We extend the $(1+1)$-dimensional Dirac-Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac-Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate $su(1,1)$ algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the $su(1,1)$ irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.
We study the Dunkl oscillator in two dimensions by the $su(1,1)$ algebraic method. We apply the Schrödinger factorization to the radial Hamiltonian of the Dunkl oscillator to find the $su(1,1)$ Lie algebra generators. The energy spectrum is found by using the theory of unitary irreducible representations. By solving analytically the Schrödinger equation, we construct the Sturmian basis for the unitary irreducible representations of the $su(1,1)$ Lie algebra. We construct the $SU(1,1)$ Perelomov radial coherent states for this problem and compute their time evolution.
We introduce the Dunkl--Klein--Gordon (DKG) equation in 2D by changing the standard partial derivatives by the Dunkl derivatives in the standard Klein--Gordon (KG) equation. We show that the generalization with Dunkl derivative of the $z$-component of the angular momentum is what allows the separation of variables of the DKG equation. Then, we compute the energy spectrum and eigenfunctions of the DKG equations for the 2D Coulomb potential and the Klein--Gordon oscillator analytically and from an ${\rm su}(1,1)$ algebraic point of view. Finally, we show that if the parameters of the Dunkl deriv…
Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential
Two Dunkl oscillator models are considered: one singular and the other with a 2:1 frequency ratio. These models are defined by Hamiltonians which include the reflection operators in the two variables x and y. The singular or caged Dunkl oscillator is second-order superintegrable and admits separation of variables in both Cartesian and polar coordinates. The spectrum of the Hamiltonian is obtained algebraically and the separated wavefunctions are given in the terms of Jacobi, Laguerre and generalized Hermite polynomials. The symmetry generators are constructed from the su(1,1) dynamical operato…
The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra generated by the constants of motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie algebra u(2) with involutions. The system admits separation of variables in both Cartesian and polar coordinates. The separated wavefunctions are …
This paper has studied the three-dimensional Dunkl oscillator models in a generalization of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. Their symmetries are obtained by the Jordan-Schwinger representations in the family of the Cayley-Klein orthogonal algebras using the creation and annihilation operators of the dynamical $sl_{-1}(2)$ algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of $so_{κ…
For a better understanding of the physical systems, even at the quantum level, the thermal quantities can be investigated. Recently, we realized that the parity of the system can also be examined simultaneously, by substituting the Dunkl operator with the ordinary differential operator in quantum mechanics. In this manuscript, we consider two relativistic Dunkl-oscillators and investigate their thermal quantities with well-known statistical methods. Besides, we establish a relationship between the Dunkl-Dirac oscillator and the Dunkl-Anti-Jayne-Cummings model by defining the Dunk creation and …
Dunkl derivative enriches solutions by discussing parity due to its reflection operator. Very recently, one of the authors of this manuscript presented one of the most general forms of Dunkl derivative that depends on three Wigner parameters to have a better tuning. In this manuscript, we employ the latter generalized Dunkl derivative in a relativistic equation to examine two-dimensional harmonic and anharmonic oscillators solutions. We obtain the solutions by Nikiforov-Uvarov and QES methods, respectively. We show that degenerate states can occur according to the Wigner parameter values.
We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the $\mathrm{S}_3$ Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system $A_2$, with corresponding Weyl group $\mathrm{S}_3$, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra $\mathfrak{so}(3)$, incorporating elements of $\mathrm{S}_3$. This was obtained using recent…
Recent studies show that deformations in quantum mechanics are inevitable. In this contribution, we consider a relativistic quantum mechanical differential equation in the presence of Dunkl operator-based deformation and we investigate solutions for two important problems in three-dimensional spatial space. To this end, after introducing the Dunkl quantum mechanics, we examine the Dunkl-Klein-Gordon oscillator solutions with the Cartesian and spherical coordinates. In both coordinate systems, we find that the differential equations are separable and their eigenfunctions can be given in terms o…
Dunkl--Pauli Oscillator in an Aharonov--Bohm Flux: Restriction of Admissible Angular Sectors and Thermodynamic Signatures
Theoretical Physics Laboratory, Department of Physics, University of Jijel, Algeria（朱杰尔大学物理系理论物理实验室）；Department of Process Engineering University of Constantine 3 –Salah Boubnider25016 Constantine, Algeria（君士坦丁三世-萨拉胡布卜尼德大学工艺工程系）
We construct the Runge-Lenz vector and symmetry algebra of rational Calogero-Coulomb problem, formulated in terms of Dunkl operators. We find that they are proper deformations of their Coulomb counterpart. Together with similar correspondence between the Calogero-oscillator and oscillator models, this observation permits to claim that most of properties of Coulomb and oscillator systems can be lifted to their Calogero-extended analogs by the proper replacement of momenta by Dunkl momenta operators.
