The spherical harmonics
Вставка
- Опубліковано 28 лип 2024
- 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FA...
📚 The spherical harmonics are the eigenstates of orbital angular momentum in quantum mechanics. As such, they feature in the wave functions of many quantum problems, including the 3D quantum harmonic oscillator and the hydrogen atom. In this video, we write down the mathematical form of the first few spherical harmonics, and we also visualize them with animated 3D graphics.
0:00 Intro
0:45 Definition of the spherical harmonics
3:09 How do we visualize spherical harmonics?
6:37 l=0 spherical harmonic
8:14 l=1 spherical harmonics
17:16 l=2 spherical harmonics
18:53 l=3 spherical harmonics
19:50 An alternative view of the spherical harmonics
23:01 Wrap-up
🐦 Follow me on Twitter: / profmscience
⏮️ BACKGROUND
Orbital angular momentum: • Orbital angular moment...
Orbital angular momentum eigenvalues: • Orbital angular moment...
Orbital angular momentum eigenfunctions: • Orbital angular moment...
⏭️ WHAT NEXT?
The 3D quantum harmonic oscillator eigenfunctions: [COMING SOON]
The hydrogen atom eigenfunctions: [COMING SOON]
👩💻 GITHUB
Plot the spherical harmonics yourself: nbviewer.org/github/profmscie...
~
Director and writer: BM
Producer and designer: MC
Never thought, I would ever like to see spherical harmonics again. You make this look so easy. Great explanation, thanks for your hard work.👍👍
Good to hear you liked the video! :)
When i first encounter spherical harmonics in Quantum Mechanics 1, i was totally confused on what they are, what they're about, and how to use them due to the formulas looking so dauntingly complicated.
It took me a while to "get" what they are, and animated visualizations like these are a big part in helping me understand them.
Not going to be questions this time, i think the material is presented clearly enough. There are things I do noticed about this.
1. I see that whenever the spherical harmonics (which relies on two numbers l and m) has m=0, they don't change when rotated around the z-axis. I believe there's more to this.
2. Sometimes having +/- m contributes a +/- sign to the complex phase factor, the same thing that will happen if you do *complex conjugation* . There's gotta be a connection somewhere.
3. In the "other" visual representation of spherical harmonics (which honestly i encounter more often in all sorts of books) it seems like the magnitude of spherical harmonics function is also scaled with the distance from the origin of the sphere, along with color to notify sign. This makes the shapes look "jutting out" from the center, the white parts lying really close to the origin (which i interpret as "color near zero" also being "distance close to zero from origin")
4. Funy that i also found mentions of spherical harmonics even outside of quantum mechanics. In electromagnetism they're related to the shapes of electromagnetic radiation coming out of a point source, and in nuclear physics they're used when a nuclei deforms from a spherical shape. Both electromagnetic and nuclear cases concern the shapes of waves with angular symmetry, what happens if you add different shapes to get new shapes, and both relate spherical harmonics to *multipole expansion* . I think i'll look more into this.
Nice one overall, hoping that more physicists use intricate animated visuals to teach complex subjects like this.
Glad you like it! A few extra thoughts:
* Spherical harmonics indeed have some nice properties as you have suggested in 1 and 2, including: a fixed parity (even functions for even l and odd functions for odd l); and a conjugation relation [Ylm(theta,phi)]*=(-1)^mY_l-m(theta,phi), which also immediately implies that for m=0 they must be real.
* Yes, this is how you would interpret the second approach: for example, as you correctly say, whites correspond to the function going to zero, and then the plot collapses to the origin along those directions.
* For point 4, spherical harmonics are indeed very useful: they form a complete basis for any (square-integrable) function of theta and phi, which means that we often expand functions in terms of spherical harmonics. As always when we use a specific basis, the question is whether its use will simplify the maths, and spherical harmonics are very useful when there is some type of spherical symmetry, e.g. scattering from a spherically symmetric potential.
I hope this helps!
1,2 is the fact that they are eigenfunction of rotation with eigenvalue exp im theta. About another axis, they mix up the same L.
The best explanation by far that I have seen on the web about spherical harmonics. This is explaining things with foundation. Thanks a lot.
