# The central mystery of quantum mechanics (according to Feynman)

1. Apr 11, 2014

Observing which slit a photon or particle goes through in the two slit experiment results in the formation of a diffraction pattern instead of an interference pattern. Were Feynmanns remarks about this made by referring to the classical experiments where there is high illumination or to the later experiments such as where there is one photon or one particle at a time or to all variations of the experiment.
By searching I have found it easy to find out what he said but have so far not been able to find out what exactly he was referring to.
Thank you

2. Apr 11, 2014

### atyy

http://www.feynmanlectures.caltech.edu/III_01.html#Ch1-S7
"In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics."

Nowadays we know that Feynman's argument was inaccurate. Non-relativistic quantum phenomena can be explained in a somewhat "classical way" with Bohmian mechanics. It is still being researched whether a Bohmian viewpoint is satisfactory for relativistic quantum mechanics. However, Bohmian trajectories are not the same as classical Hamiltonian trajectories. More importantly, Bohmian mechanics is nonlocal, and is "non-classical" or "mysterious" in that sense. So it is not so much that Bohmian mechanics is an entirely "classical way" of explaining quantum mechanics, since it is nonlocal. Rather, the double slit experiment doesn't demonstrate a violation of the Bell inequality, and so it doesn't force nonlocality on us.

The violation of the Bell inequality by quantum mechanics requires two ingredients: non-commuting observables and entanglement.

Feynman's point was probably about non-commuting observables, ie. position and momentum cannot be simultaneously well-defined. So Feynman got one of the ingredients needed for encapsulating the mystery. He didn't get the second ingredient of entanglement, so most people nowadays would say that the double slit is not the "only" mystery.

These notes by Scarani http://arxiv.org/abs/1303.3081 explain why the violation of the Bell inequality cannot be simulated classically. The comments by Fuchs and Schack on the first 3 pages of http://arxiv.org/abs/1301.3274 explain why in some sense the double slit experiment is not mysterious, and simply illustrates that different experiments produce different results (I think you can also find this point in Ballentine's book; I don't recommend the rest of the article by Fuchs and Schack or Ballentine's book, because they are rather idiosyncratic, unless one has already learnt QM the usual way, eg. Landau & Lifshitz.)

There is, however, one proposed interpretation of quantum mechanics called "Consistent Histories" in which it is said that the double slit is the only mystery.

Last edited: Apr 11, 2014
3. Apr 11, 2014

### my2cts

Keeping things a lot simpler, I think he referred to the fact that a single particle, such as an electron, appears to go through both slits. If the setup is modified so that it can be determined through which slit the electron goes no interference occurs.

4. Apr 11, 2014

### atyy

Yes, Feynman writes in the link given above "Is it true, or is it not true, that the electron either goes through hole 1 or it goes through hole 2?" The only answer that can be given is that we have found from experiment that there is a certain special way that we have to think in order that we do not get into inconsistencies.". So maybe he was talking about was what was later formalized as the Kochen-Specker theorem.

http://en.wikipedia.org/wiki/Kochen–Specker_theorem
"Hence, the KS theorem does only exclude noncontextual hidden variable theories."

Scarani also mentions that the Bell tests exclude more types of hidden variable theories than the Kochen-Specker tests.
http://arxiv.org/abs/1303.3081 (footnote 8): "Tests like "contextuality" a la Kochen Specker, or sequential measurements a la Feynman or Leggett-Garg, need a minimal amount of assumptions to prove that the outcomes do not come from a pre-established agreement. Indeed, no detailed knowledge of the system and the measurement is needed, but one must assume that the measurement device does not write in, nor reads from, other degrees of freedom than the relevant one."

Last edited: Apr 11, 2014
5. Apr 11, 2014

### my2cts

In fact if an experiment is arranged such that it can be determined if the electron goes left or right, this implies that the two possibilities are described by mutually orthogonal wavefunctions. So it is the absence of physical distinguishability that leads to the interference phenopmenon. I guess that if a system could be set up such that the wavefunctions describing the two possibility are partially but not completely overlapping, such that it would be possible to determine with say 95% certainty through wich slit the electron passed, then a strongly reduced interference pattern would be observed.

