WEBVTT 00:00:00.060 --> 00:00:02.420 The following content is provided under a Creative 00:00:02.420 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 00:00:10.130 continue to offer high quality educational resources for free. 00:00:10.130 --> 00:00:12.700 To make a donation or to view additional materials 00:00:12.700 --> 00:00:16.120 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.120 --> 00:00:17.090 at ocw.mit.edu. 00:00:23.690 --> 00:00:26.380 PROFESSOR: All right. 00:00:26.380 --> 00:00:28.050 Welcome, everyone. 00:00:28.050 --> 00:00:29.710 Hi. 00:00:29.710 --> 00:00:32.850 So today we're going to pick up where 00:00:32.850 --> 00:00:36.430 we left off last time in our study of angular momentum 00:00:36.430 --> 00:00:38.865 and rotations in quantum mechanics. 00:00:38.865 --> 00:00:40.240 Before I get started, let me open 00:00:40.240 --> 00:00:43.520 up for questions, pragmatic and physics related. 00:00:48.350 --> 00:00:49.786 Yeah? 00:00:49.786 --> 00:00:52.744 AUDIENCE: When we were solving for the 3D harmonic oscillator 00:00:52.744 --> 00:00:55.866 we solved for the energy eigenfunction that 00:00:55.866 --> 00:00:59.153 was a product of phi x, phi y, and phi z. 00:00:59.153 --> 00:01:02.604 We made an assumption that phi e was 00:01:02.604 --> 00:01:05.972 equal to phi sub n of x plus that sum. 00:01:05.972 --> 00:01:08.433 How did you get from [INAUDIBLE]?? 00:01:08.433 --> 00:01:09.100 PROFESSOR: Good. 00:01:09.100 --> 00:01:11.100 So what we had was that we had that the energy-- 00:01:11.100 --> 00:01:15.590 we wanted to find the energy of the 3D harmonic oscillator. 00:01:15.590 --> 00:01:18.345 And we wanted to find the energy eigenfunctions and eigenvalues. 00:01:18.345 --> 00:01:19.970 And they way we did this was by saying, 00:01:19.970 --> 00:01:22.540 look, the energy of the 3D harmonic oscillator, which 00:01:22.540 --> 00:01:24.730 I can think of as a function of x and px 00:01:24.730 --> 00:01:28.480 and y and py and z and pz, has this nice form. 00:01:28.480 --> 00:01:31.540 We could write it as the energy operator 00:01:31.540 --> 00:01:35.080 purely in terms of x, p squared x upon 2m 00:01:35.080 --> 00:01:37.710 plus m omega squared upon 2x squared. 00:01:37.710 --> 00:01:42.500 Plus-- so this is a single 1D harmonic oscillator energy 00:01:42.500 --> 00:01:44.150 operator in the x direction. 00:01:44.150 --> 00:01:48.730 Plus E 1D in the y direction, plus a harmonic 00:01:48.730 --> 00:01:52.510 oscillator energy 1D in the z direction. 00:01:52.510 --> 00:01:54.110 So that was the first observation. 00:01:54.110 --> 00:01:58.430 And then we said that given that this splits in this fashion, 00:01:58.430 --> 00:02:03.230 I'm going to write my energy eigenfunction, phi 00:02:03.230 --> 00:02:08.710 of x, y, and z in separated form as a product. 00:02:08.710 --> 00:02:16.620 Phi x of x, phi y of y, and phi z of z. 00:02:16.620 --> 00:02:19.830 And we used this and deduced that in order for this 00:02:19.830 --> 00:02:22.280 to be an eigenfunction of the 3D harmonic oscillator 00:02:22.280 --> 00:02:25.140 it must be true that 5x of x was itself 00:02:25.140 --> 00:02:28.520 an eigenfunction of the x harmonic oscillator 00:02:28.520 --> 00:02:32.716 equals with some energy epsilon, which I will 00:02:32.716 --> 00:02:35.870 call epsilon sub x, phi sub x. 00:02:35.870 --> 00:02:37.450 And ditto for y and z. 00:02:37.450 --> 00:02:40.770 And if this is true, if phi sub x is an eigenfunction of the 1D 00:02:40.770 --> 00:02:43.780 harmonic oscillator, then this is an eigenfunction 00:02:43.780 --> 00:02:45.272 of the 3D harmonic oscillator. 00:02:45.272 --> 00:02:46.730 But we know what the eigenfunctions 00:02:46.730 --> 00:02:47.850 are of the harmonic oscillator. 00:02:47.850 --> 00:02:49.490 The eigenfunctions of the 1D harmonic 00:02:49.490 --> 00:02:51.130 oscillator we gave a name. 00:02:51.130 --> 00:02:53.790 We call them pi sub n. 00:02:53.790 --> 00:02:56.850 So what we then said was, look, if this 00:02:56.850 --> 00:02:59.790 is an eigenfunction of the 1D harmonic oscillator 00:02:59.790 --> 00:03:04.930 in the x direction, then it's labeled by an n. 00:03:04.930 --> 00:03:06.900 And if this one is in the y direction, 00:03:06.900 --> 00:03:08.150 it's labeled by an l. 00:03:08.150 --> 00:03:10.630 And if this one is in the z direction it's labeled by a z. 00:03:10.630 --> 00:03:11.980 But on top of that we know more. 00:03:11.980 --> 00:03:13.970 We know what these energy eigenvalues are. 00:03:13.970 --> 00:03:16.270 The energy eigenvalues corresponding to these guys 00:03:16.270 --> 00:03:19.750 are if this is phi n, this is En. 00:03:19.750 --> 00:03:23.460 I can simply write this as En for the 1D harmonic oscillator. 00:03:23.460 --> 00:03:27.570 And taking this, using the fact that they're 1D eigenfunctions 00:03:27.570 --> 00:03:29.320 and plugging it into the energy eigenvalue 00:03:29.320 --> 00:03:31.040 equation for the 3D harmonic oscillator 00:03:31.040 --> 00:03:33.510 tells us that the energy eigenvalues for the 3D case 00:03:33.510 --> 00:03:40.680 are of the form E1d x plus E1d in the y 00:03:40.680 --> 00:03:44.750 plus E1d in the z, which is equal to, since these were 00:03:44.750 --> 00:03:50.600 the same frequency, h bar omega times n plus l plus m, 00:03:50.600 --> 00:03:53.820 from each of them a 1/2, so plus 3/2. 