Exact solutions of the (2+1) dimensional Dirac equation in a constant magnetic field in the presence of a minimal length

Abstract

We study the (2+1) dimensional Dirac equation in an homogeneous magnetic field (relativistic Landau problem) within a minimal length, or generalized uncertainty principle -GUP-, scenario. We derive exact solutions for a given explicit representation of the GUP and provide expressions of the wave functions in the momentum representation. We find that in the minimal length case the degeneracy of the states is modified and that there are states that do not exist in the ordinary quantum mechanics limit (β → 0). We also discuss the massless case which may find application in describing the behavior of charged fermions in new materials like Graphene.

The exact solutions of a (2 + 1)-dimensional Dirac oscillator under a magnetic field in the presence of a minimal length

“…In this context, many papers were published where a different quantum system in space with Heisenberg algebra was studied. They are: the Abelian Higgs model [15], the thermostatics with minimal length [16], the one-dimensional Hydrogen atom [17], the casimir effect in minimal length theories [18], the effect of minimal lengths on electron magnetism [19], the Dirac oscillator in one and three dimensions [20][21][22][23][24], the solutions of a two-dimensional Dirac equation in presence of an external magnetic field [25], the noncommutative phase space Schrödinger equation [26], Schrödinger equation with Harmonic potential in the presence of a Magnetic Field [27].…”

Algebraic Approach to Exact Solution of the (2 + 1)-Dimensional Dirac Oscillator in the Noncommutative Phase Space

“…Similar to [31,41] and by defining = cos , = sin , and 2 = 2 + 2 , it is seen that the (7) transform into…”

Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

“…Exact solutions of the (2 + 1)-dimensional Dirac equation for an external field in the presence of a minimal length were studied by Menculini et al [11,12].…”