Wave-Packet Techniques in WKB Theory.

Littlejoln, Robert

Robert G. Littlejohn

A practical difficulty of traditional Wentzel-Kramer-Brillouin (WKB) theory is that it suffers from non-physical infinities at caustics. A theoretical difficulty is that it does not possess any reasonable covariance or invariance properties under transformations in phase space. It turns out that the solution of one of these problems also solves the other, and leads to a version of WKB theory in phase space. The new theory is most easily implemented by using wave packets.

Jean‐Francis Bloch

Marc Boiron, M. Lombardi

We propose a semiclassical wave packet propagation method relying on classical trajectories in a complex phase space. It is based on the Schrödinger wave equation and the usual expansion with respect to ℏ, except that the amplitude of the wave packet is taken into account at the very zeroth order, unlike in the usual WKB method where it is treated as a corrective or first order term. Formally, it amounts to making both the wavelength and the width of the wave packet tend to zero with ℏ. The action and consequently the classical trajectories derived are complex. This method is tested successfully in many cases, analytically or numerically, including the bounce and even the splitting of the wave packet. Our method appears to be much more accurate than the WKB method while less computationally demanding than the Van-Vleck formula. Moreover, it has a particularly interesting property: the singularities (caustics) of the usual semiclassical theories do not appear in this formalism in all cases tested.

Edward A. Olszewski