Talk:Clebsch–Gordan coefficients

In the Symmetry section, what is j_3?

Angular momentum operators - few small changes

I have slightly changed and hopefully made clearer and less controversial the part about defining a "vector operator" $\mathbf {j}$ out of operators ${\textrm {j}}_{i}$, by showing explicitly the construction of this object.

I've also added the quotation marks since, as far as I'm aware of, there is no well established meaning of a term vector operator, besides a general linear operator like the del operator.

I've replaced the phrase "the length of $\mathbf {j}$" by an informal description and a direct construction, referring to it as the Casimir operator.

Finally, I've changed the $j_{i}$ operators symbols form italic to normal, "straight" letters (${\textrm {j}}_{i}$), to make them differ from scalars in other equations. I can also add the "hat" (${\hat {\textrm {j}}}$) to operators. Both ways are present in literature, for example in Griffiths' Introduction to Quantum Mechanics.

Lurco (talk) 20:42, 7 November 2011 (UTC)

j_1, j_2, j_3 notation:

The notation of this article is sick. j_1 and j_2 mostly refer to j quantum number of particles 1 and 2 (occasionally j_x and j_y) while j_3 mostly refers to j_z quantum number of a generic particle. I suggest labeling particles 1 and 2 by alpha and beta or A and B to clarify the notation. —Preceding unsigned comment added by 130.233.204.28 (talk) 12:19, 14 January 2011 (UTC)

I agree. I'll change that - I will change 1,2,3 -> x,y,z and retain 1,2 for particle indices. However, that will lead to an abuse of notation in then Levi-Civita symbol. --Sebastian Henckel (talk) 21:59, 27 March 2011 (UTC)

Bra-ket notation:

Does anyone think it worthwhile to mention that < | > is Diracs bra-ket notation to avoid confusion (assuming that it is!)--Light current 00:02, 15 November 2005 (UTC)

I added a sentence with a link to the bra-ket article. --agr 14:52, 15 November 2005 (UTC)

I think also under Orthogonality, bra is changed to ket in notation... and this is special to doing everything in real gauge (which is standard for SU(2)). So I think it confuses things to just say we are "defining" this, when actually it has a meaning of taking complex conjugate (which is trivial here) — Preceding unsigned comment added by 163.1.246.246 (talk) 12:14, 14 March 2022 (UTC)

Typo?

In the first paragraph there is a sentence "to perform the explicit direct sum decomposition of the tensor product of two irreducible representations into irreducible representations". Should a correction be made to read "to perform the explicit direct sum decomposition of the tensor product of two reducible representations into irreducible representations"? Njmayhall (talk) 05:05, 8 January 2008 (UTC)

The statement in the first paragraph is correct. It works like this: if you start with two irreducible representations and take the tensor product, then the result may be reducible. The Clebsch-Gordan coefficients tell you how to turn this reducible representation into a direct sum of irreducible representations - gerritg (talk) 20:53, 25 January 2008 (UTC)

I think I understand, after the explanation here, but I really think that the wording in the article itself is awfully confusing, the way irreducible shows up twice. --Qrystal (talk) 20:54, 13 May 2008 (UTC)

Notation

On November 18, 2009, Itamarhason changed the notation of Clebsch-Gordan coefficients from brackets, $\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$, to parentheses $(j_{1}m_{1}j_{2}m_{2}|JM)$. I would be in favor of changing back to the original braket notation because this notation coincides with braket notation of scalar products and it helps to understand a Clebsch-Gordan coefficient as a matrix element between coupled and uncoupled angular momentum states. Also, the change was not done everywhere, and we now have two notations on one page. gerritg (talk) 19:17, 25 November 2009 (UTC)

Mistake in "CG Properties"?

The identities in the section on the properties of Clebsch-Gordan coefficients seems to be inconsistent with those in the appendix of Preston and Bhaduri - Structure of the Nucleus. Could someone more qualified on the subject check whether a typo exists of whether there is an alternate sign convention behind this? Igor Senderovich (talk) 18:50, 27 April 2010 (UTC)

I checked the relations in the symmetry properties section numerically for a number of integer and half-integer cases and found no mistake. I also added a note on how the relations can be verified analytically by converting to 3-jm symbols. gerritg (talk) 18:19, 12 June 2010 (UTC)

Significance of CB coefficients in SU(N) group

What exactly are the CB coefficients in a Special Unitary group? Are they purely mathematical or they do have a physical significance? The way I understand is that when the coupled basis of angular momentum is expanded in the uncoupled basis, the elements of the transformation matrix forms the CB coefficient. What happens in groups? — Preceding unsigned comment added by Arkadipta Sarkar (talk • contribs) 17:22, 12 October 2014 (UTC)

Clebsch–Gordan coefficients for SU(3) YohanN7 (talk) 16:00, 14 December 2015 (UTC)

SU(N) Clebsch–Gordan coefficients

This section was added by a one-purpose account [1] by one of the authors of one of the papers cited. There is also a link to software by the authors. I don't have a big problem with it, but someone else might have, since it is a little bit of self-promotion going on. YohanN7 (talk) 12:30, 15 December 2015 (UTC)

Notational inconsistency

The last equation in Section 2 Angular momentum states now reads

$\langle j_{1}\,m_{1}|j_{2}\,m_{2}\rangle =\delta _{j_{1},j_{2}}\delta _{m_{1},m_{2}}.$

where j₁ and j₂ are understood as two values of a single angular momentum, for example the orbital angular momentum of one electron, and similarly for the z-component values m₁ and m₂.

However this is inconsistent with the rest of the article which uses j₁ and j₂ to mean two different angular momenta which are coupled together, such as the orbital angular moment and the spin of an electron. To avoid confusion therefore I will change the first equation in this comment to read

$\langle j\,m|j'\,m'\rangle =\delta _{j,j'}\delta _{m,m'}.$

This should clarify that j and j' have a different meaning from j₁ and j₂ in the rest of the article. Dirac66 (talk) 00:02, 6 January 2016 (UTC)

Good point, I did the same in the "Tensor product space" section gerritg (talk) 21:37, 15 January 2016 (UTC)

recursion

The article does mention recursion, and there are some obvious recursions to generate them. I think, though, that either this one or Table_of_Clebsch–Gordan_coefficients should mention which recursion is numerically unstable, in case one happens to try to generate them that way, numerically, and is surprised with the results. Gah4 (talk) 19:56, 19 April 2020 (UTC)