Spherical Harmonics (U2-05-05)
ฝัง
- เผยแพร่เมื่อ 5 ก.ย. 2024
- We describe the possible fundamental vibrations on a sphere in three dimensions by counting, mirroring and rotating nodal lines.
This video ist part of the online course www.quantumreflections.net dealing with quantum physics, produced by the institute for physics education research, Münster university (Germany)
For more videos on quantum physics, please visit:
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Best video on spherical harmonics yet. This video actually gives you an intuition for why those seemingly alien-looking blobs look the way they do. Mathematically, the spherical harmonics are the angular solution of Laplace's equation in spherical coordinates and form an orthogonal basis for all functions on a sphere (sort of like a generalization of a Fourier series.)
Could you explain this point more?
@@captainhd9741 Laplace's equation essentially takes on the spatial part of a wave equation, so its solutions are just like what you'd get if you solved the wave equation, but these solutions will be static (the partial with respect to time would yield zero, recovering the general form of Laplace's equation), so this is how I think of Laplace's equation giving you the set of standing waves given a set of boundary conditions. By setting up Laplace's equation in spherical coordinates and solving it via separation of variables, you find that the angular part gives you the spherical harmonics and the radial part given by the Bessel functions, which together form the set of all possible standing waves on a sphere (hence the name spherical harmonics.) By solving Laplace's equation in polar coordinates (2d analog of spherical coordinates) you find that the angular part is really just a set of sines and cosines, which form the basis of a Fourier series. Extending this notion into three dimensions, the spherical harmonics end up forming a basis of a 3-dimensional Fourier series, and just like a regular Fourier series, you can decompose any function on a sphere into a linear combination of spherical harmonics. The intuition behind this I don't fully understand, I just learned this without really getting an intuition for why it works. If you want to dive into the math, I can leave a few papers discussing this in detail:
mathworld.wolfram.com/SphericalHarmonic.html
cs.dartmouth.edu/wjarosz/publications/dissertation/appendixB.pdf
@@northernskies86 Thanks a lot for taking the time typing this up! All I am on equal footing with is that the Laplacian operator in spherical coordinates for the Schrodinger equation has solutions that are spherical harmonics. I still didn't really understand the stuff about Fourier series and I am also unfamiliar with Laplace's equation as this is a first exposure for me.
they look they way they do because they are eigen-shapes of rotations around the z-axis with eigenvalue exp(im theta).
which is very much like Fourier basis functions exp(ik x), which are eigen values of translations x -> x + dx with eigenvalue exp(ik [delta x]).
The reason spherical harmonics are more complicated is because, while (x, y, z) translations commute, general rotations do not.
I like the nodal-line break down, because it captures a major property trivially: the number of nodal lines is not changed by an arbitrary rotation, so each set of 2L + 1 shapes is closed under rotations, e.g., if you rotate the z-axis to the x-axis, the new shape is a linear combination of m = +/- 1.
From that you should convince yourself that summing the |squares|of all the "m" shape for fixed L results in a perfect spherical shell.
Also, you can see that L=1 can be mixed to represent x, y, z...that is: they are the true shape of vectors, while m=0 is a scalar (doesn't change under rotations), and from there...the 5 L=2 shapes represent that shape of rank-2 cartesian tensors that are symmetric, and traceless in all indices. This can be taken to higher N, so N=4 will give you insight into the generalized Hooke's Law, relating the rank 2 stress and strain tensors with a rank-4 "spring constant = elasticity tensor".
No, really, master Y_lm and anisotropic mechanical/dielectric materials, atomic orbital, nuclear shapes, computer game graphics, and so on will just make so much more sense.
this is one of the most profound videos i've ever seen in years of watching quantum videos
This is truly amazing. The only jump you made which I couldn’t follow was from the vibrating wave to the circle diagram.
Yeah
It was kinda going from two dimension to three..
Still wanna know
How this works in quantum.
@@qkihm Mathematically, the spherical harmonics are the set of solutions to the angular part of Laplace's equation in spherical coordinates. The angular part of the Hydrogen atom wavefunctions is the spherical harmonics because the Coulomb potential is only dependent on r, so it won't affect the angular component of psi. The angular part of Schrodinger's equation for Hydrogen is the angular part of Laplace's equation, and that as we know has the spherical harmonics as the solution set. Spherical harmonics actually show up everywhere in physics, from vibrations on a sphere to radiation patterns, and as you mentioned atomic orbitals. Somewhat intuitively speaking, the energy levels of Hydrogen must be standing waves around the nucleus, and spherical harmonics are basically just harmonics on a sphere, so that's why they show up.
@@northernskies86 Bzzzz, nerd alert!
Classical physics suggests that sine waves are created by oscillating locations of force emitters. So mass emits gravity, a charge emits electrostatic charge. As "particles" of mass or charge oscillate (move in a circle in space), a fixed point will experience a variable force over time, as the charge or mass moves closer towards it and then further away from it.
