The adiabatic theorem in the complex plane and the semiclassical calculation of nonadiabatic transition amplitudes

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Abstract
This paper is concerned with the problem of calculating amplitudes for nonadiabatic transitions induced by a time‐dependent Hamiltonian, in the semiclassical limit h/→0, with emphasis on questions relevant to semiclassical theories of electronically inelastic scattering. For this problem the semiclassical limit is mathematically equivalent to the adiabatic limit, and the adiabatic theorem says that all these transition amplitudes vanish in the limit; the question is, what is the asymptotic form of the nonadiabatic amplitudes, as they go to zero? We consider Hamiltonia that are analytic matrix functions of time. We prove a generalization of the adiabatic theorem to the complex time plane; paradoxically, the adiabatic theorem in the complex plane gives us directly the nonadiabatic amplitudes along the real time axis. We derive Dykhne’s remarkable formula for the two‐state case, which says that the limiting form of the transition amplitude depends only on the energy curves of the two states, not on the nonadiabatic coupling which is responsible for transition between them. We discuss the three‐state problem at length and show that the obvious generalization of the Dykhne formula is sometimes true, sometimes false. To indicate the scope of methods based on the adiabatic theorem in the complex plane, we give an elementary proof of the semiclassical formula for above‐barrier reflection of a one‐dimensional particle.

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