Particle with spin 1 in a magnetic field on the hyperbolic plane H_{2}

TL;DR

This paper constructs exact solutions for a spin-1 particle in a magnetic field on the hyperbolic plane, deriving energy levels and solving both nonrelativistic and relativistic equations in this curved space.

Contribution

It provides the first exact solutions and energy quantization formulas for a spin-1 particle in a magnetic field on a hyperbolic plane, extending quantum mechanics to curved spaces.

Findings

Exact solutions for spin-1 particles in hyperbolic space

Generalized energy level formulas for magnetic quantization

Solutions to both nonrelativistic and relativistic equations

Abstract

There are constructed exact solutions of the quantum-mechanical equation for a spin S=1 particle in 2-dimensional Riemannian space of constant negative curvature, hyperbolic plane, 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 motion of the vector particle in magnetic field on the 2-dimensional space H_{2} has been found, nonrelativistic and relativistic equations have been solved.