Talk:Wave function/Archive 8

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? Waleswatcher (talk) 12:55, 3 February 2016 (UTC)

The present attitude of this article is that a state vector (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. 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. 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 Ψ(x, t). 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. Waleswatcher (talk) 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). 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 ψ(r₁, r₂, t) = α₁ exp (−|r₁|²) exp (−iE₁t/ħ) + α₂ exp (−|r₂|²) exp (−iE₂t/ħ). A symbolic expression is, for example, in Dirac's notation, |z〉 = ζ₁|z₁〉 + ζ₂|z₂〉 .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. YohanN7 (talk) 09:17, 4 February 2016 (UTC)

Nipping the bud, in

${\displaystyle |\Psi \rangle =I|\Psi \rangle =\int |x\rangle \langle x|\Psi \rangle dx=\int \Psi (x)|x\rangle dx=\int |p\rangle \langle p|\Psi \rangle dp=\int \Psi (p)|p\rangle dp,}$

it is the case that

${\displaystyle |\Psi \rangle ,|x\rangle ,|p\rangle }$

are states, while

${\displaystyle \Psi (x),\Psi (p)}$

are wave functions. There is never equality between states and wave functions. 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? M∧Ŝc²ħεИ_τlk 13:29, 4 February 2016 (UTC)

(The above is to demonstrate that Chjoaygame's invention of symbolic wave functions do not stand up to inspection.) 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. 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.^[1]

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 ψ(ξ), a function of the observables ξ. A function of the ξs 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 ξs, 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 ξ, instead of eigenvalues ξ′ for those observables.

     † 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.^[2]

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.^[3]

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

... the matter wave

${\displaystyle \psi (\mathbf {r} _{2},\tau _{2})=\int K(\mathbf {r} _{2}-\mathbf {r} _{1};t_{2}-t_{1})\psi (\mathbf {r} _{1},t_{1})\,\mathrm {d} \mathbf {r} _{1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)}$

where

${\displaystyle K(\mathbf {r} ,\tau )\,\,\,=(2\pi \hbar )^{-3}\int \exp \left[{\frac {\mathrm {i} }{\hbar }}(\mathbf {p} \cdot \mathbf {r} -E\tau )\right]\mathrm {d} \mathbf {p} .}$^[4]

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, |z〉 = ζ₁|z₁〉 + ζ₂|z₂〉 .

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

${\displaystyle (26^{\prime })\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\psi _{n}(q)\,\,\,\,=e^{-{\frac {2\pi ^{2}\nu _{0}q^{2}}{h}}}H_{n}\left(2\pi q{\sqrt {\frac {\nu _{0}}{h}}}\right)}$^[5]

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

${\displaystyle \psi (\mathbf {x} )\,\,\,\,\,=R(r)Y(\theta ,\phi ),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2.1.21)}$^[6]

References

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 ξ, instead of eigenvalues 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.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. 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 waves 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? M∧Ŝc²ħεИ_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

${\displaystyle \langle x|x'\rangle =x'(x)=\delta (x-x'),}$

the position eigenstate, a wave function? It does not satisfy the Schrödinger equation in the usual sense, but (OR) it turns out that it does so in the sense of the left and the right side of the Scrödinger equation with δ(x − x′) plugged in being equal as distributions (or continuous linear functionals). When they act on wave packets composed of free waves, they yield the same result. 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: ψ(r₁, r₂, t) = α₁ exp (−|r₁|²) exp (−iE₁t/ħ) + α₂ exp (−|r₂|²) exp (−iE₂t/ħ). I think many texts display objects more or less like that, and call them wave functions.

And an example of a symbolic expression: |z〉 = ζ₁|z₁〉 + ζ₂|z₂〉.

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 (ζ₁, ζ₂). 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 (ζ₁, ζ₂) can be viewed as a function:

We may suppose the basic bras to be labelled by one or more parameters, λ₁, λ₂, ... λ_u, each of which may take on certain numerical values. The basic bras will then be written 〈λ₁ λ₂ ... λ_u| and the representative of |a〉 will be written 〈λ₁ λ₂ ... λ_u|a〉. This representative will now consist of a set of numbers, one for each set of values that λ₁, λ₂, ... λ_u may have in their respective domains. Such a set of numbers just forms a function of the variables λ₁, λ₂, ... 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.Chjoaygame (talk) 15:05, 5 February 2016 (UTC)

@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

${\displaystyle \Psi (\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})}$

from

${\displaystyle |\Psi \rangle =\sum _{s_{z\,1},\ldots ,s_{z\,N}}\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}\,\Psi (\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle }$

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

@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 set of kets (one doesn't use a basis (basic?) bra like 〈λ₁ λ₂ ... 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. M∧Ŝc²ħεИ_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?Chjoaygame (talk) 16:48, 5 February 2016 (UTC)

The modern terminology is this:

${\displaystyle \underbrace {|\Psi \rangle } _{\text{state vector (ket)}}=\underbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} _{\text{adding up}}\,\underbrace {\Psi (\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})} _{\text{wavefunction (component of state vector)}}\underbrace {|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle } _{\text{basis ket}}}$

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. M∧Ŝc²ħεИ_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. M∧Ŝc²ħεИ_τlk 13:36, 6 February 2016 (UTC)

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). YohanN7 (talk) 10:18, 6 February 2016 (UTC)

${\displaystyle \underbrace {|\Psi \rangle } _{\text{state vector (ket)}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}^{\text{discrete labels}}} \overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{adding up}}\,\underbrace {\overbrace {\Psi } ^{\text{wave function}}(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})} _{\text{component of state vector along basis state}}\underbrace {|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle } _{\text{basis state (basis ket)}}}$

YohanN7 (talk) 12:44, 6 February 2016 (UTC)

@YohanN7: Thanks for feedback, and I like your elaborated version.

