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== quantum state versus wave function ==

First off, this article is pretty bad. For instance the lede is much too long and rambling. Hopefully, it can be improved. Before making any attempt to do so, I think it's worth figuring out a basic question regarding the subject of this article - what, precisely, is the distinction between a quantum state (for which we already have an article, namely [[quantum state]]), and a wave function?

The article (to the extent it's coherent at all) defines the wave function as a "complex-valued function", and refers to a representation of the state vector in some CSCO (complete set of commuting observables). But consider a particle in 1D QM, and express the state in the energy basis <i|\psi> (where H|i> = E_i | i>). That's a discrete set of complex numbers labeled by i - it's conceptually a lot more like a vector than a function. Furthermore it doesn't satisfy anything remotely resembling a wave equation. If one instead uses the position or momentum basis, <x|\psi> or <p|\psi>, that is a function and it does satisfy an equation that's a bit more like a wave equation.

So, if we define "wave function" to mean "state vector in any representation" as is done currently, it's (a) pretty much identical to "quantum state" and (b) in some representations it's neither a wave nor a function. Perhaps we should define it instead as the position representation <x|\psi>? One problem with that is that people use "wave function" more loosely than that - for example, "momentum space wavefunction". So instead, maybe we should define it as either <x|\psi> or <p|\psi>, but not other representations? Or just as any continuous representation? Comments? <small>'''<span style="color:Olive">Waleswatcher</span>''' [[User_talk:Waleswatcher#top|''(<span style="color:green">talk</span>)'']]</small> 12:55, 3 February 2016 (UTC)

:The present attitude of this article is that a state vector ([[pure state|pure quantum state]]) is characterized by a complete set of quantum numbers (superposition allowed!) and wave functions are projections of state vectors onto a ''complete set'' of state vectors, any which one. This is a very clean definition. It does include the energy expansion (whenever it applies). All representations satisfy the relevant Schrödinger equation, including the energy representation. (To see this, just expand the energy eigenstates in the position representation.) I don't want to change this ''attitude'', because it is "all inclusive" and, besides, it is correct except perhaps terminology-wise. We could change terminology in places. But thinking of functions as vectors in this context is something one has to get used to in the long run. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 13:54, 3 February 2016 (UTC)

:Weinberg uses the terminology "coefficient" or "coefficient function" in place of wave function in the context of QFT. We could do something with that (verifiably), and toss away such things as the energy representation under that label. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 13:59, 3 February 2016 (UTC)

:Hmm..., there are situations in which the interesting dynamics lies in the spin part of the wave function. I decidely do not want to exclude such cases by limiting the article to {{math|Ψ(''x'', ''t'')}}. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 14:15, 3 February 2016 (UTC)

:: Let me ask you this - would you ever write |\psi> = \sum c_i |i> , or <i|\psi> = c_i, and then refer to the c_i as a "wave function"? I wouldn't (because, as I said above, the c_i are neither a wave nor a function in any normal usage of those terms). However, I do agree that the term "wave function" can be used loosely to mean "quantum state", in whatever representation, and come to think of it I'm pretty sure while lecturing on QM I've uttered the phase "spin part of the wave function", for instance. What bugs me is that in that usage it's basically synonymous with quantum state, so doesn't really need its own article. <small>'''<span style="color:Olive">Waleswatcher</span>''' [[User_talk:Waleswatcher#top|''(<span style="color:green">talk</span>)'']]</small> 15:31, 3 February 2016 (UTC)

:::You are right about the terminology. I wouldn't call it a wave function. We could have an introductory section where proper (at least for all acceptable) terminology is established. Then the focus should be on position and momentum space, with sections devoted to energy representation, and exotic spin wave functions where we use terminology appropriate to ''them''. Just an idea. (Landau & Lifshitz use wave function for anything b t w - edit:meaning that they don't talk much about abstract ''states'' and Hilbert spaces. They work mostly with the image of Hilbert space in any coordinates (another Hilbert space)). I'm not fond of the idea of scrapping the article. It is too much to squeeze into quantum state (that has the additional burden of non-wave functions like mixed states). [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 15:50, 3 February 2016 (UTC)

::::I think there is a case for the article to distinguish between explicit and symbolic expressions of wave functions. In older texts, wave functions were expressed explicitly as functions of the relevant ___domain, as for example {{math|''ψ''('''r'''₁, '''r'''₂, ''t'') {{=}} ''α''₁ exp (−}}|{{math|'''r₁'''}}|{{math|²) exp (−i''E''₁''t''/''ħ'') + ''α''₂ exp (−}}|{{math|'''r₂'''}}|{{math|²) exp (−i''E''₂''t''/''ħ'')}}. A symbolic expression is, for example, in Dirac's notation, |{{math|''z''}}〉 = {{math|''ζ''₁}}|{{math|''z''₁}}〉 + {{math|''ζ''₂}}|{{math|''z''₂}}〉</noinclude> .[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 22:57, 3 February 2016 (UTC)

:::::The article does distinguish. It is wave functions versus states. Waleswatcher's point (and perhaps also mine), is that the distinction allows for too much to be dignified as wave functions. Perhaps we should redefine as (coefficients/coefficient functions/coordinate expressions/your name here) versus states, and that wave functions is a conventional subset of (coefficients/coefficient functions/coordinate expressions/your name here). Not everything as it is now. "Explicit and symbolic expressions" looks like your own invention of terminology. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 09:17, 4 February 2016 (UTC)

:::::Nipping the bud, in

::::::<math>|\Psi\rangle = I|\Psi\rangle = \int |x\rangle \langle x|\Psi\rangle dx = \int \Psi(x) |x\rangle dx = \int |p\rangle \langle p|\Psi\rangle dp = \int \Psi(p) |p\rangle dp,</math>

:::::it is the case that

::::::<math>|\Psi\rangle, |x\rangle, |p\rangle</math>

:::::are states, while

::::::<math>\Psi(x), \Psi(p)</math>

:::::are wave functions. There is never equality between states and wave functions. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 11:17, 4 February 2016 (UTC)

The use of bra-ket notation to give the position/momentum representations has been introduced more and more earlier in the article. Nothing wrong with it. I'm getting from this discussion that everything from [[Wave function#Discrete and continuous bases]] up to the ontology section is too general for this article and must be deleted? [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 13:29, 4 February 2016 (UTC)

:<small>(The above is to demonstrate that Chjoaygame's invention of symbolic wave functions do not stand up to inspection.)</small> That conclusion has not been reached (yet). But at least change of terminology is probably preferred where appropriate.

:I have had a look at [[quantum state]]. In my mind, ''that'' article should deal exclusively with abstract states living in Hilbert space. ''This'' article should deal with states as viewed when projected onto a particular basis, and the machinery that comes with it. This is ''another'' Hilbert space, one for each choice of representation. Then the question is whether to limit coverage to position and momentum representations. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 13:46, 4 February 2016 (UTC)

::::::I didn't invent the term 'symbolic' in this context for this purpose. Dirac (4th edition 1958), an extract from the preface to the 1st (1930) edition:

:::::::::::With regard to the mathematical form in which the theory can be presented, an author must decide at the outset between two methods. There is the symbolic method, which deals directly in an abstract way with the quantities of fundamental importance (the invariants, etc., of the transformations) and there is the method of coordinates or representations, which deals with sets of numbers corresponding to these quantities. The second of these has usually been used for the presentation of quantum mechanics (in fact it has been used practically exclusively with the exception of Weyl's book ''Gruppentheorie und Quantenmechanik''). It is known under one or other of the two names 'Wave Mechanics' and 'Matrix Mechanics' according to which physical things receive emphasis in the treatment, the states of a system or its dynamical variables. It has the advantage that the kind of mathematics required is more familiar to the average student, and also it is the historical method.<ref>[[Paul Adrien Maurice Dirac|Dirac, P.A.M.]] (1958). ''The Principles of Quantum Mechanics'', 4th edition, Oxford University Press, Oxford UK, p. viii.</ref>

::::::Further on, Dirac writes

:::::::::::A further contraction may be made in the notation, namely to leave the symbol 〉 for the standard ket understood. A ket is then written simply as {{math|''ψ''(''ξ'')}}, a function of the observables {{math|''ξ''}}. A function of the {{math|''ξ''}}s used in this way to denote a ket is called a ''wave function''.† The system of notation provided by wave functions is the one usually used by most authors for calculations in quantum mechanics. In using it one should remember that each wave function is understood to have the standard ket multiplied into it on the right, which prevents one from multiplying the wave function by any operator on the right. ''Wave functions can be multiplied by operators only on the left.'' This distinguishes them from ordinary functions of the {{math|''ξ''}}s, which are operators and can be multiplied by operators on either the left or the right. A wave function is just the representative of a ket expressed as a function of the observables {{math|''ξ''}}, instead of eigenvalues {{math|''ξ′''}} for those observables.

:::::::::::&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;† The reason for this name is that in the early days of quantum mechanics all the examples of these functions were of the form of waves. The name is not a descriptive one from the point of view of the modern general theory.<ref>[[Paul Adrien Maurice Dirac|Dirac, P.A.M.]] (1958). ''The Principles of Quantum Mechanics'', 4th edition, Oxford University Press, Oxford UK, p. 80.</ref>

::::::Messiah (1958):

:::::::::::Of the various ways of introducing the Quantum Theory, the one which uses the general formalism is undoubtedly the most elegant and the most satisfactory. However, it requires the handling of a mathematical symbolism whose abstract character runs the risk of masking the underlying physical reality. Wave Mechanics, which utilizes the more familiar language of waves and partial differential equations, lends itself better to a first encounter. Furthermore, it is in that form that the Quantum Theory is most frequently used in elementary applications. That is why we shall begin with a general outline of Wave Mechanics.<ref>[[Albert Messiah|Messiah, A.]] (1958/1961). ''Quantum Mechanics'', volume 1, translated by G.M. Temmer from the French ''Mécanique Quantique'', North-Holland, Amsterdam, p. 48.</ref>

::::::By 'wave function' Messiah (and many other texts) means something such as for example

:::::::::::... the matter wave

:::::::::::::::<math>\psi (\mathbf r_2, \tau_2) = \int K (\mathbf r_2 - \mathbf r_1; t_2 - t_1) \psi (\mathbf r_1, t_1) \, \mathrm d \mathbf r_1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (1)</math>

:::::::::::where

:::::::::::::::<math>K(\mathbf r, \tau) \,\,\,= (2\pi \hbar)^{{-3}} \int \exp \left [\frac {\mathrm i}{\hbar}(\mathbf p \cdot \mathbf r - E \tau)\right] \mathrm d \mathbf p.</math><ref>[[Albert Messiah|Messiah, A.]] (1958/1961). ''Quantum Mechanics'', volume 1, translated by G.M. Temmer from the French ''Mécanique Quantique'', North-Holland, Amsterdam, p. 75.</ref>

::::::Messiah does not feel a need for the ordinary-language word 'explicit' here because the situation seems obvious to him from what he has written, and he is not using the Dirac notation at that point, so as to need a contrast. I think for clarity for our purpose here an ordinary language word is needed to distinguish the two forms of expression. I did not invent the more technical term 'symbolic'; that is Dirac's. A symbolic expression is, for example, in Dirac's notation, |{{math|''z''}}〉 = {{math|''ζ''₁}}|{{math|''z''₁}}〉 + {{math|''ζ''₂}}|{{math|''z''₂}}〉 .

