Talk:Spherical harmonics

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figure error[edit]

The figure at the top right, with 3D visualizations of the functions, appears to be wrong. All negative order (negative m) figures are simply the same as for the equivalent positive m, but with a sign change if m is odd. They should actually be rotated around the axis by 90 degrees. The same seems to be the case in the large colour figure at the bottom right.

CWHughes (talk) 13:51, 4 May 2012 (UTC)

The harmonics are clearly drawn wrong, although I'm not sure I agree about the exact angular error. Maybe someone has attempted to fix this already (but incompletely)?

For an obvious error, observe that the first and fifth spherical harmonics shown on the third line are currently identical. Similarly for the fourth line.

Further, the plots all seem "squashed". Are the plots supposed to be spherical polar plots? If so, the items on the second row should all look like two touching spheres. — Preceding unsigned comment added by 92.234.82.157 (talk) 16:26, 3 October 2013 (UTC)


I have implemented the correct SH by following the Wikipedia, and I got this render: https://commons.wikimedia.org/wiki/File:Spherical_Harmonics.png

I think some of the SH functions look the same because of the perspective projection. This is the code for the renderer if you want to have a look: https://www.shadertoy.com/view/lsfXWH

I think my render looks better in general, it's more readable in terms of shape and lighting? Should we replace the current picture with this one? — Preceding unsigned comment added by Inigo.quilez (talkcontribs) 02:36, 15 May 2014 (UTC)

I can confirm that several of them are incorrect and it isn't a projection problem (unless you want to say that you change your projection from m -> -m). I can't see that other thing you posted, but the l=2, m=+-2 should be opposite each other (switch the blue and yellow regions). The same goes for the l=3,m=+-2 ones. Good figure though. Jazzwhiz101 (talk) 19:27, 29 May 2014 (UTC)
The m=2 and m=-2 harmonics are actually complex conjugates of each other, and their real parts are identical. Since the visualization only shows the real components, they are correct, but they 'rotate' in opposition directions if you advance the phase. What you're suggesting with regard to switching the blue and the yellow can be achieved by negating these harmonics. So the flipped blue and yellow version isn't orthogonal to the unflipped version. It's the imaginary component that makes them orthogonal. So what you're proposing isn't incorrect, but it's not necessary, and the renderings correctly reflect the harmonics as they're presented in the article. I suppose this results from the choice to represent the harmonics according to the Condon-Shortley phase convention.

image typo[edit]

The large colour plot image near the bottom has a typo in it. The second sphere down from the top on the left is labeled (5,4) whereas it should be labeled (5,1). Whoever created the image, please fix it.

algebraic geometry[edit]

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)

Table of Harmonics[edit]

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)

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)
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)

References[edit]

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)

In the angular part of the Laplace equation the index l is shown. It seems to me natural to write down a second formula showing the index m to guide the reader. At the present the index m is quit sudden launched in the notation for the general solution and in the - l<..< m <..<+l inequality.

I might suggest that the azimutal angle name 'fi' will be replaced by 'theta' to conform to the conventions for spatial orientation.

May be the indices l and m could be given here an intuitive geometric interpretation, in advance of the QM interprations for the Schroedinger wave equation of l and m, as integer numbers related to impuls-moment and its projection in the standard ( 'z') -direction.

A last suggestion: compare the Laplace solutions with the solutions of the Schroedinger wave equation for a central potential (like the coulomb field ), to discuss the differences and common aspects for radial - cq angular parts of their solutions.

Hans van der Grift

Different Normalizations[edit]

I have added definitions for the unit power and Schmidt-semi normalized spherical harmonics. Other modifications have been made for consistency and accuracy purposes. I have also added links to software archives that are currently being developed.

Note: We need to mention if the analytic expressions for the first few spherical harmonics use the Condon-Shortley phase or not. Indeed, this should be carefully clarified on the associated Legendre functions link.

