UNCERTAIN UNCERTAINTY?

I have been hanging around physics departments, for better or worse, in sickness and in health, in poverty or semi-poverty, as an undergraduate, graduate, postdoc or faculty member, since 1957. In all that time I have rarely wavered from my opinion that the only two “concepts” of physics the average person has ever heard of are “Einstein's E = mc squared” and “Heisenberg's Uncertainty Relation.”

Here, let's stick to the Uncertainty Relations (there are actually many). The average person, when I inquire, seems to think that Uncertainty in this context means one or more of the following:

→ “Scientists don't know anything for certain, everything is uncertain.”

→“Scientists can't measure anything accurately; every measurement gives an uncertain result, no matter how careful they are.”

→“The precise outcome of any physical process is never known; scientists just have to guess at what has happened.”

and so on...

Well, none of these three statements has anything whatsoever to do with the Uncertainty Relations of Heisenberg, which are a fundamental element of quantum physics, nor do any of these three statements have much, if anything, to do with science or with physical reality.

Heisenberg pointed out that many observable quantities in physics “interfere” with one another, because the fundamental description provided by quantum physics is the probability distribution. To be specific, suppose when we make measurements over and over on the same system, always measuring its position along some axis, we find the result of each measurement, which can be as precise as we please or can contrive, is different from the results of earlier and later measurements. When plotted as a graph, the results will usually form a hill-shaped distribution, with some width Ds. Now suppose instead of measuring the position of the object, we measure its velocity. Again, each measurement can be as precise and accurate as we please or want to make. The precision of measurements depends on our care and effort, and little else. But again, each time we repeat the measurement, we will get a slightly different (but accurate) result. And again, if we plot the results on a graph, it will form a hill-shaped distribution with some width Dv.

Now, here is what Heisenberg pointed out. In this case, and many others, these observable quantities have distribution widths that are inversely related. What we mean is Dv is always proportional to 1 divided by Ds. So suppose we do something to our physical system under study, so as to guarantee that its position upon repeated measurement varies very little. In other words, suppose we try to make Ds as small as possible. This will make Dv as large as possible. In still other words, if we manipulate the system to make its position come out very nearly the same each time we measure it precisely, we pretty much destroy any information that existed in the system as regards its own velocity. Or, if we manipulate the system to make its velocity come out very nearly the same each time we measure it precisely, we pretty much destroy any information that existed in the system as regards its own position.

One of the best ways to demonstrate this in class is to use a laser and an adjustable single slit. Light consists of particles, called photons, and each photon in the laser beam has very precisely the same speed (the speed of light!) and direction of motion. Since speed plus direction is velocity, the velocity of each photon is very precisely known. Now send the laser beam through the adjustable slit, and have the slit wide open. The beam goes through unaffected; the velocity information is not disturbed. But now start closing down the slit. The images above are a link to a Java applet that will let you do this and see the result directly.

As you make the slit smaller and smaller, you are more and more precisely determining the side-to-side position of the particles in the beam. Narrowing the slit narrows Ds. The result is that the distribution of possible velocities in that direction becomes greater and greater. As we narrow the slit more and more, we see the laser beam spread out more and more along the direction of the slit.

Let's discuss some consequences and details. First, you can measure the position of a quantum system as precisely as you please and then turn right around and measure the velocity as precisely as you please. No problem at all! What Heisenberg pointed out is that you can't simultaneously discover the position and velocity of any quantum system by any imaginable physical investigation. Another way to put this is that no quantum system has a trajectory. Another way to put this is that no information exists in nature as to the simultaneous position and velocity of a quantum particle. No quantum system can be in a state where both Ds and Dv are vanishingly small.

For a historical discussion of the development of the principle, click here. It needs to be mentioned that there are many different “Uncertainty Relations” that exist in quantum physics and even in classical wave physics. A famous and useful one relates the possible spread in values of the total energy of a system, DE, to the possible spread in values for the time at which some process occurs in the system, Dt. There are uncertainty relations that connect many different observable physical quantities to others. In quantum physics, observable quantities are represented by mathematical entities called operators. When two different operators have a certain property called non-commutation, an uncertainty relation always exists between the two different observable quantities represented by the operators.

Next Science Page?

One final irony. Whenever an atom is depicted symbolically, in the media or even in elementary textbooks, it is invariably depicted as the pre-quantum Bohr atom of 1915. In Bohr's semiclassical model, electrons orbited the nucleus much as planets orbit the sun. In 1925 it was realized that the one thing electrons and other subatomic particles never do is “orbit” anything! Quantum particles have no trajectories. Why, then, is the incorrect Bohr model always depicted instead of a correct, quantum atom? My guess is that there are two aspects to the continuing problem: (1) blissful, total ignorance; and (2) artistic or congenital inability to draw a probability distribution.