The dual slit experiment seems like...well...BS!

I agree, with two caveats. First, the wavefunction doesn't just cover location and momentum; it also includes spin, polarization, and any other of they system's degrees of freedom. Second and more importantly, one needs to be careful referring to the wavefunction as a "statistical probability", because the wavefunction has phase and can interfere with itself to produce things like the double slit pattern. Probability distributions over classical variables cannot do this.

Again, I agree.

Where do you get this idea? When we look at a macroscopic light source (like a lightbulb, or a laser), there are so many photons that we don't really notice the wavefunction collapse of each individual photon. The spread-out "glow" of such a source is not how we see single photons, but a statistical average over lots of photons.

If the photon is absorbed, then it must have entered your eye. If it entered your eye, then we know where it was, and the position wavefunction has collapsed. Like I said in the previous post, if you set up many detectors over a large section of space and fire a single photon at all of them, only one detector will ever click. This remains true even if each detector is much smaller than the whole wavefunction, so the wavefunction must collapse on detection.

 

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I got that idea from Richard Feynman, from a lecture from when he discovered how to predict their probable location. They discovered that the photon goes out to every possible location, so they could then take the average of every location to find the most probable location. It just turned out that it was the same answer as just taking the extreme of two averages, and the model they used was equally valid for statistical probabilities of anything known in physics.

Then it depends on what interpretation you accept in quantum mechanics. In the Copenhagen Interpretation, that single photon exist simultaneously in every one of those location. Then in the many worlds theory, each location would then branch off to a different version of that universe. I accept the Copenhagen Interpretation, so I believe the single photon is at all possible locations in its path at the same time.

If the photon is absorbed, then that photon no longer exist. Then it would be irrelevant if the wavefunction collapsed or not. It would no longer exist.

That is exactly what I was saying that I have never heard of before, which I was trying to get someone to prove to me. If that happened and there was no interference pattern, then it would mean that the wavefunction of the photon did collapse from an act of observation; the only thing I am really doubting here.

 

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Measurement is a projection operator.

A projection operator is idempotent, which means the result of applying it once is the same as applying it more than once. Thus a projection operator has eigenvalues limited to 1 and 0 because multiplication by these numbers is idempotent: \( 1 \times 1 \times x = 1 \times x, 0 \times 0 \times x = 0 \times x\). So the shorthand of identifying a projection operators is that it satisfies \(P^2 = P\).

Since \((I - P)^2 = I - 2 P + P^2 = I - 2 P + P = I - P\) we see that if measurement is a projection operator, then in a sense non-measurement of a specific property is also a projection operator.

So in regards to the double-slit experiment, if you can measure if the electron comes through one slit then you have applied a projection operator on its position and have localized it, destroying the portion of the wave function that would have resulted in the possibility of it having past through the other slit with the predicted and observed consequences that the interference fringes disappear.

Another is a quantum Zeno effect. Zeno proposed that you can't get anywhere because you have to move half the distance before you move the whole distance. In the quantum Zeno effect, a system is put in a state and circumstances are set up that if you leave it alone it will very slowly evolve its wave function.

Say \( \psi = \cos \omega t | A > \, + \, \sin \omega t | B >\). So the probability of begin found in state B a short time after being found in state A is \(\lim \limits_{t \to 0} \sin^2 \omega t = \omega^2 \lim \limits_{t \to 0} t^2 \). So it's not linear in t, so if we measure the state frequently, with interval \(\Delta T << \omega^{-1}\), we can retard the progression to likely measuring it in state B to by a factor of \(\frac{ \Delta T}{t}\) and bring a new (and unrelated) meaning to the aphorism: "A watched pot never boils."

As confused2 pointed out, very dim sources (like starlight) and sensitive equipment ( like CCD chips or photomultiplier tubes) let you localize and count individual photons as particles. The probabilistic pattern of light through a double slit experiment does not change when it is dim like a faint star or bright like a laser, but with dim light you need apparatus to integrate the results over time to see the pattern.

// Xeno -> Zeno because Greeks used X for less cool things

 

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I'm pretty sure this has been done in the context of the double slit, but I don't know of any references for that, so instead I'll point you to a piece of Philip Walther's more recent work, http://arxiv.org/pdf/1212.2240.pdf. This is an experimental implementation of something called boson sampling, in which single photons are sent into a network of phase shifters and beam splitters to produce a probability distribution that is classically difficult to simulate. In this case, Walther uses three photons as a proof of principle, and all of their wavefunctions are spread out spatially over the output ports*. Ideally, he would see exactly 3 photons in the output with every run. In practice, photons are sometimes lost, but he still never detects more than three photons; this implies that detecting a photon at one port collapses the wavefunction and prevents it from being detected at any other ports, even ones where its wavefunction amplitude was large before the collapse.

*Technically, since the boson sampling device entangles the photons, it no longer makes sense to talk about each wavefunction as a separate entity. Rather, all three photons' locations are determined by a single, joint wavefunction that cannot be decomposed into a tensor product of single-photon wavefunctions. This is a great example of how Feynman was wrong when he said photons never interfere with each other.

Great discussion, as always! I'd point out, though, that only strong measurements are projection operators. As Farsight pointed out, there are also weak measurements, which only partially collapse the wavefunction and can continue to give information if they are applied multiple times in a row. A lot of people forget about weak measurements, but they're essential for bridging the gap between unitary time evolution and measurement-based collapse in quantum mechanics. Also, it's spelled quantum "Zeno" effect.

 

Yes, I thought Xeno might have been wrong, but I'm about a week behind on moderation duties.

