Relation Between Solutions of the Schr\"odinger Equation with Transitioning Resonance Solutions of the Gravitational Three-Body Problem

Abstract

It is shown that a class of approximate resonance solutions in the three-body problem under the Newtonian gravitational force are equivalent to quantized solutions of a modified Schr\"odinger equation for a wide range of masses that transition between energy states. In the macroscopic scale, the resonance solutions are shown to transition from one resonance type to another through weak capture at one of the bodies, while in the Schr\"odinger equation, one obtains quantized wave solutions transitioning between different energies. The resonance transition dynamics provides a classical model of a particle moving between different energy states in the Schr\"odinger equation. This methodology provides a connection between celestial and quantum mechanics.