00:03:53.820 --> 00:03:54.320 Cool? 00:03:54.320 --> 00:03:55.460 That answer your question? 00:03:55.460 --> 00:03:56.310 Excellent. 00:03:56.310 --> 00:03:57.541 Other questions? 00:03:57.541 --> 00:03:59.505 Yeah? 00:03:59.505 --> 00:04:01.469 AUDIENCE: Energy and angular momentum 00:04:01.469 --> 00:04:03.257 have to be related somehow, right? 00:04:03.257 --> 00:04:03.924 PROFESSOR: Yeah. 00:04:03.924 --> 00:04:05.397 AUDIENCE: I mean, of course. 00:04:05.397 --> 00:04:06.390 Because it's both. 00:04:06.390 --> 00:04:07.140 PROFESSOR: Indeed. 00:04:07.140 --> 00:04:09.340 AUDIENCE: [INAUDIBLE]. 00:04:09.340 --> 00:04:13.741 The thing is, we have a ladder. 00:04:13.741 --> 00:04:14.829 Is there limits on them? 00:04:14.829 --> 00:04:16.079 PROFESSOR: Very good question. 00:04:16.079 --> 00:04:17.420 So this is where we're going to pick up. 00:04:17.420 --> 00:04:18.380 Let me rephrase this. 00:04:18.380 --> 00:04:19.588 This is really two questions. 00:04:19.588 --> 00:04:20.523 Question number one. 00:04:20.523 --> 00:04:22.690 Look, there should be a relationship between angular 00:04:22.690 --> 00:04:23.780 momentum and energy. 00:04:23.780 --> 00:04:25.697 But we're just talking about angular momentum. 00:04:25.697 --> 00:04:26.510 Why? 00:04:26.510 --> 00:04:28.410 Second question, look, we've got a ladder. 00:04:28.410 --> 00:04:29.910 But is the ladder infinite? 00:04:29.910 --> 00:04:31.280 So let me come back to the second question. 00:04:31.280 --> 00:04:32.850 That's going to be the beginning of the lecture. 00:04:32.850 --> 00:04:34.900 On the first question, yes angular momentum 00:04:34.900 --> 00:04:37.108 is going to play a role when we calculate the energy. 00:04:37.108 --> 00:04:38.723 But two quick things to note. 00:04:38.723 --> 00:04:40.140 First off, consider a system which 00:04:40.140 --> 00:04:42.637 is spherically symmetric, rotationally invariant. 00:04:42.637 --> 00:04:44.970 That means that the energy doesn't depend on a rotation. 00:04:44.970 --> 00:04:47.137 If I rotate the system I haven't changed the energy. 00:04:47.137 --> 00:04:50.250 So if the system is rotationally invariant, 00:04:50.250 --> 00:04:52.550 that's going to imply some constraints on the energy 00:04:52.550 --> 00:04:55.390 eigenvalues and how they depend on the angular momentum, 00:04:55.390 --> 00:04:56.880 as we discussed last time. 00:04:56.880 --> 00:04:58.422 Let me say that slightly differently. 00:05:00.630 --> 00:05:03.620 When we talk about the free particle, 1D free particle-- 00:05:03.620 --> 00:05:05.250 we've talked about this one to death. 00:05:05.250 --> 00:05:06.850 Take the 1D free particle. 00:05:06.850 --> 00:05:08.870 We can write the energy eigenfunctions 00:05:08.870 --> 00:05:11.750 as momentum eigenfunctions, because the momentum commutes 00:05:11.750 --> 00:05:13.765 with the energy. 00:05:13.765 --> 00:05:15.723 And so the way the eigenfunctions of the energy 00:05:15.723 --> 00:05:18.240 operate are indeed e to iKX there are plane waves, 00:05:18.240 --> 00:05:20.980 they're eigenfunctions of the momentum operator as well. 00:05:20.980 --> 00:05:23.085 Similarly, when we talk about a 3D system 00:05:23.085 --> 00:05:25.210 it's going to be useful in talking about the energy 00:05:25.210 --> 00:05:29.690 eigenvalues to know a basis of eigenfunctions of the angular 00:05:29.690 --> 00:05:30.600 momentum operator. 00:05:30.600 --> 00:05:32.320 Knowing the angular momentum operator 00:05:32.320 --> 00:05:35.713 is going to allow us to write energy eigenfunctions 00:05:35.713 --> 00:05:37.130 in a natural way and a simply way, 00:05:37.130 --> 00:05:39.530 in the same way that knowing the momentum operator allowed 00:05:39.530 --> 00:05:40.905 us to write energy eigenfunctions 00:05:40.905 --> 00:05:43.700 in a simple way in the 1D case. 00:05:43.700 --> 00:05:44.717 That make sense? 00:05:44.717 --> 00:05:47.300 We're going to have a glorified version of the Fourier theorum 00:05:47.300 --> 00:05:49.090 where instead of something over e to iKX, 00:05:49.090 --> 00:05:50.150 we're going to have something over angular momentum 00:05:50.150 --> 00:05:51.072 eigenstates. 00:05:51.072 --> 00:05:53.130 And those are called the spherical harmonics. 00:05:53.130 --> 00:05:55.750 And they are the analog of Fourier expansion 00:05:55.750 --> 00:05:56.912 for this year. 00:05:56.912 --> 00:05:57.620 But you're right. 00:05:57.620 --> 00:05:58.960 We're going to have to understand how 00:05:58.960 --> 00:06:00.252 that interacts with the energy. 00:06:00.252 --> 00:06:02.160 And that'll be the topic of the next lecture. 00:06:02.160 --> 00:06:04.160 We're going to finish up angular momentum today. 00:06:04.160 --> 00:06:06.950 Other questions? 