Picturing Quantum Mechanics

They say a picture is worth a thousand words. But what if those words are wrong?

Very cool recent experiments demonstrated a chemical reaction between molecules below a millionth of a degree (in Science, subscription required). My latest story for Physical Review Focus describes theoretical modeling of this reaction. We accompanied the story with this picture from the news release issued by the Joint Quantum Institute (a partnership between the National Institute of Standards and Technology and the University of Maryland), where the work was done.

It's a pretty picture, with its superhero color scheme and all, and it satisfies our need to avoid a solid block of text. But although it might not be a bad illustration of a room-temperature chemical reaction, it distorts much of what makes these ultra-low-temperature reactions special.

It's clear in the picture that two diatomic molecules are approaching each other, with dramatic consequences in store. The details of how the artist represents the bonds connecting a potassium and a rubidium atom in each molecule don't bother me too much. It doesn't match either of the customary representations, which are ball-and-stick models and the more accurate space-filling models, but there's no perfect way to represent something that can never be seen with visible light. Of course everyone knows that potassium atoms are green, but we'll let that slide, too.

The really problematic part of this picture is very difficult to avoid: the molecules really aren't anywhere, in the sense the picture conveys.

As first shown by experiments at Bell Labs in 1927, matter acts as waves as well as particles. At temperatures below a millionth of a degree, the relevant wavelength for these molecules is hundreds of nanometers, which is much, much larger than the separation of molecules shown in the experiment. There is no meaning to saying that these molecules are separated by such a short distance. They are simultaneously close and far away.

One way to think about this is to invoke Heisenberg's uncertainty principle. According to this principle, if you know an object's momentum with very high precision, you can't, even in principle, know its position very accurately. For these ultracold molecules, the momentum is almost zero, with very high precision, so you can only know where it is to the nearest hundreds of nanometers.

There's a second problem, too. The picture shows the molecules with particular orientations in space. That may not seem strange, but the molecules in the experiment were prepared in the rotational "ground state," with the lowest possible energy. Like the s-orbitals of electrons in a hydrogen atom, this state is spherically symmetrical. This means that the molecule is equally likely to be pointing in any direction. This isn't the same thing as saying we don't know what direction it's pointing (even though it does). Quantum mechanics says that it has no direction, at least until an experiment requires it to.

So the reacting molecules really aren't at any particular distance from one another, and they don't have any particular orientation relative to each other. That's one of the things that makes this chemical reaction--and the theoretical description of it--so interesting.

But good luck drawing that.

 

Free Will and Quantum Mechanics

[NOTE: This piece is modified from one written in the spring of 2005, but never published because it was too demanding, so be warned.]

In 2004, two mathematics professors from Princeton University devised the simplest proof yet that the world really is unpredictable at a microscopic level.

Quantum mechanics has passed many experimental tests, but it generally predicts only the probabilities of various outcomes. Over the decades, many physicists, notably Einstein, have longed for a description that doesn't involve "throwing dice."

The Princeton "Free-Will Theorem" concludes that, if experimenters can make choices freely, then this unpredictable behavior of elementary particles is unavoidable. But other experts suspect that the result is another manifestation of the "spooky action at a distance," that dominates the quantum world.

"Physicists usually are not impressed much," admitted John Horton Conway, the inventor of the 1970 cellular-automaton game he called Life. "They actually believe quantum mechanics." Indeed, few dispute that quantum mechanics gives correct predictions. But Conway and his colleague Simon Kochen said that although their conclusions are familiar, they start with three axioms, called SPIN, TWIN, and FIN, that are much simpler than previous theorems, and avoid "counterfactual" experiments that can't be done.

The first axiom, called SPIN, is based on an unusual property of a "spin-one" elementary particle: Measuring the square of its angular momentum, or spin, as projected along three perpendicular directions will always yield two ones and one zero. This bizarre property is usually derived from quantum mechanics, but it could have been observed independently.

"We don't have to know what 'the square of the spin' means," Conway said. "It's really rather important that we don't, because the concept 'squared spin' that we're asking about doesn't exist-- that's one of the things that's proved."

If such a squared spin existed, then experimenters could, in principle, choose a direction to measure it along and know in advance whether it would be one or zero. But in a groundbreaking theorem published in 1967, Kochen and E.P. Specker showed that it is impossible to prepare a list beforehand that gives the required two ones and a zero for all possible sets of measurement directions. They concluded that there are no "hidden variables" that describe the "real" spin.

Later researchers, however, realized that such hidden variables could logically exist, but only if their values changed depending on which measurements were chosen, a property known as "contextuality."

To avoid this problem, Conway and Kochen analyzed pairs of particles with matched properties. Their second axiom, which they call TWIN, is that experimenters can make and separate such pairs. This ability is well established, and experiments on the pairs have confirmed the quantum prediction that measurements of their properties remain correlated long after they separate. (I described one recent experiment for Technology Review.)

In 1964, John Bell showed that the some measurements on the two particles can only be explained if each particle somehow continues to be affected by the other, even though they are far apart. This surprising "nonlocality" has since been confirmed in numerous experiments, which find that the correlation, averaged over many pairs, exceeds the maximum for any conceivable local theory.

