= Ground-state overlap after sudden oscillator displacement
{title2=$P_0=\exp[-m\omega d^2/(2\hbar)]$}
In the <sudden approximation>, an initial oscillator <ground state> overlaps the displaced ground state by $\exp[-m\omega d^2/(4\hbar)]$. Completing the square in the product of the two normalized Gaussian wavefunctions and using the <Gaussian integral> proves this amplitude. Squaring it gives the displayed probability. Only an irrelevant overall phase evolves before the sudden switch.
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