Exact solution of the two-dimensional Dirac oscillator.

In the present article we have found the complete energy spectrum and the corresponding eigenfunctions of the Dirac oscillator in two spatial dimensions. We show that the energy spectrum depends on the spin of the Dirac particle. Typeset using REVTEX 1 Recently, Moshinsky and Szczepaniak [1] have proposed a new type of interaction in the Dirac equation which, besides the momentum, is also linear in the coordinates. They called the resulting the Dirac equation the Dirac oscillator because in the non-relativistic limit it becomes a harmonic oscillator with a very strong spin-orbit coupling term. Namely, the correction to the free Dirac equation i ∂Ψc ∂t = (βγp+ βm)Ψc (1) reads p → p − imωβr (2) after substituting (2) into (1) we get an Hermitian operator linear in both p and r. Recently, the Dirac oscillator has been studied in spherical coordinates and its energy spectrum and the corresponding eigenfunctions have been obtained [2]. A generalization of the one-dimensional version of the Dirac oscillator has been proposed by Domı́nguez-Adame. In this case, the modification of the free Dirac equation, written in cartesian coordinates, is made by means of the substitution m → m−iγγV (x1). Obviously, for V (x1) = mωx1 we have the standard Dirac oscillator. Here, as well as for the three dimensional Dirac oscillator [2], bound states are present. An interesting framework for discussing the Dirac oscillator is a 2+1 space-time. The absence of a third spatial coordinate permits a series of interesting physical and mathematical phenomena like fractional statistics [4] and Chern-Simmons gauge fields among others. Since we are interested in studying the Dirac oscillator in a two-dimensional space, a suitable system of coordinates for writing the harmonic interaction are the polar ρ and θ coordinates. In this case the radial component of the modified linear momentum takes the form: pρ − imωβρ. It is the purpose of the present paper to analyze the solutions and the energy spectrum of the 2+1 Dirac oscillator expressed in polar coordinates. One begins by writing the Dirac equation (1) in a given representation of the gamma matrices. Since we are dealing with two component spinors it is convenient to introduce the following representation in terms of the Pauli matrices [5] 2 βγ1 = σ1, βγ2 = sσ2, β = σ3 (3) where the parameter s takes the values ±1 (+1 for spin up and −1 for spin down). Then, the Dirac (9) equation written in polar coordinates reads