Quantum mechanics on curved 2D systems with electric and magnetic fields

TL;DR

This paper derives the Schrödinger equation for a charged particle on curved 2D surfaces under electric and magnetic fields, revealing separability of surface and transverse dynamics and providing explicit Hamiltonians for common geometries.

Contribution

It introduces a method to incorporate electric and magnetic fields into quantum systems on curved surfaces, showing no coupling with curvature and enabling exact solutions for specific geometries.

Findings

Surface and transverse dynamics are exactly separable with proper gauge choice.

Explicit Hamiltonians derived for cylinder, sphere, and torus.

No coupling between fields and surface curvature found.

Abstract

We derive the Schroedinger equation for a spinless charged particle constrained to a curved surface with electric and magnetics fields applied. The particle is confined on the surface using a thin-layer procedure, giving rise to the well-known geometric potential. The electric and magnetic fields are included via the four-potential. We find that there is no coupling between the fields and the surface curvature and that, with a proper choice of the gauge, the surface and transverse dynamics are exactly separable. Finally, the Hamiltonian for the cylinder, sphere and torus are analytically derived.