Talk:Bra–ket notation

Recent reversion

I reverted this edit by user:Chjoaygame for the following reasons.

• A discussion of complex numbers a few sections after they are mentioned is pointless. It is already stated early in the section Background: Vector spaces that bras/kets have complex-valued coordinates (for "coordinates" in this context see coordinate vector), and later in Linearity the linear algebraic properties (linearity (addition, scalar multiplication by complex numbers)) are described better.
• Obscure terminology "conjugate imaginaries"
• A minor issue was superfluous notations, using extra brackets for complex conjugation and both overbar and star

$(c)^{*}={\bar {c}}\,,\quad ({\bar {c}})^{*}=c$

which can be crisply written as

$c=a+ib\,,\quad {\bar {c}}=a-ib\,,\quad {\bar {\bar {c}}}=c$

• Errors in the real and imaginary parts expressions which should have been (in the same notation, using overbar as complex conjugate)

$\Re (c)={\frac {c+{\bar {c}}}{2}}\,,\quad \Im (c)={\frac {c-{\bar {c}}}{2i}}$

so that

$c=\Re (c)+i\Im (c).$

• The emphasis that bras and kets cannot be added is unnecessary, it should be obvious to anyone with some mathematical maturity, but even for laymen we don't need this emphasis. Earlier in Bras as linear operators on kets it is clear they belong in different spaces, and expressed as row/column vectors it should be obvious they cannot be added.
• A vector with complex coordinates does have a notion of "real and imaginary parts", just the real part of the vector and the imaginary part. For example in 3d complex space, with the usual real Cartesian basis:

$\mathbf {c} =\mathbf {a} +i\mathbf {b} \,,\quad \mathbf {a} =a_{x}\mathbf {e} _{x}+a_{y}\mathbf {e} _{y}+a_{z}\mathbf {e} _{z}\,,\quad \mathbf {b} =b_{x}\mathbf {e} _{x}+b_{y}\mathbf {e} _{y}+b_{z}\mathbf {e} _{z}$

explicitly

$\mathbf {c} =(a_{x}+ib_{x})\mathbf {e} _{x}+(a_{y}+ib_{y})\mathbf {e} _{y}+(a_{z}+ib_{z})\mathbf {e} _{z}$

and with overbar for the complex conjugate,

${\bar {\mathbf {c} }}=\mathbf {a} -i\mathbf {b}$

(see Riemann-Silberstein vector for a physical example). This can be immediately translated into bra-ket notation.

M∧Ŝc²ħεИ_τlk 11:27, 24 March 2016 (UTC)

The undone edit intended to make a point made by Dirac, that bras and kets do not have real and imaginary parts in the same way as do complex numbers. The edit announced its purpose thus: "Bras and kets can be multiplied by complex numbers. Complex numbers have real and imaginary parts, but bras and kets do not." The above comment contradicts Dirac as cited.Chjoaygame (talk) 11:55, 24 March 2016 (UTC)

@Chjoaygame, In other words, bras and kets are adjoint. One needs to be transposed to the other's sense before you can operate (such as: add them, multiply them, ...). See the vector and matrix formalisms for more detail on the motivations for the describing a physical picture with linear algebra. --Ancheta Wis   (talk | contribs) 12:26, 24 March 2016 (UTC)

Doubtless Editor Ancheta Wis is right. Dirac was saying that bras and kets do not have real and imaginary parts in the same way as do complex numbers. The above comment contradicts Dirac as cited.Chjoaygame (talk) 12:58, 24 March 2016 (UTC)

I don't know if you are conflating complex conjugation of a complex vector (as shown above) with Hermitian conjugation (as explained in extreme detail on talk:wave function) to obtain the dual, but it is already clear you cannot add/subtract bras and kets. Citing Dirac on trivialities has no purpose in this article. M∧Ŝc²ħεИ_τlk 14:14, 24 March 2016 (UTC)

"Dirac was saying that bras and kets do not have real and imaginary parts in the same way as do complex numbers". For completeness, start from the vector

$|c\rangle =\sum _{k}c_{k}|k\rangle \,,\quad c_{k}=a_{k}+ib_{k}$

so

$|c\rangle =\sum _{k}(a_{k}+ib_{k})|k\rangle =\sum _{k}a_{k}|k\rangle +i\sum _{k}b_{k}|k\rangle$

where we have the real vectors

$|a\rangle =\sum _{k}a_{k}|k\rangle \,\quad |b\rangle =\sum _{k}b_{k}|k\rangle$

with real coordinates a_k and b_k (or "scalar components", "projections", "weights", whatever). Then the vector, its complex conjugate, and dual, are

$|c\rangle =|a\rangle +i|b\rangle \,,\quad {\overline {|c\rangle }}=|a\rangle -i|b\rangle \,,\quad |c\rangle ^{\dagger }=\langle c|=\langle a|-i\langle b|$

so obviously

${\overline {|c\rangle }}\neq |c\rangle ^{\dagger }$

and

$\Re (|c\rangle )=|a\rangle ={\frac {|c\rangle +{\overline {|c\rangle }}}{2}}\neq {\frac {|c\rangle +|c\rangle ^{\dagger }}{2}}$

So after all, there are real and imaginary parts of a complex vector (which kets are in another notation). I'd like to see you find a page or quote from Dirac or anyone contradicting the above. Whether you write the same vector in boldface or bra-ket notation doesn't matter, bra-ket notation does not automatically change the vector into something else. The operations are relevant, complex conjugation of a vector is still the same vector with i changed to -i, while Hermitian conjugation obtains the corresponding vector in the dual space, a different object altogether. M∧Ŝc²ħεИ_τlk 15:06, 24 March 2016 (UTC)

The undone edit cited Dirac as follows.

When the just foregoing comment assumes the existence of the "real" vectors

$|a\rangle =\sum _{k}a_{k}|k\rangle \,\quad |b\rangle =\sum _{k}b_{k}|k\rangle$.

it contradicts Dirac.Chjoaygame (talk) 20:52, 24 March 2016 (UTC)

No it does not. When he says on p.20 just before the part you quoted now,

"The usual method of getting the real part of the complex quantity, by taking half the sum of the quantity itself and its conjugate, cannot be applied since a bra and a ket vector are of different natures and cannot be added together."

"conjugate" (very vague) or "conjugate imaginary" (his terminology for context) is referring to "Hermitian conjugate". He uses "conjugate complex" for complex conjugate (i → −i). Then, because he used "conjugate" for a different operation, to prevent people thinking that kets and bras (Hermitian conjugate of kets) could be added/subtracted in the same way as the real/imaginary parts of a complex number, he gives that warning in the quote you presented. You should be able to figure out in modern terminology what is meant, and not blindly follow archaic terminology even if pioneers used them.

In the demonstration above, I did not "assume the existence of the real vectors", it follows from the trivial fact that the complex numbers c_k can be split into the form shown. They just can. Then the linearity of the scalar multiplication of the kets (or bras) by numbers, real or complex, allows the split into real and imaginary vectors as shown. M∧Ŝc²ħεИ_τlk 21:37, 24 March 2016 (UTC)

Tell me, are the following mathematically valid manipulations? (Ignore normalization, not needed here).

${\begin{aligned}|c\rangle &=(2+3i)|a\rangle +(6-5i)|b\rangle \\&=2|a\rangle +3i|a\rangle +6|b\rangle -5i|b\rangle \\&=2(|a\rangle +3|b\rangle )+i(3|a\rangle -5|b\rangle )\end{aligned}}$

as for the complex conjugate (or if you prefer Dirac's "conjugate complex", and not the Hermitian conjugate),

${\overline {|c\rangle }}=2(|a\rangle +3|b\rangle )-i(3|a\rangle -5|b\rangle )$

Yes? No? Ridiculous typos aside - I know they are. Perhaps you should. M∧Ŝc²ħεИ_τlk 23:49, 24 March 2016 (UTC)

Dirac's 1st edition (1930), p. 21: "... we cannot give any meaning to the splitting up of a ψ into its real and pure imaginary parts."

Dirac's 2nd edition (1935), p. 21: "One cannot have a real vector in the vector space and one cannot split up a general vector into real and pure imaginary parts." [Dirac's italics.]

Dirac's 3rd edition (1947), p. 20: "Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts."

Dirac's 4th edition (1958), p. 20: "Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts."

With $c$ an arbitrary complex number and $|\rangle$ an arbitrary ket, perhaps Dirac's view arises because $|\rangle$ and $c|\rangle$ refer to the same state? The objects of interest are sometimes regarded as points of a projective space. For a point of a projective space, homogeneous coordinates are determined only up to a non-zero scalar multiple.Chjoaygame (talk) 03:25, 25 March 2016 (UTC)

Dirac emphasized the symmetry between bras and kets. He wanted the class of pure states to be self-dual, with bras and kets in a natural one-to-one correspondence. Making kets vectors and bras their linear functionals is not necessarily quite symmetry. Kets as distributions and bras as test functions is also not necessarily quite symmetrical. Projective spaces have, however, an almost natural duality, and self-duality may be easier there. Mathematicians may have more to say on this, but perhaps not for this present article.Chjoaygame (talk) 06:46, 25 March 2016 (UTC)

As always you are sidestepping what others have to say, and resort to quotes as if I (or others) are too stupid to read them or know where to find them. I can read Dirac's 4th edition thanks. The uselessness of the section has already been explained and I will not re-explain again. Your misinterpretation of archaic terminology, and bra-ket notation which is a notation for linear algebra (and nothing more), have been explained to you over and over. Let's see if others have to comment. M∧Ŝc²ħεИ_τlk 08:29, 25 March 2016 (UTC)

It seems I need to clarify here. Editor Maschen has made several objections to my post that he undid. Some of them are sound. But I have concentrated on the most substantial here: Dirac's statement. The matter can be considered through several questions.

