Talk:Clebsch-Gordan coefficients

From Wikipedia, I changed the lead, added equation for explicit expression (plus discussion) and one special case.--Paul Wormer 03:45, 22 August 2007 (CDT)

Confusement

In the original WP text the i in j _i could stand for x, y, or z, or for "particle number" 1 or 2. In some situations this was confusing, so I introduced x, y and z.--Paul Wormer 07:32, 7 October 2007 (CDT)

Notation of eigenstates

Is there a difference between the states $|(j_{1}j_{2})JM\rangle$ and $|JM\rangle$? I guess not, because the latter notation appears in the orthogonality relation

$\langle JM|J'M'\rangle =\delta _{J\,J'}\delta _{M\,M'}$

without any comment. I find it rather confusing that both notations appear in the definition

$|(j_{1}j_{2})JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$

for the CG coeffs. -- Jitse Niesen 06:37, 30 July 2008 (CDT)

Incidentally, the book we used when learning quantum mechanics (Quantum Physics by Stephen Gasiorowicz) calls them Wigner coefficients; is it just a strange book or is that name really used or are Wigner coefficients something subtly different? -- Jitse Niesen 06:47, 30 July 2008 (CDT)

And another question: the edit history shows that the from-wp flag suddenly disappears. Was this done on purpose? -- Jitse Niesen 07:11, 30 July 2008 (CDT)

You are right both notations are in this article for the same ket $|(j_{1}j_{2})JM\rangle \equiv |JM\rangle$. But very often only J and M are important and the small j's aren't. Also in a different context the ket $|JM\rangle$ may have another origin than the coupling of two small js (e.g., only one j or more than two js).

The CG coefficients are sometimes written as

$\langle j_{1}m_{1};j_{2}m_{2}|(j_{1}j_{2})JM\rangle$

but this notation is redundant in that the same two small js are given twice in one symbol.

Some authors use indeed the term "Wigner coefficients", which historically is a much better name. But the majority of authors (especially the mathematically oriented authors) use Clebsch-Gordan for all sorts of groups, not just SO(3) and SU(2).

About the Wiki flag: is it necessary to activate it for any edit? I thought that only the first time would be sufficient, because from the history of the article is then clear that it comes from WP.

I'm melting away in my study right now, so I won't make any changes to this or any other article until the weather cools down.--Paul Wormer 08:27, 30 July 2008 (CDT)

I made some small changes according to your replies. I'm pretty sure that the Wiki flag should be activated at every edit. This does more than putting a W in the history, it also puts a text at the bottom of the article, saying that parts come from Wikipedia. I couldn't find a clear statement about this, but CZ:How to convert Wikipedia articles to Citizendium articles says

The "Content is from Wikipedia?" box is for the article as a whole, and not just the current edit.

and the (obsoleted) page CZ:The Big Cleanup says

Check the "Content is from Wikipedia?" box if any part of the article is sourced from Wikipedia.

Finally, another question. The section on the orthogonality relations start with saying that these are most clearly written down by introducing the`alternative notation $\langle JM|j_{1}m_{1}j_{2}m_{2}\rangle \equiv \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle$. But this is true by definition (using that the coefficients are real). I think stressing such a basic property of bra-ket notation may confuse people and it may be better to delete this sentence. -- Jitse Niesen 10:02, 2 August 2008 (CDT)

Jitse, you are right again, but since not too many authors stress that a CG coefficient can be seen as an inner product, this remark has some use. Often (also by Wigner in his book) CG coefficients are introduced as elements of a matrix that, through a similarity transformation, reduces an outer product representation of a group. That is, it decomposes the representation matrices into block-diagonal matrices with irreducible blocks on the diagonal. This matrix is of course unitary since only orthonormal bases come into play. I always liked this inner product (bra-ket) character of the CG coefficient, and convinced my former colleague of the usefulness of this view. My former colleague (Gerrit Groenenboom) is the main author of the WP article. --Paul Wormer 10:35, 2 August 2008 (CDT)

PS I looked at the WP history, and have to qualify: Gerrit Groenenboom is one of the main authors.--Paul Wormer 10:47, 2 August 2008 (CDT)

I see what you mean, yes. In that case, let's leave it as it is, at least until somebody has a better idea. -- Jitse Niesen 06:23, 3 August 2008 (CDT)