Classical-Limit Green's Function (Fixed-Energy Propagator) and Classical Quantization of Nonseparable Systems

On the basis of a classical-limit approximation for the Green's function, a classical quantum condition for general, nonseparable dynamical systems is obtained. Bound-state eigenvalues are seen to be associated with stable periodic trajectories, and quasibound (or metastable) states are related to unstable periodic trajectories. Application is made to ns2 states of the two-electron atom, and quite resaonable results are obtained for the energies of the 1s2 and 2s2 states of He and H−. An accurate value is also obtained for the lowest eigenvalue of three identical bosons (e.g., helium atoms) interacting through pair Morse potentials.