Cellular Automata Lecture

CINVESTAV

Professor: Harold V. McIntosh.

Oct 21

I haven't used this list of addresses in a long time, in part because the UNAM changed their e-mail service. So corrections will be included to bring it up to date if they are reported --- additions or deletions.

Two items of news.

ONE: Two issues (weeks) ago Melanie Mitchell wrote a review of ANKOS (the Wolfram book), which says pretty much what seems to be the general feeling about the book, at least in scientific circles. You can see it at at her home page, get it from Science if you have a password, or look for it in the library.

She does not, of course, verify the Universality of Rule 110, nor mention more than Matthew Cook's name. It looks like nobody is going to do this. However, she reproduces some artwork somebody did using Mathematica, which reminds me of the silliness accompanying Gardner's original article on Flexagons. Something about neckties. But that's what you'd expect if ANKOS is to be a popular rather than scientific book.

TWO: In the Complex Variables course, we have come to the topic, "Functions which Solve Differential Equations" and one of the first of the functiuons to get mentioned are the Bessel functions. In the past SERO hasn't done much with them because they belong to classical applications such as vibrating membranes, and not to quantum mechanics. In fact, I don't know of a quantum mechanical application which uses them directly, although variants such as Airy functions (Bessel of order 1/3) and spherical Bessel functions (half odd order) are pretty important.

However, looking at the differential equation in the light of the Infeld-Hull factorization method and Darboux's substitution, the equation factors which means that there should be ladder operators and even coherent states. The ladder operator part is already included as a part of the traditional identities and formulae, although it is certainly not presented in that form. Coherent states will probably turn out to be something new.

Bessel's equation arises, amongst other sources, form separating the wave equation (classical elastic waves) in polar coordinates (or cylindrical in three dimensions), multiplying functions of time, angle, and radius. Thus a circulary boundary condition, such as for the vibrationof a circular membrane (second order equation) or circular plate (fourth order equation) would invite the use of Bessel functions.

The angular coordinate implies an angular momentum, which is an eigenvalue in the theta equation and a parameter in the radial equation; it is the one which determines the order of the Bessel function. Of course the frequency of vibration, arising from separating time, fits in there too. So that gives a reason for the fact that the Bessel functions start out with some power of the radius - all those angular nodes converging at the origin don't make for very large amplitudes there. Meanwhile large angular momentum, through the centrifugal force, would keep things away from the origin.

The ladder operator works on the angular momentum index rather than the frequency, so coherent states would not be the same as in quantum mechanics. Nevertheless the idea of coherent states makes sense, since it is certainly possible to follow the evolution of a localized state. Although you don't usually pluck a drum, as with a harp or harpsichord, you most definitely strike it. The tone depends on where it is struck, nodes and antinodes of the normal modes. Of course, coupling of the sound to the air is important, and mostly depends on the fundamental where the whole drumhead is moving in the same direction at any given instant.

Coherent motion may be related to the "whispering gallery" effect. It is known that sound emanating from one focus of an elliptical ampitheatre or gallery will converge to the other focus, so that conversations can be carried on although the conversants are far apart. But there is another effect, which should be evident in circular rooms as well, where the sound clings to the walls. If they are smooth, it will travel around the perimeter.

When I say that such things haven't gotten into quantum mechanics, I should mention that there is one part, related to chaos theory, which does exactly study the propagation of particles in potentialless regions bounded by walls. Ellipses are an example in two dimensions, of course.

- hvm