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International Journal of Modern Physics A

Dunkl–Duffin–Kemmer–Petiau equation in (2 + 1) dimensions: The Dunkl–Bosonic oscillator spinless under Landau levels and Coulomb potential

Asma Merad, M. Merad, Tahar Boudjedaa

Published Apr 30, 2022• 16 citations
Publisher

Abstract

In this paper, we consider two fundamental problems in the framework of the Dunkl derivative: the Bosonic oscillator model of spin 0 under magnetic field: Landau levels and the Duffin–Kemmer–Petiau equation with the Coulomb potential. In both cases, we obtain the exact analytical solutions for the bound states in the general case for the different eigenvalues of the reflection operator.

DOI
10.1142/s0217751x22500725

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In this paper, we consider two fundamental problems in the framework of the Dunkl derivative: the Bosonic oscillator model of spin 0 under magnetic field: Landau levels and the Duffin–Kemmer–Petiau equation with the Coulomb potential. In both cases, we obtain the exact analytical solutions for the bound states in the general case for the different eigenvalues of the reflection operator.
In this study, we replace the standard partial derivatives in the Klein–Gordon equation with Dunkl derivatives and obtain exact analytical solutions for the eigenvalues and eigenfunctions of the Dunkl–Klein–Gordon equation in higher dimensions. We apply this formalism to two key quantum mechanical systems: the d-dimensional harmonic oscillator and the Coulomb potential. First, we introduce Dunkl quantum mechanics in d-dimensional hyperspherical coordinates, followed by an analysis of the d-dimensional Dunkl–Klein–Gordon oscillator. Subsequently, we derive the energy spectrum and eigenfunctions, which are expressed using confluent hypergeometric functions. Furthermore, we examine the impact of the Dunkl formalism on both the eigenvalues and eigenfunctions. In the second case, we explore both the bound-state solutions and scattering scenarios of the Dunkl–Klein–Gordon equation with the Coulomb potential. The bound-state solutions are represented in terms of confluent hypergeometric functions, while the scattering states enable us to compute the particle creation density and probability using the Bogoliubov transformation method.
Spin-0 and Spin-1 Bosonic Particles Dynamics in (1+3)(1+3)(1+3) Dimensions via the Relativistic DKP Equation with a Spatially-Dependent Exponential Mass Term
Abstract The Duffin-Kemmer-Petiau (DKP) equation is a relativistic wave equation governing the dynamics of spin-0 and spin-1 bosonic particles. In this work, we study the (1+3)(1+3)(1+3) ( 1 + 3 ) -dimensional DKP equation with a spatially dependent mass under a modified deformed exponential type potential. Using the framework of supersymmetric quantum mechanics, we derive approximate analytical solutions. A novel extension of the Pekeris approximation is introduced to treat the centrifugal barrier, enabling the complete determination of the energy spectrum. The non-relativistic limit of the DKP equation is obtained via an established transformation technique. The model is applied to diatomic molecules including lithium hydride (LiH), hydrogen chloride (HCl), 7Li2^{7}\textrm{Li}_{2}7Li2​ 7 Li 2 , Na2\textrm{Na}_{2}Na2​ Na 2 , and ScI. The energy spectra are computed for various quantum states ∣n,l,j⟩\left| n,l,j\right\rangle ∣n,l,j⟩ n , l , j using both relativistic and non-relativistic expressions. For LiH and HCl, where experimental data are scarce, our results show strong agreement with earlier theoretical predictions. For 7Li2^{7}\textrm{Li}_{2}7Li2​ 7 Li 2 , Na2\textrm{Na}_{2}Na2​ Na 2 , and ScI, the calculated energies align excellently with available experimental data. The proposed framework provides a reliable and analytically tractable tool for studying bosonic systems with position-dependent effective masses.
Abstract We consider a q -deformed extension of the Duffin–Kemmer–Petiau oscillator (DKPO) for spin-one bosons in two spatial dimensions. The corresponding problem is easily solved algebraically after realization of the q -deformed algebra in terms of operators of the ordinary Bose algebra. For states associated with the spin-projection numbers ± 1 an equivalent equation in Hamiltonian form is obtained where the Hamiltonian is pseudo-Hermitian, supporting the idea that pseudo-Hermitian models are physically relevant. Then, using the spectral method, we derived the exact expressions for the corresponding metric operator and Hermitian Hamiltonian in closed forms Notably, this Hamiltonian is found to be diagonal, thus implying that the transformation we applied to the pseudo-Hermitian Hamiltonian to work it out is a Foldy-Wouthuysen transformation.
Scalar and vector bosons in a Bonnor-Melvin-ΛΛΛ spacetime: an exact Duffin-Kemmer-Petiau analysis
We study scalar and vector bosons in the Bonnor--Melvin--ΛΛΛ spacetime within the Duffin--Kemmer--Petiau (DKP) formalism. By employing Umezawa's projection operators, we separate the physical spin-0 and spin-1 sectors and derive the corresponding exact second-order equations in the full curved spacetime, without relying on the conical approximation. For the scalar sector, the radial equation reduces to a Schrödinger-like equation with a trigonometric Pöschl--Teller effective potential. In the vector sector, the longitudinal mode is governed by the same effective potential, whereas the transverse polarizations are described by generalized trigonometric Pöschl--Teller potentials. Because the me\-tric function vanishes at a discrete set of radial points, the radial dynamics is naturally formulated as a singular Sturm--Liouville problem on a fundamental interval, with the physical radial domain fixed by the Friedrichs self-adjoint extension of the corresponding singular radial operators. As a result, all physical sectors exhibit purely discrete radial spectra, and their eigenfunctions are obtained in closed form. These results provide a unified exact treatment of scalar and vector bosons in the Bonnor--Melvin--ΛΛΛ spacetime, complement previous analyses based on the conical approximation, and clarify the role of the global geometric structure of the background in shaping confinement and spectral properties.