WEBVTT 00:00:00.135 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT Open Courseware 00:00:06.330 --> 00:00:10.690 continue to offer high quality educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.250 from hundreds of MIT courses, visit MIT Open Courseware 00:00:17.250 --> 00:00:18.210 at ocw.mit.edu. 00:00:22.110 --> 00:00:24.870 ROBERT FIELD: Today is the first of two lectures 00:00:24.870 --> 00:00:26.790 on the rigid rotor. 00:00:26.790 --> 00:00:30.300 And the rigid rotor is really our first glimpse 00:00:30.300 --> 00:00:32.340 of central force problems. 00:00:32.340 --> 00:00:35.940 And so although the rigid rotor is an exactly solved problem, 00:00:35.940 --> 00:00:38.820 it's also a problem that we-- 00:00:38.820 --> 00:00:42.160 it involves something that is universal. 00:00:42.160 --> 00:00:46.500 If we have a spherical object, then the angular part 00:00:46.500 --> 00:00:50.220 of the Hamiltonian for that spherical object 00:00:50.220 --> 00:00:52.810 is solved by the rigid rotor. 00:00:52.810 --> 00:00:57.280 And so if we understand the rigid rotor 00:00:57.280 --> 00:01:01.380 and know how to draw pictures, we're 00:01:01.380 --> 00:01:04.650 going to be able to understand the universal part 00:01:04.650 --> 00:01:07.410 of all central force problems. 00:01:07.410 --> 00:01:09.240 So it's not just a curiosity. 00:01:09.240 --> 00:01:11.910 It's a really major thing. 00:01:11.910 --> 00:01:12.420 OK. 00:01:12.420 --> 00:01:16.170 And I'm going to do a lot of unconventional stuff 00:01:16.170 --> 00:01:18.360 in this lecture and the next lecture, 00:01:18.360 --> 00:01:20.500 as far as the rigid rotor is concerned. 00:01:20.500 --> 00:01:24.750 Because it's covered to death, but with lots 00:01:24.750 --> 00:01:28.140 of equations in all textbooks. 00:01:28.140 --> 00:01:34.260 And there is very little effort to give you 00:01:34.260 --> 00:01:36.480 independent insights. 00:01:36.480 --> 00:01:40.740 And so my discussion of the vector model 00:01:40.740 --> 00:01:43.290 is something that I consider really special. 00:01:43.290 --> 00:01:45.900 And what I want you to be able to do 00:01:45.900 --> 00:01:49.290 is to draw cartoons that capture essentially 00:01:49.290 --> 00:01:51.660 all of the important points. 00:01:51.660 --> 00:01:55.305 And that you won't find anywhere except in my notes. 00:01:58.650 --> 00:02:01.080 But I think it's really important 00:02:01.080 --> 00:02:04.390 that you have your own point of view on these problems. 00:02:04.390 --> 00:02:10.229 So let me just do a quick review, with the exam in mind, 00:02:10.229 --> 00:02:14.520 of the perturbation theory stuff. 00:02:14.520 --> 00:02:17.050 So we had a harmonic oscillator. 00:02:17.050 --> 00:02:22.080 And it's perturbed by some anharmonicity terms. 00:02:22.080 --> 00:02:28.770 So the potential is going to 1/2 kQ squared plus bQ 00:02:28.770 --> 00:02:35.040 cubed plus cQ to the fourth. 00:02:35.040 --> 00:02:36.240 And we could go on. 00:02:36.240 --> 00:02:37.410 But this is enough. 00:02:37.410 --> 00:02:41.010 Because it gives a glimpse of all of the important stuff 00:02:41.010 --> 00:02:44.310 that emerges qualitatively as well as quantitatively 00:02:44.310 --> 00:02:47.460 from perturbation theory. 00:02:47.460 --> 00:02:53.130 So this is part of the H0 problem. 00:02:53.130 --> 00:02:56.960 And this is H1. 00:02:56.960 --> 00:03:02.390 And so you know now that doing second order perturbation 00:03:02.390 --> 00:03:05.060 theory is tedious and ugly. 00:03:05.060 --> 00:03:09.560 And it's easy to get overwhelmed by the equations. 00:03:09.560 --> 00:03:12.020 But what I want to do is to just review 00:03:12.020 --> 00:03:16.950 how you get the crucial structure of the problem. 00:03:16.950 --> 00:03:23.590 So the Q cubed' term-- 00:03:23.590 --> 00:03:27.680 there are no diagonal elements of the Q cubed term. 00:03:27.680 --> 00:03:31.790 And so we're always going to be talking about the second order 00:03:31.790 --> 00:03:33.650 corrections to the energy. 00:03:33.650 --> 00:03:38.000 The selection rules are delta v equals plus and minus 3 00:03:38.000 --> 00:03:42.522 and plus n minus 1. 00:03:42.522 --> 00:03:47.200 And so what I told you is, if you're going to do the algebra, 00:03:47.200 --> 00:03:50.010 which you probably don't want to ever do, 00:03:50.010 --> 00:03:52.980 you're going to want to combine the delta v of plus 3 00:03:52.980 --> 00:03:56.010 and minus 3 terms of the perturbation sum. 00:03:56.010 --> 00:03:58.650 Because the algebra simplifies. 00:03:58.650 --> 00:04:01.290 They have similar expressions. 00:04:01.290 --> 00:04:09.810 And you have a denominator, which is 3 Hc omega and minus 3 00:04:09.810 --> 00:04:10.590 Hc omega. 00:04:10.590 --> 00:04:11.440 That factors out. 00:04:11.440 --> 00:04:15.150 And you just have a plus 3 and minus 3 in the denominator. 00:04:15.150 --> 00:04:19.110 And similarly here, so when you do this, 00:04:19.110 --> 00:04:21.300 you lose the highest order term. 00:04:21.300 --> 00:04:22.830 What's the highest order term? 00:04:22.830 --> 00:04:29.160 It's v plus 1/2 to the third power. 00:04:29.160 --> 00:04:33.510 Because we have delta v of 3. 00:04:33.510 --> 00:04:35.940 We have Q cubed. 00:04:35.940 --> 00:04:39.150 And the matrix elements of Q cubed 00:04:39.150 --> 00:04:43.230 are a plus a-dagger cubed. 00:04:43.230 --> 00:04:45.440 And the selection rule for them-- 00:04:45.440 --> 00:04:49.650 or the matrix-- must go as the square root of the power 00:04:49.650 --> 00:04:51.900 of the a's. 00:04:51.900 --> 00:04:54.540 And so all of the matrix elements 00:04:54.540 --> 00:04:59.640 have the leading term v plus 1/2 to the 3/2 power. 00:04:59.640 --> 00:05:02.470 But we're squaring. 00:05:02.470 --> 00:05:06.590 And we lose the highest order because of the subtraction. 00:05:06.590 --> 00:05:11.495 So this one gives rise to terms in v plus 1/2 squared. 00:05:14.080 --> 00:05:22.100 Q4 has selection rules delta v plus or minus 4, 00:05:22.100 --> 00:05:24.820 plus or minus 2, and 0. 00:05:24.820 --> 00:05:25.770 This is important. 00:05:25.770 --> 00:05:28.950 Because this one says, we have a contribution 00:05:28.950 --> 00:05:30.840 from the first order-- 00:05:30.840 --> 00:05:33.490 in the first order of energy. 00:05:33.490 --> 00:05:35.560 And so in the first order of energy, 00:05:35.560 --> 00:05:41.745 we're going to have v plus 1/2 to the fourth power-- 00:05:46.140 --> 00:05:49.620 I'm sorry-- to the 4/2 power. 00:05:53.460 --> 00:05:56.022 Because we only have the matrix element, 00:05:56.022 --> 00:05:57.480 we don't square the matrix element. 00:06:01.630 --> 00:06:08.320 So this gives rise to a term in v plus 1/2 squared. 00:06:08.320 --> 00:06:11.980 But it's sensitive to the sign of the Q 00:06:11.980 --> 00:06:14.470 to the fourth term in the Hamiltonian. 00:06:14.470 --> 00:06:18.180 It's the only thing that's sensitive to the sine. 00:06:18.180 --> 00:06:29.740 We also have terms that are v plus 1/2 to the 4/2 times 2. 00:06:29.740 --> 00:06:39.500 So we can have matrix elements to the four-- 00:06:39.500 --> 00:06:43.670 because it's a fourth power term, we get 4/2. 00:06:43.670 --> 00:06:46.220 And because it's squared, we multiply by 2. 00:06:46.220 --> 00:06:47.780 And then we lose the highest order. 00:06:47.780 --> 00:06:54.010 So we get, from this, v plus 1/2 to the third power. 00:06:54.010 --> 00:06:58.870 So what we're hoping to get from the perturbation theory 00:06:58.870 --> 00:07:04.060 is the highest order terms in the energy expression. 00:07:04.060 --> 00:07:09.730 And so Q to the fourth gives a signed v plus 1/2 squared term. 00:07:09.730 --> 00:07:14.950 And it gives an unsigned always positive v plus 1/2 cubed term. 00:07:14.950 --> 00:07:21.790 And this one gives v plus 1/2 squared term. 00:07:24.380 --> 00:07:26.800 OK. 00:07:26.800 --> 00:07:28.390 So this would tell you, when you're 00:07:28.390 --> 00:07:32.290 organizing your work, what to expect and how to organize it. 00:07:32.290 --> 00:07:35.020 And I'm really not expecting you to do 00:07:35.020 --> 00:07:39.890 very much with this level of complexity on an exam problem. 00:07:39.890 --> 00:07:42.940 But on homework, all bets are off. 00:07:42.940 --> 00:07:44.680 OK. 00:07:44.680 --> 00:07:48.010 So that's all I want to do as far as the review is concerned. 00:07:50.890 --> 00:07:54.010 So the rigid rotor-- 00:07:54.010 --> 00:07:57.070 we have a molecule. 00:08:11.060 --> 00:08:14.900 So we have a bond of length r sub 0. 00:08:14.900 --> 00:08:16.680 And we have two masses. 00:08:16.680 --> 00:08:18.850 And if the masses aren't equal, the center of mass 00:08:18.850 --> 00:08:21.789 isn't in the middle. 00:08:21.789 --> 00:08:23.330 The center of mass-- you need to know 00:08:23.330 --> 00:08:25.670 how to calculate where the center of mass is. 00:08:25.670 --> 00:08:28.700 There are all sorts of simple algebra-- 00:08:28.700 --> 00:08:30.800 but what is happening is this guy 00:08:30.800 --> 00:08:34.914 is rotating without stretching about the center of mass. 00:08:43.549 --> 00:08:47.870 Now, what we want to do is think about this problem 00:08:47.870 --> 00:08:48.860 as if it were-- 00:08:53.710 --> 00:08:55.850 here's our 0. 00:08:55.850 --> 00:08:58.740 And this is the reduced mass. 00:08:58.740 --> 00:09:04.740 This is a motion of a fictitious particle of mass mu. 00:09:04.740 --> 00:09:16.360 Mu is m1 m2 over m1 plus m2 on the surface of a sphere. 00:09:16.360 --> 00:09:18.220 They're all the same problem. 00:09:18.220 --> 00:09:20.680 But the question is, how do we interpret 00:09:20.680 --> 00:09:23.500 what we get from the solution to this problem? 00:09:27.120 --> 00:09:32.210 So what we care about is a description of, 00:09:32.210 --> 00:09:34.910 where is the molecular axis? 00:09:34.910 --> 00:09:37.190 The molecule is rotating. 00:09:37.190 --> 00:09:41.090 And we're solving the Schrodinger equation. 00:09:41.090 --> 00:09:46.760 And we get things like the expectation value of J squared 00:09:46.760 --> 00:09:52.390 and Jz and maybe some other things where 00:09:52.390 --> 00:09:55.360 these are the angular momenta. 00:09:55.360 --> 00:10:04.270 And how do the eigenfunctions for these states-- 00:10:04.270 --> 00:10:05.620 we have a state. 00:10:05.620 --> 00:10:12.470 And we have quantum numbers Jm and their probability 00:10:12.470 --> 00:10:14.480 amplitudes and theta and phi. 00:10:14.480 --> 00:10:16.085 That's a whole lot of stuff to digest. 00:10:19.270 --> 00:10:21.580 We want to go as quickly as we can 00:10:21.580 --> 00:10:27.650 to how this is related to the thing we really care about, 00:10:27.650 --> 00:10:29.080 which is, where is the bond axis? 00:10:33.100 --> 00:10:34.970 So the molecule is rotating. 00:10:34.970 --> 00:10:40.220 And so the bond axis is moving in laboratory frame. 00:10:40.220 --> 00:10:43.450 And we want to be able to take, from that, 00:10:43.450 --> 00:10:46.525 the minimal amount of information 00:10:46.525 --> 00:10:49.730 that we memorize or remember-- 00:10:49.730 --> 00:10:50.990 I don't like "memorize." 00:10:50.990 --> 00:10:53.810 Because "memorize" doesn't involve understanding. 00:10:53.810 --> 00:10:56.060 But "remembering" does. 00:10:56.060 --> 00:11:01.120 We want to be able to, at a drop of a hat, 00:11:01.120 --> 00:11:05.240 be able to say, yes, if we pick these two quantum 00:11:05.240 --> 00:11:08.180 numbers, which are related to the eigenvalues of these two 00:11:08.180 --> 00:11:12.320 operators, we can describe the spatial distribution 00:11:12.320 --> 00:11:13.325 of the molecular axis. 00:11:17.020 --> 00:11:18.520 There is an extra complication. 00:11:25.700 --> 00:11:29.100 So we live in the laboratory. 00:11:29.100 --> 00:11:31.520 And we have a coordinate system that we 00:11:31.520 --> 00:11:34.850 care about, the lab-fixed coordinates. 00:11:34.850 --> 00:11:36.860 And they're always going to be represented 00:11:36.860 --> 00:11:39.020 with capital letters. 00:11:39.020 --> 00:11:42.350 And there's also the body frame. 00:11:42.350 --> 00:11:45.150 And things are in the body frame. 00:11:45.150 --> 00:11:47.690 That's what the molecule cares about. 00:11:47.690 --> 00:11:50.180 And how do things in the body frame 00:11:50.180 --> 00:11:51.830 relate to the laboratory frame? 00:11:51.830 --> 00:11:54.890 It's not trivial. 00:11:54.890 --> 00:11:58.970 And that's where the real effort at understanding comes. 00:11:58.970 --> 00:12:04.150 And it's kind of trivial for a diatomic molecule. 00:12:04.150 --> 00:12:10.860 But it's far from trivial when you have a nonlinear molecule, 00:12:10.860 --> 00:12:15.760 a molecule with many atoms and properties of each of the atoms 00:12:15.760 --> 00:12:19.840 that somehow combine to give things that you observe. 00:12:19.840 --> 00:12:25.990 And so these two coordinate systems are very different. 