Deformation quantization for coupled harmonic oscillators on a general noncommutative space

TL;DR

This paper applies deformation quantization to coupled harmonic oscillators in a general noncommutative space, deriving explicit Wigner functions and energy spectra, and exploring special cases with notable results.

Contribution

It provides explicit solutions for Wigner functions and energy spectra for coupled oscillators in noncommutative space, extending deformation quantization methods.

Findings

Derived all Wigner functions for the system

Obtained energy spectra considering noncommutativity

Identified special cases with significant physical implications

Abstract

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.