Mark Wieczorek, Oct. 26, 2006

Condon-Shortley phase[edit]

"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)


The article as it is now is fine by me. P.wormer 13:15, 7 November 2006 (UTC)


Sorry, hopeless math newbie here, but, (quote from article) "phase factor of (−1)m for m > 0, 1 otherwise": where does this conditional come from? i do not have access to any relevant source texts, but all occurrences i found (mostly in acoustics papers using spherical harmonics) mention a plain, unconditional (-1)^m for all m. Can somebody clarify this? Nettings (talk) 20:41, 10 January 2014 (UTC)


Now I'm confused, why is the factor (-1)^m both in the definition of the spherical harmonics in quantum mechanics and in the definition of the associated Legendre polynomials? It seems incorrect to me. 200.57.50.123 (talk) 21:11, 17 August 2015 (UTC)

Deleted images[edit]

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)

Y10 Legendre Y1 xy.png Legendre Y1 polaire.png
Y20 Legendre Y2 xy.png Legendre Y2 polaire.png
Y30 Legendre Y3 xy.png Legendre Y3 polaire.png
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)
Update. I removed the other image. Baccyak4H 19:00, 16 November 2006 (UTC)

To do: add identities[edit]

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)

Real spherical harmonics[edit]

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)

Not being an expert neither in the field nor in the (phase) conventions, I think there is an error in the definition of real spherical harmonics: for negative m the Abs(m) should be used consistently, so for nagetive m, Y_{\ell m} = {1\over i\sqrt2}\left(Y_\ell^{-m}-(-1)^{m}\, Y_\ell^{m}\right) = \sqrt{2} N_{(l,-m)} P_\ell^{-m}(\cos \theta) \sin -m\varphi . I just note the inconsistency, someone with more indepth knowledge of the field should check this. 86.61.67.116 (talk) 10:55, 1 November 2008 (UTC)

There really is an error in the definition, as mentioned above. Tried to edit it but was told to cite sources. I have no idea how to cite my own calculations. —Preceding unsigned comment added by 96.236.217.254 (talk) 20:44, 10 March 2009 (UTC)


Another question about the Real Spherical Harmonics part. Why don't we just simplify the definitions of these using the real and imaginary parts:

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)

Laplace[edit]

I seem to have landed myself with the job of cleaning-up Pierre-Simon Laplace. The section Pierre-Simon Laplace#Spherical harmonics and potential theory needs attention from an expert. There seem to me to be two issues:

  1. There is little point in explaining what spherical harmonics are, that is better done here; and
  2. The definition (which if from Rouse Ball's (1908) A Short Account of the History of Mathematics) didn't immediately strike me as a suggestive definition.

Any thoughts on how to rewrite this section?Cutler 00:03, 25 August 2007 (UTC)

Rouse Ball gives the following account:

"This is memorable for the introduction into analysis of spherical harmonics or Laplace's coefficients, ...
"If the co-ordinates of two points be (r ,μ ,ω ) and (r′,μ' ,ω' ), and if r′ ≥ r, then the reciprocal of the distance between them can be expanded in powers of r/r′, and the respective coefficients are Laplace's coefficients. Their utility arises from the fact that every function of the co-ordinates of a point on the sphere can be expanded in a series of them. It should be stated that the similar coefficients for space of two dimensions, together with some of their properties, had been previously given by Adrien-Marie Legendre in a paper sent to the French Academy in 1783. Legendre had good reason to complain of the way in which he was treated in this matter."

Does anybody recognise this as a way of defining spherical harmonics? Should go in the article if so.Cutler 10:09, 25 August 2007 (UTC)

Missing normalization on Slater coefficients?[edit]

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)

Applications[edit]

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)

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)
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)

Eigenvalue-equation for laplacian operator[edit]

I've changed

 \nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi )

into

 \nabla^2 Y_\ell^m (\theta, \varphi ) = -\frac{\ell (\ell + 1 )}{r^2} Y_\ell^m (\theta, \varphi ), because I assume that the : \nabla^2 refers to :\Delta f ( r , \vartheta , \phi ) = \frac{1}{r^2} 
\frac{\partial}{\partial r} \left( r^2 \cdot \frac{\partial f}{\partial r} \right) +
\frac{1}{r^2 \sin \vartheta} \cdot \frac{\partial}{\partial \vartheta} \left(\sin\vartheta \cdot \frac{\partial f}{\partial \vartheta} \right) +
\frac{1}{r^2 \sin^2\vartheta} \cdot \frac{\partial^2 f}{\partial \phi^2}
, otherwise we might call it :\Delta_{\vartheta,\varphi}, so that :\Delta=\Delta_{r}+\frac{1}{r^{2}}\Delta_{\vartheta,\varphi}, which would make it a lot less ambiguous. Anyway, I think that the equation as it was is not correct.