00:12:25.990 --> 00:12:30.010 And actually, the solutions of the rigid rotor 00:12:30.010 --> 00:12:37.030 Hamiltonian gives you the relationship of the body 00:12:37.030 --> 00:12:40.540 coordinates to the laboratory coordinates. 00:12:40.540 --> 00:12:42.920 And that's kind of important. 00:12:42.920 --> 00:12:47.070 So we'll see how this develops as I go on. 00:12:47.070 --> 00:12:48.970 And I feel quite passionate about this. 00:12:48.970 --> 00:12:54.820 Because one of the things that I do as an experimentalist is 00:12:54.820 --> 00:12:58.000 I observe fully resolved spectra. 00:12:58.000 --> 00:13:01.000 And the big-- the most information-rich feature 00:13:01.000 --> 00:13:04.460 of the spectrum is the rotational spectrum. 00:13:04.460 --> 00:13:09.930 And so it contains a lot of really good stuff. 00:13:09.930 --> 00:13:12.320 OK. 00:13:12.320 --> 00:13:20.630 So first of all, the Hamiltonian is just the kinetic energy. 00:13:20.630 --> 00:13:23.160 Because it's a rigid rotor, it's free. 00:13:23.160 --> 00:13:24.642 There's no potential. 00:13:27.740 --> 00:13:30.110 But that's the last nice thing about it. 00:13:30.110 --> 00:13:36.980 Because we have to go from Cartesian to spherical polar 00:13:36.980 --> 00:13:38.390 coordinates. 00:13:38.390 --> 00:13:40.850 And there are a lot of unfamiliar things 00:13:40.850 --> 00:13:44.740 in this kinetic energy expression. 00:13:44.740 --> 00:13:47.099 And when I write it down, you're not going to like it. 00:13:47.099 --> 00:13:48.890 Unless you're a mathematician, and you say, 00:13:48.890 --> 00:13:52.240 oh, yeah, I want to be able to solve these kinds of equations. 00:13:52.240 --> 00:13:56.320 Because the Schrodinger equation is, 00:13:56.320 --> 00:14:01.820 considering the simplicity of the problem, terrible. 00:14:01.820 --> 00:14:04.370 And except for mathematicians who just love it 00:14:04.370 --> 00:14:05.420 because they know-- 00:14:05.420 --> 00:14:06.620 oh, I know that equation. 00:14:06.620 --> 00:14:09.572 I know how to write down everything that I care about. 00:14:09.572 --> 00:14:11.030 But we care about different things. 00:14:14.400 --> 00:14:21.250 So we want to know the energy levels of the rigid rotor. 00:14:21.250 --> 00:14:25.255 And we also want pictures. 00:14:28.530 --> 00:14:33.670 And remember, a picture is a reduced form of all the details 00:14:33.670 --> 00:14:35.990 that you have at your disposal. 00:14:35.990 --> 00:14:38.140 And you have to really sweat to make 00:14:38.140 --> 00:14:41.980 sure you understand every detail of the pictures and the stuff 00:14:41.980 --> 00:14:44.560 that you've averaged over, or you've concealed, 00:14:44.560 --> 00:14:47.800 because you don't need it, OK? 00:14:51.631 --> 00:14:52.130 All right. 00:14:52.130 --> 00:14:55.900 So I'm going to be generating a lot of pictures. 00:14:55.900 --> 00:14:59.400 And one of the things we want to understand is-- 00:14:59.400 --> 00:15:04.460 so we have this vector, J. The J is the angular momentum 00:15:04.460 --> 00:15:07.340 of the molecule. 00:15:07.340 --> 00:15:09.760 And it's a vector as opposed to a scalar. 00:15:09.760 --> 00:15:12.850 That means it has three components. 00:15:12.850 --> 00:15:16.140 So for this part of the problem that I've just hidden over 00:15:16.140 --> 00:15:19.230 here, that's a one dimensional problem. 00:15:19.230 --> 00:15:22.740 We have a momentum and a conjugate coordinate 00:15:22.740 --> 00:15:25.230 and it's 1D. 00:15:25.230 --> 00:15:26.040 This is 3D. 00:15:26.040 --> 00:15:28.260 And there's all sorts of subtle stuff that goes on. 00:15:30.950 --> 00:15:31.670 OK. 00:15:31.670 --> 00:15:37.760 So we would like to understand how this vector moves. 00:15:37.760 --> 00:15:39.680 Well, there is a question. 00:15:39.680 --> 00:15:40.994 I said moves. 00:15:40.994 --> 00:15:43.160 We're talking about the time independent Schrodinger 00:15:43.160 --> 00:15:44.690 equation. 00:15:44.690 --> 00:15:45.560 Nothing moves. 00:15:49.750 --> 00:15:54.280 We can use our concept of motion from classical mechanics 00:15:54.280 --> 00:15:57.490 to describe certain features of the average 00:15:57.490 --> 00:16:00.470 of some quantum mechanical system. 00:16:00.470 --> 00:16:02.180 And so there is kind of a motion. 00:16:02.180 --> 00:16:04.401 But we'll see what that is. 00:16:04.401 --> 00:16:04.900 OK. 00:16:04.900 --> 00:16:07.000 We have some operators that we like. 00:16:20.580 --> 00:16:21.360 OK. 00:16:21.360 --> 00:16:24.900 Total angular momentum, projection of the angular 00:16:24.900 --> 00:16:28.350 momentum on a laboratory z-axis-- 00:16:28.350 --> 00:16:30.990 it's kind of hard to draw capital Z and small z 00:16:30.990 --> 00:16:32.610 if you haven't got anybody nearby. 00:16:32.610 --> 00:16:36.390 But anyway, this is a capital Z. And this 00:16:36.390 --> 00:16:40.230 is the thing that's analogous to the creation and annihilation 00:16:40.230 --> 00:16:40.830 operators. 00:16:43.350 --> 00:16:44.970 We need them. 00:16:44.970 --> 00:16:46.980 And it gives us a lot of insight. 00:16:46.980 --> 00:16:49.500 And that will be the subject of Monday's lecture. 00:16:52.610 --> 00:16:57.000 And so we're going to have solutions, which are described 00:16:57.000 --> 00:16:59.370 by J and m quantum numbers. 00:16:59.370 --> 00:17:06.530 And we want to understand what these things look like. 00:17:06.530 --> 00:17:07.885 Where are the nodal surfaces? 00:17:10.650 --> 00:17:14.030 The nodal surfaces give us basically everything 00:17:14.030 --> 00:17:15.199 we want to know. 00:17:18.349 --> 00:17:21.280 There is the direct correlation to where the nodes are 00:17:21.280 --> 00:17:26.230 and how many of them there are with these quantum numbers. 00:17:26.230 --> 00:17:27.790 That's one of the important things 00:17:27.790 --> 00:17:28.920 you have to come away with. 00:17:34.461 --> 00:17:34.960 OK. 00:17:34.960 --> 00:17:38.830 In my next lecture on angular momentum, 00:17:38.830 --> 00:17:42.010 I'm going to introduce a commutation rule, 00:17:42.010 --> 00:17:55.330 Ji Jj equals I H bar sum over K epsilon I Jk Jk. 00:17:55.330 --> 00:17:56.990 Isn't that a strange looking thing? 00:17:56.990 --> 00:18:01.252 Because I haven't told you what this epsilon is. 00:18:01.252 --> 00:18:02.460 There's lots of names for it. 00:18:02.460 --> 00:18:03.000 It won't help. 00:18:03.000 --> 00:18:04.833 Because I'm not going to talk about it here. 00:18:04.833 --> 00:18:08.430 But this is actually the fundamental definition 00:18:08.430 --> 00:18:10.970 of an angular momentum. 