dimensionless variables.","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"using Antique\nPT = PoschlTeller(lambda=6.0)","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Parameters:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"julia> PT.lambda\n6.0","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Eigen values:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"julia> E(PT,n=0)\n-18.0\n\njulia> E(PT,n=1)\n-12.5\n\njulia> E(PT,n=2)\n-8.0\n\njulia> E(PT,n=3)\n-4.5\n\njulia> E(PT,n=4)\n-2.0\n\njulia> E(PT,n=5)\n-0.5","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"lambda = 4.0\nPT = PoschlTeller(lambda)\n\nusing Plots\nplot(xlim=(-4,4), ylim=(-11.0,1.0), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:3\n # classical turning point:\n xE = acosh(sqrt(lambda*(lambda+1)/abs(E(PT,n=n))/2))\n # energy\n hline!([E(PT, n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-xE,xE], fill(E(PT, n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> E(PT, n=n) + ψ(PT, x,n=n), lc=n+1, lw=2, label=\"\\$n = $n\\$\")\nend\n# potential\nplot!(x -> V(PT, x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"(Image: )","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"lambda = 4.0\nPT = PoschlTeller(lambda)\n\nusing Plots\nplot(xlim=(-4,4), ylim=(-11.0,1.0), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:3\n # classical turning point:\n xE = acosh(sqrt(lambda*(lambda+1)/abs(E(PT,n=n))/2))\n # energy\n hline!([E(PT, n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-xE,xE], fill(E(PT, n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> E(PT, n=n) + ψ(PT, x,n=n), lc=n+1, lw=2, label=\"\\$n = $n\\$\")\nend\n# potential\nplot!(x -> V(PT, x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"(Image: )","category":"page"},{"location":"PoschlTeller/#Testing","page":"Pöschl-Teller Potential","title":"Testing","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"PoschlTeller/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Pöschl-Teller Potential","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":"int psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"PoschlTeller/","page":"Pöschl-Teller Potential","title":"Pöschl-Teller Potential","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | -0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | -0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 0.999999999999 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | -0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 6 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 0 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 8 | 0 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | -0.000000000000 ✔\n 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"CurrentModule = Antique","category":"page"},{"location":"InfinitePotentialWell/#Infinite-Potential-Well-(Particle-in-a-Box)","page":"Infinite Potential Well","title":"Infinite Potential Well (Particle in a Box)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"The infinite potential well (particle in a box) is the simplest model for quantum mechanical system.","category":"page"},{"location":"InfinitePotentialWell/#Definitions","page":"Infinite Potential Well","title":"Definitions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"and the Hamiltonian","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters are specified with the following struct.","category":"page"},{"location":"InfinitePotentialWell/#Parameters","page":"Infinite Potential Well","title":"Parameters","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.InfinitePotentialWell","category":"page"},{"location":"InfinitePotentialWell/#Antique.InfinitePotentialWell-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"InfinitePotentialWell/#Potential","page":"Infinite Potential Well","title":"Potential","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.V(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.V-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell/#Eigen-Values","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.E(::InfinitePotentialWell)","category":"page"},{"location":"InfinitePotentialWell/#Antique.E-Tuple{InfinitePotentialWell}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell/#Eigen-Functions","page":"Infinite Potential Well","title":"Eigen Functions","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Antique.ψ(::InfinitePotentialWell, ::Any)","category":"page"},{"location":"InfinitePotentialWell/#Antique.ψ-Tuple{InfinitePotentialWell, Any}-InfinitePotentialWell","page":"Infinite Potential Well","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"location":"InfinitePotentialWell/#Proofs","page":"Infinite Potential Well","title":"Proofs","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen Functions & Eigen Values\nNormalization","category":"page"},{"location":"InfinitePotentialWell/#Usage-and-Examples","page":"Infinite Potential Well","title":"Usage & Examples","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by InfinitePotentialWell and several parameters L, m and ℏ are set as optional arguments.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Antique\nIPW = InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Parameters:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> IPW.L\n1.0\n\njulia> IPW.m\n1.0\n\njulia> IPW.ℏ\n1.0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Eigen values:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"julia> E(IPW, n=1)\n4.934802200544679\n\njulia> E(IPW, n=2)\n19.739208802178716","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"using Plots\nplot(xlim=(0,1), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> ψ(IPW, x, n=1), label=\"n=1\", lw=2)\nplot!(x -> ψ(IPW, x, n=2), label=\"n=2\", lw=2)\nplot!(x -> ψ(IPW, x, n=3), label=\"n=3\", lw=2)\nplot!(x -> ψ(IPW, x, n=4), label=\"n=4\", lw=2)\nplot!(x -> ψ(IPW, x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"L = 1\nusing Plots\nplot(xlim=(-0.5,1.5), ylim=(-5,140), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 1:5\n # energy\n plot!([0,L], fill(E(IPW,n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(0:0.01:L, x->E(IPW,n=n) + 5*ψ(IPW,x,n=n), lc=n, lw=2, label=\"\")\nend\n# potential\nplot!([0,0,L,L], [140,0,0,140], lc=:black, lw=2, label=\"\")","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"(Image: )","category":"page"},{"location":"InfinitePotentialWell/#Testing","page":"Infinite Potential Well","title":"Testing","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"InfinitePotentialWell/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Infinite Potential Well","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 10 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 10 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 10 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 4 | 10 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 10 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 10 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 10 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 8 | 10 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n 9 | 10 | 0.000000000000 | 0.000000000000 ✔\n10 | 1 | 0.000000000000 | 0.000000000000 ✔\n10 | 2 | 0.000000000000 | 0.000000000000 ✔\n10 | 3 | 0.000000000000 | 0.000000000000 ✔\n10 | 4 | 0.000000000000 | 0.000000000000 ✔\n10 | 5 | 0.000000000000 | 0.000000000000 ✔\n10 | 6 | 0.000000000000 | -0.000000000000 ✔\n10 | 7 | 0.000000000000 | -0.000000000000 ✔\n10 | 8 | 0.000000000000 | 0.000000000000 ✔\n10 | 9 | 0.000000000000 | 0.000000000000 ✔\n10 | 10 | 1.000000000000 | 1.000000000000 ✔","category":"page"},{"location":"InfinitePotentialWell/#Eigen-Values-2","page":"Infinite Potential Well","title":"Eigen Values","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n E_n\n = int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | m | ℏ | n | analytical | numerical \n--- | --- | --- | -- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1 | 49.348021579139 | 49.348022005447 ✔\n0.1 | 0.1 | 0.1 | 2 | 197.392081461942 | 197.392088021787 ✔\n0.1 | 0.1 | 0.1 | 3 | 444.132165131018 | 444.132198049021 ✔\n0.1 | 0.1 | 0.1 | 4 | 789.568248175979 | 789.568352087149 ✔\n0.1 | 0.1 | 0.1 | 5 | 1233.700296336187 | 1233.700550136170 ✔\n0.1 | 0.1 | 0.1 | 6 | 1776.528266243334 | 1776.528792196084 ✔\n0.1 | 0.1 | 0.1 | 7 | 2418.052103857080 | 2418.053078266893 ✔\n0.1 | 0.1 | 0.1 | 8 | 3158.271745875927 | 3158.273408348594 ✔\n0.1 | 0.1 | 0.1 | 9 | 3997.187119264267 | 3997.189782441189 ✔\n0.1 | 0.1 | 0.1 | 10 | 4934.798141994514 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 1 | 4934.802157913905 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 2 | 19739.208146194214 | 19739.208802178713 ✔\n0.1 | 0.1 | 1.0 | 3 | 44413.216513101754 | 44413.219804902103 ✔\n0.1 | 0.1 | 1.0 | 4 | 78956.824817597895 | 78956.835208714852 ✔\n0.1 | 0.1 | 1.0 | 5 | 123370.029633618717 | 123370.055013616948 ✔\n0.1 | 0.1 | 1.0 | 6 | 177652.826624333364 | 177652.879219608410 ✔\n0.1 | 0.1 | 1.0 | 7 | 241805.210385707964 | 241805.307826689212 ✔\n0.1 | 0.1 | 1.0 | 8 | 315827.174587592541 | 315827.340834859409 ✔\n0.1 | 0.1 | 1.0 | 9 | 399718.711926426622 | 399718.978244118916 ✔\n0.1 | 0.1 | 1.0 | 10 | 493479.814199451241 | 493480.220054467791 ✔\n0.1 | 1.0 | 0.1 | 1 | 4.934802157914 | 4.934802200545 ✔\n0.1 | 1.0 | 0.1 | 2 | 19.739208146194 | 19.739208802179 ✔\n0.1 | 1.0 | 0.1 | 3 | 44.413216513102 | 44.413219804902 ✔\n0.1 | 1.0 | 0.1 | 4 | 78.956824817598 | 78.956835208715 ✔\n0.1 | 1.0 | 0.1 | 5 | 123.370029633619 | 123.370055013617 ✔\n0.1 | 1.0 | 0.1 | 6 | 177.652826624333 | 177.652879219608 ✔\n0.1 | 1.0 | 0.1 | 7 | 241.805210385708 | 241.805307826689 ✔\n0.1 | 1.0 | 0.1 | 8 | 315.827174587593 | 315.827340834859 ✔\n0.1 | 1.0 | 0.1 | 9 | 399.718711926427 | 399.718978244119 ✔\n0.1 | 1.0 | 0.1 | 10 | 493.479814199451 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 1 | 493.480215791390 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 2 | 1973.920814619422 | 1973.920880217871 ✔\n0.1 | 1.0 | 1.0 | 3 | 4441.321651310176 | 4441.321980490210 ✔\n0.1 | 1.0 | 1.0 | 4 | 7895.682481759791 | 7895.683520871485 ✔\n0.1 | 1.0 | 1.0 | 5 | 12337.002963361872 | 12337.005501361695 ✔\n0.1 | 1.0 | 1.0 | 6 | 17765.282662433339 | 17765.287921960840 ✔\n0.1 | 1.0 | 1.0 | 7 | 24180.521038570794 | 24180.530782668924 ✔\n0.1 | 1.0 | 1.0 | 8 | 31582.717458759253 | 31582.734083485939 ✔\n0.1 | 1.0 | 1.0 | 9 | 39971.871192642662 | 39971.897824411892 ✔\n0.1 | 1.0 | 1.0 | 10 | 49347.981419945128 | 49348.022005446779 ✔\n1.0 | 0.1 | 0.1 | 1 | 0.493480215948 | 0.493480220054 ✔\n1.0 | 0.1 | 0.1 | 2 | 1.973920815419 | 1.973920880218 ✔\n1.0 | 0.1 | 0.1 | 3 | 4.441321651944 | 4.441321980490 ✔\n1.0 | 0.1 | 0.1 | 4 | 7.895682481265 | 7.895683520871 ✔\n1.0 | 0.1 | 0.1 | 5 | 12.337002965030 | 12.337005501362 ✔\n1.0 | 0.1 | 0.1 | 6 | 17.765282661715 | 17.765287921961 ✔\n1.0 | 0.1 | 0.1 | 7 | 24.180521036064 | 24.180530782669 ✔\n1.0 | 0.1 | 0.1 | 8 | 31.582717460023 | 31.582734083486 ✔\n1.0 | 0.1 | 0.1 | 9 | 39.971871195191 | 39.971897824412 ✔\n1.0 | 0.1 | 0.1 | 10 | 49.347981417827 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 1 | 49.348021594816 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 2 | 197.392081541864 | 197.392088021787 ✔\n1.0 | 0.1 | 1.0 | 3 | 444.132165194438 | 444.132198049021 ✔\n1.0 | 0.1 | 1.0 | 4 | 789.568248126463 | 789.568352087149 ✔\n1.0 | 0.1 | 1.0 | 5 | 1233.700296503016 | 1233.700550136170 ✔\n1.0 | 0.1 | 1.0 | 6 | 1776.528266171473 | 1776.528792196084 ✔\n1.0 | 0.1 | 1.0 | 7 | 2418.052103606433 | 2418.053078266892 ✔\n1.0 | 0.1 | 1.0 | 8 | 3158.271746002275 | 3158.273408348594 ✔\n1.0 | 0.1 | 1.0 | 9 | 3997.187119519121 | 3997.189782441190 ✔\n1.0 | 0.1 | 1.0 | 10 | 4934.798141782662 | 4934.802200544679 ✔\n1.0 | 1.0 | 0.1 | 1 | 0.049348021595 | 0.049348022005 ✔\n1.0 | 1.0 | 0.1 | 2 | 0.197392081542 | 0.197392088022 ✔\n1.0 | 1.0 | 0.1 | 3 | 0.444132165194 | 0.444132198049 ✔\n1.0 | 1.0 | 0.1 | 4 | 0.789568248126 | 0.789568352087 ✔\n1.0 | 1.0 | 0.1 | 5 | 1.233700296503 | 1.233700550136 ✔\n1.0 | 1.0 | 0.1 | 6 | 1.776528266171 | 1.776528792196 ✔\n1.0 | 1.0 | 0.1 | 7 | 2.418052103606 | 2.418053078267 ✔\n1.0 | 1.0 | 0.1 | 8 | 3.158271746002 | 3.158273408349 ✔\n1.0 | 1.0 | 0.1 | 9 | 3.997187119519 | 3.997189782441 ✔\n1.0 | 1.0 | 0.1 | 10 | 4.934798141783 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 1 | 4.934802159482 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 2 | 19.739208154186 | 19.739208802179 ✔\n1.0 | 1.0 | 1.0 | 3 | 44.413216519444 | 44.413219804902 ✔\n1.0 | 1.0 | 1.0 | 4 | 78.956824812646 | 78.956835208715 ✔\n1.0 | 1.0 | 1.0 | 5 | 123.370029650302 | 123.370055013617 ✔\n1.0 | 1.0 | 1.0 | 6 | 177.652826617147 | 177.652879219608 ✔\n1.0 | 1.0 | 1.0 | 7 | 241.805210360643 | 241.805307826689 ✔\n1.0 | 1.0 | 1.0 | 8 | 315.827174600228 | 315.827340834859 ✔\n1.0 | 1.0 | 1.0 | 9 | 399.718711951912 | 399.718978244119 ✔\n1.0 | 1.0 | 1.0 | 10 | 493.479814178266 | 493.480220054468 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x","page":"Infinite Potential Well","title":"Expected Value of x","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"for only n=1.","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.050000000000 | 0.050000000000 ✔\n0.5 | 1 | 0.250000000000 | 0.250000000000 ✔\n1.0 | 1 | 0.500000000000 | 0.500000000000 ✔\n7.0 | 1 | 3.500000000000 | 3.500000000000 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-x2","page":"Infinite Potential Well","title":"Expected Value of x^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.002826727415 | 0.002826727415 ✔\n0.5 | 1 | 0.070668185378 | 0.070668185378 ✔\n1.0 | 1 | 0.282672741512 | 0.282672741512 ✔\n7.0 | 1 | 13.850964334096 | 13.850964334096 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p","page":"Infinite Potential Well","title":"Expected Value of p","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p rangle_n=1\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2fracmathrmd psi(x)mathrmdx Delta x\n + Oleft(Delta x^3right)\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n \n 2fracmathrmd psi(x)mathrmdx Delta x\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n - Oleft(Delta x^3right)\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n - fracOleft(Delta x^3right)2Delta x\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n psi(x+Delta x)\n =\n psi(x)\n + fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\n \n psi(x-Delta x)\n =\n psi(x)\n - fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.000000000003 | 0.000000000000 ✔\n0.5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.0 | 1 | 0.000000000000 | 0.000000000000 ✔\n7.0 | 1 | 0.000000000000 | 0.000000000000 ✔","category":"page"},{"location":"InfinitePotentialWell/#Expected-Value-of-p2","page":"Infinite Potential Well","title":"Expected Value of p^2","text":"","category":"section"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Reference:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" beginaligned\n langle p^2 rangle\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"InfinitePotentialWell/","page":"Infinite Potential Well","title":"Infinite Potential Well","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 986.960431582781 | 986.960440108936 ✔\n0.5 | 1 | 39.478417274195 | 39.478417604357 ✔\n1.0 | 1 | 9.869604318963 | 9.869604401089 ✔\n7.0 | 1 | 0.201420496383 | 0.201420497981 ✔\n","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"CurrentModule = Antique","category":"page"},{"location":"MorsePotential/#Morse-Potential","page":"Morse Potential","title":"Morse Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"The Morse potential is a model for inter-nuclear anharmonic vibration in a diatomic molecule.","category":"page"},{"location":"MorsePotential/#Definitions","page":"Morse Potential","title":"Definitions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH psi(r) = E psi(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"and the Hamiltonian","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"MorsePotential/#Parameters","page":"Morse Potential","title":"Parameters","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.MorsePotential","category":"page"},{"location":"MorsePotential/#Antique.MorsePotential-MorsePotential","page":"Morse Potential","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D__mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"MorsePotential/#Potential","page":"Morse Potential","title":"Potential","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.V(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.V-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Values","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.E(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.E-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Number-of-Bound-States","page":"Morse Potential","title":"Number of Bound States","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.nₘₐₓ(::MorsePotential)","category":"page"},{"location":"MorsePotential/#Antique.nₘₐₓ-Tuple{MorsePotential}-MorsePotential","page":"Morse Potential","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Eigen-Functions","page":"Morse Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.ψ(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.ψ-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials","page":"Morse Potential","title":"Generalized Laguerre Polynomials","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Antique.L(::MorsePotential, ::Any)","category":"page"},{"location":"MorsePotential/#Antique.L-Tuple{MorsePotential, Any}-MorsePotential","page":"Morse Potential","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k left(beginarrayl n+alpha n-k endarrayright) fracx^kk \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \nendaligned\n\nExamples:\n\nbeginaligned\n L_0^(0)(x) = 1 \n L_1^(0)(x) = 1 - x \n L_1^(1)(x) = 2 - x \n L_2^(0)(x) = 1 - 2 x + 12 x^2 \n L_2^(1)(x) = 3 - 3 x + 12 x^2 \n L_2^(2)(x) = 6 - 4 x + 12 x^2 \n L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"MorsePotential/#References","page":"Morse Potential","title":"References","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"P. M. Morse, Phys. Rev. 34, 57 (1929)\nJ. P. Dahl, M. Springborg, J. Chem. Phys. 88, 4535 (1988). (62), (63)\nW. K. Shao, Y. He, J. Pan, J. Nonlinear Sci. Appl., 9, 5, 3388 (2016). (1.6) \nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.12","category":"page"},{"location":"MorsePotential/#Usage-and-Examples","page":"Morse Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by MorsePotential and several parameters rₑ, Dₑ, k, µ and ℏ are set as optional arguments.","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"# Parameters for H₂⁺\n# https://doi.org/10.1002/slct.202102509\n# https://doi.org/10.5281/zenodo.5047817\n# https://physics.nist.gov/cgi-bin/cuu/Value?mpsme\nrₑ = 1.997193319969992120068298141276\nDₑ = - 0.5 - (-0.602634619106539878727562156289)\nk = 2*((-1.1026342144949464615+1/2.00) - (-0.602634619106539878727562156289)) / (2.00 - rₑ)^2\nµ = 1/(1/1836.15267343 + 1/1836.15267343)\nℏ = 1.0\n\nusing Antique\nMP = MorsePotential(rₑ=rₑ, Dₑ=Dₑ, k=k, µ=µ, ℏ=ℏ)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Parameters:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> MP.rₑ\n1.997193319969992\n\njulia> MP.Dₑ\n0.10263461910653993\n\njulia> MP.k\n0.1027265041900817\n\njulia> MP.µ\n918.076336715\n\njulia> MP.ℏ\n1.0","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Eigen values:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"julia> E(MP, n=0)\n-0.09741377794418261\n\njulia> E(MP, n=1)\n-0.08738092406760907","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(0.1:0.01:15, r -> V(MP, r), lw=2, label=\"\", xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"r\", ylabel=\"V(r)\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Wave functions:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"using Plots\nplot(xlim=(0,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> ψ(MP, x, n=0), label=\"n=0\", lw=2)\nplot!(x -> ψ(MP, x, n=1), label=\"n=1\", lw=2)\nplot!(x -> ψ(MP, x, n=2), label=\"n=2\", lw=2)\nplot!(x -> ψ(MP, x, n=3), label=\"n=3\", lw=2)\nplot!(x -> ψ(MP, x, n=4), label=\"n=4\", lw=2)\nplot!(x -> ψ(MP, x, n=5), label=\"n=5\", lw=2)","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Potential energy curve, Energy levels, Comparison with harmonic oscillator:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"MP = MorsePotential()\nHO = HarmonicOscillator(k=MP.k, m=MP.μ)\nusing Plots\nplot(xlims=(0.1,9.1), ylims=(-0.11,0.01), xlabel=\"\\$r\\$\", ylabel=\"\\$V(r), E_n\\$\", legend=:bottomright, size=(480,400), dpi=300)\nfor n in 0:nₘₐₓ(MP)\n # energy\n EM = E(MP, n=n)\n EH = E(HO, n=n) - MP.Dₑ\n plot!(0.1:0.01:15, r -> EH > V(HO, r-MP.rₑ) - MP.Dₑ ? EH : NaN, lc=\"#BC1C5F\", lw=1, label=\"\")\n plot!(0.1:0.01:15, r -> EM > V(MP, r) ? EM : NaN, lc=\"#578FC7\", lw=1, label=\"\")\nend\n# potential\nplot!(0.1:0.01:15, r -> V(HO, r-MP.rₑ) - MP.Dₑ, lc=\"#BC1C5F\", lw=2, label=\"Harmonic Oscillator\")\nplot!(0.1:0.01:15, r -> V(MP, r), lc=\"#578FC7\", lw=2, label=\"Morse Potential\")","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"(Image: )","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"where, the potential of harmonic oscillator is defined as V(r) simeq frac12 k (r - r_mathrme)^2 + V_0.","category":"page"},{"location":"MorsePotential/#Testing","page":"Morse Potential","title":"Testing","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"MorsePotential/#Generalized-Laguerre-Polynomials-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Generalized Laguerre Polynomials L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=0 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_0^(0)(x)\n = e^ - x e^x\n = 1 \n = 1\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(0)(x)\n = fracmathrmdmathrmdx x e^ - x e^x\n = 1 - x \n = 1 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=1 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_1^(1)(x)\n = frace^x fracmathrmdmathrmdx x^2 e^ - xx\n = 2 - x \n = 2 - x\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(0)(x)\n = frac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^2 e^ - x e^x\n = 1 - 2 x + frac12 x^2 \n = 1 - 2 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(1)(x)\n = fracfrac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^xx\n = 3 - 3 x + frac12 x^2 \n = 3 - 3 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=2 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_2^(2)(x)\n = fracfrac12 fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^xx^2\n = 6 - 4 x + frac12 x^2 \n = 6 - 4 x + frac12 x^2\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(0)(x)\n = frac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^3 e^ - x e^x\n = 1 - 3 x + frac32 x^2 - frac16 x^3 \n = 1 - 3 x + frac32 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(1)(x)\n = fracfrac16 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - xx\n = 4 - 6 x + 2 x^2 - frac16 x^3 \n = 4 - 6 x + 2 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(2)(x)\n = fracfrac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - x e^xx^2\n = 10 - 10 x + frac52 x^2 - frac16 x^3 \n = 10 - 10 x + frac52 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=3 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_3^(3)(x)\n = fracfrac16 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - x e^xx^3\n = 20 - 15 x + 3 x^2 - frac16 x^3 \n = 20 - 15 x + 3 x^2 - frac16 x^3\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=0 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(0)(x)\n = frac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^4 e^ - x e^x\n = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4 \n = 1 - 4 x + 3 x^2 - frac23 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=1 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(1)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^5 e^ - xx\n = 5 - 10 x + 5 x^2 - frac56 x^3 + frac124 x^4 \n = 5 - 10 x + 5 x^2 - frac56 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=2 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(2)(x)\n = fracfrac124 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^6 e^ - x e^xx^2\n = 15 - 20 x + frac152 x^2 - x^3 + frac124 x^4 \n = 15 - 20 x + frac152 x^2 - x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=3 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(3)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^7 e^ - xx^3\n = 35 - 35 x + frac212 x^2 - frac76 x^3 + frac124 x^4 \n = 35 - 35 x + frac212 x^2 - frac76 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"n=4 α=4 ✔","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n L_4^(4)(x)\n = fracfrac124 e^x fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx x^8 e^ - xx^4\n = 70 - 56 x + 14 x^2 - frac43 x^3 + frac124 x^4 \n = 70 - 56 x + 14 x^2 - frac43 x^3 + frac124 x^4\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-L_n{(\\alpha)}(x)","page":"Morse Potential","title":"Normalization & Orthogonality of L_n^(alpha)(x)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty L_i^(alpha)(x) L_j^(alpha)(x) x^alpha mathrme^-x mathrmdx = fracGamma(n+alpha+1)n delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" α | i | j | analytical | numerical \n---- | -- | -- | ----------------- | ----------------- \n0.01 | 0 | 0 | 0.994325851192 | 0.994325852936 ✔\n0.01 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 1 | 1.004269109703 | 1.004269111483 ✔\n0.01 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 1 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 2 | 2 | 1.009290455252 | 1.009290456144 ✔\n0.01 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 2 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 2 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 3 | 3 | 1.012654756769 | 1.012654758579 ✔\n0.01 | 3 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 3 | 7 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 3 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 3 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 4 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 4 | 4 | 1.015186393661 | 1.015186394564 ✔\n0.01 | 4 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 4 | 6 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 4 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 4 | 9 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.01 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 5 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 5 | 5 | 1.017216766449 | 1.017216768275 ✔\n0.01 | 5 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 5 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 6 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 6 | 4 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 6 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 6 | 1.018912127726 | 1.018912128636 ✔\n0.01 | 6 | 7 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 6 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 6 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 7 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 7 | 3 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 7 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 6 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 7 | 7 | 1.020367716480 | 1.020367717392 ✔\n0.01 | 7 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 7 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 8 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.01 | 8 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 8 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 5 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 8 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 8 | 8 | 1.021643176126 | 1.021643177967 ✔\n0.01 | 8 | 9 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.01 | 9 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.01 | 9 | 2 | 0.000000000000 | 0.000000000007 ✔\n0.01 | 9 | 3 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.01 | 