Spherical Harmonics in Atomic Orbitals and Magnetic Fields

Transcript

atomic orbitals. And yeah, the atomic orbitals are pretty much the driving force, why I'm covering this, one of the driving force, there's also magnetic fields, and also when dealing with, basically, alternative angle of physics instead of quantum mechanics, deal with the Eynaldino's Ether theory, which I... I think it's pretty next level. So interestingly, spherical harmonics resemble atomic orbitals, which describe the location and wave-like behavior of an electron in an atom, where L is often used instead of N, as I do in this video. Note that atomic orbitals are not simply radial distortions as in pure spherical harmonics. So these ones are radial distortions right here, which is basically the magnitude on the sphere. They would just make the lobe go out. So they're not proportional to the magnitude of, so they're not actually radial distortions. But they actually contain the radial term in the solution. So the atomic orbitals actually contain this part right here, basically. They actually contain the, I mean, not just this part, they actually contain the R part. They actually contain this radial term. So they actually have a, so it's a solid spherical hormone. That's how you would model them or approximate approximate. them and so on. So atomic orbitals are solutions to the Schroederner equation, which is shown for the electron and an hydrogen atom, and its solution below. As well as its solution. I'll put this as well as its solution below. Yeah, so here is the shortener equation. Basically, this is E, and this symbol is capital phi, I think. No, there's actually si, it's like they tried into one here. So it's actually si, not phi. So that's psi, like that's si, like that. So e, e si equals to negative, and then you got this is the reduced planks constant, hbar, squared divided by 2, and this is mu, and so on, this is going to be, now we have the Laplasian operator. of, uh, si minus q squared divided by 4 pi e, uh, or epsilon not, r times it by the, um, And again, with that is, I'm just going to read the Wikipedia one, I'm not going to cover too much of it, just to get the picture of it. So here, si, x of T, is a wave function, a function that assigns a complex number to each point X at each time T, and H is the reduced plank constant, which has units of action or energy multiplied by time, and then where Q, let me make this bigger, UQ is the electron charge, R is the position of the electron relative to the nucleus, and then r is equal to this right here r this is the absolute value of the r vector is the magnitude of the relative position the potential term is due to the Cuom interaction where this is epsilon not is a permittivity of free space and mu is equal to MQ times mp divided by mq plus mp is a two body reduced mass of the hydrogen nucleus just a proton so this is a p and then of mass mp and the electron of mass m p and the electron of mass mq the negative sign arises in the potential term since the proton and electron are oppositely charged so that's why they put this negative sign there. Yeah, I think that's what they meant. Is it here or here? Those two negative signs. Anyways, see what it says. Rise in the potential terms of the proton and electron are obviously charged. The reduced mass in place of the electron is used, this is apparently a reduced mass one. Since the electron and proton orbit each other about a common center of mass and constitute a two-body problem to solve, the motion of the electron is of principal interest here, so the equivalent one-body problem is the motion of the electron using the reduced. mass, interesting, so they just use that one body there. What interesting, so they're using a reduced mass there. So, anyways, so here's a summary of terms via GROC AI and their physical meanings. So E and psi is a total energy, that's E, total energy, times wave function. So that's the, again, wave function is a function that size a complex number to each point X at each time T. It's basically, yeah, basically the value of It just has a complex value on it, and somehow that's related to waves. And so on the operator is the eigenvalue equation. That's for another top of going. A sign is a note can to be negative bound states. And then right here, negative, this is the H-bar squared, divided by 2 mu, times by the Laplacian of Psi, or the Laplace operator of si, equals kinetic energy of relative motion. So that's kinetic energy. Interesting. And then now we have, this is a kinetic energy operator, and then always positive contribution. Yeah, so that's always positive, yes, this is a negative. If this is always positive, that's why they put a negative there. I think that's how they do that. And then negative pulse electrons, oh, no, and this one's also, yeah, and this one's always negative as negative as negative as always negative as negative as negative as negative as negative as negative as negative. Yeah, this is