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)

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

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.

To solve this problem, we will use perturbation theory. Perturbation theory is a mathematical method to find approximate solutions for problems that can be divided into a solvable part and a small perturbation part. In this case, the "solvable" part refers to the unperturbed Hamiltonian, which is the one-dimensional motion of an electron subject to periodic boundary conditions, and the "perturbation" is the term V(x) = e * cos(qx).

a) Let's first calculate the first-order correction to the energy levels and stationary states due to the perturbing potential V(x) = e * cos qx.

To calculate the first-order energy correction, we use the first-order perturbation theory formula:

ΔE^(1) = <n|V|x|n> / (En - En')

where n is the unperturbed state, V is the perturbation, and En and En' are the unperturbed energies of states |n> and |n'>.

Since the electron momentum is q/2, the unperturbed energy of the state |n> is given by En = (q/2)^2/(2m), where m is the mass of the electron.

The matrix element <n|V|x|n> can be calculated as follows:

<n|V|x|n> = e * ∫ ψ_n^*(x) * cos(qx) * ψ_n(x) dx

Here, ψ_n(x) is the wave function of the unperturbed state |n>. Since we have periodic boundary conditions, the wave function can be written as a Fourier series:

ψ_n(x) = (1/sqrt(L)) * e^(i*k_n*x)

where k_n = (2πn)/L and n is an integer.

The integral can be evaluated as follows:

<n|V|x|n> = (e/L) * ∫ e^(-i*k_n*x) * cos(qx) * e^(i*k_n*x) dx
= (e/L) * ∫ cos(qx) dx
= (e/L) * [sin(qx)/q] evaluated from 0 to L
= (e/L) * [sin(qL)/q]
= (e/L) * [sin(2πN)/q] (qL = 2πN)

Substituting this result into the first-order correction formula, we have:

ΔE^(1) = (e/L) * [sin(2πN)/q] / [(q/2)^2/(2m) - (q/2)^2/(2m)]

Simplifying this expression, we find the first-order correction to the energy levels.

To find the first-order correction to the stationary states, we use the formula:

|n^(1)> = Σ |m(i)> * <m(i)|V|n> / (En - Em(i))

where |m(i)> represents the degenerate states that have the same energy as the state |n>.

b) To derive the expression for the difference in the free energy of order (epsilon)^2, we need to include the second-order correction term. However, since you only provided the first-order perturbation potential V(x), I cannot provide the full expression for the second-order correction without the explicit form of the second-order perturbation potential.