Graphing Real Spherical Harmonics in Desmos Calculator

Transcript

graphing these real spherical harmonics in Desmos calculator. So here is an amazing real spherical harmonics graph in Desmos calculator that someone made. This is absolutely astonishing. Whoever made this, congrats, and thank you. So, first of all, if you go here, if you click it, what you'll end up going to is just this part right here. So here is, here it is, and here's some, it's a bit laggy on my computer here, my laptop thingy, my Surface Pro. So there's a bunch of the equations and so on, so there it is for some values on it. So, but anyway, so note the normalization constants used as well as the same formula as we obtained above. So if you go to it, it shows you this setup here, which is quite fascinating. But this is, so this is the, they call this the real part, the Y real, and that's basically when m is greater than 0. So we have the cosine m5s, and there's their... This is the associated leisandra function p m absalaya, and it's exactly the same set up there. So this is, they have two functions there, this is the cosine one, they call it the real, this is the imaginary one, I am, there, and there's the p, there's the m, this one, and number, it's L, but we used L, n, and there's cosine, and so on to some other stuff, but again, here's normalization constants. Those are the N. So that's the N one they use. So again, that's why we use proportional there. So it's a guy who again is exact same thing. They have the absolute values. They call this M-VAR. We'll explain that in a bit. So they have that M-VAR. So note the associated legerpolynomials and its giant derivative. So here it is. So this is the associated legero. There's the M-L, but ours is N, and so on. there's the negative 1 power of m, so on, 1 minus x squared and all, and so on. And then this is, and now it's a, it's a, they made as a function of the ordinary Leandro of polynomial mills. And then from here, yeah, so from here, this one, what they do now is they take the derivative, yes, they put it in here. So remember those, that giant derivative there, and it's, they basically take the derivative. of it so d derivative of the elf legend legera polynomial so that's a quite absolutely amazing stuff there and they say d d is yeah that's the variable so basically whatever the m value you use in here it gets thrown into here and then that gets into this function which is represented by d and so that's how it looks like And if you remember how this one, this is 1 divided by, well, we have 2 the power of n, this is just 1 to the n. And let's just have a look at that Legendra, general polynomial, I mean, the ordinary one. So we scroll the way back up, this was the setup here. Let's scroll the way back up, just to see our final solution there. So there is a PN, yes, so the PN of these 1 divided by 2N, there's a factorial, and then there's nth derivative of this. uh but then this one's gonna be that's p n and then we uh do the mth derivative of it and so on yes of p n there so it's was pretty fascinating put them all together it's gonna be n plus d uh plus mth derivative and so on so that's uh quite fascinating yes go back here so anyways yeah so this is the dth derivative that's the mth derivative after you already take the lth uh lezondra polynomial and in other words the elf derivative because that's why you had the n uh and plus M, so that's, and it gets something like this giant thingy then, anyway, that's, to solve it, maybe for a future video, go explain how exactly they do it, but anyways, since the spherical homonics are graph on the sphere, the radial distance needs to be stretched proportional to the magnitude of its value on the sphere, this gives the distinctive lobes. So what they do is, now they're graphing out the R terms, and they're using some of these values there, so R-sign and so on, so they're getting the radial terms. But then they have this setup here, which is using, now it's using the ML variables, the actual ones, instead of this MVAR. So they're using this MVAR, and they're basically using it as like the maximum of the, so it's using, so if m is greater than equal to zero, it's going to be plugging in the value of M inside this function, this function right here. inside this one here, Y real and Y imaginary, I'm not going to explain exactly what they're doing, but basically they're throwing these, M and N, an L, which is our N, inside there, and the theta, and so on, and then that function is using it in here. So, and they call this, they call these ones here the imaginary, they use M-Var, just because it's like a temporary one. So this is a temporary one that you throw in this one, which is actually being graphed, which is the length of it. Because you see this R max equals the maximum of one of these setups there. So this is basically the maximum lobe and so on. And some other stuff like that. But anyways, the whole idea is they throw it into this setup here. So that gets thrown into this equation and this equation. So it's fascinating. So keep going further. Let's see what else they do. And now we have some basic stuff here. So now we have the quantum, principal quantum number, and orbital angular momentum quantum number. This is the one we were dealing with. So we were dealing with this right here, L, and, yeah, we were doing this one here, this is our L, that's actually our N. So we were dealing with L and M. but these ones in quantum mechanics, they're called the setup here. So I'm going to read this all out, then I'll explain. So note the relationship between the principal, orbital, angular, momentum, and magnetic quantum number. So there's a principle quantum number N. Yes, so note the relationships between them, and the principle sets the maximum orbital angular momentum, which then sets positive and negative limits of the magnetic. So in other words, this principle is the N used in quantum mechanics. And then this sets the maximum, so this sets the maximum. orbital angular momentum is the one we were dealing with. So we didn't deal with this one, this is just basically the maximum. So where this one is between 0 and n, that's pretty much, I think that's the only place in the whole formula, the whole calculator that has it. So this one is just the maximum there. So that's