00:18:10.970 --> 00:18:12.770 And from this commutator rule, we 00:18:12.770 --> 00:18:18.140 can generate all the matrix elements of angular momentum. 00:18:18.140 --> 00:18:23.170 And there's another family of commutation rules 00:18:23.170 --> 00:18:25.880 where we have a component of angular momentum 00:18:25.880 --> 00:18:29.710 and a component of what we call a spherical tensor. 00:18:29.710 --> 00:18:33.310 Any operator can be classified according to spherical tensor 00:18:33.310 --> 00:18:34.570 rank. 00:18:34.570 --> 00:18:37.300 And that then determines all the matrix elements 00:18:37.300 --> 00:18:40.070 of that operator. 00:18:40.070 --> 00:18:42.860 And so you can imagine that this is a really powerful way 00:18:42.860 --> 00:18:44.510 of approaching stuff. 00:18:44.510 --> 00:18:51.470 And what I said before about the commutation rule of x and P sub 00:18:51.470 --> 00:18:54.800 x is that many people regard that as the foundation 00:18:54.800 --> 00:18:56.697 of quantum mechanics. 00:18:56.697 --> 00:18:58.280 And this is just an extension of that. 00:19:01.160 --> 00:19:03.580 So there is a way of getting all quantum mechanics 00:19:03.580 --> 00:19:07.092 from a few well-chosen commutation rules. 00:19:07.092 --> 00:19:08.050 And that's really neat. 00:19:11.251 --> 00:19:11.750 OK. 00:19:11.750 --> 00:19:12.875 Let's get down to business. 00:19:15.920 --> 00:19:16.920 So I already drew this. 00:19:16.920 --> 00:19:18.930 But I'll draw it again. 00:19:18.930 --> 00:19:21.790 Because I'm going to drop a companion. 00:19:21.790 --> 00:19:22.290 OK. 00:19:22.290 --> 00:19:23.730 So we have here-- 00:19:35.560 --> 00:19:36.060 OK. 00:19:36.060 --> 00:19:37.800 So we have an angular momentum. 00:19:37.800 --> 00:19:40.260 And we have the bond axis. 00:19:40.260 --> 00:19:46.270 And this angular momentum is perpendicular to the bond axis. 00:19:46.270 --> 00:19:49.150 So if we know something about how the angular momentum is 00:19:49.150 --> 00:19:52.570 distributed in space, we know something about how the bond 00:19:52.570 --> 00:19:54.460 axis is distributed in space. 00:19:54.460 --> 00:19:57.800 But it's not trivial. 00:19:57.800 --> 00:20:00.250 But once you've learned how to make those connections, 00:20:00.250 --> 00:20:02.031 it's fine. 00:20:02.031 --> 00:20:02.530 OK. 00:20:02.530 --> 00:20:03.730 So now, let's draw-- 00:20:12.250 --> 00:20:16.250 So we have a right-handed coordinate system. 00:20:16.250 --> 00:20:20.110 And we have, say, the vector J. Now, 00:20:20.110 --> 00:20:22.000 this is the laboratory frame. 00:20:22.000 --> 00:20:34.150 We have J. And we have a molecule, which 00:20:34.150 --> 00:20:40.490 is perpendicular to J. And we have 00:20:40.490 --> 00:20:45.950 the projection of J on the body axis, on the z-axis, 00:20:45.950 --> 00:20:49.470 and that's M. 00:20:49.470 --> 00:20:49.970 OK. 00:20:49.970 --> 00:20:53.900 So when we talked about the quantum number 00:20:53.900 --> 00:20:58.310 J, which is the length of this vector, and m, which 00:20:58.310 --> 00:21:00.860 is the projection of that vector on this axis, 00:21:00.860 --> 00:21:04.790 that's beginning to be how we understand how this works. 00:21:04.790 --> 00:21:06.620 Because this is perpendicular to that. 00:21:09.480 --> 00:21:15.135 Now, here we have one of the sins. 00:21:25.150 --> 00:21:28.330 So we begin by saying, well, this is a picture. 00:21:28.330 --> 00:21:36.780 And J precesses, or moves, on a cone around z. 00:21:36.780 --> 00:21:39.840 How do we have motion? 00:21:39.840 --> 00:21:43.990 This is a time independent Schrodinger equation. 00:21:43.990 --> 00:21:49.630 It's a way of saying, well, it doesn't matter where J is. 00:21:49.630 --> 00:21:52.660 It's more or less equally distributed 00:21:52.660 --> 00:21:57.170 in probability on this cone. 00:21:57.170 --> 00:21:58.734 But not an amplitude. 00:22:01.460 --> 00:22:03.130 Remember, when we have something that's 00:22:03.130 --> 00:22:06.700 moving in the time independent Schrodinger equation, 00:22:06.700 --> 00:22:09.520 we get oscillations. 00:22:09.520 --> 00:22:11.830 But we need a complex function in order 00:22:11.830 --> 00:22:13.750 to have those oscillations. 00:22:13.750 --> 00:22:18.430 If we're going to take psi star psi, there is no complex part. 00:22:18.430 --> 00:22:22.900 And you do have uniform amplitude of J. 00:22:22.900 --> 00:22:25.720 And so it's sensible to say, well, it got that way 00:22:25.720 --> 00:22:27.910 because J precesses. 00:22:27.910 --> 00:22:30.450 Maybe. 00:22:30.450 --> 00:22:32.970 There is something if you say, well, 00:22:32.970 --> 00:22:37.020 maybe if I created a system at t equal 0 00:22:37.020 --> 00:22:42.040 where J was at a particular position, 00:22:42.040 --> 00:22:45.380 and then I let that thing evolve-- 00:22:45.380 --> 00:22:46.670 this is not an eigenstate. 00:22:46.670 --> 00:22:50.330 It would be a complicated superposition. 00:22:50.330 --> 00:22:52.610 That thing would precess. 00:22:52.610 --> 00:22:55.955 And you would observe what we call polarization quantum 00:22:55.955 --> 00:22:56.780 beats. 00:22:56.780 --> 00:22:59.660 Now, that's getting way ahead of things. 00:22:59.660 --> 00:23:07.360 But it is helpful to think about J precessing on this cone. 00:23:07.360 --> 00:23:12.840 Because that gives you a sense that the length of J is 00:23:12.840 --> 00:23:14.140 conserved. 00:23:14.140 --> 00:23:17.800 And the orientation about the z-axis is not. 00:23:17.800 --> 00:23:21.610 And it's more or less uniform probability, but not 00:23:21.610 --> 00:23:23.560 uniform probability amplitude. 00:23:23.560 --> 00:23:27.340 But you don't need that for a lot of the things. 00:23:27.340 --> 00:23:29.040 If you're looking at the wave function, 00:23:29.040 --> 00:23:31.030 yeah, there's going to be some oscillation. 00:23:31.030 --> 00:23:32.740 The real part and the imaginary part 00:23:32.740 --> 00:23:36.890 will oscillate in such a way that the probability 00:23:36.890 --> 00:23:38.920 is constant about that. 00:23:38.920 --> 00:23:40.810 I finally realized that this morning. 00:23:40.810 --> 00:23:43.540 So it's not as much of a lie as you think. 00:23:43.540 --> 00:23:45.160 OK. 00:23:45.160 --> 00:23:49.750 So this is the picture. 00:23:49.750 --> 00:23:53.950 And this is what it looks like in the laboratory. 