WEBVTT 00:00:00.090 --> 00:00:01.631 FEMALE SPEAKER: The following content 00:00:01.631 --> 00:00:03.820 is provided under a Creative Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.660 To make a donation or to view additional materials 00:00:12.660 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.850 at ocw.mit.edu. 00:00:23.180 --> 00:00:25.700 PROFESSOR: So we're now starting to get 00:00:25.700 --> 00:00:28.310 into a really familiar territory, which 00:00:28.310 --> 00:00:29.451 is the hydrogen atom. 00:00:32.159 --> 00:00:40.770 And it's a short step from a hydrogen atom to molecules. 00:00:40.770 --> 00:00:41.790 And we're chemists. 00:00:41.790 --> 00:00:45.280 We make molecules. 00:00:45.280 --> 00:00:49.530 But one question that would be a legitimate question to ask 00:00:49.530 --> 00:00:52.810 is, what does a hydrogen atom have to do with molecules, 00:00:52.810 --> 00:00:55.070 because it's an atom and it's the simplest atom. 00:00:55.070 --> 00:01:00.390 And what I hope to show is that what 00:01:00.390 --> 00:01:03.090 we learn from looking at the hydrogen atom 00:01:03.090 --> 00:01:06.870 has all sorts of good non textbook stuff 00:01:06.870 --> 00:01:09.660 that prepares you for understanding stuff 00:01:09.660 --> 00:01:11.580 in molecules. 00:01:11.580 --> 00:01:15.960 And I am going to stress that because you're not 00:01:15.960 --> 00:01:19.350 going to see it in McQuarrie and you've probably never seen 00:01:19.350 --> 00:01:24.120 it elsewhere except in freshman chemistry when they expect 00:01:24.120 --> 00:01:26.790 that you understand the periodic table based 00:01:26.790 --> 00:01:28.830 on some simple concepts. 00:01:28.830 --> 00:01:31.480 And they come from the hydrogen atom. 00:01:31.480 --> 00:01:39.600 And so I'm going to attempt to make those connections. 00:01:39.600 --> 00:01:42.990 My last lecture was I told you something you're not 00:01:42.990 --> 00:01:44.110 going to be tested on. 00:01:44.110 --> 00:01:47.520 But it's about spectroscopy. 00:01:47.520 --> 00:01:49.770 That's how we learned almost everything 00:01:49.770 --> 00:01:53.730 we know about small molecules and a lot of stuff 00:01:53.730 --> 00:01:55.860 about big molecules too. 00:01:55.860 --> 00:01:59.540 The crucial approximations in that 00:01:59.540 --> 00:02:05.820 were the dipole approximation where the radiation field 00:02:05.820 --> 00:02:08.250 is such that the molecule field's 00:02:08.250 --> 00:02:11.580 a uniform but oscillating electromagnetic field. 00:02:14.620 --> 00:02:18.110 In developing the theory that I presented, 00:02:18.110 --> 00:02:24.140 we assume that only one of the initial eigenstates 00:02:24.140 --> 00:02:26.780 is populated at t equal zero. 00:02:26.780 --> 00:02:32.210 And only one final eigenstate gets tickled so 00:02:32.210 --> 00:02:34.100 that it gets populated. 00:02:34.100 --> 00:02:38.060 And so even though there is an infinity of states, 00:02:38.060 --> 00:02:43.280 the theory specifies we start with one definite one and only 00:02:43.280 --> 00:02:49.960 one final state gets selected because of resonance. 00:02:49.960 --> 00:02:52.030 And this is what we do all the time. 00:02:52.030 --> 00:02:55.120 We have an infinite dimension problem, 00:02:55.120 --> 00:02:58.780 and we discover that we only really need 00:02:58.780 --> 00:03:02.490 to worry about a very small number of states. 00:03:02.490 --> 00:03:06.420 And then we deal with that. 00:03:06.420 --> 00:03:07.920 There are a couple other assumptions 00:03:07.920 --> 00:03:11.280 that were essential for this first step 00:03:11.280 --> 00:03:14.360 into a time-dependent Hamiltonian. 00:03:14.360 --> 00:03:18.080 And that is the field is weak and so you get linear response. 00:03:18.080 --> 00:03:22.550 That means that the increase in the mixing coefficient 00:03:22.550 --> 00:03:26.600 for the final state is proportional to the coupling 00:03:26.600 --> 00:03:30.450 matrix element times time. 00:03:30.450 --> 00:03:34.930 Now mixing coefficients can't get larger than 1, 00:03:34.930 --> 00:03:40.900 and so clearly we can't use a simple theory without saying, 00:03:40.900 --> 00:03:44.200 OK, we don't care about the mixing coefficients. 00:03:44.200 --> 00:03:47.170 We care about the rate of increase. 00:03:47.170 --> 00:03:53.260 And so by going from amplitude to rates, 00:03:53.260 --> 00:03:57.940 we are able to have a theory that's generally applicable. 00:03:57.940 --> 00:04:02.830 Now what I did was to talk about a CW radiation field, 00:04:02.830 --> 00:04:05.400 continuous. 00:04:05.400 --> 00:04:10.350 Many experiments use pulsed fields. 00:04:10.350 --> 00:04:15.040 Many experiments use an extremely strong pulsed field. 00:04:15.040 --> 00:04:18.320 That I hope to revisit later in the course. 00:04:18.320 --> 00:04:23.770 But the CW probably is the best way to begin. 00:04:23.770 --> 00:04:25.540 So now let's talk about the hydrogen atom. 00:04:30.180 --> 00:04:41.780 We did the rigid rotor, and that led to some angular momenta. 00:04:41.780 --> 00:04:44.160 Well, let's just picture. 00:04:44.160 --> 00:04:48.940 So we have this rigid rotor, and it's rotating 00:04:48.940 --> 00:04:50.980 about the center of mass. 00:04:50.980 --> 00:04:53.980 And the Schrodinger equation tells us 00:04:53.980 --> 00:04:58.030 the probability amplitudes of the rotor 00:04:58.030 --> 00:05:03.790 axis relative to the laboratory frame. 00:05:03.790 --> 00:05:05.980 And we got angular momenta, which 00:05:05.980 --> 00:05:12.680 we can denote by a little l, big L, J, and many other things. 00:05:12.680 --> 00:05:16.990 And this is important because, if it's an angular momentum, 00:05:16.990 --> 00:05:18.340 you know it. 00:05:18.340 --> 00:05:21.680 You know everything about it. 00:05:21.680 --> 00:05:25.760 And you don't know yet that, if you have two angular momenta, 00:05:25.760 --> 00:05:29.194 you could know everything about the interactions between them. 00:05:29.194 --> 00:05:30.860 That's called the Wigner-Eckart theorem. 