::::::Schrödinger invented wave functions and it may be fair to give an example from him. He writes

:::::::::::<math>(26^{\prime}) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\psi_n (q) \,\,\,\,= e^{- \frac{2\pi^2\nu_0q^2}{h}} H_n \left (2\pi q \sqrt {\frac {\nu_0}{h}}\right)</math><ref>[[Erwin Schrödinger|Schrödinger, E.]] (1926). 'Quantisierung als Eigenwertproblem, Zweite Mitteilung', ''Annalen der Physik'', Series 4, '''79'''(6): 489–527.</ref>

::::::Weinberg in his ''Lectures'', on page 34, explicitly expresses a wave function as a function thus:

::::::::::::::::<math>\psi (\mathbf x) \,\,\,\,\,= R(r)Y(\theta,\phi) ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (2.1.21)</math><ref>[[Steven Weinberg|Weinberg, S.]] (2013). ''Lectures on Quantum Mechanics'', Cambridge University Press, Cambridge UK, {{ISBN|978-1-107-02872-2}}, p. 34.</ref>

'''References'''

{{Reflist}}

::::::What I am trying to draw attention to is not a distinction between two symbolic forms of expression. That distinction is, as you say, already drawn clearly by the article. The distinction I am pointing to is between a symbolic form, as labeled by Dirac, and a non-symbolic form (to use Dirac's words, "a function of the observables {{math|''ξ''}}, instead of eigenvalues {{math|''ξ′''}} for those observables") that I think ordinary language would call 'explicit'. I have now tried to indicate what I mean by two examples. If you think some ordinary-language word other than 'explicit' would be better, I have no prejudice. But I think some indicative word is needed for clarity.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 16:25, 4 February 2016 (UTC)

:::::::And I think your references are too old to guide us at all in the choice of terminology. The "ordinary language" of yours have always had me confused. It could mean pretty much anything any given time. We should stick to present day terminology. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 09:32, 5 February 2016 (UTC)

Where ''should'' we draw the line on which representations or observables are relevant to the term "wavefunction"? There are canonical transformations on position and momentum. In solid state physics there are [[Bloch wave]]s which include wavevectors. We cannot rule out discrete representations, because in particle physics a wavefunction of a quark can be split into the product of spacetime, spin, and colour wavefunctions. There are others like isospin, in condensed matter physics there is something called [[pseudospin]] (no article?), in nuclear physics the nuclear angular momenta quantum number (the total angular momentum of all the nuclei), and in chemistry there are other angular momentum related quantum numbers (they are listed in Atkin's quanta book). Shouldn't "wavefunction" in general be defined as the component of a state vector (with discrete and/or continuous representations) ''which solves'' the SE or any other relativistic wave equation? [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 11:54, 5 February 2016 (UTC)

:Perhaps we can handle this on a case by case basis. Waleswatcher's example of expansion coefficients in a countable energy representation is a good example of something rarely called a wave function. Your examples provide solid proof that there's more than the position and momentum representations called wave functions. Other than that, I am getting more and more convinced that the ''attitude'' of the present article (anything goes) is right. It is easier logically to present the full story. That said, the article really is badly organized, with a too bulky lead, and appropriate terminology can be introduced for things not usually called wave functions. By the way, is

::<math>\langle x|x'\rangle = x'(x) =\delta(x - x'),</math>

:the position eigenstate, a wave function? It does not satisfy the Schrödinger equation in the usual sense, but (OR{{smiley}}) it turns out that it does so in the sense of the left and the right side of the Scrödinger equation with {{math|''δ''(''x'' − ''x''&prime;)}} plugged in being equal as distributions (or continuous linear functionals). When they act on [[wave packet]]s composed of free waves, they yield the same result. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 12:15, 5 February 2016 (UTC)

::I did not express myself clearly, indeed, I used faulty forms of expression. But still I have something to say, if I can put my ideas better.

::Above I gave an example of my idea of an explicit wave function: {{math|''ψ''('''r'''₁, '''r'''₂, ''t'') {{=}} ''α''₁ exp (−}}|{{math|'''r₁'''}}|{{math|²) exp (−i''E''₁''t''/''ħ'') + ''α''₂ exp (−}}|{{math|'''r₂'''}}|{{math|²) exp (−i''E''₂''t''/''ħ'')}}. I think many texts display objects more or less like that, and call them wave functions.

::And an example of a symbolic expression: |{{math|''z''}}〉 = {{math|''ζ''₁}}|{{math|''z''₁}}〉 + {{math|''ζ''₂}}|{{math|''z''₂}}〉.

::The symbolic expression is not a wave function, but it shows how a wave function is conceived in symbolic terms. The wave function, conceived there in symbolic terms is {{math|(''ζ''₁, ''ζ''₂)}}. The point is that, in this symbolic frame of thought, one doesn't think in terms of explicit wave functions, such as the one I just gave as an example. One thinks of the basis in terms of the kets, keeping consistently in the symbolic frame of thought, though the wave function itself doesn't actually write the kets. I think Dirac says that {{math|(''ζ''₁, ''ζ''₂)}} can be viewed as a function:

:::::::We may suppose the basic bras to be labelled by one or more parameters, {{math|''λ''₁, ''λ''₂, ... ''λ_u''}}, each of which may take on certain numerical values. The basic bras will then be written 〈{{math|''λ''₁ ''λ''₂ ... ''λ_u''}}| and the representative of |{{math|''a''}}〉 will be written 〈{{math|''λ''₁ ''λ''₂ ... ''λ_u''}}|{{math|''a''}}〉. This representative will now consist of a set of numbers, one for each set of values that {{math|''λ''₁, ''λ''₂, ... ''λ_u''}} may have in their respective domains. Such a set of numbers just forms a ''function'' of the variables {{math|''λ''₁, ''λ''₂, ... ''λ_u''}}. Thus the representative of a ket may be looked upon either as a set of numbers or as a function of the variables used to label the basic bras.<4th edition, page 54.>

::I am sorry I didn't manage to express this clearly before now.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 15:05, 5 February 2016 (UTC)

:::{{Reply to|YohanN7}} Sure, so long as the space of wave functions is extended to distributions. Incnis tried this a while ago. I was simply thinking of these quantities

::::<math>\Psi ( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} ) </math>

:::from

::::<math>| \Psi \rangle = \sum_{s_{z\,1} , \ldots , s_{z\,N}}\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1 \, \Psi ( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} ) | \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle </math>

:::provided they solve the SE for the system, are called wavefunctions.

:::{{Reply to|Chjoaygame}} The expression you give is not dimensionally consistent in the exponential of position (you'd need to divide |'''r₁'''|² and |'''r₂'''|² each by a constant with units length² to get a number, then take the exponential). You seem to be making things quite complicated; yes a wavefunction is a complex-valued function, and the wavefunction is a function of these observables. The observables are also used to form a [[basis (linear algebra)|basis set]] of kets (one doesn't use a basis (basic?) bra like 〈{{math|''λ''₁ ''λ''₂ ... ''λ_u''}}|). What you call "representative" is really a component of the state vector (see [[coordinate vector]]). What you call "symbolic expression" seem to just be kets. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 15:42, 5 February 2016 (UTC)

::::Sorry, I just wanted to give an example of the kind of thing, not complicate it with scaling factors. Yes, kets are symbols, that's how Dirac described them. I am not clear about what you write. It seems you are saying Dirac's statements about bras and kets are wrong?[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 16:48, 5 February 2016 (UTC)

:::::The modern terminology is this:

::::::<math>\underbrace{| \Psi \rangle}_{\text{state vector (ket)}} = \underbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}_{\text{adding up}} \, \underbrace{\Psi ( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}_{\text{wavefunction (component of state vector)}} \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis ket}}</math>

:::::We seem to just be using different terminology. Also you mentioned a bra as a particular basis element, but bras are not too important unless you calculate an inner product. All you need are kets; the corresponding bras can be obtained by taking the dual (Hermitian conjugate). I can't remember what Dirac wrote, will have to check in his Principles of quantum mechanics. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 17:15, 5 February 2016 (UTC)

:::::Forgot to mention bras are also important in forming operators in a given basis, and the completeness condition for manipulating bra-ket expressions, each are in this article and the [[bra-ket notation]] article. Otherwise kets come first, then taking duals of them. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 13:36, 6 February 2016 (UTC)

::::::[[User:Chjoaygame|Chjoaygame]], you know and I know that whatever we call things in the article, you'd chose to call it something else. That you can find the word "symbolic" in a foreword to a 193x book (even one by Dirac) is not even remotely notable. Yes, Dirac would be pretty damned loopy if he called state vectors (= kets) symbols. I don't think he did. If he did, then it is complicating simple matters beyond recognition and should forever be ignored. If you need to verbatim put things in an abstract setting (particular or any state vector in specified or unspecified Hilbert space with ''unspecified'' basis), then the word "abstract" is the way to go. Just like in the article. It is even ordinary language and ''should'' therefore be to your liking (had we not used it in the article of course).

::::::[[User:Maschen]]'s description above is the clearest and most spot on exposition of the proper concepts and terminology I have ever seen. This should go into the first section after the lead in the good article to be (whether verifiable or not). [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 10:18, 6 February 2016 (UTC)

::::::<math>\underbrace{| \Psi \rangle}_{\text{state vector (ket)}} = \underbrace{\overbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}^{\text{discrete labels}}}\overbrace{\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}}}_{\text{adding up}} \, \underbrace{\overbrace{\Psi}^{\text{wave function}} ( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}_{\text{component of state vector along basis state}} \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis state (basis ket)}}</math>

::::::[[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 12:44, 6 February 2016 (UTC)

:::::::{{reply to|YohanN7}} Thanks for feedback, and I like your elaborated version.

:::::::{{reply to|Chjoaygame}} Dirac's ''Principles of Quantum Mechanics'' 4th edition p.16, he uses both: "ket vectors or simply kets" to the name the vector, and "symbol {{ket|}}" for the symbols used in the notation (so it seems). In any case, this is a little pedantic and off-topic for this article. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 13:30, 6 February 2016 (UTC)

::::::::I wrote above "If you think some ordinary-language word other than 'explicit' would be better, I have no prejudice. But I think some indicative word is needed for clarity." That goes also for bras and kets; if Editor [[User:YohanN7|YohanN7]] thinks some other word than 'symbolic' would be better, I have no prejudice or attachment to it. Evidently he prefers 'abstract'. I have no problem with that. I used the word 'symbolic' just because I read it in Dirac and found it helpful. It has been used by others, some of whom have systematically presented both modes of expression. For example, Messiah on page 48: "However, it requires the handling of a mathematical symbolism whose abstract character runs the risk of masking the underlying physical reality."

