Schrodinger equation subjected to general potential

[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 Schrodinger 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 Schrodinger 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...

 

[azntoon]

To answer question c, you can use the method of separation of variables to solve the Schrödinger equation for the time and spatial part. To do this, you need to write the wavefunction, ψ(x,t), as a product of two functions, one that depends only on x and one that depends only on t. Then, you can substitute this product into the Schrödinger equation to get two equations, one for the spatial part and one for the temporal part. By solving these equations you can determine the wavefunction for the particle in the time-dependent potential. For question d, you can make a sketch of the ground state wave function and PSI^2 in all regions of space by solving the Schrödinger equation for the general potential V (x). You can then compare your results in relation to classical physics and conclude whether or not the ground state energy can be zero as in the classical case.