Let us now consider scattering from a potential with three
disturbances, such as the three 
-functions of strength D in Figure
17. The scattering amplitudes for a potential 
, as determined from solution of the TISE, are
where 
. Because the -function is
symmetric, the scattering amplitudes are identical for both left- and
right- incidence. Figure 17 shows four diagrams for
transmission which illustrate the far richer variety of diagrams which
we now face. Although there is a much richer variety, the number of
these diagrams is still countable as they were in Figure
13. The meaning of the thick lines in the diagram
will be explained presently.
Figure 17: Transmission diagrams from a triple scatterer
Again, we seek a physical principle to organize the Feynman sum. The result will be a general form which is independent of the nature of the three scattering centers, and for which we may generate specific results by substituting in particular values for the quantum amplitudes.
The first two diagrams shown are part of an infinite subsequence of diagrams where the particle ricochets an arbitrary number of times between the first two barriers and then travels directly across to and though the third barrier. The sum of this subset of diagrams is
where the 
factor we recognize as the complete Feynman sum for
transmission across the first two barriers. The physical reason why we were
able to factor this term out is that all processes which lead to
transmission across the three barriers, no matter how complicated,
begin with a sequence which carries the particle across the first two
barriers. The sum of all of these beginning parts, , is clearly the sum
of all ways in which we may cross the first two barriers and therefore
must be the Feynman amplitude for transmission across the two first
two barriers viewed as an isolated unit, hence the notation .
All that we are doing
is changing our view of the system and its fundamental processes. We
are now beginning the consider the potential as consisting of two
scatterers, the first
being the first two -functions taken as a unit and the
second being the third 
function.
In the third and fourth diagrams of Figure 17, the
complete processes of all possible ways to transmit across the first
two barriers as a unit is indicated by a very thick line. Such a
notation is
sometimes referred to as a dressed interaction in many-body physics.
By using this notation, the diagrams in Figures 17.3 and
17.4 now serve to represent two different
infinite subsets of diagrams. The total quantum amplitudes of these
two subsets are given
by the product of the labels in the diagram, 
and 
, respectively.
The idea of summing together an infinite subset of diagrams into a unit to produce a new fundamental process with its own quantum amplitude and diagram is known as renormalization. As we shall see, renormalization is a very powerful tool in organizing infinite sets of diagrams. This idea is used to deal with certain infinities which arise in quantum electrodynamics, the theory for which Feynman shared the Nobel prize with Schwinger and Tomonaga in 1965.
The the third and fourth
diagrams in 17 begin another sequence of
diagrams which we must consider. In this sequence, after somehow
managing to penetrate through the first two -functions, the
particle reflects once off of the third -function, then
reflects from the unit consisting of the first two functions,
after some number of ricochets, and then finally travels
directly across to and though the third barrier. The sum of this
infinite collection diagrams clearly involves 
, the amplitude
for reflection from the right from the renormalized two
-function unit.
Algebraically, this part of the sum is
We may now combine the renormalized left-transmission with our
new renormalized right-reflection from the double -unit
to describe all of the transmission diagrams for the triple
-potential. So far, we have dealt with direct transmission
and transmission after one ricochet between the third
and our renormalized function pair, 
. The final diagram in Figure 17 gives the
next set of terms in this renormalized sequence. In Figure
17.5 there are two ricochets between the
renormalized double- unit and the isolated third
function. Continuing in this way, now simply generated the general
sequence of scattering between two barriers shown in Figure
15 but where where the nature of the two scatterers
is different.
This completes the renormalization process. We now have a new,
simpler set of Feynman rules for the problem at hand. The new set of
rules now has a different set of fundamental processes, namely
reflection and transmission across the first two functions as
a renormalized unit, transmission and reflection from the third
function, and propagation between them. The fact that the
Feynman formulation is not specific about the form of the fundamental
processes is what makes the renormalization process possible.
Renormalization represents a change in what is considers to be the
fundamental processes. The condition on what may be renormalized is
that any history must be able to be sequenced unambiguously into the
new fundamental processes. Grouping the first two functions
together as a unit clearly satisfies this condition, because a
scattering event from the third function can never be
interleaved with the ricocheting between the first two
functions. One could not, for instance, renormalize the
outer two functions into a single unit, because then
scattering processes within our ``fundamental process'' of scattering
between the outer functions could be interrupted by
scattering processes from the central function.
The final transmission amplitude across the three scatterers has now
been reduced to a scattering problem between two scatterers, a
renormalized unit made from the first two and the third ``bare''
scatterer. We may therefore use the result (21) with five
substitutions. First, the propagation amplitude between the two
barrier now becomes the transmission amplitude between the
renormalized unit and the bare scatterer, 
. The
transmission across the first barrier now becomes transmission across
the renormalized unit, 
. Reflection off of the
first barrier from the right now becomes reflection from the
renormalized unit, 
. Finally, the reflection
and transmission amplitudes from the second barrier become the
reflection and transmission amplitudes from the isolated bare
scatterer, 
and 
, respectively.
Alternately, we get precisely the same result by performing the Feynman sum corresponding to Figure 18,
As we have used no special properties of the amplitudes, this result
is general. As we saw in Section 3.2, the
transmission amplitudes in both directions are equal for any potential
so that the amplitude for right-incident transmission is the same, 
.
Figure 18: Complete set of renormalized transmission diagrams from a
triple scatterer
Reflections from three general scatterers may be handled in the same
way. We apply (22), where again , , , and ,
but now we need also reflection from the first barrier to become
reflection from the renormalized unit 
and
transmission from the right from the first unit to become transmission
from the right through the renormalized unit, 
, where the equality follows from the discussion in Section
3.2.2. This gives,
Again, one could obtain the same result by drawing and summing the appropriate renormalized Feynman diagrams.
Because the reflection amplitudes from the left and right need not be
equal, to maintain generality we must give also the right-incident
reflection amplitude 
. Again we use (22) where now
, transmission across the first barrier into the
scattering region is right- transmission through the bare scatterer
, reflection from the second barrier is
right-reflection from the renormalized unit 
,
direct reflection from the first barrier is right-reflection from the
bare unit 
, the ricochet reflections from the
first barrier are left-reflection from the bare scatterer 
and final transmission through the first barrier is
left-transmission through the bare unit 
. The
final result is then
Again, one could obtain the same result by drawing and summing the appropriate renormalized Feynman diagrams.
To complete the computation of the scattering amplitudes For the
specific case of the three functions in Figure
17, we take 
and r=r' and t=t' from
(25). In terms of these amplitudes, the specific
values for and come directly from (21) and
(22),