• When Dirac wrote as follows, did he mean it as it seems on its face?

Dirac's 2nd edition (1935), p. 21: "One cannot have a real vector in the vector space and one cannot split up a general vector into real and pure imaginary parts." [Dirac's italics.]?

I think so. I think this is denied by the above comment:

No it does not. When he says on p.20 just before the part you quoted now,

"The usual method of getting the real part of the complex quantity, by taking half the sum of the quantity itself and its conjugate, cannot be applied since a bra and a ket vector are of different natures and cannot be added together."

"conjugate" (very vague) or "conjugate imaginary" (his terminology for context) is referring to "Hermitian conjugate".

This question seems unsettled here.

• Was Dirac right when he wrote that?

I think so. There is an important difference between a vector and its representation by coordinates. A complex vector is complex by virtue of its being mulipliable by elements of its scalar field, not by virtue of its being resolvable into real and imaginary parts. Some complex vectors can indeed be resolved into real and imaginary parts, for example, wave functions. That is the most obvious example for the present discussion. The example of the Riemann-Silberstein vector is a little distant, referring to pre-quantum, mechanical thinking, but still valid. But that it is so for some does not imply that it is so for all.

Evidently, Editor Maschen thinks not, when he writes above:

In the demonstration above, I did not "assume the existence of the real vectors", it follows from the trivial fact that the complex numbers c_k can be split into the form shown. They just can.

I do not see that as a persuasive or sound argument to deal with the question as to whether bras and kets can be resolved into real and imaginary parts. They are abstract vectors, not fully reducible to any particular representation. That is part of how they are distinct from wave functions.

This question seems unsettled here.

With these two questions unsettled here, I think the undo remains unsettled.Chjoaygame (talk) 07:11, 27 March 2016 (UTC)

I assume you are interpreting Dirac's "conjugate imaginary" (in his context, Hermitian conjugate, hence bras) as "imaginary" (a real-valued number, real-valued vector, real-valued function, real-valued whatever, multiplied by i), hence Dirac will say that a ket cannot be split into a real plus imaginary sum. Otherwise it may be that Dirac assumes a restriction on the Hilbert space that the elements of the field (complex numbers of course) are just those, and cannot be split into real plus imaginary parts, that scalar multiplication of kets by complex numbers is not distributive over the addition of pure real and pure imaginary numbers, but I doubt that (there is no claim of this anywhere I can find). Complex numbers work as catch all, and there is little need to split kets in the context of QM or QFT into real and imaginary parts anyway. I do not intend to keep pursuing this. M∧Ŝc²ħεИ_τlk 20:51, 28 March 2016 (UTC)

Editor Maschen has made several comments here. None of them deals directly with the matter in hand. The first of them is a his saying that he assumes something about how I interpret Dirac's words. Obviously I do not so interpret those words. The next is a conjecture by Editor Maschen about Dirac's thinking. If anyone reads the rest, he may draw his own conclusions.Chjoaygame (talk) 15:12, 29 March 2016 (UTC)

Direct evidence that Dirac meant what he appears to mean is as follows.

Dirac's 2nd edition (1935), p. 21: "One cannot have a real vector in the vector space and one cannot split up a general vector into real and pure imaginary parts." [Dirac's italics.]?

Dirac defines a linear operator in this context as a linear operator that carries vectors into vectors, not vectors into numbers; i.e. in this context, his linear operators are not linear functionals. With this definition, he writes in the 4th edition on page 27: "Any linear operator may be split up into a real part and a pure imaginary part. For this reason the words 'conjugate complex' are applicable to linear operators and not the words 'conjugate imaginary'."

This contrast makes it clear that these statements by Dirac mean what they appear to mean. Dirac is not muddled about this matter.Chjoaygame (talk) 16:40, 30 March 2016 (UTC)

Dirac was not muddled, and not wrong, and I have not said Dirac was muddled or wrong as you claim in your new wall of text below.

You are the one who is obviously muddled. M∧Ŝc²ħεИ_τlk 14:09, 31 March 2016 (UTC)

The new version of my post read thus:

Conjugation

It is evident that the conjugate-transpose relation between a bra and ket is different from the complex conjugate relation between complex numbers. According to Dirac: "Our bra and ket vectors are complex quantities, since they can be multiplied by complex numbers and are then of the same nature as before, but they are complex quantities of a special kind which cannot be split up into real and pure imaginary parts."

You have undone this new version. Its first sentence states an obvious context, directly from the above parts of the article. Its second sentence is a verbatim quote from Dirac. If you accept that Dirac is correct, you have no reason to delete the section. And yet you did. Why?Chjoaygame (talk) 15:16, 1 April 2016 (UTC)

It's no different. All you keep doing is going back to the same tired quote again and again and again, because you don't know what else to say. Just because Dirac said kets cannot be split into real and imaginary parts, you unquestionably agree without convincing yourself, apparently without even bothering to self-experiment in a little mathematics. I did provide, in extremely explicit terms, the differences in the terminology of the operations, and the differences in modern and old terminology.

By the way

"I do not see that as a persuasive or sound argument to deal with the question as to whether bras and kets can be resolved into real and imaginary parts. They are abstract vectors, not fully reducible to any particular representation. "

is also vague gibberish, because I was as concrete as necessary (aside from extra mathematical formalities), and as the extensive discussions on talk:wave function show, yes you can write a quantum state in the position-spin representation, momentum-spin representation, energy eigenstates, angular momentum eigenstates, and so on and on and on...

But go right ahead and quote Dirac AGAIN. And again, and again. And whine I have not resolved the issue. Even using the same quote of course. M∧Ŝc²ħεИ_τlk 11:00, 2 April 2016 (UTC)

I recommend these lecture notes. "Complex conjugation of the expansion coefficients in a certain basis Q" is a perfectly valid (antilinear!) operator on kets, kinda useful in discussing time-reversal symmetry. Call it K_Q. Then you can have "real vectors w.r.t. Q" in the sense that K_Q sends them to themselves, and "imaginary vectors w.r.t. Q" in the sense that K_Q sends them to their negatives. But it's not a ket itself that is real or imaginary, but rather the relation between a ket and a basis. The same ket can be real, imaginary, or neither in different bases. I don't think this topic needs to be discussed in this particular article. Well, I guess I'm not totally opposed to saying something simple like "You can't add a bra to a ket, just as you can't add a row-vector to a column-vector", if this is really a common misconception. --Steve (talk) 12:51, 3 April 2016 (UTC)

"But it's not a ket itself that is real or imaginary." That's just exactly what my post quoted Dirac as saying, and is repeatedly denied by the fellow who is now seeking to have me banned for not accepting that he airbrush it out. That it is as Sbyrnes321 says here, and that it is denied hard enough to trigger a proposal for a topic ban is evidence that it is open to misconception enough to make it notable. An RfC might be more appropriate than a topic ban.Chjoaygame (talk) 14:21, 3 April 2016 (UTC)

Thanks Sbyrnes321 for the pointer to the notes, they look helpful. Although I agree this is not necessary to mention complex conjugation in this article, which is just about notation.

Chjoaygame, don't pretend that anyone is taking sides with you (I don't want or expect this for my case). You conveniently ignored The same ket can be real, imaginary, or neither in different bases.. Why I don't understand. Also this talk page is certainly not the only trigger for having you banned. You and your diatribes all over the QM talk pages have multiple people frustrated and its about time you stopped. M∧Ŝc²ħεИ_τlk 15:15, 3 April 2016 (UTC)

I am not pretending that Sbyrnes321 is taking sides with me. I am quoting words that he wrote that I think are right. As for your The same ket ...: The same ket can be represented in different ways in different bases, but it isn't thereby changed, or itself endowed with a capacity to be split into real and pure imaginary parts. It is the representatives that can be split. As for triggers: This was the actual trigger. You may have many reasons, that might be potential triggers, but there can be only one actual trigger.Chjoaygame (talk) 15:49, 3 April 2016 (UTC)

I support the deletion of the section under discussion, because I don't think this article should grow into a general intro to complex vector spaces and linear algebra. (By the same token, I've previously complained that it should not grow into a general intro to quantum mechanics.) It already has too much of that stuff. --Steve (talk) 19:30, 3 April 2016 (UTC)

undoing an edit without talk page comment

Editor YohanN7 has here undone a fair post by me. His edit summary is the one word "Gibberish". He made no talk page comment.

My edit was not gibberish.Chjoaygame (talk) 14:58, 29 March 2016 (UTC)

(I had something here but found the answer below. Suffice it to say, if you wanted to make your point, it needed to be much, much clearer than what you originally wrote.) --100.14.175.181 (talk) 11:50, 6 May 2016 (UTC)

New edit

The new edit here includes the following: "It should be kept in mind that ..." There are many other examples of such by the same editor. I do not recall the exact chapter and verse of Wiki policy for this, but I do recall that such language is deprecated in Wiki policy as condescending and didactic, in effect uncivil, not encyclopedic.Chjoaygame (talk) 19:31, 29 March 2016 (UTC)

undoing of edits that have been disputed, without talk-page comment by the undoers

Editors YohanN7 and Maschen have, more or less collusively, undone my edits here and here, without talk page discussion when it would have been appropriate, or mandatory. This was high-handed action by them.

The two undone edits were about (a) Dirac's view that bras and kets cannot be resolved into real and imaginary parts; and (b) about a few writers' view that one can view bras and kets as about initial and final states of a quantum phenomenon.