::::::::Above I quoted from Dirac [https://archive.org/stream/DiracPrinciplesOfQuantumMechanics/Dirac%20-%20Principles%20of%20quantum%20mechanics#page/n63/mode/1up where he uses bras]. I didn't think that was a worry. I was surprised that Editor [[User:Maschen|Maschen]] objected to it. It is also to be found in the current version of the article:

:::::::::::::The wave function corresponding to an arbitrary state {{ket|Ψ}} is denoted

::::::::::::::<math>\langle a, b, \ldots, l, m, \ldots|\Psi\rangle,</math>

:::::::::::::for a concrete example,

::::::::::::::<math> \Psi(x) = \langle x|\Psi\rangle.</math>

::::::::I agree that Editor Maschen's formulation is admirable, and evidently Editor YohanN7 finds it fresh. I think it agrees with the [https://archive.org/stream/DiracPrinciplesOfQuantumMechanics/Dirac%20-%20Principles%20of%20quantum%20mechanics#page/n89/mode/1up page I quoted from Dirac], and have no worry about putting it in the article. I would suggest adjusting it slightly, to make it agree with Wikipedia definitions as follows:

::::::::<math>\underbrace{| \Psi \rangle}_{\text{state vector (ket)}} = \underbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}_{\text{adding up}} \, \underbrace{\Psi ( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}_{\text{wavefunction (scalar projection of state vector)}} \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis ket}}\,.</math>

::::::::I am suggesting to use the term [[scalar projection]].

::::::::Several comments have come in here while I have been writing this one. I agree with Editor YohanN7's adjustments to the admirable expression of Editor Maschen, and think that they could include my suggestion of using the term 'scalar projection' instead of 'component' without harm.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 14:12, 6 February 2016 (UTC)

I too like Maschen's formula, but I think it is better with these labels - these are the same as YohanN7's improvement but without the separation of "wavefunction" from "component of state vector along basis state". Since those are the same thing and we are trying to define "wavefunction", I don't think we should separate them, and certainly not make it look as though \Psi and the arguments of \Psi are different objects.

<math>\underbrace{| \Psi \rangle}_{\text{state vector (ket)}} = \underbrace{\overbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}^{\text{discrete labels}}}\overbrace{\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}}}_{\text{adding up}} \, \underbrace{{\Psi}( \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}_{\text{wavefunction (component of state vector along basis state)}} \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis state (basis ket)}}</math>

As for "scalar projection" - no. If anything, just "projection" - but "component" is probably better. <small>'''<span style="color:Olive">Waleswatcher</span>''' [[User_talk:Waleswatcher#top|''(<span style="color:green">talk</span>)'']]</small> 14:35, 6 February 2016 (UTC)

:No, a function, its arguments, and a function given an argument are three different objects. I know that it is standard in physics not to distinguish functions from functions given their arguments. But the mathematically inclined reader will get allergic reactions. Since the expression we are now cooking up is somewhat pretentious and strives for precision, we should not allow for any "terminologisms" particular to a field (math, phys, etc), except those almost forced upon us (Dirac notation is really superior here, but this is more of notation than terminology).

::<math>\underbrace{| \Psi \rangle}_{\text{state vector (ket)}} = \underbrace{\overbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}^{\text{discrete labels}}}\overbrace{\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}}}_{\text{adding up}} \, \underbrace{\overbrace{\Psi}^{\text{wave function}} \overbrace{(\mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}^{\text{eigenvalues of commuting observables}}}_{\text{component of state vector (complex number)}} \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis state (basis ket)}}</math>

:[[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 09:23, 8 February 2016 (UTC)

{{reply to|Waleswatcher}} Sure, I'm not set on the exact labels and like your version better than mine.

{{reply to|Chjoaygame}} In

:<math>\langle a, b, \ldots, l, m, \ldots|\Psi\rangle </math>

what you are actually doing is start with {{ket|''a, b, ..., l, m,''}} then taking the dual to get the bra {{bra|''a, b, ..., l, m,''}} then taking the inner product of two kets {{ket|''a, b, ..., l, m,''}} and {{ket|Ψ}} (not a bra with a ket), in other words projecting {{ket|Ψ}} on {{ket|''a, b, ..., l, m,''}}, to obtain Ψ(''a, b, ..., l, m,''). This is what I wrote about above.

Now that hopefully clears up Chjoaygame's comments, it would be helpful to continue thinking about edits to improve the article. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 16:13, 6 February 2016 (UTC)

:Two ideas to get started de-cluttering and shortening the article:

:*A year ago the lead was a sensible length, and has since grown as others have noted. Maybe we could revert to an earlier version and tweak it (e.g. crisper clarification of spin, spinor, tensors, and degrees of freedom).

:*Either

:**All the cases for various numbers of particles, numbers of dimensions, no spin or spin (more generally other discrete variables) could be presented as complex valued functions and state vectors in braket notation. We need to decide other degrees of freedom are relevant for this article. Then all the general formalism of braket notation could be trimmed (most ideal, what is in [[bra-ket notation]] doesn't need to be in this article) or deleted entirely (not that I'm keen on that after writing much of it, but no matter).

:**Give the general formulation of continuous/discrete/mixed at the outset, then just give possible examples of what the continuous/discrete variables can be. (It may be compact, but unlikely to be favourable and less easy for typical readers to follow).

:[[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 16:44, 6 February 2016 (UTC)

::One can make plonking remarks about bras, relying on the mathematical point that Halmos calls the "brackets-to-parentheses revolution". The real targets of the plonking are, for example, Dirac, Weinberg, and Cohen-Tannoudji. They speak of the "scalar product". Gratifying though such plonking might be for mathematicians, it does not indicate physical understanding. The reason for distinguishing bras and kets is the physical distinction between preparation and observation of a quantum system, as indicated by Dirac.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 10:19, 7 February 2016 (UTC)

::For example, in his paper 'Derivation of the Born rule from operational assumptions', [[Simon Saunders|Saunders]] writes:

::::::The kinds of experiments we shall consider are limited in the following respects: they are repeatable; there is a clear distinction between the state preparation device and the detection and registration device; and - this the most important limitation - we assume that for a given state-preparation device, preparing the system to be measured in a definite initial state, the state can be resolved into ''channels'', each of which can be independently blocked, in such a way that when only one channel is open the outcome of the experiment is ''deterministic'' - in the sense that if there is any registered outcome at all (on repetition of the experiment) it is always the ''same'' outcome.<ref>[[Simon Saunders|Saunders, S.]] (2004). 'Derivation of the Born rule from

operational assumptions', ''Proc. Roy. Soc. A'', '''460''': 1-18.</ref>

{{Reflist}}

::[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 10:49, 7 February 2016 (UTC)

::Ignoring yet more of Chjoaygame's experiments, channels, quotes, and now this time "plonking", all off-topic for this article, I'll amend my last post to be more specific.

::*Not to revert the entire article back to a 2015 or earlier stage, but just the lead. Maybe reinstate [https://en.wikipedia.org/w/index.php?title=Wave_function&oldid=640575092 this version of the lead] and tweak it from there.

::*Move the sections [[Wave function#More on wave functions and abstract state space]] (which includes the SE) and [[Wave function#Time dependence]] higher up to somewhere in [[Wave function#Wave functions and function spaces]] now that Dirac notation has been introduced earlier

::*merge the sections

::**[[Wave function#Definition (one spinless particle in 1d)]] + [[Wave function#State space for one spin-0 particle in 1d]]

::**[[Wave function#Definitions (other cases)]] + [[Wave function#State space (other cases)]] + [[Wave function#Tensor product]]

::*have a single section gathering all things on the probability interpretation, most specifically all requirements for the interpretation, and [[Wave function#Normalized components and probabilities]] in there

::*anything left over could be further rearranged, rewritten, moved to other articles, or deleted.

::In this article, giving the bra-ket notation for the specific examples should be enough for readers to get some idea how continuous/discrete representations are formulated. Generalities should be in the articles on [[bra-ket notation]], [[quantum state]], [[identical particles]], etc. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 11:54, 7 February 2016 (UTC)

=== Proposal for dissection of definition of wave function ===

::<math>\underbrace{| \Psi \rangle}_{\text{state vector (ket)}} = \underbrace{\overbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}^{\text{discrete labels}}}\overbrace{\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}}}_{\text{adding up}} \, \underbrace{\overbrace{\Psi}^{\text{wave function}} \overbrace{(\mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} )}^{\text{eigenvalues of commuting observables}}}_{\text{component of state vector (complex number)}} \underbrace{| \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} \rangle }_{\text{basis state (basis ket)}}</math>

I am not religiously attached to the choice of any particular label, but I am a bit attached to the anatomy though, including the distinction between functions, their arguments, and their values. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 09:42, 8 February 2016 (UTC)

With these ingredients, the "local choice" in the article to always have the wave function scalar-valued as opposed to vector-valued (e.g. one entry for every spin z-component) makes a good measure of sense. It also helps making clear beyond any doubt what the ___domain and range of the wave function is (something that caused lengthy discussions here not long ago). Each of them corresponds to a brace.[[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 15:13, 8 February 2016 (UTC)

== Revision 2016-02-08 ==

Line 67 ⟶ 246:

:Likely I am missing the main point here.

[[File:Quixo-panza.jpg|Don Quixote and Sancho Panza|right|thumb|250px|Don Quixote, his horse Rocinante and his squire Sancho Panza after an unsuccessful attack on a windmill. By [[Gustave Doré]].]]

