  We investigate the properties of a quantum Robertson - Walker universe
described by the Wheeler -- DeWitt equation. The universe is filled with a
quantum Yang -- Mills uniform field. This is then a quantum mini copy of the
standard model of our universe. We discuss the interpretation of the Wheeler --
DeWitt wave function using the correspondence principle to connect
$\vert\psi\vert^2$ for large quantum numbers to the classical probability for a
radiation dominated universe. This can be done in any temporal gauge. The
correspondence principle determines the Schr\"odinger representation of the
momentum associated to the gravitational degree of freedom. We also discuss the
measure in the mini--superspace needed to ensure invariance of the quantum
description under change of the temporal gauge. Finally, we examine the
behaviour of $\vert\psi\vert^2$ in inflationary conditions.
