Circular Harmonics | Hacker News Reader

Circular Harmonics

valdes.cc

37 pointskelseyfrog26 comments

The key point which the author eventually realizes:

> Circular Harmonics are just a Fourier series

The term circular harmonics is just a nonstandard name for Fourier series.

I think the standard name is "Besell Functions" https://en.wikipedia.org/wiki/Bessel_function

Not really. In Mathematics "circle" refers to a boundary of a disk. So it's a 1D periodic manifold, and the circular harmonics/Fourier series are the basis for such periodic functions. The bessel functions give a basis for the radial part of 2D solutions to a certain class of differential equations on a 2D disk. Incidentally the radial part of those solutions are circular harmonics.

You are right.

And there are many. Zernike polynomials are a common one.

Not a wizard or anything but reading wikipedia:

Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics

Bessel functions are useful when you must solve the volume of the cylinder (or the volume outside a cylinder). If you only care about the surface of the cylinder, you must use the Fourier series in one direction and the Fourier transform in the other direction.

I (GGP comment) was wrong.

No. Bessel functions are the solution to many differential equations posed in cylindrical coordinates.

Legendre polynomials are what we are seeing here which are solutions to many differential equations posed in spherical coordinates.

Classical sine/cosine Fourier solutions are the solutions to many differential equations posed in cartesian coordinates.

So these spherical solutions should also remind people of atomic quantum "orbital" shapes because those are solutions to Schrödinger's (differential) equations in spherical coordinates.

We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started

And know the place for the first time.

T. S. Eliot - Little Gidding

Actually it's a Fourier series (or even a Laplace transform) with a coordinate frame change - spherical instead of cartesian (another is cylindrical coordinates).

Normal everyday stuff for EEs and Physicists.

Wow, that is a beautiful animation at the end there.

I would like an article that explains for dummies the exact equations to generate https://www.acs.psu.edu/drussell/demos/membranecircle/circle...

even though I made this http://particlo.org/20 for selflearning and experimentation

I still don't underestand how to simulate circular membranes

Eigenfunctions of the Laplacian on the disk.

Dude really needs to subscribe to 3Blue1Brown:

https://www.youtube.com/watch?v=spUNpyF58BY

(but anyone interested in stuff like this should be. I guess I'm surprised to see someone like this writing a blog post who isn't regularly checking out that channel)

I noticed that neural networks identified circular harmonic like patterns automatically:

https://twitter.com/BenHouston3D/status/1435621305659822083/...

Are electron orbitals spherical harmonics?

Edit: orbitals

Yes.

More details: The orbitals are the product of an Harmonic Orbital and a radial function. More details in https://en.wikipedia.org/wiki/Atomic_orbital#Types_of_orbita... and https://en.wikipedia.org/wiki/Hydrogen-like_atom#Non-relativ...

Note that this is the initial selection of orbitals. The "real"[1] orbitals are linear combinations of them. In some cases it's good enough to mix them using some easy combinations like in https://en.wikipedia.org/wiki/Orbital_hybridisation but in the general cases you must use a computer to calculate the best combination https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method .

[1] It get's more complicated.

Electrons do not orbit.

For the simple case of an electron in a hydrogen atom, energy-stable wavefunctions are products of three separate functions: exponential function, a laguerre polynomial function and a spherical harmonic.

Presumably the GP meant orbitals.

It's an important distinction. A lot of people would like to think electrons orbit the nucleus like planets orbit a star.

Yes, but someone thinking in terms of the Bohr model would not ask is the orbits are (described by) spherical harmonics.

How might spherical harmonics be relevant for quantum computing? I’m on the hunt for visualizations. (Not to be confused with the Bloch Sphere!)

Spherical harmonics are a useful mathematical basis set for eigenfunctions of a Schrodinger equation describing a hydrogen atom like system.

There's nothing particularly special about them, and they fail to be a good basis set for solutions of other types of quantum systems. Conversely, they can be useful for non-quantum systems - any sort of wave equation in a sphere.

It's not an intrinsically quantum mechanical thing.

Idk… seems intrinsic in so far as waves are intrinsic to Quantum Mechanics.

Check out these visuals from “the orbitron”: https://winter.group.shef.ac.uk/orbitron/

I can see only two images in the article:

Shape of the first 6 Circular Harmonic bands

A series of CH approximating a pulse with more and more bands

The other “images” only have captions.

Is this a (mobile) safari problem?

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