Consider a particle of unit mass in a one-dimensional square well potential

$$V(x)=0 \text { for } 0 \leqslant x \leqslant \pi,$$

with $V$ infinite outside. Find all the stationary states $\psi_{n}(x)$ and their energies $E_{n}$, and write down the general normalized solution of the time-dependent Schrödinger equation in terms of these.

The particle is initially constrained by a barrier to be in the ground state in the narrower square well potential

$$V(x)=0 \quad \text { for } \quad 0 \leqslant x \leqslant \frac{\pi}{2}$$

with $V$ infinite outside. The barrier is removed at time $t=0$, and the wavefunction is instantaneously unchanged. Show that the particle is now in a superposition of stationary states of the original potential well, and calculate the probability that an energy measurement will yield the result $E_{n}$.