Talk:Wave function/Archive 8

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Revision 2016-02-08

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)

Maximal commuting sets and common eigenfunctions

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.

Don Quixote, his horse Rocinante and his squire Sancho Panza after an unsuccessful attack on a windmill. By Gustave Doré.

— 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 ${\mathfrak {V}}$ 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 $n$ members, they must be linearly dependent. The number $n$ is then called the dimension of ${\mathfrak {V}}$." 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, 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)

Out of date term?

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

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

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: $f$ denotes a function; the domain of $f$ is $\mathbb {C}$; the range of $f$ is $\mathbb {R}$; $f$ is into, not onto; $f:\mathbb {C} \rightarrow \mathbb {R}$ ; $f:z\mapsto |z|$ .

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

$f:D\rightarrow R$

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)#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)

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. Boris Tsirelson (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. M∧Ŝc²ħεИ_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? Boris Tsirelson (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. M∧Ŝc²ħεИ_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 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". Boris Tsirelson (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. Boris Tsirelson (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. M∧Ŝc²ħεИ_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. Boris Tsirelson (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. Boris Tsirelson (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? Boris Tsirelson (talk) 13:31, 17 February 2016 (UTC)

There already is an ontology section in this article, and 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. M∧Ŝc²ħεИ_τlk 15:09, 17 February 2016 (UTC)

Indeed... maybe I am too pessimistic. Boris Tsirelson (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

" Inner product

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 |Ψ₁⟩ is a state in the above basis with components c₁, c₂, ..., c_n and |Ψ₂⟩ is another state in the same basis with components z₁, z₂, ..., z_n, the inner product is the complex number: ..."

???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. Boris Tsirelson (talk) 21:09, 17 February 2016 (UTC)

The edit material was introduced by this edit.Chjoaygame (talk) 21:51, 17 February 2016 (UTC)Chjoaygame (talk) 23:56, 18 February 2016 (UTC)

What is the puzzle? M∧Ŝc²ħεИ_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"?

Boris Tsirelson (talk) 22:16, 18 February 2016 (UTC)

@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. M∧Ŝc²ħεИ_τlk 16:46, 25 February 2016 (UTC)

scalar product

I read in the article

... The state space is postulated to have an inner product, denoted by

$\langle \Psi _{1}|\Psi _{2}\rangle ,$

that is (usually, this differs) linear in the first argument and antilinear in the second argument. The dual vectors are denoted as "bras", ⟨Ψ|. These are linear functionals, elements of the dual space to the state space. The inner product, once chosen, can be used to define a unique map from state space to its dual, see Riesz representation theorem. this map is antilinear. One has

$\langle \Psi |=a^{*}\langle \psi |+b^{*}\langle \phi |\leftrightarrow a|\psi \rangle +b|\phi \rangle =|\Psi \rangle ,$

where the asterisk denotes the complex conjugate. For this reason one has under this map

$\langle \Phi |\Psi \rangle =\langle \Phi |(|\Psi \rangle ),$

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

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. 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

The state space of kets is postulated to have an inner product, denoted by

$(|\Psi _{1}\rangle ,|\Psi _{2}\rangle ).$

The inner product is (usually, this differs) linear in the first argument and antilinear in the second argument. The dual vectors are denoted as "bras", ⟨Ψ|. These are linear functionals, elements of the dual space to the state space. The inner product, once chosen, can be used to define a unique map from state space to its dual, see Riesz representation theorem. this map is antilinear. One has

$\langle \Psi |=a^{*}\langle \psi |+b^{*}\langle \phi |\,\,\,\,\leftrightarrow \,\,\,\,a|\psi \rangle +b|\phi \rangle =|\Psi \rangle ,$

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

$\langle \Phi |\Psi \rangle =(|\Phi \rangle ,|\Psi \rangle ).$

As I read it, Wikipedia posts what reliable sources say, in context. Dirac would have a fair chance of being a reliable source on this topic. He says "scalar product".<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<Gottfried, K., Tung-Mow Yan (2003), Quantum Mechanics: Fundamentals, 2nd edition, Springer, New York, ISBN 978-0-387-22023-9,[1], p. 31: "to define the scalar products as being between bras and kets."> .

