A mapping is obtained relating analytical radial Coulomb systems in any dimension greater than one to analytical radial oscillators in any dimension. This mapping, involving supersymmetry-based quantum-defect theory, is possible for dimensions unavailable to conventional mappings. Among the special cases is an injection from bound states of the three-dimensional radial Coulomb system into a three-dimensional radial isotropic oscillator where one of the two systems has an analytical quantum defect. The issue of mapping the continuum states is briefly considered.
It is shown that the Hurwitz transformation connects the eight-dimensional repulsive oscillator problem with the five-dimensional Coulomb problem for continuous spectrum. The hyperspherical and parabolic bases for this system are calculated. The quantum mechanical scattering problem of charged particles in the 5-dimensional Coulomb field is solved.
The Levi-Civita transformation is applied in the two-dimensional (2D) Dirac and Klein-Gordon (KG) equations with equal external scalar and vector potentials. The Coulomb and harmonic oscillator problems are connected via the Levi-Civita transformation. These connections lead to an approach to solve the Coulomb problems using the results of the harmonic oscillator potential in the above-mentioned relativistic systems.
Schrödinger equation for two center Coulomb plus harmonic oscillator potential is solved by the method of ethalon equation at large intercenter separations. Asymptotical expansions for energy term and wave function are obtained in the analytical form.
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 analyze the conclusions of the influence of a Coulomb-type potential on the Klein-Gordon oscillator. We show that the truncation method proposed by the authors do not yield all the eigenvalues of the radial equation but just one of them for a particular value of a model parameter. Besides, the existence of allowed oscillator frequencies that depend on the quantum numbers is an artifact of the truncation method.
Quantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The $D$ dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb-Sturmian basis, and ca…
We show that and how the Coulomb potential can be regularized and solved exactly at the imaginary couplings. The new spectrum of energies is real and bounded as expected, but its explicit form proves totally different from the usual real-coupling case.
In this paper we construct generalizations to spheres of the well known Levi-Civita, Kustaanheimo-Steifel and Hurwitz regularizing transformations in Euclidean spaces of dimensions 2, 3 and 5. The corresponding classical and quantum mechanical analogues of the Kepler-Coulomb problem on these spheres are discussed.
Utilizing an appropriate ansatz to the wave function, we reproduce the exact bound-state solutions of the radial Schrodinger equation to various exactly solvable sextic anharmonic oscillator and confining perturbed Coulomb models in $D$-dimensions. We show that the perturbed Coulomb problem with eigenvalue $E$ can be transformed to a sextic anharmonic oscillator problem with eigenvalue E. We also check the explicit relevance of these two related problems in higher-space dimensions. It is shown that exact solutions of these potentials exist when their coupling parameters with $k=D+2\ell $ appea…
We have obtained the uncertainty relations for arbitrary states of the hydrogen atom. It is shown that the minimal value of the uncertainty relation is attained for the circular Rydberg states.
A mapping is obtained relating radial screened Coulomb systems with low screening parameters to radial anharmonic oscillators in N-dimensional space. Using the formalism of supersymmetric quantum mechanics, it is shown that exact solutions of these potentials exist when the parameters satisfy certain constraints.
First a set of coherent states a la Klauder is formally constructed for the Coulomb problem in a curved space of constant curvature. Then the flat-space limit is taken to reduce the set for the radial Coulomb problem to a set of hydrogen atom coherent states corresponding to both the discrete and the continuous portions of the spectrum for a fixed \ell sector.
We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb-like potentials) and compare their spectra and the sets of eigenfunctions. We construct all self-adjoint Schrodinger operators for these theories and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving spectral problems, we follow the Krein's method of guiding functionals. We show, that there …
We analyze the calculation of bound states for a nonrelativistic spin-half neutral particle under the influence of a Coulomb-like potential induced by Lorentz symmetry breaking effects. We show that the truncation condition proposed by the authors only provides one energy eigenvalue for a particular model potential and misses all the other bound-state energies. The dependence of the cyclotron frequency on the quantum numbers is a mere artifact of the truncation condition that is by no means necessary for the existence of bound states.
