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InfinitePotentialWell.md

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CurrentModule = Antique

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

Potential

Antique.V(::InfinitePotentialWell, ::Any)

Eigenvalues

Antique.E(::InfinitePotentialWell)

Eigenfunctions

Antique.ψ(::InfinitePotentialWell, ::Any)

Proofs

Usage & Examples

[Install Antique.jl](@ref Install) for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. 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)
; #hide

Parameters:

IPW.L
IPW.m
IPW.ℏ

Eigenvalues:

E(IPW, n=1)
E(IPW, n=2)

Wave functions:

using CairoMakie

# settings
f = Figure()
ax = Axis(f[1,1], xlabel=L"$x$", ylabel=L"$\psi(x)$")

# plot
w1 = lines!(ax, 0..1, x -> ψ(IPW, x, n=1))
w2 = lines!(ax, 0..1, x -> ψ(IPW, x, n=2))
w3 = lines!(ax, 0..1, x -> ψ(IPW, x, n=3))
w4 = lines!(ax, 0..1, x -> ψ(IPW, x, n=4))
w5 = lines!(ax, 0..1, x -> ψ(IPW, x, n=5))

# legend
axislegend(ax, [w1, w2, w3, w4, w5], [L"n=1", L"n=2", L"n=3", L"n=4", L"n=5"], position=:lb)

f

Potential energy curve, Energy levels, Wave functions:

using CairoMakie

# settings
f = Figure()
ax = Axis(f[1,1], xlabel=L"$x$", ylabel=L"$V(x),~E_n,~\psi_n(x) \times 5 + E_n$", aspect=1, limits=(-0.5,1.5,0,140))
# hidespines!(ax)
# hidedecorations!(ax)

# plot
for n in 1:5
  # energy
  lines!(ax, [0,IPW.L], fill(E(IPW,n=n),2), color=:black, linewidth=2)
  # wave function
  lines!(ax, 0..IPW.L, x -> E(IPW,n=n) + 5*ψ(IPW,x,n=n), linewidth=2)
end

#potential
lines!(ax, [0,0,IPW.L,IPW.L], [140,0,0,140], color=:black, linewidth=2)

f
save("assets/fig/InfinitePotentialWell.png", f) # hide
; # hide

Testing

Unit testing and Integration testing were done using numerical integration (QuadGK.jl). The test script is here.

using Markdown
using Antique
Markdown.parse(Antique.load("../../test/result/InfinitePotentialWell.log"))