00:06:06.950 --> 00:06:11.360 OK, so from last time, these are the commutation relations 00:06:11.360 --> 00:06:13.450 which we partially derived in lecture 00:06:13.450 --> 00:06:16.630 and which you will be driving on your problem set. 00:06:16.630 --> 00:06:18.140 It's a really good exercise. 00:06:18.140 --> 00:06:19.230 Commit these to memory. 00:06:19.230 --> 00:06:20.840 They're your friends. 00:06:20.840 --> 00:06:23.130 Key thing here to keep in mind. 00:06:23.130 --> 00:06:26.270 h bar has units of angular momentum, so this makes sense. 00:06:26.270 --> 00:06:29.270 Angular momentum, angular momentum, angular momentum. 00:06:29.270 --> 00:06:34.220 So when you see an h bar in this setting, 00:06:34.220 --> 00:06:37.520 its job, in some sense, is to make everything dimensionally 00:06:37.520 --> 00:06:39.330 sensible. 00:06:39.330 --> 00:06:42.545 So the important things here are that lx and ly do not commute. 00:06:42.545 --> 00:06:44.840 They commute to lz. 00:06:44.840 --> 00:06:47.070 Can you have a state with definite angular momentum 00:06:47.070 --> 00:06:49.445 in the x direction and definite angular momentum in the y 00:06:49.445 --> 00:06:50.622 direction simultaneously? 00:06:50.622 --> 00:06:52.330 No, because of this commutation relation. 00:06:52.330 --> 00:06:55.760 It would have to vanish. 00:06:55.760 --> 00:06:58.065 This is, say, the x component of the angular momentum. 00:06:58.065 --> 00:07:00.190 Can you have a state with definite angular momentum 00:07:00.190 --> 00:07:03.850 in the x direction and total angular momentum all squared? 00:07:03.850 --> 00:07:04.380 Yes. 00:07:04.380 --> 00:07:04.880 OK, great. 00:07:04.880 --> 00:07:06.570 That's going to be important for us. 00:07:06.570 --> 00:07:08.410 So in order to construct the eigenfunctions 00:07:08.410 --> 00:07:09.993 it turns out to be useful to construct 00:07:09.993 --> 00:07:12.140 these so-called raising and lowering operators, 00:07:12.140 --> 00:07:15.167 which are Lx plus iLy. 00:07:15.167 --> 00:07:16.750 They have a couple of nice properties. 00:07:16.750 --> 00:07:19.260 The first is, since these are built up out of Lx and Ly, 00:07:19.260 --> 00:07:20.802 both of which commute with L squared, 00:07:20.802 --> 00:07:24.280 the L plus minuses commute with L squared. 00:07:24.280 --> 00:07:25.370 So these guys commute. 00:07:25.370 --> 00:07:29.990 So if we have an eigenfunction of L squared, 00:07:29.990 --> 00:07:34.700 acting with L plus does not change its eigenvalue. 00:07:34.700 --> 00:07:37.130 Similarly, L plus commuting with Lz 00:07:37.130 --> 00:07:42.360 gives us h bar L plus or minus, with a plus or minus out front. 00:07:42.360 --> 00:07:47.570 This is just like the raising and lowering operators 00:07:47.570 --> 00:07:49.060 for the harmonic oscillator. 00:07:49.060 --> 00:07:52.850 But instead of the energy we have the angular momentum. 00:07:52.850 --> 00:07:57.260 So this is going to tell us that the angular momentum 00:07:57.260 --> 00:07:59.970 eigenvalues, the eigenvalues of Lz, 00:07:59.970 --> 00:08:02.330 are shifted by plus or minus h bar 00:08:02.330 --> 00:08:04.140 when we raise or lower with L plus or L 00:08:04.140 --> 00:08:07.200 minus, just like the energy was shifted-- 00:08:07.200 --> 00:08:12.130 for the 1d harmonic oscillator the energy was shifted, 00:08:12.130 --> 00:08:15.990 plus h bar omega a dagger, was shifted 00:08:15.990 --> 00:08:17.800 by h bar omega when we acted with a plus 00:08:17.800 --> 00:08:18.860 on an energy eigenstate. 00:08:18.860 --> 00:08:19.370 Same thing. 00:08:22.580 --> 00:08:24.930 Questions on the commutators before we get going? 00:08:24.930 --> 00:08:27.090 In some sense, we're going to just to just take 00:08:27.090 --> 00:08:28.757 advantage of these commutation relations 00:08:28.757 --> 00:08:30.820 and explore their consequences today. 00:08:30.820 --> 00:08:40.360 So our goal is going to be to build the eigenfunctions 00:08:40.360 --> 00:08:47.347 and eigenvalues of the angular momentum operators, 00:08:47.347 --> 00:08:49.680 and in particular of the most angular momentum operators 00:08:49.680 --> 00:08:51.980 we [INAUDIBLE] complete set of commuting observables, 00:08:51.980 --> 00:08:53.310 L squared and Lz. 00:08:53.310 --> 00:08:54.810 You might complain, look, why Lz? 00:08:54.810 --> 00:08:55.310 Whoops. 00:08:55.310 --> 00:08:56.550 I don't mean a commutator. 00:08:56.550 --> 00:08:57.710 I mean the set. 00:08:57.710 --> 00:08:59.480 You might say, why Lz? 00:08:59.480 --> 00:09:00.620 Why not Lx? 00:09:00.620 --> 00:09:03.247 And if you call this the z direction then 00:09:03.247 --> 00:09:04.830 I will simply choose a new basis where 00:09:04.830 --> 00:09:06.380 this is called the x direction. 00:09:06.380 --> 00:09:08.110 So it makes no difference whatsoever. 00:09:08.110 --> 00:09:08.950 It's just a name. 00:09:08.950 --> 00:09:10.490 The reason we're going to choose Lz 00:09:10.490 --> 00:09:13.480 is because that coordinate system plays nicely 00:09:13.480 --> 00:09:16.710 with spherical coordinates, just the conventional choice 00:09:16.710 --> 00:09:19.260 of spherical coordinates where theta equals 0 is the up axis. 00:09:19.260 --> 00:09:20.760 But there's nothing deep about that. 00:09:20.760 --> 00:09:24.670 We could have taken any of these. 