00:05:30.860 --> 00:05:32.820 That's not in this course. 00:05:32.820 --> 00:05:35.450 But the important thing is, if you've got angular momenta, 00:05:35.450 --> 00:05:41.280 you're on solid ground and you can do stuff that doesn't ever 00:05:41.280 --> 00:05:44.140 need to be repeated. 00:05:44.140 --> 00:05:47.650 And so one of the things that we learned about the angular 00:05:47.650 --> 00:05:53.350 momentum was that we have-- 00:05:53.350 --> 00:05:58.180 even though it's going to be L as the angular momentum mostly 00:05:58.180 --> 00:06:02.050 in the hydrogen atom, I'm going to switch to J, 00:06:02.050 --> 00:06:03.850 which is my favorite notation. 00:06:03.850 --> 00:06:08.740 So J squared operating on a state-- 00:06:08.740 --> 00:06:13.600 now I can denote it like this or I can denote it like this. 00:06:13.600 --> 00:06:16.180 It's the same general idea. 00:06:16.180 --> 00:06:18.080 These are not quite equivalent things. 00:06:22.620 --> 00:06:30.450 And you get H bar squared J A plus one JM. 00:06:30.450 --> 00:06:36.440 And we have JZ operating on JM. 00:06:36.440 --> 00:06:41.100 And you get H bar M JM. 00:06:41.100 --> 00:06:43.930 And we can have J plus minus operating on JM. 00:06:46.600 --> 00:06:49.620 And that's H bar times this more complicated 00:06:49.620 --> 00:06:57.327 looking thing, J plus 1 minus M, M plus or minus 1. 00:06:57.327 --> 00:06:59.160 So that's the only bit you have to remember. 00:07:02.440 --> 00:07:04.900 And you might say, well, is it plus or minus 1 00:07:04.900 --> 00:07:07.480 or minus or plus 1? 00:07:07.480 --> 00:07:09.360 And so if you choose-- 00:07:09.360 --> 00:07:17.160 M is equal to J. Then you know that if you 00:07:17.160 --> 00:07:20.010 have M is equal to J, this is going 00:07:20.010 --> 00:07:25.030 to be 0 if we're doing a raising operator. 00:07:25.030 --> 00:07:29.260 And so there's nothing to remember 00:07:29.260 --> 00:07:32.530 if you are willing to go to an extreme situation 00:07:32.530 --> 00:07:35.560 and decide whether it's plus minus or minus or plus. 00:07:43.990 --> 00:07:49.160 Now the rigid rotor is a universal problem, 00:07:49.160 --> 00:07:50.020 which is solved. 00:07:50.020 --> 00:07:53.510 It deals with all central force problems. 00:07:53.510 --> 00:07:56.800 Everything that's round, the rigid rotor 00:07:56.800 --> 00:07:59.360 is a fantastic starting point. 00:07:59.360 --> 00:08:03.250 So one-electron atoms for sure-- 00:08:03.250 --> 00:08:05.620 many-electron atoms, maybe. 00:08:05.620 --> 00:08:07.860 We'll see. 00:08:07.860 --> 00:08:11.220 But anything that's round, the rigid rotor 00:08:11.220 --> 00:08:14.910 is a good zero point for dealing with the angular 00:08:14.910 --> 00:08:17.840 part of the problem. 00:08:17.840 --> 00:08:21.580 So for the hydrogen atom, we have a potential. 00:08:24.390 --> 00:08:25.990 It looks like this. 00:08:28.850 --> 00:08:32.679 And so the radial potential is new. 00:08:32.679 --> 00:08:36.440 For that rigid rotor, the radial potential was-- 00:08:36.440 --> 00:08:42.970 it's just 0 for the R is equal to R zero. 00:08:42.970 --> 00:08:47.270 And not zero-- then the potential is infinite. 00:08:47.270 --> 00:08:49.840 Now here we have potential, which 00:08:49.840 --> 00:08:53.200 doesn't go to infinity here and it does something strange here 00:08:53.200 --> 00:08:56.700 because you can't get to negative R 00:08:56.700 --> 00:08:59.270 in spherical polar coordinates. 00:08:59.270 --> 00:09:02.200 So is it a boundary condition or is it 00:09:02.200 --> 00:09:09.430 just an accident of the way we use coordinates? 00:09:09.430 --> 00:09:13.510 So we're going to apply what we know about angular momenta 00:09:13.510 --> 00:09:15.280 to the hydrogen atom. 00:09:15.280 --> 00:09:18.310 So our potential for the hydrogen atom 00:09:18.310 --> 00:09:23.860 is going to be expressed in terms 00:09:23.860 --> 00:09:27.230 of the distance of the electron from the nucleus 00:09:27.230 --> 00:09:30.220 and then the theta phi coordinates, which you already 00:09:30.220 --> 00:09:30.970 know. 00:09:30.970 --> 00:09:33.480 And the theta phi part is universal, 00:09:33.480 --> 00:09:37.900 and the R part is special to each problem that 00:09:37.900 --> 00:09:39.970 has physical symmetry. 00:09:39.970 --> 00:09:43.960 And there are different kinds of approximations you use 00:09:43.960 --> 00:09:45.790 to be able to deal with them. 00:09:45.790 --> 00:09:49.540 The nice thing about it is it's one dimensional, 00:09:49.540 --> 00:09:53.830 and we're very good at thinking about one dimensional problems. 00:09:53.830 --> 00:09:56.950 And even if the problem isn't exactly one dimensional 00:09:56.950 --> 00:09:59.470 or it has some hidden stuff to it, 00:09:59.470 --> 00:10:04.960 we can extend what we know from one dimensional problems 00:10:04.960 --> 00:10:06.700 and get a great deal of insight. 00:10:06.700 --> 00:10:09.710 And if you have a one dimensional problem, 00:10:09.710 --> 00:10:14.380 it's very easy to describe the potential in one dimension 00:10:14.380 --> 00:10:18.430 and the eigen functions in one dimension. 00:10:18.430 --> 00:10:22.698 And so we can begin to really understand everything. 00:10:26.180 --> 00:10:31.410 So we're going to have a wave function, which is 00:10:31.410 --> 00:10:32.660 going to have quantum numbers. 00:10:38.690 --> 00:10:40.770 And we're going to be able to write it 00:10:40.770 --> 00:10:43.185 as a product of two parts. 00:10:51.270 --> 00:10:55.950 Well, this is the same thing we had before for the rigid rotor. 00:10:55.950 --> 00:10:59.580 So we're going to be able to take this rotary equation 00:10:59.580 --> 00:11:04.610 and separate the wave function into two parts, a radial part 00:11:04.610 --> 00:11:06.030 and an angular part. 00:11:06.030 --> 00:11:07.620 And this is old and this is new. 00:11:13.800 --> 00:11:19.350 Because we've got this, we can use all of that stuff 00:11:19.350 --> 00:11:21.150 without taking a breath, except maybe 00:11:21.150 --> 00:11:25.575 getting the right letter L, J, S, whatever. 