Free energy of a charged oscillator in a magnetic field and coupled to a heat bath through the momentum variables

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.