Weinberg (2013) also speaks of the "scalar product".

As does Messiah (1961).

Also, mostly Auletta, Fortunato, and Parisi (2009).

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

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

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

And Jauch (1968).

And Kemble (1937).

And Zettili (2009).

Bransden & 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 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 Von Neumann (1932/1955) and Schiff (1949), though not Weinberg, use "inner product".

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. Boris Tsirelson (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 terminologyChjoaygame (talk) 23:54, 18 February 2016 (UTC)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? Boris Tsirelson (talk) 06:19, 19 February 2016 (UTC)

I am hardly understanding what is really the fuss 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. Boris Tsirelson (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.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 – and once again embark on a ridiculous rant. The physics lies in the Born rule. 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".Chjoaygame (talk) 09:48, 19 February 2016 (UTC)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 ⟨a|b⟩, is Weinberg (Lectures on Quantum Mechanics, 2013).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. Boris Tsirelson (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. I guess your native culture is humanities (or medicine?) rather than hard science. 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. Boris Tsirelson (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'.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. Boris Tsirelson (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.Chjoaygame (talk) 11:48, 19 February 2016 (UTC)

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

Thank you.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 ⟨a|b⟩, is Weinberg (Lectures on Quantum Mechanics, 2013).Chjoaygame (talk) 15:18, 19 February 2016 (UTC)

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

β.       Scalar product

          With each pair of kets |φ⟩ and |ψ⟩, taken in this order, we associate a

complex number, which is their scalar product, (|φ⟩,|ψ⟩), ...

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

$\left(|\phi \rangle ,|\psi \rangle \right)\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2.10)$

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

$\langle \phi |\psi \rangle =\langle \psi |\phi \rangle ^{*}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2.15)$

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 (x₁,x₂) and a pairing such as ⟨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.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. Chjoaygame (talk) 17:24, 19 February 2016 (UTC)Chjoaygame (talk) 17:52, 19 February 2016 (UTC)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 φ's and ψ's. The sum of a φ and a ψ has no meaning and will never appear in the analysis." And "In the vector picture we can take the number φψ to be the scalar product of the two vectors φ and ψ. ... The vector picture, however, allows us also to form the products φ₁φ₂ and and ψ₁ψ₂. Thus we again find the vector picture giving more properties to the ψ's and φ's and than required in quantum mechanics." Is this his saying that the products such as φ₁φ₂ are not required in quantum mechanics because the metric is already supplied by the scalar product? 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 φ's and ψ's ..."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 φ corresponding to any |A⟩ may be looked upon as the scalar product of that |A⟩ with some new vector, there being one of these new vectors for each linear function of the ket vectors |A⟩." The same sentence appears in the 4th edition (1958).Chjoaygame (talk) 18:58, 19 February 2016 (UTC)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' αf with α a complex number and 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 φψ. I guess Dirac would have been well aware of all this.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 φ and ψ have a product, which is a number, in general complex. This product must always be written φψ, i.e. the φ must be on the left-hand side and the ψ on the right. Products such as ψφ, ψ₁ψ₂, φ₁φ₂, 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.Chjoaygame (talk) 11:07, 21 February 2016 (UTC)

In the second (1935) edition, on page 23, we read: "symbolic products of the type ψ_aψ_b or φ_aφ_b never occur in the theory."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.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 |u⟩ is represented by the symbol ⟨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.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. Boris Tsirelson (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. YohanN7 (talk) 12:04, 22 February 2016 (UTC)

Editor 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.Chjoaygame (talk) 13:05, 22 February 2016 (UTC)

functions of space

I read in the article:

• The idea that quantum states are vectors in an abstract vector space (technically, a complex 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 $\mathbf {Q} =Q_{x}\times Q_{y}\times Q_{z}$ has a wave function $\phi \,:\mathbf {Q} \rightarrow \mathbb {C}$. 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.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 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 $\rangle$ for the standard ket understood. A ket is then written simply as $\psi (\xi )$, a function of the observables $\xi$. A function of the $\xi$s used in this way to denote a ket is called a wave function."

