Talk:Wave function/Archive 10

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Referencing

Chjoaygame, you are messing up the referencing. I put in quite some effort to introduce streamlined reusable referencing a while ago. Please fix. YohanN7 (talk) 15:05, 1 March 2016 (UTC)

article topic is the wave function

Editor YohanN7 has made this edit.

The edit converts a sub-section that was directly about the topic of the article, Wave function, into a sub-section leading with a statement about the quantum state. This edit goes against the intention of the text that it changed. The overwritten intention was to state directly the facts about wave functions as such, the topic of the article. The new edit would perhaps be suitable for the article on the Quantum state, but I think it has degraded the present article, because it is primarily about that other topic, the quantum state.

The new sub-section perhaps is readily intelligible to sophisticated experts, for example those who were introduced to quantum mechanics only after several courses in mathematics and now have degrees in the subject. I think few such experts will come to Wikipedia to learn. But many readers of Wikipedia are not such experts. Editor Maschen has a good knowledge of this topic and has asked several questions in this edit above. I think those questions are sensible and reasonable, and will occur to many readers. The new edit has overwritten the former sub-section, that was an attempt to answer those questions directly in terms that a newcomer could relate to. The new edit has hidden the answers in a sophisticated expression that I think that newcomers would not easily grasp. The new edit is valid, and perhaps would be good as an addition to, but not as a total overwrite of, the former sub-section.

The new edit would perhaps be suitable for the article on the Quantum state, but I think it has degraded the present article.Chjoaygame (talk) 20:27, 1 March 2016 (UTC)

The new edit, in its current form, is in need of clarification. It writes of $x$ as "a basis element (thought of as variable over the complete domain)". The reader needs to be told here that such an $x$ is thought of in two distinct ways: as the label of a ket, and as the value of a variable in the domain. While the expression used in the new edit is conventional, it might be confusing for newcomers, as noted here in Wikipedia. The former version was intended to make such things directly evident.Chjoaygame (talk) 21:25, 1 March 2016 (UTC)

Another unclarity of the new edit is that it uses and chiefly relies on, without definition, the Dirac bra–ket notation before it has been otherwise introduced into the article.Chjoaygame (talk) 23:05, 1 March 2016 (UTC)

The new edit says "The Dirac way is a generalization of the Schrödinger wave functions to abstract Hilbert space." I think a preferable wording would be 'In the Dirac way, the state vector Ψ appears in two forms, known as the bra, ⟨Ψ|, and the ket, |Ψ⟩, which are elements of abstract vector spaces. The Schrödinger and Dirac formulations are intertranslatable.'Chjoaygame (talk) 23:50, 1 March 2016 (UTC)

The new edit unnecessarily leaves the reader in the dark as to the question asked above on this talk page by mathematician Editor Tashiro: What are the range and domain of the wave function? At the end of that conversation, Editor YohanN7 asked Editor Tashiro the following "Tashiro, do you find that the new edits to the lead and the section Wave functions and function spaces answer your questions?" Tashiro did not reply. I think it means he had given up trying to find out the answer to his questions. I think the new edit suffers from the very same superconcision that made mathematician Tashiro give up. The article also had respected and expert Editors Vaughan Pratt and W puzzled. I don't recall exactly VP's bio, but I do recall that he knows a lot about physics. Editor W says he is a lecturer on quantum mechanics.Chjoaygame (talk) 13:55, 2 March 2016 (UTC)

First off, what is wrong with, in a section labeled "Dirac and Schrödinger formulations" (correctly) describing the relation between the two?

I made the edit because some of what you wrote were nonsense. Dirac state vectors appear in one and only one form. The bras are not state vectors. They belong to the dual space, which is not the same space as the space of state vectors. Then what you call "wave function in the Dirac tradition" is just a (Schrödinger) wave function period.

No need to go to near incomprehensible detours of "infinite array of complex numbers" → "array of complex number components can be recognized as a table of values of a function" → "recognition of that table is that it belongs to a differentiable function of multiple real variables, expressible as an analytic formula" → "solution of the Schrödinger equation for the specific system". (I'd like to see you rigorously justify these steps ). Do you really think anyone will understand what you are doing? You start with two things that are by definition the same. Then you "prove" that they are equal.

You have been conducting original research. YohanN7 (talk) 14:32, 2 March 2016 (UTC)

It may comfort you (and perhaps others) that this article is now off of my watch list. I decided yesterday that the edit I made was to be the last. You are like a freight train. Totally impossible to stop when you gain some speed. I'll make one last edit. I'll change names appropriately. There is no such thing as "Schrödinger wave functions" and "Dirac wave functions" that I now see that you intended. There are wave functions and quantum states. It is probably undue to give Dirac ALL credit for generalizing "Schrödinger wave functions" to states. YohanN7 (talk) 14:42, 2 March 2016 (UTC)

To learn about the contribution of Dirac, one way is to read what he wrote. He is a reliable source, recommended by Heisenberg and Einstein amongst many others.Chjoaygame (talk) 16:20, 2 March 2016 (UTC)

Refering to YohanN7's last sentence, Wikipedia , History_of_quantum_mechanics , says: "Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics and the non-relativistic Schrödinger equation as an approximation to the generalised case of de Broglie's theory. Schrödinger subsequently showed that the two approaches were equivalent." Schrödinger showed the isomorphism between the two theories, I was always taught. (Of course here we are speaking of the non-relativitic cases.) Since this article is about wave functions, maybe more of Schrödinger's approach would be appropiate, and at end mention the equivalence to Heisenberg matrix mechanics. I think this article is trying to cover too much. GangofOne (talk) 01:36, 3 March 2016 (UTC)

This article is all about Schrödinger's approach to QM. YohanN7 (talk) 08:16, 3 March 2016 (UTC)

Editor YohanN7 has made some valuable comments that I will think over when I have time. Right now, I am sorry to say, I have other urgent fish to fry. Also Editor GangofOne's comment is valuable.Chjoaygame (talk) 08:37, 3 March 2016 (UTC)

It may be useful here to quote Weinberg's Lectures, p. 53: "... the wave functions that we have been using to describe physical states in wave mechanics should be considered as the set of components ψ(x) of an abstract vector Ψ, known as the state vector, in an infinite-dimensional space in which we happen to choose coordinate axes that are labeled by all the values that can be taken by the position x."Chjoaygame (talk) 21:00, 3 March 2016 (UTC)

I will start commenting in more detail on the foregoing remarks by Editor YohanN7.

First off, what is wrong with, in a section labeled "Dirac and Schrödinger formulations" (correctly) describing the relation between the two?

This is argumentum ad verecundiam. No reply called for.

I made the edit because some of what you wrote were nonsense. Dirac state vectors appear in one and only one form. The bras are not state vectors. They belong to the dual space, which is not the same space as the space of state vectors.

Editor YohanN7 is mistaken here. This is because he is not looking at what Dirac wrote, but is instead giving views of others. What I wrote is pretty nearly verbatim from Dirac, not nonsense as Editor YohanN7 claims. I am not saying the views of others that he is putting are wrong; I am saying that they do not make my report of Dirac's views nonsense. My report is accurate.

Then what you call "wave function in the Dirac tradition" is just a (Schrödinger) wave function period.

I am trying to draw attention to the difference in presentation between the Dirac and Schrödinger traditions. They are intertranslatable, but not the same. Dirac starts with states as abstract vectors and develops waves functions from there, without concern about their nature as functions, while Schrödinger thinks immediately of them as functions.

No need to go to near incomprehensible detours of "infinite array of complex numbers" → "array of complex number components can be recognized as a table of values of a function" → "recognition of that table is that it belongs to a differentiable function of multiple real variables, expressible as an analytic formula" → "solution of the Schrödinger equation for the specific system". (I'd like to see you rigorously justify these steps ). Do you really think anyone will understand what you are doing? You start with two things that are by definition the same. Then you "prove" that they are equal.

More argumentum ad verecundiam. No reply.

... It is probably undue to give Dirac ALL credit for generalizing "Schrödinger wave functions" to states.

This is a valid point, and valuable. That's what I meant when I commented above on Editor YohanN7's remarks. I will bear it in mind. I think it is not a primary concern for the present article, which is about wave functions, not primarily Hilbert spaces, vector spaces, or quantum states. Dirac's work was pretty original, but he was not the only one to do useful work on this topic.Chjoaygame (talk) 14:04, 4 March 2016 (UTC)

To clarify one of the above points:

I made the edit because some of what you wrote were nonsense. Dirac state vectors appear in one and only one form. The bras are not state vectors. They belong to the dual space, which is not the same space as the space of state vectors.

I will cite Dirac:

Dirac 1st edition (1930), pp. 19–20: " We now introduce another set of symbols $\phi _{1}$, $\phi _{2}$, ... also denoting states. Any state denoted by a $\psi$-symbol $\psi _{r}$ can be equally well denoted by a $\phi$-symbol $\phi _{r}$ having the same suffix."

Dirac 2nd edition (1935), p. 22: "Thus the space of $\phi$'s provides a representation of the states of our dynamical system just as well as the space of $\psi$'s, each state being associated with one direction in the space of $\phi$'s. There is, in fact, perfect symmetry between the $\phi$'s and $\psi$'s, which symmetry will survive all through the theory."

Dirac (1939) p. 418: "any expression containing an unclosed bracket symbol $\langle$ or $\rangle$ is a vector in Hilbert space, of the nature of a $\phi$ or $\psi$ respectively."

Dirac 4th edition (1958), p. 21: "On account of the one-one correspondence between bra vectors and ket vectors, any state of our dynamical system at a particular time may be specified by the direction of a bra vector just as well as by the direction of a ket vector. In fact the whole theory will be symmetrical in its essentials between bras and kets."

