= First-order excitation in a linearly expanding infinite well
{title2=$P_{1\to2}\simeq(16/9)|\int_0^t e^{-i\alpha/s}ds/s|^2$}
For $L(t)=vt$ and $\alpha=3\pi^2\hbar/(2mv^2)$, 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 $P_{1\to2}\sim16t^2/(9\alpha^2)$ as $t/\alpha\to0$. Large excitation invalidates the first-order approximation.
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