00:11:29.330 --> 00:11:33.910 So really the hydrogen atom is just one thing 00:11:33.910 --> 00:11:37.585 with some curve balls thrown at you in the latter stages. 00:11:45.350 --> 00:11:48.620 One of the things you also want to be able to do-- 00:11:48.620 --> 00:11:52.310 because nodes are so important in determining 00:11:52.310 --> 00:11:57.290 both the names of the states and how they behave 00:11:57.290 --> 00:12:05.240 in various situations, including external fields and excitation 00:12:05.240 --> 00:12:07.830 by electromagnetic radiation-- 00:12:07.830 --> 00:12:10.730 you want to be able to understand the nodal surfaces. 00:12:10.730 --> 00:12:14.230 And you already know this one. 00:12:14.230 --> 00:12:16.770 So how many nodal surfaces are there 00:12:16.770 --> 00:12:18.630 if you have a particular value of L? 00:12:21.440 --> 00:12:21.950 Yes. 00:12:21.950 --> 00:12:22.640 AUDIENCE: L. 00:12:22.640 --> 00:12:27.505 PROFESSOR: Right, and if you have a particular value of M, 00:12:27.505 --> 00:12:29.770 how many nodes are there in the xy plane? 00:12:37.790 --> 00:12:39.223 You're hot-- 00:12:39.223 --> 00:12:42.980 AUDIENCE: I mean, I know if M is 0, 00:12:42.980 --> 00:12:47.150 the entire angular momentum is tipped into the xy plane. 00:12:47.150 --> 00:12:52.760 So that means that the axis of the rotor 00:12:52.760 --> 00:12:56.655 has to be orthogonal to the angular momentum. 00:12:56.655 --> 00:12:58.670 So that means that the probability density is 00:12:58.670 --> 00:13:02.252 oriented along Z, I think. 00:13:02.252 --> 00:13:03.710 PROFESSOR: Now you're saying things 00:13:03.710 --> 00:13:05.168 that I have to stop and think about 00:13:05.168 --> 00:13:07.760 because you're not telling me what I expected to know. 00:13:07.760 --> 00:13:13.670 So if M is equal to 0, L is perpendicular 00:13:13.670 --> 00:13:19.685 to the quantization axis, and there are no nodes. 00:13:19.685 --> 00:13:21.310 AUDIENCE: Well, it depends on if you're 00:13:21.310 --> 00:13:25.000 talking about the nodes of the probability 00:13:25.000 --> 00:13:28.750 density of the rotor, where the axis is. 00:13:28.750 --> 00:13:32.410 PROFESSOR: That's true, but where the axis 00:13:32.410 --> 00:13:36.670 is determined by theta. 00:13:36.670 --> 00:13:42.100 And when theta is equal to pi over 2, you're in the xy plane. 00:13:42.100 --> 00:13:48.430 And the number of nodes in the xy plane 00:13:48.430 --> 00:13:58.370 is M, or absolute value of M. The phi part of the rigid rotor 00:13:58.370 --> 00:13:59.420 is simple. 00:13:59.420 --> 00:14:03.300 It's a differential equation that everybody can solve. 00:14:03.300 --> 00:14:05.570 We already know that one, and we know what 00:14:05.570 --> 00:14:06.810 the wave functions look like. 00:14:15.240 --> 00:14:26.550 We can write the hydrogen atom Schrodinger equation, 00:14:26.550 --> 00:14:32.500 and it's very quick to show separation of variables. 00:14:32.500 --> 00:14:34.480 And I'll do that in a minute. 00:14:34.480 --> 00:14:43.230 And then we have the pictures of the separated parts-- 00:14:43.230 --> 00:14:48.140 RNL of R and YLM of theta phi. 00:14:50.900 --> 00:14:55.670 And essential in these pictures is the number of nodes. 00:14:55.670 --> 00:14:57.920 In the spacing between nodes, remember 00:14:57.920 --> 00:15:01.210 the semiclassical approximation. 00:15:01.210 --> 00:15:04.620 We know that Mr. DeBroglie really hit it out of the park 00:15:04.620 --> 00:15:12.070 by saying that the wavelength is H over P. 00:15:12.070 --> 00:15:13.600 For every one dimensional problem, 00:15:13.600 --> 00:15:16.250 we know what to do with that. 00:15:16.250 --> 00:15:19.970 And we're going to discover that, for the radial problem, 00:15:19.970 --> 00:15:22.640 we have a very simple way of determining 00:15:22.640 --> 00:15:25.770 what the classical momentum is. 00:15:25.770 --> 00:15:29.790 And so we know everything about the nodes and the node spacing 00:15:29.790 --> 00:15:32.250 and the amplitudes between nodes and how 00:15:32.250 --> 00:15:39.220 to evaluate every integral of some power of R or Z. 00:15:39.220 --> 00:15:41.580 And so there's just an enormous amount 00:15:41.580 --> 00:15:45.270 gotten from the semiclassical picture 00:15:45.270 --> 00:15:48.450 once you are familiar with this. 00:15:48.450 --> 00:15:55.560 And so we can say that we have the classical wavelength, 00:15:55.560 --> 00:15:57.480 or the semiclassical wavelength. 00:15:57.480 --> 00:16:12.460 It has an index R. So we have a momentum, a linear momentum 00:16:12.460 --> 00:16:14.596 with a quantum number on it for L. See, 00:16:14.596 --> 00:16:15.595 that's a little strange. 00:16:18.500 --> 00:16:20.960 But that tells us what the potential is going to be 00:16:20.960 --> 00:16:22.860 and that tells us what to use in order 00:16:22.860 --> 00:16:25.500 to determine the wavelength. 00:16:25.500 --> 00:16:27.890 And this is really the core of how 00:16:27.890 --> 00:16:30.230 we can go way beyond textbooks. 00:16:30.230 --> 00:16:34.105 We can do-- in our heads or on a simple piece of paper, 00:16:34.105 --> 00:16:34.730 we can do this. 00:16:34.730 --> 00:16:41.645 And we can draw pictures, and we can evaluate matrix elements. 00:16:45.040 --> 00:16:49.210 Without any complicated integral tables, 00:16:49.210 --> 00:16:54.310 you can make estimates that are incredibly important. 00:16:54.310 --> 00:17:03.960 And from that you can get expectation values and also 00:17:03.960 --> 00:17:05.910 off-diagonal matrix elements of integer 00:17:05.910 --> 00:17:07.871 powers of the coordinate. 00:17:10.460 --> 00:17:13.079 That's an enormous amount of stuff that you can do. 00:17:13.079 --> 00:17:15.700 Now you can't do it yet. 00:17:15.700 --> 00:17:19.240 But after Wednesday's lecture, you will. 00:17:19.240 --> 00:17:23.869 And so this will be Wednesday. 00:17:23.869 --> 00:17:31.655 And then we'll have evidence of electron spin. 00:17:34.940 --> 00:17:39.130 And I will get to that today. 00:17:39.130 --> 00:17:40.430 So this is the menu. 00:17:40.430 --> 00:17:43.130 Now let's start delivering some of this stuff. 00:17:48.360 --> 00:17:51.510 I have to write some big equations. 00:17:51.510 --> 00:17:57.920 So the Hamiltonian is kinetic energy plus potential energy, 00:17:57.920 --> 00:18:00.510 and kinetic energy was-- 00:18:00.510 --> 00:18:01.790 I'm sorry. 00:18:01.790 --> 00:18:06.710 For the rigid rotor, V was zero. 00:18:06.710 --> 00:18:08.650 And everything was in the kinetic energy. 00:18:08.650 --> 00:18:10.660 Well, it's not quite true. 00:18:10.660 --> 00:18:15.420 And so, for hydrogen-- so we know 00:18:15.420 --> 00:18:21.720 this is P squared over 2 mu, and mu for the hydrogen atom, 00:18:21.720 --> 00:18:23.130 reduced mass. 00:18:23.130 --> 00:18:32.750 And we know this is just Coulomb's Law, minus e 00:18:32.750 --> 00:18:42.240 squared over 4 pi epsilon zero R, H bar squared. 00:18:45.750 --> 00:18:51.570 I'm sorry-- it was supposed to be e squared. 00:18:51.570 --> 00:18:52.800 So this is the classical-- 00:18:55.610 --> 00:18:57.820 So these are the parts. 00:18:57.820 --> 00:19:00.470 But the since it's spherical, we're 00:19:00.470 --> 00:19:04.840 not going to work in Cartesian coordinates. 00:19:04.840 --> 00:19:09.560 And in fact, that's a very strong statement. 00:19:09.560 --> 00:19:12.270 If you have a spherical problem, don't 00:19:12.270 --> 00:19:14.190 start in Cartesian coordinates because it's 00:19:14.190 --> 00:19:18.720 a horrible mess transforming to spherical polar coordinates. 00:19:18.720 --> 00:19:22.820 Just remember the spherical polar coordinates. 00:19:22.820 --> 00:19:24.990 So you know how spherical polar coordinates work. 00:19:24.990 --> 00:19:26.031 I'm not going to draw it. 00:19:28.260 --> 00:19:36.850 So the kinetic energy term is-- 00:19:44.690 --> 00:19:51.590 and this is the Laplacian, and that's a terrible thing. 00:19:51.590 --> 00:19:58.070 And Del squared-- it looks like it's 00:19:58.070 --> 00:20:02.428 going to be a real nightmare partial with respect to R, 00:20:02.428 --> 00:20:08.550 R squared partial with respect to R. And we have 1 over R 00:20:08.550 --> 00:20:13.460 squared sine squared theta partial with respect 00:20:13.460 --> 00:20:21.130 to theta, sine theta partial with respect to theta. 00:20:21.130 --> 00:20:23.560 And then a third term-- 00:20:23.560 --> 00:20:28.640 1 over R squared sine squared. 00:20:28.640 --> 00:20:30.100 I got a square here. 00:20:30.100 --> 00:20:32.170 That's wrong. 00:20:32.170 --> 00:20:34.570 And this is sine squared theta. 00:20:34.570 --> 00:20:36.340 I heard a mumble over there. 00:20:43.420 --> 00:20:48.005 Yeah, but I'm just doing Del squared. 00:20:48.005 --> 00:20:50.130 AUDIENCE: With your first Del, it's H plus squared. 00:20:50.130 --> 00:20:50.921 PROFESSOR: My what? 00:20:50.921 --> 00:20:53.410 AUDIENCE: Your first time you used the Del operator, it's H 00:20:53.410 --> 00:20:55.280 plus squared. 00:20:55.280 --> 00:20:56.662 PROFESSOR: Oh, yes, yes. 00:20:56.662 --> 00:20:58.078 AUDIENCE: And then the second one, 00:20:58.078 --> 00:21:00.165 if you could clarify that that's a Del-- 00:21:00.165 --> 00:21:01.040 PROFESSOR: I'm sorry. 00:21:01.040 --> 00:21:01.920 AUDIENCE: The second Del squared, 00:21:01.920 --> 00:21:03.764 if you could clarify that's a Del squared. 00:21:03.764 --> 00:21:05.120 It looks like a-- 00:21:05.120 --> 00:21:06.870 PROFESSOR: It looks like a terrible thing. 00:21:11.640 --> 00:21:18.690 And the last part is a second derivative with respect to phi. 00:21:18.690 --> 00:21:22.710 This looks like a terrible thing to build on. 00:21:22.710 --> 00:21:27.670 But with a little bit of trickery-- 00:21:27.670 --> 00:21:32.900 and that is, suppose we multiply this equation by R squared-- 00:21:32.900 --> 00:21:41.340 then we have killed the R squared terms here and here. 00:21:41.340 --> 00:21:44.020 And we're going to be able to separate it. 00:21:44.020 --> 00:21:50.980 And so we are able to write an equation which 00:21:50.980 --> 00:21:53.230 has the separability built in-- 00:21:53.230 --> 00:21:56.784 and so partial with respect to R, 00:21:56.784 --> 00:22:05.300 R squared partial with respect to R plus L 00:22:05.300 --> 00:22:23.980 squared plus 2 mu H R squared V of R minus E psi. 00:22:23.980 --> 00:22:26.150 That's a Schrodinger equation. 00:22:26.150 --> 00:22:32.020 So we have an R dependent term and another R dependent term 00:22:32.020 --> 00:22:36.510 and a theta phi dependent term all in this one, nice operator 00:22:36.510 --> 00:22:37.570 that we've understood. 00:22:41.380 --> 00:22:44.780 So now there's one more trick. 00:22:44.780 --> 00:22:47.330 In order for the shorter equation 00:22:47.330 --> 00:22:51.860 to be separated into a theta phi part and an R part 00:22:51.860 --> 00:22:54.230 is we need a commutator. 00:22:54.230 --> 00:23:00.170 And so that commutator is this. 00:23:00.170 --> 00:23:05.000 What is a commutator between L squared and any function of R? 00:23:08.330 --> 00:23:08.910 Yes. 00:23:08.910 --> 00:23:11.243 AUDIENCE: Their operators depend on different variables? 00:23:11.243 --> 00:23:13.250 PROFESSOR: Absolutely, that's really important. 00:23:13.250 --> 00:23:16.310 We often encounter operators that 00:23:16.310 --> 00:23:18.780 depend on different variables. 00:23:18.780 --> 00:23:21.420 And when they do, they commute with each other, 00:23:21.420 --> 00:23:24.200 which is an incredibly convenient thing because that 00:23:24.200 --> 00:23:26.690 means we can set up the problem as a product 00:23:26.690 --> 00:23:31.238 of the eigenfunctions of the different operators. 