Introduction to Spherical Harmonics(puye.blog)
puye.blog
Introduction to Spherical Harmonics
https://puye.blog/posts/SH-Introduction-EN/
8 comments
I can't tell if it's all tongue-in-cheek, or it's accidental due to not being a native English speaker.
For example, this line is also pretty funny:
> If you have learned Taylor Expansion or Fourier Transform in Primary School, you will be familiar with Basic Functions.
But then it uses "Basic" everywhere when it means "Basis", and I cannot see how that's intentional.
For example, this line is also pretty funny:
> If you have learned Taylor Expansion or Fourier Transform in Primary School, you will be familiar with Basic Functions.
But then it uses "Basic" everywhere when it means "Basis", and I cannot see how that's intentional.
The author of this post seems to have put their own watermark these images which they did not create or cite? For example I recognize some made by inigo quilez.
Spherical Harmonics became extremely trendy in the graphics literature for dynamic lighting and pre-baked light maps of some sort, and completely dropped off at some point. Still remember some discussions and questions regarding how they're not easy to "orient".
They're an excellent solution for dynamic objects w/o too much hardware reqs(PS3 era HW should handle it easily), I think they're still used in tons of places.
It's just that most usage is through finished impls on lower end HW like Unity, Godot,etc and those that implement their own engines these days probably skip them for more global methods directly or just go for good-enough simpler solutions to get "gi-like" appearance like screen space ambient occlusion and/or cubemaps.
Also I think once people finds Greens walkthrough they can get much from it (or give up).
https://www.cse.chalmers.se/~uffe/xjobb/Readings/GlobalIllum...
It's just that most usage is through finished impls on lower end HW like Unity, Godot,etc and those that implement their own engines these days probably skip them for more global methods directly or just go for good-enough simpler solutions to get "gi-like" appearance like screen space ambient occlusion and/or cubemaps.
Also I think once people finds Greens walkthrough they can get much from it (or give up).
https://www.cse.chalmers.se/~uffe/xjobb/Readings/GlobalIllum...
This page seems to use "basis functions" and "basic functions" interchangeably, which is confusing.
The feeling you get from deriving these for the first time from Schrodinger's Equation the after 100 sheets of calculations... Pardon my French, but it's almost a mental orgasm
You are not alone: had a physics professor who appeared to have a mental orgasm whenever spherical harmonics arose in lecture. Perhaps he also once did the "100 sheets of calculations...".
Given a difficult problem, we would joke the solution was obvious - use an expansion of spherical harmonics - voila!
Given a difficult problem, we would joke the solution was obvious - use an expansion of spherical harmonics - voila!
Nice to see how these functions are used to describe the familiar s-, p-, d-, f-orbitals of chemistry:
https://en.wikipedia.org/wiki/Atomic_orbital
https://en.wikipedia.org/wiki/Atomic_orbital
Why are there 2 colors of lobes in the pictures? One seems to be the same as the other but with a coefficient of -1, but why show that?
Because as you sweep an angle about the center, the points where the function is negative "skips over" to the other side. Its unclear visually that you are not getting 4 flower leafs as the angle rotates, but rather a figure 8 pattern.
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Isn’t this a good model for the hydrogen atom?
But then not for other elements?
But then not for other elements?
Spherical harmonics are a good model for any spherical density function. You could describe the electron probability densities of a particular energy level of any atom in terms of spherical harmonic coefficients.
I think it just happens that hydrogen atoms electron clouds have particularly simple sets of coefficients so their density functions look a lot like the raw harmonic bases.
That’s more because hydrogen atoms are simple than because of anything profound.
Like, when something vibrates in 1D, if it’s simple (like a mass on a spring) it will move in a simple sine wave - the Fourier coefficients describing its behavior will be simple, because it is simple.
But if it’s more complicated (like, a lump of jelly on a linked set of springs) it will move in a complex wave, with complex Fourier coefficients. Its behavior will be complex because it is complex.
That hydrogen electron clouds look like these lobed shapes then is as unsurprising - or possibly as surprising, depending how you feel about the unreasonable power of mathematics I guess - as the fact that a weight on a spring follows a sine wave.
I think it just happens that hydrogen atoms electron clouds have particularly simple sets of coefficients so their density functions look a lot like the raw harmonic bases.
That’s more because hydrogen atoms are simple than because of anything profound.
Like, when something vibrates in 1D, if it’s simple (like a mass on a spring) it will move in a simple sine wave - the Fourier coefficients describing its behavior will be simple, because it is simple.
But if it’s more complicated (like, a lump of jelly on a linked set of springs) it will move in a complex wave, with complex Fourier coefficients. Its behavior will be complex because it is complex.
That hydrogen electron clouds look like these lobed shapes then is as unsurprising - or possibly as surprising, depending how you feel about the unreasonable power of mathematics I guess - as the fact that a weight on a spring follows a sine wave.
It has to do with representations and quantum mechanics. The energy eigenvalues of the hydrogen atom are tied to L values of the spherical harmonics.
If a quantum system is spherically symmetric, like the hydrogen atom is, then its energy states are gonna correspond to certain groups of spherical harmonics. That's why they're used when you study the hydrogen atom.
This comes out of a branch of math called representation theory.
This comes out of a branch of math called representation theory.
There's always perturbation theory for that
Basically "JPEG in 3D polar coordinates"
"This article tries to explain Spherical Harmonics in simple words without mathematical terms."
Immediately followed by: Taylor Expansion, Fourier Transform, Polynomial Basis Functions, Trigonometric Basis Functions, Square Wave Function