To check information of this kind, one may read what Dirac wrote.Chjoaygame (talk) 06:01, 6 March 2016 (UTC)

Also it may be useful to clarify another point. Editor YohanN7 writes above "This article is all about Schrödinger's approach to QM." There he is distinguishing the wave mechanics of Schrödinger from the matrix mechanics of Heisenberg. The concern of the sub-section that is affected by edit in question is about the distinction between Schrödinger's way and Dirac's way.Chjoaygame (talk) 06:15, 6 March 2016 (UTC)

These quotes are relevant. I'll interpret them. Dirac says that there is a one-to-one-correspondence between bras and kets. He's right. The article refers to this fact as well (Riesz representation theorem). He also says that the bra's constitute a representation of the kets. This is indeed almost so due to the above-mentioned one-to-one correspondence. The "almost" qualifier is due to the conjugate-linear nature of the correspondence. It is also true that bras and kets are elements of some Hilbert space. But it is not true that they are elements of the same space. Dirac doesn't say so either. The space of states is the space of kets. The space of bras is the space dual to that of the space of kets. Thus

$|\Psi \rangle \in {\mathcal {H}}\Rightarrow \langle \Psi |\in {\mathcal {H}}^{*}.$

But

$\langle \Psi |\notin {\mathcal {H}}.$

Consider for instance a Hilbert finite-dimensional Hilbert space (could be the spin part of a system). If kets there in a representation correspond to column vectors,

$|\Psi \rangle ={\begin{bmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{bmatrix}},$

then

$\langle \Psi |={\begin{bmatrix}a_{1}^{*},a_{2}^{*},\cdots ,a_{n}^{*}\end{bmatrix}},$

in other words, the one-to-one-correspondence is conjugate transpose. Thus

$|\Psi \rangle +\langle \Psi |={\text{nonsense}},$

and they can obviously not belong to the same space, at least some sort of vector space without becoming extraordinarily contrived. You can find this material (low-dim example) in any modern treatment. I can recommend Shankar (listed in article ref section) Chapter 1.

One remedy is to abandon the Dirac notation (it is notation, there is no "extra physics"). It is observed over an over again that it leads to misunderstanding (albeit harmless such as described in the article) of the sort demonstrated here. YohanN7 (talk) 10:10, 7 March 2016 (UTC)

Dirac does call elements of H^∗ "states" (first quote). He is very explicit, and this is surely fine within the context of his book. This is utterly misleading when elevated to "truth" and should not find its way into the article. Though Dirac knows what he is doing, the average reader will be confused. Lack of precision in statements of this sort is not a virtue and should be a thing of the past and is not a good tradition to carry on, even if it is "verifiable". YohanN7 (talk) 10:26, 7 March 2016 (UTC)

Dirac had in mind that a full experiment that gives a datum for a probability estimate consists of two views of the state: the state as prepared, before reduction of the wave function, and the state as observed, after reduction of the wave function. The literaure is clear about this. Dirac said that one can take either the ket to denote the prepared state and the bra as the detected state, or vice versa. He emphasizes that the theory is symmetrical between the two views. L&L and Feynman both recognize this. It gives the bra–ket distinction a physical meaning. The system passes through the experimental apparatus with its identity intact, but appearing twice, as a prepared state, and as a detected state. It is the reason why Dirac identifies the dual pairs before defining the scalar product. The identification is primary and physical and experimentally based. The scalar product is derived from the identification, rather than the identification being derived from the scalar product. Dirac puts the physical meaning first. That's because his topic of interest is physics. Mathematicians, whose topic is mathematics, do it their way, defining the inner product first. But the inner product is not physically observable. Dirac writes in the first edition "Products such as ψφ, ψ₁ψ₂, φ₁φ₂, have no meaning and will never appear in the analysis." Obviously, later, the outer product is defined in the bra–ket notation, and |ψ⟩⟨φ| does get a meaning, though of course not as a scalar product. Editor YohanN7 helpfully above reproduces Dirac's careful account of how a bra cannot be added to a ket. Editor YohanN7 recommends Shankar as a source for this, but Dirac himself is clear enough on the point.

Personally, I find Dirac's physical view of his notation more helpful than evidently does Editor YohanN7. I think readers who come to Wikipedia to learn would also find it helpful. It is not that Dirac is "imprecise" as Editor YohanN7 proposes. It is that Dirac is primarily interested in the physics, and puts it first. Editor YohanN7 seems to deprecate the bra–ket notation, but the article is full of it, and many writers use it. Wikipedia readers are capable of following a grand master such as Dirac if they are given a fair account of what he wrote. They don't need to be given only a censored version. Dirac's work is not mathematically faulty as Editor YohanN7 suggests. It is just sound mathematics deliberately and specifically constructed (Editor YohanN7 says "contrived") for the purpose of expressing physical ideas. That mathematicians have different purposes is their privilege. It is not, however, a reason to censor Dirac, as Editor YohanN7 would like us to do. We may observe also that von Neumann did it in the mathematicians' way. Wikipedia reports the several viewpoints, it doesn't impose a single viewpoint.Chjoaygame (talk) 13:36, 7 March 2016 (UTC)

You are evading the topic by wall-of-texting (and rather rudely so putting words in my mouth and thoughts in my head that aren't there). The topic is whether

$\langle \Psi |\in {\mathcal {H}}$

or

$\langle \Psi |\notin {\mathcal {H}}.$

Which is it? YohanN7 (talk) 15:58, 7 March 2016 (UTC)

Editor YohanN7 wants the topic to be collapsed to the just above question that he has invented. I think the topic is the value of his edit. As things have gone here, a major aspect of that is his deprecatory view of Dirac's approach to and presentation of quantum mechanics. Against that deprecatory view is that Heisenberg wrote that Dirac's 4th edition was his go-to place for mathematical questions on quantum mechanics, and that Einstein wrote that Dirac's was the most logically perfect presentation of quantum mechanics. Perhaps Editor YohanN7 has improved on that, but I remain to be convinced of it. Editor YohanN7's question presupposes that ${\mathcal {H}}$ is the one and only manifestation of the quantum state. That is not how Dirac saw it.Chjoaygame (talk) 19:53, 7 March 2016 (UTC)

Instead of the usual off-topic ramble, why not simply answer Yohan's question?

Or do you not understand what a vector space is, what its dual space is, what ${\mathcal {H}}$ and ${\mathcal {H}}^{*}$ are (in this context), and what $|\Psi \rangle$ and $\langle \Psi |$ actually are?

Just a thought, do $|\Psi \rangle$ and $\langle \Psi |$ correspond two different physical things? Or do they correspond to the same quantum state?

You have ignored Tsirelson's extensive explanations of what these things are, and maybe my brief comments also. M∧Ŝc²ħεИ_τlk 21:11, 7 March 2016 (UTC)

Thank you for this comment.

You ask "do $|\Psi \rangle$ and $\langle \Psi |$ correspond two different physical things? Or do they correspond to the same quantum state?" According to Dirac they refer to different aspects of the same quantum state, as it is prepared, and as it is observed, two different physical things.

As for Editor Tsirel: For me, Dirac's views are central to the present topic. Editor Tsirel writes above "About Dirac, I do not know."Chjoaygame (talk) 21:30, 7 March 2016 (UTC)

---

Insinuations: I don't have a deprecatory view of Dirac's presentation of quantum mechanics. I have not said that Dirac is imprecise. I have said that Dirac becomes imprecise (even utterly misleading) when he is quoted out of context.

Name-dropping (noun): The introduction into one's conversation, letters, etc., of the names of famous or important people as alleged friends or associates in order to impress others. Surely, Dirac, Heisenberg, Einstein, Landau & Lifshitz, Feyman, ..., I think this speaks for itself.

Fallacious references to old threads: Chjoaygame, you also have a way of referring to old threads that is not really meant to put me in a good light. You speak of "...respected and expert Editors Vaughan Pratt and W", and implicitly suggess..., well, I don't know what.

• W questions the scope of the article (specifically length of lead) and in particular the general way in which we define wave function here. So what?

• Then VP's question: position and momentum wave functions represent the "same" object? I let the link speak for itself.

If you want to claim in the article that bras are state vectors you better make that precise. Precise as hell. Therefore,

$\langle \Psi |\in {\mathcal {H}}$

or

$\langle \Psi |\notin {\mathcal {H}}.$

Which is it? The first? The second? No name-dropping, not even an innocent little quote, no fallacious references to earlier threads, no insinuations about me vetting all known Nobel laureates, no mention of "physics" as opposed to "mathematics".

I also allow for "I actually don't know", which would be pretty honest by you.

Over the past year you have cost me lots of time because I benevolently assumed, contrary to enormous piles of evidence to the contrary, that you could actually contribute. My patience with you is gone.

Therefore, if you fail to answer the very simple question about the real issue without babbling, I'll never discuss with you again if I can help it. You might find that a relief. I'd find it a much better way to waste my time. So go on: Start babbling! YohanN7 (talk) 10:47, 8 March 2016 (UTC)

symmetry

Dirac's Principles of quantum mechanics (primarily)

The immediately foregoing section ends with a disorderly posting. It would be out of order for me to reply to it.

On page 19 of the first edition (1930) of his text, Dirac writes:

We now introduce another set of symbols φ₁, φ₂, ... also denoting states. Any state denoted by a ψ symbol ψ_r can be equally well denoted by a φ symbol φ_r having the same suffix.

In the second edition (1935) he writes of his two kinds of vector, the ψ's and the φ's:

... Instead of picturing the ψ's and φ's as vectors in two different vector spaces, we may picture them as two different kinds of vector associated with the same space. The relation between these two kinds of vector is then just the one well known in differential geometry as the relation between covariant and contravariant vectors.

In his 1939 introduction of his bra–ket notation for those same objects, he writes:

... any expression containing an unclosed bracket symbol   $\langle$   or   $\rangle$   is a vector in Hilbert space, of the nature of a φ or ψ respectively. As names for the new symbols   $\langle$   and   $\rangle$   to be used in speech, I suggest the words bra and ket respectively.

On page 20 of the first edition (1930), he writes:

... 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.

       The introduction of a second set of symbols to denote the states may appear to be superfluous, but actually it is necessary when one allows complex coefficients c_r in order to preserve the symmetry between the two roots of −1.

