# -*- coding: utf-8 -*-
# Python simulation of an electron in a 1d infinite box potential
# Integrate time independent SE using the Verlet method
# Boundary conditions are found by shooting
# MW 190402
from pylab import *
import numpy as np
import matplotlib.pyplot as plt
from bisection import bisection
# Constants
a=1.0e-9 # well width a=1 nm
hbar=1.0545718e-34 # Plancks constant
m=9.10938356e-31 # electron mass
e=1.60217662e-19 # electron charge=-e
c=2.0*m/hbar**2 # constant in Schrödinger equation
# EeV = 0.3 # input energy in eV: test 0.3 , 0.4 , 0.3760 , 1.5
#print('Exact solution for infinite box potential')
# print('E1=',(hbar*np.pi/a)**2/(2.0*m)/e,'eV')
# print('E2=',(hbar*5.0*np.pi/a)**2/(2.0*m)/e,'eV')
# potential energy function
# V0=10*e
def V(x):
y = 0.0
return y
def verletmethod(EeV, N): #, N=1000):
E = EeV*e # input energy in J
# N=1000 # number of mesh points
dx=a/N # step length
dx2=dx**2 # step length squared
# initial values and lists
x = 0 # initial value of position x
psi = 0.0 # wave function at initial position
dpsi = 1.0 # derivative of wave function at initial position
x_tab = [] # list to store positions for plot
psi_tab = [] # list to store wave function for plot
x_tab.append(x/a)
psi_tab.append(psi)
for i in range(N) :
d2psi = c*(V(x)-E)*psi
d2psinew = c*(V(x+dx)-E)*psi
psi += dpsi*dx + 0.5*d2psi*dx2
dpsi += 0.5*(d2psi+d2psinew)*dx
x += dx
x_tab.append(x/a)
psi_tab.append(psi)
print('E=',EeV,'eV , psi(x=a)=',psi)
# return x_tab, psi_tab
return psi
def plot_line(x_tab, psi_tab):
plt.close()
plt.figure(num=None, figsize=(8,8), dpi=80, facecolor='w', edgecolor='k')
plt.plot(x_tab, psi_tab, linewidth=1, color='red')
#plt.xlim(0, 1)
#limit=1.e-9
#plt.ylim(0, limit)
#plt.ylim(-limit, limit)
#plt.autoscale(False)
plt.xlabel('x/a')
plt.ylabel('$\psi$')
#plt.savefig('psi.pdf')
plt.show()
def error_plot():
dpsi_a = []
EeV1 = (hbar*np.pi/a)**2/(2.0*m)/e
N = [1000,10000,100000]
for n in N:
psi = verletmethod(EeV1, n)
dpsi_a.append(psi)
print(dpsi_a)
print(N)
plt.close()
plt.figure(num=None, figsize=(8,8), dpi=80, facecolor='w', edgecolor='k')
plt.loglog(N, dpsi_a, 'r-x', linewidth=1, color='red')
plt.xlabel('N')
plt.ylabel('$\Delta \Psi(x=a)$')
plt.grid(True, which="both")
plt.show()
def main():
# x_tab, psi_tab =
# psi = verletmethod(1.5)
# plot_line(x_tab, psi_tab)
# print(bisection(verletmethod, 9.3, 9.5, 1e-9))
error_plot() # Correct
pass
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