Hermitian and Gauge-Covariant Hamiltonians for a particle in a magnetic field on Cylindrical and Spherical Surfaces

TL;DR

This paper develops Hermitian and gauge-covariant Hamiltonians for particles confined to cylindrical and spherical surfaces in magnetic fields, using an intuitive approach that avoids complex differential geometry.

Contribution

It introduces a straightforward method to derive Hermitian and gauge-covariant surface Hamiltonians without advanced geometric techniques.

Findings

Derived Hermitian surface Hamiltonians for particles in magnetic fields.

Identified physical radial momentum and set it to zero for confinement.

Surface Hamiltonians are naturally Hermitian and gauge-covariant.

Abstract

We construct the Hermitian Schr\"{o}dinger Hamiltonian of spin-less as well as the gauge-covariant Pauli Hamiltonian of spin one-half particles in a magnetic field that are confined to cylindrical and spherical surfaces. The approach does not require the use of involved differential-geometrical methods and is intuitive and physical, relying on the general requirements of Hermicity and gauge-covariance. The surfaces are embedded in the full three-dimensional space and confinement to the surfaces is achieved by strong radial potentials. We identify the Hermitian and gauge-covariant (in the presence of a magnetic field) physical radial momentum in each case and set it to zero upon confinement to the surfaces . The resulting surface Hamiltonians are seen to be automatically Hermitian and gauge-covariant. The well-known geometrical kinetic energy also emerges naturally.