6. Apr 12, 2014

Thank you atyy and my2cts. My question was enquiring about the history of interference and I wanted to know what variation(s) of the two slit experiment Feynman was referring to.
I intended to look at the theory in greater detail but was thrown a bit by the above responses. To me the suggestion seems to be that Bohmian mechanics can explain the results of those interference experiments where which way information can be obtained. Is that so, can Bohmian mechanics even explain the results of quantum eraser type experiments of the type where which way information is not actually obtained but where there is the potential to do so?
Clarification would be greatly appreciated.
Thank you

7. Apr 12, 2014

### stevendaryl

Staff Emeritus
I have a slightly different perspective on Bell's inequality. I don't think that its violation in experiments is a mystery about quantum theory itself. Instead it dashes the hopes for replacing quantum theory by a future, less mysterious theory. So to me, it's not so much that Bell introduced any new mysteries of quantum theory, he just made it clear that we are pretty much stuck with all the old mysteries.

8. Apr 12, 2014

Richard Feynman on the Double Slit Paradox: Particle or Wave?

Ask atyy about initial conditions and the Born rule.

9. Apr 12, 2014

Agree 100%. :thumbs:

10. Apr 12, 2014

### atyy

I haven't worked through BM for a quantum eraser experiment, but my understanding is that the consensus is that BM is the only established interpretation of non-relativistic quantum mechanics, apart from Copenhagen (other leading approaches to interpretation are many-worlds and consistent histories, but it is not universally acknowledged that they solve all problems). So BM should be able to handle the quantum eraser experiment.

I agree, and that's what I intend to say: the violation of the Bell inequality encapsulates the mysteries of quantum mechanics.

11. Apr 13, 2014

Thank you DevilsAvocado. When I clicked on the link I remembered having seen that video in the past. I probably just scanned through it. I will take another look

12. Apr 14, 2014

Hello atty.My feeling at present is that all interpretatiuons of QM have some or even the same conceptual difficulties, for example in terms of anti intuitive results, when dealing with quantum eraser type experiments. If at any time you take a look at the experiments I would be interested to hear of any opinions you may form. Thank you again.

13. Apr 14, 2014

### Staff: Mentor

The interesting thing I have found about QM is to start with it seems very anti intuitive, but after a while, when you understand some of the more modern ways of viewing it, such as the most reasonable probability theory that allows continuous transformations, things seem a lot more reasonable.

The argument goes something like this. Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense out of pure states that are complex numbers. There is really only one reasonable way to do it - by the Born rule (you make the assumption of non contextuality - ie the probability is not basis dependant, plus a few other things no need to go into here) - as shown by Gleason's theorem. But it can also be done without such high powered mathematical machinery:
http://arxiv.org/pdf/quant-ph/0101012.pdf

In my opinion intuition has a lot to do with understanding, familiarity and the way you view things.

Thanks
Bill

14. Apr 15, 2014

Thank you Bill,
What I'm particularly interested in at the moment are the quantum eraser type experiments. I can see that theory can succesfully predict the variety of interference/diffraction patterns that can be observed but can theory explain the circumstances that are necessary to observe opposite type of patterns? In other words why does the potential to get which way information result in diffraction only even when, in some circumstances, the paths are not actually marked?

Last edited: Apr 15, 2014
15. Apr 15, 2014

### Staff: Mentor

Well if the eraser experiments were not explainable by QM that would be BIG news. They are of course so really your worry is what's going on.

Its not difficult really. In principle you can unscramble dehoherence but its impossible at a practical level when a large number degrees of freedom are involved. All the eraser experiments show its possible to do that in simple cases when there is a small number.

See:

Thanks
Bill

16. Apr 16, 2014

Thank you but does the explanation really conform to intuition? To quote professor Jim Al Khalili:

"If you can explain this using common sense and logic do let me know because there is a Nobel Prize waiting for you."

The remark was probably a bit tongue in cheek but it referred to the conceptual difficulties of the phenomenom.

(See the mini U tube presentation "Double slit explained by! Jim Al Khalili")

The professor seemed to be referring to just the basic one photon at a time experiments and not to, what I think is the more challenging, one photon at a time quantum eraser experiments.

17. Apr 16, 2014

### Staff: Mentor

Stand on a platform that can rotate with a spinning bicycle wheel in your hand. Rotate the wheel and you start to spin around. When most people first see it they are dumbfounded - gyroscopic effects are actually quite puzzling at first. But once you understand it its pretty ho hum really.

