  Process of formation of the universe with its further expansion in the first
evolution stage is investigated in the framework of Friedmann-Robertson-Walker
metrics on the basis of quantum model, where a new type of matter is
introduced, which energy density is dependent on velocity of the expansion. It
is shown that such an improvement of the model forms potential barrier for the
flat universe at $k=0$ (in contrast with generalized Chaplygin gas model).
Peculiarities of wave function are analyzed in details, which is calculated by
fully quantum (non-semiclassic) approach, for the different barrier regions and
stages of evolution. Resonant influence of the initial and boundary conditions
on the barrier penetrability is shown (in contrast with Vilenkin and Hawking
approaches). In order to perform a comparative analysis, how much quickly the
universe is expanded by different models, new quantum definitions of velocity
and Hubble function are introduced. These notions allow us to study dynamics of
evolution of universe in quantum cosmology both in the first stage, and in
later times.
