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

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.