Hello everyone welcome back to exploring quantum physics, I'm Charles Clark. We're going to continue our discussion of the dynamics of, rotating polar molecule in electric field. Okay, so were just pursuing perturbation theory the usual form and we found the order Eigen functions, we knew those already. We evaluated the first zero and so now we continue to the next order. So again, what we did in this example so far was determine the zeroth-order wave function, the zeroth-order energy and then the first-order energy which is again, it's just a problem of quadrature. So whatever difficulty you have in doing integrals, integration is in many respects a mechanical procedure. You can easily find the miracle techniques in many cases and will do for if we don't have to evaluate. Now the next step is to get the first order wave function side. One, and that is rather different undertaking. So here is the equation that defines the 1st order wave function. And, I prefer to write it in this way here. This is the defining equation of psi 1 and so you see it satisfies An inhomogeneous. Sorry, inhomogeneous linear equation. Which depends upon sign on. Now important point to know that we're going to go on calculate sci one for a particular case in a moment just to show you how that's done but once you have sci one and hand once you have all those hand. ANd then let's see to try to go through this without getting too bogged down in it. This is the second order equation. That is the first equation in which e two appears for the second order, energy and you see that there's a second order wave function as well as a first order zero over m zero throw it away function. But once again, when we apply the bra sine aught to both side of this equation, then I hope you can see that this term here and that term Cancel. They're the same because of the formation property. We're left with this term. This term vanishes or let me just say This term is necessarily orthogonal because of the orthogonal sign ons [INAUDIBLE] or if we have E1=0. Then it turns out that this term goes away. And so we're left with an equation where E2, which that that E2 passes through and sine 0 is normalized. Here's the important equation for the second-order energy, it's just equal to the matrix element the potential operating on the first order wave function. Projected back onto the 0th order. So this is the second order energy, is something that defines response functions, generally useful, and that's our target for today. Now, here's how we're going to solve the equation for the first-order perturbe function which we need in order to get the lowest non-vanishing energy. Since we saw that the first-order energy vanished. We have to get up to find something interesting. So once again, here's the 0th order Hamiltonian, the potential. 0th order energy and the 0th order wave function, and we've established that the 1st order energy also vanishes. So now it's relatively straight forward to write down the equation for sine one because e0 is 0. And this the left hand side of the equation is the angular momentum operator apply to sine one, is this ratio of the strengths of the two interactions, which comes from these two terms respectively. And then here you see there is an angle, the cos of theta and the normalization, 1 / square root of 4 pi. Well this means, you see since the, the angular momentum operator is spherically symmetric. It's rotationally symmetric. When it operates on side one it returns a particular angle. That means that we should expect the angular behavior of the wave function to be embedded in psi one. It's only going to be revealed by the angular momentum operator, which is spherically symmetric. It's not going to be imposed by it. And, indeed, it turns out that cosine theta is a spherical harmonic with M = 1. I think you can see that. That's just because a cosine theta is x three over r so it satisfies our neumatic, x three is a solution to cross the equation as is x three of r. And it's, this actually, this L, L is the polynomial order of that expression that we were looking at earlier first part of this lecture. So then, it then follows that the first order of way function is [INAUDIBLE] four. So you see in this case the existence of the understanding of angular momentum. Enables us to solve the first order equation in a relatively straight forward way. Okay, so we've solved for the first order wave function and we can now compute the second order energy by this simple formula and well, okay, so this number, maybe this number doesn't mean anything in and of itself. But let's just ask what it means from the standpoint of classical electrodynamics. As we'll see, this describes a very important quality, the polarizability of a molecule. So in electrostatics you learn about the properties of material objects when their expose to electromagnetic fields and one of the classic problems concerned a conducting sphere of some finite radius which you put in an electric field and then that result in the sphere acquire in an electric dipole moment. So there is a re distribution of charges on the sphere with the positive charges on one side and the negative charges on the other. And this is an empirical long. There was understood about the electrical properties matter in ancient time before there was any known quantitative theory, how matter response to electricity. And so it turns out that there's a very useful convention. The relationship of the, for weak fields as a linear response to the dipole moment induced by an applied electric field Is just proportional to this constant that's characteristic of the object for which for a conducting sphere is just the cube of its radius. And that where we use the c.g.s. Units for the electric field charges. Now, as it happens it's easy to show this from the defining relations. The relationship and the polarizability and the second order energy. Is just given in these simple terms. So this means we can now use quantum mechanics to describe the polarized ability of matter. It's really the first and to date only fundamental theory that allows us to understand electrical properties from first principles. So when we apply it to this potassium molecule, we find that the polarized ability in the ground state is given by this expression and this again has the dimensions of volume, so it looks like a conducted sphere of that. 25. Two and a half nanometers in radius. I'm just using this as a measure of a number. I want you to know the following interesting Interesting to me. The polarizability which is the ratio of the induced dipole moment to the applied field strength, actually diverges as one goes to the classical limit. Taking h bar and going to zero, so for a rotating molecule, if we, if we lived in a classical world, a polarizability would be infinite and that I think you can see is for a rotating molecule of the If there were no energy cost of localization, Which is due to the uncertainty principle, then the the dipole moment will be localized Instantly by even the smallest field. So this shows the least as concerns the behavior of a rotating polar molecule, the electrical polarized ability of matter is definitely a quantum effect. That is, it ceases exist when and h bar goes to 0. We're going to see another example of that, actually very different type in the next part of the slide.
