API reference
Antique.DeltaPotential — TypeDeltaPotential(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).
Antique.HarmonicOscillator — TypeHarmonicOscillator(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).
Antique.HydrogenAtom — TypeHydrogenAtom(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).
Antique.InfinitePotentialWell — TypeInfinitePotentialWell(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).
Antique.MorsePotential — TypeMP = 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).
Antique.PoschlTeller — TypePoschlTeller(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. $E_\mathrm{char} = \frac{\hbar^2}{2 m x_0^2}$.
Antique.E — MethodE(model::DeltaPotential)
\[E = - \frac{m\alpha^2}{2\hbar^2}\]
Antique.E — MethodE(model::HarmonicOscillator; n=0)
\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]
where $\omega = \sqrt{k/m}$ is the angular frequency.
Antique.E — MethodE(model::HydrogenAtom; n=1)
\[E_n +
API reference
Antique.DeltaPotential — TypeDeltaPotential(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).
Antique.HarmonicOscillator — TypeHarmonicOscillator(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).
Antique.HydrogenAtom — TypeHydrogenAtom(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).
Antique.InfinitePotentialWell — TypeInfinitePotentialWell(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).
Antique.MorsePotential — TypeMP = 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).
Antique.PoschlTeller — TypePoschlTeller(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. $E_\mathrm{char} = \frac{\hbar^2}{2 m x_0^2}$.
Antique.E — MethodE(model::DeltaPotential)
\[E = - \frac{m\alpha^2}{2\hbar^2}\]
Antique.E — MethodE(model::HarmonicOscillator; n=0)
\[E_n = \hbar \omega \left( n + \frac{1}{2} \right),\]
where $\omega = \sqrt{k/m}$ is the angular frequency.
Antique.E — MethodE(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.
Antique.E — MethodE(model::InfinitePotentialWell; n=1)
\[E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]
Antique.E — MethodE(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.
Antique.E — MethodE(model::PoschlTeller; n=0)
\[E_n = -\frac{\mu^2}{2},\]
where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.
Antique.H — MethodH(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.
Antique.E — MethodE(model::InfinitePotentialWell; n=1)
\[E_n = \frac{\hbar^2 n^2 \pi^2}{2 m L^2}\]
Antique.E — MethodE(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.
Antique.E — MethodE(model::PoschlTeller; n=0)
\[E_n = -\frac{\mu^2}{2},\]
where $\mu = \mu(n) = n_\mathrm{max}-n+1$, and $n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1$.
Antique.H — MethodH(model::HarmonicOscillator, x; n=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} H_{n}(x) &:= (-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \mathrm{e}^{-x^2} \\ &= n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m}{m! (n-2m)!}(2 x)^{n-2m}. @@ -17,7 +17,7 @@ H_{8}(x) &= 1680 - 13440 x^{2} + 13440 x^{4} - 3584 x^{6} + 256 x^{8}, \\ H_{9}(x) &= 30240 x - 80640 x^{3} + 48384 x^{5} - 9216 x^{7} + 512 x^{9}, \\ &\vdots -\end{aligned}\]
Antique.L — MethodL(model::HydrogenAtom, x; n=0, k=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} +\end{aligned}\]
Antique.L — MethodL(model::HydrogenAtom, x; n=0, k=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} L_n^{k}(x) &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} L_n(x) \\ &= \frac{\mathrm{d}^k}{\mathrm{d}x^k} \frac{1}{n!} \mathrm{e}^x \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left( \mathrm{e}^{-x} x^n \right) \\ @@ -40,7 +40,7 @@ L_4^3(x) &= 4 - x, \\ L_4^4(x) &= 1, \\ \vdots -\end{aligned}\]
Antique.L — MethodL(model::MorsePotential, x; n=0, α=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} +\end{aligned}\]
Antique.L — MethodL(model::MorsePotential, x; n=0, α=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ @@ -62,7 +62,7 @@ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots -\end{aligned}\]
Antique.P — MethodP(model::HydrogenAtom, x; n=0, m=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} +\end{aligned}\]
Antique.P — MethodP(model::HydrogenAtom, x; n=0, m=0)
Rodrigues' formula & closed-form:
\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ @@ -84,22 +84,22 @@ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots -\end{aligned}\]
Antique.P — MethodP(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 $(-1)^\m$.
