

Phase
Transitions and How Energy related to SHO
How can you model the transition from liquids to
gases? What is the difference between gases and liquids (at the molecular
level)?
Let's look at the potential energy stored in the
system of two atoms. One is taken to be fixed at the origin and the
other oscillates about it.

For 1D systems, one can usually use the analogy that
the potential energy curve causes behavior just like a roller coaster with
the same shape. (This works because gravitational potential energy
is proportional to height.)

To determine the motion, graphically add the total energy
to the potential energy graph.

If E and U cross then you have a point where K is 0
a "turning point".

If there are no turning points between where the particle
is at any given moment and infinity, then the particle is free to travel
to infinity, in other words it is "unbound".

If a particle is trapped between turning points (it
simply oscillates between these points), then the particle is "bound".

Any place where the particle does oscillate you can
approximate the potential by a parabola. This means that, for small
oscillations about the equilibrium point, the motion can be treated as
simple harmonic oscillations. (Essentially this is what we've already
done when we looked at ideal springs. We said we were going to approximate
their complicated behavior, by a relatively simple potential/ force.)

If E is below U then this tells you that K must be less
than zero, which is impossible so this is an area that is forbidden.
(Well,... this is no longer true when working with small, quantum systems.
There it is possible for particles to exist in classically "forbidden regions";
this is what's known as tunneling.)

Notice that the slope of the potential curve gives you
the force on the particle. A steeper slope means a greater force
on the particle. (This about how a car behaves on a roller coaster
track.) This force points "down hill" if the slope is positive,
then the force is in the negative direction. (F=  dU/ dx)
Notice how we can approximate the valley of this
complicated curve by a parabola. This is what we try to do with other
complicated systems. Often if we only have small oscillations about
an equilibrium point, we are able to describe that portion of the potential
energy curve by a parabola. We know the solution to this problem;
therefore we approximate the complicated system with the simpler simple
harmonic oscillator.
This is the beauty/ power of simple harmonic oscillations.
Any oscillation can be approximated by this solvable model.