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…
Three-Dimensional Eigenvalues of Harmonic Oscillator- and Coulomb-type Potentials from One-Dimensional Generalized Morse Potential: Perturbative Analyse based on Generalized Laguerre Polynomials
We consider the Dirac equation, written in polar formalism, in presence of general Coulomb-like potentials, that is potentials arising from the time component of the vector potential and depending only on the radial coordinate, in order to study the conditions of integrability, given as some specific form for the solution: we find that the angular dependence can always be integrated, while the radial dependence is reduced to finding the solution of a Riccati equation so that it is always possible at least in principle. We exhibit the known case of the Coulomb potential and one special generali…
Remote entanglement of massive oscillators via wire-mediated Coulomb interaction
Technical University of Vienna（维也纳工业大学）；Institute for Theoretical Physics and Vienna Center for Quantum Science and Technology, Technical University of Vienna（维也纳大学理论物理研究所与维也纳量子科学与技术中心）；Institute of Science and Technology Austria（奥地利科学技术研究所）
We propose a scheme to modulate the entanglement between two oscillators separated in space via the squeezing cavity field generated by the optical parametric amplifier instead of injecting the squeezing field directly with the assistance of Coulomb interaction. We show that the Coulomb interaction between the oscillators is the essential reason for the existence of entanglement. Due to the gain of the optical parametric amplifier and the phase of the pump driving the optical parametric amplifier can simultaneously modulate the squeezing cavity field, the radiation pressure interaction between…
A useful and universal formula for the expectation value of the radial operator in the presence of the Aharonov-Bohm flux and the Coulomb Field is established. We find that the expectation value $< r^λ>$ $(-\infty \leq λ\leq \infty)$ is greatly affected due to the non-local effect of the magnetic flux although the Aharonov-Bohm flux does not have any dynamical significance in classical mechanics. In particular, the quantum fluctuation increases in the presence of the magnetic flux due to the Aharonov-Bohm effect. In addition, the Virial theory in quantum mechanics is also constructed for the s…
The explicit semiclassical treatment of logarithmic perturbation theory for the bound-state problem within the framework of the radial Klein-Gordon equation with attractive real-analytic screened Coulomb potentials, contained time-component of a Lorentz four-vector and a Lorentz-scalar term, is developed. Based upon $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As…
By introducing the scalar potential as modification in the mass term of the Klein-Gordon equation, the influence of a Coulomb-type potential on the Klein-Gordon oscillator is investigated. Relativistic bound states solutions are achieved to both attractive and repulsive Coulomb-type potentials and the arising of a quantum effect characterized by the dependence of angular frequency of the Klein-Gordon oscillator on the quantum numbers of the system is shown.
In this paper we establish a relation between Coulomb and oscillator systems on $n$-dimensional spheres and hyperboloids for $n\geq 2$. We show that, as in Euclidean space, the quasiradial equation for the $n+1$ dimensional Coulomb problem coincides with the $2n$-dimensional quasiradial oscillator equation on spheres and hyperboloids. Using the solution of the Schrödinger equation for the oscillator system, we construct the energy spectrum and wave functions for the Coulomb problem.
We show that the oscillators on a sphere and pseudosphere are related, by the so-called Bohlin transformation, with the Coulomb systems on the pseudosphere: the even states of an oscillator yields the conventional Coulomb system on pseudosphere, while the odd states yield the Coulomb system on pseudosphere in the presence of magnetic flux tube generating half spin. In the higher dimensions the oscillator and Coulomb(-like) systems are connected in the similar way. In particular, applying the Kustaanheimo-Stiefel transformation to the oscillators on sphere and pseudosphere, we obtained the preu…
We consider a background of the violation of the Lorentz symmetry determined by the tensor $\left( K_{F}\right)_{μναβ}$ which governs the Lorentz symmetry violation out of the Standard Model Extension, where this background gives rise to a Coulomb-type potential, and then, we analyse its effects on a relativistic quantum oscillator. Furthermore, we analyse the behaviour of the relativistic quantum oscillator under the influence of a linear scalar potential and this background of the Lorentz symmetry violation. We show in both cases that analytical solutions to the Klein-Gordon equation can be …
Bound state solutions of the Schrödinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods
University of Batna I（巴特纳第一大学）；University of Biskra（比斯克拉大学）；University of Khenchela
Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence…
In this paper, we investigate the quantum mechanical dynamics of the massive and relativistic Feshbach-Villars oscillator in cosmic dislocation space-time induced by a coulomb-type potential. The first-order Feshbach-Villars version of the Klein-Gordon equation is used to find movement equations. Wave functions and associated energy have been calculated (both in the free case and in the interaction case). We analyse the impact of dislocation topology on this interaction. As a result, the effect of the dislocation on the quantum system under study is examined.
