  We provide a fully quantum description of a mechanical oscillator in the
presence of thermal environmental noise by means of a quantum Langevin
formulation based on quantum stochastic calculus. The system dynamics is
determined by symmetry requirements and equipartition at equilibrium, while the
environment is described by quantum Bose fields in a suitable non-Fock
representation which allows for the introduction of temperature. A generic
spectral density of the environment can be described by introducing its state
trough a suitable P-representation. Including interaction of the mechanical
oscillator with a cavity mode via radiation pressure we obtain a description of
a simple optomechanical system in which, besides the Langevin equations for the
system, one has the exact input-output relations for the quantum noises. The
whole theory is valid at arbitrarily low temperature. This allows the exact
calculation of the stationary value of the mean energy of the mechanical
oscillator, as well as both homodyne and heterodyne spectra. The present
analysis allows in particular to study possible cooling scenarios and to obtain
the exact connection between observed spectra and fluctuation spectra of the
position of the mechanical oscillator.
