## Abstract

This paper deals with non-adiabatic processes (i.e. processes excluded by the adiabatic theorem) from the geometrical (group-theoretical) point of view. An approximate 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, respectively. A comparison with analytic solutions for simple solvable models is provided.

Original language | American English |
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Pages (from-to) | 1741-1752 |

Number of pages | 12 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 34 |

Issue number | 8 |

DOIs | |

State | Published - 2 Mar 2001 |

Externally published | Yes |