# Talk:Wave function

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## "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∧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∧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 12 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)

• 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∧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.
"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 fineis 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.
• 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∧Ŝc2ħεИτlk 22:47, 29 January 2015 (UTC)
In the above, replace the stroked-out segments by adjacent underlined segments. M∧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.
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∧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)

## state or species ?

How should this article define the term 'quantum state', and consequently the 'wave function? Below are some findings from possibly reliable sources.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Born

"A knowledge of ψ enables us to follow the course of a physical process in so far as it is quantum-mechanically determinate; not in a causal sense, but in a statistical one. Every process consists of elementary processes which we are accustomed to call transitions or jumps; the jump itself seems to defy all attempts to visualize it, and only its result can be ascertained. This result is, that after the jump, the system is in a different quantum state. The function ψ determines the transitions in the following way: every state of the system corresponds to a particular characteristic solution, an Eigenfunktion, of the differential equation; for example the normal state the function ψ1, the next state ψ2, etc."<Born, M. (1927). Physical aspects of quantum mechanics, Nature 119: 354–357.>

It seems that Born thought of ascertained results of determinate physical processes in terms of probabilistic successions of jumps between quantum states as physical objects that correspond with mathematical entities called eigenfunctions.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Bohr

"As a more appropriate way of expression, one may advocate limitation of the use of the word phenomenon to refer to observations obtained under specified circumstances, including an account of the whole experiment."<Bohr, N. (1948). On the notions of complementarity and causality, Dialectica 2: 312–319.>

Bohr's thought continued to develop long after the early days. He eventually settled on the idea of a 'phenomenon'. He refers to the just-quoted paper in his celebrated attack on Einstein in the 1949 Schilpp book. Here below, Rosenfeld and Wheeler note this culminating concept. In ordinary language, I would say that by 'phenomenon', Bohr means 'process observed and described'. He is not referring to what I would think of as Einstein's idea of a natural process that happens whether or not someone later observes it. Obviously, an account or description of an unobserved process is to a large extent a theoretical speculation. Quantum mechanics is a method of description of experiments. Bohr thinks it ineluctably involves preparation and detection as ingredients of phenomena. The preparation is specified by a generation of an initial 'state' and the detection determines the specification of a final 'state'. Sometimes they are the same. The quantum 'states' are specified in terms of appropriate 'configuration' spaces. Unlike classical mechanics using states in phase space, quantum mechanics using 'configuration' space 'states' cannot in general support deterministic predictions, although the Schrödinger equation itself is deterministic as noted by von Neumann, and a 'phenomenon' is a determinate actual physical entity.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Heisenberg

"But that a revision of kinematical and mechanical concepts is necessary seems to follow directly from the basic equations of quantum mechanics. .... But what is wrong in the sharp formulation of the law of causality, "When we know the present [state] precisely, we can predict the future," is not the conclusion but the assumption. Even in principle we cannot know the present [state] in all detail."<Heisenberg, W. (1927). The physical content of Quantum kinematics and mechanics, Zeit. Phys.43: 172–198, translated in Wheeler, Zurek (1983) pp. 62–84.>

I have inserted the items [state] to bring out the relevance of Heisenberg's remarks here to the notion of quantum state. Also the kinematics are the description of the 'state'. As Dirac points out below, what we can know is determined by our mode of construction of the artificial state (e.g. our necessary choice of momentum space or configuration space, or whatever) that we observe. That is the ineluctable limitation on knowledge of state to which Heisenberg is referring, imposed by the quantum mechanical kinematics. A quantum phenomenon becomes determinate only when it has been detected, as pointed out below by Rosenfeld. Its initial condition as specified by quantum kinematics does not determine it. This contrasts with the classical kinematics which allow a state description that supports exact determination of the future.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Rosenfeld

"It is only too true that, isolated from their physical context, the mathematical equations are meaningless: but if the theory is any good, the physical meaning which can be attached to them is unique. .... The wholeness of quantal processes necessitates a revision of the concept of phenomenon. Since the concepts which in classical theory describe the state of a physical system are actually subject to mutual limitations, they can no longer be regarded as denoting attributes of the system. Their true logical function is rather to express relations between the system and certain apparatus of entirely classical (i.e. directly controllable) character which serve to fix the conditions of observation and register the results. A phenomenon is therefore a process (endowed with the characteristic quantal wholeness) involving a definite type of interaction between the system and the apparatus."<Rosenfeld, L. (1957). Misunderstandings about the foundations of quantum theory, pp. 41–45 in Observation and Interpretation, edited by S. Körner, Butterworths, London.>

To make a definite actual physical entity, a phenomenon, quantum physics requires that both initial and final conditions be determinate. Quantum kinematics defines a quantum 'state' that supplies only the initial, not the final, condition. That enforces its probabilistic character. (Perhaps I may remark that Einstein was not sure that Nature works by preparing pure states and detecting final states as required by quantum mechanics. Indeed, it is obvious that Nature supplies only mixed states.)Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Kramers

"A physical situation which is characterised by a solution of the Schrödinger equation of the form ψ = φ exp (−iEt/ħ) with normalizable ψ and which thus in accordance with the quantum postulate E = corresponds to a well defined energy of the system under consideration is called a stationary state of the system."<Kramers, H., (1937/1956). Quantum Mechanics, translated by D. ter Haar, North-Holland, Amsterdam, pp. 58–59.>

