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

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.

Algebraic approach for the one-dimensional Dirac-Dunkl oscillator

“…This result is in full agreement with that obtained previously in Ref. [24], where the energy spectrum was obtained using the so-called Wigner-Dunkl algebra.…”

“…Finally, we point out that in Ref. [24] the (1 + 1) Dirac-Moshinsky oscillator was studied in terms of the Wigner-Dunkl algebra. However, in that work, the authors only obtained the energy spectrum of the problem.…”

“…Recently, the Dunkl derivatives were also used to study some relativistic problems, such as the Dirac equation and the Dirac harmonic oscillator in one and two dimensions [24,25]. The aim of the present work is to introduce an SU(1, 1) algebraic approach to obtain the eigenfunctions and energy spectrum of the (1 + 1)-dimensional Dirac-Dunkl oscillator.…”

Thermal Properties of Relativistic Dunkl Oscillators

“…A Dunkl operator, D, can be defined as a linear superposition of differential and difference operators [5] for use in various problems in mathematics [6][7][8] and theoretical physics [9][10][11][12][13][14]. Especially in recent years, we observe that this interest has increased in the context of relativistic and non-relativistic quantum mechanical problems [15][16][17][18][19][20][21][22][23][24][25][26].…”

The Dirac equation in a class of topologically trivial flat Gödel-type space-time backgrounds

“…The obtained eigenvalues of energy in these different classes of Gödel-type space-times are found different and the results are enough significant [71,76]. Other works are the quantum dynamics of Klein-Gordon scalar field subject to Cornell potential [82], survey on the Klein-Gordon equation in a class of Gödel-type space-times [83], the Dirac-Weyl equation in graphene under a magnetic field [84], effects of cosmic string framework on thermodynamical properties of anharmonic oscillator [85], study of bosons for three special limits of Gödel-type space-times [86], the Klein-Gordon oscillator in the presence of Cornell potential in the cosmic string space-time [87], the covariant Duffin-Kemmer-Petiau (DKP) equation in the cosmic-string space-time with interaction of a DKP field with the gravitational field produced by topological defects investigated in [88], the Klein-Gordon field in spinning cosmic string space-time with the Cornell potential [89], the relativistic spin-zero bosons in a Som-Raychaudhuri space-time investigated in [90], investigation of the Dirac equation using the conformable fractional derivative [91], effect of the Wigner-Dunkl algebra on the Dirac equation and Dirac harmonic oscillator investigated in [92], investigation of the relativistic dynamics of a Dirac field in the Som-Raychaudhuri space-time, which is described by Gödel-type metric and a stationary cylindrical symmetric solution of Einstein's field equations for a charged dust distribution in rigid rotation [93], investigation of relativistic free bosons in the Gödel-type spacetimes [94], investigation of relativistic quantum dynamics of a DKP oscillator field subject to a linear interaction in cosmic string space-time to understand the effects of gravitational fields produced by topological defects on the scalar field [95], the behaviour of relativistic spin-zero bosons in the space-time generated by a spinning cosmic string investigated in [96], relativistic spin-0 system in the presence of a Gödel-type background space-time investigated in [97], study of the Duffin-Kemmer-Petiau (DKP) equation for spin-zero bosons in the space-time generated by a cosmic string subject to a linear interaction of a DKP field with gravitational fields produced by topological defects investigated in [98], the information-theoretic measures of (1 + 1)dimensional Dirac equation in both position and momentum spaces are investigated for the trigonometric Rosen-Morse and the Morse potentials investigated in [99], analytical bound and scattering state solutions of Dirac equation for the modified deformed Hylleraas potential with a Yukawat...…”