Talk:Wave–particle duality/Archive 2

Objects

 * JA: The word "object" does not imply tangibility. If you want to imply tangibility you say "tangible object".  An object is an "object of discussion and thought" (OODAT).  Some folks would add "object of consistent discussion and thought" (OOCDAT).  So, for example, numbers are objects, even imaginary numbers.  Jon Awbrey 20:45, 17 February 2006 (UTC)


 * object –n. 1 a material thing that can be seen or touched.
 * The Concise Oxford Dictionary of Current English, 8th ed., Ed. R.E. Allen, Oxford: Clarendon (1990).
 * --Srleffler 01:47, 19 February 2006 (UTC)


 * JA: Oh, surely it's not that concise. Jon Awbrey 04:08, 19 February 2006 (UTC)


 * Of course not. There are several more meanings. My point, though, was that the first meaning, the most common one, specifically implies tangibility contrary to what you asserted. In general usage it is not necessary to add "tangible" to distinguish from other meanings. The usage in philosophy might be otherwise, but this article is about physics and the common usage of object is just fine here.--Srleffler 05:07, 19 February 2006 (UTC)


 * JA: (A => 1 or 2 or 3) <=/=> (A => 1). If they are still using logic in physics, and if they are still using numbers in physics, then object does not imply tangible, even in physics.  Jon Awbrey 05:14, 19 February 2006 (UTC)
 * Sorry, but that just doesn't make any sense. It's perfectly valid to use a word in its most common sense, notwithstanding the fact that the word may have other senses.--Srleffler 06:19, 19 February 2006 (UTC)


 * JA: Yes, but the word does not "imply" the meaning that its most common use has. It implies only the meaning that all of its uses have in common.  Jon Awbrey 06:26, 19 February 2006 (UTC)


 * OK, so then what was your point? Why should we care what the word "implies"? The word is used correctly in the article.--Srleffler 06:30, 19 February 2006 (UTC)


 * JA: That was my point. Someone expressed some doubts, and I was trying to soothe them.  That's evidently more difficult than it sounds.  Jon Awbrey 06:38, 19 February 2006 (UTC)

Probability density
The magnitude of the square of the wavefunction is a probability density, not a probability. Probability densities can exceed 1. I believe the article should reflect this mathematical fact - probability density is not that difficult and there is an entire article article on it for those unfamiliar with the term. - JustinWick 05:06, 23 February 2006 (UTC)
 * I don't believe probability densities can exceed 1. What gave you that idea?--Srleffler 12:26, 23 February 2006 (UTC)


 * Certainly they cannot exceed 1, but this is merely a result of semantic nitpicking. Probability densities have the dimensions of inverse distance, and so may not even be compared in magnitude with a dimensionless number.  However, for all intents and purposes, there is really nothing wrong with saying they can exceed 1, as there is nothing to stop a probability density from being as large as you like, e.g., one can have a probability density of something like "one billion per meter", as long as this density extends over no more than a nanometer. - Expensivehat 04:41, 6 March 2006 (UTC)

good article?
I've removed this article's good article status. The reason is that the only theoretical explanation that this article provides of wave-particle duality is that quantum systems are solutions to differential equations. That condition is neither necessary (there are quantum systems which exhibit the duality which are not solutions to diff eqs) nor sufficient (there are classical systems which are solutions to diff eqs which do not exhibit the duality) as an explanation of the phenomenon. A historical treatise on the evolution of whether light was modeled by waves or particles is interesting, but certainly not the primary reason purpose for this article. As such, not only is this not a good article, it is a bad article. -lethe talk [ +] 23:01, 7 March 2006 (UTC)


 * Hi, well, as I'm repsponsible for the current state of the article, this remark catches my attention. I am rather totally unclear on what you are trying to say here. Is there any sort of definition of wave-partcile duality aside from random handwaving and appeals to history? To the best of my knowledge, there is no such thing as a "theoretical explanation of wave-particle duality", beyond the usual overheated argumentas about the interpretation of quantum mechanics and local realism. What do you have in mind? linas 00:45, 8 March 2006 (UTC)
 * Observables in quantum mechanics are represented by operators on a Hilbert space. If two observables do not commute, then the spectra of the two operators exhibit complementarity.  We understand waves as momentum eigenstates and particles as position eigenstates, then wave particle duality is described by the noncommutativity of X and P.  In relativistic field theory, there is no position operator, and instead we will talk about localization operators.  None of this is mentioned in the article, instead there is talk of differential equations, the relevance of which I fail to see entirely.  I also don't see the relevance of any discussion of local realism or interpretations of quantum mechanics.  As far as I know, all formulations of quantum mechanics exhibit wave particle duality, which shows that this is not dependent on the interpretation. -lethe talk [ +] 01:29, 8 March 2006 (UTC)


 * Complementarity has nothing to do with wave-particle duality. Complementarity just says things with momentum p have a wave number hbar k. But what's a particle? To talk about particles in QM, you have to talk about "wave function collapse", and that's thin ice. There are questions: when I measure a wave, why don't I ever see two half-particles? At its heart, QM is a wave theory, and no more. At least in QFT, you have the idea of creation and anhilation operators, and so finally you get back to something that can be called a particle. Anyway, you are right; the theory section is lacking; and someone should revise it. I have just about zero interest in writing a "theory" section for this article, mostly because I can't think of anything lucid to say about the theory aside from "shut up and calculate" (or rather, shut up and hit the books). linas 01:43, 10 March 2006 (UTC)


 * Perhaps you should take a look at complementarity. It is not the principle p = h/λ.  I don't agree with much of your comments.  I suppose the best solution would be for me to undertake to rewrite the article.  Perhaps someday I will. -lethe talk [ +] 04:14, 10 March 2006 (UTC)


 * I agree; I think Linas is referring to the de Broglie hypothesis. I've added a section on the Heisenberg uncertainty principle, explaining that it follows from the application of de Broglie to classical field theory, which may answer some of your earlier concerns about misleading statements about its derivation. --Michael C. Price talk 14:23, 29 November 2006 (UTC)

Wavelengths of human sized objects.
The article says "One does not observe the wave-like quality of everyday objects because the associated wavelengths of people-sized objects are exceedingly small." But can a human sized object be described by a single wave? There's no object of that kind of mass that is composed of a single fundamental particle. So wouldn't the non-wavelikeness of large objects be due to the fact that they are made up of lots and lots waves with wavelengths much smaller than the physical size of the object?
 * You don't have to be a fundamental particle to exhibit a wavefunction. For example, protons have wavefunctions, and atoms, and even some molecules exhibit a little bit of quantum interference.  Furthermore, for certain systems like Bose-Einstein condensates and superfluids and superconductors, quantum effects are manifest even though there are a macroscopic number of particles.  However, you're still right.  The presense of a whole bunch of particles tied together greatly suppresses the quantum effects, if the particles are thermalized, meaning they are randomly arranged with no possibility of coherence of their wavefunctions.  In this case, the large numbers make the objects behave classically, and this phenomenon is known as decoherence.  So macroscopic objects do not exhibit quantum effects for two reasons: 1. h is too small, the wavelengths are completely unobservable, and 2. because of the large number of thermal particles, all quantum effects get averaged out through decoherence. -lethe talk [ +] 05:15, 27 March 2006 (UTC)

Standard Quantum states that there is only *one* “wave function” for an ensemble of particles, not a sum of individual postulated wave functions of each particle.. This immediately dispenses with the daft idea that particles are also “waves”. QM states the probability of finding *particles* in regions of space. End of story. The “wave nature” of particles is simply that fact that particles don’t obey Newtonian Mechanics. Waves, as clearly shown manner times, are just the statistical collection of individual particles. 217.155.191.217 12:50, 8 February 2007 (UTC)

Mathematically Possible
If at all, how is the dual nature of light mathematically possible?

--203.200.220.71 11:20, 7 August 2006 (UTC) My question is why do we have to introduce photons for explaining the photoelectric effect? would it not be sufficient to say that only light waves of a certain frequency and above can knock off electrons? --203.200.220.71 11:20, 7 August 2006 (UTC)


 * I have expanded the explanation at Photoelectric_effect. --Michael C. Price talk 12:08, 7 August 2006 (UTC)


 * The introduction of photons, or quantized light energy, to explain the photoelectric effect, is historical and conventional. Einstein proposed it, got the Nobel prize for it, and it worked.  But it is not the only way.  One can explain quantum transitions between electron energy levels (which the photoelectric effect is a special case of) by the quantum electromagnetic interaction of the electrons; the transition involves a mixture of states, so the wavefunctions of the two states, differing in frequency by the frequency that corresponds to the energy difference, will make a "beat" in the electron distribution, which will allow it to couple electromagnetically to other subsystems of the universe that are at zero space-time interval (i.e. with other atoms that can absorb the photon when it gets there, if you want to put it back into photon terms).  The details of this view of quantum transitions is explained by Carver Mead in his book Collective Electrodynamics: Quantum Foundations of Electromagnetism. Einstein was unhappy with the "random" results of his proposal, i.e. the Copenhagen interpretation that led only to probabilities; but he did not make the bold step of retracting the quantization of light, putting all the quantization into electron eigenstates instead, and finding the alternate view that the apparent randomness is due to all the other bits of the universe that every photon event is potentially coupled to. Wheeler and Feynman did a bit of that, but backed off; Mead picked it up later, as did Cramer, in his transactional interpretation of QM. Dicklyon 19:51, 7 August 2006 (UTC)

Revert by afshar
Ashfar removed my comment in the intro about how photons were the first to be seen as both waves and particles. I gave the explanation that they are "both", and that in a double slit experiment, a detector detects points - and over time diffraction patterns show up.

Anyways, I'm not sure what is wrong about it. I skimmed the article on Complementarity (physics), and it doesn't seem to contradict me. Whats specifically wrong with the paragraph? I thought it would help clarify what wave-particle duality means, and especially what it means for something to "sometimes act like a wave and sometimes like a particle. Fresheneesz 10:17, 23 April 2006 (UTC)


 * Dear Fresheneesz, you must be careful not to say that photons are "both" unless you qualify that by saying "but never at the same time." Bohr's Principle of Complementarity posits that both wave and particle natures of quantum systems (including photons) are equally important but are not both present in the same experiment. Although, I have performed an experiment that violates Bohr's assertion, I would not support such declarations without mentioning the controversial nature of my findings. My results are still being critically studied, and until the dust settles one way or another, it would be irresponsible to declare simultaneous presence of wave and particle behaviors for photons or any other particle. It is however allowed to talk about wave-particle duality in the sense presented by the de Broglie relation in which both wave (wavelength) and particle (momentum) properties show up. I hope this explanation helps. Regards.-- Prof. Afshar 03:48, 25 April 2006 (UTC)


 * Hmm, what experiment have you done that you mentioned? I believe that slit-experiments involving a single particle at a time imply that photons are in fact both waves and particles at the same time. A particle will hit somewhere on a detector, but throughout the length of the experiment, the particles will eventually hit in a pattern that is predicted by wave diffraction. Saying that the photon acts like a wave sometimes, and a particle sometimes - doesn't quite work in this case. The effects wouldn't show up unless the photon is both, at the same time. Fresheneesz 03:31, 26 April 2006 (UTC)

The experiment shows that discrete particles have a characteristic number, that is usually associated with “waves”, conventionally named lambda (wavelength). This lambda number can be calculated by examining the statistics of the locations of the particle spots. That is, an average “wavelength”, with a standard deviation for that wavelength, can be calculated. This wavelength can be nominally assigned to a particle, but it only has real meaning, when discussing an ensemble of particles. Thus, the experiment indeed shows particle and “wave” nature together. There is of course, no real wave in any medium. The wave is an illusion, just as a water wave is an illusion, brought about by the statistics of large number of water molecules. Wave-particle duality is completely bogus. Standard QM is about particulate objects that follow non Newtonian laws of motion. End of story. Kevin aylward
 * If wave-duality is such a load of shit then please describe how a radio antenna measures the signal frequency in terms of radio particles. (And please time-date stamp your signature for further replies.)  --Michael C. Price talk 18:31, 28 October 2006 (UTC)


 * See Afshar experiment for Afshar's experiment. He claims that he measured a violation.  I agree with Afshar that such a claim doesn't belong at the top of this article, and I appreciate his modesty. -lethe talk [ +] 04:11, 26 April 2006 (UTC)


 * Wow, very interesting. I'm honored. I still don't understand why a single photon double-slit experiment also doesn't contradict complimentarity. Fresheneesz 06:55, 26 April 2006 (UTC)


 * Basically, complementary says you can not both see the diffration pattern and determine which slit the particle went through. In the two-slit experiment with both slits open, you get the diffraction pattern, but no idea which slit a particle went through (because it had to go through "both" to interfere); with one slit closed, you know which slit the detected particles when through, but you lose the diffraction pattern.  That's all the duality is.  Afshar's experiment can be interpreted as saying that the hole through which the photon went is determined, at the same time as a wave pattern is determined (somewhat indirectly); exactly how it relates to the formal principal of complementarity is subject to considerable dispute.  Dicklyon 23:41, 30 May 2006 (UTC)


 * so the short answer, Fresheneesz, is that the single photon double-slit experiment does contradict complementarity as defined by Afshar, since the photon exhibits wave behaviour by going through both slits and particle behaviour when it "collapses" onto a single sliver-bromide emulsion grain. Of course most people do not accept Afshar's definition of complementarity he gave above as implying that wave and particle properties "are not both present in the same experiment".  Rather the conventional view is that pure wave and particle behaviour may not be present at the same space-time event, as quantified by the Heisenberg uncertainty constraints.  --Michael C. Price talk 18:23, 22 June 2006 (UTC)


 * Dear Michael, Once again you have misrepresented the facts. I have never said that the single photon interference experiment violates Complementarity. In fact I have specifically mentioned on page 3 of my original paper (if you care to read before making uninformed statements) that: " It is noteworthy to mention that quantum mechanics does not forbid the presence of non-complementary wave and particle behaviours in the same experimental setup. What is forbidden is the presence of sharp complementary wave and particle behaviours in the same experiment. Such complementary observables are those whose projection operators do not commute [1]." Projection operators for sharp which-way information and interference qualify as complementary observables. A simple interference pattern contains no which-way information, thus it is not subject to Complementarity. Please read the paper by [1] Bandyopadhyay, Phys. Lett. A 276 (2000) 233, you may learn a thing or two on the subject.


 * As regards Bohr's statement concerning Complementarity, here's what he himself has said: "...we are presented with a choice of either tracing the path of the particle, or observing interference effects…we have to do with a typical example of how the complementary phenomena appear under mutually exclusive experimental arrangements."[2] N. Bohr, in: Albert Einstein: Philosopher-Scientist, P. A. Schilpp, Ed. (Library of Living Philosophers, Evanston, Illinois, 1949). There is no mention of the requirement you impose above that the measurements be made "at the same space-time event." In fact, it is impossible to see an interference pattern if the measurement is supposed to be made at the same spacetime event, as the pattern only builds up over time, which means it requires great many space-time events to register as interference pattern. Sorry to burst your bubble, but you really need to read up on the subject before you embarrass yourself any further. You can start with my paper, and references therein. On the other hand, if you are deliberately spreading disinformation on my work, you should expect to see some repercussions. Regards.-- Prof. Afshar 16:11, 28 October 2006 (UTC)
 * If you're saying that the Schrodinger wave equation and complementarity are incompatible then just come straight out and say it; get the claim published in a peer rewiewed credible journal and I'd be very surprised. What Bohr meant by complementarity is explained over at complementarity; there's no evidence that he doubted the Schrodinger equation as a non-relativistic model. PS please give the full Bohr quote.  --Michael C. Price talk 18:31, 28 October 2006 (UTC)

Apparent paradox?
The paradox article says "A paradox is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition." What then is an apparent paradox? Is the article wrong? Is there a source/reference for Lethe's interpretation "paradoxes are apparently true but lead to contradiction. apparent paradoxes look like they should lead to contradition, but are actually only counterintuitive"? Dicklyon 17:21, 21 June 2006 (UTC)


 * The second part ("which defies intuition") makes my case weaker, so I won't revert if you remove the addition again. I restored the word "apparent" because the person who removed it a month ago only did so in order that it would support an argument he was having with me, and I don't that's a good reason to change the article.  I am having an rfc with him, and so was reviewing his edits. -lethe talk [ +] 19:02, 21 June 2006 (UTC)


 * OK, I found other definitions that agree (not just copies of the wiki def), so I'll take it out again. Don't want to step in that other mess... Dicklyon 04:16, 22 June 2006 (UTC)

Particle and its definition
The opening sentence did not have an internal link to the articles known as "elementary particles" or "composite particles" so I added the needed brackets. Be that as it may, my quick read of the article on "elementary particle" is one that is internally cyclic and does not really define what constitutes a "particle". Is anyone ready to tackle this task? Bvcrist 03:57, 14 August 2006 (UTC)

This point concerns me as well. The article doesn't seem to say explicitly what it means to "behave like a particle". I don't know if my thoughts can be worked in somehow, but here they are. To me, "behaving like particles" means the amount of that thing is quantized. You can have one photon in a box as a standing wave, but you can't have a quarter that much energy. I think the confusion arises because a lot of people hear "particle" and they (understandedly) think "localized". It's impossible to reconcile "wave" and "localized", so that leaves people confused. I say emphasize that "particle" refers to the quantization but does not imply the thing is a small dot. A single particle can extend over a large volume. Spiel496 05:20, 14 October 2006 (UTC)

Modern interpretation
Someone has been putting in and taking out the attempt at a "modern interpretation":
 * Quantum mechanics was and is still a controversial sicence. It "evolved" throughout the years, but somehow, old interpretations are still used and being taught.
 * Light and matter seem to both exist as particles. What behaves like a wave is the probability of being at a certain place. The wave function is said to be to be a mere mathematical abstaction.

The big problem with this attempt is that it represents only one POV, and doesn't say whose it is. Phrases like "is controversial" and "is said to be" are non-encyclopedic. If an interpretation is be discussed, we need to know whose it is, and we need to include other ones as well. Dicklyon 16:17, 20 August 2006 (UTC)

Kevin aylward 12:53, 25 October 2006 (UTC) The big problem with your reply Michael, is that this modern interpretation IS the correct modern impetration. The whole idea of wave particle duality is completely bogus. All graduate level academic physics text books state quite clearly that QM is about the probability of, for example, locating a particle in a specific volume. Do a web search on “postulates of quantum mechanics”. This interpretation is not controversial in the slightest, and is universally held by professional physicists.


 * And more appropriate at interpretation of quantum mechanics. --Michael C. Price talk 18:37, 20 August 2006 (UTC)

Uses section implies that wave / particle duality is confirmed
I am uncomfortable with the wording in the Uses section. It seems to imply that the examples listed depend on (and thus prove) wave / particle duality. Instead, I think they exploit the wave nature of the things listed, but do not require or demonstrate a particle nature.

The things described can also be explained by, for example, QFT (all is waves / fields), and for example, the collapse of a wave on impact with another wave of the right type (e.g., the nucleus of an atom (or some part thereof, e.g. a proton, neutron, quark, or ??).

I haven't thought much about how to reword the section, but hope that I can manage to revisit the page in a week or so and reword it.

To summarize, I think those examples exploit and demonstrate the wave nature of the items mentioned, without demonstrating, proving, or requiring a particle nature.

If on the other hand, those things do prove the wave / particle duality, well, then physics is further ahead than I thought.

Rhkramer (talk) 14:15, 18 November 2021 (UTC)


 * Note: I have removed your note from the article, as it provides an unsourced claim—see wp:RS and wp:NOR. The place to discuss is here. If nobody seems to respond here, you can leave a request for comment at Wikipedia talk:WikiProject Physics. Cheers. - DVdm (talk) 16:36, 27 November 2021 (UTC)
 * Note: Reinserting the text and calling it a fact in your edit summary, is not the best way forward—see wp:Edit warring and wp:BRD. If it is indeed a fact, then it should be easy to find a wp:reliable source for it. - DVdm (talk) 19:43, 27 November 2021 (UTC)
 * Just for the record, the first time I made that edit, I prefaced it with the word "Note:", and when it was reverted, the reason given was that it was because it was a note. So, the obvious (to me) solution was to remove the word "Note:".
 * I still believe the Uses section is misleading (as I discussed above), and hope that someday, I or someone else will make that less misleading.
 * Note that the article itself, in the section "Alternative Views" makes it clear that the wave | particle duality is not universally agreed.
 * Rhkramer (talk) 13:41, 20 May 2022 (UTC)

two images are simply incorrect.
"A quantum particle is represented by a wave packet" and the similar one claiming to show particle interference are not correct physics for wave particle duality. No theory or experiment shows such a thing. Johnjbarton (talk) 21:53, 4 July 2023 (UTC)


 * Agreed. Both should be replaced by standard versions of the two. Ldm1954 (talk) 14:03, 5 July 2023 (UTC)
 * I removed both images.
 * Wave packets of light can be created, but the animation seems unlikely to be accurate even of this case. More important these images on a page about 'wave-particle duality' confuse the topic. Duality is almost never about packets as was famously discovered by E. Schrodinger vs N. Bohr. The story around Schrodinger's harmonic oscillator packets should be included in this article.
 * Johnjbarton (talk) 02:13, 6 July 2023 (UTC)

lead sentence is misleading
The current lead sentence makes a statement of fact that is contrary to, for example, Bohr's point of view.

"Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave."

