Semiclassical quantization and the Langer modification

Abstract

A recent proposal by Collins of a method for obtaining the classical-limit quantization of momenta conjugate to angles is examined. Among other results, this method gives half-integral quantization for the angular momentum pφ in circular cylinder or spherical polar coordinates, in contradistinction to the integral quantization given by quantum mechanics. Collins’ method is shown to give incorrect results because of an incorrect use of the stationary-phase approximation. However, departures from integral quantization are not generalized Langer corrections, because the Langer modification from l(l+1) to (l+1/2)2 in the expression for the square of the angular momentum in the radial quantization condition does not correspond to a change of angular-momentum quantization but rather to the inclusion of a higher-order correction which improves the quality of the zero-order energies and wave functions for certain classes of potentials. The question of whether the Langer modification should be used in higher-order RKR calculations for diatomic molecules is discussed, and it is suggested that no advantage is achieved in calculations of this type.