Talk:Wave function/Archive 10

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, $\langle$Ψ$|$, and the ket, $|$Ψ$\rangle$, 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 $ψφ$, $ψ_{1}ψ_{2}$, $φ_{1}φ_{2}$, have no meaning and will never appear in the analysis." Obviously, later, the outer product is defined in the bra–ket notation, and $|$ψ$\rangle$$\langle$φ$|$ 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'&and;Ŝc2ħεИτlk 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)

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 $φ_{1}$, $φ_{2}$, ... 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}$\rangle$ 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 $|Ψ\rangle$ for the system in question, and $$\mathcal{H}_\langle$$ (in other symbols $$\mathcal{H}_\rangle^{*}$$) the dual space with elements $\langleΨ|$ 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 $|A\rangle$, then the derivative must act on $\langleψ|$ not $|A\rangle$). 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'&and;Ŝc2ħεИτlk 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.
 * $ and $ 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 $|Ψ\rangle$ for the system in question, and $\mathcal{H}_\langle$ (in other symbols $\mathcal{H}_\rangle^{*}$) the dual space with elements $\langleΨ|$ 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 $φ_{r}ψ_{s}$. 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$\rangle$ and $\langle$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 $|A\rangle$, then the derivative must act on $\langleψ|$ not $|A\rangle$). 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 $\langlen|$f$|i\rangle$ as you mention, the observed state happens to be a bra $\langlen|$, the initial state is a ket $|i\rangle$. But this is a mathematical expression for an integral/sum. If you have $\langlen|$ 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'&and;Ŝc2ħεИτlk 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 $\langlen|$f$|i\rangle$ as you mention, the observed state happens to be a bra $\langlen|$, the initial state is a ket $|i\rangle$. But this is a mathematical expression for an integral/sum. If you have $\langlen|$ 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$\rangle$ is concerned, there is just one space, the Hilbert space, in which it resides." He does not explicitly say that the corresponding bra $\langle$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 $\langle$ψ_{1}$|$ is called a "bra" and $|$ψ_{2}$\rangle$ 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)

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)