To avoid the need for statistical averages, Conway and Kochen applied the Kochen-Specker theorem to a single, matched, spin-one pair. If researchers measure the same component of the squared spin for both particles, they should always find either both zeroes or both ones. (Rutgers student Douglas Hemmick also derived this result in his 1996 doctoral thesis.) A similar "Bell's theorem without inequalities" was described in 1989 by Daniel Greenberger, Michael Horne, and Anton Zeilinger, using three "spin-1/2" particles.

Conway and Kochen's third axiom, FIN, is grounded in special relativity, and says that information travels no faster than, say, the speed of light. In spite of appearances, they say, nonlocal effects do not exceed this speed limit, because they describe only coincidence between two measurements, not causation. In fact, special relativity makes it meaningless to say that either of two widely separated measurements occurred "first," so it makes no sense to talk of information passing between the two.

Combining these ingredients, Conway and Kochen imagine that the squared spin is measured in all three directions for one member of a pair. If an experimenter measures the spin of the other member along any of these directions, the result must agree with the one for the first member.

But if the experimenter is free to choose which direction to measure, then because of FIN, that information is not available to the first particle. The result of the first measurement can't depend on her choice, but since there is no way to consistently anticipate all possible measurements, Conway and Kochen conclude that no hidden variable could have predicted the outcome. The only way to avoid this unpredictability, they say, is if the experimenter wasn't really free to choose which experiment to do.

Tim Maudlin, who heard Conway present the work in a colloquium in November 2004, disputes that conclusion. A philosophy professor at Rutgers University and author of "Quantum Non-Locality & Relativity," Maudlin remarked that saying the behavior of a particle cannot be determined by information in its own past "is just what we mean by non-locality," which is already clearly established. "You've taken the contextuality and stretched it out" to include both members of the pair, he asserts.

Conway and Kochen published their Free Will Theorem in 2008, and the Princeton Alumni Weekly has posted videos of lectures by Conway. But it appears that other scientists are free to choose whether to believe it.

Forbidden Questions

To navigate the quantum world, you have to know what questions not to ask.

In the everyday world, we get along fine assuming that a baseball, for example, had a certain momentum even before we whacked it and felt the effects. But at the quantum level, an observable effect like the push on a bat does not give us permission to regard the ball's earlier momentum as having been a "real" quantity, independent of the swinging bat. Talking about such unobserved properties is a recipe for trouble.

This takes a lot of getting used to.

I ran head on into this problem in my latest story for Physical Review Focus. I first titled the story "How Long is a Photon?" and described the experiments as measuring the "duration of individual photons." That description was wrong, and came from asking forbidden questions.

Optics experts often measure the duration of pulses that are only a few femtoseconds (10^-15 seconds) long. This is much too fast for direct electronic measurements, so they do it by making two similar pulses and measuring whether they overlap. Delaying one of the pulses by more than their length stops them from overlapping. Actually the researchers repeat the experiment with millions of pairs of pulses, each with a particular delay, to build up a picture of how the overlap varies with delay. For pulses consisting of many photons, it is natural to regard the overlap time as reflecting the length of the underlying pulses.

The new experiments look a lot like this. But the difference is critical.

Kevin O'Donnell, at CICESE in Baja California, built on earlier experiments from the Weizmann Institute in Israel. Instead of pairs of pulses, however, these groups measure pairs of photons. They get the pairs by shining a steady green laser into a special crystal, which splits about one green photon in ten million into two infrared photons. Because these two photons are created as a pair in a single quantum-mechanical process, they are "entangled": properties deduced from measurements on one will always be related to properties deduced from measurements on the other, even if the measurements are done far apart.

The nature of this connection is one of the central oddities of quantum mechanics. In fact, we could save a lot of trouble by not talking about the individual photons at all, because in a profound sense they do not exist as separate entities, even after "they" move away from each other. But our language makes it hard to talk about a pair without think of it as a pair of something.

As in the experiments on pulses, O'Donnell delays one photon with respect to the other and measures their overlap. (I can say that without saying photon, but it gets a lot more complicated.) But as he explained to me, it is not meaningful to relate this overlap to the "length" of the photons. Instead, the result of the overlap experiment at different delays is a property of the combined state of the two photons. The final story, "The Overlap of Two Photons," takes pains to describe that correctly, at the cost of clunkier language and probably losing some readers.

As another example, a researcher who measures the energy of one photon in a pair can be assured that the energy of the other will be just right, so that their combined energy equals that of the original green photon. But that doesn't mean that the photon "had" that energy before the measurement was made. More complex experiments, in fact, show that the unmolested photon does not have any particular energy.

In the current experiment, you don't go too far wrong by imagining (incorrectly) that it measures the length of a photon. But this "bad habit," as described by David Mermin in Physics Today (subscribers only, or you can google the title), of conferring reality on properties that aren't or can't be measured, is the root of much confusion. More importantly, thinking (and talking) precisely about what actually exists is key to understanding the nature of the quantum world we inhabit.