00:23:31.238 --> 00:23:38.880 So this means that we can write a theta phi term, which 00:23:38.880 --> 00:23:44.190 we completely know, and an R term, which we don't know 00:23:44.190 --> 00:23:46.200 and contains all of the interesting stuff. 00:23:50.450 --> 00:23:52.890 So let's forget about theta and phi. 00:23:52.890 --> 00:23:54.760 Let's just look at the R part. 00:24:02.710 --> 00:24:04.750 Well, the way we separated variables, 00:24:04.750 --> 00:24:06.670 we had the Schrodinger equation and we 00:24:06.670 --> 00:24:10.360 divided by the wave function, which would be 00:24:10.360 --> 00:24:17.210 of R of R, Y L M of theta phi. 00:24:17.210 --> 00:24:22.790 And on one side of the equality, we have only the angle part. 00:24:22.790 --> 00:24:25.400 And so when we do that, when we divide by R, 00:24:25.400 --> 00:24:27.680 we kill the R part on this side. 00:24:27.680 --> 00:24:30.900 And on the other side, it's the opposite. 00:24:30.900 --> 00:24:33.710 And so we get two pieces-- 00:24:33.710 --> 00:24:36.230 one is only dependent on theta and phi 00:24:36.230 --> 00:24:38.330 and one is only dependent on R. So they both 00:24:38.330 --> 00:24:40.040 have to be a constant. 00:24:40.040 --> 00:24:42.680 So we get two separate differential equations. 00:24:42.680 --> 00:24:45.570 And we've already dealt with one of them. 00:24:45.570 --> 00:24:57.430 So what you end up getting is 1 over R of R times stuff 00:24:57.430 --> 00:25:09.131 times R of R is equal to 1 over YLM L squared YLM-- 00:25:09.131 --> 00:25:09.630 sorry. 00:25:13.700 --> 00:25:15.710 So this is the separation. 00:25:15.710 --> 00:25:17.900 This is all R stuff. 00:25:17.900 --> 00:25:21.960 This is all theta stuff and separation constant. 00:25:21.960 --> 00:25:24.350 Well, this one is inviting a separation concept 00:25:24.350 --> 00:25:27.830 because L squared operating on YLM 00:25:27.830 --> 00:25:30.679 is H bar squared, LL plus 1. 00:25:30.679 --> 00:25:31.970 That's the separation constant. 00:25:38.320 --> 00:25:44.140 So now let's look for the only time 00:25:44.140 --> 00:25:48.350 at the R part of the differential equation. 00:25:48.350 --> 00:25:53.080 And so this writing out all the pieces honestly-- 00:26:31.600 --> 00:26:33.070 we have-- there's one more. 00:26:40.090 --> 00:26:43.269 So that's the Schrodinger equation for the radial part. 00:26:43.269 --> 00:26:44.560 It looks a little bit annoying. 00:26:47.680 --> 00:26:52.600 The important trick is that we've 00:26:52.600 --> 00:26:58.410 taken the separation constant, and it has an R dependence 00:26:58.410 --> 00:27:04.020 but when we divide through by 2 mu HR squared. 00:27:04.020 --> 00:27:07.650 And we say, oh, well, let's call these two things together-- 00:27:07.650 --> 00:27:15.510 VL of R. This is the effect of potential, 00:27:15.510 --> 00:27:18.790 and it depends on the value of L. 00:27:18.790 --> 00:27:22.390 So this is just like an ordinary one dimensional problem 00:27:22.390 --> 00:27:24.250 except now, for every value of L, 00:27:24.250 --> 00:27:27.140 we have a different potential. 00:27:27.140 --> 00:27:30.520 And this potential is-- 00:27:30.520 --> 00:27:36.470 OK, when L is not equal to zero, this potential 00:27:36.470 --> 00:27:40.350 goes to infinity at R equals 0, which 00:27:40.350 --> 00:27:44.340 is bad, except it's good because it keeps the particle ever 00:27:44.340 --> 00:27:46.425 from getting close to the nucleus. 00:27:49.710 --> 00:27:51.870 And that's what Mr. Schrodinger-- 00:27:51.870 --> 00:27:53.970 I'm sorry what Mr. Bohr was thinking about, 00:27:53.970 --> 00:27:56.150 that we have only circular orbits 00:27:56.150 --> 00:28:03.570 or we have only orbits that are away from the place 00:28:03.570 --> 00:28:08.740 where the Coulomb interaction would be infinite. 00:28:08.740 --> 00:28:11.130 And I mean, there are several things that's 00:28:11.130 --> 00:28:12.390 wrong with Bohr's picture. 00:28:12.390 --> 00:28:15.580 One is that we have orbits. 00:28:15.580 --> 00:28:20.280 And the other is that we don't include L equals 0, 00:28:20.280 --> 00:28:28.020 but this is telling you a lot of really important stuff 00:28:28.020 --> 00:28:32.490 because the value of L determines 00:28:32.490 --> 00:28:36.930 the importance of this thing that goes to infinity at R 00:28:36.930 --> 00:28:39.490 equals 0. 00:28:39.490 --> 00:28:44.450 And that has very significant consequences as far as 00:28:44.450 --> 00:28:47.270 which orbital angular momentum states we're dealing with. 00:28:54.420 --> 00:29:01.440 So we're going to get the usual LML quantum numbers, 00:29:01.440 --> 00:29:04.200 and we're going to get another one from the radial part. 00:29:04.200 --> 00:29:08.100 And since this is a 1D equation, there's 00:29:08.100 --> 00:29:10.420 only one quantum number. 00:29:10.420 --> 00:29:12.790 And we're going to call it L. And now 00:29:12.790 --> 00:29:17.180 there's this word principal, and there's really two words-- 00:29:17.180 --> 00:29:19.835 principle with an LE and principal with an AL. 00:29:22.880 --> 00:29:27.200 Which one do you think is appropriate? 00:29:27.200 --> 00:29:28.680 I'm sorry. 00:29:28.680 --> 00:29:30.229 I can't hear. 00:29:30.229 --> 00:29:30.770 AUDIENCE: AL. 00:29:30.770 --> 00:29:32.710 PROFESSOR: AL is the appropriate one. 00:29:32.710 --> 00:29:36.570 Principle has to do with something fundamental. 00:29:36.570 --> 00:29:40.980 Principal has to do with something that's important. 00:29:40.980 --> 00:29:44.720 And I can't tell you how many times 00:29:44.720 --> 00:29:48.498 people who should know better use the wrong principal. 00:29:52.470 --> 00:29:55.620 You'll never do that now because it was like, what's nu? 00:29:55.620 --> 00:29:56.460 C over lambda. 00:29:59.730 --> 00:30:01.590 So this is the right principal. 00:30:10.464 --> 00:30:18.110 If we do something clever and we have the radial part 00:30:18.110 --> 00:30:20.240 and we say, let us replace it by 1 00:30:20.240 --> 00:30:29.900 over R times this new function chi L of R, well, 00:30:29.900 --> 00:30:35.380 when we do that, this equation becomes really simple. 