Talk:Angular momentum operator

Useful links

I found a couple useful links which are helpful as additional resources http://quantummechanics.ucsd.edu/ph130a/130_notes/node216.html http://www.ifi.unicamp.br/~maplima/fi001/2007/aula21b.pdf —Preceding unsigned comment added by 76.65.22.185 (talk) 22:18, 15 April 2010 (UTC)

Propose page move

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

Not moved. I'm not sure that there is consensus. There is discussions about rewriting and merging in material. So maybe this discussion should way for the article changes to happen. If in rewriting, the article was split if that was appropriate, then this discussion would not be needed. So let the experts have more time to decide on the best approach. Vegaswikian (talk) 23:43, 28 January 2012 (UTC)

Angular momentum operator → Angular momentum (quantum mechanics) – Relisted, awaiting comments from the WT:PHYS crowd. Favonian (talk) 13:38, 21 January 2012 (UTC) I propose changing the page name from Angular momentum operator to Angular momentum (quantum mechanics) because it more clearly and fully and accurately describes the scope of the article. Any thoughts? --Steve (talk) 15:24, 14 January 2012 (UTC)

• Just dropped a note about this discussion at WT:PHYS – hopefully that will generate some comments. Jenks24 (talk) 06:30, 21 January 2012 (UTC)

I think so too. -- F = q(E + v × B) 07:38, 21 January 2012 (UTC)

• Agree. It used to be all about the operator, but someone apparently expanded it and then the name "angular momentum operator" could not hold all this content. However it needs a rewrite after that: the structure is a mess now, and many important aspects are missing (for example, raising and lowering operators and the coupling of several angular momenta).--Netheril96 (talk) 11:14, 21 January 2012 (UTC)

• Comment That was me. Maybe you're thinking, "First Steve edited the article to be on a different topic, and now he wants to change the title to match his new topic." Well, I don't mean to be deceptive. My thought process was (1) There really should be an overview article on angular momentum in quantum mechanics, (2) All the content I added truly is intimately related to the angular momentum operator (that content was partly written by me, partly copied out of the main angular momentum article) (3) More generally, I can't think of any topic related to angular momentum in quantum mechanics that is not equally tightly related to the angular momentum operator. (On the other hand, orbital angular momentum operator could legitimately be a separate article, e.g. giving the expression in spherical coordinates.) It's true, this article needs expansion, not only the topics you mentioned but also everything in the "See Also" section. Actually I'm working right now on expanding this article. Luckily brief summaries and links to other articles are sufficient for many topics. :-) --Steve (talk) 18:10, 21 January 2012 (UTC)

• Comment: You might consider merging stuff from vector model of the atom.-- F = q(E + v × B) 22:30, 21 January 2012 (UTC)

• Disagree: I would disagree with the move as the article is about the angular momentum operator, angular momentum operator does describe the scope of the article. Every section discusses the angular momentum operator and related operators. IRWolfie- (talk) 11:36, 25 January 2012 (UTC)

• Comment: The question is not whether "angular momentum operator" is horribly bad, only whether it is better or worse or equal to "angular momentum (quantum mechanics)". (For my part, I don't think that "angular momentum operator" is particularly "bad" or "wrong", but I do think it's not as good/clear/accurate as "angular momentum (quantum mechanics)".) Anyway, can you say more directly how you would compare these two options? --Steve (talk) 20:12, 25 January 2012 (UTC)

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Re-directs

FYI: I just created two re-directs to this article:

• Angular momentum in quantum mechanics
• Angular momentum (quantum mechanics)

which are surely reasonably frequent phrases for the contents of this article. AM in QM is ultimately ruled by the operator. F = q(E+v×B) ⇄ ∑_ic_i 09:53, 28 March 2012 (UTC)

Commutation relations involving vector magnitude

The statement in this section of the article regarding the derivation of L² commuting with L's individual components L_x, L_y and L_z is somewhat disingenuous. It states that the derivation starts from the commutation relations [ L_ℓ, L_m ] given in the preceding section ("Commutation relations between components"). While true, the derivation might puzzle a reader unaware of the following commutator identity:

