Question

An electron is moving in one dimension (x)subject to the periodic boundary conditions that the wave function reproduces itself after a length L(L is large_
a)add a perturbation V(x)=ecos qx
where qL=2piN (N is large integer) recalculate the energy levels and stationary states to first order in epsilon for electron momentum of q/2
b)Derive the expression for the difference in the free energy of order (epsilon)^2 to the above answer (a)

Answer 1

I have not looked at this subject for years. Sorry, perhaps WLS can help.

Answer 2

It's been 48 years for me. I would have to combine a review of notes or text chapters on "particle in a box" and perturbation theory.

I lack the time or motivation to do that. Life is too short. I suggest instead looking at a host of websites like

http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Acond-mat%2F9906241

There are some recent journal papers on the subject.