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!

 

[Jhero]

Hello! As a fellow scientist, I wanted to offer some feedback on your solution attempt.

Firstly, your use of the formula for a harmonic oscillator is correct. However, I would suggest including the explicit values for m, ω, and ℏ in your calculations, as this will make it easier to plug in values later on.

Additionally, it seems like you may have mixed up the variables in your integral. The integral should be with respect to time (t) instead of position (x), since we are calculating the expectation value of the position operator. So the integral should be:

\int_{- \infty}^{\infty} \psi_0 x \psi_1^* exE_0 \sin{\omega_L t} dt

Also, don't forget to include the complex conjugate of \psi_1 in the integral, since it appears in the expectation value as \psi_1^*.

Lastly, while using Mathematica is a great idea for computing the integral, I would also suggest double checking your calculation by hand to make sure everything is correct.

Hope this helps! Good luck with your problem.