Note that by the matching of boundary conditions of value and slope at each of the interfaces, the entire solution at a given energy E is determined by the solution in one of the regions (in this case, the solution in region where we started). This is just a reflection of the fact that the solution of a second order differential equation is completely determined by any two conditions, in this case the value and slope of the solution at any point. The entire solution we have built up is
First note that there a single degree of freedom A left. This
freedom is just a constant multiplying the overall solution. This
degree of freedom cannot be determined from the TISE alone because the
TISE is a linear equation and any solution to it may be multiplied by
an overall constant. Ultimately, A will be determined by the
physical condition of normalization of 
as a quantum
mechanical wavefunction. For the solution to be normalizable, we must
also avoid exponentially growing solutions in region III. Because, the
differential equation completely determines the form of 
once
we set the behavior in region I, we are only be able to meet the
condition that there is no growing part to our solution in region III
when k and 
have a very special relationship. Because
and 
are
ultimately defined in terms of E, it is precisely this relationship
between k and 
which will determine the physically allowed
eigenenergies of our system, the energies of the pure energy states of
the system.
