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In this article https://mrtrix.readthedocs.io/en/dev/concepts/spherical_harmonics.html the following statement is given:

Spherical harmonics are special functions defined on the surface of a sphere. They form a complete orthonormal set and can therefore be used to represent any well-behaved spherical function.

I have two questions:

1. What does it means "functions defined on the surface of a sphere"? Functions whose domain is given by $(\theta,\phi)$ with $\theta \in [0, \pi]$ and $\phi \in [0,2\pi]$?

2. Why spherical harmonics form an orthonormal basis for the space of functions defined on the sphere? Since spherical harmonics are also simultaneous eigenfunctions of operators $L^2$ and $L_z$, don't they form a basis for any function?

Salmon
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  • "Since spherical harmonics are also simultaneous eigenfunctions of operators $L^2$ and $L_z$, don't they form a basis for any function?" - Why do you think so? – Filippo Feb 17 '22 at 17:18
  • @Filippo If 2 operators commute, they have a set of simultaneous eigenfunctions which is also a basis. – Salmon Feb 17 '22 at 17:26
  • Yes, but why do you believe that the spherical harmonics are not only simultanous eigenfunctions, but even a basis for the functions on $\mathbb R^3$? – Filippo Feb 17 '22 at 17:36
  • Since spherical harmonics aren't functions defined on $\mathbb R^3$, this doesn't even make sense. – Filippo Feb 17 '22 at 17:37
  • @Filippo Ok so this is the point, spherical harmonics are a basis only for functions defined on the sphere? – Salmon Feb 17 '22 at 17:39
  • Yes. But the spherical harmonics are also useful to construct a basis for the functions on $\mathbb R^3$. If $Y=Y(\theta,\phi)$ is a function on the sphere and $R=R(r)$ is defined on the positive real numbers, then $f(r,\theta,\phi):=R(r)Y(\theta,\phi)$ is a function on $\mathbb R^3$ (via spherical coordinates). – Filippo Feb 17 '22 at 17:48
  • @Filippo Ok thank you. Can you help with my first question? What do we mean by functions defined on the sphere? – Salmon Feb 17 '22 at 18:06
  • I will add an answer. – Filippo Feb 17 '22 at 18:50
  • Possible duplicate in the Related sidebar: Spherical harmonics for dummies. – march Feb 17 '22 at 18:51
  • I need to know how much you already know. You know about spherical coordinates, don't you? – Filippo Feb 17 '22 at 19:42
  • @Filippo yes of course. I know about Laplacian in spherical coordinates, I know how to obtain spherical harmonics from laplace equation. – Salmon Feb 17 '22 at 20:45

1 Answers1

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What does "functions defined on the surface of a sphere" mean?

You can literally define the spherical harmonics on a sphere:

In your article, the following formula is given: $$Y_l^m(\theta,\phi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos \theta) e^{im\phi}$$ Clearly, $Y(\theta,\phi)$ is well defined for all $(\theta,\phi)\in\mathbb R\times\mathbb R$. On the other hand\begin{align}F\colon[0,\pi]\times\mathbb R&\to S\\(\theta,\phi)&\mapsto\begin{pmatrix}\sin\theta\cos\phi\\\sin\theta\sin\phi\\\cos\theta\end{pmatrix}\end{align} is a surjective function to the sphere. Thus, if a spherical harmonic $Y$ is constant on the level sets of $F$, we can define $Y$ on the sphere by requiring that $$(Y\circ F)(\theta,\phi)=Y(\theta,\phi)$$for all $(\theta,\phi)\in[0,\pi]\times\mathbb R$ (note the slight abuse of notation).

Well, it turns out that spherical harmonics are indeed constant on level sets of $F$: For the north pole and the south pole, you can find the explanation here and for the other points on the sphere - i.e. the points with $0<\theta<\pi$ - this is easy to prove.

