quite a number of conceptual and computational advantages. For instance, the standard description of the hard wall stationary states would decompose these into plane wave components interpreted as “incident” and “reflected” waves. This language suggests a process, or change over time—i.e. a state that is initially incident, at some later time is somehow transformed into a reflected state. Nothing in the standard description, however, would seem to render transparent the usage of such terminology, i.e. $\Psi(x)$ is stationary, and so the reflected and transmitted components are in fact [*both*]{} present for all times. Of course, a localized superposition of stationary states, i.e. a wavepacket, may well exhibit such an explicit transformation over the course of the time evolution, as such a state is decidedly non-stationary. Indeed, wavepackets are relied upon by the more rigorous formulations of scattering theory, in order to justify the use of terms such as “reflected wave,” even in a stationary context.[@taylor] Such formulations, though certainly legitimate, seem always to require a clever use of limits, the subtle distinction between unitary and isometric transformations, and other esoteric mathematical tricks.

On the other hand, the time-dependent bipolar quantum trajectory hopping picture presented above provides a physicality to such 