 

This is ingenious! Of course I am familiar with the projection operator in the general sense.

For, suppose the operator \(A:V \to V\) then the operator \(P:V \to V_A\), say, sends all \(v \in V\) to the subspace \(V_A \subsetneq V\) for which the characteristic eqn \((A-I\lambda)v=0\) is satisfied. (Sorry, just showing off)

It never occurred to me that the projection operator has its own eigenvalues and eigenvectors - but of course it does

Are you suggesting that \(I - P\) corresponds to a "non-measurement"? I don't doubt you, I am having trouble seeing it[/QUOTE]

 

OK -- you have 2 slits where the electron can come through and you set up magnetic induction loops at one slit.

When the electron comes through the slit you are watching, its position state collapses into an electron localized at that slit. That's the operation of projection operator P.

But when an electron reaches the screen without having set off your device, you have "effectively" localized its position to the other slit at the time it would have set off your detector at one slit. That's the operation of projection operator I - P.

And with only one detector in place, the pattern you get on the screen is the sum of the one-slit pattern of the first slit and the one-slit patter of the second slit.

That's the sense where I wanted to indicate non-measurement can collapse the wave function of the electron you would have measured.

 

Ah, OK. Let me state this in a slightly different form, remembering that to make a measurement on a system is to "select" one of its eigenvalues.

The identity operator \(I\) acting on any system - any state vector - has eigenvalue = 1. Ignoring multiplicities I write \(\lambda_I =1\)

As pointed out, and using the same notation, the projection operator has eigenvalues \(\lambda_P=1\) and \(\lambda_P=0\).

Then obviously \(\lambda_I - \lambda_P=0\) if \(\lambda_P=1\) otherwise this sum is 1.

This is surely just a fancy way to restate the superposition principle - if a system in state S always has measurements that are sometimes definitely a or sometimes definitely b and nothing else, one says that S is a superposition of states A and B, each of which always have measurements a and b respectively

 

I, too, agree.

 

You're in good company, rustyw, a lot of people have trouble with this.

The mundane answer is, yes, in fact, if you want to figure out which slit a particle went through, you have to hit it with another particle, but that's a very glib and only partly correct answer. There is a tendency to make movies with little balls representing the particles, but the fact of the matter is, there isn't a bath of light like there is in the movie. The real situation in the world of the very small is that it's dark. We're so used to seeing things lit up that it biases the way we think, and this introduces confusion when people try to think about quantum mechanics. Quantum mechanics is more like a game of baseball in the dark, where you find the baseball by trying to bounce other baseballs off it, than it is like those movies with the little balls.

It's immaterial where the photons came from. The photons came from some source, bounced off the object, and then went in your eye. Lots of other photons prolly bounced off it too, but those weren't the ones you saw. The only ones you saw were the ones that happened to be going in the right direction to go into your eye.

To think of this the way physicists do, think of the whole thing as an experiment: there's a light source, the photons it emits, the object you're observing, your eye, and you. By "you" I mean the guy who lives in your brain. So you can see from this how a physicist sees things when he talks about "you" "observing something."

Suppose the light source is a flashlight you're holding in your hand; then, in fact, the source of the photons is "you."

Guaranteed, if you saw something, you saw it because photons came from some source, hit it, bounced off, and landed in your eye. The only exception is if you're actually looking at the light source, in which case you saw the source.

Actually, that's not quite right. If there is any way of determining which slit the particle went through, then there will be no interference pattern (what you are calling the "wave pattern"). It doesn't matter whether someone actually looks to see which slit it went through; just that there is some way of finding out. It's very easy to think that the collision is what destroys the interference pattern, but some very tricky people have made some very tricky experiments where it's not necessary to bounce a particle off the one going through the slits, you determine which slit by splitting the original photon in two, with the information on which slit the original photon went through being implicit in the second photon, and sending that second photon off somewhere else to be measured to see which slit it went through. If they look at that second photon with a detector, then the interference goes away; if they scramble it so the encoding is lost, then the interference is observed. This experiment is called the Delayed Choice Quantum Eraser, and I can describe it if you like, but it's pretty convoluted. When you first look at it, it appears as if it means that, yes, when the human watching looks away, there is interference, and when she looks, there is no interference, but it's actually much more complicated than that (and you really need all the math to fully understand it).

This is a general property of quantum mechanics; the logic it follows isn't the classical logic of the world around you. Quantum particles can do things that ordinary material objects you encounter in everyday life cannot. When they're not interacting with any other particles, they don't have to be in a particular location; it's only when they interact that they have to be in a particular place. They cruise along in a sort of cloud of probability of their position, but when they get close to another particle and interact with it, then the probability they were at the exact location of that interaction becomes 100% and the probability they were anywhere else becomes 0%. Richard Feynman said that if you don't think quantum mechanics is outrageous, you haven't understood it, by which he meant that the logic quantum mechanics uses violates all sorts of assumptions you make based on your experience of the world around you, as you see it.

You can actually interpret the DCQE as doing exactly that: retroactively erasing the observation. That's not the right way to see it, but it's an easy mistake to make. It's quite an interesting experiment and I very much recommend it to your attention. I understood a great deal more about QM after I understood it properly than I had before.

You are definitely going to have to dig deeper. I think I can help, if your style of learning fits with my style of explaining.

I've tried not only not to talk over your head, but not to talk down to you either. I hope I succeeded. I'm sure you'll have more questions.

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As posted in the starting post by the starter rustyw:

Of course there resides the right explanation of the experiment.