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

(a) We know that the probability distribution for the particlein a stationary state is time-independent. Let us now prepare, attime 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)|^2for this state. Is it periodic in the sense that after some time Tit 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 3or 4 moments of time t between 0 and T that would nicely displaythe qualitative features of the changes, if any.

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

Ψ(x, T) = Ψ(x, 0)

(perhaps with accuracy to an inconsequential overall phasefactor)? 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 systemfor an arbitrary state?

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

Need help with Part C!