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Modern Physics Letters A

Effect of the Wigner–Dunkl algebra on the Dirac equation and Dirac harmonic oscillator

Sepideh Sargolzaeipor, HASSAN HASSANABADI, Won Sang Chung

Published Jul 31, 2018• 41 citations
Publisher

Abstract

In this work, we study the Dirac equation and Dirac harmonic oscillator in one-dimensional via the Dunkl algebra. By using Dunkl derivative, we solve the momentum operator and Hamiltonian that include the reflection symmetry. Based on the concept of the Wigner–Dunkl algebra and the functional analysis method, we have obtained the energy eigenvalue equation and the corresponding wave function for Dirac harmonic oscillator and Dirac equation, respectively. It is shown all results in the limit state satisfied what we had expected before.

DOI
10.1142/s0217732318501468

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In this work, we study the Dirac equation and Dirac harmonic oscillator in one-dimensional via the Dunkl algebra. By using Dunkl derivative, we solve the momentum operator and Hamiltonian that include the reflection symmetry. Based on the concept of the Wigner–Dunkl algebra and the functional analysis method, we have obtained the energy eigenvalue equation and the corresponding wave function for Dirac harmonic oscillator and Dirac equation, respectively. It is shown all results in the limit state satisfied what we had expected before.
We extend the (1 + 1)-dimensional Dirac–Moshinsky oscillator by changing the standard derivative by the Dunkl derivative. We demonstrate in a general way that for the Dirac–Dunkl oscillator be parity invariant, one of the spinor component must be even, and the other spinor component must be odd, and vice versa. We decouple the differential equations for each of the spinor component and introduce an appropriate su(1, 1) algebraic realization for the cases when one of these functions is even and the other function is odd. The eigenfunctions and the energy spectrum are obtained by using the su(1, 1) irreducible representation theory. Finally, by setting the Dunkl parameter to vanish, we show that our results reduce to those of the standard Dirac-Moshinsky oscillator.
Abstract We discuss quantum mechanical systems with Dunkl derivatives by constructing the Dunkl-Heisenberg relation in the momentum representation by means of the reflection operator for momentum and we obtain the corresponding position quantum eigenfunction. We examine the one-dimensional Dunkl oscillator in the momentum space in terms of ν -deformed Hermite polynomials. We obtain the energy levels as well as the ground-state and excited wave functions in terms of the ν -deformed Hermite polynomials. We also describe some properties of the ν -deformed Hermite polynomials. We apply the method to the construction of coherent states.
Abstract In this paper, we extend the path integral formalism for the Dirac oscillator in (1+1) dimension by replacing the spatial derivative with the Dunkl derivative. Utilizing representations in position space-time coordinates, we precisely calculate the propagator, expressed in terms of generalized Hermite polynomials. The energy eigenvalues of the electron, along with their corresponding wave functions, are determined. In special cases, we can precisely evaluate the non-relativistic energy eigenvalues and wave functions, even in the absence of Dunkl parameters.
In this research, we investigate the extended Dunkl quantum mechanics with two parameters. We apply this formalism to study general quantum phenomena. Using cartesian coordinate, we calculate the solution to problem of a spinless particle confined to a box in three dimensions. Then, we derive the creation and annihilation operators to analyze harmonic oscillator in three dimensions via algebraic approach. Subsequently, utilizing the same operators and the generators of [Formula: see text] Lie algebra, we examine the Dirac harmonic oscillator in [Formula: see text] dimension. Finally, we construct the hydrodynamical formulation of the extended Dunkl quantum mechanics by deriving continuity and quantum Hamilton–Jacobi equations.
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 parameters on the thermodynamical quantities for even parity are discussed. The case of the odd parity can be easily obtained by substitution of the b → − b and γ → − γ . All results in the limit case are reduced to ordinary statistical mechanics.