  Careful monitoring of harmonically bound (or as a limiting case, free) masses
is the basis of current and future gravitational wave detectors, and of
nanomechanical devices designed to access the quantum regime. We analyze the
effects of stochastic localization models for state vector reduction, and of
related models for environmental decoherence, on such systems, focusing our
analysis on the non-dissipative forced harmonic oscillator, and its free mass
limit. We derive an explicit formula for the time evolution of the expectation
of a general operator in the presence of stochastic reduction or
environmentally induced decoherence, for both the non-dissipative harmonic
oscillator and the free mass. In the case of the oscillator, we also give a
formula for the time evolution of the matrix element of the stochastic
expectation density matrix between general coherent states. We show that the
stochastic expectation of the variance of a Hermitian operator in any
unraveling of the stochastic process is bounded by the variance computed from
the stochastic expectation of the density matrix, and we develop a formal
perturbation theory for calculating expectation values of operators within any
unraveling. Applying our results to current gravitational wave interferometer
detectors and nanomechanical systems, we conclude that the deviations from
quantum mechanics predicted by the continuous spontaneous localization (CSL)
model of state vector reduction are at least five orders of magnitude below the
relevant standard quantum limits for these experiments. The proposed LISA
gravitational wave detector will be two orders of magnitude away from the
capability of observing an effect.