The same with QM once you understand its conceptual core.

I can and have given you some links that do it. But I doubt if I could ever collect because part of it is what is common-sensical changes as you understand more. Gyroscopic behaviour isn't common-sensical either until you understand it - but when you do - it's quite easy.

The issue with QM is it's not understandable in terms of everyday experience. Extend that experience a bit and the barriers fall away - the same with many areas really.

We have met the enemy and he is us - Pogo

Thanks
Bill

Last edited: Apr 16, 2014
18. Apr 16, 2014

I think I get the point you're making. Something that might seem difficult and weird loses its difficulty when you understand it. It can lose its weirdness also.
I don't think,however, that an understanding of the presently available knowledge and theories of QM necessarily results in a loss of the apparent weirdness of some parts of the subject. In his day Feynman knew more about QM than the majority but still pointed out a "mystery". Now we have people like Khalili still pointing out the logical and common sense problems with the subject even though their knowledge is far broader than was available to Feynman.

19. Apr 16, 2014

### Staff: Mentor

Like I said:
I personally feel very comfortable with it - maybe with time you will to.

Thanks
Bill

20. Apr 18, 2014

In my case I can get to understand certain things, according to currently available knowledge, and I can get to feel comfortable with certain things. However, when I get to think more deeply about some of those things I often realise how little I really understand. That can make it more interesting.

Thank you for your comments Bill. They were much appreciated.

21. Apr 20, 2014

### jk22

I read somewhere that Feynman considers the superposition principle as the mystery of QM. In some sense we could say that this principle is a way of expressing ignorance more precisely : if we don't know which result it is, then it is a superposition of the possible result with given probabilities. This way of expressing ignorance seems to have very accurate prediction abilities.

In some sense QM is going from the classical local-causal laws to a non-causal law from the superposition to the effective results.

On this other mystery, I find Bell's theorem strange, since it computes : <AB-AB'+A'B+A'B'>=<A(B-B')>+<A'(B+B')>.

But we know that the measurement results of the sum is not the sum of the measurement results, since $$\nu(B-B')=\pm\sqrt{2}\neq\nu(B)-\nu(B')$$ hence the quantity above becomes :

$$(\underbrace{\pm 1}_A)( \underbrace{\pm\sqrt{2}}_{B-B'})+(\underbrace{\pm 1}_{A'})( \underbrace{\pm\sqrt{2}}_{B+B'})\leq 2\sqrt{2}$$.

As we see this is a sum of results that are product of result in A place and B place, which gives greater than two. So what does it mean to add the measurement results of B and B' ? Even if it is mathematically correct, it does not correspond to any measurement of B+B'...which could be seens as the superposition of the measurement operators ?

Last edited: Apr 20, 2014
22. Apr 20, 2014

### Staff: Mentor

I have zero idea what you are trying to get at.

But in QM one of the central mysteries is that for operators the sum of the measurement results of any two operators is linear in the sense of expectation values ie if A and B are observables then E(aA+bB) = aE(A) + bE(B). Its pretty intuitive. This in fact implies the Born rule. A = ∑yi |bi><bi| E(A) = ∑ yi E(|bi><bi|). Let P = ∑ E(|bi><bi|) |bi><bi|. E(A) = Trace (PA) - which is the Born Rule. Since from the definition of observables E(|bi><bi|) must be positive and sum to one P is a positive operator of unit trace. For pure states this immediately implies the principle of superposition because they form a vector space, and all the rest of the QM formalism.

The key is in fact linearity. It took physicists a little while to see this very intuitive assumption doesn't always apply for hidden variable theories because they can be contextual. The above argument is essentially Von Neumans famous proof against hidden variable theories - but turned on its head to deduce QM. Gleason's Theorem is basically a stronger version of it based on much weaker assumptions.

Thanks
Bill

23. Apr 24, 2014

### jk22

I wondered if there exist a matrix operation let's name it &, such that A & B has as eigenvalues the sum of the eigenvalues of A and B, in particular if A and B are 2 nxn matrices, then, A & B should be a n^2 x n^2 matrix.

If this exist then this special "sum" of operator would correspond to make the sum of the eigenvalues, and hence there would be a structure in the comparison inside Bell's theorem. But I fear there exist no such operation &.

24. Apr 25, 2014