Hello all! It has been a very long time since I last wrote; I have been going to and from CERN as we’re preparing for the LHC. I’m in Geneva right now, and I just came back from watching En Pleine Nature (Into the Wild). Without giving away too much plot, this movie contains a bear that did not eat anyone. It made me think of Grizzly Man from a few years ago, in which another bear did. What impressed me most about Grizzly Man is that the probability of being eaten by a bear, should we find ourselves face-to-face, is not a simple 20%. It depends a great deal on who the bear is, what he thinks about humans, how hungry he is, what I smell like, the weather, his mood, etc. There’s a whole space of parameters, and some regions of this space are filled with nearly 100%, others with nearly 0%.
Let’s say we’re doing a scientific study of bears eating people. In our first experiment, we put 100 people in the woods and just count how many get eaten. Then we’d like to get more grant money, so we do a more in-depth analysis by controlling for several variables: some of our volunteers are smeared in honey barbeque sauce, others aren’t. Our sequence of studies slowly sharpen the focus on the bear-eating parameter space, identifying the high-probability regions and the low-probability regions. Where does this process end? If we could do infinitely many studies, would we find that each point is either 100% or 0% (deterministic bears)? Or not (uncertain bears)?
Framing a discussion of quantum mechanics in this way illustrates a feature that is often missed: the connection between quantization and uncertainty. Usually these two topics just fall out of the postulates with little indication that they are related. As it turns out, quantization makes fundamental uncertainty possible.
To move this discussion toward quantum mechanics, let’s consider a simpler system: the motion of a pendulum. The state of a pendulum at a given time has two parameters, the position and the momentum. Given two initial conditions, the pendulum will follow a curve in the space of momentum and position (phase space).
If we were studying this system with no a priori knowledge about pendulums, we might first block off a region and ask, “What fraction of the points in this region does the pendulum go through?” If this is an empirical study, there will be round-off error (note that the above image is a bitmap) and we will get some small fraction, greater than zero. (If you like, you can put in the epsilons necessary to make sense of this.) The result of our first study in the Journal of Pendulums and Bears might go something like this: “With such-and-such initial conditions, we sampled 1000 points in region of parameter space, and 5 of them came within of the path of the pendulum. We therefore conclude that if you are in region , you have a 0.50% 0.22% probability of catching a ride on a pendulum. Follow-up studies are underway.”
The next study involves two subregions, and , which result in 0.7% and 0.1%. The third generation involves even more regions, and after a while, we reproduce the entire bitmapped image above, where some pixels have 0% pendulum crossings and some have high probabilities.
The story gets more interesting when we consider a chaotic system with a phase space diagram like the following.
Here, some regions might actually be dense (ask a dynamicist), and we can dispense with our . We can map this with the same scientific technique of subdividing regions and sampling probabilities. Just as for the bears, and especially for the pendulum and the chaotic system, the probability sharpens as we narrow our focus, and if we could go all the way down to individual points, the answer would either be “Yes, the path did pass through this point,” or “No, it did not.” 100% and 0%. Deterministic bears.
All of the above was pre-quantum mechanics, in that I assumed we could infinitely subdivide our region. On the contrary, quantum systems are restricted to discrete sets. You’ve probably heard of the discrete energy levels of the Hydrogen atom: electrons can only take values of -13.6 eV, -3.4 eV, -1.5 eV, -0.85 eV, etc., and nothing in between. Freely-propagating electron waves can occupy a continuum of momentum states, yet there is a sense in which they, too, are discretized.
Heisenberg’s uncertainty principle puts a lower limit on the area of phase space we can consider. The famous relation is
where is an interval of space, is an interval of momentum, and is a fundamental constant of nature. On our phase space diagram, this could be a small rectangle with sides and , or it could be a blob with the same area. A freely-propagating electron occupies at least one of these blobs, and that is the sense in which it is discrete. This is a rather fluid definition of discretization, as we get to choose the shapes of our blobs, but all of quantum mechanics works this way. (In the Hydrogen atom, energy is a particularly natural variable, but we could have worked with regions in energy-time.)
If we now study a quantum system, such as the phase space of a free electron, we start by asking, “Is the electron somewhere in the universe?” The answer is “yes, 100%.” We go on with our scientific technique, until we cover the universe with pixels, each with area . Now we can go no further. We’re left with a map of probabilities, or 100% in one pixel and 0% in the rest.
If we could continue our subdivision process, as in the classical case, we could hold out for the possibility that the 30% grey regions are really populated with black and white in a pattern that only looks grey from a distance. Or we might discover that our 100% point has a shape or is fuzzy. But in a quantized system, this is impossible. We have to stop when we get to the atomic unit of phase space.
Thus, you can never be sure whether the classical system you’re studying is deterministic or uncertain, but quantum systems are certainly uncertain. This uncertainty is different from measurement error because you have already fully measured everything that you can; it is a fundamental uncertainty deriving from quantization. Without quantization, there would be no fundamental uncertainty, only a measurement error that can be reduced by a notch on the next round of experiments.
Thus, we don’t need to worry about whether a certain parameter in Saturn’s orbit is rational or irrational, because it’s quantized anyway.
We do, however, need to worry if bears are irrational.