Bohr and many others to this day argue that no description is justified. Duality refers to the possible outcomes of experiments, not possible descriptions. Johnjbarton (talk) 22:48, 21 July 2023 (UTC)


 * Agreed. Ldm1954 (talk) 23:01, 21 July 2023 (UTC)
 * I will add that I think this article will confuse someone who does not know the topic -- it rambles. Ldm1954 (talk) 15:58, 31 July 2023 (UTC)
 * Unfortunately topic is controversial with several hard held points of view. But we should take this one up soon. Johnjbarton (talk) 17:09, 31 July 2023 (UTC)

Restructure
I think this page needs significant restructuring, particularly the history. I suggest:
 * History before quantization
 * Light as a wave
 * What is already there
 * Electrons as particles
 * Borrow from electron diffraction and similar
 * The advent of quantization
 * Light as quantized particles
 * Planck, Einstein
 * Electrons as quantized waves
 * Diffraction

After this would come other stuff, what we can decide later. Exact titles can change, but I think this structure flows, and lends itself to summaries before to guide the reader. Ldm1954 (talk) 01:50, 1 August 2023 (UTC)


 * Your outline would definitely be better. Here is another alternative to consider.
 * A chronological focus could lead the article towards a "history of wave particle duality". That's a fine subject but not one that highlights the physics. This is especially a problem in wave particle duality because much of the history dips in to philosophy.
 * As an alternative, consider an example driven or phenomenon driven article. Small sections, each of which describe an experiment and the physics, ending with a short history section linking to the History of quantum mechanics article.
 * The higher level organization could be (top)Wave: (second)light/electron/neutron/molecule followed by (top)Particle: (second)light/electron/neutron/molecule or it could be Light: wave/particle, Electron: wave/particle, etc.
 * In addition to avoiding the sea of dates and philosophy this approach makes wave-particle duality manifest in the outline. Rather than a series corrected mistakes, the story become one of overwhelming evidence. Johnjbarton (talk) 14:36, 1 August 2023 (UTC)
 * If you mean delete the whole "History" section and only focus on current/established science, then I am happy with that. I think a plan/sketch would be needed first as I am not sure about the example-driven approach. It will also to give others time to complain/edit/suggest. Maybe, please expand/edit:
 * Waves as particles
 * Photons
 * Collective excitations (phonons, plasmons)
 * Particles as waves
 * Electrons
 * Atoms
 * Everything as everything
 * Quantum field theory?
 * Ldm1954 (talk) 15:01, 1 August 2023 (UTC)
 * I created Draft:Revised_wave-particle_duality with an outline. That Talk page can be for development. If you like the idea we can post to the Physics Talk page. Johnjbarton (talk) 17:28, 1 August 2023 (UTC)
 * I am not enthused, in part because I don't exactly see the structure. To me it should be more "These we all routinely think of as waves/particles -- but aha, what normally we think of as particles can behave as waves, and also waves as particles." Maybe spelt out more I will understand better. (I added some comments, but these reflect my uncertainty.) Ldm1954 (talk) 17:45, 1 August 2023 (UTC)
 * Sorry I was unclear. By "like the idea" I meant using the Draft namespace to agree on an outline/revised article. Otherwise this topic will get quite deep and off track to the current page. Johnjbarton (talk) 17:50, 1 August 2023 (UTC)
 * Sorry I was unclear. By "like the idea" I meant using the Draft namespace to agree on an outline/revised article. Otherwise this topic will get quite deep and off track to the current page. Johnjbarton (talk) 17:50, 1 August 2023 (UTC)

Is the Newton-Young-Planck-Einstein history really about duality?
The current history section recounts all of wave physics, but "duality" had no part in that discussion until Bohr's complementarity. Before Bohr's work, physics wanted one or other other. So the history of wave/particle duality should be much shorter, with just enough background to demonstrate the dilemma that gave rise the to the concept covered in the article. Johnjbarton (talk) 18:47, 1 August 2023 (UTC)


 * 👍 Ldm1954 (talk) 20:42, 1 August 2023 (UTC)
 * I'll move some of this to the main History of quantum mechanics Johnjbarton (talk) 21:34, 1 August 2023 (UTC)
 * I started my hatchet job on the history, but now I think the next step is to create a proper history of duality so we know how much of the old-quantum theory history is needed. Then the remaining history sections can be boiled down to appropriate summaries. Johnjbarton (talk) 00:03, 2 August 2023 (UTC)
 * I've added a section on Bohr's 1926 Como lecture and used it to summarize the dilemmas of the previous 20 years. Please review.
 * My next step would be to check that the content in the remaining history is covered elsewhere and then delete the remaining history. Then we can build forward from 1926 with Feynman / quantum eraser etc. Johnjbarton (talk) 17:40, 2 August 2023 (UTC)
 * I copied everything to User:Ldm1954/Sandbox/Duality. I added a couple of sentences before the Como lecture material (which I like), to give a gentler entry. I added some comments. I suggest editing this until it is decent before making the massive changes. Ldm1954 (talk) 18:35, 2 August 2023 (UTC)
 * I'm ok slowing down the pace here. A disadvantage of using the sandbox is a lack of transparency, then boom a new article. So if we do that route we'd need a review period. I guess that's ok. Johnjbarton (talk) 22:35, 2 August 2023 (UTC)
 * OK, no problem as I am juggling multiple things anyway. What I think is important is to have a coherent partial section with refs ready, it does not have to be everything. Interim versions should still be readable. Currently refs 10 & 11 are defined but not used.
 * Maybe we can just decide/agree on the lead? Ldm1954 (talk) 22:53, 2 August 2023 (UTC)
 * So it's ok to say: I should have drafted the history changes rather than just add Como. Yes, my bad as they say these days.  Feel free to revert now that you have a copy.
 * Perhaps a middle ground is to transfer the draft in sections with Talk commentary. (OTOH draft will get out of sync with article changes as has already happened).
 * Your deletions in the Draft lead are fine, though I want to check if the Heisenberg paragraph could be in history. I think we should wait on the rest as the lead implies an outline and I don't think its completely clear yet. Johnjbarton (talk) 15:45, 3 August 2023 (UTC)
 * I think the way to do it is to complete in a Sandbox something close, at least the lead & history. Then start to put this in with entries in the Talk page to go with it. I will check the main page now as I have a little time, and cure any nasties that I see. Ldm1954 (talk) 15:56, 3 August 2023 (UTC)

Visualization is about uncertainty principle not wave-particle duality.
One of the issues with this article is its conflation of uncertainty principle wave-particle duality. To be fair, Bohr's charter paper on what came to be called wave-particle duality talked about both. But uncertainty principle is a wave property; the Visualization section is about Fourier transforms of waves.

I propose to move the section directly into uncertainty principle, minus the random comment at the top of the section. Johnjbarton (talk) 22:40, 2 August 2023 (UTC)


 * Copied the content of the Visualization section with edits to Uncertainty principle. Johnjbarton (talk) 23:01, 2 August 2023 (UTC)
 * I concur that the "Visualization" section does not belong here. XOR&#39;easter (talk) 18:44, 3 August 2023 (UTC)

Importance section is incorrect or superfluous.
The Importance section seems designed to make wave-particle duality appear unimportant.

The first paragraph is about Schrodinger solutions with and without mass. It uses the words 'particle' and 'wave' that seems to be the only connection to the article.

The second paragraph is about measurement but it attempts to discuss measurement from the interpretation point of view and thus is doomed to fail. Completely misses the wave-particle aspect of measurement.

The third paragraph announces the QFT solves wave-particle duality, news to physics.

Each of the topics could find a home in the article, but not as written. I think "Importance" as a subject section would need some significant references from Big Names in Physics. I'd skip that myself. Johnjbarton (talk) 23:16, 2 August 2023 (UTC)

Duplicate sections in "Alternatives"
Bohm model is discussed twice in the Alternatives and they should be merged.

The Buchanan ref for "significant minority" is on a completely different topic.

The Couder visualization is cool but its about pilot waves, not duality. It is refed in pilot-wave theory. Johnjbarton (talk) 16:06, 3 August 2023 (UTC)


 * The "Alternative views" section is a trainwreck that conflates so many different topics I doubt it's worth the effort to pull them apart. The debate over whether QFT more supports a particle or field ontology is not the same thing as the debate over QM interpretations. Nobody can learn anything from a disjointed collection of garbled takes about various gripes over the past century. Nuke it from orbit. XOR&#39;easter (talk) 18:39, 3 August 2023 (UTC)
 * There are several reasons I would like to save this section (re titled as eg Interpretations)
 * 1) As contrast. I think some readers will come to this topic expecting interpretation discussions. Having an Interpretations section allows the topic to be set against the "above the fray" duality notion.
 * 2) Interpretations have historically been a driving force in QM foundations, and duality is part of foundations.
 * 3) The Interpretations of quantum mechanics article is a menagerie and the wave/particle/both/neither outline here is a cute slice that won't fit any where else.
 * 4) The section allows a lot of contextual links to other parts of wikipedia.
 * "Save" means some work but I think worth a try. Johnjbarton (talk) 02:58, 4 August 2023 (UTC)
 * I see no text in it that is worth saving. Rewriting something new from scratch would be simpler. XOR&#39;easter (talk) 15:50, 4 August 2023 (UTC)
 * Ok I've come up with another way to discuss "interpretations" in the context of wave-particle duality, with good refs. Both Feynman and Schrodinger (in later life) discuss the role of unverifiable physical models. So this section can go also.
 * Johnjbarton (talk) 23:00, 6 August 2023 (UTC)

"Uses" section is about uncertainty principle.
The third bullet point refs a (bogus) press release article about some cool images from EPFL fast TEM but the rest is not useful here. Johnjbarton (talk) 16:10, 3 August 2023 (UTC)


 * The first bullet point is also wrong. Electron microscopes and diffraction depend upon Schroedinger's equation, too many amateur sources invoke de Broglie, which is wrong. The key explanation was by Hans Bethe, see the Electron diffraction page. Ldm1954 (talk) 19:43, 3 August 2023 (UTC)

Well produced animation is incorrect in several ways.
The .mov file labeled "Animation showing the wave–particle duality" is very nicely produced but several things about it are incorrect.

The central and biggest issue is that it is "ontological": it shows tiny things doing stuff. That is exactly what wave-particle duality says you need not do. We do not know what tiny stuff looks like so animating it is necessarily incorrect.

The video claims a quantum object is a wave and a particle at the same time. There is no physical evidence to support this claim.

Also bad: the non-tiny bullets are not correctly demonstrated. They wander about as if in some kind of wind, resulting in a uniform background on the detector. This is so wrong. Ray trajectories are straight lines and consequently the pattern on the screen would be two sharp shadows. This has been known for centuries.

The wave part is ok, it does show the wave "starting" but I would let that slide.

The quantum part is I guess a wave packet, perpetuating the idea that because wave packets are possible they are also some kind of explanation.

The observer part is also mythical, leaving the impression that the observer converts the wave to a particle mid-flight. There is no experimental or theoretical justification for that view.

I like the perky music track.

Johnjbarton (talk) 16:03, 4 August 2023 (UTC)

"A dramatic series of experiments" are not about wave-particle duality.
The paragraph about neutron diffraction detecting the gravitational phase shift of neutron matter waves is about QM, gravity, and interferometers. But it does not add anything to the story of duality.

I added a paragraph to Neutron interferometer based on the reference in the paragraph. Johnjbarton (talk) 21:35, 8 August 2023 (UTC)

Yve Couder paragraph misrepresents their work.
The cool visual work of Yve Couder et al. is presented as related to wave-particle duality. However, these scientists do not say that in their published work. They use words like "Classical Wave-Particle Association". The only connection to the article is the Science Channel video (narrated by Morgan Freeman!) Johnjbarton (talk) 21:44, 8 August 2023 (UTC)

My images are right
@Quondum, he or she wrote :

@Thierry Dugnolle:, maybe you could correct what seems like a qualitative error in this GIF? Specifically, for a massive particle the phase velocity (the speed of the phase waves) is always greater than the speed of light, whereas its group velocity (the speed of the wave packet) is always less than the speed of light. It would be helpful to animate a solution to a quantum wave equation rather than just making a crude image from a conception, so that gross errors like this are not made. —Quondum 16:46, 18 May 2023 (UTC)


 * It is not a crude image. It is a solution from a quantum equation, the quantum Gaussian wave packet. The phase velocity is not always superior to the group velocity :
 * $$E=\hbar\omega$$
 * $$p=\hbar k$$
 * $$E=p^2/2m=\hbar^2k^2/2m$$
 * $$\omega=E/\hbar=\hbar k^2/2m$$
 * The phase velocity $$v_p=\omega/k=\hbar k/2m$$
 * The group velocity $$v_g=\partial\omega/\partial k=\hbar k/m$$
 * TD (talk) 11:53, 12 August 2023 (UTC)


 * How does one even start arguing here? TD simply states their own (erroneous) premises and derivation as fact.  For example, the equation $$E=p^2/2m$$ has no basis, and deriving results without a source for those results is not permitted.  Since I have raised the issue and the response is evidently not one of interested engagement, I will feel free to remove clearly erroneous material without discussion.  —Quondum 13:10, 12 August 2023 (UTC)
 * Are you ignorant ?
 * E=1/2 mv²
 * p=mv
 * Hence
 * E=p²/2m
 * Don't you know ? TD (talk) 13:35, 12 August 2023 (UTC)
 * Can I suggest that we tone this down a little. The relevant details can be found in Matter wave. The form being used by Quondum includes the rest mass, that by TD does not. Different definitions, Ldm1954 (talk) 13:36, 12 August 2023 (UTC)
 * Is it a reason to state that the equation E=p²/2m has no basis. I know it is non-relativistic, but it is still one of the fundamental equations of physics. TD (talk) 13:39, 12 August 2023 (UTC)
 * Chill please. I agree that the statement by Quondum was a bit too aggressive. Ldm1954 (talk) 13:44, 12 August 2023 (UTC)
 * I assume this discussion is about "File:Wavelet.gif" in the context of the article wave-packet duality.
 * The image claims to be a Gaussian wave packet. It would be really helpful if the equations used to prepare the image where included in the description. But it seems like it could be ok.
 * My objection to the image in the context of wave-particle duality does not relate to the quality or correctness of the image. Rather my issue is the implication. Showing this image implies that wave-particle duality is somehow "explained" by wave packets. That is a persistent myth in pop-science articles and discussions.
 * In the context of wave packet the image may be fine. My primary concern in that application is that the image shows no scale and no dispersion. Among the reasons that wave packet are interesting is their scale relative to scattering potentials and their rapid spreading that quickly makes mince meat of any idea of localization. Johnjbarton (talk) 14:21, 12 August 2023 (UTC)
 * This is not a myth in pop-science articles, this is in almost every textbook about quantum mechanics. Can you quote a textbook which agrees with you ? — Preceding unsigned comment added by Thierry Dugnolle (talk • contribs) 14:34, 12 August 2023 (UTC)
 * I'm unsure what "this is not a myth" refers to?
 * Waves can be formed into wave packets, but quantum systems do not spontaneously do so. As your image of the localized spherical wave spread shows, the apparent localization of the packet rapidly gives way to waves. Even the most localized cases, like alpha particles, the packet starts with initial width.
 * Wave packet have no role in explaining double slit experiments for many reasons. The experiments carefully create monochromatic sources, meaning coherent wave trains exactly opposite to wave packets. The theoretical analysis of double slit experiments are "time-independent", the waves are probability and do no propagate. The observed time dependence (dots on a screen) are probability not trajectories.
 * As for wave-particle duality, it is a concept that measurements will show one or the other behavior. It is not a model of QM. The closest model for wave packets in QM might be pilot wave theory, but they don't use packets either.
 * Depending on what aspect concerns you I can find references. Feynman has a long section on double slit issues: https://www.feynmanlectures.caltech.edu/III_01.html. Johnjbarton (talk) 15:05, 12 August 2023 (UTC)
 * I studied physics with the Feynman Lectures a long time ago. I don't understand why you think Feynman denies the relation between wave packets and wave-particle duality. A wave which passes through a slit cannot be purely monochromatic. The calculation with purely monochromatic waves is only an approximation. A wave we observe cannot be time-independent because it does not last forever. Gaussian wave packets are also a simplification, but a more realistic one than purely monochromatic waves. TD (talk) 15:16, 12 August 2023 (UTC)
 * May be you will prefer this animation:TD (talk) 14:29, 12 August 2023 (UTC)
 * For what purpose? This image is also not about wave-particle duality. However is it very cool! What does it show? Johnjbarton (talk) 14:32, 12 August 2023 (UTC)
 * A solution to the Schrödinger equation for an initially very localized free particle. TD (talk) 14:35, 12 August 2023 (UTC)
 * Great! So we just need a reference describing a physical situation where a localized free particle appears and a page to place the image on. I would be interested to help with that. Maybe photoemission in a gas? Or alpha particle emission? I'm pretty sure that will work. Johnjbarton (talk) 14:42, 12 August 2023 (UTC)
 * Isn't the Schrödinger equation important ? Isn't it important to show one of its simplest solution ? Why do you think that the wave-particle duality has nothing to do with wave packets ? TD (talk) 14:46, 12 August 2023 (UTC)
 * The Schrödinger equation is a differential equation and the process of differentiation "loses" information. For example if you compute the velocity of a car you can get km/hr, but you lose the information on the car location. So solutions to differential equations require additional information to create a full solution.
 * Do you suggest that we cannot know both the velocity and the position of a car?TD (talk) 15:48, 12 August 2023 (UTC)
 * No. I was trying to explain why differential equations need initial conditions and thus why plane waves or spherical waves may be adequate math solutions but short on physics. Johnjbarton (talk) 15:59, 12 August 2023 (UTC)
 * We solve differential equations with initial conditions. The process of differentiation does not lose any information. We calculate plane and spherical waves because they are the simplest solutions. With complicated initial conditions, we need computers. But plane and spherical waves, and others, are very good pedagogical examples. We need them to understand our equations. We cannot do theoretical physics if we reject them. TD (talk) 16:11, 12 August 2023 (UTC)
 * Sorry we are getting mixed up. I don't "reject" plane waves or spherical waves. But these are just "basis functions". To create a full solution you need to combine them. A monochromatic wave in free space with one axis of linear momentum is well represented by a plane wave. They are super useful because this simple case needs only one wave solution.
 * The wave created by a slit in the double slit experiment can be represented by two spherical waves. This model will give excellent agreement with experiment. It does not use any packets. (This model, which you probably know is Huygen's construction, is also not wave-particle duality, because again duality is a concept about measurement not about models.)
 * The packet-interference idea for double slit is simply incorrect. It implies that a quantum particle travels with group velocity, magically gets through the slits, then crosses over and interferes. Schrodinger's equation does not say that. More important no experiments show that. Any attempt to measure what goes through the slits will destroy the interference. That is the meaning of wave-particle duality. See quantum eraser. Johnjbarton (talk) 16:41, 12 August 2023 (UTC)
 * I don't pretend that Gaussian wave packets represent exactly what happens in a double slit experiment. What happens exactly I don't know, because I don't know the initial conditions. Gaussian wave packets are only a theoretical and pedagogical example which shows how a particle can interfere with itself. This interference is a lesson of Feynman. He taught us that interference happens with a single particle. There is only one particle and it passes through both slits at the same time. TD (talk) 16:52, 12 August 2023 (UTC)
 * Great, we agree that Feynman is a reliable source! I encourage you to go back and read what he wrote. Wave interference and single-particle events are what we observe. Self-interference matches the observations. But we have no observations of anything between the source and detection event. Zero. And no expectations that we ever will.
 * The world's greatest physicists have worked for 100 years to find a detailed interpretation for QM. Particles, waves, wavicles, pilot waves, solitons, wave packets, the list goes on and on. None of these models have shown any value in physics. If you read Feynman's philosophy section he says having any of these models in mind is fine. But none of them solve the mystery of QM.
 * Here is a direct quote from Feynman vIII chapter 1:
 * "It is all quite mysterious. And the more you look at it the more mysterious it seems. Many ideas have been concocted to try to explain the curve for in terms of individual electrons going around in complicated ways through the holes. None of them has succeeded."
 * You have a personal model that works for you. Great. Many physicists have their own personal model. We all get regularly surprised by actual facts. I encourage you to have an open mind and to read more, especially the quantum entanglement work of the last 30 years. It takes the mystery of wave interference and particle events to a whole new level.
 * Johnjbarton (talk) 18:55, 12 August 2023 (UTC)
 * I just gave a simple theoretical and pedagogical solution to explain why a quantum particle can interfere with itself. It is the ABC of theoretical physics, not my fantasy. If you are interested in quantum entanglement, I suggest you read my chapter in Quantum theory of observation/Entanglement TD (talk) 19:17, 12 August 2023 (UTC)
 * The spherical wave is part of the solution to the Schrödinger equation in free space, but you also need initial conditions. Those conditions will match the solution to the physical problem. These conditions will distinguish spherical vs plane waves for example.
 * I don't know of any physical problem where the initial conditions enforce wave packets.
 * Wave packets have nothing to do with duality because duality is a statement about measurement, not about mathematical solutions.
 * The wave-packet model is heavily used in discussions of the uncertainty principle, but those discussions use hypothetical or unusual experiments. The packet is just a visual device to show the optimal trade off of position and momentum. The optimum is just on possible initial condition, not one commonly seen and certainly not one related to double slit interference. Johnjbarton (talk) 15:21, 12 August 2023 (UTC)
 * Thanks for your explanation. Your point of view seems to be very uncommon, not the one we usually read in quantum mechanics textbooks. TD (talk) 15:26, 12 August 2023 (UTC)
 * I would like to learn which QM text books you read so I can understand your point of view. My primary sources are 1) Messiah, Albert (1976). Quantum mechanics. 1 (22. print ed.). Amsterdam: North-Holland. ISBN 978-0-471-59766-7. 2) Schiff, Leonard I. (1995). Quantum mechanics. International series in pure and applied physics (3. ed., 29. print ed.). New York: McGraw-Hill. ISBN 978-0-07-055287-6. 3) Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. ISBN 978-0-465-08061-8. 4) Penrose, Roger (2006). The road to reality: a complete guide to the laws of the universe (8. printing ed.). New York, NY: Knopf. ISBN 978-0-679-45443-4. 5) Feynman, Richard P.; Robert B. Leighton; Matthew Sands (1965). "Quantum Behavior". The Feynman Lectures on Physics, Vol. 3. Addison-Wesley. pp. 1.1–1.8. ISBN 978-0201021189.
 * Baggott book on history of QM also has a lot of material on duality. Baggott, J. E. (2013). The quantum story: a history in 40 moments (Impression: 3 ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-965597-7.
 * Johnjbarton (talk) 15:38, 12 August 2023 (UTC) Johnjbarton (talk) 15:38, 12 August 2023 (UTC)
 * I replied at the bottom of this page. TD (talk) 15:46, 12 August 2023 (UTC)