On page 22 of the second edition (1935), he writes:

Each vector ψ_a in the space of ψ's determines uniquely a vector |φ_a⟩ in the space of φ's and vice versa. Thus the space of φ's provides a representation of the states of our dynamical system just as well as the space of ψ's, each state being associated with one direction in the space of φ's. There is, in fact, perfect symmetry between the φ's and ψ's, which symmetry will survive all through the theory.

On page 43 of the fourth edition (1958), he writes:

... Also it is easily seen that the whole theory of functions of an observable is symmetrical between bras and kets and that we could equally well work from the equation

$\langle \xi ^{\prime }|f(\xi )=f(\xi ^{\prime })\langle \xi ^{\prime }|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(38)$

instead of from (34).

On page 40 of the 4th edition, he wrote:

        The space of bra or ket vectors when the vectors are restricted to be of finite length and to have finite scalar products is called by mathematicians a Hilbert space. The bra and ket vectors that we now use form a more general space than a Hilbert space.

He wasn't over-chatty about Hilbert spaces.

For the peace of mind of some, I may add the following piece of original research. If Dirac had been in the habit of denoting his vector spaces by mathcal symbols (which he wasn't), I guess he might perhaps have written

$\langle \Psi |\in {\mathcal {H}}_{\langle }$        and        $\langle \Psi |\notin {\mathcal {H}}_{\rangle }$        and        $|\Psi \rangle \in {\mathcal {H}}_{\rangle }$        and        $|\Psi \rangle \notin {\mathcal {H}}_{\langle }.$

He would perhaps have added that $\langle \Psi |$ and $|\Psi \rangle$ are mutually dual. He said they were a conjugate imaginary pair. Also, I guess he might have said that ${\mathcal {H}}_{\langle }$ and ${\mathcal {H}}_{\rangle }$ are mutually dual.

There is plenty of literature commentary on this, but I think that should do for now.Chjoaygame (talk) 05:18, 10 March 2016 (UTC)

It really does.not.matter.

"If Dirac had been in the habit of denoting his vector spaces by mathcal symbols (which he wasn't),".

The fact that you write

"I guess he might perhaps have written"

in nonstandard and obscure notation the Hilbert space and its dual, followed by

"He would perhaps have added that $\langle \Psi |$ and $|\Psi \rangle$ are mutually dual. He said they were a conjugate imaginary pair. Also, I guess he might have said that ${\mathcal {H}}_{\langle }$ and ${\mathcal {H}}_{\rangle }$ are mutually dual.

suggests you do not actually know what you are defining, and resort to what others have said. ${\mathcal {H}}_{\rangle }$ is the Hilbert space of all allowable states |Ψ⟩ for the system in question, and ${\mathcal {H}}_{\langle }$ (in other symbols ${\mathcal {H}}_{\rangle }^{*}$) the dual space with elements ⟨Ψ| right? Then you could have said so. On top of your reply last section, that

"According to Dirac they refer to different aspects of the same quantum state, as it is prepared, and as it is observed, two different physical things"

suggests you also misunderstand what the bra-ket notation is for, because p.21 in the 4th edn of Dirac's Principles of QM:

"On account of the correspondence between bra vectors and ket vectors, any state of our dynamical system at a particular time may be specified by the direction of a bra vector just as well as a ket vector. In fact the whole theory will be symmetrical in its essentials between bras and kets."

A ket, and its corresponding bra, describe the same state as you correctly say, but one does not correspond to the quantum state "as prepared", and the other to the quantum state "as observed". Where does Dirac say this? I can't find it in his book, please point to other papers or books. Given a bra-ket equation, you can take the Hermitian conjugate of it and the meaning of the equation is the same. Mathematically: the bras become kets and vice versa, and the operators are replaced by their Hermitian conjugates, for example

$|\psi \rangle \rightarrow \langle \psi |\,,\quad \langle \phi |{\hat {\Omega }}|\psi \rangle \rightarrow \langle \psi |{\hat {\Omega }}^{\dagger }|\phi \rangle$

and each expression has the same meaning (retains the same information about the physical system) before and after Hermitian conjugation. Hermitian conjugation is a mathematical operation, not physical.

Suppose that bras really did correspond to "observed" and the kets to "prepared". Start from the SE of any quantum state

$i\hbar {\frac {d}{dt}}|\psi \rangle ={\hat {H}}|\psi \rangle$

Now take the Hermitian conjugate.

$-i\hbar \langle \psi |{\frac {d}{dt}}=\langle \psi |{\hat {H}}^{\dagger }$

where in the conjugate equation, the derivative acts to the left (because if this bra equation premultiplies a new ket |A⟩, then the derivative must act on ⟨ψ| not |A⟩). What happens? Is the original SE the equation for the prepared state, and the second the observed state?

At least... you (seem to) know correctly that bras and kets are mutually dual. M∧Ŝc²ħεИ_τlk 11:06, 13 March 2016 (UTC)

It really does.not.matter.

"If Dirac had been in the habit of denoting his vector spaces by mathcal symbols (which he wasn't),".

The fact that you write

"I guess he might perhaps have written"

in nonstandard and obscure notation the Hilbert space and its dual, followed by

My non-standard and obscure notation is, as I observed, the dreaded OR. I need to invent it because notation of this kind seems necessary here. The point is that the notation that Editor Yohan7 above tried to make me use does not express my meaning. The non-standard notation does so. The notation ${\mathcal {H}}$ and ${\mathcal {H}}^{*}$ that Editor Yohan7 tried to make me use is misleading because it wrongly gives priority to ${\mathcal {H}}$ for kets as the Hilbert space, with ${\mathcal {H}}^{*}$ for bras secondary to it. This hides Dirac's intended symmetry. As you observe, when you copy my many times repeated quotes of Dirac, the theory is symmetrical between bras and kets. My non-standard notation, ${\mathcal {H}}_{\langle }$ for bras and ${\mathcal {H}}_{\rangle }$ for kets, is adapted to express that symmetry.

"He would perhaps have added that $\langle \Psi |$ and $|\Psi \rangle$ are mutually dual. He said they were a conjugate imaginary pair. Also, I guess he might have said that ${\mathcal {H}}_{\langle }$ and ${\mathcal {H}}_{\rangle }$ are mutually dual. Dirac wasn't over-chatty about Hilbert spaces, so I can only guess what he might have said if he had used the mathcal symbols.

suggests you do not actually know what you are defining, and resort to what others have said. Argumentum ad verecundiam, no reply. ${\mathcal {H}}_{\rangle }$ is the Hilbert space of all allowable states |Ψ⟩ for the system in question, and ${\mathcal {H}}_{\langle }$ (in other symbols ${\mathcal {H}}_{\rangle }^{*}$) the dual space with elements ⟨Ψ| right? No, wrong, as I have indicated just above, in that they are mutually dual, and your prejudice in favor of the kets as the states is misleading. Dirac writes in the 1930 edition on page 62: "We can without inconsistency suppose that each fundamental φ and the conjugate imaginary fundamental ψ of an orthogonal representation are to be both pictured by the same real vector." Then you could have said so. Why would I hide my meaning by using the misleading notation that you advocate and that expresses your prejudice? But yes, you have it right that ${\mathcal {H}}_{\langle }^{*}={\mathcal {H}}_{\rangle }$. On top of your reply last section, that

"According to Dirac they refer to different aspects of the same quantum state, as it is prepared, and as it is observed, two different physical things"

suggests you also misunderstand what the bra-ket notation is for, because p.21 in the 4th edn of Dirac's Principles of QM:

"On account of the correspondence between bra vectors and ket vectors, any state of our dynamical system at a particular time may be specified by the direction of a bra vector just as well as a ket vector. In fact the whole theory will be symmetrical in its essentials between bras and kets."}} At last you are registering what I have repeatedly quoted from Dirac.

A ket, and its corresponding bra, describe the same state as you correctly say, but one does not correspond to the quantum state "as prepared", and the other to the quantum state "as observed". Where does Dirac say this? I can't find it in his book, please point to other papers or books. My partial mistake. Though he says something close to it in the first edition, Dirac doesn't exactly say this in later editions. It is many years since I learnt from Feynman's text, that I will shortly quote, that kets refer to the starting state or the state as prepared, and bras to the final state or state as observed. In reading Dirac I read it into the text from habitual thinking following Feynman. For example, it is a natural reading of the following sentences, from pages 23–24 of the first edition of Dirac: "Our introduction of products of φ's with ψ's has so far been entirely a mathematical question, with no physical implications. A physical meaning will now be given to the product φ_rψ_s. Consider that maximum observation of the state φ_r for which there is a certainty of a particular result being obtained. We have seen that such a maximum observation always exists. Suppose now this maximum observation to be made on the system in the state ψ_s. There will be a certain probability of the same result being obtained, which we call the probability of agreement of ψ_s with φ_r. It is a number that depends only on the two states ψ_s and φ_r." In Dirac's bra–ket notation, it is an arbitrary convention to choose to denote (a), states as prepared, by kets, and then (b), states as observed, by bras. Feynman says: "In other words, the two brackets 〈 〉 are a sign equivalent to “the amplitude that”; the expression at the right of the vertical line always gives the starting condition, and the one at the left, the final condition." Auletta, Fortunato, Parisi say: "In this context, we see that kets may be thought of as input states, whereas bras as output states of a certain physical evolution or process." Landau and Lifshitz 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).

By Dirac's symmetry, it would be valid, though it is not customary, to make the alternative choice, with kets as output states, and bras as input states mutatis mutandis. Perhaps I need to say initial state = input state = starting condition = state as prepared, and final state = output state = final condition = state as observed. Dirac does distinguish preparation and observation, as is customary. For example, for observation: "... for any state there must exist one maximum observation which will for a certainty lead to one particular result, and conversely, if we consider any possible result of a maximum observation, there must exist a state of the system for which this result for the observation will be obtained with certainty." For preparation: "In practice the conditions could be imposed by a suitable preparation of the system, consisting perhaps in passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being left undisturbed after the preparation." The important physics here is that, as I tried recently to convey, the preparation-observation set-up is symmetrical. One can interchange the oven and the anti-oven, and get the same results if the the filters and so forth commute. To be a good quantum analyzer, something such as a prism or crystal must have a sort of Helmholtz reciprocity. That means it can be turned 180 degrees and work the same. That's why bras and kets are symmetrical and why compatible degrees of freedom are encoded by commuting operators. They say "simultaneously measured" but they mean 'observed through contiguous analyzers'. "Simultaneously measured" is an abuse of language almost characteristic of quantum mechanics.