@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 |⟩" for the symbols used in the notation (so it seems). In any case, this is a little pedantic and off-topic for this article. M∧Ŝc²ħεИ_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 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 where he uses bras. I didn't think that was a worry. I was surprised that Editor 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 |Ψ⟩ is denoted

${\displaystyle \langle a,b,\ldots ,l,m,\ldots |\Psi \rangle ,}$

for a concrete example,

${\displaystyle \Psi (x)=\langle x|\Psi \rangle .}$

I agree that Editor Maschen's formulation is admirable, and evidently Editor YohanN7 finds it fresh. I think it agrees with the 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:

${\displaystyle \underbrace {|\Psi \rangle } _{\text{state vector (ket)}}=\underbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} _{\text{adding up}}\,\underbrace {\Psi (\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})} _{\text{wavefunction (scalar projection of state vector)}}\underbrace {|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle } _{\text{basis ket}}\,.}$

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.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. ${\displaystyle \underbrace {|\Psi \rangle } _{\text{state vector (ket)}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}^{\text{discrete labels}}} \overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{adding up}}\,\underbrace {{\Psi }(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})} _{\text{wavefunction (component of state vector along basis state)}}\underbrace {|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle } _{\text{basis state (basis ket)}}}$

As for "scalar projection" - no. If anything, just "projection" - but "component" is probably better. Waleswatcher (talk) 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).

${\displaystyle \underbrace {|\Psi \rangle } _{\text{state vector (ket)}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}^{\text{discrete labels}}} \overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{adding up}}\,\underbrace {\overbrace {\Psi } ^{\text{wave function}}\overbrace {(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N})} ^{\text{eigenvalues of commuting observables}}} _{\text{component of state vector (complex number)}}\underbrace {|\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}\rangle } _{\text{basis state (basis ket)}}}$

YohanN7 (talk) 09:23, 8 February 2016 (UTC)

@Waleswatcher: Sure, I'm not set on the exact labels and like your version better than mine.

@Chjoaygame: In

${\displaystyle \langle a,b,\ldots ,l,m,\ldots |\Psi \rangle }$

what you are actually doing is start with |a, b, ..., l, m,⟩ then taking the dual to get the bra ⟨a, b, ..., l, m,| then taking the inner product of two kets |a, b, ..., l, m,⟩ and |Ψ⟩ (not a bra with a ket), in other words projecting |Ψ⟩ on |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. M∧Ŝc²ħεИ_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).

M∧Ŝc²ħεИ_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.Chjoaygame (talk) 10:19, 7 February 2016 (UTC)

For example, in his paper 'Derivation of the Born rule from operational assumptions', 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.^[1]

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 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. M∧Ŝc²ħεИ_τlk 11:54, 7 February 2016 (UTC)

I think this revision by Chjoaygame is an improvement. However, this statement,

There is at least one such maximal set of observables for which the state is a simultaneous eigenstate,

needs elaboration. Think of a gaussian wave packet. It is not an eigenstate of momentum nor is it of position. Yet, the statement is (now I am actually guessing) true. Arrange for a projection operator projecting out the one-dimensional subspace spanned by that wave packet. Then build a Hermitian operator with that projection operator as an ingredient, ... This sort of statement most definitely requires a citation of its own. YohanN7 (talk) 09:57, 8 February 2016 (UTC)

Yes, indeed a citation would be good. The question about the gaussian wave packet is good. I suppose some operator exists that has it as an eigenfunction. I mentioned above the notion of "exotic" operators and crystals. I vaguely recall, subject to correction, that there might be such a thing as the square root of the Fourier transform? Definitely one for a mathematician!Chjoaygame (talk) 10:15, 8 February 2016 (UTC)

The mathematical community has been pinged. YohanN7 (talk) 13:44, 8 February 2016 (UTC)

No need to invent "strange" operator for the gaussian wave packet; it is well-known to be the ground state of the Hamiltonian of a harmonic oscillator. Widely used in quantum optics, in relation to coherent states.

On the other hand, mathematically it is absolutely evident that every vector (of norm 1) is an eigenvector of some Hermitian operator. Moreover, vectors could not be different in this respect, since the Hilbert space in invariant under the unitary group, and every unit vector is transformed to every other unit vector by some unitary operator.

