On a Dirac particle in an uniform magnetic field in 3-dimensional spaces of constant curvature

TL;DR

This paper derives exact solutions for the Dirac equation of a spin-1/2 particle in curved 3D spaces with constant curvature under magnetic fields, providing generalized energy level formulas for both hyperbolic and spherical geometries.

Contribution

It presents new exact solutions and generalized energy level formulas for Dirac particles in curved spaces with magnetic fields, extending quantum mechanics in non-Euclidean geometries.

Findings

Exact solutions for Dirac equation in hyperbolic and spherical spaces.

Generalized energy level formulas for particles in curved magnetic fields.

Quantization of particle motion in curved geometries.

Abstract

There are constructed exact solutions of the quantum-mechanical Dirac equation for a spin S=1/2 particle in Riemannian space of constant negative curvature, hyperbolic Lobachevsky space, in presence of an external magnetic field, analogue of the homogeneous magnetic field in the Minkowski space. A generalized formula for energy levels, describing quantization of the transversal motion of the particle in magnetic field has been obtained. The same problem is solved for spin 1/2 particle in the space of constant positive curvature, spherical Riemann space. A generalized formula for energy levels, describing quantization of the transversal and along the magnetic field motions of the particle on the background of the Riemann space geometry, is obtained.