Advances in High Energy Physics

Volume 2017 (2017), Article ID 9371391, 12 pages

https://doi.org/10.1155/2017/9371391

## The Statistical Properties of the -Deformed Dirac Oscillator in One and Two Dimensions

^{1}Laboratoire de Physique Appliquée et Théorique, Université Larbi Tébessi, Tébessa, Algeria^{2}Physics Department, Shahrood University of Technology, Shahrood, Iran

Correspondence should be addressed to Abdelmalek Boumali

Received 23 January 2017; Accepted 11 June 2017; Published 16 July 2017

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2017 Abdelmalek Boumali and Hassan Hassanabadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the behavior of the eigenvalues of the one and two dimensions of -deformed Dirac oscillator. The eigensolutions have been obtained by using a method based on the -deformed creation and annihilation operators in both dimensions. For a two-dimensional case, we have used the complex formalism which reduced the problem to a problem of one-dimensional case. The influence of the -numbers on the eigenvalues has been well analyzed. Also, the connection between the -oscillator and a quantum optics is well established. Finally, for very small deformation , we (i) showed the existence of well-known -deformed version of Zitterbewegung in relativistic quantum dynamics and (ii) calculated the partition function and all thermal quantities such as the free energy, total energy, entropy, and specific heat. The extension to the case of Graphene has been discussed only in the case of a pure phase ().

#### 1. Introduction

Quantum groups and quantum algebras have attracted much attention of physicists and mathematicians during the last eight years. There had been a great deal of interest in this field, especially after the introduction of the -deformed harmonic oscillator. Quantum groups and quantum algebras have found unexpected applications in theoretical physics [1]. From the mathematical point of view they are -deformations of the universal enveloping algebras of the corresponding Lie algebras, being also concrete examples of Hopf algebras. When the deformation parameter is set equal to 1, the usual Lie algebras are obtained. The realization of the quantum algebra SU in terms of the -analogue of the quantum harmonic oscillator [2, 3] has initiated much work on this topic [4–6]. Biedenharn and Macfarlane [2, 3] have studied the -deformed harmonic oscillator based on an algebra of -deformed creation and annihilation operators. They have found the spectrum and eigenvalues of such a harmonic oscillator under the assumption that there is a state with a lowest energy eigenvalue.

Recently, the theory of the -deformed has become a topic of great interest in the last few years, and it has been finding applications in several branches of physics because of its possible applications in a wide range of areas, such as a -deformation of the harmonic oscillator [7], a -deformed Morse oscillator [8], a classical and quantum -deformed physical systems [9], Jaynes-Cummings model and the deformed-oscillator algebra [10], -deformed supersymmetric quantum mechanics [11], for some modified -deformed potentials [12], on the thermostatistic properties of a -deformed ideal Fermi gas [13], -deformed Tamm-Dancoff oscillators [14], -deformed fermionic oscillator algebra, and thermodynamics [15], and finally on the fermionic deformation and its connection to thermal effective mass of a quasiparticle [16].

The relativistic harmonic oscillator is one of the most important quantum systems, as it is one of the very few that can be solved exactly. The Dirac relativistic oscillator (DO) interaction is an important potential both for theory and for application. It was for the first time studied by Itô et al. [17]. They considered a Dirac equation in which the momentum is replaced by , with being the position vector, the mass of particle, and the frequency of the oscillator. The interest in the problem was revived by Moshinsky and Szczepaniak [18], who gave it the name of DO because, in the nonrelativistic limit, it becomes a harmonic oscillator with a very strong spin-orbit coupling term. Physically, it can be shown that the DO interaction is a physical system, which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field [19, 20]. The electromagnetic potential associated with the DO has been found by Benitez et al. [21]. The DO has attracted a lot of interests both because it provides one of the examples of the Dirac’s equation exact solvability and because of its numerous physical applications [22–27]. Finally, Franco-Villafañe et al. [28] have exposed the proposal of the first experimental microwave realization of the one-dimensional DO.

