= 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> $\psi_Z=(\beta_Z^3/\pi)^{1/2}e^{-\beta_Zr}$, $\beta_Z=Z/a_0$, gives
$$
\langle1s,Z_f|1s,Z_i\rangle=\frac{8(\beta_i\beta_f)^{3/2}}{(\beta_i+\beta_f)^3},\qquad \boxed{P_{1s}=\frac{64(Z_iZ_f)^3}{(Z_i+Z_f)^6}.}
$$
This follows by integrating $4(\beta_i\beta_f)^{3/2}r^2e^{-(\beta_i+\beta_f)r}$ 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.
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