Probing spin as a degree of freedom in two- and three-dimensional Dirac equations for two-dimensional motion with external magnetic fields

Abstract

When describing quantum relativistic spin-1/2 particles moving in a plane one can in principle use Dirac matrices realized by either $2\times 2$ matrices (2-component spinors) or $4\times 4$ matrices (4-component spinors) in the quantum equation of motion. However, when spin is a relevant degree of freedom for particle dynamics, as when na external magnetic field is applied, it is important to assess how this is taken into account in those two realizations. In the iterature, thissometimes is not so clear as far as the $2\times 2$ realization is concerned. To reply to this question, and at the same time assess the effect of spinor dimensionality on an energy spectrum which depends on spin, we compute the bound solutions of those Dirac equations when it is applied a constant magnetic field perpendicular to the plane of motion. The conclusions arerelevant to several physics domainswhere the Dirac equation is applied for 2-dimensional relativisitic quantum motion, including graphene systems. (AU)