Talk:Spherical harmonics

message: Edited out this:
'Very closely related to this expansion is the algebraic geometric interpretation of a 2-sphere as the real commutative algebra generated by a,b and c subject to the relation a2+b2+c2-1=0 or the C* interpretation as the C* algebra generated by the self-adjoint elements a,b and c subject to a2+b2+c2-1=0.'
I don't see the close relation. Spherical harmonics do have an upmarket explanation from representation theory; but this isn't (yet) it.
Charles Matthews 12:52, 23 Oct 2003 (UTC)
thread_title: algebraic geometry

message: Please consider creating a distinct article, possibly Table of spherical harmonics that will hold the list of explicit expressions for l>4. While such a table is not overwhelmingly useful, it can come in handy if one just wants to glance at something and doesn't want to write software to work out some particular case.  linas 22:55, 16 August 2005 (UTC)
replies: text: I had already made the table, just had commented most of it out, so the hardest part was getting it to save without an error message like "Something didn't return a response to your request" error message. Κσυπ Cyp   06:38, 20 August 2005 (UTC)
replies: text: Yeah, WP has been spitting out that error message recently.  I've discovered that quite often, the changes did make it into the database, but the web page didn't get rendered. Thus, if you open a second browser, and load the page, you'll (usually) see that your changes made it.  So you don't have to hit the submit button over and over.  linas 16:54, 20 August 2005 (UTC)
thread_title: Table of Harmonics

message: I added the reference of [Varshalovich, et al.]. This book contains hundred pages of tables and formulas about spherical harmonics and could be very usefull.
Alain Michaud 16:47, 31 Oct 2005 (UTC)
thread_title: References

message: "The phase of Y_\ell^{m} necessary for the validity of this complex conjugation relation is the so-called Condon and Shortley phase."
I don't think that I agree with this. It is my understanding the the Condon Shortely phase has nothing to do with the complex conjugation relationship--It comes from the identity for the negative order associated Legendre functions.
------
I don't agree, another phase, the so-called Racah phase, can be recognized by (Y_\ell^{m})^* = (-1)^{\ell+m}Y_\ell^{-m}\,, which agrees with the usual definition of time-reversal. Racah
multiplied Ylm with (i)^\ell. Both phase conventions give easy to remember formulas for step up/down operations. Perhaps I should have stated: the CS convention can be recognized by this complex conjugation relation. P.wormer
 
"In quantum mechanically oriented texts the Condon and Shortley phase (-1)^m\, is usually introduced in the spherical harmonic functions. That is to say, the associated Legendre functions are defined without phase. The Wikipedia article defines the latter functions including the phase (-1)^m\,, which is why it is correctly absent in the definition of the spherical harmonic function in the present article." 
----
See Edmonds, Messiah (vol I), Brink and Satchler, Biedenharn and Louck (and I could mention more references) for Plm's without phase. My favorite definition of a spherical harmonic is one that does not need Plm's with negative m, because a Plm with negative m is somewhat unnatural and requires a new definition. See Altmann S. L.;  Herzig P. Point-Group Theory Tables; Clarendon: Oxford, 1994. 
P.wormer
I think that we need to have a section devoted to the Condon-Shortley phase convention, because this becomes very confusing when discussing across disciplines. Some physicists put the phase within the Legendre function, whereas some append it the the definition of the spherical harmonic functions instead. Some disciplines (like geodesy and magnetics) do not use the CS phase at all. Honestly, I am not sure why people would want to use this phase, and the reaason for using it should be explained here. I think I heard that it simplifies operations involving ladder operators, but I am not sure. 
----
Yes ladder (=step up/down) operations become easier, they become the same for positive and negative
m. Without CS (or Racah) phase one must distinguish  the sign of m. P.wormer
Finally, we need to note if the analytic expressions for the Ylm's use the CS phase or not. Since I am guessing this section was written by a physicist, it probably does, be someone needs to check. Lunokhod 04:18, 6 November 2006 (UTC)
thread_title:  Condon-Shortley phase 