$[{\hat {A}},{\hat {B}}{\hat {C}}]={\hat {B}}[{\hat {A}},{\hat {C}}]\ +\ [{\hat {A}},{\hat {B}}]{\hat {C}}$

which appears in the article on commutation relations and is easily verified. It is then straightforward, if tedious, to work out that, for example,

${\begin{array}{lcl}[L^{2},L_{x}]&=&L^{2}L_{x}+L_{x}L^{2}\\&=&[L_{x}^{2},L_{x}]\ +\ [L_{y}^{2},L_{x}]\ +\ [L_{z}^{2},L_{x}]\\&=&L_{y}[L_{y},L_{x}]\ +\ [L_{y},L_{x}]L_{y}\ +\ L_{z}[L_{z},L_{x}]\ +\ [L_{z},L_{x}]L_{z}\\&=&i\hbar (-L_{y}L_{z}-L_{z}L_{y}+L_{z}L_{y}+L_{y}L_{z})\\&=&0\end{array}}$

An alternative derivation begins with the anticommutation relations:

$\{L_{l},L_{m}\}\ \equiv \ L_{l}L_{m}+L_{m}L_{l}\ =\ {\frac {1}{2}}\hbar ^{2}\delta _{lm}.$

(1)

Equality follows from the expression of L 's components in base eigenkets $|L_{z};\ \pm \rangle$ (denoted $|\pm \rangle$ for brevity) in the usual way :

${\begin{array}{lcl}L_{x}\ =\ {\frac {\hbar }{2}}[\ \ \ \,(|+\rangle \langle -|)\ +\ \ (|-\rangle \langle +|)]\\L_{y}\ =\ {\frac {\hbar }{2}}[-i(|+\rangle \langle -|)\ +\ i(|-\rangle \langle +|)]\\L_{z}\ =\ {\frac {\hbar }{2}}[-i(|+\rangle \langle +|)\ +\ i(|-\rangle \langle -|)]\end{array}}$

(2)

Then from (1) we have

$\{L_{l},L_{l}\}=L_{l}^{2}+L_{l}^{2}=2L_{l}^{2}$

(3)

which leads at once to $L^{2}={\frac {3}{4}}\hbar ^{2}$ by virtue of its definition $L^{2}\ \triangleq \ L_{x}^{2}+L_{y}^{2}+L_{z}^{2}$, whence commutation with its components is immediate.

The use of braces $\{,\}$ for the Poisson bracket in the article is perhaps an unfortunate choice of notation since this classical concept corresponds in quantum mechanics to the commutator, and thus square brackets $[,]$ are preferred by many authors of physics texts, mathematicians tending to use parentheses $(,)$. The article on commutation relations uses braces for the anticommutator in the section on Ring theory.

It is probably worthwhile to add definition (1) to the preceding section in the article because of its central importance in deriving the Dirac equation, together with the reference to Sakurai and Napolitano given above. It is then probably unnecessary to include (2). Finally, (3) can be added to either section along with the one sentence derivation of the commutation of L² with its components added to the present section. With the introduction of the anticommutator into the article, the notation for the Poisson bracket should be changed to square brackets to avoid confusing the anticommutator with it.

I added the commutation-relation derivation. I changed the text to indicate that it is one possible proof. I didn't mean to suggest that it was the only possible proof!

The anticommutator formula $\{L_{l},L_{m}\}={\frac {1}{2}}\hbar ^{2}\delta _{lm}$ can't possibly be true in general ... for example, for a 1s electron, $L_{x}|1s\rangle =L_{y}|1s\rangle =L_{z}|1s\rangle =0$. Therefore, $\{L_{x},L_{x}\}|1s\rangle =0\neq {\frac {1}{2}}\hbar ^{2}|1s\rangle$. Right?? I think that anticommutator formula is specific to a spin-1/2 system, whereas the commutator formulas (and every other formula in the whole article right now) are general operator equalities that are true regardless of what kind of system they are operating on. (But sorry if I am being stupid and confused!)

