  A new fully quantum method describing penetration of packet from internal
well outside with its tunneling through the barrier of arbitrary shape used in
problems of quantum cosmology, is presented. The method allows to determine
amplitudes of wave function, penetrability $T_{\rm bar}$ and reflection $R_{\rm
bar}$ relatively the barrier (accuracy of the method: $|T_{\rm bar}+R_{\rm
bar}-1| < 1 \cdot 10^{-15}$), coefficient of penetration (i.e. probability of
the packet to penetrate from the internal well outside with its tunneling),
coefficient of oscillations (describing oscillating behavior of the packet
inside the internal well). Using the method, evolution of universe in the
closed Friedmann--Robertson--Walker model with quantization in presence of
positive cosmological constant, radiation and component of generalize Chaplygin
gas is studied. It is established (for the first time): (1) oscillating
dependence of the penetrability on localization of start of the packet; (2)
presence of resonant values of energy of radiation $E_{\rm rad}$, at which the
coefficient of penetration increases strongly. From analysis of these results
it follows: (1) necessity to introduce initial condition into both
non-stationary, and stationary quantum models; (2) presence of some definite
values for the scale factor $a$, where start of expansion of universe is the
most probable; (3) during expansion of universe in the initial stage its radius
is changed not continuously, but passes consequently through definite discrete
values and tends to continuous spectrum in latter time.