Antique.R — MethodR(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 $0\leq r \lt \infty$.
Antique.V — MethodV(model::DeltaPotential, x)
\[V(x) = -\alpha \delta(x).\]
Antique.V — MethodV(model::HarmonicOscillator, x)
\[V(x) +\end{aligned}\]
Antique.P — MethodP(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 $(-1)^\m$.
Antique.R — MethodR(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 $0\leq r \lt \infty$.
Antique.V — MethodV(model::DeltaPotential, x)
\[V(x) = -\alpha \delta(x).\]
Antique.V — MethodV(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 and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.
Antique.V — MethodV(model::HydrogenAtom, r)
\[\begin{aligned} += \frac{1}{2} \hbar \omega \xi^2,\]
where $\omega = \sqrt{k/m}$ is the angular frequency and $\xi = \sqrt{\frac{m\omega}{\hbar}}x$.
Antique.V — MethodV(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 $0\leq r \lt \infty$.
Antique.V — MethodV(model::InfinitePotentialWell; x)
\[V(x) = +\end{aligned}\]
The domain is $0\leq r \lt \infty$.
Antique.V — MethodV(model::InfinitePotentialWell; x)
\[V(x) = \left\{ \begin{array}{ll} \infty & x \lt 0, L \lt x \\ 0 & 0 \leq x \leq L \end{array} -\right.\]
Antique.V — MethodV(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 $0\leq r \lt \infty$.
Antique.V — MethodV(model::PoschlTeller, x)
\[\begin{aligned} +\right.\]
Antique.V — MethodV(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 $0\leq r \lt \infty$.
Antique.V — MethodV(model::PoschlTeller, x)
\[\begin{aligned} V(x) &= -\frac{\lambda(\lambda+1)}{2} \mathrm{sech}(x)^2 &= -\frac{\lambda(\lambda+1)}{2} \frac{1}{\mathrm{cosh}(x)^2}. -\end{aligned}\]
Antique.Y — MethodY(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
\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]
Antique.nmax — Methodnmax(model::PoschlTeller)
\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]
Antique.nₘₐₓ — Methodnₘₐₓ(model::MorsePotential)
\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]
where $\omega = \sqrt{k/µ}$ is defined.
Antique.ψ — Methodψ(model::DeltaPotential, x)
\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]
Antique.ψ — Methodψ(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.
Antique.ψ — Methodψ(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 $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.
Antique.ψ — Methodψ(model::InfinitePotentialWell, x; n=1)
\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]
Antique.ψ — Methodψ(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 $0\leq r \lt \infty$.
Antique.ψ — Methodψ(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.
Settings
This document was generated with Documenter.jl version 1.3.0 on Thursday 28 March 2024. Using Julia version 1.10.2.
Antique.Y — MethodY(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
\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]
Antique.nmax — Methodnmax(model::PoschlTeller)
\[n_\mathrm{max} = \left\lfloor \lambda \right\rfloor - 1.\]
Antique.nₘₐₓ — Methodnₘₐₓ(model::MorsePotential)
\[n_\mathrm{max} = \left\lfloor \frac{2 D_e - \omega}{\omega} \right\rfloor,\]
where $\omega = \sqrt{k/µ}$ is defined.
Antique.ψ — Methodψ(model::DeltaPotential, x)
\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]
Antique.ψ — Methodψ(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.
Antique.ψ — Methodψ(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 $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.
Antique.ψ — Methodψ(model::InfinitePotentialWell, x; n=1)
\[\psi_n(x) = \sqrt{\frac{2}{L}} \sin \frac{n\pi x}{L}\]
Antique.ψ — Methodψ(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 $0\leq r \lt \infty$.
Antique.ψ — Methodψ(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.