As to (a). Editor Maschen has tried on the talk page to blow away Dirac's view, that my edit quoted. I think his undo is based in part on his belief that his try was valid or successful. I think I have above presented enough argument to dispel his try. Editor Maschen's undo had the edit summary "Remove Ch's personal misinterpretations once again, that bras and kets are "physically distinct"." My edit just quoted Dirac's view, not my interpretation of it. The problem here is that Editor Maschen thinks that Dirac was mistaken. I think not, but the point here is that Editor Maschen is confusing his rejection of Dirac's view with his thought that it is my view that is what I posted. I am accurately reporting Dirac's view, not inventing a view of my own. As it happens, I think Dirac is right, but that does not mean that I invented it. So far as I can see, Editor YohanN7 did not involve himself in this matter.

As to (b). Editor YohanN7 involved himself in this by naming me in his edit summary "Addressing Chjoaygame's perpetual concern". Editor Maschen did the actual undo. My post reported in brief summary four writers' views. The writers are respectable secondary sources, giving their opinions, which happen to be nearly enough concordant. Editor YohanN7 has expressed his deprecation of Feynman's Lectures, but I think Feynman's view deserves bring reported, since it agrees closely with that of Landau and Lifshitz. Again, I did not invent these views. I learnt Feynman's long ago, and recently found the others' concordance. My post made it clear that their view is not widely expressed. Editor YohanN7 has replaced my post with his own view in the lead, about S-matrices, not bras and kets as used by my four sources. Editor YohanN7 is competent to express views about S-matrices, but they are not directly about bras and kets as was my post. The topic of the article is bra-and-ket notation, not S-matrices. Editor Maschen has not very much engaged on this topic, beyond just now undoing my edit here on it.

I can hardly overcome this kind of attack. I can, however, say that I think it unethical. Both Editor YohanN7 and Editor Maschen have recently said they will not further engage on a talk-page with me. This does not entitle either of them to undo my edits without normal talk-page engagement. I think Editor Maschen's withdrawal is due to his failure to support his mistaken case with argument. Editor YohanN7 withdrew from talk about the traditional term "measurement". I think a large part of his difficulty there is in his not having read much literature on this topic. The kind of measurement considered by Dirac and the other relevant sources is essentially about many times repeated phenomena, not just one-off ones such as L&L discuss. Editor YohanN7 is right that a one-off "measurement" of the kind discussed by his source Landau nd Lifshitz does not fit with the traditional language used by the sources that are directly relevant to the topic that was being considered. But that does not mean that those relevant sources can be dismissed, and my post with them. I think Editor YohanN7 was significantly in error there, and has withdrawn because of that. Both of my present edits here took careful and adequate account of the objections of those editors against my previous versions of my edits. In short, I think Editors YohanN7 and Maschen are not entitled to avoid talk-page discussion of their present collusive undoing of my present edits.Chjoaygame (talk) 11:36, 31 March 2016 (UTC)

If you want to make statements about the nature of states going further than their just being states, then you need additional context. I use the same context as did you when I exemplified with the S-matrix. You talked about kets and bras "distinguishing initial and final conditions of a phenomenon", undoubtedly much more to your taste since it sounds more like the kind of word-poop you prefer. I gave a concrete and very general example. YohanN7 (talk) 12:03, 31 March 2016 (UTC)

The reason for my talk page silence is that one and a half year of interaction with you on the various talk pages has taught me that I do not get through to you. It doesn't matter what the particular topic may be. As far as I am concerned, you made your mind up long ago. You read your prophet literally, and time has stood still since 1930. I am no longer trying to convince you. It is a waste of time. Hence my silence and lack of talk page invitations to discuss. YohanN7 (talk) 12:11, 31 March 2016 (UTC)

The just foregoing comment by Editor YohanN7 is disorderly and continues his unethical conduct.Chjoaygame (talk) 15:22, 1 April 2016 (UTC)

There is a standard procedure: It probably translates, for the current situation, to refrain from editing until the editors of the page identify themselves as interested parties in the article.

Please assume good faith, for starters. There is a whole set of editorial guidelines which we can use to guide the development of the article. --Ancheta Wis   (talk | contribs) 21:00, 1 April 2016‎ (UTC)

The just foregoing outdented unsigned fragment comment was added here by the editor indicated.Chjoaygame (talk) 02:05, 2 April 2016 (UTC)

Ancheta Wis, you or anyone one else from the physics community should not write on my nose about "assuming good faith" when it comes to editor Cjhoaygame. Unlike the rest of the community, I have endured wading through endless rivers of bullshit from this individual, actually assuming good faith. You and the rest have just walked off into the night with the probable excuse that it isn't your business or that you have other things to do. Like someone in the math team noted, the sort of perpetuating misconceptions of mislead laymen represented by Cjhoaygame's edits wouldn't be allowed to occur for long in a math article. The physicists attitude is different: "If I ignore it, I don't see it, and if I don't see it, it doesn't exist. In particular, I don't have to deal with it." Fine with me, but don't write on my nose about assuming good faith on part of Chjoaygame. He is playing a game and is damned well knowing what he is doing. Either keep quiet or have an opinion on the subject. YohanN7 (talk) 11:38, 2 April 2016 (UTC)

@Chjoaygame, It appears that we need to back away from this article. May I ask that you first use the talk page to make your case for any changes you may wish to introduce. If the community agrees, you are free to make your changes. Otherwise, please forbear from changes to the article.

@YohanN7, We do not enjoy the same interests in the same articles. We have only AGF and community to fall back onto for guidelines. The respect of other editors is a high currency, which we expend one-to-one, editor to editor; perhaps you might be interested in the proposal below? --Ancheta Wis   (talk | contribs) 12:48, 2 April 2016 (UTC)

I agree with the deletion of both those sections: The conjugation section is unnecessary and off-topic (while it's true that complex vectors don't have real and imaginary parts in the same way that complex numbers do, that has nothing to do with bra-ket notation), and the "interpretations" section is all wrong (for example, there are references in which people are describing mnemonics for reading this or that particular equation, but this is misunderstood as a description of bras and kets in general). Bra-ket notation is just a notation for linear algebra, it doesn't have or need any "interpretation" beyond that. --Steve (talk) 19:43, 3 April 2016 (UTC)

Thank you for this support of the correctness of my quote from Dirac. He thought it relevant enough to repeat, with explicit reference to bras and kets in the four editions, that they don't have real and imaginary parts in the same way that complex numbers do. Wave functions usually do have such parts. The difference is notable in the sense that it is strenuously and persistently denied here by a respected Wikipedia editor. With respect, bras and kets are a notation for the scalar product of Dirac's transformation theory, not quite linear algebra in general. A good number of high-quality linear algebra texts prefer to stay with the (·,·) notation for the inner product. For the duality pairing, Halmos uses [·,·], not ⟨·|·⟩ . It is unfortunate that the Wikipedia article on the inner product uses ⟨·|·⟩, a relic of early days of Wikipedia that has not been looked at critically. The preparation-observation interpretation has now been indirectly incorporated in the lead in terms of S-matrices.Chjoaygame (talk) 21:06, 3 April 2016 (UTC)

Dirac was writing a textbook on quantum mechanics in general, not a description of bra-ket notation in particular. Dirac wrote all kinds of things that do not belong in this article. Random pedagogical comments countering this or that potential misunderstanding that someone might have about complex vector spaces: That's a great example of something that does not belong in this article. You say "bras and kets are a notation for the scalar product of Dirac's transformation theory, not quite linear algebra in general". Your evidence is that some people use different notations. Yes they do. Nobody said that bra-ket notation was the only possible notation for linear algebra in general. Dirac was discussing and using complex vector spaces. "The scalar product of Dirac's transformation theory" is the very same scalar product as the one discussed in the article Inner product space. Dirac may have invented the notation and used it in a particular context but he could not prevent people from spending the next 50 years using it in whatever other contexts they want.

I think the addition to the lead about S-matrices was wrong and I intend to delete it. (I think it was just put in to appease you?) Bras and kets are a general-purpose tool, and the interpretation depends entirely on the context in which it is used. A good analogy is line segment. It has a meaningful "interpretation" when it shows up in a Feynman diagram, a different interpretation in a flowchart, and a third different interpretation when it's between the numerator and denominator of a fraction. Notice that the line segment article discusses none of these. I love scattering matrix elements but they are far far from the only interpretation of bras and kets. There are hundreds or thousands of other equally-important uses of bra-ket notation, and in each they are interpreted a different way. --Steve (talk) 22:04, 3 April 2016 (UTC)

Fair comment.Chjoaygame (talk) 00:48, 4 April 2016 (UTC)

The addition was indeed put there in response to Chjoaygame's persistence. My original edit [1] just noted the absence of "physics" in the notation. I agree that the paragraph doesn't read well, but what should we do? There are other examples of articles (see observer effect (physics)) where fairly non-encyclopedic formulations are used to underline that there are severe misconceptions floating around. I clearly list the S-matrix element as an example, not the use if the inner product. It is probably mostly in this particular context that bra's get the status of "observed states" and kets "prepared states". What is the alternative? I don't agree however that there are hundreds and thousands of "equally important" uses of the inner product. Know the S-matrix and you have solved the theory as far as all computations go. YohanN7 (talk) 07:52, 4 April 2016 (UTC)

No problem there. According to Dirac, the theory itself is symmetrical between bras and kets. That means that they can swap roles and the results are the same. Fine to have bras as states as prepared and kets as observed. What matters physically is that to study phenomena you have to have an actual source of systems and an actual destination for them. The physics isn't in the notation. It's in the lab. No amount of new notation would introduce new physics. Just that to be better for physics, new notation has to follow the lab physics better. Within one form of the theory, bras and kets show antilinear symmetry, so also within the other form.Chjoaygame (talk) 08:23, 4 April 2016 (UTC)

To close up my comments to this page, and (attempt to) satisfy Chjoaygame as he unearths an old thread here, I admit blindly thinking a complex vector could be generally split into real and imaginary parts exactly as for complex numbers, for any basis. However, for the a + ib example above, I did state what the basis was (e_x, e_y, e_z), so the vector could be written that way. If another basis was used then the components would be different. Me admitting this mistake still does not excuse him from a ban. M∧Ŝc²ħεИ_τlk 14:33, 4 April 2016 (UTC)

Thank you, Editor Maschen, for this magnanimous comment.Chjoaygame (talk) 00:09, 5 April 2016 (UTC)

Coming back to (b) above, the 'two-aspects of a state' question of physics. I am here to learn. Editor Sbyrnes has my respect, but seems to reject the idea that I find in Feynman and in Landau & Lifshitz, that it is naturally symbolized in the bra-ket distinction.