message: I deleted this table of images. I find it very confusing, and am not even sure what cooridinate is being plotted. Lunokhod 18:41, 16 November 2006 (UTC)
replies: text: No objection — between the little animation at the top and the graphic showing the partitioning of the spheres' surfaces by sign, I think one can get quite a good feel for these functions.  There is that one graphic showing cross-sections of the sphere which generate the images you deleted; unless your images are brought back (which I would object to), I cannot see the purpose of this image either.  I suggest deleting it too.  Baccyak4H 18:58, 16 November 2006 (UTC)

text: Update.  I removed the other image.  Baccyak4H 19:00, 16 November 2006 (UTC)
thread_title:  Deleted images 

message: I think that this page is finally in decent shape. However, one thing that might be useful would be to have a list of commonly used identities, such as the integral of three harmonics expressed in wigner 3j coefficients, etc. I'll start this when I have the time, but this is low on my priority list. Is there anything else that is missing? Lunokhod 18:53, 16 November 2006 (UTC)
thread_title:  To do: add identities 

message: The transformation (given in this article) from complex to real spherical harmonics, will, if applied to Condon-Shortley spherical harmonics, give an m-dependent phase in the real harmonics. This is unpleasant. If this article were only meant for quantum mechanicians I would change it. But in another fields people have different opinions.--P.wormer 15:06, 6 July 2007 (UTC)
replies: text: Y_{\ell m} = \begin{cases}
\sqrt{2}\, \Re (Y_{\ell}^m) = \sqrt{2} N_{(\ell,m)} P_\ell^m(\cos \theta) \cos m\varphi 
& \mbox{if } m>0 \\
Y_\ell^0 & \mbox{if } m=0\\
\sqrt{2}\, \Im (Y_{\ell}^m) = \sqrt{2} N_{(\ell,|m|)} P_\ell^{|m|}(\cos \theta) \sin |m|\varphi 
&\mbox{if } m<0.
\end{cases}
Monsterman222 (talk) 03:29, 5 March 2013 (UTC)
thread_title:  Real spherical harmonics 

message: In Eq. (13-17) of Quantum Theory of Atomic Structure, Vol I, Slater defines the coefficient normalized as
:c^k(l,m,l,'m')=\sqrt{\frac{4 \pi}{2k + 1}} \int d^2\Omega \ Y_l^m(\Omega)^* Y_{l'}^{m'}(\Omega) Y_k^{m-m'}(\Omega) 
Bethe and Jackiw use this normalization in Eq. (4-48) of Intermediate Quantum Mechanics (1985),
and Condon and Shortley also use it on p. 175 of The Theory of Atomic Spectra (1959).
-Brian  —Preceding unsigned comment added by 128.97.23.22 (talk) 16:22, 14 July 2008 (UTC)
thread_title:  Missing normalization on Slater coefficients? 

message: I'm thinking of adding an applications section. So far, I'm thinking about spherical harmonic representations in geophysics, such as:
* Gravity
* Topography
* Planetary magnetic fields
If anyone else has anything that they can think of as an application, please leave a note here, so the section doesn't become biased towards my area of knowledge. Awickert (talk) 04:56, 3 May 2009 (UTC)
replies: text: It's heavily used in the simulation of global lighting and a few other areas in 3D computer graphics.  Classic computer graphics considers light sources such as the sun and street lights as infinitely small points - but to get the best realism, you need to include the 'ambient' light scattered from the sky.  To do that, you need to understand what area of the sky is visible from each point on the surface of each object.  That area reduces (for example) when a large object moves close to the point we're considering - so we can't simply calculate the total solid-angle of the sky that's visible - we need to know which parts - the shape of the sky visible from that point.  Describing that 'sky shape' is a function on an imaginary sphere surrounding that point.  However, a FULL description of that shape would require insanely large amounts of data at every point on the surface of every object...but spherical harmonics allows us to approximate that shape using much less data.  Math in action! SteveBaker (talk) 02:25, 4 May 2009 (UTC)
replies: text: That's really cool! Thanks. Maybe we could have a couple of images - one of 3D computer graphics, and maybe another one of Mars' magnetic field or something. Awickert (talk) 06:27, 4 May 2009 (UTC)
thread_title:  Applications 

message: Why does "Spherical Harmonics" redirect to "Spherical Harmonics"?  —Preceding unsigned comment added by 76.185.73.179 (talk) 05:38, 12 July 2009 (UTC)
thread_title:  Redirection? 

