Entangled state representations in noncommutative quantum mechanics

TL;DR

This paper introduces entangled state representations for noncommutative quantum mechanics, providing new tools to analyze quantum systems with noncommuting coordinates and demonstrating their application through an exactly solved two-dimensional harmonic oscillator.

Contribution

It presents novel entangled state representations and derives transformations to standard forms, enabling new analytical approaches in noncommutative quantum mechanics.

Findings

Derived eigenfunctions of key operators in the new representations

Established unitary transformations between representations

Solved a two-dimensional harmonic oscillator with noncommutative couplings

Abstract

We introduce new representations to formulate quantum mechanics on noncommutative coordinate space, which explicitly display entanglement properties between degrees of freedom of different coordinate components and hence could be called entangled state representations. Furthermore, we derive unitary transformations between the new representations and the ordinary one used in noncommutative quantum mechanics (NCQM) and obtain eigenfunctions of some basic operators in these representations. To show the potential applications of the entangled state representations, a two-dimensional harmonic oscillator on the noncommutative plane with both coordinate-coordinate and momentum-momentum couplings is exactly solved.