Hamilton-Jacobi treatment of a non-relativistic particle on a curved space

TL;DR

This paper applies the Hamilton-Jacobi formalism to analyze a non-relativistic particle on a curved hypersurface and demonstrates its equivalence with the Dirac formalism, providing insights into the energy spectrum of a multidimensional rotator.

Contribution

It introduces a Hamilton-Jacobi approach to curved space quantum mechanics and shows its equivalence with Dirac formalism in different coordinate systems.

Findings

Energy spectrum matches Laplace-Beltrami operator results

Equivalence between Hamilton-Jacobi and Dirac formalisms demonstrated

No additional curvature-dependent constants in the spectrum

Abstract

In this paper a non-relativistic particle moving on a hypersurface in a curved space and the multidimensional rotator are investigated using the Hamilton-Jacobi formalism. The equivalence with the Dirac Hamiltonian formalism is demonstrated in both Cartesian and curvilinear coordinates. The energy spectrum of the multidimensional rotator is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere.