Talk:Wave function/Archive 6

"wave function" vs "quantum state" ?, domain and codomain?
The lead sentence of the article say that a "wave function" represents "the" quantum state of a system. It would be better to say that a wave function represents a "type" or "kind" of "quantum state".

A "quantum state" is represented by a vector in a Hilbert space (according to the current Wikipedia article on "quantum state). Functions can certaily be vectors in a Hilbert space of functions, but my understanding is that not all "quantum states" are wave functions. The term wave function only applies when variables in the domain of the function include position, momentum, and time.  Other sorts of functions and vectors can be "quantum states", but they are not called wave functions.

To a mathematician, if someone says "I have a function", the question "What are its domain and codomain" should have a completely straightforward answer. It would be nice to have a concise and precise statement of the domain and codomain of a wave function (or an admission that "wave function" refers to a jumble of different sorts of functions!). I would infer from the current article that the domain is a finite dimensional vector space over the real numbers and that the codomain is a finite dimensional vector space over the complex numbers.

Those accustomed to thinking of "states" in terms of classical physics, tend to think of a "state" as the condition of a system at a particular instant of time. it would be useful to point out that the "quantum state" represented by the wave function contains information about the current and future condition of the system as well as information about its past. The information about the future is valid only so long as the system is not disturbed. The information about the past is only valid back to the time when the wave function was created by some disturbance to a previous wave function.

Tashiro (talk) 16:42, 11 January 2015 (UTC)


 * It's not unusual to "isolate", say, the spin part of a wave function and call it the spin wave function. In this case the domain is a finite set and the range is ℂ. Formally one can proceed using tensor products. The same applies for other degrees of freedom. Stuff like this could go into a footnote (of the "visible" kind). This is what "n-tuples of complex numbers" hide for good and for bad. YohanN7 (talk) 17:36, 11 January 2015 (UTC)


 * Sorry to Tashiro for running off-topic, I split my previous post into the next section.
 * I thought "quantum state" refers to a collection of variables that characterize the (quantum) system, which involves coordinates or momenta and quantum numbers.
 * In any case "wave function" refers to the actual quantity solved from the Schrödinger equation. Since the wavefunction is a function of position or momentum, and the set of all wavefunctions for a particlar system can be conveniently enumerated by quantum numbers, the wavefunction can be thought of as "describing" the quantum state.
 * For this article, "the quantum state" instead of "a quantum state" is used since "the" refers to the system in question. For now I don't know how to change the wording... M&and;Ŝc2ħεИτlk 23:13, 11 January 2015 (UTC)


 * Please be kind to mathematicians! If there is ambiguity or variety in the domain or codomain of "the" wave function then let this be pointed out in the article, even if it is not discussed in detail. (For example, if a wave function is a "function of position and momentum", does it become a function of time?  Is it a function of time via the fact that both position and momentum are functions of time?  Or is time an independent variable?)  From a mathematical point of view, its baffling that there is an important function, but no statement (perhaps no agreement?) about the domain and codomain of the function. Tashiro (talk) 17:27, 12 January 2015 (UTC)


 * As you can see below, we intend to clarify the domain and codomain throughout, but in general it is a complex-valued function of all the position coordinates of all the particles (or momentum components), and time, and the spin projection quantum number for each particle along some direction. To answer your question about time, it is an independent variable in the wavefunction along with the position coordinates (or momentum components), and the observables (position or momentum) do not depend on time. This is the Schrödinger picture, the main theme of the article. You can move the time dependence around, see Dynamical pictures (quantum mechanics). Hope this helps, M&and;Ŝc2ħεИτlk 17:55, 12 January 2015 (UTC)


 * Caveates: (To maybe remember when editing article)
 * There may be more than one discrete variable (isospin...).
 * Sometimes the discrete variables are ignored, and then the wave function is truly vector-valued as a function of configuration space (see below). This is mostly a notational matter, but potentially extremely confusing.
 * Equally potentially confusing is that $x, y, z$ shouldn't be taken as points in space ⊂ spacetime, at least not when several particles are involved. The domain can generally be thought of (now ignoring discrete variables) as tensor products (Cartesian product works too AFAIK) of an appropriate number $n$ of space, not spacetime, together with one copy of time. (This is configuration space.) Likewise for momentum, the latter wave function is just a $3n$-dimensional Fourier transform.
 * For some applications the discrete variables are ignored as an approximation. (E.g. the usual Schrödinger equation for spin $1/2$ particles.) Then the result is a scalar (complex valued).
 * Other domains are possible, the appropriate ones are related to the original ones by canonical transfromations.
 * YohanN7 (talk) 18:20, 12 January 2015 (UTC)


 * In other words: There is a Hilbert space of allowable states, and a wavefunction is a point in that space. But the space is different for different systems. For one spin-0 particle in 3D space, the space is (something like) $$L^2(\mathbb{R}^3)$$ (see Lp space, but actually it may be a different function space than L2, I don't remember). For three spin-0 particles in 2D space, the space is (something like) $$L^2(\mathbb{R}^2) \otimes L^2(\mathbb{R}^2) \otimes L^2(\mathbb{R}^2)$$ where $$\otimes$$ is tensor product. For two spin-7/2 particle in 3D space, it's $$(L^2(\mathbb{R}^3) \otimes \mathbb{C}^{7}) \otimes (L^2(\mathbb{R}^3) \otimes \mathbb{C}^{7})$$. The term "wavefunction" is more-or-less a synonym of "pure quantum state", except that you don't normally call it a "wavefunction" if the Hilbert space is finite-dimensional. --Steve (talk) 20:17, 12 January 2015 (UTC)


 * I see that the term wave function is widely and perhaps variously used. I think some effort should go into surveying the possible sources and reporting the best.Chjoaygame (talk) 11:53, 13 January 2015 (UTC)


 * In the commonest or default usage, think a wave function is a solution of the Schrödinger equation for its defining Hamiltonian, with domain the Cartesian product of quantum configuration space and a time interval, and range the complex numbers. This ensures that it represents a pure state of a quantum system, and other goodies. (By the way, the Wikipedia articles that one would expect to provide definitions of the latter are respectively in need of repair and appalling. It is very far from obvious that this article should refer to them until they have been put into good shape.) Quantum configuration space has various forms and manifestations. I think it best to define the wave function simply in general terms as having it as domain. The specification of the quantum configuration space, in its own right, deserves a paragraph, section or article, and I think should, for clarity, be well separated from the definition of the wave function.


 * Perhaps some more general dynamical specification than 'a solution of the Schrödinger equation' may be mentioned. The Schrödinger equation is one way of stating quantum dynamics, but there are others. A wave function must be defined with respect to specified dynamics.


