GitHub

Delta Potential

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

Definitions

Antique.DeltaPotential — Type

Model

This model is described with the time-independent Schrödinger equation

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

and the Hamiltonian

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

Parameters are specified with the following struct:

DP = 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).

References

source

Potential

Antique.V — Method

V(model::DeltaPotential, x)

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

source

Eigenvalues

Antique.E — Method

E(model::DeltaPotential)

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

source

Eigenfunctions

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(), wave function ψ() and potential V() will be exported. 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.α1.0
julia> DP.m1.0
julia> DP.ℏ1.0

Eigenvalues:

julia> E(DP)-0.5

Wave functions:

using CairoMakie

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

# plot
w = lines!(ax, -5..5, x -> ψ(DP, x))

f
Example block output

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=(-5,5,-0.6,0.6))
# hidespines!(ax)
# hidedecorations!(ax)

# energy
hlines!(ax, E(DP), color=:black, linewidth=1, linestyle=:dash)

# wave function
lines!(ax, -5..5, x -> E(DP) + ψ(DP,x), linewidth=2)

#potential
lines!(ax, [-5,0,0,0,5], [0,0,-1,0,0], color=:black, linewidth=2)

f

Testing

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

Normalization of $\psi(x)$

\[\int_{-\infty}^{\infty} \psi^\ast(x) \psi(x) ~\mathrm{d}x = 1\]

  α |   m |   ℏ |     analytical |      numerical 
--- | --- | --- | -------------- | -------------- 
0.1 | 0.1 | 0.1 |    1.000000000 |    1.000000000 ✔
0.1 | 0.1 | 1.0 |    1.000000000 |    1.000000000 ✔
0.1 | 0.1 | 7.0 |    1.000000000 |    1.000004676 ✔
0.1 | 1.0 | 0.1 |    1.000000000 |    1.000000000 ✔
0.1 | 1.0 | 1.0 |    1.000000000 |    1.000000000 ✔
0.1 | 1.0 | 7.0 |    1.000000000 |    1.000000000 ✔
0.1 | 7.0 | 0.1 |    1.000000000 |    1.000000000 ✔
0.1 | 7.0 | 1.0 |    1.000000000 |    1.000000000 ✔
0.1 | 7.0 | 7.0 |    1.000000000 |    1.000000000 ✔
1.0 | 0.1 | 0.1 |    1.000000000 |    1.000000000 ✔
1.0 | 0.1 | 1.0 |    1.000000000 |    1.000000000 ✔
1.0 | 0.1 | 7.0 |    1.000000000 |    1.000000000 ✔
1.0 | 1.0 | 0.1 |    1.000000000 |    1.000000000 ✔
1.0 | 1.0 | 1.0 |    1.000000000 |    1.000000000 ✔
1.0 | 1.0 | 7.0 |    1.000000000 |    1.000000000 ✔
1.0 | 7.0 | 0.1 |    1.000000000 |    1.000000000 ✔
1.0 | 7.0 | 1.0 |    1.000000000 |    1.000000000 ✔
1.0 | 7.0 | 7.0 |    1.000000000 |    1.000000000 ✔
7.0 | 0.1 | 0.1 |    1.000000000 |    1.000000000 ✔
7.0 | 0.1 | 1.0 |    1.000000000 |    1.000000000 ✔
7.0 | 0.1 | 7.0 |    1.000000000 |    1.000000000 ✔
7.0 | 1.0 | 0.1 |    1.000000000 |    1.000000000 ✔
7.0 | 1.0 | 1.0 |    1.000000000 |    1.000000000 ✔
7.0 | 1.0 | 7.0 |    1.000000000 |    1.000000000 ✔
7.0 | 7.0 | 0.1 |    1.000000000 |    1.000000000 ✔
7.0 | 7.0 | 1.0 |    1.000000000 |    1.000000000 ✔
7.0 | 7.0 | 7.0 |    1.000000000 |    1.000000000 ✔

Eigenvalues

\[\begin{aligned} E_n &= \int_{-\infty}^{\infty} \psi^\ast(x) \hat{H} \psi(x) ~\mathrm{d}x \\ &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ \hat{V} + \hat{T} \right] \psi(x) ~\mathrm{d}x \\ &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ -\alpha\delta(x) - \frac{\hbar^2}{2m} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} \right] \psi(x) ~\mathrm{d}x \\ &= \int_{-\infty}^{\infty} \psi^\ast(x) \left[ -\alpha\delta(x) - \frac{\hbar^2}{2m} (\kappa^2 -2\kappa \delta(x))\right]\psi(x) ~\mathrm{d}x \\ \end{aligned}\]

where the $\kappa=m\alpha/\hbar^2$ and the integration with the delta function yeild a term proportional to the wave function at the origin $|\psi(0)|^2$.

  α |   m |   ℏ |     analytical |      numerical 
--- | --- | --- | -------------- | -------------- 
0.1 | 0.1 | 0.1 |   -0.050000000 |   -0.050000000 ✔
0.1 | 0.1 | 1.0 |   -0.000500000 |   -0.000500000 ✔
0.1 | 0.1 | 7.0 |   -0.000010204 |   -0.000010204 ✔
0.1 | 1.0 | 0.1 |   -0.500000000 |   -0.500000000 ✔
0.1 | 1.0 | 1.0 |   -0.005000000 |   -0.005000000 ✔
0.1 | 1.0 | 7.0 |   -0.000102041 |   -0.000102041 ✔
0.1 | 7.0 | 0.1 |   -3.500000000 |   -3.500000000 ✔
0.1 | 7.0 | 1.0 |   -0.035000000 |   -0.035000000 ✔
0.1 | 7.0 | 7.0 |   -0.000714286 |   -0.000714286 ✔
1.0 | 0.1 | 0.1 |   -5.000000000 |   -5.000000000 ✔
1.0 | 0.1 | 1.0 |   -0.050000000 |   -0.050000000 ✔
1.0 | 0.1 | 7.0 |   -0.001020408 |   -0.001020408 ✔
1.0 | 1.0 | 0.1 |  -50.000000000 |  -50.000000000 ✔
1.0 | 1.0 | 1.0 |   -0.500000000 |   -0.500000000 ✔
1.0 | 1.0 | 7.0 |   -0.010204082 |   -0.010204082 ✔
1.0 | 7.0 | 0.1 | -350.000000000 | -350.000000000 ✔
1.0 | 7.0 | 1.0 |   -3.500000000 |   -3.500000000 ✔
1.0 | 7.0 | 7.0 |   -0.071428571 |   -0.071428571 ✔
7.0 | 0.1 | 0.1 | -245.000000000 | -245.000000000 ✔
7.0 | 0.1 | 1.0 |   -2.450000000 |   -2.450000000 ✔
7.0 | 0.1 | 7.0 |   -0.050000000 |   -0.050000000 ✔
7.0 | 1.0 | 0.1 | -2450.000000000 | -2450.000000000 ✔
7.0 | 1.0 | 1.0 |  -24.500000000 |  -24.500000000 ✔
7.0 | 1.0 | 7.0 |   -0.500000000 |   -0.500000000 ✔
7.0 | 7.0 | 0.1 | -17150.000000000 | -17150.000000000 ✔
7.0 | 7.0 | 1.0 | -171.500000000 | -171.500000000 ✔
7.0 | 7.0 | 7.0 |   -3.500000000 |   -3.500000000 ✔