Two Coupled Harmonic Oscillators on Non-commutative Plane

TL;DR

This paper analyzes a system of two coupled harmonic oscillators on a non-commutative plane, revealing how non-commutativity affects eigenstates, eigenvalues, and quantum properties like uncertainty and entropy, with implications for quantum Hall systems.

Contribution

It introduces a diagonalization method for coupled oscillators on a non-commutative plane and explores how non-commutativity influences quantum states and physical quantities.

Findings

Eigenstates and eigenvalues depend on the non-commutativity parameter .

Uncertainty relations are shifted by a ^2 term, indicating quantum corrections.

Entropy remains unchanged despite non-commutativity effects.

Abstract

We investigate a system of two coupled harmonic oscillators on the non-commutative plane \RR^2_{\theta} by requiring that the spatial coordinates do not commute. We show that the system can be diagonalized by a suitable transformation, i.e. a rotation with a mixing angle \alpha. The obtained eigenstates as well as the eigenvalues depend on the non-commutativity parameter \theta. Focusing on the ground state wave function before the transformation, we calculate the density matrix \rho_0(\theta) and find that its traces {\rm Tr}(\rho_{0}(\theta)) and {\rm Tr}(\rho_0^2(\theta)) are not affected by the non-commutativity. Evaluating the Wigner function on \RR^2_{\theta} confirms this. The uncertainty relation is explicitly determined and found to depend on \theta. For small values of \theta, the relation is shifted by a \theta^2 term, which can be interpreted as a quantum correction. The calculated entropy does not change with respect to the normal case. We consider the limits \alpha=1 and \alpha={\pi\over 2}. In first case, by identifying \theta to the squared magnetic length, one can recover basic features of the Hall system.