Glad you found it useful! :)
I had encountered Spherical Harmonics, and was very intimidated reading up on it. This video made it much simpler to understand. Thanks for the good work in content, presentation and animation.
Glad you found it helpful! :)
Amazing explanation for beginners and curious people. Thanks Prof. M
Glad you like it!
finallly I was waiting for it🎉🎉🎉🎉🎉... thankyou soo much for your hardwork..you really awesome job in explaining things🖤
Glad you like it! :)
If your stuck on what’s a tensor, spherical harmonics are a great tool. They are things that are eigenvalues of rotation….Y_lm that is…but so are tensors. How you get from N squared rank N Cartesian tensors to 2N + 1 symmetric trace free tensors and some left overs is a long story though. For example, the well known inertia tensor is just the L=2 moments of a shape, while Y_1,0 is r cos theta which is “Z”….it is a vector. More so than an arrow is.
The non zero M values are complex combination of X hat and y hat, such they they are eigenvectors of Z rotations.
Wow! I was totally engaged, especially towards the end when I saw the orbital shapes that I recognized in high-school chemistry. I had always wondered about the mathematics to describe these orbital shapes! This is beautiful and meaningful to me, Thank You!
Thanks for your message, really glad to hear this worked for you! :)
What a fantastic video. Thanks for making this!
Glad you like it! :)
This is the best explanation I have ever seen on spherical harmonics!
Glad you like it!
I appreciate this so much! This helped a lot with my understanding, I was feeling really insecure in my classes having just learned about this.
Glad you found our video helpful!
Wow 😲 ! I mean this is the best video i have ever seen on the spherical harmonics
I mean absolutely amazing 👏👏👏
This video has everything in it,(mathematical formula, visualisation, interpretation, that's exactly what a person need to understand rather than just deriving equation)
Salute to your hardwork 👏👏👏
Thanks for your kind words, and glad you like it! :)
You guys are pure gold.
Glad you like the videos! :)
Thank you for this video, great details and visualization for computer graphics!
Also love the way you often say "take some times to convince yourself that this is true" xD
Glad you like our videos!
I am half-way through. But I wanted to Thanks before watching the second half. Excellent Explanation. Thanks a lot.
Glad to hear, and hope you liked the second half too! :)
Amazingly Explained
Glad you like it! :)
Man this channel got to be more popular
Thanks for your support! :)
Great explanation! Very helpful for some Computer Graphics problems too!
Glad you liked it, and it is always great to learn about other areas where a particular piece of maths is being used :)
Thank You Processor for such a good illustration.
Glad you like the illustrations!
This video saved my life! Thank you so much! Very clearly explained!! with amazing figures!
Glad to hear! :)
If this saved your life, you fell in with the wrong group. I hope it was nt the Lie Group, those thugs are everywhere.
Love to watch your videos, can you make a video on coupled and uncoupled representation in angular momentum. Also try to solve any related problem from liboff
Thanks for the suggestions!
Even that i want to learn to use spherical harmonics for geodesy and geophysics, I have found interesting and useful your video. I am very fascinated how you define surface spherical harmonics which is useful for hydrogen atom modeling among other aplications regarding quantum mechanics. Thanks a lot and greetings from Perú¡¡¡
Glad you like the video! :)
Nice video! Could you explain Wigner's 3j symbols and Clebsch-Gordan coefficients and their utilities in a future video? Thank you very much for your effort in explaining these topics!
Glad you like it, and thanks for the suggestions! We will cover addition of angular momentum (after we've covered spin) and touch on these topics!
Thanks for this clear presentation. The P_l^m(u) are only polynomials for even m, as they contain a factor (1-u^2)^(m/2). Hence the terms "associated Legendre functions" is preferred. Many authors, including reputable ones, are sloppy in this respect.
Thanks for your comment!
Thank you very much, this helped me a lot.
Glad it helped!
Superb! Thank you
Glad you like it!
Great video! Thank you!
Glad you liked it!
Thank you! That was fun!
Glad you enjoyed it!
Best video on spherical harmonics and delivers a new insight
Plz do something for wiegner D function
Glad you like it, and thanks for the suggestion!
