Semiclassical estimation of the radiative mean lifetimes of hydrogenlike states

Based on the correspondence principle, a simple semiclassical derivation of a universal formula for the radiative mean lifetime {τ}(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 {τ}(n,l){}τ₀{n}³$l(l+1)/${Z}⁴$, where ${{{τ}}}₀= 3/(2{{{α}}}⁵μ{c}²)93.42×{10}^{{{-}}12}$ s and {μ} 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.