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

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

2000-11-30

arxiv.org/abs/quant-ph/0011121v1

Physics

Quantum Physics

15 pages, 2 figures

Scientific paper

10.1088/0305-4470/34/8/317

This article deals with non-adiabatic processes (i.e. processes excluded by the adiabatic theorem) from the geometrical (group-theoretical) point of view. An approximated formula for the probabilities of the non-adiabatic transitions is derived in the adiabatic regime for the case when the parameter-dependent Hamiltonian represents a smooth curve in the Lie algebra and the quantal dynamics is determined by the corresponding Lie group evolution operator. We treat the spin precession in a time-dependent magnetic field and the over-barrier reflection problem in a uniform way using the first-order dynamical equations on SU(2) and $SU(1.1)$ group manifolds correspondingly. A comparison with analytic solutions for simple solvable models is provided.

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