  A fundamental prediction of quantum theory that is derived from the
"projection postulate" is that under continuous measurement, the state of a
system traces out a "quantum trajectory" in time that depends upon its
measurement record, and that in certain situations, the quantum state of a
system stochastically collapses at a finite rate toward a random eigenstate. On
the other hand, quantum theory also predicts that environmental coupling
produces the opposite effect of spreading the wave function into a mixed state.
Studying the effect of these two competing processes, as the ratio of their
strengths is varied, on the distribution of a quantum state is of fundamental
interest in any setting, but especially so in the mechanical realm. A key
issue, however, is that a fundamental study of such physics can not entail an a
priori assumption of the projection postulate itself, and Bayesian inference
may therefore not be employed to observe the quantum state. In this theoretical
work we propose a feasible way to perform such a fundamental test with a
mechanical wave function in the energy eigenbasis. The scheme is based on an
optomechanical setup wherein optical fields provide a direct probe of the
energy variance of the mechanical wave function. Such an energy variance probe
is unprecedented in any type of system. Numerical simulations that incorporate
the projection postulate predict a monotonic decrease in the time-averaged
energy variance as the ratio of continuous measurement strength to dissipation
is increased. The measurement strength in the proposed scheme is tunable in
situ, and the behavior predicted by the simulations therefore implies that the
scheme amounts to a way to verifiably control in situ the time-averaged
variance of a mechanical wave function over the course of a single quantum
trajectory.
