Analytical and numerical analysis of a rotational invariant D=2 harmonic oscillator in the light of different noncommutative phase-space configurations
Everton M. C. Abreu, Mateus V. Marcial, Albert C. R. Mendes, Wilson Oliveira
In this work we have investigated some properties of classical phase-space with symplectic structures consistent, at the classical level, with two noncommutative (NC) algebras: the Doplicher-Fredenhagen-Roberts algebraic relations and the NC approach which uses an extended Hilbert space with rotational symmetry. This extended Hilbert space includes the operators $θ^{ij}$ and their conjugate momentum $π_{ij}$ operators. In this scenario, the equations of motion for all extended phase-space coordinates with their corresponding solutions were determined and a rotational invariant NC Newton's second law was written. As an application, we treated a NC harmonic oscillator constructed in this extended Hilbert space. We have showed precisely that its solution is still periodic if and only if the ratio between the frequencies of oscillation is a rational number. We investigated, analytically and numerically, the solutions of this NC oscillator in a two-dimensional phase-space. The result led us to conclude that noncommutativity induces a stable perturbation into the commutative standard oscillator and that the rotational symmetry is not broken. Besides, we have demonstrated through the equations of motion that a zero momentum $π_{ij}$ originated a constant NC parameter, namely, $θ^{ij}=const.$, which changes the original variable characteristic of $θ^{ij}$ and reduces the phase-space of the system. This result shows that the momentum $π_{ij}$ is relevant and cannot be neglected when we have that $θ^{ij}$ is a coordinate of the system.
We study the entanglement entropy of harmonic oscillators in noncommutative phase space. We propose a new definition of quantum Rényi entropy based on Wigner functions in noncommutative phase space. Using the Rényi entropy, we calculate the entanglement entropy of the ground state of the 2D isotropic harmonic oscillators. We find that for some values of the noncommutative parameters, the harmonic oscillators can be entangled in noncommutative phase space. This is a new entanglement-like effect caused by the noncommutativity of the phase space.
Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the correspond…
Noncommutative oscillators are first-quantized through an abelian Drinfel'd twist deformation of a Hopf algebra and its action on a module. Several important and subtle issues making possible the quantization are solved. The spectrum of the single-particle Hamiltonians is computed. The multi-particle Hamiltonians are fixed, unambiguously, by the Hopf algebra coproduct. The symmetry under particle exchange is guaranteed. In d=2 dimensions the rotational invariance is preserved, while in d=3 the so(3) rotational invariance is broken down to an so(2) invariance.
When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.
We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras $sl(2,\mathbb{R})$ or $su(2)$ according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some…
The two dimensional set of canonical relations giving rise to minimal uncertainties previously constructed from a q-deformed oscillator algebra is further investigated. We provide a representation for this algebra in terms of a flat noncommutative space and employ it to study the eigenvalue spectrum for the harmonic oscillator on this space. The perturbative expression for the eigenenergy indicates that the model might possess an exceptional point at which the spectrum becomes complex and its PT-symmetry is spontaneously broken.
When a non-hermitian hamiltonian has a certain symmetry, such as the PT pseudo-hermiticity, it is still possible that the hamiltonian has a real spectrum. In this note, by adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a PT pseudo-hermitian hamiltonian and obtain its real spectrum. In addition, we find that the reality of the energy spectrum remains when the model is extended to the canonical noncommutative space.
A generalized harmonic oscillator on noncommutative spaces is considered. Dynamical symmetries and physical equivalence of noncommutative systems with the same energy spectrum are investigated and discussed. General solutions of three-dimensional noncommutative harmonic oscillator are found and classified according to dynamical symmetries. We have found conditions under which three-dimensional noncommutative harmonic oscillator can be represented by ordinary, isotropic harmonic oscillator in effective magnetic field.
