June 28, 2006

The Uncertainty Principle

Werner Heisenberg, On the Physical Content of Quantum Kinematics and Mechanics, 1927

This is probably the single most misunderstood and misinterpreted paper in the history of physics. This is partly because you can sum up one of the paper's observations in a single (incorrect) sentence that is easily remembered and (mis)understood by the average high-school physics student: "You can't know the position and velocity of a particle at the same time."

A more accurate description of what the paper actually says is: An electron's position and momentum cannot, even in principle, be measured at the same time and with perfect accuracy by bouncing light off of it. In addition, there are a number of other experiments where similar restrictions hold on pairs of properties to be measured, including energy and time, and angular momentum and angle.

So, first, you can know the position and momentum of a particle at the same time, just not with perfect accuracy. If you want to know the position more accurately, the accuracy of your knowledge about the momentum will decrease correspondingly, and vice versa. Second, this uncertainty applies only to electrons when you measure their position or momentum by bouncing light off of them, not necessarily to every particle with every method of measurement. Third, there are other pairs of properties which have a similar relationship when you measure them in particular ways.

Scientists have always acknowledged that there is a degree of error in every measurement. However, until quantum mechanics was worked out, it was thought that you could, in principle, take the time and trouble to make your measurements as accurate as necessary. Heisenberg's uncertainty principle is, in this way, very novel: he has shown that you cannot, even in principle, make infinitely accurate measurements of certain pairs of properties in certain ways.

The bit about "in certain ways" is actually pretty important. Einstein, Podolsky, and Rosen pointed out that you can measure the position or momentum of an electron without actually bouncing light off of it. In the jargon of quantum mechanics, when two particles are "entangled", taking a measurement of one can tell you something about the other. For example, if an event produces two electrons travelling in opposite directions, but with equal speed, then measuring the speed of one will automatically tell you what the speed of the other is. (Or an event that produces two electrons with opposite spin. Or two photons with opposite (or identical) phase. All these are called "entangled" particles.) The so-called EPR "Paradox" is an argument that either quantum mechanics can best be explained as the result of a "hidden variable" model, or entanglement is causing spooky action-at-a-distance, which is impossible. In other words, E, P, and R think there's a simple explanation for what's actually going on down at the quantum scale, but because we can't measure anything that small directly, we have quantum mechanics as a model instead, which describes the much more complicated results of the measurements we can actually take.

I've made an attempt to be neutral about everything I've said so far. Now I'm going to complain about the level of jargon used to describe quantum mechanics in the popular press and in high-school textbooks, a problem which has reached such a level that otherwise reputable sources can be caught speculating about teleportation, faster-than-light communication, and computers that can parallelize their computations through the use of parallel universes. Take the word "entanglement", for example. The word has associated physical imagery which is totally incompatible with its technical meaning in quantum mechanics, in much the same way that the word "tree" is used in computer science to describe something that is much, much more abstract than a physical tree. Nonetheless, it's easy to find news articles about quantum entanglement that talk about using entangled particles to communicate faster than light. This has been going on for so long that science fiction novels now use entanglement as their standard explanation for FTL communication. I've read 3 using this device this month alone, including Iron Sunrise, by Charles Stross.

In the defense of the people writing these news articles and science fiction stories, Einstein and co. did show that either there's a hidden variable model, or else there are non-local effects (i.e., FTL effects), and modern physicists generally say that there's no need for a hidden variable model. But, everyone already knew at the time they made their argument, that entanglement cannot be used to transmit information faster than light. If you ask any professional physicist about entanglement, he or she will tell you that FTL communication using entangled particles is impossible.

Quantum teleportation, quantum computing, quantum superposition, and a number of other terms are equally badly misinterpreted by non-physicists (and even by non-quantum physicists), to the point that a lot of science fiction these days is as much science fantasy as the 1930s Astounding Stories about psychic powers or native venusians.

Okay, rant over. I'm done.

Since Heisenberg wrote his essay about uncertainty, the concept has been generalized and expanded quite a bit. Heisenberg dealt with certain pairs of properties being measured for one particle, while modern physicists can calculate the uncertainty relation between any two measurements on a system. This is one of the reasons that the EPR "Paradox" is generally disregarded by modern physicists: although non-locality is impossible, entanglement still can't be used to measure a particle's position without any uncertainty, because the whole system has uncertainties. E.g., the position of an event that produces two "entangled" particles is uncertain, so measuring the position of one can't tell you the position of the other to the same degree of precision.

Personally, I think there's something to the whole idea of a hidden variable model. There's Bell's Inequality to argue against that view, but it doesn't seem to be a watertight argument. However, I'm barely qualified to ask these questions, much less understand the answers. On the other hand, I'm not the only one who suspects that modern physics might have run off the rails at some point in the past couple of decades.


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