9 | 5 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 6 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 7 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 8 | 0.000000000000 | 0.000000000028 ✔\n0.01 | 9 | 9 | 1.022778335210 | 1.022778336127 ✔\n0.05 | 0 | 0 | 0.973504265563 | 0.973504267703 ✔\n0.05 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 8 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 0 | 9 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 1 | 1.022179478841 | 1.022179479980 ✔\n0.05 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 1 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 5 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 6 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 7 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 1 | 8 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 1 | 9 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 2 | 2 | 1.047733965812 | 1.047733966390 ✔\n0.05 | 2 | 3 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 4 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 2 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 6 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 2 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 2 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 3 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 3 | 3 | 1.065196198575 | 1.065196199813 ✔\n0.05 | 3 | 4 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 5 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 3 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 3 | 8 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 3 | 9 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.05 | 4 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 2 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 4 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 4 | 4 | 1.078511151058 | 1.078511152326 ✔\n0.05 | 4 | 5 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 4 | 7 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 8 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 4 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 5 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 5 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 3 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 5 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 5 | 5 | 1.089296262568 | 1.089296263862 ✔\n0.05 | 5 | 6 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 5 | 7 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 5 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 5 | 9 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.05 | 6 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 6 | 2 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 6 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 4 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 6 | 5 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 6 | 6 | 1.098373731423 | 1.098373732739 ✔\n0.05 | 6 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 6 | 9 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 1 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 7 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 3 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 7 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 7 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 7 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 7 | 1.106219258076 | 1.106219258720 ✔\n0.05 | 7 | 8 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 7 | 9 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 8 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 8 | 1 | 0.000000000000 | 0.000000000004 ✔\n0.05 | 8 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 8 | 3 | 0.000000000000 | 0.000000000016 ✔\n0.05 | 8 | 4 | 0.000000000000 | 0.000000000017 ✔\n0.05 | 8 | 5 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 7 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 8 | 8 | 1.113133128439 | 1.113133129790 ✔\n0.05 | 8 | 9 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 0 | 0.000000000000 | 0.000000000002 ✔\n0.05 | 9 | 1 | 0.000000000000 | 0.000000000007 ✔\n0.05 | 9 | 2 | 0.000000000000 | 0.000000000008 ✔\n0.05 | 9 | 3 | 0.000000000000 | 0.000000000015 ✔\n0.05 | 9 | 4 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 5 | 0.000000000000 | 0.000000000034 ✔\n0.05 | 9 | 6 | 0.000000000000 | 0.000000000035 ✔\n0.05 | 9 | 7 | 0.000000000000 | 0.000000000036 ✔\n0.05 | 9 | 8 | 0.000000000000 | 0.000000000074 ✔\n0.05 | 9 | 9 | 1.119317201375 | 1.119317202034 ✔\n0.10 | 0 | 0 | 0.951350769867 | 0.951350771636 ✔\n0.10 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 0 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 0 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 1 | 1 | 1.046485846854 | 1.046485847852 ✔\n0.10 | 1 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 1 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 1 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 2 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 2 | 1.098810139196 | 1.098810140297 ✔\n0.10 | 2 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 2 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 2 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 8 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 2 | 9 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 3 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 3 | 3 | 1.135437143836 | 1.135437145012 ✔\n0.10 | 3 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 3 | 6 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 7 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 3 | 8 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 3 | 9 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.10 | 4 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 2 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 4 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 4 | 1.163823072432 | 1.163823073667 ✔\n0.10 | 4 | 5 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 4 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 7 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 4 | 8 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 4 | 9 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 5 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 3 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 5 | 4 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 5 | 5 | 1.187099533881 | 1.187099535166 ✔\n0.10 | 5 | 6 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 5 | 7 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 5 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 5 | 9 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 1 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 6 | 2 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 6 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 6 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 5 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 6 | 6 | 1.206884526112 | 1.206884527440 ✔\n0.10 | 6 | 7 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 8 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 6 | 9 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 7 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 7 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 7 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 3 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 7 | 4 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 7 | 5 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 7 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 7 | 7 | 1.224125733628 | 1.224125734265 ✔\n0.10 | 7 | 8 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 7 | 9 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 8 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 8 | 1 | 0.000000000000 | 0.000000000003 ✔\n0.10 | 8 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 8 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 8 | 4 | 0.000000000000 | 0.000000000014 ✔\n0.10 | 8 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 6 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 8 | 7 | 0.000000000000 | 0.000000000066 ✔\n0.10 | 8 | 8 | 1.239427305298 | 1.239427306699 ✔\n0.10 | 8 | 9 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 0 | 0.000000000000 | 0.000000000001 ✔\n0.10 | 9 | 1 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 2 | 0.000000000000 | 0.000000000006 ✔\n0.10 | 9 | 3 | 0.000000000000 | 0.000000000013 ✔\n0.10 | 9 | 4 | 0.000000000000 | 0.000000000029 ✔\n0.10 | 9 | 5 | 0.000000000000 | 0.000000000030 ✔\n0.10 | 9 | 6 | 0.000000000000 | 0.000000000065 ✔\n0.10 | 9 | 7 | 0.000000000000 | 0.000000000031 ✔\n0.10 | 9 | 8 | 0.000000000000 | 0.000000000067 ✔\n0.10 | 9 | 9 | 1.253198719802 | 1.253198721234 ✔\n0.50 | 0 | 0 | 0.886226925453 | 0.886226925863 ✔\n0.50 | 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 0 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 1 | 1.329340388179 | 1.329340389103 ✔\n0.50 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 2 | 1.661675485224 | 1.661675485734 ✔\n0.50 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 3 | 1.938621399428 | 1.938621400123 ✔\n0.50 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 3 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 3 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 4 | 4 | 2.180949074356 | 2.180949075236 ✔\n0.50 | 4 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 8 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 4 | 9 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 5 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 5 | 2.399043981792 | 2.399043982856 ✔\n0.50 | 5 | 6 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 7 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 5 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 5 | 9 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 6 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 6 | 6 | 2.598964313608 | 2.598964314050 ✔\n0.50 | 6 | 7 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 8 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 6 | 9 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 7 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 7 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 7 | 7 | 2.784604621723 | 2.784604622230 ✔\n0.50 | 7 | 8 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 7 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 8 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 4 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 8 | 5 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 6 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 8 | 7 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 8 | 8 | 2.958642410581 | 2.958642412199 ✔\n0.50 | 8 | 9 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 0 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n0.50 | 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n0.50 | 9 | 3 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 4 | 0.000000000000 | 0.000000000003 ✔\n0.50 | 9 | 5 | 0.000000000000 | 0.000000000001 ✔\n0.50 | 9 | 6 | 0.000000000000 | 0.000000000008 ✔\n0.50 | 9 | 7 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 8 | 0.000000000000 | 0.000000000009 ✔\n0.50 | 9 | 9 | 3.123011433391 | 3.123011435194 ✔\n1.00 | 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n1.00 | 0 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 1 | 2.000000000000 | 2.000000000000 ✔\n1.00 | 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 2 | 3.000000000000 | 3.000000000000 ✔\n1.00 | 2 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 3 | 4.000000000000 | 4.000000000000 ✔\n1.00 | 3 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 3 | 9 | 0.000000000000 | -0.000000000001 ✔\n1.00 | 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 4 | 5.000000000000 | 4.999999999999 ✔\n1.00 | 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 5 | 6.000000000000 | 6.000000000000 ✔\n1.00 | 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 5 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 6 | 6 | 7.000000000000 | 7.000000000000 ✔\n1.00 | 6 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 4 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 7 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 7 | 8.000000000000 | 8.000000000000 ✔\n1.00 | 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 8 | 8 | 9.000000000000 | 9.000000000000 ✔\n1.00 | 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 3 | 0.000000000000 | -0.000000000001 ✔\n1.00 | 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 5 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n1.00 | 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n1.00 | 9 | 9 | 10.000000000000 | 10.000000000002 ✔","category":"page"},{"location":"MorsePotential/#Normalization-and-Orthogonality-of-\\psi_n(r)","page":"Morse Potential","title":"Normalization & Orthogonality of psi_n(r)","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"int_0^infty psi_i^ast(r) psi_j(r) mathrmdr = delta_ij","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | -0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | 0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | -0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000002 ✔\n 0 | 8 | 0.000000000000 | -0.000000000026 ✔\n 0 | 9 | 0.000000000000 | -0.000000000104 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | -0.000000000000 ✔\n 1 | 3 | 0.000000000000 | 0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | -0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | 0.000000000001 ✔\n 1 | 8 | 0.000000000000 | -0.000000000022 ✔\n 1 | 9 | 0.000000000000 | -0.000000000067 ✔\n 2 | 0 | 0.000000000000 | -0.000000000000 ✔\n 2 | 1 | 0.000000000000 | -0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000009 ✔\n 2 | 9 | 0.000000000000 | -0.000000000030 