Time-dependent Dunkl-Pauli Oscillator

TL;DR

This work addresses the time-dependent dynamics of a two-dimensional Dunkl-Pauli oscillator, incorporating a time-dependent mass $m(t)$, a time-varying oscillator scale via $\\Omega(t)$, and a magnetic field through the cyclotron frequency $\\omega_c(t)=\\frac{eB(t)}{m(t)c}$. The authors formulate the Dunkl-Pauli Hamiltonian first in Cartesian coordinates and then in polar coordinates, introducing the Dunkl angular operator $\\mathcal{J}_{\\theta}$ and solving the angular part exactly; the radial part yields Laguerre-type functions, with the spectrum tied to reflection sectors. Using the Lewis-Riesenfeld invariant method, they construct an $\\mathrm{SL}(2,\\mathbb{R})$-based invariant and the Ermakov-Pinney equation for a scale parameter $\\rho(t)$ to obtain exact time-dependent eigenfunctions and a calculable phase, which reduces to the stationary spectrum in the appropriate limit. The results illustrate how Dunkl deformed symmetries shape both spectra and dynamics of quantum systems, and provide a robust framework for exploring quantum mechanics on spaces with reflected and deformed symmetries.

Abstract

This study explores the time-dependent Dunkl-Pauli oscillator in two dimensions. We constructed the Dunkl-Pauli Hamiltonian, which incorporates a time-varying magnetic field and a harmonic oscillator characterized by time-dependent mass and frequency, initially in Cartesian coordinates. Subsequently, we reformulated the Hamiltonian in polar coordinates and analyzed the eigenvalues and eigenfunctions of the Dunkl angular operator, deriving exact solutions using the Lewis-Riesenfeld invariant method. Our findings regarding the total quantum phase factor and wave functions reveal the significant impact of Dunkl operators on quantum systems, providing precise expressions for wave functions and energy eigenvalues. This work enhances the understanding of quantum systems with deformed symmetries and suggests avenues for future research in quantum mechanics and mathematical physics.