00:30:35.380 --> 00:30:40.530 So what we get when we make that substitution is H bar squared 00:30:40.530 --> 00:30:52.040 over 2 mu H second derivative with respect to R plus VL of R 00:30:52.040 --> 00:30:57.950 minus E chi L of R is equal to 0. 00:30:57.950 --> 00:31:02.250 That looks like a differential equation we've seen before. 00:31:02.250 --> 00:31:04.820 It's a simple one dimensional differential equation-- 00:31:04.820 --> 00:31:06.710 kinetic energy, potential energy. 00:31:06.710 --> 00:31:07.690 But it's not. 00:31:07.690 --> 00:31:13.140 There's some kinetic energy hidden in the potential energy. 00:31:13.140 --> 00:31:16.580 But this is simple, and we can deal with this a lot. 00:31:16.580 --> 00:31:18.080 I'm not going to. 00:31:18.080 --> 00:31:20.709 But if you're going to actually do stuff, 00:31:20.709 --> 00:31:23.000 you're going to be wanting to look at this differential 00:31:23.000 --> 00:31:24.680 equation. 00:31:24.680 --> 00:31:26.975 But I'm going to forego that pleasure. 00:31:31.100 --> 00:31:34.810 One of the problems with the radial equation 00:31:34.810 --> 00:31:39.130 is the fact that R is a special kind of coordinate. 00:31:39.130 --> 00:31:41.860 It can't go negative. 00:31:41.860 --> 00:31:48.730 And so treating the boundary condition for R is equal to 0 00:31:48.730 --> 00:31:49.840 is a little subtle. 00:31:53.280 --> 00:31:58.920 And it turns out that when L is equal to 0, then R of 0 00:31:58.920 --> 00:31:59.880 is not 0. 00:32:02.570 --> 00:32:09.340 But for all other values of L, R of 0 is 0. 00:32:09.340 --> 00:32:18.340 And all of NMR depends on L being 00:32:18.340 --> 00:32:25.130 0 because the electron feels the nucleus. 00:32:25.130 --> 00:32:29.450 It doesn't experience an infinite singularity. 00:32:29.450 --> 00:32:30.950 It feels the nucleus. 00:32:30.950 --> 00:32:34.910 And when L is not equal to 0, then it's 00:32:34.910 --> 00:32:38.030 sensing the nucleus at a distance. 00:32:38.030 --> 00:32:41.060 And that leads to some very small splittings 00:32:41.060 --> 00:32:43.580 called hyperfine. 00:32:43.580 --> 00:32:48.180 And so there's different kinds of hyperfine structure. 00:32:48.180 --> 00:32:52.200 But anyway, this is a really subtle and important point. 00:32:57.757 --> 00:33:01.230 How much time left? 00:33:01.230 --> 00:33:02.350 Yes? 00:33:02.350 --> 00:33:04.340 No. 00:33:04.340 --> 00:33:07.151 So now pictures of orbitals-- 00:33:09.740 --> 00:33:13.970 So the problem with pictures of orbitals 00:33:13.970 --> 00:33:18.710 is now we have a function of three variables 00:33:18.710 --> 00:33:23.460 and it's equal to some complex number. 00:33:23.460 --> 00:33:25.880 So we need two degrees of freedom 00:33:25.880 --> 00:33:29.910 to present a complex number and we have three variables. 00:33:29.910 --> 00:33:34.940 And so representing that on a two dimensional sheet of paper 00:33:34.940 --> 00:33:35.990 is horrible. 00:33:35.990 --> 00:33:38.150 But we have this wonderful factorization 00:33:38.150 --> 00:33:42.380 where we have RNL of R. And we could draw that easily. 00:33:42.380 --> 00:33:44.460 We don't need any special skill. 00:33:44.460 --> 00:33:48.380 And we have YLM of theta phi. 00:33:48.380 --> 00:33:50.120 And we've got lots of practice with that, 00:33:50.120 --> 00:33:51.830 although what we've practiced with 00:33:51.830 --> 00:33:54.140 may not be completely understood yet. 00:33:58.630 --> 00:34:00.760 So we have ways of representing these. 00:34:00.760 --> 00:34:05.770 And so we go through the understanding 00:34:05.770 --> 00:34:09.030 of the hydrogen atom by looking at these two things separately. 00:34:12.330 --> 00:34:16.365 Now for the radial part, the energy levels-- 00:34:27.409 --> 00:34:29.429 where this is the Rydberg constant. 00:34:29.429 --> 00:34:32.540 It's a combination or fundamental constants. 00:34:32.540 --> 00:34:37.070 And for hydrogen, the Rydberg constant 00:34:37.070 --> 00:34:46.440 is equal to 109,737.319-- 00:34:46.440 --> 00:34:49.770 there's actually more digits, wave numbers-- 00:34:49.770 --> 00:34:56.590 times mu H over mu infinite. 00:34:56.590 --> 00:34:59.260 Well, this is actually the Rydberg constant 00:34:59.260 --> 00:35:00.355 for an infinite mass. 00:35:09.900 --> 00:35:20.610 And so mu H is equal to the mass of the electron 00:35:20.610 --> 00:35:25.969 times the mass of the proton over the mass of the electron 00:35:25.969 --> 00:35:27.135 plus the mass of the proton. 00:35:30.180 --> 00:35:33.230 Now the mass of the proton is much bigger 00:35:33.230 --> 00:35:34.790 than the mass of the electron. 00:35:34.790 --> 00:35:37.610 And so you can use this as a trick. 00:35:37.610 --> 00:35:39.310 You can say, well, what is it? 00:35:39.310 --> 00:35:41.430 Well, we know what this is. 00:35:41.430 --> 00:35:42.509 It's easy to calculate. 00:35:42.509 --> 00:35:43.550 But there are two limits. 00:35:43.550 --> 00:35:47.810 What is the smallest possible reduced mass? 00:35:47.810 --> 00:35:49.265 And that would be for-- 00:35:55.790 --> 00:36:00.240 if the mass of the proton is infinite, 00:36:00.240 --> 00:36:02.560 then we just get the mass of the electron. 00:36:02.560 --> 00:36:05.790 That's the biggest reduced mass. 00:36:05.790 --> 00:36:08.310 And the smallest is, if we have positronium, 00:36:08.310 --> 00:36:13.380 where we have an electron bound to a proton-- 00:36:13.380 --> 00:36:17.680 and then when we do that, we get half. 00:36:17.680 --> 00:36:18.180 I'm sorry. 00:36:18.180 --> 00:36:19.520 It's not a proton. 00:36:19.520 --> 00:36:24.590 It's a positively charged particle, 00:36:24.590 --> 00:36:27.020 which we call a positron. 00:36:27.020 --> 00:36:34.010 And so each of the terms here is the mass of the electron. 00:36:34.010 --> 00:36:43.820 And so the range is from 1/2 me to me 00:36:43.820 --> 00:36:47.490 depending on what particles you're dealing with. 00:36:47.490 --> 00:36:50.690 And so that's a useful thing. 