I have no opinion about what notation is best for poisson bracket. Just tell me what notation to use, and I will change it. Or you can change it yourself. --Steve (talk) 15:40, 27 January 2014 (UTC)

Sorry, I should have mentioned that the anticommutator proof is good only for spin 1/2 systems, and Sakurai and Napolitano make that quite clear, pointing out that "for spins higher than 1/2, $S^{2}$ is no longer a multiple of the identity operator.". They're treating only spin, and so I shouldn't be using L above in the first place. Mea culpa. Still, I think it worthwhile to include the spin proof as these systems are studied extensively at the undergraduate level, providing much of the curriculum content to illustrate fundamental QM principles. It could be added, for example, at the bottom of the section where you give the commutation relations for spin and total angular momentum.

With your blessing I will change the Poisson bracket notation, and also add a reference to Goldstein et al who present a terrific exposition of the subject (IMHO) beginning p. 388.Edgeorge (talk) 18:52, 28 January 2014 (UTC)Duncan

When you write $|+\rangle$ and $|-\rangle$ you are presupposing that there are two states that are spin-1/2 (i.e. that are eigenvalues of S^2) and that are also eigenvalues of S_z. Which means that you are already presupposing that S^2 and S_z commute (at least in the spin-1/2 space). So it seems to me, that it is inappropriate to use this kind of argument as a basis for proving that S^2 and S_z commute.

Of course topics like Spin-½ and Pauli matrices are important ... it might be worth saying more about that somewhere. But in this article I think it should be put into the larger context, i.e. discussing not only the spin-1/2 hilbert space, but also spin-0, spin-1, etc.

(Not to mention that real systems may be in more complicated spaces, like a molecule can be in "the direct sum of a spin-2 and a spin-4 space" or an electron would be in "the direct product of a spin-1/2 space and the space of complex functions on R^3"...) :-D --Steve (talk) 20:47, 28 January 2014 (UTC)

Steve - I've changed the Poisson bracket notation, added a little context and appropriate reference in both sections. Take a look, and let me know what you think. Edgeorge (talk) 00:42, 29 January 2014 (UTC)Duncan

The basis kets $|\pm \rangle$ for $S_{z}$ are introduced to describe the experimental results of Stern-Gerlach experiments on spin 1/2 particles. The spin z-component being an observable is represented by the operator $S_{z}$, the basis is chosen to match the vector space implied by the results, and $S_{z}$'s action on the basis adjusted to reflect the measured values $\pm {\frac {\hbar }{2}}$ after invoking completeness and introducing the identity operator. I realize you will of course be familiar with this, but I repeat it here to emphasize that the sequence of reasoning begins with the experimental measurements of the individual components of spin, and that is precisely because they do not commute. Next, analysis reveals how to construct the x- and y-components in terms of the z basis kets, following which it is then straightforward to derive the commutation relationships

$[S_{l},S_{m}]=i\hbar \varepsilon _{lmn}S_{n}$,

and the anticommutator ones (1) above. It is at this point that $S^{2}$ is introduced, and indeed it is a triviality to show it commutes with each of its components. This, however, depends on the availability of either of the two kinds of relations (take your pick). Finally, once we have two commuting operators, it is straightforward to show they do indeed share one set of eigenstates (ignoring questions of degeneracy), albeit with not necessarily the same eigenvalues. And, indeed, analysing the eigenvalue spectrum for spin particles using the raising and lowering operators (also termed ladder operators) $S_{\pm }$, intrinsic spin s is associated with $S^{2}$ and components such as $S_{\pm }$ with $s,s-1,...-s+1,-s$ which, for spin 1/2 particles, reduces to $\pm {\frac {1}{2}}$ .