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | 0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000001 ✔\n 3 | 8 | 0.000000000000 | -0.000000000002 ✔\n 3 | 9 | 0.000000000000 | -0.000000000006 ✔\n 4 | 0 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000001 ✔\n 4 | 9 | 0.000000000000 | 0.000000000001 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | -0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | -0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000001 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | -0.000000000001 ✔\n 6 | 0 | 0.000000000000 | -0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | -0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000002 ✔\n 6 | 9 | 0.000000000000 | -0.000000000003 ✔\n 7 | 0 | 0.000000000000 | 0.000000000002 ✔\n 7 | 1 | 0.000000000000 | 0.000000000001 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000001 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000001 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000004 ✔\n 8 | 0 | 0.000000000000 | -0.000000000026 ✔\n 8 | 1 | 0.000000000000 | -0.000000000022 ✔\n 8 | 2 | 0.000000000000 | -0.000000000009 ✔\n 8 | 3 | 0.000000000000 | -0.000000000002 ✔\n 8 | 4 | 0.000000000000 | -0.000000000001 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000002 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 0.999999999995 ✔\n 8 | 9 | 0.000000000000 | 0.000000000000 ✔\n 9 | 0 | 0.000000000000 | -0.000000000104 ✔\n 9 | 1 | 0.000000000000 | -0.000000000067 ✔\n 9 | 2 | 0.000000000000 | -0.000000000030 ✔\n 9 | 3 | 0.000000000000 | -0.000000000006 ✔\n 9 | 4 | 0.000000000000 | 0.000000000001 ✔\n 9 | 5 | 0.000000000000 | -0.000000000001 ✔\n 9 | 6 | 0.000000000000 | -0.000000000003 ✔\n 9 | 7 | 0.000000000000 | 0.000000000004 ✔\n 9 | 8 | 0.000000000000 | 0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000015 ✔","category":"page"},{"location":"MorsePotential/#Eigen-Values-2","page":"Morse Potential","title":"Eigen Values","text":"","category":"section"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" beginaligned\n E_n\n = int psi^ast_n(r) hatH psi_n(r) mathrmdx \n = int psi^ast_n(r) left hatV + hatT right psi(r) mathrmdx \n = int psi^ast_n(r) left V(r) - frachbar^22m fracmathrmd^2mathrmd r^2 right psi(r) mathrmdx \n simeq int psi^ast_n(r) left V(r)psi(r) -frachbar^22m fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2 right mathrmdx\n endaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\n 2psi(r)\n + fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n + Oleft(Delta r^4right)\n =\n psi(r+Delta r)\n + psi(r-Delta r)\n \n fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n =\n psi(r+Delta r)\n - 2psi(r)\n + psi(r-Delta r)\n - Oleft(Delta r^4right)\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n - fracOleft(Delta r^4right)Delta r^2\n \n fracmathrmd^2 psi(r)mathrmd r^2\n =\n fracpsi(r+Delta r) - 2psi(r) + psi(r-Delta r)Delta r^2\n + Oleft(Delta r^2right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":"beginaligned\npsi(r+Delta r)\n= psi(r)\n+ fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n+ frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\n\npsi(r-Delta r)\n= psi(r)\n- fracmathrmd psi(r)mathrmd r Delta r\n+ frac12 fracmathrmd^2 psi(r)mathrmd r^2 Delta r^2\n- frac13 fracmathrmd^3 psi(r)mathrmd r^3 Delta r^3\n+ Oleft(Delta r^4right)\nendaligned","category":"page"},{"location":"MorsePotential/","page":"Morse Potential","title":"Morse Potential","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | -0.097482629904 | -0.097482629943 ✔\n0.1 | 1 | -0.087576629073 | -0.087576629208 ✔\n0.1 | 2 | -0.078201265005 | -0.078201265359 ✔\n0.1 | 3 | -0.069356537702 | -0.069356538266 ✔\n0.1 | 4 | -0.061042447162 | -0.061042448777 ✔\n0.1 | 5 | -0.053258993386 | -0.053258996131 ✔\n0.1 | 6 | -0.046006176374 | -0.046006177829 ✔\n0.1 | 7 | -0.039283996126 | -0.039283997743 ✔\n0.1 | 8 | -0.033092452642 | -0.033092467851 ✔\n0.1 | 9 | -0.027431545922 | -0.027431467792 ✔\n0.2 | 0 | -0.095387461081 | -0.095387461144 ✔\n0.2 | 1 | -0.081689100176 | -0.081689100427 ✔\n0.2 | 2 | -0.069052012799 | -0.069052013380 ✔\n0.2 | 3 | -0.057476198949 | -0.057476199867 ✔\n0.2 | 4 | -0.046961658628 | -0.046961660317 ✔\n0.2 | 5 | -0.037508391834 | -0.037508393202 ✔\n0.2 | 6 | -0.029116398568 | -0.029116400340 ✔\n0.2 | 7 | -0.021785678830 | -0.021785684062 ✔\n0.2 | 8 | -0.015516232619 | -0.015516237539 ✔\n0.2 | 9 | -0.010308059937 | -0.010308062755 ✔\n0.3 | 0 | -0.093795214605 | -0.093795214695 ✔\n0.3 | 1 | -0.077310338322 | -0.077310338694 ✔\n0.3 | 2 | -0.062417372330 | -0.062417373167 ✔\n0.3 | 3 | -0.049116316630 | -0.049116318029 ✔\n0.3 | 4 | -0.037407171221 | -0.037407173073 ✔\n0.3 | 5 | -0.027289936105 | -0.027289938027 ✔\n0.3 | 6 | -0.018764611280 | -0.018764613693 ✔\n0.3 | 7 | -0.011831196747 | -0.011831198102 ✔\n0.3 | 8 | -0.006489692505 | -0.006489694275 ✔\n0.3 | 9 | -0.002740098556 | -0.002740100893 ✔\n0.1 | 0 | -0.097413777944 | -0.097413777967 ✔\n0.1 | 1 | -0.087380924068 | -0.087380924205 ✔\n0.1 | 2 | -0.077893174789 | -0.077893175145 ✔\n0.1 | 3 | -0.068950530107 | -0.068950530660 ✔\n0.1 | 4 | -0.060552990023 | -0.060552989095 ✔\n0.1 | 5 | -0.052700554537 | -0.052700557255 ✔\n0.1 | 6 | -0.045393223648 | -0.045393222818 ✔\n0.1 | 7 | -0.038630997356 | -0.038631017157 ✔\n0.1 | 8 | -0.032413875662 | -0.032413886246 ✔\n0.1 | 9 | -0.026741858566 | -0.026742018376 ✔\n","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"CurrentModule = Antique","category":"page"},{"location":"HydrogenAtom/#Hydrogen-Atom","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"The hydrogen atom is the simplest 2-body Coulomb system.","category":"page"},{"location":"HydrogenAtom/#Definitions","page":"Hydrogen Atom","title":"Definitions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH psi(pmbr) = E psi(pmbr)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"and the Hamiltonian","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" hatH = - frachbar^22mu fracmathrmd^2mathrmdr ^2 + V(r)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"where mu=left(frac1m_mathrme+frac1m_mathrmpright)^-1 is the reduced mass of electron mathrme and proton mathrmp. mu = m_mathrme holds in the limit m_mathrmprightarrowinfty. Parameters are specified with the following struct.","category":"page"},{"location":"HydrogenAtom/#Parameters","page":"Hydrogen Atom","title":"Parameters","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.HydrogenAtom","category":"page"},{"location":"HydrogenAtom/#Antique.HydrogenAtom-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HydrogenAtom/#Potential","page":"Hydrogen Atom","title":"Potential","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.V(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.V-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Values","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.E(::HydrogenAtom)","category":"page"},{"location":"HydrogenAtom/#Antique.E-Tuple{HydrogenAtom}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Eigen-Functions","page":"Hydrogen Atom","title":"Eigen Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.ψ(::HydrogenAtom, ::Any, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Radial-Functions","page":"Hydrogen Atom","title":"Radial Functions","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.R(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.R-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)\n\nwhere Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Laguerre-Polynomials","page":"Hydrogen Atom","title":"Associated Laguerre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.L(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.L-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\n L_0^0(x) = 1 \n L_1^0(x) = 1 - x \n L_1^1(x) = 1 \n L_2^0(x) = 1 - 2 x + 12 x^2 \n L_2^1(x) = 2 - x \n L_2^2(x) = 1 \n L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^1(x) = 3 - 3 x + 12 x^2 \n L_3^2(x) = 3 - x \n L_3^3(x) = 1 \n L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_4^2(x) = 6 - 4 x + 12 x^2 \n L_4^3(x) = 4 - x \n L_4^4(x) = 1 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Spherical-Harmonics","page":"Hydrogen Atom","title":"Spherical Harmonics","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.Y(::HydrogenAtom, ::Any, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.Y-Tuple{HydrogenAtom, Any, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#Associated-Legendre-Polynomials","page":"Hydrogen Atom","title":"Associated Legendre Polynomials","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Antique.P(::HydrogenAtom, ::Any)","category":"page"},{"location":"HydrogenAtom/#Antique.P-Tuple{HydrogenAtom, Any}-HydrogenAtom","page":"Hydrogen Atom","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\nendaligned\n\nwhere Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.\n\nExamples:\n\nbeginaligned\n P_0^0(x) = 1 \n P_1^0(x) = x \n P_1^1(x) = left(+1right)sqrt1-x^2 \n P_2^0(x) = -12 + 32 x^2 \n P_2^1(x) = left(-3 xright)sqrt1-x^2 \n P_2^2(x) = 3 - 6 x \n P_3^0(x) = -32 x + 52 x^3 \n P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n P_3^2(x) = 15 x - 30 x^2 \n P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n P_4^4(x) = 105 - 420 x + 420 x^2 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HydrogenAtom/#References","page":"Hydrogen Atom","title":"References","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"cpprefjp, legendre, assoc_legendre, laguerre, assoc_laguerre\nThe Digital Library of Mathematical Functions (DLMF), 18.3 Table1, 18.5 Table1, 18.5.16, 18.3 Table1, 18.5 Table1, 18.5.17, 18.3 Table1, 18.5 Table1, 18.5.12\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965), p.598 (c.1), p.598 (c.4), p.603 (d.13), p.603 (d.13)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968), p.79 (14.12), p.93 (16.19)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999), p.493 (B.72), p.494 Table, p.493 (B.72), p.483 (B.12), p.483 (B.12)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994), p.83 (4), p.83 (5), p.149 (21)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995), p.126 (4.28), p.96 Table3.1, p.126 (4.27), p.139 (4.88), p.140 Table4.4, p.139 (4.87), p.140 Table4.5\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997), p.195 Table6.1, p.196 (6.26), p.196 Table6.2, p.207 Table6.4\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008), p.234\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021), p.245 Problem 3.30.b, ","category":"page"},{"location":"HydrogenAtom/#Usage-and-Examples","page":"Hydrogen Atom","title":"Usage & Examples","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HydrogenAtom and several parameters Z, Eₕ, mₑ, a₀ and ℏ are set as optional arguments.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Antique\nH = HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Parameters:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> H.Z\n1\n\njulia> H.Eₕ\n1.0\n\njulia> H.mₑ\n1.0\n\njulia> H.a₀\n1.0\n\njulia> H.ℏ\n1.0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eigen values:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"julia> E(H, n=1)\n-0.5\n\njulia> E(H, n=2)\n-0.125","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Wave length (n=2rightarrow1, the first line of the Lyman series):","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Eₕ2nm⁻¹ = 2.1947463136320e-2 # https://physics.nist.gov/cgi-bin/cuu/CCValue?hrminv\nprintln(\"ΔE = \", E(H,n=2) - E(H,n=1), \" Eₕ\")\nprintln(\"λ = \", ((E(H,n=2)-E(H,n=1))*Eₕ2nm⁻¹)^-1, \" nm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"ΔE = 0.375 Eₕ\nλ = 121.50227341098497 nm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Hyperfine Splitting:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"# constants: https://doi.org/10.1103/RevModPhys.93.025010\ne = 1.602176634e-19 # C https://physics.nist.gov/cgi-bin/cuu/Value?e\nh = 6.62607015e-34 # J Hz-1 https://physics.nist.gov/cgi-bin/cuu/Value?h\nc = 299792458 # m s-1 https://physics.nist.gov/cgi-bin/cuu/Value?c\na0 = 5.29177210903e-11 # m https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0\nµ0 = 1.25663706212e-6 # N A-2 https://physics.nist.gov/cgi-bin/cuu/Value?mu0\nµB = 9.2740100783e-24 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mub\nµN = 5.0507837461e-27 # J T-1 https://physics.nist.gov/cgi-bin/cuu/Value?mun\nge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem\ngp = 5.5856946893 # https://physics.nist.gov/cgi-bin/cuu/Value?gp\n\n# calculation: https://doi.org/10.1119/1.12733\nδ = abs(ψ(H,0,0,0))^2\nΔE = 2 / 3 * µ0 * µN * µB * gp * ge * δ * a0^(-3)\nprintln(\"1/π = \", 1/π)\nprintln(\"<δ(r)> = \", δ, \" a₀⁻³\")\nprintln(\"<δ(r)> = \", δ * a0^(-3), \" m⁻³\")\nprintln(\"ΔE = \", ΔE, \" J\")\nprintln(\"ν = ΔE/h = \", ΔE / h * 1e-6, \" MHz\")\nprintln(\"λ = hc/ΔE = \", h*c/ΔE*100, \" cm\")","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"1/π = 0.3183098861837907\n<δ(r)> = 0.3183098861837908 a₀⁻³\n<δ(r)> = 2.1480615849063944e30 m⁻³\nΔE = 9.427622831641132e-25 J\nν = ΔE/h = 1422.8075794882932 MHz\nλ = hc/ΔE = 21.070485027063118 cm","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h},~E_n/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nplot!(0.1:0.01:15, r -> V(H,r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Potential energy curve, Energy levels:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlims=(0.0,15.0), ylims=(-0.6,0.05), xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$V(r)/E_\\\\mathrm{h}\\$\", legend=:bottomright, size=(480,400))\nfor n in 0:10\n plot!(0.0:0.01:15, r -> E(H,n=n) > V(H,r) ? E(H,n=n) : NaN, lc=n, lw=1, label=\"\") # energy level\nend\nplot!(0.1:0.01:15, r -> V(H,r), lc=:black, lw=2, label=\"\") # potential","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Radial functions:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"using Plots\nplot(xlabel=\"\\$r~/~a_0\\$\", ylabel=\"\\$r^2|R_{nl}(r)|^2~/~a_0^{-1}\\$\", ylims=(-0.01,0.55), xticks=0:1:20, size=(480,400), dpi=300)\nfor n in 1:3\n for l in 0:n-1\n plot!