Let us now see how the expressions (17) which we determined for the transmission and reflection amplitudes fit within the Feynman formulation. We first focus on the amplitude for transmission T(k). Figure 12 shows the generation of two of the terms which contribute to the sum which gives the final transmission amplitude. According to Postulate (2) of the Feynman formulation, each of these terms corresponds to an individual history leading to transmission across the barrier.
Figure 13: List of Histories Leading to a Transmission Event
From the form of the amplitude associated with the first term, 
, we know from Postulate (3) that three fundamental processes make
up the first history. These processes (and associated quantum
amplitudes) are 1) transmission across the first step ( 
),
propagation across the barrier (p), 3) transmission across the second
step ( 
). This sequence of fundamental processes is clearly one
possible history which we associate with the final transmission of a
particle across the barrier. Figure 13.1 shows a
short-hand for representing this
sequence of fundamental events. We now connect the
arrows from Figure 13 and place the factor associated
with each fundamental process on the diagram near where the
corresponding process occurs. The amplitude associated with the
Feynman diagram for this first history is simply the product of all
factors written near it, where the values of these and the factors
used below are given by
(11,14,15,16).
The next term contributing to the transmission amplitude from Figure
12 is 
. This term was generated
as we satisfied boundary conditions in the sequence of events which
Figure 13.2 illustrates: transmission through the
first step ( ), propagation across the barrier (p), reflection
from the second step ( 
), propagation back across the barrier
(p), reflection from the first step as approached from the right
( 
), propagation across the barrier (p) and transmission across
the second step . This again is another history which we
associate with transmission across the barrier as we generate
reflections and transmissions at all points for which there is a
change in the potential and therefore the possibilities of a
reflection. We also include factors for the processes of crossing the
regions in between these scattering points. Clearly, as we continue to
patch up unbalanced boundary conditions we may generate an arbitrary
number of reflections back and forth within the barrier before the
final transmission process occurs. The third order term in the
sequence is also indicated in Figure 13.
The same logic applies for the terms leading to reflection. The
Feynman diagrams for reflection from the barrier are shown in
Figure 14. The first history, Figure
14.1, is just reflection
from the first step ( ) and corresponds to the first term contributing to
the b(x) wave function in Figure 12. The next history
involves transmission through the first step ( ), propagation
across the barrier (p), reflection from the second step ( 
),
propagation back across the barrier (p), and transmission through
the first step approaching now from the right ( 
). Successively
higher terms then involve more ricochets within the barrier before the
final transmission back into Region (s). Figure
14 also gives the digram for the third term.
Figure 14: List of Histories Leading to a Reflection Event