  Recently, the quantum counterpart of energy equipartition theorem has drawn
considerable attention. Motivated by this, we formulate and investigate an
analogous statement for the free energy of a quantum oscillator linearly
coupled to a passive heat bath consisting of an infinite number of independent
harmonic oscillators. We explicitly demonstrate that the free energy of the
Brownian oscillator can be expressed in the form $F(T) = \langle f(\omega,T)
\rangle $ where $f(\omega,T)$ is the free energy of an individual bath
oscillator. The overall averaging process involves two distinct averages: the
first one is over the canonical ensemble for the bath oscillators, whereas the
second one signifies averaging over the entire bath spectrum of frequencies
from zero to infinity. The latter is performed over a relevant probability
distribution function $\mathcal{P}(\omega)$ which can be derived from the
knowledge of the generalized susceptibility encountered in linear response
theory. The effect of different dissipation mechanisms is also exhibited. We
find two remarkable consequences of our results. First, the quantum counterpart
of energy equipartition theorem follows naturally from our analysis. The second
corollary we obtain is a natural derivation of the third law of thermodynamics
for open quantum systems. Finally, we generalize the formalism to three spatial
dimensions in the presence of an external magnetic field.