pretty much sets the energy level, the maximum is the maximum value of n. And then the magnetic quantum number basically is equal, is now this one sets the limits, positive and negative. is negative L, negative, I'm positive n. In other words, n and negative n. Yeah, and then this one here is the maximum, this was using our setup. Yeah, this is the L. Again, that's why I don't like using the L, it looks like a one or something. They use it like that, and this is negative L. I'll say negative L, positive L. And then where this one here is the maximum is N. Yeah, so then these have maximums of negative n, positive n. Now here's some values here, so if you have n equals to 5, L equals to 5m is 0, you get this setup here, that's the default 1, which is quite fascinating. And then also here for going further, so here are the real spherical harmonics for various values of the quantum numbers. So if you put n is 0, and then L automatically become 0, these all become 0. If this n is 0, we just get a sphere. And they all have exact same values, if n is 1, and L is maximum, make it 1. Otherwise, if you set this as zero, it will just be the same thing as one. And then you get M is zero, you get this, if you make it the same thing, but M is 1, it rotates, M is 2, it gets it on this y -axis. But then interestingly, if you make n is 2, and then this one, L2, you get this setup, but then if you, and then if you're Y, M is 0, but if you have M is 1, it turns like this, and, um, M is 2, it still rotates, and so on, so that's pretty interesting. When it's zero, it acts differently than when it's not zero. So that's zero. This is M is two. So it's not just a simple rotation as in this one here. So that's pretty fast. And let's go back into the graph and see what we can come across. So again, here's the principle, quantum number. Let's change it to, yeah, but anyways, before that, again, you can click the spherical monics. You can see a whole lot of stuff they're doing. But the general stuff is, again, we have, this is what I want to get to, and then everything else is just the extra. This is the real one. This is the Y, and this is the this is the m is greater than zero, m is less than zero. They call that imaginary. But the same thing, there's a sign, and there's absolute value. So that's the real ones we set up. And there's the associated Lezondra polynomials, and then D throat of the elf, Lezondra polynomial. It's fascinating stuff here, and again, a whole bunch of other stuff. But I will close this, and then you can also toggle access here and on. This is pretty cool, so you can change it. Anyways, so I'm going to make this like this. I'm going to go to n is equal to 0, so let's see what happens. N is 0, L and M becomes 0. And then if you make this as 1, again, that's pretty much, I think that's the only place where it's dependent, just this one here. But if it's still 0, it still 0, so you make it N is 1. And then this one, you get modified, and so on. And then it also takes a while to load, because there's a lot of other stuff in there. And I think if I go to sphericalomonics, some of the this one has like a 684 element list so it's like literally taking solutions at each different degree or something and that's amazing stuff here so you can click this and see elements holy okay so let's close this now uh see even more so that's uh pretty fascinating just close that so i don't know okay that's how you close let's go back into here and i'm gonna make this into two yeah so notice what happens if you do two becomes this the one we just showed and then if you, oh, this is at m equals 1, m equals to 0, and again notice those all integers, it comes like that. And then you go to, let's do a giant one, see what happens. Whoops, slide this here. Okay, so that's two. Let's go to 10. So now we're going to go into another level 10. So that just sets the maximum you could do on the N. So there it is. And that becomes like a butterfly. Let's go to zero. Holy, yeah, okay, this is taking a lot of time to render. and it might blow up my computer. So there it is, that's how it looks like there. So it's always that weird shape like that. And then let's go here. Okay, let's do this one, and I wanna take a screenshot, maybe. Oh, it's going back, it's taking a while. Yeah, so that's 10. That's pretty ridiculous. Anyways, I'm gonna make it like this. Yeah, that's pretty cool. So that's a bunch of... stuff there, but you can also change graph options. You can do lines, do less lines. And it's taken a while to load. Do you change the width? Yeah, so it's a little notice, less lines, looks like that. Go to 77 or whatever. Okay, so there it is. Opacity. Make this 100. All the lines, no opacity. Looks like that. Okay, now that's the opacity, so you can't see through it. Let's make it a bit less. All right, let's see what happens here. It's still taking a while to load. This is pretty ridiculous. Okay, I think it's done it. Just go back here. And I want to take a screenshot. Go back up to here, principal quantum numbers. And yeah, that's what it looks like, and I will do it. Make this closer. Can it go over then? Okay, it's good enough. And I'll take a screenshot and add it. Alrighty, so I'm going to add this to it. basically putting a whole bunch of stuff together and see what else. Yeah, so that is that, and now there is the maximum quantum number. Alright, so absolutely, fascinating stuff.

Comments

More videos on 3Speak

• process of creating portrait: Drawing, charcoal and conte pencil on cardboard by @rishehri
• Trailer for MES Livestream 141: John Hutchison shows Tom Sky levitation footage he missed by @mes
• Problems Plus 6: Cable wound around a spool by @mes
• MES Math Q/A 72: What is a Ground Rod and how to find it in your house? by @mes
• Billiards enthusiasts by @epicdrama
• 🐝🍯 一只小蜜蜂，对 HIVE 满满的爱——请投票支持这位见证人！ by @hive3
• Billiards enthusiasts by @epicdrama
• 📽🎨Dibujando Día de Paz por Fútbol /Drawing Day of Peace Through Football (Esp/Eng)⚽️📊🔥 by @reyghart04
• hive3 — The Channel Trailer: A Bee's Journey Through the Hive by @hive3
• Problems Plus 4: Curvature from Parametric Equations that involve Integrals by @mes
• Pokémon Rips - 10 Packs of Ascended Heroes by @bulliontools
• Just downloaded Combat Cruiser to pass some time, and now I'm totally hooked! by @gamingnature