00:36:50.690 --> 00:36:55.040 And so the Rydberg constant for hydrogen 00:36:55.040 --> 00:36:59.491 is smaller than the reduced mass of the infinite. 00:36:59.491 --> 00:36:59.990 I'm sorry. 00:36:59.990 --> 00:37:05.270 It's smaller than that for the infinitely mass nucleus. 00:37:05.270 --> 00:37:11.810 And it is the value that made Mr. Bohr very happy-- 00:37:11.810 --> 00:37:15.890 679. 00:37:15.890 --> 00:37:19.430 So the important thing is this mass scaled, 00:37:19.430 --> 00:37:25.296 or reduced mass scaled, Rydberg constant explains to a part 00:37:25.296 --> 00:37:28.970 in 10 to the 10th all the energy levels of one-electron 00:37:28.970 --> 00:37:29.870 systems-- 00:37:29.870 --> 00:37:34.010 hydrogen, helium plus, lithium 2 plus. 00:37:34.010 --> 00:37:34.540 That's it. 00:37:37.725 --> 00:37:39.200 And that's fantastic. 00:37:45.560 --> 00:37:48.740 Now we have to talk to become really familiar 00:37:48.740 --> 00:37:53.500 with this Rnl of our function. 00:38:06.670 --> 00:38:09.200 And one of the questions is, how many radial nodes. 00:38:15.710 --> 00:38:21.570 And you know that for 1S there aren't any nodes. 00:38:24.990 --> 00:38:28.510 And you know for 2P there aren't any radial nodes. 00:38:31.110 --> 00:38:34.230 And so what we need is something that 00:38:34.230 --> 00:38:38.870 goes like n minus L minus 1. 00:38:38.870 --> 00:38:41.740 The number of radial nodes, which 00:38:41.740 --> 00:38:45.100 is all you need to know about the radial A function, 00:38:45.100 --> 00:38:50.070 is how many nodes are and how far are they apart 00:38:50.070 --> 00:38:53.710 and what's the amplitude of each loop between nodes. 00:38:53.710 --> 00:38:57.770 Semi classical theory gives you all of that. 00:38:57.770 --> 00:39:01.970 And often when you're calculating an integral, 00:39:01.970 --> 00:39:04.550 all you care about is the amplitude in the first loop. 00:39:07.190 --> 00:39:11.880 And so instead of having to evaluate an integral, 00:39:11.880 --> 00:39:16.770 you just figure out what is the envelope function based 00:39:16.770 --> 00:39:18.810 on the classical momentum function. 00:39:24.950 --> 00:39:27.410 Now the thing that everybody's been waiting for. 00:39:33.280 --> 00:39:33.780 Spin. 00:39:39.660 --> 00:39:49.950 So remember, we know an angular momentum is R cross P, 00:39:49.950 --> 00:39:53.177 and we know that there isn't any internal structure. 00:39:53.177 --> 00:39:55.260 Or at least we don't know about internal structure 00:39:55.260 --> 00:39:58.930 of the electron or a proton. 00:39:58.930 --> 00:40:04.850 And so we can't somehow say, well, it's R cross P. 00:40:04.850 --> 00:40:06.260 So we have the Zeeman effect. 00:40:14.530 --> 00:40:17.400 So we can look at the Zeeman effect 00:40:17.400 --> 00:40:20.000 for an atom in a magnetic field. 00:40:20.000 --> 00:40:23.860 And we have several things that we know. 00:40:23.860 --> 00:40:28.950 So we have the magnetic moment of the electron 00:40:28.950 --> 00:40:37.470 is equal to minus the charge on the electron times 2ME times 00:40:37.470 --> 00:40:40.850 L. Well, that's an angular momentum. 00:40:48.980 --> 00:40:57.070 So if we had circulating charge, well, that circulating charge 00:40:57.070 --> 00:41:00.680 will produce a magnetic moment, which has a magnitude which 00:41:00.680 --> 00:41:02.330 is related to the velocity. 00:41:04.890 --> 00:41:09.120 And what is L divided by mass? 00:41:09.120 --> 00:41:11.660 It's a velocity. 00:41:11.660 --> 00:41:13.530 And we have the charge here. 00:41:13.530 --> 00:41:17.440 So this is a perfectly reasonable thing. 00:41:17.440 --> 00:41:22.940 And that's good because we know that, if we have electrons 00:41:22.940 --> 00:41:25.520 with some kind of internal structure, 00:41:25.520 --> 00:41:26.900 we know what this is going to do. 00:41:29.630 --> 00:41:36.390 Now the magnetic potential is equal to minus 00:41:36.390 --> 00:41:40.080 the magnetic moment times the external magnetic field. 00:41:42.840 --> 00:41:51.920 And so what we have is E, BC, LZ over 2 NE. 00:41:55.110 --> 00:41:56.970 Well, this is another very good thing 00:41:56.970 --> 00:42:02.440 because, not only do we know what this is, 00:42:02.440 --> 00:42:06.370 we know that it has only diagonal elements. 00:42:06.370 --> 00:42:12.000 And so we can do first order perturbation theory. 00:42:15.330 --> 00:42:18.960 If this is so easy, if this is the Hamiltonian, 00:42:18.960 --> 00:42:22.410 we can just tack that on and it adds an extra splitting 00:42:22.410 --> 00:42:23.820 to our energy levels. 00:42:27.230 --> 00:42:32.350 The only tricky thing is, we don't know that it's LZ. 00:42:32.350 --> 00:42:35.360 We just know that it has a magnetic moment 00:42:35.360 --> 00:42:39.640 or it could have a magnetic moment. 00:42:39.640 --> 00:42:41.450 And so this wonderful experiment-- 00:42:41.450 --> 00:42:45.920 suppose we start with a 1S state, 00:42:45.920 --> 00:42:48.880 and you go to a 2P state. 00:42:48.880 --> 00:42:53.800 Now this implies that we know something 00:42:53.800 --> 00:42:58.422 about the selection rules for electromagnetic transitions. 00:42:58.422 --> 00:43:00.130 And so for an electric dipole transition, 00:43:00.130 --> 00:43:02.500 we go from 1S to 2P. 00:43:02.500 --> 00:43:05.470 Now this is not how it was done initially 00:43:05.470 --> 00:43:09.820 because the frequency of this transition 00:43:09.820 --> 00:43:12.370 is well into the vacuum ultraviolet. 00:43:12.370 --> 00:43:16.090 And in the days when quantum mechanics was being developed, 00:43:16.090 --> 00:43:18.510 that was a hard experiment. 00:43:18.510 --> 00:43:24.750 So one used an S to P transition on some other atom, 00:43:24.750 --> 00:43:31.580 like mercury, but let's pretend it was hydrogen. 00:43:31.580 --> 00:43:35.380 So this is an angular momentum of 0. 