Amazing explanation and visuals.... Thanks for the python code!
Glad you liked it, and hope you enjoy the code! :)
I am studying Spherical Harmonics in Electrostatic in Bsc 1st year..you can feel the agony I was having while going through the notes..this video is a bless to me
So glad to hear you found this useful! May we ask where you study?
Awsome video!!. Also do you guys have any plans on making a Statistical Mechanics lecture series as well? That'll be really helpful.
We do hope to be able to do much more beyond quantum mechanics, including statistical mechanics. But it will take a while before we get there...
@@ProfessorMdoesScience Sounds good 👍
In the animations following 18:50, hopefully I understand correctly that the animations represent a change of camera position and not a time feature of the function. Regardless, the animation would be greatly enhanced if it included the axes as a frame of reference. Especially since the rotation animations are not synchronized, so a key feature of the multiple cases (look similar but rotated by certain amount) is lost.
Indeed it is a change of camera position, and thanks for the suggestion!
Great explanation!
Thank u so much ......u are the great man
Not quite! ;)
Hello! Thanks for the great content! It's really interesting and helpful. Do you have a reference to a textbook somewhere on these topics?
For the quantum mechanics content we like standard textbooks such as those by Sakurai, Cohen-Tannoudji, or Shankar. We don't yet have many videos on the more mathematical front (this one on spherical harmonics being an exception), and we haven't been using any specific text for these. I hope this helps!
Crystal clear!
Glad you like it!
thank you very much 🎉 please I want all your lectures in angular momentum
Glad you like this! We have a series on angular momentum, you can follow it here: ua-cam.com/play/PL8W2boV7eVfmm5SZRjbhOKNziRXy6yIvI.html
@@ProfessorMdoesScience please I want pdf slide for all your lectures
@@drabd3863 unfortunately we don't have that at the moment, we are however working on sharing more material in the future (including problems+solutions).
Now I see why these are called spherical harmonics.
It does become clear when we visualize them, right?
@@ProfessorMdoesScienceyess!
This one is awesome
Glad you like it! :)
Awesome Lecture. Please how do you write and present math equations as nice and fast hand writing?
Glad you like it! We accelerate/slow the writing at the video editing stage.
I see pepsi logo... Wait that's pePSI as in Psi = (the radial function)x(the spherical harmonics)😲
Good one! :)
LOL!
I was learning how Unity game engine saves light values and they used the term SH in document which meant spherical harmonics and got sidetracked to here.. I wasn't expecting to understand the term after seeing all the mathematical definitions but with your explanation, I think I got the point! thanks
Nice journey, glad you found us! :)
13:40 I just had the biggest aha moment of my entire life! Ok. So the harmonics for each quantized value of angular momentum has a white line along the surface of the sphere for the real and imaginary components. So for Y1,-1(θ,Φ) when θ=0,π and Φ=π/2, 3π/2 both the real and imaginary components of the spherical harmonic = 0. This is super important because it must mean that the operators L^2 and Lz are being applied to a wavefunction that is = to 0 and L^2(0) = 0. It literally means that the particle has a 0% chance of being found at those coordinates! THAT'S A NODE ISN'T IT!?!?! THE SPHERICAL HARMONICS ARE 3D PROBABILITY WAVES!! I literally started screaming "OHHH!!!! OHH!HH!!!" at my computer (dw I wasn't in the library) as I realized this. We use angular momentum as a value to describe the shape of orbitals because we can use angular momentum to tell us where nodes are!! Some part of this is probably wrong because I still don't know wtf I'm doing but I'm hyped out of my mind right now. ok ok. I'm gonna finish the video now.
Y1,0(θ,Φ) IS A P ORBITAL!!!!!
Glad to see that our videos cause some "aha" moments :)
p orbitals indeed refer to states of angular momentum quantum number l=1 :)
Kindly make some detailed video lectures on spin also.
Thanks for the suggestion, it is on our list and should come out in the next few months!
This video restored my faith in humanity. Breathtaking!!
Glad you like it! :)
Hi. Thanks for a wonderful playlist! On the internet I found this question: "Why in the ground state of any QM system (s state) is the angular momentum always zero?" I think of this like n = 1, so l = 0 and L = 0. But can the electron in the ground state of a H-atom ever have zero angular momentum? Please exclude the spin-part in your answer. Thank you!
