I decomposition is essentially identical to the present decomposition, as is worked out in detail in the Appendix.

In the second interpretation, the effect of the hard wall at $x=0$ is to cause instantaneous elastic reflection of a $\Psi_+(x)$ LM trajectory momentum, from $p = p_+ = +\sqrt{2 m E}$ to $p = p_- =
-\sqrt{2 m E}$. Afterwards, the reflected trajectory propagates uniformly backward, along the $\Psi_-(x)$ LM. In this interpretation, the trajectories never leave the allowed configuration space, $x\le0$. However, wavepacket reflection is essentially achieved via trajectory [*hopping*]{} from one LM to the other—not unlike that previously considered, e.g., in the context of non-adiabatic transitions.[@tully71] The trajectory hopping interpretation is adopted in the present paper, and in paper III, but the first interpretation will also be reconsidered in later publications. Note that for discontinuous potentials—and indeed more generally[@poirier05bohmIII]—one can regard trajectory hopping as the [*source*]{} of $\Ppm(x)$ interaction coupling.

For the hard wall case, trajectory hopping only manifests at $x=0$, the sink of all $\Psi_+(x)$ LM trajectories, and the source of all $\Psi_-(x)$ LM trajectories. If these trajectories are to be regarded as one and the same via hopping, then a unique field transformation for $r$, $s$, and 