language that is immediately apparent. Over the course of the time evolution, although the wavefunction as a whole is stationary, each individual [*trajectory*]{} is first incident from the left, then collides with the hard wall, and is subsequently reflected back towards the left (i.e. towards $x\ra-\infty$). The bipolar quantum trajectories are all classical, as the bipolar quantum potentials, $q_\pm(x)$, are zero everywhere except at the wall itself. Interference arises naturally from the superposition of the two LMs—i.e., from the trajectories that have already progressed to the point of reflecting, vs. those that have not reflected yet. In contrast, since $\Psi(x)$ itself exhibits very substantial interference, and an infinite number of nodes, the traditional unipolar QTM treatment would be very ill-behaved, i.e. $R(x)$ would oscillate wildly in the large $k$ limit, and $Q(x)$ would be numerically unstable near the nodes. Apart from these important pragmatic drawbacks, the incident/reflected interpretation of the quantum trajectories would also be lost.

The bipolar quantum trajectory description of the hard wall system is very reminiscent of ray optics, as used to describe the reflection of electromagnetic waves off of a perfectly reflecting surface.[@jackson] Indeed, much can be gained from applying a ray optics analogy to quantum 