In the olden days they tried to define their terms. Kramers distinguished the physical situation from its mathematical characterisation.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Messiah

"When the system is in a state represented by a wave of type II.34, it is said to be in a stationary state of energy E; the time-independent wave function ψ is usually called the wave function of the stationary state, although the true wave function differs from the latter by a phase factor exp (−iEt/ħ)." <Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam, page 72.>

As I read this, Messiah has in mind two entities, a physical object in a quantum state, and a mathematical object that lives in a function space. He thinks the mathematical object "represents" the physical object.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Weinberg

"The viewpoint of this book is that physical states are represented by vectors in Hilbert space, with the wave functions of Schrödinger just the scalar products of these states with the basis states of definite position. This is essentially the approach of Diracs's ″transformation theory″."<Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-1-107-02872-2, page xvi.>

Evidently, Weinberg agrees with the view of Messiah that there are two kinds of object, physical and mathematical. He calls the physical ones "states" and the mathematical ones "vectors" or "wave functions". The mathematical ones "represent" the physical ones. It seems he has important points of agreement with Dirac.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### Dirac

"A state of a system may be defined as an undisturbed motion that is restricted by as many conditions or data as are theoretically possible without mutual Interference or contradiction. In practice, the conditions could be imposed by a suitable preparation of the system, consisting perhaps of passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being undisturbed after preparation. The word 'state' may be used to mean either the state at one particular time (after the preparation), or the state throughout the whole of the time after the preparation. To distinguish these two meanings, the latter will be called a 'state of motion' when there is liable to be ambiguity."<Dirac, P.A.M. (1940). The Principles of Quantum Mechanics, fourth edition, Oxford University Press, Oxford UK, pages 11–12.>

No mention here of mathematical objects. Dirac is referring to physical objects. He distinguishes between an instantaneous state and a state with an indeterminate duration in time. The state so defined is physically indeterminate because it it not actually observed by detection. That is the meaning of 'undisturbed'. An indeterminate state does not define a physical phenomenon, such as is intended by Wheeler in his well-known aphorism

"Had quantum mechanics stopped here, its deepest lesson would have escaped attention: ″No elementary quantum phenomenon is a phenomenon until it is a registered (observed) phenomenon.″"<Wheeler, Zurek (1983), page xvi.>

Here Wheeler is referring to statements such as the following by Bohr:

"... every atomic phenomenon is closed in the sense that its observation is based on registrations obtained by means of suitable amplification devices with irreversible functioning such as, for example, permanent marks on the photographic plate caused by the penetration of electrons into the emulsion ..."<Bohr (1958) quoted by Wheeler, Zurek (1983), page viii.>

Evidently, for Wheeler and Bohr, a quantum mechanical phenomenon is an actual physical entity, a fully determinate process, with a finite time duration, with no remaining unrealized potential possibility. Such is not a quantum state as defined by Dirac.

A physical entity that is indeterminately defined can have future adventures only probabilistically. Being indeterminate, it cannot have a determined future. This contrasts with a determinate classical physical object, which can have a determined future. That is one difference between Dirac's quantum 'state' and a classical ordinary language physical state.

Nevertheless, Dirac's state is defined as restrictively as is theoretically possible for a quantum system. This makes it a pure state. The pure state is not that of a raw natural object, such a an atom of silver vapour escaping through a small hole in an oven wall. No, it is an artificially prepared state. Even though it is not yet observed, it is still causally conditioned by the observer, not in a native state. For example, it might have been prepared in a definite state of uniform motion in a nearly straight line if it is observed in a place in space where there is nearly no gravity. Then it is in a momentum eigenstate. It has no definite position. Its momentum can be measured by its angle of deflection by a diffraction grating and detection by a suitably placed device.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

### present article

The present article says

• Wave functions corresponding to a state are not unique. This has been exemplified already with momentum and position space wave functions describing the same abstract state.
• The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is analogous to a vector space without a specified basis.
• The wave functions of position and momenta, respectively, can be seen as a choice of basis yielding two different, but entirely equivalent, explicit descriptions of the same state.
• Corresponding to the two examples in the first item, to a particular state there corresponds two wave functions, Ψ(x, Sz) and Ψ(p, Sy), both describing the same state.

As I read it, this part of the present article flatly contradicts the consensus of Born, Bohr, Heisenberg, Rosenfeld, Kramers, Messiah, Weinberg, and Dirac.

As I read it, the present article uses the term "state", where ordinary language would speak of 'species of quantum system', or 'species of quantum entity', or 'kind of particle', or some such, and would say that a species of quantum system can be prepared in several different states, respectively pure with respect to several observables = quantum analysers = operators, each pure state with its own respective wave function. And, as I read it, the present article rejects the idea that a pure quantum state is described by a particular eigenfunction of the operator with respect to which it is pure. Rather, for the present article, the "state" extends over all possible operators for the species or kind of particle. The notion of a mixed state seems to have faded out.

An example of a species of system would be typified by an unfiltered atom emerging from a hole in the wall of an oven containing metal vapour. It is in a mixed state. Filtering it with some device such as a Stern-Gerlach magnet will split the beam into several sub-beams in states respectively pure for that filter.

The confusion may perhaps arise by using the phrase "abstract state" to mean 'class of pure states in which a species of system can be prepared'. Alternatively, perhaps the just above eight cited authors are obsolete, and quantum mechanics has changed since their days?

