Pauli equation for a charged spin particle on a curved surface in an electric and magnetic field

TL;DR

This paper derives the Pauli equation for a charged spin particle on curved surfaces in electromagnetic fields, revealing geometric and spin connection potentials, and applies the theory to various surfaces.

Contribution

It introduces a comprehensive derivation of the surface Pauli equation incorporating geometric and spin connection potentials, extending previous results to curved geometries.

Findings

Geometric potential $V_g$ is confirmed on curved surfaces.

Spin connection geometric potentials are generated by curvilinear derivatives.

Only the normal Pauli matrix appears in the surface equations.

Abstract

We derive the Pauli equation for a charged spin particle confined to move on a spatially curved surface $\mathcal{S}$ in an electromagnetic field. Using the thin-layer quantization scheme to constrain the particle on $\mathcal{S}$, and in the transformed spinor representations, we obtain the well-known geometric potential $V_g$ and the presence of $e^{-i\varphi}$, which can generate additive spin connection geometric potentials by the curvilinear coordinate derivatives, and we find that the two fundamental evidences in the literature [Giulio Ferrari and Giampaolo Cuoghi, Phys. Rev. Lett. 100, 230403 (2008).] are still valid in the present system without source current perpendicular to $\mathcal{S}$. Finally, we apply the surface Pauli equation to spherical, cylindrical, and toroidal surfaces, in which we obtain expectantly the geometric potentials and new spin connection geometric potentials, and find that only the normal Pauli matrix appears in these equations.