But some operators are physically much more feasible than others. Many operators could be implemented only by a quantum computer. (Also, superselection sectors may prevent it.) Boris Tsirelson (talk) 14:22, 8 February 2016 (UTC)

By the way (partially off-topic): a linear combination of two states prepared by two different devices can be prepared by a combination of these two devices and a third device that activates one of the two with given amplitudes. All that must be made coherently (preventing decoherence), which may be a hard challenge for a laboratory, but should be possible in principle (unless prevented by superselection). Boris Tsirelson (talk) 14:30, 8 February 2016 (UTC)

Both ways are perfectly fine. Choose freely.Chjoaygame (talk) 11:05, 11 February 2016 (UTC)

Perhaps I am mistaken, but I think it is impossible in principle to couple laboratory devices=apparatuses coherently enough to produce a pure state; perfect coherence would be needed. My understanding is that that is why one can't exactly observe position and momentum at once on one degree of freedom.Chjoaygame (talk) 11:20, 11 February 2016 (UTC)

The last remark is something for quantum state, and is, I think, relevant to Chjoaygame's efforts to make a physical definition of "pure state". YohanN7 (talk) 14:38, 8 February 2016 (UTC)

With respect, I wrote not 'strange', but "exotic". I am wondering if one could physically start with, for example, a beam of quantum systems in an eigenstate of momentum, and from it generate a pure beam of systems with gaussian wave function in that degree of freedom? Does that question even make sense? Is there a square root of the Fourier transform? I am comforted that you confirm "that every vector (of norm 1) is an eigenvector of some Hermitian operator." Is there a convenient and suitable source that could be cited here for that?

I guess, no such source. It is like to seek a source for the claim "for every vector in 3-dim space there exists a vector field with this vector at the origin, and vanishing on infinity". Both claims are too evident. Their notability (in math) is less than 0.001 (where 1 means the minimal notability for being mentioned in literature). Boris Tsirelson (talk) 15:24, 8 February 2016 (UTC)

An apparatus could just absorb the incoming particle and at this moment (coherently) activate a process that prepares the output state. Not elegant, but should work. It seems, in nonlinear quantum optics such processes are in use; for example, two incoming photons together excite an atom, and then the atom emits one photon of twice the energy. Boris Tsirelson (talk) 15:29, 8 February 2016 (UTC)

Above I suggested a way for making beams in some simple non-standard states, but I think it would be difficult or impossible to make it work for a gaussian weighting.Chjoaygame (talk) 14:55, 8 February 2016 (UTC)

The role of the Fourier transform is the same as in a previous discussion. The Hermite functions (eigenfunctions of the FT) include the pure Gaussian. YohanN7 (talk) 15:17, 8 February 2016 (UTC)

The square root of the Fourier transform exists, and is well-known. This is again about harmonic oscillator and coherent states. Fourier transform is the evolution operator (for the harmonic operator) during 1/4 of the period. Now take 1/8 of the period. Boris Tsirelson (talk) 15:20, 8 February 2016 (UTC)

Thank you.Chjoaygame (talk) 17:26, 8 February 2016 (UTC)

See also Fourier_transform#Eigenfunctions (the last paragraph) for refs. Boris Tsirelson (talk) 17:54, 8 February 2016 (UTC)

Yes, the Fourier point is a bit on the side. What Editor YohanN7 was really asking for was a source for the statement that there is at least one maximal compatible set for which a wave function is a simultaneous eigenfunction. I think the state is called 'degenerate' if there is more than one such set? The nearest I have found is on page 49 of Dirac 4th edition: "If they do not commute a simultaneous eigenstate is not impossible, but is rather exceptional. On the other hand, if they do commute there exist so many simultaneous eigenstates that they form a complete set, as will now be proved. [Dirac's italics.]" I think that is a sort of converse of what he asking for? And on page 50: "The idea of simultaneous eigenstates may be extended to more than two observables and the above theorem and its converse still hold, i.e. if any set of observables commute, each with all the others, their simultaneous eigenstates form a complete set, and conversely." And on page 52 he concludes: "From the point of view of general theory, any two or more commuting observables may be counted as a single observable, the result of a measurement of which consists of two or more numbers. The states for which this measurement is certain to lead to one particular result are the simultaneous eigenstates." [Dirac's italics.] Does this mean that any wave function is a member at least of one rather special complete set? This is getting into deep water.Chjoaygame (talk) 20:58, 8 February 2016 (UTC)

As usual, we speak different languages and therefore do not understand each other. When I speak about existence of operator for a given vector, I mean just this: a vector is given in an "empty" Hilbert space; that is, nothing is given in addition, just this vector. But you speak about degenerate vectors etc. This shows that you are not in an "empty" Hilbert space, but in a Hilbert space endowed with some additional structure. Then please specify this structure. Do you mean that a commuting set is already chosen? If so, how should I take it into account? Did you ask for an operator commuting with these? Or what? A question to a mathematician should be formal enough. I cannot think in terms of quotations (from Dirac or whoever), but only in terms of mathematical objects (and structures, and symmetries). And I repeat: in the "empty" Hilbert space all vectors (of norm 1) are "equal" in their properties (due to the evident symmetry). Just like all points of the 3-dim Euclidean affine space. They may differ only with respect to something (a coordinate system or whatever). Boris Tsirelson (talk) 21:37, 8 February 2016 (UTC)

Thank you for your care in this. Sorry to take up your time. This isn't really a problem to ask you about. I think we three agree on the truth of the statement. The problem for a Wikipedia editor is to find a suitable source to cite. Please don't spend more time on this; you have other things to do.Chjoaygame (talk) 05:34, 9 February 2016 (UTC)

