# Schrodinger equation subjected to general potential

1. Apr 10, 2013

### ace1719

There are few questions on an assignment (taken from an old exam, YIKES!) that are confounding me and I was wondering if you in the physics community could help. Here is the question;

Given the wavefunction, ψ(x,t)=∫PHI(p)*exp[i((px-E(p)t)/hbar)]dp For the case where a quantum particle is subjected to a general potential V , show that from E = p^2/2m + V(x) you can construct an equation for the wavepacket given above. This equation is the Schrodinger equation.

Here are the follow up questions;

c. Consider a generic time-dependent potential V = V (x; t) = V0cos(OMEGA*t) where OMEGA is a generic angular frequency. Can you use the method of separation of variables to solve the Schrodinger equation for the time and spatial part? If yes, do the calculations. If not, provide
arguments.

d. Consider a general potential V (x) as shown below, and a particle with a total energy E1. Make a sketch of the ground state wave function you expect in all regions of space, i.e. for x < x1, x > x2 and x1 < x < x2. Make a sketch of PSI^2. What do you conclude? Compare your results in relation to classical physics. Could the ground state energy be zero as in the classical case?

I really don't know where to start. It makes me shiver to think that this may be on our exam in a week and a half...

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Last edited: Apr 10, 2013