 * The wave function can be considered more abstractly, as a mathematical entity in its own right, not as a function as such. Then it can be viewed as a point in an abstract vector space (a function space as it happens, but that isn't the present focus of interest right here). That abstract vector space happens to have the structure of a Hilbert space, give or take some more details about Hilbert spaces. In the sense that it is a point in a vector space, one can call it a vector. There are other mathematical representations of quantum states, ways which go more directly to the view that they are points in an abstract vector space, without going into their structure as wave functions. I think it may be useful to say that this article will not call those other modes of representation by the name 'wave function', though often enough one encounters the latter usage in a loose way.


 * Configuration space for an n-particle system has 3n kinematic degrees of freedom or dimensions, which may be various, but are usually specified by 3n real numbers that have meanings pretty much the same as for classical mechanics, respectively three per particle, and it also has for each particle a spin degree of freedom, which is in general a spinor, but may be presented in a less general way. A spinor is a kind of object not encountered in classical mechanics, and may be represented in various ways. Configuration space is subject to transformations, which I suppose deserve an article of their own.Chjoaygame (talk) 11:53, 13 January 2015 (UTC)


 * Don't think we need to go very deep to sort out what domain and codomain is for a function. The literature also doesn't tell the obvious in each case. YohanN7 (talk) 12:45, 13 January 2015 (UTC)


 * Classical physics admits two basic types of object, particle and wave/field. A particle system has a trajectory that is a path in configuration space, without a spin. A wave/field has a physically valued displacement at every point in ordinary space, and that for every instant of time. It is customary to say that waves diffract and particles don't, but that distinction is made obsolete by the discovery of quantal transfer of energy/momentum, even in the old quantum theory without quantum mechanics. Once one has quantal momentum transfer, particles diffract. Many standard texts prefer to hide this elementary fact, for sociological reasons. For physics, what makes a wave is that it has a physical displacement at every point in 'space' at each instant of time. What makes a particle is that it is all at just one point, leaving the rest of space empty, at each instant of time. No interpretation is needed to distinguish classical wave from particle. Diffraction has nothing to do with it.


 * A quantum mechanical system of particles is not like a classical wave, because it is not specified by a physical displacement at every point in ordinary space. It is specified by an abstract scarcely physical displacement at every point in configuration space at every instant of time. There is at face value no hint of particulate character. Neither classical particle nor classical wave. Talk of wave-particle duality is sociological. The quantum concept of 'wave' is pure interpretation. The "wave" is imagined as a blurring of the pattern of particle detections. At present the lead says "just like" a classical wave. That is inaccurate and misleading, and is in the article for sociological not physical reasons. Quantum mechanical contact with physical reality is by calculations that predict particle counts in suitably placed detectors.Chjoaygame (talk) 15:21, 13 January 2015 (UTC)


 * Chjoaygame -- Despite your quotation marks, the words "just like" are not in the lead. It says "The wave function behaves qualitatively like other waves", which I think is fair and helpful and accurate, in the sense that "behavior" includes things like refraction, diffraction, interference, etc. It doesn't say "The wave function behaves unlike particles in the old quantum theory", and it also doesn't say "The wave function is fundamentally like other waves". --Steve (talk) 19:54, 13 January 2015 (UTC)


 * Sbyrnes -- thank you for this correction. Yes, you are right, I should have checked, it's the article Matter wave that has the objectionable phrase "just like", not this one as I mistakenly wrote just above. Still I think the present "qualitatively like" is misleading, and that the reasons refraction, diffraction, interference, etc. are sociological not physical, for the reason I gave.Chjoaygame (talk) 21:24, 13 January 2015 (UTC)

domain of the wave function
Further confusion about the (or "a") domain and codomain of the wave function, is caused by the sentence "For a given system, the wave function is a complex-valued function of the systems degrees of freedom, continuous as well as discrete." This sentence has a link to the "Degrees of Freedom (mechanics)" Wikipedia article that says "degrees of freedom" is an integer that tells the number of state variables. By that definition "degrees of freedom" is not the set of state variables, it is the cardinality of that set. It would clearer to say that a wave function for a physical system is function of its state variables. However, that seems to be at odds with idea that the wave function represents the state. Do the state variables represent the state? If so, why is a function of the state variables needed to represent the state? Tashiro (talk) 18:33, 22 January 2015 (UTC)


 * Editor Tashiro, I think you are right to ask this question. Perhaps it is better that I do not expatiate on why I say that. But I may offer some pointers towards a partial answer.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)


 * The wave function is defined with respect to a definite physical context. As a purely mathematical entity without physical context, it is either undefined or meaningless, as you please.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)


 * Evidently here, the context is partly set by the words "For a given system". But those words are little more than a syntactic place filler. As far as I can see, we do not here have a useful definition for them. How is the system given? If the words are to contribute to setting the physical context, they must have physical content. Physical content necessarily involves empirical or experimental information. Physically, a system is given by describing, primarily in ordinary language, an experimental or observational activity. For example: "Place the telescope at the North Pole of the earth, and, at 00:00 UTC 31 Jan 2015, point it at the zenith. Take a photograph in the visible spectral range. Show us the photo, and tell us what you see in it." At least, something like that.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)


 * Here, is the "system" specified by a one-off time interval of observation for a specified apparatus? Its result is a count from a detector. Perhaps the count is zero in that time interval. Or is the system specified somehow else? Etc..Chjoaygame (talk) 23:08, 22 January 2015 (UTC)


 * A wave function is a solution of the Schrödinger equation for a specified quantum Hamiltonian. The Schrödinger equation now needs to be defined. Let us write it as f(ψ) = 0. What kind of object is f ? What kind of object is ψ ? Chjoaygame (talk) 23:08, 22 January 2015 (UTC)


 * In order to define these terms, we need to recognize that quantum configuration space (the domain of ψ) depends on the system. This will determine the kind of operator that is needed to express f. Sometimes f can take scalar values, sometimes it has to take spinor values. Etc. That will determine the range of ψ.

I am asking questions here.Chjoaygame (talk) 23:08, 22 January 2015 (UTC)


 * Tashiro and/or Chjoaygame, to answer your questions:
 * Interpretations of wave functions are open, there is no single one, the most common one in non-relativistic QM is probability amplitude.
 * "For a given system" is not a "filler" the wave function depends on the system.
 * "system" as in "system of particles which may or may not interact with each other in all space or a region of it". What more needs to be said? Any issues of measurements and apparatus are irrelevant in the definition of the wave function, measurement and collapse is a separate topic (and postulate).
 * For "f(ψ) = 0"... we have an article on the SE, it is an operator equation (the operator being f), no need to "define" or elaborate on what f is - this article is about ψ. Also, the SE is not always the best starting point for determining what ψ can be (tensor or spinor), you have to look at the relevant Lorentz group theory and relativistic wave equations.
 * For the zillionth time already... the wave function is a complex valued function of the variables characterizing the system dynamics. This is written to death in the article, several times over, and many more times on this talk page. In the context of mechanics, "degree's of freedom" are those variables which are characteristic of motion, in classical mechanics they are just position (or if you prefer - points in configuration space), in QM it is the same, along with other non-classical quantum numbers. Where is the difficulty or disagreement or controversy in this aspect of the definition of the wave function??
 * M&and;Ŝc2ħεИτlk 10:40, 27 January 2015 (UTC)


 * There are two different definitions of "degrees of freedom" in the current Wikipedia. The link used the current Wave Function article is the link to https://en.wikipedia.org/wiki/Degrees_of_freedom_%28mechanics%29, which defines "degrees of freedom" to be the cardinality of the set of state variables.  The article https://en.wikipedia.org/wiki/Degrees_of_freedom_%28physics_and_chemistry%29 defines "degrees of freedom" to be the state variable themselves.  I suggest changing the link used in this article to the second of those links.


 * If there is no general agreement among physicists on the definition of the wave function as a mathematical function, then the proper thing to do is to state this in the article and when examples of various of wave functions are given, state the domain and codmain for each particular example.


 * Perhaps the project of finding a unifying mathematical defintion of "Wave function" is an area of research rather than established convention. Tashiro (talk) 17:46, 26 January 2015 (UTC)


 * No, it is, on the contrary, so trivial that it is not treated in depth in the literature. You are supposed to be able to figure out yourself if you attempt a QM book. Feel free to change the wikilink. YohanN7 (talk) 18:10, 26 January 2015 (UTC)


 * As to "degrees of freedom". I find that the present use of expression a bit awkward, even though it may perhaps have a standard meaning stated in Wikipedia for physicists and chemists.


 * Reading the article Degrees of freedom (physics and chemistry), I think it would be a bad mistake to link this present article to it. That article is poorly written so as to leave the reader thoroughly confused between phase space and configuration space. The reader would, I think likely, come from that article with the idea that the degrees of freedom are the variables that define phase space, just the wrong idea for the present article.


 * The domain of the wave function for spinless particles corresponds with the configuration space of classical mechanics, or with a canonical transformation of that space; I shall for brevity here speak of 'the quantum configuration space'. (The classical phase space has twice as many variables per particle.)


 * The word 'state' is violently and brutally abused in quantum mechanics, if ordinary language and classical mechanics are regarded as the norms for usage. A quantum phenomenon requires for its specification both its initial and its final conditions, specified by their respective possibly different quantum configuration spaces. For a single particle that means six variables, three initial and three final. A single classical particle is considered to have all its characteristics defined by its state at every instant. That is six variables, three for position and three for momentum. Or a canonical transformation of the foregoing. That is phase space, twice as many variables as configuration space.


 * In quantum physics, no physical apparatus can prepare a particle in a supposedly specified state defined by a point in phase space; to speak or think of such is forbidden by the Heisenberg indeterminacy principle. The most exhaustively specified 'quantum state' that can be physically prepared is determined by a point in quantum configuration space. A particle in such a specified 'quantum state', if left unaffected by any causally efficacious influence, is either in a stable condition or else in a metastable condition. If in a stable condition, it will certainly remain in that 'quantum state' for ever. If in a metastable condition, it will jump to another state with Poisson distributed delay. The state to which it will jump is determined only up to a probability. This is the natural consequence of the restricted definition provided by quantum configuration space, three variables only. Full non-probabilistic determination belongs only to the six-variable specification of classical phase space. There the initial condition by itself fully specifies the state of the particle in the ordinary language sense of the word. So also does the final condition. Not for quantum state: there both initial and final quantum configurations are needed.


 * It is an unnecessary complication to consider spin right here on this page right now. Nevertheless the article itself must consider it, of course.


 * In short, this present article should make it clear that the quantum state of a particle is determined by the values of its quantum configuration space coordinates. This article would invite confusion if it relied on another Wikipedia article for this.


 * I think the definition of the wave function is not merely trivial. I think it needs to be done properly in this article, based on a good survey of reliable sources. I think it is not a matter of research. It is a matter just of diligent editorial work.Chjoaygame (talk) 21:59, 26 January 2015 (UTC)


 * I think the mathematical aspects of wave functions can be stated in a simple manner. Assuming a reader takes the mathematical definition of a function seriously, "the" wave function is not a specific function and the term "wave function" doesn't even refer to a member of one particular family of functions. The codomain of a wave function is the compex numbers. The domain is whatever mathematical structures are chosen to represent the independent variables involved.


 * I suggest the sentences:
 * The codomain of a wave function is the complex numbers and the domain is whatever type of mathematical structure is needed to represent variables that describe the physical system being modeled. Sets of wave functions that have the same domain form a mathematical structure known as a Hilbert Space. For this reason a wave function is often defined as an element of a Hilbert Space.  A Hilbert Space is a type of abstract vector space, so a wave function is often called a "vector in a Hilbert Space of functions".


 * Further mathematical questions:
 * Must the variables used to describe the physical system include spatial position and time in order for a function to be called a wave function?
 * Must a function satsify a wave equation (https://en.wikipedia.org/wiki/Wave#Wave_equation) in order to be called a wave function? (This is a frequently asked question on the web, so it would be helpful to provide the answer.)


 * The sentence in the article:


 * For a given system, once a representation, or basis, corresponding to a maximal set of commuting observables is chosen, the wave function is a complex-valued function of the systems degrees of freedom corresponding to the chosen basis, continuous as well as discrete.


 * only makes sense to a specialist who wouldn't need to consult the Wave Function article in the first place. The technical terms involved need to be explained. Is there less technical language that conveys the physical concepts? Tashiro (talk) 21:56, 29 January 2015 (UTC)