• Auletta, G., Fortunato, M., Parisi, G. (2009). Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 9781107665897.
• Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics, Volume 3, Addison-Wesley, Reading MA, available here.
• Landau, L., Lifshitz, E. (1974/1977). Quantum Mechanics: Non-Relativistic Theory, 3rd edition, translated from Russian into English by J.B. Sykes, J.S. Bell, Pergamon, Oxford UK, ISBN 0-08-020940-8.

Given a bra-ket equation, you can take the Hermitian conjugate of it and the meaning of the equation is the same. Mathematically: the bras become kets and vice versa, and the operators are replaced by their Hermitian conjugates, for example

$|\psi \rangle \rightarrow \langle \psi |\,,\quad \langle \phi |{\hat {\Omega }}|\psi \rangle \rightarrow \langle \psi |{\hat {\Omega }}^{\dagger }|\phi \rangle$

and each expression has the same meaning (retains the same information about the physical system) before and after Hermitian conjugation. Hermitian conjugation is a mathematical operation, not physical.

Suppose that bras really did correspond to "observed" and the kets to "prepared". Start from the SE of any quantum state

$i\hbar {\frac {d}{dt}}|\psi \rangle ={\hat {H}}|\psi \rangle$

Now take the Hermitian conjugate.

$-i\hbar \langle \psi |{\frac {d}{dt}}=\langle \psi |{\hat {H}}^{\dagger }$

where in the conjugate equation, the derivative acts to the left (because if this bra equation premultiplies a new ket |A⟩, then the derivative must act on ⟨ψ| not |A⟩). What happens? Is the original SE the equation for the prepared state, and the second the observed state?}} The Schrödinger equation describes the evolution of the isolated system, not referring to the contact with the outside world through the oven and anti-oven, i.e. producer and destroyer. These two opposite contacts are two different physical processes, both necessary for physical experiments, not described by the Schrödinger equation. The evolution of the isolated system is not real evolution unless the Hamiltonian is explicitly time-dependent. This is because the oven and anti-oven can be interchanged without affecting the results. There are two signs for the square root of minus one. Time can run backwards or forwards. That's why people make muddles on causality in quantum mechanics.

Perhaps it may be useful to add the Dirac's first edition (1930) was considered too abstract for many readers, and he responded by making later edition less abstract, though he commented that readers who like abstraction for its own sake might still prefer the 1930 edition.

At least... you (seem to) know correctly that bras and kets are mutually dual. Argumentum ad verecundiam. No reply.Chjoaygame (talk) 17:57, 13 March 2016 (UTC)

Please clarify/define what "with ${\mathcal {H}}^{*}$ for bras secondary to it" followed by "No, wrong, as I have indicated just above, in that they [ ${\mathcal {H}}_{\rangle }$ and ${\mathcal {H}}_{\langle }$ ] are mutually dual" mean.

It isn't that I have "only just" acknowledged what Dirac has written, because I don't "dislike"/disregard what he has written, never have or will. That symmetry quote refers to the Hermitian conjugate as the correspondence between kets and bras, something you seem to keep avoid discussing.

Please understand this. A quantum state as a bra or ket is not "as measured/observed". It just is what it is - and corresponds to the same quantum state, the same physical entity. The correspondence between the two is mathematical, not physical.

For someone who relies so heavily on Dirac's quotes, don't you realize that the Dirac quote on the symmetry between bras and kets, AND your insistence that bras and kets are physically different, is a contradiction?

This may be useful (certainly more useful than your unintelligible "ovens" and "anti-ovens"): In the expression ⟨n|f|i⟩ as you mention, the observed state happens to be a bra ⟨n|, the initial state is a ket |i⟩. But this is a mathematical expression for an integral/sum. If you have ⟨n| on its own, then this does not automatically make it an observed state.

The causality issue has nothing to do with the present topic. Nevertheless... Time reversal is just negating the time parameter. Complex/Hermitian conjugation only approximately corresponds to time reversal, because time often just happens to be multiplied by i, e.g. the phase factor e^−iEt/ħ for stationary states.

Finally - "Argumentum ad verecundiam" is exactly what you are doing. Deferring to the pioneers. M∧Ŝc²ħεИ_τlk 12:39, 15 March 2016 (UTC)

Please clarify/define what "with ${\mathcal {H}}^{*}$ for bras secondary to it" followed by "No, wrong, as I have indicated just above, in that they [ ${\mathcal {H}}_{\rangle }$ and ${\mathcal {H}}_{\langle }$ ] are mutually dual" mean. I read '${\mathcal {H}}$ is the state space and ${\mathcal {H}}^{*}$ its dual' as intending that kets have priority that negates Dirac's symmetry. It seems to rule out that bras might be points in the state space with kets merely their duals. "${\mathcal {H}}_{\rangle }$ and ${\mathcal {H}}_{\langle }$ are mutually dual" means '${\mathcal {H}}_{\rangle }^{*}={\mathcal {H}}_{\langle }$ and ${\mathcal {H}}_{\langle }^{*}={\mathcal {H}}_{\rangle }$', as I wrote above. The sentence '${\mathcal {H}}$ is the state space and ${\mathcal {H}}^{*}$ its dual' gives the impression of denying that bras denote states with kets merely their duals. I don't see you rejecting that denial.

It isn't that I have "only just" acknowledged what Dirac has written, because I don't "dislike"/disregard what he has written, never have or will. That symmetry quote refers to the Hermitian conjugate as the correspondence between kets and bras, something you seem to keep avoid discussing. I wrote above "You have it right that ${\mathcal {H}}_{\langle }^{*}={\mathcal {H}}_{\rangle }$".

Please understand this. A quantum state as a bra or ket is not "as measured/observed". It just is what it is - and corresponds to the same quantum state, the same physical entity. The correspondence between the two is mathematical, not physical. I am persuaded by Feynman, by L&L, and by Auletta, G., Fortunato, M., & Parisi, G..

For someone who relies so heavily on Dirac's quotes, don't you realize that the Dirac quote on the symmetry between bras and kets, AND your insistence that bras and kets are physically different, is a contradiction? Symmetry isn't necessarily identity.

This may be useful (certainly more useful than your unintelligible "ovens" and "anti-ovens"): In the expression ⟨n|f|i⟩ as you mention, the observed state happens to be a bra ⟨n|, the initial state is a ket |i⟩. But this is a mathematical expression for an integral/sum. If you have ⟨n| on its own, then this does not automatically make it an observed state. No comment.

The causality issue has nothing to do with the present topic. Nevertheless... Time reversal is just negating the time parameter. Complex/Hermitian conjugation only approximately corresponds to time reversal, because time often just happens to be multiplied by i, e.g. the phase factor e^−iEt/ħ for stationary states. Too complicated to pursue here.

Finally - "Argumentum ad verecundiam" is exactly what you are doing. Deferring to the pioneers. By argumentum ad verecundiam I mean that you try to shame me. Wikipedia editing requires careful comparison of sources to find reliable ones. Dirac is a candidate.Chjoaygame (talk) 04:13, 16 March 2016 (UTC)

Other sources

Shankar (2nd edition, 1994), on page 11, discusses two ways of seeing the inner or scalar product. One is as between two vectors of the same space, with no involvement of a dual space. The other is between two vectors from dual spaces. Of the latter, he writes "Inner products are really defined only between bras and kets and hence from elements of two distinct but related vector spaces." Unlike Dirac, Shankar does not emphasize the symmetry between bras and kets. But he does not explicitly deny it. On page 121, Shankar seems to depart from Dirac. He writes "As far as the state vector |ψ⟩ is concerned, there is just one space, the Hilbert space, in which it resides." He does not explicitly say that the corresponding bra ⟨ψ| is not also a state vector. Nevertheless, one could easily get the impression from Shankar that he does not make the point made by Dirac, L&L, and Feynman, that bras and kets represent states as observed and as prepared. I think there is a case for going against Shankar, with Dirac, L&L, and Feynman. They have a physical reason for their way, while Shankar is silent as to his reason, if he has one. I don't think 'modernity' is a useful reason. Looking at other sources: Merzbacher (2nd edition, 1970) is also silent on the point, so far as I have seen; Sakurai & Napolitano (2nd edition, 2011) are also silent on the point, as far as I have seen, though on page they express deference to "the master" Dirac, and profess to "follow" him. Zettili (2nd edition, 2009) is also silent on the point. Even the unusually careful Messiah (1961) is silent, so far as I have seen on flicking through his text. It seems that to follow Dirac closely here, one would need to rely on quality against quantity. I favor Dirac, L&L (3rd edition, 1977, p. 35), and Feynman, with their physical reason, which I find convincing, against the others with their silence. Perhaps a further survey of sources might be useful, but not by me right now.Chjoaygame (talk) 11:50, 10 March 2016 (UTC)

On page 69, Walter Greiner <(2001), Quantum Mechanics: An Introduction, 4th edition, Springer, Berlin, ISBN 3-540-67458-6> writes: "... the element ⟨ψ₁| is called a "bra" and |ψ₂⟩ is called a "ket" ... Both are vectors (state vectors) in a linear vector space."

Dirac, his notation, and his book

In physics we have a number of heroes. Dirac was a hero, and should be treated as such. But in physics, we do not have prophets. It would be ridiculous to go to Einstein to seek answers to present day questions about general relativity. It is likewise ridiculous to assume that every formulation in Dirac's pioneering book should be taken and the final formulation, word by word, of topics quantum mechanics. It is widely acknowledged that Dirac's theory is the correct one. See for example Weinberg ("The Quantum Theory of Fields" vol 1, and also "Lectures on Quantum Mechanics"), and Weinberg's knowledge of QM is a lifetime deeper than Dirac's, so he should know.

