Quantum particle relaxation in a magnetic field

The Fokker-Planck equation for the wigner function of a quantum system with given arbitrary linear equations of motion for average values of coordinates and momenta is considered. The relations between the diffusion coefficients ensuring the non-negative definiteness of the density matrix for all moments of time are obtained. The problem of quantum description of damped isotropic two-dimensional harmonic oscillator in a uniform magnetic field is studied in detail in the case of arbitrary relations between the proper oscillator frequency, the cyclotronic frequency, the damping coefficients and temperature. Various limit cases of this problem are studied in detail