Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section A / A3

In the sudden approximation, the nuclear charge changes before the electronic wavefunction has time to evolve. Immediately afterward the state is still the original hydrogenic 1s wavefunction, now expanded in the eigenstates of the new Coulomb Hamiltonian. The final-ground-state probability is the squared overlap, rather than one minus an energy change.

For initial and final charges $Z_1,Z_2$, the normalized 1s overlap isUsing $Z_1=1,Z_2=2$ givesThe remaining probability includes higher bound $ns$ states and the $s$-wave ionization continuum. In this central, instantaneous model the initial orbital has $l=0,m=0$, and angular orthogonality excludes states with $l\ne 0$. The electronic spin is unchanged. Nuclear recoil and interactions with the emitted decay particles are outside this leading sudden-charge-change approximation.