The bi-partite Gaussian state, corresponding to an anisotropic harmonic oscillator in a noncommutative-space, is investigated with the help of the Simon's separability condition (generalized Peres-Horodecki criterion). It turns out that, in order to exhibit the entanglement between the noncommutative co-ordinates, the parameters (mass and frequency) have to satisfy an unique constraint equation. Exact solutions for the system are obtained after diagonalizing the model, keeping the intrinsic symplectic structure intact. It is shown that, the identification of the entangled degrees of freedom is…
We construct the photon added coherent states of a noncommutative harmonic oscillator associated to a $q$-deformed oscillator algebra. Various nonclassical properties of the corresponding system are explored, first, by studying two different types of higher order quadrature squeezing, namely the Hillery-type and the Hong--Mandel-type and, second, by testing the sub-Poissonian nature of photon statistics in higher order with the help of the correlation function and the Mandel parameter. By comparing our results with those of the usual harmonic oscillator, we notice that the quadratures and phot…
Isotropic oscillator on a plane is discussed where both the coordinate and momentum space are considered to be noncommutative. We also discuss the symmetry properties of the oscillator for three separate cases when both the noncommutative parameters Θand \barΘ satisfy specific relations. We compare the Landau problem with the isotropic oscillator on noncommutative space and obtain a relation between the two noncommutative parameters with the magnetic field of the Landau problem.
We consider the noncommutative algebra which is rotationally invariant. The hydrogen atom is studied in a rotationally invariant noncommutative space. We find the corrections to the energy levels of the hydrogen atom up to the second order in the parameter of noncommutativity. The upper bound of the parameter of noncommutativity is estimated on the basis of the experimental results for 1s-2s transition frequency.
Noncommutative algebra which is rotationally invariant, time reversal invariant and equivalent to noncommutative algebra of canonical type is considered. Perihelion shift of orbit of a particle in Coulomb potential in the rotationally-invariant noncommutative phase space is found up to the second order in the parameters of noncommutativity. Applying the result to the case of Mercury planet and using observable results for precession of its orbit we find upper bounds on the parameters of noncommutativity in the rotationally-invariant noncommutative phase space. The obtained upper bound for the …
Noncommutative space which is rotationally invariant is considered. The hydrogen atom is studied in this space. We exactly find the leading term in the asymptotic expansion of the corrections to the $ns$ energy levels over the small parameter of noncommutativity.
We study the Dirac and the klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics of a particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.
We derive the partition function of the one-body and two-body systems of classical noncommutative harmonic oscillator in two dimensions. Then, we employ the path integral approach to the quantum noncommutative harmonic oscillator and derive the partition function of the both systems at finite temperature.
We solve explicitly the two-dimensional harmonic oscillator and the harmonic oscillator in a background magnetic field in noncommutative phase-space without making use of any type of representation. A key observation that we make is that for a specific choice of the noncommutative parameters, the time reversal symmetry of the systems get restored since the energy spectrum becomes degenerate. This is in contrast to the noncommutative configuration space where the time reversal symmetry of the harmonic oscillator is always broken.
The noncommutative harmonic oscillator in arbitrary dimension is examined. It is shown that the $\star$-genvalue problem can be decomposed into separate harmonic oscillator equations for each dimension. The noncommutative plane is investigated in greater detail. The constraints for rotationally symmetric solutions and the corresponding two-dimensional harmonic oscillator are solved. The angular momentum operator is derived and its $\star$-genvalue problem is shown to be equivalent to the usual eigenvalue problem. The $\star$-genvalues for the angular momentum are found to depend on the energy …
Classical and quantum anharmonic noncommutative oscillators with quartic self-interacting potential are considered and the effect of self-interaction term on the free energy and partition function of both models is calculated to first order in coupling constant.
We consider a system of two particles in noncommutative space which is rotationally invariant. It is shown that the coordinates of the center-of-mass position and the coordinates of relative motion satisfy noncommutative algebra with corresponding effective tensors of noncommutativity. The hydrogen atom is studied as a two-particle system. We find the corrections to the energy levels of the hydrogen atom up to the second order over the parameter of noncommutativity.
We use the path integral approach to a two-dimensional noncommutative harmonic oscillator to derive the partition function of the system at finite temperature. It is shown that the result based on the Lagrangian formulation of the problem, coincides with the Hamiltonian derivation of the partition function.
Harmonic oscillator in noncommutative two dimensional lattice are investigated. Using the properties of non-differential calculus and its applications to quantum mechanics, we provide the eigenvalues and eigenfunctions of the corresponding Hamiltonian. First we consider the case of ordinary quantum mechanics, and we point out the thermodynamic properties of the model. Then we consider the same question when both coordinates and momentums are noncommutative.
