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International Journal of Modern Physics A

The Feynman propagator within three parameters generalized Dunkl derivative: One-dimensional case

H. Benzair, Tahar Boudjedaa, M. Merad

Published Oct 30, 2024• 2 citations
Publisher

Abstract

In this paper, the Feynman path integral approach within three parameters Dunkl derivative is investigated in the one-dimensional case of nonrelativistic quantum mechanics. We successfully derive the propagator in Cartesian coordinates and give its exact expression for the free particle, inverse square, harmonic oscillator and pure Coulomb potentials. The energy eigenvalues and their corresponding wave functions are determined.

DOI
10.1142/s0217751x2450146x

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In this paper, the Feynman path integral approach within three parameters Dunkl derivative is investigated in the one-dimensional case of nonrelativistic quantum mechanics. We successfully derive the propagator in Cartesian coordinates and give its exact expression for the free particle, inverse square, harmonic oscillator and pure Coulomb potentials. The energy eigenvalues and their corresponding wave functions are determined.
Abstract In this paper, we extend the path integral formalism for the Klein–Gordon oscillator by replacing the standard derivative with the Dunkl derivative in 1D, 2D, and even in d − dimensions. Using space-time Cartesian coordinates, we derive the propagators and solve the problems exactly. We determine the energy eigenvalues and corresponding wave functions for the spinless particle. In limiting cases, where the Dunkl derivative parameters are set to zero, our results converge appropriately to those found in the literature for these problems.
Abstract A central coordinate system is investigated within the framework of the Wigner-Dunkl deformation of quantum mechanics, treated using Feynman’s path integral approach. This approach involves performing path integration of a Lagrangian in polar coordinates, with an effective potential term that depends on both radial and angular coordinates. The success in separating variables within the two-dimensional Dunkl system facilitated obtaining exact expressions for the propagators associated with the free particle, inverse square potential, harmonic oscillator, and Coulomb potential. The energy eigenvalues and their corresponding wave functions are also determined.
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.
Abstract The Green’s functions for relativistic oscillators with spin-0 and spin-1/2, within the framework of a new generalization of the Dunkl derivative with three parameters, are calculated using the path integral approach. For simplicity, this derivative is reformulated to depend on only two parameters. As a result, the energy eigenvalues and their corresponding wave functions are determined in both cases as functions of the three deformed parameters. In the limiting cases of the deformed parameters, the standard Dunkl derivative system is recovered, even in the absence of the Dunkl parameters.
This study addresses a significant gap in the literature by extending the path integral formalism to time-dependent systems within the Wigner–Dunkl framework, deriving exact propagator solutions for analytically solvable cases. By employing generalized canonical transformations, we reformulated the path integral to develop an explicit expression for the propagator. This formalism is applied to specific cases, including a Dunkl-harmonic oscillator with time-dependent mass and frequency. Solutions for the Dunkl–Caldirola-Kanai oscillator and a model with a strongly pulsating mass are derived, providing exact propagator expressions and corresponding wave functions. These findings extend the utility of Dunkl operators in quantum mechanics, offering new insights into the dynamics of time-dependent quantum systems and possibly find application in quantum optics, plasma physics, and other fields.