  Radiation-filled Friedmann-Robertson-Walker universes are quantized according
to the Arnowitt-Deser-Misner formalism in the conformal-time gauge. Unlike
previous treatments of this problem, here both closed and open models are
studied, only square-integrable wave functions are allowed, and the boundary
conditions to ensure self-adjointness of the Hamiltonian operator are
consistent with the space of admissible wave functions. It turns out that the
tunneling boundary condition on the universal wave function is in conflict with
self-adjointness of the Hamiltonian. The evolution of wave packets obeying
different boundary conditions is studied and it is generally proven that all
models are nonsingular. Given an initial condition on the probability density
under which the classical regime prevails, it is found that a closed universe
is certain to have an infinite radius, a density parameter $\Omega = 1$
becoming a prediction of the theory. Quantum stationary geometries are shown to
exist for the closed universe model, but oscillating coherent states are
forbidden by the boundary conditions that enforce self-adjointness of the
Hamiltonian operator.