Thank you so much!!! This video is 👍 👌
Glad you like it! :)
@@ProfessorMdoesScience very much so. I was going back over my quantum mechanics notes and this really helped develop the intuition that I couldn't back when I was first going over the material some time ago. So this video was genuinely empowering.
@@Rebel8MAC This is great to hear! May I ask where you are studying?
@@ProfessorMdoesScience University of Houston at the main campus. I had to pause because of a life event but I only needed 2 upper level math courses and 2 physics courses for my bachelors. Since I'm saving up money right now, I'm using the fact that I love doing math on my free time to get even more intuition and practice.
This is a fantastic video, kudos! A question that I can't seem to find an answer to is, are the spherical harmonics inextricably linked to quantum mechanics? Or are they, in general, a set of special functions that are defined along the surfaces of spheres?
Since the equations need the quantum numbers (m, l) I would think the answer is yes, but I'm not too sure. Was this math developed specifically to describe QM or is it just an application?
Edit: I'd also love an explanation on how these harmonics relate to energy levels of electrons and how we categorize orbitals using them!
The spherical harmonics are not tied to quantum mechanics, they are a general set of functions defined on a surface of a sphere that in fact form a complete set. As such, they appear in the solution of many differential equations, not only those of quantum mechanics. Examples are as varied as electromagnetism, gravity, and 3D computer graphics!
And to explore the spherical harmonics in action in the area of quantum mechanics, I recommend you take a look at our playlists on:
* Central potentials: ua-cam.com/play/PL8W2boV7eVfkqnDmcAJTKwCQTsFQk1Air.html
* Hydrogen atom: ua-cam.com/play/PL8W2boV7eVfnJbLf-p3-_7d51tskA0-Sa.html
In these two playlists we explore the quantum mechanics of a 3D quantum harmonic oscillator and of a hydrogen atom, both of which make use of the spherical harmonics. And we directly address your question about electron energy levels and associated orbitals, particularly in the hydrogen atom playlist.
I hope this helps!
This is really great! I work in diffusion MRI where people are also using SH. Can you also show me how you the code about how you draw the alternative view of SH, which is a more wide-used representation in my field? Thanks a lot!
Glad you like it! We have now updated the code to also show the spherical harmonics with the alternative view: you can control which view you use by changing the second argument of the "createAllHarmonics" function at the end. For example, to generate the l=1 spherical harmonics in the original view you would use:
createAllHarmonics(1,'sphere')
and for the alternative view you would use:
createAllHarmonics(1,'radial')
I hope his helps, and let us know how you get along!
@@ProfessorMdoesScience Thank you! It works really well!
@@zxbian Great, hope it is useful! :)
I was just wondering that how were you able to understand these conepts without animation ? Because in old days, there were no animations, how people were able to imagine such complicated topics ?
Animations do certainly help :)
Well done and an enjoyable video. Thank you best wishes continued success. Daniel J Blatecky USA
Glad you like it!
I'm a 3D artist and I am trying to learn what spherical harmonics are, because they are used in real-time 3D graphics to encode directional lighting information. It's called directional light maps. Unfortunately it seems that outside of white papers, there's no resources that describe how do spherical harmonics work in the context of computer graphics.
I'm starting to suspect that they work similarly to the DCT (Discreet Cosine Transform) lossy image encoding technique, but instead of being on a plane, it's mapped onto a sphere. Outside of that though - it's a mystery to me, and I am not sure if even that is true.
Definitely not an expert in real-time 3D graphics. However, let me just point out that you should be able to explore the basic mathematics of spherical harmonics in most maths for scientists or maths for engineers textbooks, as they are used in many applications. I hope this helps!
@@ProfessorMdoesScience Thanks :)
Wow, Its amazing
Thanks! :)
@@ProfessorMdoesScience I recently started working on radiation heat transfer in particles. It make sense why we need this approximation for the emissivity distribution on a spherical surface.
this is the beast exampel
:)
Thanks a lot for this video, helped me clearly understand Spherical Harmonics for this first time!
