= Ground-state overlap after sudden expansion of a square well
{title2=$P=\left[\frac{4\eta^{3/2}\cos(\pi/(2\eta))}{\pi(\eta^2-1)}\right]^2$}
For an <infinite square well> initially on $[-a,a]$, a normalized <ground state> is $a^{-1/2}\cos(\pi x/(2a))$. If the endpoints change instantaneously to $\pm\eta a$, with $\eta>1$, the <sudden approximation> leaves the <wave function> unchanged, extended by zero outside the old well. Its overlap with the new normalized <ground state> $(\eta a)^{-1/2}\cos(\pi x/(2\eta a))$ is $4\eta^{3/2}\cos(\pi/(2\eta))/[\pi(\eta^2-1)]$. The <Born rule> squares this <probability amplitude>; for $\eta=2$ the result is $64/(9\pi^2)$, and the limit as $\eta\downarrow1$ is $1$.
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