Nonrelativistic quantum dynamics on a cone with and without a constraining potential

TL;DR

This paper explores quantum bound states on conical surfaces, comparing approaches with and without a constraining potential, revealing differences in bound state properties and angular momentum dependence.

Contribution

It introduces a comparative analysis of quantum bound states on cones using two different theoretical frameworks, highlighting the impact of the constraining potential.

Findings

Bound states depend on the approach used (with or without potential).

Presence of bound states varies with the inclusion of the mean curvature potential.

Differences in angular momentum dependence between the two theories.

Abstract

In this paper we investigate the bound state problem of nonrelativistic quantum particles on a conical surface. This kind of surface appears as a topological defect in ordinary semiconductors as well as in graphene sheets. Specifically, we compare and discuss the results stemming from two different approaches. In the first one, it is assumed that the charge carriers are bound to the surface by a constraining potential, while the second one is based on the Klein-Gordon type equation on surfaces, without the constraining potential. The main difference between both theories is the presence/absence of a potential which contains the mean curvature of a given surface. This fact changes the dependence of the bound states on the angular momentum $l$. Moreover, there are bound states that are absent in the Klein-Gordon theory, which instead appear in the Schr\"{o}dinger one.