  The aim of this paper is to compute transitions amplitudes in quantum
cosmology, and in particular pair creation amplitudes and radiative
transitions. To this end, we apply a double adiabatic development to the
solutions of the Wheeler-DeWitt equation restricted to mini-superspace wherein
gravity is described by the scale factor $a$. The first development consists in
working with instantaneous eigenstates, in $a$, of the matter Hamiltonian. The
second development is applied to the gravitational part of the wave function
and generalizes the usual WKB approximation. We then obtain an exact equation
which replaces the Wheeler-DeWitt equation and determines the evolution, i.e.
the dependence in $a$, of the coefficients of this double expansion. When
working in the gravitational adiabatic approximation, the simplified equation
delivers the unitary evolution of transition amplitudes occurring among
instantaneous eigenstates. Upon abandoning this approximation, one finds that
there is an additional coupling among matter states living in expanding and
contracting universes. Moreover one has to face also the Klein paradox, i.e.
the generation of backward waves from an initially forward wave. The
interpretation and the consequences of these unusual features are only sketched
in the present paper. Finally, the examples of pair creation and radiative
transitions are analyzed in detail to establish when and how the above
mentioned unitary evolution coincides with the Schr\" odinger evolution.
