A particle of mass m is prepared in the ground state of an infinite-potential box of size a extending from x = 0 to x = a. Suddenly, the wall at x = a is moved to x = 2a within a time Δt doubling the box size. You may assume that the wavefunction is the same immediately after the change, if the change happens fast enough.

(a) How fast is fast enough?

(b) What is the probability that the particle is in the second (n = 2) state of the new well, immediately after the change? Note that the wavelength within the well, and hence the energy, for this state is the same as for the initial state in the old well. Make sure that you properly normalized wavefunctions for your calculations.

(c) What is the probability that the particle would be found in the ground state of the sudden expansion?

(d) What is the expectation value of the energy of the particle before and after the sudden expansion?