the present paper II will focus exclusively on the step potential and related discontinuous potential systems, for which $V(x) = \text{const}$ in between successive steps. Discontinuous potentials are important for chemical physics, because they model steep repulsive wells, and are used in statistical theories of liquids. Moreover, they hold a special significance for QTM methods, for which they serve as a “worst-case scenario” benchmark. Indeed, conventional QTM techniques [*always*]{} fail when applied to discontinuous potentials. To date, The only such calculations that have been performed[@holland] have computed the quantum potential from a completely separate time-dependent fixed-grid calculation (the “analytical approach”)[@wyatt] rather than directly from the quantum trajectories themselves. Even if one [*could*]{} propagate trajectories for discontinous systems using a traditional QTM, the trajectories that would be generated would be very kinky and erratic,[@holland] and a great many time trajectories and time steps would thus be required.

Second, since the new $\Ppm(x)$ do [*not*]{} satisfy condition (1), unlike the paper I CPWM decomposition, the time evolution of these two component functions is clearly not that of the time-dependent  equation. Moreover, since the $|\Ppm|^2$ are constant over time \[because $|\Psi|^2$ itself is stationary, and is presumed unique\], [*the two $\Ppm(x,t)$ time evolutions 