There is a reason that books have been written the past 80 years. Dirac's presentation can be improved significantly on. In particular, Dirac's text is a graduate level text and requires considerable mathematical background to interpret correctly. This is very clear, if not from anything else, from this talk page. "Dirac wasn't over-chatty about Hilbert spaces." Right. Then don't try to learn about Hilbert spaces from Dirac. You'll get things wrong. You'll get quantum mechanics (the physical interpretation and its mathematical formulation) wrong. The same goes for Feynman (another of the few true heroes) and his (pretty awful) lecture notes.

In particular, you (Chjoaygame) read things (physics not found elsewhere) into the Dirac notation. They aren't there.

Modern educational methods are better than those of the 1930:s. Modern ways of presenting physics are better than those of the 1930:s. Mathematics has become the irreversibly final tool and language in expressing physics. You may not like it, but you can't turn the flow of time around.

Filling the article, and in particular this talk page, with hundreds of Dirac quotes based on misunderstandings makes Dirac look like babbling fool. I don't like that at all. YohanN7 (talk) 13:30, 15 March 2016 (UTC)

For instance,

... Instead of picturing the ψ's and φ's as vectors in two different vector spaces, we may picture them as two different kinds of vector associated with the same space. The relation between these two kinds of vector is then just the one well known in differential geometry as the relation between covariant and contravariant vectors.

This is interpretable by a mathematician or a mathematically inclined physicist (and actually also by me). Do you really think this is the way to learn about Hilbert spaces and quantum mechanics? That "same space" to which bras and kets are associated is not what you'd think. It is not at all bra's and kets lumped together in the same Hilbert space or anything of the sort. YohanN7 (talk) 13:47, 15 March 2016 (UTC)

I'm not suggesting putting bras and kets in the same Hilbert space. I have explicitly written the contrary above, and Dirac, as I cited him above, explicitly says they cannot be added together. Bras and kets are heads and tails of coins that are the mathematical signifiers of the state space.Chjoaygame (talk) 22:25, 15 March 2016 (UTC)Chjoaygame (talk) 03:21, 16 March 2016 (UTC)Chjoaygame (talk) 04:24, 16 March 2016 (UTC)

The time when the exact nature of your interpretations could be worth discussing is long gone. No doubt you will go on and on and on about this and somehow have it look like that you were perfectly right all the time. This is not the issue. YohanN7 (talk) 12:43, 16 March 2016 (UTC)

Perhaps it may help when I say that for the sake of good will I have just now looked over Chapters 1–4 of Shankar. I have also recently looked at that text. Also I have in the not-too-distant past several times read Weinberg's view on interpretation in his Lectures.Chjoaygame (talk) 06:38, 16 March 2016 (UTC)

Does "for the sake of good will" mean that you step down from the Dirac pedestal temporarily and try to see things from the inferior perspective? It sure sounds that way. But good for you. We might even be able in the far future to agree on a complete sentence if we share some common understanding of the subject. YohanN7 (talk) 12:43, 16 March 2016 (UTC)

Citations

"When a maximum observation is made on a system, its subsequent state is completely determined by the result of the observation and is independent of its previous state. This may be considered as an axiom, or as a more precise definition of a state."

This is actually not correct. A precise momentum measurement of a particle at any one time leaves you in the dark about its momentum at a later time.

Another problem is that the vast majority of states cannot be prepared in practice. I personally question whether all states can be prepared even in principle. Then it also needs to be added that there's nothing such as a truly isolated state. For instance, fixing by measurement the state of a hydrogen atom to the 2s-state, doesn't mean it is in the 2s-state a split second later.

"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."

True, but this is off topic, and has nothing to do with the "maximal set" under discussion. This is a mathematical statement about one operator (and a definition of maximality in that context). YohanN7 (talk) 10:37, 12 March 2016 (UTC)

Done. I have removed the off-topic instance.Chjoaygame (talk) 23:58, 12 March 2016 (UTC)

On looking more closely, I think the definition by von Neumann of a maximal operator that I removed is on topic. Extension of a Hermitian operator here is virtually adding a linearly independent and commuting observable to make it supply a more comprehensive observation. That means, for example, adding a degree of freedom, until the maximum possible number of degrees of freedom has been reached. Then the operator is maximal in von Neumann's terms. I think this is also how Dirac goes about it on pages 49–52 of the 4th edition.Chjoaygame (talk) 07:21, 13 March 2016 (UTC)

Thank you for these enlightening comments. I think it would be helpful if you would very kindly give more detail.Chjoaygame (talk) 13:59, 12 March 2016 (UTC)

On the von Neumann quote: I don't have his book. But, I am almost 100% sure (> 99%) he is talking mathematics of unbounded operators such as the momentum operator. That is, the domain on which it is defined, a linear subspace of Hilbert space, is under discussion. Self-adjointness of such operators on infinite-dimensional spaces is more subtle than on finite-dimensional spaces. (Mathematicians speak in terms of symmetric and self-adjoint operators on these spaces, and distinguish the terms. Physicists usually only speak of Hermitian operators, where Hermitian means either of symmetric or self-adjoint because they rarely have reason to delve into the mathematical intricacies – at least not in elementary treatments. But see Shankar early on in chapter 1 for some discussion on the topic.)

On the Dirac quote (my first objection only): What I mean here is that a momentum measurement is, in principle, non-repeatable. Landau and Lifshitz devotes a complete section (44) to this. The main result is

$|v'_{x}-v_{x}|\Delta p_{x}\approx \hbar /\Delta t,$

where

${\begin{aligned}&\Delta p_{x}&{\text{is uncertainty in measured value of momentum,}}\\&\Delta t&{\text{is duration of measurement,}}\\&v_{x}&{\text{is velocity of particle }}{\textit {before}}{\text{ measurement,}}\\&v'_{x}&{\text{is velocity of particle }}{\textit {after}}{\text{ measurement.}}\\\end{aligned}}$

Note that v′_x, an unknown quantity, is related to particle momentum after the measurement. The formula, and its interpretation, is attributed to Niels Bohr (1928). Dirac was of course aware of this. Therefore, I think it is unfortunate to use the quote, which I dare say is profoundly misleading when taken out of context. Why not bring it up in quantum state, accompanied with a proper discussion? YohanN7 (talk) 10:29, 14 March 2016 (UTC)

On the von Neumann quote: I don't have his book. But, I am almost 100% sure (> 99%) he is talking mathematics of unbounded operators such as the momentum operator. That is, the domain on which it is defined, a linear subspace of Hilbert space, is under discussion. Self-adjointness of such operators on infinite-dimensional spaces is more subtle than on finite-dimensional spaces. (Mathematicians speak in terms of symmetric and self-adjoint operators on these spaces, and distinguish the terms. Physicists usually only speak of Hermitian operators, where Hermitian means either of symmetric or self-adjoint because they rarely have reason to delve into the mathematical intricacies – at least not in elementary treatments. But see Shankar early on in chapter 1 for some discussion on the topic.)

I spent time carefully reading the relevant parts of the English translation of von Neumann's classic text for the present purpose. It has a fair number of typographical errors that I corrected for by checking against the German original. I stand by my reading. If someone likes to check the text, I will be interested to learn what they think.

On the Dirac quote (my first objection only): What I mean here is that a momentum measurement is, in principle, non-repeatable. Landau and Lifshitz devotes a complete section (44) to this. The main result is

$|v'_{x}-v_{x}|\Delta p_{x}\approx \hbar /\Delta t,$

where

${\begin{aligned}&\Delta p_{x}&{\text{is uncertainty in measured value of momentum,}}\\&\Delta t&{\text{is duration of measurement,}}\\&v_{x}&{\text{is velocity of particle }}{\textit {before}}{\text{ measurement,}}\\&v'_{x}&{\text{is velocity of particle }}{\textit {after}}{\text{ measurement.}}\\\end{aligned}}$

Note that v′_x, an unknown quantity, is related to particle momentum after the measurement. The formula, and its interpretation, is attributed to Niels Bohr (1928). Dirac was of course aware of this. Therefore, I think it is unfortunate to use the quote, which I dare say is profoundly misleading when taken out of context. Why not bring it up in quantum state, accompanied with a proper discussion?

There are two main kinds of quantum mechanical measurement. The one the Dirac chooses for his presentation of his approach to the wave function is not the one that is referred to just above. The context is Dirac's presentation. Again, I will be interested to learn the thoughts of someone who likes to check what Dirac wrote.Chjoaygame (talk) 12:01, 14 March 2016 (UTC)

Then what is that other kind of measurement that can do the trick of measuring particle momentum without interacting with the particle? I think it would be helpful if you would very kindly give more detail. YohanN7 (talk) 09:07, 15 March 2016 (UTC)

The topic is how Dirac did it. Editor Yohan7 is evidently asking what kind of measurement Dirac had in mind. Editor YohanN7 suggests that Dirac's method would be "measuring particle momentum without interacting with the particle". I guess from this suggestion that Editor YohanN7 intends to make a fool of me. I guess Editor YohanN7 is quite familiar with Dirac's thinking. Nevertheless, perhaps as to Editor YohanN7's suggestion of measurement without interaction, I should respond that a reading of Dirac would show that Dirac does not think so. Dirac considers the kind of "measurement", that I have repeatedly said should better be called observation, not measurement, in which the system interacts with a measuring device, and is thereby cast into an eigenstate of the relevant observable. An immediately subsequent repeat of the same interaction will produce the system again in the same eigenstate. This is customarily called 'reduction of the wave function', as I think is common knowledge. It is unfortunate, but cast in stone in the quantum mechanical literature, that 'measurement' is the word used for this procedure. I cannot change that, though I would like to. Pauli tells of the two kinds, the other one being that, instead of the subsequent state of the system being determined by the "measuring" interaction, it is the prior state that is determined by it. Of Dirac's kind of "measurement", on page 7, Pauli<General Principles of Quantum Mechanics (1958/1980), translated by P. Achuthan and K. Venkatesan, Springer, Berlin, ISBN 3-540-09842-9> writes: "Hence, it is correct to say that here the measurement of the system leads to a new state." Of the other kind of measurement, which he calls of the second kind, on page 75 Pauli writes: "On the other hand it can also happen that the system is changed but in a controllable fashion by the measurement — even when, in the state before the measurement, the quantity measured had with certainty a definite —, value. In this method, the result of a repeated measurement is not the same as that of the first measurement. But still it may be that from the result of this measurement, an unambiguous conclusion can be drawn regarding the quantity being measured for the concerned system before the measurement. Such measurements, we call the measurements of the second kind." The calculation of Editor YohanN7 I think refers to a hybrid between the two that Pauli names.Chjoaygame (talk) 11:43, 15 March 2016 (UTC)

I don't agree.

