#!/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)
=#