Dirac defines a state by the most restrictive possible set of conditions, making it a pure state, while the present article's "abstract state" seems to extend over the least restrictive possible range of conditions.Chjoaygame (talk) 23:31, 21 January 2015 (UTC)

For the record, the present article states
• Basic states are characterized by a set of quantum numbers. This set is a set of eigenvalues of a maximal set of commuting observables. I'd say it is pretty much spot on. YohanN7 (talk) 15:05, 23 January 2015 (UTC)
Thank you for this comment. I will think it over.Chjoaygame (talk) 00:32, 24 January 2015 (UTC)
This edit significantly addresses the concerns that prompted my just foregoing remarks. I am still thinking this over.Chjoaygame (talk) 18:35, 2 February 2015 (UTC)
Thinking it over, I find the above comment on "basic states" is hardly a direct response to the concern I raised in this section. So I will not go further with it in this section. My concern, expressed in this section, is still alive, and I have addressed and extended it in a new section below, for ease of editing.Chjoaygame (talk) 16:43, 6 February 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."<Born. M. (1969). Atomic Physics, eighth edition, translated by J. Dougall, R.J. Blin-Stoyle, J.M. Radcliffe, Blackie & Son, London, p. 130.>

Likely Born is referring to <Dirac, P.A.M. (1964). Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, New York.>

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#time dependence.
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∧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<Razavy, M. (2011). Heisenberg's Matrix Mechanics, World Scientific, Singapore, ISBN 978-981-4304-10-8.> 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.<plenty of references for this if appropriate>. But they are not quite so, as pointed out by N.R. Hanson.<Hanson, N.R. (1961). Are wave mechanics and matrix mechanics equivalent theories?, Czech. J. Phys. 11: 693–708.> 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)
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∧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∧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)

### 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)

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 X2, X3, ... . 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, Sz (position and spin z-component) on your list, then you can add to it S2, (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 = 1 + 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";[1] 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.)[2]
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."[3]
1. ^ Merzbacher, E. (1961/1970). Quantum Mechanics, second edition, Wiley, New York.
2. ^ Gottfried, K. (1966). Quantum Mechanics, volume 1, Fundamentals, W.A. Benjamin, New York, p. 329.
3. ^ Gottfried, K., Yan, T.-M. (2003). Quantum Mechanics: Fundamentals, 2nd edition, Springer, New York, ISBN 978-0-387-22023-9, p. 382.
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∧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∧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)

## 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)

## concern

In this edit, I wrote:

### "present article

The present article says
• Wave functions corresponding to a state are not unique. This has been exemplified already with momentum and position space wave functions describing the same abstract state.
• The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is analogous to a vector space without a specified basis.
• The wave functions of position and momenta, respectively, can be seen as a choice of basis yielding two different, but entirely equivalent, explicit descriptions of the same state.
• Corresponding to the two examples in the first item, to a particular state there corresponds two wave functions, Ψ(x, Sz) and Ψ(p, Sy), both describing the same state.
As I read it, this part of the present article flatly contradicts the consensus of Born, Bohr, Heisenberg, Rosenfeld, Kramers, Messiah, Weinberg, and Dirac.
As I read it, the present article uses the term "state", where ordinary language would speak of 'species of quantum system', or 'species of quantum entity', or 'kind of particle', or some such, and would say that a species of quantum system can be prepared in several different states, respectively pure with respect to several observables = quantum analysers = operators, each pure state with its own respective wave function. And, as I read it, the present article rejects the idea that a pure quantum state is described by a particular eigenfunction of the operator with respect to which it is pure. Rather, for the present article, the "state" extends over all possible operators for the species or kind of particle. The notion of a mixed state seems to have faded out.
An example of a species of system would be typified by an unfiltered atom emerging from a hole in the wall of an oven containing metal vapour. It is in a mixed state. Filtering it with some device such as a Stern-Gerlach magnet will split the beam into several sub-beams in states respectively pure for that filter.
The confusion may perhaps arise by using the phrase "abstract state" to mean 'class of pure states in which a species of system can be prepared'. Alternatively, perhaps the just above eight cited authors are obsolete, and quantum mechanics has changed since their days?
Dirac defines a state by the most restrictive possible set of conditions, making it a pure state, while the present article's "abstract state" seems to extend over the least restrictive possible range of conditions."Chjoaygame (talk) 16:35, 6 February 2015 (UTC)
Well, you read it wrong. It is clear from the very first sentence in the lead that we are talking about one system, not an ensemble. Hence a pure state. From where do you get the idea that the article speaks about particle species and not states? Besides, that atom of yours is in a pure state, not a mixed state. YohanN7 (talk) 06:13, 7 February 2015 (UTC)
It is good to have a response from you.
It seems there are problems of communication here. Perhaps with patience and goodwill they can be solved.
I have no problem in accepting the obvious, that we are talking about one system that may consist of many particles, as stated in the sentence of the lead "There is one wave function containing all the information about the entire system, not a separate wave function for each particle in the system." I was not thinking in terms of ensembles. Perhaps I should add that your introduction here of the term 'ensemble' might need some attention if it becomes relevant. There may perhaps be potential for a communication difficulty in it.
You write "Hence a pure state." It my be here that there is a communication problem. Perhaps the Stern-Gerlach experiment may help to clarify. For the present I will limit my response to this, for the sake of comfort of discussion.
As I read the Stern-Gerlach experiment, they have an oven containing silver vapour. The vapour consists of single atoms. There is a small hole in the wall of the oven, through which atoms emerge in a beam, labelled here beam 0. The atoms in the beam travel independently of one another. Each atom is, for the present purposes, a single system. The beam passes through a Stern-Gerlach magnetic field. Some of the atoms go up, others down. These emerging sub-beams are here labelled beam 1u and beam 1d. Following The Feynman Lectures on Physics, each sub-beam is passed to another respective Stern-Gerlach magnet that has the same orientation as the primary one. From the 'up' secondary magnet, only one beam emerges, a second 'up' sub-beam, labelled beam 2uu. From the 'down' secondary magnet, only one beam emerges, a second 'down' sub-beam, labelled beam 2dd.
Then beam 0 is mixed with respect to the chosen Stern-Gerlach magnet orientation, because it will emerge from the primary Stern-Gerlach magnet in two nearly equal sub-beams, beam 1u and beam 1d.
Also, beam 1u is pure with respect to the chosen Stern-Gerlach magnet orientation, because it will emerge as a single beam 2uu from its second Stern-Gerlach magnet. Likewise, beam 1d is pure with respect to the chosen Stern-Gerlach magnet orientation, because it will emerge as a single beam 2dd from its second Stern-Gerlach magnet.
To judge from your above remarks, it may be better that I stop at this stage, to let you reply to what I have just written.Chjoaygame (talk) 07:26, 7 February 2015 (UTC)

