A new kind of representations on noncommutative phase space

TL;DR

This paper introduces novel representations for quantum mechanics on noncommutative phase space, revealing entanglement properties and providing explicit formulas for Wigner functions, with an exact solution for a two-dimensional harmonic oscillator.

Contribution

The paper presents new representations for quantum mechanics on noncommutative phase space, highlighting entanglement and deriving explicit Wigner functions and operators.

Findings

Explicit expressions for Wigner function and operator in new representations

Exact solution of a 2D harmonic oscillator on noncommutative phase space

Demonstration of entanglement between different degrees of freedom

Abstract

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.