It is shown, that oscillators on the sphere and the pseudosphere are related, by the so-called Bohlin transformation, with the Coulomb systems on the pseudosphere. The even states of an oscillator yield the conventional Coulomb system on the pseudosphere, while the odd states yield the Coulomb system on the pseudosphere in the presence of magnetic flux tube generating spin 1/2. A similar relation is established for the oscillator on the (pseudo)sphere specified by the presence of constant uniform magnetic field $B_0$ and the Coulomb-like system on pseudosphere specified by the presence of the …
We define the oscillator and Coulomb systems on four-dimensional spaces with U(2)-invariant Kahler metric and perform their Hamiltonian reduction to the three-dimensional oscillator and Coulomb systems specified by the presence of Dirac monopoles. We find the Kahler spaces with conic singularity, where the oscillator and Coulomb systems on three-dimensional sphere and two-sheet hyperboloid are originated. Then we construct the superintegrable oscillator system on three-dimensional sphere and hyperboloid, coupled to monopole, and find their four-dimensional origins. In the latter case the metri…
We solve a generalized Klein-Gordon oscillator (KGO) in the presence of a uniform magnetic field including quantum flux under the effects of a scalar and vector potentials of Coulomb-types in the static cosmic string space-time. We obtain the energy eigenvalues and corresponding eigenfunctions, and analyze a relativistic analogue of the Aharonov-Bohm effect for bound states
We define Sturmian basis functions for the harmonic oscillator and investigate whether recent insights into Sturmians for Coulomb-like potentials can be extended to this important potential. We also treat many body problems such as coupling to a bath of harmonic oscillators. Comments on coupled oscillators and time-dependent potentials are also made. It is argued that the Sturmian method amounts to a non-perturbative calculation of the energy levels, but the limitations of the method is also pointed out, and the cause of this limitation is found to be related to the divergence of the potential…
The relativistic quantum dynamics of an electrically charged particle subject to the Klein-Gordon oscillator and the Coulomb potential is investigated. By searching for relativistic bound states, a particular quantum effect can be observed: a dependence of the angular frequency of the Klein-Gordon oscillator on the quantum numbers of the system. The meaning of this behaviour of the angular frequency is that only some specific values of the angular frequency of the Klein-Gordon oscillator are permitted in order to obtain bound state solutions. As an example, we obtain both the angular frequency…
It is shown that each one of the Lie algebras su(1,1) and su(2) determine the spectrum of the radial oscillator. States that share the same orbital angular momentum are used to construct the representation spaces of the non-compact Lie group SU(1,1). In addition, three different forms of obtaining the representation spaces of the compact Lie group SU(2) are introduced, they are based on the accidental degeneracies associated with the spherical symmetry of the system as well as on the selection rules that govern the transitions between different energy levels. In all cases the corresponding gen…
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…
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…
The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrödinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Ke…
The relation between the R- and P-matrix approaches and the harmonic oscillator representation of the quantum scattering theory (J-matrix method) is discussed. We construct a discrete analogue of the P-matrix that is shown to be equivalent to the usual P-matrix in the quasiclassical limit. A definition of the natural channel radius is introduced. As a result, it is shown to be possible to use well-developed technique of R- and P-matrix theory for calculation of resonant states characteristics, scattering phase shifts, etc., in the approaches based on harmonic oscillator expansions, e.g., in nu…
In this work, we obtain bound states for a nonrelativistic spin-half neutral particle under the influence of a Coulomb-like potential induced by the Lorentz symmetry breaking effects. We present a new possible scenario of studying the Lorentz symmetry breaking effects on a nonrelativistic quantum system defined by a fixed space-like vector field parallel to the radial direction interacting with a uniform magnetic field along the z-axis. Furthermore, we also discuss the influence of a Coulomb-like potential induced by Lorentz symmetry violation effects on the two-dimensional harmonic oscillator…
In this article, in the framework of an analytical approach and with help of the generalized version of the Hurwitz transformation the five--dimensional bound system composed of the Yang monople coupled to a particle of the isospin by SU(2) and Coulomb interaction is constructed from the eight-dimensional quantum oscillator. The generalized Runge-Lentz vector and the SO(6) group of the hidden symmetry are established. It is also shown that group of hidden symmetry makes it possible to calculate the spectrum of system by a pure algebraic method.
We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb like potentials) on a plane and compare their spectra and the sets of eigenfunctions. All self-adjoint Schrodinger operators for these theories are constructed and rigorous solutions of the corresponding spectral problems are presented. The first part of the problem is solved by using a method of specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving spectral problems, we follow the Krein's method of guiding functionals.…
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.
The Hamiltonian of the radial Coulomb Klein-Gordon and second order Dirac equations are shown to possess an infinite symmetric tridiagonal matrix structure on the relativistic Coulomb Sturmian basis. This allows us to give an analytic representation for the corresponding Coulomb Green's operators in terms of continued fractions. The poles of the Green's matrix reproduce the exact relativistic hydrogen spectrum.