00:09:24.670 --> 00:09:26.220 OK. 00:09:26.220 --> 00:09:27.740 So this is our goal. 00:09:27.740 --> 00:09:28.670 So let's get started. 00:09:28.670 --> 00:09:32.902 So first, because of these commutation relations 00:09:32.902 --> 00:09:34.360 and in particular this one, we know 00:09:34.360 --> 00:09:36.485 that we can find common eigenfunctions of L squared 00:09:36.485 --> 00:09:37.480 and Lz. 00:09:37.480 --> 00:09:41.610 Let us call those common eigenfunctions by a name, 00:09:41.610 --> 00:09:47.720 Y sub lm such that-- so let these guys 00:09:47.720 --> 00:09:56.460 be the common eigenfunctions of L squared and Lz. 00:09:56.460 --> 00:10:02.120 I.e., L squared Ylm is equal to-- well first off, units. 00:10:02.120 --> 00:10:04.220 This has units of angular momentum squared, 00:10:04.220 --> 00:10:05.040 h bar squared. 00:10:05.040 --> 00:10:06.590 So that got rid of the units. 00:10:06.590 --> 00:10:08.490 And we want our eigenvalue, lm. 00:10:08.490 --> 00:10:11.495 And because I know the answer, I'm 00:10:11.495 --> 00:10:13.120 going to give-- instead of calling this 00:10:13.120 --> 00:10:16.210 a random dimensionless number, which would be the eigenvalue, 00:10:16.210 --> 00:10:19.950 I'm going to call it a very specific thing, l, l plus 1. 00:10:19.950 --> 00:10:21.880 This is a slightly grotesque thing to do, 00:10:21.880 --> 00:10:24.120 but it will make the algebra much easier. 00:10:24.120 --> 00:10:28.080 So similarly, so that's what the little l is. 00:10:28.080 --> 00:10:30.080 Little l is labeling the eigenvalue of L squared 00:10:30.080 --> 00:10:31.747 and the actual value of that eigenvalue. 00:10:31.747 --> 00:10:33.668 I'm just calling h bar squared ll plus 1. 00:10:33.668 --> 00:10:35.460 That doesn't tell you anything interesting. 00:10:35.460 --> 00:10:37.430 This was already a real positive number. 00:10:37.430 --> 00:10:40.660 So this could have been any real positive number as well, 00:10:40.660 --> 00:10:43.260 by tuning L. 00:10:43.260 --> 00:10:48.140 Similarly, Lz Ylm-- I want this to be an eigenfunction-- 00:10:48.140 --> 00:10:49.795 this has units of angular momentum. 00:10:49.795 --> 00:10:50.670 So I'll put an h bar. 00:10:50.670 --> 00:10:52.640 Now we have a dimensionless coefficient Ylm. 00:10:52.640 --> 00:10:55.870 And I'll simply call that m. 00:10:55.870 --> 00:10:57.690 So if you will, these are the definitions 00:10:57.690 --> 00:11:02.440 of the symbols m and little l. 00:11:02.440 --> 00:11:05.600 And Ylm are just the names I'm giving to the angular momentum 00:11:05.600 --> 00:11:06.820 eigenfunctions. 00:11:06.820 --> 00:11:07.830 Cool? 00:11:07.830 --> 00:11:09.330 So I haven't actually done anything. 00:11:09.330 --> 00:11:11.120 I've just told you that these are the eigenfunctions. 00:11:11.120 --> 00:11:13.493 What we want to know is what properties do they have, 00:11:13.493 --> 00:11:15.910 and what are the actual allowed values of the eigenvalues. 00:11:18.680 --> 00:11:30.470 So the two key things to note are first that L plus and minus 00:11:30.470 --> 00:11:32.160 leave the eigenvalue of L squared alone. 00:11:32.160 --> 00:11:33.077 So they leave l alone. 00:11:36.350 --> 00:11:40.890 So this is the statement that L squared on L plus Ylm 00:11:40.890 --> 00:11:45.230 is equal to L plus on L squared Ylm 00:11:45.230 --> 00:11:50.180 is equal to h bar squared ll plus 1, 00:11:50.180 --> 00:11:56.760 the eigenvalue of L squared acting on Ylm, L plus Ylm. 00:11:56.760 --> 00:12:00.120 And so L plus Ylm is just as much an eigenfunction of L 00:12:00.120 --> 00:12:05.960 squared as Ylm was itself, with the same eigenvalue. 00:12:05.960 --> 00:12:06.850 Two. 00:12:06.850 --> 00:12:07.932 So that came from-- 00:12:10.710 --> 00:12:14.200 where are we--- this commutation relation. 00:12:14.200 --> 00:12:23.620 OK, so similarly, L plus minus raise or lower m by one. 00:12:26.420 --> 00:12:29.290 And the way to see that is to do exactly the same computation, 00:12:29.290 --> 00:12:37.530 Lz on L plus, for example, Ylm is equal to L plus-- 00:12:37.530 --> 00:12:39.750 now we can write the commutator-- 00:12:39.750 --> 00:12:50.110 Lz, L plus, plus L plus Lz Ylm. 00:12:50.110 --> 00:12:53.160 But the commutator of Lz with L plus we already have, 00:12:53.160 --> 00:12:54.820 is plus h bar L plus. 00:12:54.820 --> 00:12:58.340 So this is equal to h bar L plus. 00:12:58.340 --> 00:13:01.210 And from this term, L plus Lz of Ylm, Lz acting on Ylm 00:13:01.210 --> 00:13:08.360 gives us h bar m, plus h bar m L plus Ylm is equal to, 00:13:08.360 --> 00:13:11.800 pulling this out, this is h bar times m plus 1 times L plus. 00:13:11.800 --> 00:13:16.810 h bar m plus 1 L plus Ylm. 00:13:16.810 --> 00:13:21.390 So L plus has raised the eigenvalue m by one. 00:13:21.390 --> 00:13:24.270 This state, what we get by acting on Ylm with the raising 00:13:24.270 --> 00:13:26.987 operator is a thing with m greater by one. 00:13:26.987 --> 00:13:28.820 And that came from the commutation relation. 