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#!/usr/bin/env julia
"""Find eigenstates and eigenenergies of central potential problems"""
using LinearAlgebra
using Plots
N = 5000 # number of lattice points
L = 20.0 # r runs from 0 to L
dr = L / N
D = zeros(N, N) # discrete radial 2nd derivative 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(r, ℓ = 0)
""" The potential energy"""
#return 0.5 * ell * (ℓ+1.0) * pow(r, -2.0) # V=0: Free particle in spherical coordinates
return -1.0 / r + 0.5 * ℓ * (ℓ + 1.0) * r^(-2.0) # Hydrogen atom
#return -r^(-1.1) + 0.5 * ℓ * (ℓ+1.0) * r^(-2.0) # modified Coulomb potential
end
for i in 1:N
r = (i + 0.5) * dr # radial coordinates of lattice points
V[i, i] = potential(r, 0)
end
H = -0.5 * dr^(-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])
plot(potential)
plot(v[:, 1])
#plot(v[:,2])
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