# Perturbation Theory

Please help me try to understand this problem. It deals with the quantum-confined Stark effect in nanoparticles.

For odd n, n = 1, 3, 5, ...
$$\psi_{n}(x) = \sqrt{\frac{2}{a}} \cos (\frac{n \pi x}{a})$$

and for even n = 2, 4, 6, ...
$$\psi_{n}(x) = \sqrt{\frac{2}{a}} \sin (\frac{n \pi x}{a})$$

and the zeroth order energy levels are

$$E_{n} = \frac{h^2 \pi^2 n^2}{2ma^2}$$

The external field pertubation, H' = -qFx , where q is the charge and F is the applied electric field strength.

Now here's my work for the first order correction to the energy levels.

For odd n:
$$E_{n} = < \sqrt{\frac{2}{a}} \cos (\frac{n \pi x}{a})| H' | \sqrt{\frac{2}{a}} \cos (\frac{n \pi x}{a})> = 0$$

For even n, I still get 0 for the first order correction. I just know that isn't right, and I think I know why:

Am I treating H' = -qFx correctly by assuming q and F are constants and x as the operator?

Thanks for the help. :shy:

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