  The free energy of a quantum oscillator in an arbitrary heat bath at a
temperature T is given by a "remarkable formula" which involves only a single
integral. This leads to a corresponding simple result for the entropy. The low
temperature limit is examined in detail and we obtain explicit results both for
the case of an Ohmic heat bath and a radiation heat bath. More general heat
bath models are also examined. This enables us to determine the entropy at zero
temperature in order to check the third law of thermodynamics in the quantum
regime