Quantized space described by time reversal invariant and rotationally invariant noncommutative algebra of canonical type is studied. A particle in uniform field is considered. We find exactly the energy of a particle in uniform field in the quantized space and its wavefunctions. It is shown that the motion of the particle in the field direction in time reversal invariant and rotationally invariant noncommutative space is the same as in the ordinary space (space with the ordinary commutation relations for operators of coordinates and operators of momenta). Noncommutativity of coordinates has in…
In this paper we study the (2 + 1)-dimensional Dirac oscillator in the noncommutative phase space and the energy eigenvalues and the corresponding wave functions of the system are obtained through the sl(2) algebraization. It is shown that the results are in good agreement with those obtained previously via a different method.
In this study, we investigate the Pauli oscillator in a noncommutative space. In other words, we derive wave function and energy spectrum of a spin half non-relativistic charged particle that is moving under a constant magnetic field with an oscillator potential in noncommutative space. We obtain critical values of the deformation parameter and the magnetic field, which counteract the normal and anomalous Zeeman effects. Moreover, we find that the deformation parameter has to be smaller than $2.57\times 10^{-26}m^2$. Then, we derive the Helmholtz free energy, internal energy, specific heat, an…
We study the Wigner Function in non-commutative quantum mechanics. By solving the time independent Schrödinger equation both on a non-commutative (NC) space and a non-commutative phase space, we obtain the Wigner Function for the harmonic oscillator on NC space and NC phase space respectively.
We find transformation matrices allowing to express non-commutative three dimensional harmonic oscillator in terms of an isotropic commutative oscillator, following ``philosophy of simplicity'' approach. Non-commutative parameters have physical interpretation in terms of an external magnetic field. Furthermore, we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation, whose transformation matrices are far more complicated. We indicate a way to obtain the more complex solutions from the simple ones.
Energy spectrum of isotropic oscillator as a function of noncommutativity parameter theta is studied. It is shown that for a dense set of values of theta the spectrum is degenerated and the algebra responsible for degeneracy can be always chosen to be sU(2). The generators of the algebra are constructed explicitely.
We study influence of noncommutativity of coordinates and noncommutativity of momenta on the motion of a particle (macroscopic body) in uniform and non-uniform gravitational fields in noncommutative phase space of canonical type with preserved rotational symmetry. It is shown that because of noncommutativity the motion of a particle in gravitational filed is determined by its mass. The trajectory of motion of a particle in uniform gravitational field corresponds to the trajectory of harmonic oscillator with frequency determined by the value of parameter of momentum noncommutativity and mass of…
We show that 2D noncommutative harmonic oscillator has an isotropic representation in terms of commutative coordinates. The noncommutativity in the new mode, induces energy level splitting, and is equivalent to an external magnetic field effect. The equivalence of the spectra of the isotropic and anisotropic representation is traced back to the existence of SU(2) invariance of the noncommutative model.
We consider a problem of the consistent deformation of physical system introducing a new features, but preserving its fundamental properties. In particular, we study how to implement the noncommutativity of space-time without violation of the rotational symmetry in quantum mechanics or the Lorentz symmetry in f{i}eld theory. Since the canonical (Moyal) noncommutativity breaks the above symmetries one should work with more general case of coordinate-dependent noncommutative spaces, when the commutator between coordinates is a function of these coordinates. F{i}rst we describe in general lines h…
From a study of an oscillator in a $4D$ NC spacetime, we establish the Hamilton equations of motion. The formers are solved to give the oscillator position and momentum coordinates. These coordinates are used to build a metric similar to that describing a cosmic string. On this basis, Dirac and Klein-Gordon oscillators are investigated. Their spectrum and dynamics are analysed giving rise to novel interesting properties.
Using the momentum space representation, we study the (2 +1)-dimensional Duffin-Kemmer-Petiau oscillator for spin 0 particle under a magnetic field in the presence of a minimal length in the noncommutative space. The explicit form of energy eigenvalues are found, the wave functions and the corresponding probability density are reported in terms of the Jacobi polynomials. Additionally, we also discuss the special cases and depict the corresponding numerical results.
We consider the influence of a noncommutative space on the Klein-Gordon and the Dirac oscillators. The nonrelativistic limit is taken and the $θ$-modified Hamiltonians are determined. The corrections of these Hamiltonians on the energy levels are evaluated in first-order perturbation theory. It is observed a total lifting of the degeneracy to the considered levels. Such effects are similar to the Zeeman splitting in a commutative space.
