Relativistic Quantum Dynamics of a Neutral Dirac Fermion in the Presence of an Electromagnetic Field

In this work, we study the (2 + 1)-dimensional Dirac equation for a neutral fermion with magnetic dipole moment in the presence of an electromagnetic field. Next, we explicitly determine the eigenfunctions and the relativistic energy spectrum of the fermion. As result, we verified that these eigenfunctions are written in terms of the generalized Laguerre polynomials and the energy spectrum depends on the quantum numbers, n = 0,1,2,… and m_j = 0,± 1,± 2,…, homogeneous magnetic field B, and of the cyclotron frequency ω_AC generated by the electric field. Besides that, this energy spectrum may have finite or infinite degeneracy depending on the values of m_j. In particular, we also verified that in the absence of the electric field (ω_AC = 0), the energy spectrum reduces to a physical quantity (energy) that depends on the rest mass of the fermion and antifermion and of the magnetic field, already in the absence of the magnetic field (B = 0), the energy spectrum still remains quantized in terms of the quantum numbers n and m_j; on the other hand, in the absence of the electromagnetic field (ω_AC = B = 0), we get the rest energy of the fermion and antifermion. Finally, we compare our results with the literature, where we observe a similarity in some results of the Dirac oscillator.