(0:0.01:20, r->r^2*R(H,r,n=n,l=l)^2, lc=n, lw=2, ls=[:solid,:dash,:dot,:dashdot,:dashdotdot][l+1], label=\"\\$n = $n, l=$l\\$\")\n end\nend\nplot!()","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"(Image: )","category":"page"},{"location":"HydrogenAtom/#Testing","page":"Hydrogen Atom","title":"Testing","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HydrogenAtom/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Hydrogen Atom","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"int_0^L psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | 0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | -0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | -0.000000000000 ✔\n 1 | 10 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | 0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | 0.000000000000 ✔\n 2 | 9 | 0.000000000000 | -0.000000000000 ✔\n 2 | 10 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | -0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | 0.000000000000 ✔\n 3 | 9 | 0.000000000000 | -0.000000000000 ✔\n 3 | 10 | 0.000000000000 | 0.000000000000 ✔\n 4 | 1 | 0.000000000000 | 0.000000000000 ✔\n 4 | 2 | 0.000000000000 | 0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | -0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | 0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 4 | 10 | 0.000000000000 | 0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | -0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | -0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 10 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | -0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | -0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | -0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 10 | 0.000000000000 | -0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | -0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | -0.000000000000 ✔\n 7 | 9 | 0.000000000000 | -0.000000000000 ✔\n 7 | 10 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | 0.000000000000 ✔\n 8 | 3 | 0.000000000000 | 0.000000000000 ✔\n 8 | 4 | 0.000000000000 | 0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | -0.000000000000 ✔\n 8 | 7 | 0.000000000000 | -0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 8 | 10 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | -0.000000000000 ✔\n 9 | 2 | 0.000000000000 | -0.000000000000 ✔\n 9 | 3 | 0.000000000000 | -0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | -0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔\n 9 | 10 | 0.000000000000 | 0.000000000000 ✔\n10 | 1 | 0.000000000000 | 0.000000000000 ✔\n10 | 2 | 0.000000000000 | 0.000000000000 ✔\n10 | 3 | 0.000000000000 | 0.000000000000 ✔\n10 | 4 | 0.000000000000 | 0.000000000000 ✔\n10 | 5 | 0.000000000000 | 0.000000000000 ✔\n10 | 6 | 0.000000000000 | -0.000000000000 ✔\n10 | 7 | 0.000000000000 | -0.000000000000 ✔\n10 | 8 | 0.000000000000 | 0.000000000000 ✔\n10 | 9 | 0.000000000000 | 0.000000000000 ✔\n10 | 10 | 1.000000000000 | 1.000000000000 ✔","category":"page"},{"location":"HydrogenAtom/#Eigen-Values-2","page":"Hydrogen Atom","title":"Eigen Values","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n E_n\n = int_0^L psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int_0^L psi^ast_n(x) left 0 - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | m | ℏ | n | analytical | numerical \n--- | --- | --- | -- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1 | 49.348021579139 | 49.348022005447 ✔\n0.1 | 0.1 | 0.1 | 2 | 197.392081461942 | 197.392088021787 ✔\n0.1 | 0.1 | 0.1 | 3 | 444.132165131018 | 444.132198049021 ✔\n0.1 | 0.1 | 0.1 | 4 | 789.568248175979 | 789.568352087149 ✔\n0.1 | 0.1 | 0.1 | 5 | 1233.700296336187 | 1233.700550136170 ✔\n0.1 | 0.1 | 0.1 | 6 | 1776.528266243334 | 1776.528792196084 ✔\n0.1 | 0.1 | 0.1 | 7 | 2418.052103857080 | 2418.053078266893 ✔\n0.1 | 0.1 | 0.1 | 8 | 3158.271745875927 | 3158.273408348594 ✔\n0.1 | 0.1 | 0.1 | 9 | 3997.187119264267 | 3997.189782441189 ✔\n0.1 | 0.1 | 0.1 | 10 | 4934.798141994514 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 1 | 4934.802157913905 | 4934.802200544678 ✔\n0.1 | 0.1 | 1.0 | 2 | 19739.208146194214 | 19739.208802178713 ✔\n0.1 | 0.1 | 1.0 | 3 | 44413.216513101754 | 44413.219804902103 ✔\n0.1 | 0.1 | 1.0 | 4 | 78956.824817597895 | 78956.835208714852 ✔\n0.1 | 0.1 | 1.0 | 5 | 123370.029633618717 | 123370.055013616948 ✔\n0.1 | 0.1 | 1.0 | 6 | 177652.826624333364 | 177652.879219608410 ✔\n0.1 | 0.1 | 1.0 | 7 | 241805.210385707964 | 241805.307826689212 ✔\n0.1 | 0.1 | 1.0 | 8 | 315827.174587592541 | 315827.340834859409 ✔\n0.1 | 0.1 | 1.0 | 9 | 399718.711926426622 | 399718.978244118916 ✔\n0.1 | 0.1 | 1.0 | 10 | 493479.814199451241 | 493480.220054467791 ✔\n0.1 | 1.0 | 0.1 | 1 | 4.934802157914 | 4.934802200545 ✔\n0.1 | 1.0 | 0.1 | 2 | 19.739208146194 | 19.739208802179 ✔\n0.1 | 1.0 | 0.1 | 3 | 44.413216513102 | 44.413219804902 ✔\n0.1 | 1.0 | 0.1 | 4 | 78.956824817598 | 78.956835208715 ✔\n0.1 | 1.0 | 0.1 | 5 | 123.370029633619 | 123.370055013617 ✔\n0.1 | 1.0 | 0.1 | 6 | 177.652826624333 | 177.652879219608 ✔\n0.1 | 1.0 | 0.1 | 7 | 241.805210385708 | 241.805307826689 ✔\n0.1 | 1.0 | 0.1 | 8 | 315.827174587593 | 315.827340834859 ✔\n0.1 | 1.0 | 0.1 | 9 | 399.718711926427 | 399.718978244119 ✔\n0.1 | 1.0 | 0.1 | 10 | 493.479814199451 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 1 | 493.480215791390 | 493.480220054468 ✔\n0.1 | 1.0 | 1.0 | 2 | 1973.920814619422 | 1973.920880217871 ✔\n0.1 | 1.0 | 1.0 | 3 | 4441.321651310176 | 4441.321980490210 ✔\n0.1 | 1.0 | 1.0 | 4 | 7895.682481759791 | 7895.683520871485 ✔\n0.1 | 1.0 | 1.0 | 5 | 12337.002963361872 | 12337.005501361695 ✔\n0.1 | 1.0 | 1.0 | 6 | 17765.282662433339 | 17765.287921960840 ✔\n0.1 | 1.0 | 1.0 | 7 | 24180.521038570794 | 24180.530782668924 ✔\n0.1 | 1.0 | 1.0 | 8 | 31582.717458759253 | 31582.734083485939 ✔\n0.1 | 1.0 | 1.0 | 9 | 39971.871192642662 | 39971.897824411892 ✔\n0.1 | 1.0 | 1.0 | 10 | 49347.981419945128 | 49348.022005446779 ✔\n1.0 | 0.1 | 0.1 | 1 | 0.493480215948 | 0.493480220054 ✔\n1.0 | 0.1 | 0.1 | 2 | 1.973920815419 | 1.973920880218 ✔\n1.0 | 0.1 | 0.1 | 3 | 4.441321651944 | 4.441321980490 ✔\n1.0 | 0.1 | 0.1 | 4 | 7.895682481265 | 7.895683520871 ✔\n1.0 | 0.1 | 0.1 | 5 | 12.337002965030 | 12.337005501362 ✔\n1.0 | 0.1 | 0.1 | 6 | 17.765282661715 | 17.765287921961 ✔\n1.0 | 0.1 | 0.1 | 7 | 24.180521036064 | 24.180530782669 ✔\n1.0 | 0.1 | 0.1 | 8 | 31.582717460023 | 31.582734083486 ✔\n1.0 | 0.1 | 0.1 | 9 | 39.971871195191 | 39.971897824412 ✔\n1.0 | 0.1 | 0.1 | 10 | 49.347981417827 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 1 | 49.348021594816 | 49.348022005447 ✔\n1.0 | 0.1 | 1.0 | 2 | 197.392081541864 | 197.392088021787 ✔\n1.0 | 0.1 | 1.0 | 3 | 444.132165194438 | 444.132198049021 ✔\n1.0 | 0.1 | 1.0 | 4 | 789.568248126463 | 789.568352087149 ✔\n1.0 | 0.1 | 1.0 | 5 | 1233.700296503016 | 1233.700550136170 ✔\n1.0 | 0.1 | 1.0 | 6 | 1776.528266171473 | 1776.528792196084 ✔\n1.0 | 0.1 | 1.0 | 7 | 2418.052103606433 | 2418.053078266892 ✔\n1.0 | 0.1 | 1.0 | 8 | 3158.271746002275 | 3158.273408348594 ✔\n1.0 | 0.1 | 1.0 | 9 | 3997.187119519121 | 3997.189782441190 ✔\n1.0 | 0.1 | 1.0 | 10 | 4934.798141782662 | 4934.802200544679 ✔\n1.0 | 1.0 | 0.1 | 1 | 0.049348021595 | 0.049348022005 ✔\n1.0 | 1.0 | 0.1 | 2 | 0.197392081542 | 0.197392088022 ✔\n1.0 | 1.0 | 0.1 | 3 | 0.444132165194 | 0.444132198049 ✔\n1.0 | 1.0 | 0.1 | 4 | 0.789568248126 | 0.789568352087 ✔\n1.0 | 1.0 | 0.1 | 5 | 1.233700296503 | 1.233700550136 ✔\n1.0 | 1.0 | 0.1 | 6 | 1.776528266171 | 1.776528792196 ✔\n1.0 | 1.0 | 0.1 | 7 | 2.418052103606 | 2.418053078267 ✔\n1.0 | 1.0 | 0.1 | 8 | 3.158271746002 | 3.158273408349 ✔\n1.0 | 1.0 | 0.1 | 9 | 3.997187119519 | 3.997189782441 ✔\n1.0 | 1.0 | 0.1 | 10 | 4.934798141783 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 1 | 4.934802159482 | 4.934802200545 ✔\n1.0 | 1.0 | 1.0 | 2 | 19.739208154186 | 19.739208802179 ✔\n1.0 | 1.0 | 1.0 | 3 | 44.413216519444 | 44.413219804902 ✔\n1.0 | 1.0 | 1.0 | 4 | 78.956824812646 | 78.956835208715 ✔\n1.0 | 1.0 | 1.0 | 5 | 123.370029650302 | 123.370055013617 ✔\n1.0 | 1.0 | 1.0 | 6 | 177.652826617147 | 177.652879219608 ✔\n1.0 | 1.0 | 1.0 | 7 | 241.805210360643 | 241.805307826689 ✔\n1.0 | 1.0 | 1.0 | 8 | 315.827174600228 | 315.827340834859 ✔\n1.0 | 1.0 | 1.0 | 9 | 399.718711951912 | 399.718978244119 ✔\n1.0 | 1.0 | 1.0 | 10 | 493.479814178266 | 493.480220054468 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-x","page":"Hydrogen Atom","title":"Expected Value of x","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle x rangle_n=1\n= int_0^L psi_1^ast(x) hatx psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"for only n=1.","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.050000000000 | 0.050000000000 ✔\n0.5 | 1 | 0.250000000000 | 0.250000000000 ✔\n1.0 | 1 | 0.500000000000 | 0.500000000000 ✔\n7.0 | 1 | 3.500000000000 | 3.500000000000 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-x2","page":"Hydrogen Atom","title":"Expected Value of x^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle x^2 rangle_n=1\n= int_0^L psi_1^ast(x) hatx^2 psi_1(x) mathrmdx\n= frac2(2a)^2pi^3 left( fracpi^36 - fracpi4 right)","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.002826727415 | 0.002826727415 ✔\n0.5 | 1 | 0.070668185378 | 0.070668185378 ✔\n1.0 | 1 | 0.282672741512 | 0.282672741512 ✔\n7.0 | 1 | 13.850964334096 | 13.850964334096 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-p","page":"Hydrogen Atom","title":"Expected Value of p","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle p rangle_n=1\n= int_0^L psi_1^ast(x) hatp psi_1(x) mathrmdx\n= 0","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n langle p rangle_n=1\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -ihbarfracmathrmdmathrmd x right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -ihbar fracpsi(x+Delta x) - psi(x-Delta x)2Delta x right mathrmdx\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n 2fracmathrmd psi(x)mathrmdx Delta x\n + Oleft(Delta x^3right)\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n \n 2fracmathrmd psi(x)mathrmdx Delta x\n = \n psi(x+Delta x)\n - psi(x-Delta x)\n - Oleft(Delta x^3right)\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n - fracOleft(Delta x^3right)2Delta x\n \n fracmathrmd psi(x)mathrmdx\n = \n fracpsi(x+Delta x)- psi(x-Delta x)2Delta x\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n psi(x+Delta x)\n =\n psi(x)\n + fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\n \n psi(x-Delta x)\n =\n psi(x)\n - fracmathrmd psi(x)mathrmdx Delta x\n + frac12 fracmathrmd^2 psi(x)mathrmdx^2 Delta x^2\n + Oleft(Delta x^3right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 0.000000000003 | 0.000000000000 ✔\n0.5 | 1 | 0.000000000000 | 0.000000000000 ✔\n1.0 | 1 | 0.000000000000 | 0.000000000000 ✔\n7.0 | 1 | 0.000000000000 | 0.000000000000 ✔","category":"page"},{"location":"HydrogenAtom/#Expected-Value-of-p2","page":"Hydrogen Atom","title":"Expected Value of p^2","text":"","category":"section"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"langle p^2 rangle\n= int_0^L psi_1^ast(x) hatp^2 psi_1(x) mathrmdx\n= fracpi^2hbar^2L^2","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Reference:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"LibreTexts PHYSICS, 6.4: Expectation Values, Observables, and Uncertainty","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" beginaligned\n langle p^2 rangle\n = int_0^L psi^ast_n(x) hatp psi_n(x) mathrmdx \n = int_0^L psi^ast_n(x) left -hbar^2fracmathrmd^2mathrmdx^2 right psi(x) mathrmdx \n simeq int_0^L psi^ast_n(x) left -hbar^2 fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HydrogenAtom/","page":"Hydrogen Atom","title":"Hydrogen Atom","text":" L | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 1 | 986.960431582781 | 986.960440108936 ✔\n0.5 | 1 | 39.478417274195 | 39.478417604357 ✔\n1.0 | 1 | 9.869604318963 | 9.869604401089 ✔\n7.0 | 1 | 0.201420496383 | 0.201420497981 ✔\n","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"CurrentModule = Antique","category":"page"},{"location":"DeltaPotential/#Delta-Potential","page":"Delta Potential","title":"Delta Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"The Delta potential is one of the simplest models for quantum mechanical system in 1D. It always has one bound state and its wave function has a cusp at the origin.","category":"page"},{"location":"DeltaPotential/#Definitions","page":"Delta Potential","title":"Definitions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"and the Hamiltonian","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters are specified with the following struct.","category":"page"},{"location":"DeltaPotential/#Parameters","page":"Delta Potential","title":"Parameters","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.DeltaPotential","category":"page"},{"location":"DeltaPotential/#Antique.DeltaPotential-DeltaPotential","page":"Delta Potential","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"DeltaPotential/#Potential","page":"Delta Potential","title":"Potential","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.V(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.V-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Values","page":"Delta Potential","title":"Eigen Values","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.E(::DeltaPotential)","category":"page"},{"location":"DeltaPotential/#Antique.E-Tuple{DeltaPotential}-DeltaPotential","page":"Delta Potential","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Eigen-Functions","page":"Delta Potential","title":"Eigen Functions","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Antique.ψ(::DeltaPotential, ::Any)","category":"page"},{"location":"DeltaPotential/#Antique.ψ-Tuple{DeltaPotential, Any}-DeltaPotential","page":"Delta Potential","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"DeltaPotential/#Usage-and-Examples","page":"Delta Potential","title":"Usage & Examples","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by DeltaPotential and several parameters α, m and ℏ are set as optional arguments.","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"using Antique\nDP = DeltaPotential(α=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Parameters:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"julia> DP.