#### definitions

I suggest you use the definitions in quantum state instead of experiments of thought. By "pure" is not meant "pure with respect to an observable" (whatever that means). There is not a trace of mixed states (as defined in the referenced article) in this article, and I consider the topic being off. YohanN7 (talk) 08:08, 7 February 2015 (UTC)

"Whatever that means." Feynman <Lectures, volume III, page 5–5>: "The answer is this: If the atoms are in a definite state with respect to S, they are not in the same state with respect to T—a +S state is not also a +T state. There is, however, a certain amplitude to find the atom in a +T state—or a 0T state or a T state."[Feynman's format of —]Chjoaygame (talk) 03:04, 5 November 2015 (UTC)
A more general definition would be that measurement of (observable) S with certainty would yield a certain value s. Mathematically, the state is an eigenstate of (the Hermitian operator) S. (Using the same symbol for observable quantity and corresponding operator is standard.) I don't mind using the terminology "state pure with respect to observable" as long as we define it and don't confuse it with "pure state", which is defined elsewhere in Wikipedia. YohanN7 (talk) 11:10, 5 November 2015 (UTC)
Thank you for your reply. I think it would be valuable to actually put an explicit version of that 'elsewhere' definition right here, or very much better, in your own words, for clarity of comparison, if you have time. I am asking not for a mathematical abstract expression, such as that of David J. Griffiths, who says he intends not to tell what quantum mechanics means, but rather to tell "how to do it"; "shut up and calculate", to quote Mermin. I am asking for a physical expression, like Feynman's. Wikipedia is not a reliable source.Chjoaygame (talk) 14:18, 5 November 2015 (UTC)
What do you want a physical expression for? "Pure state" or "Pure state with respect to observable"? If you are looking for "pure state" as opposed to "mixed state", then I'd suggest you go to the literature if you don't trust Wikipedia. I'm not suggesting Wikipedia as a reference for the article, but this is a talk page you know. I can only say that a superposition of eigenstates (more than one) of an observable is not a mixed state, but rather a pure state. But it is not a pure state with respect to the observable in question (as I am thinking about it here in this discussion). It may, however, be a pure state with respect to another observable. This is what Feynman alludes to. His S and T do not commute, hence if pure w r t S, then it is "mixed w r t T" (but not a mixed state). YohanN7 (talk) 12:19, 16 November 2015 (UTC)
I was looking for a physical definition of a pure state. I have spent some effort on the literature, looking for a physical definition, but I find only mathematical ones, if you don't count 'pure with respect to a specified observable' as 'pure'. For definiteness, also I may refer to Wikipedia. There I am led to the article Quantum state, which is mathematical, not physical.
You advise me to "use the definitions in the article quantum state instead of experiments of thought". As I read this advice, it is to use mathematical definitions, not physical ones. My interest is in physics. Mathematics is useful for expression of physical ideas, but mathematics is not physics. I think physical ideas need expression in experimental terms as well as in mathematical terms. Without the experimental terms, they are undefined physically. I think experiments of thought are often near-enough to experimental expressions. With no experimental expression, we are not talking physics.
You write that "... a superposition of eigenstates (more than one) of an observable ... may, however, be a pure state with respect to another observable."
I agree. But I would go further and say that for a state to be pure, it must have an observable with respect to which it is pure. I am saying that a every pure state is pure with respect to some observable. Evidently you think I am mistaken in that?
There may be a problem here about the word 'ensemble'. It might have different meanings, depending on whether it was used in a physical or a probabilistic sense. In my view, a probability usually refers to an ensemble in a virtual sense used by probability theorists. Each single particle in an occasion of experiment has probabilities associated with it. Usually the probabilities are found in experiment by repeated, on occasion after occasion of experiment, practically identical preparations of single particles. By 'practically identical', I mean that definite identical practical steps are taken to prepare, on occasion after occasion of experiment, single particles. If there remain untaken further practical steps, by which the repeatedly prepared single particles could be further purified, then they are in a mixed state. If all possible further practical steps of purification have been taken, then the single particles are each in the same pure state. The repeatedly, on occasion after occasion of experiment, prepared single particles, constitute a probability theorist's virtual ensemble. A single occasion of preparation of a single particle does not provide experimental data for the determination of the relevant probabilities. In fact we cannot ensure that the practically identical particles are ultimately identical, whatever that might mean. I understand 'practically identical' in this sense to refer to single particles.
The single particles so prepared have not yet been observed. For observation, more is needed. In general, that 'more' is as follows. The single particles are passed, one by one, through some sorting process, and then detected by an array of particle detectors lying in the several output channels. The sorting process may be trivial, not actually doing anything. For a pure state, there is a definite sorting process, embodied in a definite physical device, symbolically designated by a definite linear operator, such that only one detector, of the several in the array, actually detects particles.
This is my idea of a physical definition of a pure state.
As I read you, the just-above story is, in your view, wrong. As I read you, you have not yet here proposed a physical definition of a pure state. Instead you advise me to be content with a mathematical one.Chjoaygame (talk) 22:00, 16 November 2015 (UTC)
Let us continue this discussion on either your or my talk page. You are welcome at my place.
I have not read the just-above story (I will, but not now). I can't see how you can argue with definitions. Definitions are there to establish terminology. You can't complain because you think the English word "pure" in "pure state" (through the definition) is misused in some linguistic or physical sense. You wrote,
I agree. But I would go further and say that for a state to be pure, it must have an observable with respect to which it is pure. I am saying that a every pure state is pure with respect to some observable. Evidently you think I am mistaken in that?.
You may be mistaken in that, depending on what you want. Mathematically you are right, because you can construct a Hermitian operator = observable with the state in question as an eigenvector. But its eigenvalues will in general have little to do with physically measurable quantities, which is what I guess you are after, in which case you are wrong.
Here is one attempt at definition of "pure state": Every vector in Hilbert space represent a pure state. (Mixed states are described not by state vectors, but by density matrices.) YohanN7 (talk) 10:11, 17 November 2015 (UTC)
Thank you for this. Following your suggestion, I will reply on my talk page.Chjoaygame (talk) 10:44, 17 November 2015 (UTC)

