  The recent remarkable developments in quantum optics, mesoscopic and cold
atom physics have given reality to wave functions. It is then interesting to
explore the consequences of assuming ensembles over the wave functions simply
related to the canonical density matrix. In this note we analyze a previously
introduced distribution over wave functions which naturally arises considering
the Schroedinger equation as an infinite dimensional dynamical system. In
particular, we discuss the low temperature fluctuations of the quantum
expectations of coordinates and momenta for a particle in a double well
potential. Our results may be of interest in the study of chiral molecules.
