Entropy of a quantum oscillator coupled to a heat bath and implications for quantum thermodynamics

The free energy of a quantum oscillator in an arbitrary heat bath at 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 explicit results are obtained both for the case of an Ohmic heat bath and a radiation heat bath. More general heat bath models are also examined. In all cases it is found that the entropy vanishes at zero temperature, in conformity with the third law of thermodynamics (Nernstʹs theorem).