You cannot solve a routine mathematical exercise by quoting great minds. Boris Tsirelson (talk) 05:49, 9 February 2016 (UTC)

Of course you are right. But the game here is not to think through to the truth. It is to fill in one of those pesky <ref></ref> thingoes. Please don't waste your time on this.Chjoaygame (talk) 06:36, 9 February 2016 (UTC)

Well, try the book: K.R. Parthasarathy, "An Introduction to Quantum Stochastic Calculus", page 8: "The unitary group ... acts ... and the action is transitive on the set of pure states ... and on the set of atomic events." This means that "all pure states are created equal" as long as we deal with "empty" Hilbert space. Boris Tsirelson (talk) 07:12, 9 February 2016 (UTC)

Thank you for your kind care in this. I have got a copy of Parthasarathy and read (not very word) through to page 8 and there indeed I found your quote. I will think it over. Thank you again.Chjoaygame (talk) 15:44, 9 February 2016 (UTC)

Probably, no proof there. If you'll want a proof, just say so. Boris Tsirelson (talk) 16:31, 9 February 2016 (UTC)

Really and truly, I don't want to waste your time. We three agree that it is true. That is not in question. The problem is to comply with the Wikipedia rule against synthesis. A citation has to say exactly what it is supporting; not provide reason to believe it true, nor reasons which combine to prove it true. Also (though one may consider this trivial) the article defines a state as being pure, with no suggestion of the possibility of a mixed state. The Parthasarathy book defines a state as in general mixed, with purity an exceptional special case, of probability zero. This is in a sense trivial, but it is a problem for sourcing in conformity with the Wikirule against synthesis. This is all trivial. It is not appropriate at the place where sentence is, to expand. Expansion belongs in the body of the article. If the idea finds its way into the body of the article, it can be expanded on and even proved there, and sourced more easily there. That's why I have been saying don't waste your time on it. By the way, I have now seen the physical reason why a pure state has a family of natural siblings. In many cases (with exceptions), I think (perhaps mistakenly) that a pure state is a simultaneous eigenstate of only one maximal set of compatible observables. But for that set, there are very many simultaneous eigenstates, even many more than are needed to form a basis. I won't bore you with the physical reason. Dirac proves it mathematically. I really appreciate the care you have given here. Thank you. I think I will eventually accidentally stumble on the right citation now that I know what is needed.Chjoaygame (talk) 18:44, 9 February 2016 (UTC)Chjoaygame (talk) 21:09, 9 February 2016 (UTC)

Wow! Either physics articles on Wikipedia are treated as seriously as political articles, or it is your own attitude. If this were the case for math articles, I would not participate at all. Fortunately, mathematicians are much less stringent; indeed, a routine mathematical exercise cannot be sourced, very often. Well... I return to the better world of math. Boris Tsirelson (talk) 19:15, 9 February 2016 (UTC)

I am not clued up on this, but I think there are special rules for maths, that routine calculations are permitted to be assumed, or somesuch. Maths is in principle perfectly logical, while empirical subjects are mostly illogical, so in an empirical topic, if something seems logical, it can still be wrong, for example because the terminology is ambiguous or ill-defined, or the supposed logic is only approximate when exactitude would be needed to get it right. Anyway, your posts were helpful, thank you. They will help guide my adventures.Chjoaygame (talk) 20:46, 9 February 2016 (UTC)

For example, in quantum mechanics they talk a lot about "measurement", but really they never do measurements in the ordinary sense of the word. Huge muddles arise from this, leading to endless nonsense.Chjoaygame (talk) 21:00, 9 February 2016 (UTC)

The environment at certain physics articles is such that not only every sentence, but every single word may require a citation. This has been the case with this article the past year or so.

Thus if H₁, H₂, … H_n is a maximal set of commuting observables and |Φ⟩ is a common eigenvector, then if U is a unitary operator taking |Φ⟩ to (a multiple of) our general state |Ψ⟩, then the set UH₁U⁻¹, UH₂U⁻¹, … UH_nU⁻¹ is a maximal set of commuting observables with common eigenvector |Ψ⟩, right? This can be put in an "nb" if we wish. (I can see one potential technical problem. What if the state is only "normalizable to a delta function"?) YohanN7 (talk) 09:34, 10 February 2016 (UTC)

Then how do you characterize a "maximal commuting set of observables"? It seems reasonable to me that such a set generates a maximal abelian algebra of Hermitian operators. The set should also, for economical reasons, be a minimal set of generators of that algebra. YohanN7 (talk) 09:42, 10 February 2016 (UTC)

Likely I am missing the main point here.

— Preceding unsigned comment added by 194.68.82.241 (talk) 13:55, 10 February 2016 (UTC) Re-posted by 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?Chjoaygame (talk) 11:05, 10 February 2016 (UTC)

No you don't, except that you end up with too much. If A, B are "in", then every polynomial in them is "in" too. Therefore finish off with a minimal generating set. YohanN7 (talk) 11:33, 10 February 2016 (UTC)

How about this:

One starts building the set with some choice of observable. Then one chooses another that is linearly independent of members of the set. If the new one commutes with all members if the set, it stays; if it doesn't, it's out. Repeat until one can't find any more that commute with all.

I think 'maximal' intends that one can't add any more members that qualify.

