Reconciling semiclassical and Bohmian mechanics. III. Scattering states for continuous potentials

In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition psi = psi1 + psi2 was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large-action limit. The corresponding bipolar quantum trajectories, as defined in the usual Bohmian mechanical formulation, are classical-like and well behaved, even when Psi has many nodes or is wildly oscillatory. A modification for discontinuous potential stationary scattering states was presented in a second, companion paper [C. Trahan and B. Poirier, J. Chem. Phys.124, 034115 (2006), previous paper], whose generalization for continuous potentials is given here. The result is an exact quantum scattering methodology using classical trajectories. For additional convenience in handling the tunneling case, a constant-velocity-trajectory version is also developed.