  Inspired by quantum cosmology, in which the wave function of the universe is
annihilated by the total Hamiltonian, we consider the internal dynamics of a
simple particle system in an energy eigenstate. Such a system does not possess
a uniquely defined time parameter and all physical questions about it must be
posed without reference to time. We consider in particular the question, what
is the probability that the system's trajectory passes through a set of regions
of configuration space without reference to time? We first consider the
classical case, where the answer has a variety of forms in terms of a phase
space probability distribution function. We then consider the quantum case, and
we analyze this question using the decoherent histories approach to quantum
theory, adapted to questions which do not involve time. When the histories are
decoherent, the probabilities approximately coincide with the classical case,
with the phase space probability distribution replaced by the Wigner function
of the quantum state. For some initial states, decoherence requires an
environment, and we compute the required influence functional and examine some
of its properties. Special attention is given to the inner product used in the
construction (the induced or Rieffel inner product), the construction of class
operators describing the histories, and the extent to which reparametrization
invariance is respected. Our results indicate that simple systems without an
explicit time parameter may be quantized using the decoherent histories
approach, and the expected classical limit extracted. The results support, for
simple models, the usual heuristic proposals for the probability distribution
function associated with a semiclassical wave function satisfying the
Wheeler-DeWitt equation.
