Quantum mechanics of fermion confined to a curved surface in Foldy-Wouthuysen representation

TL;DR

This paper derives an effective quantum mechanics framework for fermions on curved surfaces using Foldy-Wouthuysen representation, revealing spin effects, curvature-induced energy shifts, and relativistic corrections.

Contribution

It introduces a novel approach combining thin-layer quantization with Foldy-Wouthuysen representation to analyze fermions on curved surfaces, highlighting relativistic spin effects and curvature contributions.

Findings

Spin effects due to confined potential are relativistic corrections.

Curvature-dependent spin connection causes a Zeeman-like gap.

Confined potential induces a curvature-independent energy shift.

Abstract

In Foldy-Wouthuysen representation, we deduce the effective quantum mechanics for a particle confined to a curved surface by using the thin-layer quantization scheme. We find that the spin effect caused by confined potential as the results of relativistic correction in the non-relativistic limit. Furthermore, the spin connection appeared in curved surface which depends on curvature contributes a Zeeman-like gap in the relativistic correction term. In addition, the confined potential also induces a curvature-independent energy shift, which is from the zitterbewegung effect. As an example, we apply the effective Hamiltonian to torus surface, in which we obtain expectantly the spin effects related to confined potential. Those results directly demonstrate the scaling of the uncommutation of the non-relativistic limit and the thin-layer quantization formalism