Talk:Table of spherical harmonics

Sources

This article is almost entirely unsupported by sources and several small edits have recently been made. User:Loudandras has just supported a small correction with "Quantum Theory of Angular Momentum by D.A. Varshalovich, A.N. Moskalev, V.K. Kershonskii (World Scientific 1988))" mentioned in the edit summary. Can anyone support that this reference supports the content of all or part of this article, so it can be added as a suitable reference? I do not have access to it. Other references would also be welcome and particularly welcome if there were more easily available. --Bduke (Discussion) 22:14, 21 January 2013 (UTC)

Mathworld is an web-accessible source that can be taken to reliable. Xxanthippe (talk) 23:15, 21 January 2013 (UTC).

OK, so add the appropriate specific link to the article as a reference. I am not familiar with Mathworld but it looks interesting. --Bduke (Discussion) 23:38, 21 January 2013 (UTC)

Putting it in the article as an external link, as you have done, is fine, but I do not think it can be used as a source for the material. How would mathworld be used to get the material? --Bduke (Discussion) 05:04, 22 January 2013 (UTC)

By going to the Mathworld home page and searching for "spherical harmonic". Xxanthippe (talk) 09:57, 22 January 2013 (UTC).

OK, so as you seem to know more about this than I do, put some specific links as references to support specific parts of this article. A general link is not enough. --Bduke (Discussion) 10:40, 22 January 2013 (UTC)

I added cited references to an accessible article and the Varshalovich-book. The former gives real spherical harmonics up to l=2, the latter complex ones up to l=5. I checked every single harmonic on the page, and added references to the sections corresponding to the checked l's. The spherical-to-cartesian transcriptions are also fine for complex harmonics. I added the citations at the end of section titles, what might be frowned upon, but I couldn't think of anything better. I also added a direct link to Wolfram Alpha. Sorry if some cleanup is needed after my edit, I tried to be as efficient as possible.--Loudandras (talk) 22:08, 23 January 2013 (UTC)

Fixed transcription error for l=10, m=0 to agree with http://www.wolframalpha.com/input/?i=SphericalHarmonicY%5B10%2C0%2Ctheta%2C0%5D . ThaeliosActual 14:21, 25 January 2013 (UTC)

Found a decent source for real spherical harmonics, unfortunately g orbitals are not tabulated there. I checked every single function and their transformation from complex spherical harmonics, all are according to the Chisholm book and the other cited article.--Loudandras (talk) 14:03, 1 February 2013 (UTC)

Great. I can find the Chisholm book. I even thought I had it, but it seems not, but it will be in the university library. It is a good source for chemists. --Bduke (Discussion) 19:59, 1 February 2013 (UTC)

Splendid work. Xxanthippe (talk) 00:02, 2 February 2013 (UTC).

Contraction of +-m harmonics

I just noticed that anonymous 193.175.8.58 made some edits in the middle of November, contracting the formulae of spherical harmonics with $l=2$. While this notation is indeed more concise, I have some objections. Firstly, the choice of only $l=2$ for the edit seems arbitrary. Why not fix the other ones as well? Secondly, I think the expanded version would be much more useful in the case of this specific article. As the article is a table, it doesn't matter that 2 (or $l$) more lines are used for these functions. However, when users access the page and want to use/check these functions, the expanded notation (I mean having separate $+|m|$ and $-|m|$ harmonics) is much more transparent, and provides much less room for errors. If it was up to me, I would revert to a preceding version, but others might argue to change all the other spherical harmonics instead. Either way, I'm hoping for some input from others. --Loudandras (talk) 20:36, 3 January 2014 (UTC)

I noticed that set of edits back then but was too busy to think about it. I agree that all sections should be consistent on this point. I have a small preference for reverting the edits as I agree that would be more transparent. --Bduke (Discussion) 20:54, 3 January 2014 (UTC)

Thanks for the feedback! I decided to revert the edits by IP. --Loudandras (talk) 16:20, 8 January 2014 (UTC)

Direction cosines

I see the listing 'drys up' at l=5 (going from x,y,z to theta, phi. Philip Cotterell did produce the direction cosine (x,y,z) equations upto l=6. They are at <http://ambisonics.ch/standards/channels/ACN25> etc.,but would need some converting to match your 'style'. (They were republished at 'AMBISONICS SYMPOSIUM 2009 JUNE 25–27, GRAZ' "Symmetries of Spherical Harmonics: applications to ambisonics" Michael Chapman (pages 2 and 4), if you need a more formal source. Various copies of the PDF are on the Web.) Hope this helps...? — Preceding unsigned comment added by 92.157.70.113 (talk) 04:44, 10 January 2014 (UTC)

Real harmonics, notation and order

Regarding recent changes to the section about real spherical harmonics: I think that the orbital notation was really helpful, I don't really see why this was removed. There is no actual meaning of the $m$ quantum number of real harmonics, I mean there can be no a priori distinction between real harmonics with $+|m|$ and $-|m|$. Furthermore, the atomic orbitals are actually used to denote these functions, for example in various chapters of solid state physics. Is there a particular reason to remove the orbital notation? And on a similar note: was there any reason to switch the two real harmonics with $l=1,m=\pm 1$? It shouldn't make any difference, choice of indexing is arbitrary. I'm just arguing that the orbital notation should also be kept.

OK, I reread the article and the one on spherical harmonics. The new notation here seems to be in accordance with that article, and this also gives a reason to the change in the harmonics for $l=1$. Maybe some more extensive edit comments would have been helpful with the latter. Still, I believe that the orbital notation should be kept. And perhaps the functions should be checked now to see whether the notation is completely consistent with the spherical harmonics article. -- Loudandras (talk) 19:18, 17 January 2014 (UTC)

I came here to make essentially the same comments. I think the atomic orbital notation should be added back, so both notations are there. I have this on my watchlist because, as a computational chemist, I have used them. I suspect most people who will use this page, at least for the section on the real spherical harmonics, will be chemists. --Bduke (Discussion) 21:21, 17 January 2014 (UTC)

Sure, it can be restored.. but I think the normal indexes should still have the priority, since real expansions often go far beyond the orbital notation. So what I suggest is that a separate section be added on what specific names have been given to the Y_{lm} in the physics and chemistry communities. This can be either a subsection for each l value, below the equations for the Y_{lm}, or a wholly separate section by itself in e.g. the Spherical harmonics page. Susilehtola (talk) 08:00, 18 January 2014 (UTC)

I put in the term symbols as well in the table. Is this OK for you guys? To reiterate, I think the normal indexes should appear first since they're used everywhere for performing spherical expansions, while atomic orbital (AO) symbols have rather limited use outside quantum chemistry. (I'm actually a quantum chemist myself and have never used the AO symbols since they are cumbersome in computer implementations.) Susilehtola (talk) 09:35, 18 January 2014 (UTC)

Yes, I am OK with that. I was going to suggest it. Now, could anyone add the real spherical harmonics for l=5 and l=6 as these are used in quantum chemistry quite a lot these days? --Bduke (Discussion) 10:01, 18 January 2014 (UTC)

Thanks a lot, this is what I had in mind as well. Someone suggested some work on l=5 and 6 real harmonics in the "Direction cosines" section of the talk page, and I found a corresponding pdf here. But citing those would involve checking the prefactors, converting the notation, adding a factor of 1/r^l, and linking each harmonic to the corresponding complex ones to keep the syntax of l=1...4. So... maybe some other source?:) --Loudandras (talk) 10:38, 18 January 2014 (UTC)