Semi - classical Propagation on j log hj - Time - Scales

We study the quantum evolution of a coherent state originally located in phase space on the unstable fixed point of a multiple well potential in one dimension. We show that for sufficiently long times the coherent state “splits” into two distinct components, each of which localizes semi-classically on a neighbouring unstable fixed point. We show moreover that during the transition the coherent state is distributed along the classical trajectory joining the two unstable points according to a density which we compute explicitly. At the times we explore the quantum dynamics of the coherent state it is no longer determined by the classical dynamics of its center, but rather by the classical dynamics of the trajectories that are at a distance p h of its center.