I think one of prime fundamentals of the physics of quantum mechanics is that experiments are conducted by the contiguous placement of a source and a destination device (or eleborations thereof), which are often recognized as preparative and observational respectively. Repeated replicas of the quantum system are envisaged as passing from source to destination. I won't bore you with literature support for this. May I ask the assembled company of experts: in the mathematical formalism of quantum mechanics, how, if at all, is this fundamental physical distinction recognized?Chjoaygame (talk) 03:19, 5 April 2016 (UTC)

This is an "Oh, by the way" statement which needs to be placed in another venue. --Ancheta Wis   (talk | contribs) 18:38, 5 April 2016 (UTC)

I do not know how properly to respond to this comment of Editor Ancheta Wis. If I remove the question-statement to which he refers, I could be accused of removing a talk-page comment of mine that has been replied to. If I remove his reply I could be accused of removing a talk-page statement by another editor. Since my question-statement is a continuation of the start of this section, item (b), I am not quite sure that it is "Oh, by the way." I doubt that his suggestion of placing it elsewhere would be practical. Perhaps Editor Ancheta Wis has a suggestion for this. If he wishes simply to delete either or both posts, I will not complain.Chjoaygame (talk) 20:58, 5 April 2016 (UTC)

Your question about " preparative and observational".. "source to destination"... is answered by pointing out the initial state is a ket and so is the final state. The apparatus is represented by an operator, let's call is A. The inital state(ket) is |phi> and final state(ket) is A|phi>. That's how the "fundamental physical distinction [js] recognized", If I understand your question. In differing notation, A could be |foo><foo|, so your final state, your final ket is A|phi>, in other words |foo><foo|phi>. Notice how the bra slipped into the notation there, and maybe that is what is confusing you about the bra in the final state. Reference: Cohen-Tannoudji page 115+. If that didn't answer the question, try rewording it. GangofOne (talk) 21:23, 5 April 2016 (UTC)

Agree with GangofOne. If we're talking about evolution from state A to state B, you don't need to communicate this fact by having a special notation that makes A and B look superficially different. You can just say "I'm talking about evolution from state A to state B", i.e. use the English language!

There are undoubtedly millions of times that people have written <B|U|A> where A is the initial state and B is the final state. But there are even more times that people have written final states as kets, or used bras and kets when describing some concept where there is no initial or final state, or used density matrices in which both the initial and final states are ket*bra outer products. As just one example, people very often write things like |B> = U|A> where |A> is the initial state, |B> is the final state, and U is the time evolution operator. --Steve (talk) 23:33, 5 April 2016 (UTC)

It seems this section is the one in which to continue. My reply in the next section thus probably belongs here.

Thank you, Editor Sbyrnes, for your response. My reply to it is pretty much the same as my reply to Editor GangofOne's. I will signal this by re-casting the section division (reminding me of re-casting the preparation-observation division). It is a matter here of what is intended by 'initial' and 'final'. The intention that I see in Feynman and in Landau & Lifshitz is that the 'initial' state label refers to the ultimate origin in the oven, furnace, sun, whatever, while the 'final' state label refers to the ultimate destination in the detector. The transition is referred to by Heisenberg's term 'reduction'. The transformations to which I read you as referring by 'initial' and 'final' are not those that I see F and L&L referring to. Your transitions, from ket to ket (bra to bra) do not involve Heisenberg's 'reduction', as I read you.Chjoaygame (talk) 00:39, 6 April 2016 (UTC)

Chjoaygame: Are you saying that Feynman is saying that kets always represent "ultimate origin" and bras always represent "ultimate destination"? If you look again at Feynman you'll find lots and lots of examples where that is not the case. --Steve (talk) 13:16, 6 April 2016 (UTC)

Thank you. That is a good comment. I am just quoting one statement, that I found helpful. I will look again as you advise, for contrary instances. That may take a little time. He doesn't use the words "ultimate origin" or "ultimate destination", but I think his words that I cite have the meaning of "as prepared" and "as observed", or equivalent. I just used "ultimate origin" and "ultimate destination" rhetorically to sharpen the difference from ordinary evolution according to the Schrödinger equation, which takes bras to bras and kets to kets, and goes nowhere near the ends. Quite likely he isn't strictly consistent about it, but I will look. L&L, whom I also cite, routinely don't use bras and kets.Chjoaygame (talk) 16:35, 6 April 2016 (UTC)

Well, you are right. I have had a look. Feynman doesn't actually use the bras as I imagined. At Feynman III, 8—8, he writes:

For consistency we will always use the ket, writing |ψ⟩, to identify a state. (It is, of course an arbitrary choice; we could equally well have chosen to use the bra, ⟨ψ|.)

For Wikipedia, that settles it. My post, that was undone without talk-page response, was as follows:

Interpretation

Most writers do not mention a physical distinction between bras and kets, but a few interpret them as distinguishing initial and final conditions of a phenomenon. The theory is symmetrical between bras and kets, so that it is merely conventional as which of bras or kets is taken as initial or final.

Because people don't actually write bras separately as states, my post about bra-ket/prep-obs fails. Thank you, Editor Sbyrnes for your valuable talk-page response.Chjoaygame (talk) 21:17, 6 April 2016 (UTC)

Copy-and-paste from my previous post in a separate section:

Thank you, Editor GangofOne, for your kind reply.

There is some syntactics and semantics here. I see the transformation of a ket to a ket (or a bra to a bra) as an evolution, within the fully developed preparative device, of an unobserved prepared state, or as an evolution within the observational device before detection. The step from prepared state to observed state is the one that I see Feynman and Landau & Lifshitz as indicating by the bra-to-ket change. Thus I do not see a ket-to-ket (or bra-to-bra) transformation as indicating the prepared-to-observed step. If a pure beam ψ is passed through a prism that will split it into sub-beams, but the sub-beams are not subjected to intervention, then the beam is still regarded by Dirac as in the state ψ. It may be conceptually analyzed as for example ψ = φ₁ + φ₂, and is thereby said to be in a superposition. This unobserved process I see as a ket-to-ket (or bra-to-bra) step. Re-assembly of the original beam is still possible. If an intervention, for example a detector, is put into one of the sub-beams, the superposition is broken because the original beam can no longer be re-assembled. This is said, in Heisenberg's word, to 'reduce' the state to the one (φ₁ or φ₂) that is pure with respect to the prism. I see such a 'reduction' as indicated by the ket-to-bra (or bra-to-ket) step. That is my reading of Feynman and of Landau & Lifshitz (and of a few others). Cohen-Tannoudji et al. do not mention this reading, as I read them. I am strongly driven by Dirac's view that the state has two equally ranking formal symbols, bra and ket. I do not see him as privileging the ket as the state. They are mutually dual. The state space is self-dual.Chjoaygame (talk) 22:18, 5 April 2016 (UTC)

ok. I was talking about an operator operating on a state, giving a new state, such as the propagator operator giving time evolution of state; now I see you are talking about a measurement. I.e. a "reduction", a "wave function colapse (if you will allow such terminology)", and now you are in an eigenstate of the observable you just measured, and you get a probability amplitude, that's what <b|a> is, a complex number, which can lead to a probability, and you are in eigen state |b> with probablity |<b|a>|^2, so what's the confusion? What's the question you are trying to ask? GangofOne (talk) 01:59, 6 April 2016 (UTC)

Thank you for this response. I don't see Feynman and Landau & Lifshitz as confused. I see them as viewing the expression <b|a> as composed from a bra <b| and a ket |a>, with the bra denoting the detected state and the ket the prepared state (or vice versa). You ask what question I am trying to ask. I am asking, if the only kind of state to be symbolized should be of the ket kind, how should the notation distinguish the reduced from the prepared state? I am saying that Feynman and Landau & Lifshitz see the bra as denoting the reduced state when the prepared state state is denoted by the ket. In terms of your comment, in the F—L&L view, one would say that after reduction 'you are in eigen state <b| with probablity |<b|a>|^2'. As I put in my post, many writers ignore this question. But I don't see that the Feynman—Landau–Lifshitz minority viewpoint should be excluded from Wikipedia, provided that it is clearly marked as in my post, a minority viewpoint. It seems, however, that we may be developing here a consensus to exclude that viewpoint.Chjoaygame (talk) 02:33, 6 April 2016 (UTC)

User:GangofOne, you should be aware that physically flawed editor Chjoaygame thinks bras and kets are physically different, that bras are "observed states" and kets are "prepared states" (or vice versa), and his last post is his inability to understand how to form an inner product to reflect this "fact". Then its a puzzle that "many writers ignore this question" and LL and Feynman are the minority of authors who supposedly agree with this "fact" (they don't, but never mind). M∧Ŝc²ħεИ_τlk 07:53, 6 April 2016 (UTC)

