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In this work, based on the generalized Dunkl derivative in quantum mechanics we study the one-dimensional Schrödinger equation with a harmonic oscillator potential and obtain the energy eigenvalues. The principal thermodynamical properties including the Helmholtz free energy, mean energy and entropy are carried out. The effects of the Dunkl paramet...
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... To obtain the solution of any quantum system for a given radial potential function V(r) , the radial Schrödinger equation with a nonrelativistic energy E n, , reduced Planck's constant ℏ, reduced mass , the radial wave function R n, (r), is written as [35][36][37][38] To deal with the centrifugal term in the radial equation given above, the following approximation scheme is applied [35,39] Plugging Eq. (1) and Eq. (27) ...
The study investigates the combination of the Coulomb potential with itself (standard Coulombic potential) under the nonrelativistic wave equation. The energy equation and the corresponding unnormalized radial wave function are obtained using the parametric Nikiforov–Uvarov method. This is achieved by applying a Green–Aldrich approximation scheme to the centrifugal term. The resulting energy equation is utilized to calculate the partition function, from which thermodynamic properties such as mean energy, specific heat capacity, entropy, and free energy are derived. Numerical results are generated for the standard Coulombic potential and its special cases, including Coulomb potential with negative potential strength and Coulomb potential with positive potential strength. The study reveals that the system’s energy is fully bounded. Notably, the two special cases, representing Coulomb–Coulomb potentials with positive and negative potential strengths, yield equal results when the strengths are equal but opposite in sign. The thermodynamic properties align with existing literature but exhibit some unique behaviors.
... To obtain the solution of any quantum system for a given radial potential function ( Rr is written as [34][35][36][37] ( ) ...
The Combination of Coulomb potential with itself(standard Coulombic potential) is studied under the non-relativistic wave equation. The energy equation and its corresponding un-normalized redial wave are obtained using parametric Nikiforov-Uvarov method by applying a Green-Aldrich approximation scheme to the centrifugal term. The energy equation obtained was used to calculated the partition function from where the thermodynamic properties such as the mean energy, specific heat capacity, entropy and free energy are calculated. Numerical results are generated for the standard Coulombic potential and its special cases. The special cases are Coulomb potential with negative potential strength and the other is also Coulomb potential with positive potential strength. The study showed that the energy of the system is fully bounded. It is noted that the two special cases which are Coulomb-Coulomb potentials with positive and negative potential strengths are equal provided the strength are equal but opposite in sign. The thermodynamic properties aligned with those of the literature but has some unique behaviours.
... One may quote, for instance, the Dunkl oscillators in one, two, and three dimensions [20][21][22][23][24], the Dunkl-Coulomb problem in two and three dimensions [25][26][27], as well as the one-dimensional infinite [28] and finite [29] square wells. Coherent states [30], a generalization of shape invariance in supersymmetric quantum mechanics [31], Dunkl derivatives with two and three parameters [32,33], and some relativistic systems [34,35] have also been investigated. ...
It is shown that the extensions of exactly-solvable quantum mechanical problems connected with the replacement of ordinary derivatives by Dunkl ones and with that of classical orthogonal polynomials by exceptional orthogonal ones can be easily combined. For such a purpose, the example of the Dunkl oscillator on the line is considered and three different types of rationally-extended Dunkl oscillators are constructed. The corresponding wavefunctions are expressed in terms of exceptional orthogonal generalized Hermite polynomials, defined in terms of the three different types of Xm-Laguerre exceptional orthogonal polynomials. Furthermore, the extended Dunkl oscillator Hamiltonians are shown to be expressible in terms of some extended Dunkl derivatives and some anharmonic oscillator potentials.
... One may quote, for instance, the Dunkl oscillators in one, two, and three dimensions [20,21,22,23,24], the Dunkl-Coulomb problem in two and three dimensions [25,26,27], as well as the one-dimensional infinite [28] and finite [29] square wells. Coherent states [30], a generalization of shape invariance in supersymmetric quantum mechanics [31], Dunkl derivatives with two and three parameters [32,33], and some relativistic systems [34,35] have also been investigated. ...