Who removed that correction?! The current formulation 'on the unit sphere' is simply not enough. It might happen that people just look up the equation and use it for a calculation without the important factor of \frac{1}{r^2}. In fact that was what happened to me and took my pretty long to find the error.



I think there's another problem in that line, too (and also it took me pretty long to understand what had been done and what had gone wrong). The equation

 r^2\nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi ).

made me think that the double nabla refers to the standard Laplacian, which it does not. I think the equation would become correct if one writes \Delta_{\theta,\varphi} instead, defining it analogously to how the person above this post has defined it.

Redirection?[edit]

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

solutions in intro[edit]

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)

EDIT: I realized that r^l.Y_l^m is a solution to Laplace's equation. The statement "Y_l^m is the angular part of a set of solution to Laplace's equation" is thus true. However, the fact that they are eigen functions of the angular part of the laplacian seems even more interesting to report in the intro.... --Nbonneel (talk) 17:23, 24 July 2009 (UTC)

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)

Unfortunate restriction in the very beginning[edit]

In

Laplace's spherical harmonics[edit]

it says

Consider the problem of finding solutions of the form ƒ(r,θ,φ) = rnΘ(θ)Φ(φ), where n is a non-negative integer.


followed by the foot note:

Generally, one can consider solutions of the form R(r)Θ(θ)Φ(φ) and derive the form of R from that of Θ by solving a Sturm-Liouville equation with appropriate boundary conditions afterwards, but assuming that R has the required form in advance simplifies the exposition.


The solutions with n as a negative integer are just as present and these functions that go to zero when r goes to infinity are very important in mathematical physics. In fact, in general only these solutions with negative n are of use in physics!

Stamcose (talk) 11:33, 24 December 2009 (UTC)

A few remarks. First of all, the (surface) spherical harmonics obtained by negative homogeneity are identical with the ones obtained for positive homogeneity. Secondly, negative homogeneous solutions do not come from solving the same Sturm-Liouville eigenvalue problem because they are obviously not regular at r = 0. Instead, you need to solve the problem that comes from assuming decay at infinity. Obviously, given that the two approaches ultimately produce identical results, it is a matter of taste which of these approaches one wants to take. For me at least, having nice functions that do not blow up is more understandable. Thirdly, the treatment here is from Courant and Hilbert's Methods of mathematical physics which is a very authoritative modern source by two of the leading mathematical physicists of the first half of the twentieth century. Sławomir Biały (talk) 12:51, 24 December 2009 (UTC)


Also for me Courant-Hilbert is a favourite. Nevertheless in physics I only meet those that go to zero in infinity!

I have another comment,though:


Making the approach for a general   R(r) one gets to the differential equation

 \frac {d} {dr}\left(r^2 \frac {dR} {dr}\right) = \lambda  R

with the general solution

 c_1 r^{l_1} +  c_2 r^{l_2}

where  c_1 and  c_1 are constants of integration and  l_1 and  l_2 are the two solutions to the equation

 l \,  ( l +1) = \lambda

1) This has nothing to do with "Sturm-Liouville", the connection to "Sturm-Liouville" comes later in the analysis

2) Here one restricts oneself to integer values of  l , i.e only to those   \lambda that are of the form  \lambda = l \,  ( l +1)  are considered further

3) In mathematical physics one is in general only interested in the negative value for the integer  l (i.e. what was said already)

Stamcose (talk) 13:50, 24 December 2009 (UTC)

