LECTURE 1. FINITE MODELS 339Exercise 2. Repeat the experiment that produced Figure 1 using uniformly distrib-
uted matrices rather than normally distributed ones. This is a problem best done
with Matlab which has commands rand(50) and randn(50) giving a random and a
normal random 50 x 50 matrix, respectively.
So physicists h ave devoted more attention to histograms of level spacings rather
than levels. This means that you arrange the energy levels (eigenvalues) Ei in
decreasing order:
E1 2 E2 2 · · · 2 En.
Assume that the eigenvalues are normalized so that the mean of the level spacings
I E i - Ei+ 1 I is 1. Then one can ask for the shape of the histogram of the normalized
level spacings. There are (see Sarnak [67]) two main sorts of answers to this ques-
tion: Poisson, meaning e-x, and GOE (see Mehta [59]) which is more complicated
2
to describe exactly but looks like ~xe_rr~ (the Wigner surmise). Wigner (see [91])
conjectured in 1957 that the level spacing histogram for levels having the same val-
2
ues of all quantum numbers is given by ~xe-rr~ if the mean spacing is 1. In 1960,
Gaudin and Mehta found the correct distribution function which is surprisingly
close to Wigner's conjecture but different. The correct graph is labeled GOE in
Figure 4. Note the level r epulsion indicated by the vanishing of the function at the
origin. Also in Figure 4, we see the Poisson density which is e-x.
.. Er l•l
108 spacings05lbl
lllE
1726 spocift9sFigure 4. (from Bohigas, Haq, and Pandey [12]) Level spacing histogram for
(a) 166 Er and (b) a nuclear data ensemble.See the next section for more information on the Gaudin-Mehta distribution.
You can find a Mathematica program to compute this function at the website in
[30]. Sarnak [67], p. 160 says: "It is now believed that for integrable systems
the eigenvalues follow the Poisson behavior while for chaotic systems they follow