Quantum Mechanics
Program Example
Time evolution of a wave packet

As an example of program , we use the time evolution of a wave packet. 
 
In this program, We can:

1. Choose any arbitrary initial un-normalized wave function : psi(x,0)
Example : Gaussian wave  packet

 
2.  Choose a potantial energy  U(X): 

a) Free particle  : U = 0;

b) Quadratic harmonic oscillator potential :  U = 1/2*100*x.^(2);

c) Quartic potential: U = 1/2*x.^(4); 

d) Step potential  located at x=L/2: U = heaviside(x-(L/2)); 

e) Use expectation value of energy   = 
avgE = phi’*0.5*diag(k.^2,0)*phi*dk/(phi’*phi*dk);
 For example avgE = 1 or 2 ,.. 5.

f) Wavepacket scattering off the step potential: 
U = avgE*heaviside(x-(L/2)); 

g) Finite square well of width 2w and depth d: 
U = -d*(heaviside(x+ w )-heaviside(x-w));
 For example w = 0.1, and d = 200 
 
h) Make your own.