By giving us a short-hand notation, Feynman diagrams are very convenient for proving identities in quantum mechanics. In this section we apply the diagrams to investigate how the transmission and reflection probabilities and time delays compare when particles approach a collision potential from all possible directions. This will not only give us interesting insights into the physics of the scattering process but will also provide us with important identities for our later general discussion. The results we find here are not necessary in our construction of the Feynman formulation, but appear here to help familiarize the student with representing wave functions as diagrams.
We first compare what happens when we approach an arbitrary collision potential from the left to what happens when we approach the mirror-image of that potential from the right, as in Figures 4b and 4c. By symmetry and physical reasoning, the transmission and reflection amplitudes and delays must be the same in both cases. As these two physical quantities determine both the magnitude and phase derivatives of the quantum amplitudes, the quantum amplitudes for reflection and transmission in these two situations may differ by at most a constant phase factor. Below in Section 3.2.1 we give a formal proof for the interested student using Feynman diagrams that the quantum amplitudes are precisely equal for both cases.
Figure 4: Scattering from a general potential from various directions: a)
Initial approach to the first scatterer from the left ( 
), b)
left-approach to the second, reflected scatter scatterer ( 
), c) approach to the first
scatterer from the right, d) time-reversal of c). The two scatterers
have width a.
We then turn to what happens when we approach the same potential but from two different directions, as in Figures 4a and 4c. Now, when the overall potential is not symmetric, the situation is not so clear, and the results somewhat surprising. It turns out the the probabilities for transmission and reflection are in fact always identical, regardless of the direction from which we approach the potential! Although the time delays for reflection need not be identical, the time delays for transmission are always equal. Finally, the sum of the transmission delays for both directions must equal the sum of the two transmission delays. For the interested student, we use Feynman diagrams to prove these intriguing properties in Section 3.2.2 below.
Note on notation: Because the reflection amplitudes across a barrier are different in the two directions, we shall denote them by r and r', where a ``''' denotes right-incidence and the lack of one denotes left-incidence. Because a theorem shows that the transmission amplitudes must always be identical in both directions, it is not necessary is to make a distinction between t and t'. Nonetheless, we will at times use both t and t' to remind us of the physical origin of the different factors.