Calculating Probabilities of Particle in a L-box After Wall Movement

[mcheung4]

Homework Statement

A particle is in the ground state of a L-box. At t = 0 the wall at x = L is suddenly moved to 2L.

(a) If an energy measurement is made after the wall is moved, what is the probability of measuring the energy to be that of the ground state of the new box? (given solution ≈ 0.36)

(b) What is the probability of measuring the energy to be that of the first excited state of the new box? (given solution = 1/2)

*L-box is the infinite square well with zero potential for 0≤x≤L.

Homework Equations

eigenstates ψ$_{n}$ = √(2/L) sin (n*pi*x/L)

eigenvalues (energy) E$_{n}$ = ((n*pi*ℏ)^2)/(2mL^2)

The Attempt at a Solution

So initially the ground state (of L-box) ψ$_{1}$ = √(2/L) sin (pi*x/L) and E$_{1}$ = ((pi*ℏ)^2)/(2mL^2), so I think this energy is the maximum energy that the particle can have, before and after the change in size of the box.

For the 2L-box its ground state ψ'$_{1}$ = √(1/L) sin (pi*x/2L) and E'$_{1}$ = ((pi*ℏ)^2)/(8mL^2); its 1st excited state ψ'$_{2}$ = √(1/L) sin (pi*x/L) and E'$_{2}$ = ((pi*ℏ)^2)/(2mL^2).

So the particle can only be in either of these states (it doesn't have enough energy to move to other states). But since I wasn't given the wave function in this new box, how am I supposed to calculate the probablities?

My guess was that there should be no preference on the 2 states in the new box, so the probabilities are both 1/2, but which are not exactly correct according to the solution

 

[vela]

It's not the energy that remains constant; it's the wave function that's the same immediately before and after the box is lengthened.