α\n1.0\n\njulia> DP.m\n1.0\n\njulia> DP.ℏ\n1.0","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Eigen values:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"julia> E(DP)\n-0.5","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Wave functions:","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"DP = DeltaPotential(α=0.1, m=0.5, ℏ=0.1)\nx = LinRange(-2,2,500);\n\nusing Plots\nplot(x, x->ψ(DP,x), linewidth=3)\nplot!(xlim=[-2,2], ylim=[0,2.5], legend=false)\nplot!(xlabel=\"x\", ylabel=\"ψ(x)\", title=\"Delta Potential\")","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"(Image: )","category":"page"},{"location":"DeltaPotential/#Testing","page":"Delta Potential","title":"Testing","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"DeltaPotential/#Normalization-of-\\psi(x)","page":"Delta Potential","title":"Normalization of psi(x)","text":"","category":"section"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":"int_-infty^infty psi^ast(x) psi(x) mathrmdx = 1","category":"page"},{"location":"DeltaPotential/","page":"Delta Potential","title":"Delta Potential","text":" α | m | ℏ | analytical | numerical \n--- | --- | --- | ----------------- | ----------------- \n0.1 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 0.1 | 7.0 | 1.000000000000 | 1.000004676239 ✔\n0.1 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 1.0 | 7.0 | 1.000000000000 | 0.999999999999 ✔\n0.1 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n0.1 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 0.1 | 7.0 | 1.000000000000 | 0.999999999999 ✔\n1.0 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 1.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n1.0 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 0.1 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 1.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔\n7.0 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔\n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"CurrentModule = Antique","category":"page"},{"location":"HarmonicOscillator/#Harmonic-Oscillator","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The harmonic oscillator is the most frequently used model in quantum physics.","category":"page"},{"location":"HarmonicOscillator/#Definitions","page":"Harmonic Oscillator","title":"Definitions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"This model is described with the time-independent Schrödinger equation","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH psi(x) = E psi(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"and the Hamiltonian","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" hatH = - frachbar^22m fracmathrmd^2mathrmdx ^2 + V(x)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters are specified with the following struct.","category":"page"},{"location":"HarmonicOscillator/#Parameters","page":"Harmonic Oscillator","title":"Parameters","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.HarmonicOscillator","category":"page"},{"location":"HarmonicOscillator/#Antique.HarmonicOscillator-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"HarmonicOscillator/#Potential","page":"Harmonic Oscillator","title":"Potential","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.V(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.V-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency and xi = sqrtfracmomegahbarx.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Values","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.E(::HarmonicOscillator)","category":"page"},{"location":"HarmonicOscillator/#Antique.E-Tuple{HarmonicOscillator}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Eigen-Functions","page":"Harmonic Oscillator","title":"Eigen Functions","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.ψ(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.ψ-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are defined.\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Hermite-Polynomials","page":"Harmonic Oscillator","title":"Hermite Polynomials","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Antique.H(::HarmonicOscillator, ::Any)","category":"page"},{"location":"HarmonicOscillator/#Antique.H-Tuple{HarmonicOscillator, Any}-HarmonicOscillator","page":"Harmonic Oscillator","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\nendaligned\n\nExamples:\n\nbeginaligned\n H_0(x) = 1 \n H_1(x) = 2 x \n H_2(x) = -2 + 4 x^2 \n H_3(x) = -12 x + 8 x^3 \n H_4(x) = 12 - 48 x^2 + 16 x^4 \n H_5(x) = 120 x - 160 x^3 + 32 x^5 \n H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n H_7(x) = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n H_9(x) = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"HarmonicOscillator/#Reference","page":"Harmonic Oscillator","title":"Reference","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"DLMF 18.5.18\ncpprefjp\nThe Digital Library of Mathematical Functions (DLMF) 18.3 Table1, 18.5 Table1, 18.5.13, 18.5.18\nL. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon Press, 1965) p.595 (a.4), (a.6)\nL. I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, 1968) p.71 (13.12)\nA. Messiah, Quanfum Mechanics (Dover Publications, 1999) p.491 (B.59)\nW. Greiner, Quantum Mechanics: An Introduction Third Edition (Springer, 1994) p.152 (7.22)\nD. J. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 1995) p.41 Table 2.1, p.43 (2.70)\nD. A. McQuarrie, J. D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, 1997) p.170 Table 5.2\nP. W. Atkins, J. De Paula, Atkins' Physical Chemistry, 8th edition (W. H. Freeman, 2008) p.293 Table 9.1\nJ. J. Sakurai, J. Napolitano, Modern Quantum Mechanics Third Edition (Cambridge University Press, 2021) p.524 (B.29)","category":"page"},{"location":"HarmonicOscillator/#Usage-and-Examples","page":"Harmonic Oscillator","title":"Usage & Examples","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. In this system, the model is generated by HarmonicOscillator and several parameters k, m and ℏ are set as optional arguments.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Antique\nHO = HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Parameters:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> HO.k\n1.0\n\njulia> HO.m\n1.0\n\njulia> HO.ℏ\n1.0","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Eigen values:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"julia> E(HO, n=0)\n0.5\n\njulia> E(HO, n=1)\n1.5","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(-5:0.1:5, x -> V(HO, x), lw=2, label=\"\", xlabel=\"x\", ylabel=\"V(x)\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5,5), xlabel=\"x\", ylabel=\"ψ(x)\")\nplot!(x -> ψ(HO, x, n=0), label=\"n=0\", lw=2)\nplot!(x -> ψ(HO, x, n=1), label=\"n=1\", lw=2)\nplot!(x -> ψ(HO, x, n=2), label=\"n=2\", lw=2)\nplot!(x -> ψ(HO, x, n=3), label=\"n=3\", lw=2)\nplot!(x -> ψ(HO, x, n=4), label=\"n=4\", lw=2)","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Potential energy curve, Energy levels, Wave functions:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"using Plots\nplot(xlim=(-5.5,5.5), ylim=(-0.2,5.4), xlabel=\"\\$x\\$\", ylabel=\"\\$V(x),~E_n,~\\\\psi_n(x)\\\\times0.5+E_n\\$\", size=(480,400), dpi=300)\nfor n in 0:4\n # energy\n hline!([E(HO, n=n)], lc=:black, ls=:dash, label=\"\")\n plot!([-sqrt(2*HO.k*E(HO, n=n)),sqrt(2*HO.k*E(HO, n=n))], fill(E(HO, n=n),2), lc=:black, lw=2, label=\"\")\n # wave function\n plot!(x -> E(HO, n=n) + 0.5*ψ(HO, x,n=n), lc=n+1, lw=2, label=\"\")\nend\n# potential\nplot!(x -> V(HO, x), lc=:black, lw=2, label=\"\")","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"(Image: )","category":"page"},{"location":"HarmonicOscillator/#Testing","page":"Harmonic Oscillator","title":"Testing","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.","category":"page"},{"location":"HarmonicOscillator/#Hermite-Polynomials-H_n(x)","page":"Harmonic Oscillator","title":"Hermite Polynomials H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=0 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_0(x)\n = e^ - x^2 e^x^2\n = 1 \n = 1\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=1 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_1(x)\n = - e^x^2 fracmathrmd e^ - x^2mathrmdx\n = 2 x \n = 2 x\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=2 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_2(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -2 + 4 x^2 \n = -2 + 4 x^2\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=3 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_3(x)\n = - fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx e^x^2\n = - 12 x + 8 x^3 \n = - 12 x + 8 x^3\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=4 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_4(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 12 - 48 x^2 + 16 x^4 \n = 12 - 48 x^2 + 16 x^4\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=5 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_5(x)\n = - fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx e^x^2\n = 120 x - 160 x^3 + 32 x^5 \n = 120 x - 160 x^3 + 32 x^5\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=6 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_6(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n = -120 + 720 x^2 - 480 x^4 + 64 x^6\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=7 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_7(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = - 1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n = - 1680 x + 3360 x^3 - 1344 x^5 + 128 x^7\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=8 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_8(x)\n = e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"n=9 ✔","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n H_9(x)\n = - e^x^2 fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmdmathrmdx fracmathrmd e^ - x^2mathrmdx\n = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-H_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of H_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int_-infty^infty H_j(x) H_i(x) mathrme^-x^2 mathrmdx = sqrtpi 2^j j delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.772453850906 | 1.772453850906 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000001 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 3.544907701811 | 3.544907701811 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | -0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000014 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 14.179630807244 | 14.179630807244 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | 0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000011 ✔\n 2 | 9 | 0.000000000000 | -0.000000000002 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 85.077784843465 | 85.077784843465 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | -0.000000000000 ✔\n 3 | 7 | 0.000000000000 | -0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000139 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 680.622278747718 | 680.622278747718 ✔\n 4 | 5 | 0.000000000000 | -0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000002 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000063 ✔\n 4 | 9 | 0.000000000000 | 0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | 0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | -0.000000000000 ✔\n 5 | 5 | 6806.222787477181 | 6806.222787477180 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000009 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000001339 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | -0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000002 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 81674.673449726179 | 81674.673449726135 ✔\n 6 | 7 | 0.000000000000 | 0.000000000004 ✔\n 6 | 8 | 0.000000000000 | 0.000000000397 ✔\n 6 | 9 | 0.000000000000 | -0.000000000087 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | -0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000009 ✔\n 7 | 6 | 0.000000000000 | 0.000000000004 ✔\n 7 | 7 | 1143445.428296166472 | 1143445.428296166705 ✔\n 7 | 8 | 0.000000000000 | -0.000000000007 ✔\n 7 | 9 | 0.000000000000 | 0.000000011649 ✔\n 8 | 0 | 0.000000000000 | -0.000000000001 ✔\n 8 | 1 | 0.000000000000 | -0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000011 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000063 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000397 ✔\n 8 | 7 | 0.000000000000 | -0.000000000007 ✔\n 8 | 8 | 18295126.852738663554 | 18295126.852738667279 ✔\n 8 | 9 | 0.000000000000 | 0.000000001630 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000014 ✔\n 9 | 2 | 0.000000000000 | -0.000000000002 ✔\n 9 | 3 | 0.000000000000 | 0.000000000139 ✔\n 9 | 4 | 0.000000000000 | 0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000001339 ✔\n 9 | 6 | 0.000000000000 | -0.000000000087 ✔\n 9 | 7 | 0.000000000000 | 0.000000011649 ✔\n 9 | 8 | 0.000000000000 | 0.000000001630 ✔\n 9 | 9 | 329312283.349295914173 | 329312283.349295675755 ✔","category":"page"},{"location":"HarmonicOscillator/#Normalization-and-Orthogonality-of-\\psi_n(x)","page":"Harmonic Oscillator","title":"Normalization & Orthogonality of psi_n(x)","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"int psi_i^ast(x) psi_j(x) mathrmdx = delta_ij","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" i | j | analytical | numerical \n-- | -- | ----------------- | ----------------- \n 0 | 0 | 1.000000000000 | 1.000000000000 ✔\n 0 | 1 | 0.000000000000 | 0.000000000000 ✔\n 0 | 2 | 0.000000000000 | 0.000000000000 ✔\n 0 | 3 | 0.000000000000 | 0.000000000000 ✔\n 0 | 4 | 0.000000000000 | -0.000000000000 ✔\n 0 | 5 | 0.000000000000 | -0.000000000000 ✔\n 0 | 6 | 