My current progress is that Dirac doesn't use the term. I think I must have picked it up from reading the Wikipedia articles, but I will keep checking. I am pretty sure I didn't invent it!!Chjoaygame (talk) 11:45, 10 February 2016 (UTC)

Now you miss the point. You get an algebra (with infinitely many elements) that way. The right algebra, but what is needed is a minimal generating set (usually a finite set) of that algebra. YohanN7 (talk) 11:57, 10 February 2016 (UTC)

Fair enough. Next item in my survey: Messiah doesn't seem to use the term. On page 203, he writes "More generally, one says that the observables A, B, ..., L form a complete set of commuting observables if they possess one and only one common basis."Chjoaygame (talk) 12:06, 10 February 2016 (UTC)

Next item. This doesn't justify the use of the term, but may give a clue as to how it arose? London & Bauer 1939: "Elles ne touchent pas la précision avec laquelle l'état du système est actuellement connu ; celle-ci est maximum lorsque la fonction ψ est donnée." Wheeler & Zurek 1983 translate this as "They do not affect the precision with which the state of the system is currently known; thus it is already maximal when the ψ function is given."Chjoaygame (talk) 12:37, 10 February 2016 (UTC)

This may divert off topic but may help with understanding: Chjoaygame, do you understand what a basis set of a vector space is, in the context of linear algebra? (No need to answer explicitly). If so, it should not be hard to understand what a minimal generating set means. M∧Ŝc²ħεИ_τlk 12:48, 10 February 2016 (UTC)

Thank you Editor Maschen for this helpful comment.

Next item. Another use of the word 'maximal', that may hint, but doesn't justify. Wheeler & Zurek 1983 (p. 154) reprint of translation by Trimmer of Schrödinger 1935: "If through a well-chosen, constrained set of measurements one has gained that maximal knowledge of an object which is just possible according to A, then the mathematical apparatus of the new theory provides means of assigning, for the same or for any later instant of time, a fully determined statistical distribution to every variable, that is, an indication of the fraction of cases it will be found at this or that value, or within this or that small interval (which is also called probability.)"Chjoaygame (talk) 13:01, 10 February 2016 (UTC)

Clebsch–Gordan coefficients for SU(3)#The maximally commuting set of operators. I didn't invent it. I was just intending to use what seemed to be the local language. I don't know exactly where I picked it up. I will keep looking. I have no attachment to the term.Chjoaygame (talk) 13:16, 10 February 2016 (UTC)

The term appears above on this page at Talk:Wave function#Physical interpretation of "basis, corresponding to a maximal set of commuting observables". I guess I picked it up from there or a related source.Chjoaygame (talk) 13:24, 10 February 2016 (UTC)

Yes, it seems I copied it from the just previous version, that read "For a given system, once a representation corresponding to a maximal set of commuting observables and a suitable coordinate system is chosen, the wave function is a complex-valued function of the system's degrees of freedom corresponding to the chosen representation and coordinate system, continuous as well as discrete."Chjoaygame (talk) 13:29, 10 February 2016 (UTC)

Is this the edit that introduced it?Chjoaygame (talk) 13:37, 10 February 2016 (UTC)

Next item. I didn't find it in Cohen-Tannoudji et al. (1973/1977).Chjoaygame (talk) 13:43, 10 February 2016 (UTC)

Next item. Weinberg's 2013 Lectures don't seem to use it. On page 70, he writes "Recall that if a system is in a state represented by a normalized Hilbert space vector Ψ, and we perform a measurement (say, of a set of observables represented by commuting Hermitian operators) which puts the system in any one of a complete set of states represented by orthonormal state vectors Φ_i, ..."Chjoaygame (talk) 14:08, 10 February 2016 (UTC)

Found something. Von Neumann, J. (1932/1955), Mathematical Foundations of Quantum Mechanics, translated by R.T. Beyer, Princeton University Press, Princeton NJ, on pp. 153–154:

... An operator which possesses no proper extensions -- which is already defined at all points where it could be defined in a reasonable fashion, i.e., without violation of its Hermitian nature -- we call a maximal operator. Then, by the above, a resolution of the identity can belong only to maximal operators.

           On the other hand, the following theorem holds: each Hermitian operator can be extended to a maximal Hermitian operator.

I think this may be the original source for the term. Von Neumann continues with relevant material.Chjoaygame (talk) 15:54, 10 February 2016 (UTC)

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."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 ${\displaystyle {\mathfrak {V}}}$ 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 ${\displaystyle n}$ members, they must be linearly dependent. The number ${\displaystyle n}$ is then called the dimension of ${\displaystyle {\mathfrak {V}}}$." He doesn't use it in that sense elsewhere in that book.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 {{isbn}}: Check isbn value: checksum (help), 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."Chjoaygame (talk) 20:32, 10 February 2016 (UTC)

Auletta, G., Fortunato, M., 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 |ψ〉 contains the maximal information about a quantum system."Chjoaygame (talk) 20:48, 10 February 2016 (UTC)

I Googled the phrase 'maximal set of commuting observables', and found this, and also this, and moreover this, and yet again this, and now this.Chjoaygame (talk) 04:18, 11 February 2016 (UTC)