@Johnjbarton, he wrote : "A quantum particle is represented by a wave packet" and the similar one claiming to show particle interference are not correct physics for wave particle duality. No theory or experiment shows such a thing. Johnjbarton (talk) 21:53, 4 July 2023 (UTC)


 * Don't you know the double-slit experiment and its quantum interpretation ? TD (talk) 12:03, 12 August 2023 (UTC)
 * I find the coloring confusing here. According to image description, phase angle is represented by white-red-yellow colors, and probability amplitude by lightness. However, with full lightness, red is darker than white and yellow, which makes it look as if the probability amplitude of a gaussian packet would have fringes. In my opinion, it would be better to show only the probability amplitude. For single packet, the illustration would not look like anything, but with two packets, one would at least see fringes forming, something that is currently obscured by the phase information. Jähmefyysikko (talk) 13:22, 12 August 2023 (UTC)
 * I agree that the colors are confusing. A quick look would give the impression that the wavepacket has probability oscillations, which is not right of course. I agree with Jähmefyysikko. (I am OK with neglecting the dispersion etc for illustrative purposes.) Ldm1954 (talk) 13:50, 12 August 2023 (UTC)


 * This is an exact representation of a solution to the Schrödinger equation for a free particle. The phase is indicated by the color. The probability density is indicated by the intensity of the color. There is no simplification for illustrative purposes and the image gives full information about the solution.TD (talk) 13:59, 12 August 2023 (UTC)


 * I have no problem with the accuracy. However, like others, I think it will confuse novices. KISS please. Ldm1954 (talk) 14:09, 12 August 2023 (UTC)
 * The novices can read the description on the WikiCommons page. What KISS means in this context ? TD (talk) 14:12, 12 August 2023 (UTC)
 * KISS Ldm1954 (talk) 14:16, 12 August 2023 (UTC)
 * Is "Keep It Simple, Stupid" a polite way for discussion? TD (talk) 19:43, 12 August 2023 (UTC)
 * Sorry, you should not use an illustration that is not self contained/self explanatory and needs readers to go to another source to understand it. That is true in everything from undergraduate term papers to scientific papers or commercial documents. Ldm1954 (talk) 14:20, 12 August 2023 (UTC)
 * Are you sure of this rule you just invented? When I published the following animation with the commentary "Magnus effect in a 2D liquid of hard disks" I had the following reply : "02:56, 7 April 2023‎ Volker Siegel  talk contribs‎  24,016 bytes +11‎   Describing the invisible inner workings of the simulation ("hard disks") was confusing and not important."
 * [[File:Magnus effect.gif]] TD (talk) 19:39, 12 August 2023 (UTC)
 * Also, please look at WP:1AM. I, and others consider that it will confuse. Ldm1954 (talk) 14:13, 12 August 2023 (UTC) to
 * My films have been on this page for at least ten years, and no one objected. Where are the many ? TD (talk) 14:21, 12 August 2023 (UTC)
 * I can't explain why no one looked into the content on this page for a long time. But the content was not correct physics and it remains dubious and confusing. We fixing it now.
 * Your images are really very cool, and I would encourage you to contribute more images as we need them. However, the physics needs to be correct. No wave packet belongs on the wave-particle duality page. Johnjbarton (talk) 14:38, 12 August 2023 (UTC)
 * The physics is correct. Could you justify your arbitrary statement "No wave packet belongs on the wave-particle duality page" with a reference to a textbook ? TD (talk) 14:41, 12 August 2023 (UTC)
 * No of course I can't find any such textbook because a quality textbook would not mix wave packets with wave-particle duality discussion. The subjects are "apples and oranges". You might find wave packets and the uncertainty principle however.
 * I think we might have better luck in understanding each other if you could point to the sources that lead you to want wave packets on wave-particle duality. Johnjbarton (talk) 15:26, 12 August 2023 (UTC)
 * It is in all textbooks. They teach us how to calculate the solutions of the Schrödinger equation, waves, and how to interpret them, the probability density of the detection of a particle. Isn't it wave-particle duality ? Wave packets are realistic solutions to the Schrödinger equation, why don't you like them ? TD (talk) 15:33, 12 August 2023 (UTC)
 * I realize that the conjunction of wave solutions and probability density for particle detection might logically lead you to wave-packets. It did lead Erwin Schrodinger to wave packets back in 1926, in his second paper on wave mechanics. But that direct connection failed. Wave packets only work for a few specialized problems. They work for harmonic potentials. (the hypothetical potential Schrodinger used). They do solve the minimum uncertainty condition . That's about it. Nature does not need to obey our desires for simple answers. In the case of QM 90 years of research has shown that no, this is not at all a simple thing.
 * The narrow scope of wave-packets in QM is not the primary reason to leave them off of wave-particle duality. The primary reason is that duality is not about theoretical interpretations of Schrodinger's equation. Duality is a completely different thing. Duality is Bohr's expression of a deeply fundamental concept: what ever quantum stuff is, we only know about through experiments and the details of each experiment dictate whether we will see particle or wave behavior. It has nothing to do with models. Johnjbarton (talk) 15:54, 12 August 2023 (UTC)
 * This is your point of view. It is not a common one. TD (talk) 15:56, 12 August 2023 (UTC)
 * Hey TD: It's not helpful, in being asked for your source, to say "all textbooks". You have been offered, and yourself have provided, sources that do not attempt to describe the DSEs in a manner consistent with your animation. It is only polite then to try to provide an RS that does so when asked. If you're sure you saw it somewhere but can't remember, maybe check your old textbooks. If that'll take a while to do, no matter -- the discussion continues when you find it. If you end up realizing you may have synth'd it in your mind, there's no shame in that -- it happens to all of us. SamuelRiv (talk) 22:38, 12 August 2023 (UTC)
 * The question about sources was about the assertion "No wave packet belongs on the wave-particle duality page", not about the double slit experiment. My animation is only a simple theoretical example (a solution of the Schrödinger equation) which shows how a particle can interfere with itself. It is not in any textbook I know. TD (talk) 02:44, 13 August 2023 (UTC)

The wave-particle duality is a direct consequence of the quantum superposition principle.
‎ @Headbomb

Please, could you justify on this talk page why you deleted my contribution:

The wave-particle duality is a direct consequence of the quantum superposition principle. Since any particle can be in any localized state $$|x\rangle$$ it can also be in a superposition of these states $$\sum_x \psi(x) |x\rangle$$, or $$\int \psi(x) |x\rangle dx$$ where $$\psi(x)$$, is the wave function of the particle. TD (talk) 10:21, 13 August 2023 (UTC)


 * Wikipedia require sources which directly backs the material. You didn't provide one. &#32; Headbomb {t · c · p · b} 10:26, 13 August 2023 (UTC)
 * This is the ABC of quantum physics. Read any textbook. In Quantum superposition they write "The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:$$\sum_x \psi(x) |x\rangle$$" and they don't give any source, because there are too many. For example, "if $$\phi$$ and $$\psi$$ are vectors in the space (often called "state vectors") then so is $$\xi\phi + \eta\psi$$, for arbitrary complex numbers $$\xi,\eta$$." (Steven Weinberg, The Quantum theory of fields, p.49) TD (talk) 10:42, 13 August 2023 (UTC)
 * That just talks about superposition in general. And a reference is apparently given, with Weinberg.
 * If 'any textbook' mentions that quantum superposition = particle-wave duality, then find one and cite it. I suspect you won't find any, because particle-like phenomena, like the photo-electric effect have nothing to do with quantum superposition. &#32; Headbomb {t · c · p · b} 10:47, 13 August 2023 (UTC)
 * Precisely, I said quantum superposition implies wave-particle duality, not that it is equal to it. There are other consequences of quantum superposition (spin and entanglement). In almost all textbooks (Cohen-Tannoudji, Diu and Laloë, for example) they begin with the quantum superposition principle (the main principle of quantum physics) and then they introduce wave functions. TD (talk) 10:54, 13 August 2023 (UTC)
 * We never found any exception to the quantum superposition principle. All physical phenomena can be explained with it, even classical physics, through decoherence. The modern theory of photons, and of electrons in metals, is based on quantum physics, hence on its main principle. The photoelectric effect is not an exception.TD (talk) 11:12, 13 August 2023 (UTC)

I will restore my contribution (quoted above) to the Wave-particle page. If any wikipedian has an objection, please give it on this talk page.TD (talk) 11:29, 13 August 2023 (UTC)


 * Reverted, again. Do not restore this material without proper citation. &#32; Headbomb {t · c · p · b} 13:12, 13 August 2023 (UTC)
 * Since most libraries are closed in August, you will have to wait for a proper citation. I don't own many books. But I recall you that when it is the ABC of science, wikipedians usually don't ask for citations. Didn't you read the article on quantum superposition? Do you need a citation for 2+2=4? The quantum superposition principle is not very different. TD (talk) 13:43, 13 August 2023 (UTC)
 * I announced that I would restore my contribution. I invited wikipedians to give any objection. Why didn't you object before the restoration, only immediately after it? Why do you ask here for a proper citation, and not for quantum superposition? Is it fair? Are these good manners? Don't you think you discourage us to contribute to Wikipedia? Since you don't give any proper justification for your deletion, I will feel free to restore my contribution again. If you or any other wikipedian has any objection, please give it on this talk page.
 * On your user page you write "don't for a moment think you have nothing to contribute, because together we help the Internet not suck." Are your acts in agreement with your words? TD (talk) 15:28, 13 August 2023 (UTC)

@Headbomb, what will you do if I restore my modest contribution, which teaches the ABC of quantum mechanics? Will you delete it without proper justification? You didn't answer to my objections. Do you think that your more than 400000 edits on Wikipedia give you the right to behave like a tyrant? I gave good references. Isn't it enough to deserve to be published? Asking for a proper citation for my contribution is like asking for a proper citation for 2+2=4, can you find one? TD (talk) 18:14, 13 August 2023 (UTC)
 * Careful, TD. You are butting up against a core Wikipedia policy here: Verifiability. If someone challenges a statement and asks for a reliable source, you must provide one to reinsert that information. It doesn't matter how obvious the statement is to you. Once the statement is challenged, it cannot be reinserted without a source. (Editors are discouraged from abusing this; it is expected that challenges will be issued in good faith.)--Srleffler (talk) 18:53, 13 August 2023 (UTC)
 * Thank you for your good advice. I think I will find what I want in Dirac (Principles of quantum mechanics) or in Cohen-Tannoudji, Diu and Laloë (Quantum mechanics) but I have to wait for the libraries to be opened. TD (talk) 19:18, 13 August 2023 (UTC)
 * Why didn't you object before the restoration, only immediately after it? First, I had already objected. Second, you gave less than two hours of notice. I have other things to do with my life than refresh this talk page every 30 seconds to keep up to date with your personal demands.
 * what will you do if I restore my modest contribution? I will revert it again and get the page protected, and issue you a formal warning against edit warring (see WP:EW) and the repeated addition of unsourced material (see WP:V).
 * I gave good references You gave none. On this talk page, you gave references about quantum superposition. You gave no reference that link quantum superposition to wave-particle duality.
 * If being asked to provide sources is tyranny to you (and see WP:NPA here), you have lived an extremely sheltered life. &#32; Headbomb {t · c · p · b} 19:26, 13 August 2023 (UTC)
 * I gave a reference to a Wikipedia article, quantum superposition, that you didn't challenge :"a particle can have any position, so that there are different configurations which have any value of the position $x$. These are written: $$|x\rangle$$. The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients: $$\sum_x \psi(x) |x\rangle$$. This sum is defined only if the index $x$ is discrete. If the index is over $$\reals$$, then the sum is replaced by an integral. The quantity $$\psi(x)$$ is called the wavefunction of the particle." TD (talk) 19:41, 13 August 2023 (UTC)
 * Congratulations, you copy-pasted something from the superposition principle. You have made zero connection to wave-particle duality, nor provided a reference (i.e. a source, e.g. a journal article, a book, etc...) that made such connection. &#32; Headbomb {t · c · p · b} 19:47, 13 August 2023 (UTC)
 * From the superposition principle, we infer that there is a wave function for any particle. Isn't it wave-particle duality? TD (talk) 19:51, 13 August 2023 (UTC)
 * You wrote "I have other things to do with my life than refresh this talk page every 30 seconds". How did you do to delete my contribution after one minute?TD (talk) 20:13, 13 August 2023 (UTC)
 * Since you are clearly unfair, I will feel free to restore my fair contribution.TD (talk) 20:27, 13 August 2023 (UTC)
 * "I will feel free to restore my fair contribution" I strongly advise against that course of action. &#32; Headbomb {t · c · p · b} 03:10, 14 August 2023 (UTC)
 * Paul Adrien Maurice Dirac was the first to derive all of quantum mechanics from the quantum superposition principle, in his famous book : The principles of quantum mechanics (1930, first edition). The chapter 1 of this book is "The superposition principle". Its §1 is "Waves and particles". In this § he states than any particle, not only photons, can behave like a wave, and that the theory will "go into the relations between waves and particles" as far as is required (p.2). Another reference : "the wave functions $$\psi(x)$$ that we have been using to describe physical states in wave mechanics should be considered as the set of components of an abstract vector $$\psi$$ known as the state vector" (Steven Weinberg, Lectures on quantum mechanics, p.53). The quantum superposition principle means that physical states are represented by vectors in a complex vector space.TD (talk) 12:35, 21 August 2023 (UTC)
 * Both duality and superposition are aspects of quantum mechanics: they are bound to be related.
 * I believe many editors are approaching duality as "the essence of quantum mechanics" requiring a synopsis of their preferred interpretation of QM as the primary content. I don't think this is helpful for a couple of reasons. Duality is not unique in being labeled as the "essence" ("mystery" etc.). I've seen that label for superposition, measurement, collapse, entanglement, etc.  This labeling is a kind of "star" or yellow liner thing: it only means duality important without telling us anything at all about what it is to physics. Duality is not fully equivalent to QM and therefore the article does not call for a full investigation of QM.
 * Duality is a topic of its own, with its own history and role. Johnjbarton (talk) 14:54, 21 August 2023 (UTC)
 * What do you mean by wave-particle duality ? Isn't it that any particle (or system of particles) can behave like a wave?
 * Please, keep to the point of discussion. I said nothing about the essence, or the mystery, of quantum mechanics.TD (talk) 15:37, 21 August 2023 (UTC)
 * I will respond on another topic, as this one concerns superposition. Johnjbarton (talk) 15:55, 21 August 2023 (UTC)

The quantum superposition principle is the first principle of quantum physics. This is not an arbitrary or mysterious statement, because from this principle we can infer all the rest of quantum physics : Hence, the quantum superposition principle is rightly stated as the first principle of quantum mechanics. This is so in many textbooks (Dirac, Cohen-Tannoudji, Diu and Laloë, Weinberg, Le Bellac, and others), sometimes in a slightly different (and almost equivalent) form, that the space of states is a (complex) Hilbert space.TD (talk) 16:15, 21 August 2023 (UTC)
 * Wave-particle duality, because the wave function of a particle is the set of components of the state vector in a position basis.
 * The quantization of spin, because it can be obtained from the representation theory of rotations (Wigner).
 * Quantum entanglement, because for two unentangled states $$|a1,a2\rangle$$ and $$|b1,b2\rangle$$ the state $$|a1,a2\rangle + |b1,b2\rangle$$ is entangled, if a1 and b1 are two orthogonal states of the particle 1, and a2, b2 two orthogonal states of the particle 2.
 * Even the Schrödinger equation can be derived from the quantum superposition principle (with the Born rule) because the conservation of probabilities imposes that the evolution operator is unitary, and from here we deduce the existence of a Hermitian operator, the Hamiltonian.

Third opinion
I'm writing here because asked for a third opinion on the Physics Wikiproject. I think the main cause of the dispute is confusion about what wave-particle duality even is (which, unfortunately, this article doesn't help clarify). There are two widespread notions:

One, more common in the lay media, is that it's a duality about what quantum systems are: sometimes they behave as particles, sometimes as waves, so what are they? Bohmian mechanics even has a hybrid ontology that combines wave and particle. That's a very unpopular view, though, there's widespread consensus that a quantum system is represented as a wavefunction, there is no particle aspect to its ontology. In the ancient times this was not the case: the fact that interference disappears in a double-slit experiment when which-path information is added was seen as mysterious. Today we know it's a straightforward consequence of entanglement and decoherence. In this sense 's illustrations represent very well the mainstream point of view: a wavefunction ontology is enough to reproduce simple aspects such as having a well-localized quantum system that can interfere with itself. The fact that it's a simple, idealized description is not a problem, we do use such idealizations for pedagogical purposes all the time.

The other notion, more common in the technical literature, is that wave-particle duality is about measurement: how do we reconcile the fact quantum systems are represented by wavefunctions and evolve according to a wavefunction with the fact that they show up as discrete points when measured, like classical particles? Now there's no end of controversy about this, the solution is the main point of the various interpretations. In this sense 's illustrations are not useful, they don't address the issue at all.

Therefore, I think we should keep his illustrations given the proper context, with the caveat that they don't tell the whole story. They are well-made. In addition, I'd like to ask everyone to cool down. Everybody here is acting in good faith with the purpose of improving the article. A bit of civility and respect for each other's knowledge goes a long way towards having a productive discussion instead of a battle. Tercer (talk) 12:19, 13 August 2023 (UTC)


 * If I understand your proposal, you are suggesting that we have a section describing what popular media say about wave-particle duality, and this section would include the wave-packet images.


 * As I said earlier, the images are very well made and they appear to illustrate wave packets. That is a strong case for their appearance on wave packet. A claim that they illustrate any quantum mechanical concept would need to sourced. Johnjbarton (talk) 15:44, 13 August 2023 (UTC)
 * It seems you don't know that wave packets are solutions of the Schrödinger equation, and that this equation is the fundamental equation of quantum mechanics. TD (talk) 16:05, 13 August 2023 (UTC)
 * You clearly haven't understood anything I wrote. I regret trying to help, what I get is being insulted as a flat-earther. Next time you post anything on WT:PHYSICS I will just ignore it. Tercer (talk) 16:12, 13 August 2023 (UTC)
 * I don't understand. Did I insult you ? Did I write something untrue? I never posted anything on WT:PHYSICS.TD (talk) 16:19, 13 August 2023 (UTC)
 * I wasn't replying to you, but to . Tercer (talk) 16:42, 13 August 2023 (UTC)
 * Please forgive me for the misunderstanding. Thanks for your help. TD (talk) 16:45, 13 August 2023 (UTC)
 * @Tercer I'm sorry, I did not at all mean to imply you are a flat-earther! I was only trying to explain why I oppose combining an incorrect description of wave-particle duality sources only to media articles with a description based on physics references. Again I did not mean to insult, sorry. Johnjbarton (talk) 16:43, 13 August 2023 (UTC)
 * I removed the paragraph I posted that offended. Johnjbarton (talk) 16:55, 13 August 2023 (UTC)
 * Why do you still offend me? I don't rely on media articles. My references are good textbooks: Cohen-Tannoudji, Weinberg and others. TD (talk) 17:08, 13 August 2023 (UTC)
 * Apologies accepted. I don't think it is fair to declare this "popular" description of wave-particle duality as just incorrect. I think it's actually closer to what the ancient papers were arguing about than the modern description. I'm afraid the modern description is reducing "wave-particle duality" to only the aspects that we still consider mysterious.
 * In any case, if a misconception is widespread enough, our readers will come here wanting to read about it, and if we say nothing they'll left even more confused than then came. Tercer (talk) 15:27, 14 August 2023 (UTC)

Why does Johnjbarton deny that wave packets illustrate quantum mechanical concepts whereas they are solutions to the Schrödinger equation? What are the criteria to distinguish good, physical, solutions from bad ones (wave packets)?TD (talk) 16:54, 13 August 2023 (UTC)


 * The historical attempts to use wavepackets to explain quantum entities is covered in:
 * Kragh, H. (2009). Wave Packet. In: Greenberger, D., Hentschel, K., Weinert, F. (eds) Compendium of Quantum Physics. Springer, Berlin, Heidelberg. https://doi-org.wikipedialibrary.idm.oclc.org/10.1007/978-3-540-70626-7_232
 * It can be accessed through the Wikipedia library Springer link.
 * It outlines Schrodinger's attempt to use wave packets in atoms and the observations of Lorentz and Heisenberg that the results only apply to harmonic potentials. Then the article says "The papers by Schr¨odinger and Heisenberg were discussed by several physicists in 1927–1928, including George Darwin, Earle Kennard and Arthur Ruark, who all recognized that electrons cannot be represented just as wave packets." and later "wave packets would not do as representations of subatomic particles".
 * Wave packets have had and do have a role in illustrating quantum mechanical concepts, especially the uncertainty principle which is closely related to wave-particle duality. This role is not to represent "particles" but to explain things like uncertainty principle, minimum uncertainty, coherent states, states of harmonic potentials, and dispersion. These are all areas where we could use the nice images that don't fit here. Johnjbarton (talk) 22:04, 13 August 2023 (UTC)
 * May be there is a misunderstanding here. I didn't pretend that all quantum particles shall be represented by wave packets, only that a wave packet can represent one. TD (talk) 03:09, 14 August 2023 (UTC)
 * Let me ask a question here: The 'particle' aspect in the illustration is the localization of the packet. Can this way of illustrating the particle nature be extended to, say, to a bound electron in hydrogen atom or is there a difference in how wave-particle duality manifests for bound and free electrons? Jähmefyysikko (talk) 03:43, 14 August 2023 (UTC)
 * The same kind of representation works for a single particle in a potential, hence it works for the hydrogen atom. More precisely, in the hydrogen atom, there are two entangled particles, the proton and the electron, and I don't know how to visualize entanglement. But we usually reason as if the electron were in a potential, because the wave function can be separated in two parts, the center of mass (which is approximately the proton) and the other part (which is approximately the electron). And there is another difficulty: the hydrogen atom is in 3D. But it would be interesting to visualize a particle in a 2D potential. I think about it. Thanks for your question.TD (talk) 04:14, 14 August 2023 (UTC)

A largely cuncurring third opinion: The animations by nicely show wavepackets of a single particle evolving according to the Schrodinger equation, but cast no light on wave-particle duality. I recommend removing the animations. Even if they are kept, the surely don't belong in the Experimental confirmation section as they don't depict any experimental measurements. The animations are well-done and could be helpful in an article discussing wave packets as suggested by, or in related articles. All linear systems including linear wave equations exhibit superposition, so that is nothing unique to quantum mechanics. Linear wave equations exhibit wave packets, so that's not unique to QM either, and it does not imply particle behavior. If it did, then wave-particle duality would have been a hot topic in the 1800's when classical electrodynamics, sound, and water waves were understood and seen to show wave packets that can interfere. Wave equations also show the uncertainty priciple, which is also not wave-particle duality even though it is sometimes lumped in with it as aspects of complementarity. Uncertainty relations apply in classical waves and signals in the form of the Gabor limit, a general theorem about Fourier transforms.

