Son of “Notes Toward ‘Weird Quantum Phenomenon'”

A continuation of the lecture transcription/ working out of idea for Boskone that I started in the previous post. There’s a greater chance that I say something stupid about quantum measurement in this part, but you’ll have to look below the fold to find out…

At the end of the previous post, I wrote:

We can verify this by doing the experiment with single particles, and what we see is exactly the prediction of quantum theory. If we send one electron at a time toward a set of slits, and detect the electron position on the far side, we see individual electrons arriving one at a time, in an apparently random pattern. If we repeat this many, many times, and add up the results, though, we recover the interference pattern. While the individual particles show up as discrete spots, there are places in the pattern where the probability of finding an electron is absolutely zero, and other spots where the probability is quite high. The electrons are detected as individual particles, but their arrangement shows signs that they were in a superposition state along the way– the interference shows that each electron took two different paths to detection point.

Picking up from there, we come to:

V) This is where the idea of measurement first enters the picture. When we do the double-slit experiment with electrons, we detect the electrons as discrete particles, in a single position. Something must be happening that causes the wavefunction of the system to change from a weird, spread-out superposition state to a single outcome, determining the position of the electron to be a single point on the far side of the screen.

This is a complicated case of quantum measurement, but whatever the physical situation, the results are the same: when you make a measurement of the state of the system, you change the wavefunction from a weird quantum superposition of states to a single result. To take a simple example, if we set up a very small detector near one of the slits, and detect which slit the electron went through, we destroy the superposition state– we go from a wavefunction that is a sum of the wavefunctions for the two individual slits to a wavefunction that is only one or the other (depending on whether the electron was detected passing through the slit we are monitoring).

This has an immediate and dramatic consequence for our experiment: the interference pattern goes away. As soon as we take the electrons out of a superposition state, we go from having a quantum interference pattern on the far side of the slits to just the simple two-lump pattern we expect for classical particles.

You can understand this process in terms of the uncertainty principle, if you like– when you detect that the electron has gone through one slit, you make the uncertainty in its position very small, which in turn makes the uncertainty in the momentum very large. The wavelength of the electrons is determined by their momentum, so a big momentum spread gives you a big wavelength spread, and the interference of waves with very different wavelengths will wash out to give a classical-looking pattern. You can even come up with semi-classical explanations of why this happens, in terms of the physical mechanism used to measure the position– Bohr was an absolute genius at this sort of thing.

That sort of explanation works pretty well for a beam of electrons, but it’s not so good for the single electron. And, ultimately, everything comes back to the single-electron case. In order to understand that, you really have to think in terms of superposition states and the destruction thereof. When you don’t measure the path, the electron goes both ways, and you get interference. When you do measure the path, it only takes one path, and everything looks classical.

VI) Of course, it gets even weirder (quantum mechanics almost always gets even weirder). It’s not actually necessary for you to do the measurement of which path the electron took– simply arranging things so you could measure it is enough to destroy the superposition, whether you record the path information or not.

You can also get the pattern back, by doing something to remove your ability to distinguish paths, after the fact. If you imagine doing the double-slit experiment with photons rather than electrons, you could (for example) cover one slit with a horizontal polarizer, and the other with a vertical polarizer. If you put a horizontal polarizer over your detector, you’ll detect photons only from one slit, and see a classical pattern. If you put a vertical polarizer in front of your detector, you’ll detect only photons from the other slit, and see a classical pattern.

But, if you put a polarizer in front of your detector that is oriented at 45 degrees from the vertical, and thus is equally likely to pass either horizontal or vertical polarizations, you’ll see interference again. Once you can no longer distinguish between the two slits, you get back the original superposition state, and you see interference again. If you want to be really tricky, you can even decide which sort of detection you’re going to do while the photons are in flight, and past the slits (imagine rotating your polarizer really quickly, based on some sort of random number generator), and you’ll still get the appropriate type of pattern, when you add up the measurements later on.

These “quantum eraser” experiments are some of the most dramatic demonstrations that reality is much, much weirder than we think it ought to be.

