We study the ($2+1$)-dimensional Dirac equation in a homogeneous magnetic field (relativistic Landau problem) within a minimal length or generalized uncertainty principle scenario. We derive exact solutions for a given explicit representation of the generalized uncertainty principle 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 ($\beta \to 0$). We also discuss the massless case, which may find application in describing the behavior of charged fermions in new materials like graphene.

• Received 12 November 2012

DOI:https://doi.org/10.1103/PhysRevD.87.065017