 * I apologize if I sounded rude above, but have been finding it hard how the wave function is not clearly or precisely defined in this article.
 * Your wording
 * "The codomain of a wave function is the complex numbers and the domain is whatever type of mathematical structure is needed to represent variables that describe the physical system being modeled."
 * seems fine is too vague, since "whatever type of mathematical structure is needed" does not even say what the domain of the wave function (for the system) is, but to then then to say the rest
 * "Sets of wave functions that have the same domain form a mathematical structure known as a Hilbert Space." etc.
 * is not much of an improvement, since the reasons that wave functions are elements of Hilbert spaces are not presented. Your starting sentence could be placed before this (in the second paragraph of the current article),
 * "Wave functions can be added together and multiplied by complex numbers to form new wave functions, and hence are elements of a vector space. This vector space is endowed with an inner product such that it is a complete metric topological space with respect to the metric induced by the inner product. In this way the set of wave functions for a system form a function space that is a Hilbert space. "
 * This seems reasonably readable and accurate for anyone.
 * To answer your questions (which are either in the lead or article at least somewhere):
 * No, the wave function does not need to be a function of position, the momenta of the particles could be used instead of positions. Whether to place the time dependence in the wave function or not depends on which dynamical picture of QM is used.
 * Yes, the wave equation is the Schrödinger equation, or other relativistic wave equations.
 * Finally, about the "commuting observables" etc. in the lead, I didn't write that but agree it could be written clearer using fewer unfamiliar QM concepts, but we should including something along these lines since it is important. Will have a look tomorrow.
 * Cheers, M&and;Ŝc2ħεИτlk 22:47, 29 January 2015 (UTC)
 * In the above, replace the stroked-out segments by adjacent underlined segments. M&and;Ŝc2ħεИτlk 11:47, 30 January 2015 (UTC)
 * (EC - Have not read what Maschen replied. Here's my reply for comparison.)
 * No, we are certainly not going to replace ... corresponding to a maximal set of commuting observables... with whatever is needed. I thought you were the one wanting to know what the domain is. If you don't understand what either of "set", "maximal" or "commuting observable" is, then I suggest you look it up. At least "observable" is blue linked. Better yet, read the article. All of your questions are answered in it. The lead is supposed to summarize the article, not be the article.
 * For your question,
 * Must the variables used to describe the physical system include spatial position and time in order for a function to be called a wave function?,
 * the answer (no) is given clearly in the article. If you are unhappy about calling a function not depending on time a wave function, then don't call it a wave function. The other question is likewise a non-issue. In the article, there is an example of a function in the space of consideration that clearly does not satisfy any wave equation. Whether you call it a wave function or not is up to you. YohanN7 (talk) 23:15, 29 January 2015 (UTC)

Dirty work
Having full references inline sucks. It is highly uneconomical and strongly discourages (at least me) people to put in new citations. We probably need few new sources, but more inline citations. I'll do the dirty work of moving all references to a reference section and introduce Harvard citations (I think they are called so). This makes it so much simpler to cite and avoids senseless duplication. No time table. You are allowed, even encouraged, to help. YohanN7 (talk) 21:43, 17 January 2015 (UTC)


 * Not sure what triggered this? Not even clear what you mean by your above remarks? I just put the references here above where I did to avoid format complications on the talk page. Surely you have more urgent things to do?Chjoaygame (talk) 21:51, 17 January 2015 (UTC)


 * On looking at what you done so far, on bended knee I beg you to stop, and undo it.


 * Though Editor Maschen feels strongly for them, I think footnotes are very undesirable in Wikipedia. I am not a Wikilawyer, but I have a feeling I am not alone in this. Footnotes open the way for all kinds of slipshod editing and admission of material of dubious relevance and other abuses. I think the present (before your latest) arrangement is good. The template method has advantages and disadvantages. Horses for courses.Chjoaygame (talk) 22:08, 17 January 2015 (UTC)
 * I'm neutral if anything on footnotes, and don't feel strongly on them. A few don't cause any harm, but then the article should the details. My rule of thumb would be: only digressions which disrupt the main flow of text should be in the footnotes. Otherwise it should be in the main text. M&and;Ŝc2ħεИτlk 10:11, 18 January 2015 (UTC)
 * What on earth are you talking about? The visual appearance is identical as far as relevant information goes. That is what and in where . The reader eager to see details about the publication can click on the appropriate link and get taken to the reference section. It is far superior. (Who the hell wants to see isbn numbers and doi's in a popup?) It is just easier to maintain and use for further inline literature references. YohanN7 (talk) 22:33, 17 January 2015 (UTC)


 * Ok, ok. Now I see what you intend. Panic no longer.Chjoaygame (talk) 22:41, 17 January 2015 (UTC)


 * At first your intention was not clear to me. Now looking at what you are doing. Yes, I very much agree with your plan for separate sections for the specific citations (page x, pages y–z, etc.), and for the full bibliographic details for the source books or articles for the citations. I have done it on several articles. It makes it much easier to give many different specific citations from one book or article.


 * I have to admit I am only lukewarm for the citation template method, as distinct from handwork. The templates can be too rigid for some complicated citations or unusual sources.


 * I do not know of standards in this. Briefly glancing around, I see diverse ways of doing things.


 * But what very much concerns me is the risk of opening of the floodgates for 'notes'. That is why I posted the heading 'citations' instead of 'notes'. I am strongly of the view that if something is worth a place, it is worth a place in the body of the text. If it doesn't fit right there, it usually means that the article structure needs fixing. A footnote is in my view a bad way to deal with such problems. I seem to recall that I learnt this from some good Wikipedia source, but I don't recall the detail, and I am not a Wikilawyer. That is why I was upset at the header 'notes'. I still think it is an invitation to abuse.Chjoaygame (talk) 23:20, 17 January 2015 (UTC)


 * We already have a dedicated section for what you dislike, namely the remarks section. These are footnotes in the traditional sense. Footnotes are standard in good articles when appropriate, as here. The biggest structural problem with the article at present is a one mile long figure caption in the lead. This is not standard. YohanN7 (talk) 23:36, 17 January 2015 (UTC)


 * Yes, this article does have footnotes, which I think is a bad idea. But it seemed at first to me that your present plan (that I now see as good) to separate citations and bibliography looked as if you were introducing a second footnote scheme. As for what is standard, I don't know much about that. I am inclined to do what seems best for the particular article, and let the standard follow that good lead. I am sorry you find my new caption offensive.Chjoaygame (talk) 02:34, 18 January 2015 (UTC)

This is done now. I think I didn't lose any information, but can't be sure. Those who have put in full (inline) references to begin with might want to check that all is still there. But I did add a lot of missing stuff, like ISBN numbers and publication years.

One advantage of this is that it is apparent who is in the list of refs and who isn't. Einstein is there (perhaps too much, 4 entries) but where is Dirac? YohanN7 (talk) 17:23, 30 January 2015 (UTC) And where is Schrödinger? YohanN7 (talk) 17:40, 30 January 2015 (UTC)

Max Born on Dirac
According to Max Born


 * "It may be mentioned in conclusion that the fundamental idea behind Heisenberg's work was worked out by Dirac (1925) in a very original way, and that in 1964 he put views to the effect that although Heisenberg's and Schrödinger's approaches are perfectly equivalent in ordinary (non-relativistic) quantum mechanics, this is not the case in quantum field theory. Here Heisenberg's method turns out to be more fundamental."