(1)-(2) Well, you will get no argument from me that this is the general solution of the equation (incidentally, a Sturm-Liouville equation, regardless of whether we want to pretend otherwise). However, Sturm-Lioville is used to justify confining attention to solutions of the form considered (with non-negative integer exponent) as these are the only solutions that are regular at the origin.Struck because it is regularity of Θ, not R, that gives the form of the eigenvalue. In the text, we have just assumed that the solution has the required form in advance. The purpose of the footnote is to indicate that this assumption is justified without going into the details of the justification, but apparently this purpose is unclear, as it has lead to the above misunderstanding. Perhaps it is better to include more details in the footnote. Would that satisfy your objection? (3) Do you have any reference indicating that one is only interested in spherical harmonics Y^m_\ell for negative values of \ell? This certainly disagrees with most treatments of spherical harmonics that I have seen (including Courant and Hilbert). Tables of spherical harmonics always have \ell=0,1,2,.... I should also say that I agree that many physicists are primarily interested in irregular solid harmonics for things like multipole expansions, but the surface harmonics are exactly the same in both cases. In mathematics, a spherical harmonic is a harmonic homogeneous polynomial (in any dimension). The present article has the virtue of agreeing with this convention, and it seems to be supported by most of the available literature. Sławomir Biały (talk) 14:24, 24 December 2009 (UTC)

Ok, I have included almost full details. I don't find this improves readability much (especially for someone who doesn't know the theory already), but at least it should shore up any objections to the text. Sławomir Biały (talk) 16:00, 24 December 2009 (UTC)


I am still not completely happy with the text

1)

\frac{1}{\Phi(\varphi)} \frac{d^2 \Phi(\varphi)}{d\varphi^2} = -m^2

Why \ -m^2 ?

One must write the general analytical solution to linear second order differential equation and clearly explain that only with

\frac{1}{\Phi(\varphi)} \frac{d^2 \Phi(\varphi)}{d\varphi^2} = -\mu where  \mu = m^2 for an integer m is this a unique function in 3-dimensional space defined in spherical coordinates.

2) The argument why the parameter λ has to be of the form λ = ℓ(ℓ+1) for some integer ℓ is also not clear from the text. The argument must come from the analysis of R(r).

3) It is not Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ(ℓ+1) for some integer ℓ. The Sturm–Liouville problem is the solution of the differential equation for the Legendre polynomial (see this article!)

Stamcose (talk) 22:57, 24 December 2009 (UTC)

1) See the fundamental theorem of algebra. 2) The reason is regularity of Θ. It has nothing to do with R (unless we impose regularity at the origin, which we then do, but not because it gives the form of λ). 3) Yes, this is a classic Sturm-Liouville eigenvalue problem: imposing regularity of solutions at boundary values. I'm away from home at the moment, later I can give precise references. Sławomir Biały (talk) 23:34, 24 December 2009 (UTC)
I have now made a careful effort to ensure that the content closely matches with the cited source (Courant & Hilbert). Unfortunately, it now also includes quite a lot of detail, which I had wanted to avoid, but that seemed to be necessary given the above disagreements and confusion. I hope all that this settles some of the issues that you point out. Thanks, Sławomir Biały (talk) 16:27, 25 December 2009 (UTC)

++++++++++++++++++


I would propose a text like this. It is quite more detailed and explicit and therefore in my opinion more user friendly for the average Wikipedia user then Courant-Hilbert that addresses professional mathematicians


Laplace's spherical harmonics[edit]

Real (Laplace) spherical harmonics Y_{\ell m}, for \ell=0 to 4 (top to bottom) and m=0 to 4 (left to right). The negative order harmonics Y_{\ell,-m} are rotated about the z axis by 90^\circ/m with respect to the positive order ones.

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

{d \over dx} \left[ (1-x^2) {d \over 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)


I disagree with the proposed revision. The version in the article agrees with Courant and Hilbert, to which it is currently sourced. If there are other sources of a similarly high quality that do things in a similar way, then we can discuss the merits. However, as I have already indicated, the radial equation is of only peripheral importance. Spherical harmonics are the eigenfunctions of the angular part of the Laplacian, which definitely does require solving a non-trivial eigenvalue problem (see associated Legendre function). Moreover, the proposed revision fails to explain why l must be an integer, which seems to be a serious limitation given that the original objection was that the article failed to explain precisely this point. Also, this is not the place for long tables of solid spherical harmonics. We have solid spherical harmonics and table of spherical harmonics for such things. Finally, I also generally disagree that using negative-homogeneity solid harmonics improves the presentation, but at any rate I have already said that the radial part is peripheral, and there is now a footnote to the article explaining the different conventions. Sławomir Biały (talk) 22:26, 25 December 2009 (UTC)

