Path Integral Approach for for Quantum Motion on Spaces of Non-constant Curvature According to Koenigs: Three Dimensions

TL;DR

This paper develops a path integral method for quantum motion on three-dimensional Koenigs spaces with non-constant curvature, incorporating curvature terms into the Hamiltonian, and derives complex energy equations for bound states.

Contribution

It introduces a novel path integral approach for quantum mechanics on Koenigs spaces derived from superintegrable potentials, including curvature effects.

Findings

Derived equations for bound states up to twelfth order in energy

Constructed quantum Hamiltonians with curvature terms for Koenigs spaces

Analyzed quantum motion on non-constant curvature spaces

Abstract

In this contribution a path integral approach for the quantum motion on three-dimensional spaces according to Koenigs, for short``Koenigs-Spaces'', is discussed. Their construction is simple: One takes a Hamiltonian from three-dimensional flat space and divides it by a three-dimensional superintegrable potential. Such superintegrable potentials will be the isotropic singular oscillator, the Holt-potential, the Coulomb potential, or two centrifugal potentials, respectively. In all cases a non-trivial space of non-constant curvature is generated. In order to obtain a proper quantum theory a curvature term has to be incorporated into the quantum Hamiltonian. For possible bound-state solutions we find equations up to twelfth order in the energy E.