# Time-dependent perturbation of an infinite square well

Suppose we have an infinite square well extending from $0<x<a$ and a particle in this wells ground state at $t \to -\infty$. We apply a time dependent perturbation of the form $$V(x,t)=-xF_0\exp(-t^2/\tau^2)$$ which, according to my understanding, means we apply a linearly $x$-depending perturbation that is strongest around $t=0$ and symmetrical, as well as characterized in rapidity of change by $\tau$.

We now want to calculate the probabilities for the particle to be in any excited state $|n\rangle$ at $t \to \infty$ in first order time-dependent perturbation theory.

This means we need to calculate the matrix elements $$\langle n|V|0\rangle$$ as we only have $V$ as perturbation. I know that the unperturbed wave eigenfunctions of the infinite square well of this form take the form (due to the boundary conditions) $$\psi_n = \sqrt{\frac{2}{a}}\sin(\frac{n\pi x}{a})$$ (in this form, the ground state has $n=1$, of course). I get $$\langle n | V | 0 \rangle = \frac{2}{a}\int_0^a \sin(\frac{n\pi x}{a}) \sin(\frac{\pi x}{a}) (-xF_0\exp(-t^2/\tau^2)) dx$$ at which point I'm not sure how to proceed, as so far we have only covered perturbations that didn't have time-dependency. Do I need to integrate w.r.t. time as well? What assumptions can/do I need to make? Feel free to ask for any additional information you might need.

• It is not clear to me at which times $V$ and $V'$ act. It only says "later". – noah May 10 '17 at 20:37
• Oh, sorry. V is given for exercise a), V' for exercise b) (so 2 different scenarios) – John W. May 11 '17 at 1:07