Phase-Integral Approximation in Momentum Space and the Bound States of an Atom. II

The phase-integral approximation of the Green's function in momentum space is investigated for a particle of negative energy (bound state) which moves in a spherically symmetric potential. If this potential has a Coulomb-like singularity at the origin, it is shown that any two momenta can be connected by an infinity of classical trajectories with a fixed energy. The summation of the usual phase and amplitude factors over these trajectories is the approximate Green's function. If there is orbital precession, there are not only poles along the negative energy axis, but also weaker singularities which are not examined in detail. The poles are found at the energies which are given by the semiclassical quantum conditions: angular momentum = (l + ½)ℏ and action integral for the radial motion = (n + ½)2πℏ, where l and n are integers ≥ 0. The residues at these poles give the approximate bound-state wavefunctions as a product of the asymptotic formula for Legendre polynomials with the asymptotic solution of the radial Schrödinger equation. It is conjectured that the occurrence of poles in the approximate Green's function is a direct consequence of the periodic character of the classical motion.