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I've learned about spherical harmonics (Legendre polynomials $\longrightarrow$ Associated Legendre polynomials $\longrightarrow$ orthogonality relations $\longrightarrow$ normalization coefficient(s) $\longrightarrow$ laplace's equation in spherical coordinates [not exactly in this order, and with other tidbits here and there]). However, I've only done a few spherical harmonics expansions, such as simple symmetric functions, functions explicitly given by a finite number of spherical harmonics (in cartesian or cylindrical coordinates usually), etc.

Given this, I'd say my knowledge/understanding of spherical harmonics is minimal. I wanted to know, how important is a thorough understanding of spherical harmonics for working physicists/cosmologists/scientists? Are they of much use outside of working problems in an electromagnetism and/or quantum mechanics class? Would it be worthwhile to study and work them in a lot more detail?

(I didn't know if this question should be tagged as a soft-question, so please forgive me if it should)

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closed as primarily opinion-based by ACuriousMind, Gert, user36790, John Rennie, Kyle Kanos Nov 17 '15 at 11:13

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Yes, They are very much worth studying. They even find applications in heat conduction problems. $\endgroup$ – Gert Nov 17 '15 at 2:38
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    $\begingroup$ Do you mean Legendre polynomials instead of Laguerre? $\endgroup$ – Ryan Unger Nov 17 '15 at 2:41
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    $\begingroup$ Somebody should talk about the decomposition of the CMB spectrum into spherical harmonics, as an example. $\endgroup$ – Physics_Plasma Nov 17 '15 at 3:58
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    $\begingroup$ They're the basis functions which diagonalize the angular part of the Laplacian in spherical coordinates. Any time you care about solving anything having to do with waves on a sphere you are lead to the spherical harmonics because they make the math easier. $\endgroup$ – DanielSank Nov 17 '15 at 4:04
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    $\begingroup$ This question (v2) seems to be a list question. $\endgroup$ – Qmechanic Nov 17 '15 at 8:38
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In electrodynamics spherical harmonics and the multipole expansion helps you figure out radiation from antennas. You sort of have to know this if you're going to have a serious theoretical conversation about electrodynamics.

If you're looking at the finer details of Earth's gravitational field, storing information in the form of spherical harmonic coefficients is useful.

If you're trying to understand small atoms or molecules, you are usually going to have a lot of spherical harmonics around in the electron orbitals, because the single electron hydrogen atom can be solved in terms of spherical harmonics. So it's a jumping off point there.

In general they fit into the much broader class of Sturm-Liouville problems, where the whole orthogonality relation/normalization coefficient method is in pretty nice generality. This would be in any mathematical methods of physics class (undergraduate and graduate physics level) and many partial differential equations courses. For good reason! These apply even when you want to use numerical methods.

See also: https://en.wikipedia.org/wiki/Multipole_expansion#Applications_of_multipole_expansions

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