Wave packets for a single particle provide no explanatory benefit wave-particle duality, except perhaps as a carefully worded preliminary to prepare readers that this is not what is meant by particle behavior. As Tercer notes, wave-particle duality is about measurement, entanglement, and decoherence. These are not yet illustrated here. (The illustration that is now in Visualization shows the uncertainty principle but not particle properties, as noted in the Talk section on Visualization above. It's already present in Uncertainty principle, and recently added it to Wavepacket, where I agree it is helpful.)

An illustration that depicts the essentials of this wave-particle duality would be great, but it's a challenge. A higher-dimensional wavefunction is required to describe more than one particle, so at least two dimensions are needed to show the joint wavefunction: one dimension for particle to be measured and one for a particle that serves in the first step of the measurement process, and some scattering step that entangles them, which and precludes further interference in particle 1 ( aside from practically difficult steps that essentially reverse that entanglment process). –MadeOfAtoms (talk) 00:32, 14 August 2023 (UTC)


 * Do it if you can. Represent the wave function of two entangled particles. I thought about it, but I never found a good idea to make it. TD (talk) 03:21, 14 August 2023 (UTC)
 * I suppose there is a misunderstanding here. I didn't pretend that wave packets are the last and final words about quantum particles. They are only particular solutions of the Schrödinger equation for a single particle. There are many other solutions. Of course I know that there are wave packets in classical physics, but they are real-valued functions. Quantum wave functions are complex-valued. TD (talk) 03:39, 14 August 2023 (UTC)


 * Agreed, that quantum wavefunctions are different in being complex-valued; other wave functions operate over a variety of scalar and vector fields and have important different properties, as you know, but I hoped to make the point that interference and superposition aren't enough to get to wave-particle duality. (It's a elegant trick that the Schrodinger equation uses the real and imaginary parts to make a first-order differential equation act like a second-order real one. I hope don't offend anyone reading this thread by calling it a trick.) –MadeOfAtoms (talk) 05:44, 14 August 2023 (UTC)

Resource for developing a two-particle graphic
Regarding the discussion above about graphics by intending to illustrate duality, I was not successful in finding a clear video on the web showing the evolution of the joint wavefunction ψ(r1,r2) that goes from a product state of wave packets to an entangled one by scattering off each other. But I just stumbled onto this fantastic interactive website that let's us tune the parameters for dynamics of a packet for two particles colliding in a 1D box, and for the low energy eigenstates of two interacting particles in a 1D box. Super-fun, and food for thought. Turn up the interaction parameters to make it a good mirror. Maybe we can work out a variant that makes a case for showing the particle behavior in a wave equation solution – something like a pair of wave packets for one particle, which gets entangled with another by colliding it, serving to "measure" which packet it was in. Then tracing over the second particle will decohere the first, leaving it in an incoherent superposition of it's two packet positions. Or just leave it coherent but show that particle 1 won't show interference fringes anymore. On the other hand, we can't get too creative about interpreting it as duality, and it needs to be a straightforward rendition of what an RS says or it will appropriately get reverted as OR. –MadeOfAtoms (talk) 06:25, 14 August 2023 (UTC)


 * If you are looking for hints, I suggest you look for WKB approximation. These kinds of simulations or similar animations would be great in the WKB page, with suitable context explaining when they are useful. They are not an interpretation of QM but rather a visualization of classical waves. The simulations don't show any collisions because these waves don't interact, they are just superimposing. They can't show anything about wave-particle duality because duality is not about "things", its about the duality of properties measured on quantum systems. Johnjbarton (talk) 14:09, 14 August 2023 (UTC)
 * Your point of view is interesting, but there isn't any consensus about this. Here is my point of view (I don't pretend to impose it, it's just another point of view). Wave-particle duality is a direct consequence of the quantum superposition principle, because from it, we deduce that there is a wave function for any particle or system of particles. The quantum superposition principle is about reality, "things". It does not depend on a point of view. Measurements are real, they are "things" too. Bohr was wrong to disregard the Everett analysis of measurement. Measurements are quantum phenomena. Quantum theory, with the superposition principle, explains everything, measurements included. If you want to know more about this point of view, you can read my book Quantum theory of observation TD (talk) 14:23, 14 August 2023 (UTC)

Experimental confirmation
@MadeOfAtoms, you wrote "I recommend removing the animations. Even if they are kept, they surely don't belong in the Experimental confirmation section as they don't depict any experimental measurements."

I published three animations:

The first one shows a simulation of an experimental result. The last one gives a theoretical explanation of the interference of a particle with itself, hence of the experimental result. The middle one is a preliminary step for the understanding of the last one. I agree that the choice of colors was not a good one. Next time, I will try to make a better choice.TD (talk) 12:38, 14 August 2023 (UTC)


 * A small heads-up: if I understand correctly Ldm1954's and Johnjbarton's plan for this page, the first one will soon be replaced by this gif. The simulated one has been a good substitute, but nothing beats real experimental data. Jähmefyysikko (talk) 14:25, 14 August 2023 (UTC)
 * This gif is magnificent. I made the change because "nothing beats real experimental data". Thanks. TD (talk) 14:33, 14 August 2023 (UTC)

Cross purposes
At the expense of perhaps creating more angst, I think it should be recognized that much of this discussion has been at cross purposes. There are two ways to approach duality: These two are quite different. The animation on the left of the Experimental verification kindoff has this right, although it is not obvious. If you look really carefully you will see that in the first part the electrons are coming out across the source. This is classic large and incoherent illumination, rather like a light bulb. Again, if you look carefully, later on the electrons are coming from more of a point, coherent illumination. The standard approach is to think in terms of probabilities, rather than wave packets. There is nothing wrong with wavepackets, but the are not the focus. The classic quote from Einstein is
 * Via wave packets and the uncertainty principle. If momentum (wavevector) is specified or has only a small range, then a single wavepacket is delocalized, close to a wave. When the range is large it approaches a discrete entity or particle.
 * Via probability and coherence which can either be described using optical mutual coherence or density matrices. These are approaches dealing not with one electron, but with many.
 * God does not play dice

To which Bohr responded
 * Einstein, stop telling God what to do.

Probability. N.B., there was a crossing comment by Tercer. I agree that we should cover both the wavepacket and probability approaches respectfully. Ldm1954 (talk) 15:39, 14 August 2023 (UTC)


 * I'm not sure I understand you. How do we calculate probabilities without wave functions? Both come together (the Born rule).
 * I think that the animation Wave-particle duality.ogv is very bad. From a classical point of view, a double-slit experiment with particles should give two separated spots (Feynman lectures), not this mess. I would like to delete this animation, because it is confusing. TD (talk) 15:52, 14 August 2023 (UTC)
 * I agree that the animation on the left is flakey, but I am not so concerned about it currently.
 * Here is an expansion for the two slit experiment. Suppose that a single-electron wavefunction can be written as $$\psi(r)$$. Consider the two slits at $$r_1$$and $$r_2$$. If we write
 * $$\psi(r_1)=|\psi(r_1)|\exp(i\phi_1)$$
 * $$\psi(r_2)=|\psi(r_2)|\exp(i\phi_2)$$
 * with phase terms $$\psi_1$$ and $$\psi_2$$, the mutual coherence is defined as
 * $$\Gamma(r_1,r_2)=\psi(r_1)\psi^*(r_2) = |\psi(r_1)\psi(r_2)|\exp(i(\phi_1-\phi_2))=|\psi(r_1)\psi(r_2)|\exp(i\Delta\phi)$$
 * The question now is whether the phase difference $$\Delta\phi$$ is a constant for every single electron (or photon, atom) or not when we consider millions being detected. If it is constant, ideally $$\Delta\phi=0$$, then the system is coherent and the oscillating fringes detailed above are observed. If, instead, the phase difference is statistically random then the system is incoherent and the fringes are not observed, just a smoothly varying intensity. The same type of approach can also be used within quantum mechanics, and is called the density matrix formalism.
 * The distance over which there is coherence is called the coherence length in classical electromagnetic theory, and in quantum mechanics the quantum coherence. The light from a laser is highly coherent, with a coherence length of centimeters or more; that from a normal light bulb is almost completely incoherent. When the coherence length is very small then there is no interference behavior, and effectively pure particle behavior; when it is large then wave-like behaviour dominates. The coherence length of the detector also matters, since in most cases it is very small, for instance the "dots" in the video . Whether the behavior is that of particles or a waves depends on experimental details, the probabilistic wave-particle duality via coherence/incoherence.
 * N.B., this is the optical approach, as that is what is used in electron microscopy. I will let someone else (if needed) construct this in density matrix language. Ldm1954 (talk) 16:09, 14 August 2023 (UTC)
 * I'm not sure I understand you. Do you suggest that we need a laser to obtain an interference pattern with light? TD (talk) 16:27, 14 August 2023 (UTC)
 * A laser works. Alternatively you use small apertures, which is what is done in most high school optics labs. The coherence length is determined by the source and/or aperture sizes. It is in Born and Wolf and perhaps elsewhere. You can find a short derivation at http://www.numis.northwestern.edu/460/Notes/Partial%20Coherence.docx ; if someone knows of a good source please let me know. It is a big tricky, and I have seen many people have difficulty with coherence. Ldm1954 (talk) 16:35, 14 August 2023 (UTC)
 * I'm not an expert in this field, hence I don't understand your point. The Born rule is part of the ABC of quantum mechanics: to calculate probabilities, calculate the squared modulus of the wave function. TD (talk) 16:40, 14 August 2023 (UTC)
 * It is not just one probability, it is for millions or trillions of electrons/photons/atoms/samosa/hamburgers. Perhaps think of it like a gas, for instance pressure is an average over a very large of single molecule events. Ldm1954 (talk) 16:44, 14 August 2023 (UTC)
 * I still don't understand your point. Now we know how to make experiments with single quantum events, Haroche's experiments for example, or quantum computers. What do you want to prove? TD (talk) 16:48, 14 August 2023 (UTC)
 * As the video shows, the double slit result is due to millions of single electrons. It is a statistical result. Ldm1954 (talk) 16:51, 14 August 2023 (UTC)
 * You exaggerate. There are only two millions pixels on my screen. Hence I can't see millions of single electrons. And you didn't answer to my question. What do you want to prove? Is it that quantum mechanical calculations can only give statistical results? TD (talk) 16:56, 14 August 2023 (UTC)
 * Observations are statistical. Ldm1954 (talk) 17:42, 14 August 2023 (UTC)
 * Not always. If we observe a pure state which is also an eigenstate of the measurement operator, the result is certain. TD (talk) 18:23, 14 August 2023 (UTC)
 * Not in the double slit experiment, which is the key point of this discussion. Ldm1954 (talk) 18:24, 14 August 2023 (UTC)
 * An explanation about this rather obscure discussion: a density matrix if for a mixed state. This means that we don't know the initial and/or limit conditions, we only know their statistics. A wave function is for a pure state. This means that we know the initial and limit conditions. Quantum mechanics is basically about wave functions, pure states. But when we know how to calculate with pure states, we also know how to calculate with mixed states. Until a few decades ago, we didn't know how to prepare experimentally pure states, but now we know (Haroche and others). Since classical physics is deterministic, there can't be probabilities with pure states (this is a lesson of Poincaré). But quantum mechanics is not deterministic, and very paradoxical. There can be probabilities even with pure states. A wave function (a pure state) don't always give us the means to calculate the result of an experiment, only its probability.TD (talk) 17:41, 14 August 2023 (UTC)
 * This discussion was unfortunately sidetracked. I find that Ldm1954 has a point here, as did Tercer above: The concept does have multiple meanings, as illustrated in the review by Bhatta (2020). A later paper, Bhatta (2021), also reviews some of the literature about the interpretations of the concept. I think the article manages does fine by focusing on the experiments. Jähmefyysikko (talk) 07:56, 4 September 2023 (UTC)

Is Einstein a good reference on wave-particle duality?
Like many others, I want to improve this page, which does not meet the standards of a good encyclopedia. To begin with Einstein seems to me to be a mistake, because Einstein never accepted modern quantum mechanics. From Einstein's point of view, there are two theories, one for particles and one for waves, but from a modern point of view, there is only one theory, which describes both particles and waves.TD (talk) 18:44, 14 August 2023 (UTC)


 * I like the quote because it expresses the facts and the frustration of a realist. However, I do agree that some reader might come away thinking that somehow two models would work. (Einstein accepted QM but believed it was somehow incomplete. He was committed to physics and never denied QM results. What he considered a minor work (EPR) created yet another entire new field of quantum computing). Johnjbarton (talk) 19:26, 14 August 2023 (UTC)
 * Your correction is right. I should have written: Einstein never accepted that QM was a correct description of reality. EPR is a beautiful error in the history of science. Einstein thought he proved QM was incomplete, because he thought quantum entanglement could not exist, and he discovered with this error an extraordinary phenomenon, quantum entanglement. TD (talk) 19:38, 14 August 2023 (UTC)
 * That's incorrect. Einstein believed that quantum mechanics was incomplete because he believed that quantum entanglement existed. His argument was that entanglement caused nonlocality, and that it could only be fixed by hidden variables. Tercer (talk) 19:54, 14 August 2023 (UTC)
 * It seems to me that we are both right and wrong: entanglement with hidden variables is not really quantum entanglement, only a phenomenon in search of a classical explanation. TD (talk) 20:01, 14 August 2023 (UTC)

Another suggestion for a new version
The article could begin with a summary of Feynman's lectures on the double-slit experiment. The electrons are detected one by one, like particles, but the interference pattern shows that a single electron behaves like a wave, that it passes through both slits at the same time. Then we could show the experimental confirmation. Then we give the standard theoretical explanation, that wave-particle duality is a direct consequence of the quantum superposition principle. After that, we give a summary of other experimental confirmations and of the current controversies on this very controversial subject. TD (talk) 19:49, 14 August 2023 (UTC)


 * I prefer the other way around. Show the experiment first. Historically, duality summarized observations. Observations continue to uphold duality. Physics is about explaining experimental results and proposing new experiments based on the explanation. It's not about confirming theories, that's the dull part.
 * Saying duality is a direct consequence of (fill in the blank) is not I think a good way to proceed. This topic is one of the central points of discussion in QM. If it is a consequence of any thing it is a consequence of 20 years of confusing experiments. It's an empirical "fact" that any theory must conform it.
 * So the wave equation agrees with duality, but this isn't a consequence but a necessity of their common origin in experimental results. Johnjbarton (talk) 20:55, 14 August 2023 (UTC)
 * I like that you give your point of view, because it is an interesting discussion, but I can't agree with you. You seem to say that in physics, experiments are always first, and theories always second. It is not always true. Radio waves, black holes, quantum entanglement, gravitational waves and many other phenomena were first discovered theoretically. The experimental confirmation came after. Please don't be offending. Theories are not always dull. TD (talk) 21:08, 14 August 2023 (UTC)
 * I did not say experiments are always first. I did not say any thing about theories being dull. Johnjbarton (talk) 22:02, 14 August 2023 (UTC)
 * Here is a citation of Dirac: "I think that there is a moral to this story, namely that it is more important to have beauty in one's equations that to have them fit experiment. If Schrödinger had been more confident of his work, he could have published it some months earlier, and he could have published a more accurate equation. It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory." Scientific American, May 1963.TD (talk) 21:16, 14 August 2023 (UTC)

A precision about my point of view: I'm a realist about quantum wave functions: they are correct descriptions of reality. I think that Everett is right (the so-called many-worlds interpretation of quantum mechanics) and that it is not an interpretation, but simply the pure truth. I know this point is very controversial. I don't want to impose it, but I think it deserves to be defended among others in an encyclopedia.TD (talk) 20:21, 14 August 2023 (UTC)

How to visualize wave functions
I want to make better animations of wave functions. It could improve this article on wave-particle duality. For now I can't, because my computer is hacked by ... who want to prevent me from doing my scientific work. But you too can make such animations. Here is the idea:

The phase of a wave function can be represented by a continuum of different colors. The probability density can be represented by the intensity of the color. In this way we can give full information about any 2D wave function. There is a problem of contrast. If the probability density of one part of the image is too little than the one on another part, it won't be seen. But there is a simple solution : the intensity of the color shall depend logarithmically, not linearly, on the probability density. This is a mistake I made in my previous animations (linear dependence, not logarithmic). I just thought about it. I will make use of this thought in my future animations. I hope also that others will make use of this idea. With it we can visualize many quantum wave functions. TD (talk) 09:16, 16 August 2023 (UTC)


 * I am also away from a desktop computer for some time, so cannot do anything else than to offer a suggestion: If one wants to clearly separate the phase information from the amplitude, the article on Domain coloring and the references it contains has some good ideas. Jähmefyysikko (talk) 09:27, 16 August 2023 (UTC)
 * If you want to know what happens to me read: RAM Motherboards and Call for testimonials. If I'm not mistaken, the french police prevents me from loading any developer software, hence I can't do my work. TD (talk) 03:27, 20 August 2023 (UTC)
 * I am opposed to any visualization on this page and I will back up that stand with sources. Duality is not visualizable: that is exactly what the most famous quantum physicists have discovered through many decades of effort.
 * Wave packets are visualizable and fun. There are plenty of places to put better wave-packet visualizations.
 * The biggest issue I have seen with wave-packet visualizations concerns physically meaningful scale. Almost all visualizations are minimum-uncertainty non-dispersive packets. These are very rare in nature which makes the visualizations inappropriate for nature. The visualizations that juxtapose packets and double slits are especially perverse because they imply localization ("particles") but are drawn as macroscopic.
 * Packets in scattering are related to wave coherence, a complex advanced topic. Visualization of various aspects of coherence (lifetime to longitudinal coherence; source geometry to lateral coherence; role of uncertainty principle in quantum slits for examples) would really be neat. Johnjbarton (talk) 15:15, 21 August 2023 (UTC)
 * I don't understand. Who are the most famous quantum physicists who discovered through many decades of effort that duality can't be visualized? The experimental result of a double-slit experiment with electrons is a visualization of wave particle duality. Wave packets are not only fun. They are exact and very interesting solutions of the Schrödinger equation. Any solution to the Schrödinger equation illustrates wave-particle duality. TD (talk) 17:53, 21 August 2023 (UTC)

What is wave-particle duality?
I've been reading a lot about duality and discussing it. A lot of things I thought I understood a few months ago were not correct. And of course as a popular topic it has multiple meanings.

Here is Albert Messiah's version:

"Universal Character of the Wave-Corpuscle Duality

We conclude from all this that microscopic objects have a very general property: they appear under two apparently irreconcilable aspects, the wave aspect on the one hand, exhibiting the superposition property characteristic of waves, and the corpuscular aspect on the other hand, namely localized grains of energy and momentum, There exists a universal relationship between these two aspects, given by equations (II.5)."

His reference is to this equation:

$$E=h\omega\ \ \ \boldsymbol{p}=h\boldsymbol{k}\ \ $$(II.5)

is described as "...relations between dynamical variables of the particle and characteristic quantities of the associated wave."

This text (page 59 of the English edition) is positioned after de Broglie and before Schrodinger.

My conclusion, consistent with other things I've read, is that duality is both a summary of experimental findings and a name for the combination of de Brogile's matter wave and Einstein's photon particle hypothesis, the combination expressed in the above equations. Duality is not solely the hypothesis since without the observations the hypothesis has no meaning or purpose.

So duality is a summary of contradictory experimental findings and the hypothesis linking wave and particle properties of quantum systems. It expresses a failure of classical wave or particle models.

I encourage you to imagine I ended the last sentence with "Full Stop", as they say. Taking this limited view, as Messiah does, is both correct and leaves plenty of scope for a great article. Duality lives at the boundary between the Old Quantum theory and modern QM. Things that come after are quantum mechanics and we have lots of other articles to work on.

