In [1]:

    

import numpy as np
import matplotlib.pyplot as plt
from numpy import linalg as LA
%matplotlib inline


    

In [21]:

    

def V(x, w=.0001, l=10., V_0=200.):
    return -V_0 / 2. * (np.tanh((x + l / 2) / w) \
                      - np.tanh((x - l / 2) / w)) + V_0


    

In [22]:

    

x = np.linspace(-20, 20, 1000)
for k in - np.arange(-1, 2):
    plt.plot(x, V(x, 10. ** k, 10, 10),\
             label='w = 10^'+str(k))
plt.legend()

plt.figure()
for k in np.arange(1, 10, 3):
    plt.plot(x, V(x, 1, 10, k),\
             label='V_0 = '+str(k))
plt.legend()

plt.figure()
for k in np.arange(1, 10, 3):
    plt.plot(x, V(x, 1, k, 4),\
             label='l = '+str(k))
plt.legend()


    

    
Out[22]:





<matplotlib.legend.Legend at 0x7f3457336ef0>






    













    













    

In [17]:

    

# define the grid
N = 2 ** 10 # number of grid points
x0 , x1 = -20. , 20. # grid boundaries
dx =(x1 - x0)/(N -1) # grid spacing
x = np.linspace (x0 , x1 , N) # grid points
# setup the Hamiltonian matrix
H = -1 / (2 * dx ** 2) * (np.diag(np.ones(N - 1) , -1) - 2 * np.diag(np.ones(N)) + \
    np.diag(np.ones(N - 1), 1)) + np.diag(V(x))
# compute eigenvalues
E = LA.eigvalsh(H)
# plot first eigenvalues
n =100
ns =  np.array (range (0 , n ))
plt.plot( ns, np.sort (E)[0: n ] ,'+', label ='eigenvalues of discretized Hamiltonian ')
plt.plot(ns[E_teo(ns) < 200], E_teo(ns)[E_teo(ns) < 200])
plt.grid()
sum(E < 200)


    

    
Out[17]:





66






    

In [20]:

    

E , Psi_E = LA.eigh(H)
Psi_E /= np.sqrt (dx)
for n in np.arange(2):
    c = np.zeros ( n +1)
    c [n]=1
    plt.plot (x ,- Psi_E [: , n ] , label ='$n =% i$ ' % n )
    
plt.legend()
def E_teo(n):
    return n** 2 * np.pi ** 2 / (2 * (10**2))
print (E[:5])
print (E_teo(np.arange(5))/78)


    

    




[ 0.04821121  0.19283743  0.43385644  0.77123123  1.20490998]
[ 0.          0.00063267  0.00253067  0.005694    0.01012267]






    

In [23]:

    

plt.plot(x, V(x))


    

    
Out[23]:





[<matplotlib.lines.Line2D at 0x7f345926a8d0>]






    

In [ ]: