Displaced quantum harmonic oscillator ($E_n=\hslash \omega (n+1/2)-F^2/(2m{\omega }^2)$)

Adding $-Fx$ to the quantum harmonic oscillator potential gives $\frac{1}{2}m{\omega }^2(x-d{)}^2-F^2/(2m{\omega }^2)$, where $d=F/(m{\omega }^2)$. Translation preserves the oscillator spectrum, apart from the displayed constant shift. Eigenfunctions are translated by $d$ and the energy spacing is unchanged.

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.