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This article deals with non-adiabatic processes (i.e. processes excluded bythe adiabatic theorem) from the geometrical (group-theoretical) point of view.An approximated formula for the probabilities of the non-adiabatic transitionsis derived in the adiabatic regime for the case when the parameter-dependentHamiltonian represents a smooth curve in the Lie algebra and the quantaldynamics is determined by the corresponding Lie group evolution operator. Wetreat the spin precession in a time-dependent magnetic field and theover-barrier reflection problem in a uniform way using the first-orderdynamical equations on SU(2) and $SU(1.1)$ group manifolds correspondingly.  A comparison with analytic solutions for simple solvable models is provided.Comment: 15 pages, 2 figure

journal of physics. a, mathematical and general/journal of physics. a, mathematical and general

A geometrical approach to non-adiabatic transitions in quantum theory: applications to NMR, over-barrier reflection and parametric excitation of quantum oscillator