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…
The bound-state solutions and the su(1,1) description of the $d$-dimensional radial harmonic oscillator, the Morse and the $D$-dimensional radial Coulomb Schrödinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrödinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a $d$-dimensional radial harmonic oscillator i…
The Bertrand's theorem is extended, i.e. closed orbits still may exist for other central potentials than the power law Coulomb potential and isotropic harmonic oscillator. It is shown that for the combined potential $V(r)=W(r)+b/r^2$ ($W(r)=ar^ν$), when (and only when) $W(r)$ is the Coulomb potential or isotropic harmonic oscillator, closed orbits still exist for suitable angular momentum. The correspondence between the closeness of classical orbits and the existence of raising and lowering operators derived from the factorization of the radial Schrödinger equation is investigated.
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.
An overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces is presented. It turns out that each of these systems can be considered either as an oscillator or as a Kepler-Coulomb Hamiltonian. We show that two possible quantization prescriptions for all these curved systems arise if we impose that superintegrability is preserved after quantization, and we prove that both possibilities are gauge equivalent.
We discuss the separability of the position-dependent mass Hamiltonian in cylindrical coordinates in the framework of a radial power-law position-dependent mass. We consider two particular radial mass settings; a harmonic oscillator type, and a Coulombic type. We subject the radial harmonic oscillator type mass to a radial harmonic oscillator potential and the radial Coulombic mass to a radial Coulombic potential. Azimuthal symmetry is assumed and spectral signatures of various z-dependent interaction potentials are reported.
The (3+1)-dimensional Dirac equation with position dependent mass in 4-vector electromagnetic fields is considered. Using two over-simplified examples (the Dirac-Coulomb and Dirac-oscillator fields), we report energy-levels crossing as a spectral property or as an effect of the hidden supersymmetric quantum mechanical language and/or quasi-parity signatures. Under different settings of the related interactions' way-of-coupling into Dirac equation, it is observed that the two ultimate/effective descendents, Dirac-Coulomb and Dirac-oscillator, exhibit different conditions on the energy-levels cr…
The oscillator representation method is presented and used to calculate the energy spectra for a superposition of Coulomb and power-law potentials and for Coulomb and Yukawa potentials. The method provides an efficient way to obtain analytic results for arbitrary values of the parameters specifying the above-type potentials. The calculated energies of the ground and excited states of the quantum systems in question are found to comply well with exact results.
An impermeable barrier at $r=r_{cl}$ in the effective potential of the relativistic Schrödinger-type equation leads to exclusion of the range $0 \leq r < r_{cl}$ from the wave function domain. Based on duality of the Schrödinger-type equation and the Dirac equation, a similar exclusion should be made in the wave functions domain of the Dirac equation. As a result, we obtain new solutions to the Dirac equation in the Coulomb repulsive field. Calculations show that depending on working parameters, at distances of several fractions or units of the Compton wavelength of the fermion from $r=r_{cl}$…
Bound states of the generalized spiked harmonic oscillator potential are calculated accurately by using the generalized pseudospectral method. Energy eigenvalues, various expectation values, radial densities are obtained through a nonuniform, optimal spatial discretization of the radial Schrödinger equation efficiently. Ground and excited states corresponding to arbitrary values of $n$ and $\ell$ are reported for potential parameters covering a wide range of interaction. Both weak and strong coupling is considered. The effect of potential parameters on eigenvalues and densities are discussed. …
The eigenvalue problem of the spherically symmetric oscillator Hamiltonian is revisited in the context of canonical raising and lowering operators. The Hamiltonian is then factorized in terms of two not mutually adjoint factorizing operators which, in turn, give rise to a non-Hermitian radial Hamiltonian. The set of eigenvalues of this new Hamiltonian is exactly the same as the energy spectrum of the radial oscillator and the new square-integrable eigenfunctions are complex Darboux-deformations of the associated Laguerre polynomials.
The Shannon entropy, the desequilibrium and their generalizations (Rényi and Tsallis entropies) of the three-dimensional single-particle systems in a spherically-symmetric potential $V(r)$ can be decomposed into angular and radial parts. The radial part depends on the analytical form of the potential, but the angular part does not. In this paper we first calculate the angular entropy of any central potential by means of two analytical procedures. Then, we explicitly find the dominant term of the radial entropy for the highly energetic (i.e., Rydberg) stationary states of the oscillator-like sy…
In this paper, We analyze a spin-zero relativistic quantum oscillator in the presence of the Aharonov-Bohm (AB) magnetic flux in a space-time background produced by a point-like global monopole (PGM). Afterward, we introduce a static Coulomb-type scalar potential and subsequently with the same type of vector potential in the quantum system. We solve the generalized Klein-Gordon oscillator analytically for different functions (e.g. Coulomb- and Cornell-type functions) and obtain the bound-states solutions in each case. We discuss the effects of topological defects associated with the scalar cur…
A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to $k$th-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a ($k-1$)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape in…