Mapped from the perspective of the fixed point, the force of the oscillating mass or charge becomes a sine wave. Therefore, if we have a sine wave, we may extrapolate backwards, and suggest the source is an oscillating "thing".
Just so, if we measure peculiar arrangements of harmonics in charge, which we do, we can extrapolate backwards and infer that these harmonics are generated by oscillating particles exhibiting charge.
In other words, spherical harmonics are a real thing, in the world of atomic charge, and from such patterns we extrapolate the idea of many oscillating particles of charge, moving in a certain way.
It's very odd, because as I have said, the idea that waves are caused by oscillating "things" is very much classical physics, not quantum mechanics. Therefore spherical harmonics are actually a wave function based on classical physics, and not deduced by the mathematics of quantum mechanics.
Quantum mechanics has come to accommodate spherical harmonics (the actual existence of them, which cannot be denied) by describing the oscillating movement of particles, but this is, on a philosophical level, a refutation of the core idea of quantum mechanics, which is that wave harmonics are actually just the probabilities of discreet particles colliding with each other.
Spherical harmonics in charge suggest that waves are not just mathematical representations of probabilities.
I do not believe that a better visualization exists on this topic, and we are graced to have access to this free of charge. Thank you!
finally, these shapes as the harmonics of a sphere make sense. thank you!
Hahahaha I had previously watched a MUCH LONGER VIDEO about this but ended up with VERY LITTLE IDEA of what these things really looked like.
This short video has made things SO MUCH CLEARER!
Welcome to QuantumVisions. Ma aim is to show key ideas only.
@@quantumvisionsumunster8208 that's a tall order. What are the key ideas? For some ppl, it's nodes and waves on a sphere, for others is irreducible representations of SO(3), for others it's an eigenvalue-of-rotations problem! idk, maybe those are all equivalent, but they sure sound different.
@@DrDeuteron I just plot the equations for complete Fourier system on the sphere S2. Its the usual (boring) stuff, but I tried to make explicit the idea of nodes. Euqations are standard, see e.g. en.wikipedia.org/wiki/Spherical_harmonics. For more general representations, see www.quantenspiegelungen.de/en/subdimension-line-u3/topology-of-the-quantum-dimension/topology-and-quantum-nodes/
This is the clearest explanation I've seen to date.
Amazing contribution to understand the geometry behind spherical harmonics!!!
Thank you I've never understand spherical armonics and this videio it's so intuitive and easy to understand.
I wonder such an awesome video has so less views. Thanks for this.
Good question. But on the long run, I think high-quality work will remain visible anyway.
the algo giveth and the algo taketh away. I've been watching physics vids since yt started, and it only chose today to show me this.
So this is where quantum numbers come from…. Finally makes sense
Best visual explanations I've ever seen!!!
This is a pure pleasure to watch! You deserve more subs!
Just subscribed immediately after watching few seconds of video! What a clear and crisp explanation!
Thanks and welcome to QuantumVisions!
This is an extremely useful video! Conceptually, I am trying to visualize how each spherical harmonic can be a superimposition of traveling waves to form the standing wave oscillation on the sphere. For example, I can visualize how in the m=0, l=1 harmonic, this could be visualized in two dimensions as two single wavelength traveling waves originating at the north pole, traveling on opposite sides of a circle and crossing each other at the south pole to return to the the origin, and the average wave intensity would be similar to the oscillation you show with the node at the equator. What seems harder to visualize is how real traveling waves could form the m=1, l=2 scenario where the nodal lines are perpendicular on the sphere. I read for example that this might be like circularly polarized traveling waves on the sphere with the 90 degree rotated components of the waves accounting for the two 90 degree separated nodal lines, although I am struggling to determine if that is the correct way to visualize it or if it is actually due to some other pattern of traveling waves. Any chance you could comment on the correct way to visualize the m=1, l=2 as traveling waves?
Just plug in the equations. This is what I did. Good luck!
sooo perfect this is the bright side of humanity creating and explaining the world !! ( one and only )
I really like this visualization, thank you.
I started thinking about this after watching a singer shatter a wine glass by singing at it using its resonant frequency. Putting more energy into it than it could handle. A bit like a photon escaping. Then I considered how the glass could flex, and how physical limitations would cause any oscillation that wasn't at a resonant frequency to be self canceling, since the ends of the oscillation wouldn't meet. Then I imagined the limitations in 3 dimensions and how with something as small as an atom, unaffected by the macro universe, there would be basically no damping, so this "wine glass" could ring virtually forever.
So good. Period. Thank you so much.
This is amazing, please continue making such amazing videos! This surely deserves more watches
Finally someone explained why orbitals looks like those weird shaped blobs!!❤
Great video, but in the end, why are only the m=0 spheres pulsating and all the other ones are just rotating without any deformation?
In the video, I show the real part of Y_lm exp(i w t), which either has a cos(+|m| \phi + w t) (right moving) or cos( - |m| \phi + w t) (left moving) term, which leads in the time dependence to the "rotating bubble" without further deformation, since time only appears in this term. Thanks for the question!