By modifying and generalizing known supersymmetric models we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of $\frac{d}{dx}$. Also, by changing the derivative $\frac{d}{dx}$ by the Dunkl derivative in the creation and annihilation-type operators of …
We construct a Darboux transformation for a class of two-dimensional Dirac systems at zero energy. Our starting equation features a position-dependent mass, a matrix potential, and an additional degree of freedom that can be interpreted either as a magnetic field perpendicular to the plane or a generalized Dirac oscillator interaction. We obtain a number of Darbouxtransformed Dirac equations for which the zero energy solutions are exactly known.
We explore the effect of the external magnetic and Aharonov-Bohm (AB) flux fields on the energy levels of Dirac particle subjects to mixed scalar and vector anharmonic oscillator field in the two-dimensional (2D) space. We calculate the exact energy eigenvalues and the corresponding un-normalized two-spinor-components wave functions in terms of the chemical potential parameter, magnetic field strength, AB flux field and magnetic quantum number by using the Nikiforov-Uvarov (NU) method.
This Letter is based on the $κ$-Dirac equation, derived from the $κ$-Poincaré-Hopf algebra. It is shown that the $κ$-Dirac equation preserves parity while breaks charge conjugation and time reversal symmetries. Introducing the Dirac oscillator prescription, $\mathbf{p}\to\mathbf{p}-imωβ\mathbf{r}$, in the $κ$-Dirac equation, one obtains the $κ$-Dirac oscillator. Using a decomposition in terms of spin angular functions, one achieves the deformed radial equations, with the associated deformed energy eigenvalues and eigenfunctions. The deformation parameter breaks the infinite degeneracy of the D…
Three dimensional Dirac oscillator was considered in deformed space obeyed to deformed commutation relations known as Snyder-de Sitter algebra. Snyder-de Sitter commutation relations gives rise to appearance minimal uncertainty in position as well as in momentum. To derive energy spectrum and wavefunctions of the Dirac oscillator supersymmetric quantum mechanics and shape invariance technique was applied.
The main goal of this work is to study the Dirac oscillator as a quantum field using the canonical formalism of quantum field theory and to develop the canonical quantization procedure for this system in $(1+1)$ and $(3+1)$ dimensions. This is possible because the Dirac oscillator is characterized by the absence of the Klein paradox and by the completeness of its eigenfunctions. We show that the Dirac oscillator field can be seen as constituted by infinite degrees of freedom which are identified as decoupled quantum linear harmonic oscillators. We observe that while for the free Dirac field th…
It is shown that a spin-1/2 fermion coupled to the axially symmetric electromagnetic vector potential has the same matrix structure as that one for the planar Dirac oscillator. In particular, the planar Dirac oscillator can be interpreted as a charged particle minimally coupled to a transverse homogeneous magnetic field.
We study generalized Dirac oscillators with complex interactions in $(1+1)$ dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are $η$ pseudo Hermitian with respect to certain metric operators $η$. Exact solutions of the generalized Dirac Oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with Anti Jaynes Cummings type model and by spin flip it can also be identified with Jaynes Cummings type model.
In this Letter, 2D Dirac oscillator in the quantum deformed framework generated by the $κ$-Poincaré-Hopf algebra is considered. The problem is formulated using the $κ$-deformed Dirac equation. The resulting theory reveals that the energies and wave functions of the oscillator are modified by the deformation parameter.
