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
— 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).
Potential
Antique.V
— MethodV(model::DeltaPotential, x)
\[V(x) = -\alpha \delta(x).\]
Eigen Values
Antique.E
— MethodE(model::DeltaPotential)
\[E = - \frac{m\alpha^2}{2\hbar^2}\]
Eigen Functions
Antique.ψ
— Methodψ(model::DeltaPotential, x)
\[\psi(x) = \frac{\sqrt{m\alpha}}{\hbar} \mathrm{e}^{-m\alpha |x|/\hbar^2}\]
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
julia> DP.m
1.0
julia> DP.ℏ
1.0
Eigen values:
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
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 | 1.000000000000 | 1.000000000000 ✔
0.1 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔
0.1 | 0.1 | 7.0 | 1.000000000000 | 1.000004676239 ✔
0.1 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔
0.1 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔
0.1 | 1.0 | 7.0 | 1.000000000000 | 0.999999999999 ✔
0.1 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔
0.1 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔
0.1 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔
1.0 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔
1.0 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔
1.0 | 0.1 | 7.0 | 1.000000000000 | 0.999999999999 ✔
1.0 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔
1.0 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔
1.0 | 1.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔
1.0 | 7.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔
1.0 | 7.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔
1.0 | 7.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔
7.0 | 0.1 | 0.1 | 1.000000000000 | 1.000000000000 ✔
7.0 | 0.1 | 1.0 | 1.000000000000 | 1.000000000000 ✔
7.0 | 0.1 | 7.0 | 1.000000000000 | 1.000000000000 ✔
7.0 | 1.0 | 0.1 | 1.000000000000 | 1.000000000000 ✔
7.0 | 1.0 | 1.0 | 1.000000000000 | 1.000000000000 ✔
7.0 | 1.0 | 7.0 | 1.000000000000 | 1.000000000000 ✔
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 ✔