A recent conjecture of Morita predicts a lower bound in temperature $T$ of a chaotic system, $T\geq (\hbar/2π)Λ$, $Λ$ being the Lyapunov exponent, which was demonstrated for a one dimensional inverse harmonic oscillator. In the present work we discuss the robustness of this demonstration in an extended version of the above model, where the inverse harmonic oscillator lives a in two dimensional noncommutative space. We show that, without noncommutativity, Morita's conjecture survives in an essentially unchanged way in two dimensions. However, if noncommutativity is switched on, the noncommutati…
We show that (2+1) dimensional noncommutative Dirac oscillator in an external magnetic field is mapped onto the same but with reduced angular frequency in absence of magnetic field. We construct the relativistic Landau levels by solving corresponding Dirac equation in (2+1) dimensional noncommutative phase space. We observe that lowest Landau levels are exactly same as in commutative space and independent of non-commutative parameter. All the Landau levels become independent of noncommutative parameter for a critical value of the magnetic field. Several other interesting features along with th…
We consider the quantum dynamics of a harmonic oscillator in noncommutative space under the influence of linearized gravitational waves (GW) in the long wave-length and low-velocity limit. Following the prescription in \cite{ncgw1} we quantize the system. The Hamiltonian of the system is solved by using standard algebraic iterative methods. The solution shows signatures of the coordinate noncommutativity via alterations in the oscillation frequency of the harmonic oscillator system from its commutative counterpart. Moreover, it is found that the response of the harmonic oscillator to periodic …
We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be constructed explicitly, we derive the commutation relations for the dynamical variables in noncommutative space-time. We compute minimal areas resulting from these relations, i.e. finitely extended regions for which it is impossible to resolve any substructure in form of measurable knowledge. The size of the regions we find is determined by the noncommutativ…
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…
We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.
We establish existence and stabilty results for solitons in noncommutative scalar field theories in even space dimension $2d$. In particular, for any finite rank spectral projection $P$ of the number operator ${\mathcal N}$ of the $d$-dimensional harmonic oscillator and sufficiently large noncommutativity parameter $θ$ we prove the existence of a rotationally invariant soliton which depends smoothly on $θ$ and converges to a multiple of $P$ as $θ\to\infty$. In the two-dimensional case we prove that these solitons are stable at large $θ$, if $P=P_N$, where $P_N$ projects onto the space spanned …
In this paper we have obtained the exact eigenstates of a two dimensional damped harmonic oscillator in time dependent noncommutative space. It has been observed that for some specific choices of the damping factor and the time dependent frequency of the oscillator, there exists interesting solutions of the time dependent noncommutative parameters following from the solutions of the Ermakov-Pinney equation. Further, these solutions enable us to get exact analytic forms for the phase which relates the eigenstates of the Hamiltonian with the eigenstates of the Lewis invariant. We then obtain exp…
We conjecture an embedding operator which assigns, to any 2n+1 hermitian matrices, a 2n-dimensional hypersurface in flat (2n + 1)-dimensional Euclidean space. This corresponds to precisely defining a fuzzy D(2n)-brane corresponding to N D0-branes. Points on the emergent hypersurface correspond to zero eigenstates of the embedding operator, which have an interpretation as coherent states underlying the emergent noncommutative geometry. Using this correspondence, all physical properties of the emergent D(2n)-brane can be computed. We apply our conjecture to noncommutative flat and spherical spac…
We investigate effects of noncommutativity of phase space generated by two scalar fields conformally coupled to curvature in FRW cosmology. We restrict deformation of minisuperspace to noncommutativity between scalar fields and between their canonical conjugate momenta. The investigation is carried out by means of comparative analysis of mathematical properties of time evolution of variables in classical model and wave function of universe in quantum level. We find that impose of noncommutativity causes more ability in tuning time solutions of scalar fields and hence, has important implication…
Quantum entanglement, induced by spatial noncommutativity, is investigated for an anisotropic harmonic oscillator. Exact solutions for the system are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp's shift to the noncommuting degrees of freedom. Employing Simon's separability criterion, we find that the states of the system are entangled provided a unique function of the (mass and frequency) parameters obeys an inequality. Entanglement of Formation for this system is also computed and its relation to the degree of anisotropy is discusse…
Noncommutative phase space of an arbitrary dimension is considered. The both of operators coordinates and momenta in noncommutative phase space may be noncommutative. In this paper, we introduce momentum-momentum noncommutativity in addition to coordinate-coordinate noncommutativity. We find an exact form for the linear coordinate transformation which relates a noncommutative phase space to the corresponding ordinary one. As an example, the Hamiltonian of a three-dimensional harmonic oscillator is examined.