#### With respect

With respect, Wikipedia is not a reliable source. I am not willing to accept it as such for the present discussion. I have offered what I think is a suitable paradigm for a definition of a pure state. It seems you think it not a good definition. I think the next step in our discussion would best be that you would say in your own words how you would define a pure state. Also I should perhaps back up my definition. I will do so in due course.
I am a little puzzled that you seem to object to my using the Stern-Gerlach experiment as I have done here. I am working from Chapter 6 of volume III of Feynman, Leighton, Sands, as I noted in my comment. A rather similar approach is taken by B.-G. Englert's posthumous edition of Schwinger's Quantum Mechanics: Symbolism of Atomic Measurements, Springer (2001).Chjoaygame (talk) 09:54, 7 February 2015 (UTC)
From the article that you don't trust:
Mathematically, a pure quantum state is represented by a state vector in a Hilbert space over complex numbers, ... .Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 0-13-111892-7:93–96 If this Hilbert space is represented as a function space, then its elements are called wave functions.
There's your pure state with reference complete with page number and all. If you worry about what this has to do with physics, then you can turn to the first postulate of QM that says precisely that (pure) physical states are represented by vectors in Hilbert space.
For mixed states, I suggest you look up paragraph 14 in L&L. (They put quotation marks around "mixed". (B t w, Landau invented the density matrix afaik)). You can also read p 68 and on in Weinberg (he doesn't baptize mixed states, but his description is as any good description).
Your "definitions" are homemade and completely unintelligible to me. I'll not discuss them. YohanN7 (talk) 10:42, 7 February 2015 (UTC)
Yes, you give mathematical definitions, which are about mathematical representations. But my concern here is as to their physical meaning, which neither the Wikipedia article you cite, nor your definition, make any attempt to address. According to Rosenfeld, a reliable source on Bohr's thinking and Copenhagenism in general, without physical context, which you avoid, the mathematical formulas are physically meaningless. Since you avoid considering the physical meaning, and I am largely concerned with it in combination with the formulas, it is hardly surprising that that you find my thoughts not worth discussing. I am sorry about this.Chjoaygame (talk) 13:38, 7 February 2015 (UTC)

#### "species"

Chjoaygame, for one thing, "species" in this physical/chemical context usually means "which subatomic particle/composite subatomic particle/atom/molecule/compound", e.g. electrons, hydrogen atoms, helium nuclei, silver atoms in a beam, etc. . What does this term have anything to do with "wave function", considering all the particles in the system regardless of what they are, are described by the wave function? And no, this article should not be about ensembles and mixed states. M∧Ŝc2ħεИτlk 14:52, 7 February 2015 (UTC) I added the underlined segments, typed the above very fast. M∧Ŝc2ħεИτlk 16:45, 7 February 2015 (UTC)

