Lewis-Riesenfeld quantization and SU(1,1) coherent states for 2D damped harmonic oscillator

TL;DR

This paper investigates a 2D damped harmonic oscillator using Lewis-Riesenfeld invariants, constructs exact quantum solutions, and develops SU(1,1) coherent states to analyze their properties and uncertainty relations.

Contribution

It introduces a novel application of Lewis-Riesenfeld invariants to a 2D damped oscillator and constructs SU(1,1) coherent states in this context.

Findings

Exact solutions for the time-dependent Schrödinger equation were obtained.

Generalized Heisenberg uncertainty relations were verified.

SU(1,1) coherent states were constructed and analyzed.

Abstract

In this paper we study a two-dimensional [2D] rotationally symmetric harmonic oscillator with time-dependent frictional force. At the classical level, we solve the equations of motion for a particular case of the time-dependent coefficient of friction. At the quantum level, we use the Lewis-Riesenfeld procedure of invariants to construct exact solutions for the corresponding time-dependent Schr\"{o}dinger equations. The eigenfunctions obtained are in terms of the generalized Laguerre polynomials. By mean of the solutions we verify a generalization version of the Heisenberg's uncertainty relation and derive the generators of the $su(1,1)$ Lie algebra. Based on these generators, we construct the coherent states $\grave{\textrm{a}}$ la Barut-Girardello and $\grave{\textrm{a}}$ la Perelomov and respectively study their properties.