"An immediately subsequent repeat of the same interaction will produce the system again in the same eigenstate."

This is wrong in the case of momentum. Momentum measurements are non-repeatable period. Paragraph 44 of Landau and Lifshitz and the formula above apply to measurements whether you call them "observations" using "measuring devices" or anything else. Sorry if I sound offensive, but a flood of words can't change this.

What you can determine (using experiment/observation/measurement/whatever) with arbitrary precision in principle (in the non-relativistic theory only, things get worse with relativity) is the momentum the particle had (Δp_x above) when being experimented upon/observed/measured/whatevered, given that you allow for the experiment/observation/measurement/whatever to take an arbitrarily long time. Then the subsequent state can also be predicted to the arbitrary precision (|v′_x − v_x| above). This follows from the formula above, and is also spelled out in L & L. Actually, what you have with Δt finite is a trade-off between knowing Δp_x or |v′_x − v_x| well. I don't even see a guarantee that both can be made small, even when Δt is very large, just their product.

Basing a definition of "state vector" on such shaky grounds without further comment is, as I have said, profoundly misleading and far far far from a more "precise" definition (as you have it in the article). I think you should take this to the relevant article, state vector, if you want to discuss further. This article is not the place. YohanN7 (talk) 12:31, 15 March 2016 (UTC)

It turns out that the part "this may be considered as an axiom, or as a more precise definition of a state" probably isn't due to Dirac. This is a relief. Citation, and addition to it, removed. YohanN7 (talk) 15:07, 15 March 2016 (UTC)

"...probably isn't due to Dirac." Editor YohanN7 guessed rightly about the meaning of my von Neumann quote, but not about the content of this Dirac quote. The words that are not italicized are still Dirac's, as one may verify from the original text. My quote was accurate. Talk page comment is needed.Chjoaygame (talk) 22:35, 15 March 2016 (UTC)

Between you and me, who did the guesswork about von Neumann's quote? You have the book, have supposedly read it, not once but twice, and still got it wrong.

No doubt that you would have italicized Dirac's word if it were an actual quote, and not one of your personal interpretations. It still doesn't belong in the article or on the talk page. You need to put the parts it in quantum measurement, quantum state and axioms of quantum mechanics, and provide a context. These issues are highly nontrivial, and even controversial. I could conceivably discuss the axiomatic status of (the relevant part) of your statement if you bring it up in the right place. But not if you even once more address me as Editor YohanN7. YohanN7 (talk) 12:19, 16 March 2016 (UTC)

Between you and me, who did the guesswork about von Neumann's quote? You have the book, have supposedly read it, not once but twice, and still got it wrong. Between you and me, both of us guessed. I mistakenly from carelessness and relative lack of skill, you from experience. You got it right, and I got it wrong until you put me right and I recognized my mistake. On your first protest I promptly took it down, and thereby rectified the text of the article. And I brought it to the talk page. This proves that you are far more familiar than I with the details of functional analysis.

No doubt that you would have italicized Dirac's word if it were an actual quote, and not one of your personal interpretations. The italicizing and non-italicizing of his words was Dirac's. It was not mine, as you mistakenly guessed. It isn't clear whether you have recognized that my quote was accurate in every word, not in any part my personal invention or fabrication. It still doesn't belong in the article or on the talk page. It is about the sentence "There is at least one representation for which the state is a simultaneous eigenstate." You need to put the parts it in quantum measurement, quantum state and axioms of quantum mechanics, and provide a context. Dirac was offering his opinion about its axiomatic status. Not something that I would lightly dismiss. These issues are highly nontrivial, yes, and even controversial. One can make controversy about much in quantum mechanics. It didn't occur to me that this was controversial, because it is pretty much established thinking. The problem is that Dirac was using the somewhat ridiculous terminology that I have deprecated many times on these pages, but that is nevertheless set in stone in much of the literature. I could conceivably discuss the axiomatic status of (the relevant part) of your statement if you bring it up in the right place. Not necessary. Dirac likes to base his thinking in experimental facts when that is possible and appropriate, though he often enough gives preference to mathematics when the experimental situation is unclear. You like to base it in mathematics. But not if you even once more address me as Editor YohanN7. You might try to guess why I did that.Chjoaygame (talk) 02:09, 17 March 2016 (UTC)

As to my above comments about von Neumann's maximal operators. I have looked again and now see that Editor YohanN7 is right on this point. Von Neumann was there not engaged in his version of Dirac's building up of Dirac's comprehensive observable that is maximal in the sense that no more independent degrees of freedom exist that can be included in it, so that it comprises a complete set of commuting observables. I did not find von Neumann following Dirac in that exercise. In the passage I cited, indeed, as Editor YohanN7 says, von Neumann was concerned with the eigenvectors of a single observable.Chjoaygame (talk) 12:01, 15 March 2016 (UTC)

Thank you. He is talking about conditions under which expressions like

$\int |p\rangle \langle p|dp=\mathrm {Id}$

are true. YohanN7 (talk) 12:31, 15 March 2016 (UTC)

more acceptable quote from Dirac

Instead of the two sentences that I previously posted from Dirac (1930), I have now posted their preceding sentence. Dirac's notion of compatible observations assumes Dirac's censored terminology.Chjoaygame (talk) 03:49, 17 March 2016 (UTC)

Dirac's next sentence after the censored one reads: "The state of a system after a maximum observation has been made on it is such that there exists a maximum observation (namely, an immediate repetition of the maximum observation already made) which, when made on the system in this state, will for a certainty lead to one particular result (namely, the previous result over again)." It is evident that Dirac was in no doubt about his meaning. Further sentences in Dirac amplify this, but I will not quote them here. Interested editors can consult the original text of Dirac.Chjoaygame (talk) 03:53, 17 March 2016 (UTC)

Sentences by Dirac filtered out and interpreted by you weigh little compared with mathematics of quantum mechanics in Landau and Lifshitz, derived by Bohr on physical grounds. YohanN7 (talk) 07:36, 17 March 2016 (UTC)

I think it was Heisenberg, not Bohr, who discovered the Heisenberg uncertainty principle, and did the physical derivations for it.Chjoaygame (talk) 08:30, 18 March 2016 (UTC)

explanation for the curious

How can it be that there is such disagreement about "measurement" here?

In a nutshell, the kind of measurement that is mathematically analyzed above in detail measures all comers by interacting materially and substantially with them. It is a measurement in a more obvious and thorough sense than the one that Dirac used. It emphasizes Heisenberg uncertainty for this reason. The kind of "measurement", to use the term that is used, for example in discussions of "collapse", and was used by Dirac, rejects any but the desired eigenstate, and passes unscathed the desired eigenstate effectively by selection rather than substantial interaction. Only the desired eigenstate is "measured". All other comers are absorbed into the side walls of the device, or by some like mechanism. In this way Heisenberg uncertainty is manifest in virtually infinite error in the rejected particles: 'no result because of no passage' may be counted as infinite error. There is still some room for finite error for the particles that pass, but it is manageable. It comes under Dirac's heading of "sorting apparatus, such as slits". Dirac, 4th edition (1958), pp. 11–12: "A state of a system may be defined as an undisturbed motion that is restricted by as many conditions or data as are theoretically possible without mutual interference or contradiction. In practice the conditions could be imposed by a suitable preparation of the system, consisting perhaps in passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being left undisturbed after the preparation." An arrangement of choppers would count here as 'sorting apparatus'.Chjoaygame (talk) 06:53, 17 March 2016 (UTC)

On thinking about it overnight, I see my Heisenberg story just above is wrong. I just wrote the just above off the top of my head. The proper story is of course as follows. The method considered by Dirac produces a beam of systems, each system in a pure state. If the pure state degree of freedom considered is the momentum in the direction of the beam, that has been produced by an arrangement of choppers, the value of the momentum will be narrowly defined by the choppers, nearly an eigenstate. But the position in the same direction will be practically undefined, with a practically infinite uncertainty. It is the position that is the conjugate variable to the momentum that goes into the uncertainty product.Chjoaygame (talk) 22:22, 17 March 2016 (UTC)

Heisinberg is correct. Narrowly defined by the choppers, nearly an eigenstate is possible indeed.

On pages 132–133, Shankar (2nd edition, 1980/1994) reads: "In our earlier discussion on how to produce well-defined states |ψ⟩ for testing quantum theory, it was observed that the measurement process could itself be used as a preparation mechanism: if the measurement of Ω on an arbitrary, unknown initial state given a result ω, we are sure we have the state |ψ⟩ =  |ω⟩. ... To prepare a state for studying quantum theory then, we take an arbitrary initial state and filter it by a sequence of compatible measurements till it is down to a unique, known vector. Any nondegenerate operator, all by itself, is a "complete set.""Chjoaygame (talk) 23:39, 17 March 2016 (UTC)

Shankar is incorrect. But he, on the other hand assumes the artifact of quantum ideal measurement using arbitrarily ultra-soft photons with precise energy. These do not exist because the quantum states producing them would have to have infinite lifetime. (The actual lifetime of these states provide an other means of deriving Bohrs formula of above.)

