Global uniform semiclassical approximation for Clebsch-Gordan coefficients

Abstract

Semiclassical integral representations, analogous to initial value expressions for the propagator, are presented for the Clebsch-Gordan angular momentum coupling coefficients. Two forms (L and R types) of the approximation are presented. For each form, new non-Gaussian expressions, which are specifically adapted to the nature of angular momentum variables, are proposed in place of the familiar Gaussian coherent state functions. With these non-Gaussian kernels, it is found that the present treatments are capable of accuracy similar to that obtained from a uniform Airy approximation. Although the present semiclassical approximations involve only real-valued angle variables, associated with sets of angular momenta that are related by ordinary, real, classical transformations, the treatments produce accurate results not only for classically allowed choices of quantum numbers but also for very strongly classically forbidden values.