00:13:28.820 --> 00:13:30.530 And if we had done the same thing with minus, 00:13:30.530 --> 00:13:33.060 if you go through the minus signs, it just gives us this. 00:13:36.660 --> 00:13:39.460 What does that tell us? 00:13:39.460 --> 00:13:41.460 What this tells us is we get a ladder of states. 00:13:44.773 --> 00:13:46.190 Let's look at what they look like. 00:13:46.190 --> 00:13:50.110 Each ladder, for a given value of L, if you raise with L plus 00:13:50.110 --> 00:13:52.180 and lower with L minus, you don't change L 00:13:52.180 --> 00:13:53.670 but you do change m. 00:13:53.670 --> 00:13:56.890 So we get ladders that are labeled by L. So for example, 00:13:56.890 --> 00:13:59.910 if I have some value L1, this is going to give me 00:13:59.910 --> 00:14:03.330 some state labeled by m. 00:14:03.330 --> 00:14:04.790 Let me put the m to the side. 00:14:04.790 --> 00:14:08.590 I can raise it to get m plus 1 by L plus. 00:14:08.590 --> 00:14:13.930 And I can lower it to get m minus 1 by L minus. 00:14:13.930 --> 00:14:15.855 So I got a tower. 00:14:18.500 --> 00:14:21.240 And if we have another value, a different value of L, 00:14:21.240 --> 00:14:25.370 I'll call it L2, we got another tower. 00:14:25.370 --> 00:14:28.050 You get m, m plus 1, m plus 2, dot, dot, dot, minus 00:14:28.050 --> 00:14:29.890 1, dot, dot, dot. 00:14:29.890 --> 00:14:34.270 So we have separated towers with different values of L squared. 00:14:34.270 --> 00:14:37.310 And within each tower we can raise and lower by L plus, 00:14:37.310 --> 00:14:38.110 skipping by one. 00:14:41.840 --> 00:14:43.830 OK, questions? 00:14:43.830 --> 00:14:47.260 So that was basically the end of the last lecture, 00:14:47.260 --> 00:14:48.710 said slightly differently. 00:14:52.500 --> 00:14:54.767 Now here's the question. 00:14:54.767 --> 00:14:56.850 So this is the question that a student asked right 00:14:56.850 --> 00:14:59.102 at the beginning. 00:14:59.102 --> 00:15:00.060 Is this tower infinite? 00:15:05.225 --> 00:15:05.850 Or does it end? 00:15:10.890 --> 00:15:12.140 So I pose to you the question. 00:15:12.140 --> 00:15:15.410 Is the tower infinite, or does it end? 00:15:15.410 --> 00:15:16.010 And why? 00:15:19.209 --> 00:15:21.040 AUDIENCE: [INAUDIBLE] direction. 00:15:21.040 --> 00:15:23.498 PROFESSOR: OK, so it's tempting to say it's infinite in one 00:15:23.498 --> 00:15:25.290 direction, because--? 00:15:25.290 --> 00:15:28.020 AUDIENCE: There are no bounds to the angular momentum. 00:15:28.020 --> 00:15:29.520 PROFESSOR: OK, so it's because there 00:15:29.520 --> 00:15:31.800 are no bounds to the angular momentum one can have. 00:15:31.800 --> 00:15:32.467 That's tempting. 00:15:34.950 --> 00:15:37.557 AUDIENCE: But at the same time, when 00:15:37.557 --> 00:15:40.735 you act a raising or lowering operator on L, 00:15:40.735 --> 00:15:44.550 the eigenvalue of L squared remains the same. 00:15:44.550 --> 00:15:48.327 So then you can't raise the z [INAUDIBLE] 00:15:48.327 --> 00:15:50.410 of the angular momentum above the actual momentum. 00:15:50.410 --> 00:15:50.950 PROFESSOR: Thank you. 00:15:50.950 --> 00:15:51.520 Exactly. 00:15:51.520 --> 00:15:52.320 So here's the statement. 00:15:52.320 --> 00:15:53.153 Let me restate that. 00:15:53.153 --> 00:15:54.380 That's exactly right. 00:15:54.380 --> 00:15:57.450 So look, L squared is the eigenvalue of the total angular 00:15:57.450 --> 00:15:57.950 momentum. 00:15:57.950 --> 00:16:00.870 Roughly speaking, it's giving you precisely in the state Ylm 00:16:00.870 --> 00:16:03.730 it tells you the expected value of the total angular momentum, 00:16:03.730 --> 00:16:04.900 L squared. 00:16:04.900 --> 00:16:06.510 That's some number. 00:16:06.510 --> 00:16:08.890 Now, if you act with L plus you keep increasing 00:16:08.890 --> 00:16:10.525 the expected value of Lz. 00:16:10.525 --> 00:16:12.692 But if you keep increasing it and keep increasing it 00:16:12.692 --> 00:16:16.010 and keep increasing it, Lz will eventually 00:16:16.010 --> 00:16:19.505 get much larger than the square root of L squared. 00:16:19.505 --> 00:16:21.990 That probably isn't true. 00:16:21.990 --> 00:16:23.502 That sounds wrong. 00:16:23.502 --> 00:16:24.460 That was the statement. 00:16:24.460 --> 00:16:24.960 Excellent. 00:16:24.960 --> 00:16:25.720 Exactly right. 00:16:25.720 --> 00:16:27.400 So let's make that precise. 00:16:27.400 --> 00:16:28.930 So is the tower infinite? 00:16:28.930 --> 00:16:29.540 No. 00:16:29.540 --> 00:16:31.810 It's probably not, for precisely that reason. 00:16:31.810 --> 00:16:32.935 So let's make that precise. 00:16:32.935 --> 00:16:34.560 So here's the way we're going to do it. 00:16:34.560 --> 00:16:35.980 This is a useful trick in general. 00:16:35.980 --> 00:16:37.940 This will outlive angular momentum 00:16:37.940 --> 00:16:41.400 and be a useful trick throughout quantum mechanics for you. 00:16:41.400 --> 00:16:43.090 I used it in a paper once. 00:16:43.090 --> 00:16:45.630 So here's the nice observation. 