The Dunkl–Coulomb problem in three-dimensions: energy spectrum, wave functions and <i>h</i>-spherical harmonics

Ghazouani, Sami; Sboui, Insaf; Amdouni, M A; Rhouma, Mounir Ben El Hadj

doi:10.1088/1751-8121/ab0d98

Cited by 46 publications

(30 citation statements)

References 25 publications

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“…which is in full agreement to that reported in Ref. [16] for the Dunkl-Coulomb problem in three dimensions.…”

Section: S(ssupporting

confidence: 93%

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Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential

Mota,

Ojeda-Guillén

2021

Preprint

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We solve exactly the Schrödinger equation for the free-particle, the pseudo-harmonic oscillator and the Mie-type potential in three dimensions with the Dunkl derivative. The equations for the radial and angular parts are obtained by using spherical coordinates and separation of variables. The wave functions and the energy spectrum for these potentials are derived in an analytical way and it is shown that our results are adequately reduced to those previously reported when we remove the Dunkl derivative parameters.

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“…which is in full agreement to that reported in Ref. [16] for the Dunkl-Coulomb problem in three dimensions.…”

Section: S(ssupporting

confidence: 93%

“…[13,14], it has been introduced the su(1, 1) Lie algebra and its irreducible representation to obtain the radial solutions and their coherent states. Also, the Dunkl-Coulomb and the harmonic oscillator problems for the Schrödinger equation in 3D have been solved and its superintegrability and dynamical symmetry have been studied [15,16].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential

Mota,

Ojeda-Guillén

2021

Preprint

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“…Besides, Chung et al discussed one dimensional Dunkl-quantum mechanics with the simplest example a confined particle in one-dimensional box in [27,28]. Kim et al extended electrostatics to Dunkl formalism in [29], while Ghazouani et al solved Dunkl-Coulomb problem in three dimensions [30]. Mota et al derived the Landau energy levels of the relativistic Dunkl oscillator in [31].…”

Section: Introductionmentioning

confidence: 99%

Dunkl-Klein-Gordon equation in three-dimensions: The Klein-Gordon oscillator and Coulomb Potential

Hamil¹,

Lütfüoğlu²

2021

Preprint

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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 of the associate Laguerre and Jacobi polynomials. We observe how the Dunkl formalism is affecting the eigenvalues as well as the eigenfunctions. As a second problem, we examine the Dunkl-Klein-Gordon equation with the Coulomb potential. We obtain the eigenvalue, their corresponding eigenfunctions, and the Dunkl-fine structure terms.

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“…In addition to these results, some works in this direction have studied the Dunkl-Coulomb problem based on algebra so(n), one of the simplest of them is the review of the Dunkl-Coulomb problem in the plane [7] in terms of Dunkl operators. Elsewhere, the Dunkl-Laplacian operator is related to Z 3 2 reflection group in the realization of so(1, 2) algebra and it is discussed h-spherical harmonics [8][9][10]. Also, in generalization of the Dunkl oscillator in the plane (1), are singular ones associated to the su(1, 1) algebra with a special case of Askey-wilson algebra AW(3) by reflection involutions [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

Dong,

Najafizade,

Panahi

et al. 2021

Preprint

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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 1 κ 2 (4) with reflections, which is known as the Jordan-Schwinger-Dunkl algebra jsd κ 1 κ 2 (4). Hence, this model is shown to be maximal superintegrable. On the other hand, the superintegrability of the three-dimensional Dunkl oscillator model is studied from the factorization approach viewpoint. The spectrum of this system is derived through the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

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The Dunkl–Coulomb problem in three-dimensions: energy spectrum, wave functions and h-spherical harmonics

Abstract: The Dunkl–Coulomb system in three-dimensions is introduced. The energy spectrum and the wave functions of the system are solved by means of spectrum generating algebra techniques based on the Lie algebra. An explicit h-spherical harmonics basis is given in terms of Jacobi polynomials.

Cited by 46 publications

References 25 publications

Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential

Exact solutions of the Schrödinger Equation with Dunkl Derivative for the Free-Particle Spherical Waves, the Pseudo-Harmonic Oscillator and the Mie-type Potential

Dunkl-Klein-Gordon equation in three-dimensions: The Klein-Gordon oscillator and Coulomb Potential

Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

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