The system is composed initially of some species or generalized species in some state, and the particular wave function of interest is specific for that. Yes, I agree this article should not be about ensembles and mixed states.Chjoaygame (talk) 16:02, 7 February 2015 (UTC)
How is the wave function "specific" for this or that species? It isn't. What does matter are quantities which affect statistics, most obviously the spin of the particles, and other quantum numbers, but also the number of particles, and the number of spatial/momenta dimensions (e.g. 1d, 2d, and 3d and higher dimensions, all have various effects on statistics), and there could be others. M∧Ŝc2ħεИτlk 16:45, 7 February 2015 (UTC)
The wave functions for the hydrogen atom in its various states are different from those for the hydrogen molecule in its various states. That's what I mean by specific.Chjoaygame (talk) 18:26, 7 February 2015 (UTC)
What about hydrogen-like atoms or ions? The family of wave functions is not so different to that of the hydrogen atom is it? Similarly for analogues of the hydrogen molecule, what about systems which have particles of different masses and charges but the same spin and statistics and are subject to the same potential as for the hydrogen molecule? M∧Ŝc2ħεИτlk 18:43, 7 February 2015 (UTC)

Back to your original point, it seems you are the one who has conflated "state" with "species" while reading the article. The article doesn't say, in what you term "ordinary language", that a state refers to "species of quantum system", or "species of quantum entity", or "kind of particle", "or some such". It also doesn't say "species of quantum system can be prepared in several different states". It is also completely irrelevant, circular, and meaningless to say "a pure quantum state is described by a particular eigenfunction of the operator with respect to which it is pure". M∧Ŝc2ħεИτlk 18:53, 7 February 2015 (UTC)

### Key to my concern

The key to my concern here was and still is that, in the traditional or older conception, a wave function represents a pure state, while there is no state pure with respect both to to position and to momentum. Position and momentum purity are physically, or in actuality, incompatible, and cannot co-exist in nature; their mathematical expressions as operators do not commute. Thus a 'state', to which "there corresponds two wave functions, Ψ(x, Sz) and Ψ(p, Sy), both describing the same state", is an abstract kind of object, at a level of logical abstraction further from actuality than the older or traditional wave function that represents a particular physical state. We seem perhaps to be somewhere near what Alfred North Whitehead, co-author of Principia Mathematica, called 'the fallacy of misplaced concreteness'.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

States are no more abstract than wave functions as explained in the article. They are only clearer as what what is meant. There is a one-to-one correspondence between states and wave functions in different representations (as also explained in the article). Wave functions have not changed one bit since the old days. They (each) provide a complete description of a system in a given state.YohanN7 (talk) 06:19, 7 February 2015 (UTC)

### Further edits

Since my post copied above, there has been extensive editing which might be construed as related to my post copied above. Read as they stand, these edits seem to relate to it, but they seem to me not to resolve the problem altogether.

For comparison, the article currently reads:

• Wave functions corresponding to a state are accordingly not unique. This has been exemplified already with momentum and position space wave functions describing the same abstract state. This non-uniqueness reflects the non-uniqueness in the choice of a maximal set of commuting observables.
• The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is the same as saying that no choice of maximal set of commuting observables has been given. This is analogous to a vector space without a specified basis.
• The wave functions of position and momenta, respectively, can be seen as a choice of representation yielding two different, but entirely equivalent, explicit descriptions of the same state for a system with no discrete degrees of freedom.
• Corresponding to the two examples in the first item, to a particular state there corresponds two wave functions, Ψ(x, Sz) and Ψ(p, Sy), both describing the same state. For each choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation that is associated to a function space of wave functions.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)
It is very nice of you to quote the article in addition to old posts by yourself. But no, your old post was just too confused to have influenced the editing. YohanN7 (talk) 06:22, 7 February 2015 (UTC)

### Consequences

I think this indicates a radical problem in the article, as follows. I will start by referring to

<Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-1-107-02872-2.>

#### Weinberg on Dirac's notation

Apart from the short section on pages 57–58 about Dirac's bra-ket notation, throughout the book, Weinberg uses the notation ( . , . ), not Dirac's bra-ket notation.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

So? Each to his own. YohanN7 (talk) 06:23, 7 February 2015 (UTC)

#### A distinction drawn by Weinberg

Weinberg writes on page xvi:

"The viewpoint of this book is that physical states are represented by vectors in Hilbert space, with the wave functions of Schrödinger just the scalar products of these states with the basis states of definite position."

Weinberg is at fault in logic here. He is conflating [physical] states with their [mathematical] representatives. Properly speaking, there is no inner product of states, because they are physical, not mathematical, objects. There are inner products between vectors, and dual vectors, which are mathematical objects. That it is proper for Wikipedia to fault Weinberg in logic is apparent to a careful Wikipedia editor reading Weinberg's Section 3.7 on "Interpretation of Quantum Mechanics", in which on page 95 he writes "We can live with the idea that the state of a physical system is described by a vector in Hilbert space, rather than by numerical values of the positions and momenta of all the particles in the system, but it is hard to live with no description of physical states at all, only an algorithm for calculating probabilities." Here Weinberg does not conflate. The aforementioned conflation is thus seen as a mistake by Weinberg. Part of a Wikipedia editor's mandate is to ensure reliability of sources. This may require survey of many candidate sources.

Nevertheless, Weinberg distinguishes clearly between wave functions and state vectors. He writes on page 52:

"... wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical variable? For instance, we might want to describe the states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than position. ... The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space."

Weinberg's distinction between wave functions and his own state vectors is clear. He writes on page 59:

"... a wave function of Schrödinger's wave mechanics is nothing but the scalar product

ψ(x) = (Φx,Ψ)."

