# Talk:Bra–ket notation

## More embarresingly + trivial

In section Notation used by mathematicians, it was quite amusing/odd to read:

"Moreover (and more embarrassingly, although this is essentially trivial), mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one..."

I don't remember reading that from recently when the page was re-written, because that section was left alone and almost everything else it was re-written... Anyway is there any reason for embarrassment? 18:23, 6 June 2012 (UTC)

## New templates introduced

See {{langle}} and {{rangle}}. Hope people like the change. 16:44, 12 June 2012 (UTC)

For now, in spite of extensive labour over these templates, there seems to be a slight bug in them. I'll remove them later, then maybe reinstate them. Its so effing annoying after all that time and energy...... =( 23:53, 12 June 2012 (UTC)
The templates have been reverted/tweaked back to the older version that actually worked. 08:59, 13 June 2012 (UTC)

## Pronunciation

How do you say <A|B> when speaking? Is it standardized, or are there both lengthy and brief ways to read it? This should be included in the lede, just like the pronunciation of bra and ket. 130.60.6.54 (talk) 20:25, 10 June 2013 (UTC)

The last sentence in the lede actually already gives the longwinded way of saying it: "the probability amplitude for the state A to collapse into the state B." A lecturer who's just introduced bra-ket notation will describe every inner product this way until everyone gets used to it.
Later on, the shorthand you'll hear, especially when people are talking as they write, is simply "A B". Laura Scudder | talk 20:59, 10 June 2013 (UTC)
These two extremes of pronunciation (I agree they exist) make the formulas in the "Unit operator" section for example either difficult to say or difficult to understand.
Is there a concise yet clear standard form for purely spoken usage, when there is not a shared visual formula to point to? For example, how would one say (on the phone, to another person familiar with quantum mechanics notation) the expression ${\displaystyle \langle v|e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|w\rangle }$? For the case of one careful physicist speaking to another physicist, or giving a formal spoken talk to physicists, I would expect something like "The amplitude of ${\displaystyle v}$ from ${\displaystyle e_{i}}$, times the amplitude of ${\displaystyle e_{i}}$ from ${\displaystyle e_{j}}$, times the amplitude of ${\displaystyle e_{j}}$ from ${\displaystyle w}$." Is there standard terminology at this level (e.g. "the amplitude of ${\displaystyle v}$ from ${\displaystyle w}$", or "the amplitude from ${\displaystyle w}$ to ${\displaystyle v}$", or "the amplitude of ${\displaystyle v}$ given ${\displaystyle w}$", or "the amplitude for ${\displaystyle v}$ from ${\displaystyle w}$", etc.) that is concise yet clear, or, as the evidence below suggests, does every physicist "roll their own"?
The same question goes for standalone bra and ket vectors appearing in formulas.
Feynman speaks of "the amplitude for [some event]", but I do not have an example of him saying "${\displaystyle \langle A|B\rangle }$". Oxford professor Binney in his youtube video (lecture 2) at 9:30 says that "${\displaystyle \langle f|}$" should be read as "bra f". He often says "ket ${\displaystyle \psi }$" similarly. However, he is less consistent for the full "${\displaystyle \langle j|i\rangle }$", which he often reads as "j of i" (10:56, 11:04, 11:07, 15:26), presumably due to his emphasis (e.g. 8:00-10:00) that the bra is a function acting on the ket, which is slightly different from this article's expositional approach. He also uses "the probability amplitude of j given i" (38:40). Of course, he is always also using a visual reference, so the spoken form does not need to be so clear, and his audience is non-expert, so he is expressing things in multiple ways to try to reinforce the meaning. In later lectures he gravitates towards the simple form you mention, "j i", relying on the accompanying written form for clarity. Stanford professor Susskind on the other hand reads it backwards as "i j" (video, 1:17:50, 1:18:00) or, also in later lectures, "the inner product of i with j".
From these sources, it is not clear that there is any standardized way of speaking the full bra-ket form. 130.60.6.54 (talk) 17:40, 11 June 2013 (UTC)

## Vector spaces over ℂ

In the section introducing vector spaces over ℂ I amended the phrase 'though the coordinates and vectors are now all complex-valued' to read instead simply 'though the coordinates are now all complex-valued'. Although I see what the original author meant, it's not necessary for the vectors in a vector space over ℂ to in any way 'contain' complex numbers - it's simply necessary that the vectors (whatever they are) to be able to be scaled by elements in ℂ. For example, you can give ℝ2 the structure of a vector space over ℂ [when you scale (x,y) by a+ib you end up with (xa-yb,xb+ya)] but in this case the vectors (elements of ℝ2) don't 'contain' complex numbers. Felix116 (talk) 21:31, 14 October 2013 (UTC)

