First-order excitation in a linearly expanding infinite well ($P_{1\to 2}\simeq (16/9)|{\int }_0^t{e}^{-i\alpha /s}ds/s{|}^2$)

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.