Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S3 / 2 / 3 / Solution

A sudden wall displacement leaves the initial wavefunction unchanged: it is $\sqrt{2/L}\sin (\pi x/L)$ on $[0,L]$ and zero on $(L,2L]$. It is not an eigenfunction of the new Hamiltonian, so there is no single certain new energy. Its energy expectation immediately afterward remains $E_1^{\mathrm{old}}$, as seen from ${\hslash }^2\int |{\psi }^{\prime }{|}^2/(2m)$; the measurement outcomes are $E_n^{\mathrm{new}}=n^2{E}_1^{\mathrm{old}}/4$.

The new ground state is $L^{-1/2}\sin (\pi x/(2L))$. Its overlap is $4\sqrt{2}/(3\pi )$, soThe discontinuity of the derivative at the former wall does not invalidate the finite quadratic-form expectation, although it matters when applying the differential Hamiltonian pointwise. Adiabatic expansion would instead keep the state in the ground level and reduce its energy by four.