ar X iv : h ep - t h / 03 03 14 2 v 1 16 M ar 2 00 3 One - dimensional Dirac oscillator in a thermal bath

The name Dirac oscillator was first introduced by Moshinky and Szczepaniak [1] for a Dirac equation in which the momentum p is replaced by p → p−imωβr with r being the position vector, m the mass of the particle and ω the frequency of the oscillator. The name was given because the determination of the spectrum and eigenstate of a Dirac oscillator [1] was obtained from that of an ordinary oscillator with a very strong spin-orbit coupling term. The concept, though not the name, had been discussed previously first by Ito et al. [2] and later by Cook [3] and Ui and Takeda [4] and possibly others. Besides intrinsic mathematical interest the study of the Dirac oscillator has invoked much attention because of its various physical applications. For instance, the last three chapter of a book of Moshinsky and Smirnov [5] deal with the Dirac oscillator relativistic interactions between systems of one, two and three particles. Moreno and Zentella showed [6] that it could be the object of an exact Foldy-Wouthuysen transformation. Benitez et al. [7] found the electromagnetic potential associated with the Dirac oscillator, and showed that this exactly soluble problem has a hidden supersymmetry, which is responsible for the special properties of its energy spectrum. They also calculated the related superpotential and discussed the implications of this supersymmetry on the stability of the Dirac sea. Rozmej and Arvieu [8] have shown a very interesting analogy between the relativistic Dirac oscillator and the JaynesCummings model. They showed that the strong spin-orbit coupling of the Dirac oscillator produces the entanglement of the spin with the orbital motion similar to what is observed in the model of quantum optics. More recently Nogami and Toyama [9] and Toyama et al [10] have studied the behaviour of wave packets of the Dirac oscillator in the Dirac representation in (1+1) dimensions. The aim of these authors was to study wave packets which could possibly be coherent. This reduction of the dimension was brought as an attempt to get rid of spin effects and to concentrate on the relativistic effects. As a matter of fact, the one-dimensional relativistic Dirac equation has been largely used to treat physical problems where relativistic effects could play an important role. Indeed, this treatment gives enlightening information about more realistic models in solid state physics [11], particularly in semiconductor theories [12]. In spite of the great number of papers that has been recently published concerning the solution and properties of the Dirac oscillator, as far as we know no one has reported on its thermodynamics properties. In order to overcome this lack of information, in this letter we study the one-dimensional Dirac oscillator in a thermal bath. Let us consider the energy spectrum for a one-dimensional Dirac oscillator [13]