Time-Dependent Perturbation Theory - Two-level System

[DanielFaraday]

Homework Statement

See attached. The problem is labeled "Peatross 1". Don't worry, it's short. I just didn't feel like retyping it.

Homework Equations

Included in attempt.

The Attempt at a Solution

I'm not sure if I am doing this correctly, but here it goes.

I'll just do it for $$H'_{10}$$, since the method will be the same for both.

I think all I have to do is calculate $$\psi_0$$ and $$\psi_1$$ using the following formula (for a harmonic oscillator):

$$\left\langle x | \psi_n \right\rangle = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot \exp \left(- \frac{m\omega x^2}{2 \hbar} \right) \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right)$$

Then just compute the resulting integral (probably in Mathematica):

$$\int_{- \infty}^{\infty} \psi_0 exE_0 \sin{\omega_L t\psi_1^*} dx$$

Is this all there is to it, or am I missing something?

Thanks for your help!