Hundreds of papers and maybe that many books discuss duality. Almost all of these sources are based on interpretations of Schrodinger's or similar QM equations. The most reliable sources and almost all of the modern work that mentions duality focus on Bohr's complementarity as the manifestation of duality in QM. We should definitely include some of that material, but the article should treat this material as summary. Otherwise we very quickly end up with an article about something else altogether. Johnjbarton (talk) 16:37, 21 August 2023 (UTC)


 * Our points of view are very different. Yours (if I understand you correctly): wave-particle duality is a chapter in the history of physics (the old quantum theory). It begins with Planck and Einstein and ends with de Broglie. Modern quantum mechanics (from Heisenberg, Schrödinger and Dirac to the present) is about another subject. Mine: wave-particle duality is simply that any particle (or system of particles) can behave like a wave. The old quantum theory postulated it, but didn't explain it, and couldn't give any consistent theory about it. Modern quantum theory explains it (with the quantum superposition principle) and gives a consistent theory about it. TD (talk) 16:52, 21 August 2023 (UTC)
 * Duality is absolutely not part of the Old Quantum theory. de Broglie's hypothesis ended the Old Quantum theory. Historically duality as a concept arrives at the boundary. That's why I said boundary.
 * While duality appeared in history it does not remain there. Duality remains valid to this day as a summary of experimental findings. Duality has a limited role as a theory, simply that all future experiments will also be contradictory. Duality is not a mechanics, that came with Schrodinger.
 * A claim like "any particle (or system of particles) can behave like a wave" seems similar to de Broglie's hypothesis, but it presupposes a quantum system contains "particles" which then can "behave". No such evidence exists. I know this may sound pedantic but duality is exactly about such careful distinctions.
 * The explanation in modern quantum theory, such as it is, traces back to Bohr's complementarity. Zeilinger, a recent Nobel laureate, has worked on modern experiments related to duality. In between there is a lot of work we could discuss in the article. That was what I originally intended to propose.
 * The problem is that people don't want to hear about complementarity. Everyone wants realism all the way down. Since the reliable sources (reluctantly) deny realism, we will struggle.
 * My pragmatic solution is to summarize complementarity and put the modern material in complementarity (physics). That way we can have a correct article about duality even if it feels incomplete. Everyone will feel it is incomplete in different ways and take comfort that Einstein also thought as much. Johnjbarton (talk) 18:01, 21 August 2023 (UTC)
 * I don't understand: do you deny that quantum physics is about particles and systems of particles?
 * You seem to recognize that your point of view is a minority one. Such a point of view has a place in Wikipedia, among others, but it shouldn't be the law for a whole article.
 * Do you suggest that any source who doesn't deny realism is not reliable?
 * I won't struggle. This discussion deprives me of the will to work on this article.TD (talk) 18:22, 21 August 2023 (UTC)
 * Sorry, you have two questions about my views (I think), but I think we should focus on the sources.
 * As sources go, Nobel physicists have to be counted as exceptionally reliable. Their work been vetted so thoroughly, both in peer reviews and in prize reviews. My summary of their point of view is that wave or particle properties are a consequence of measurement.
 * Zeilinger is typical:
 * "The most sensible position, according to quantum mechanics, is to assume that no such waves preexist before any measurement. Zeilinger, Anton (1999-03-01). 'Experiment and the foundations of quantum physics'. Reviews of Modern Physics. 71 (2): S288–S297."
 * There is an entire subfield of experiments around Wheeler's delayed-choice experiment related to these questions. The simplest interpretation is no interpretation: properties are a consequence of measurement.
 * You may recall that my understanding changed after reading actual sources. Now I understand why Einstein and Bohr debated so long, why Feynman speaks of "mystery". QM is deeply weird.
 * Should we go into all of this in the article on duality? I think it will just be a fight. Johnjbarton (talk) 19:03, 21 August 2023 (UTC)
 * This is an old philosophical debate : is there a reality independent of our observations? Here is my point of view: observations are as real as what is observed. We can observe observers. This is a real possibility: an atom can be used to observe the electromagnetic field in a cavity (Haroche and others), hence it is an observer, and it can then be observed. Therefore it seems rather incoherent to declare that only observations are real and that what is observed is not, before it is observed.
 * Quantum physics is about real particles, atoms, molecules... Particles are really particles, because they can be detected one by one.
 * About this article, I think we should present the standard point view: particles behave like waves, and then give a summary of the current controversies, the opposition realism/antirealism included. There is no need for a fight. Different points of view can be defended in the same article.TD (talk) 19:37, 21 August 2023 (UTC)
 * Thanks, that is encouraging. I actually think most will agree to present the standard point of view. Personally I am only interested in well source views, with the standard view identified clearly.
 * I'm not a philosopher but I think there are two kinds of "realism" in the QM discussions: 1) are there tiny real thingys? 2) is "reality independent of our observations"? The standard point of view "I don't know" on both counts. The standard view is that there is no reason to predict measurements you don't perform. For the duality article only #1 matters, that is what I was referring to earlier, and that is the one related to complementarity. To be clear I only mentioned realism in the context of how we discuss, not in terms of the article.
 * Particle can absolutely be detected. Particles do not behave like waves, that's, "rather incoherent". Particles behave like particles; Waves behave like waves. Experiments on quantum systems can give results consistent with either one, depending on the experiment. The standard view does not involve tiny things switching between particles and waves. Johnjbarton (talk) 20:40, 21 August 2023 (UTC)
 * According to classical physics, a particle can only be at one place at one time. According to modern quantum physics, a particle can be at many places at the same time, this is why it can behave like a wave. That a particle is at many places at the same time doesn't mean that it isn't a particle. There is no incoherence in modern quantum theory.
 * Do you suggest that particles, atoms and molecules are not real?
 * The standard view is the one that is taught in almost all textbooks and by almost all teachers. There is a consensus about it. Not all textbooks state that the quantum superposition principle is the first one of quantum physics, but many do. Schiff for example doesn't teach it in this way, but he is still in agreement with Dirac on everything, or nearly so.TD (talk) 21:09, 21 August 2023 (UTC)
 * Let's check that we do agree on some things. The standard model views solutions to the wave equation as probability amplitudes to be squared, giving probability of particle observations via the Born rule. I suppose we will agree on this point. Whether one will observe wave interference or particle trajectories depends on the experiment. That is what duality summarizes and I guess is not at question.
 * We don't agree on a special connection between superposition and duality. Quantum mechanics is governed by a wave equation: superposition of eigenfunctions is built in. But earlier you spoke of particles being detected; that is after measurement. Such detected particles not in many places, they are where we detected them.  A superposition of particles stands directly contradictory to duality: superposition is a wave property.
 * Based on sources I have tracked down, the part of QM that duality most closely relates to is complementarity. Here is Messiah again:
 * "18. Wave-Corpuscle Duality and Complementarity
 * If one adopts the principle of complementarity, the wave-corpuscle duality ceases to be paradoxical: the wave aspect and the corpuscular aspect are two complementary aspects which are exhibited only in mutually exclusive experimental arrangements. Any attempt to reveal one of the two aspects requires a modification of the experimental set-up which destroys any possibility of observing the other aspect"
 * And here are the opening paragraphs of Scully, Marian O.; Englert, Berthold-Georg; Walther, Herbert (1991-05-01). "Quantum optical tests of complementarity". Nature. 351 (6322): 111–116. https://itp.uni-frankfurt.de/~giacosa/neqm/radierer/scully.pdf
 * COMPLEMENTARITY distinguishes the world of quantum phenomena from the realm of classical physics. The lion's share of the credit for teaching us to accept complementarity as a fact and for insisting that we have to learn to live with it belongs to Niels Bohr. In 1927, when he was reviewing the subject at Como¹ in a speech delivered in honour of Count Alessandro Volta (1745-1827), quantum theory as we know it today was still new, and all examples used to illustrate complementarity referred to the position (particle-like) and momentum (wave-like) attributes of a quantum mechanical object, be it a photon or a massive particle. This is the historical reason why complementarity is often superficially identified with the 'wave-particle duality of matter'.
 * Richard Feynman, discussing the two-slit experiment in his admirable introduction to quantum mechanics², notes that this wave-particle dual behaviour contains the basic mystery of quantum mechanics. In fact, he goes so far as to say: "In reality it contains the only mystery."
 * In Scully, Marian O.; Englert, Berthold-Georg; Walther, Herbert (1991-05-01). "Quantum optical tests of complementarity". Nature. 351 (6322): 111–116. https://itp.uni-frankfurt.de/~giacosa/neqm/radierer/scully.pdf
 * If you have sources that claim duality is something else or connected to QM in some other way it would be helpful to discuss them. Johnjbarton (talk) 23:39, 21 August 2023 (UTC)
 * If there are three trajectories for an atom at the same time (like in section 5-3, Stern-Gerlach filters in series, of The Feynman Lectures of physics) is it a wave or a particle?
 * Superposition in not only a wave property. It works for spin and entanglement.
 * Quantum superposition is the theoretical foundation of quantum computers. And we don't need waves to make them.TD (talk) 09:11, 22 August 2023 (UTC)
 * The lines in the Feynman lecture image do not represent waves or particles. These lines are "high probability rays", lines orthogonal to probability amplitudes selected to be representative. Any attempt to interpret these lines as particle trajectories would have to explain how a pair of slits, crossing the beam, would cause an interference pattern in the intensity of measured particles. The explanation in terms of complementarity is simple: adding the slits changes the experiment and thus the expected outcome from particle behavior to wave behavior.
 * Quantum superposition is a consequence of the wave equation. Particles don't obey wave equations. Ergo no superposition of particles. The observation of interference provides evidence of superposition.
 * Since you seem quite keen on particles you may be interested in Ballentine, Leslie E. "The statistical interpretation of quantum mechanics." Reviews of modern physics 42.4 (1970): 358. http://www.psiquadrat.de/downloads/ballentine70.pdf Ballentine starts by observing that QM is fundamentally a statistical (ensemble) theory, not a theory that predicts individual events, but he insists that particles cannot be doubted. His work can be used as a source here and I believe it can be used to discuss an alternative to some aspects of complementarity. Johnjbarton (talk) 16:01, 22 August 2023 (UTC)
 * Quantum theory is not only a statistical theory. We now know how to make experiments with pure states. We can predict individual events.
 * Is the quantization of spin a consequence of the wave equation? Is quantum entanglement a wave property?TD (talk) 16:50, 22 August 2023 (UTC)
 * As the Scully reference points out, pure states have a role in wave-particle duality. But discussing the meaning of experiments on pure states seems like it would lead us off topic. Theoretically, if we prepare a momentum eigenfunction we can predict its momentum. States are always statistical, some have values with probability one. If we prepare a million particles in an eigenfunction of momentum most will not make it into the measurement apparatus and we don't know when. We cannot measure the initial position and momentum of an individual particle let alone predict its motion. So "predict individual event" is predicated on an ensemble. I don't see how this level of detail helps. Johnjbarton (talk) 17:49, 22 August 2023 (UTC)

Animations of wave packets
The wave function is $$\psi(x,y,t)=\psi(x,t) \psi(y,t)$$, $$k_{0x}= k_0$$, $$k_{0y}=0$$, $$ \psi (x,t) = (\frac{2a ^2}{\pi}) ^{1/4} \frac{e ^{i \phi}}{(a ^4 + \frac{4 \hbar ^2 t ^2}{m ^2}) ^{1/4}} e ^{i k_0 x} exp [ - \frac{(x - \frac{\hbar k_0}{m} t ) ^2 }{a ^2 + \frac{2 i \hbar t}{m} } ]$$ where $$ \phi = - \theta - \frac{\hbar k_0 ^2}{2 m} t$$ and $$tg (2 \theta) = \frac{2 \hbar t}{m a ^2}$$ (source: Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, complement GI, §3-a)

The Python files which generated my animations are here.TD (talk) 17:08, 25 August 2023 (UTC)
 * Thanks! I see you put this info in the File description, that's great.
 * Do you have a reference to support the caption: "This wave packet represents a quantum particle."?
 * This, in view of other references, cannot be correct. The entire point of 'wave-particle duality' is that we have no representations of quantum entities. Physics would be so much simpler if we did! But then we would not need wave-particle duality. We would just call it the theory of wave-packets and be done.  No mystery, no debates between Nobel laureates, no endless stream of articles about double-slit experiments.
 * You could say "one possible solution to Schrodinger's equation". But then one has it ask: what does this image say for the topic of the article? I say it is misleading. It misleads readers into believing wave-particle duality is about wave packets. But I don't know of any experiment or theory which claims wave packets represent wave-particle duality, and several which claim they do not.
 * To reiterate my other comments, I like the image and I appreciate your documentation for it. I just do not agree that it belongs on this particular page.
 * Johnjbarton (talk) 18:28, 25 August 2023 (UTC)
 * Cohen-Tannoudji, Diu and Laloë write explicitly at the beginning of this complement GI: "Consider the case of a free particle" (my translation from french).
 * Do you deny that the proton and the electron are particles? The wave function of the electron in the hydrogen atom is also a wave packet.TD (talk) 18:51, 25 August 2023 (UTC)
 * Based on our previous discussions I assume you agree with my observations here.
 * We've known the properties and equations of motion for particles for over 400 years. The equations do not predict particle interference and none has been detected. The interference observed with electrons conclusively demonstrate they are not particles.
 * The quantum equations of motion for electrons are wave equations and thus predict interference in agreement with experiment. Again we must conclude electrons are not particles.
 * There are scenarios where particle models of electrons are effective as well as scenarios where these models fail. That is why we have wave-particle duality.
 * Since I think you accept the premises here I cannot understand how you can avoid the conclusion. What am I missing?
 * Wave packets in the same form as you animation were shown to fail to represent electrons in atoms already in 1927, see Darwin, Charles Galton. "Free motion in the wave mechanics." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117.776 (1927): 258-293. https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1927.0179?download=true
 * This is related to my comment earlier about scale. If you start with a Gaussian packet around an atom in diameter its width increases linearly with time, doubling in $$10^{-16}s$$, see https://farside.ph.utexas.edu/teaching/315/Waves/node91.html Johnjbarton (talk) 23:35, 25 August 2023 (UTC)
 * Clearly, I can't agree with you. We didn't know the equation of motion for particles, before Heisenberg, Dirac and Schrödinger found them. That you deny that electrons are particles is very amazing. Almost all physicists agree that electrons and other elementary particles are particles.TD (talk) 00:34, 26 August 2023 (UTC)
 * The quantum pioneers built upon the work of classical era physics like Lagrangian mechanics and Hamiltonian mechanics. Schrodinger in particular developed his equation from Hamilton's optico-mechanical analogy which relates geometrical optics (trajectories as rays) to wave equations. You can read his first paper online,"Quantisation as an eigenvalue problem", http://ofp.cosmo-ufes.org/uploads/1/3/7/0/13701821/quantisation_as_an_eigenvalue_problem.pdf
 * In a poll of physicists, only 3% said unequivocal yes to: "Do you believe that physical objects have their properties well defined prior to and independent of measurement?" A majority allows that some cases would be "yes", but almost half say "No". See "A Snapshot of Foundational Attitudes Toward Quantum Mechanics" https://doi.org/10.48550/arXiv.1301.1069
 * Sadly it is true the "particle physics" uses the word "particle" for collections of measured properties, including bound states and resonances observed in scattering experiments. The early work in scattering was all down with waves before the advent of quantum field theory in which "particles" are field excitations.
 * I read through a bit of Cohen-Tannoudji, Diu and Laloë. It has many excellent features, but like many textbooks it tries to avoid discussion of interpretation (see the end of Chapter 1 where we are referred to a reading list). The end of Chapter 1 has this sentence: "The dynamic state of a particle, at a given time, is characterized by a wave function." As they explain, this state is not a trajectory. "The structure and behavior of matter on an atomic level are incomprehensible in the framework of classical mechanics." I think their language is loose on aspects they don't care much about, but I doubt they say "electrons are particles".
 * If you believe electrons are particles, then why do you even care about wave-particle duality? Johnjbarton (talk) 02:42, 26 August 2023 (UTC)
 * Because particles move like waves. TD (talk) 02:59, 26 August 2023 (UTC)
 * Frank Laloë wrote a very good book about the interpretation of quantum mechanics : Do We Really Understand Quantum Mechanics? TD (talk) 12:49, 26 August 2023 (UTC)
 * At the expense of inflaming this worse, I think you are both partially right, and wrong. It is all probability. Millions of particles, when detected, behave statistically as waves. Measured individually they have point -like properties.
 * Does that mean that electrons, for instance, are just particles or waves -- no. The probability, called after scaling by the electron charge the charge density, is measurable, e.g. x-ray scattering factors, polarizability and many more. As shown in the Kohn–Sham equations if we know the change density then all ground state properties are defined.
 * God does play with dice Mr Einstein. Ldm1954 (talk) 13:14, 26 August 2023 (UTC)
 * One interpretation of quantum mechanics, initiated by Born and strongly advocated by Einstein and Ballentine holds that the state in Schrodinger's equation represents an ensemble. Born's summary was "the motion of the particle follows the laws of probability, but the probability itself propagates in accord with causal laws" (the other two make no such claim). This viewpoint was the core objection Einstein held against QM. Ballentine has developed the view consistently and has strong arguments but of course no evidence.
 * This interpretation is quite interesting in that it avoids wavefunction collapse and provides alternative views for measurement and Schrodinger's cat. One difficult aspect is that the meaning of probability underlying QM, challenged by Karl Popper and developed by Ballentine, is key to understanding ensemble interpretations.
 * However this interpretation is only relevant to the wave-particle duality article to the extent it has a direct, reference-able connection to it. To place the ensemble interpretation, or really any interpretation other than Copenhagen, in the main position of the wave-particle duality article would be historically incorrect as well as scientifically unsupportable since no particular interpretation has been shown correct. Johnjbarton (talk) 15:40, 26 August 2023 (UTC)
 * I disagree. If your model is inconsistent with DFT, x-ray scattering factors etc then it is wrong. The electron double slit data is ensemble. The loss of fringes in the double slit experiment due to incoherence is ensemble. Sorry, I really think you are making everything way more complicated than it is. There is masses of evidence, I think some of these philosophical concepts
 * are obfuscating. If you only focus on a single electron, photon, brownie then it is easy to go wrong - but in reality they are of little relevance. As a simple example, include correlation in your thinking, Slater determinants etc.
 * N.B., if there is any issue about the relevance of coherent and incoherent wavepackets, let me mention that many millions are used daily in STEM instruments. There is no ambiguity on this. Ldm1954 (talk) 16:09, 26 August 2023 (UTC)
 * "The loss of fringes in the double slit experiment due to incoherence is ensemble." I agree and I confident we can find references to support a version of this claim. Loss of fringe visibility is however not an important topic for wave-particle duality.
 * "The electron double slit data is ensemble." You will need to have a source for this. I can provides reliable sources which contradict it.
 * Wave-particle duality is a concept of quantum foundation theory. It deals with the presence or absence of interference.
 * Coherence is a complex, important practical effect that reduces the degree of interference. Coherence is necessary but not sufficient to produce interference. In the real workday world of physics coherence is much more important than wave-particle duality. They are not equal. Johnjbarton (talk) 16:30, 26 August 2023 (UTC)
 * The experimental video is ensemble! Interference is ensemble. Coherence always leads to ensemble interference -- put in a lens. Ldm1954 (talk) 16:42, 26 August 2023 (UTC)
 * Regarding the book by Frank Laloë, a shorter version with the same title and author and similar outline is available online: https://arxiv.org/abs/quant-ph/0209123. It is mostly about entanglement and does not discuss duality directly. Many different interpretations are discussed.
 * At the top of page 8 Laloë reprises the Schrodinger wave-packet-do-not-work-after-all story I recited earlier.
 * Laloë clearly considers the issue of interpretation unsettled and he does not say electrons are particles. Johnjbarton (talk) 16:08, 26 August 2023 (UTC)
 * I would like to understand the scale of the animation.
 * Based on the Python code, atomic units are used with $$\hbar=1$$. Time runs from 0 to 300*.04 units, so the range is $$3\times 10^{-16}$$s. During this time the center of mass moves 12 units, so $$6\times 10^{-10}$$m, 6 times the diameter of hydrogen. The velocity comes out to $$2\times 10^{6}$$m/s, a small part of the speed of light. The diameter of the packet then is around the diameter of hydrogen.
 * I could be completely off, but this scale means the wave packet cannot fill two slits in any real double slit experiment, and thus cannot explain wave-particle duality. Of course if I am mistaken and the scale turns out to be comparable to the distance between the slits, then I will complain that the packet does not represent a particle but rather some partially coherent state (even that is in dispute).
 * The animation can't both explain wave particle duality and be a representation of a "particle". Thus either the caption is wrong or it's in the wrong article. Johnjbarton (talk) 18:09, 26 August 2023 (UTC)
 * The animation "Interference of a particle with itself" doesn't pretend to represent what happens in a double-slit experiment with electrons. It's only theory. With a very simple and theoretical example, it explains how a particle can interfere with itself. About the real experiment, I don't know. TD (talk) 18:39, 26 August 2023 (UTC)
 * To have any relevance on wave-particle duality (which is based on experimental observations) and especially in a section titled Experimental confirmation, the figure should be related to some experiment which shows wave-particle duality, like Feynman's two-slit experiment. I haven't seen any references which would indicate that wave packets are relevant for such an experiment. Therefore, I would strongly prefer to remove the wave packet animations here and allow the reader to focus on the experimental animation. Jähmefyysikko (talk) 22:13, 26 August 2023 (UTC)
 * Done. TD (talk) 22:17, 26 August 2023 (UTC)
 * Thank you. I found our discussions of these question very helpful. Johnjbarton (talk) 22:24, 26 August 2023 (UTC)
 * I didn't.
 * The interference of a particle with itself is the heart of the matter, about wave-particle duality.TD (talk) 22:33, 26 August 2023 (UTC)
 * These animations reappeared in another section. I deleted them. They were re-inserted with a comment:
 * "Please discuss on the talk page before deleting these animations, which are now in the right section".
 * We have discussed these animation extensively. There is no "right section" in this article for these animations. They are improper and misleading in the context of wave-particle duality. They imply physical ideas which have limited scope while wave-particle duality is broadly applicable and fundamental. The captions for the images are flatly and simply incorrect in the context of wave-particle duality.
 * We've been all through this.
 * If you can produce a reliable source that says wave-particle duality is caused by or frankly in any significant way related to wave packets, post it for discussion. I will post 5  discuss wave-particle duality in detail and do not mention packets. I have already posted reliable references that wave packets are not a fundamental concept in quantum mechanics. Johnjbarton (talk) 18:05, 28 August 2023 (UTC)
 * Your arguments were not conclusive. I replied with reliable sources (Weinberg, Cohen-Tannoudji and others). The motion of a quantum particle is represented by a wave function. Wave packets are wave functions. TD (talk) 18:09, 28 August 2023 (UTC)

What Feynman doesn't say.
The very well written section on Feynmam ends with an extrapolation:

", and it seems that an electron can be at several positions at the same time, that it passes through both slits, just like if it was spreading in space, like a wave."

As far as I am aware Feynman does not say this, and wouldn't as it is not sensible. Particles by nature are in one position. If we want things to be in several position at the same time, we need things other than particles.

