Ground-state overlap after sudden oscillator displacement ($P_0=\exp [-m\omega d^2/(2\hslash )]$)

In the sudden approximation, an initial oscillator ground state overlaps the displaced ground state by $\exp [-m\omega d^2/(4\hslash )]$. 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.