interesting, this always, oh, no, no, no, no, no, no, that's always positive. So in other words, yeah, yeah, yeah, so they put a negative there, so that, because inside terms somehow becomes negative, and it becomes positive. Anyways, that's for, uh, wrap my head around that. Anyways, so the main point about this is, you take this right here, and then using separation of variables and spherical coordinates, uh, you get, uh, you get this setup right here, again, exactly what we were covering. So you get, si of r, theta, phi, is equal to the r of r. This is the equation of the radius term only, and then times it by the spherical harmonic component, the angular component, Y, L of M, no, LM of theta and phi, equals to now R right here, and you have this symbol right here. Yeah, and this term right here, yeah, this is actually just capital theta theta, so theta capitals like that, interesting, it looks like that. or like this, something like that. That's pretty fascinating stuff. And then there is the Phi, and this is capital Phi in this lowercase, Phi. Sometimes it's written like that, and I just like this one, right? I just like that. But anyways, that's similar to the capital one like this. That's interesting stuff. Anyways, so where R are radial functions, and Y, L, N, L, M of Theta, and Phi are spherical harmonics of degree L and order M. So that's what you call them. This would be N in our case. So degree and L, order M. This is the only atom. So again, this is this is the equation for, if it's shown for the electron and a hydrogen atom or a hydrogen like atom. I'll put that in, actually, I'll put it in here, here's in Wikipedia. So a shorter equation for the electron in a hydrogen item or hydrogen like atom. I think that's pretty important. I'll put that right here. Whoops, and I'll paste it. Yeah. As well as the solution below, so basically the solution to this one here, you will get, so using separation of variables in spherical monics, we get this set up right here, and then we solve these. Yeah, and now, yeah, basically we solve these, we get the here. This is the only atom for which the shorthy equation has been solved exactly. otherwise you can need approximate solutions. So multi-electron atoms require approximate methods. The family of solutions are, so this is the, for the electron around hydrogen atom. So we have psi of n-lm, so now we have three terms there, this is R, theta, phi, and this is going to be square as a normalization factor, square root, and then we have two divided by N -A-0. or n a 0 power of 3 times n minus l minus 1 so this n is not the same one we use this is like some other operate that's why they use n there the one we were using is this l is going to be our n and then they're using some other term there for the actual term so n is a separate one so ours are l so we have n minus l minus 1 divided by n is factorial divided by 2n and then times it by n plus l factorial times it by e to the r divided by n a knot, and this is the r component, actually this r component keeps going here, times by 2 r, divided by n, a knot, divided the power of l. So this is the r component, this is n, this is r of r. And now we have, this is the new spherical harmonics setup, what it looks like, so no, there's still an r there. This is now this is an L, 2L plus 1 of uh yeah this is n minus l minus 1 so this is the new um a new angular spherical harmonic operator so now we have a 2r divided by n a knot r it is massive so where a not equals to make this bigger so a not equals to 4 pi uh this is epsilon uh not hbar squared divided by mq q squared is the bore radius and then l2l plus 1 again this is the associated Lagrangea now it's called the generalized laguar polynomials of degree n minus l minus 1 okay this is like every time I think I grasp it they it gets next level and this is not covering it in this video maybe a future one so n lm are the principal as mythal and magnetic quantum numbers, respectively, which take the values n equals 1, 2,3, L equals 0,0,0,0, n minus 1, and m equals to negative L. So again, in our situation, L is N, and we don't have one for this one. Again, that's because this is a solid, basically using actual R values. All right, and now we'll take a look at this here. This is the hydrogen wave function here, and I just want to make a quick note. I'll draw this in a different color. This uses, it's pretty much exact same formula here. This is the psi nlm, and there it is, and si nlm. I'll zoom in like this. Yeah, there it is. It's pretty much the same thing, but they have this row sign. So, and that's because they put this L associated Lagrange function with the row instead of this one where we had this. So basically they remove that. They add this one, and then they pull out the spherical harmonics yL on M-term. So basically just rearrange it, but it's pretty much all the same thing here. Alright, now that we covered that, just go back over here. So basically, this figure right