ro2m+@andrew.cmu.edu (Randy O'Reilly) (02/12/91)
I have a wacky theory about quantum wave functions that I would be interested in getting your opinions on. It goes something like this: What if all those probability functions obeying Schrodinger's equation that Quantum Mechanics (QM) is based on *actually exist*, and further that the mechanism which is creating these waves is a local, continuous-valued CA whose mesh size (cell size) and rate of update are somehow related to Plank's constant. By continuous-valued CA, I mean a CA system that has a real number in each cell, and some function operating on the neighbors that updates the value of each cell in such a way as to obey the various constraints imposed by Schrodinger's EQ? Has anyone ever modeled wave behavior of any kind using such a setup? It seems likely that constraints such as linearity and superposition would significantly restrict the number of possible functions. My best hunch without delving into it is that something like an average function would have the right qualities (eg. Cn = (Cn-1 + Cn+1) / 2). I plan to attempt a mathematical analysis of a simple 1-dimensional wave function and see what kinds of locally computed, real-valued functions would do the trick, that is unless anyone out there knows of, or has already done such a thing. Also, if this is a trivial exercise, please let me know ASAP (by email, ;-). I have been reluctant to sit down and figure out these complex wave functions, so any pointers to relevant material would be appreciated... -Randy (ro2m+@andrew.cmu.edu)
kmc@netcom.COM (Kevin McCarty) (02/13/91)
Randy O'Reilly asks about wave phenomena in CA systems with continuous state variables, but discrete space and time. This kind of system was mentioned in a general framework in a recent article I saw that dealt with the general phenomenon of pattern formation in the following kinds of systems: Model Space Time State ----- ----- ---- ----- partial diff. eq'ns C C C iterated functional eqn's C D C oscillator chains D C C lattice dynamical systems D D C cellular automata D D D D = discrete C = continuous J. P. Crutchfield, K. Kaneko, "Phenomenology of Spatio-Temporal Chaos", in "Directions in Chaos", Hao Bai-lin, ed., World Scientific, 1987. The lattice dynamical systems are what numerical analysts really study when they discretize P.D.E.'s for computation. Autonomous, non-quiescent behavior of such systems is a botheration to numerical analysts though, and they strive to avoid it by quantizing 'appropriately'. If you're looking for self-organizing wave phenomena, why then begin with a numerical analysis text and learn about 'numerical instability'. The above-referenced article looks at the topic from a fresh perspective though, and presents an interesting catalog of lattice dynamical system phenomena.