Quantum mechanics for a vector particle in the magnetic field on 4-dimensional sphere

TL;DR

This paper investigates the quantum mechanics of a spin-1 particle in a magnetic field on a 4D sphere, solving some differential equations and reducing the problem to a system of linked equations.

Contribution

It introduces a wave equation for a spin-1 particle in curved space with magnetic field and performs variable separation, advancing understanding of quantum particles in non-Euclidean geometries.

Findings

Differential equations in r solved via hypergeometric functions

Reduced z-dependence to a system of linked differential equations

Partial analytical solutions obtained for the wave functions

Abstract

Quantum-mechanical wave equation for a particle with spin 1 is investigated in presence of external magnetic field in spaces with non-Euclidean geometry with constant positive curvature. Separation of the variable is performed; differential equations in the variable r are solved in hypergeometric functions. T he study o f z-dependence of the wave function has been reduced to a system of three linked ordinary differential 2-nd order equations; till now the system in z variable is not solved.