Ground-state overlap after sudden expansion of a square well ($P={\left[\frac{4{\eta }^{3/2}\cos (\pi /(2\eta ))}{\pi ({\eta }^2-1)}\right]}^2$)

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 $\eta \downarrow 1$ is $1$.