Quantum Mechanics, Chaos Theory, and More

I’m on another one of my math/science sprees, where for a short measure of time my learning efforts are more concentrated toward disciplines of a rigorous mathematical nature (although rigor does indeed succumb to the constraints of the practical where science is concerned). Due to some fascinating things that I have begun to learn in astronomy class, and some things that I have investigated on my own time, I am becoming increasingly interested in the wild and wonderful realm of knowledge where advanced math and science intersect. I like science of a mathematical nature (in harmony with my natural tendency to favor the general and abstract over the specific and concrete), and have finally found some areas of scientific inquiry where math is applied beyond mundane uses of multiplication and addition. Quantum physics deals with wave functions, partial differential equations, and even lie algebras, a topic from modern algebra. Meanwhile, chaos theory busies itself with recursive functions and fractals. All brilliant, all fascinating!

Chaos theory is a real treat. Theoretically speaking, it deals with deterministic processes (math has yet to produce the truly random!) However, even in deterministic processes, the systems can be so complex and touchy that it is impossible to predict specific phenomena beyond a certain level of accuracy. Consider the weather—it is so dependent on minute fluctuations in air currents and slight changes in temperature that we might never be able to know for certain ahead of time whether or not it will rain. Chaos theory instead looks for patterns that are threaded through the chaos, patterns that make it possible to discover astonishing and bizarre facts that arise out of seemingly meaningless data.

For instance, some things, such as stock market fluctuations, were shown by Mandelbrot to exhibit fractal behavior! That is, the fluctuations of a day to day basis were rough copies of the fluctuations on a month to month basis. Another interesting relationship between fractals and chaotic systems is shown in bifurcation diagrams. Depending on the growth rate of an animal population, it might stabilize, or oscillate between two, three, or more values. Or, it might descend into “chaos”. Also, chaos theory deals with things called “attractors”, states to which chaotic systems tend to if they get close enough.

I’m also beginning to understand a bit more about the strange and wonderful world of quantum mechanics. We’ve encountered quantum mechanics multiple times in astronomy class, and I think that I may finally be beginning to attain a glimmer of understanding as to what Heisenberg’s uncertainty principle really means. It’s confusing, because it’s often explained with metaphors that make it sound as if the uncertainty principal is merely a facet of our tendency toward human error, rather than a property of quantum mechanics itself. But it seems that it really is a property of physics, at least insofar as it is described by current quantum mechanical theory.

Two things helped me understand—the double slit experiment, and the properties of wave addition. It is all very complex, and I am still struggling to wrap my mind around it, but it seems that practically speaking, measurement is inseparable from something being restricted, an action that effects the properties of the thing measured. For instance, if a particle travels through a small slit, it’s momentum at the point of the slit will be quite uncertain. Total momentum will be known, but its maximum direction, or its value in any particular direction, remains unknown. Thus, you don’t know where the particle will hit a sheet on the opposite side of the slit.