In the case of a free particle (V(x)=0, Figure 3),
energies E;SPMlt;0 correspond to solutions everywhere classically
forbidden. Such solutions will grow exponentially either as
or 
and place ever-growing
probabilistic weight of finding the particle at infinite distances.
Such wave functions do not correspond to probability distributions and
thus do not describe physically allowed states. We reject them.
Figure 3: Potential and total energy for a free particle
For E;SPMgt;0, on the other hand, the entire space becomes classically allowed, and we find two linearly independent solutions of the form
where k may take either value 
, and
A is a
normalization constant. These states we recognize as physical; they
are the pure states of momentum 
.
