Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2003-09-10
Int.J.Mod.Phys.A20:1515-1529,2005
Physics
High Energy Physics
High Energy Physics - Theory
15 pages, 1 figure
Scientific paper
10.1142/S0217751X05020835
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.
El Kinani El Hassan
Jellal Ahmed
Schreiber Michael
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