The quantum model of homogeneous and isotropic universe filled with the uniform scalar field is considered .
The time-independent equation for the wavefunction has the solutions which describe the universe in quasistationary states .
The evolution of the universe is realized in the form of transitions between such states .
In the first stage in the early universe scalar field slow rolls into a vacuum-like state with a vanishing energy density .
In the second stage this field begins to oscillate near the minimum of its potential energy density and becomes a source of creation of matter/energy in the universe .
The quantum model predicts effective inverse square-law dependence of the mean total energy density \overline { \rho } on the expectation value of cosmological scale factor \langle a \rangle where the averaging is performed over the state with large quantum numbers .
Such a law of decreasing of \overline { \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 \langle a \rangle .
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 / \langle a \rangle calculated from observed positions of individual supernovae are in good agreement with their estimations in quantum theory .
Proportionality of the average value \langle a \rangle to total quantity of matter/energy in the universe on the one hand and to its age on the other hand predicted by the quantum model agrees with the present-day cosmological observables .
Possible consequences from the conclusions of quantum theory are discussed .