Probability of staying in same state after time-dep perturbation?

[vincebs]

Let's say that you've got a time-dependent perturbation to your potential (say, the particle-in-a-box to make things simple). Say you start in energy eigenstate #3. What's the probability that the particle will stay in eigenstate 3 after time T?

This is not a homework problem. I'm not understanding the equation for the probability which seems to give a number larger than 1.

 

[genneth]

I take you're staring at Fermi's Golden Rule. That rule is a first order time dependent perturbation result, and so will not give the correct answer at large times or for large perturbations.