Relaxation of a quantum particle in a magnetic field

The Fokker–Planek equation for the Wigner function of a quantum system with given arbitrary linear equations of motion for the mean values of the coordinates and momenta is considered. Conditions on the matrix of diffusion coefficients are found that guarantee the preservation in time of the non-negative definiteness of the density matrix. A detailed study is made of the quantum description of a damped isotropic two-dimensional harmonic oscillator in a homogeneous magnetic field for arbitrary relationships between the intrinsic and cyclotron frequencies, the coefficient of friction, and the temperature; various limiting cases of this problem are also considered.