# Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few sorry excuses for "derivations" usually just depend on counter intuitive generalized notions after spherical harmonics were found.

There is not a single article, page, document on the entire Internet which actually writes this up in a meaningful and intuitive way and everyone who is using spherical harmonics pretty much has no idea how they work, only what they can do for them.

I am not interested in proofs of pre-existing "definitions" as if they fell out of the sky and we just had to convince ourselves of them. I desire a profound understanding of them in a way which is natural to humans. I am not immediately interested in all the rigorous mathematical underpinnings and the incessant generalizations which only make me hate the subject, where the derivations depend on recreating the before known in artificial ways.

I just wish to understand how spherical harmonics tick, the simplest and most important strokes. How to start with, possibly, an issue in physics and derive spherical harmonics from it... Someone went through the exhilarating process of converting the Laplace's equation from cartesian to spherical coordinates, so it had to have been really inspired. I did it once, I think it took an entire landscape A4 page. I dread to think someone just poked about until it popped out.

I've searched this site for an answer, none has offered anything useful. It's all convolved restating of the obvious.

Please, help me understand this. Is there a book which doesn't start the chapter on spherical harmonics with "Spherical harmonics are defined as..."? An online article hidden deep on the Interwebs?

One that just doesn't plug in the associated Legendre polynomials, but gives a strong intuitive rationale and a way to derive them. Legendre sure didn't wake up one day with them in his mind.

Could a brilliant soul derive it here in the simplest possible way?

• "Historically accurate", "intuitive" and "simplest possible" are usually mutually incompatible. – Robert Israel Apr 7 '15 at 23:29
• Simplest possible may be as complex as it has to be, but not more complex than that. And I would prefer intuitive, if it clashes with historically accurate. – James Reid Apr 7 '15 at 23:30
• Anything you guys (and gals) can do to help would be appreciated! – James Reid Apr 7 '15 at 23:32
• Start with the wave equation in three dimensions, impose spherical constraints, and perform the derivatives, as outlined here: eng.fsu.edu/~dommelen/quantum/style_a/nt_soll2.html. I developed my intuition of them (in graduate quantum mechanics) by considering limiting cases, e.g., a purely radial solution, or a purely angular solution. – David G. Stork Apr 7 '15 at 23:35
• Thanks, I'll take a look! If anyone has more valuable material to add, just drop it inside here. I don't want to impose or have you do the work for me, I just require some help/guidance to get to the solution. – James Reid Apr 7 '15 at 23:39