message: The introduction states that spherical harmonics are a set of solutions of the laplacian. However, they are not solutions but 
eigenfunctions (see the discussion above or see http://en.wikipedia.org/wiki/Laplace_operator ).
I find it very misleading... --Nbonneel (talk) 12:36, 20 July 2009 (UTC)
replies: text: Well, they are both things, aren't they?  "Solutions of Laplace's equation" is the classical (i.e., 19th century) way of thinking about them.  However, I agree that the "eigenfunctions" approach is a bit underemphasized in the introductory paragraph.  173.75.156.204 (talk) 01:11, 21 October 2009 (UTC)
thread_title: solutions in intro

message: Laplace's equation in spherical coordinates is:
: \nabla^2 f = {1 \over r^2}{\partial \over \partial r}\left(r^2 {\partial f \over \partial r}\right) 
  + {1 \over r^2\sin\theta}{\partial \over \partial \theta}\left(\sin\theta {\partial f \over \partial \theta}\right) 
  + {1 \over r^2\sin^2\theta}{\partial^2 f \over \partial \varphi^2} = 0
(see also del in cylindrical and spherical coordinates). Consider the problem of finding solutions of the form ƒ(r,θ,φ) = R(r)Θ(θ)Φ(φ).  By separation of variables, two differential equations result by imposing Laplace's equation:
:\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) = \lambda,\qquad \frac{1}{\Theta\sin\theta}\frac{d}{d\theta}\left(\sin\theta \frac{d\Theta}{d\theta}\right) + \frac{1}{\Phi\sin^2\theta}\frac{d^2\Phi}{d\varphi^2} = -\lambda
where λ is a constant.  
The first of these equations takes the form
: \frac {d} {dr}\left(r^2 \frac {dR} {dr}\right) = \lambda  R
i.e. it is a regular differential equations of second order. Given initial values for the function R and for its derivative \frac{dR}{dr} at a some radius  r = r_0 this differential equation uniquely determines the value of R(r) for any r
A function 
: r^l 
is a solution to this differential equation if and only if 
: l \,  ( l +1) = \lambda 
For any  \lambda > -\frac {1}{4} this equation has two solutions  l_1 , l_2 and the function
: R(r) = c_1 r^{l_1} +  c_2 r^{l_2}
then satisfies the differential equation for any constants  c_1 , c_2. Given any values for the function R\,\, and for its derivative \frac{dR}{dr}\,\, at a given radius r=r_0\, the constants  c_1 , c_2 can be selected to fit these initial values, i.e. the general form of the solution has been found.
In the following we will only consider integer values (positive and negative!) for  l in  r^l and therefore only    \lambda = l \,  ( l +1) for integer values of  l, i.e. for    \lambda = 1,2,6,.... To each    \lambda  value there is one negative and one positive value for  l, for example    \lambda = 2  corresponds both to  l =-2 and to  l = 1.  With   l \ge 1 one has that  R(r) goes to infinity when r goes to infinity, with  l \le -1  one has a singularity at  r =0 .
Applying separation of variables again to the second equation gives way to the pair of differential equations
:\frac{1}{\Phi(\varphi)} \frac{d^2 \Phi(\varphi)}{d\varphi^2} = -\mu 
:l \, ( l +1)\sin ^2(\theta) + \frac{\sin(\theta)}{\Theta(\theta)} \frac{d}{d\theta} \left [ \sin(\theta) \frac{d\Theta}{d\theta} \right ] = \mu 
for some integer l. As we are looking for a continuous and differentiable function satisfying Laplace's equation in 3 dimensional space it is a priori clear that  \Phi(\varphi)  must be periodic with the period  2 \pi . And if  \Phi(\varphi)  is not constant then \Theta(\theta) must be zero for \theta=0 and for \theta=2 \pi, otherwise there would be a singularity at the polar axis of the spherical coordinate system.
