# Talk:Spherical harmonics

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 Definition:  A series of harmonic basis functions that can be used to describe the boundary of objects with spherical topology. [d] [e]
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This article would definitely benefit from some images. Fredrik Johansson 11:04, 23 August 2007 (CDT)

• Although I am still working on this article (one of the things I want to do is type in explicit expressions), I had not planned any illustrations. I teak my lead from Whittaker and Watson: no diagrams :-) (joke). Do you have any suggestions? In the past I made some 3D-like plots of s, p, d, etc. orbitals (by means of Maple). They looked nice but, in my view, you don't learn anything from it.--Paul Wormer 11:49, 23 August 2007 (CDT)

## Illustrations?

I'm afraid I don't have any graphs to add myself. Those shown here are nice. There's also this image on Wikimedia Commons, which is free. Fredrik Johansson 08:38, 2 September 2007 (CDT)
I uploaded Spher_harm_10.svg‎ and tried to show it at the bottom of the article, but no luck! Can somebody tell me what I do wrong?--Paul Wormer 09:46, 2 September 2007 (CDT)
I got it to work by copying the file name from the image description page, but I don't understand why that helped; the lines look identical to me. The image could use some cropping, by the way. Fredrik Johansson 10:59, 2 September 2007 (CDT)
You mean that you removed the underscores in the file name? Can I crop here, or do I have to do that at the PostScript level? What does thumb do? Thank you for your help.--Paul Wormer 11:06, 2 September 2007 (CDT)
Underscores or no underscores shouldn't matter; it works if I insert underscores in the current version. The MediaWiki software doesn't support cropping as far as I know, so you'll have to crop the file yourself and reupload it. "Thumb" places a frame around the image that makes the caption visible (it also resizes the image to 180px unless you specify a custom size). Fredrik Johansson 11:18, 2 September 2007 (CDT)

## Opening paragraph and definition

While I like the definition on the subpage very much, I think that the first paragraph of the article itself is to theoretical. How about incorporating the Spherical harmonics/Definition into the first few sentences of the article? -- Alexander Wiebel 23:53, 30 January 2009 (UTC)

I see this article deteriorate in a manner very reminiscent of WP. The newly added section Lie algebra was added by somebody who did not read (or understand) the rest of the article.

The relevant Lie algebra is already introduced in the article: it is the algebra of orbital momentum operators (the so-called Lie derivatives):  ${\displaystyle \{L_{i}\}\,}$. Neither the algebra nor the Levi-Civita tensor have to be introduced a second time (at least not without back reference and motivation).

The matrices representing (irreducibly) the ${\displaystyle \{L_{i}\}\,}$ are directly and easily obtained (by differentiation at zero angles) from the Wigner D-matrices that are introduced in the article. The newly added section has this uninformative babble about them:

In particular, one can construct sets of square matrices of various dimensions that satisfy these commutation rules; each set is a so-called representation of the rules. One finds that there are many such sets, but they can be sorted into two kinds: irreducible and reducible. The reducible sets of matrices can be shown to be equivalent to matrices with smaller irreducible matrices down the diagonal, so that the rules are satisfied within these smaller constituent matrices, and the entire matrix is not essential. The irreducible sets cannot be arranged this way.

In the equations (taken from the section Lie algebra):

${\displaystyle J^{2}|j,\ m>=j(j+1)|j,\ m>\ ,}$
${\displaystyle J_{z}|j,\ m>=m|j,\ m>}$

the kets | j, m > are nothing but the spherical harmonics ${\displaystyle Y_{\ell }^{m}}$ (the subject of the article) and the J operators are identical to what everywhere else in the article is indicated by L. The reason that this is not stated can be that the identical piece of text is used in magnetic moment, that the person making the addition didn't realize it, or both.

Once more we see a failure of the Citizendium's system of experts. Signed by a sad --Paul Wormer 07:33, 25 December 2010 (UTC)

Hi Paul: I'm sorry to have disturbed you with my addition of the Lie Algebra section. I thought it added a simple introduction accessible to a novice, and some useful references. Perhaps you would like to engage in some discussion? John R. Brews 15:12, 25 December 2010 (UTC)
As an aside, my objective here was to lighten the burden somewhat in discussing the commutation rules in magnetic moment. I was not as successful as I had hoped. Maybe some further development could accomplish that end?
As I understand the matter, feel free to correct me on this, the Lie Algebra although connected to group theory, only overlaps it, and is not equivalent to it. So a separate discussion is not out of place, particularly as it is so easy to introduce (in my opinion, of course). John R. Brews 15:26, 25 December 2010 (UTC)
Looking further, perhaps the best approach to this is to include angular momentum (quantum) in the discussion. For purposes beyond atomic physics, the treatment of angular momentum using J, L, S is too restricted, being tied to spherical symmetry. Maybe symmetry and angular momentum should be treated separately? What could be done here? John R. Brews 16:25, 25 December 2010 (UTC)
I have moved the explanation of the Levi-Civita symbol to its first occurrence, and added two internal cross links tying the Lie algebra discussion and the orbital angular momentum discussions together. John R. Brews 16:49, 27 December 2010 (UTC)
I've written an introductory article for Levi-Civita symbol and deleted wording describing its meaning in the Spherical harmonics article. John R. Brews 05:50, 2 January 2011 (UTC)