  We apply a recent proposal for defining states and observables in quantum
gravity to simple models. First, we consider a Klein-Gordon particle in an ex-
ternal potential in Minkowski space and compare our proposal to the theory ob-
tained by deparametrizing with respect to a time slicing prior to quantiza-
tion. We show explicitly that the dynamics of the deparametrization approach
depends on the time slicing. Our proposal yields a dynamics independent of the
choice of time slicing at intermediate times but after the potential is turned
off, the dynamics does not return to the free particle dynamics. Next we apply
our proposal to the closed Robertson-Walker quantum cosmology with a massless
scalar field with the size of the universe as our time variable, so the only
dynamical variable is the scalar field. We show that the resulting theory has
the semi-classical behavior up to the classical turning point from expansion to
contraction, i.e., given a classical solution which expands for much longer
than the Planck time, there is a quantum state whose dynamical evolution
closely approximates this classical solution during the expansion. However,
when the "time" gets larger than the classical maximum, the scalar field be-
comes "frozen" at its value at the maximum expansion. We also obtain similar
results in the Taub model. In an Appendix we derive the form of the Wheeler-
DeWitt equation for the Bianchi models by performing a proper quantum reduc-
tion of the momentum constraints; this equation differs from the usual one ob-
tained by solving the momentum constraints classically, prior to quantization.