The first of the two equations takes the form 
: \frac{d^2 \Phi(\varphi)}{d\varphi^2} = -\mu \Phi(\varphi)
This linear differential equation of second order with constant coefficients has a periodic solution with period 2 \pi if and only if  \mu = m^2 for an integer   m in which case the general solution is
:\Phi(\varphi = A \cos(m \phi) + B \sin(m \phi) 
where  A ,\, B  are constants.
The second of the equations takes the form:
: \frac {1}{\sin(\theta)} \frac{d}{d\theta} \left [ \sin(\theta) \frac{d\Theta}{d\theta} \right ] + \left [l \, ( l +1) - \frac{m^2 }{\sin ^2(\theta)}\right ] \Theta(\theta) = 0  
and after the variable substitution x=\cos (\theta) it reads
: \frac{d}{dx} \left[ (1-x^2) \frac{d}{dx} \Theta(x) \right] + \left [ l(l+1)- \frac{m^2}{1-x^2} \right]\Theta(x) = 0 
This is a Sturm-Liouville differential equation. For m=0, i.e. for the case that \Phi(\phi) is constant, this is the differential equation to  which the Legendre polynomials are the solutions. If m \ge  1 this is the differential equation to  which the Associated Legendre functions are the solutions.
The first few Legendre polynomials are:
l
P_l(x)\,
0
1\,
1
x\,
2
\begin{matrix}\frac12\end{matrix} (3x^2-1) \,
3
\begin{matrix}\frac12\end{matrix} (5x^3-3x) \,
4
\begin{matrix}\frac18\end{matrix} (35x^4-30x^2+3)\,
5
\begin{matrix}\frac18\end{matrix} (63x^5-70x^3+15x)\,
6
\begin{matrix}\frac1{16}\end{matrix} (231x^6-315x^4+105x^2-5)\,
7
\begin{matrix}\frac1{16}\end{matrix} (429x^7-693x^5+315x^3-35x)\,
8
\begin{matrix}\frac1{128}\end{matrix} (6435x^8-12012x^6+6930x^4-1260x^2+35)\,
9
\begin{matrix}\frac1{128}\end{matrix} (12155x^9-25740x^7+18018x^5-4620x^3+315x)\,
10
\begin{matrix}\frac1{256}\end{matrix} (46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63)\,
The scaling factor is by convention selected such that the absolute value of the Legendre polynomials take the value 1 for x=-1 and x=1.
The spherical harmonics corresponding to l=-3\,,\,\lambda=6\,,\,m=0 is
:\frac{P_2(cos(\theta))}{r^3}=\frac{1}{r^3}\frac{1}{2}(3 \cos^2-1)
Using a rectangular coordinate system this harmonics takes the form
:\frac{1}{r^5}\frac{1}{2}(3 z^2-r^2)
The first few associated Legendre functions P_{l}^{m}are:
:P_{1}^{1}(x)=-(1-x^2)^{1/2}
:P_{2}^{1}(x)=-3x(1-x^2)^{1/2}
:P_{2}^{2}(x)=3(1-x^2)
:P_{3}^{1}(x)=-\begin{matrix}\frac{3}{2}\end{matrix}(5x^{2}-1)(1-x^2)^{1/2}
:P_{3}^{2}(x)=15x(1-x^2)
:P_{3}^{3}(x)=-15(1-x^2)^{3/2}
:P_{4}^{1}(x)=-\begin{matrix}\frac{5}{2}\end{matrix}(7x^3-3x)(1-x^2)^{1/2}
:P_{4}^{2}(x)=\begin{matrix}\frac{15}{2}\end{matrix}(7x^2-1)(1-x^2)
:P_{4}^{3}(x)= - 105x(1-x^2)^{3/2}
:P_{4}^{4}(x)=105(1-x^2)^{2}
These associated Legendre functions all have that P_{l}^{m}(-1)=P_{l}^{m}(1)=0 which is prerequisite for the function 
:R(r) \Theta(\theta) \Phi(\phi) 
to be continuous and differentiable at the polar axis of the spherical coordinate system
As an example, for l=-3 and for m=1 the two spherical harmonics are
:\frac{1}{r^3} 3 \cos(\theta) \sin(\theta) \cos(\phi) 
:\frac{1}{r^3} 3 \cos(\theta) \sin(\theta) \sin(\phi) 
Using rectangular coordinates they are
:\frac{1}{r^5} 3 z x 
:\frac{1}{r^5} 3 z y 
i.e they are continuous and differentiable everywhere except for the singularity at r=0
Stamcose (talk) 20:58, 25 December 2009 (UTC)
thread_title: Laplace's spherical harmonics