VII) Another interesting wrinkle on the whole measurement issue is seen from the EPR gedankenexperiment. Imagine taking a pair of particles, and putting them into a quantum state in which some property of each particle is correlated. For example, you take a pair of photons, and arrange so that one is vertically polarized, and the other horizontally polarized. Then you send them off in opposite directions, without measuring which is which.

According to quantum mechanics, that’s a superposition state, and when you finally measure the state of one particle, you instantaneously determine the state of the other, no matter how far apart they are. Einstein really hated this part of quantum mechanics, but you can do some very clever experiments to show that this is really the way things work– the first such experiment is the final entry in the Top Eleven, to be posted tomorrow.

The really interesting thing here is that the state of the photons in flight is indeterminate. It’s not just that you don’t know which photon is horizontal and which is vertical– they don’t know it either. The state of each photon is both horizontal and vertical at the same time, right up until the instant of measurement. That’s a deeply odd and non-classical state of affairs.

VIII) So, what causes all this? Good question.

Sadly, I don’t have a good answer. One of the maddening but fascinating things about quantum theory is that we’re sort of stuck in that classic Sydney Harris cartoon, where Step Two is “Then a miracle occurs.” Something happens during the process of measurement that changes the wavefunction, but it’s not clear exactly what.

A lot of excellent work has been done on a process known as “decoherence,” which uses interactions with the environment to destroy quantum superpositions. The basic idea is that as a particle moves around, it interacts with lots of other particles, and if we don’t keep track of the states of all those other particles, we lose information about the state. That information loss amounts to the destruction of the superposition state, and we move into a scenario where the electron has followed a single definite path. (This is a little hard to explain in the usual wavefunction language, but I’m not about to introduce the density matrix formalism here…)

Decoherence gets you a lot of the way there, but it isn’t a full explanation. It can get rid of the interference between terms of the wavefunction, and get you to a situation where you have a simple choice between two options, but it doesn’t explain how you end up with one or the other. There are also some arrangements of systems where you can avoid decoherence entirely, even for some fairly complex systems, so there’s work yet to be done.

There are various “interpretations” of quantum mechanics that deal with measurement in different ways. The grand-daddy of them all is the Copenhagen Interpretation, which has the killer “then a miracle occurs” statement. It holds that something in the process of observation causes a “collapse” of the wavefunction into only one of the possible states. What that something is is pretty nebulous– some versions hold that sentient observers are required, which opens all sorts of annoying questions about who counts as an observer, and what counts as an observation, and if a tree falls in a forest does it make the sound of one hand clapping? It’s great late-night dorm-room bull session stuff, but not too terribly satisfying.

Then there’s the “Many Worlds Interpretation,” which holds that there is no collapse of the wavefunction into a single state– instead, the universe splits into two or more parallel universes, one for each of the possible outcomes. The wavefunctions continue on their merry way, evolving smoothly without a miraculous step, and it just happens that we only perceive one of the possible branches.

This isn’t a whole lot better than the Copenhagen Interpretation in terms of actual answers. It really just sort of pushes the question back a level, as it’s not clear why we see only one branch. Again, you have a host of questions to blow the minds of really stoned college students, about what happens to all those other universes, and why we only see one of them, and if what I see as “green” is really the same thing that other people see as “Red,” can I get out of that traffic ticket?

Both of these interpretations have led to a wealth of vaguely annoying science fiction stories and pop-science books. They don’t exhaust the possibilities by any stretch, though– there are lots of other interpretations, each with its passionate supporters, and each with a host of unanswered questions.

IX) At the end of the day, most physicists end up with some form of the “Shut Up and Calculate” interpretation. What causes the weird behavior of quantum measurement is an open question at this point, but if you just try not to think about it, and roll with the theory as it exists, it’s brilliantly successful. Using quantum mechanics, and the various outgrowths of it, you can predict some physical quantities to twelve or thirteen decimal places, and confirm those predictions by experiment. It’s probably the most precisely tested theory in the history of science.

And yet, there’s this fundamental weirdness at the heart of it, where systems will happily sit in two (or more) states at the same time, and stop being in superposition states for reasons that we don’t fully understand. It’s an indication that for all our success in using the theory, we still don’t fully understand it.