why the wave function

RUTA

You solve the KG eqn for psi, use psi*(operator)psi to obtain the current, and psi is the amplitude whose non-relativistic limit is found via the SE. So, this doesn't strike me as a promising method for producing a theory of probabilities a la classical physics.

akhmeteli

I did not say anything about "promising" or "not-promising" methods. TBN asked: "how can probability waves interfere destructively?", so I only discussed possibility or impossibility. Specifically, I said about the non-relativistic Schrödinger equation in electromagnetic field that it can be rewritten in terms of probabilities only as follows: for each solution \psi of the Schrödinger equation in electromagnetic four-potential A^\mu you can build (using a gauge transform) a real solution \phi of the Schrödinger equation in electromagnetic potential B^\mu, where A^\mu and B^\mu produce the same electromagnetic field. Therefore, the non-relativistic Schrödinger equation in electromagnetic field is generally equivalent to the non-relativistic Schrödinger equation in electromagnetic field for a real wave function (to prove it, you just need to use the unitary gauge where the wave function is real). In the resulting equation, you can express the real wave function via its square, which is probability density (I emphasized that the equation for probability density is nonlinear). That will work, at least locally. Whether this is "promising" or not, I don't know and I don't care for now :-) Let me note that this approach does not use the Klein-Gordon equation in any way. Again, I just offered some information about little-known mathematical results. If you think the mathematics is flawed, please advise. As for interpretation of these results... That is a different story.

akhmeteli

I'm not sure the double slit experiment creates the weirdness in QM; however, if it does, this weirdness can be reproduced in classical mechanics. You may wish to look at this article: Yves Couder, Emmanuel Fort, Single-Particle Diffraction and Interference at a Macroscopic Scale, Phys. Rev. Lett. 97, 154101 (2006), where, in particular, two-slit diffraction is successfully modeled by classical objects. I guess the following recent article of the same authors is publicly available: http://iopscience.iop.org/1742-6596/...1_1_012001.pdf (I don't want to copy their abstract here or rephrase what they did - it would be better if you look at the original articles, if you're not aware of them yet). Nobody doubts that Feynman was a greatest physicist, but your quote is 50 years old. My understanding is nowadays people tend to see quantum weirdness in experiments with more than one particle, such as experiments on the Bell inequalities, rather than in experiments with one particle, such as double-slit experiments.

akhmeteli

I told RUTA how the non-relativistic Schrödinger could be rewritten in terms of probability density only (my posts 10 and 19). Does that contradict what you're saying? I don't know. First, the resulting equation would be nonlinear, so there are no superpositions, second, there may be some difficulties with the square root, third, there may be some subtleties with the gauge choice. But it seems the equation can be written. Again, I just mentioned some mathematical results. You choose how to interpret them (I tend to think they allow different interpretations).

RUTA

My response was to your reference to the KG equation and the possibility of it providing probability interference per TBN's question. My statement stands.

akhmeteli

I may be somewhat confused about which statement you have in mind, but if it is "one would expect to obtain probabilities from the Klein-Gordon solutions by squaring", then again, this statement does stand, but only as an approximation.

RUTA

The twin-slit experiment does not "create the weirdness of QM," it serves as an example of it, i.e., the interference of probability amplitudes. This is what distinguishes QM probability from CM probability, a fact many in the foundations community would like explained 50 years after Feynman's quote.

The experiment you allude to will not explain the interference of amplitudes in QM or QFT, nor does it map onto the SE, since the particle can only receive updates for changing boundary conditions at the finite wave speed of the vibrating liquid. In that sense it has less explanatory power than DBB (admittedly a relatively popular interpretation of QM).

You are correct, there are other things that foundationalists want explained, e.g., entanglement.

akhmeteli

I am not trying to draw more profound conclusions from this experiment than warranted. I am just saying that it emulates some unusual features of the quantum double-slit experiment. It is not an explanation of everything under the sun. However, the Feynman's quote is about the double slit experiment, as far as I understand, not about "interference of probability amplitudes" in general, and Couder's experiment does tell us something new about the double slit experiment. Furthermore, as far as I understand, there is no positive experimental evidence of particles receiving updates for changing boundary conditions at infinite speed so far, whereas theoretical proof of the violations of the Bell inequalities has its share of problems as well, as we discussed elsewhere.

bohm2

I know this might be slightly off-topic but there have been some attempts to use Couder's experiments to model QM including an understanding of non-locality and entanglement:
And they offer some ideas in this direction:
A Classical Framework for Nonlocality and Entanglement
http://lanl.arxiv.org/pdf/1210.4406.pdf

rewtnode

Could you explain with a few words or in a nutshell what the "gauge mechanism" is?

RUTA

Feynman is using the twin-slit experiment as a model of the point he wants to make, not as something to be explained in and of itself. See (1) - (3) of the Summary:

(1) The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number phi which is called the probability amplitude.

(2) When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately.

(3) If an experiment is performed so that it's possible to determine which alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost.

I'm sticking to mainstream physics. Although PF permits me to answer the original question per my own published interpretation (Foundations of Physics, Classical and Quantum Gravity and Studies in the History and Philosophy of Modern Physics), I think it best to answer this OP per QM texts. I'm very interested in alternatives such as you have presented, that's why I attend conferences in foundations of physics. But, I don't want to lead the OP to believe such interpretations are mainstream. We're in a very tiny minority.

