  The quantum model of homogeneous and isotropic universe filled with the
uniform scalar field is considered. This model predicts effective inverse
square-law dependence of the mean total energy density <\rho> on the
expectation value of cosmological scale factor <a> where the averaging is
performed over the state with large quantum numbers. Such a law of decreasing
of <\rho> during the expansion of the universe allows to describe the observed
coordinate distances to type Ia supernovae and radio galaxies in the redshift
interval z = 0.01 - 1.8. A comparison with phenomenological models with the
cosmological constant (\Lambda CDM) and with zero dark energy component
(\Omega_{M} = 1) is made. It is shown that observed small deviations of the
coordinate distances to some sources from the predictions of above mentioned
simple quantum model can be explained by the fluctuations \delta a of the scale
factor about the average value <a>. These fluctuations can arise due to finite
widths of quasistationary states in the early universe. During expansion the
fluctuations \delta a grow with time and manifest themselves in the form of
observed relative increase or decrease of coordinate distances. The amplitudes
of fluctuations \delta a/<a> calculated from observed positions of individual
supernovae are in good agreement with their estimations in quantum theory.
Possible consequences from the conclusions of quantum theory are discussed.