Weinberg writes on page 57:

"In Dirac's notation, a state vector Ψ is denoted  |Ψ〉. ... In the special cases where  Ψ is identified as a state with a definite value a for some observable  A, the ket in Dirac's notation is frequently written  |a〉."

Thus it appears that  |Ψ〉 denotes a state that is not identified with a definite observable that would make it a particular actual state. This indicates the abstract character of Weinberg's Hilbert space vectors, which comprise all possible relevant particular states of the species of particle.

See also Cohen-Tannoudji et al. on page 147, as below. And Zettili 2009.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

There is a world of understanding (and a Nobel prize) between you and Weinberg. The second to last paragraph above pinpoints exactly the misunderstanding of yours. You do not need to express a state in terms of a particular observable to make it a "particular" state. All information abut a state is contained in the state vector so it is already "particular". An electron doesn't give a damned whether you express it's state abstractly or as a wave function of momentum or as a wave function of position.
Besides, regarding your first paragraph. The inner product on Hilbert space (any which one, the abstract state space or a Hilbert space of wave functions) is modeled on physical reality. The fact that two states (or wave functions if you want) are orthogonal has physical interpretation. YohanN7 (talk) 06:35, 7 February 2015 (UTC)
Might also add here that the inner product is between vectors, not between vectors and dual vectors as you believe. See Wave function#More on wave functions and abstract state space. YohanN7 (talk) 12:26, 15 March 2015 (UTC)
According to Kurt Gottfried:
Although we shall frequently say that  $|\alpha\rangle$ and  $|\beta\rangle$ are orthogonal if  $\langle \beta|\alpha\rangle\,=\,0$, it should be remembered that the scalar product is only defined between a vector and a dual vector.[1]
In a later edition, we read:
It is necessary, first, to associate a dual vector to every ket in a one-to-one manner, called a bra, denoted by the symbol  $\langle\,|$; and second, to define the scalar products as being between bras and kets.[2]
1. ^ Gottfried, K. (1966). Quantum Mechanics, volume 1, Fundamentals, W.A. Benjamin, New York, p. 202.
2. ^ Gottfried, K., Yan, T.-M. (2003). Quantum Mechanics: Fundamentals, 2nd edition, Springer, New York, ISBN 978-0-387-22023-9, p. 31.
Is Gottfried a reliable source on this point?Chjoaygame (talk) 06:53, 13 September 2015 (UTC)Chjoaygame (talk) 15:27, 30 September 2015 (UTC)
Never heard of him. Perhaps reliable, but decidedly wrong. It is a standard mistake made by many, and this is pointed out strongly in some math texts, e. g. Lie Groups, Lie Algebras, and their representations by Brian C. Hall. See also Inner product. Please do not push this. As I recall, the article explains well why one gets away with confusing the concepts. YohanN7 (talk) 12:26, 29 September 2015 (UTC)
That is to say, the left argument in
$\langle \Psi_1 | \Psi_2 \rangle,$
is a member of the space, not the dual. This is so even if
$\langle \Psi_1 |$
is a member of the dual. Confusing perhaps, but decidedly true (and explained in the article). Perhaps not well enough. YohanN7 (talk) 12:34, 29 September 2015 (UTC)
I am puzzled by your request "Please do not push this." I thought it trivial until you drew attention to it, I suppose because it is significant.
Gottfried's being decidedly wrong on this point would seem to imply that he is not a Wikipedia-reliable source on this point.
You say it is perhaps confusing. Perhaps, therefore, you may very kindly be willing to help me along here. In the article, I find this:
The state space is postulated to have an inner product, denoted by
$\langle \Psi_1 | \Psi_2 \rangle,$
that is (usually, this differs) linear in the first argument and antilinear in the second argument. The dual vectors are denoted as "bras", Ψ|. These are linear functionals, elements of the dual space to the state space. The inner product, once chosen, can be used to define a unique map from state space to its dual, see Riesz representation theorem. this map is antilinear. One has
$\langle \Psi | = a^{*} \langle \psi | + b^{*} \langle \phi | \leftrightarrow a|\psi\rangle + b|\phi\rangle = |\Psi\rangle,$
where the asterisk denotes the complex conjugate. For this reason one has under this map
$\langle \Phi|\Psi\rangle = \langle \Phi| (|\Psi\rangle),$
and one may, as a practical consequence, at least notation-wise in this formalism, ignore that bra's are dual vectors.
Is this one of the passages in the article to which you intend to draw my attention? Or should I be looking elsewhere? And perhaps you may also very kindly be willing to say what, if any, is the significance of this?Chjoaygame (talk) 15:45, 29 September 2015 (UTC)
Yes, that is the passage. First off, the whole issue is notational, and it is peculiar to the Dirac bra-ket notation. The significance is that many (including physicists) gets tricked into believing that the inner product marries a bra and a ket. That is, they think that we really have the action of a linear functional on a vector. In the case of the inner product, we do not. If that were the case, we wouldn't have a Hilbert space to begin with (postulate one of QM would be ruined).
The point of the passage is that the misconception, while regrettable, is harmless. You would never have the same issue if you used another notation for the inner product, say like
$(\Phi, \Psi),$
(like e.g. Weinberg) because no elements of the dual space can be suspected here. I hope this helps. Let me know otherwise. (I'll not be very much at the computer the nearest future.) YohanN7 (talk) 13:10, 30 September 2015 (UTC)
Aha! I have looked closer at Gottfried's second statement:
It is necessary, first, to associate a dual vector to every ket in a one-to-one manner, called a bra, denoted by the symbol  $\langle\,|$; and second, to define the scalar products as being between bras and kets.
He is falling into the trap just like I thought. He gets the order wrong. The thing is this. Without an inner product, there is no way of canonically identifying kets and bras. You must have an inner product first.. Then you can make the identification. It is done precisely as described in "the passage". When this has been done, there is a canonical identification between bras and kets. Thereafter, any misconception about what the left argument is in
$\langle \Psi_1 | \Psi_2 \rangle,$
really is is completely harmless (because you can think of it either way under the identification) - but regrettable (e.g. his book spreads the misconception). YohanN7 (talk) 13:25, 30 September 2015 (UTC)
Thank you for this.Chjoaygame (talk) 16:17, 30 September 2015 (UTC)