I looked back through the revision history and found that the term 'maximal set' was introduced by this edit. The full expression 'maximal set of commuting observables' was introduced by this one. In the current version of the article, the full expression is used at least six times by editors other than me. I didn't invent this term. I think the immediately foregoing five external links show that it is a pretty nearly standard term.Chjoaygame (talk) 08:48, 14 February 2016 (UTC)

I suggested above the term 'scalar projection', and this was duly rejected. I suggested it while feeling that it didn't seem right, but was the best I could find in Wikipedia for the purpose at the time. Now my memory has eventually at last produced what I was really looking for, and it occurs to me that it may be useful here. Here it is, for the consideration of interested editors:

I read in the article, for example, "Now take the projection of the state Ψ onto eigenfunctions of momentum ...". . I have felt uncomfortable with, and even baffled by, such expressions. I have worked out that it intends to refer to what I would think of as a list of projections, at least when there are only countably many components (a continuum of projections in the uncountable case). Probably the usage is conventional?

What has come back to my memory is the term resolution. Perhaps it is far-out obsolete, I don't know.

As I recall, perhaps mistakenly, the old term for the 'scalar projection' was 'resolute'. And, using the word-root in a slightly different way, I would feel comfortable with the wording 'Now take the resolution of the state Ψ into eigenfunctions of momentum ....' I would instantly know what that intended, pretty nearly, while the term 'projection' there leaves me feeling a little bemused. For the present specific context, I would also feel comfortable with the wording 'Now take the resolution of the state Ψ into a superposition of eigenvectors of momentum ...' or with 'Now take the resolution of the state Ψ into a superposition of eigenstates of momentum ....' There is still some slip-room in this, but perhaps something could be done about it.

As I remember, the projection is a vector, and the accompanying resolute is its magnitude. Resolution means analysis into components. This terminology is not perfectly logical, but I seem to remember it as conventional.

Googling, I find this. It gives three alternative terms, scalar projection = scalar resolute = scalar component. My memory lights on simply 'resolute', not needing to add that it means the magnitude. But it seems my memories are out of date?Chjoaygame (talk) 16:00, 12 February 2016 (UTC)

Apologies for not answering this adequately before (more so for absent mindlessly replying about vector projection than scalar projection), but you could have prompted an earlier reply by asking earlier. Also, maybe you would like to decide why it is not correct (or at least misleading) to use "scalar projection". The scalar projection of a vector along the direction of another vector is a specific number. A component of a vector is any one of the scalar projections along a basis. In the state vectors above, the wavefunction Ψ(x₁, x₂, ..., s_z1, s_z2, ...) is a component of the vector at a given configuration of the system, and is not a particular component of a vector like Ψ((2,3,6)₁, (9/2,−4,32)₂, ..., 1/2, −3/2, ... ). For this example, the scalar projection in Dirac notation is

${\displaystyle \langle (2,3,6)_{1},(9/2,-4,32)_{2},\ldots ,1/2,-3/2,\ldots |\Psi \rangle }$

As for other terms like "resolution" or "resolute", I'm not keen on them. Can't we just use "component"? It is a term used from the end of high-school (for kids just learning vector algebra) and beyond. M∧Ŝc²ħεИ_τlk 08:44, 13 February 2016 (UTC)

Thank you for this. I think the term 'scalar projection' is good. I suggested it because I thought (perhaps mistakenly) it seemed to be the Wikipedia term. My impression was that a certain editor had effectively knocked it off. My use of the word 'duly' was satirical, not literally intended. To repeat, I think (subject to correction) that the term 'scalar projection' seems to be the Wikipedia term. I have no serious objection to it.

As for your preferred term 'component'. I am not clued up on this topic. I am not familiar with the current customs. If it is current custom to read 'component' as meaning the same as 'scalar projection', then I suppose (subject to correction) that its consisting of one word makes it preferable to the two-word term 'scalar projection'. Somehow in the back of my mind is the perhaps mistaken idea that the default meaning of 'component' is 'vector component'. My only reason (perhaps not valid) for suggesting 'resolute' was that it is one-word, and that it seemed to ring a bell in my memory.

Perhaps it may help to say explicitly in the article just what the terms mean, to take out the guesswork for the reader as to the default meaning.

My just foregoing three paragraphs refer to the singular case, of one scalar resulting from one projection.

On the other hand, I am quite keen on the wording 'Now take the resolution of the state Ψ into a superposition of eigenstates of momentum ....' I think (perhaps mistakenly) that many vector projections are involved here, one vector projection per degree of freedom. The several vector projections, duly weighted, superpose to reproduce the state vector of interest. The wave function is the family (list or continuum) of scalar projections that belongs. I feel confused, mistaken or muddled about this. (This is not to advocate 'resolute' as a word for 'scalar projection' = 'component'.)

I have read your above comment, and I feel confused, I think partly because you use the term 'at a given configuration of the system', partly because I am a bear of little brain. I am not at all suggesting you are wrong, I am just saying I don't understand. The term 'at a given configuration of the system' seems an import here. Perhaps you will explicitly define it. I think perhaps relevant here is Editor YohanN7's above comment about the value of a function. The traditional notation y = f(x) is I think unsuitable for some tasks required for our present purposes. Is f(x) a value or a function? Perhaps it may help to use the more modern protocol, more or less along the following lines: ${\displaystyle f}$ denotes a function; the domain of ${\displaystyle f}$ is ${\displaystyle \mathbb {C} }$; the range of ${\displaystyle f}$ is ${\displaystyle \mathbb {R} }$; ${\displaystyle f}$ is into, not onto; ${\displaystyle f:\mathbb {C} \rightarrow \mathbb {R} }$ ; ${\displaystyle f:z\mapsto |z|}$ .

