Quantum particle confined to a thin-layer volume: Non-uniform convergence toward the curved surface

TL;DR

This paper refines the thin-layer quantization method for quantum particles on curved surfaces, incorporating surface thickness effects into the geometric potential and kinetic terms, with applications to toroidal systems.

Contribution

It introduces a refined formalism that includes first-order terms in surface thickness, extending the traditional thin-layer quantization approach.

Findings

Modified geometric potential accounting for surface thickness

Altered kinetic term in the quantum equation due to thickness effects

Application to toroidal systems demonstrating the formalism's impact

Abstract

We clearly refine the fundamental framework of the thin-layer quantization procedure, and further develop the procedure by taking the proper terms of degree one in $q_3$ ($q_3$ denotes the curvilinear coordinate variable perpendicular to curved surface) back into the surface quantum equation. The well-known geometric potential and kinetic term are modified by the surface thickness. Applying the developed formalism to a toroidal system obtains the modification for the kinetic term and the modified geometric potential including the influence of the surface thickness.