message: The present text says:
:\frac{1}{\Phi(\varphi)} \frac{d^2 \Phi(\varphi)}{d\varphi^2} = -m^2
and then
A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, 
No reason to complicate things with complex numbers.  The harmonic functions are functions in R^3. And the general solution to the differential equation 
: \frac{d^2 \Phi(\varphi)}{d\varphi^2} = -\mu \Phi(\varphi)
for positive and negative \mu is well known from school mathematics without using complex numbers
The present text says:
Imposing this regularity in Θ at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ(ℓ+1) for some non-negative integer with ℓ ≥ |m|.
This is simply wrong as I already wrote a few times.
That λ is of the form λ = ℓ(ℓ+1) for some non-negative integer should be (ℓ ≥ m because m is by definition greater or equal to 0). comes instead from the differential equation 
: \frac {d} {dr}\left(r^2 \frac {dR} {dr}\right) = \lambda  R
as I write in the correct version! Has nothing to do with the Sturm–Liouville problem!
It is probably better to use the convention that l is a non-negative integer but stressing that both R=r^l and R=\frac{1}{r^{l+1}} are equally valid.
Without going into this detail I am also not convinced about the rest of the text.  This material is complicated enough for the reader also with an absolutely clear and correct text.  And I think it is important that some concrete examples of spherical harmonics are presented to make the matter less abstract.
Stamcose (talk) 11:29, 26 December 2009 (UTC)
replies: text: I'm sorry, but I must disagree yet again.  First of all, "no need to complicate things with complex numbers" seems like strange advice, given that the solutions are complex exponentials anyway and the article says this in the next sentence.  At any rate I changed the text to this as a conciliatory gesture already because you suggested that there was some problem writing the number m2; but to me at least introducing a new constant just to refer to this only to be immediately dispensed with seems to be more complicated than just writing m2 at the outset (since this is obviously possible).  Second, I see that you are still unconvinced that this approach is correct and elegant.  A good way to reach a resolution is, per my above suggestion, present reliable sources that are of a similarly high quality to that already in the text (Courant–Hilbert) in which it is instead the radial equation that gives the proper eigenvalue.  Comparing your source with the one I have given may then help to resolve our dispute.  Thirdly, there already are examples of spherical harmonics further down in the text, including pictures, illustrations of their nodal domains, etc.  There is no need for this individual section to duplicate content with the rest of the article.  Fourthly, regarding your belief that "This is simply wrong as I already wrote a few times", you have written this and I have responded to it as well, but my responses have all gone unanswered (the discussion seems to be one way, as though we were both talking past each other).  Once again, Courant and Hilbert is a good introduction to Sturm–Liouville problems; they present the general Legendre equation as an example of exactly this sort of problem (§V.10. Problems of the Sturm–Liouville type.  Singular boundary points.).  Finally, the latest version (which I encourage you read and try to understand until the very end) downplays the role of the radial equation altogether, and at least indicates that both r^\ell and r^{-\ell-1} are valid.  Sławomir Biały (talk) 13:11, 26 December 2009 (UTC)
thread_title: talk

message: There seems to be a missing \frac{1}{Y}  in the second equation (in the second row), obtained by  by seperation of variables in the "Laplace's spherical harmonics" section. Adding that to equation.  —Preceding unsigned comment added by Pocav (talk • contribs) 19:36, 31 January 2010 (UTC)
thread_title:  Missing <math>\frac{1}{Y}</math> in "Laplace's spherical harmonics" 

