Classical versus quantum calculation of radiative electric quadrupole transition rates for hydrogenic states

The semiclassical Kepler-Coulomb problem and the quantum-mechanical Schr\"odinger-Coulomb problem are compared for their predictions of quadrupole E2 transitions. The semiclassical treatment involves an extension of previous work for the electric dipole transitions (Physical Review A 71, 020501), and rates are derived for $\Delta \ell= 0, \pm 2$ transitions on the basis of the multipolar properties of the emitted radiation. For the quantum case a derivation is presented within the Schr\"odinger framework without reference to spin. Comparison of the E2 rates shows reasonable agreement, but not as good as was found for the electric dipole case.