In memory of Marcos Moshinsky, who promoted the algebraic study of the harmonic oscillator, some results recently obtained on an infinite family of deformations of such a system are reviewed. This set, which was introduced by Tremblay, Turbiner, and Winternitz, consists in some Hamiltonians $H_k$ on the plane, depending on a positive real parameter $k$. Two algebraic extensions of $H_k$ are described. The first one, based on the elements of the dihedral group $D_{2k}$ and a Dunkl operator formalism, provides a convenient tool to prove the superintegrability of $H_k$ for odd integer $k$. The se…
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 …
We demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasi-exactly solvable systems. Electric field configurations permitting quasi-exact solvability of the system based on the $sl(2)$ symmetry are discussed separately in spherical, cylindrical, and Cartesian coordinates. Some exactly solvable field configurations are also exhibited.
We study the $(1+1)$ dimensional generalized Dirac oscillator with a position-dependent mass. In particular, bound states with zero energy as well as non zero energy have been obtained for suitable choices of the mass function/oscillator interaction. It has also been shown that in the presence of an electric field, bound states exist if the magnitude of the electric field does not exceed a critical value.
The dynamics of wave packets in a relativistic Dirac oscillator is compared to that of the Jaynes-Cummings model. 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. The collapses and revivals of the spin which result extend to a relativistic theory our previous findings on nonrelativistic oscillator where they were known under the name of `spin-orbit pendulum'. There are important relativistic effects (lack of periodicity, zitterbewegung, negative energy states). Many o…
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 Sturm-Liouville problem, exists.
In this paper, we study the influence of noninertial effects on the Dirac oscillator in the cosmic string spacetime background. We discuss the behaviour of the oscillator frequency in a noninertial system that allows us to obtain relativistic bound state solutions. We also discuss the influence of the topology of the cosmic string spacetime on the relativistic energy levels, and obtain the Dirac spinors for positive-energy solutions. Furthermore, by taking the nonrelativistic limit of the energy levels, we compare the nonrelativistic energy levels to the confinement of a spin-half particle to …
We obtain exact solutions of the (2+1) dimensional Dirac oscillator in a homogeneous magnetic field within the Anti-Snyder modified uncertainty relation characterized by a momentum cut-off ($p\leq p_{\text{max}}=1/ \sqrtβ$). In ordinary quantum mechanics ($β\to 0$) this system is known to have a single left-right chiral quantum phase transition (QPT). We show that a finite momentum cut-off modifies the spectrum introducing additional quantum phase transitions. It is also shown that the presence of momentum cut-off modifies the degeneracy of the states.
In this work the dynamics of a 2D Dirac oscillator in the spacetime of a magnetic cosmic string is considered. It is shown that earlier approaches to this problem have neglected a $δ$ function contribution to the full Hamiltonian, which comes from the Zeeman interaction. The inclusion of spin effects leads to results which confirm a modified dynamics. Based on the self-adjoint extension method, we determined the most relevant physical quantities, such as energy spectrum, wave functions and the self-adjoint extension parameter by applying boundary conditions allowed by the system.
We examine the non-inertial effects of a rotating frame on a Dirac oscillator in a cosmic string space-time with non-commutative geometry in phase space. We observe that the approximate bound-state solutions are related to the biconfluent Heun polynomials. The related energies cannot be obtained in a closed form for all the bound states. We find the energy of the fundamental state analytically by taking into account the hard-wall confining condition. We describe how the ground-state energy scales with the new non-commutative term as well as with the other physical parameters of the system.
