In general, scattering states are pure states of energy
(solutions to the TISE) where there is a classically allowed region at
either 
or 
, or both.
States with energies 
or 
in the Asymmetric
Finite Square Well in Figure
6, for instance, are scattering states. On the other
hand, a state with energy 
would not be a scattering state
but rather a bound state because both regions 
are forbidden. Finally, a state with energy 
is
forbidden from all regions, must always curve away from the
x-axis and thus grows
exponentially in at least one of the limits
and would never be acceptable physically.
Figure 6: Ranges of disallowed, bound and scattering states in an
Asymmetric Finite Square Well
The TISE is a second order equation, we thus expect its general
solutions to involve two degrees of freedom, one of which corresponds
to the normalization of the wave function, leaving only one remaining
degree of freedom in the solution. For bound state energies such as
, we impose two boundary conditions, that the exponentially
component of the wave function be zero in decay exponentially in both
forbidden regions . It is in general
impossible to satisfy both of these conditions with the one degree of
freedom remaining to our solutions. This is why we find solutions for
bound states only under very special circumstances, only at the
allowed energies.
For scattering states such as where exactly one region of the
two regions at infinity is forbidden, we need to impose only one
boundary condition, exponential decay in the forbidden region. (The
oscillatory form of the solutions in the classically allowed region
always results in physically acceptable behavior.) The one boundary
condition which we now impose may always be satisfied with the one
remaining degree of freedom in the normalized solution of the TISE.
The solution is thus completely determined,
and there is exactly one solution for each energy in this range.
Finally, when we move to energies in the range of , the states
naturally remain oscillatory in both allowed regions as . The one remaining degree of freedom corresponds to the
fact that there are two linearly independent physical solutions to the
TISE for each energy in this range which we may then mix together
arbitrarily. This is precisely what we found for particles
propagating in free space in Section 2.3.