To implement the Feynman Formulation as described in Section 2, we need rules for deciding what are the valid histories which we must consider in our sum for Postulate (2) and determining the fundamental events and their quantum amplitudes for Postulate (3). This section extracts general lessons from the specific rules which we determined in the discussion of the previous section of scattering from the barrier of Figure 1.
In Section 3.4 the histories which we considered for transmission or reflection from a collision potential arose from all ways in which a final transmission or reflection is eventually generated after all sequences of transmissions or reflections from the points in space for which we must generate reflections to match the proper boundary conditions. As we know, such reflections are generated wherever there is a disturbance in the potential. This then completes the specification of the histories which must be considered for Postulate (2) for any one-dimensional scatting problem:
The most common such disturbances include sudden steps
(discontinuities) in the potential as in Figure 1 and
also often consider reflection generating terms in our potential such
Dirac 
functions. In Section 6 we consider
problems involving smoothly varying potentials.
With the allowed histories specified, we must then complete Postulate (3) by specifying the fundamental events which make up the histories and the associated quantum amplitudes. As we saw in Section 3.4, the final quantum amplitude for a given history consists of the product of the reflection and transmission amplitudes associated with each boundary condition and propagation factors between matching boundary conditions. The fundamental events are thus scattering from points of disturbance in potential and propagation across regions of constant potential.
The quantum amplitude for propagation is simple in form, we have seen
that it is simply 
, where a is the length of the region
and k is the constant wave vector across the region. The quantum
amplitudes for scattering from steps and 
-functions are simple
to describe but must be solved for on an individual basis. As we saw
in Section 3.4 the quantum amplitude for each
fundamental reflection and transmission process is just what is
required to satisfy the boundary conditions in the Schrödinger
approach, and are therefore the same quantities which we originally
defined as the quantum amplitudes for scattering from that disturbance
in isolation.
Apart from the discussion of Feynman diagrams for classically forbidden regions in Section 3.6 (which is optional material provided for the interested student), this, completes the specification of Feynman rules for scattering in one dimension. We here state the final set of rules for Postulate (3) in their full general form including the treatment of classically forbidden regions.