message: The image 'Harmoniki.png' shown in the box at the top right of the article is wrong (or at least misleading) - the functions shown are not linearly independent. For example, on the second row of functions (L=1, looking like a 2p orbital), from left to right, the plot shows p_x, p_z, (-p_x) rather than p_x, p_z, p_y. [In trig formulae: sin(theta)cos(phi), cos(theta), -sin(theta)cos(phi) whereas the last one should be sin(theta)sin(phi)]  —Preceding unsigned comment added by Gingekerr (talk • contribs) 15:20, 16 November 2010 (UTC)
replies: text: Good point.  Someone should fix this.  It looks like whoever generated the image used an incorrect rotation.  Sławomir Biały (talk) 16:11, 16 November 2010 (UTC)
replies: text: Not only that, but this image is used for both spherical and cubic harmonics, which I'm assuming should not be the same. — Parsa (talk) 01:20, 23 September 2011 (UTC)
replies: text: The article cubic harmonics is very bad.  It says that a cubic harmonic is a spherical harmonic times the "radial part of the wavefunction", but it never says what the radial part of the wavefunction is.  I think these are the same things as solid spherical harmonics, but possibly I'm wrong.  At any rate, it's clear to me that these images are definitely not accurate visual depictions of cubic harmonics.  Cubic harmonics are functions in space (i.e., functions of three variables), not just functions on the sphere.  These images are graphs of functions on the sphere, with color denoting sign.  Sławomir Biały (talk) 12:45, 23 September 2011 (UTC)
thread_title:  Harmoniki.png 

message: I removed a template from the external links section that was unsupported by any discussion as to why the external links were either inappropriate or excessive. An editor using such a template is well advised to support his/her objections with appropriate discussion that might lead to article improvement. Otherwise he/she is just adding noise. Dlw20070716 (talk) 18:28, 15 July 2011 (UTC)
replies: text: Good point.  I've removed the external links from the article, since they seem to add unnecessary noise.  Please restore them individually if you feel that a link adds value and complies with WP:ELNO.  Here is the list of links.  Best, Sławomir Biały (talk) 19:03, 15 July 2011 (UTC)

text: Wikipedia is not a linkfarm.  Normally, we should not include links to software, unless the article is specifically about a software package.  There are other published lists of relevant software packages on the web, but that's not our job here.  As for the others, these are mostly people's personal websites or sites lacking in any peer review, and so also of dubious value to an encyclopedia such as this.  Encyclopedic external links are, of course, appropriate.  Sławomir Biały (talk) 12:02, 3 November 2011 (UTC)
thread_title:  Removed unsupported template from External links 

message: I think in this equation
:\frac{1}{4 \, \pi} \int_\Omega  f(\Omega)^2\, d\Omega = \sum_{\ell=0}^\infty S_{f\!f}(\ell),
the factor 
:\frac{1}{4 \, \pi}
should be dropped, given the normalization adopted for the spherical harmonics, or instead added in
:S_{f\!f}(\ell) = \frac{1}{4 \, \pi} \sum_{m=-\ell}^\ell  f_{\ell m}^2 
for consistency, but this is not my preferred solution.  — Preceding unsigned comment added by F. Mignard (talk • contribs) 12:56, 18 April 2012 (UTC)
thread_title:  Power  in signal processing 

message: I have a feeling that this article is simply not comprehensible to anyone who doesn't already have an understanding of the material. In my view, the result is that the article isn't particularly useful. I have no suggestions regarding how to remedy this, but it's an issue that I observe with virtually every math-heavy article on Wikipedia, and my perspective is not that of a layperson. Snrrub (talk) 18:48, 1 May 2012 (UTC)
replies: text: It's true that this is not exactly an introductory article, but the topic of spherical harmonics is not exactly one that lends itself to introductory treatment in encyclopedia form.  There are plenty of textbooks and tutorials that fill this need.  Sławomir Biały (talk) 21:13, 1 May 2012 (UTC)
replies: text: Shouldn't any encyclopedia still be able to explain what something is to anyone looking at it though?  This doesn't even have a summary that helps most people.  The kind of people that understand this article already know what SH are.  This is an encyclopedia, not a social club.  — Preceding unsigned comment added by 205.201.101.153 (talk) 17:57, 8 March 2015 (UTC)
replies: text:  The principal aim of this article is to include lots of information about spherical harmonics.  That is generally the purpose of an encyclopedia: to contain lots of information, usually organized into some helpful form, so that researchers can then use this information.  I am not aware of any social clubs for which this is the primary purpose.  I would not necessarily object to a more introductory account (provided it is not wrong), but what could such an account look like?  Spherical harmonics are the angular parts to solutions of Laplace's equation.  Equivalently, they are the eigenstates of angular momentum.  Both of these are derived more or less from first principles in the second section of the article, assuming only a familiarity with vector calculus.  So claiming that the article lacks introductory material here feels like laziness on the part of the reader.  And comparing the article to a social club seems rather like trolling.  Sławomir Biały (talk) 22:56, 8 March 2015 (UTC)