RUTA

Their statements are speculative. I'm trying to answer the post concerning probability amplitudes in textbook fashion. See my post immediately supra.

akhmeteli

Your words do not contradict mine. The quote is still about the double slit experiment, and the Summary is several pages apart from the quote.

I guess my approach is much less noble:-) I am not too shy to plug in articles that I admire or my own published articles, if some of them seem relevant. On the one hand, as you said, PF rules do allow that, on the other hand, why should I hide Schrödinger’s work, which is highly relevant, from OP? For example, hypothetically, OP could take up nanosiborg’s challenge (post 3 in this thread) and waste a lot of time, trying to invent a bicycle:-) Unfortunately, Schrödinger’s beautiful work is not mentioned in quantum texts, although it fully deserves it.

Thank you very much. I did not think you might be interested, as your approach is so different. I should note though that I did not present any interpretation in this thread, I just discussed some little-known mathematical and experimental facts.

I respect your point of view – it is certainly reasonable, but other considerations can also be important in specific situations.

RUTA

The Summary contains no mention of twin-slit. He used twin-slit to illustrate, as he states explicitly therein, the manner by which one produces probability in QM (which holds in QFT as well). The reason I introduced the quote was in response to the question, "how can probability waves interfere destructively?" In other words, I'm pointing out that per quantum physics, they don't. If your answer is the Couder experiment, then I infer that you're claiming quantum physics is severly flawed. In my dealings therein, the foundations community does not believe quantum physics is flawed. If anything, the community is moving towards an "all in" position as evidenced by widespread interest in quantum information theory and Many Worlds. [15 years ago there was some interest in using classical methods to replace quantum physics, but it's rare these days.]

I could be wrong. Perhaps there is a large or growing subset of foundationalists who want to see quantum physics replaced by classical methods. Do you believe this to be the case?

akhmeteli

This is just your interpretation of the quote. The Summary does not equal the quote. The quote stands (or at least can stand) on its own. Anyway, whatever Feynman meant, your specific use of his quote ("Exactly, the twin-slit experiment “has in it the heart of quantum mechanics. In reality, it contains the only mystery.” ") left no doubt that you were talking about "the twin-slit experiment." It turns out you had in mind something else. Well, I am no neurosurgeon, I could not read your thoughts:-)

I gave my opinion in post 10 in this thread (question 5).

As I said, I am not trying to deduce too much from the Couder experiment, but I think it is an instructive classical analogy of the quantum experiment. As a result, people who believe that features of the quantum two-slit experiment cannot be reproduced in classical mechanics (and there are a lot of such people) have to offer more sophisticated arguments. I should say that my approach to interpretation of quantum theory is quite different: I only consider systems of interacting matter and electromagnetic fields. On the other hand, this approach gave a surprising interpretation-independent spin-off: it turned out that the Dirac equation in an arbitrary electromagnetic field is generally equivalent to a fourth-order PDE for just one (complex or real) component.
I don’t know about “severely”, but yes, I think, strictly speaking, standard quantum theory is indeed flawed, and, judging by your posts elsewhere, you know that it is so: there is the measurement problem, in particular, the contradiction between unitary evolution and the theory of quantum measurements, e.g., the projection postulate. You know very well that I did not discover or invent this problem:-).

Well, I also go to conferences on quantum foundations (I have been at four this year:-) , but this was a fluctuation:-) ), but it is difficult for me to say what the community believes or does not believe. Your assessment is probably correct, but that does not necessarily mean that standard quantum theory is immaculate:-) Thanks god, I don’t need to explain to you that the measurement problem does exist:-) You said that we're in a very tiny minority, but it looks to me that everybody is in minority in quantum foundations: I’d say there are at least three mainstream interpretations (Copenhagen, Many Worlds, and “shut up and calculate”) and a lot of non-mainstream ones. This situation suggests that there is no satisfactory interpretation so far.

Probably not, although there were and there are quite a few outstanding physicists who are not happy about mainstream interpretations. Again, the issue of interpretation cannot be decided by a popular vote. There is some steady incremental progress in foundations, both in theory and experiment, and I don’t think the final destination is a sure bet. We all will have to live with future progress, whether we’ll like what we’ll see in the future or not.

bhobba

I think the emerging modern view is its the simplest probabilistic theory that allows continuous transformations between pure states which when you think about it is what is really needed to model physical systems:
http://arxiv.org/pdf/quant-ph/0101012v4.pdf

Still doesn't really make it less weird and its not my personal approach which is based on invariance but does seem to capture quite a bit of its essence.

Thanks
Bill

DiracPool

Wow, I can't believe this thread is still going. Here's the original question in case anyone forgot...

Shrodinger attempted to describe how an electron interacts with a proton within an atom. He did this by applying the Hamiltonian operator to the wave equation and came up with the SE, which described the behavior (e.g., position, momentum, energy) of that electron or, in fact, any particle, by a wave "function." The particluar function depended upon the particulars of the system being analyzed.

Upon reviewing the solutions to the SE that manifested these wave functions, everyone was puzzled as to what they meant. It was only when Max Born came along and said, hey, if we multiply this wave function by its complex conjugate, it looks like we can use this to determine the probability of the particle or "system" being at some small interval of space d-tau.

This method has worked for 80 years so why do we want to change it now? It's kind of like asking the Swanson company why they cook their frozen dinners before they package them. We could try to make the argument to them that they could save time by simply packaging them while they cook them at the same time. That might not come out so well, though. Squaring the result is one more step in equations that can run many pages long. Why not play it safe and cook the meal before you package it?