Also, I notice that the separation of variables is never justified. The proposed revision therefore also fails to give any indication why the radial eigenfunctions are spanned by Laplace's spherical harmonics Y_\ell^m. Sławomir Biały (talk) 22:36, 25 December 2009 (UTC)

talk[edit]

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)

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)

Missing \frac{1}{Y} in "Laplace's spherical harmonics"[edit]

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 (talkcontribs) 19:36, 31 January 2010 (UTC)

Harmoniki.png[edit]

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 (talkcontribs) 15:20, 16 November 2010 (UTC)

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)
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)
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)

I added a few comments at File_talk:Harmoniki.png. The image in the lead is beautiful, and could easily be corrected. Does anyone have the source code? 70.116.95.66 (talk) 00:40, 15 April 2013 (UTC)

Removed unsupported template from External links[edit]

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)

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)
Software

Can't understand why you removed software and other links. Them are very useful for me and may be for others.--Mikeev (talk) 12:44, 26 October 2011 (UTC)

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)

Power in signal processing[edit]

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 (talkcontribs) 12:56, 18 April 2012 (UTC)

Comprehensibility[edit]

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)

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)
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)
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)
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)

style issue and image placement[edit]

image in the article is the most confusing one in the whole article because spherical harmonics are defined on a sphere of fixed radius and therefore do not depend on radius, but this image makes it look like they are functions of radius. I propose switching the placement of this first image with the image lower down that shows the spherical harmonics on a surface of a sphere. 129.63.129.196 (talk) 19:38, 11 October 2012 (UTC)

For what its worth, the first image is a visualization of r=Y(φ,θ). Since this visualization apparently has other issues (see previous comments; IIRC I've tried to resolve this issue on pl: as well), I have no problem with the reconfiguration of images on this page. Although I feel like an adequate caption, and possibly a correction to the original first image is ideal. Sławomir Biały (talk) 00:09, 12 October 2012 (UTC)
129.63.129.196 is right, File:Harmoniki.png would mislead a reader who did not make a significant effort to understand the representation used. The pictures File:Spherical harmonics.png and File:Harmoniques spheriques positif negatif.png in the Visualization of the spherical harmonics section are the clearest. However, a radial plot could also be clear if it showed small-amplitude deviations from a basically spherical surface.
The File:Rotating spherical harmonics.gif animation is not bad, but it would be better if it showed oscillating standing waves, rather than equatorially travelling waves. Even the most casual reader could then see that waves were represented, and it wasn't the case that spheres were simply bodily rotating. --catslash (talk) 12:01, 12 October 2012 (UTC)

Orbital angular momentum[edit]

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)

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)

Boy, do I feel stupid, but thanks.Linkato1 (talk) 16:00, 19 October 2012 (UTC)

Lets put the correct calculation here so that others see how L_-L_+ = \mathbf{L}^2 - L_z^2 -L_z is derived.


2L_z = [L_+,L_-] = L_+L_- - L_-L_+ = (L_x + iL_y)(L_x - iL_y) - (L_x - iL_y)(L_x + iL_y) = L_x^2 - iL_xL_y + iL_yL_x + L_y^2 - L_x^2 - iL_xL_y + iL_yL_x - L_y^2


= -2iL_xL_y + 2iL_yL_x) = -2i(L_xL_y - L_yL_x)


=> i(L_xL_y - L_yL_x) = - L_z


L_-L_+ = (L_x - iL_y)(L_x + iL_y) = L_x^2 + iL_xL_y - iL_yL_x + L_y^2


= \mathbf{L}^2 - L_z^2 + i(L_xL_y - L_yL_x) = \mathbf{L}^2 - L_z^2 - L_z


Linkato1 (talk) 09:47, 20 October 2012 (UTC)

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

Typo in real (tesseral) spherical harmonics?[edit]

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)