In the context of some deformed canonical commutation relations leading to isotropic nonzero minimal uncertainties in the position coordinates, a Dirac equation is exactly solved for the first time, namely that corresponding to the Dirac oscillator. Supersymmetric quantum mechanical and shape-invariance methods are used to derive both the energy spectrum and wavefunctions in the momentum representation. As for the conventional Dirac oscillator, there are neither negative-energy states for $E=-1$, nor symmetry between the $l = j - {1/2}$ and $l = j + {1/2}$ cases, both features being connected …
We obtain exact solutions of the (2+1) dimensional Dirac oscillator in a homogeneous magnetic field within a minimal length ($Δx_0=\hbar \sqrtβ$), or generalised uncertainty principle (GUP) scenario. This system in ordinary quantum mechanics has a single left-right chiral quantum phase transition (QPT). We show that a non zero minimal length turns on a infinite number of quantum phase transitions which accumulate towards the known QPT when $β\to 0$. It is also shown that the presence of the minimal length modifies the degeneracy of the states and that in this case there exist a new class of st…
The Dirac equation with both scalar and vector couplings describing the dynamics of a two-dimensional Dirac oscillator in the cosmic string spacetime is considered. We derive the Dirac-Pauli equation and solve it in the limit of the spin and the pseudo-spin symmetries. We analyze the presence of cylindrical symmetric scalar potentials which allows us to provide analytic solutions for the resultant field equation. By using an appropriate ansatz, we find that the radial equation is a biconfluent Heun-like differential equation. The solution of this equation provides us with more than one express…
We obtain the solution of the Dirac equation in (2+1) dimensions in the presence of a constant magnetic field normal to the plane together with a two-dimensional Dirac-oscillator potential coupling. We study the energy spectrum of graphene quantum dot (QD) defined by electrostatic gates. We give discussions of our results based on different physical settings, whether the cyclotron frequency is similar or larger/smaller compared to the oscillator frequency. This defines an effective magnetic field that produces the effective quantized Landau levels. We study analytically such field in gate-tuna…
The Dirac equation is considered in the background of potentials of several types, namely scalar and vector-potentials as well as "Dirac-oscillator" potential or some of its generalisations. We investigate the radial Dirac equation within a quite general spherically symmetric form for these potentials and we analyse some exactly and quasi exactly solvable properties of the underlying matricial linear operators.
We study the algebraic structure of the one-dimensional Dirac oscillator by extending the concept of spin symmetry to a noncommutative case. An SO(4) algebra is found connecting the eigenstates of the Dirac oscillator, in which the two elements of Cartan subalgebra are conserved quantities. Similar results are obtained in the Jaynes--Cummings model.
We analyze recent results on a Dirac oscillator. We show that the truncation of the Frobenius series does not yield all the eigenvalues and eigenfunctions of the radial equation. For this reason the eigenvalues reported by the authors are useless and the prediction of allowed oscillator frequencies meaningless.
The Dirac oscillator in a homogenous magnetic field exhibits a chirality phase transition at a particular (critical) value of the magnetic field. Recently, this system has also been shown to be exactly solvable in the context of noncommutative quantum mechanics featuring the interesting phenomenon of re-entrant phase transitions. In this work we provide a detailed study of the thermodynamics of such quantum phase transitions (both in the standard and in the noncommutative case) within the Maxwell-Boltzmann statistics pointing out that the magnetization has discontinuities at critical values of…
The $D$-dimensional $(β, β')$-two-parameter deformed algebra introduced by Kempf is generalized to a Lorentz-covariant algebra describing a ($D+1$)-dimensional quantized space-time. In the D=3 and $β=0$ case, the latter reproduces Snyder algebra. The deformed Poincaré transformations leaving the algebra invariant are identified. It is shown that there exists a nonzero minimal uncertainty in position (minimal length). The Dirac oscillator in a 1+1-dimensional space-time described by such an algebra is studied in the case where $β'=0$. Extending supersymmetric quantum mechanical and shape-invari…
$1+1$-dimensional Dirac oscillator with minimal uncertainty in position and maximal in momentum is investigated. To obtain energy spectrum SUSY QM technique is applied. It is shown that the Dirac oscillator has two branches of spectrum, the first one gives the standard spectrum of the Dirac oscillator when the parameter of deformation goes to zero and the second branch does not have nondeformed limit. Maximal momentum brings an upper bound for the energy and it gives rise to the conclusion that the energy spectrum contains a finite number of eigenvalues. We also calculate partition function fo…
We study generalized (1+1)-dimensional Dirac oscillator in nonuniform electric field. It is shown that in the case of specially chosen electric field the eigenvalue equation can be casted in the form of supersymmetric quantum mechanics. It gives a possibility to find exact solution for the energy spectrum of the generalized Dirac oscillator in nonuniform electric field. Explicit examples of exact solutions are presented. We show that sufficiently large electric field destroys the bounded eigenstates.
We analyze the one-dimensional Dirac oscillator in a thermal bath. We found that the heat capacity is two times greater than the heat capacity of the one-dimensional harmonic oscillator for higher temperatures.