Likely Born is referring to 

Perhaps this may help in planning the conceptual structure of this or other articles.Chjoaygame (talk) 11:46, 25 January 2015 (UTC)


 * Is this about the Schrödinger picture and the Heisenberg pictureYohanN7 (talk) 12:58, 25 January 2015 (UTC)


 * I ought to have given some context. It is about Schrödinger's wave mechanics and Heisenberg's matrix mechanics.Chjoaygame (talk) 13:07, 25 January 2015 (UTC)


 * Yes. It would help tremendously if you supplied some context to your mountain of quotes. Taken in isolation, they are useless, especially the older ones from times when terminology (and knowledge and interpretation) was different from today.
 * Maybe something for the history section.
 * The Schrödinger picture and the Heisenberg picture must definitely go into the article. In one case the wave function is time dependent, in the other it is not. In QFT the Heisenberg equation in the Heisenberg picture is taken as the master equation. The resulting equation for the field operator Schrödinger picture is a derived quantity, often, but not always, the same equation as obtained using Schrödinger picture RQM for the wave functions. (This can be taken as further evidence of RQM being an both incorrect and incomplete reconciliation of QM and SR.) YohanN7 (talk) 13:29, 25 January 2015 (UTC)


 * The Schrödinger, Heisenberg, and Dirac (interaction) pictures are mentioned in the article wave function.
 * Sorry once again for negligence, there is a huge amount of talk to read through, and still more to rewrite, I'll catch up and try some over the next few days... M&and;Ŝc2ħεИτlk 13:59, 25 January 2015 (UTC)

Conceptual structure
This clears up slight mystery for me. It has seemed in the past to me that Heisenberg's matrix mechanics just disappeared into the ether of history. And it has seemed a little puzzling that the Heisenberg picture somehow appeared out of nowhere. Now I see what happened. The Heisenberg matrix formalism went into the Dirac theory, where it appeared as the Heisenberg picture. It is common enough to read that Heisenberg's matrix mechanics and Schrödinger's wave mechanics were equivalent.. But they are not quite so, as pointed out by N.R. Hanson. and later by Dirac, and then by Born, as noted above.

The "quantum mechanics all the way" noted by User:YohanN7 is Heisenberg's scheme, first in the guise of his matrix mechanics, then in the guise of the Heisenberg picture in the Dirac 〈bra|ket〉 formulation. It seems that Schrödinger's wave mechanics did not make it all the way.Chjoaygame (talk) 19:36, 25 January 2015 (UTC)Chjoaygame (talk) 20:14, 25 January 2015 (UTC)


 * The difference between the Heisenberg and Schrödinger pictures is, in the context, just a mathematical triviality (with practical computational consequences of course). If there is any real difference between matrix mechanics and Schrödinger's wave mechanics, I couldn't tell since I don't know matrix mechanics. I have acquired Dirac's 1925 paper though. Even his early papers feel more modern in touch and style than most papers from much later times (meaning readable). YohanN7 (talk) 20:39, 25 January 2015 (UTC)


 * The referenced paper abstract questions the validity of the proof that MM and WM are equivalent. It says too that Born proved them equivalent as physical theories. YohanN7 (talk) 20:48, 25 January 2015 (UTC)


 * Interesting.


 * Matrix mechanics is quantum mechanics pretty much as Heisenberg invented it in 1925. But Heisenberg was no pure mathematician and had never heard of matrices. One could say he invented them all over again by himself for the purpose. Born and Jordan were in close touch with him and were familiar with the matrix as a mathematical object, and recognized it in his work. Then they wrote joint papers. Some practical calculations were done with matrix mechanics in its original form, but very soon Schrödinger's wave mechanics appeared and was much more congenial to work with. It was generally accepted that Schrödinger had satisfactorily /demonstrated-proven-established/ that his wave mechanics and Heisenberg's matrix mechanics were equivalent. In a sense then the Heisenberg matrix version disappeared from sight. But it was still there as the Heisenberg picture in the Dirac 〈bra|ket〉 formulation. (By 'Dirac 〈bra|ket〉 formulation' I do not mean 'Dirac picture'.)


 * In the Dirac 〈bra|ket〉 formulation, the equivalence of the Schrödinger and Heisenberg pictures is as you say pretty much a mathematical triviality. But by 1964, Dirac had thought things over some more, and wrote what Born interpreted as a statement that the Heisenberg version is "more fundamental" (in my quick read of it I didn't find those exact words in Dirac 1964). The mathematically trivial proof that the Schrödinger and Heisenberg pictures are equivalent works for the non-relativistic case, but there is as you rightly say no valid relativistic version of Schrödinger's wave mechanics, so no question of relativistic equivalence arises. It seems Hanson saw signs of this but it wasn't made clear till Dirac 1964. It was the 1964 Dirac lectures that made Born become aware of it and see that his former view of equivalence worked only for the non-relativistic case. Enough for now.Chjoaygame (talk) 00:11, 26 January 2015 (UTC)


 * This is, you say, in Dirac's QM book? Do you know in which chapter? (It's against my principles, but I know a (very probably illegal) copy of it floating around on the net, and I'll sneak a peek nonetheless.) YohanN7 (talk) 00:30, 26 January 2015 (UTC)


 * After 'edit conflict' message.
 * Looking quickly at the fourth edition 1958 of Dirac's Principles of Quantum Mechanics, I see him on pages 111 and 112 apparently inventing the terms 'Schrödinger picture' and 'Heisenberg picture'. The 1964 Lectures on Quantum Mechanics is what Born was referring to.Chjoaygame (talk) 00:51, 26 January 2015 (UTC)
 * It seems you are saying above that there is a valid Schrödinger picture also for the valid relativistic case, which is the quantum theory of fields. The Schrödinger picture in the quantum theory of fields cannot be validly derived from a Schrödinger-like "relativistic quantum theory", because there is none. It must be derived from the Heisenberg picture, which is the primary and only reliable way to construct new formulas in the quantum theory of fields. The Schrödinger picture will be different for different inertial reference frames.Chjoaygame (talk) 15:01, 26 January 2015 (UTC)Chjoaygame (talk) 15:08, 26 January 2015 (UTC)
 * You read me wrong and get more than one other thing wrong. It seems clear that you do not know what the Heisenberg picture and the Schrödinger picture (or for that matter the interaction picture) are. These are different concepts from matrix mechanics and wave mechanics respectively. The Schrödinger picture is not the same as the Schrödinger equation, or any other wave equation, relativistic or not. YohanN7 (talk) 16:22, 26 January 2015 (UTC)


 * Thank you for your kind and helpful advice and helpful bold font.


 * I think I do know what the Schrödinger, Heisenberg, and interaction, pictures are in non-relativistic quantum mechanics. That is set out in all standard elementary texts that I have read. In a nutshell, the time dependence is in the wave function and its dual in the Schrödinger picture, but in the observable operator in the Heisenberg picture. The difference is easily expressed in the Dirac 〈bra|ket〉 formulation.


 * I do not think the Schrödinger picture is "the same as the Schrödinger equation". Nevertheless, the Schrödinger equation of standard elementary texts, in terms of continuous functions on configuration space, is very often seen as working in the Schrödinger picture.


 * Wave mechanics is the form of quantum mechanics invented by Schrödinger. The observables are represented as operators that act on continuous functions of points in configuration space, including differential operators.


