In quantum systems, we find that the
distribution of
energy is not continuous. Rather, energy is distributed in
discrete
elements. The plot depicts the spacing of these discrete energy
elements
for three different situations. The allowed values for the energy are
plotted
as lines with increasing values from bottom to top. The total number of
allowed
energies is the same in all three situations, but the type of spacing
looks
rather different for each.
On the far left, the energy level spacing corresponds to a completely
random
distribution. There are clusters of levels very close together, as well
as
areas where the levels are very far apart. For this case, one
just
"rolls dice" to determine the energy levels. This way, it is
easy to see that the areas where the spacing appears clustered arise
from the
fact that some numbers are arbitrarily rolled more than once.
On the far right is the exact opposite situation. Here, it is
evident
that the energy spacing is extremely regular. Each of the levels
is spaced
evenly. This type of spacing could result from a familiar
situation such
as an ideal harmonic oscillator.
The plot on the left and the plot on the right result from situations
that can
be observed in nature. However, the middle plot is even more
consistent
with nature. Note that the spacing here is not as random as the
left, but
also not as regular as the right. This spacing distribution is
commonly
found in actual many-body quantum systems such as nuclei.
A more mathematical understanding of the above plots can be obtained by
analyzing the equations that correspond to each one. The far left
yields
spacing plot known as the Poisson spacing distribution. When the
spacing
distribution is analyzed more thoroughly, by taking the actual spacing
divided
by the average spacing, the resulting graph is similar to the function
exp(-x),
where x is the level spacing. This reaches a maximum when x = 0,
meaning that
the energy levels have a predisposed tendency to cluster together. This kind of spacing distribution is
directly related to an element of weak interaction within the system. It is also evidence that a regular system
will yield the chaotic Poisson spacing distribution.
When the spacings of the middle plot are analyzed in the same manner,
the resulting
graph follows a function called the Wigner spacing distribution. Unlike
the
left plot, this plot results from stronger interaction within the
system. Any element of interaction will
cause the
energy levels to have a tendency to repel each other like opposite
charges. The right plot is representative of an ideal repellent
situation. The middle plot, however, shows what happens when
randomness
and regularity fight against each other. The randomness dictates
the
levels to be clustered. But instinctively, the levels also want
to repel
each other. The spacing distribution of the middle plot is thus a
combination of two ideal cases: completely random and completely
regular. Also, this distribution is
evidence that a chaotic system will yield the regular Wigner spacing
distribution.
This webpage was written by Catherine
Kennedy as part of
her work in the 2004 Research Experience for Undergraduate (REU)
program at
MSU.