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#!/usr/bin/env julia
"""Find eigenstates and eigenenergies of 1-D quantum problems"""
using LinearAlgebra
using Plots
#println("\nLattice Laplacian operator")
#println(D)
function potential(x, n)
""" 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
return abs(x)^n
end
#println("\nMatrix elements of Hamiltonian = ")
#println(H)
function map_n_to_energies(n)
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
for i in 1:N
x = (i + 0.5) * dx - 0.5 * L # coordinates of lattice points
V[i, i] = potential(x, n)
end
H = -0.5 * dx^(-2.0) * D + V # Hamiltonian. Here m = hbar = 1
e, v = eigen(H) # diagonalize Hamiltonian
println("\n n = ", n)
println("Ground state energy = ", e[1])
println("1st excited state energy = ", e[2])
println("2nd excited state energy = ", e[3])
println("3rd excited state energy = ", e[4])
println("4th excited state energy = ", e[5], "\n")
return e
end
n_s = collect(0:1:18)
n_to_e = [map_n_to_energies(n) for n in n_s]
# plot e[0] for all N
ground_state = []
excited_1 = []
excited_2 = []
excited_3 = []
for i in 1:length(n_to_e)
push!(ground_state, n_to_e[i][1])
push!(excited_1, n_to_e[i][2])
push!(excited_2, n_to_e[i][3])
push!(excited_3, n_to_e[i][4])
end
plot(ground_state, label = "groud state energy for n", xlabel = "n (level)", ylabel = "energy", title = "excited energy levels for V(n) = abs(x)^n", marker = :circle)
plot!(excited_1, label = "1st excited state", marker = :circle)
plot!(excited_2, label = "2nd excited state", marker = :circle)
plot!(excited_3, label = "3rd excited state", marker = :circle)
# plot the energies for an inifinite square well as a horizontial line
# function excited_state_to_energy_inf_square_well(n)
# return n^2 * pi^2 / 2
# end
# ground_state_inf_square_well = [excited_state_to_energy_inf_square_well(1) for i in 1:length(n_s)]
# excited_1_inf_square_well = [excited_state_to_energy_inf_square_well(2) for i in 1:length(n_s)]
# excited_2_inf_square_well = [excited_state_to_energy_inf_square_well(3) for i in 1:length(n_s)]
# excited_3_inf_square_well = [excited_state_to_energy_inf_square_well(4) for i in 1:length(n_s)]
# plot!(ground_state_inf_square_well, label = "ground state energy for infinite square well")
# plot!(excited_1_inf_square_well, label = "1st excited state for infinite square well")
# plot!(excited_2_inf_square_well, label = "2nd excited state for infinite square well")
# plot!(excited_3_inf_square_well, label = "3rd excited state for infinite square well")
savefig("hw8/10-3.png")
# 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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