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.
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.
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.
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 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.
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.
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.
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…
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 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.
The generalized Dirac oscillator as one of the exact solvable model in quantum mechanics was introduced in 2+1-dimensional world in this paper. What is more, the general expressions of the exact solutions for these models with the inverse cubic, quartic, quintic and sixtic power potentials in radial Dirac equation were further given by means of the Bethe ansatz method. And finally, the corresponding exact solutions in this paper were further discussed.
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…
$1+1$-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 present the first experimental microwave realization of the one-dimensional Dirac oscillator, a paradigm in exactly solvable relativistic systems. The experiment relies on a relation of the Dirac oscillator to a corresponding tight-binding system. This tight-binding system is implemented as a microwave system by a chain of coupled dielectric disks, where the coupling is evanescent and can be adjusted appropriately. The resonances of the finite microwave system yields the spectrum of the one-dimensional Dirac oscillator with and without mass term. The flexibility of the experimental set-up a…
In this contribution, we study the Dirac oscillator under the influence of noninertial effects of a rotating frame in the cosmic string spacetime. We show that both noninertial effects and the topology of the cosmic string spacetime restrict the physical region of the spacetime where the quantum particle can be placed, and discuss two different cases of bound states solutions of the Dirac equation by analysing the behaviour of the Dirac oscillator frequency.
The dynamics of wavepackets in a relativistic Dirac oscillator (DO) is considered. A comparison to nonrelativistic spin-orbit pendulum effect is discussed. Particular relativistic effects, like Zitterbewegung in spin motion, are found in Dirac representation. This trembling motion disappears in Foldy-Wouthuysen representation. A substantial difference between the dynamics of wavepackets corresponding to circular and linear orbits of a particle is obtained and discussed.
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…
We have presented an elegant high energy quantum problem, namely, the full Dirac oscillator under axial magnetic field with its full solution. We have found the energy spectrum which is rich and at the same time has a novel structure. The quantized energy levels show coupling of the oscillator frequency with the Larmor frequency in the 2D surface where the electrons under consideration follow a 2D oscillator. The axis in which magnetic field is pointed, the electrons follow a 1D oscillator. There is also coupling between spin and orbital motion and also a coupling between a resultant effect of…
We determine the energy spectrum and the corresponding eigenfunctions of a 2D Dirac oscillator in the presence of Aharonov-Bohm (AB) effect . It is shown that the energy spectrum depends on the spin of particle and the AB magnetic flux parameter. Finally, when the irregular solution occurs it is shown that the energy takes particular values. The nonrelativistic limit is also considered.
The system of two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem already addressed by many authors that we present here in some fresh points of view and carry on smoothly a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac's canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As result the quantum system is simply modeled by the original quantum Hamiltonian.
The Dirac hydrogen atom with spin symmetry is shown has a SO(4) symmetry. The generators are derived, and the corresponding Casimir operator leads to the energy spectrum naturally. This type hydrogen atom is connected to a four-dimensional Dirac system with equal scalar and vector harmonic oscillator potential, by the Kustaanheimo-Stiefel transformation with a constraint.
We show that 2+1 dimensional Dirac oscillators in an external magnetic field is mapped onto the same with reduced angular frequency in absence of magnetic field. This can be used to study the atomic transitions in a radiation field. Relativistic Landau levels are constructed explicitly. Several interesting features of this system are discussed.
In this work, we study the Aharonov-Casher effect in the $(2+1)$-dimensional Dirac oscillator coupled to an external electromagnetic field. We set up our system in two different scenarios: in the Minkowski spacetime and the cosmic string spacetime. In both cases, we solve exactly the Dirac oscillator and we determine the energy spectrum and the eigenfunctions for the bound states. We verify that in the Minkowski spacetime, the Dirac oscillator spectrum depends linearly on the strength of the magnetic field $B$, and on the Aharonov-Casher phase. In addition, we explicitly obtain the corrections…
We investigate the energy spectrum and the corresponding eigenfunctions of a 2D Dirac oscillator confined by an antidot potential in the presence of a magnetic field and Aharonov-Bohm flux field. Analytical solutions are obtained and compared with the results of the Schrödinger equation found in the literature. Further, the dependence of the spectrum on the magnetic quantum number and on the repulsive potential is discussed.
We introduce a Dirac equation which reproduces the usual radial sextic oscillator potential in the non-relativistic limit. We determine its energy spectrum in the presence of the magnetic field. It is shown that the equation is solved in the context of quasi-exactly-solvable problems. The equation possesses hidden $sl_{2}$-algebra and the destroyed symmetry of the equation can be recovered for a specific values of the magnetic field which leads to exact determination of the eigenvalues.