One thing I am confused about though is that the plots are usually presented as the probability of finding an electron around the nucleus in a particular shell right (unless I am mistaken about it?)
However, these plots are just the Real or imaginary parts of the wave functions. I thought probability should be the absolute value squared of the wave function? What am I missing here?
Glad the video was helpful! As to your question, spherical harmonics are indeed helpful in understanding the electron wave function in, say, the hydrogen atom. However, they only provide the angular part of the wave function, for the full solution you also need the radial part. We are working on a series on the hydrogen atom where this should all become very clear. For the point about real and imaginary parts vs. absolute value squared, you are correct that if we want to explore the probability of finding an electron at a particular point, we need the latter. Although we haven't visualized these in the video, you should be able to use the code in the Jupyter notebook to re-plot the absolute value squared of the spherical harmonics. I hope this helps!
Regarding these as part of the wave function: the wave function is not the probability of finding an electron at a coordinate, is the probability amplitude to do so….which a purely quantum concept. But even that is pushing it, since you can’t actually measure the position of the electron, and it’s not like the e is a point the moves around…it’s the thing described by the wave function. A bit abstract, but it won’t lead to wrongthink.
Anyway, about the real an imaginary part, you can multiple all solutions by a global complex phase factor, and nothing changes, so what is real and what is imaginary is a matter of convention and has no physical meaning. Moreover, the time evolution of an eigenstate is just multiplication by a clobal phase factor with phase increasing linearly with rate energy / hbar….which should convince you it’s not physical….e.g you can a constant to the potential, it will increase faster…..but adding a constant to the Hamiltonian doesn’t affect physics.
Thanks
Thanks for watching!
which are the eigenfunctions for the other 117 chemical elements (hydrogen, etc.)?
What about the videos on spin angular momentum? When will they come?
🙂
We are currently working on videos covering the hydrogen atom, and after that we should move to spin.
Oh now i see this is actually describing orbit of atom.plz show us diagram and picture of g orbital or l=4.i always wandered what g orbitals looks like.
You can follow the link in the description to the Jupyter notebook, and there you can plot the l=4 case yourself! Let me know if it is not clear, and I can guide you through it.
can you also do a series on spin and the corresponding angular Momentum please
Thanks for the suggestion! We do plan to publish a series on spin over the coming months.
@@ProfessorMdoesScience that would be really great 👍
Where I get full details calculation of full form of spherical harmonics
We introduce them in the context of the eigenfunctions of orbital angular momentum, which you can find in this video:
ua-cam.com/video/Gk2XNmIHVwo/v-deo.html
I hope this helps!
This made my life easier
Glad it was helpful!
Very clever Pepsi product placement
Is it that obvious? ;)
Why can't all the three components of angular momentum be quantised? I get that it comes from uncertainty principle but I don't know how to phrase it.
For example, (standard deviation of L_z)* standard deviation L_y) >= hbar ⟨L_x⟩ /2
Since if we have quantised the L_z (m * hbar) therefore standard deviation of L_z is zero which implies ⟨L_x ⟩ =0 (which from my understanding says that if we perform experiment to calculate L_x I would get values which will average out to zero) but what does it tells about quantisation?
But in Griffiths it says "You have missed the point. It is not merely that you don't know all three components of L; there just aren't three components."
Please help me out.
I think there are a few different concepts here, so let me make some comments:
1. All angular momentum components are quantized. What this means is that the possible eigenvalues of Lx, Ly, and Lz can only take a discrete set of values.
2. A different point is that Lx, Ly, and Lz do not commute with each other. This implies that they are not compatible observables, and this implies that, for example, they obey an uncertainty relation.
We have a full series on angular momentum where we look into all these concepts in some detail, so I recommend you take a look: ua-cam.com/play/PL8W2boV7eVfmm5SZRjbhOKNziRXy6yIvI.html
I hope this helps!
@@ProfessorMdoesScience So is this implies that all three can't be simultaneously quantised?
Like if I try to measure L_x I would get m* h bar but L_y and L_z would have uncertainty ?
Professor, I have watched the full playlist on angular momentum. It is very good and I truly appreciate you replying to these questions I have.
