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, $$\langle j_1 m_1 j_2 m_2|J M\rangle$$, to parentheses $$(j_1 m_1 j_2 m_2|J M)$$. 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 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)