00:16:45.630 --> 00:16:47.365 Suppose it's true that the tower ends. 00:16:47.365 --> 00:16:49.365 Just like for the raising and lowering operators 00:16:49.365 --> 00:16:51.852 for the harmonic oscillator in one dimension, 00:16:51.852 --> 00:16:53.560 that tells us that in order for the power 00:16:53.560 --> 00:16:57.210 to end that state must be 0 once we raise it. 00:16:57.210 --> 00:17:02.360 The last state, so Yl, and I'll call this m plus, must be 0. 00:17:02.360 --> 00:17:03.740 There must be a max. 00:17:03.740 --> 00:17:06.150 Oh, sorry. 00:17:06.150 --> 00:17:08.480 Let me actually, before I walk through exactly 00:17:08.480 --> 00:17:09.859 this statement-- 00:17:09.859 --> 00:17:11.859 So let me make, first, this, the no, 00:17:11.859 --> 00:17:13.560 slightly more obvious and precise. 00:17:13.560 --> 00:17:16.420 So let's turn that argument into a precise statement. 00:17:16.420 --> 00:17:18.420 L squared is equal to, just from the definition, 00:17:18.420 --> 00:17:22.780 Lx squared plus Ly squared plus Lz squared. 00:17:22.780 --> 00:17:24.630 Now let's take the expectation value 00:17:24.630 --> 00:17:29.940 in this state Ylm of both sides of this equation. 00:17:29.940 --> 00:17:34.130 So on the left-hand side we get h bar squared ll plus 1. 00:17:34.130 --> 00:17:36.770 And on the right-hand side we get the expectation value of Lx 00:17:36.770 --> 00:17:40.470 squared plus the expectation value of Ly squared 00:17:40.470 --> 00:17:42.653 plus the expectation value of Lz squared. 00:17:42.653 --> 00:17:44.820 But we know that the expectation value of Lz squared 00:17:44.820 --> 00:17:46.520 is h bar squared, m squared. 00:17:46.520 --> 00:17:49.850 But the expectation of Lx squared and Ly squared 00:17:49.850 --> 00:17:52.070 are strictly positive. 00:17:52.070 --> 00:17:54.410 Because this can be written as a sum 00:17:54.410 --> 00:17:57.932 over all possible eigenvalues of Lx squared, 00:17:57.932 --> 00:17:59.890 which is the square of the possible eigenvalues 00:17:59.890 --> 00:18:03.170 of Lx times the probability distribution. 00:18:03.170 --> 00:18:05.570 That's a sum of positive, strictly positive [INAUDIBLE].. 00:18:05.570 --> 00:18:07.830 These are positive [INAUDIBLE]. 00:18:07.830 --> 00:18:10.580 So ll plus 1 is equal to positive 00:18:10.580 --> 00:18:12.700 plus positive plus h bar squared m squared. 00:18:12.700 --> 00:18:14.283 In particular that tells you that it's 00:18:14.283 --> 00:18:18.690 greater than or equal to h bar squared m squared. 00:18:18.690 --> 00:18:20.620 Maybe these are 0. 00:18:20.620 --> 00:18:22.350 So the least it can be is-- 00:18:22.350 --> 00:18:24.270 so the most m squared can possibly 00:18:24.270 --> 00:18:26.340 be is the square root of L plus 1. 00:18:26.340 --> 00:18:28.310 Or the most m can be. 00:18:28.310 --> 00:18:30.120 So m is bounded. 00:18:30.120 --> 00:18:33.800 There must be a maximum m, and there must be a minimum m. 00:18:33.800 --> 00:18:34.800 Because this is squared. 00:18:34.800 --> 00:18:36.160 The sign doesn't matter. 00:18:36.160 --> 00:18:37.160 Everyone cool with that? 00:18:39.920 --> 00:18:42.770 OK, so let's turn this into, now, a precise argument. 00:18:42.770 --> 00:18:44.710 What are the values of m plus and m minus? 00:18:44.710 --> 00:18:47.770 What is the top of the value of each tower? 00:18:47.770 --> 00:18:50.388 Probably it's going to depend on total L, right? 00:18:50.388 --> 00:18:52.930 So it's going to depend on each value of L. Let's check that. 00:18:52.930 --> 00:18:54.030 So here's the nice trick. 00:18:54.030 --> 00:18:57.300 Suppose we really do have a maximum m plus. 00:18:57.300 --> 00:19:01.190 That means that if I try to raise the state Ylm plus, 00:19:01.190 --> 00:19:05.420 I should get the state 0, which ends the tower. 00:19:05.420 --> 00:19:07.120 So suppose this is true. 00:19:07.120 --> 00:19:10.210 In that case, in particular here's the nice trick, 00:19:10.210 --> 00:19:12.830 L plus Ylm, if we take its magnitude, 00:19:12.830 --> 00:19:15.450 if we take the magnitude of this state, the state is 0. 00:19:15.450 --> 00:19:16.450 It's the zero function. 00:19:16.450 --> 00:19:18.690 So what's its magnitude? 00:19:18.690 --> 00:19:19.360 Zero. 00:19:19.360 --> 00:19:22.370 You might not think that's all that impressive an observation. 00:19:22.370 --> 00:19:24.660 OK, but note what this is. 00:19:24.660 --> 00:19:26.125 We know how to work with this. 00:19:26.125 --> 00:19:27.625 And in particular this is equal to-- 00:19:27.625 --> 00:19:30.860 I should have done this, dot, dot, dot-- 00:19:30.860 --> 00:19:33.140 I'm now going to use the Hermitian adjoint 00:19:33.140 --> 00:19:35.110 and pull this over to the right-hand side. 00:19:35.110 --> 00:19:36.970 The adjoint of L plus is L minus. 