An observable $\xi$ is an operator on a vector space. The domain $\mathbf {Q}$ of the above wave function $\phi$ is not a set of operators such as $\xi$ on a vector space; it is a set of points in configuration space.

Accordingly, the objects such as $\psi (\xi )$ and $\phi$ 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.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 Ψ is denoted |Ψ⟩, and the scalar product (Φ,Ψ) of two state vectors is written ⟨Φ|Ψ⟩. The symbol ⟨Φ| is called a “bra,” and |Ψ⟩ is called a “ket,” so that ⟨Φ|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.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.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. M∧Ŝc²ħεИ_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

$\langle n|f|m\rangle$.                         (11.17)

This symbol is written so that it may be regarded as "consisting" of the quantity f and the symbols |m⟩ and ⟨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.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.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. M∧Ŝc²ħεИ_τlk 12:33, 27 February 2016 (UTC)

I think a patient reader would manage.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 vectors 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. M∧Ŝc²ħεИ_τlk 15:37, 27 February 2016 (UTC)

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 this, this, and this. I guess 'standard' is a variable thing. It is not very evident in Wikipedia, though it occurs here and here. Though Wikipedia does not seem to determine standard usage for us.

In the article Euclidean vector I find

Decomposition

For more details on this topic, see Basis (linear algebra).

As explained above a vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the 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.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."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 Maschen to produce the following:

$\underbrace {|\Psi \rangle } _{\text{state ket}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}} ^{{\text{discrete}} \atop {\text{labels}}}\overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{superposing weighted basis kets}}\,\underbrace {\overbrace {\Psi } ^{{\text{wave}} \atop {\text{function}}}(\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ argument of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,})} _{{{{\text{component of state ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \atop {\text{= value of scalar product }}\langle \,{\text{basis bra }}|\,\Psi \,\rangle \,\,} \atop {\text{= value of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \atop {\text{= complex number weight of basis ket}}\,\,\,\,\,\,\,\,\,\,}\underbrace {|\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ label of basis ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\rangle } _{\text{basis ket}}$

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 = .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':

$\underbrace {|\Psi \rangle } _{\text{state ket}}=\underbrace {\overbrace {\sum _{s_{z\,1},\ldots ,s_{z\,N}}} ^{{\text{discrete}} \atop {\text{labels}}}\overbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} ^{\text{continuous labels}}} _{\text{superposing weighted basis kets}}\,\underbrace {\overbrace {\Psi } ^{{\text{wave}} \atop {\text{function}}}(\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ argument of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,})} _{{{{\text{component of state ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \atop {\text{= value of scalar product }}\langle \,{\text{basis bra }}|\,\Psi \,\rangle \,\,} \atop {\text{= value of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \atop {\text{= weight of basis ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\underbrace {|\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},s_{z\,1},\ldots ,s_{z\,N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ label of basis ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\rangle } _{\text{basis ket}}$

For the spinless case this version may be ok:

$\underbrace {|\Psi \rangle } _{\text{state ket}}=\underbrace {\int \limits \limits _{R_{N}}d^{3}\mathbf {r} _{N}\cdots \int \limits \limits _{R_{1}}d^{3}\mathbf {r} _{1}} _{{\text{superposing}} \atop {\text{weighted basis kets}}}\,\underbrace {\overbrace {\Psi } ^{{\text{wave}} \atop {\text{function}}}(\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ argument of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,})} _{{{{\text{component of state ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \atop {\text{= value of scalar product }}\langle \,{\text{basis bra }}|\,\Psi \,\rangle \,\,} \atop {\text{= value of wave function}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \atop {\text{= complex number weight of basis ket}}\,\,\,\,\,\,\,\,\,\,}\underbrace {|\overbrace {\mathbf {r} _{1},\ldots ,\mathbf {r} _{N}} ^{{\text{eigenvalues of basis observables}} \atop {\mathord {\sim }}{\text{ label of basis ket}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\rangle } _{\text{basis ket}}$

Chjoaygame (talk) 16:42, 29 February 2016 (UTC)