Spherical Harmonics • Hacker News Board

Re: Spherical Harmonics

#61

Post by dahart » Mon, Dec 23, 2024, 3:54 PM UTC

Applejinx wrote: post #40

xeonmc wrote: post #33

Earlier quoted context omitted.

a vibrating string make sine waves, a vibrating bubble make spherical harmonics.

Love it, that's a fantastic image for these. As a musician intimately aware of how vibrating strings work, the vibrating bubble is a perfect analogy, it seems.

Musicians: see also radial planar circular harmonics, e.g. drums. This is a halfway point in between vibrating strings and vibrating bubbles. It’s highly applicable in music and is easier to understand and visualize than 3d basis functions; maybe a nice stepping stone.

https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membr...

Re: Spherical Harmonics

#62

Post by Koshkin » Mon, Dec 23, 2024, 4:25 PM UTC

quantum_state wrote: post #32

ghostpepper wrote: post #7

Anyone know a good explanation of what spherical harmonics are?

They are just eigen states of the angular momentum operator.

Did you mean that they are basis functions for irreducible representations of SO(3)?

Re: Spherical Harmonics

#63

Post by momoschili » Mon, Dec 23, 2024, 7:32 PM UTC

ajross wrote: post #57

momoschili wrote: post #30

Earlier quoted context omitted.

not quite as they are missing the radial dependence

It's actually more confusing IMHO, because these graphs overload the radial dimension to show probability as "distance from the origin". You have to multiply that by the radial function to get an actual probability distribution, which kinda/sorta looks like these pictures but not really. Really the harmonics are best understood as something like "wave height on the surface of a sphere". They tell you how the electron…

Simply speaking, "l" describes the number of nodes. In the same sense that a particle in a box with sin(nx) wave function has more nodes the higher energy (or momentum) state it is in.

As for why l==0 has no rotation going on at all, one would say that this should be expected. Qualitatively, the symmetric sphere does not change with rotation, so how would we tell if it is rotating or not? And perhaps the next step is controversial, but if there is no way to tell, maybe there is no dependence? This is a similar argument to why the electric field of an infinite plane is constant with respect to distance from it.

Re: Spherical Harmonics

#64

Post by dagss » Mon, Dec 23, 2024, 10:16 PM UTC

TL;DR about spherical harmonics: It is what you use instead of Fourier transforms if what you transform is on the surface of a sphere.

My experience is from cosmology (CMB) where they are heavily used just like Fourier transforms, I think they are also used in meteorology.

Re: Spherical Harmonics

#65

Post by nextaccountic » Fri, Dec 27, 2024, 7:40 AM UTC

ghostpepper wrote: post #7

Anyone know a good explanation of what spherical harmonics are?

An one dimensional wave can be decomposed as a sum of sines/cosines. So sines form a vector basis in the space of periodic functions

A 3D wave coming from a single point can be decomposed as a sum of spherical harmonics. It's a basis for waves, but ones expanding radially in 3D space