00:19:36.970 --> 00:19:43.330 So this gives us Ylm, L minus, L plus Ylm. 00:19:43.330 --> 00:19:45.517 This also doesn't look like much of an improvement, 00:19:45.517 --> 00:19:47.600 until you notice from the definition of L plus and 00:19:47.600 --> 00:19:50.140 L minus that what's L minus L plus? 00:19:50.140 --> 00:19:53.870 Well, L minus L plus, we're going to get an Lx squared. 00:19:53.870 --> 00:20:01.470 So Ylm, we get an Lx squared plus an Ly squared. 00:20:01.470 --> 00:20:04.890 I'm going to get my sign right, if I'm not careful. 00:20:04.890 --> 00:20:06.560 Plus-- well, let's just do it. 00:20:06.560 --> 00:20:08.240 So we have L minus L plus. 00:20:08.240 --> 00:20:12.770 So we're going to get a Lx iLy minus i, iLyLx. 00:20:12.770 --> 00:20:15.410 So plus i commutator of Lx with Ly. 00:20:18.930 --> 00:20:19.980 Good. 00:20:19.980 --> 00:20:23.050 So that's progress, Ylm. 00:20:23.050 --> 00:20:25.820 But it's still not progress, because the natural operators 00:20:25.820 --> 00:20:28.900 with which to act on the Ylm's are L squared and Lz. 00:20:28.900 --> 00:20:31.350 So can we put this in the form L squared and Lz? 00:20:31.350 --> 00:20:31.850 Sure. 00:20:31.850 --> 00:20:35.970 This is L squared minus Lz squared. 00:20:35.970 --> 00:20:41.790 Ylm L squared minus Lz squared. 00:20:41.790 --> 00:20:47.430 And this is i h bar Lz times i is minus h bar Lz. 00:20:47.430 --> 00:20:51.270 So minus h bar Lz, Ylm. 00:20:53.920 --> 00:20:56.750 And this must be equal to 0. 00:20:56.750 --> 00:20:58.510 But this is equal to-- 00:20:58.510 --> 00:21:04.160 from L squared we get h bar squared, ll plus 1. 00:21:04.160 --> 00:21:08.100 From the Lz squared we get minus m squared h bar squared. 00:21:08.100 --> 00:21:10.530 And from here we get minus h bar, h bar m, 00:21:10.530 --> 00:21:13.395 so minus m each bar squared. 00:21:13.395 --> 00:21:14.770 Notice that the units worked out. 00:21:14.770 --> 00:21:17.970 And all of this was multiplying Ylm, Ylm. 00:21:17.970 --> 00:21:20.850 But Ylm, Ylm, if it's a properly normalized eigenstate, which 00:21:20.850 --> 00:21:24.310 is what we were assuming at the beginning, is just 1. 00:21:24.310 --> 00:21:27.310 So we get this times 1 is 0. 00:21:27.310 --> 00:21:28.080 Aha. 00:21:28.080 --> 00:21:29.730 And notice that in all of this, this was m plus. 00:21:29.730 --> 00:21:31.313 We were assuming, we were working here 00:21:31.313 --> 00:21:35.740 with the assumption that L plus annihilated this top state. 00:21:35.740 --> 00:21:38.488 And so grouping this together, this says therefore h bar 00:21:38.488 --> 00:21:40.780 squared times-- pulling out a common h bar squared from 00:21:40.780 --> 00:21:41.620 of all this-- 00:21:41.620 --> 00:21:52.740 times l, l plus 1 minus m plus, m plus, plus 1 is equal to 0. 00:21:52.740 --> 00:21:57.120 And this tells us that m plus is equal to l. 00:21:57.120 --> 00:22:01.870 And if you want to be strict, put a plus sign. 00:22:01.870 --> 00:22:03.160 Everyone cool with that? 00:22:03.160 --> 00:22:04.480 This is a very useful trick. 00:22:04.480 --> 00:22:06.130 If you know something is 0 as function, 00:22:06.130 --> 00:22:07.220 you know its norm is 0. 00:22:07.220 --> 00:22:09.500 And now you can use things like Hermitian adjoints. 00:22:09.500 --> 00:22:10.300 Very, very useful. 00:22:13.400 --> 00:22:14.925 OK questions about that? 00:22:14.925 --> 00:22:16.410 Yeah. 00:22:16.410 --> 00:22:20.755 AUDIENCE: [INAUDIBLE] 00:22:20.755 --> 00:22:21.630 PROFESSOR: Excellent. 00:22:21.630 --> 00:22:23.463 Where this came from is I was just literally 00:22:23.463 --> 00:22:26.600 taking L plus and L minus, taking the definitions, 00:22:26.600 --> 00:22:27.680 and plugging them in. 00:22:27.680 --> 00:22:30.320 So if I have L minus L plus, L minus 00:22:30.320 --> 00:22:31.820 is, just to write it out explicitly, 00:22:31.820 --> 00:22:33.090 L minus is equal to Lx. 00:22:33.090 --> 00:22:34.990 Because the [INAUDIBLE] are observables, 00:22:34.990 --> 00:22:37.115 and so they're Hermitians, so they're self-adjoint, 00:22:37.115 --> 00:22:38.470 the minus iLy. 00:22:38.470 --> 00:22:41.270 So L minus L plus gives me an LxLx. 00:22:41.270 --> 00:22:43.020 It gives me an LxiLy. 00:22:43.020 --> 00:22:45.020 It gives you a minus iLyLx. 00:22:45.020 --> 00:22:49.370 And it gives me a minus ii, which is plus 1 Ly squared. 00:22:49.370 --> 00:22:50.140 Cool? 00:22:50.140 --> 00:22:51.400 Excellent. 00:22:51.400 --> 00:22:53.650 I'm in a state of serious desperation here. 00:22:58.720 --> 00:22:59.290 Good. 00:22:59.290 --> 00:23:00.468 Other questions? 00:23:00.468 --> 00:23:01.782 Yeah. 00:23:01.782 --> 00:23:04.640 AUDIENCE: [INAUDIBLE] How do you get from that to that? 00:23:04.640 --> 00:23:05.640 PROFESSOR: This to here? 00:23:05.640 --> 00:23:07.010 OK, good. 00:23:07.010 --> 00:23:08.463 Sorry, I jumped a step here. 00:23:08.463 --> 00:23:09.880 So here what I said is, look, I've 00:23:09.880 --> 00:23:12.210 got this nice set of operators acting on Ylm. 00:23:12.210 --> 00:23:15.240 I know how each of these operators acts on Ylm. 00:23:15.240 --> 00:23:19.060 Lz gives me an h bar m, minus some h bar squared. 00:23:19.060 --> 00:23:20.360 Lz squared, L squared. 