Quantum analysis of the effects of coordinate noncommutativity on bi-dimensional harmonic motion under parametric variations

TL;DR

This paper investigates how coordinate noncommutativity affects two-dimensional quantum harmonic motion with time-dependent parameters, using advanced quantum methods to derive solutions and analyze nonstationary effects in a noncommutative phase space.

Contribution

It extends the analysis of 2D quantum harmonic oscillators to include coordinate noncommutativity and parameter variations, providing new solutions using Lewis-Riesenfeld invariants.

Findings

Derived quantum solutions for noncommutative 2D oscillators with static parameters

Extended solutions to time-dependent parameters using invariant theory

Enhanced understanding of noncommutativity effects in nonstationary quantum systems

Abstract

In high-energy physics, coordinate noncommutativity represents the core idea that space itself can be quantized, as expressed through the frameworks of string theory and noncommutative field theory. Influence of such a noncommutativity on 2D quantum oscillatory motion, which undergoes parameter variations, is investigated. We first derive quantum solutions of the system described with time-independent parameters considering the noncommutativity of coordinates as a preliminary step. And then, we extend our study, framed with noncommutative phase-space formalism, to obtain relevant solutions of the system with time-dependent parameters. This system, which we focus on, is nonstationary due to variation of parameters in time. While the left and right circular annihilation and creation operators are utilized in the quantal management of the basic stationary system, the Schr\"odinger equation of the nonstationary system is solved using the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation procedure. The outcome of our analysis is useful in understanding the effects of noncommutativity from quantum perspectives, especially in conjunction with the impact of parameter variations.