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We have found the complete energy spectrum and the corresponding eigenfunctions of the recently proposed Dirac oscillator. We found the electromagnetic potential associated with its interaction term.This exactly soluble problem has a hidden supersymmetry, responsible for the special properties of its energy spectrum. We calculate the related superpotential and~e discuss the implications of this supersymmetry on the stability of the Dirac sea.PACS numbers: 11.10. Qr, 11.30.Pb, 12.40. gq In a recent work, Moshinsky and Szczepaniak' introduced a very interesting potential in the Dirac equation. Following the original Dirac procedure they introduce a system whose Dirac Hamiltonian is linear in both p and r; consequently, its nonrelativistic approximation is quadratic in r as in the case of the harmonic oscillator. This property is in fact the origin of the name Dirac oscillator. The system is then, excepting for a strong spinorbit coupling term, the square root of the harmonic oscillator in the same sense as the Dirac equation is the square root of the Klein-Gordon equation. The Hamiltonian of the system is obtained by introducing in a nonminimal way the external potential with the substitution i(rilir/rlt) =Hiir= [a (p imrupr)-+mpl lir.(2)The interaction introduced by Eq. (1) has been shown to correspond to an anomalous magnetic interaction. In fact, if we select a frame-dependent vectorthen the interaction term in Eq.(2) can be put in the form imcoa r =cr"'x"u, ;(4) consequently, Eq.(2) can be written in the manifestly covariant form [y"p"-m+ (ke/4m )cr"'F", l @=0, with (s) F""= (u "x"u'x"), p p i mrupr, where m is the mass of the particle and co is the oscillator frequency. We follow the usual conventions concerning the Dirac matrices. The Dirac equation for the system is then (11'i c =1) Dirac oscillator. The electromagnetic potential can be cast in the form (7) but, taking advantage of the gauge invariance of the electromagnetic interaction and selecting the gauge l= -, ' x (u x), we can write Eq. (7) in the equivalent forms For either selection the electromagnetic field takes the form E=mror and 8 0, expressions which are, of course, frame dependent. In this way we have shown that the interaction occurs as if it were produced by an infinite sphere carrying a uniform charge-density distribution, resulting in a linearly growing electric field. The interaction term introduced by Eq. (1) is interesting in its own right, but also has applications in quantum chromodynamics (QCD). For example, if we think of the construction of quark-confinement models, the linearly growing interaction characteristic of the system can be regarded as an effective chromoelectric field, at least if we think that the color-field lines are constrained to strings of constant volume instead of constrained to constant cross section, as is usually supposed. Also, it can be useful for estimating the quark masses.These considerations make the Dirac oscillator important to QCD models, as we will discuss in a forthcoming paper. Moshinsky and Sz...

The conformal invariance properties of a Dirac oscillator are established. A set of operators is constructed whose algebra shows that it can be considered as a conformal system. The operators are then used to solve the problem using algebraic techniques. The superconformal generalization of the algebra is also worked out, and some consequences of these invariances for the properties of the model are mentioned.

The Dirac oscillator is an exactly soluble model recently introduced in the context of many particle models in relativistic quantum mechanics. The model has been also considered as an interaction term for modelling quark confinement in quantum chromodynamics. These considerations should be enough for demonstrating that the Dirac oscillator can be an excellent example in relativistic quantum mechanics. In this paper we offer a solution to the problem and discuss some of its properties. We also discuss a physical picture for the Dirac oscillator's non-standard interaction, showing how it arises on describing the behaviour of a neutral particle carrying an anomalous magnetic moment and moving inside an uniformly charged sphere. ResumenEl oscilador de Dirac es un modelo exactamente resoluble que ha sido introducido recientemente en el contexto de la mecánica cuántica relativista de muchos cuerpos.El problema ha sido también explorado como posible fuente de un término de interacción para modelar confinamiento en cromodinámica cuántica. Estas consideraciones establecen sin lugar a dudas que el oscilador de Dirac puede servir como ejemplo interesante en mecánica cuántica relativista. Este artículo ofrece una solución al problema y la discusión de algunas de sus propiedades. También discutimos una imágen física que se ha introducido para el término de interacción del oscilador, mostrando como surge de considerar el comportamiento de una partícula neutra con momento magnético anómalo que se mueve dentro de una esfera cargada uniformemente.