## Falling p.5

### Making Sense of these Results

What can one make out of these results for "falling" bodies
which show the position probabilities don't change with time?
One way of making some sense out of the QM results is to
ask: "What would classical mechanics tell us if the only information
we had about the particle was its energy?" Looked at this way,
it's not surprising the position probabilities don't change with time:
we always have exactly the same information (only the given constant energy)
so we must have the same probabilities. We can say that the particle
must be bouncing around in the classically allowed region:
0<*z*<*z*_{max}=*E/mg*.
All subdivisions within the classically allowed region are not
equally probable, as the particle is whizzing through the parts near
the ground, while spending more time in the parts through which
its slowly moving. The time (*dt*) a particle spends in a little
region *dz* depends on its speed: *dt*=*dz*/|*v*|.
The probability (*dP*) of finding the particle in *dz* is then proportional
to the time it spends there: *dP*=*dt*/*T*,
where *T* is the period: the time it takes the particle to
complete a bounce-cycle. In the below, the red line shows the classical
probability density, and the black the exact QM result for *n*=32.

You can see that the classical result is essentially the
average of the quantum mechanical result. Saying it differently:
the main difference between results is QM has "waves".
(Under many conditions the wavelength is so short, that any
position probability measurement lumps together lots of waves,
and hence just measures the average, i.e., the classical result.)

The wave nature of everything is the main surprising result
of QM. De Broglie first argued for it seeking a unified
picture of everything. So if part of everything (e.g., light)
shows wave properties, all of everything must show wave properties.
He said the wavelength of a particle was given by:

=*h/p*

where *h* is Planck's constant and *p* is the
particles momentum. Of course, in quantum mechanics particles
don't have *a* momentum, they have a distribution of
of momentum, i.e., a superposition of simple sinusoidal waves.
The peak-to-peak distance is essentially a measure of the
most common momentum at that location [not a totally correct
statement, but good enough for this document].

Thus where the particle is moving quickly--like near the ground--
expect short wavelengths, where the particle is moving slowly--like
near *z*_{max} expect long wavelengths.

There is one other definite difference between the classical
result and the quantum result. Classically you *never*
find the particle above *z*_{max}. In QM there is
a small chance of finding the particle in the classically disallowed region,
and the chance gets ever less as you move further and further
into the classically disallowed region.

Here is one last look at a wavefunction. Notice:
(1) smaller amplitude where the particle is moving fast (2) shorter
wavelength where the particle is moving fast (3) small and decreasing
probability of finding the particle beyond the classically allowed region
[which in this case ends at about *z'*=28, see the red dot].

The mathematical version of the above three rules is called
the WKB approximation.

Next