I would not say that they cannot be "quantized" simultaneously, they are quantized. Something you could say is that they cannot be "measured" simultaneously, in the sense that if you start with an eigenstate of say Lz, and then you measure Lx or Ly, then the state after the measurement will no longer be an eigenstate of Lz. I think this may be what you mean by "measuring Lx and then Ly and Lz having uncertainty". I hope this helps!
@@ProfessorMdoesScience My professor asked this question : "why can't all the components of Angular Momentum be quantised simultaneously?"
I said , " because of uncertainty relation between the components."
Professor: "look for a better answer"
And I told her whatever I stated above, but she is still not satisfied.
@@vaibhavshukla6926 I just think your professor is using a somewhat different nomenclature; I would not say that the components of angular momentum cannot be quantized simultanously, as I am not sure that means anything with the meaning of "quantized" we use.
When I assign a complex number to each point on the surface of a sphere, do I actually assign a vector to it, because two-dimensional vectors behave the same as complex numbers? I am trying to visualize it without distinguishing between a sphere with real and a sphere with imaginary values. If it is possible, how would I visualize spinors? Thanks a lot
Complex numbers and two-dimensional vectors certainly obey some isomorphisms (e.g. in their addition). However, they are not the same thing, so if you represent a complex number by a vector you should always be careful to keep that distinction clear!
I got a simple question. Why do you plot the imaginary component in the diagrams? I thought only the real components are relevant for our real world or do you need to convert the whole equation as a real equation to plot them. Some time ago, I was quite happy with the explanation of the square trick to get real values for probability measures.
Great question! Both real and imaginary parts of complex numbers are important in many areas of physics. Taking quantum mechanics as an example, the wave function is in general complex, and both real and imaginary parts matter. You are correct that physically observable quantities must be real, but these are obtained in ways that require both the real and imaginary parts of complex numbers, not just theri real part. One way in which this appears in quantum mechanics is when we calculate the absolute value squared of the (complex) wave function, which gives the probability density of the particle position. In this case, the absolute value squared of a complex number is |a + ib|^2 = (a+ib)(a-ib)=a^2+b^2, which indeed depends on both real a and imaginary b parts. I hope this helps!
@@ProfessorMdoesScience
Thx but my query is why did you plot the imaginary part of equation as blue on the diagram. I suppose the real part is only red in color. ( unless i misunderstood the clip).
Probably this is "correct" explanation. One could get rid of the imaginary part by plotting the probability function and then infer the real locations from the probability function. In this way, one obtains both the red and blue portions. Another explanation is to invoke the eigenvalues are real criteria ( are the function eigenvalues of eigenfunctions ?) and impose the real nature of the equation. OR you somehow used the Euler relations to convert complex factors into real factors OR just ignored the phase factor term as unity.
I have yet to figure out how the imaginary part is derived /appears in the equation but i think a bit more maths knowledge is required on my part.
What is the "d" in the long formula for Y(theta,phi)?
"d" is just the symbol for derivative, as in df(x)/dx being the derivative of the function f(x) with respect to x. In the video, we have d^{l-m}/d(cos(theta))^{l-m}, which means that we are taking the (l-m)th derivative with respect to cos(theta) of the quantity that follows, which in our case is "sin(theta)^{2l}". I hope this helps!
@@ProfessorMdoesScience awesome thank you very much for the explanation, really appreciate it Prof M. Was not sure that I would get a response but thank you and thanks for the video.
can you guys do a video or more on coupling of momenta? clebsch gordon coeff. etc
Thanks for the suggestion! Regarding angular momentum, our plan is to first cover spin, and then look at combining different angular momenta together, so we will indeed get to this, hopefully over the next few months!
@@ProfessorMdoesScience Sounds cool^^ in QM lectures its kind of sad that more often than not the profs skip certain aspects or proofs which makes upcoming stuff less intuitive. These videos really help filling these gaps
@@elomnusk7656 Really glad to hear you find our videos helpful!
Do You maybe have a video about spherical harmonics triple tensor?
