  We obtain an exact formula for the equilibrium free energy of a charged
quantum particle moving in a harmonic potential in the presence of a uniform
external magnetic field and linearly coupled to a heat bath of independent
quantum harmonic oscillators through the momentum variables. We show that the
free energy has a different expression than that for the coordinate-coordinate
coupling between the particle and the heat-bath oscillators. For an
illustrative heat-bath spectrum, we evaluate the free energy in the
low-temperature limit, thereby showing that the entropy of the charged particle
vanishes at zero temperature, in agreement with the third law of
thermodynamics.
