 Moving-wall derivative coupling

For a well on $0<x<L(t)$ with real normalized sine modes, the displayed coupling holds for $n\ne p$ and the diagonal coupling vanishes. Differentiate both the normalization and the sine at fixed position, then integrate on the instantaneous interval. The moving boundary contributes no endpoint term because the modes vanish there. The resulting real antisymmetric matrix preserves total probability in the instantaneous eigenstate expansion.