Filippo
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  • I didn't understand anything. What is S? What is F? Why spherical harmonics are defined on $R\times R$ and not on $[0,π]\times[0,2π]$? And I still can't get the definition of functions defined on a sphere. as far as I've understood, a functions is defined over a sphere if it is a functions of type: $f(\theta,\phi):[0,π]\times[0,2π] \rightarrow \mathbb{R}$, but I can't understand what your answer is about. – Salmon Feb 18 '22 at 13:53
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    @Salmone $S$ is the sphere with radius $1$:$$S:={x\in\mathbb R^3:|x|=1}$$ – Filippo Feb 18 '22 at 15:42
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    @Salmone Of course, we can consider spheres centered in different points and with different radii. But given some sphere, it makes sense to choose its center as the origin of our coordinate system and its radius as our unit of length, so that we always end up working with $S$. – Filippo Feb 18 '22 at 15:50
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    @Salmone "And I still can't get the definition of functions defined on a sphere" - When I say that a function $f$ is defined on the sphere, I mean it, i.e. the sphere $S$ is the domain of $f$. The set $[0,\pi]\times[0,2\pi]$ is a rectangle, not a sphere. – Filippo Feb 18 '22 at 15:53
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    @Salmone Is everything clear up to this point? – Filippo Feb 18 '22 at 16:04
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    Yep, thank you. – Salmon Feb 18 '22 at 17:16
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    @Salmone I'm glad to hear that. To put it into a nutshell: If $$F\colon [0,\pi]\times\mathbb R\to S$$ was bijective, then given a spherical harmonic $Y\colon [0,\pi]\times\mathbb R\to\mathbb C$ we could simply take the composition $Y\circ F^{-1}\colon S\to\mathbb C$ to obtain a function defined on $S$. However, $F$ is not injective, this is why we need to verify that $Y$ is constant on the level sets of $F$: Since $F$ is surjective, it induces a bijective function $\tilde{F}$ on the set of all level sets, and since $Y$ is constant on the level sets, – Filippo Feb 18 '22 at 17:47
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    it induces a function $\tilde{Y}$ on the set of all level sets. Thus, we can consider the composition $\tilde{Y}\circ\tilde{F}^{-1}\colon S\to\mathbb C$. – Filippo Feb 18 '22 at 17:47
  • No, I can't understand anything of this. Why do I need to know all of these things? I just wanted to know the definition of functions defined on a sphere. – Salmon Feb 18 '22 at 17:53
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    @Salmone As you pointed out, the article says that spherical harmonics are defined on the sphere, but the formula that is given in that article does not define the spherical harmonics on the sphere, but on $\mathbb R\times\mathbb R$. I explained how you can use that formula do define the spherical harmonics on the sphere. Unfortunately, this involves a lot of technicalities. Anyways, the conclusion is that you can indeed think of the spherical harmonics as being defined on the sphere. – Filippo Feb 18 '22 at 18:01
  • I think I'll go with that because the rest is too hard for me. Thank you. – Salmon Feb 18 '22 at 18:03
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    @Salmone Thank you for the discussion - when I had my QM lecture, I had similar issues and it was nice to revisit that stuff. I guess this was too much at once, but if you have any questions later on, don't hesitate to ask :) One more comment (I hope that you'll find it intriguing and understand it properly later on): The reason for all this complications is essentially that the sphere is nontrivial manifold - we can not cover it by a single chart. But it is "nearly" a trivial manifold: If we restrict $F$ to the open rectangle $]0,\pi[\times]0,2\pi[$, then $F$ is injective and – Filippo Feb 18 '22 at 19:12
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    "nearly" surjective, in the sense that the image contains all points of the sphere apart from the line connecting the north pole and the south pole with $\phi=2\pi$. But if integrate over the sphere, this doesn't really matter (since the area of a line is zero), that's why this is swept under the rug. – Filippo Feb 18 '22 at 19:19
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    Thank for your time and kindness. Maybe in ten years I'll try to understand "trivial manifold". – Salmon Feb 18 '22 at 19:58