here, this is cross sections of atomic orbitals of the electron in a hydrogen atom at different energy levels. The probability of finding the electron is given by the color, as shown in the key at upper right. And know that the negative and plus indicate the phase of the wave function. So this darker ones, it just means that the phase is going to be negative, and this bright ones, it's positive. But these can go either direction. It's going to be the same. probability or similar probability of high probability of finding an electron. For example, in this region here, this would be high probability, and this would be a, the outer one will also be high, but in a different phase. And then these super bright ones would be on this phase of the wave function. There, so you have different probabilities, and then the black regions are the ones, you're not going to have, you're going to have very low probability of finding any in these darker black stuff there. It's pretty interesting setup there, and you got all these different energy levels, different lobes, and so on. All right, and going further. So here's another figure. This figure is 3D views of some hydrogen -like, basically just any atom or ion with a single electron atomic orbitals showing probability density and phase. These G orbitals and higher are not shown. Basically, yeah, you have this S-orbital, and you have this set up here. and it goes on, yes, then you have the other next one, and so on. So it's pretty interesting stuff. And the table below shows the real hydrogen -like wave functions for all atomic orbitals up to 7S, and therefore covers the occupied orbitals in the ground state of all elements in the periodical table, up to radium. I guess I get, I think the bigger ones have even bigger, more complex, the electron configurations, and more different layers on them. and so on, but anyways, here's just the idea of this, you get these, just to get the idea of how spherical harmonics are used to basically help plot these orbitals or the probability density function or the wave function of just electrons around an atom, so that's pretty, or a electron around in hydrogen or other types of atoms. So that's pretty fascinating stuff here. and you get different shapes, and so on, and then here's the more complex one going over here. And then, again, in this case, this is N, and our cases would be N, but it has another term, this is the energy level N, and so on, and then they use L pretty much everywhere. I just use N for this video, just for derivation. That's pretty epic stuff. So, no elements with 6F, 7D, or 7F, I just want to read these notes, because it's pretty interesting. 7F electrons have been discovered yet. So that's over here. So there's on the F, this is on the L or N is equal 3, and goes over to here. So this, yes, so there is no 6F, so none of these, 6 or 7, none of them have been discovered. And elements with 7p electrons haven't discovered, but their electron configurations are only predicted. Save the exceptional LR, which fills 7P1 instead of 6D1. And yeah, this LR, I quickly just went over here to Wikipedia. So LR, or Larencium, is a synthetic chemical element as a simple LR, formerly LW, and Atomic Number 103, is named after Ernest Lorenz, inventor of the cyclotron. Interesting, a device used to discover many artificial radioactive elements, and so on, that's pretty fascinating stuff. How does it even look like? Is there any images of it? All right, so that's quite a fascinating stuff anyways, that's for another day. Yeah, actually, again, I just wanted to Google it. So here's Laurentium. Silver solid at 25 degrees there. That's his appearance looks like silver and solid at 25 degrees Celsius. It's pretty interesting setup there. And let's see right here. The most stable is the half-life of 11 hours, but the shorter-lived is 2 .67 is most commonly used chemistry because it can be produced on a large scale. So does that mean it just like radioactive decays in two minutes? That's quite wild. So, all right, let's continue further. All right, and for the elements whose high is occupied orbital is 6D orbital, only some electron configurations have been confirmed as using MT, DSRG, and CN are still missing. Interesting, this is other elements there. This is 6D, for elements who's highest occupied orbital 6D, that's over here. So you go to the D, you go to the 6th, right there. Yeah, so this one, these ones here, the ones that is the maximum there, and it just says, only some have been confirmed, and then these ones are still missing. All right, these are the real value orbitals commonly used in chemistry. only the m equals zero orbitals where our eigenstates of the orbital angular momentum operator LZ. Interesting, and again, yeah, these are the real valued set up, keep that in mind. Is the real valued? So it's ignoring that complex rotation there, and we'll go over that in a bit. So these are the real orbitals, so it's kind of like a scalar, take a