Dynamics of uncertainties for bound one-dimensional semiclassical wave packets

We study the time evolution of the uncertainties Delta x and Delta p in position and momentum, respectively, associated with the semiclassical propagation of certain Gaussian initial states. We show that these quantities behave generically as vertical bar P-1(t) + tP(2)(t)vertical bar, where P-1 and P-2 are periodic in time with period that of an underlying classical trajectory. We also show that, despite the overall (generically quadratic) growth in time, the uncertainty product Delta x Delta p achieves its minimum of h/2 at arbitrarily large times. Published by AIP Publishing.