## Bracket characters

I reluctantly reverted the edit of User:131.231.242.221 today, who complained: "The characters "〈" and "〉" appear on my computer as rectangles with little numbers in. Is there an alternative?" Indeed, older systems lack one or both pairs of html characters. Every few months, there a complaint of this sort somewhere or elsewhere in WP. There is also the alternate set of ⟨ψ|z⟩, I strongly suspect equally invisible/missing-character'd to him/her. Supplanting with <.> (lessthan-greater than) as a workaround gives the text an amateur look, and TeXing in text, ${\displaystyle \langle .\rangle }$ gives that distinctly ransom note look in those very systems. I have no good ideas, not even on how to advise this user to reset his/her preferrences to render formulas better. Cuzkatzimhut (talk) 11:17, 20 October 2013 (UTC)

I wrote on the IP's talk page here asking if some alternatives may work: the template:bra, template:ket, template:bra-ket, or just the angular brackets in the lower editing toolbar (the one just below the edit window): ⟨ ⟩ . M∧Ŝc2ħεИτlk 14:29, 20 October 2013 (UTC)
Indeed, the templates use these characters, most likely the ones associated with z and ψ above. I can't speak for him/her, but, in fact, it is these brackets that are missing-boxed in early systems, about half a dozen years old. (The characters he actually complained about, appeared more robust to me. Check these explicit HTML-character numbered 〈 and 〉, & # 9 0 0 1 ; and & # 9 0 0 2 ; in the edit source.) Ultimately, such complaints will cease with computers modernizing, but I fear it is too much to punish everybody with in-text TeX, just to assist a few readers missing characters. In my ignorance, I wonder if there were a diagnostic of the "do you see this well" type associated with one setting their preferences−−assuming they were registered to do that. Cuzkatzimhut (talk) 15:54, 20 October 2013 (UTC)

If it helps I saw bra-ket symbol issue viewing the page with google chrome browser, but not with firefox or mirosoft internet explorer. — Preceding unsigned comment added by 67.249.102.221 (talk) 21:30, 16 November 2013 (UTC)

## Background is wrong.

This section has a number of errors (or simplifications to the point of excessive inaccuracy). A R3 point is NOT a vector. You do not add points. You do not multiply points. You don't take their dot (inner) products. A vector might be a entity with only magnitude and direction, but it is common in physics to also locate the vector in 2 or 3-d space. (ie. there are two distinct technical meanings for the term, (I'm not sure if vector fields can be used as equivalent to a located vector?)). Only vectors originating from the origin can be described with three scalars in 3-d space (n scalars in n-space). That is only vectors which only have magnitude and direction (not position) are the subject of this section.

ALSO: The coordinates are NOT "the number of basis vectors in each direction". Who wrote this??!!216.96.79.61 (talk) 22:19, 21 January 2014 (UTC)

The initial paragraphs of this section need a rewrite.

## Wrong notation

kets and the vectors that represent kets are not equal! A ket is related to a vector that represents the ket, like a poster of a star is related to the star. A "coordinate vector" needs the reference to a basis, whereas the ket does not. See Modern Quantum Mechanics Revised Revision, Sakurai, p. 20. Therefore the equal sing between a ket and a vector needs to be changed to another symbol since the mathematical objects on both sides are not equal. For example instead of ${\displaystyle |B\rangle ={\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}}$ something like ${\displaystyle |B\rangle {\overset {.}{=}}{\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}}$ should be used (with ${\displaystyle {\overset {.}{=}}}$ like Sakurai for "is represented by" or ${\displaystyle \leftrightarrow }$ like Shankar) --Biggerj1 (talk) 12:27, 2 February 2014 (UTC)

I agree. This kind of shorthand should be avoided in any reference material for two reasons: it requires clarification before use, and it potentially leads a wide audience to believe that they are the same thing. This kind of misidentification is quite common in linear algebra (vectors, tensors etc.), but unfortunately hinders a more natural and complete understanding. Cleaning this up would be a definite improvement. —Quondum 17:35, 2 February 2014 (UTC)
I tried to present the problem. --Biggerj1 (talk) 19:20, 17 February 2014 (UTC)