I see above that Chjoaygame is trying to create a different view of what his edits were about, changing history, so to speak, a bit. No, the reason you were reverted was not because of the only sane sentence in your edits (quoted in blue above by Chj). The reason was primarily this. It is so full of nonsense that it is nearly impossible to pinpoint exactly the nature of the misconceptions. Does this change in tactics to try to speak sanely have something to do with the present ANI proposal of a topic ban? YohanN7 (talk) 09:18, 7 April 2016 (UTC)

For you benefit, here it is verbatim:

Dirac invented the bra–ket notation specifically for the purposes of quantum mechanics. In quantum mechanics, for a physical reason, the directly physically measurable quantity is the value of the duality pairing ⟨φ|ψ⟩, not the inner product (|φ⟩,|ψ⟩). Moreover, for the same physical reason, the duality map is directly obtained without resort to the inner product, by saying that the duals belong to the same state. Consequently, Dirac used the duality map to establish overlap and orthogonality. He did not use the inner product, because it is not directly physically measurable. The physical reason is that bras and kets refer to distinct aspects of the state, as prepared and as observed. The theory is symmetrical with respect to bras and kets, again for a physical reason. The sorting devices are in general required to obey some sort of Helmholtz reciprocity principle. Physicists tend to follow this procedure without comment. In contrast, von Neumann, being primarily a mathematician, preferred the customary mathematical procedure, of first establishing the inner product, and then using that to construct the duality map. Mathematicians tend to follow this procedure.

For starters,

...the directly physically measurable quantity is the value of the duality pairing ⟨φ|ψ⟩, not the inner product (|φ⟩,|ψ⟩)

is nonsense. They are equal. None is measurable (complex quantities aren't physically observable). Next,

The sorting devices are in general required to obey some sort of Helmholtz reciprocity principle.

is so goofy that it is beyond analysis. Sorting devices? It is meaningless to discuss nonsense with Chjoaygame. Reversion without comment saves time in the long run. YohanN7 (talk) 10:04, 7 April 2016 (UTC)

I agree with YohanN7 that that paragraph above (Dirac & von Neumann) is nonsense. --Steve (talk) 11:54, 7 April 2016 (UTC)

Editor YohanN7 alleges above that "Chjoaygame is trying to create a different view of what his edits were about, changing history, so to speak, a bit." What is going on there? As announced at its beginning, this present talk-page section is not about the edit that Editor YohanN7 has brought in here. As announced at its beginning it is about two quite other edits, thus:

Editors YohanN7 and Maschen have, more or less collusively, undone my edits here and here, without talk page discussion when it would have been appropriate, or mandatory. This was high-handed action by them.

How does that make me "trying to create a different view of what his edits were about, changing history, so to speak, a bit." It seems on the face of it that Editor YohanN7 is practically accusing me of lying. No further comment from me on that.

Now to defend the italicized paragraph above that Editor YohanN7 says is "full of nonsense" and that Editor Sbyrnes agrees is nonsense.

The paragraph that Editor YohanN7 italicized above had a heading which was material to its meaning, not copied here by Editor YohanN7. The key to my meaning has been omitted by Editor YohanN7 for the sake of his attack. The uncopied header was

Dirac and von Neumann approaches

Without the key provided by the header, indeed the subsection that I posted was indeed hard to read, and might well seem nonsense. My post was undone by Editor YohanN7 with the edit summary "Gibberish", without talk page comment. I have not tried again to reconstruct or re-post this edit. I do not however, think it nonsense, as I shall try to indicate here.

Dirac's first edition came out in 1930. The treatise by von Neumann came out in German in 1932. von Neumann was to some extent responding to Dirac. For example, von Neumann was concerned about the validity of Dirac's delta function. The particular aspect that my post referred to was the difference between the ways von Neumann and Dirac handled the inner product or scalar product. One would have little clue to von Neumann's relevance without the omitted header. That omission was thus to be expected to make my post seem nonsense. Dirac at that time was not using the bra-ket notation, of course. He used a different mathematical approach from von Neumann's. Dirac went straight to what he called the scalar product. This was a duality pairing. von Neumann did not use the duality pairing concept, but used instead the kind of inner product used by Schrödinger, without the more abstract or symbolic method of Dirac. I shall now use the notation that Dirac used in 1930. Dirac wrote then that he did not use such forms as φ₁φ₂. Instead he used his pairing such as φψ. The two symbols φ andψ were duals for for the same state, as we would now recognize them as a bra-ket pair. Dirac said he used the duality scheme for physical reasons. I think that meant that one cannot do an experiment with only a preparative device, one needs also a detective device. That is of course standard quantum mechanical thinking. Thus φ₁φ₂ is a juxtaposition of symbols, but does not refer to any experiment, because it consists of two preparative expressions with no detective expression. On the other hand Dirac did use experimentally accessible quantities such as φψ. (Editor YohanN7 has a good point when he seems to say that the quantity actually directly experimentally accessible is the probability density |φψ|². He could have reasonably corrected me on that point.) Thus Dirac did not use von Neumann's usual mathematical procedure of taking an inner product of two vectors from the same space. Dirac used instead a duality pairing of two different kinds of vector, dual to each other. Dirac did not find the the duals from the inner product within the primary vector space, which is the usual mathematical procedure. von Neumann did not proceed in Dirac's way. He used an inner product as Schrödinger did, by integration of a wave function multiplied by its complex conjugate. In short, Dirac and von Neumann used different mathematical procedures. Dirac used his new abstract theory of vector spaces, while von Neumann used Schrödinger wave functions in their native form.

The term 'sorting device' is Dirac's. It refers to such objects as slits, polarizers, choppers, and prisms. These devices have a Helmholtz reciprocity property, that one can turn them back to front and they will perform in the same way. This kind of apparatus is the only one referred to directly in Dirac's initial account. Evidently Dirac thinks it sufficient for the purpose. I do. Its reciprocity property is a major factor in why quantum mechanics uses Hermitian matrices to express such apparatus.

Nowadays, usage varies. I did a small survey and found that physicists tend to speak of the scalar product while mathematicians more often speak of the inner product, though there is wide variation. But Dirac's method of establishing orthogonality through the duality pairing, not the product of two vectors from what he considered to be of the same space, is not used today, and I think hardly even recognized. This doesn't mean Dirac was wrong. It just means that there are various options for dealing with vectors, and he chose one different from the one usually chosen by mathematicians, because he thought his method was more physically meaningful.

My post was not nonsense. To see that, one can check Dirac's first edition. Editor YohanN7 comments above "Reversion without comment saves time in the long run." That's just what he did and I did not try again at to post on that subject. He may hold me up to ridicule, but I did no more than post a reasonable post, evidently on a topic unfamiliar to him. He undid it with the edit summary "Gibberish". I did comment on the talk page that this had happened, but did not proceed further with it.Chjoaygame (talk) 15:23, 7 April 2016 (UTC)

I had in fact seen the section header. I stand by my statement that the paragraph is nonsense. If there is any difference between "inner products", "scalar products", and "duality pairings", I believe that it is entirely superficial, or else I don't understand what it is. (Despite reading Chjoaygame's texts over and over.)

See Riesz representation theorem: Every inner product is by definition a duality pairing (in the sense of "an isomorphism between a vector space and its dual space"), and every duality pairing is by definition an inner product. They're different ways of describing the same concept.

Mathematicians can define the set of irrational numbers via Dedekind cuts or Cauchy sequences or dozens of other definitions but they are the same numbers regardless of how you get there. Likewise, some linear algebra courses start with inner products and derive dual spaces, others start with dual spaces and derive inner products, but at the end everyone is talking about the same thing. --Steve (talk) 16:23, 7 April 2016 (UTC)

@Chjoaygame, please give this a rest. Stop. Enough. --Ancheta Wis   (talk | contribs) 18:15, 7 April 2016 (UTC)

Editor Sbyrnes is entitled to a second statement of his opinion, which he has just posted. Editor Ancheta Wis, it seems, is asking me not to defend myself against a fresh attack, that virtually said I was trying to re-write history, and that said I was talking nonsense.Chjoaygame (talk) 20:33, 7 April 2016 (UTC)

Another standard Chjoaygame tactic is now employed. Write a wall of text, addressing everything else in this world except the issue, which is the nonsense that constitutes the actual Chjoaygame paragraph. Adding a header makes little difference, except that Dirac is coming through even more as an idiot. Dirac wasn't an idiot, but he becomes one when bowdlerized by Chjoaygame. Dirac was a Hero (though not a prophet). I could accept a well-written history section. I don't accept nonsense in the articles. Nor should anyone else. YohanN7 (talk) 09:42, 8 April 2016 (UTC)

Editor Sbyrnes seems to accept that I am talking about different ways of approaching things. But he is unhappy about my undone post, that tried to point out just such a difference between Dirac and von Neumann. He volunteers the Dedekind/Cauchy difference to make his point. That seems to me to indicate that he knows precisely that such is the nature of my concern in my post. His opinion is nevertheless that my post and my just above defence are nonsense. That's his privilege. I think it nevertheless is a virtual concession that my post makes a valid point, however badly expressed it may be. As for Editor YohanN7, he also thinks my post was nonsense. That is his privilege. Exactly why he thinks my post makes Dirac look an idiot, I don't understand. My aim, now abandoned, was to post about what Dirac wrote. I am not yet convinced that Editor YohanN7 has found time to read the part of the 1930 edition that I am referring to. I have previously suffered an occasion on which Editor YohanN7 mistakenly alleged that a verbatim exact quote that I posted from Dirac was a fabrication by me; Editor YohanN7 has not, so far as I recall, confessed that he was wrong about that; I suppose he made his mistake by guessing instead of checking Dirac's original words. At least we can agree, I think, that Dirac was a hero and not an idiot. There is another thing here: Editor YohanN7 has accused me of trying to change history, so to speak. That is about proceedings here, not about physics. His comment seems to indicate that he expects me to pretend he hasn't done that, and ignore it.Chjoaygame (talk) 11:54, 8 April 2016 (UTC)Chjoaygame (talk) 01:51, 9 April 2016 (UTC)

@Chjoaygame, please stop. I thought you got the message when you mentioned your "post fails" ( 21:17, 6 April 2016) in reply to S Byrne. But something is impelling you to keep writing. Are you seeking validation from a community? Why here? Why do you not frequent a history of physics site, instead. This is supposed to be an encyclopedia and not a forum. --Ancheta Wis   (talk | contribs) 03:04, 9 April 2016 (UTC)

User:Ancheta Wis, on top of his POV pushing, Chjoaygame is indeed seeking validation, support, attention, and sympathy from the community, even having the nerve to speak for certain others as if they give it to him. Everyone should stop fueling the fire (here and elsewhere and on the admin page) by ceasing to respond to him.

