Three-body problem in the ground-state representation

Abstract

The ground-state probability density of a three-body system is used to construct a classical potentialU whose minimum coincides exactly with the ground-state energy. The spectrum of excited states may approximately be obtained by imposing quasiclassical quantization conditions over the classical motion inU. We show nontrivial one-dimensional models in which either this quantization condition is exact or considerably improves the usual semiclassical quantization. For three-dimensional problems, the small-oscillation frequencies in states with total angular momentumL=0 are computed. These frequencies could represent an improvement over the frequencies of triatomic molecules computed with the use of ordinary quasiclassics for the motion of the nuclei in the molecular term. By providing a semiclassical description of the first excited quantum states, the sketched approach rises some interesting questions such as, for example, the relevance (once again) of classical chaos to quantum mechanics.