and $s_\sc(x)$, do not in and of themselves correspond to any actual quantum scattering solutions $\Psi(x)$. In light of the bipolar decomposition ideas introduced in paper I, however, our perspective is the reverse one: for any [*actual*]{} quantum $\Psi(x)$, can one determine an decomposition such that the resultant $\Ppm(x)$ resemble their well-behaved semiclassical counterparts, and is such a decomposition unique? Among other properties,[@poirier04bohmI] the $\Ppm(x)$ LM’s should become identical to the semiclassical LM’s in the large action limit, so as to satisfy the correspondence principle. Based on the considerations of the previous paragraph it is clear that the paper I decomposition does not achieve this goal, when applied to stationary scattering states.

We defer a full accounting of these issues—in the context of completely arbitrary continuous potentials $V(x)$—to paper III, wherein it will be demonstrated how to compute exact quantum reflection and transmission probabilities (and stationary scattering states) using only classical trajectories, and without the need for explicit numerical differentiation of the wavefunction. In the present paper, we lay the foundation for paper III, by focusing attention onto two key aspects whose development comprises an essential prerequisite.

First, as the paper III approach treats $V(x)$ as a sequence of steps,[@poirier05bohmIII] 