This may seem all too much, and for someone familiar with it, very likely it is so. But perhaps it may help newcomers? This is topic not child's play.Chjoaygame (talk) 11:56, 13 February 2016 (UTC)

I can understand why you want to use "resolute" as in "resolve a vector into components", but "component" is certainly common enough to use.

Maybe my "at a given configuration of the system" is poorly worded but it just means specify the configuration, then get a complex number (value of the wavefunction). Maybe it's better to say "at any configuration", since you can plug in any allowed position coordinates and spin projection quantum numbers.

About your next topic on functions "f(x)", x is a number, and f is the rule (not a number itself) taking x and assigning another number f(x) to this x. In physics, it is usual to abuse notation and just abbreviate f(x) by f, so f is effectively conflated with a quantity (from the context, one should be able to tell what f is).

Writing

${\displaystyle f:D\rightarrow R}$

with D the domain and R the range does explicate the domain and range, but is not helpful for typical readers because they have to look up the notation, or we have to waste space explaining it when f(x) will do (last year I did think it may be good to clarify the domain and range, but scrapped the idea for this reason). The notation f(x) for wavefunctions is perfectly fine and standard, and it would be clumsy to use the colon-arrow notation . M∧Ŝc²ħεИ_τlk 12:43, 13 February 2016 (UTC)

Writing f(x) for functions is fine as long as the distinction between f(x) and f is not needed. Se my proposal for "definition and anatomy" above. YohanN7 (talk) 12:52, 13 February 2016 (UTC)

With respect, do I detect a typo here? Did you accidentally omit a 'not' from the first sentence of this comment? Did you intend 'Writing f(x) for functions is fine as long as the distinction between f(x) and f is not needed.' .?Chjoaygame (talk) 21:53, 13 February 2016 (UTC)

Yes. Now changed. YohanN7 (talk) 08:37, 15 February 2016 (UTC)

Thank you for this. Still, I am a bear of little brain. Your use of the word 'configuration' here is not self-explanatory. I don't know what you mean by it. It isn't part of the current-context vocabulary. I think the present topic is rather special, and has special needs. The difference between wave functions and state vectors is not easy to grasp for a newcomer. I don't find Weinberg's presentation easy to grasp, nor Dirac's. I think they ought to have included some use of the colon-arrow notation (at least Weinberg; it probably wasn't common currency in Dirac's day). I think someone who is expected to deal with Hilbert space notions would be familiar with the colon-arrow notation, and would find it very helpful for the present purpose. It would be so for me. Both together would be good, considering the difficulty of the topic. We are really concerned with distinguishing functions from values, and suchlike.

I am not advocating 'resolute' as an alternative for 'component' meaning 'scalar projection'. Nevertheless, I am rather keenly advocating the language 'Now take the resolution of the state Ψ into a superposition of eigenstates of momentum ....' This does not rely on or imply the word 'resolute'. The wording currently in the article is 'Now take the projection of the state Ψ onto eigenfunctions of momentum ....' What exactly does that intend? I can partly understand 'Now take the projection of the state Ψ onto such-and-such an eigenfunction of momentum'. But 'projection onto many eigenfunctions'? That looks at face value to be intending a family of projections onto a family of eigenstates and somehow extracting a family of functions therefrom. It may be standard telescoped terminology, but if so, I think the reader deserves explicit notice of it. Likely I am muddled here?

I don't intend to change the topic by saying here that it comes to mind that it may help to say something along the lines of 'A state vector is an equivalence class of wave functions.' This is part of the topic of how to present this distinction in an easily graspable way.Chjoaygame (talk) 13:32, 13 February 2016 (UTC)

Though I am a bear of little brain, I think I am not the only kind of person to find it hard to get a really clear idea of what is going on. A certain other editor has informed us elsewhere that he is smarter than the average bear: indeed he tells us "I am a professor of physics that teaches QM at both the undergraduate and PhD level, and uses it every day in research. If I cannot understand the section, there is a big problem." And above here he writes

.... 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?

I think that is evidence that this topic is not easily conveyed. I think this is an argument that that article should be generous in availing itself of helpful modes of expression, and of repetition of ideas in different formats, if that will help. (In making this quote I am not endorsing its content. I am just using it to support my claim that this topic is not easily conveyed.)

I would like to repeat my suggestion that it may help to say that a state vector is an equivalence class of wave functions, and to explicate that statement a little.Chjoaygame (talk) 21:45, 13 February 2016 (UTC)

This discussion on nitpicking individual choices of words and notation is getting nowhere, and tiresome. You are saying repeatedly that the standard jargon (component, projection, basis, ket etc. etc.) is likely not to help the reader, and perpetually propose alternative terminology which turns out to be less standard in this context (resolution, representative) or more clumsy notation (alike the colon-arrow notation for functions), which would be even more confusing.

