Centrifugal terms in the WKB approximation and semiclassical quantization of hydrogen

Abstract

A systematic semiclassical expansion of the hydrogen problem about the classical Kepler problem is shown to yield remarkably accurate results. Ad hoc changes of the centrifugal term, such as in the WKB approximation and semiclassical quantization of hydrogen, where the factor $l(l+1)$ is replaced by ${(l+1/2)}^{2}$, are avoided. Expanding systematically in powers of \ensuremath{\Elzxh}, the semiclassical energy levels are shown to be exact to first order in $\ensuremath{\Elzxh}$ with all higher-order contributions vanishing. The wave functions and dipole matrix elements are also discussed.