00:43:35.380 --> 00:43:40.490 So we would expect it would be ML equals 0. 00:43:40.490 --> 00:43:46.970 And here we have it could split into three components. 00:43:46.970 --> 00:43:56.530 And this is an L equals 1, 0 minus 1, because that's minus-- 00:43:56.530 --> 00:43:59.350 there with a minus sign somewhere. 00:43:59.350 --> 00:44:01.700 That's what we expect. 00:44:01.700 --> 00:44:05.920 And so being naive, you might expect transitions like this. 00:44:09.070 --> 00:44:12.070 Well, the transitions where you do not 00:44:12.070 --> 00:44:14.260 change the projection quantum number 00:44:14.260 --> 00:44:18.040 are done with Z polarized radiation. 00:44:18.040 --> 00:44:23.360 And so if it's Z polarized, you get only delta ML equals 0. 00:44:23.360 --> 00:44:25.890 And if you have x and y, you have delta 00:44:25.890 --> 00:44:29.030 ML equals plus and minus 1. 00:44:29.030 --> 00:44:32.380 So depending on how the experiment was done, 00:44:32.380 --> 00:44:35.680 you would expect to see one, two, 00:44:35.680 --> 00:44:38.694 or three Zeeman components. 00:44:41.480 --> 00:44:43.610 And they did the experiment, and they 00:44:43.610 --> 00:44:45.980 saw more than three components. 00:44:45.980 --> 00:44:48.440 They saw five components. 00:44:48.440 --> 00:44:51.780 And that's possibly for a number of reasons. 00:44:51.780 --> 00:44:53.770 But they saw five components. 00:44:53.770 --> 00:44:57.040 So they knew that there was something else going on. 00:44:57.040 --> 00:45:09.110 And so you have to try to collect enough information 00:45:09.110 --> 00:45:13.490 to have a simple minded picture that will explain it all. 00:45:16.300 --> 00:45:18.090 So one thing you might do is say, OK, 00:45:18.090 --> 00:45:24.180 suppose there is another quantum number, another thing. 00:45:24.180 --> 00:45:26.920 And we're going to call it spin. 00:45:26.920 --> 00:45:29.890 And we don't know whether spin is integer or half integer. 00:45:29.890 --> 00:45:33.910 We know from our exercise with computation rules 00:45:33.910 --> 00:45:37.390 that both half integer and integer angular 00:45:37.390 --> 00:45:39.350 momentum are possible. 00:45:39.350 --> 00:45:43.820 And so we can say we have a spin. 00:45:43.820 --> 00:45:45.430 And it could be 1/2. 00:45:45.430 --> 00:45:46.150 It could be 1. 00:45:46.150 --> 00:45:47.600 It could be anything. 00:45:47.600 --> 00:45:50.290 And then we start looking at the details of what we observe. 00:45:53.130 --> 00:45:58.500 And what we find is, all we need is spin 1/2 00:45:58.500 --> 00:46:00.660 to account for almost everything. 00:46:03.540 --> 00:46:08.590 However, if it were spin 1/2, then you get two here. 00:46:08.590 --> 00:46:14.830 And you'd get two here, two, here and two here. 00:46:14.830 --> 00:46:15.790 And that's six. 00:46:15.790 --> 00:46:18.510 That's bigger than five. 00:46:18.510 --> 00:46:21.350 So you need to know something else. 00:46:21.350 --> 00:46:23.810 And one thing is quite reasonable-- you can say, 00:46:23.810 --> 00:46:28.790 well, the spin thing is mysterious, 00:46:28.790 --> 00:46:32.810 and electromagnetic radiation acts on the spatial 00:46:32.810 --> 00:46:34.584 coordinates. 00:46:34.584 --> 00:46:36.250 And there aren't any spatial coordinates 00:46:36.250 --> 00:46:39.100 of the internal structure of the electron. 00:46:39.100 --> 00:46:43.390 And so we have a selection rule delta MS equals 0. 00:46:47.809 --> 00:46:48.850 That still doesn't do it. 00:46:51.500 --> 00:46:53.630 You still need something else, and that 00:46:53.630 --> 00:46:58.940 is that the proportionality concept between the angular 00:46:58.940 --> 00:47:06.320 momentum and the energy, which is called the G factor. 00:47:06.320 --> 00:47:10.940 For orbital angular matter, the G factor is 1, 00:47:10.940 --> 00:47:12.620 or the proportionality constant is 1. 00:47:12.620 --> 00:47:18.070 So the G factor is L. And the G factor for the electron 00:47:18.070 --> 00:47:20.570 turns out to be 2-- 00:47:20.570 --> 00:47:23.330 not exactly 2, just a little more than 2-- 00:47:23.330 --> 00:47:26.510 Nobel Prize for that. 00:47:26.510 --> 00:47:30.360 And so with those extra little things, 00:47:30.360 --> 00:47:35.390 then every detail of the Zeeman effect is understood. 00:47:35.390 --> 00:47:38.540 And we say, oh, well, the electron has a spin of 1/2, 00:47:38.540 --> 00:47:40.480 and it like it acts like an angular momentum. 00:47:40.480 --> 00:47:44.580 It obeys the angular momentum computation rules. 00:47:44.580 --> 00:47:48.480 It also obeys the computation rules. 00:47:48.480 --> 00:47:50.600 Well, I won't say that. 00:47:50.600 --> 00:47:52.940 So I should stop-- 00:47:52.940 --> 00:47:56.660 there are other things that provide us information 00:47:56.660 --> 00:47:58.700 that there is something else. 00:47:58.700 --> 00:48:03.980 And one, I'm called by some people the spin orbit kid 00:48:03.980 --> 00:48:07.550 because I've made a whole lot of mileage on using spin orbits 00:48:07.550 --> 00:48:08.315 splittings. 00:48:08.315 --> 00:48:13.690 And the spin orbit Hamiltonian has the form L.S 00:48:13.690 --> 00:48:19.700 And this gives rise to splittings of a doublet P 00:48:19.700 --> 00:48:21.290 state, for example, as what you would 00:48:21.290 --> 00:48:24.620 see in the excited state of hydrogen 00:48:24.620 --> 00:48:27.110 into two components at zero field. 00:48:29.620 --> 00:48:33.310 And so there's all sorts of really wonderful stuff. 00:48:33.310 --> 00:48:36.070 And I'm going to cheat you out of almost all of it 00:48:36.070 --> 00:48:39.580 because we're going to go over to the semiclassical picture 00:48:39.580 --> 00:48:41.660 next time. 00:48:41.660 --> 00:48:45.410 And we'll understand everything about Rydberg states 00:48:45.410 --> 00:48:48.560 and about how do we estimate everything 00:48:48.560 --> 00:48:52.110 having to do with the radial part of the wave function. 00:48:52.110 --> 00:48:54.640 And there are some astonishing things.