Moving-wall derivative coupling ($⟨n|\dot{p}⟩=(\dot{L}/L)(-1{)}^{n+p}2np/(n^2-p^2)$)

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.

For $L(t)=vt$ and $\alpha =3{\pi }^2\hslash /(2m{v}^2)$, first-order excitation out of the instantaneous ground state has the displayed oscillatory integral. Its lower endpoint means the limit of a ground-state preparation at positive time and positive width; no normalized state exists in a zero-width well. Substitution $u=\alpha /s$ makes convergence explicit, and $P_{1\to 2}\sim 16t^2/(9{\alpha }^2)$ as $t/\alpha \to 0$. Large excitation invalidates the first-order approximation.