text: I tried the make the lead a bit lower level at the start; if nothing else, perhaps the gentle lay reader could glean that these are useful functions defined on a sphere. See what you think. --Mark viking (talk) 00:00, 9 March 2015 (UTC)
thread_title: Comprehensibility

message: I have a problem with this section. It says:
\mathbf{L}^2 = L_x^2+L_y^2+L_z^2
\begin{align}
L_+ &= L_x + iL_y\\
L_- &= L_x - iL_y
\end{align}
and
L_-L_+ = \mathbf{L}^2 - L_z^2 -L_z
but my math says:
L_-L_+ = (L_x - iL_y)(L_x + iL_y) = L_x^2 + L_y^2 = \mathbf{L}^2 - L_z^2
So where doest the term -L_z come from?
Linkato1 (talk) 09:22, 19 October 2012 (UTC)
replies: text: You assumed that L_x L_y - L_y L_x = 0 but that's not correct ... you're composing operators, which is not necessarily commutative ... see Canonical commutation relation etc. etc. :-) --Steve (talk) 12:09, 19 October 2012 (UTC)

text: Maybe could write "For a detailed derivation, see Ladder operator#Angular momentum --Steve (talk) 14:00, 20 October 2012 (UTC)
thread_title:  Orbital angular momentum  

message: Something seems wrong in the the expression for real spherical harmonics, written in terms the Legendre polynomials: for negative m, I get different result when compared to the definition based on Y_l^m. I guess that the order of Legendre polynomial should be m (and not Abs[m]) in the definition of real spherical harmonics for negative m.
For positive m, the result is the same and is consistent with complex spherical harmonics.
JiriVejrazka (talk) 10:28, 10 July 2014 (UTC)
thread_title:  Typo in real (tesseral) spherical harmonics? 

message: I don't know if anyone else agrees, but I would personally be interested in seeing a section listing the specific uses of spherical harmonics in different fields. For example: I've seen some similar-looking diagrams used in sound recording (to explain XY and mid-side recording techniques) and in antenna arrays for long-range radio-wave broadcasting (beamforming). It also seems applicable to musical instrument design, such for an ocarina's higher, non-Helmholtz resonance-derived harmonics (see p.4 of Daniel A. Russell's paper "Basketballs as Spherical Acoustic Cavities" to see some familiar diagrams, and Jared Kearns' "Hole Size in a Spherical Resonator"). 
Alternately, if anyone knows of a good overview of these various applications somewhere, please leave a link here. Esn (talk) 02:55, 31 December 2016 (UTC)
thread_title: Real-world applications?

message: It could be useful to include at least one figure showing visualizations of the complex (rather than real) spherical harmonics. The complex phase can be visualized as on a colorwheel while the amplitude can be visualized as saturation or brightness. One can then get a sense for geometric similarities/differences between the complex and real spherical harmonics. I'm coming from this as a physicist interested in atomic orbitals presently. There is a little bit of clarification which could be made about complex and real atomic orbitals on that page: Atomic_orbital#Complex_orbitals which could be aided by more recognition/visualization of the complex spherical harmonics here. 
Jagerber (talk) 05:21, 17 July 2018 (UTC)
replies: text: I added lots of visualizations for real and complex spherical harmonics at Table of spherical harmonics. Twistar48 (talk) 09:46, 5 November 2021 (UTC)
thread_title:  Visualizations of Complex Spherical Harmonics? 

message: The term Tesseral Harmonics is redirected to the page of Spherical Harmonics, whereas the Cubic Harmonics have their own extra page. The page on the site of Cubic Harmonics does not discuss the eigen states of cubic symmetry, but discusses the Tesseral Harmonics. I would suggest to make a separate page for the Tesseral Harmonics and the Cubic Harmonics and am willing to fill these. (This would result in the same structure as one has on the German Wiki site) 
Maurits W. Haverkort (talk) 08:23, 21 May 2020 (UTC)
thread_title:  Tesseral, Cubic, and Spherical Harmonics