This is awesome. The semi-circle transformation kinda confused me. I thought it was sliding at first, but looking at one quadrant shows it to be streching and shrinking
We just used the mathematical expressions for the spherical harmonics- and this is what we get.
Thank you for pointing this out 😭😭 I was so lost
good catch. This is why mass distributions don't have a "dipole moment"--it's just a translation of the center-of-mass.
also, since z is arbitrary, can you figure out which combinations of of m = +/- 1 are shifts in the x and y directions? (Hint: ask chemist).
Waah, the best 👍
Excellent presentation
It deserves billion likes ,♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️♥️
I like that, thank you!
Fantastic clarity and right on point.. thanks !
Thanks for the clear and concise conceptual explanation! Additionally, when somebody can't fit into their favourite pair of jeans anymore, we could say "you're looking a bit too 2nd spherical harmonic for those". :D
it depends on the phase (prolate or oblate). Note that the L=2 harmonics of a shape *are* it's moment-of-inertia tensor, so it captures both your weight and, wait for it, "how your body carries the weight").
@@DrDeuteron Haha cool! Yeah I learnt just enough about it to implement for diffuse global illumination from an environment map, but one day I'll circle back and try to understand it fully. :)
@@leighkite1164 another comment points out the similarity to Fourier series, where you project an arbitrary function onto pure sine and cosine functions,-standard in so many field. It really is the same thing, just on a sphere instead of a string or a plane.
Have fun with it.
See "Wigner d-Matrix" for a deep dive.
Wow just amazing so this is what math looks like
Thank you so much this is amazingly helpful
this is awesome!
This is the best explanation. thank you !
While spherical harmonics are the eigenfunctions of the hydrogen atom to describe its state, which are the eigenfunctions for the other 117 chemical elements (helium, etc.)?
Thank you so much!!
Thank you. This truly helped me alot. Thank you
It's a truly amazing video!
Fascinating
thank you so much!
What a representation!! Wonderful.
Beautiful....🙏thank you so much...
thank you
The question whether Laplace had thought of them like this and how he had visualized them ?
A genius like Laplace just used the equations, I believe.
I have never seen the nodal analysis before, and I thought I'd seen *everything* about Y_lm.
More general: www.quantenspiegelungen.de/en/subdimension-line-u3/topology-of-the-quantum-dimension/ see also my technical comments on www.quantenspiegelungen.de/wp-content/uploads/QuantumVisions.pdf (starts at p. 89)
@@quantumvisionsumunster8208 very cool. My German is pretty bad, even after spending 5 years at DESY.
Fantastic
Thanks for this great video. Which software do you use for creating the visualization?
Great...!
thanks much
I have never used the following phrase before, but I think here it is quite fitting. What the actual fuck!?
Very impressive video..
Do you have research for dynamical behaviour of the earth? Do see oval shapes in objects in space so ...they perhaps can be round/sphere on another time frame. (perhaps in thousand years). Do see a lot of static anal;yses of the earth and also mostly local. Do not see many global analyses of the earth and its shape for instance ....volume changes in relation to volcansim. Earthquakes and how they move along the earth and insde the earth.
Vid : Harmonics go burr...
My Brain : Ha ha Squishy Squoshy Bloby Bloby lol
i studied mindlin plate theory at uni last year, i wonder if these shapes are dictated by the boundary conditions. i just know this is going to be a rabbit hole for me 😂
Just look into you textbook. Indeed, spherical harmonics give a complete basis on the sphere. Not much choice left. Generalization to spin is shown here: www.quantenspiegelungen.de/en/subdimension-line-u3/topology-of-the-quantum-dimension/topology-and-quantum-nodes/
amazing
Finally THANKSS ALOTTTT
Watching your video, I was able to find the surface equation of l=0,1,2.... for m=0 but not for all values m, how can I get that ?
here. www.quantumreflections.net/subdimension-line-u2/spherical-vibrations/spherical-harmonics/
Awesome
Am I the only one who couldn't get anything in the first watch.??
rotating the nodal lines was a bit hard to follow visually, thus conceptually, but great content.
So there are 2l+1 modes of vibration but which one we call the first overtone, second overtone, and so on.
PLEASE HELP
We have 2l+1 modes with l nodal lines in TWO dimensions (surface of a sphere). Overtones are related to vibrations of a string in ONE dimension. The groud tone is given by l=1, the first overtone l=2, etc. See
www.quantumreflections.net/subdimension-line-u2/standing-waves/spectrum-of-a-guitar-string/
@@quantumvisionsumunster8208
Ohh thank you!!
see "Chladni Plates" for a 2D generalization .
What is the software used to make these videos?
boy if that doesnt look like orbitals
are you /sarc-ing? If not: they don't just look them, they *are* them!
It's an atom
This loos like some Hollywood movie.
Thanks a lot... in contrast to Hollywood, we just want to teach physics. Not more and not less.
Holy fuck.