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1-D Schroedinger.py
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1-D Schroedinger.py
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#Computational physics- written by tao pang - Chapter 4 - section 9
#An example of solving the eigenvalue problem of the one-dimensional Schroedinger equation via the secant and Numerov methods..
#packages
import matplotlib.pyplot as plt
import numpy as np
#initial values for definig equation and calculation
nx=500
m=10
ni=10
delta=1e-6
e=2.4
de=0.1
x1=-10
x2=10
h=float((x2-x1)/500.)
#initial values
ql=[{}]*(nx+1)
qr=[{}]*(nx+1)
s=[0]*(nx+1)
u=[{}]*(nx+1)
#------------------------------Simpson method for finding the estimated amount for our initial values--------------------------
def simpson(y,h):
n=int(len(y)-1)
s0=0
s1=0
s2=0
for i in range(1,n):
s0 += y[i]
s1 += y[i-1]
s2 += y[i+1]
s=float((s1+4*s0+s2)/3.)
if (n+1)%2==0:
return h*(s+(5*y[n]+8*y[n-1]-float(y[n-2]/12.)))
else:
return h*s
#----------------------------------------------------------------
#Method to provide the given potential in the problem.
#----------------------------Potential---------------------------
import math
def v(x):
alpha = 1
lamb =4
return alpha*alpha*lamb*(lamb-1)*float(0.5-float(1/(math.pow(math.cosh(alpha*x),2))))/2.
#---------------------------------------------------------------
#----------------------------Secant method : Find the eigenvalue via the secant search-----------------------------
def Sacant(n,delta,x,dx):
k=0
x1=x+dx
while abs(dx)>delta and k<n :
x2 = x1-f(x1)*float(((x1-x)/(f(x1)-f(x))))
x=x1
x1=x2
dx=x1-x
k+=1
return x1
#-------------------------------------------------------------------
#-------------------------------Numerov method-----------------------------
#which is an extremely accurate scheme for linear differential equations without the first-order derivative term
#defined in textbook
def numerov(m,h,u0,u1,q,s):
u=[{}]*m
u[0]=u0
u[1]=u1
g=float((h*h)/12)
for i in range(1,m-1):
c0 = 1+g*q[i-1]
c1 = 2-10*g*q[i]
c2 = 1+g*q[i+1]
d = g*(s[i+1]+s[i-1]+10*s[i])
u[i+1] = float((c1*u[i]-c0*u[i-1]+d)/c2)
return u
#-------------------------------------------------------------------
#definition the function which is suppose to calculate its root to find the matching points between potential and wavefunction
#----------------------------------F(x)-----------------------------
def f(E):
y=[{}]*(nx+1)
u0=0
u1=0.01
for i in range(0,nx+1):
x = -10+i*h
ql[i] = 2*(E-v(x))
qr[nx-i] = ql[i]
im=0
for i in range(nx):
if (ql[i]*ql[i+1]<0 and ql[i]>0):
im=i
nl=im+2
nr=nx-im+2
c0 = 0
c1 = 0
c2 = 0
ul=[{}]*nl
ul[0]=0
ul[1]=0.01
g=float((h*h)/12)
for i in range(1,nl-1):
c0 = 1+g*ql[i-1]
c1 = 2-10*g*ql[i]
c2 = 1+g*ql[i+1]
d = g*(s[i+1]+s[i-1]+10*s[i])
ul[i+1] =float((c1*ul[i]-c0*ul[i-1]+d)/c2)
c0 = 0
c1 = 0
c2 = 0
ur=[{}]*nr
ur[0]=u0
ur[1]=u1
g=float((h*h)/12)
for i in range(1,nr-1):
c0 = 1+g*qr[i-1]
c1 = 2-10*g*qr[i]
c2 = 1+g*qr[i+1]
d = g*(s[i+1]+s[i-1]+10*s[i])
ur[i+1] = float((c1*ur[i]-c0*ur[i-1]+d)/c2)
#left Wave function
ratio = float(ur[nr-2]/ul[im])
#print ratio
for i in range(im+1):
u[i] = ratio*ul[i]
y[i] = u[i]*u[i]
#right wave function
for i in range(nr-1):
u[i+im] = ur[nr-i-2]
y[i+im] = u[i+im]*u[i+im]
#normalize the wave function
Sum=simpson(y,h)
Sum = math.sqrt(Sum)
#print Sum
for i in range(nx+1):
u[i] /= Sum
f0 = ur[nr-1]+ul[nl-1]-ur[nr-3]-ul[nl-3]
return float(f0/(2*h*ur[nr-2]))
#main section
e = Sacant(ni,delta,e,de)
#plot potetial and wave function :
x=x1
V = [{}]*(nx+1)
alpha = 1
lamb =4
for i in range(nx+1):
x = x1 + i*h
V[i] = alpha*alpha*lamb*(lamb-1)*float(0.5-float(1/(math.pow(math.cosh(alpha*x),2))))/2.
x=np.linspace(-10,10,501)
plt.plot(x,u,x,V)
plt.grid()
plt.title("The eigenvalue Schrodinger equation")
plt.xlabel("$x$")
plt.ylabel("$V(x),U(x)$")
plt.show()