00:23:53.950 --> 00:24:02.650 And now, I'm going to prepare you for-- 00:24:02.650 --> 00:24:06.040 so suppose you had a molecule. 00:24:06.040 --> 00:24:14.670 And you have a little person standing on the molecule frame. 00:24:14.670 --> 00:24:17.875 And now, you have somebody out here observing as the molecule 00:24:17.875 --> 00:24:18.375 rotates. 00:24:22.982 --> 00:24:26.790 Well, and here's J. All information that you're 00:24:26.790 --> 00:24:30.800 allowed to know from outside the molecule 00:24:30.800 --> 00:24:33.620 comes from the projection of whatever 00:24:33.620 --> 00:24:41.650 is in the molecule frame on J. And that's 00:24:41.650 --> 00:24:42.950 what the vector model does. 00:24:42.950 --> 00:24:44.920 It tells you how to take stuff you 00:24:44.920 --> 00:24:48.160 know about the individual atoms and project it 00:24:48.160 --> 00:24:51.974 on the thing that communicates with the outside world. 00:24:51.974 --> 00:24:53.390 There are a couple of other things 00:24:53.390 --> 00:24:54.765 that I'm going to say about this. 00:24:54.765 --> 00:24:57.190 But now, let's get down to the business 00:24:57.190 --> 00:25:00.430 of actually doing a little bit of the solution 00:25:00.430 --> 00:25:01.718 of the Schrodinger equation. 00:25:14.180 --> 00:25:18.170 So for our free rotor, the potential 00:25:18.170 --> 00:25:25.220 is r equals r0 theta phi. 00:25:25.220 --> 00:25:29.630 And it's equal to 0 if r equals r0. 00:25:29.630 --> 00:25:32.060 And it's equal to infinity if it's not. 00:25:32.060 --> 00:25:35.810 So this is very much like a particle in an infinite box. 00:25:35.810 --> 00:25:40.810 In fact, you can do a really cheap solution 00:25:40.810 --> 00:25:44.740 if you say, well, let's consider a particle 00:25:44.740 --> 00:25:48.020 in an infinite circular box. 00:25:48.020 --> 00:25:51.620 Well, you can solve this problem just using the de Broglie 00:25:51.620 --> 00:25:53.380 wavelength. 00:25:53.380 --> 00:25:57.250 And you get that the energy levels for this circular box 00:25:57.250 --> 00:26:01.800 problem go as the quantum number squared. 00:26:01.800 --> 00:26:04.870 And that's almost exactly like the solutions 00:26:04.870 --> 00:26:09.980 to the free rotor. 00:26:09.980 --> 00:26:14.390 The difference is the energies for this go 00:26:14.390 --> 00:26:16.040 as the quantum number squared. 00:26:16.040 --> 00:26:18.530 And the energies for this go as the quantum number 00:26:18.530 --> 00:26:20.476 times the quantum number plus 1. 00:26:20.476 --> 00:26:21.350 It's almost the same. 00:26:24.250 --> 00:26:25.870 OK. 00:26:25.870 --> 00:26:31.930 So because the potential is constant as long as r is fixed, 00:26:31.930 --> 00:26:35.800 the Hamiltonian is just theta-- 00:26:42.540 --> 00:26:43.040 OK. 00:26:43.040 --> 00:26:47.520 And so we have to understand this kinetic energy. 00:26:54.080 --> 00:26:57.960 For 1D problems, the kinetic energy 00:26:57.960 --> 00:27:04.618 is P squared over 2 mu, the linear momentum. 00:27:04.618 --> 00:27:08.260 I should put a hat on this. 00:27:08.260 --> 00:27:12.430 Well, we have motion in three dimensions, 00:27:12.430 --> 00:27:14.150 or two dimensions, theta and phi. 00:27:16.810 --> 00:27:21.860 And so we want to do something. 00:27:21.860 --> 00:27:24.880 We want to generate a kinetic energy Hamiltonian, which 00:27:24.880 --> 00:27:28.480 is analogous to this, some kind of momentum squared 00:27:28.480 --> 00:27:30.520 over some kind of a mass. 00:27:30.520 --> 00:27:32.050 And so we can go by analogy. 00:27:32.050 --> 00:27:34.310 Or we can go back to classical mechanics. 00:27:34.310 --> 00:27:40.190 We know that the orbital angular momentum is r cross P. 00:27:40.190 --> 00:27:42.340 So these are two vectors. 00:27:42.340 --> 00:27:44.320 Cross product is a vector. 00:27:44.320 --> 00:27:45.550 We know all that stuff. 00:27:45.550 --> 00:27:48.280 We know how to write the cross product 00:27:48.280 --> 00:27:51.017 in terms of a matrix involving unit vectors 00:27:51.017 --> 00:27:51.850 and stuff like that. 00:27:51.850 --> 00:27:55.390 You've done that before. 00:27:55.390 --> 00:27:56.580 OK. 00:27:56.580 --> 00:28:09.020 So now, we have an angular motion, omega. 00:28:09.020 --> 00:28:17.210 And we would like to know what the velocity of the mass points 00:28:17.210 --> 00:28:19.130 are on a sphere. 00:28:19.130 --> 00:28:26.750 Or if we think of this as just a particle of mass mu rotating, 00:28:26.750 --> 00:28:28.550 then we want to know its velocity. 00:28:28.550 --> 00:28:31.670 And to get from the angular velocity 00:28:31.670 --> 00:28:36.800 to the linear velocity, the linear velocity is r omega. 00:28:44.790 --> 00:28:49.810 So we can say, all right, knowing-- 00:28:49.810 --> 00:28:55.180 well, for a vector cross product, if the two things-- 00:28:55.180 --> 00:28:57.700 the r and the P-- 00:28:57.700 --> 00:29:07.010 are always orthogonal, well, then we just do r times P. 00:29:07.010 --> 00:29:11.970 And in fact, for motion on this sphere, here's r. 00:29:11.970 --> 00:29:15.750 And the motion is always orthogonal to r. 00:29:15.750 --> 00:29:24.900 And so we can write that the angular momentum is m1 r1 00:29:24.900 --> 00:29:33.560 squared omega plus m2 r2 squared times omega. 00:29:33.560 --> 00:29:36.980 Or it's just I omega. 00:29:43.250 --> 00:29:44.840 OK. 00:29:44.840 --> 00:29:52.970 And so we can write the kinetic energy term just 00:29:52.970 --> 00:29:58.840 goes as L squared over 2I where I 00:29:58.840 --> 00:30:05.530 is the sum of the individual masses M sub r I squared. 00:30:09.190 --> 00:30:10.490 OK. 00:30:10.490 --> 00:30:15.900 So we have an angular momentum operator, 00:30:15.900 --> 00:30:18.560 which is the guts of the kinetic energy. 00:30:22.340 --> 00:30:24.230 OK. 00:30:24.230 --> 00:30:28.060 Now, we're thinking Cartesian. 00:30:28.060 --> 00:30:30.680 And this is a spherical problem. 00:30:30.680 --> 00:30:33.070 And so we want to go from Cartesian 00:30:33.070 --> 00:30:37.338 coordinates to spherical polar coordinates. 00:30:37.338 --> 00:30:40.290 And that's a non-trivial problem. 00:30:40.290 --> 00:30:43.620 And it's also an extremely boring problem. 00:30:43.620 --> 00:30:45.300 And what you get is also something 00:30:45.300 --> 00:30:48.600 that's not terribly rewarding, except it's the differential 00:30:48.600 --> 00:30:50.170 equation you have solve. 00:30:50.170 --> 00:30:54.990 So T, when you go to spherical polar coordinates, 00:30:54.990 --> 00:31:06.150 is minus h squared over 2I 1 over sine theta 00:31:06.150 --> 00:31:10.380 partial with respect to theta times 00:31:10.380 --> 00:31:19.200 sine theta partial with respect to theta plus 1 over sine 00:31:19.200 --> 00:31:25.990 squared theta second partial with respect to phi. 00:31:25.990 --> 00:31:30.360 So that's the kinetic energy operator. 00:31:30.360 --> 00:31:35.760 So first of all, you say, what am I going to do? 00:31:35.760 --> 00:31:39.410 And the first thing you say is, separate the variables. 00:31:39.410 --> 00:31:41.080 So you do that. 00:31:41.080 --> 00:31:43.960 And you do the standard trick for separating variables. 00:31:43.960 --> 00:31:46.630 And you get two differential equations-- the theta 00:31:46.630 --> 00:31:48.250 equation and the phi equation. 00:31:51.230 --> 00:31:51.950 I'm sorry? 00:31:51.950 --> 00:31:53.140 AUDIENCE: H bar! 00:31:53.140 --> 00:31:56.380 ROBERT FIELD: When I wrote it, I said, why is that not H bar? 00:31:56.380 --> 00:31:58.790 OK. 00:31:58.790 --> 00:32:05.370 All right, so when you separate variables, 00:32:05.370 --> 00:32:06.820 we have this result-- 00:32:06.820 --> 00:32:10.450 1 over phi of phi. 00:32:10.450 --> 00:32:14.430 This is the phi part of the equation. 00:32:22.185 --> 00:32:23.740 Whoops, yeah, that's right. 00:32:30.470 --> 00:32:33.330 m squared is the separation constant. 00:32:33.330 --> 00:32:37.280 So we arrange to have a differential equation that 00:32:37.280 --> 00:32:43.130 depends only on phi and another that depends only on theta. 00:32:43.130 --> 00:32:45.360 And they're equal to each other. 00:32:45.360 --> 00:32:48.660 And so we call the thing that they're equal to a constant. 00:32:48.660 --> 00:32:50.090 And we call it m squared. 00:32:50.090 --> 00:32:53.370 We like that because m can be positive or negative. 00:32:53.370 --> 00:32:57.200 And the sign of m determines whether you 00:32:57.200 --> 00:33:01.820 have an oscillating function or an exponential function. 00:33:01.820 --> 00:33:05.490 You're all familiar with that. 00:33:05.490 --> 00:33:07.320 And so this is the phi equation. 00:33:09.880 --> 00:33:12.340 And this one, you can solve easily. 00:33:12.340 --> 00:33:14.530 You know the solution to this. 00:33:14.530 --> 00:33:17.500 You could, with a little bit of thought, 00:33:17.500 --> 00:33:23.450 write down the solution and have normalized functions. 00:33:23.450 --> 00:33:24.400 So I'll just do that. 00:33:29.580 --> 00:33:36.290 So the phi part of the solution is-- 00:33:42.210 --> 00:33:46.870 it's not the order in which I wrote my notes-- 00:33:46.870 --> 00:33:52.050 1 over the square root of 2 pi times e to the i m phi. 00:33:52.050 --> 00:33:56.560 And m is equal to 0 plus and minus 1 plus and minus 00:33:56.560 --> 00:33:58.450 2 et cetera. 00:33:58.450 --> 00:34:02.490 Quantization comes from imposition 00:34:02.490 --> 00:34:06.410 of what we call periodic boundary conditions. 00:34:06.410 --> 00:34:11.590 The wave function has to be the same for phi and phi 00:34:11.590 --> 00:34:14.600 plus 2 pi and phi plus 4 pi. 00:34:14.600 --> 00:34:17.590 And so that gives quantization. 00:34:17.590 --> 00:34:18.820 So this is the phi part. 00:34:18.820 --> 00:34:19.530 That's simple. 00:34:19.530 --> 00:34:20.139 That's simple. 00:34:20.139 --> 00:34:20.764 It's wonderful. 00:34:20.764 --> 00:34:23.949 We understand that perfectly. 00:34:23.949 --> 00:34:26.110 And one of the nice things about the vector model 00:34:26.110 --> 00:34:28.480 is mostly that's what you're focused on. 00:34:37.060 --> 00:34:41.550 So immediately, we can draw some pictures. 00:34:41.550 --> 00:34:47.610 And we can look at the nodes in the xy plane. 00:34:51.080 --> 00:35:00.760 So m equals 0, there are no nodes. 00:35:03.930 --> 00:35:12.910 m equals 1, first of all, let's draw where phi is 0. 00:35:12.910 --> 00:35:14.200 So this is phi. 00:35:14.200 --> 00:35:19.969 And so there can be a nodal plane like that. 00:35:19.969 --> 00:35:21.760 And there could be a nodal plane like that. 00:35:29.450 --> 00:35:34.320 And already, you know this looks like an S orbital. 00:35:34.320 --> 00:35:37.810 And this looks like a Px orbital and a Py orbital. 00:35:40.830 --> 00:35:45.660 And then we could also have-- 00:35:45.660 --> 00:35:56.150 yeah, m equals 2, we could have this, and this, and that-- 00:35:56.150 --> 00:35:59.550 no node, one node, two nodes. 00:35:59.550 --> 00:36:00.650 I'm sorry, yeah. 00:36:05.040 --> 00:36:06.630 What am I doing here? 00:36:06.630 --> 00:36:08.160 This is one node. 00:36:12.890 --> 00:36:16.790 I have to be a little bit more careful here. 00:36:16.790 --> 00:36:20.687 So what I wrote down is not true here. 00:36:20.687 --> 00:36:22.020 There are going to be two nodes. 00:36:22.020 --> 00:36:23.853 So they're probably like this and like that. 00:36:27.000 --> 00:36:32.550 But it's clear that what I have in my handwritten notes-- 00:36:32.550 --> 00:36:34.640 I'm not sure what the printed notes say. 00:36:34.640 --> 00:36:36.290 But there are going to be two nodes. 00:36:36.290 --> 00:36:40.850 And so immediately, you know something about-- 00:36:40.850 --> 00:36:42.560 if you draw a reduced picture and you 00:36:42.560 --> 00:36:47.990 show the nodal structure in the xy plane, 00:36:47.990 --> 00:36:54.500 you can tell what the projection of the angular momentum quantum 00:36:54.500 --> 00:36:58.049 number is, which is really important. 00:37:11.530 --> 00:37:13.900 There's the other differential equation. 00:37:13.900 --> 00:37:17.145 And that's the theta differential equation. 00:37:20.460 --> 00:37:25.880 And that is complicated. 00:37:25.880 --> 00:37:30.170 It is an exactly solved differential equation. 00:37:30.170 --> 00:37:32.160 But one of the things it gives you, 00:37:32.160 --> 00:37:35.150 which is easily memorized-- remembered, I'm sorry-- 00:37:35.150 --> 00:37:39.170 is that the energy levels have-- 00:37:46.260 --> 00:37:49.780 where this is the overlying momentum. 00:37:49.780 --> 00:37:51.370 So you can solve this. 00:37:51.370 --> 00:37:53.950 And you can get the eigenenergies. 00:37:53.950 --> 00:37:58.690 And the eigenenergies do not depend on m. 00:37:58.690 --> 00:38:01.519 They only depend on the L quantum number. 