On the quantum dynamics of a point particle in conical space

TL;DR

This paper investigates the quantum behavior of a particle on a conical surface, revealing how boundary conditions influence bound states and exploring potentials that naturally arise in such geometries.

Contribution

It demonstrates that the common boundary condition at the cone's tip is arbitrary and shows how self-adjoint extension methods reveal more bound states in conical quantum systems.

Findings

The usual boundary condition at the cone tip is arbitrary.

Self-adjoint extension yields additional bound states.

The model relates to dipole motion in conical space.

Abstract

A quantum neutral particle, constrained to move on a conical surface, is used as a toy model to explore bound states due to both a inverse squared distance potential and a $\delta$-function potential, which appear naturally in the model. These pathological potentials are treated with the self-adjoint extension method which yields the correct boundary condition (not necessarily a null wavefunction) at the origin. We show that the usual boundary condition requiring that the wavefunction vanishes at the origin is arbitrary and drastically reduces the number of bound states if used. The situation studied here is closely related to the problem of a dipole moving in conical space.