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Transition amplitudes between instantaneous eigenstates of quantum two-levelsystem are evaluated analytically on the basis of a new parametrization of itsevolution operator, which has recently been proposed to construct exactsolutions. In particular, they are estimated when the Hamiltonian variesinfinitesimally slowly. The results, not only confirm the adiabatic theorem inthe adiabatic limit, but also bring us with an analytic estimation of theadiabatic approximation. The condition under which no transition betweendifferent instantaneous eigenstates is allowed is also clarified.

Analytic estimation of transition between instantaneous eigenstates of quantum two-level system