0.000000000000 | 0.000000000000 ✔\n 0 | 7 | 0.000000000000 | 0.000000000000 ✔\n 0 | 8 | 0.000000000000 | -0.000000000000 ✔\n 0 | 9 | 0.000000000000 | 0.000000000000 ✔\n 1 | 0 | 0.000000000000 | 0.000000000000 ✔\n 1 | 1 | 1.000000000000 | 1.000000000000 ✔\n 1 | 2 | 0.000000000000 | 0.000000000000 ✔\n 1 | 3 | 0.000000000000 | -0.000000000000 ✔\n 1 | 4 | 0.000000000000 | -0.000000000000 ✔\n 1 | 5 | 0.000000000000 | 0.000000000000 ✔\n 1 | 6 | 0.000000000000 | 0.000000000000 ✔\n 1 | 7 | 0.000000000000 | -0.000000000000 ✔\n 1 | 8 | 0.000000000000 | 0.000000000000 ✔\n 1 | 9 | 0.000000000000 | 0.000000000000 ✔\n 2 | 0 | 0.000000000000 | 0.000000000000 ✔\n 2 | 1 | 0.000000000000 | 0.000000000000 ✔\n 2 | 2 | 1.000000000000 | 1.000000000000 ✔\n 2 | 3 | 0.000000000000 | -0.000000000000 ✔\n 2 | 4 | 0.000000000000 | -0.000000000000 ✔\n 2 | 5 | 0.000000000000 | -0.000000000000 ✔\n 2 | 6 | 0.000000000000 | 0.000000000000 ✔\n 2 | 7 | 0.000000000000 | 0.000000000000 ✔\n 2 | 8 | 0.000000000000 | -0.000000000000 ✔\n 2 | 9 | 0.000000000000 | 0.000000000000 ✔\n 3 | 0 | 0.000000000000 | 0.000000000000 ✔\n 3 | 1 | 0.000000000000 | -0.000000000000 ✔\n 3 | 2 | 0.000000000000 | -0.000000000000 ✔\n 3 | 3 | 1.000000000000 | 1.000000000000 ✔\n 3 | 4 | 0.000000000000 | -0.000000000000 ✔\n 3 | 5 | 0.000000000000 | 0.000000000000 ✔\n 3 | 6 | 0.000000000000 | 0.000000000000 ✔\n 3 | 7 | 0.000000000000 | 0.000000000000 ✔\n 3 | 8 | 0.000000000000 | -0.000000000000 ✔\n 3 | 9 | 0.000000000000 | 0.000000000000 ✔\n 4 | 0 | 0.000000000000 | -0.000000000000 ✔\n 4 | 1 | 0.000000000000 | -0.000000000000 ✔\n 4 | 2 | 0.000000000000 | -0.000000000000 ✔\n 4 | 3 | 0.000000000000 | -0.000000000000 ✔\n 4 | 4 | 1.000000000000 | 1.000000000000 ✔\n 4 | 5 | 0.000000000000 | 0.000000000000 ✔\n 4 | 6 | 0.000000000000 | 0.000000000000 ✔\n 4 | 7 | 0.000000000000 | 0.000000000000 ✔\n 4 | 8 | 0.000000000000 | -0.000000000000 ✔\n 4 | 9 | 0.000000000000 | -0.000000000000 ✔\n 5 | 0 | 0.000000000000 | -0.000000000000 ✔\n 5 | 1 | 0.000000000000 | 0.000000000000 ✔\n 5 | 2 | 0.000000000000 | -0.000000000000 ✔\n 5 | 3 | 0.000000000000 | 0.000000000000 ✔\n 5 | 4 | 0.000000000000 | 0.000000000000 ✔\n 5 | 5 | 1.000000000000 | 1.000000000000 ✔\n 5 | 6 | 0.000000000000 | 0.000000000000 ✔\n 5 | 7 | 0.000000000000 | 0.000000000000 ✔\n 5 | 8 | 0.000000000000 | 0.000000000000 ✔\n 5 | 9 | 0.000000000000 | 0.000000000000 ✔\n 6 | 0 | 0.000000000000 | 0.000000000000 ✔\n 6 | 1 | 0.000000000000 | 0.000000000000 ✔\n 6 | 2 | 0.000000000000 | 0.000000000000 ✔\n 6 | 3 | 0.000000000000 | 0.000000000000 ✔\n 6 | 4 | 0.000000000000 | 0.000000000000 ✔\n 6 | 5 | 0.000000000000 | 0.000000000000 ✔\n 6 | 6 | 1.000000000000 | 1.000000000000 ✔\n 6 | 7 | 0.000000000000 | -0.000000000000 ✔\n 6 | 8 | 0.000000000000 | 0.000000000000 ✔\n 6 | 9 | 0.000000000000 | 0.000000000000 ✔\n 7 | 0 | 0.000000000000 | 0.000000000000 ✔\n 7 | 1 | 0.000000000000 | -0.000000000000 ✔\n 7 | 2 | 0.000000000000 | 0.000000000000 ✔\n 7 | 3 | 0.000000000000 | 0.000000000000 ✔\n 7 | 4 | 0.000000000000 | 0.000000000000 ✔\n 7 | 5 | 0.000000000000 | 0.000000000000 ✔\n 7 | 6 | 0.000000000000 | -0.000000000000 ✔\n 7 | 7 | 1.000000000000 | 1.000000000000 ✔\n 7 | 8 | 0.000000000000 | 0.000000000000 ✔\n 7 | 9 | 0.000000000000 | 0.000000000000 ✔\n 8 | 0 | 0.000000000000 | -0.000000000000 ✔\n 8 | 1 | 0.000000000000 | 0.000000000000 ✔\n 8 | 2 | 0.000000000000 | -0.000000000000 ✔\n 8 | 3 | 0.000000000000 | -0.000000000000 ✔\n 8 | 4 | 0.000000000000 | -0.000000000000 ✔\n 8 | 5 | 0.000000000000 | 0.000000000000 ✔\n 8 | 6 | 0.000000000000 | 0.000000000000 ✔\n 8 | 7 | 0.000000000000 | 0.000000000000 ✔\n 8 | 8 | 1.000000000000 | 1.000000000000 ✔\n 8 | 9 | 0.000000000000 | -0.000000000000 ✔\n 9 | 0 | 0.000000000000 | 0.000000000000 ✔\n 9 | 1 | 0.000000000000 | 0.000000000000 ✔\n 9 | 2 | 0.000000000000 | 0.000000000000 ✔\n 9 | 3 | 0.000000000000 | 0.000000000000 ✔\n 9 | 4 | 0.000000000000 | -0.000000000000 ✔\n 9 | 5 | 0.000000000000 | 0.000000000000 ✔\n 9 | 6 | 0.000000000000 | 0.000000000000 ✔\n 9 | 7 | 0.000000000000 | 0.000000000000 ✔\n 9 | 8 | 0.000000000000 | -0.000000000000 ✔\n 9 | 9 | 1.000000000000 | 1.000000000000 ✔","category":"page"},{"location":"HarmonicOscillator/#Virial-Theorem","page":"Harmonic Oscillator","title":"Virial Theorem","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"The virial theorem langle T rangle = langle V rangle and the definition of Hamiltonian langle H rangle = langle T rangle + langle V rangle derive langle H rangle = 2 langle V rangle = 2 langle T rangle.","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"2 int psi_n^ast(x) V(x) psi_n(x) mathrmdx = E_n","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.1 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.1 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.1 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.1 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.1 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.1 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.1 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.1 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.1 | 9 | 9.500000000000 | 9.500000000000 ✔\n0.5 | 0 | 0.500000000000 | 0.500000000000 ✔\n0.5 | 1 | 1.500000000000 | 1.500000000000 ✔\n0.5 | 2 | 2.500000000000 | 2.500000000000 ✔\n0.5 | 3 | 3.500000000000 | 3.500000000000 ✔\n0.5 | 4 | 4.500000000000 | 4.500000000000 ✔\n0.5 | 5 | 5.500000000000 | 5.500000000000 ✔\n0.5 | 6 | 6.500000000000 | 6.500000000000 ✔\n0.5 | 7 | 7.500000000000 | 7.500000000000 ✔\n0.5 | 8 | 8.500000000000 | 8.500000000000 ✔\n0.5 | 9 | 9.500000000000 | 9.500000000000 ✔\n1.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n1.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n1.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n1.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n1.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n1.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n1.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n1.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n1.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n1.0 | 9 | 9.500000000000 | 9.500000000000 ✔\n5.0 | 0 | 0.500000000000 | 0.500000000000 ✔\n5.0 | 1 | 1.500000000000 | 1.500000000000 ✔\n5.0 | 2 | 2.500000000000 | 2.500000000000 ✔\n5.0 | 3 | 3.500000000000 | 3.500000000000 ✔\n5.0 | 4 | 4.500000000000 | 4.500000000000 ✔\n5.0 | 5 | 5.500000000000 | 5.500000000000 ✔\n5.0 | 6 | 6.500000000000 | 6.500000000000 ✔\n5.0 | 7 | 7.500000000000 | 7.500000000000 ✔\n5.0 | 8 | 8.500000000000 | 8.500000000000 ✔\n5.0 | 9 | 9.500000000000 | 9.500000000000 ✔","category":"page"},{"location":"HarmonicOscillator/#Eigen-Values-2","page":"Harmonic Oscillator","title":"Eigen Values","text":"","category":"section"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" beginaligned\n E_n\n = int psi^ast_n(x) hatH psi_n(x) mathrmdx \n = int psi^ast_n(x) left hatV + hatT right psi(x) mathrmdx \n = int psi^ast_n(x) left V(x) - frachbar^22m fracmathrmd^2mathrmd x^2 right psi(x) mathrmdx \n simeq int psi^ast_n(x) left V(x)psi(x) -frachbar^22m fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2 right mathrmdx\n endaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"Where, the difference formula for the 2nd-order derivative:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\n 2psi(x)\n + fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n + Oleft(Delta x^4right)\n =\n psi(x+Delta x)\n + psi(x-Delta x)\n \n fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n =\n psi(x+Delta x)\n - 2psi(x)\n + psi(x-Delta x)\n - Oleft(Delta x^4right)\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n - fracOleft(Delta x^4right)Delta x^2\n \n fracmathrmd^2 psi(x)mathrmd x^2\n =\n fracpsi(x+Delta x) - 2psi(x) + psi(x-Delta x)Delta x^2\n + Oleft(Delta x^2right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"are given by the sum of 2 Taylor series:","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":"beginaligned\npsi(x+Delta x)\n= psi(x)\n+ fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n+ frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\n\npsi(x-Delta x)\n= psi(x)\n- fracmathrmd psi(x)mathrmd x Delta x\n+ frac12 fracmathrmd^2 psi(x)mathrmd x^2 Delta x^2\n- frac13 fracmathrmd^3 psi(x)mathrmd x^3 Delta x^3\n+ Oleft(Delta x^4right)\nendaligned","category":"page"},{"location":"HarmonicOscillator/","page":"Harmonic Oscillator","title":"Harmonic Oscillator","text":" k | n | analytical | numerical \n--- | -- | ----------------- | ----------------- \n0.1 | 0 | 0.158113883008 | 0.158113879883 ✔\n0.1 | 1 | 0.474341649025 | 0.474341633410 ✔\n0.1 | 2 | 0.790569415042 | 0.790569374409 ✔\n0.1 | 3 | 1.106797181059 | 1.106797102928 ✔\n0.1 | 4 | 1.423024947076 | 1.423024818987 ✔\n0.1 | 5 | 1.739252713093 | 1.739252522506 ✔\n0.1 | 6 | 2.055480479109 | 2.055480213500 ✔\n0.1 | 7 | 2.371708245126 | 2.371707891950 ✔\n0.1 | 8 | 2.687936011143 | 2.687935558100 ✔\n0.1 | 9 | 3.004163777160 | 3.004163211450 ✔\n0.5 | 0 | 0.353553390593 | 0.353553374944 ✔\n0.5 | 1 | 1.060660171780 | 1.060660093649 ✔\n0.5 | 2 | 1.767766952966 | 1.767766749878 ✔\n0.5 | 3 | 2.474873734153 | 2.474873343556 ✔\n0.5 | 4 | 3.181980515339 | 3.181979874817 ✔\n0.5 | 5 | 3.889087296526 | 3.889086343463 ✔\n0.5 | 6 | 4.596194077713 | 4.596192749665 ✔\n0.5 | 7 | 5.303300858899 | 5.303299093519 ✔\n0.5 | 8 | 6.010407640086 | 6.010405374197 ✔\n0.5 | 9 | 6.717514421272 | 6.717511593266 ✔\n1.0 | 0 | 0.500000000000 | 0.499999968773 ✔\n1.0 | 1 | 1.500000000000 | 1.499999843774 ✔\n1.0 | 2 | 2.500000000000 | 2.499999593764 ✔\n1.0 | 3 | 3.500000000000 | 3.499999218732 ✔\n1.0 | 4 | 4.500000000000 | 4.499998718747 ✔\n1.0 | 5 | 5.500000000000 | 5.499998093755 ✔\n1.0 | 6 | 6.500000000000 | 6.499997343602 ✔\n1.0 | 7 | 7.500000000000 | 7.499996468887 ✔\n1.0 | 8 | 8.500000000000 | 8.499995468843 ✔\n1.0 | 9 | 9.500000000000 | 9.499994343445 ✔\n5.0 | 0 | 1.118033988750 | 1.118033832523 ✔\n5.0 | 1 | 3.354101966250 | 3.354101184969 ✔\n5.0 | 2 | 5.590169943749 | 5.590167912524 ✔\n5.0 | 3 | 7.826237921249 | 7.826234014984 ✔\n5.0 | 4 | 10.062305898749 | 10.062299492494 ✔\n5.0 | 5 | 12.298373876249 | 12.298364344997 ✔\n5.0 | 6 | 14.534441853749 | 14.534428572309 ✔\n5.0 | 7 | 16.770509831248 | 16.770492175222 ✔\n5.0 | 8 | 19.006577808748 | 19.006555152416 ✔\n5.0 | 9 | 21.242645786248 | 21.242617504750 ✔\n","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = Antique","category":"page"},{"location":"#Antique.jl","page":"Home","title":"Antique.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Self-contained, Well-Tested, Well-Documented Analytical Solutions of Quantum Mechanical Equations.","category":"page"},{"location":"#Install","page":"Home","title":"Install","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install this package, run the following code in your Jupyter Notebook:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg; Pkg.add(\"Antique\")","category":"page"},{"location":"#Usage-and-Examples","page":"Home","title":"Usage & Examples","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ(), potential V() and some other functions are suppoted. Here are examples in hydrogen-like atom. The analytical notation of energy (eigen value of the Hamiltonian) is written as","category":"page"},{"location":"","page":"Home","title":"Home","text":"E_n = -fracZ^22n^2 E_mathrmh","category":"page"},{"location":"","page":"Home","title":"Home","text":"Hydrogen atom has symbol mathrmH and atomic number 1 (Z=1). Therefore the ground state (n=1) energy is -frac12 E_mathrmh.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antique\nH = HydrogenAtom(Z=1)\nE(H)\n# output> -0.5","category":"page"},{"location":"","page":"Home","title":"Home","text":"Helium cation has symbol mathrmHe^+ and atomic number 2 (Z=2). Therefore the ground state (n=1) energy is -2 E_mathrmh.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Antique\nHe⁺ = HydrogenAtom(Z=2)\nE(He⁺)\n# output> -2.0","category":"page"},{"location":"","page":"Home","title":"Home","text":"There are more examples on each model page.","category":"page"},{"location":"#Supported-Models","page":"Home","title":"Supported Models","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
\n
\n \n \"InfinitePotentialWell\"/\n \n InfinitePotentialWell\n
\n
\n \n \"HarmonicOscillator\"/\n \n HarmonicOscillator\n
\n
\n \n \"MorsePotential\"/\n \n MorsePotential\n
\n
\n \n \"HydrogenAtom\"/\n \n HydrogenAtom\n
\n
\n \n \"DeltaPotential\"/\n \n DeltaPotential\n
\n
\n \n \"PoschlTeller\"/\n \n PoschlTeller\n
\n