It is shown that the extensions of exactly-solvable quantum mechanical problems connected with the replacement of ordinary derivatives by Dunkl ones and with that of classical orthogonal polynomials by exceptional orthogonal ones can be easily combined. For such a purpose, the example of the Dunkl oscillator on the line is considered and three different types of rationally-extended Dunkl oscillators are constructed. The corresponding wavefunctions are expressed in terms of exceptional orthogonal generalized Hermite polynomials, defined in terms of the three different types of $X_m$-Laguerre exceptional orthogonal polynomials. Furthermore, the extended Dunkl oscillator Hamiltonians are shown to be expressible in terms of some extended Dunkl derivatives and some anharmonic oscillator potentials.
By replacing the spatial derivative with the Dunkl derivative, we generalize the Fokker-Planck equation in (1+1) dimensions. We obtain the Dunkl–Fokker–Planck eigenvalues equation and solve it for the harmonic oscillator plus a centrifugal-type potential. Furthermore, when the drift function is odd, we reduce our results to those of the recently developed Wigner–Dunkl supersymmetry.
In this study, we investigated the impact of a topological defect ( λ ) on the properties of heavy quarkonia using the extended Cornell potential. We solved the fractional radial Schrödinger equation (SE) using the extended Nikiforov-Uvarov (ENU) method to obtain the eigenvalues of energy, which allowed us to calculate the masses of charmonium and bottomonium. One significant observation was the splitting between nP and nD states, which attributed to the presence of the topological defect. We discovered that the excited states were divided into components corresponding to 2 l + 1 , indicating that the gravity field induced by the topological defect interacts with energy levels like the Zeeman effect caused by a magnetic field. Additionally, we derived the wave function and calculated the root-mean radii for charmonium and bottomonium. A comparison with the classical models was performed, resulting in better results being obtained. Furthermore, we investigated the thermodynamic properties of charmonium and bottomonium, determining quantities such as energy, partition function, free energy, mean energy, specific heat, and entropy for P-states. The obtained results were found to be consistent with experimental data and previous works. In conclusion, the fractional model used in this work proved an essential role in understanding the various properties and behaviors of heavy quarkonia in the presence of topological defects.
It is shown that the Dunkl harmonic oscillator on the line can be generalized to a quasi-exactly solvable one, which is an anharmonic oscillator with n + 1 known eigenstates for any n ∈ ℕ. It is also proved that the Hamiltonian of the latter can also be rewritten in a simpler way in terms of an extended Dunkl derivative. Furthermore, the Dunkl isotropic oscillator and Dunkl Coulomb potentials in the plane are generalized to quasi-exactly solvable ones. In the former case, potentials with n + 1 known eigenstates are obtained, whereas, in the latter, sets of n + 1 potentials associated with a given energy are derived.
In continuation of our earlier work on the nonextensive form of the Gross–Pitaevskii equation (GPE) [M. Maleki, H. Mohammadzadeh and Z. Ebadi, Int. J. Geom. Methods Mod. Phys. 20 (2023) 2350216], we now delve into its [Formula: see text]-deformed counterpart. GPE is a type of nonlinear partial differential equation that is specifically designed to describe the behavior of a group of particles with Bose–Einstein statistics, such as atoms in a superfluid or a Bose–Einstein condensate (BEC). In some systems, the standard Bose–Einstein or Fermi–Dirac statistics may not apply, and generalized statistics may be needed to describe the behavior of the particles. Therefore in this paper, we investigate the dynamics of a system with particle obeying [Formula: see text]-deformed statistics described by the [Formula: see text]-deformed GPE. First, we use the oscillator algebra and [Formula: see text]-calculus to obtain the well-known Schrödinger equation. By selecting an appropriate Hamiltonian for the condensate phase and minimizing the free energy, we derive the [Formula: see text]-deformed time-independent GPE.
In this research, we shall present the Klein-Gordon and Dirac oscillators in the framework of the generalized Dunkl derivative with two parameters by using the Cartesian coordinates, the eigenvalues of energy and eigenfunctions are obtained. The thermodynamic properties are discussed and plotted graphically.