An important phenomena which we will discuss here is the connection between DO with the Jaynes-Cummings JC model, an archetypical quantum optical system: the Jaynes-Cummings model (JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light (in the form of a bath of electromagnetic radiation that can cause spontaneous emission and absorption) [29, 30]. The JCM is of great interest in atomic physics, quantum optics, and solid-state quantum information circuits, both experimentally and theoretically. This model describes well the DO interaction and, following Bermudez et al. [31], the exact mapping of the 2 + 1 Dirac oscillator onto the Jaynes-Cummings model well established. In this paper, the extension of this connection in the presence of deformation will be our goal.

The -deformed oscillator systems have attracted much attention and have been considered in many papers (see [32] and references therein). The representation theory of the quantum algebras has led to the development of -deformed oscillator algebra. Since there have been an increasing interest in the study of physical systems using -oscillator algebra, it has found applications in several branches of physics such as vibrational spectroscopy, nuclear physics, and many body theory and quantum optics. The -analogue of the one-dimensional nonrelativistic harmonic oscillator has been studied by several authors [2, 3, 33, 34]. Realizations of the quantum algebra via the one-dimensional -harmonic oscillator were suggested by Chaichian [35]. The representation theory of quantum algebras with a single deformation parameter, , has led to the development of tile now well-known -deformed harmonic oscillator algebra.

In the nonrelativistic case, the investigation of thermodynamic functions of some types of potentials, such as Morse and improved Manning-Rosen potentials and improved Rosen-Morse and Tiez oscillators, through a partition function and its derivatives with respect to temperature, was an important field of research in the literature (see [36–39]). The extension of the nonrelativistic -harmonic oscillator to the relativistic case, to the best of our knowledge, is not available in the literature. In this context and in order the overcome this lack in the literature, the principal aim of this paper will be the study of -deformed DO in one and two dimensions. The concept of -deformation is also applied to the investigation of the connection of -deformed DO with quantum optics and the existence of the well-known Zitterbewegung in relativistic quantum dynamics of the problem in question. In addition, for small deformation , we have evaluated various thermodynamic quantities such as partition function, entropy, and internal energy. Such studies are expected to be relevant when we want to extended them to the case of Graphene. To prove existence of such a system, we want to indicate some papers including applications of -deformations such as study of electronic conductance in disordered metals and doped semiconductors [40], analyzing of the phonon spectrum in [41], and expression of the oscillatory-rotational spectra of diatomic and multiatomic molecules [42–46].

The structure of this paper is as follows: Section 2 is devoted to the case of the standard -harmonic oscillator. The extension to the -deformed DO will be treated in Section 3. Different numerical results about the thermal properties of -deformed DO are discussed in Section 4. Finally, Section 5 will be the conclusions.

#### 2. -Deformed One-Dimensional Harmonic Oscillator

The Hamiltonian of the -deformed harmonic oscillator iswhere the -momentum () and -position () operators are directly written in terms of the -boson operators and aswithBoth -factorial and -numbers are defined, respectively, byHere is the number operator. The eigenvalues of the -deformed one-dimensional harmonic oscillator are [47]Following (5) and (6), two cases can be distinguished.(i)When we put , the eigenvalues become(ii)Now, when we write , the form of the spectrum, in this case, isIn both cases, when the well-known relationis recovered.

Thus, according to (7) and (8), one can see that for real () the energy eigenvalues increase more rapidly than the ordinary case, and the spectrum, in this case, gets expanded. In contrast, when is a pure phase (), the eigenvalues of the energy increase less rapidly than the ordinary case; that is, the spectrum is squeezed [47].

In what follows, we treat the case of the one- and two-dimensional -deformed DO.

#### 3. Solutions of a -Deformed Dirac Oscillator

##### 3.1. One-Dimensional -Deformed Dirac Oscillator

The one-dimensional DO iswith , , and . In this case, (10) becomeswithby introducing the usual annihilation and creation operators of the -deformed harmonic oscillatorthis Hamiltonian transforms intowith being the coupling strength between orbital and spin degrees of freedom, and is a parameter which controls the nonrelativistic limit. It is an important parameter that specifies the importance of relativistic effects in the DO.

