Antique.jl

Self-contained, well-tested, well-documented Analytical Solutions of Quantum Mechanical Equations.

Install

Run the following code on the REPL or Jupyter Notebook to install this package.

]add Antique

Or specify the version like ]add Antique@0.11.1 to install a specific version. The version of this package can be found at ]status Antique.

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use.

using Antique

The energy E(), the wave function ψ(), the potential V() and some other functions will be exported. There are two ways to avoid function name conflicts. Run import Antique instead of using Antique, and use the energy Antique.E(), the wave function Antique.ψ() and the potential Antique.V(). Or try giving other function names like using Antique: V as potential, E as energy, ψ as wavefuntion, HydrogenAtom. Here are examples for the hydrogen-like atom. The analytical notation of the energy (the eigen value of the Hamiltonian) is written as

$$E_n = -\frac{Z^2}{2n^2} E_\mathrm{h}.$$

The Hydrogen atom has the symbol $\mathrm{H}$ and atomic number 1 ($Z=1$). Therefore the ground state ($n=1$) energy is $-\frac{1}{2} E_\mathrm{h}$.

H = HydrogenAtom(Z=1)
E(H, n=1)
# output> -0.5

The Helium cation has the symbol $\mathrm{He}^+$ and atomic number 2 ($Z=2$). Therefore the ground state ($n=1$) energy is $-2 E_\mathrm{h}$.

He⁺ = HydrogenAtom(Z=2)
E(He⁺, n=1)
# output> -2.0

There are more examples on each model page.

Supported Models

• Delta Potential DeltaPotential
• Infinite Potential Well InfinitePotentialWell
• Harmonic Oscillator HarmonicOscillator
• PoschlTeller PoschlTeller
• Morse Potential MorsePotential
• Rigid Rotor RigidRotor
• Infinite PotentialWell 3D InfinitePotentialWell3D
• Spherical Oscillator SphericalOscillator
• Hydrogen Atom HydrogenAtom
• Coulomb 2-Body System CoulombTwoBody

Demonstration

This is an example of a variational calculation for the hydrogen atom based on Thijssen(2007). We check the accuracy of the numerical solution by comparison with the analytical solution. Comparing wavefunctions can be difficult, but Antique.jl makes it easy. You can extend it to excited states ($n>1$) as well as the ground state ($n=1$). Thus, Antique.jl is useful for testing numerical methods. We hope many numerical methods to be developed using Antique.jl.

# calculations based on Thijssen(2007) https://doi.org/10.1017/CBO9781139171397
using LinearAlgebra
α = [13.00773, 1.962079, 0.444529, 0.1219492] 
nₘₐₓ = length(α)
S = [(pi/(α[i]+α[j]))^(3/2) for i=1:nₘₐₓ, j=1:nₘₐₓ]
T = [3*pi^(3/2)*α[i]*α[j]/(α[i]+α[j])^(5/2) for i=1:nₘₐₓ, j=1:nₘₐₓ]
V = [-2*pi/(α[i]+α[j]) for i=1:nₘₐₓ, j=1:nₘₐₓ]
H = T + V
E, C = eigen(Symmetric(H),Symmetric(S))

# norm & energy
import Antique
HA = Antique.HydrogenAtom(Z=1, Eₕ=1.0, a₀=1.0, mₑ=1.0, ℏ=1.0)
println("Norm")
println("  numerical : ", transpose(C[:,1]) * S * C[:,1])
println("  analytical: ", 1)
println("Energy")
println("  numerical : ", E[1])
println("  analytical: ", Antique.E(HA,n=1))

# wave function
using CairoMakie
f = Figure(size=(420,300), fontsize=11.5)
ax = Axis(f[1,1], xlabel=L"$r$", ylabel=L"$\psi(r,0,0)$", limits=(0,4,0,0.6), ylabelsize=16.5, xlabelsize=16.5)
l1 = lines!(ax, 0:0.01:10, r -> sum(C[:,1] .* exp.(-α*r^2)))
l2 = lines!(ax, 0:0.01:10, r -> real(Antique.ψ(HA,r,0,0)), color=:black, linestyle=:dash, label="Antique.jl")
axislegend(ax, [l1,l2], ["Numerical, Thijssen(2007)","Analytical, Antique.jl"], position=:rt)
f

Norm
  numerical : 0.9999999999999997
  analytical: 1
Energy
  numerical : -0.49927840566748566
  analytical: -0.5

Future Works

The candidate models are listed on the Wikipedia page of List of quantum-mechanical systems with analytical solutions. Please submit your requests and suggestions as issues on GitHub.

Developer's Guide

This is the guideline for adding new models. Adding a new model may take from a few days to a few weeks due to reference search, test implementation, and writing documentation.

1. First, please submit a new issue or comment here. I will assign you to the issue. We need to find orthodox references (textbooks or papers, not Wikipedia) for the analytical solutions (eigenvalues and eigenfunctions) before the development. This will take more time than you think.
2. Fork the repository on GitHub.
3. Clone the forked repository to your local machine by Git.
4. Please create 3 files:

files	comments
src/ModelName.jl	Write the source codes and docstrings in this file. The most helpful examples are the harmonic oscillator for one-dimensional systems and the hydrogen atom for three-dimensional systems. We recommend that you copy these files. First create a structure struct ModelName with the same name as the model name (The best way is Find & Replace). Because the function names conflict, you must always give the struct ModelName as the first argument to V, E, ψ and other functions. Multi-dispatch avoids conflicts. We recommend using Revice.jl while coding. Run include("./dev/revice.jl") on the REPL or use dev.ipynb.
test/ModelName.jl	Write test code in this file. At a minimum, please check the normalization and the orthogonality of the eigenfunctions using QuadGK.jl. Please also do tests for the eigenvalues (for example, calculate the expectation values of the Hamiltonian (energy) using the eigenfunctions and check that these values match the eigenvalues).
docs/src/ModelName.md	Write documentation in this file. Include at least the definition of the Hamiltonian and the analytical solutions (eigenvalues and eigenfunctions). Call a docstring in the source code (src/ModelName.jl) .

5. Please rewrite 5 files:

files	comments
src/Antique.jl	Add the new model name :ModelName to the models = [...] array in this file. : is required at the beginning.
docs/make.jl	Add the new model into pages=[...] in this file.
test/runtests.jl	Change for model in [...] in this file. Please test all models before pull requests.
README.md	Add the new model to the list of supported models.
docs/index.md	Add the new model to the list of supported models.

6. Execute include("./dev/test.jl") to run tests. It will take few minutes to complete.
7. Execute include("./dev/docs.jl") to compile documents. HTML files (docs/build/*.html) will be generated. Please check them with Chrome or any other web browsers.
8. Commit and Push the codes.
9. Submit a pull request on GitHub.

Acknowledgment

Thanks to all contributors. This package was named by @KB-satou and @ultimatile. @MartinMikkelsen contributed to writing docstrings. Special thanks to @hyrodium for his help with managing the documentation and advice on coding style. @lhapp27 implemented 2 models, and @ajarifi implemented 3 models.