III, our approach will be to apply a similar modification to the exact quantum decomposition (actually, a [*reverse*]{} modification) such that the correspondence principle remains satisfied, and the two theories thus reconciled, even for scattering systems.

It will be shown the modified CPWM gives rise to bipolar Bohmian LMs that are [*identical*]{} to the semiclassical LMs, regardless of whether or not the action is large. Put another way, this means that the bipolar quantum potentials $q$ effectively [*vanish*]{}, so that the resultant quantum trajectory evolution is [*completely classical*]{}. Moreover, the resultant component wavefunctions, $\Psi_1(x)$ and $\Psi_2(x)$, correspond asymptotically to the familiar “incident,” “transmitted,” and “reflected” waves of traditional scattering theory. Thus, the modified CPWM implementation of the bipolar Bohmian approach provides a natural generalization of these conceptually fundamental entities [*throughout all of configuration space*]{}, not just in the asymptotic regions, as is the case in conventional quantum scattering theory.

The above conclusions will be demonstrated for both discontinuous and continuous potential systems, in papers II and III, respectively. Discontinuous potentials—e.g. the hard wall, the step potential, and the square barrier/well—serve as a useful benchmark for the modified CPWM approach, because the scattering component waves (e.g. “incident wave,” etc.) in this 