:::::<small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/194.68.82.241|194.68.82.241]] ([[User talk:194.68.82.241|talk]]) 13:55, 10 February 2016 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot--> Re-posted by [[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 14:19, 10 February 2016 (UTC)

:But anyway, here's a start. "How do you characterize a "maximal commuting set of observables"?" I think this is standard phrasing, at least in some places. One starts with some choice of observable. Then one chooses another. If they commute, it stays; if they don't, it's out. Repeat until one can't find any more that commute. I suppose that seems rather rough and ready, and hardly convincing. I will forthwith have a look to check this. Or is this utterly missing the point?[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 11:05, 10 February 2016 (UTC)

Line 114 ⟶ 293:

:::::::German original (1932/1996), ''Mathematische Grundlagen der Quantenmechanik'', Springer, Berlin, ISBN-13: 978-3-642-64828-1, p. 79: "Einen Operator, der keine echten Fortsetzungen besitzt — der also an allen Stellen, wo er vernünftigerweise, d. h. ohne Durchbrechung des Hermiteschen Charakters, definiert werden könnte, auch schon definiert ist — nennen wir maximal. Wir haben also gesehen: nur zu maximalen Operatoren kann eine Zerlegung der Einheit gehören."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 04:41, 11 February 2016 (UTC)

:::::::Newton, R.G. (2002), in ''Quantum Physics: a Text for Graduate Students'', Springer, New York, {{ISBN |0-387-95473-2}}, writes on page 317: "Suppose that <math>\mathfrak V</math> is such that there is a maximal number of linearly independent vectors in it, i.e., given any set of non-zero vectors with more than <math>n</math> members, they must be linearly dependent. The number <math>n</math> is then called the dimension of <math>\mathfrak V</math>." He doesn't use it in that sense elsewhere in that book.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 16:31, 10 February 2016 (UTC)

:::::::Bransden, B.H., Joachain, C.J. (1989/2000), ''Quantum Mechanics'', second edition, Pearson–Prentice–Hall, Harlow UK, {{ISBN |978-0-582-35169-7}}, p. 641: "Until now we have considered quantum systems which can be described by a ''single'' wave function (state vector). Such systems are said to be in a ''pure state''. They are prepared in a specific way, their state vector being obtained by performing a ''maximal measurement'' in which all values of a complete set of commuting observables have been ascertained. In this chapter we shall study quantum systems such that the measurement made on them is not maximal. These systems, whose state is incompletely known, are said to be in ''mixed states''."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:32, 10 February 2016 (UTC)

:::::::[[Gennaro Auletta|Auletta, G.]], Fortunato, M., [[Giorgio Parisi|Parisi, G.]] (2009), ''Quantum Mechanics'', Cambridge University Press, Cambridge UK, {{ISBN |978-0-521-86963-8}}, p. 174: "From Sec. 1.3 and Subsec. 2.3.3 we know that the state vector |{{math|''ψ''}}〉 contains the maximal information about a quantum system."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:48, 10 February 2016 (UTC)

:::::::I Googled the phrase 'maximal set of commuting observables', and found [https://books.google.com.au/books?id=4ZwKCAAAQBAJ&pg=PA16&lpg=PA16&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=ihL-ouMAoN&sig=iXg5Sm2CzApf0ZEKLZC0OsNO4Xk&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIMTAG#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false this], and [https://books.google.com.au/books?id=lJaX2PsTxNoC&pg=PT94&lpg=PT94&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=N7mHtcmXS9&sig=lIFMSkB-3JiMp3XG_yMjMaGqBAg&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEILjAF#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false also this], and [https://books.google.com.au/books?id=Xg2NZD73b4cC&pg=PA107&lpg=PA107&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=XNpZbW3V-K&sig=hjtTtngz9OkXfbzrTtv2A4EeZA4&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIKjAE#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false moreover this], and yet [https://books.google.com.au/books?id=eWdDAAAAQBAJ&pg=PA131&lpg=PA131&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=5ICKajbc5X&sig=UHUeWc7xUNMAAFvJnAQlfR9MhMM&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIOTAI#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false again this], and [https://books.google.com.au/books?id=v1owGsfiJcoC&pg=PA4&lpg=PA4&dq=maximal+set+of+commuting+observables+quantum+mechanics&source=bl&ots=k9hKAHG4-4&sig=wGW8ASjUSj6u_x9XrRjxzHZpvts&hl=en&sa=X&ved=0ahUKEwiQg8GC6u7KAhVEkZQKHQ4qB_IQ6AEIPDAJ#v=onepage&q=maximal%20set%20of%20commuting%20observables%20quantum%20mechanics&f=false now this].[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 04:18, 11 February 2016 (UTC)

Line 201 ⟶ 380:

:"And how is "a state vector is an [[equivalence class]] of wave functions"?" I would have thought that was a standard way of expressing the situation. I learnt it when I studied algebra. It seems to be assumed as common mathematical parlance by the writer of this sentence: "Assuming that the unchanging reading of an ideal thermometer is a valid "tagging" system for the equivalence classes of a set of equilibrated thermodynamic systems, then if a thermometer gives the same reading for two systems, those two systems are in thermal equilibrium, and if we thermally connect the two systems, there will be no subsequent change in the state of either one." The sentence was posted in [https://en.wikipedia.org/w/index.php?title=Zeroth_law_of_thermodynamics&diff=next&oldid=667219209 this] edit by respected Editor [[User:PAR|PAR]]. My usage intends that all the wave functions that belong to a particular state are interconvertible by a group of one-to-one mathematical transformations. That makes them members of an equivalence class. (The equivalence class has the structure of a Hilbert space, more or less.) I find this form of expression helpful to show the relation between wave functions and state vectors. It may or may not be so for others.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 09:39, 14 February 2016 (UTC)

:Well, it seems that I have led myself astray by looking in Wikipedia and Google. Looking at a textbook on my shelves that I forgot I had, I find that indeed, as you say, a component is there defined as a scalar. Bloom, D.M. (1979), ''Linear Algebra and Geometry'', Cambridge University Press, Cambridge UK, {{ISBN |0-521-21959-0}}, p. 98. I hardly need say this makes me look silly. I am sorry. I can only say I misled myself by looking in Wikipedia and Google. That's a lesson. Well, I can only say I am sorry. My only excuse can be that I wrote "I would suggest adjusting it slightly, to make it agree with Wikipedia definitions as follows: .... I am suggesting to use the term [[scalar projection]]." Evidently that was a mistake. Now checking more in Wikipedia, I find at [[Basis (linear algebra)]] that I did not look in right place in Wikipedia. Just for clarity here, I will repeat, I now agree that 'component' is suitable. I guess a link to [[Basis (linear algebra)]] might be a good idea.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 12:37, 14 February 2016 (UTC)

:See [[Talk:Scalar projection#This article has gravely misled me, and helped to make me look foolish, because I thought that on such a simple matter, an article like this could be trusted.]][[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 12:59, 14 February 2016 (UTC)

:Also [[Talk:Basis (linear algebra)/Archive 1#customary terminology not clear in Wikipedia; local editors, heads up]].[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 18:17, 14 February 2016 (UTC)

:Perhaps I went overboard with the ''mea culpa''. Looking a bit further, I get the impression that customs vary.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 19:26, 14 February 2016 (UTC)

Line 226 ⟶ 405:

:::Though, a reader interested only in [[Linear combination of atomic orbitals]] needs less... [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 12:53, 16 February 2016 (UTC)

== Do we need this article at all? ==

We have (a) wave mechanics well described in [[Schrödinger equation]], (b) [[Matrix mechanics]], and (c) their synthesis in [[Mathematical formulation of quantum mechanics#Postulates of quantum mechanics]] and [[Matrix mechanics#Wave mechanics]]. Is anything still missing? Sure, textbooks contain more detailed information, but we are not a textbook. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 06:51, 17 February 2016 (UTC)

:This was raised by [[user:Waleswatcher]] many reams ago.

:I would say ''yes'', since a lot of people coming to WP would expect to see an article about wavefunctions (at least the basics on interpretations, ontology, and examples).

:Then again this article has a long history of extensive rewriting and people still tend to feel unhappy about it. So if people think there is no need for this article it could redirect to [[quantum state]]. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 09:06, 17 February 2016 (UTC)

::Does it mean that here they need very basic explanations for beginners, plus links to other articles on more advanced topics? [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 10:20, 17 February 2016 (UTC)

:::If this article is to stay, then yes. Ideally this article would take the reader from the popular science level (lots of people will come across the term "wavefunction" from something they have read) to undergraduate level (in physics or chemistry, when wavefunctions are first introduced), and little more to examples the reader may not expect (examples can be drawn from condensed matter and particle physics). At the same time, it should be formal enough and not vague. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 10:35, 17 February 2016 (UTC)

::::That is, to fill the gap between [[Introduction to quantum mechanics]] and harder articles. Nice. In the spirit of [[Linear combination of atomic orbitals|LCAO]]. But at the same time "be formal enough and not vague"? Is this possible? I guess, it ''must'' say many times something like this: "but this is only a fragment of the truth; deeper discussion of this matter needs both a good mathematical background and a lot of cogitation toward the interpretation". [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 11:12, 17 February 2016 (UTC)

::::In this case, I guess, the only "complete system of commuting observables" should be, the three Cartesian coordinates (implicitly, of course). And the only interpretation should be, the squared absolute value. And, of course, pointers to more advanced articles. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 11:17, 17 February 2016 (UTC)

:::::Actually no, not just the position representation but momentum and spin also. Other observables can be listed. No, not to explicitly keep saying "but this is only a fragment of the truth; deeper discussion of this matter needs both a good mathematical background and a lot of cogitation toward the interpretation", the scope of the article should be implicit from the context. "By formal enough and not vague", just using the minimum amount of mathematics correctly without abuses of terminology or concepts.

:::::There is still no agreement on what should be in this article. This is what I think the scope should be:

::::::*"status" of wavefunctions in QM past and present, and their position in the postulates of QM,

::::::*Nonrelativistic QM: wave particle duality, position and momentum representations, Fourier transforms, probability interpretation (and requirements for it to hold), spin, many particle systems, the Pauli principle, implications from them

::::::*Prototypical examples in physics (potential well, harmonic oscillator, hydrogen atom), in chemistry (atomic and molecular orbitals), more realistic examples in physics (particle physics, nuclear physics, condensed matter),

::::::*wavefunctions as spinors or tensors for particles of any spin, occurrence in relativistic QM and QFT

::::::*ontology and philosophy

:::::all in [[WP:Summary style]] as much as possible. What's wrong with that? If people want to insist on scrapping this article and redirecting elsewhere, that's up to them. [[user:YohanN7]] and I and others have tried our best to make the article decent. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 11:43, 17 February 2016 (UTC)

::::::Well, if you can do it... I could not. Such a large fragment of QM has too large boundary, and you'll get again the problem, how to cut it from the environment. As a result, the article will be long, not so accessible to beginners, overlap other articles, and editors will war along the boundary, forever. My idea was rather, to say this is a small and not self-contained fragment of QM, from which it is impossible to make any far-reaching conclusions. Spinors! -- hard math! Ontology and philosophy! -- in summary style! No, this is not for me. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 11:57, 17 February 2016 (UTC)

::::::To make it worse: I am very skeptical about any decent "ontology and philosophy" without contemporary achievements of quantum technology around quantum computation (cavity electrodynamics, ion traps etc). For example: what do you think about a generic pure state of 1000 qubits (say, spins-1/2)? I can prove easily that such state cannot be prepared at all (and I claim no credit, experts know this). Well, and Bell theorem, surely... "Progetto grandioso". I'll be very surprised if you'll succeed. I was puzzled by the "Don Quixote" picture inserted above by some anon, but now I start to understand it. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 12:17, 17 February 2016 (UTC)

::::::And, are you ready to answer such questions of ontology, as: does the wave function describe the system, or our knowledge about the system, or ensemble of systems, or the preparation process, or what? [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 13:31, 17 February 2016 (UTC)

:::::::There already is an [[Wave function#Ontology|ontology section]] in this article, and [[Interpretations of quantum mechanics|another article]] of its own.

