Greenʹs function and position correlation function for a charged oscillator in a heat bath and a magnetic field

We formulate, in the framework of the generalized quantum Langevin equation approach, the retarded Greenʹs functions and the symmetrized position correlation functions for the motion of a charged quantum-mechanical particle in a spatial harmonic potential, coupled linearly to a passive heat bath, and subject to a constant homogeneous magnetic field. General conclusions can then be reached by using only those properties of the generalized susceptibility tensor imposed by fundamental physical principles. Explicit calculations are made for the Ohmic heat bath. We next investigate the Brownian motion of a charged particle in an external magnetic field. We continue by proving general relations between the retarded Greenʹs functions and displacement correlation functions in the limit of long times at both absolute zero and nonzero temperatures, and further evaluate the long-time asymptotic behaviors of the two functions, for both the Ohmic and a rather general class of heat baths discussed extensively in the literature.