So, in sum, I don't find the argument circular. To your other remark, I understand your point about spin 1/2, Pauli matrices etc. not being especially relevant to the article per se. As I noted earlier, spin 1/2 systems play a crucial pedagogical role and the anticommutator is a key element of the Dirac equation. As such, I thought this would provide both a helpful example as well as referring to the Dirac equation article. However, it's your call - Cheers! Oh, and by the way, I do think you've written a very good encyclopedia article, taking a single topic and touching on its many applications and properties, especially the concepts of generator, Lie groups, conservation, coupling and, of course, the ladder operator derivation of the quantization rules. And, by way of which, now that it occurs to me, do you think it would be helpful to the reader to introduce a short proof in the section "Derivation using ladder operators" that $J^{2}$ and $J_{z}$ have the same eigenbasis, and explicitly show the action that the $J_{\pm }$ operators have on the basis states $|j,m\rangle$ showing the raising or lowering by $\hbar$? E.g.

$\langle j,m'|J_{+}|j,m\rangle ={\sqrt {j(j+1)-m(m+1)}}\hbar \delta _{m',m+1}$.

Townsend has a most accessible treatment of this on pp. 70, 77, resp. Sakurai and Napolitano treat the simultaneous eigenbasis description slightly more abstractly on p.29. Edgeorge (talk) 00:08, 30 January 2014 (UTC) Duncan

About ladder operators -- I think you're right, I added more details....

About poisson brackets -- looks good thanks :-D

About spin-1/2: I can't follow all the details of what you wrote (too sleepy) but I'm open-minded that you're correct. I still think that this sort of discussion isn't going to be very helpful here, unlike spin (physics) or Spin-½ where it's essential. There is one proof of [J^2, Jz] = 0 from a more basic starting point, and I don't think a second proof is really necessary. Anyway you're welcome to edit the article, don't let me discourage you ..... :-D --Steve (talk) 04:13, 30 January 2014 (UTC)

Re: spin-1/2, after considering for a few days what you say Steve regarding adding additional spin-1/2 related material, I've come to agree with your point of view. The Spin-½ article seems to me a little thin, so what I might do is go into it, add the derivation I cited above, then install a short sentence referencing that in your article. I might also put a reference to it in the spin (physics) article where it is perhaps more understandably absent.Edgeorge (talk) 19:30, 5 February 2014 (UTC) Duncan

Comment on one of the references in Further reading

McMahon's book is the Cole's notes of QM for undergraduates. It's claim to fame is the large number of worked examples and, indeed, these will be quite helpful for those encountering QM for the first time. The book covers the standard introductory curriculum with some advanced undergraduate topics touched on. I am just finishing up reading the second edition, and can reliably report that it is one of the most poorly edited scientific texts I have ever encountered. While none of the material McMahon has written is in any way incorrect, the book is filled with typographical errors and omissions. At times this makes for almost impenetrably obtuse reading unless you otherwise know your way around the subject. Ten examples chosen at random:

   p. 204, fourth equality at the top: '+' sign missing between two matrices;
   p. 204 after par. (c): factor of √½ on RHS is missing;
   p. 164.  Fourth equation from bottom of page  "1 = <a2|a1> = (a - a*) ..." should read "1 = <a2|a1> = (a* - a*) ..."  (The asterisk is missing on the first 'a'.)
   p. 114, Fig. 407: Axes and origin are not labelled;
   p. 114, LHS of first equation: this is the integrated expression.  Drop the integral sign on the left and the 'dx' on the right sides, and add a vertical bar on the right side of the expression with 'a' and '0' respectively signifying the two end points at which the expression is to be evaluated;
   p. 114, third equation, second equality, RHS: the upper limit to the integral should be 'a', not '12';
   p. 133, second line from the top: second equality '=' should be addition '+' and the term should be <ψ|φ>*, not <φ|ψ>*, as in "<ψ|φ>+ <ψ|φ>* = ..." 
   p. 165, Ex. 6-14, par (b): The initial condition | Ψ(0)> = |0> should be preceded by the word 'where' to signify what it is;
   p. 93, sixth line from the bottom: should read "An even function for which ψ(-x) = ψ(x) is seen" ...;  then two lines down should read "for which ψ(-x) = -ψ(x) corresponds to the ..."  
   p. 54, fifth equality from the top: factor of '30' missing on RHS.

and on and on.

