In this region, 
is a constant. To remind ourselves
that this constant is positive, we shall rewrite (2)
as
where 
is a real constant defined by
By convention we always choose 
itself to be positive.
(3) is a second order linear differential equation with
constant coefficients. Such linear, constant coefficient
differential equations always possesses solutions of the form 
for
some constants 
and A. The above equation is satisfied so
long as 
. There are thus
have two linearly independent solutions 
and 
. Because the TISE is linear, we
may add these two solutions to produce a more general solution 
. Finally, because
(3 is a second order equation, there are at most two
linearly independent solutions and thus we have found the most general
solution to the TISE in region I:
The behavior of any physical state in
region I must be of the form (5). However, the
converse of this statement is not true. Just because a function obeys
the TISE in region I does not mean that it is a physically acceptable
solution. For instance, if 
in 5, then
would grow exponentially as 
, meaning
that as we go further and further to the left of the well, the
probability of finding the
particle becomes greater and greater. 
for all 
meaning that there is an infinite
weight of finding the particle at a infinite distance from the well.
Physically, we see that if 
, 
cannot represent the
state of the particle in the system we are studying.
Mathematically, we must reject this solution because the probability
cannot be normalized. These considerations lead us to the first
restriction which we place on allowable wavefunctions on physical grounds,
We thus insist that for any physically acceptable solution, 
.
To reduce the complexity of the algebra, we define 
so that our final form for the behavior of the
wavefunction in region I is
One may arrive at (7) directly by observing that it is the only solution which obeys the TISE for region I (3) and is consistent with the boundary condition (6).
