Re:Circular Harmonics
by cshimmin on Friday October 29, 2021 @08:07PM
(#29044878)
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.
Re:Circular Harmonics
by gus_massa on Saturday October 30, 2021 @01:42PM
(#29050714)
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.
Re:Circular Harmonics
by xyzzy21 on Saturday October 30, 2021 @09:44AM
(#29048700)
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.
Re:Circular Harmonics
by xyzzy21 on Saturday October 30, 2021 @09:42AM
(#29048674)
Actually it's a Fourier series (or even a Laplace transform) with a coordinate frame change - spherical instead of cartesian (another is cylindrical coordinates).
(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)
Re:Circular Harmonics
by frutiger on Friday October 29, 2021 @06:30PM
(#29044258)
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.
Re:Circular Harmonics
by robotsteve2 on Friday October 29, 2021 @07:37PM
(#29044683)
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.
> Circular Harmonics are just a Fourier series
The term circular harmonics is just a nonstandard name for Fourier series.
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
Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics
I (GGP comment) was wrong.
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.
Normal everyday stuff for EEs and Physicists.
even though I made this http://particlo.org/20 for selflearning and experimentation
I still don't underestand how to simulate circular membranes
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)
https://twitter.com/BenHouston3D/status/1435621305659822083/...
Edit: orbitals
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.
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.
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.
Check out these visuals from “the orbitron”: https://winter.group.shef.ac.uk/orbitron/
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?