Quantum particle on a surface: Catenary surface and Paraboloid of revolution

TL;DR

This paper derives and solves the Schrödinger equation for a quantum particle confined to curved surfaces, specifically catenary and paraboloid of revolution, highlighting the role of curvature in quantum confinement.

Contribution

It provides a clearer formulation of the Schrödinger equation on curved surfaces and offers exact solutions for specific geometries like catenary and paraboloid surfaces.

Findings

Exact solutions for a particle on a catenary surface.

Exact solutions for a particle on a paraboloid of revolution.

Emphasizes the influence of principal curvatures on quantum confinement.

Abstract

We revisit the Schr\"{o}dinger equation of a quantum particle that is confined on a curved surface. Inspired by the novel work of R. C. T. da Costa [1] we find the field equation in a more convenient notation. The contribution of the principal curvatures in the effective binding potential on the surface is emphasized. Furthermore, using the so-called Monge-Gauge we construct the approximate Schr\"{o}dinger equation for a flat surface with small fluctuations. Finally, the resulting Schr\"{o}dinger equation is solved for some specific surfaces. In particular, we give exact solutions for a particle confined on a Catenary surface and a paraboloid of revolution.