Talk:Wave function/Archive 9

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Archive 7

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 ∧Ŝ c²ħε И_τlk 09:06, 17 February 2016 (UTC)

Does it mean that here they need very basic explanations for beginners, plus links to other articles on more advanced topics? 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 ∧Ŝ c²ħε И_τlk 10:35, 17 February 2016 (UTC)

That is, to fill the gap between Introduction to quantum mechanics and harder articles. Nice. In the spirit of 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 ∧Ŝ c²ħε И_τlk 11:43, 17 February 2016 (UTC)

Well, if you can do it... I could not. Such a large fragment of QM has too large boundary, and you'll get again the problem, how to cut it from the environment. As a result, the article will be long, not so accessible to beginners, overlap other articles, and editors will war along the boundary, forever. My idea was rather, to say this is a small and not self-contained fragment of QM, from which it is impossible to make any far-reaching conclusions. Spinors! -- hard math! Ontology and philosophy! -- in summary style! No, this is not for me. 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 ∧Ŝ c²ħε И_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 | $Ψ 1$ ⟩ is a state in the above basis with components $c 1, c 2, ..., c n$ and | $Ψ 2$ ⟩ is another state in the same basis with components $z 1, z 2, ..., z n$ , 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 ∧Ŝ c²ħε И_τlk 21:04, 18 February 2016 (UTC)

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

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

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 ∧Ŝ c²ħε И_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.^[1] 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 ).

^ Sources for this:

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 $φ 1 φ 2$ and and $ψ 1 ψ 2$ . 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 $φ 1 φ 2$ 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

,

ψ 1 ψ 2

,

φ 1 φ 2

, have no meaning and will never appear in the analysis.

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

In the second (1935) edition, on page 23, we read: "symbolic products of the type $ψ a ψ b$ or $φ a φ b$ 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 ∧Ŝ c²ħε И_τlk 16:49, 25 February 2016 (UTC)

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

\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 ∧Ŝ c²ħε И_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 ∧Ŝ c²ħε И_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)

[1] Sources for this:

[1]