 * Matrix mechanics is a form of quantum mechanics as it was originally invented by Heisenberg, and quickly recognized as using matrices and developed by Born and Jordan. The observables are represented as matrices. It is hardly ever seen in standard elementary texts (none that I can recall right now), but is extensively treated in Razavy's 2011 monograph cited above. I think it works in what, since Dirac's Principles, is called the Heisenberg picture; that is what I find newly revealed to me in the present conversation. It uses the Heisenberg equation of motion. I am saying it slipped out of sight when the Schrödinger formulation arrived, but appeared again in a different notation in the Dirac 〈bra|ket〉 formulation, newly called the Heisenberg picture.


 * What I don't know is the exact status of the Schrödinger picture in the quantum theory of fields. I am fairly sure that it is not the most fundamental representation in that theory. You and Born agree about that. You say that it may nevertheless be derived from the Heisenberg picture in that theory. I see that it often appears in Weinberg's The Quantum Theory of Fields. I would like to know if you think that such a derivation is always valid and available in that theory. (In non-relativistic theory, it is agreed by all that the Schrödinger picture is always valid and available.)Chjoaygame (talk) 20:29, 26 January 2015 (UTC)


 * It may help to not use the term "derived" or "more fundamental" in the context of the "dynamical pictures", you can get from one picture to another by an appropriate unitary transformation. The Schrödinger picture isn't used in QFT since it is inconvenient to use wave functions as presented in the article, especially for applications to N-body theory (when QFT is in a way superior to QM) - would you solve the SE for a wave function containing N particles where N is a multiple of Avagadro's number, while the observables are time-independent? No one does. M&and;Ŝc2ħεИτlk 11:06, 27 January 2015 (UTC)


 * Just checking: you are saying in effect that Born's reading of Dirac is unhelpful?Chjoaygame (talk) 20:19, 30 January 2015 (UTC)


 * I don't know what "Born's reading of Dirac" is, but it doesn't matter anyway. Did they solve the Schrödinger equation for so many particles? What else would motivate field theory? M&and;Ŝc2ħεИτlk 16:51, 7 February 2015 (UTC)


 * The Heisenberg picture and the Schrödinger picture (and for that matter the interaction picture) are unitarily equivalent – including in QFT. In QFT, an important part of the job is to pick the picture in which it is easiest to solve the problem at hand. The unitary equivalence ensures that a solution obtained in any of these pictures ensures the validity in any picture. See any introductory QFT text. The Heisenberg and interaction pictures are undoubtedly the most useful in QFT due to the way in which it emerges from canonical quantization of fields, aka second quantization of wave functions in case of a non-classical field. Field theoretical Poisson brackets (linked article doesn't cover these) are replaced by commutators of (time-dependent) operators. Their equation of motion is the Heisenberg equation. This does not invalidate the Schrödinger picture (or the interaction picture). YohanN7 (talk) 21:30, 26 January 2015 (UTC)


 * Also, please see the article Heisenberg picture. The lead and the first section echoes exactly what I just wrote. YohanN7 (talk) 21:37, 26 January 2015 (UTC)


 * Or perhaps an echo follows rather than precedes in time?Chjoaygame (talk) 20:19, 30 January 2015 (UTC)


 * Actually, no I don't rely on Wikipedia for facts. Was surprised myself how well the article described it and how it manages to get in all points already in the lead and the first section. (But really, these are all trivialities, they should naturally be in the article forming the backbone. Had they NOT been there, then you should be surprised.) This is why I wrote you this pointer. Besides, every point has already been explained to you, piecemeal, by me, in this thread, so this is nothing new. YohanN7 (talk) 21:15, 30 January 2015 (UTC)


 * It wasn't that you might or might not rely on this or that. It was that 'echo' was not the right word for the context.Chjoaygame (talk) 22:37, 30 January 2015 (UTC)


 * Don't lie. You were insinuating things. Besides, you don't chose the right word for me to use. YohanN7 (talk) 01:46, 31 January 2015 (UTC)


 * You missed the point that I was suggesting. I was not insinuating that you had taken the prior text as your source; it was obvious you had not. I was, however, suggesting that you were inappropriately and unconsciously driven when you used a word that unintentionally implied that the prior text had followed you. The word echo was wrong for context. So, you see, you are mistaken to tell me I was lying; in so doing, you made another unintentional mistake of the same unconsciously driven kind. I don't choose the words you use, but I am free to point out when they are wrong for their context.Chjoaygame (talk) 03:24, 31 January 2015 (UTC)


 * Here is one more point that I honestly think is quite useful. When you read the phrase "quantum mechanics", be very open-minded about what the author is referring to. It is most often not limited to the Schrödinger equation of relativistic versions of it. (At least it should not be.) It then refers to the foundational framework, which is valid in general whenever a modern theory incorporates it. This is the case with the pictures of above, but it is not the case, for example, with the probability interpretation of the wave function. We have discussed this at length before. YohanN7 (talk) 22:14, 26 January 2015 (UTC)


 * Am I reading your meaning aright if I am prompted to ask please would you very kindly give a link here to where a non-probabilistic interpretation of the wave function is discussed?Chjoaygame (talk) 20:19, 30 January 2015 (UTC)


 * Sure. Walter Greiner, Relativistic Quantum Mechanics, page 1, item 3. One more: Landau & Lifshitz, Quantum Electrodynamics pp 1-4. (L&L kills it off pretty good.)
 * The probability interpretation simply doesn't work well with relativity. You can have it only when applying relativistic equations (e.g. Dirac) to essentially non-relativistic problems (e,g hydrogen atom). I do not intend to explain to you why. YohanN7 (talk) 21:15, 30 January 2015 (UTC)


 * With much respect for your erudition in the mathematical direction, I have given some thought to your proposal that "Landau & Lifshitz, Quantum Electrodynamics pp 1-4. (L&L kills it off pretty good.)" It is the Born rule that is said to be killed off. I think the scattering matrix is a formulation that keeps account of data found in experimental set-ups, but I think the fundamental physics of observation is still given by the Born rule. That rule tells how to relate an actual physical particle detection to some mathematical wave functions. The scattering matrix keeps account of such detections. The scattering matrix is in the phyicist's notebook or computer. The particle detections are in the laboratory apparatus. Also I have looked at your cited place in Greiner's Relativistic Quantum Mechanics. I don't think this kills the Born rule. It means one must reconsider, but not discard the Born rule. More precisely, that item is about probability density, not the Born rule as such, though closely related.
 * And my question is 'what alternative is there to the Born rule?' I think neither of those cited items offers an alternative.Chjoaygame (talk) 17:35, 9 January 2016 (UTC)

Probability interpretation
Thank you for the pointers. I didn't ask you to explain why.Chjoaygame (talk) 22:37, 30 January 2015 (UTC)Chjoaygame (talk) 04:37, 31 January 2015 (UTC)

Following your lead on the probability interpretation.

First a side comment. I think it fair to cite Born, not the Copenhagen interpretation, as the primary source of the probability interpretation of the wave function.