In this paper we have obtained the exact eigenstates of a two dimensional damped harmonic oscillator in the presence of an external magnetic field varying with respect to time in time dependent noncommutative space. It has been observed that for some specific choices of the damping factor, the time dependent frequency of the oscillator and the time dependent external magnetic field, there exists interesting solutions of the time dependent noncommutative parameters following from the solutions of the Ermakov-Pinney equation. Further, these solutions enable us to get exact analytic forms for the…
We study non Hermitian quantum systems in noncommutative space as well as a \cal{PT}-symmetric deformation of this space. Specifically, a \mathcal{PT}-symmetric harmonic oscillator together with iC(x_1+x_2) interaction is discussed in this space and solutions are obtained. It is shown that in the \cal{PT} deformed noncommutative space the Hamiltonian may or may not possess real eigenvalues depending on the choice of the noncommutative parameters. However, it is shown that in standard noncommutative space, the iC(x_1+x_2) interaction generates only real eigenvalues despite the fact that the Ham…
Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, an…
We have calculated the hydrogen atom spectrum on curved noncommutative space defined by the commutation relations $\left[ \hat {x}^{i},\hat{x}^{j}\right] =iθ\hatω^{ij}\left( \hat {x}\right) $, where $θ$ is the parameter of noncommutativity. The external antisymmetric field which determines the noncommutativity is chosen as $ω^{ij}(x) =\varepsilon^{ijk}{x}_{k}f\left( {x_i}x^{i}\right) $. In this case the rotational symmetry of the system is conserved, preserving the degeneracy of the energy spectrum. The contribution of the noncommutativity appears as a correction to the fine structure. The cor…
A system of $N$ non-canonical dynamically free 3D harmonic oscillators is studied. The position and the momentum operators (PM-operators) of the system do not satisfy the canonical commutation relations (CCRs). Instead they obey the weaker postulates for the oscillator to be a Wigner quantum system. In particular the PM-operators fulfil the main postulate, which is due to Wigner: they satisfy the equations of motion (the Hamiltonian's equations) and the Heisenberg equations. One of the relevant features is that the coordinate (the momentum) operators do not commute, but instead their squares d…
We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives obeying the nonlinear Dolan-Grady relations. This restricts the structure function of the deformed oscillator algebra to a quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R) algebras are investigated in detail. Reducing the Hamiltonian to 1D finite-difference quasi-exactly solvable operators, we demonstrate partial algebraization…
We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang-Mills-type effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative $ϕ^4$-theory on Moyal space ensures renormalisability. The expression of a possi…
In this paper we present a harmonic oscillator realization of the most degenerate discrete series representations of the SU(2,1) group and the deformation quantization of the coset space $D=SU(2,1)/U(2)$ with the method of coherent state quantization.
We provide an explicit construction of entangled states in a noncommutative space with nonclassical states, particularly with the squeezed states. Noncommutative systems are found to be more entangled than the usual quantum mechanical systems. The noncommutative parameter provides an additional degree of freedom in the construction by which one can raise the entanglement of the noncommutative systems to fairly higher values beyond the usual systems. Despite of having classical-like behaviour, coherent states in noncommutative space produce little amount of entanglement and therefore they posse…
We propose a phase-space path integral formulation of noncommutative quantum mechanics, and prove its equivalence to the operatorial formulation. As an illustration, the partition function of a noncommutative two-dimensional harmonic oscillator is calculated.
We carry out an investigation into the possibility of developing a Bohmian interpretation based on the continuous motion of point particles for noncommutative quantum mechanics. The conditions for such an interpretation to be consistent are determined, and the implications of its adoption for noncommutativity are discussed. A Bohmian analysis of the noncommutative harmonic oscillator is carried out in detail. By studying the particle motion in the oscillator orbits, we show that small-scale physics can have influence at large scales, something similar to the IR-UV mixing.
In this paper, we investigate the bound-state solutions of the noncommutative Dirac oscillator with a permanent electric dipole moment in the presence of an electromagnetic field in (2+1)-dimensions. We consider a radial magnetic field generated by anti-Helmholtz coils, and the uniform electric field of the Stark effect. Next, we determine the bound-state solutions of the system, given by the two-component Dirac spinor and the relativistic energy spectrum. We note that this spinor is written in terms of the generalized Laguerre polynomials, and this spectrum is a linear function on the potenti…