:::::::I didn't claim I will write everything, the above points were just an outline. I will try later in the next few days to reorganize the article. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 15:09, 17 February 2016 (UTC)

::::::::Indeed... maybe I am too pessimistic. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 17:48, 17 February 2016 (UTC)

== a puzzle ==

In the article I read "One therefore talks about an abstract Hilbert space, '''state space''', where the choice of basis is left undetermined." Further on I read

"<big><u> '''Inner product''' </u></big>

Physically, the nature of the inner product is dependent on the basis in use, because the basis is chosen to reflect the quantum state of the system.

If {{ket|{{math|Ψ₁}}}} is a state in the above basis with components {{math|''c''₁, ''c''₂, ..., ''c''_''n''}} and {{ket|{{math|Ψ₂}}}} is another state in the same basis with components {{math|''z''₁, ''z''₂, ..., ''z''_''n''}}, the inner product is the complex number: ..."

???[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:51, 17 February 2016 (UTC)

:Mathematically, the inner product is independent of the basis in use; about "physically" ask a physicist. :-) We mathematicians define a Hilbert space as given with inner product (but not with basis; bases exist, but no one is chosen a priori). "basis is chosen to reflect the quantum state"? Strange. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 21:09, 17 February 2016 (UTC)

::The <s>edit</s> material was introduced by [https://en.wikipedia.org/w/index.php?title=Wave_function&diff=next&oldid=590355949 this] edit.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 21:51, 17 February 2016 (UTC)[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 23:56, 18 February 2016 (UTC)

:::''What'' is the puzzle? [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 21:04, 18 February 2016 (UTC)

::::First puzzle: why "Physically, the nature of the inner product is dependent on the basis in use"?

::::Second puzzle: why "the basis is chosen to reflect the quantum state of the system"?

::::[[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 22:16, 18 February 2016 (UTC)

:::::{{replyto|Tsirel}}Thanks for clarifying and sorry for a late reply. The first sentence I didn't write and have no idea what it means. The second was probably me, a bad way of describing the basis in some chosen representation. Both statements should be deleted as being opaque. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 16:46, 25 February 2016 (UTC)

== scalar product ==

I read in the article

{{Font color|blue|... The state space is postulated to have an [[inner product]], denoted by

:<math>\langle \Psi_1 | \Psi_2 \rangle,</math>

that is (usually, this differs) linear in the first argument and [[antilinear]] in the second argument. The [[dual vector]]s are denoted as "bras", {{bra|{{math|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}}

:<math>\langle \Psi | = a^{*} \langle \psi | + b^{*} \langle \phi | \leftrightarrow a|\psi\rangle + b|\phi\rangle = |\Psi\rangle,</math>

{{Font color|blue|where the asterisk denotes the complex conjugate. For this reason one has under this map

:<math>\langle \Phi|\Psi\rangle = \langle \Phi| (|\Psi\rangle),</math>

and one may, as a practical consequence, at least notation-wise in this formalism, ignore that bra's are dual vectors.}}

I am very happy to observe that this illustrates the indubitable fact the certain Wikipedia editors, with mathematical inclinations, are very good at mathematics.

But I think that doesn't entitle them to flout the rules of Wikipedia (no source cited, evidently no fair source survey for the physical context) and appropriate Dirac's notation. The harm in this, I think, is that they mistakenly feel it justifies that their editing of this article should deliberately downplay the role of bras. Dirac thought it was important from a physical point of view. Instead of talking about inner products, Dirac, Gottfried, Cohen-Tannoudji, and Weinberg talk of scalar products.<ref>'''<big><u>Sources for this:'''</u></big></ref> I don't think this means that these authors do not know what an inner product is. I think it means that for the physics, they are more interested in their scalar product.

Therefore I am very keen that the article should use the easily understood, recognizably distinct, and to some extent customary mathematical notation (·,·) for the inner product,<Fabian, M., Habala, P., Hájek, P., Santalucía, V.M., Pelant, J., Zizler, V. (2001), ''Functional Analysis and Infinite-Dimensional Geometry'', Springer, New York, ISBN 0-387-95219-5, p. 16.> and leave the Dirac notation for the scalar product that Dirac invented it for. Yes, plenty of mathematics texts use the angle brackets, as well as plenty of others that use the parentheses. The bra has an important physical significance, routine neglect of which has generated a lot of rubbishy pseudo-metaphysics and drivel. So I would like to change the above to read

{{Font color|green|The state space of kets is postulated to have an [[inner product]], denoted by

:<math>(| \Psi_1 \rangle, | \Psi_2 \rangle).</math>

The inner product is (usually, this differs) linear in the first argument and [[antilinear]] in the second argument. The [[dual vector]]s are denoted as "bras", {{bra|{{math|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}}

:<math>\langle \Psi | = a^{*} \langle \psi | + b^{*} \langle \phi |\,\,\,\, \leftrightarrow \,\,\,\, a|\psi\rangle + b|\phi\rangle = |\Psi\rangle,</math>

{{Font color|green|where the asterisk denotes the complex conjugate. For this reason, using Dirac's bra–ket notation for the '''scalar product''', one has under this map

:<math>\langle \Phi|\Psi\rangle = (| \Phi \rangle, | \Psi \rangle).</math>

}}

{{Reflist}}

As I read it, Wikipedia posts what reliable sources say, in context. [[Paul Dirac|Dirac]] would have a fair chance of being a reliable source on this topic. He says "scalar product".<[[Paul Adrien Maurice Dirac|Dirac, P.A.M.]] (1958). ''The Principles of Quantum Mechanics'', 4th edition, Oxford University Press, Oxford UK, p. 20: "The bra vectors, as they have been here introduced, are quite a different kind of vector from the kets, and so far there is no connexion between them except for the existence of a scalar product of a bra and a ket.">

So does [[Kurt Gottfried]]<[[Kurt Gottfried|Gottfried, K.]], [[Tung-Mow Yan]] (2003), ''Quantum Mechanics: Fundamentals'', 2nd edition, Springer, New York, ISBN 978-0-387-22023-9,[https://books.google.com/books?id=8gFX-9YcvIYC&pg=PA31&vq=%22to%20define%20the%20scalar%20products%20as%20being%20between%20bras%20and%20kets.%22], p. 31: "to define the scalar products as being between bras and kets."> .

[[Steven Weinberg|Weinberg]] (2013) also speaks of the "scalar product".

As does [[Albert Messiah|Messiah]] (1961).

Also, mostly [[Gennaro Auletta|Auletta]], Fortunato, and [[Giorgio Parisi|Parisi]] (2009).

Ballentine (1998) sees 'inner' and 'scalar' as alternatives.

Beltrametti and Cassinelli (1982) speak of the "scalar" product.

As do [[Claude Cohen-Tannoudji|Cohen-Tannoudji]], Diu, and Laloë, F. (1973/1977).

And [[Josef-Maria Jauch|Jauch]] (1968).

And [[Edwin C. Kemble|Kemble]] (1937).

And Zettili (2009).

Bransden & [[Charles J. Joachain|Joachain]]'s ''Physics of Atoms and Molecules'' (1983/1990) routinely uses 'scalar product', though it once mentions (in parentheses) 'inner product' as an alternative. Their ''Quantum Mechanics'' (2nd edition 2000) uses only 'scalar product'.

David (2015) uses 'scalar product'.

Davydov (1965) uses 'scalar product'.

Robinett (2006) mixes Dirac notation with the {{math|''ψ''(''x'', ''t'')}} notation, and uses "inner product".

Busch, Lahti & Mittelsteadt (''The Quantum Theory of Measurement'', 2nd edition 1991/1996) uses the Dirac notation and 'inner product'.

De Muynck (''Foundations of Quantum Mechanics, an Empiricist Approach'', 2004) uses 'inner product'.

D.J. Griffiths (1995) uses Dirac notation and 'inner product'.

R.B. Griffiths (2002) uses Dirac notation and 'inner product'.

Some authors who do not use the Dirac bra–ket notation, such as [[John von Neumann|Von Neumann]] (1932/1955) and [[Leonard Schiff|Schiff]] (1949), though not Weinberg, use "inner product".

[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 19:33, 18 February 2016 (UTC)

:Indeed, sometimes physicists and mathematicians ''deliberately'' differ in terminology; in such cases I shrug: sovereign states. A mathematician would probably say: "bra" and "ket" are a [[dual pair]]. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 20:00, 18 February 2016 (UTC)

::The problem here is not about terminology. It is about emphasis and reliable sourcing. It is clear that the article wants to teach the physicists a lesson, about the supposed unimportance of the distinction between bras and kets. The article says, as above, "one may, as a practical consequence, at least notation-wise in this formalism, ignore that bra's are dual vectors." The standard physics texts don't do that. When I first raised this with a leading editor, citing Gottfried, he replied that he had never heard of Gottfried and that Gottfried was wrong. You may read above on this page a deprecatory remark about bras, made by another editor. Gottfried's text is recommended by J.S. Bell on a par with Landau & Lifshitz. Dare I say it, the wave function is a topic in physics, and it is not up to mathematically inclined Wikipedia editors, no matter how clever and well qualified they may be, to over-rule respected physical sources on the grounds that such editors think sources such as I have cited above are wrong or misleading.

::Endless drivel is manufactured from the term "wave function collapse", invented by David Bohm to make the Copenhagen people look silly. It works for the drivel manufacturers because they ignore or downplay the distinction between bras and kets. Dare I say it, Dirac was no fool. He thought the bras were importantly different from kets from a physical point of view, and his notation distinguishes them. It is not the mandate of Wikipedia editors to over-rule him. One of the relevant editors wrote somewhere here that he had for the first time read an early Dirac paper, and found Dirac fresher than many writers, a having a modern approach. It is not easy then to dismiss Dirac when his term is used by such writers as Weinberg and Cohen-Tannoudji. Maybe Dirac is a voice from the past, but that is not so for Weinberg and Cohen-Tannoudji.

::You can read people saying that von Neumann wrote about "collapse". No he didn't. You can easily check that. I have looked in the English translation of von Neumann's book (and now have checked the German). My impression is that he uses neither Heisenberg's word 'reduce' nor the questioned word "collapse", nor a near substitute. As far as I have so far seen, the translator simply says there are two forms of "intervention", what the translator calls "arbitrary changes by measurement" (German: "die willkürlichen Veränderungen durch Messungen"), and what he calls "automatic changes which occur with the passage of time" (German: "die automatischen Veränderungen durch den Zeitablauf"). Personally, I wouldn't count evolution in time of an isolated system as a form of "intervention" (German: "Eingriffen"), but that word is not crucial.