Regrettably, McGraw-Hill Education appears to have farmed out some of the publishing functions to a company called Conveo Publisher Services with distressing results. I have contacted Conveo regarding the condition of the published work, and am in the process of contacting McGraw-Hill too.

For an ex-science undergrad seeking to refresh his long ago university days, a book such as this would be a terrific asset, and that is precisely why I bought a copy. Having been through a couple of QM courses I had enough knowledge to be able to work through the typos. But for someone with, say, just one or two years of college, possibly not at the honours level, this book could be a disastrous waste of time. Edgeorge (talk) 23:58, 23 January 2014 (UTC)Duncan George

I'm very happy that you are trying to get the book fixed up. For the purposes of this wikipedia article, do you suggest that we make any change? Should we remove the reference entirely? Is there an errata page online that we can link to? --Steve (talk) 15:40, 27 January 2014 (UTC)

Steve - in the reference you might want to just add a note of caution to the reader to expect typographical errors. Many are very obvious, but they slow the reader down. Others are not so obvious, and after much hunting and pecking around to determine why one is so stupid as not to understand, one realizes the bloody text is bad! I did get a response from Conveo, and they are taking the matter up with McGraw-Hill Education, the subsidiary which published the book. Let's wait and see what they say.Edgeorge (talk) 19:22, 28 January 2014 (UTC)Duncan

Hmmm, if there are so many errors, why read it? There are dozens or hundreds of books on this topic, surely we can make a list of a few Further Reading references where none of them is full of typos. I just went ahead and deleted the reference. --Steve (talk) 17:35, 29 January 2014 (UTC)

Fair enough. I'll let you know if I hear anything further. I think McMahon's idea was a good one because it attempted to bridge the gap between Scientific American style articles where you get no math, and the fully rigorous training that honours or graduate physics students receive, too onerous for the casual reader. It is most unfortunate the execution so diminished QMD's value. There is a discussion group of generally disgruntled readers, and a partial errata sheet that everyone seems to agree is incomplete. McMahon has also written a similar book on QFT which I've only peaked at.Edgeorge (talk) 18:30, 29 January 2014 (UTC) Duncan

Section entitled: "Analogy to Poisson brackets in classical physics"

This section is not very well explained and is hard to understand. 178.38.88.134 (talk) 21:28, 27 February 2015 (UTC)

Here is an old version with my take on the Poisson bracket discussion. You will see that I dedicated only one sentence to the subject of Poisson brackets! In my (biased) opinion, it was better that way! If other people agree with me, then we can delete that section.

I just don't think the section is really relevant to the article ... Well, maybe the topic is relevant in principle, but the way it's written now, I don't think readers will learn anything relevant from it. --Steve (talk) 01:04, 28 February 2015 (UTC)

That there is a classical correspondence to quantum angular momentum operator commutation relations is worth mentioning, but I agree that is doesn't need its own section. I think your single sentence is fine, with the addition of pointing out that in that sentence, $L_{n}$ represents the classical angular momentum operator. --Mark viking (talk) 04:03, 28 February 2015 (UTC)

I deleted the section. --Steve (talk) 19:22, 28 February 2015 (UTC)

Looks good, thanks. --Mark viking (talk) 19:49, 28 February 2015 (UTC)

Should title be plural?

Should this article be renamed Angular momentum operators in the plural? The introduction already makes clear that there are several: total, orbital and spin. In addition for each of these we have the magnitude (L², S², J²), three components and two ladder operators. Really the article considers the whole family of angular momentum operatorS. Dirac66 (talk) 01:09, 9 January 2016 (UTC)

@Dirac66, over 4 years later, I think I agree, it might be helpful. Though the standalone term "angular momentum operators" does not seem to be used much, it seems like we always just refer to them one at a time like separate objects. A fully appropriate title would be "Angular momentum operators in quantum mechanics", but it may be too long. Footlessmouse (talk) 05:56, 26 August 2020 (UTC)