Saturday, January 19
Cellular Automata Lecture
Professor: Harold V. McIntosh.


To have some discussion between meetings I am going to try ''e-mail teaching.'' It isn't really teaching, but is a way to suggest things to read preparing for the next meeting, and mention exercises that ought to be carried out. Besides it gives a way to work on the notes peice by piece. The most interesting news is that Andrew Wuensche, for whatever reason, intends to be in Puebla for a visit, and is willing to meet with the class on Saturday. He has worked on cellular automata, spends his time between England and the Santa Fe Institute, and has coauthored an Atlas of Cellular Automata which is one of our regular references. After dividing (2,1) automata into symmetry classes, they give sample evolutions and basin diagrams for all the classes. He has also writen a simulation program, somewhat after the style of the LCAU series, and will demonstrate it on Saturday, if we can find a projector for his laptop. As far as general preparations are concerned:

1) for Wuensche's visit, look over "Linear Cellular Automata" and the cellular automata articles, particularly the one in which his book was reviewed.

2) about differential equations, read the material on SERO, the Complex Variable Notes , and The Summer of 1999.

3) for wave packets, look around the internet for waves, phase and group velocity, visual quantum mechanics, and so on. There are more waves in quantum mechanics, but there are also mechanical waves, waves in electrical circuits, hydrodynamic waves, waves in optics and wave guides, and there are probably sites with articles and demonstratioins.

4) the concrete example I mentioned. Add two sine waves, and note the beats. It is simple algebra or trigonometry to turn a sum of waves into a product, and also some simple symbolic manipulation to add a whole batch of waves with gaussian or poisson amplitudes, and compare the results. ---

The Physical Society has finally got a century of Physical Review on line, and I have finally gotten a subscription to most of it. So I have been going after old articles, or articles missing from the issues in the library. It always seems that the most interesting article is the one that is missing. The bibliography has grown to about 250 articles, but probably only 10% of that are things which we should copy and study. The rest are useful to know about and for having a complete bibliography, but many are redundant and not all are directly pertinent to the subject.

- hvm


Here is another try at making up a maioing list

Meanwhile I have been looking over making up wave packets, and there doesn't actually seem to be much information available. Of course, all the quantum mechanics books mention the Gaussian wave packet for a free particle with the Schroedinger equation. There is very little on a Dirac particle except for the assumption that you make up a packet the same way.

There are three cases (maybe four). The ordinary wave equation has second derivatives both in x and t, and so wave numbers are proportional to frequencies, and there is no dispersion. The Schroedinger equation has a second x-derivative and first time derivative, so (wave number)squared goes with energy; Schroedinger packets do disperse.

The Dirac equation has first x derivative and first time derivative, but now the fact that there are two components makes (wave number) squared = mass squared - energy squared (or some permutation of this. The fourth case would be the Klein-Gordon equation, which is back to looking like the wave equation although there is a mass there besides.

The reason for a Gaussian wave packet is that it is such a nice algebraic trick --- completing the square --- to evaluating the sum (integral) over wave numbers to get the reciprocal gaussians for position and for wave number. It would be interesting to try to track down who was the first to do this --- Schroedinger, maybe. But surely something similar must hav4e been familiar to people who were already working with wave equations, in hydrodynamics, say. From then on, the result has probably just been copied from one treatment to another.

The problem here, as I mentioned in the last class, is that boundary conditions are being completely ignored. True, complex exponentials of either sign are solutions, and that there are two such solutions is a fundamental result of using 2x2 matrices to look at second order equations. But from the point of a basis, they are sort of redundant. That is, you could ask for left-propagating waves, right propagating waves, standing waves, or some other mixture, but do you need all of them?.

That is why I suggested graphing some packets using only one sign of wave numbers. Otherwise you get these mixtures of phase velocity going in one direction and group velocity in the other, and so on. I hope everyone has looked at the internet demonstration of phase and group velocity.

When it comes to the Dirac equation, it is even harder to find explicit examples. One thing that happens is that both signs of the wave number (momentum, in quantum mechanics) are associated with both signs of the energy (frequency), so that it is possible to complete the Gaussian integral over wave numbers by combining left-moving wave functions of positive energy and right-moving functions of negative energy (except for the fact that they really also move left because of the way vave numbers combine with energy in the time-dependent solutions).

If that isn't bad enough, the fact that you have to use both components of a two-component wave function means that there are many mixtures of the components having the same absolute value, which is what gives the probabilities. Apparently trying to understand this is what is behind the Foldy-Wouthuysen transformation, and the Newton-Wigner analysis. And the Zitterbewegung.

Bernd Thaller seems to be abother of these authors whose book is forever forthcoming. He has this book ``Visual Quantum Mechanics'' with a lot of the usual illustrations, but his web page shows some Dirac packets and claims that they will appear in Volume 2. But it is now two years, and it doesn't seem to have been published yet (maybe not even finished writing?). He has wave packets squirming around in different ways as they undergo Zitterbewegung, but no text to describe just how and why. He also has a series showing the wave packet in different Lorentz frames, which makes nice pictures, but one wants to understand it better.

It is kind of like those Star Trek pictures where they try to represent the appearance of the sky when making their hyperspace jumps. There is a whole set of physics demonstrations to be found in various places as to what familiar objects would look like when moving past them at relativistic velocities.

It is remarkable that a mere plane wave creates all this complication. I'll have copies of some of the original articles on Saturday.

- hvm