How do wave packets spread? Time evolution on Ehrenfest time scales

We derive an extension of the standard time-dependent WKB theory, which can be applied to propagate coherent states and other strongly localized states for long times. It in particular allows us to give a uniform description of the transformation from a localized coherent state to a delocalized Lagrangian state, which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator that is used to modify the initial state in a way that the standard time-dependent WKB theory can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed. (i) For scattering of a wave packet on a barrier near the critical energy, we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time; this is illustrated with the kicked harmonic oscillator.