Harmonic oscillator in noncommuting two-dimensional space

In the present paper, we study a two-dimensional harmonic oscillator in a constant magnetic field in noncommuting space. We use the following Hamiltonian H = 1/2m ((p) over cap (2)(1) + (p) over cap (2)(2) + 1/m m omega(2)(1)(q) over cap (2)(1) + 1/2 m omega(2)(2)(q) over cap (2)(2)) with commutation relations [(q) over cap (1), (q) over cap (2)] = i theta, [(p) over cap (1), ($) over cap (2)] = i lambda and [(q) over cap (j), (q) over cap (k)] = ih delta(jk.) The parameter lambda expresses the presence of the magnetic field. We find the exact propagator of the system and the time evolution of the basic operators. We prove that the system is equivalent to a two-dimensional system where the operators of momentum and coordinates of the second dimension satisfy a deformed commutation relation [(Q) over cap (2), (P) over cap (2)] = ih mu. The deformation parameter, mu, depends on lambda and theta, and is independent of the Hamiltonian. Finally, we investigate the thermodynamic properties of the system in Boltzmann statistics. We find the statistical density matrix and the partition function, which is equivalent to that of a two-dimensional harmonic oscillator with two deformed frequencies Omega(1) and Omega(2)-