Sadly, content disputes continue on the admin page. M∧Ŝc²ħεИ_τlk 12:38, 9 April 2016 (UTC)

After I wrote here the words "my post about bra-ket/prep-obs fails", an editor here virtually accused me of lying, literally "trying to create a different view of what his edits were about, changing history, so to speak, a bit." Am I to blame for responding to that?Chjoaygame (talk) 03:24, 9 April 2016 (UTC)

Your next and natural step is to ,when everything else fails, admit your edit was bad and subsequently shut up. but no, you will now escalate this now to the personal level. YohanN7 (talk) 09:28, 9 April 2016 (UTC)

---

For the historical record: Now topic banned editor Chjoaygame (again and again) wrote the last few miles above that Landau and Lifshitz (and naturally a bunch of other Nobel prize winners) support Chjoaygame's views about bras, kets, initial states, prepared states, furnaces, etc. I noticed that just now, since I after a Chjoaygame-free year came back to read this again to see what the hell all was about, and if I could understand it from a new perspective. (Chjoaygame did cost me some time, not only here).

Landau and Lifshitz do not use the bra-ket notation. Not at all. Except, in a section (one of more than a hundred) devoted to a very mathematical subject, Clebsch-Gordan decomposition, where it is appropriate and there are no prepared or final states in miles sight. They presented things in their own way, with very, very little of what occurs very, very much (bras, kets ovens, detectors, etc) in Chjoaygame's posts.

I'd hate to see a newcomer here get the wrong impression about L&L. Hence this post. Only too bad no-one caught it early when the thread was going. I myself didn't at that point waste any time to read and decipher Chjoaygame's post.

Maybe best to archive all this? It can only serve as entertainment or a horror thrill or just plain torture (depending on reader), and is too off-topic to be kept much longer. (Bra-ket notation any-one?) YohanN7 (talk) 14:00, 1 August 2017 (UTC)

Operators

For me, at least, the linear operators section of the article seems have something missing, such as that which is covered in the 'Operators revisited' part of this physics course. --Ancheta Wis   (talk | contribs) 08:55, 1 April 2016 (UTC)

For starters, the applications of the notation seem wider than the history might suggest. As an example, the delta functions are definitely in use in at least 3 widely diverse areas extending beyond mathematics.

Perhaps other editors might suggest more applications.--Ancheta Wis   (talk | contribs) 20:59, 1 April 2016 (UTC)

Could you give specific examples for the topics covered in the notes, and for the applications? I scanned the PDF you link to quickly and they look good,, will have to check in detail. Thanks, M∧Ŝc²ħεИ_τlk 08:08, 2 April 2016 (UTC)

I can offer a stream-of-consciousness list triggered by the pdf:

• The p, or momentum operator, as a derivative
• The derivative of a step function is a delta function
• The use of delta functions as selection operators to simplify expressions
• The use of delta functions and step functions in circuit design (I think at Lincoln Laboratory, MIT)
• The use of bra-ket notation in HTML markup (invented at CERN).
• The modelling of activation potentials as delta functions (in neuroscience)

--Ancheta Wis   (talk | contribs) 13:22, 2 April 2016 (UTC)

The article already gives sufficient context about the p operator as a derivative, and should not heavily weigh on delta functions just because they can be written in bra-ket notation. Applications of step or delta functions are for those articles. This article should just be about the notation as a notation for linear algebra, with its applications (most obviously quantum mechanics). HTML characters are inessential (though a quick mention may not hurt), all this article needs to do is say what characters are used (vertical bar, angular brackets) and link to them. M∧Ŝc²ħεИ_τlk 16:34, 2 April 2016 (UTC)

Thank you. I will now step back, but if you need anything, please ping me. --Ancheta Wis   (talk | contribs) 16:44, 2 April 2016 (UTC)

Culture of physics

In the interest of peace in this community (i.e., the readers of this page), can I ask that we all step back and refrain from editing, until our reflex actions cease to feed the spectacle, which is turning into entertainment, I suppose.

I found a citation in arXiv which addresses "the Culture of physics"; we are witnessing a culture clash, I think. Let's all relax a bit.

See also the following techniques:

• Bottom Line Up Front
• SBAR

-- Ancheta Wis   (talk | contribs) 12:19, 9 April 2016 (UTC)

'Misleading Uses' section's use of 'Abuses'

The later parts of this section seem odd to me. Certainly using one symbol as a label, a variable, and as an operator are confusing to newbies. No argument there. It is much less clear to me that it's actually an abuse of notation. When you use a variable name in a label, what you're doing is labeling by a quantity, and getting that quantity from the variable. So if you put an algebraic expression as a label, you're labeling the state with a quantity, and getting that quantity from the algebraic expression.

What else are you supposed to do - declare an entire new set of labels with an algebraic relation to the old labels and then use those? Use the same old labels and Dirac delta a difference between the label's value (which to be consistent must have a different symbol than the label itself) and the value we want? Either of those would be far more confusing, and I am not at all sure that either would actually solve the problem.

And naming the operator after the variable it extracts seems only natural. It has a hat - it's not the same thing. --100.14.175.181 (talk) 12:10, 6 May 2016 (UTC)

But abuse of notation is a well-introduced term in mathematics, meaning that existing notation is expanded in scope or used in a less strict sense. I think you're taking the word abuse too literally. It doesn't mean that it's reprehensible. It just needs explaining when you do it for the first time, and optimally a rigorous definition of the expanded use case later. In fact it's a necessary technique because existing notation is often insufficient to annotate new ideas.--92.208.44.56 (talk) 12:02, 3 January 2017 (UTC)

Bra-Ket notaion

What is significance of <B|A|C> A(inner part) in bra-ket notation. Also what does |P,Q> means?

Sorry if this is inappropriate place to ask.. But I was unable to find this on wiki and there is no proper explanation for "inner part" also in there. VaibhaW (talk) 01:48, 13 October 2016 (UTC)

Good question. The article should definitely answer the first of these. The second is simpler. It simply means the description of the state (vector) is broken into two parts. I have never seen the combination P, Q before, but others like n, l, m are common. Whatever is in a ket, it is supposed to completely characterize a vector (quantum state). For the first question, A is an operator supposed to act on the vector represented by C, or to the left, acting on the state represented by B, but now via the Hermitean adjoint of A. (Both variants yield the same complex number in the end.) YohanN7 (talk) 08:43, 13 October 2016 (UTC)

Scalar and Vector quantities

I'm new to quantum mechanics and dirac notation and I come to this page to learn. However, dirac notation is varied and taught differently depending on the type of background one comes from. I will be less confused if the vector quantities could have arrows placed over the characters. Thanks to whoever accomplishes this! Also, this my first post on Wikipedia--let me know if I'm doing something wrong! — Preceding unsigned comment added by Murphaid (talk • contribs) 23:24, 1 November 2016 (UTC)

Physicists don't put arrows over quantum states (or kets), even though quantum states are (abstract mathematical) vectors in a (abstract mathematical) vector space. Physicists reserve arrows for 3D vectors in real-world 3D space (or sometimes related situations like real-world 2D space, or 3D reciprocal space, or pseudovectors, etc.). They also reserve the word "vector" similarly, for real-world-3D (or related) situations. I wish there was more consistent notation in the world but we Wikipedia editors are not able to change widespread conventions, and would cause confusion by going our own way. :-/ --Steve (talk) 13:39, 2 November 2016 (UTC)

$=$ or $\doteq$?

The $\doteq$ notation may need further explanation. Coming from mathematics, I wasn't familiar with it and had to search for a while (other Wikipedia articles say it can mean "approximately equal" or "goes toward the limit" which isn't very helpful). It's an important point that a bra/ket is not identical to the actual vector but a coordinate-free representation of it. I like how the German article explains it. Also, is $\doteq$ necessary in $\mathbf {A} \doteq \!\,A_{x}\mathbf {e} _{x}+A_{y}\mathbf {e} _{y}+A_{z}\mathbf {e} _{z}$? This looks like an equality to me, and in the figure an equals sign is used.--92.208.44.56 (talk) 12:40, 3 January 2017 (UTC)

The "doteq" operator is also not explained in the German article. It does not appear in the lists of mathematical symbols, or if it does, it has a different meaning. In mathematics, there is usually no distinction between a vector and its representation by some basis. So apparently, there is a thing like a "representation operator", but unfortunately, nowhere is there a definition for this. It certainly is an inconvenient habit to introduce new and unusual symbols without defining them. Might be due to insufficient understanding of the authors themselves. --95.91.210.84 (talk) 15:02, 18 November 2022 (UTC)

Abstracting matrix/operators (to transformations) and vectors/functions (to states)

The Dirac notation was the unification of Heisenberg's matrix and Schrödinger's differential theory into one cohesive whole.

