  The dynamics of the expanding universe is analyzed in terms of the quantum
geometrodynamical model. It is shown that the equations of quantum theory in
the form of the eigenvalues equation similar to the stationary Schr\"{o}dinger
equation complemented by the equations of motion for the momentum operator and
its time derivative in Heisenberg's form reduce to the Einstein equations with
an additional source of the gravitational field of quantum nature. The
spatially closed universe with cosmological constant, originally filled with a
uniform scalar field and radiation, is considered as quantum cosmological
system. The perfect fluid in the form of radiation defines the material
reference frame. The properties of the averaged scalar field which acts like
ordinary matter are investigated. After averaging over its quantum states, the
free scalar field turns into the Weyssenhoff fluid characterized by the energy
density, pressure, and spin of constituent particles. The cases when the
contribution of the quantum effects into the gravitational interaction becomes
significant on macroscopic scale are analyzed. It is demonstrated that, unless
the whole, at least a part of such matter-energy constituents as dark matter
and dark energy may have a quantum origin.
