Sudden hydrogenic charge change

In the sudden approximation, changing the nuclear charge of a one-electron hydrogen atom from $Z_{i}e$ to $Z_{f}e$ leaves its electronic state unchanged during the change. Its subsequent probability to occupy a final state is the squared overlap with that state. Using normalized $1s$ wavefunctions ${\psi }_Z=({\beta }_Z^3/\pi {)}^{1/2}{e}^{-{\beta }_{Z}r}$, ${\beta }_Z=Z/a_0$, givesThis follows by integrating $4({\beta }_i{\beta }_f{)}^{3/2}{r}^2{e}^{-({\beta }_i+{\beta }_f)r}$ over positive $r$. A spherically symmetric initial state has zero overlap with every final state of nonzero orbital angular momentum, by the orthogonality of the spherical harmonics. Nuclear recoil and interactions with emitted particles can invalidate the perfectly spherical sudden-change model.