Quantum mechanics of a constrained electrically charged particle in the presence of electric currents

TL;DR

This paper explores the quantum dynamics of a charged particle constrained on a surface with electric currents, revealing how geometry and boundary conditions influence its behavior and coupling to electromagnetic fields.

Contribution

It introduces a detailed analysis of how a charged particle couples to surface curvature and electromagnetic potentials, highlighting the effects of boundary constraints on separability of equations.

Findings

Charged particles couple to mean curvature and electromagnetic potentials.

Neumann constraints lead to non-separable equations of motion.

Electric currents generally cause non-separability regardless of boundary conditions.

Abstract

We discuss the dynamics of a classical spinless quantum particle carrying electric charge and constrained to move on a non singular static surface in ordinary three dimensional space in the presence of arbitrary configurations of time independent electric currents. Starting from the canonical action in the embedding space we show that a charged particle with charge $q$ couples to a term linear in $qA^3M$, where $A^3$ is the transverse component of the electromagnetic vector potential and $M$ is the mean curvature in the surface. This term cancels exactly a curvature contribution to the orbital magnetic moment of the particle. It is shown that particles, independently of the value of the charge, in addition to the known couplings to the geometry also couple to the mean curvature in the surface when a Neumann type of constraint is applied on the transverse fluctuations of the wave function. In contrast to a Dirrichlet constraint on the transverse fluctuations a Neumann type of constraint on these degrees of freedom will in general make the equations of motion non separable. The exceptions are the equations of motion for electrically neutral particles on surfaces with constant mean curvature. In the presence of electric currents the equation of motion of a charged particle is generally non separable independently of the coupling to the geometry and the boundary constraints.