","category":"page"},{"location":"#Future-Works","page":"Home","title":"Future Works","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"List of quantum-mechanical systems with analytical solutions","category":"page"},{"location":"#Developer's-Guide","page":"Home","title":"Developer's Guide","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This is the guideline for adding new models.","category":"page"},{"location":"","page":"Home","title":"Home","text":"First, please add a new issue here. We need to find a reference for the definition and analytical solutions (eigenvalues and eigenfunctions) before the development.\nFork the repository on GitHub.\nClone the forked repository to your local machine by Git.\nAdd the new model name :ModelName to the models = [...] array in src/Antique.jl. : is required at the beginning.\nAdd the file src/ModelName.jl with the same name as the model name. The most helpful code examples are harmonic oscillators for one-dimensional systems and hydrogen atoms for three-dimensional systems. We recommend that you copy these.\nWrite the code in that file. First we need to create a structure struct ModelName with the same name as the model name (The best way is Find & Replace). Create V, E, ψ and other functions. Because the function names conflict, you must always give the structure as an argument. Multi-dispatch avoids conflict. We recommend using Revice.jl while coding. Run include(\"./developer/revice.jl\") on the REPL or use dev.ipynb.\nAdd test code test/ModelName.jl. At a minimum, it is recommended to check the normalization and the orthogonality of wavefunction using QuadGK.jl. All tests will be executed by executing include(\"./developer/test.jl\"). It will take about 2 minutes to complete.\nAdd documentation. Add either docs/ModelName.md or docs/jmd/ModelName.jmd (if you have a jmd file, the md file will be automatically generated). Include at least the definition of the Hamiltonian and the analytical solutions (eigenvalues and eigenfunctions).\nAdd the new model into pages=[...] in docs/make.jl.\nExecute include(\"./developer/docs.jl\") to compile. Please check docs/build/*.html in your browser.\nPush the code.\nSubmit a pull request on GitHub.","category":"page"},{"location":"#Acknowledgment","page":"Home","title":"Acknowledgment","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package was named by @KB-satou and @ultimatile.","category":"page"},{"location":"API/","page":"API reference","title":"API reference","text":"CurrentModule = Antique","category":"page"},{"location":"API/#API-reference","page":"API reference","title":"API reference","text":"","category":"section"},{"location":"API/","page":"API reference","title":"API reference","text":"Order = [:type, :function]\nModules = [Antique]","category":"page"},{"location":"API/#Antique.DeltaPotential","page":"API reference","title":"Antique.DeltaPotential","text":"DeltaPotential(α=1.0, m=1.0, ℏ=1.0)\n\nalpha is the potential strength, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HarmonicOscillator","page":"API reference","title":"Antique.HarmonicOscillator","text":"HarmonicOscillator(k=1.0, m=1.0, ℏ=1.0)\n\nk is the force constant, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.HydrogenAtom","page":"API reference","title":"Antique.HydrogenAtom","text":"HydrogenAtom(Z=1, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)\n\nZ is the atomic number, m_mathrme is the electron mass, a_0is the Bohr radius, E_mathrmh is the Hartree energy and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.InfinitePotentialWell","page":"API reference","title":"Antique.InfinitePotentialWell","text":"InfinitePotentialWell(L=1.0, m=1.0, ℏ=1.0)\n\nL is the length of the box, m is the mass of particle and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.MorsePotential","page":"API reference","title":"Antique.MorsePotential","text":"MP = MorsePotential(rₑ=2.0, Dₑ=0.1, k=0.1, µ=918.1, ℏ=1.0)\n\nr_mathrme is the equilibrium bond distance, D__mathrme is the the well depth , k is the force constant, mu is the reduced mass and hbar is the reduced Planck constant (Dirac's constant).\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.PoschlTeller","page":"API reference","title":"Antique.PoschlTeller","text":"PoschlTeller(lambda=1.0)\n\nlambda determines the potential strength. This model is defined dimensionless, i.e. x is given in units of a characteristic length x_0, and E in units of a characteristic energy, e.g. E_mathrmchar = frachbar^22 m x_0^2.\n\n\n\n\n\n","category":"type"},{"location":"API/#Antique.E-Tuple{DeltaPotential}","page":"API reference","title":"Antique.E","text":"E(model::DeltaPotential)\n\nE = - fracmalpha^22hbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HarmonicOscillator}","page":"API reference","title":"Antique.E","text":"E(model::HarmonicOscillator; n=0)\n\nE_n = hbar omega left( n + frac12 right)\n\nwhere omega = sqrtkm is the angular frequency.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{HydrogenAtom}","page":"API reference","title":"Antique.E","text":"E(model::HydrogenAtom; n=1)\n\nE_n\n= -fracm_mathrme e^4 Z^22n^2(4pivarepsilon_0)^2hbar^2\n= -fracZ^22n^2 E_mathrmh\n\nwhere E_mathrmh is the Hartree energy, one of atomic unit. About atomic units, see section 3.9.2 of the IUPAC GreenBook. In other units, E_mathrmh = 27211386245988(53)mathrmeV from here.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{InfinitePotentialWell}","page":"API reference","title":"Antique.E","text":"E(model::InfinitePotentialWell; n=1)\n\nE_n = frachbar^2 n^2 pi^22 m L^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{MorsePotential}","page":"API reference","title":"Antique.E","text":"E(model::MorsePotential; n=0)\n\nE_n = - D_mathrme + hbar omega left( n + frac12 right) - chi hbar omega left( n + frac12 right)^2\n\nwhere omega = sqrtkµ and chi = frachbaromega4D_mathrme are defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.E-Tuple{PoschlTeller}","page":"API reference","title":"Antique.E","text":"E(model::PoschlTeller; n=0)\n\nE_n = -fracmu^22\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.H-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.H","text":"H(model::HarmonicOscillator, x; n=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n H_n(x)\n = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 \n = n sum_m=0^lfloor n2 rfloor frac(-1)^mm (n-2m)(2 x)^n-2m\nendaligned\n\nExamples:\n\nbeginaligned\n H_0(x) = 1 \n H_1(x) = 2 x \n H_2(x) = -2 + 4 x^2 \n H_3(x) = -12 x + 8 x^3 \n H_4(x) = 12 - 48 x^2 + 16 x^4 \n H_5(x) = 120 x - 160 x^3 + 32 x^5 \n H_6(x) = -120 + 720 x^2 - 480 x^4 + 64 x^6 \n H_7(x) = -1680 x + 3360 x^3 - 1344 x^5 + 128 x^7 \n H_8(x) = 1680 - 13440 x^2 + 13440 x^4 - 3584 x^6 + 256 x^8 \n H_9(x) = 30240 x - 80640 x^3 + 48384 x^5 - 9216 x^7 + 512 x^9 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.L","text":"L(model::HydrogenAtom, x; n=0, k=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\nL_n^k(x)\n = fracmathrmd^kmathrmdx^k L_n(x) \n = fracmathrmd^kmathrmdx^k frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right) \n = sum_m=0^n-k (-1)^m+k fracnm(m+k)(n-m-k) x^m \n = (-1)^k L_n-k^(k)(x)\nendaligned\n\nwhere Laguerre polynomials are defined as L_n(x)=frac1nmathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right).\n\nExamples:\n\nbeginaligned\n L_0^0(x) = 1 \n L_1^0(x) = 1 - x \n L_1^1(x) = 1 \n L_2^0(x) = 1 - 2 x + 12 x^2 \n L_2^1(x) = 2 - x \n L_2^2(x) = 1 \n L_3^0(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^1(x) = 3 - 3 x + 12 x^2 \n L_3^2(x) = 3 - x \n L_3^3(x) = 1 \n L_4^0(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 512 x^4 \n L_4^1(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_4^2(x) = 6 - 4 x + 12 x^2 \n L_4^3(x) = 4 - x \n L_4^4(x) = 1 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.L-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.L","text":"L(model::MorsePotential, x; n=0, α=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n L_n^(alpha)(x)\n = fracx^-alphae^xn fracd^ndx^nleft(x^n+alphae^-xright) \n = sum_k=0^n(-1)^k left(beginarrayl n+alpha n-k endarrayright) fracx^kk \n = sum_k=0^n(-1)^k fracGamma(alpha+n+1)Gamma(alpha+k+1)Gamma(n-k+1) fracx^kk \nendaligned\n\nExamples:\n\nbeginaligned\n L_0^(0)(x) = 1 \n L_1^(0)(x) = 1 - x \n L_1^(1)(x) = 2 - x \n L_2^(0)(x) = 1 - 2 x + 12 x^2 \n L_2^(1)(x) = 3 - 3 x + 12 x^2 \n L_2^(2)(x) = 6 - 4 x + 12 x^2 \n L_3^(0)(x) = 1 - 3 x + 32 x^2 - 16 x^3 \n L_3^(1)(x) = 4 - 6 x + 2 x^2 - 16 x^3 \n L_3^(2)(x) = 10 - 10 x + 52 x^2 - 16 x^3 \n L_3^(3)(x) = 20 - 15 x + 3 x^2 - 16 x^3 \n L_4^(0)(x) = 1 - 4 x + 3 x^2 - 23 x^3 + 124 x^4 \n L_4^(1)(x) = 5 - 10 x + 5 x^2 - 56 x^3 + 124 x^4 \n L_4^(2)(x) = 15 - 20 x + 152 x^2 - 1 x^3 + 124 x^4 \n L_4^(3)(x) = 35 - 35 x + 212 x^2 - 76 x^3 + 124 x^4 \n L_4^(4)(x) = 70 - 56 x + 14 x^2 - 43 x^3 + 124 x^4 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.P","text":"P(model::HydrogenAtom, x; n=0, m=0)\n\nRodrigues' formula & closed-form:\n\nbeginaligned\n P_n^m(x)\n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m P_n(x) \n = left( 1-x^2 right)^m2 fracmathrmd^mmathrmdx^m frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right \n = frac12^n (1-x^2)^m2 sum_j=0^leftlfloorfracn-m2rightrfloor (-1)^j frac(2n-2j)j (n-j) (n-2j-m) x^(n-2j-m)\nendaligned\n\nwhere Legendre polynomials are defined as P_n(x) = frac12^n n fracmathrmd^nmathrmdx ^n left left( x^2-1 right)^n right. Note that P_l^-m = (-1)^m frac(l-m)(l+m) P_l^m for m0. (It is not compatible with P_k^m(t) = (-1)^mleft( 1-t^2 right)^m2 fracmathrmd^m P_k(t)mathrmdt^m caused by (-1)^m.) The specific formulae are given below.\n\nExamples:\n\nbeginaligned\n P_0^0(x) = 1 \n P_1^0(x) = x \n P_1^1(x) = left(+1right)sqrt1-x^2 \n P_2^0(x) = -12 + 32 x^2 \n P_2^1(x) = left(-3 xright)sqrt1-x^2 \n P_2^2(x) = 3 - 6 x \n P_3^0(x) = -32 x + 52 x^3 \n P_3^1(x) = left(32 - 152 x^2right)sqrt1-x^2 \n P_3^2(x) = 15 x - 30 x^2 \n P_3^3(x) = left(15 - 30 xright)sqrt1-x^2 \n P_4^0(x) = 38 - 154 x^2 + 358 x^4 \n P_4^1(x) = left(- 152 x + 352 x^3right)sqrt1-x^2 \n P_4^2(x) = -152 + 15 x + 1052 x^2 - 105 x^3 \n P_4^3(x) = left(105 x - 210 x^2right)sqrt1-x^2 \n P_4^4(x) = 105 - 420 x + 420 x^2 \n vdots\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.P-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.P","text":"P(model::PoschlTeller, x; n=0, m=0)\n\nAssociated Legendre polynomials are the associated Legendre functions for integer indices. Please note here, that for the Poschl-Teller potential we use a slightly different notation of the associated Legendre functions as compared to the model HydrogenAtom. Here we have an additional factor (-1)^m.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.R-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.R","text":"R(model::HydrogenAtom, r; n=1, l=0)\n\nR_nl(r) = -sqrtfrac(n-l-1)2n(n+l) left(frac2Zn a_0right)^3 left(frac2Zrn a_0right)^l exp left(-fracZrn a_0right) L_n+l^2l+1 left(frac2Zrn a_0right)\n\nwhere Laguerre polynomials are defined as L_n(x) = frac1n mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right), and associated Laguerre polynomials are defined as L_n^k(x) = fracmathrmd^kmathrmdx^k L_n(x). Note that replace 2n(n+l) with 2n(n+l)^3 if Laguerre polynomials are defined as L_n(x) = mathrme^x fracmathrmd^nmathrmdx ^n left( mathrme^-x x^n right). The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::DeltaPotential, x)\n\nV(x) = -alpha delta(x)\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.V","text":"V(model::HarmonicOscillator, x)\n\nV(x)\n= frac12 k x^2\n= frac12 m omega^2 x^2\n= frac12 hbar omega xi^2\n\nwhere omega = sqrtkm is the angular frequency and xi = sqrtfracmomegahbarx.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{HydrogenAtom, Any}","page":"API reference","title":"Antique.V","text":"V(model::HydrogenAtom, r)\n\nbeginaligned\n V(r)\n = - fracZe^24pivarepsilon_0 r \n = - frace^24pivarepsilon_0 a_0 fracZra_0\n = - fracZra_0 E_mathrmh\nendaligned\n\nThe domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.V","text":"V(model::InfinitePotentialWell; x)\n\nV(x) =\nleft\n beginarrayll\n infty x lt 0 L lt x \n 0 0 leq x leq L\n endarray\nright\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.V","text":"V(model::MorsePotential, r)\n\nV(r) = D_mathrme left( mathrme^-2a(r-r_e) - 2mathrme^-a(r-r_e) right)\n\nwhere a = sqrtfrack2Dₑ is defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.V-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.V","text":"V(model::PoschlTeller, x)\n\nbeginaligned\n V(x)\n = -fraclambda(lambda+1)2 mathrmsech(x)^2\n = -fraclambda(lambda+1)2 frac1mathrmcosh(x)^2\nendaligned\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.Y-Tuple{HydrogenAtom, Any, Any}","page":"API reference","title":"Antique.Y","text":"Y(model::HydrogenAtom, θ, φ; l=0, m=0)\n\nY_lm(thetavarphi) = (-1)^fracm+m2 sqrtfrac2l+14pi frac(l-m)(l+m) P_l^m (costheta) mathrme^imvarphi\n\nThe domain is 0leq theta lt pi 0leq varphi lt 2pi. Note that some variants are connected by \n\ni^m+m sqrtfrac(l-m)(l+m) P_l^m = (-1)^fracm+m2 sqrtfrac(l-m)(l+m) P_l^m = (-1)^m sqrtfrac(l-m)(l+m) P_l^m\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nmax-Tuple{PoschlTeller}","page":"API reference","title":"Antique.nmax","text":"nmax(model::PoschlTeller)\n\nn_mathrmmax = leftlfloor lambda rightrfloor - 1\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.nₘₐₓ-Tuple{MorsePotential}","page":"API reference","title":"Antique.nₘₐₓ","text":"nₘₐₓ(model::MorsePotential)\n\nn_mathrmmax = leftlfloor frac2 D_e - omegaomega rightrfloor\n\nwhere omega = sqrtkµ is defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{DeltaPotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::DeltaPotential, x)\n\npsi(x) = fracsqrtmalphahbar mathrme^-malpha xhbar^2\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HarmonicOscillator, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HarmonicOscillator, x; n=0)\n\npsi_n(x) = A_n H_n(xi) expleft( -fracxi^22 right)\n\nwhere omega = sqrtkm, xi = sqrtfracmomegahbarx, A_n = sqrtfrac1n 2^n sqrtfracmomegapihbar, H_n(x) = (-1)^n mathrme^x^2 fracmathrmd^nmathrmdx^n mathrme^-x^2 are defined.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{HydrogenAtom, Any, Any, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::HydrogenAtom, r, θ, φ; n=1, l=0, m=0)\n\npsi_nlm(pmbr) = R_nl(r) Y_lm(thetavarphi)\n\nThe domain is 0leq r lt infty 0leq theta lt pi 0leq varphi lt 2pi.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{InfinitePotentialWell, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::InfinitePotentialWell, x; n=1)\n\npsi_n(x) = sqrtfrac2L sin fracnpi xL\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{MorsePotential, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::MorsePotential, r; n=0)\n\npsi_n(r) = N_n z^lambda-n-12 mathrme^-z2 L_n^(2lambda-2n-1)(xi)\n\nN_n = sqrtfracn(2lambda-2n-1)aGamma(2lambda-n), lambda = fracsqrt2mu D_mathrmeahbar, a = sqrtfrack2Dₑ, L_n^(alpha)(x) = fracx^-alpha mathrme^xn fracmathrmd^nmathrmd x^nleft(mathrme^-x x^n+alpharight), xi = 2lambdamathrme^-a(r-r_e) are defined. The domain is 0leq r lt infty.\n\n\n\n\n\n","category":"method"},{"location":"API/#Antique.ψ-Tuple{PoschlTeller, Any}","page":"API reference","title":"Antique.ψ","text":"ψ(model::PoschlTeller, x; n=0)\n\npsi_n(x) = P_lambda^mu(mathrmtanh(x)) sqrtmufracGamma(lambda-mu+1)Gamma(lambda+mu+1)\n\nwhere mu = mu(n) = n_mathrmmax-n+1, and n_mathrmmax = leftlfloor lambda rightrfloor - 1 and P_lambda^mu are the associated Legendre functions.\n\n\n\n\n\n","category":"method"}] }