Do you mean products of three spherical harmonics as they appear in the theory of addition of angular momentum? If so, we do plan to go over that theory in the future :)
@@ProfessorMdoesScience I think so - if that refers to Wigner 3-j symbols or Clebsch-Gordan coefficients then that is the thing :D
I was trying to use it in computer graphics to combine two SH vectors of coefficients into a third one. I was wondering whether I've got everything right and a video about the theory would be helpful I guess.
@@ProfessorMdoesScience BTW - 17:09 Are the real valued visualizations of Y correct here? I can recall the ones for m = 1 and m = -1 to be a version of each other rotated by pi/2. Or is it like so when you get the square of amplitude of Y (to eliminate the imaginary part)?
@@kadlubom Indeed, so our plan is to cover spin first, and then look at addition of general angular momenta :)
@@kadlubom The real parts of the m=1 and m=-1 spherical harmonics are the negative of each other (hence the red vs blue infront). It is the real vs imaginary parts that are rotated by pi/2 with respect to each other (and they are the same for m=1 and m=-1). Does this help?
Hii everyone this is Professor M does Science gives me New energy
Thanks for your support! :)
Wouldn't it make more sense to visualize the modulus for physical intuition?
The modulus can be useful, for example when considering observable quantities in quantum mechanics, but the full complex nature of the spherical harmonics is essential to get things right, so this is way we decided to go this way. But may I encourage you to try visualize the modulus yourself? Could be a nice exercise :)
Well if you look at the animations at the end, you may notice that she says she’s rotating them about the Z axis by theta, but it is identical to leaving them fixed and multiplying them by a phase equal to theta times m….
l could be ½, right?
Nice!
Thanks!
What if you started from an ellipsoid rather than a sphere? Would closed-form eigenfunctions still exist?
The spherical harmonics are the eigenfunctions of the orbital angular momentum operators. Could you please specify at which step you would consider introducting an ellipsoid?
@@ProfessorMdoesScience At the beginning: the mode shapes of an ellipsoidal shell. A numerical (FEA) solution would be possible but might there be any hope of a closed-form solution?
@@warrengibson7898 Not something I have thought about, but may be worth trying. Do let us know if you get anywhere! :)
Was expecting this , also @beena
Hope you like it! :)
@@ProfessorMdoesScience ofcourse! Also is there anyway i can know how to derive these solutions in a trivial way? Anywhere i look up, they just write it down as the solution by differentiating legendre polynomials m times. I tried deriving it through frobenius method but the solutions do not come in a closed form.
i def needed this! #_#
As far as I know there is no simple way to write these down. I think it is good to go over a few derivations at least once, as we do in the video, and then what we do is to simply look them up in relevant tables...
@@ProfessorMdoesScience thanks alot. i was just hoping to get the intuition of the first person who ever came up with such a solution.
Where is the radial variable in the function?
The spherical harmonics have no radial variable, only angular variables. In the plots, we just use a sphere of radius 1 for illustration purposes. I hope this helps!
@@ProfessorMdoesScience Is this true no radial dependence on true in the application of spherical harmonics in the appropriate places quantum mechanics where they are applied or is true of spherical harmonics in general (mathematically) without any physics.
I wonder what happens when we convert the normal cartesian coordinates to spherical coordinates will we get spherical harmonics with a perhaps a different form (perhaps with some radial dependence) (perhaps an example of using spherical coordinates somewhere it's not the best choice)
@@SidharthGat-kf4cb If you check our videos on the hydrogen atom, you will see how the spherical harmonics can be combined with radial functions (functions that only depend on the radial coordinate) to build the eigenstates of the hydrogen atom. I hope this helps!
Thank you so (non countable infinity) much
This is a big thank you :)
👍👍
Very interesting, but why use a white background..?, its blinding, watch it another time with my sunglasses
Thanks for the feedback, we'll look into this!
❤❤❤❤❤❤
This is a subliminally coded pepsi commercial innit
Good one! :)
Dr.Iam from in Iraq
Ken W sent me.
Welcome!
i came from a funny video with weird dancing shapes into this. I know some of those words.
Oh, sorry, I'm in the wrong classroom. Can you please point me to Underwater Basket-weaving 101?
Patty Jenkins
Great video - thank you!
Glad you like it!