::These muddles arise because people work with words, not thinking of their physical meaning. Over-ruling the physical sources because it seems more mathematically stream-lined is an example of that, not permitted by Wikipedia. It's got a special Wikipedia name, expressing disapproval, but I don't want to get too polemical by writing that name here and now.

:: You write above "A mathematician would probably say: "bra" and "ket" are a [[dual pair]]." Of course you are right that he would say it. And the mathematician is right to say it. And it is not to be dismissed. Dirac invented a notation that made it clear for good physical reason. It is the physical reason that matters, not the mere terminology[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 23:54, 18 February 2016 (UTC)[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 02:13, 19 February 2016 (UTC)

::: Now I am puzzled. "Wave function collapse", invented by David Bohm?? to make the Copenhagen people look silly?? In [[Wave function collapse#History and context]] I read: The concept of wavefunction collapse, under the label 'reduction', not 'collapse', was introduced by Werner Heisenberg in his 1927 paper on the uncertainty principle. Is this wrong? Or is there an important difference between reduction and collapse? [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 06:19, 19 February 2016 (UTC)

::: I am hardly understanding what is really <strike>the fuss</strike> your point; but anyway, I feel that it is not specific to a basis, and therefore, it is about a state vector rather than wave function. If so, you'd better raise your point there; and there, hopefully, you'll face a more competent and interested physical community than here. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 08:19, 19 February 2016 (UTC)

::::I now reply to "Now I am puzzled. "Wave function collapse", invented by David Bohm?? to make the Copenhagen people look silly??"

::::It is a subtle but powerful point of language. A 'collapse' is a dramatic, even catastrophic, event. 'Reduction' is a relatively modest word, hardly an event. "The concept of wavefunction collapse, under the label 'reduction', not 'collapse', was introduced by Werner Heisenberg in his 1927 paper on the uncertainty principle." Yes, I wrote that. Heisenberg did not think of it in dramatic terms. So far as I have been able to find, it was Bohm who lit it up with the dramatic term 'collapse'. Now people make out that it somehow means that something has 'happened to the wave function'. Bohm wanted to highlight his new interpretation, that appears to endorse the idea of instantaneous propagation of a quantum potential. The use of the word 'collapse' makes Copenhagenism look silly. One reads that Bohr believed in 'collapse'. Nonsense, he didn't use the word at all, so far as I can find out. No serious student of Bohr says he used the word. Born didn't bother to use even the word 'reduction'. He was just beginning to think about it. Heisenberg called it 'reduction'. These words, in the pens of pseudo-metaphysicians, spawn industries of drivel.

::::I guess you are tired of my repeating that we are talking about physics here. Born first, then Heisenberg, talked about it in terms of collision between particles. The incoming particle is described by a wave function or state vector that tells how it came on the scene. It collides and its momentum changes. It is as if this 'prepared' it afresh and so after the collision it has a fresh wave function. Alternatively, but much less easily, one could also describe this in terms of a joint wave function (tensor product) including the incoming–outgoing particle and the target particle jointly. But in the simple way, of just considering the incoming–outgoing particle as 'the particle' and forgetting the quantum nature of the target particle, one sees an abrupt transition in the wave function. Nothing happened to the wave functions. What happened was a collision of particles. The physicist changed his focus of interest from the incoming wave function to the outgoing wave function. This is transmogrified into "collapse" of the wave function, and an industry is born, to "explain" this metaphysical miracle. The target particle can be considered in two ways. One is as a heavy thing that behaves more or less (near enough) classically (put into the Hamiltonian if you like). The other is as a quantum object that needs to be treated as having a wave function. The 'collapse' story treats it pseudo-classically, ignoring the quantum aspect. This story is somewhat hidden by the Copenhagenism that makes it a crime to think about what happens in the innards of the apparatus. Perhaps that is enough chatter from me for now about that.

::::You suggest that I should raise my point elsewhere. With respect, this point is about this article. It is unsourced and misleading in this article. It should be fixed here. It is written here in terms of bras and kets, which denote state vectors. True, this article is written from a condescending viewpoint, that makes wave functions look like country cousins beside the more sophisticated state vectors. It is almost the case that this article, though headed 'wave function', is dominated by the state vector, with the wave function as a footnote. This makes the authors of the article look sophisticated. But the problem is in this article and should be fixed in this article.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 09:37, 19 February 2016 (UTC)

:Scalar product and inner product are synonyms. Take two vectors and produce a number according to a set of rules. ''I find it mildly shocking that you do not know this &ndash; and once again embark on a ridiculous rant.'' The physics lies in the Born rule. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 09:27, 19 February 2016 (UTC)

::If you think they are synonymous, it would seem that you would be indifferent as to which is used. If so, I guess you will not mind using the one that is most used in reliable physics sources, namely, scalar product, since this is a physics article. Dirac makes a point that bras and kets are different, vectors and dual vectors. He states that the theory is symmetrical between them, but not that they are the same thing. He thinks that the scalar product is between vectors and dual vectors. That is not the same as the inner product, which is between vectors. You are trying to de-emphasize that. It is not right to de-emphasize in Wikipedia what reliable sources emphasize. It is not polite to say that my comments are "a ridiculous rant".[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 09:48, 19 February 2016 (UTC)[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 09:53, 19 February 2016 (UTC)

::An example of a physics writer who has a good claim to be a reliable source who uses the notation that I am recommending for the inner product, namely (·,·) , and who uses the term 'scalar product', that I am recommending for such objects as our article writes {{bra-ket|''a''|''b''}}, is Weinberg (''Lectures on Quantum Mechanics'', 2013).[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 15:09, 19 February 2016 (UTC)

::: "Between vectors and dual vectors", it is neither scalar nor inner product, it is duality pairing, unable to lead to any metric (on either of the two mutually dual spaces). At least, this is the mathematical terminology. About Dirac, I do not know. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 10:15, 19 February 2016 (UTC)

::: Now about "a subtle but powerful point of language". Yes, you can throw away the collapse. No problem. This is done long ago, and is called the [[many-worlds interpretation]]. No one was able to avoid both collapse and many-world. <small>I guess your native culture is humanities (or medicine?) rather than hard science.</small> The choice of a name is so much important for you... but it is important only if it leads to different physical predictions. In which case it is a different theory rather than a different interpretation of the quantum theory. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 10:22, 19 February 2016 (UTC)

:::::The choice of a name is indeed important for me. Names are important in guiding people's thinking. 'Collapse' suggests a process in nature. 'Reduction' is less committed than 'collapse', and is more compatible with the real situation, that what changes is the descriptive framework as distinct from the facts. Your opposition of 'collapse' vs 'many worlds' is evidence of the importance of names. Both of those ideas are way off beam, though words makes them seem compatible with each other. The nonsense of 'many worlds' is the offspring of the misleading word 'collapse'.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 08:31, 25 February 2016 (UTC)

::: The founding fathers, naturally, were more than happy to succeed in predictions about colliding particles, atomic transitions etc. It was not the time to think about [[macroscopic quantum phenomena]], [[Bose–Einstein condensate]], [[decoherence]], [[squeezed vacuum]], [[quantum computing]], [[false vacuum]], [[Hawking radiation]] (the more so, [[quantum gravity]]). Now it is another century. It does not mean that we should mention these in the article. It only means that the article should not smell of mold. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 11:21, 19 February 2016 (UTC)

::::With respect, this is Wikipedia about physics. In a sense you rule yourself out of order by saying "About Dirac, I do not know." It is an important part of Wikipedia editing to know something of reliable sources. Dirac has a fair claim to be a reliable source. Heisenberg wrote to Dirac that he went to his 4th edition for the soundest mathematical presentation. Einstein wrote that Dirac's presentation was the most logically perfect he had found. This is fair reason to consider Dirac as a possible reliable source. In his 2013 text, Weinberg wrote "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 basis states of definite position. This is essentially the approach of Dirac’s “transformation theory.” I do not use Dirac’s bra-ket notation, because for some purposes it is awkward, but in Section 3.1 I explain how it is related to the notation used in this book." These are reasons to consider Dirac as a possible reliable source. But as a potential Wikipedia editor on this topic you write "About Dirac, I do not know." I have no doubt, obviously, that you are a towering intellect, and of course I very much respect that. But this is Wikipedia, which has its policies. Amongst its prime policies is reliable sourcing.

::::Of course you and I know that the many worlds story is fanciful at best. Collapse is lazy talk, not physics. I will not continue more about the rest of your comments.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 11:48, 19 February 2016 (UTC)

::::: Happy sourcing this nearly orphaned article. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 12:34, 19 February 2016 (UTC)

::::::Thank you.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 12:51, 19 February 2016 (UTC)

::::::As for your worry lest the article smell of mould, an example of a physics writer who has a good claim to be a reliable source who uses the notation that I am recommending for the inner product, namely (·,·) , and who uses the term 'scalar product', that I am recommending for such objects as our article writes {{bra-ket|''a''|''b''}}, is Weinberg (''Lectures on Quantum Mechanics'', 2013).[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 15:18, 19 February 2016 (UTC)

::::::On page 109, Cohen-Tannoudji ''et al.'' write:

:::::::::β.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''Scalar product''

:::::::::&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;With each pair of kets {{ket|''φ''}} and {{ket|''ψ''}}, taken in this order, we associate a

:::::::::complex number, which is their scalar product, ({{ket|''φ''}},{{ket|''ψ''}}), ...

