  For a quantum system, a density matrix rho that is not pure can arise, via
averaging, from a distribution mu of its wave function, a normalized vector
belonging to its Hilbert space H. While rho itself does not determine a unique
mu, additional facts, such as that the system has come to thermal equilibrium,
might. It is thus not unreasonable to ask, which mu, if any, corresponds to a
given thermodynamic ensemble? To answer this question we construct, for any
given density matrix rho, a natural measure on the unit sphere in H, denoted
GAP(rho). We do this using a suitable projection of the Gaussian measure on H
with covariance rho. We establish some nice properties of GAP(rho) and show
that this measure arises naturally when considering macroscopic systems. In
particular, we argue that it is the most appropriate choice for systems in
thermal equilibrium, described by the canonical ensemble density matrix
rho_beta = (1/Z) exp(- beta H). GAP(rho) may also be relevant to quantum chaos
and to the stochastic evolution of open quantum systems, where distributions on
H are often used.
