  The exactly solvable quantum model of the homogeneous, isotropic and closed
universe in the matter-energy production epoch is considered. It is assumed
that the universe is originally filled with a uniform scalar field and a
perfect fluid which defines a reference frame. The stationary state spectrum
and the wave functions of the quantum universe are calculated. In this model
the matter-energy in the universe has a component in the form of a condensate
of massive zero-momentum excitation quanta of oscillations of primordial scalar
field. The mean value of the scale factor of the universe in a given state is
connected with the mass of a condensate by a linear relation. The nucleation
rate of the universe from the initial cosmological singularity point is
calculated. It is demonstrated that the process of nucleation of the universe
can have an exponential (explosive) nature. The evolution of the universe is
described as transitions with non-zero probabilities between the states of the
universe with different masses of a condensate.
