Here we gather summarize the main results of what follows in this section.
To determine the scattering properties of a particular potential, one
first divides space into three regions as in Figure 7.
The potential is taken to be constant in the ``source'' region s,
extending by convention from 
to x=0. This is the region
from which particles from an external source come to interact
with the potential. It is also this region into which particles
reflect back. The potential is also be taken constant in the
``transmitted'' region t, extending by convention from x=L to
. This is the region into which particles originating from
the source in Region s may be transmitted. Finally, in Region
c, where the collisions generating the scattering takes place,
the wave functions may take on arbitrarily complicated forms. Note
that one may generalize our results to situations where the particles
are incident onto the potential from the right by changing the
direction the x-axis.
Once the potential is determined, the first step in analyzing the
problem is to solve the TISE with left-incident boundary
conditions, which state that the form of the wave function in Region t
is just some pure beam state 
of particles traveling to
the right. This determines the entire solution to the
TISE, which, by multiplying through by a normalization constant, may
always be put into the following general form,
where 
is the wave vector in Region t written as a function
of the incoming wave vector in Region s.
Once the quantum amplitudes r(k) and t(k) for the
reflected and transmitted beams, respectively, are determined,
all of the relevant issues in scattering may be studied.
Probability of Reflection and Transmission- The magnitudes of the scattering amplitudes give the probabilities of a particle reflecting or transmitting,
respectively, where 
is the wave vector about which the incoming wave
packet is centered.
Time delays- The time delays for transmission and reflection after the source packet collides with Region c are determined by the phases of the quantum amplitudes, which are defined through
The reflected packet emerges into Region s a
time 
after the source packet
collides with Region c, and the transmitted packet emerges into Region
t a time 
after the source packet
collides with Region c,
Here, is the wave vector about which the incoming wave
packet is centered and 
is the classical
velocity expected of a particle propagating in Region s.
