Hamiltonian for a particle in a magnetic field on a curved surface in orthogonal curvilinear coordinates

TL;DR

This paper derives gauge-invariant Hamiltonians for spin-zero and spin-half particles confined to curved surfaces in electromagnetic fields, providing explicit formulas for common geometries using a novel, gauge-independent approach.

Contribution

It introduces a new method to obtain Hermitian, gauge-invariant surface Hamiltonians by removing the transverse velocity component, applicable to Schrödinger and Pauli cases without approximations.

Findings

Derived explicit Hamiltonians for cylindrical, spherical, and toroidal surfaces.

Established gauge invariance and Hermiticity of the surface Hamiltonians.

Provided a general approach applicable to various curved geometries.

Abstract

The Schr\"odinger Hamiltonian of a spin zero particle as well as the Pauli Hamiltonian with spin-orbit coupling included of a spin one-half particle in electromagnetic fields that are confined to a curved surface embedded in a three-dimensional space spanned by a general Orthogonal Curvilinear Coordinate (OCC) are constructed. A new approach, based on the physical argument that upon squeezing the particle to the surface by a potential, then it is the physical gauge-covariant kinematical momentum operator (velocity operator) transverse to the surface that should be dropped from the Hamiltonian(s). In both cases,the resulting Hermitian gauge-invariant Hamiltonian on the surface is free from any reference to the component of the vector potential transverse to the surface, and the approach is completely gauge-independent. In particular, for the Pauli Hamiltonian these results are obtained exactly without any further assumptions or approximations. Explicit covariant plug-and-play formulae for the Schr\"odinger Hamiltonians on the surfaces of a cylinder, a sphere and a torus are derived.