::::::[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 06:50, 21 February 2016 (UTC)

::::::According to Abers, E.S. (2004), ''Quantum Mechanics'', Pearson, Upper Saddle River NJ, ISBN 0-13-146100-1, p. 25:

:::::::::... A straightforward notation for the scalar product would be

::::::::::::::::<math>\left (|\phi\rangle,|\psi \rangle \right )\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2.10)</math>

:::::::::... I will follow the standard physics tradition and use a notation introduced by Dirac. We write

:::::::::::::::<math>\langle\phi|\psi \rangle=\langle\psi|\phi \rangle^*\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2.15)</math>

::::::[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:38, 29 February 2016 (UTC)

==vectors and dual vectors==

Your above comment ""Between vectors and dual vectors", it is neither scalar nor inner product, it is duality pairing, unable to lead to any metric (on either of the two mutually dual spaces). At least, this is the mathematical terminology" is very interesting to me. I understand the difference between an inner product such as ('''x'''₁,'''x'''₂) and a pairing such as {{bra-ket|'''x'''|''ξ''}}. Halmos introduces the dual spaces on page 20. He waits till page 118 to introduce inner products. Physically one cannot directly compare vectors except by observing pure states that come out of distinct channels of the analyzing device, and then one says they are orthogonal, because they are perfectly distinct. Such an observation requires detection, which is signified by a bra if one follows the custom of taking the ket as the prepared but not yet detected beam. One gets the bra–ket link physically by saying that the detection of a beam straight from the preparation device identifies the detected bra with the prepared ket. Dirac doesn't talk separately about the inner product. I think he derives the metric by looking at the pairing rather than the inner product, because the inner product does not correspond to a direct observation. This isn't how math texts proceed. One can observe a pairing directly. Does this make sense to you? Still it's my best effort to describe what I read Dirac as doing.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 17:17, 19 February 2016 (UTC)

The symmetry between bras and kets arises because typical quantum analyzers satisfy some version of the Helmholtz reciprocity principle. That means you can interchange the source and the detector and still get the same result. That's why the observables are required to be Hermitian. If you can't do that with a proposed potential analyzer, it fails the test and doesn't provide a proper observation. For example, a prism can be turned back-to-front and it looks unchanged. It is also why the observables of a basis set must commute. [[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 17:24, 19 February 2016 (UTC)[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 17:52, 19 February 2016 (UTC)[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 18:15, 19 February 2016 (UTC)

In the 1st edition (1930), Dirac hasn't yet invented the bra–ket notation. He writes instead: "The theory will throughout be symmetrical between the {{math|''φ''}}'s and {{math|''ψ''}}'s. The sum of a {{math|''φ''}} and a {{math|''ψ''}} has no meaning and will never appear in the analysis." And "In the vector picture we can take the number {{math|''φψ''}} to be the scalar product of the two vectors {{math|''φ''}} and {{math|''ψ''}}. ... The vector picture, however, allows us also to form the products {{math|''φ''₁''φ''₂}} and and {{math|''ψ''₁''ψ''₂}}. Thus we again find the vector picture giving more properties to the {{math|''ψ''}}'s and {{math|''φ''}}'s and than required in quantum mechanics." Is this his saying that the products such as {{math|''φ''₁''φ''₂}} are not required in quantum mechanics because the metric is already supplied by the scalar product? [[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 18:23, 19 February 2016 (UTC)

In the 2nd edition (1935), he continues with this notation: "Also it is easily seen that the whole theory is symmetrical between {{math|''φ''}}'s and {{math|''ψ''}}'s ..."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 18:50, 19 February 2016 (UTC)

By the 3rd edition (1947) he has invented the bra–ket notation. He writes: "Then the number {{math|''φ''}} corresponding to any {{ket|{{math|''A''}}}} may be looked upon as the scalar product of that {{ket|{{math|''A''}}}} with some new vector, there being one of these new vectors for each linear function of the ket vectors {{ket|{{math|''A''}}}}." The same sentence appears in the 4th edition (1958).[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 18:58, 19 February 2016 (UTC)[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 19:07, 19 February 2016 (UTC)

Dirac, poor fellow, would not have made the grade as a Wikipedia editor! He gives no references that I can see. Terrible. On the other hand, one may guess that perhaps in 1935 he had read von Neumann's mighty work of 1932. Von Neumann there writes of the 'Hermitian inner product' (·,·) and the 'scalar product' {{math|''αf''}} with {{math|''α''}} a complex number and {{math|''f''}} an element of 'abstract Hilbert space'. Von Neumann notes that he has read, but does not copy, Dirac's 1930 ''Principles'', which he says is "scarcely to be surpassed in brevity and elegance". That, as noted above, uses the term 'scalar product' for such duality pairings as {{math|''φψ''}}. I guess Dirac would have been well aware of all this.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 01:41, 20 February 2016 (UTC)

Dirac invented the bra–ket notation in a gradual development. On page 21 of the first (1930) edition we read:

:::::''We now suppose that any {{math|φ}} and {{math|ψ}} have a product, which is a number, in general complex. This product must always be written {{math|φψ}}, i.e. the {{math|φ}} must be on the left-hand side and the {{math|ψ}} on the right.'' Products such as {{math|''ψφ''}}, {{math|''ψ''₁''ψ''₂}}, {{math|''φ''₁''φ''₂}}, have no meaning and will never appear in the analysis.

He did not yet recognize the tensor product, and held that the inner product had no meaning. I think it has marginal physical meaning.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 11:07, 21 February 2016 (UTC)

In the second (1935) edition, on page 23, we read: "symbolic products of the type {{math|''ψ_aψ_b}} or {{math|φ_aφ_b''}} never occur in the theory."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 11:20, 21 February 2016 (UTC)

In Dirac 1926a we read: "In order to be able to get results comparable with experiment from our theory, we must have some way of representing q-numbers by means of c-numbers, so that we can compare these c-numbers with experimental values."<''Proc. Roy.

''Soc. A'', '''110''': 561–579.> Dirac is looking to experimental results to build his calculus.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 17:33, 21 February 2016 (UTC)

Messiah on page 247 of volume 1 is explicit that he derives the metric from the duality pairing:

:::::In order to introduce a metric in the vector space we have just defined, we make the hypothesis that there exists a one-to-one correspondence between the vectors of this space and those of the dual space. Bra and ket thus associated by this one-to-one correspondence are said to be conjugates of each other and are labelled by the same letter (or the same indices). Thus the bra conjugate to the ket {{ket|{{math|''u''}}}} is represented by the symbol {{bra|{{math|''u''}}}}.

Messiah has announced that he is following Dirac. Thus it appears that Dirac's rejection of the inner product is accompanied by his use of his scalar product to provide the metric in a way differing from that of mathematics textbooks.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 15:05, 23 February 2016 (UTC)

== Nearly orphaned ==

If this "wave function" article becomes a mini-encyclopedia of non-relativistic quantum mechanics, then inevitably it attracts controversy.

A burst of controversy occurs sometimes also around a mathematical article; see, for example, [[Talk:Complex affine space#Requested move 13 October 2015]]; but there, a content dispute is solved effectively by a reasonably large, competent and interested community.

Here I see that the physical community is nearly silent (and apparently expresses its attitude via the Don Quixote picture above). If so, then this page is an unsuccessful project, alas. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 07:23, 19 February 2016 (UTC)

:This page is utterly unsuccessful. But the topic at hand is though decidedly "notable". For instance, L&L mentions nowhere Hilbert space, but use "wave function" throughout. [[User:YohanN7|YohanN7]] ([[User talk:YohanN7|talk]]) 12:04, 22 February 2016 (UTC)

::Editor [[User:YohanN7|YohanN7]] makes a very good and well-thought point. L&L is broadly speaking a reliable source and "mentions nowhere Hilbert space, but use "wave function" throughout". As he says in consequence, "... the topic at hand is ... decidedly "notable"." Two important aims for writing Wikipedia articles are (1), as noted by Editor YohanN7, notability, and (2), as mostly achieved by various editors including especially Y and M, reliability. As usual, however, in my deviationist and counter-revolutionary way, I regretfully depart from the semi-consensus of respected editors W, T, and Y: judging by average Wikipedia standards, it is true neither that "First off, this article is pretty bad" (editor W), nor that it is "utterly unsuccessful" or "an orphan" (editors Y and T). There are many articles that, in my opinion, are significantly worse. I have had some experience with Editor W. Believe it or not, occasionally I have even agreed with him. I think the main factor that made him say that this article is pretty bad was the length of the lead. Yes, it was too long, but that is something fairly easily remedied. I am sorry I have caused such anguish by my mistake about the symbolic approach of Dirac. It remains that there are some things about the article that I think need revision. I guess I may not be the only one who thinks so.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 13:05, 22 February 2016 (UTC)

== functions of space ==

I read in the article:

*{{fontcolor|blue|The idea that [[quantum state]]s are vectors in an abstract vector space (technically, a complex [[projective space|projective]] [[Hilbert space]]) is completely general in all aspects of quantum mechanics and [[quantum field theory]], whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.}}

I think this would be well amended, as follows. I think it is, properly speaking, never true that "quantum states are complex-valued "wave" functions of space." The nearest would be a point particle with no spin, and then the wave function would be a function of its configuration space, not of space simple. One can say "Oh, such a configuration space is isomorphic with space simple." But on such an important matter, I think near enough is not good enough.

There is an important and widely used sense in which such a particle, with configuration space <math>\mathbf Q = Q_x \times Q_y \times Q_z</math> has a wave function <math>\phi \, :\mathbf Q \rightarrow \mathbb{C}</math>. I would like to ask experts is there a precisely corresponding usage in the quantum theory of fields? It is my impression, subject to correction by experts, that there is not. My impression is that the sense of the term 'wave function' in the quantum theory of fields is a notable generalization of the just now stated sense of the term. I think the article should make this clear, but does not currently do so.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 23:45, 24 February 2016 (UTC)

==terminology==

I read above "A wave function is the projection of a state vector onto a specific set of coordinate axes. I. e. it is a coordinate vector. See {{harvtxt|Weinberg|2013}} ..."

In his 2013 text, Weinberg wrote "The right way to combine relativity and quantum mechanics is through the quantum theory of fields, in which the Dirac wave function appears as the matrix element of a quantum field between a one-particle state and the vacuum, and not as a probability amplitude. ... 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 basis states of definite position. This is essentially the approach of Dirac’s “transformation theory.” I do not use Dirac’s bra-ket notation, because for some purposes it is awkward, but in Section 3.1 I explain how it is related to the notation used in this book."

This present article is partly written unsourced. Nevertheless, some sourcing may be considered. Dirac on his notation is a fair candidate for some parts of the sourcing.

Another candidate for part of the sourcing is Schrödinger. He is mentioned in the history section, but in the rest of the article one one would not get the message that the wave function is his invention. For example, one of the few mentions that links him specifically with wave functions reads: "The Heisenberg picture wave function is a snapshot of a Schrödinger picture wave function, representing the whole spacetime history of the system."

On page 80 of the 4th edition, Dirac writes: "A further contraction may be made in the notation, namely to leave the symbol <math>\rangle</math> for the standard ket understood. A ket is then written simply as <math>\psi (\xi)</math>, a function of the observables <math>\xi</math>. A function of the <math>\xi</math>s used in this way to denote a ket is called a ''wave function''."

An observable <math>\xi</math> is an operator on a vector space. The ___domain <math>\mathbf Q</math> of the above wave function <math>\phi</math> is not a set of operators such as <math>\xi</math> on a vector space; it is a set of points in configuration space.

Accordingly, the objects such as <math>\psi(\xi)</math> and <math>\phi</math> are of different natures. One is a ket and the other is not. Accepting Dirac's omission of the ket symbol as a contraction of notation, they both seem to claim to be 'wave functions'. I think that such a Wikipedia article as this one, specifically about wave functions, ought not allow a potential muddle such as this.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 23:45, 24 February 2016 (UTC)

== wave function as scalar product ==

In his 2013 text, Weinberg wrote "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 basis states of definite position. This is essentially the approach of Dirac’s “transformation theory.” I do not use Dirac’s bra-ket notation, because for some purposes it is awkward, but in Section 3.1 I explain how it is related to the notation used in this book."

It is a mouthful to say that a function is or is the value of a scalar product. We are looking at a telescoped form of expression. A function is a scalar? Or is the value of the wave function the scalar? According to Dirac the value of a scalar product is a number. This suggests the reading that a value of the scalar product is the value of the pertinent wave function.