For example, see the comments by Roy Glauber, 2005 Nobel laureate, in a one page article https://doi.org/10.1119/1.17790

"Things that interfere in quantum mechanics are not particles. They are probability amplitudes for certain states." Johnjbarton (talk) 01:24, 27 August 2023 (UTC)


 * Thanks for your comment ("very well written"). Please forgive me. I'm too often too impulsive. Our discussion is very instructive for me too. I will think about it and hope that a good work can result from this.TD (talk) 01:59, 27 August 2023 (UTC)
 * I removed the unsourced content. "Particles in multiple places" is pseudoscience. Johnjbarton (talk) 15:36, 28 August 2023 (UTC)
 * Why did you delete my animations? I deleted them a few days ago because I've been told they were unwanted in the experimental confirmation section. But in this section about superposition, they are clearly a the right place.TD (talk) 17:29, 28 August 2023 (UTC)
 * Illustrations of superposition of wave packets belong on the page about wave packet or about superposition. They have nothing to do with wave-particle duality. Johnjbarton (talk) 18:08, 28 August 2023 (UTC)
 * Interference with particles is wave-particle duality. TD (talk) 18:11, 28 August 2023 (UTC)
 * The figures do not illustrate the particle aspect. I think the movement would be distracting from the main point in the superposition page also (some static figure would work better). The pictures should go to wave packet page only. Jähmefyysikko (talk) 18:20, 28 August 2023 (UTC)
 * A wave function represents the motion of a quantum particle. TD (talk) 18:27, 28 August 2023 (UTC)
 * How do I see that from the figure? One could make a similar animation for classical wave packets propagating in some medium. Something extra, like the Born rule, is needed to give the wave function the correct physical interpretation as a probability of finding a particle at a certain location. Jähmefyysikko (talk) 19:05, 28 August 2023 (UTC)
 * What do you want to see? The figure represents a wave function of a particle. That a wave function represents the motion of a particle (or system of particles) is standard quantum theory. TD (talk) 19:08, 28 August 2023 (UTC)
 * Compare with the experimental figure. There you have the particle aspect clearly visible, as the particles arrive as quantized packets to the detector pixels. You also have the wave aspect as the interference fringes. Jähmefyysikko (talk) 19:10, 28 August 2023 (UTC)
 * You're right. But this is not a reason to exclude a visualization of wave functions from an article on wave particle duality. TD (talk) 19:12, 28 August 2023 (UTC)
 * I think it is. It is confusing to have a section that is not relevant for the topic of the article. And there is also the danger that a reader who does not read the caption carefully could be
 * lef twith the impression that the wave packet represents the particle side of the duality. Jähmefyysikko (talk) 19:29, 28 August 2023 (UTC)
 * If i understand you correctly, you deny that the superposition principle is relevant to the topic of wave-particle duality. Why?
 * A wave packet is a wave.TD (talk) 19:36, 28 August 2023 (UTC)
 * No, that second point was about the wave packet specifically since it looks somewhat like a particle with a finite size. That criticism would not apply to e.g. two circular waves interfering. Jähmefyysikko (talk) 19:59, 28 August 2023 (UTC)
 * I suppose that you deny that a wave packet is a wave! Why? A wave packet has a finite size because its spreading in space becomes negligible at a distance, but it is still a wave. TD (talk) 20:04, 28 August 2023 (UTC)
 * Sorry, but you misunderstand me. Let me restate: A wave packet is a wave. But it is also a "small localized object" (see Particle). However, that property of a wave packet is not the particle-like aspect wave-particle duality refers to, and is potentially confusing. Jähmefyysikko (talk) 20:25, 28 August 2023 (UTC)
 * I understand better your point. You're worried because you think a beginner could think that wave-particle duality means wave packet. Am I right? TD (talk) 20:37, 28 August 2023 (UTC)
 * Yes, that's my worry. Jähmefyysikko (talk) 20:50, 28 August 2023 (UTC)
 * I think about it. I will try to fix this problem. TD (talk) 20:52, 28 August 2023 (UTC)

@Jähmefyysikko If you delete again this section you prevent other wikipedians from participating to this discussion.TD (talk) 21:08, 28 August 2023 (UTC)


 * Other editors are able to read the history just fine. For the benefit of others, here is a link to Cohen-Tannoudji, Ch.III, Sec. E, which you quote. After this sentence, within the same paragraph, probabilities are discussed, something which is missing from your discussion. Jähmefyysikko (talk) 21:25, 28 August 2023 (UTC)
 * The problem of the difference between the size of a wave packet and the size of a particle is now fixed. Isn't it? TD (talk) 21:52, 28 August 2023 (UTC)
 * Please don't bring the figures back. There is no reason to show a wave packet, and then explain that its not the wave packet one should be looking at. Besides, it is still irrelevant to the duality, only shows the wave aspect. Jähmefyysikko (talk) 22:14, 28 August 2023 (UTC)
 * The citation of Cohen-Tannoudji is clearly a good reference about the relation between wave functions and wave-particle duality. Wave packets are wave functions. Why do you deny this point? TD (talk) 22:17, 28 August 2023 (UTC)
 * It is a single sentence taken out of context. The context is a discussion of a probabilitic nature of measurement. Jähmefyysikko (talk) 22:19, 28 August 2023 (UTC)

Measurement defines duality.
I want to propose this short phrase "measurement defines duality", not for the article, but just to summarize the boundary of duality. Most importantly this phrase helps distinguish what duality is not: it is not an interpretation of quantum mechanics nor a shorthand for a physical model of the quantum entities.

Here are the statements that "measurement defines duality" summarizes: Nothing else is needed to provide an encyclopedic discussion of wave-particle duality.
 * Carefully designed measurements show interference; interference is characteristic of wave behavior.
 * Different careful measurements also show trajectories consistent with particle behavior.
 * Measurements are the only means to study quantum entities.
 * A wave equation accurately predicts quantum measurements when combined with a quantum measurement hypothesis.
 * No interpretation of the wave equation beyond the measurement hypothesis has experimental support.

Wave-particle duality is not about anything below the level of measurement.

One addition statement could be added based on many sources: In some sources view this as part of orthodox QM. I would leave this out of an article on duality because it is not essential. That is, I would put this statement outside the boundary for an article on wave-particle duality.
 * The properties of quantum entities are caused by measurements.

Wave-particle duality is not an interpretation, part of an interpretation, or a shorthand for any aspect of an interpretation. We have many pages devoted to interpretation of quantum mechanics for that material. For the area of interpretation, duality describes a problem, not a solution. There is no need to discuss the "solution" for this "problem" because there isn't one.

I propose this boundary for the content in the page. Johnjbarton (talk) 19:00, 28 August 2023 (UTC)


 * Sounds good to me. Focusing on measuments seems like a natural approach since duality originates from seemingly incompatible experimental results. Jähmefyysikko (talk) 20:31, 28 August 2023 (UTC)
 * In this discussion, wave-particle duality is understood in two different ways. 1) Duality is a consequence of measurement. 2) A wave function represents the real motion of a particle, or of a system of particles. Among physicists there isn't any consensus in favor of one or the other understanding. Hence in this article, both shall be explained. TD (talk) 14:05, 29 August 2023 (UTC)
 * The statement 1) can be found in most of the mainstream sources. The phrasing above is a bit peculiar, but I assume you mean a similar statement to the one currently found in the introduction of this article. To give a single popular source, here's Britannica.
 * The statement 2) refers to the wave function, a theoretical contruction, and seems to me more of an explanation for the physical meaning of the Schrödinger equation than a statement of wave-particle duality. Can you provide a source which discusses wave-particle duality primarily from this point of view, and not from the first one? Jähmefyysikko (talk) 15:24, 29 August 2023 (UTC)
 * There is actually no contradiction between the two viewpoints as long as the measurement process in the wave mechanics is also discussed (through Born rule). The discussion about how the wave-particle duality is implemented in the fully developed quantum theory definitely falls within the boundary drawn by Johnjbarton and should be discussed in the article. Jähmefyysikko (talk) 15:56, 29 August 2023 (UTC)

Core concept of wave-particle duality: the measurement matters.
In the current article is the statement:

"Any wave function can represent the motion of a quantum particle, or of a system of such particles."

This sentence is incorrect. Wave functions represent probability amplitude; they evolve in phase but otherwise they have no motion and do not represent motion.

We have had to assume for close to 100 years that wave functions represent probability of measurements. Consequently a wave function alone has no meaning. A wave function gains meaning in measurement. If the measurement is configured for wave properties, the appropriate operator will produce interference probability distributions. If the measurement is configured for particle properties, no interference will observed. That is what wave-particle duality is about.

As a slightly more technical level, most QM simple experiments are "stationary", without a time-varying potential. Thus the wave function is stationary. Concretely the predicted probability distribution is time-independent. Clearly this cannot "represent the motion" of anything. Johnjbarton (talk) 22:14, 28 August 2023 (UTC)


 * Similarly
 * This is not correct. The referenced sentences which follow this one refer to state, not to particles and not to motion. Johnjbarton (talk) 22:18, 28 August 2023 (UTC)
 * A state is a state of a particle or of a system of particles. A motion is the evolution of a state. TD (talk) 22:21, 28 August 2023 (UTC)
 * "A motion is the evolution of a state." Prove it. Johnjbarton (talk) 22:23, 28 August 2023 (UTC)
 * The Schrödinger equation is the fundamental equation of motion in quantum mechanics. With it, we can calculate how the wave function (the state) changes with time, that is, its evolution. TD (talk) 22:27, 28 August 2023 (UTC)
 * Please read Cohen-Tannoudji, Diu and Laloë - Quantum Mechanics (vol. I, II and III, 2nd ed.) page 13:
 * "Note the important difference between the concepts of classical states and quantum states. The classical state of a particle is determined at time t by the specification of six parameters characterizing its position and its velocity at time t: L,Y, Z% Vg, Vy, Vz. The quantum state of a particle is determined by an infinite number of parameters: the values at the various points in space of the wave function (r,t) which is associated with it. For the classical idea of a trajectory (the succession in time of the various states of the classical particle), we must substitute the idea of the propagation of the wave associated with the particle. Consider, for example, Young’s double-slit experiment, previously described for the case of photons, but which in principle can also be performed with material particles such as electrons. When the interference pattern is observed, it makes no sense to ask through which slit each particle has passed, since the wave associated with it passed through both.'" Johnjbarton (talk) 22:33, 28 August 2023 (UTC)
 * Same book page 11
 * (ii) $$\Psi(r,t)$$ is interpreted as a probability amplitude of the particle’s presence. Johnjbarton (talk) 22:37, 28 August 2023 (UTC)
 * Page 24:
 * A wave function of the form (D-7) is called a stationary solution of the Schrodinger equation: it leads to a time-independent probability density.  Johnjbarton (talk) 22:39, 28 August 2023 (UTC)
 * Page 6:
 * Moreover, as the photons arrive one by one, their impacts on the screen gradually build up the interference pattern. This implies that, for a particular photon, we are not certain in advance where it will strike the screen. Now these photons are all emitted under the same conditions. Thus another classical idea has been destroyed: that the initial conditions completely determine the subsequent motion of a particle. We can only say, when a photon is emitted, that the probability of its striking the screen at x is proportional to the intensity I(x) calculated using wave theory, that is, to $$|E(zx)|^2$$. Johnjbarton (talk) 22:51, 28 August 2023 (UTC)
 * The quotations and the texts inserted between and after them are discussing completely different things, and there is no logical structure in this section. I'm not sure why motion is being discussed in the text, perhaps to have an excuse to insert the figures of wave packets? But even as an excuse it is not very good: wave packets are not needed to represent motion in QM, as even plane waves can represent currents. Jähmefyysikko (talk) 22:49, 28 August 2023 (UTC)
 * Wave packets are very useful to represent motion of particles in an accelerator, like the LHC. TD (talk) 09:11, 29 August 2023 (UTC)
 * Perhaps they are (I see no source). They are also useful in femtochemistry. But how is that relevant for the duality? Jähmefyysikko (talk) 09:27, 29 August 2023 (UTC)
 * The theory of scattering is made with wave functions and particles are detected. TD (talk) 09:30, 29 August 2023 (UTC)
 * So why invoke accelerators? The double-slit experiment is a much cleaner setting to demonstrate duality. Jähmefyysikko (talk) 10:14, 29 August 2023 (UTC)
 * I answered to your assertion "wave packets are not needed to represent motion in QM, as even plane waves can represent currents." TD (talk) 10:17, 29 August 2023 (UTC)
 * Accelerators, electron microscopes, and most optical systems can be modeled with geometrical optics. This is a classical (or semiclassical if the phase is tracked) model that is a consequence of approximations to the wave equation. Schrodinger used this concept in reverse to develop his equation, see Hamilton's optico-mechanical analogy. You can read more about it under Fresnel equations.
 * To model finite sized sources and finite-lifetime emitters, a range of initial conditions, typically in the shape of a sausage, are processed by the geometrical optics code. The given range of initial conditions, propagated through the (virtual) instrument produce a range of final values predicting images or cross sections.
 * These are entirely wave effects converted to "rays" corresponding to lines perpendicular to constant phase. If the phase along the ray is tracked, the approximate phase difference can be used to estimate interference. See Fresnel diffraction.
 * Despite the visual similarity, rays are not trajectories for particles, as can be demonstrated with the double slit experiment. This is a fascinating subject but a large detour for wave-particle duality. Johnjbarton (talk) 16:12, 29 August 2023 (UTC)
 * Despite the visual similarity, rays are not trajectories for particles, as can be demonstrated with the double slit experiment. This is a fascinating subject but a large detour for wave-particle duality. Johnjbarton (talk) 16:12, 29 August 2023 (UTC)

Section "Wave Packets."
I added a review reference to the section on wave packets that describes the failure of packets to prove useful in understanding quantum systems.

I'm ok with removing the section as it has nothing to do with wave-particle duality. However, if it remains, other content related to wavepackets should remain together so readers can understand their limited value. Johnjbarton (talk) 23:18, 28 August 2023 (UTC)


 * Wave packets are very useful to represent the motion of particles in many experiments. Think to the Stern-Gerlach experiment for example.TD (talk) 09:15, 29 August 2023 (UTC)
 * We would need sources for the Stern-Gerlach experiment with wave packets, and its relevance for duality is also questionable.
 * Here is a review article on wave packet dynamics: https://iopscience.iop.org/article/10.1088/0034-4885/58/4/0 According to the introduction (p.367):
 * Sadly, wave-packets have been a side issue in quantum mechanics for a long time. Their use in theory has been limited to simple textbook examples.
 * Thus, they do not seem very relevant for the theory or description of experiments. Their preparation also requires special techniques, since such packets do not occur naturally in general systems:
 * For a long time the wave-packets had no real practical use because their preparation seemed impossible. However, recent advances in the physics and chemistry of laser interactions with atoms and molecules have brought the wave-packets and their dynamics into the limelight. For example, it became possible to prepare wave-packets by exciting atoms to Rydberg states with short pulses.
 * The recent interest on wave packet dynamics comes largely from femtochemistry (Zewail's 1999 Nobel). Jähmefyysikko (talk) 09:46, 29 August 2023 (UTC)
 * If in an experiment a particle has one or a few trajectories, the simplest theoretical way to represent these trajectories is to reason on one or a few wave packets. TD (talk) 09:51, 29 August 2023 (UTC)
 * The wave functions of an electron in an hydrogen atom are also wave packets.TD (talk) 09:55, 29 August 2023 (UTC)
 * Without sources, this is all original research. Jähmefyysikko (talk) 10:00, 29 August 2023 (UTC)
 * A wave packet is by definition a localized wave. This means that its spreading in space is negligible at a distance. Do you need a source for this simple point? TD (talk) 10:03, 29 August 2023 (UTC)
 * I think it is not useful to call the (eigen?)states of a hydrogen atom a wave packet, and would like to know who uses such term in this setting. From Wave packet: In physics, a wave packet is a short burst of localized wave action that travels as a unit. They do not travel. Perhaps one can contruct some orbiting wave packets in a r^2 potential, but they are very artificial solutions. Jähmefyysikko (talk) 10:11, 29 August 2023 (UTC)
 * Atoms travel. TD (talk) 10:14, 29 August 2023 (UTC)
 * A gaussian wave packet doesn't necessarily travel.TD (talk) 06:45, 31 August 2023 (UTC)
 * I agree that it is probably not useful to call the (eigen?)states of a hydrogen atom a wave packet. One could write a wave-packet solution for a hydrogen atom's center-of-mass degrees of freedom, say in a scattering scenario, but nobody calls the bound eigenstates of the electron-proton relative degrees of freedom "wave packets". XOR&#39;easter (talk) 18:07, 29 August 2023 (UTC)
 * I stand to the definition: a wave packet is a localized wave. TD (talk) 18:12, 29 August 2023 (UTC)
 * Except that's not the definition, because it's not how physicists use the words. Nor would it be a definition that makes sense to adopt. If anything other than a plane wave of infinite extent is a "wave packet", then "wave packet" is just synonymous with "properly normalizable wavefunction". That is, your proposed definition of "wave packet" makes it mean the same thing as "wavefunction", instead of what physics books mean when they say "wave packet" &mdash; roughly, a wavefunction explicitly considered to be a superposition of different-frequency waves that themselves often do not belong to the Hilbert space of square-integrable functions. There's extra conceptual baggage to the term "wave packet" that makes it not apply to, e.g., the energy eigenstates of a particle in a box, even though the latter are obviously localized. XOR&#39;easter (talk) 19:33, 29 August 2023 (UTC)
 * What is the extra conceptual baggage? TD (talk) 19:47, 29 August 2023 (UTC)
 * You stated exactly what I think: everything is a wave packet, because everything is a square-integrable function.TD (talk) 20:12, 29 August 2023 (UTC)
 * That is a highly nonstandard use of the term "wave packet" which is likely to confuse the people you are trying to communicate with. XOR&#39;easter (talk) 22:04, 29 August 2023 (UTC)

@Johnjbarton It seems that you don't make the difference between two very different theses: "the motion of a particle can be represented by the motion of a wave packet, at least in some cases" and "all motions of all particles shall always be represented by wave packets". I defend the first thesis, not the second.TD (talk) 13:44, 29 August 2023 (UTC)


 * These are not different, they are both incorrect. Motions cannot be represented in quantum mechanics. That is why many full-time, career geniuses of physics (Einstein, Bohr, de Broglie, Bohm, Bell, Schwinger, Wheeler, von Neumann, Feynman, Glauber, a thousand others) discussed alternatives for decades. The idea that these folks some how just missed the boat is completely ludicrous.
 * Solutions to Schrodinger's equation in free space can be combined to produce a wave packet with an amplitude that moves in space. But wave-particle duality is about the results of measurements. You can't make measurements in free space. You also cannot make any predictions from wave equation solutions in free space. Predictions are predictions about measurements. And measurements produce either particle or wave behaviors depending on the measurement. Measurements never produce wave packets.
 * If you want to predict the detailed structure of a double slit interference pattern, you can combine a set of patterns across the range of size and emission lifetimes of the source and include effects like vibrations. The effect can be compactly described as a packet of initial conditions. This is not a fundamental quantum effect related to duality, just a practical matter of experimental physics. Johnjbarton (talk) 16:31, 29 August 2023 (UTC)
 * These two theses are very different. Your references are always against the second one. The first one is standard quantum mechanics. TD (talk) 18:39, 29 August 2023 (UTC)

A summary of the present discussion
A few arguments of my detractors:
 * If we know through which slit the electron passes, we can still observe an interference pattern, with a coherent detector -- I don't think such detectors exist beyond paper. Ldm1954 08:36, 4 September 2023 (UTC)
 * Solve for a Hamiltonian H for slit one, and H+exp(i*phi) for slit 2, where "phi" is constant and represents a fictitious coherent detector that shifts the phase but nothing else. Ldm1954 09:08, 4 September 2023 (UTC)
 * Hamiltonian combine operators, vectors and scalars. For instance the Coulomb potential from a nuclear potential is a scalar. Ldm1954 09:19, 4 September 2023 (UTC)
 * The equation $$E = p ^2 / 2 m $$ has no basis. Quondum 13:10, 12 August 2023 (UTC)
 * Superposability is only a secondary aspect of states being elements in a vector space. XOR&#39;easter 22:03, 31 August 2023 (UTC)
 * The interference observed with electrons conclusively demonstrate they are not particles. We must conclude electrons are not particles. Johnjbarton 23:35, 25 August 2023 (UTC)
 * States interfere, not particles. Johnjbarton 14:19, 3 September 2023 (UTC)
 * Motions cannot be represented in quantum mechanics. Johnjbarton 16:31, 29 August 2023 (UTC)
 * Wave functions have no motion and do not represent motion. Johnjbarton 22:14, 28 August 2023 (UTC)
 * Wave functions refer to state, not to particles and not to motion. Johnjbarton 22:18, 28 August 2023 (UTC)
 * Particle-like phenomena, like the photo-electric effect have nothing to do with quantum superposition. Headbomb 13 August 2023 (UTC)
 * Duality is not visualizable. Johnjbarton 15:15, 21 August 2023 (UTC)
 * The problem is that people don't want to hear about complementarity. Everyone wants realism all the way down. Johnjbarton 18:01, 21 August 2023 (UTC)
 * Particles do not behave like waves, that's, "rather incoherent". Particles behave like particles; Waves behave like waves. Experiments on quantum systems can give results consistent with either one, depending on the experiment. The standard view does not involve tiny things switching between particles and waves. Johnjbarton 20:40, 21 August 2023 (UTC)
 * Particles don't obey wave equations. Johnjbarton 16:01, 22 August 2023 (UTC)
 * The entire point of 'wave-particle duality' is that we have no representations of quantum entities. Johnjbarton 18:28, 25 August 2023 (UTC)
 * I don't know of any experiment or theory which claims wave packets represent wave-particle duality. Johnjbarton 18:28, 25 August 2023 (UTC)
 * To place any interpretation other than Copenhagen, in the main position of the wave-particle duality article would be historically incorrect as well as scientifically unsupportable since no particular interpretation has been shown correct. Johnjbarton 15:40, 26 August 2023 (UTC)
 * The animation can't both explain wave particle duality and be a representation of a "particle". Johnjbarton 18:09, 26 August 2023 (UTC)
 * Particles by nature are in one position. If we want things to be in several position at the same time, we need things other than particles. Johnjbarton 01:24, 27 August 2023 (UTC)
 * "Particles in multiple places" is pseudoscience. Johnjbarton 15:36, 28 August 2023 (UTC)
 * Wave packets are not needed to represent motion in QM, as even plane waves can represent currents. Jähmefyysikko 22:49, 28 August 2023 (UTC)
 * Wave packets are not useful in understanding quantum systems. Johnjbarton 23:18, 28 August 2023 (UTC)

