Question

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?

Need help with Part C!

Answer 1

c) The classical revival time is just the time period of a classical particle moving in a box of length L with velocity v

To return to the same point, the particle has to cover 2L distance with velocity v. The classical time period is then

p = mv is the momentum

Also, the total kinetic energy of the particle is equal to the total energy of the particle since the potential energy inside the box is 0

Therefore

Hence

Therefore

Now suppose this particle is quantized and its energy is given by quantum mechanical energy of its state in a stationary state with quantum number n.

Then

or

Therefore

Now consider a non-stationary state which is the superposition of two stationary states with quantum numbers n1 and n2.

The quantum revival time is

where

as you may have found in part a) and b)

This is the frequency of non-stationary state composed of a superposition of two stationary states with quantum numbers n1 and n2

Therefore the quantum revival time is

where

and are the energy of the stationary states with quantum numbers n1 and n2

The difference in energy is

Therefore

For this quantum revival time to be equal to the classical revival time

or

This is possible if

and

Then

Since n2 is very large we can neglect +1 term and

Therefore the classical revival behavior is equivalent to quantum revival behavior when the quantum number is very large.

This is called the correspondence principle in quantum mechanics.