  A modified semi-classical method is used to construct both ground and excited
state solutions to the canonically quantized vacuum Bianchi IX (Mixmaster)
cosmological models. Employing a modified form of the semi-classical Ansatz we
solve the relevant Wheeler-DeWitt equation asymptotically by integrating a set
of linear transport equations along the flow of a suitably chosen solution to
the corresponding Euclidean-signature Hamilton-Jacobi equation. For the
Moncrief-Ryan (or `wormhole') Hamilton-Jacobi solution, we compute the ground
state quantum correction term associated with operator ordering ambiguities and
show how higher order correction terms can be computed. We also determine the
explicit, leading order forms of a family of excited states and show how to
compute their quantum corrections as smooth, globally defined functions on the
Bianchi IX minisuperspace. These excited-state solutions are peaked away from
the minisuperspace origin and are labeled by a pair of positive integers that
can be plausibly interpreted as graviton excitation numbers for the two
independent anisotropy degrees of freedom. The modified semi-classical method
used here is applicable to more general models, representing a significant
progress in the Wheeler-DeWitt approach to quantum gravity.