Shankar isn't talking about "arbitrarily ultra-soft photons with precise energy." He is talking about "measurement" by selection in the same sense as do L&L, Cohen-Tannoudji et al., and Dirac.Chjoaygame (talk) 21:14, 18 March 2016 (UTC)

On page 235, Cohen-Tannoudji, Diu & Laloë (2nd edition, 1973/1977) write: "Similarly, we can construct devices, intended to prepare a quantum system, in such a way that they only allow the passage of one state, corresponding to a particular eigenvalue of each of the observables of the complete set chosen."Chjoaygame (talk) 23:51, 17 March 2016 (UTC)

Incorrect in general for reasons stated.

The correct treatment is found in section 44 or Landau and Lifshitz with the result that

$|v'_{x}-v_{x}|\Delta p_{x}\approx \hbar /\Delta t,$

where

${\begin{aligned}&\Delta p_{x}&{\text{is uncertainty in measured value of momentum,}}\\&\Delta t&{\text{is duration of measurement,}}\\&v_{x}&{\text{is velocity of particle }}{\textit {before}}{\text{ measurement,}}\\&v'_{x}&{\text{is velocity of particle }}{\textit {after}}{\text{ measurement.}}\\\end{aligned}}$

Read also section 7 as a preparation. Chjoaygame, why do you chose not to quote L&L this time? They are the ones treating this. YohanN7 (talk) 08:53, 18 March 2016 (UTC)

People often talk about "simultaneous" measurement. They usually don't mean it literally and exactly. More precisely they usually mean compatible serial measurement, not envisaging literal simultaneity. One might say instead perhaps 'joint measurement', or 'coexistent measurement'. But 'simultaneous' seems the customary term. The idea is that each system, on its way from oven to anti-oven (= detector), passes through several sorting devices in tandem (meaning 'at length', not in parallel). These discussions, if referring to several degrees of freedom, in many cases seem more about idealized scenarios than exactly and practically real ones. In practice, it is not easy to chain together too many slits, polarizers, choppers, beam-splitters, prisms, Stern–Gerlach magnets, calcite crystals, and what not. The key concept is commutativity.Chjoaygame (talk) 09:09, 18 March 2016 (UTC)

No one is contradicting L&L. The relevant devices considered by the writers I cite are designed to make one aspect of the degree of freedom rather closely defined, while the conjugate aspect has huge latitude of error.Chjoaygame (talk) 09:15, 18 March 2016 (UTC)

Page 5 of the 3rd edition of L&L (Pergamon 1977): "We shall now formulate the meaning of a complete description of a state in quantum mechanics. Completely described states occur as a result of the simultaneous measurement of a complete set of physical quanti­ties."Chjoaygame (talk) 09:27, 18 March 2016 (UTC)

This is written before reading your last addition:

You are mixing in irrelevancies, and do not lecture on what people mean. I know what they mean. My original objection is that you cannot

• Obtain a complete measurement an arbitrary state precisely at any given time

AND

• Expect a repeated measurement to yield the same result.

Note the AND. It is capitalized for you so that you surely note it. In particular, you cannot use the reasoning to define quantum states more "precisely". This is what you did in the article, and this is what found profoundly misleading.

Regarding L&L. Among the math, they, in several places, take note that non-repeatability of precise momentum measurements is fundamental to quantum mechanics. (Don't take this as a literal quote. I don't have my book in my lap atm.) I dare say that L&L are more reliable than Dirac, especially when it comes to isolated one-liners, since they do the math and Landau was just as clever as Dirac, if not cleverer. The book has also been revised into modern times. YohanN7 (talk) 08:53, 18 March 2016 (UTC)

This is written after reading your last addition:

Now you try to make it as if you were right al the time. L&L has gone from "do not apply" to be "maybe true, I don't care because I am right anyway". YohanN7 (talk) 09:33, 18 March 2016 (UTC)

---

I have now left the building. I intended to do that when I retracted my first reply to let you have your final word. Now I must do it. It takes too much of my time and is too frustrating. The article is all yours. YohanN7 (talk) 09:40, 18 March 2016 (UTC)

<Greiner, W. Quantum Mechanics: An Introduction, 4th edition, Springer, Berlin, ISBN 3-540-67458-6> p. 438: "Axiom 1: As a result of the measurement of an observable, only one of the eigenvalues of the corresponding operator can be found. After the measurement, the system occupies that state which corresponds to the measured eigenvalue."

Pure states and von Neumann's projection postulate

A state of a system may be defined as an undisturbed motion that is restricted by as many conditions or data as are theoretically possible without mutual interference or contradiction. In practice, the conditions could be imposed by a suitable preparation of the system, consisting perhaps of passing it through various kinds of sorting apparatus, such as slits and polarimeters, the system being undisturbed after preparation.

When we measure a real dynamical variable ξ, the disturbance involved in the act of measurement causes a jump in the state of the dynamical system. From physical continuity, if we make a second measurement of the same dynamical variable ξ immediately after the first, the result of the second measurement must be the same as that of the first. Thus after the first measurement has been made, there is no indeterminacy in the result of the second.

von Neumann (1932) makes his famous projection postulate on pp. 200–201:

( P.) The probability that in the state φ the quantities with the operators R₁, ... ,R_l take on values from the respective intervals I₁ ... I_l is

|| E₁(I₁) ... E_l(I_l) φ ||²

where E₁(λ), ..., E_l(λ) are the resolutions of the identity belonging to R₁, ... ,R_l respectively.

...we postulate P. for all commuting R₁, ... ,R_l . Then the E₁(I₁), ..., E_l(I_l) commute, and therefore E₁(I₁) ... E_l(I_l) is a projection (THEOREM 14. in II. 4.), and the probability in question becomes

P  =  || E₁(I₁) ... E_l(I_l) φ ||²  =  (E₁(I₁) ... E_l(I_l) φ , φ)

According to Weinberg on page 26:

...Thus if a system is in a state represented by a wave function ψ, and we make a measurement that puts the system in any one of a set of states represented by orthonormal wave functions ψ_n (which may or may not be energy eigenfunctions) then the probability that the system will be found to be in a particular state represented by the wave function ψ_m is

$P(\psi \rightarrow \psi _{m})=\left|\int \psi _{m}^{*}\psi \right|^{2}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1.5.18)$

This can be taken as the fundamental interpretive postulate of quantum mechanics.

Weinberg in 2013 is using the traditional term "measurement" as Dirac used it in 1958, to express the von Neumann projection postulate.

Greiner, W. (2001), Quantum Mechanics: An Introduction, 4th edition, Springer, Berlin, ISBN 3-540-67458-6

P. 438:

Axiom 1: As a result of the measurement of an observable, only one of the eigenvalues of the corresponding operator can be found. After the measurement, the system occupies that state which corresponds to the measured eigenvalue.

P. 469:

Now we ask how measurement influences the system. Let the measurement have the result q_l. Immediately after the measurement, we assume it to be repeated. If the measurements are to make physical sense, we have to claim that the experimental result does not change: the second measurement also has to result in the value q_l.

Greiner in 2001 is using the traditional term "measurement" as Dirac used it in 1958, to express the von Neumann projection postulate.

Page 5 of the 3rd edition of L&L (Pergamon 1977):

We shall now formulate the meaning of a complete description of a state in quantum mechanics. Completely described states occur as a result of the simultaneous measurement of a complete set of physical quanti­ties.

L&L in 1977 used the traditional term "measurement" as Dirac used it in 1958, to express the meaning of a complete description of a state.

On page 235, Cohen-Tannoudji, Diu & Laloë (2nd edition, 1973/1977) write:

Similarly, we can construct devices, intended to prepare a quantum system, in such a way that they only allow the passage of one state, corresponding to a particular eigenvalue of each of the observables of the complete set chosen.

These authors do not here use the traditional term "measurement" for their preparation procedure, though they rely on devices which prepare by selecting or filtering.

The traditional term "measurement", as used by Dirac, Greiner, Landau & Lifshitz, and Weinberg, is, regrettably, an oft-encountered abuse of language. A better term would be 'purification by selection or filtration', or Dirac's "sorting apparatus".

This proposed better term is not original research. For example, Park & Band in 1992 wrote:

P. 658:

It is not unusual to find excellent didactical presentations of the quantal algorithm embedded in a kind of structureless philosophical void which offers the student no clue as to the empirical significance of the formalism. ... We believe that the student of quantum theory can successfully weather the above-mentioned and other common affronts to his intellect if at some point in the educational process, preferably early, he is introduced to the preparation-measurement format of experimental science and then immediately taught the particular manner in which quantum theory copes with physical problems by employing that framework.

P. 659:

Nevertheless, to this day the preparation process is still often misnamed "measurement" ...

P. 662:

A further selection may be imposed ... We call this an almost "pure" state because the result of any subsequent momentum measurement has an almost foregone conclusion. ... When we have completed this selective preparation of a pure state, we have, according to conventional wisdom, already performed the measurement. ... Conventional discussions based on the von Neumann mathematical scheme stopped at the pure state preparation and asserted that the measurement act had left the particle in the pure state corresponding to the result of the measurement. This is really nothing worse than a semantic error.

Perhaps this may be useful.Chjoaygame (talk) 13:25, 30 March 2016 (UTC)

""""measurement"""" as state preparation ???

As I have repeatedly written in this talk page, the word 'measurement' in quantum mechanics is a trap for young players. I have advocated 'observation' instead simply because it is less fraught with prejudice, and more likely to get the reader to have a critical attitude.

Without prejudice, the following quote is from a textbook that is free on the Internet, by J.D. Cresser, talking about the so-called 'von Neumann projection postulate':

This postulate is almost stating the obvious in that we name a state according to the information that we obtain about it as a result of a measurement. But it can also be argued that if, after performing a measurement that yields a particular result, we immediately repeat the measurement, it is reasonable to expect that there is a 100% chance that the same result be regained, which tells us that the system must have been in the associated eigenstate. This was, in fact, the main argument given by von Neumann to support this postulate. Thus, von Neumann argued that the fact that the value has a stable result upon repeated measurement indicates that the system really has that value after measurement.

