Question

# In this problem we are interested in the time-evolution of the states in the infinite square...

In this problem we are interested in the time-evolution of the states in the infinite square potential well. The time-independent stationary state wave functions are denoted as ψn(x) (n = 1, 2, . . .).

(a) We know that the probability distribution for the particle in a stationary state is time-independent. Let us now prepare, at time t = 0, our system in a non-stationary state

Ψ(x, 0) = (1/√( 2)) (ψ1(x) + ψ2(x)).

Study the time-evolution of the probability density |Ψ(x, t)|^2 for this state. Is it periodic in the sense that after some time T it will return to its initial state at t = 0? If so, what is T? Sketch, better yet plot (by using some software), |Ψ(x, t)|^2 for 3 or 4 moments of time t between 0 and T that would nicely display the qualitative features of the changes, if any.

(b) Let us now prepare the system in some arbitrary non-stationary state Ψ(x, 0). Is it true that after some time T, the wave function will always return to its original spatial behavior, that is,

Ψ(x, T) = Ψ(x, 0)

(perhaps with accuracy to an inconsequential overall phase factor)? If so, what is this quantum revival time T? Compare to (a). And why do you think it was possible to have it in this system for an arbitrary state?

(c) Think now about the revival time for a classical particle of energy E bouncing between the walls. Assuming the positive answer to (b), if we were to compare the classical revival behavior to the quantum revival behavior, when these times would be equal?