Schrödinger’s Cat Paradox

photoWritten by Samuel J. Ferguson, a mathematics PhD student and friend of the Belin-Blank Center, on the occasion of Erwin Schrödinger’s birthday (check out the Google doodle on his famous cat paradox).

A cat is placed in a steel chamber, together with the following hellish contraption…. In a Geiger counter there is a tiny amount of radioactive substance, so tiny that maybe within an hour one of the atoms decays, but equally probably none of them decays. If one decays then the counter triggers and via a relay activates a little hammer which breaks a container of cyanide.

-Erwin Schrödinger, trans. Josef Jauch

Just like Schrödinger’s cat, your potential relationship with Leonard right now can be thought of as both good and bad. It is only by opening the box that you’ll find out which it is.

-Sheldon, speaking to Penny, The Big Bang Theory

If a machine repeatedly throws tennis balls at a wall, they’ll hit about the same spot. Ordinary-sized objects go where they’re sent. What about the tiny electrons that orbit atoms? Here are images by Akira Tonomura from a 1989 experiment in Japan. A certain machine repeatedly threw electrons at a wall, one at a time, through a vertical opening.

image_electrons

It’s difficult to say in advance where an electron is going, but here there’s a pattern. It’s likely to end up in one of the vertical bands forming in the pictures.

To make things simpler, imagine that there are just two same-sized bands, with nothing in between (magnets might help corral the electrons into the same general area). As an electron flies from the machine to the bands, you can’t tell which band it’s going to—all you can say is that it’s equally likely to hit either.

electron_fifty_fifty

Between the machine and the wall, the electrons do not act like tennis balls. Why? The electrons are prepared exactly the same way in the experiment, so if they acted like tennis balls they would all hit the wall at about the same spot. There would be only one band in the images.

If our electrons have the same properties, then, as there are two bands, at every moment of transit they all have some left-band-strikingness and some right-band-strikingness. What this means is that until an electron hits the wall it has some likelihood of striking either band. What causes the electron to actually strike the left or the right band? Not the electrons by themselves—they’re the same, yet they hit the wall at different bands regardless.

If we insist on speaking about causes, it seems reasonable to say that it’s the interaction with the wall that makes the electron “take a stand,” ending up in the left band or the right. If this sounds strange, try thinking of it as a shorthand for the fact that although an electron is equally likely to strike either band, it only actually strikes one of them, similar to how a flipped coin has to come out heads or tails. But be careful—coins don’t act this strange! If a machine flipped another coin precisely the same way, it’d get the same result—whereas another electron launched exactly the same way may not.

The Cat Paradox

What does this have to do with a cat in a box? Schrödinger wanted to show that applying this thinking sloppily would lead to confusion. He did this by linking the life or death of a cat with the outcome of a 50-50 situation involving atoms with electron-like behavior. The situation is a little like a bomb with a lit fuse—with a 50% chance of the fuse being snuffed. This was an imagined experiment, not an actual one.

Schrödinger leaves his cat in a box for an hour. During that time either a radioactive atom runs down—and cyanide kills the cat—or else it doesn’t, and the cat lives. Similar to the electron having left-band-strikingness and right-band-strikingness, during that hour the atom has run-down-ness and stability (not-run-down-ness). At the end of the hour, an observed cat would be dead or alive, depending on the atom. If we treat the cat in the unopened box at the hour’s end like an electron in transit, it has some deadness and some liveness. When Schrödinger wants to determine whether the cat lives, he opens the box. He forces it to be either alive or dead—in the latter case, his curiosity killed the cat!

This cat paradox of Schrödinger intends for you to be taken aback. Something is wrong, and we need to trace back the culprit. This is how a paradox is supposed to work: the author trips you up intentionally. When you play detective and find faulty reasoning, you avoid that pitfall in the future.

The most widely accepted solution to Schrödinger’s cat paradox runs as follows. It’s hard to swallow the idea that observing the cat killed it. Experience suggests that either the cyanide killed the cat or it didn’t—that the cat never has some deadness and some liveness simultaneously. Now let’s look back to find the error.

We started by assuming there’s a 50-50 chance of the atom running down, so the atom indeed has both run-down-ness and stability at first. But you can’t treat the cat in the unopened box at the end of the hour like an electron in transit (nor does the observer play the role of the wall). The reason the cat doesn’t end up sharing two starkly different fates—dead and alive—is that the role of the “wall” is played by the first interaction between the atom and an ordinary-sized object. In this case, you need a Geiger counter to detect the shock of energy given off by the dying atom. Only then can, say, a hammer use this information to bust the cyanide container and imperil the cat. In this view, the interaction with the Geiger counter at the hour’s end requires that the atom be burst or not, so there is never a point where the cat is both alive and dead.

To fix ideas, here’s a simpler, less dramatic situation. We have the same atom as before alone inside a blue box. However, the walls of the box are so sensitive that they’ll turn red instantly if the atom runs down. The paradox then asks whether my opening the box makes it blue or red. The answer is that the interaction with the box determines whether it’s blue or red, because the box plays the Geiger counter, telling you whether the atom has disintegrated. The key is not that the atom must be “observed,” but that it must be “measured,” and its impressions or measurements could be left on any suitable regular-sized object.

Is it possible at all, perhaps in some other experiment, for a cat to be both alive and dead? Our answer does not tackle this directly because it’s tricky. What we’re asking is, basically, whether the whole cat—all of its atoms in concert—could be in an uncertain state as the traveling electron is in the experiment. The last word has not been said on whether this is possible, but it doesn’t seem to matter. Statistically, it’s improbable that the cat’s composition would be organized in this two-faced manner. The parts of large objects tend to align themselves in a way that prevents radical discrepancies in their overall state—there is no zombie cat. Electrons are just different from bigger objects.

Beyond the Electron

Do all objects behave either like regular-sized objects or electrons? Extremely dense, massive objects—such as neutron stars or black holes—bend light and warp time. They are better understood with Einstein’s relativity theory than in terms of “regular-sized” objects. For example, early 20th-century astronomers couldn’t understand why Mercury was “running slow” until Einstein’s relativity showed that the lag in its orbit was due to the sun’s great girth. For many, now there was something else they couldn’t understand.

What if we have an object so dense that relativity is called for—yet so small that it’s electron-sized? Then Einstein’s rules and the electron’s disagree, and that calls for more experiments. But where can we get objects like that to experiment with? Are there any? Great excitement was sparked recently by an object that might qualify, a “Higgs boson.” In March its detection was tentatively confirmed by CERN, whose Large Hadron Collider was originally built with Higgs particles in mind. Although much was gleaned from the encounter, it’s still unclear whether there’s more than one kind of these particles. For now, their behavior—and just how the theories of physics will be unified—remains a mystery.

One response to “Schrödinger’s Cat Paradox

  1. Leland ferguson

    Thanks. Well done explanation of the classic analogy–and more.

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