This is referring to the same kind of """measurement""" as considered above as cited from Dirac, Cohen-Tannoudji et al., Shankar, Feynman, and page 5 (but not necessarily section 44) of L&L. I think the essence of this kind of """measurement""" is that it works by selection rather than by interaction. This contrasts with what Pauli cited above calls 'measurement of the second kind', which tells about the state before measurement, and which involves 'kicking' the system.

Without prejudice, it my be useful to cite von Neumann. Page 380 of the Princeton English translation says:

... Therefore, each measurement on a state is irreversible , unless the eigenvalue of the measured quantity (i.e., this quantity in the given state) has a sharp value, in which case the measurement does not change the state at all.

Though not the only ideas to consider, these cannot be just dismissed as irrelevant, mistaken, or nonsensical.Chjoaygame (talk) 23:13, 19 March 2016 (UTC)

Drilling into the details of measurement is indeed off topic for this article. There are other articles for these things. The way you are going, the article will just be another article on quantum states or even QM itself.

A few weeks ago I intended to stay away, yet I have been silly enough to engage with you on this talk page. This will be my last reply. M∧Ŝc²ħεИ_τlk 23:36, 19 March 2016 (UTC)

bras and kets

Without prejudice, here, in Baaquie, B.E. (2013), The Theoretical Foundations of Quantum Mechanics, Springer, New York, ISBN 978-1-4614-6223-1, on page 222, one may find the following:

The initial state |ψ⟩ is the “history state” of the transition amplitude, and the final state ⟨χ| is its “destiny state.” In quantum theory both the history state and the destiny state can be independently specified.

We may consider the possible worth of this, in the light of the above cited statements of Feynman, of L&L, and of Auletta, G., Fortunato, M., & Parisi, G..Chjoaygame (talk) 23:39, 19 March 2016 (UTC)

Also we may consider another writer, Schwartz, M.D. (2014), Quantum Field Theory and the Standard Model, Cambridge University Press, Cambridge UK, ISBN 978-1-107-03473-0, here on page 56:

We can write such inner products as ⟨f;t_f|i;t_i⟩, where |i;t_i⟩ is the initial state we start with at time t_i and ⟨f;t_f| is the final state we are interested in at some later time t_f.

Also on page 70:

The state |i⟩ is the initial state at t = −∞ and ⟨f| is the final state at t = +∞.

Perhaps others may turn up? Chjoaygame (talk) 00:30, 20 March 2016 (UTC)

Schwinger distinguished bras and kets as creation and destruction symbols. He went contary to the usual custom of seeing kets as creation and bras as destruction symbols, but this choice is allowed by the symmetry of the theory. He wrote (as reported by Englert in a posthumously assembled text):

... certainly ⟨a'|, symbolizing a creation act, cannot be equated to |a'⟩, representing an act of destruction, (reading L → R)."<Schwinger, J. (2001). Quantum Mechanics: Symbolism of Atomic Measurements, edited by B.-G. Englert, Springer, Berlin, ISBN 3-540-41408-8, p. 38.>

Chjoaygame (talk) 04:03, 22 March 2016 (UTC)

Not qualifying as a reliable source is this:

|Ψ⟩ represents a system in the state Ψ and is therefore called the state vector. The ket can also be interpreted as the initial state in some transition or event. The bra ⟨ | represents the final state or the language in which you wish to express the content of the ket | ⟩.

Chjoaygame (talk) 05:24, 20 March 2016 (UTC)

A deprecatory view of the proposed preparation/observation interpretation of kets/bras is given by Peres, A. (2002), Quantum Theory: Concepts and Methods, Kluwer, New York, ISBN 0-792-33632-1, on page 77:

... (... also fruitless attempts to attribute different physical meanings to the two types of vectors, such as preparation states and observation states).

Evidently, opinion is not uniform.Chjoaygame (talk) 02:29, 20 March 2016 (UTC)

a representation defined by a complete set of bras

In the 4th edition (1958), on page 5, Dirac writes:

The problem we must now consider is how to fit in these ideas with the known facts about the resolution of light into polarized components and the recombination of these components.

On page 53:

To set up a representation in a general way, we take a complete set of bra vectors, i.e. a set such that any bra can be expressed linearly in terms of them (as a sum or an integral or possibly an integral plus a sum). These bras we call the basic bras of the representation. They are sufficient, as we shall see, to fix the representation completely.

The beam of systems passes through the representation's characteristic quantum analyzer and comes out in its several output channels with the appropriate weighting or probability distribution. The representation's basic bras denote the several output channels, with a detector in each. The analyzer has resolved the beam into components for the representation. If there were no detectors in the output channels to fix the systems as bras, they would still be coherently available as kets for interference or re-assembly into a replica of the original beam from the source.

The analyzer acts as the representation's resolution of the identity:

$\sum _{\xi ^{\prime }}|\xi ^{\prime }\rangle \langle \xi ^{\prime }|=1$

and

$\int |\xi ^{\prime }\rangle d\xi ^{\prime }\langle \xi ^{\prime }|=1$.

Chjoaygame (talk) 16:40, 20 March 2016 (UTC)

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Normalized, or not?

It seems, position-space and momentum-space wave functions are normalized, but others are not; and every linear combination of wave functions is a wave function... a mess? Boris Tsirelson (talk) 14:26, 30 April 2016 (UTC)

It says in the inner product section when a wave function is normalized, orthogonal, orthonormal, or not. M∧Ŝc²ħεИ_τlk 14:30, 30 April 2016 (UTC)

Yes, and still... even in that section, the interpretation of the inner product is given before introducing normalization. Does this interpretation of the inner product apply to non-normalized wave functions? Boris Tsirelson (talk) 15:45, 30 April 2016 (UTC)

"...this general requirement a wave function must satisfy is called the normalization condition..." So, really, must satisfy? Throughout the article? Or, depending on the context, sometimes? How does the reader know, when it must? Boris Tsirelson (talk) 15:49, 30 April 2016 (UTC)

Where does the article say "this general requirement a wave function must satisfy is called the normalization condition...". I can't find it anywhere. I thought the wave function must be normalizable for the probability interpretation to work, and collapse assumes normalized wave functions because the formula given is for the transition probability. M∧Ŝc²ħεИ_τlk 16:26, 30 April 2016 (UTC)

The lead, the paragraph before the last:

In Born's statistical interpretation,[8][9][10] the squared modulus of the wave function, | ψ |2, is a real number interpreted as the probability density of measuring a particle's being detected at a given place, or having a given momentum, at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation, this general requirement a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured. Its value does not in isolation tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.

Boris Tsirelson (talk) 16:47, 30 April 2016 (UTC)

The problem is, that the article is for now the mix of statements that are true when "wave function" means "normalized" (and wrong otherwise), and statements that are true when "wave function" means "not just normalized" (and wrong otherwise). Boris Tsirelson (talk) 16:51, 30 April 2016 (UTC)

Ack, careless me, missed the lead. Just tried rearranging more sections in the article to make the flow easier to follow. It would be helpful to separate off all content on the probability interpretations in its own section, and maybe rearrange statements when normalization is necessary and unnecessary. Maybe you would like to try? M∧Ŝc²ħεИ_τlk 17:10, 30 April 2016 (UTC)

Hmmm... let us think. The lead uses the probability interpretation, and this is necessary, since otherwise it is just some math. First of all, a wave function describes a quantum state, and here it must be normalized. But later, probably, it appears that physicists do a lot of calculations, postponing interpretation (till the end of calculation); and in this process they are, somehow, temporary, effectively, mathematicians... and tolerate non-normalized, and even non-normalizable, functions... and do not hesitate to call them wave functions... right? A kind of abuse of language. Surely, the superposition principle does not mean that a linear combination of wave functions is a wave function even if it is identically zero (which really could happen)! But, who cares... Really, not many centuries ago, mathematicians got true equalities after hard calculations with divergent series, without bothering too much... These intermediate divergent series were probably called "functions", but they were not...

But no, I am reluctant to edit physical articles myself. Since my text smells of math, inevitably, I know. And another, no less important reason: I never read elementary textbooks on physics, and so, I do not know, in which form all that is written there. Boris Tsirelson (talk) 17:58, 30 April 2016 (UTC)

The observation

...physicists do a lot of calculations, postponing interpretation (till the end of calculation); and in this process they are, somehow, temporary, effectively, mathematicians... and tolerate non-normalized, and even non-normalizable, functions...

is absolutely correct. This does not provide an excuse for us to do the same in this article, at least not before we tell the reader what is about to happen. I'll try to do my bit in due time. YohanN7 (talk) 16:43, 2 May 2016 (UTC)

Well, we must follow the physics that stretches outside Wikipedia... that is, "do the same"; but indeed, we should tell the reader what is about to happen. At least, it is done this way in math; for instance, "Baire set". Boris Tsirelson (talk) 18:12, 2 May 2016 (UTC)

On a side note: Gieres, F. (2000). "Mathematical surprises and Dirac's formalism in quantum mechanics". Rep. Prog. Phys. 63: 1893–1931. arXiv:quant-ph/9907069. YohanN7 (talk) 16:54, 2 May 2016 (UTC)

Woooooow! Boris Tsirelson (talk) 18:12, 2 May 2016 (UTC)

Rearrangement of sections

On one hand, it is nice to pin down all mathematical and physical properties/requirements first thing (after the history).

On the other hand, it would be easier for typical readers to just see what the wave function is with examples, and the mathematical details later. Also, since the probability interpretation is what most sources present that section could be moved further up. The Hydrogen atom content in the function spaces section has been moved to the section of that example.

I tried to rearrange to see how it looks, and will not make further edits. Anyone is free to revert if they disagree. M∧Ŝc²ħεИ_τlk 06:50, 3 May 2016 (UTC)

Well, this happened. But anyone can revert to this version. M∧Ŝc²ħεИ_τlk 12:37, 3 May 2016 (UTC)

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