Two Coupled Harmonic Oscillators on Non-commutative Plane

We investigate a system of two coupled harmonic oscillators on the non-commutative plane \RR^2_θ 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 α. The obtained eigenstates as well as the eigenvalues depend on the non-commutativity parameter θ. Focusing on the ground state wave function before the transformation, we calculate the density matrix ρ_0(θ) and find that its traces {\rm Tr}(ρ_{0}(θ)) and {\rm Tr}(ρ_0^2(θ)) are not affected by the non-commutativity. Evaluating the Wigner function on \RR^2_θ confirms this. The uncertainty relation is explicitly determined and found to depend on θ. For small values of θ, the relation is shifted by a θ^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 α=1 and α={π\over 2}. In first case, by identifying θto the squared magnetic length, one can recover basic features of the Hall system.

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