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#!/usr/bin/env julia
"""Find eigenstates and eigenenergies of 1-D quantum problems"""
using LinearAlgebra
using Plots
N = 1000 # number of lattice points
L = 20.0 # x runs from -L/2 to L/2
dx = L / N
D = zeros(N, N) # discrete laplacian operator
V = zeros(N, N) # potential
for i in 1:N
D[i, i] = -2.0
end
for i in 1:N-1
D[i, i+1] = 1.0
D[i+1, i] = 1.0
end
#println("\nLattice Laplacian operator")
#println(D)
function potential(x)
""" The potential energy"""
#return 0.0 # particle in a box
#return 0.5 * x^2 # SHO with the spring constant k = 1
#return -6.0 * x^2 + 8.0 * x^6 # potential with zero ground state energy
#return 0.1 * x^4 - 2.0 * x^2 + 0.0 * x # double-well potential
return 8 * x^6 - 8 * x^4 - 4 * x^2 + 1 # another double-well potential
end
for i in 1:N
x = (i + 0.5) * dx - 0.5 * L # coordinates of lattice points
V[i, i] = potential(x)
end
H = -0.5 * dx^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1
#println("\nMatrix elements of Hamiltonian = ")
#println(H)
e, v = eigen(H) # diagonalize Hamiltonian
println("\nGround state energy = ", e[1])
println("\n1st excited state energy = ", e[2])
println("\n2nd excited state energy = ", e[3])
println("\n3rd excited state energy = ", e[4])
println("\n4th excited state energy = ", e[5])
gs(x) = exp(-0.5 * x^2) # Gaussian that is exact ground state of SHO
plot(potential)
plot(v[:, 1])
#plot(v[:,2])
#plot(gs)
#=
eList = zeros(0)
for i in 1:20
push!(eList, e[i])
end
bar(eList, orientation = :horizontal)
=#
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