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Spherical Harmonics

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Re: Spherical Harmonics

#41

Post by ziotom78 »

Scene_Cast2 wrote: post #23

If anyone is curious about applications - these can be used to approximate low-frequency components of a point's surroundings. They were used in Halo 3 to do real-time HDRI lighting and shadowing (see "Lighting and Material of Halo 3" from Siggraph 2008). After the success of this method, there was a fairly long stretch of researchers looking for a better orthonormal basis (such as 2D Haar wavelents, as spherical har…

They are used also to characterize the statistical properties of fields over the sphere. A notable example is the pattern of hot/cold spots in the Cosmic Microwave Background Radiation (CMBR, [1]). They are distributed stochastically, and the best way to fit cosmological models against the measurements is to decompose the temperature/polarization fields into spherical harmonics and compute the power spectrum associated with each ℓ (which plays the role of a “spatial frequency” over the sky sphere).

[1] https://en.wikipedia.org/wiki/Cosmic_microwave_background

Re: Spherical Harmonics

#42

Post by dustingetz »

Scene_Cast2 wrote: post #23

If anyone is curious about applications - these can be used to approximate low-frequency components of a point's surroundings. They were used in Halo 3 to do real-time HDRI lighting and shadowing (see "Lighting and Material of Halo 3" from Siggraph 2008). After the success of this method, there was a fairly long stretch of researchers looking for a better orthonormal basis (such as 2D Haar wavelents, as spherical har…

in QM the spherical harmonics are more of a basis space for electronic state and not the actual electronic state, right? So does that mean there are other ways to think about electron configurations that satisfy Shrodinger etc?

Re: Spherical Harmonics

#45

Post by setopt »

ajkjk wrote: post #18
ghostpepper wrote: post #7

Anyone know a good explanation of what spherical harmonics are?

Just for fun... They are relatively easy to understand if you already understand Fourier transforms. In a Fourier transform you can write some (suitably well-behaved) function f(x) as a sum of a bunch of sinusoids of different frequencies: f(x) = a_0 + a_1 cos(x) + a_2 cos(2x) + ... + b_1 sin(x) + b_2 sin(2x) + ... Or more generally a sum over all real values, f(x) = ∫ a(k) cos(kx) + b(k) sin(kx) dk, since signals ca…

Excellent explanation.

> So the spherical harmonic functions are also the shape of the various electron orbitals that you see in a chemistry textbook.

Minor nitpick: Chemistry textbooks usually use the “cubic harmonics” instead of the “spherical harmonics”. They are real-valued linear combinations of the standard spherical harmonics, with the additional benefit that the basis set respects the Cartesian symmetries.

For example, the “p_z orbital” is a l=1 spherical harmonic and a cubic harmonic. But the cubic harmonics then add p_x and p_y orbitals as basis functions, whereas the spherical harmonics choose the chiral "p_x ± ip_y orbitals” as its basis instead.

Re: Spherical Harmonics

#46

Post by setopt »

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.

That’s a very quantum physics-centric explanation though (username checks out…).

A more general definition is that they are eigenfunctions of the Laplacian operator on a sphere, which arise in many contexts.

Re: Spherical Harmonics

#47

Post by JeremyHerrman »

JeremyHerrman wrote: post #13

For those of you curious about WHY these shapes look like the do (e.g. "why does l=0, m=0 have a donut in the middle of two lobes?"), this video from Münster University finally gave me an intuitive understanding of how these shapes arise. https://youtu.be/Opufc3onVow

typo here, l=2,m=0 is the orbital with the donut

Re: Spherical Harmonics

#48

Post by ajkjk »

setopt wrote: post #45
ajkjk wrote: post #18

Earlier quoted context omitted.

Just for fun... They are relatively easy to understand if you already understand Fourier transforms. In a Fourier transform you can write some (suitably well-behaved) function f(x) as a sum of a bunch of sinusoids of different frequencies: f(x) = a_0 + a_1 cos(x) + a_2 cos(2x) + ... + b_1 sin(x) + b_2 sin(2x) + ... Or more generally a sum over all real values, f(x) = ∫ a(k) cos(kx) + b(k) sin(kx) dk, since signals ca…

Excellent explanation. > So the spherical harmonic functions are also the shape of the various electron orbitals that you see in a chemistry textbook. Minor nitpick: Chemistry textbooks usually use the “cubic harmonics” instead of the “spherical harmonics”. They are real-valued linear combinations of the standard spherical harmonics, with the additional benefit that the basis set respects the Cartesian symmetries. Fo…

Oh thanks, I didn't know that (I studied physics and only know the chemistry part superficially). But I did know, and wondered about, the fact that the spherical harmonics have an e^imϕ factor yet there's nothing about the explanation that involves complex numbers per se (besides that writing out sine/cosine series is way more tedious than exponentials). Makes sense that they just get factored.

Re: Spherical Harmonics

#49

Post by jasomill »

setopt wrote: post #46
quantum_state wrote: post #32

Earlier quoted context omitted.

They are just eigen states of the angular momentum operator.

That’s a very quantum physics-centric explanation though (username checks out…). A more general definition is that they are eigenfunctions of the Laplacian operator on a sphere, which arise in many contexts.

More generally, see also the Laplace-Beltrami and Laplace-de Rham operators, both defined on (pseudo-)Riemannian manifolds not necessarily embedded in ℝ³.

Even more generally, see the huge body of beautiful mathematics that has arisen from the study of elliptic differential operators in general (de Rham cohomology, Hodge theory, the Atiyah-Singer index theorem, …).

Re: Spherical Harmonics

#50

Post by rsfern »

dustingetz wrote: post #42
Scene_Cast2 wrote: post #23

If anyone is curious about applications - these can be used to approximate low-frequency components of a point's surroundings. They were used in Halo 3 to do real-time HDRI lighting and shadowing (see "Lighting and Material of Halo 3" from Siggraph 2008). After the success of this method, there was a fairly long stretch of researchers looking for a better orthonormal basis (such as 2D Haar wavelents, as spherical har…

in QM the spherical harmonics are more of a basis space for electronic state and not the actual electronic state, right? So does that mean there are other ways to think about electron configurations that satisfy Shrodinger etc?

Yes - for example when modeling solid materials it’s common to use a plane wave basis set for the electronic wave function instead of atomic orbitals
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