We consider a two-dimensional Dirac oscillator in the presence of magnetic field in noncommutative phase space in the framework of relativistic quantum mechanics with minimal length. The problem in question is identified with a Poschl-Teller potential. The eigenvalues are found and the corresponding wave functions are calculated in terms of hypergeometric functions.
An elegant method to circumvent quantum measurement backaction is the use of quantum mechanics free subsystems (QMFS), with one approach involving the use of two oscillators with effective masses of opposite signs. Since negative energies, and hence masses, are a characteristic of relativistic systems a natural question is to what extent QMFS can be realized in this context. Using the example of a one-dimensional Dirac oscillator we investigate conditions under which this can be achieved, and identify Zitterbewegung or virtual pair creation as the physical mechanism that fundamentally limits t…
Scattering discussion due to Double Dirac Equation in Quaternionic version of relativistic quantum mechanics has been studied in this paper in details. In such a quantum mechanics Dirac equation in presence vector and scalar potential has been considered. Then a Quaternionic double Dirac delta potential comes to our considered system which causes to scatter the particles. Scattering states of the particles have been derived as well as reflected and transmission coefficients are calculated.
We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncom- mutative space where the noncommutativity is induced by a shift of the dynamical variables with generators of SL(2;R) in a unitary irreducible representation. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, finding no constraints between coordinates and…
In this work, we study of the (2+1)-dimensional Dirac oscillator in the presence of a homogeneous magnetic field in an Aharonov-Bohm-Coulomb system. To solve our system, we apply the $left$-$handed$ and $right$-$handed$ projection operators in the Dirac oscillator to obtain a biconfluent Heun equation. Next, we explicitly determine the energy spectrum for the bound states of the system and their exact dependence on the cyclotron frequency $ω_c$ and on the parameters $Z$ and $Φ_{AB}$ that characterize the Aharonov-Bohm-Coulomb system. As a result, we observe that by adjusting the frequency of t…
In this manuscript, we investigate the influence of the Aharonov-Casher effect on the generalized Dirac oscillator containing the Coulomb-type potential function related to a relativistic neutral particle having a permanent magnetic dipole moment interacting with the external electromagnetic fields in (1+2)-dimensional cosmic string space-time. The eigenfunctions and energy eigenvalues of such a Dirac oscillator are derived by using the Nikifornov-Uvarov method. We indicate that implementing the scenario gives the relativistic modified exact analytical solutions. In this way, we can see that t…
In this paper, we study the influence of the Aharonov-Casher effect [Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984).] on the Dirac oscillator in three different scenarios of general relativity: the Minkowski spacetime, the cosmic string spacetime and the cosmic dislocation spacetime. In this way, we solve the Dirac equation and obtain the energy levels for bound states and the Dirac spinors for positive-energy solutions. We show that the relativistic energy levels depend on the Aharonov-Casher geometric phase. We also discuss the influence of curvature and torsion on the relativist…
We study the dynamics of the 2+1 Dirac oscillator exactly and find spin oscillations due to a {\it Zitterbewegung} of purely relativistic origin. We find an exact mapping of this quantum-relativistic system onto a Jaynes-Cummings model, describing the interaction of a two-level atom with a quantized single-mode field. This equivalence allows us to map a series of quantum optical phenomena onto the relativistic oscillator, and viceversa. We make a realistic experimental proposal, at reach with current technology, for studying the equivalence of both models using a single trapped ion.
We investigate the feasibility of performing quantum non-demolition (QND) measurements in relativistic quantum systems, using the one-dimensional Dirac oscillator as a specific example. We derive general expressions for its QND observables and find that they are intricate combinations of the position, momentum, and spin operators, which makes them challenging to realize experimentally in general. However, the situation is considerably simplified in both the weakly and strongly relativistic limits, where their experimental realization will be possible.
We develop the equivalence between the two-dimensional Dirac oscillator and the anti-Jaynes-Cummings model within a q-deformed scenario. We solve the Hamiltonian spectrum and the time evolution for number and coherent initial states, focusing on the appearance of the Zitterbewegung effect. We show the lack of preservation of the q-deformed versions of the total angular momentum that reproduces a collapse-revival structure. We provide suitable relations for the non relativistic limit.
$C_λ$-extended oscillator algebras are realized as generalized deformed oscillator algebras. For $λ= 3$, the spectrum of the corresponding bosonic oscillator Hamiltonian is shown to strongly depend on the algebra parameters. A connection with cyclic shape invariant potentials is noted. A bosonization of PSSQM of order two is obtained.
In this paper, we show, by using the approach of effective mass, that the model of a two-dimensional Dirac oscillator can be used to describe the thermal properties of graphene under an uniform magnetic field. All thermal quantities of graphene, such as the free energy, the mean energy, the entropy and the specific heat, have been found by using an approach based on the zeta function.
