1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
|
#!/usr/bin/env julia
using Plots
using LinearAlgebra
using DifferentialEquations
k_0 = 700.0
# useful functions:
function H(psi, dx, k = 0, width = 0.1) # action of Hamiltonian on wavefunction
Hpsi = zeros(ComplexF64, size(psi))
# -(1/2) * laplacian(psi) (m = hbar = 1)
Hpsi[2:end-1] = 0.5 * (2.0 * psi[2:end-1] - psi[3:end] - psi[1:end-2]) / (dx * dx)
# add the potential function V_0 = 2 * k_0^2 for specific region
for i in 1:length(psi)
x = i * dx
if x >= 30.0 && x <= 30.0 + width
Hpsi[i] += 2 * k^2
# else
# Hpsi[i] += -0.01
end
end
# periodic boundary conditions
#Hpsi[1] = 0.5 * (2.0*psi[1] - psi[2] - psi[end])/(dx*dx)
#Hpsi[end] = 0.5 * (2.0*psi[end] - psi[end-1] - psi[1])/(dx*dx)
return Hpsi
end
derivative(psi, dx) = -1.0im * H(psi, dx, k_0)
function initialWavefunction(x::Vector{Float64}, x0 = 10.0, Delta = 1.0, k = 4.0)
Delta2 = Delta^2
return exp.(-(x .- x0) .^ 2 / Delta2) .* exp.(1.0im * k * x)
end
function normalization(psi, dx) # normalization of wavefunction
n2 = dot(psi, psi) * dx
return sqrt(n2)
end
prob(psi) = real(psi .* conj(psi))
# The actual simulation
N = 5000 # number of lattice points
L = 40.0 # x runs from 0 to L
dx = L / N
x = range(0.0, L, N) |> collect # lattice along x-axis
println("dx = ", dx)
# initial wavefunction has position (x0), width (Delta), and momentum (k)
psi0 = initialWavefunction(x, 10.0, 0.05, k_0)
# normalize wavefunction
n = normalization(psi0, dx)
psi0 = psi0 / n
println("norm = ", normalization(psi0, dx))
# plot initial wavefunction and probability density
plot(prob(psi0))
# integrate forward in time
tf = 3.0
dt = 5e-7
tspan = (0.0, tf)
# function timeEvolve(psi0, tf, dt) # second order Runge-Kutta algorithm
# psi = psi0
# for t in range(0, stop = tf, step = dt)
# psiMid = psi + 0.5 * dt * derivative(psi, dx)
# psi = psi + dt * derivative(psiMid, dx)
# # println("t = ", t, " norm = ", real(normalization(psi, dx)))
# if real(normalization(psi, dx)) < 0.999 || real(normalization(psi, dx)) > 1.010
# println("dt = ", dt, " t = ", t, " norm = ", normalization(psi, dx))
# println("Normalization deviated too far from 1.0")
# break
# end
# end
# return psi
# end
tendency(psi, dx, t) = derivative(psi, dx) # use ODE solver
problem = ODEProblem(tendency, psi0, tspan, dx) # specify ODE
sol = solve(problem, Tsit5(), reltol = 1e-14, abstol = 1e-14) # solve using Tsit5 algorithm to specified accuracy
# compare initial and final wavefunction probabilities
# psi = timeEvolve(psi0, tf, dt)
psi = sol[:, end]
times = sol.t
# check that normalization hasn't deviated too far from 1.0
println("norm = ", normalization(psi, dx))
plot(prob(psi0))
savefig("10-17-1.png")
# plot a vertical line where the barrier is
# barrier_on_lattice = 6.5 / dx
barrier_on_lattice = 30.0 / dx
plot(prob(psi), label = "final (t=$tf)", xlabel = "x", ylabel = "|ψ|^2", title = "Initial and final probability densities k_0=$k_0", lw = 1.5)
plot!([barrier_on_lattice, barrier_on_lattice], [0.0, 0.3], color = :red, label = "barrier (width .2)", lw = 1.5, linestyle = :dash)
savefig("10-17-width-.1.png")
# plot([prob(psi0), prob(psi)], label = ["initial" "final (t=$tf)"], xlabel = "x", ylabel = "probability density", title = "Initial and final probability densities k_0=$k_0", lw = 1.5)
# plot!([barrier_on_lattice, barrier_on_lattice], [0.0, 2.0], color = :red, label = "barrier (width .2)", lw = 1.5, linestyle = :dash)
# savefig("10-17-width-.2.png")
|