00:38:01.519 --> 00:38:02.727 And they have this nice form. 00:38:06.000 --> 00:38:06.540 OK. 00:38:06.540 --> 00:38:09.690 Now, in my notes and in all the textbooks, 00:38:09.690 --> 00:38:15.750 there are these horrendously beautiful detailed expressions 00:38:15.750 --> 00:38:20.790 of the solution, the mathematical form 00:38:20.790 --> 00:38:24.240 to the solution of the theta equation. 00:38:24.240 --> 00:38:26.910 And it's the Legendre equation. 00:38:26.910 --> 00:38:29.070 And there's Legendre polynomials. 00:38:29.070 --> 00:38:31.500 And remember, with the harmonic oscillator, I told you, 00:38:31.500 --> 00:38:34.180 you don't ever want to look at these things. 00:38:34.180 --> 00:38:35.560 The computer will look at them. 00:38:35.560 --> 00:38:37.185 And if you have integrals, the computer 00:38:37.185 --> 00:38:38.790 will know how to do those integrals. 00:38:38.790 --> 00:38:40.530 Because you told it how. 00:38:40.530 --> 00:38:42.630 And you don't have to keep that in your head. 00:38:42.630 --> 00:38:46.870 You've got better things to do with your limited attention. 00:38:46.870 --> 00:38:49.390 So there is a solution. 00:38:49.390 --> 00:38:51.810 And it has a form, which I have decided 00:38:51.810 --> 00:38:55.050 that I won't talk about. 00:38:55.050 --> 00:38:59.280 Because you get much more insight into real problems 00:38:59.280 --> 00:39:01.530 from the vector model. 00:39:01.530 --> 00:39:07.360 Now, the vector model is really only 00:39:07.360 --> 00:39:12.150 marginally useful for a diatomic molecule 00:39:12.150 --> 00:39:17.250 in an electronic state, which you don't know about yet, which 00:39:17.250 --> 00:39:22.100 is called sigma sigma plus, where there is no angular 00:39:22.100 --> 00:39:25.120 momentum associated with the electron 00:39:25.120 --> 00:39:27.160 and when there is no angular momentum associated 00:39:27.160 --> 00:39:30.760 with the nuclei. 00:39:30.760 --> 00:39:34.680 So this is just practice for real life 00:39:34.680 --> 00:39:39.780 when you have to understand other things about what is 00:39:39.780 --> 00:39:42.310 living in the molecular frame. 00:39:42.310 --> 00:39:48.450 But this is hard enough to understand completely that it's 00:39:48.450 --> 00:39:50.340 a worthwhile investment. 00:39:50.340 --> 00:39:52.680 And you will be doing most of the understanding 00:39:52.680 --> 00:39:55.710 of more complicated problems on your own 00:39:55.710 --> 00:39:58.500 if you ever do anything with diatomic molecules 00:39:58.500 --> 00:40:00.520 or polyatomic molecules. 00:40:00.520 --> 00:40:03.040 And so, OK. 00:40:08.300 --> 00:40:09.910 So what is the vector model? 00:40:16.970 --> 00:40:20.540 Well, the things we want to know about the vector model 00:40:20.540 --> 00:40:23.870 is, how long is J? 00:40:23.870 --> 00:40:28.640 So if we have a state with J and m quantum numbers, 00:40:28.640 --> 00:40:32.320 how long is the vector associated with J? 00:40:32.320 --> 00:40:36.190 And what is the angle-- 00:40:36.190 --> 00:40:42.910 what is the projection of J on the laboratory z-axis? 00:40:42.910 --> 00:40:45.670 So we want to know the length of J. 00:40:45.670 --> 00:40:55.200 And we want to know the projection of J on z. 00:40:55.200 --> 00:40:56.280 OK. 00:40:56.280 --> 00:41:03.260 And we want to know the angle between the z-axis and J. 00:41:03.260 --> 00:41:04.700 That's the beginning of a picture. 00:41:07.830 --> 00:41:12.700 And again, we stick with this idea that J-- 00:41:12.700 --> 00:41:14.750 so here's the z-axis. 00:41:14.750 --> 00:41:19.850 And J precesses about z. 00:41:23.970 --> 00:41:27.860 And here is an angle we want to know. 00:41:27.860 --> 00:41:31.950 This theta is a different theta from the theta in-- 00:41:31.950 --> 00:41:35.860 in fact, I should call it alpha. 00:41:35.860 --> 00:41:36.360 OK. 00:41:40.000 --> 00:41:42.370 So some things we know-- 00:41:42.370 --> 00:41:49.570 we know that the eigenvalues of the rigid rotor equation 00:41:49.570 --> 00:41:51.430 are this. 00:41:51.430 --> 00:42:00.010 And we know that Jz operator operating on y Lm 00:42:00.010 --> 00:42:07.584 gives H bar m y Lm. 00:42:11.900 --> 00:42:21.590 And J squared operating on y Lm gives H bar squared J J plus 1. 00:42:25.070 --> 00:42:27.180 So what do we do with these two things? 00:42:27.180 --> 00:42:31.640 Well, one is to say, all right, the length of J 00:42:31.640 --> 00:42:34.640 is going to be the square root-- 00:42:34.640 --> 00:42:37.700 times y Lm. 00:42:37.700 --> 00:42:43.790 So the length of J is going to be the square root 00:42:43.790 --> 00:42:47.390 of H bar squared J J plus 1. 00:42:51.980 --> 00:42:58.109 So that becomes H bar and approximately J plus 1/2. 00:42:58.109 --> 00:42:59.650 Because the square root of J J plus 1 00:42:59.650 --> 00:43:01.390 is almost exactly J plus 1/2. 00:43:06.810 --> 00:43:09.180 When you get really high J, it's exactly. 00:43:09.180 --> 00:43:12.420 At low J, It's a little bit less than exact. 00:43:12.420 --> 00:43:15.470 So we know the length of J is this. 00:43:15.470 --> 00:43:19.670 And we know the length of m is H bar m. 00:43:19.670 --> 00:43:21.790 The length of J z is H bar m. 00:43:34.930 --> 00:43:43.280 So now, we know that this is H bar m. 00:43:43.280 --> 00:43:48.120 And this is H bar times J plus 1/2. 00:43:51.750 --> 00:43:57.300 And so this is alpha. 00:43:57.300 --> 00:44:03.210 So we know how to calculate the cosine of alpha. 00:44:03.210 --> 00:44:07.230 So the cosine of alpha is equal to H 00:44:07.230 --> 00:44:20.150 bar m over H bar J plus 1/2, or m over J-- 00:44:20.150 --> 00:44:24.170 oh, right, plus 1/2, really small. 00:44:24.170 --> 00:44:27.320 So the angle, the cosine of the angle 00:44:27.320 --> 00:44:35.180 that j makes with the z-axis is m over J. 00:44:35.180 --> 00:44:37.820 And all of a sudden, now, we have lengths 00:44:37.820 --> 00:44:40.250 of things and angles of things. 00:44:40.250 --> 00:44:41.810 And we can do all sorts of stuff. 00:44:41.810 --> 00:44:46.730 Remember, what we care about is, if this is J, 00:44:46.730 --> 00:44:50.210 the body axis is like that. 00:44:50.210 --> 00:44:52.190 So we now know where the body axis 00:44:52.190 --> 00:44:55.460 is relative to the laboratory. 00:44:55.460 --> 00:45:00.320 So we have the body axis is perpendicular to J. 00:45:00.320 --> 00:45:03.670 And so we can then calculate where everything is. 00:45:12.560 --> 00:45:15.870 All right, so special cases-- 00:45:15.870 --> 00:45:25.250 if m is equal to plus or minus J, then 00:45:25.250 --> 00:45:27.590 what does that mean for this picture? 