He's idea was that there were (are) many representations of a state or a transformation. This article does not convey Dirac's abstraction.

The misconception of the article: There is belief that p is the operator -i hbar gradient. This is only true in the coordinate basis. It happens to be p in the momentum basis and is some matrix in a given discrete basis. Dirac saw this. He invented the ket as an abstract state that when projected onto a basis would act like a vector (discrete) or function (continuous). Operators were abstract transformations of the states that when projected onto a basis would act like a matrix (discrete) or a differential operator (continuous). He invented the Dirac delta in order to generalized linear algebra to a continuous basis. (Although the Dirac delta is still looked at as NOT making "sense as a mathematical object" by too many mathematicians. See Delta Function talk page!)

These ideas are all in Dirac's book on Quantum mechanics but I think Feynman may have covered it even better.

"The Feynman Lectures on Physics" Feynman, Leighton, Sands. See Chapters 16 and 20. In particular 16-5.

Sure, he was not rigorous (Was Newton? or Leibniz?). But Dirac was a genius. I believe Dirac deserves more credit than the mathematics community on Wikipedia is willing to give. In many aspects he was ahead of the mathematics community and it took decades for mathematicians to catch up. In some ways mathematics still hasn't... — Preceding unsigned comment added by 131.252.127.172 (talk) 03:31, 6 August 2017 (UTC)

I don't see anywhere in the article that "there is belief that p is the operator -i hbar gradient". Which part of which section do you find misleading? I only see p discussed in the "Spinless position–space wave function" section, which is explicitly discussing the position basis. Again, if something in that section is misleading, can you elaborate?

For the rest of your comment, I don't really disagree with anything you said, but I'm not sure what aspects of the article you're disagreeing with or how you think it could be improved. --Steve (talk) 15:24, 6 August 2017 (UTC)

Two things I find misleading:

1) The pedagogy: Since so many students use Wikipedia to help learn concepts that are new to them, I think it is important that Wikipedia editors do their best to convey the ideas in a way that imparts the intended usage of the concept. Sure, this is all just linear algebra but Dirac generalized the idea into a more general algebra to unify Quantum Mechanics. And its not just any mathematical algebra, this algebra describes the world we live in very accurately.

2) The Machinery: In the section "Spinless position–space wave function", it has the p Psi(r) =def "bra r" p "ket Psi". That is not what Dirac defined. It is better to say -i hbar gradient psi(r) = integral "bra r" p "ket r' " "bra r' " "ket psi" d^3x' where "bra r" p "ket r' " = -i hbar psi(r') delta(r- r'). And the delta function and integral "cancel" each other out. Understanding the need for the delta function and integral is important once students begin applying QM (to ideas such as scattering) where we cannot just define the correct relationships but rather we need to calculate them.

This is done very clearly in Feynmans QM lectures 16-5. I hope the editors understand the difference between what the article states and what is explained by Feynman.

Also, it is traditional (standard?) to use lowercase psi for the position wavefunctions and capital Psi for the position and time wavefunctions. — Preceding unsigned comment added by 131.252.127.172 (talk) 17:42, 6 August 2017 (UTC)

If you're saying that the article is poorly written and hard to read, I agree. But I'm not sure I agree with your specific complaints.

You wrote: "this is all just linear algebra but Dirac generalized the idea into a more general algebra". It seems to me like "this is all just linear algebra" is contradicting "more general algebra". You mean, "more general" than linear algebra? But it's "all just linear algebra". I don't get what you're saying. My opinion is: Bra-ket notation is a notation for "just linear algebra", and everything in quantum mechanics is "just linear algebra", and there is no "more general algebra" in this context.

I can imagine the article saying something to the effect of:

Hey, readers, when you learned linear algebra, you probably learned about finite-dimensional vector spaces etc., and I bet you think of calculus/analysis as a different topic unrelated to linear algebra, with a different professor and a different textbook. But you're wrong! For example, there's an infinite-vector space whose vectors are functions and where differentiation is a linear operator, and lots of other aspects of calculus and analysis can be viewed as examples of concepts in linear algebra! And kudos to Dirac for knowing this, and hence recognizing that wave mechanics and matrix mechanics both fit in the exact same linear-algebra framework!! And bra-ket notation is specifically tailored for this framework, the standard framework which is now universally accepted as the basis for quantum mechanics.

I think saying something like this (but better written of course) is a good idea, as long as it's very short, and it doesn't grow out of control into a whole introduction to quantum mechanics, or into a whole introduction to linear algebra—which would be very much off-topic in this particular article on "bra-ket notation". (The specific thing that I consider on-topic is the fact that a linear-algebra notation can be wielded for talking about calculus.) (BTW I consider that the article has lots of off-topic stuff in it right now and hope to edit it down at some point.)

Next, you talked about the difference between

$p\Psi (r)\equiv <r|p|\Psi >$ vs

$-i\hbar \nabla \psi (r)=\int <r|p|r'><r'|\psi >d^{3}r'$ where $<r|p|r'>=-i\hbar \psi (r')\delta (r-r')$.

I haven't read Dirac and don't know what he defined. Remember, this is an article on "bra-ket notation", not "bra-ket notation as used by Dirac". To me, these two lines look awfully consistent with each other, and I'm still not sure what your complaint is with the first line. My best guess is that you object to the part of the section that says:

It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by

$A\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |A|\Psi \rangle .$

Do you disagree that this is a "customary" thing to do? Or is your complaint something else? --Steve (talk) 18:24, 6 August 2017 (UTC)

Did you even read Feynman? Your response seems like a knee jerk reaction! — Preceding unsigned comment added by 131.252.127.172 (talk) 18:28, 6 August 2017 (UTC)

I have had mathematics students make statements like, "Isn't this all just linear algebra." Or "Quantum Mechanics is just Group Theory." True statements but these students are being mislead by the mathematicians. For example, High school physics uses basic algebra. Just because a student knows basics algebra does not mean he/she understands all of High School physics. The are concepts to be learned!

Read Dirac, Read Feynman. There is a difference. The difference is physics. In math you can define. In physics we need to agree with the experiments. Understanding the concepts are important.

Try to understand. — Preceding unsigned comment added by 131.252.127.172 (talk) 18:43, 6 August 2017 (UTC)

The last psi(r') should have been a gradient! — Preceding unsigned comment added by 131.252.127.172 (talk) 18:47, 6 August 2017 (UTC)

Yes I have read Feynman many times. Yes I agree that the psi should have been a gradient (that was your typo, not mine, for the record. I just copied what you wrote and fixed the notation. But I'm confident that we both understood it correctly, and we both read it too fast to see the missing derivative.) I am trying in good faith to understand what you are saying. That's why I am asking you all these follow-up questions!

I don't disagree that "understanding the concepts is important". The statement that quantum mechanics is "just linear algebra" is false, because you also need to the interpretation, i.e. the relationship between the math and the real world. I do believe that bra-ket notation is a notation for linear algebra, a notation which is especially (but not exclusively) useful when linear algebra is being used for quantum mechanics calculations.

Please remember that this should be an article about bra-ket notation, not an introduction to quantum mechanics, nor an introduction to linear algebra. Aspects of these things might come up in the course of explaining bra-ket notation, but they should be narrowly-tailored towards that purpose. Do you agree with that principle? --Steve (talk) 19:14, 6 August 2017 (UTC)

Yes and no. Isn't one of Wikipedia purposes to help young people become better educated? We need to remember that students use Wikipedia to look up these concepts. Yes, we should make the concepts as simple/straightforward as possible. “Everything should be made as simple as possible, but no simpler.” I think we disagree about where that line is.

Dirac notation is primarily a physics notation, invented to ease physics calculations. Students coming to the page are going to be primarily physics students. Sure, it could be used in linear algebra but that was not its intended purpose and I haven't seen it mentioned in any linear algebra textbook.

"We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations." - Richard P. Feynman 131.252.127.172 (talk) 21:13, 6 August 2017 (UTC)

I just remembered a book that I think is a good compromise in the treatment of Dirac notation.

"Explorations in Mathematical Physics" Don Koks p. 41-71

He starts with the notion of the inner product of linear algebra and only mentions how it is applied to QM in last few pages at the end of the chapter. Most everything is there. Quite a bit would have to be cut out in order to be more concise. I think is shows the machinery without getting too much into its QM interpretations. Although, I still believe the best way to learn Dirac notation is to use it in the context of QM, I am willing to try to do so without it. If anyone knows of any other books that do something similar, then I am willing to read them. — Preceding unsigned comment added by 131.252.127.172 (talk) 00:32, 7 August 2017 (UTC)

You complained above about the "Spinless position–space wave function", I didn't understand your complaint and asked follow-up questions, and you never answered them. Can you answer them please? Or did you decide it was OK after all?