And how is "a state vector is an equivalence class of wave functions"? This is certainly going to confuse more readers. A state vector is a vector. A vector can be expressed in a convenient basis. The components of the vector are elements of a field (in the case of wavefunctions, complex numbers). This is about as standard and modern as anyone could expect. M∧Ŝc²ħεИ_τlk 08:32, 14 February 2016 (UTC)

user:YohanN7 was right. Whatever words everyone else chooses, you just have to pick something else. Also, I am aware Yohan also explained about the abuse of f(x) and f, but decided to give a second input while responding. M∧Ŝc²ħεИ_τlk 08:38, 14 February 2016 (UTC)

I am sorry to be a nuisance. Of course such is not my intention. In a complicated area such as the present one, people in Wikipedia have different presuppositions.

Before my initial comment on the term 'component', I looked it up in Wikipedia, supposing that it would be part of a standard Wikipedia terminology. (I don't recall exactly, but I quite likely also Googled it.) I found it used as a vector, not a scalar, for example in the article Tangential and normal components. I accept that Wikipedia is not a reliable source, but I did assume that basic things like this would be standard. Evidently not. Perhaps you will find in Wikipedia a usage that supports yours? I think if you Google 'component of a vector' you will find that I am not alone in my reading of the default meaning of 'component' as a vector.

"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 this edit by respected Editor 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.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.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.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.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.Chjoaygame (talk) 19:26, 14 February 2016 (UTC)

At my university, quantum mechanics was introduced only after two courses in linear algebra, two courses in analysis, one course in complex analysis, courses in vector analysis and ordinary differential equations, in addition to courses in basic and analytical mechanics (and other irrelevant courses). At least I did not have this gruesome trouble of understanding basic notation, terminology and concepts. Mathematics is a prerequisite for physics, and certainly should be expected on part of the reader of this article. I'd go as far as saying that physical interpretation (or physical content if you want) is not possible without the prerequisite mathematical training. It might be possible in very basic mechanics and thermodynamics, but not so in quantum mechanics for the reason that no-one is born with the correct intuition about the subject. But people are born with, or develop, mathematical intuition. This intuition and knowledge of the mathematical setting is simply necessary to treat quantum mechanics.

My editing at Wikipedia has, apart from pure pleasure, been aimed at building a bridge between mathematics and physics. Physics texts, especially the older ones are downright appalling when it comes to presenting mathematics, the prime example being group theory. I prefer to tilt physics articles towards better mathematical precision. But this does not include endless discussions about the choice of particular words (all standard, all presumably meaning the same thing). It involves fewer words. It involves splashing up the correct equations. Then the reader can chose to call things whatever he or she is accustomed to calling it. It doesn't matter. It also doesn't matter that the doctrine used to be that physics must be described in "ordinary language" eighty years ago. Ordinary language is fine, but is no substitute for precision. If the reader is unable to extract physical content (from a good precise presentation (i.e. not this one)) because of lack of basic mathematics, then he will not be able to properly digest an account given in "ordinary language". Popular science magazines provide a better source that Wikipedia for such things.

I'll stay away from this article for a while, because I don't want to force my intentions upon it until there is some sort of agreement of what should and shouldn't be here. Whaleswatcher's topic got lost all together in new walls of text. This is unfortunate. YohanN7 (talk) 11:21, 15 February 2016 (UTC)

Partially off-topic, but I cannot resist. My mathematical education is described, very briefly, in the first paragraph here: "...Kruglov taught mathematical analysis (from the definition of a metric space till spectral theory of operators in Hilbert spaces)..." But Kruglov also emphasized that his goal is, our ability to understand the quantum theory! Thanks to him, I have no idea of what the quantum theory looks like for a person without such mathematical preparation. Boris Tsirelson (talk) 16:39, 15 February 2016 (UTC)

Also: a large-scale perfect consistency of terminology is a must for large proof assistant-based projects. Indeed, a computer (for now) cannot understand the matter otherwise. Accordingly, computer-assisted verification of a nontrivial mathematical result takes many months of hard work quite similar to programming. Fortunately, we humans are very different. We grasp the idea and do not stumble on a poorly fitting technical details. The price we pay for this is, some small probability of error, alas. But a programmer usually errs more often than a mathematician. Boris Tsirelson (talk) 19:12, 15 February 2016 (UTC)

A lot of quotations being already here, let me add some.

"The apparent enormous complexities of nature, with all its funny laws and rules ... are really very closely interwoven. However, if you do not appreciate the mathematics, you cannot see, among the great variety of facts, that logic permits you to go from one to the other." Feynman "The Character of Physical Law" Sect. 2.

Boris Tsirelson (talk) 19:36, 15 February 2016 (UTC)

I should add that the humble background (listed above) I had when embarking on OM the first time was inadequate. In parallel, we studied transform methods (Fourier, Laplace, Z-transform, etc) and probability theory. But this is still by far not enough. Group theory taken to the level of representation theory of Lie groups and functional analysis taken at least to the point to the full-blown version of the spectral theorem is desirable. (Parts of a second course were incomprehensible to me (while simple had I known!) because group theory was lacking.) There is no end to it. But there is a bottom line.YohanN7 (talk) 11:31, 16 February 2016 (UTC)

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