00:45:27.590 --> 00:45:30.210 Well, we have the z-axis. 00:45:30.210 --> 00:45:37.150 And we have J. And J is almost exactly along the z-axis. 00:45:37.150 --> 00:45:39.795 It would be exactly along if there wasn't this plus 1/2. 00:45:47.100 --> 00:45:51.690 And so what that means is the bond 00:45:51.690 --> 00:45:56.770 axis is almost exactly rotating in the xy plane. 00:45:59.930 --> 00:46:03.590 So if we choose m equals plus or minus J, 00:46:03.590 --> 00:46:07.420 the molecule is rotating in the xy plane. 00:46:07.420 --> 00:46:10.160 That's kind of nice to know. 00:46:10.160 --> 00:46:15.010 The other easy case is m equals 0. 00:46:15.010 --> 00:46:21.150 If m equals 0, J is perpendicular to the z-axis. 00:46:21.150 --> 00:46:28.720 And so the body axis is in the yz plane. 00:46:28.720 --> 00:46:31.700 Or if we're coming around here, it's in the xz plane. 00:46:31.700 --> 00:46:36.330 So we have xz yz. 00:46:36.330 --> 00:46:39.010 And together, what it says is the body 00:46:39.010 --> 00:46:44.690 axis is more along z than along anything else. 00:46:44.690 --> 00:46:46.760 That's a lot of insight. 00:46:46.760 --> 00:46:51.310 And so we can go to our friendly extreme cases, special cases. 00:46:51.310 --> 00:46:54.806 And we could say, where's the body axis? 00:46:54.806 --> 00:46:56.180 In the laboratory, which is where 00:46:56.180 --> 00:46:57.470 we're making the observations. 00:47:00.410 --> 00:47:03.580 That should make you happy. 00:47:03.580 --> 00:47:06.680 Because there's no equations here. 00:47:06.680 --> 00:47:10.940 There's just a lot of pictures that you can understand 00:47:10.940 --> 00:47:12.815 and develop your intuition. 00:47:18.040 --> 00:47:21.331 Now, I want a blackboard. 00:47:21.331 --> 00:47:21.830 OK. 00:47:21.830 --> 00:47:25.370 Suppose you have a laboratory in which you 00:47:25.370 --> 00:47:26.570 can make a molecular beam. 00:47:30.932 --> 00:47:32.140 And that's easy enough to do. 00:47:32.140 --> 00:47:34.360 You squirt molecules out of a pinhole. 00:47:34.360 --> 00:47:38.350 And you have some kind of an aperture. 00:47:38.350 --> 00:47:44.770 And so you have a directed flow of molecules in one direction. 00:47:44.770 --> 00:47:46.570 And you could also say, all right, 00:47:46.570 --> 00:47:48.760 I'm going to do something with my laser. 00:47:48.760 --> 00:47:50.950 And I'm going to make the molecules, which 00:47:50.950 --> 00:47:58.390 are like a helicopter, in other words, 00:47:58.390 --> 00:48:04.420 the molecular axis is rotating in a circle 00:48:04.420 --> 00:48:08.620 about the propagation axis of the beam, so a helicopter. 00:48:12.390 --> 00:48:14.620 And we can also do this. 00:48:14.620 --> 00:48:21.210 And so these are two ways that you can prepare molecules 00:48:21.210 --> 00:48:21.780 in this beam. 00:48:24.630 --> 00:48:28.710 And you could say, oh, I've got a whole bunch of molecules 00:48:28.710 --> 00:48:30.120 that are not in the beam. 00:48:30.120 --> 00:48:32.910 And these molecules are going to collide 00:48:32.910 --> 00:48:39.015 with the molecules outside the beam and get scattered. 00:48:39.015 --> 00:48:43.650 Well, which one is going to be scattered more? 00:48:43.650 --> 00:48:47.480 The one that sweeps out a big volume or the one that 00:48:47.480 --> 00:48:54.640 sweeps out something only essentially one dimension? 00:48:54.640 --> 00:48:55.265 That's insight. 00:48:57.824 --> 00:48:59.240 There's lots of other things, too. 00:48:59.240 --> 00:49:01.280 Now, let's do something which is actually 00:49:01.280 --> 00:49:02.420 slightly exam relevant. 00:49:06.480 --> 00:49:08.670 Suppose we have a diatomic molecule 00:49:08.670 --> 00:49:14.450 which is positive on one end and negative on the other, OK? 00:49:14.450 --> 00:49:17.500 And we're going to apply an electric field. 00:49:17.500 --> 00:49:20.640 And if the electric field is in this direction 00:49:20.640 --> 00:49:23.890 and the molecule is doing this, the molecule doesn't care. 00:49:27.380 --> 00:49:30.130 But if the electric field is in that direction 00:49:30.130 --> 00:49:34.450 and the molecule is doing this, the energy levels-- 00:49:34.450 --> 00:49:38.142 the energy goes up and down as the molecule rotates. 00:49:38.142 --> 00:49:39.850 Well, how is that? 00:49:39.850 --> 00:49:42.580 The molecule was free to precess. 00:49:42.580 --> 00:49:45.860 And we knew the amplitude along this axis. 00:49:45.860 --> 00:49:50.230 But if there's some angle dependence, what we're doing 00:49:50.230 --> 00:49:55.870 is we're mixing in a different J. 00:49:55.870 --> 00:50:00.750 And so that's telling you that, if you apply an electric field, 00:50:00.750 --> 00:50:03.500 there will be a Stark effect. 00:50:03.500 --> 00:50:07.490 And it will affect the m equals 0 levels differently 00:50:07.490 --> 00:50:10.440 from the m equals J levels. 00:50:10.440 --> 00:50:15.480 Now, I'm not going to tell you which one is more affected. 00:50:15.480 --> 00:50:18.330 But it's clear that one is hardly affected. 00:50:18.330 --> 00:50:21.780 And the other is profoundly affected. 00:50:21.780 --> 00:50:24.870 And that gives rise to a splitting of the energy 00:50:24.870 --> 00:50:27.790 levels in an electric field, which 00:50:27.790 --> 00:50:31.540 depends on J at m in a way that you would recognize. 00:50:35.410 --> 00:50:36.390 Time to quit. 00:50:36.390 --> 00:50:39.300 I did what I was hoping I could do. 00:50:39.300 --> 00:50:42.720 Now, there is some stuff at the end of the notes 00:50:42.720 --> 00:50:45.620 which is standard textbook material about polar plots 00:50:45.620 --> 00:50:47.670 of the wave functions. 00:50:47.670 --> 00:50:49.620 These polar plots are useful. 00:50:49.620 --> 00:50:52.860 But most of the time, when you first encounter them, 00:50:52.860 --> 00:50:56.910 you have no idea what they mean, except that they 00:50:56.910 --> 00:51:00.109 express where the nodes are. 00:51:00.109 --> 00:51:01.650 Now, you want to go back and you want 00:51:01.650 --> 00:51:05.280 to actually think about what these polar plots mean 00:51:05.280 --> 00:51:10.500 and whether you can use them as an auxiliary 00:51:10.500 --> 00:51:12.720 to this vector picture. 00:51:12.720 --> 00:51:13.220 OK. 00:51:13.220 --> 00:51:17.615 So I will then lecture, and it's exam relevant, 00:51:17.615 --> 00:51:21.890 on commutation rules on Monday. 00:51:21.890 --> 00:51:27.280 And then have a good weekend.