snapshot or projection, and get that value. Yes, so only, and it says here, only the M equals zero orbitals where are eigenstates of the orbital angular momentum operator to this. The columns, that's M equals zero. Yes, so these are the M equals zero. Again, this is just beyond the scope of this, today's video, or my head for now, until I understand what's going on here for now. I'll be another for another video, again, this is just for deep diving into. the, basically the motivation for going over to spherical harmonics. Anyways, the columns with m equals plus minus one and plus or minus two are combinations of two eigenstates. See comparison to the following picture. Anyways, I'm not going to go over that. So this is plus or minus one. So that's these ones, plus or minus one, etc. Those are combinations, and I think this would be combinations of even more eigenstates. Here's a quote from Mookie. So one should remember that these orbital, quote, states as described here, are merely eigenstates of an electron in its orbit. An actual electron exists in a superposition of states, which is like a weighted average, but with complex number of weights. And the quantum state, after measurement, is in an eigenstate corresponding to that measurement and the value measured. All right, here I quickly just bolded this eigenstate there, because you see that a lot, you see that eigenvalues, eigenstate, and so on. So it's also from Wiki, quote, numerical or analytic solutions and quantum mechanics can be expressed as pure states. These solution states are called eigenstates, are labeled with quantized values, typically quantum numbers. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant pure states are identified by the... principle quantum number N, the angle momentum quantum number L, the magnetic quantum number M, and the spin Z component SZ. So basically, I think what it's just trying to say is the eigenstates are just like snapshots into the full solutions. You have the full wave function, so on, and you take a snapshot with quantized numbers, and that will be your eigenstate, one of the eigenstates, where you have these specific quantum numbers, L, N, M, and the spin, and so on. So when you quantize, put a specific value in them, you get a solution, and those are called eigenstates. So instead of the, and now this is also from Wiki, so instead of the complex orbitals described earlier, so these ones actually real, typically all the ones that graphs are just real orbitals, or real spherical harmonics. So instead of the ones we discussed here, this one's actually uses the imaginary setup there. The imaginary number I is probably somewhere hidden in there, I'm assuming. Because you have, yeah, you have. you're going to have an imaginary component inside here, and so on. Let's go further. Yeah, you actually go back here. It's either in there, or it might be in here, and so on it's, I don't know, this is the full generalizer, like, horrible, no, so you're going to have imaginary numbers somewhere in there. It was going further here. So instead of using that complex atomic orbital, so instead of the complex... orbitals described earlier, it is common, especially in the chemistry literature to use real atomic orbitals. These real orbitals arise from these simple linear combinations of complex orbitals. Real orbitals are related to complex orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. So here's the si-reel, you put the superscript real, then you put n-lm, and then you have these multiples, this constant squared. 2, this negative 1, power of m, times it by now you take the imaginary part of si, this is the wave function there, the imaginary part, and this is 4m is less than 0, and again, notice here, if you have an imaginary, if this one has an imaginary number I, and then you're multiplying, but this is a negative 1 to power of m, well, we know that i is this equal to, well, negative 1, yes, square of negative 1, so in other words, squaring this equals to negative 1. throw on some negative ones in there, you're going to eventually cancel out the I. So the whole idea is to cancel out that extra I there. So you have no complex setup there. And then for M is equal zero, it's the same thing. Nothing changes. And then for m is greater than zero, you get this right here. You take the real component and you multiply by this setup there. And then for this part right here, and this simplifies all down to this giant. thingy there, so that's interesting. So I, divided by square 2, and tens by this, and then for different values, absolute value, negative, positive M, and so on. So again, very, very similar to the same setup we had all the way back in our real spherical harmonics. And that's this setup here. So for the real spherical harmonics, we got this set up here. So those the real, and then, but now we have the extra term, we have NLM, the other one, we just had an M, but…

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