Past exam of the theoretical physics II course of the University of Cambridge / 2014 / Paper 1 / 1 / c / Solution

Use normalized instantaneous modes ${\phi }_p(x,t)=\sqrt{2/L}\sin (p\pi x/L)$ and expand $\Psi ={\sum }_p{a}_p{e}^{-i{\vartheta }_p}{\phi }_p$. Differentiating at fixed $x$ givesFor $n\ne p$, integration usesso the moving-wall derivative coupling isFor $n=p$ the normalization derivative and the remaining integral cancel exactly, giving zero. The moving integration endpoint contributes no boundary term because every mode vanishes there. ThereforeThe real coupling matrix is antisymmetric, as required by conservation of total probability.