Remarks on the Dirac oscillator in (2 + 1) dimensions

Abstract

-In this work the Dirac oscillator in (2 + 1) dimensions is considered. We solve the problem in polar coordinates and discuss the dependence of the energy spectrum on the spin parameter s and angular momentum quantum number m. Contrary to earlier attempts, we show that the degeneracy of the energy spectrum can occur for all possible values of sm. In an additional analysis, we also show that an isolated bound state solution, excluded from SturmLiouville problem, exists.The Dirac oscillator, first introduced in [1] and after develop in [2], has been an usual model for studying the physical properties of physical systems in various branches of physics. In the context of theoretical contributions, the Dirac oscillator has been analyzed under different aspects such as the study of the covariance properties and FoldyWouthuysen and Cini-Touschek transformations [3], as a special case of a class of chiral solutions to the automorphism gauge field equations [4], and hidden supersymmetry produced by the interaction iM ωβr, where M is the mass, ω the frequency of the oscillator and r is the position vector, when it plays a role of anomalous magnetic interaction [5] (see also Refs. [3,6]).Recently, the one-dimensional Dirac oscillator has had its first experimental realization [7], which made the system more attractive from the point of view of applications. The Dirac oscillator in (2 + 1) dimensions, when the third spatial coordinate is absent, has also been studied in Refs. [8][9][10]. Additionally, this system was proposed in [11] to describe some electronic properties of monolayer an bylayer graphene. For a detailed approach of the Dirac oscillator see the Refs. [12,13].In this Letter, we address the Dirac oscillator in (2 + 1) dimensions. In [10], it was argued that the energy eigenvalues are degenerated only for negative values of k ϑ s, where k ϑ represents the angular momentum quantum number and s the spin projection parameter. This result, however, is not correct, as properly shown in this work. Additionally, an isolated bound state solution for the Dirac oscillator in (2 + 1) is worked out.We begin by writing the Dirac equation in (2 + 1) dimensions ( = c = 1)where p = (p x , p y ) is the momentum operator and ψ is a two-component spinor. The Dirac oscillator is obtained through the following nonminimal substitution [2]:where r = (x, y) is the position vector and ω stands for the Dirac oscillator frequency. Thus, the relevant equation isIn three dimensions the γ matrices are conveniently defined in terms of the Pauli matrices [14]where s is twice the spin value, with s = +1 for spin "up" and s = −1 for spin "down". In this manner, eq. (3) can be written aswhere π j = p j − iM ωσ z r j .p-1

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