I am strongly in favor of "helping young people become better educated". (And middle-aged and old people too!) But there is a big difference between an effective encyclopedia article and an effective textbook. (Wikipedia is not a textbook. See, it's right here: WP:NOTTEXTBOOK. :-D) They are complementary, both should exist, they serve different purposes. Most people look at a wikipedia article on X because they don't understand X and want to understand X better as quickly as possible. So everything in the article should serve the purpose of making people experts on X. We do in fact have an article Introduction to quantum mechanics (as well as Quantum mechanics, Wavefunction, Quantum state, Matrix mechanics, etc.) I expect that most people who want an introduction to quantum mechanics will successfully find their way to the Introduction to quantum mechanics article, and not get stuck here. If you disagree, we can talk about how to clarify the links / navigation. --Steve (talk) 01:32, 7 August 2017 (UTC)

Why are you so upset? I am only trying to help improve the article. I have not made any edits to the article. I have tried my best to be understanding of your point of view. I realize I might not be a clever young Wikipedia editor (I am new to this...) but I am trying to compromise with you. It seems like you don't even read the books I cite. If you had, at least you haven't had much time ponder their ideas.

I had two issues. One was how the article lacks Dirac's motivation and treats his notation as just another way to do linear algebra. I felt like that was a bit of an oversimplification (We might disagree here. I understand that.). I gave you examples of mathematics students make statements like, "Isn't this all just linear algebra." Or "Quantum Mechanics is just Group Theory." The second issue was the lack of examples of how the machinery is used. Instead of defining the answer, it would be better for students know how the machinery is used. This is important when the answers are not known like when they are calculating scattering amplitudes. I found a book that shows the machinery (without reference to QM!) so students can learn how it works. Please, don't just try to come back with some quick witty retort with immature emojis. Be respectful, try to read the books. Think about the ideas. Think about the students that read these articles. How can the article be improved?

131.252.127.172 (talk) 02:26, 7 August 2017 (UTC)

I'm not upset at all. The :-D emoticon was a way for me to show you that I am not upset. It's a big sincere smile on my face!

I've probably read 15 or 20 intro quantum mechanics textbooks in my life. Feynman is one of them, but not Koks and not Dirac. I will endeavor to read them, but it's time-consuming, and I won't get to it any time soon. You are obviously welcome to edit the article yourself. It definitely has room for improvement. --Steve (talk) 03:33, 7 August 2017 (UTC)

Meaning of "vector" in physics

I wrote: "In mathematics, the term "vector" is used to refer generally to any element of any vector space. In physics, however, the term "vector" is much more specific: "Vector" then refers almost exclusively to quantities like displacement or velocity, which have three components that relate directly to the three dimensions of the real world." YohanN7 edited to say "In introductory physics", with the comment "usage far from universal". Well, I had thought it was universal or at least near-universal. What are the counterexamples?

Even if there are counterexamples that I haven't thought of (which is entirely possible), I more confidently object to the term "introductory". For example this usage is rampant in quantum field theory (vector particles, vector fields, as opposed to pseudovector or tensor etc.) and other advanced QM courses (spherical tensor operators, Wigner-Eckart theorem etc.), GR, etc. etc. Indeed, I didn't really fully appreciate this point until taking graduate physics courses. (OK sure, if I had read Feynman more carefully, I would have appreciated it sooner!) So if it's not universal, I vote for saying "often in physics" or "generally in physics" but I don't think we should say "in introductory physics". :-D --Steve (talk) 12:46, 14 September 2017 (UTC)

Introductory now sounds wrong to me too. I'll edit that per suggestion. What I had in mind is something like this: At some point in physics education the mathematical definition of a vector space is unavoidable. This typically comes (if no sooner) on first encounter of quantum mechanics. Then vectors are no longer only arrows in 3d. A typical example is Shankar's QM book. Suddenly vectors can be matrices and functions (even operators). I dare say that all these do not universally go under the label ket. On the contrary, I think ket is more or less reserved for state vectors in QM, and that the notion of vector is that of the mathematician after this first encounter with the general definition. YohanN7 (talk) 13:27, 14 September 2017 (UTC)

I reverted to the previous formulation. But I am not entirely happy about it. The main point is that "ket" would be a catch-all. For one thing, it presupposes bra-ket notation. This notation is not free from objections and, e.g. Weinberg refuses to use it. YohanN7 (talk) 13:47, 14 September 2017 (UTC)

OK, that makes sense, I tried changing it again, what do you think? --Steve (talk) 16:38, 14 September 2017 (UTC)

Looks fine. YohanN7 (talk) 13:25, 15 September 2017 (UTC)

Pronunciation/reading?

How do you read <a|b>? "Bracket a b", "a times b", something else? --Sigmundur (talk) 08:39, 8 March 2018 (UTC)

When the symbol inside the ket is not the name of a vector per se

The lead says "A ket looks like " | v ⟩ ". Mathematically it denotes a vector, v , "

... from which one would expect that the symbol found inside a ket would be the name of a vector. That is, take that symbol outside the ket, stick an arrow on top (or style it bold), and you have a vector in conventional notation.

However, evidently there are additional ket conventions. One is apparently to place an integer inside the ket. This is obviously not the name of a vector per se. Apparently it indicates one of the orthonormal basis vectors. So |2> indicates the second basis vector, perhaps j-hat (or e_y).

An example; https://quantum.phys.cmu.edu/CQT/chaps/cqt03.pdf page 32

Taking this further, some authors apparently may place a variable inside the ket, such as | n >, to indicate the nth basis vector, perhaps useful in a summation. But that is indistinguishable syntactically from the normal ket usage, and mistakenly suggests that n might be a vector.

At any rate, because a reader may well be visiting the page to try to interpret the bra-ket notation they've newly encountered written somewhere, it would be helpful if variants such as the one I just described could be discussed in the article in an orderly fashion. Gwideman (talk) 15:54, 22 February 2021 (UTC)

I strongly believe this would be overkill, but, of course, if somebody proposed redress here, on this page, before editing the page, its adequacy or defects could be vetted. IMO, a sensible, perceptive, well-meaning reader could not be confused, and could adapt the notation as circumstances dictated. This article is not a tutorial, and is meant to summarize/refresh basic facts one has already been taught at some level, e.g. in a course that then bears the onus of miscommunication. There is an extreme and present danger of burdening the article with lots of fussing distractions making it unreadable to all but the dedicated pedant reader, thereby furthering the degradation process... Cuzkatzimhut (talk)

Thanks for responding, but I am very puzzled by your remarks. I do not understand how, in an article titled "Bra-ket notation", the subject of common variations in the syntax of the notation could possibly be overkill. Your second sentence ("IMO...") I do not understand at all -- are you referring to the page as currently, or as potentially revised? And what does "well meaning" have to do with the current discussion? Sure, articles on Wikipedia are not tutorials. They are pages in an encyclopedia. But then you go on to say it's meant to review facts that the reader is presumed to have learned previously. No, it's a page in an encyclopedia. Wikipedia:Wikipedia is an encyclopedia You conjure up some previous course in which there was miscommunication -- apropos of what? I might concur that there's a risk of articles becoming cluttered with distractions. Right now it's chock full of applications and examples in which Bra-ket notation is used, related only secondarily to the notation per se (but not unwelcome), yet somehow you are resistant to the idea of bolstering the description of the notation itself, which would actually improve the signal-to-noise ratio. Gwideman (talk) 23:19, 25 February 2021 (UTC)

I'm never resistant to the idea of improving the article! How could I be? I would like to be resistant to its routine abuses: of course, I've been watching with horror its runaway descent into mush, by "bold" hit-or-miss "improvers". Notation like $|n\rangle$ is self-explanatory, since what goes in the ket, most of the time, is the eigenvalue of the operator specifying the basis, like position, momentum, etc, as clarified in section 3.1, just about the only meaningful section in the article; in this case, the number operator N, s.t. $N|n\rangle =n|n\rangle$. If you wished to think of it as the cardinal index of a basis vector, of course it would tag the n+1-th vector since the counting starts from n = 0. In any case, the power of kets is maintaining flexibility and abstraction, and not merely rewriting vectors, for which there exist excellent notations. Dirac introduced them precisely so detailed pedantic questions about them could be bypassed, once their formal properties are completely specified. Physics has never looked back. Cuzkatzimhut (talk) 01:34, 26 February 2021 (UTC)

Well, this discussion is related to the heading explaining that "the mathematicians" do it all so differently. It is true that mathematicians rather redefine the scope of existing notation, so that it retains its familiar meaning vis a vis the basic stuff (e.g. linear algebra 101) but also works in the more advanced stuff (linear functional analysis 101), depending on the spaces specified. But most mathematicians who do functional analysis (and adjacent such as integral eqns, PDEs etc) are quite conversant with Dirac notation as well and may even use it when the mood strikes. A distinct drawback of conventional notation is that sometimes the index is the salient part and the variable glyph itself an unimportant dummy - Dirac allows you, effectively, to insert the index inside the | > moreover, Dirac notation has the Riesz-Frechet theorem and the notion of an observation as being an operator baked right into it. One can see why physicists like this very much and wax (somewhat too) lyrically about its "power". Others may prefer to establish everything in conventional notation - just to make sure there is no sleight of hand! 2A01:CB0C:CD:D800:C5C0:901B:9EDE:87D8 (talk) 12:44, 17 March 2021 (UTC)

Outer product

Interesting the way |ψ><ψ| has a crossing X like ⊗ in the middle. Starple (talk) 23:10, 26 March 2021 (UTC)

Removed Request For Simplified Lead In

Removed request after providing the information.

Hopefully the simplification is acceptable because the topic loses something important if any further condensing is done. Astrojed (talk) 03:07, 5 February 2023 (UTC)

Provided a lead section and removed the notice of it missing.

This is a highly technical topic used in some quantum mechanics. A general practitioner of science may understand it if somewhat knowledgeable about quantum mechanics, complex numbers, vector products, and matrix algebra. A well qualified editor might make the improvements that are requested. Astrojed (talk) 08:46, 11 February 2023 (UTC)