At t < 0 we have an unperturbed infinite square well. At 0 < t < T, a small perturbation is added to the potential: V(x) + V'(x), where V'(x) is the perturbation. At t > T, the perturbation is removed. Suppose the system is initially in the tenth excited state if the unperturbed potential.
What are the possible results of energy measurements at t = T/2, 3T/2, and infinity. Explain how you would calculate the probability of each result. Indicate whether or not these probabilities should depend on time.
ψn = A sin(nπx/a)
φ(x,t) = c(t)φ(x)e-iEt/ħ
E = (nπħ)2/2ma2.
The Attempt at a Solution
Since the system begins only in the tenth excited state of the unperturbed potential, there is no time dependence present via the wavefunction. And because the perturbation itself has no time dependence I believe that in all cases the probabilities should not depend on time for any t > 0. As t approaches infinity, the perturbation is long since removed, and I think the energy should be the same as it would be for any stationary wave function in the square well: E = (10πħ)2/2ma2.
What I'm not certain of is exactly how to determine the possible results of the energy measurements of the system for t = T/2 and 3T/2, or what these results would be. I am also not sure if my time dependence reasoning above is correct.