In his Section 3.1, Weinberg writes: "This is a good place to mention the “bra-ket” notation used by Dirac. In Dirac’s notation, a state vector {{math|Ψ}} is denoted {{ket|{{math|Ψ}}}}, and the scalar product ({{math|Φ}},{{math|Ψ}}) of two state vectors is written {{bra-ket|{{math|Φ}}|{{math|Ψ}}}}. The symbol {{bra|{{math|Φ}}}} is called a “bra,” and {{ket|{{math|Ψ}}}} is called a “ket,” so that {{bra-ket|{{math|Φ}}|{{math|Ψ}}}} is a bra-ket, ..."

This looks like a difference between two Nobel Prize winners. Dirac thinks his scalar product is what Tsirel rightly calls a duality pairing. Weinberg thinks Dirac's bra-ket is an inner product between two vectors of the same space with different respective notations. I think Dirac should be declared the winner here. Weinberg is not an addict to Dirac's notation, and may not care too much about its finer points. If so, we are not compelled to read Weinberg's ''verba ipsissima'' as gospel on every aspect of the terminology here.

It seems to me that the continuum of values of a wave function may be regarded as a continuum of values of a scalar product, numbers according to Dirac.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 09:20, 25 February 2016 (UTC)

More explicitly, a wave function in the present Dirac tradition is an expression of the resolution of a state vector into a superposition of appropriate orthogonal basis kets weighted by complex number components that are the scalar products of the state vector's ket with the eigenbras of the orthogonal basis that specifies a chosen representation and coordinate system. That expression can also be recognized as a table of values of a function with ___domain the degrees of freedom of the representation, and range appropriate to the specific system, in the spinless case just the set of complex numbers. This is well exhibited by the above dissections by Editors Maschen and YohanN7. Further recognition, in the spinless case, of that table is as its belonging to a function expressed as an analytic formula such as is usual for wave functions in the Schrödinger tradition. The latter, by the way, could be made a little more visible in the article.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 14:27, 25 February 2016 (UTC)

:You write

::''"More explicitly, a wave function in the present Dirac tradition is an expression of the resolution of a state vector into a superposition of appropriate orthogonal basis kets weighted by complex number components that are the scalar products of the state vector's ket with the eigenbras of the orthogonal basis that specifies a chosen representation and coordinate system."''

:I cannot believe how complicated you are making things. A lot of people agreed above that the terminology was clarified. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 16:49, 25 February 2016 (UTC)

:::Thank you for this comment. I was amongst those who praised your admirable formula above. It helps with what I have all along been interested in: distinguishing and tying together the Dirac and the Schrödinger conceptions of the wave function. On page 35, L&L <''Quantum Mechanics: Non-Relativistic Theory'', 3rd edition, Pergamon, Oxford UK, (1977)> write

::::::::::::<math>\langle n|f|m\rangle</math>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(11.17)

:::::::This symbol is written so that it may be regarded as "consisting" of the quantity {{math|''f''}} and the symbols {{ket|{{math|''m''}}}} and {{bra|{{math|''n''}}}} which respectively stand for the initial and final states as such (independently of the representation of the wave functions of the states)."

:::After all, many people, I guess, still think of the wave function as Schrödinger's invention. L&L did.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 18:47, 25 February 2016 (UTC)

:::Revised version: A wave function in the present Dirac tradition is an expression of the resolution of a state vector into a superposition of kets of an orthogonal basis that specifies a chosen representation and coordinate system, weighted by complex number components that are the scalar products of the state vector's ket with the eigenbras of the orthogonal basis.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 11:28, 27 February 2016 (UTC)

::::Please read out loud the part I quoted from your original post in this section, or the revised version you just wrote here.

::::Even with punctuation to break up the long sentence, it is unreadable and impenetrable to anyone. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 12:33, 27 February 2016 (UTC)

:::::I think a patient reader would manage.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 13:36, 27 February 2016 (UTC)

::::::No they would not. It is a flood of words introduced all in one go. The reader, patient or impatient, has to connect everything together.

::::::What does ''"is an expression of the resolution of a state vector into a superposition of kets of an orthogonal basis"'' mean?? It may make sense to you, but a typical reader would wonder what "resolution" means in this context (you still insist), and will have no idea what is going on:

::::::*is the wave function a component of this state vector (in which case a complex number)?

::::::*Or the superposition of kets (the state vector itself, which is not the same thing as its components, and if this was the case then "wave function" and "state vector" are synonymous so the definition is circular/meaningless)?

::::::*Is it collectively all of the components of the state vector?

::::::*Is it ''any'' single component of the state vector (the observables do not have have given values), or ''a'' specific given component (the observables have definite values)?

::::::They would also get the idea that a basis must be orthogonal (it does not have to be). An [[orthonormal]] (normalized i.e. [[unit vector]]s and orthogonal) basis set is convenient to work with because the inner products are very simple. In general a set of vectors in a vector space qualifies as a basis if every vector in the space can be written as a unique [[linear combination]] (standard technical term) of the vectors. A basis simply requires [[linear independence]], and not orthogonality, not normalized, nor even orthonormality. [[user:Maschen|'''M''']][[User:Maschen/sandbox|'''&and;''Ŝ''''']][[special:contributions/Maschen|''c''²''ħ''ε]][[user talk:maschen|''И_τlk'']] 15:37, 27 February 2016 (UTC)

{{Outdent|7}}

Thank you for these helpful comments.

Weinberg leaves a bit to the imagination when he writes as cited at the beginning of this section. The list of questions you provide pretty nearly summarizes those I raised at the start of this section. My intention was to answer them in what I wrote.

The term 'resolution into components' is pretty standard, though not universal. Some examples are [http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Resolution this], [http://physics.info/vector-components/practice.shtml this], and [http://study.com/academy/lesson/vector-resolution-definition-practice-problems.html this]. I guess 'standard' is a variable thing. It is not very evident in Wikipedia, though it occurs [[Vector projection|here]] and [[Bessel's inequality|here]]. Though Wikipedia does not seem to determine standard usage for us.

In the article [[Euclidean vector]] I find

:::::<big><u>'''Decomposition'''</u></big>

:::::''For more details on this topic, see [[Basis (linear algebra)]].''

:::::As explained [[Euclidean vector#Representations|above]] a vector is often described by a set of vector components that [[#Addition and subtraction|add up]] to form the given vector. Typically, these components are the [[Vector projection|projections]] of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be ''decomposed'' or ''resolved with respect to'' that set.

Since we are talking quantum mechanics, it seemed a good idea to remind the reader that superposition is at work here.

Like YohanN7, I think your admirable formula above would go well near the front of the article, and would clarify these points.[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 20:06, 27 February 2016 (UTC)

Resolving things into components is pretty much ordinary language. For example, in the article on the [[Stern–Gerlach experiment]], I read : "As the particles pass through the Stern–Gerlach device, they are being observed by the detector which resolves to either spin up or spin down."[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 01:28, 29 February 2016 (UTC)

== captions for formula ==

I have been fiddling with the captions to the admirable and excellent formula of Editor [[User:Maschen|Maschen]] to produce the following:

:<math>\underbrace{| \Psi \rangle}_{\text{state ket}} = \underbrace{\overbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}}^{\text{discrete}\atop\text{labels}}\overbrace{\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}}}_{\text{superposing weighted basis kets}} \, \underbrace{\overbrace{\Psi}^{\text{wave}\atop\text{function}} (\overbrace{\mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} }^{\text{eigenvalues of basis observables}\atop\mathord{\sim}\text{ argument of wave function}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,})}_{ {{{\text{component of state ket}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\atop\text{= value of scalar product } \langle \,\text {basis bra }|\,\Psi\,\rangle\,\,}}\atop\text{= value of wave function}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\atop\text {= complex number weight of basis ket}\,\,\,\,\,\,\,\,\,\,}\underbrace{|\overbrace { \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N}}^{{\text{eigenvalues of basis observables}}\atop\mathord{\sim}\text { label of basis ket}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, }\rangle }_{\text{basis ket}}</math>

The eigenvalues appear in two guises. One is as labels for the bras and kets, the other is as quantities that are arguments for the wave function considered as a function. That is why the sign ~ is shown instead of the sign = .[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 15:40, 29 February 2016 (UTC)

Considering that the spinor/vector/tensor character of the spin variables is not correctly expressed by that version. To deal with that perhaps it may be easier to omit the words 'complex number':

:<math>\underbrace{| \Psi \rangle}_{\text{state ket}} = \underbrace{\overbrace{\sum_{s_{z\,1} , \ldots , s_{z\,N}}}^{\text{discrete}\atop\text{labels}}\overbrace{\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}^{\text{continuous labels}}}_{\text{superposing weighted basis kets}} \, \underbrace{\overbrace{\Psi}^{\text{wave}\atop\text{function}} (\overbrace{\mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N} }^{\text{eigenvalues of basis observables}\atop\mathord{\sim}\text{ argument of wave function}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,})}_{ {{{\text{component of state ket}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\atop\text{= value of scalar product } \langle \,\text {basis bra }|\,\Psi\,\rangle\,\,}}\atop\text{= value of wave function}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\atop\text {= weight of basis ket}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\underbrace{|\overbrace { \mathbf{r}_1, \ldots, \mathbf{r}_N , s_{z\,1} , \ldots , s_{z\,N}}^{{\text{eigenvalues of basis observables}}\atop\mathord{\sim}\text { label of basis ket}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, }\rangle }_{\text{basis ket}}</math>

For the spinless case this version may be ok:

:<math>\underbrace{| \Psi \rangle}_{\text{state ket}} = \underbrace{\int\limits\limits_{R_N} d^3\mathbf{r}_N \cdots \int\limits\limits_{R_1} d^3\mathbf{r}_1}_{\text{superposing}\atop\text {weighted basis kets}} \, \underbrace{\overbrace{\Psi}^{\text{wave}\atop\text{function}} (\overbrace{\mathbf{r}_1, \ldots, \mathbf{r}_N}^{\text{eigenvalues of basis observables}\atop\mathord{\sim}\text{ argument of wave function}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,})}_{ {{{\text{component of state ket}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\atop\text{= value of scalar product } \langle \,\text {basis bra }|\,\Psi\,\rangle\,\,}}\atop\text{= value of wave function}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\atop\text {= complex number weight of basis ket}\,\,\,\,\,\,\,\,\,\,}\underbrace{|\overbrace { \mathbf{r}_1, \ldots, \mathbf{r}_N}^{{\text{eigenvalues of basis observables}}\atop\mathord{\sim}\text { label of basis ket}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, }\rangle }_{\text{basis ket}}</math>

[[User:Chjoaygame|Chjoaygame]] ([[User talk:Chjoaygame|talk]]) 16:42, 29 February 2016 (UTC)