Phase integral theory, coupled wave equations, and mode conversion

Abstract

Phase integral or WKB theory is applied to multicomponent wave equations, i.e., wave equations in which the wave field is a vector, spinor, or tensor of some kind. Specific examples of physical interest often have special features that simplify their analysis, when compared with the general theory. The case of coupled channel equations in atomic or molecular scattering theory in the Born-Oppenheimer approximation is examined in this context. The problem of mode conversion, also called surface jumping or Landau-Zener-Stuckelberg transitions, is examined in the multidimensional case, and cast into normal form. The group theoretical principles of the normal form transformation are laid out, and shown to involve both the Lorentz group and the symplectic group.