scattering applications, especially where discontinuous potentials are concerned. One can construct a simple gedankenexperiment as follows. Let $x_L<0$ denote some effective left edge of the system, well to the left of the interaction region. At some initial time $t=0$, all trajectories on the positive LM lying to the right of $x_L$ are [*ignored*]{}, as is the negative LM altogether. One then evolves the retained trajectories over time, and monitors the contribution that just these trajectories make to the total wavefunction. In some respects, it is as if the point $x_L$ were serving as the initial wavefront for some incoming wave, that at $t=0$ had not yet reached the hard wall/reflecting surface. Of course, if the actual wave were in fact truncated in this fashion, then the discontinuity in the field functions at the wavefront would result in a very non-trivial propagation over time, owing to the high-frequency components implicitly present. For actual waves, the precise nature of the wavefront is known to have a tremendous impact on the resultant dynamics.[@jackson; @brillouin14] We avoid such complicating details by always interpreting the “actual wave” to be the full stationary wave itself, i.e. the truncation is conceptual only.

In the ray optics analogy, 