Cosine perturbation to potential well

[unscientific]

Homework Statement

Part (b): Find the perturbed energy.

Homework Equations

The Attempt at a Solution

I've solved everything, except part (b).
I got an answer of 0 for part (b) for all orders, which is kind of strange, as one would expect some perturbation.

\Delta E_n = \langle \psi_n |U|\psi_n\rangle
= \frac{2}{L}U_0\int_0^L cos(\frac{2\pi}{L}x)sin^2(\frac{n\pi x}{L})
= 0

Integrating this leads to zero.

All the other orders give zero as well.

 

[unscientific]

I'm thinking since the period is L, integrating a function throughout its entire period gives 0 area?

 

[BruceW]

not generally. for example, think of the function cos(x)+5, the curve lies above the x-axis for all values of x, so it definitely does not integrate to zero. Anyway, you should try to do the integration for the three values of $n$ and see what you get.