Now my main follow up. Heisenberg's matrix mechanics and the S-matrix have respectively fairly direct interpretations as descriptions of experiment. This source/destination line of thinking is reflected in theoretical work such as in Ludwig, G. (1954/1983), Foundations of Quantum Mechanics $I$, translated by C.A. Hein, Springer, New York. Like Bohr, Ludwig there thinks of quantum mechanics as describing things in terms of a preparative and a registrative device. That lends itself to an idea of registration by particle counting, and thus to counting frequency idea of probability. In contrast, it is not apparent to me how such pictures as the currently displayed coloured diagram of the hydrogen atom's electron orbitals fit that source/destination paradigm. It seems to belong to a different paradigm? I have in mind your L&L point that it hardly makes sense to try to pinpoint the locality of an electron bound in an orbital. In that sense, the probability interpretation there seems less obvious? But what else is there?Chjoaygame (talk) 05:05, 31 January 2015 (UTC)

variable number of particles
Planck 1900 didn't know about light travelling as particles, but he didn't need relativity to work out that light is created and destroyed when his heuristic virtual oscillators lose and gain quanta of energy. A theory that counts light as particulate inevitably has variable particle number, without needing relativity to account for it. When an electron passes from an orbital of high energy to one of low energy, a particle of light is created; and the reverse. Conservation of energy demands this. Of course relativity provides a fuller explanation and account.Chjoaygame (talk) 19:58, 30 January 2015 (UTC)


 * You have been asked to stay on topic and keep it short. You got one out of two things right this time, a considerable improvement. YohanN7 (talk) 21:22, 30 January 2015 (UTC)


 * From this I get the message that I should be reproved. But beyond that, your meaning here is not apparent to me; perhaps you may clarify?


 * Perhaps I should give a context for my comment here. It was your recent edit that said "Relativity makes it inevitable that the number of particles in a system is not constant."Chjoaygame (talk) 04:26, 31 January 2015 (UTC)

Physical interpretation of "basis, corresponding to a maximal set of commuting observables"
It would be useful to give a physical interpretation of the above phrase. The link to the current Wikipedia article on Observable doesn't accomplish that goal. Some questions about the physical interpretation are:

1) Is an observable a set of possible values for a physical measurement that produces a real number? If so, is there any dimensional constraint on the values in this set?  For example, can some of the values be in meters and others in joules? Or is it understood that an observable is the possible set of outcomes that are all for "the same type" of physical measurement?

2) What makes a set of observables maximal? For example, if I can measure the weight of a box and its volume, I could invent all sorts of hybrid measurements derived from that data like (volume times weight), (weight divided by volume), (sqare root of weigh times cube root of volume) etc.  Would a set of possible values of such a hybrid quantity also be observable?  Do we define the maximum set of observables by listing a finte set of observables and then including any set of measurements that can be defined as a mathematical function of those in the finite set?

3) Is it important to say a basis "corresponding" to the maximal set - i.e. does this mean something different than "a basis for the maximal set" in the sense of "a basis for a vector space"

4) Is it understood that the values in the observable represent definite outcomes? Or can they represent probabilities? One concept of an experiment is to measure something once and get a single real number. A more complicated idea of an experiment is to make many measurements of the same property on a population and determine the probability that some given event occurs. Tashiro (talk) 08:36, 1 February 2015 (UTC)


 * Overall, good points. That the article observable isn't in good shape does not necessarily mean we should fill out all detail here. Effort should go in there, but some could go in here.


 * 1) Roughly, an observable is a set of possible outcomes. Technically, an observable is a Hermitian operator on the Hilbert space. They have real eigenvalues. These eigenvalues are the possible outcomes. There is no dimensional constraint. If you measure the energy, then the observable is the Hamiltonian operator whose eigenvalues are the possible energy levels of the system, and the possible outcomes are numbers to be interpreted as joules (or whatever system of units you have). If you measure position, then it is the position operator, and the outcome is interpreted in meters. If it is the spin z-projection, then the outcome is a half-integer in units of the Planck constant (angular momentum).


 * 2.) A set of observables is maximal if you can add no more observables to the list that are linearly independent from (operators are elements of a vector space too) and commutes with the ones already in the list.


 * Take the linear independence with a pinch of salt, because if you already have $X$, then you can add $X^{2}$, $X^{3}$, ... . this is not really the place to make this precise. The important thing is that they all commute and that you can add no more operator representing something not derivable from the ones on the list that commute with all on the list. You get the point.


 * For example, if you have $X, S_{z}$ (position and spin z-component) on your list, then you can add to it $S^{2}$, (total spin (squared)) because it commutes with the ones on the list. You cannot add $P$ (momentum) because it doesn't commute with $X$. The significance of a maximal set is that is what you can measure in the same experiment.


 * 3.) We are going to change from "basis" to "representation" when it comes to maximal commuting sets of observables for exactly the reason of risk of confusion with vectors space bases. A "representation" does not really fix a basis completely. If you chose "position representation", then you'll still have "position representation" after, say, a rotation of the coordinate axes. (Saying "a basis for the maximal set" is, well, wrong, or at least dubious.)


 * 4.) The eigenvalues of the observable represent definite outcomes. The possible outcomes are not probabilities. For the last statement, simply yes. To wit, if you know $Ψ = aψ_{1} + bψ_{2}$, then you can repeat an expriment many times to find out what $a$ and $b$ are. YohanN7 (talk) 09:27, 1 February 2015 (UTC)


 * Perhaps the electrons are required to find their respective independent ways through a rotating spiral tube of some known length and bore. Each electron thus has a known velocity therefore in the z direction. One knows thus its momentum in the z direction. But one has no control over when it travels, so one cannot know its position along the z axis. It hits a photographic plate that is normal to the z axis. One can deduce its x and y momenta. We are looking at 3 momenta, a momentum "representation"; it has fixed the quantum configuration space. We can draw the x and y axes on the plate as we please, but we will still be measuring momenta, say now with axes x' and y'. We have a mathematical relation between (x,y) and (x',y'). That is a transformation of the coordinates of the quantum configuration space, but it leaves us still in that same quantum configuration space. Alternatively, we might turn the photo plate through a right angle about a line perpendicular to the z axis, and move the plate so as to catch the particle at a measured distance along the z axis at a known time. This will give us some position information at a known time, with loss of momentum information. This changes the quantum configuration space to a different one. It is not a change of coordinates in a fixed quantum configuration space. They call this a change of "representation", bless their hearts. With each respective fixed quantum configuration space, if the electrons all land in the same spot on the plate, they are in a pure state with respect to that quantum configuration space, and they have a wave function that characterises that pure state. If they land every which way, they are in a mixed state, and they don't have a characterising wave function. A statistical matrix is needed. That's my story.Chjoaygame (talk) 12:17, 1 February 2015 (UTC)


 * I think this article should be self-supporting in defining quantum configuration space. Other articles, unless we here deliberately take them in hand, don't care a rap about the notion. Just look.


