  We consider a minisuperspace model for a closed universe with small and
positive cosmological constant, filled with a massive scalar field conformally
coupled to gravity. In the quantum version of this model, the universe may
undergo a tunneling transition through an effective barrier between regions of
small and large scale factor. We solve numerically the minisuperspace
Wheeler--De Witt equation with tunneling boundary conditions for the wave
function of the universe, and find that tunneling in quantum cosmology is quite
different from that in quantum mechanics. Namely, the matter degree of freedom
gets excited under the barrier, provided its interaction with the scale factor
is not too weak, and makes a strong back reaction onto tunneling. In the
semiclassical limit of small values of cosmological constant, the matter energy
behind the barrier is close to the height of the barrier: the system ``climbs
up'' the barrier, and then evolves classically from its top. These features are
even more pronounced for inhomogeneous modes of matter field. Extrapolating to
field theory we thus argue that high momentum particles are copiously created
in the tunneling process. Nevertheless, we find empirical evidence for the
semiclassical-type scaling with the cosmological constant of the wave function
under and behind the barrier.