Writing that , this equation can be solved algebraically. Following the above section, when is real, the spectrum of energy isNow, if is complex, it becomesIn both cases, when the well-known relationis recovered [25]. The eigensolutions of a two-dimensional DO, in both cases, can be written asUsing the creation and annihilation operators and the raising and lowering operators Dirac Spinor one can rewrite the previous equation asThis Hamiltonian is exactly the JCM Hamiltonian in quantum optics [48]. Thus the one-dimensional DO maps exactly onto the Jaynes-Cummings (JC), provided that one identifies the isospin with the atomic system and the spatial degrees of freedom with the cavity mode. Thus, as a result, the relativistic Hamiltonian of a -deformed one-dimensional DO can be mapped onto a -deformed Jaynes-Cummings (JC).

##### 3.2. Two-Dimensional -Deformed Dirac Oscillator

###### 3.2.1. Complex Formalism

In terms of complex coordinates and its complex conjugate, we haveThe operators momenta and , in the Cartesian coordinates, are defined byWhen we use , we getwith . These operators obey the basic commutation relationsThe usual creation and annihilation operators, and , withcan be reformulated, in the formalism complex, as follows:These operators, also, satisfy the habitual commutation relationsNow, In the case of -deformed DO, the creation and annihilation operators and satisfy the commutation relationwhere is the number operator, satisfying

###### 3.2.2. The Solutions

The two-dimensional DO iswith , , and . With definitions of Dirac matrices, (8) can be decoupled in a set of equations as follows:and so (9) readsThis last form of Hamiltonian of Dirac can be written, in the complex formalism, byThus the problem is transformed to the one-dimensional case with a complex variable .

Now, following (9) and (10), the wave functions and can be rewritten in the language of the complex annihilation-creation operators asWhen we write the component in terms of the quanta bases, , these equations can be simultaneously diagonalized, and the energy spectrum can be described bywhen is real and byif is complex. In both cases, when the well-known relationis recovered. Our results are in good agreement with those obtained by Hatami and Setare [49].

In order to establish the connection between a two-dimensional -deformed DO and quantum optics, the Dirac Hamiltonian can be rewritten into another form as

This form of the Hamiltonian corresponds to the -deformed Anti-Jaynes-Cummings (AJC) model. Here are the spin arising and lowering operators, and is a detuning parameter. As a result, the relativistic Hamiltonian of a -deformed two-dimensional DO can be mapped onto a couple of -deformed Anti-Jayne-Cummings (AJC) which describes the interaction between the relativistic spin and bosons.

We further observe that the Zitterbewegung frequency for the -deformed (2 + 1)-dimensional DO depends on the parameter of deformation . To show this we, first, start with the following eigensolutions of a two-dimensional DO:Here, for real (or complex), respectively. The eigenstates can be expressed transparently in terms of two-component Pauli spinors and [31]:where (or ) and (or ) are real. Following these equations, the eigenstates present entanglement between the orbital and spin degrees of freedom. To clarify this, we start with some initial pure state at ,This equation shows that the starting initial state is a superposition of both the positive and negative energy solutions, which is the fundamental ingredient that leads to Zitterbewegung in relativistic quantum dynamics.

The evolution of this initial state can be expressed aswherefor real andfor complex. In both cases, describes the frequency of oscillations: the frequency oscillation between positive and negative energy solutions.

If we consider very small deformation and neglect all terms proportional to , we havewith being the frequency of a two-dimensional DO without deformation, and the sign + denotes the case for real and the sign − denotes the case of complex. In the approximation of very small , the final form will bewhereThe sign (−) denotes the case of real and (+) denotes the other case (pure phase). This equation shows the oscillatory behavior between the states and which is exactly similar to atomic Rabi oscillations occurring in the JC and AJC models. The -deformed Rabi frequency, , follows (46) for both cases.

##### 3.3. Discussions

This section is devoted to study the influence of -deformed algebra on the eigenvalues of the DO in one and two dimensions. This influence has been well established through the parameter with . In Figure 1, we present the eigenvalues of the -deformed DO in one and two dimensions versus the quantum number with different values of parameter in both cases of real and complex. In order to discuss the results, we use the same explication used by Neskovic and Urosevic [50] in their study of the statistical properties of quantum oscillator: thus, the energy levels of the -oscillator are not uniformly spaced for . The behavior of the energy spectra is completely different in the cases and . When is real , the separation between the levels increases with the value of ; that is, the spectrum is extended. On the other hand, when is a pure phase, the separation between all the levels decreases with increasing ; that is, the spectrum is squeezed.