 * With respect, I think it might be useful to comment on "Roughly, an observable is a set of possible outcomes. Technically, an observable is a Hermitian operator on the Hilbert space." It uses one word with two very different meanings. True, the meanings correspond, but the correspondence is a specialized one, not at all obvious to a newcomer. Telescoping the meanings is handy for an expert, but I think confusing for a newcomer. Dare I say it, we mean "a quantum analyser is a carefully designed physical device with a suitable set of possible outcomes". And we mean "an observable is a mathematical object represented by a Hermitian operator on the Hilbert space". Why is it Hermitian? Because it has a kind of reciprocity: a suitably designed quantum analyser's input and output can be interchanged without altering its functioning; for example, a prism works in the same way, whichever face is taken as input; this is necessary to make interference intelligible, because interference is re-assembly after disassembly by a quantum analyser. Contrary to the omniscient Feynman, this is not at all mysterious; it is mysterious to him because he forgets that re-assembly makes physical sense only after dis-assembly into pure states; nature does not usually supply pure states. (The real mysteries are why are there stable and metastable quantum states and why and how are there jumps between them?) The theory uses the observable to represent the quantum analyser. I admit that the term 'quantum analyser' is not standard, but it is easy to find (e.g. Merzbacher 2nd edition p. 219, "analysers and slits"; and many more) literature usage of the term 'analyser' or 'polarization analyser', of which 'quantum analyser' is an obvious generalization useful for the present pedagogical purpose, apparently almost immediately intelligible to a mathematician asking about this subject. Dare I say it.Chjoaygame (talk) 18:17, 1 February 2015 (UTC)


 * I was a bit thrown by this edit. I felt I was being accused of non-standard usage, perhaps of incomprehensibility. It seemed perhaps from this by respected editor Tsirel that I was not incomprehensible in that usage, but I wasn't sure if other editors agreed. So I follow it up now.


 * Besides the just above reference to Merzbacher, another use of the term 'analyzer' as if it were standard, not calling for on-the-spot definition:


 * The scattered beam is scattered a second time from an identical target through $θ$ in the same plane both to the right and left. (Now the second target acts as an analyzer.)


 * The scattered beam is then scattered again through the same angle to the right and left from an identical target, which now acts as the analyzer."


 * I think my use of the term 'analyzer' (however spelt) is near enough to standard or routine.Chjoaygame (talk) 04:22, 27 September 2015 (UTC)

Tashiro, do you find that the new edits to the lead and the section Wave functions and function spaces answer your questions? YohanN7 (talk) 03:23, 4 February 2015 (UTC)

Real or complex values?
I think it should be mentioned in what conditions the wave function as a mathematical concept can be real valued.--188.26.22.131 (talk) 15:39, 6 February 2015 (UTC)
 * This can happen for certain potentials in one dimension (see Landau and Lifshitz) and for some exotic spin representations of the Lorentz group (see Majorana spinor). Feel free to add to the article, I think this has low priority a t m, but I do agree we shoud at some point put it in. (Complex valued works as a catch-all.) YohanN7 (talk) 08:44, 7 February 2015 (UTC)


 * Without disrespect, who actually cares about the case when the wave function is real, and why? There is no priority in mentioning this in the article, although if there are good reasons then feel free to briefly mention this. M&and;Ŝc2ħεИτlk 14:44, 7 February 2015 (UTC)
 * Why not? For comprehensiveness of the article!--5.15.187.165 (talk) 23:10, 5 May 2015 (UTC)

I think the case when the wave function is real is important in connection to De Broglie-Bohm theory wave function.--86.125.188.87 (talk) 14:15, 4 August 2015 (UTC)


 * Is it usually important if the result is real since the probability, which is of interest, is the value times its complex conjugate which is always real? RJFJR (talk) 14:33, 4 August 2015 (UTC)


 * No, not really. Why would this be important? M&and;Ŝc2ħεИτlk 14:39, 4 August 2015 (UTC)


 * It has some conceptual representation-theoretic importance. The number of real dimensions of the representation space in which the wave function takes its values drops by a factor of two. The best we have regarding this is the (excellent) spin representation article, which is "beyond the scope" of this article, but could potentially be linked in any discussion about real-valuedness. Whether real-valuedness has any physical implications such as that the amplitude actually could have physical existence (like the EM field) is better left to the philosophy department. I still think this issue has priority level low in that this is not the place to discuss it. The neutrino wave function may be a Majorana spinor, see also here where some links are provided. YohanN7 (talk) 15:55, 4 August 2015 (UTC)


 * Philosophy department? Why is physical existance of amplitude better left to it?--5.15.21.207 (talk) 22:37, 3 November 2015 (UTC)
 * Because nobody really knows. Some may think they know, others will also think they know but will think differently. What is known is that wave function concept is an expression of a mathematical formalism that does a good of describing reality. Whether wave functions are reality is another matter. It is a bit of a stretch to ascribe a complex-valued field physical reality. It is less of a stretch if the field is real-valued. It is to murky to speculate in this article about such things. YohanN7 (talk) 14:42, 4 November 2015 (UTC)
 * Speculate? If these considerations are presented in some RS available, we should have no hesitation to present the reasonings in the article.--193.254.231.34 (talk) 06:36, 29 January 2016 (UTC)
 * No, there are articles dedicated to such things, like Copenhagen interpretation. We could perhaps note in the article that e.g. the wave function of a $π^{0}$ is real and for $π^{+}$ and $π^{–}$, they are complex. But this doesn't really mean anything other than that in the mathematical formalism, it happens to turn out that way. YohanN7 (talk) 10:49, 29 January 2016 (UTC)

Archival settings
I think that the archival settings of this talk page need to be changed because the most archives are too short. Archives 2, 3, 4, 5 could be combined in a single archive. A 14d archival frequency is not necessary.--188.26.22.131 (talk) 15:44, 6 February 2015 (UTC)
 * Sofixit. But a 14d archival frequency has been necessary the last couple of months. YohanN7 (talk) 08:41, 7 February 2015 (UTC)

Adding clarification about the asterisk
I am adding information that the asterisk next to the wave function means "the complex conjugate of the wave function" to make the article easier to read. If I am wrong about that interpretation I apologize and please revert the change.75.128.143.229 (talk) 16:38, 1 October 2015 (UTC)

When I went to look I noticed that that clarification is already there, so I left the page unchanged.75.128.143.229 (talk) 16:39, 1 October 2015 (UTC)

For a good time, having discussions
https://www.physicsforums.com/forums/general-physics.111/ For all who would like to discuss phyical topics with a wider group that those few who happen to edit this page; discussion is very welcome there. GangofOne (talk) 06:05, 28 December 2015 (UTC)