#### Problem in the article

Though it is interesting and informative, the present article has an element of ambivalence. The present lead writes "For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is ..." This is not far from the wording of Bransden & Jochain <1983, page 69> "Once a given basis — also called a representation — has been chosen, Ψ is completely specified by its components cj ." This wording is strongly suggestive that a wave function refers to a definite pure state, as opposed to an abstract class of states, which seems to be the concept of the rest of the article.

Besides this there is a larger problem. Evidently, along with Cohen-Tannoudji et al. and with Zettili, Weinberg regards Schrödinger's wave mechanics as less general than Weinberg's own approach in terms of Hilbert space. And he regards the wave function as belonging to Schrödinger's wave mechanics. Weinberg is one of the main sources cited by the article. But Weinberg's distinction is largely rejected by the present version of the article, which mostly conflates wave functions with abstract vectors. The citations are therefore inaccurate or misleading. The viewpoint of the present version of the article, conflating wave functions with Hilbert space vectors, is thus original research, or perhaps merely erroneous. Such is not permitted by Wikipedia policy.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

Which citation is inaccurate or misleading? Wave function are Hilbert space vectors as explained in both the lead and in the section Wave functions and function spaces. This you can find in every reliable source (at the basic level). It is taught in the kindergarten of quantum mechanics. The Schrödinger equation is linear. Therefore its solutions can be added and multiplied with scalars. There is an inner product on this function space (given in every reliable source). YohanN7 (talk) 06:46, 7 February 2015 (UTC)

#### Possible remedies

To this unfortunate error, two remedies suggest themselves. One is to give the article some new title that accurately reflects its deeply abstract content, such as perhaps Hilbert space vectors. Then a fresh article should be written that is truly titled Wave function. The other remedy that suggests itself is a thorough rewrite of the present article, to make it conform its advertised title, Wave function, as distinct from 'Abstract Hilbert space vectors'.

It seems that the present error arose partly because of a desire to build the article on the most abstract lines. The error arose partly also because the article was written largely with apparent disregard for the usual Wikipedia policy that in-line sources should be well supplied. Care would be needed in the rewrite to give accurate in-line reliable sources.

There is imaginable a possibility that a temptation might be felt to cast the new Wave function article again according to the abstract conceptual structure of the present article, arguing that the concept of the wave function is partly outdated. That this might happen is suggested by the fact that the present article already has a subsection that proposes that the topic Wave function is not a leading idea of the quantum theory of fields, which prefers to work with field operators. Obviously, experts in quantum field theory would be in a good position to create a new article entitled Quantum field operators or some such. That would leave the title Wave function comfortably free to concentrate on that topic, unconstrained to try to cover or be based on the distinctly more abstract concept of Hilbert space vectors.

A question still perhaps remains unresolved. Is the [generalised] 'state' of a system to be defined in an abstract modern way as the set of all possible [particular] states of the species of particle, or is the state to be defined in the older traditional directly particular way, by the actual initial condition of the particles as they emerge from the chosen preparatory device into the experimental registratory field, pure (and described by a wave function) or mixed (and described by a density operator) as the case may be? I would favour the choice that the article entitled Wave function should take as its primary focus the traditional wave function that refers to an actually possible particular state, with perhaps a short section explaining how the traditional concept has been more or less superseded by a more sophisticated abstract concept of a class of possible states.Chjoaygame (talk) 16:35, 6 February 2015 (UTC)

The article puts wave functions into context, namely quantum mechanics of yesterday and today. You seem to disapprove of that. Regarding inline citations, well, there could be more. But, and this is important, the subject of the article is basic and in the reference list there are more than six classic textbooks (I have read only six of them in the reference list) on quantum mechanics where the reader can look up for himself. If you want inline citations added, well, add them.
Again, your last paragraph, the very last sentence in fact, again (see also further up, same blunder) illustrates your misunderstanding of the subject. This is not meant as an insult, it is actually well meant. Please, read up on the subject. Read any book, not just the history sections. It is really not very advanced what is treated in the article, and you have understood wrong that a state is referring to something else than (what you call) "wave function that refers to an actually possible particular state". Right now you are just wasting your own time, and hinder people from touching the article. If I would have known that this was coming, then I wouldn't have touched the article with a barge pole to begin with.
If there are many readers like you, then it is absolutely necessary to include material about the context of wave functions (precisely the things you dislike) to ward off the kind of misinterpretations that you have demonstrated that you have adopted. YohanN7 (talk) 07:24, 7 February 2015 (UTC)