Semiclassical estimation of the radiative mean lifetimes of hydrogenlike states

Abstract

Based on the correspondence principle, a simple semiclassical derivation of a universal formula for the radiative mean lifetime \ensuremath{\tau}(n,l) of hydrogenlike states is presented. (The possibilities of stimulated emission and collisional deexcitation are neglected.) Within the nonrelativistic dipole approximation the mean lifetime for an electron in a state characterized by quantum numbers n and l in the field of a nucleus of charge Z is given by \ensuremath{\tau}(n,l)\ensuremath{\approxeq}${\mathrm{\ensuremath{\tau}}}_{0}$${\mathit{n}}^{3}$l(l+1)/${\mathit{Z}}^{4}$, where ${\mathrm{\ensuremath{\tau}}}_{0}$= 3\ensuremath{\Elzxh}/(2${\mathrm{\ensuremath{\alpha}}}^{5}$\ensuremath{\mu}${\mathit{c}}^{2}$)\ensuremath{\approxeq}93.42\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}12}$ s and \ensuremath{\mu} is the reduced mass. The formula is accurate to at least 6% for the lowest states and to a much higher degree of accuracy for highly excited states. The semiclassical result is expected to be valid to leading order in n and l. However, the very simple derivation yields results of an accuracy comparable to several approximate quantum-mechanical and semiclassical results that have been published. The approach is based primarily upon a treatment of the rate of loss of angular momentum, not of energy. A clear physical interpretation of some aspects of the radiative decay process, which is somewhat obscure in the quantum evaluation, emerges naturally from the angular momentum approach.