Path Integration in Conical Space

TL;DR

This paper investigates quantum mechanics in conical space using path integrals, revealing how curvature induces an effective potential and relates to modified Schrödinger equations, with implications for understanding quantum behavior in curved geometries.

Contribution

It demonstrates that the radial path integral in conical space can be transformed to flat space form by adjusting angular momentum, linking curvature effects to effective potentials in quantum mechanics.

Findings

Curvature induces an effective potential proportional to mean curvature.

Radial path integral in conical space reduces to flat space form with non-integral angular momentum.

Path integral results align with a modified Schrödinger equation incorporating curvature effects.

Abstract

Quantum mechanics in conical space is studied by the path integral method. It is shown that the curvature effect gives rise to an effective potential in the radial path integral. It is further shown that the radial path integral in conical space can be reduced to a form identical with that in flat space when the discrete angular momentum of each partial wave is replaced by a specific non-integral angular momentum. The effective potential is found proportional to the squared mean curvature of the conical surface embedded in Euclidean space. The path integral calculation is compatible with the Schr\"odinger equation modified with the Gaussian and the mean curvature.