We derive an extension of the standard time dependent WKB theory which can be
applied to propagate coherent states and other strongly localised states for
long times. It allows in particular to give a uniform description of the
transformation from a localised coherent state to a delocalised Lagrangian
state which takes place at the Ehrenfest time. The main new ingredient is a
metaplectic operator which is used to modify the initial state in a way that
standard time dependent WKB 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.Comment: 30 pages, 3 figure
