Graphical solutions of the Schrödinger-equation for an idealised H_2 potential well

h2.py

# encoding: utf-8
# Needs the python2 version of maplotlib because of a bug in the python3 version :(

from pylab import *

coth = lambda x: cosh(x)/sinh(x)

#  ∞              ∞ 
#   |            |
#   |            |
#   |     __     |  _
#   |    |  |    |   | V0
#   |____|  |____|  _|
#
#        |__|
#         2a

def mkplot(V0, a):
    # For plotting E on the x-axis
    #left = lambda E: a*tan(a*sqrt(E))
    #right1 = lambda E: sqrt(E)/sqrt(V0-E)*coth(a*sqrt(V0-E))
    #right2 = lambda E: sqrt(E)/sqrt(V0-E)*tanh(a*sqrt(V0-E))

    # for plotting sqrt(E) on the x-axis
    left = lambda sqrtE: a*tan(a*sqrtE)
    right1 = lambda sqrtE: sqrtE/sqrt(V0-sqrtE**2)*coth(a*sqrt(V0-sqrtE**2))
    right2 = lambda sqrtE: sqrtE/sqrt(V0-sqrtE**2)*tanh(a*sqrt(V0-sqrtE**2))

    xs = linspace(0, V0, 40000)
    xs_filtered = ma.masked_where(abs(left(xs)) > 70, xs)
    
    plot(xs_filtered, left(xs_filtered), 'b-', label=r'$\mathrm{tan}(\alpha)$')
    
    plot(xs, right1(xs), 'r-', label=r'$\frac{\alpha}{\beta}\,\mathrm{cotanh}(\beta a)$ (symmetric)')
    plot(xs, right2(xs), 'g-', label=r'$\frac{\alpha}{\beta}\,\mathrm{tanh}(\beta a)$ (asymmetric)')
    
    xlim(0, sqrt(V0))
    ylim(0, 10)
    xlabel(r'$\sqrt{E}\ / \ \sqrt{\mathrm{J}}$')
    
    legend(loc=2)

def V0_sweep():
    for V in logspace(1, 3, 200):
        mkplot(V, 1)
        title('V = {}'.format(V))
        savefig('V0_sweep_%020.7f.png' % V)
        clf()

def a_sweep():
    for a in linspace(0.1, 10, 200):
        mkplot(40, a)
        title('a = {}'.format(a))
        savefig('a_sweep_%011.7f.png' % a)
        clf()

def assignment():
    mkplot(50, 1)
    title('small potential $V_0$')
    savefig('V0_small.pdf')
    clf()
    mkplot(500, 1)
    title(u'large potential $V = 10 V_0$')
    savefig('V0_large.pdf')
    clf()
    mkplot(20, 2)
    title('small distance $a_0$')
    savefig('a_small.pdf')
    clf()
    mkplot(20, 10)
    title(u'large distance $a = 5 a_0$')
    savefig('a_large.pdf')
    clf()

assignment()