One argument for the conservation of these animations:
 * In this sense Thierry Dugnolle's illustrations represent very well the mainstream point of view: a wavefunction ontology is enough to reproduce simple aspects such as having a well-localized quantum system that can interfere with itself. The fact that it's a simple, idealized description is not a problem, we do use such idealizations for pedagogical purposes all the time. Tercer 12:19, 13 August 2023 (UTC)

TD (talk) 09:59, 4 September 2023 (UTC)


 * The wavefunction ontology works well in predicting the (static) probability distribution for measurements that produce interference. To put the same thing another way, the wavefunction ontology works for measurements where wave behavior is expected.
 * The wavefunction ontology does not predict the dots on the screen and it does not predict anything like a "well-localized quantum system". The localization is a completely separate effect, the nature of which is an area of on-going research. Nothing in the wave function equations predict localization; we have to add on the Born "rule" or hypothesis together with some hand waving about state reduction to get localization.
 * I'm strongly in favor of simple idealized descriptions for core concepts. Ideal means to strip away all that is inessential. In the case of the double-slit, the Gaussian envelop in the images is inessential. Its inclusion creates an illusion of localization contrary to science. The envelop is not ideal but confusing.
 * Wave-particle duality is about opposites, particles vs waves, local vs delocalized, collisional vs superposable. It's not about a combination.
 * Wave packets are great for idealized descriptions of the uncertainty principle. The uncertainty principle is a purely wave effect that (ideally) does not involve Born rule issues. The trade-off in the uncertainty principle is also momentum vs position but it is about how close they can get in hypothetical scenarios. There are a few sources that mix the two concepts and Bohr used uncertainty principle ideas to analyze double slit experiments. But in my opinion the broadest and simplest idea of duality should be the content of this article. Johnjbarton (talk) 18:24, 29 August 2023 (UTC)
 * I don't see what the animations contribute, and to be honest, I don't think the "Wave packets" section as it stands is very helpful at all. It seems to confuse Schrödinger's original hope, that some quantity derived from the solutions to his wave equation could be the physical charge density of the electron, and how the term "wave packet" is actually used in modern parlance, i.e., a type of wavefunction that, like all other wavefunctions, is interpreted probabilistically. XOR&#39;easter (talk) 19:54, 29 August 2023 (UTC)
 * I didn't create this section. I didn't want it. And I still don't want it. It's a very bad one. My animations were in the section 'Wave-particle duality and the quantum superposition principle' and were presented very differently: Interference of a particle with itself. This new section on wave packets has been created against my will.TD (talk) 20:08, 29 August 2023 (UTC)
 * That section doesn't contribute anything either. It's disjointed quotations from different textbooks, of which the second just recapitulates the first, and the third says that the superposition principle, when combined with other ideas, leads (somehow) to "interference effects" that are related (somehow) to wave-particle duality. No one is going to learn anything from that. XOR&#39;easter (talk) 20:26, 29 August 2023 (UTC)
 * This section (Wave-particle duality and the quantum superposition principle) has been censured. TD (talk) 20:28, 29 August 2023 (UTC)
 * What do you mean by "censured"? It's right there in the current version of the article. And it conveys no information. XOR&#39;easter (talk) 20:33, 29 August 2023 (UTC)
 * Read the old version, before another wikipedian deprived it of its logic. TD (talk) 20:36, 29 August 2023 (UTC)
 * This version? Sorry, but I don't think that's any better. If anything, language like "a particle can be at many positions at the same time" just makes it more confusing. Who is the intended audience? People who know what "complex vector space" means don't need vague vulgarizations; readers who don't understand the mathematics will balk before they even get to the verbal (over)simplification. XOR&#39;easter (talk) 21:00, 29 August 2023 (UTC)
 * With a realist interpretation of the wave function (the state vector) it is obvious that a particle can be at many places at the same same time. In the double-slit experiment, an electron passes through both slits, hence it can be at in the two slits at the same time. TD (talk) 21:05, 29 August 2023 (UTC)
 * Taking an interpretation-dependent statement, oversimplifying to the point where its meaning is almost lost, and sandwiching the resulting vulgarization between mathematical statements does not benefit any reader that I can imagine. XOR&#39;easter (talk) 21:12, 29 August 2023 (UTC)
 * I will try to explain this point better. But it's heavy work. Without good references, I can't publish on Wikipedia. TD (talk) 21:15, 29 August 2023 (UTC)
 * The intended audience is anyone who wants to understand quantum mechanics, and who wants to work for it, beginners included.TD (talk) 21:24, 29 August 2023 (UTC)
 * How is a "beginner" served by two different quotations that are both dependent upon knowledge of the abstract, axiomatic definition of a vector space? XOR&#39;easter (talk) 21:35, 29 August 2023 (UTC)
 * We are all beginners, before we work. Even a Nobel prize can be a beginner. TD (talk) 21:37, 29 August 2023 (UTC)
 * I don't see how that answers my question. XOR&#39;easter (talk) 21:39, 29 August 2023 (UTC)
 * Do you think that the axiomatic definition of a vector space is only for a few, not for everyone who wants to understand science? TD (talk) 21:41, 29 August 2023 (UTC)
 * If the reader does not already know the axiomatic definition of a vector space, they will get absolutely nothing from the section you wrote. If the reader does already know the axiomatic definition of a vector space, they will not need the popularized platitudes about particles being in multiple places at once. The section you seem to be so insistent upon benefits no one, as far as I can tell. I remain completely in the dark about what you are trying to achieve. XOR&#39;easter (talk) 22:03, 29 August 2023 (UTC)
 * Wave-particle duality is a direct consequence of superposition. Isn't it clear? TD (talk) 04:15, 30 August 2023 (UTC)
 * Not really. Any linear wave theory such as classical electrodynamics has superposition property. Jähmefyysikko (talk) 06:47, 30 August 2023 (UTC)
 * Wave-particle duality is a direct consequence of superposition in a complex vector space, if we reason about the vector states of particles. TD (talk) 06:55, 30 August 2023 (UTC)
 * Not without extra assumptions which establish the probabilistic interpretation for the wave function. If you consider e.g. Gamow's tunneling model of alpha decay, you see with Schrödinger equation that the wave function is leaking out of the potential well. But why should you have discrete decay events? It's not explained by mere superposition. Jähmefyysikko (talk) 07:04, 30 August 2023 (UTC)
 * Wave-particle duality is a direct consequence of quantum superposition combined with the Born rule, if we reason about the state vectors of particles. TD (talk) 07:07, 30 August 2023 (UTC)
 * Great, so now we agree on how wave-particle duality is modeled in quantum mechanics. Jähmefyysikko (talk) 07:20, 30 August 2023 (UTC)
 * Thanks for this discussion. It helped me to improve my explanations. TD (talk) 07:23, 30 August 2023 (UTC)

Thanks to all who contributed to this long discussion. It was great help to improve my knowledge, my animations and my explanations.TD (talk) 21:40, 30 August 2023 (UTC)


 * Thanks. Unfortunately, a discussion on a talk page does not give a permission to post WP:Original research. I removed your additions. See diff. Jähmefyysikko (talk) 03:32, 31 August 2023 (UTC)
 * This is not original research but simply standard quantum mechanics. What is original here? TD (talk) 06:14, 31 August 2023 (UTC)
 * All of the material in the above diff. The burden of demonstrating that your additions are not WP:OR falls on you. From WP:OR: To demonstrate that you are not adding original research, you must be able to cite reliable, published sources that [...] directly support the material being presented. Also see WP:SYNTH. Jähmefyysikko (talk) 06:32, 31 August 2023 (UTC)
 * I did. Don't you think the reference to Cohen,-Tannoudji, Diu and Laloë is sufficient? TD (talk) 06:41, 31 August 2023 (UTC)
 * They are not sufficient. For example, the first sentence Wave-particle duality is a direct consequence of the quantum superposition principle combined with the Born rule. is not explicitly stated in the sources. From WP:SYNTH: "Do not combine material from multiple sources to reach or imply a conclusion not explicitly stated by any source." Jähmefyysikko (talk) 06:47, 31 August 2023 (UTC)
 * Do we really need references for the following elementary statements? "Wave packets are waves which are localized. Their spreading in space is negligible at a distance. When the motion of a particle is represented by a wave packet, this doesn't mean that the size of the particle is the size of the wave packet: the size of an elementary particle (like an electron) is not measurable (zero or nearly so) but the size of a wave packet may be very large. Theoretically there isn't any limit of size for wave packets, but particles are usually detected in tiny places whose size depend on the detector." Is it original research? According to me, this is simply standard quantum mechanics. TD (talk) 07:53, 31 August 2023 (UTC)
 * Same questions for the following statement: "The motion of a particle is represented by a wave function, just like if it could be at many places at the same time, but the particle is always detected at a single place."TD (talk) 07:59, 31 August 2023 (UTC)
 * Basically for every statement, a reference is needed. And please do not quote large sections, like Cohen-Tannoudji, Ch.III, Sec.E, but indicate the page on which the statement can be found. Jähmefyysikko (talk) 08:01, 31 August 2023 (UTC)
 * These rules you just invented are almost never respected in Wikipedia. TD (talk) 08:09, 31 August 2023 (UTC)
 * WP:PROVEIT: The cited source must clearly support the material as presented in the article. Cite the source clearly, ideally giving page number(s). The burden of proof is still on you. If it is not obvious which page is used, the citation may be challenged. Jähmefyysikko (talk) 08:20, 31 August 2023 (UTC)
 * "though sometimes a section, chapter, or other division may be appropriate instead" TD (talk) 08:25, 31 August 2023 (UTC)
 * Some sources have no page numbers. If your sources truly state what you say they do, please provide precise locations. Jähmefyysikko (talk) 08:53, 31 August 2023 (UTC)
 * Born doesn't make use of the expression 'wave-particle duality'. But the whole article is clearly about it. It is generally considered as the basis of the understanding of wave-particle duality in standard quantum mechanics. I already gave a precise reference to the quotation of Cohen-Tannoudji, Diu and Laloë. I only have the french edition of this book.
 * My contribution is clearly not original research. Are you sure that the verifiability of standard quantum mechanics deserves to be challenged? TD (talk) 09:15, 31 August 2023 (UTC)
 * It is the idiosyncratic interpretation of QM and its concepts that I challenge. Jähmefyysikko (talk) 15:42, 31 August 2023 (UTC)
 * What was idiosyncratic here? TD (talk) 15:46, 31 August 2023 (UTC)
 * For example, the interpretation that the particle exists in multiple places at once until it is measured. (I may misunderstand your intended meaning) Also calling the whole of postulate I from here the "superposition principle" is an example of idiosyncratic use of language. Talking about "motion" is another. I agree with Johnjbarton on his point, but would phrase this it differently: that word is not part of quantum vocabulary. It has no precise meaning even if it occurs in textbooks from time to time. One talks about time evolution, classical trajectories etc. instead. Jähmefyysikko (talk) 20:46, 31 August 2023 (UTC)
 * I'm not the only one to adopt a realist interpretation of the wave function. With such an interpretation, a particle exists at multiple places at the same time. That's what I think but not what I wrote: "just as if the particle could be at many places at the same time". My "idiosyncratic" use of language (principle of superposition) is largely shared: Dirac, Cohen-Tannoudji, Diu, Laloë, Le Bellac, Fortunato, Auletta, Parisi, and many others. Motion and evolution are almost synonymous. If this is an idiosyncrasy, I will say evolution instead of motion, from now on.TD (talk) 21:05, 31 August 2023 (UTC)
 * Cohen-Tannoudji et al. don't say that the first postulate is "the superposition principle"; they say that it "implies a superposition principle" (III.B.1, page 215). Later, they repeat that states being "linearly superposable" is a consequence of an idea they take to be more basic (III.E, p. 252). Superposability is only one aspect of states being elements in a vector space. Then they write, One of the important consequences of the first postulate, when it is combined with the others, is the appearance of interference effects such as those which led us to wave-particle duality (chap. I). Your insertion of "the quantum superposition principle" after "the first postulate" gets their meaning wrong. XOR&#39;easter (talk) 21:29, 31 August 2023 (UTC)
 * This use of the expression "the quantum superposition principle" or "the principle of superposition" is still very ordinary, from 1930 (the first edition of Dirac's book) to the present. My insertion was here only to give a precision to the reader about their first principle. Is the name we give to it so important? TD (talk) 21:37, 31 August 2023 (UTC)
 * Your statement about what their first principle is was incorrect. It conflated two related but distinct ideas. And it was in service to a goal that I still cannot understand: why single out the linear superposability of quantum states as the explanation for wave-particle duality, separate from all the other ideas which as the cited source says have to be included to reach that conclusion? XOR&#39;easter (talk) 21:45, 31 August 2023 (UTC)
 * What are the two related but distinct ideas? And I wrote that the principle of superposition with the Born rule led to wave-particle duality. TD (talk) 21:49, 31 August 2023 (UTC)
 * As I said: Superposability is only one aspect of states being elements in a vector space. Their first postulate is the latter, not the former, which they explicitly regard as a secondary consequence. Saying that Wave-particle duality is a consequence of the quantum superposition principle combined with the other principles of quantum mechanics just amounts to saying that wave-particle duality is a consequence of the principles of quantum mechanics. What does that contribute? Not a lot. It's nearly redundant with the first line of the article. XOR&#39;easter (talk) 22:03, 31 August 2023 (UTC)
 * I don't understand. What are the other aspects of being elements in a vector space? TD (talk) 22:06, 31 August 2023 (UTC)
 * The principle of superposition is a necessary and sufficient condition for a space to be a vector space.TD (talk) 22:26, 31 August 2023 (UTC)
 * No, it isn't; for example, the state space could be a module and satisfy superposition without being a vector space. (People who wonder whether quantum mechanics can be developed over the quaternions instead of the complex numbers care about this sort of thing.) And, chasing down how Cohen-Tannoudji et al. define their terms, they actually presume that states belong to an inner product space (this is most clear in II.B). They really do presume more structure than just the ability to take linear combinations. XOR&#39;easter (talk) 22:32, 31 August 2023 (UTC)
 * The principle of superposition here is with complex numbers.  TD (talk) 22:37, 31 August 2023 (UTC)
 * OK, let's work over the complex numbers, and suppose that our linear combinations are commutative (not explicitly stated in your Weinberg quote, but let's assume it). A vector space over $$\mathbb{C}$$ doesn't have to be equipped with an inner product, let alone one that makes the vector space into a complete metric space; Cohen-Tannoudji et al. assume that it does. There's more than linear combinations going on in their first postulate. And even taking all six of their postulates, of which the Born rule is the fourth, doesn't get us all the way to wave-particle duality. For that, we still need to introduce some mathematical meaning of "wave" and "particle". That will mean invoking something like the canonical commutation relation &mdash; something that ties the abstract formalism of inner-product spaces to experimentally meaningful quantities that, moreover, form a complementary pair. XOR&#39;easter (talk) 23:03, 31 August 2023 (UTC)
 * Even with the field of quaternions, it is still a vector space, not only a module (if I'm not mistaken. I'm not an expert about abstract algebra).TD (talk) 22:56, 31 August 2023 (UTC)
 * Multiplication in the quaternions is not commutative, so they form a skew field rather than a true-blue field, meaning that the strict definition of vector space can't be satisfied, though some authors gloss over this quickly so they can stick with terminology they find more familiar. XOR&#39;easter (talk) 23:18, 31 August 2023 (UTC)
 * I don't know. It seems to me that this abstract algebra is not relevant to quantum mechanics. TD (talk) 23:21, 31 August 2023 (UTC)
 * It's relevant if one wants to discuss the algebraic premises on which quantum mechanics is built and what possibilities are allowed or forbidden by each premise. XOR&#39;easter (talk) 23:26, 31 August 2023 (UTC)
 * @Thierry Dugnolle the statement is self contradictory and incorrect. the motion of a particle is not represented in quantum mechanics. waves do not represent particles. particles by nature are in a location. Johnjbarton (talk) 14:11, 31 August 2023 (UTC)
 * Didn't you read the reference? "physical states are represented by rays in Hilbert space" (Weinberg). The state of a particle is a physical state. Each wave function at a given time defines a ray in Hilbert space. Hence the state of a particle is represented by a wave function at a given time. The motion of a particle is a continuous succession of states. Hence the motion of a particle is represented by a wave function. This is the ABC of standard quantum mechanics. TD (talk) 15:54, 31 August 2023 (UTC)
 * @Johnjbarton If a particle was always at a single position at every time, then its speed and momentum would also be definite at every time. But it is forbidden by the uncertainty (or indeterminacy) principle of Heisenberg.TD (talk) 17:49, 31 August 2023 (UTC)
 * @Thierry Dugnolle Exactly my point. That is why we cannot describe particle motion. That was the source of great frustration for generations of physicists. You can't make up a story to get around this aspect of duality. Johnjbarton (talk) 18:06, 31 August 2023 (UTC)
 * This is a very non-standard thesis (like another of your theses: electrons are not particles). According to standard quantum mechanics, the Schrödinger equation is the fundamental equation of motion for particles and systems of particles. TD (talk) 18:11, 31 August 2023 (UTC)
 * @Thierry Dugnolle I think you would really learn a lot if you read the first 60 pages of A. Messiah. He has the most to say about wave particle duality and you can find it online. trying to patch together references to support your made up story is just a waste of time. Johnjbarton (talk) 18:53, 31 August 2023 (UTC)
 * I already read it. I read most of good textbooks about quantum mechanics. TD (talk) 18:55, 31 August 2023 (UTC)
 * Much progress has been made since this book (1959). If you're interested in a much more modern point of view, I suggest you read my book Quantum theory of observation.TD (talk) 19:17, 31 August 2023 (UTC)
 * @Thierry Dugnolle yes, original research. matter wave packets disperse, their spreading in space is not negligible. matter wave packets are not particles, they are waves. they do not represent particle motion. I posted reliable references earlier. Johnjbarton (talk) 14:00, 31 August 2023 (UTC)
 * What is original here? I gave reliable references too. And my animation shows clearly that a wave packet grows with time. TD (talk) 14:05, 31 August 2023 (UTC)
 * Their spreading in space is negligible means here: at a given time. TD (talk) 14:13, 31 August 2023 (UTC)
 * The whole section is still a confusing mess that needs to go. What is the purpose of stringing together quotations like that? It says, in effect, "Wave-particle duality (which the reader does not yet understand) is a consequence of the superposition principle (which the reader does not yet understand) combined with the Born rule (which the reader is just now hearing about) and other rules of quantum mechanics (with which the reader is unfamiliar)". It's talking to itself, mumbling in the corner about state vectors rather than communicating. It exemplifies the complaints I have heard time and time again about Wikipedia's physics articles being self-indulgent to the point of opacity. XOR&#39;easter (talk) 19:44, 31 August 2023 (UTC)
 * An article is not a textbook. And this section is not the first of this article. The section before it (Feynman lectures on the double slit experiment) and the introduction give to the reader a few hints about wave-particle duality. Not all readers are unfamiliar with the rules of quantum mechanics. Obviously, if readers want to understand quantum mechanics, they have to read much more than this article. Don't ask for the impossible. This article is not supposed to be a first course on QM. TD (talk) 19:56, 31 August 2023 (UTC)
 * Oh, the Feynman section is bad for its own reasons. But "Wave-particle duality, the quantum superposition principle and the Born rule" is not saved in the least by there being "a few hints" elsewhere. It's disjointed, repetitive, and unclear. Plucking random statements out of books and then handwaving away the confusion by saying that one has to read the rest of the books too is not a good way to construct an encyclopedia article. XOR&#39;easter (talk) 20:01, 31 August 2023 (UTC)
 * I agree with you that this article is not very good. User:Johnjbarton/sandbox/duality is much better. But I still think this section is important because it goes to the heart of the matter: where does wave-particle duality come from? How is it manifested? The quantum superposition principle is the answer to the first question, the Born rule, to the second.TD (talk) 20:16, 31 August 2023 (UTC)
 * All the section manages to do is say that duality comes from the math of quantum mechanics. It uses a lot of words and mathematical symbols to say that, but that's all it says. XOR&#39;easter (talk) 20:23, 31 August 2023 (UTC)
 * "The existence of a wave function for a particle is a direct consequence of the quantum superposition principle: since any particle can be in any localized state $$|x\rangle$$ it can also be in a superposition of these states $$\sum_x \psi(x) |x\rangle$$, or $$\int \psi(x) |x\rangle dx$$ where $$\psi(x)$$, is the wave function of the particle." Isn't it clear?TD (talk) 20:29, 31 August 2023 (UTC)
 * No. It. Is. Not. Clear. You've managed to introduce yet another unexplained notation, while also adding a physically incorrect statement: a "position eigenstate" $$|x\rangle$$ cannot belong to the Hilbert space of physical states. XOR&#39;easter (talk) 20:37, 31 August 2023 (UTC)
 * A localized position state here means that the particle is between x and x +dx. The sum is then a discrete sum. The integral is for the limit, when dx approaches zero. TD (talk) 20:43, 31 August 2023 (UTC)
 * You're asking the reader to guess what you mean by a statement that at best has an indirect bearing on the topic of the article when combined with other unspecified ideas. XOR&#39;easter (talk) 20:50, 31 August 2023 (UTC)
 * The new version is explicit. TD (talk) 06:23, 1 September 2023 (UTC)

Consensus?
@Jähmefyysikko I didn't ignore the discussion with you, Johnjbarton and XOR'easter. I answered carefully to these arguments. Are you sure there is a consensus about the arguments which have been advanced against my animations and explanations, summarized at the top of the previous section? TD (talk) 06:38, 1 September 2023 (UTC)


 * Your "summary" is just your own misrepresentation of others' arguments. It is not neutral, and has been ignored.
 * Now, the section you introduced. Apart from you, there is no support for it. Even for you it is just an excuse to present the figures you made. Those who have commented it (me, Johnjbarton and XOR'easter) are strongly against it. This is consensus. You differ, but consensus is not unanimity. Jähmefyysikko (talk) 06:54, 1 September 2023 (UTC)
 * I usually copy/pasted the arguments. Where is the not neutral misinterpretation?


