The above diagram plots various the energies involved
as a function of position (*x*). The
line representing "Total Energy" is flat, because total energy is
conserved (i.e., is constant). Potential energy,
*U(x)*, is plotted in red: for |*x*|<*a*
*U(x)*=0 and for |*x*|>*a*
*U(x)*=*U*_{0}.
Total energy is the sum of
kinetic energy and potential energy: thus kinetic energy
is what you have to add to potential energy to get the
constant total energy. Kinetic energy must be positive,
so the only places the particle can actually go are where
*U(x)*<*E*. The points on the edge of
the allowed region (where *U(x)*=*E*, i.e.,
*x*=±*a* are called turning
points. There the kinetic energy is zero, and so the velocity
of the particle is zero. Just prior to reaching *x*=*a*,
the particle is moving to the right (positive velocity). Just after reaching
*a* the particle is moving to the left (negative
velocity). At *a* the particle is momentarily at rest.
The below plots the position and velocity as functions of time.

We will also find it useful to set up a square well with one
edge at *x*=0 and the other at *x*=*L*
(i.e., 2*a*=*L*). Clearly this shift in our choice
of origin has no effect on the physics, but it occasionally will make
the formulas a bit simpler.

It makes no difference to classical bound-state problems, but
an even simpler version is the "infinite square well" in which
the potential for |*x*|>*a* (*U*_{0})
is taken to be infinite.

For the infinite square well, classically the particle is
always confined to |*x*|<*a*, whereas for the
finite square well, if the particles energy *E* exceeds
*U*_{0} the particle will travel in the
region |*x*|>*a* but at a reduced speed. In
|*x*|<*a* the particle's speed is:

*v*_{max}=[2*E*/*m*]^{½}

whereas in |*x*|>*a* it would be:

[2(*E*-*U*_{0})/*m*]^{½}