must be coupled together*]{}. It is essential that the nature of this coupling be completely understood, in order that the present approach may be generalized to non-stationary state situations—e.g. to wavepacket scattering, as will be discussed in future publications. The ramifications for QTMs are equally important. Accordingly, the present paper focuses on the QTM propagation of the wavefunction and its bipolar components—with a keen eye towards generality and physical interpretation—even though the states involved are stationary. This approach leads to a pedagogically useful reinterpretation of “incident,” “transmitted,” and “reflected” waves—very reminiscent of ray optics in electromagnetic theory—which is applicable much more generally than traditional usage might suggest.

Basic applications {#timeevol}
------------------

The necessary theory will be developed over the course of a consideration of various model application systems of increasing complexity.

### free particle system {#free}

Let us first consider the simplest case imaginable, the free particle system, $V(x)=0$. In this case, the exact solutions $\Ppm(x) = \Ppm^\sc(x)$ clearly satisfy the conditions of Sec. \[bipolar\], and the bipolar quantum potentials $q_\pm(x)$ are zero everywhere. Thus, the bipolar decomposition developed for bound states in paper I can be used directly with this continuum system, requiring only the slight modification that arbitrary 