  Transition amplitudes between instantaneous eigenstates of quantum two-level
system are evaluated analytically on the basis of a new parametrization of its
evolution operator, which has recently been proposed to construct exact
solutions. In particular, they are estimated when the Hamiltonian varies
infinitesimally slowly. The results, not only confirm the adiabatic theorem in
the adiabatic limit, but also bring us with an analytic estimation of the
adiabatic approximation. The condition under which no transition between
different instantaneous eigenstates is allowed is also clarified.
