There is a standard choice for the two linearly independent solutions
of the TISE in cases where the particle is classically allowed at
infinite distances in both directions. Note that in the case of
particles moving in free space, the solutions 
for k;SPMgt;0
contain only currents moving to the right as 
. This is an appropriate state for when one imagines
particles originating from a source on the far left of the
system. The boundary condition that there be no left-moving currents
for is termed the left-incident boundary
condition. Setting this condition removes all freedom and specifies a
unique solution to the TISE, once a convention for normalization have
been specified. The right-incident
boundary condition, that there be no right-moving currents as 
, specifies another, linearly
independent solution.
These
boundary conditions are sketched in Figure 7. As
indicated in the figure, specifying either condition on one side of
the system will in general generate solutions
traveling in both directions on the other side.
Figure 7: Left- (top) and Right- (bottom) Incident Boundary Conditions:
The source, reflected and transmitted beams in either case are
indicated by s, r and t, respectively. (We label the region x;SPMlt;0
``Region s'' because our convention is to work with left-incident
boundary conditions.)
As indicated in Figure 7, we may generally divide the
problem into three regions. A scattering or collision region
(Region c in the
figure) where there is a disturbance in the potential, and two regions
(Regions s and t), where particles propagate normally. Although the
form of the wave function in Region c may be complicated, the
solutions to the TISE in Regions s and t will be linear combinations
of plane waves with wave vectors given by 
, where 
is the value
of the (constant) potential in the corresponding region. The precise
form of the linear combination will be determined by the solution of
the TISE in the scattering region.
For the left-incident case, which we will associate with values k;SPMgt;0, in general we will find solutions to the TISE of the form,
where we have taken care to follow our usual practice and centered our
functions on the boundaries of the regions in which they are defined.
In 11, k refers to the wave vector in region of the source, Region
s. This wave vector
determines the energy of the state and thus also the wave vector in Region
t, which we have written as a function of the source wave vector, 
.
The relation 
determines
this function to be
where 
is a measure of the
difference in potential going from Region s to Region t. This
sign in the definition of 
is generally chosen so as to make
a real number.
We have left the form of the solution in Region c unspecified because the form of the wave function in this region is not directly needed to determine the behavior of the wave packets entering and leaving the scattering region - only the final form of the solution to the TISE in the regions s and t. Currently, the factors S, R and T are undetermined because we have yet to choose a normalization convention for such states. Finally, we note that in the case where Region t classically forbidden, we insist upon exponentially decaying solutions, and only the factors S and R then play explicit roles in the scattering of particles.
