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

TL;DR

This paper compares classical and quantum methods for calculating electric quadrupole transition rates in hydrogenic states, highlighting their similarities and differences in predictions.

Contribution

It extends previous semiclassical work to quadrupole transitions and provides a quantum derivation within the Schrödinger framework without spin.

Findings

Semiclassical and quantum rates show reasonable agreement.

Discrepancies are larger than in electric dipole transitions.

The work clarifies the applicability of classical and quantum models for E2 transitions.

Abstract

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.