Probability of violation of the Ehrenfest principle in fast passage

ADIABATIC passage through an almost degenerate intermediate state is a common experimental technique. It is ba sed on the Ehrenfest principle which states that if the external conditions for a system are changed slowly enough, the change of the system can be predicted by computing qu asi statically the modification of its initial quantum state. The Ehrenfest principle is not a law of na ture beca use the system has an alternate path it can follow, which might appropriately be termed 'inertial*. This path is favored by a rapid change of the external variables. Among the two paths, the adiabatic one has a privileged position. Thus, it has been shown re cently that if the external change is measured by a "slowness parameter" inversely proportional to the time ta ken to effect the change then the Ehrenfest principle is valid to all orders in th is para meter [l]. The following lines are to derive a rigorous formula for the probability that the adiabatic path is not followed in the simple case of the near crossing of two quantum levels.