In practice, the rules for formulating the Feynman approach within a given context are often derived from another established formulation. The overall procedure we shall follow here for scattering problems in one dimension is the same as employed in far more complicated contexts. The procedure for constructing Feynman rules is very intuitive and usually precedes by generalizing from specific examples. The example with which we shall work is scattering from a potential barrier as pictured in Figure 1.
The first stage in the process is agreeing on a pictorial notation for the relevant mathematical functions. This we do in Section . We then immediately, in Section 3.2, make use of our new diagrams to prove some very interesting theorems about how the transmission and reflection probabilities and delays compare when we approach a potential from both the left and the right.
Once we are familiar with the diagrams, we then use the diagrams to compute in a new way, but still within the Schrödinger formulation, the scattering amplitudes for a specific problem. This new approach (Section 3.3) to computing the scattering amplitudes is sufficiently close to the postulates of the Feynman formulation that in Section 3.4 we are able to prove that the Feynman formulation is indeed equivalent to the Scrödinger approach, for the specific problem which we have taken up. Section 3.5 then generalizes our results from the specific problem to arbitrary problems in one dimensional scattering theory. As the last topic in this section we then complete our discussion by specifying specific Feynman rules for scattering problems involving forbidden regions.
After developing the Feynman rules for one dimensional scattering, we then take up in Section 4 the mathematics of performing the resulting Feynman sums.