 * For example, the summary does not state who wrote which comments, and does not include my arguments. I assume its purpose is to make the arguments advanced against your figures look random and incoherent, and to emphasize the single positive comment which is quoted at more length. FYI: I will not continue the discussion about the summary, imo it is best ignored. Jähmefyysikko (talk) 07:29, 1 September 2023 (UTC)
 * I included one of your arguments. Now the summary states who wrote which comments.TD (talk) 09:12, 1 September 2023 (UTC)
 * It seems that your opinion fluctuates: "Wave-particle duality is a direct consequence of quantum superposition combined with the Born rule, if we reason about the state vectors of particles." TD 07:07, 30 August 2023 (UTC) "Great, so now we agree on how wave-particle duality is modeled in quantum mechanics." Jähmefyysikko 07:20, 30 August 2023 (UTC) TD (talk) 09:34, 1 September 2023 (UTC)
 * Does it? Again, for the context, see the initial comment that I replied to. The above comment I gave as I felt that we had dispelled some misunderstandings, even if the final statement was still vague, and definitely not something to be added into the article. Jähmefyysikko (talk) 11:29, 1 September 2023 (UTC)
 * There were good objections which led me to improve my animations and my explanations, and less good ones, which are listed at the top of the previous section. TD (talk) 11:48, 1 September 2023 (UTC)

At present, all my detractors, except three, are silent, since more than a week, may be because I conclusively answered to their objections. Three is not a consensus. TD (talk) 19:14, 3 September 2023 (UTC)


 * People are just tired of this endless discussion. But three may be enough, if the number of people actively engaged in the discussion is four. Jähmefyysikko (talk) 19:55, 3 September 2023 (UTC)
 * This is your interpretation. I recall you that @Tercer was in favor of the conservation of my animations. Three against two is not conclusive. TD (talk) 20:40, 3 September 2023 (UTC)
 * I got the impression that he may not want to participate in this discussion anymore and avoided discussing his arguments. Jähmefyysikko (talk) 20:49, 3 September 2023 (UTC)
 * I don't know. But you should let others give their opinions, if they want, before deleting a section without a good reason. TD (talk) 20:56, 3 September 2023 (UTC)
 * Indeed, I have no wish to participate in the discussion anymore. I thought I could help the participants see each other's perspective. I was mistaken. Please don't count my !vote as for or against anything. Tercer (talk) 21:01, 3 September 2023 (UTC)
 * Received. Three against one. TD (talk) 21:03, 3 September 2023 (UTC)

A different kind of animation for wave-particle duality?
This image from a paper showing electron double slit results shows intensity individual slits ("which way" form) and from the two slit form. The movie from the paper, showing the dots building up the interference, is already in the article. The article also has a simulation of the dots.

I think an effective and interesting animation might come out of a combination. Imagine the single slit probability distribution drawn then filled in by piles of pixels adding on top of each other eventually matching the distribution. Similarly the double slit result. Stack them on top of each other. The dramatic dips in the double slit result and lack of particle pile up there will illustrate just how different these two cases are. Johnjbarton (talk) 18:36, 29 August 2023 (UTC)

A fresh start proposal for Wave-particle duality.
I have a draft of an article on wave-particle duality here User:Johnjbarton/sandbox/duality.

The presentation is qualitative, empirical, and brief. The rough outline is All sections are brief and the scope is purposefully limited to focus on just duality.
 * What are particles and waves.
 * Observations that demonstrate duality.
 * Impact on quantum mechanics
 * History.

I welcome comments. I would prefer suggestions specific to my draft on User_talk:Johnjbarton/sandbox/duality.

I won't be editing again for a week so take your time and keep an open mind.

Johnjbarton (talk) 22:47, 29 August 2023 (UTC)


 * I went ahead and brought the draft over. XOR&#39;easter (talk) 20:47, 31 August 2023 (UTC)
 * To delete a whole article and replace it by another seems to me to be contrary to the rules, just as if you didn't want us to discuss about all these deletions. TD (talk) 21:12, 31 August 2023 (UTC)
 * It's entirely within the rules. XOR&#39;easter (talk) 21:17, 31 August 2023 (UTC)
 * It looks like a coup. The improvement of an article should be done step by step, so that we have time to discuss it, not in this way. TD (talk) 21:21, 31 August 2023 (UTC)
 * We have been discussing it for days. The article that you have reinstated is a train wreck. XOR&#39;easter (talk) 23:28, 31 August 2023 (UTC)
 * I agree that this article is not very good. But we should improve it step by step. "Be bold" is a good rule, but it seems to me that you misinterpret it. TD (talk) 23:31, 31 August 2023 (UTC)
 * For heaven's sake. You said that the new draft was an improvement. The only objection you raised was that the section you wrote was omitted. No one other than you has argued in favor of that section. I don't want to accuse you of trying to own the article, but I have to admit, you're making it hard for me. Surely the better course of action would be to use the version that you agreed was much better and then decide if a section with more mathematical details ought to be added? XOR&#39;easter (talk) 23:38, 31 August 2023 (UTC)
 * No. I wrote to Johnjbarton that we shall make a fusion between the old and the new version, that some parts of the old version might be conserved. This is not only about my contributions. There other parts which might be conserved. TD (talk) 23:44, 31 August 2023 (UTC)
 * You wrote, I agree with you that this article is not very good. User:Johnjbarton/sandbox/duality is much better. I don't understand how reverting to a worse version is a way to make progress. XOR&#39;easter (talk) 23:51, 31 August 2023 (UTC)
 * Because there are good things in the old version which might be conserved. Go step by step. It's a better way. There isn't any emergency. TD (talk) 23:54, 31 August 2023 (UTC)
 * I've read the page. It needs to be melted down to slag. XOR&#39;easter (talk) 23:56, 31 August 2023 (UTC)
 * This is your opinion. Let others discuss about it. TD (talk) 23:59, 31 August 2023 (UTC)
 * Going section-by-section: "The Feynman lectures on the double slit experiment" &mdash; unencyclopedic tone, meandering, essay-like, giving undue prominence to one particular textbook. "Wave-particle duality, the quantum superposition principle and the Born rule": disjointed quotations, unexplained notation, misrepresentation of at least one source, irrelevant pictures. "History": we have History of quantum mechanics. "Photoelectric effect": already covered, guess where, at Photoelectric effect. The same goes for de Broglie hypothesis and Uncertainty principle. "Wave–particle duality in a single experiment": covered in far more depth at double-slit experiment. "Importance" and "Alternative views" are a mess of vaguely-related ideas. Cleanup is years overdue. XOR&#39;easter (talk) 00:06, 1 September 2023 (UTC)
 * This is your opinion. Here is another: "very well written" (johnjbarton about the Feynman lectures on the double slit experiment, a section I wrote). TD (talk) 00:10, 1 September 2023 (UTC)
 * Feynman invited us &mdash; who is "us"? to think about double slit experiments &mdash; term not introduced yet, likely to be unfamiliar to the lay reader.  This is the mystery of wave-particle duality. POV writing, suitable for a pop-science story but not a Wikipedia article.  We're an encyclopedia, not part of the Feynman-industrial complex. Feynman didn't originate the double-slit experiment. He just made use of it. Crediting ideas to Feynman that date back to the Bohr–Einstein debates is not helpful to anyone. XOR&#39;easter (talk) 00:16, 1 September 2023 (UTC)
 * "Us" is everyone who wants to understand quantum mechanics. Feynman is a good reference. He taught to many a new way to make science (clear and generous, even the beginner can understand the most difficult questions about theoretical physics). This section might be deleted. But we can take some time to think about it. TD (talk) 00:24, 1 September 2023 (UTC)
 * That's not a reasonable use of "us" in an encyclopedia article. The Feynman Lectures on Physics have a reputation for being hard to learn from, better suited to inspiring people who already know physics than teaching physics for the first time (for example). Even if we refer to them, it's not Feynman's experiment. The current presentation is physics fanboy hero-worship, not education. XOR&#39;easter (talk) 00:35, 1 September 2023 (UTC)
 * Incidentally, the term "duality" doesn't appear in the Feynman Lectures, or in The Character of Physical Law, or in Quantum Mechanics and Path Integrals. Going by the passing mention in a footnote in QED: The Strange Theory of Light and Matter (p. 23), he seems to regard the term as having mainly historical interest. No reasonable model could explain this fact, so there was a period for a while in which you had to be clever: You had to know which experiment you were analyzing in order to tell if light was waves or particles. This state of confusion was called the "wave-particle duality" of light, and it was jokingly said by someone that light was waves on Mondays, Wednesdays, and Fridays; it was particles on Tuesdays, Thursdays, and Saturdays; and on Sundays, we think about it! "Duality" is just a heuristic dating to the time before the confusion was resolved, for practical purposes, by the maturation of quantum theory. If we were actually going by Feynman, this article wouldn't exist. "Wave-particle duality" would be a section in History of quantum mechanics, perhaps, but certainly not a stand-alone page. The other presentations I've been checking over the past few days tend to support this as well. Messiah's textbook, for example, doesn't give "wave-particle duality" a full chapter of its own. Instead, chapter 4 covers it as one of a cluster of related ideas: it's an example of the broader idea of complementarity, the uncertainty principle can be considered a consequence of it, it is "intimately connected with the atomism of action", and it is mathematically deduced using the statistical interpretation of wavefunctions. Pulling out "wave-particle duality" and trying to discuss it in isolation fails to respect the logic of the subject and how the references cover it. XOR&#39;easter (talk) 17:28, 2 September 2023 (UTC)
 * For another example: Griffiths, whose textbook is as standard as one can get, relegates "wave-particle duality" to a disparaging footnote in the Afterword. The so-called wave-particle duality, which Niels Bohr elevated to the status of a cosmic principle (complementarity), makes electrons sound like unpredictable adolescents, who sometimes behave like adults, and sometimes, for no particular reason, like children. I prefer to avoid such language. XOR&#39;easter (talk) 18:29, 2 September 2023 (UTC)
 * I wish to make my opinion clear for those who won't read between the lines. I agree with XOR'easter's assessment here: a clean start is needed, and it can be done in a "single" step. In reality, it is not a single sudden step, as the plan was announced beforehand some time ago, and a lot of work has alread been done by Johnjbarton and others. Jähmefyysikko (talk) 06:28, 1 September 2023 (UTC)
 * Apparently, XOR'easter changed his mind. Now he suggests that this article could be suppressed. TD (talk) 06:55, 1 September 2023 (UTC)
 * Again, for heaven's sake. Speculating that merging a topic into a larger article might be better organization is not suppressing anything. It's a natural question prompted by the fact that just about everything substantial in this article is already discussed elsewhere. I haven't changed my mind about the replacement draft being better than the current version and good enough that further work on it should happen in article space. I just wanted to put a marginally more radical possibility on the table. XOR&#39;easter (talk) 14:11, 1 September 2023 (UTC)
 * Sorry for the misunderstanding. TD (talk) 15:41, 1 September 2023 (UTC)
 * We've had enough discussion. There is a clear WP:CONCENSUS in favor of the new page without wave packets, even is it not WP:UNANIMOUS. Jähmefyysikko (talk) 21:19, 31 August 2023 (UTC)
 * This is not the point in discussion (and I don't know the consensus you're talking about). TD (talk) 21:23, 31 August 2023 (UTC)

I tried a fusion between the old and the new version.TD (talk) 16:45, 1 September 2023 (UTC)


 * I fixed a broken reference, but I still don't see what "Wave-particle duality, the quantum superposition principle and the Born rule" contributes. XOR&#39;easter (talk) 17:14, 2 September 2023 (UTC)
 * It's the heart of the explanation of wave-particle duality in modern quantum mechanics. TD (talk) 17:36, 2 September 2023 (UTC)
 * As written, the section is a pile of conflicting notations, poor phrasing, asides of unclear relevance, and an image gallery that is 2/3 irrelevant and 1/3 misplaced. It is actually less helpful than the blunt injunction "read a textbook and then this will make sense". XOR&#39;easter (talk) 17:47, 2 September 2023 (UTC)
 * The notations aren't conflicting. They are standard (all textbooks). From my point of view, the logic is very clear. What is not clear from your point of view? Why doesn't it make sense to you? TD (talk) 17:51, 2 September 2023 (UTC)
 * Bold or not bold? Dirac or not Dirac? Textbooks don't flop about from one sentence to the next! The entirety of the section is unclear. A person who doesn't already know the mathematics will learn nothing; a person who does already know it will be puzzled about why the idea of "state vector" is introduced twice, why the point that we can factor out a physically irrelevant phase is stressed, why it struggles so much with localization, why normalization is thrown in as a parenthetical late in the story, why a wavefunction is being said to represent "the evolution of" a state rather than a state itself... The passage beginning The existence of a wave function for a particle is a direct consequence of the quantum superposition principle reads like circular logic. It presumes that a ket $$|x\rangle$$, an element of an abstract Hilbert space, can be said to be "localized". This has no meaning unless a notion of localization has already been introduced, i.e., that $$|x\rangle$$ is somehow a function of a position variable that is only nonzero in a small region. In other words, it says that we can represent a ket by a function of $$x$$ because we can represent kets by functions of $$x$$; we have wavefunctions because we have wavefunctions. XOR&#39;easter (talk) 18:06, 2 September 2023 (UTC)
 * Furthermore, it doesn't actually explain where the "wave" idea comes from, or the "duality". It just talks about locating particles! Let's say that $$\psi_1(x)$$ and $$\psi_2(x)$$ are two square-integrable functions on $$\mathbb{R}$$ that are properly normalized so that $$P_1(x) = |\psi_1(x)|^2$$ and $$P_2 = |\psi_2(x)|^2$$ are probability density functions. Suppose that $$\psi_1(x)$$ and $$\psi_2(x)$$ have non-overlapping support. Then $$\psi_1^*(x) \psi_2(x) = 0$$, meaning that there is no interference term when we evaluate $$|a\psi_1(x) + b\psi_2(x)|^2$$. So, $$|a|^2 P_1(x) + |b|^2 P_2(x)$$ is a probability density function as long as we choose $$a$$ and $$b$$ appropriately. We're just classically mixing probability densities, entirely in accord with the corpuscular thinking that the particle is either over here or over there. Talking about duality requires two bases. Even if the only observable we measure is position, if we want anything wavelike we need a non-vanishing Hamiltonian, i.e., we need to introduce $$p$$ so we can have a $$p^2/2m$$ term generating some time evolution. Otherwise, our nicely localized functions will just sit there, nicely non-overlapping with each other, and we'd just be talking about probability densities with extra steps. XOR&#39;easter (talk) 19:02, 2 September 2023 (UTC)
 * A wave function at a given time represents the state of the physical system at that time. A wave function at all times represents the evolution of the state.
 * When we calculate probabilities we make use of normalized wave functions because the sum of all probabilities must be equal to one. TD (talk) 14:38, 3 September 2023 (UTC)
 * A physical state is represented by a ray in Hilbert space, because we don't always need normalized wave functions, if we don't calculate probabilities, and because a global phase factor hasn't any physical meaning, since it doesn't change the probabilities of measurements. TD (talk) 15:17, 3 September 2023 (UTC)
 * Yes, I know basic quantum mechanics. But what is the rationale for including those specific facts in this specific section? XOR&#39;easter (talk) 16:33, 3 September 2023 (UTC)
 * So that readers can understand better the definition of the quantum superposition principle, and the Born rule, if they already know a little about quantum mechanics and linear algebra. TD (talk) 16:37, 3 September 2023 (UTC)
 * So, the rationale is that if readers already know basic quantum mechanics, they can learn basic quantum mechanics? XOR&#39;easter (talk) 16:48, 3 September 2023 (UTC)
 * No, the reason is to give the crux of the matter about wave-particle duality in an article about wave-particle duality, as it should be in an encyclopedia, which is different from a textbook. TD (talk) 16:58, 3 September 2023 (UTC)

Superposition and the Born rule.
The current article contains a confusing and partial incorrect section on superposition and the Born rule. It also misses some important connections to duality, specifically that states interfere, not particles, and that the Born rule applies to measurements. If we must have this section, then we will need to explain that states are not particles, outline what a state means, explain and simplify the notation, explain how position measurements exclude wave measurements, and so on. Instead of a summary of the introductory chapter of QM texts, we will summarize half the book. Is that the direction we want to go? there are connections to duality but enough to justify this level of detail? Johnjbarton (talk) 14:19, 3 September 2023 (UTC)


 * A state is a state of a particle or of a system of particles. TD (talk) 14:27, 3 September 2023 (UTC)
 * @Thierry Dugnolle yes that is an example of the difficulties: circular and ambiguous definitions. Johnjbarton (talk) 14:36, 3 September 2023 (UTC)
 * "The state of a quantum particle is determined by an infinity of parameters: the values at various points in space of the wave function $$\psi($$r$$,t)$$ associated with it." (Cohen-Tannoudji, Diu and Laloë, Quantum Mechanics, chapter I, section B-2, p.14 in the french new edition) Hence the state of a quantum particle is determined by a wave function at a given time. I don't understand why you contest this very elementary point. TD (talk) 14:49, 3 September 2023 (UTC)
 * I find it hard to believe that you only have the french edition, yet you are able to quote the english edition verbatim. Jähmefyysikko (talk) 15:44, 3 September 2023 (UTC)
 * Google translation (slightly modified): TD (talk) 15:45, 3 September 2023 (UTC)
 * Remarkable. By coincidence it matches the english edition, p.21 I am not accusing you of anything, but this is a bit weird. Jähmefyysikko (talk) 16:16, 3 September 2023 (UTC)
 * This is not a coincidence. This automatic translator is trained to match translations already made by human beings. This translation is probably a part of the training of the automatic translator. TD (talk) 16:21, 3 September 2023 (UTC)
 * That must the be case here. Thanks for pointing this out. Jähmefyysikko (talk) 16:53, 3 September 2023 (UTC)
 * L'état d'une particule quantique est déterminé par une infinité de paramètres : les valeurs aux divers points de l'espace de la fonction d'onde qui lui est associée.
 * What is confusing or incorrect in this section? It is standard quantum mechanics, with the best references. Hence it deserves to remain. Your objections are very non-standard. TD (talk) 14:57, 3 September 2023 (UTC)
 * @Thierry Dugnolle exactly my point: the section can be boiled down to a sentence linking the QM article. Johnjbarton (talk) 15:36, 3 September 2023 (UTC)
 * This is the explanation of particle-wave duality in standard QM (reference at the beginning of the section). Why do you want to delete it from this article? TD (talk) 15:39, 3 September 2023 (UTC)
 * Because it's a confused jumble of quotes and asides that is only comprehensible to people who already know quantum mechanics and who are willing to puzzle through it to make a guess at what it is trying to explain. XOR&#39;easter (talk) 16:35, 3 September 2023 (UTC)
 * Even if one manages to piece together some meaning from these disconnected sentences, that meaning is not present in the cited texts, but requires WP:SYNTHESIS. Already on that basis, the section should be removed. Jähmefyysikko (talk) 17:57, 3 September 2023 (UTC)
 * Where is the original research here? This is only standard quantum mechanics. This not a synthesis of different arguments: Weinberg and Cohen-Tannoudji, and others are fully in agreement on the points explained in this section. TD (talk) 18:03, 3 September 2023 (UTC)
 * This section can be divided into two parts. The first part discusses some random pieces of quantum mechanics which are possibly related to wave-particle duality. This part cites Refs. [15-19], which do not discuss wave packets in the sections that are referred to. The second part consists of only the last paragraph, and suddenly invokes wave packets. The only reference here is Ref. [20] (Cohen-Tannoudji, complement G-I), which discusses the time-evolution of wave packets, but makes no mention of duality. By the juxtaposition of these two parts, there is an implication that wave packets have something to do with wave-particle duality. Such implication is not found in the references.
 * Such implied meanings are forbidden in Wikipedia by policy. Compare with examples from No_original_research/Examples. Jähmefyysikko (talk) 18:49, 3 September 2023 (UTC)
 * Gaussian wave-packets are simple examples of wave functions. They are presented here to illustrate the general principles with pedagogical animations: the wave function of a particle, the interference of a particle with itself (the crux of the matter about wave-particle duality) and the Born rule in action. TD (talk) 19:02, 3 September 2023 (UTC)
 * Ok, I agree on this. There is no SYNTH, because concepts are just randomly thrown around, and nothing is implied. The purpose of the section seems to be to introduce all of quantum mechanics, and say that this is what wave-particle duality is. Its not useful for anyone. Jähmefyysikko (talk) 19:52, 3 September 2023 (UTC)
 * The purpose of this section is to explain its first sentence, that is, to explain why the quantum superposition principle, combined with the others (the Born rule suffices) leads to wave-particle duality. The assertions are not disconnected. They give precision to the meaning of the quantum superposition principle, the wave function of a particle and the Born rule. That's all. I don't understand why you don't understand the logic, which from my point of view is very clear. Obviously, this is not an introduction to all of quantum mechanics. This can't done in a few words. TD (talk) 20:04, 3 September 2023 (UTC)
 * As it stands, your section flips between using $$|x\rangle$$ to mean a state of finite width and, when you write $$\psi(x) = \langle x | \psi\rangle$$, a generalized function of zero width and infinite norm. Section II.B.2 of Cohen-Tannoudji et al. points out that freely interchanging ket and bra vectors in the latter context doesn't work. One ends up either introducing "generalized kets" that don't actually belong to the Hilbert space, or working with states of finite width that only approximately form a basis. Weinberg has a fairly detailed (for a physics textbook) history, on pages 1–30 of the 2015 edition, but he doesn't use the term duality anywhere. XOR&#39;easter (talk) 18:59, 3 September 2023 (UTC)
 * To get the wave function, we need dx to tend to zero. The states of finite width are only a preliminary step to show that the existence of a wave function results from the quantum superposition principle. This is elementary quantum mechanics. That Weinberg and others don't make use of an expression doesn't prove that they disagree with the standard conclusions of this section : the motion of a particle is represented by a wave function (at all times) but the particle is always detected at a single place. TD (talk) 19:09, 3 September 2023 (UTC)
 * Instead of a summary of the introductory chapter of QM texts, we will summarize half the book. I share this concern. Is that the direction we want to go? I don't think it's a good direction to go in. XOR&#39;easter (talk) 16:46, 3 September 2023 (UTC)