00:23:20.360 --> 00:23:23.290 And then overall that was a just some number times Ylm, which 00:23:23.290 --> 00:23:24.950 I can pull out of the inner product. 00:23:24.950 --> 00:23:25.450 Cool? 00:23:25.450 --> 00:23:26.498 Yeah? 00:23:26.498 --> 00:23:28.513 AUDIENCE: [INAUDIBLE] 00:23:28.513 --> 00:23:29.180 PROFESSOR: Good. 00:23:29.180 --> 00:23:30.750 This subscript plus meant that, look, 00:23:30.750 --> 00:23:32.250 there was a maximum value of m. 00:23:32.250 --> 00:23:34.990 So m squared had to be less than or equal to this. 00:23:34.990 --> 00:23:37.630 There's a maximum value and a minimum value. 00:23:37.630 --> 00:23:38.630 Maybe they're different. 00:23:38.630 --> 00:23:39.172 I don't know. 00:23:39.172 --> 00:23:42.470 I'm just going to be open-minded about that. 00:23:42.470 --> 00:23:42.970 Others? 00:23:42.970 --> 00:23:43.850 Yeah? 00:23:43.850 --> 00:23:45.975 AUDIENCE: Will that mean, then, that you can't ever 00:23:45.975 --> 00:23:48.758 have all the angular momentum [INAUDIBLE]?? 00:23:48.758 --> 00:23:50.300 PROFESSOR: Yeah, awesome observation. 00:23:50.300 --> 00:23:51.490 Exactly. 00:23:51.490 --> 00:23:53.282 We were going to get there in a little bit. 00:23:53.282 --> 00:23:53.782 I like that. 00:23:53.782 --> 00:23:54.700 That's exactly right. 00:23:57.920 --> 00:24:00.100 Let me go through a couple more steps, and then I'll 00:24:00.100 --> 00:24:01.400 come back to that observation. 00:24:01.400 --> 00:24:02.655 And I promise I will say so. 00:24:05.200 --> 00:24:07.560 So if we go through exactly a similar argument, 00:24:07.560 --> 00:24:10.710 let's see what happens if we did L minus on Ylm minus, 00:24:10.710 --> 00:24:12.330 just to walk through the logic. 00:24:12.330 --> 00:24:15.660 So if this were-- you lower the lowest one, you get 0. 00:24:15.660 --> 00:24:18.170 Then we'd get the same story, L minus L minus. 00:24:18.170 --> 00:24:21.800 When we take the adjoint we'd get L plus, L minus. 00:24:21.800 --> 00:24:23.200 And L plus L minus, what changes? 00:24:23.200 --> 00:24:24.742 The only thing that's going to change 00:24:24.742 --> 00:24:27.790 is that you get this commutator the other direction. 00:24:27.790 --> 00:24:29.720 And there are minus signs in various places. 00:24:29.720 --> 00:24:33.470 The upshot of which is that m minus is equal to minus l. 00:24:35.990 --> 00:24:37.240 So this is quite parsimonious. 00:24:37.240 --> 00:24:38.440 It's symmetric. 00:24:38.440 --> 00:24:40.118 If you take z to minus z, if you switch 00:24:40.118 --> 00:24:42.660 the sign of the angular momentum you get the same thing back. 00:24:42.660 --> 00:24:43.940 That's satisfying, perhaps. 00:24:48.580 --> 00:24:50.790 But it's way more than that. 00:24:50.790 --> 00:24:53.460 This tells us a lot about the possible eigenvalues, 00:24:53.460 --> 00:24:55.330 in the following way. 00:24:55.330 --> 00:24:56.980 Look back at our towers. 00:24:56.980 --> 00:25:01.242 In our towers we have that the angular momentum is 00:25:01.242 --> 00:25:02.450 raised and lowered by L plus. 00:25:02.450 --> 00:25:05.450 So the Lz angular momentum is raised and lowered by L plus. 00:25:05.450 --> 00:25:06.970 l remains the same. 00:25:06.970 --> 00:25:10.230 But there's a maximum state now, which is plus l. 00:25:10.230 --> 00:25:12.450 And there's a minimum stay in here, l1. 00:25:12.450 --> 00:25:14.527 And here there's a minimum state, minus l1. 00:25:14.527 --> 00:25:16.110 Similarly for the other tower, there's 00:25:16.110 --> 00:25:20.000 a minimum state minus l2 and a maximum state, l2. 00:25:20.000 --> 00:25:23.127 So how big each tower is depends on the total angular momentum. 00:25:23.127 --> 00:25:24.460 That kind of makes sense, right? 00:25:24.460 --> 00:25:26.150 If you've got more angular momentum 00:25:26.150 --> 00:25:28.080 and you can only step Lz by one, you've 00:25:28.080 --> 00:25:30.535 got more room to move with a large value of l 00:25:30.535 --> 00:25:32.500 than with a small value of l. 00:25:32.500 --> 00:25:33.000 OK. 00:25:35.600 --> 00:25:38.370 So what does that tell us about the values of m? 00:25:38.370 --> 00:25:47.530 Notice that m spans the values from its minimal value, 00:25:47.530 --> 00:25:53.220 minus l to l in integer steps, in unit steps. 00:25:58.140 --> 00:26:04.895 So if you think about m is l, then m is l minus 1. 00:26:04.895 --> 00:26:09.490 And if I keep lowering I get down to m is minus l plus 1. 00:26:09.490 --> 00:26:11.976 And then m is equal to minus l. 00:26:15.240 --> 00:26:18.435 So the difference in the Lz eigenvalue between these guys, 00:26:18.435 --> 00:26:19.310 the difference is 2l. 00:26:23.070 --> 00:26:24.930 But the number of unit steps in here 00:26:24.930 --> 00:26:30.230 is one fewer, because one, dot, dot, dot 2l minus 1. 00:26:30.230 --> 00:26:33.090 So this is some integer, which is 00:26:33.090 --> 00:26:35.330 the number of states minus 1. 00:26:35.330 --> 00:26:40.940 So if these are n states, and I'll call this N sub l states, 00:26:40.940 --> 00:26:44.570 then this difference twice l is N sub l minus 1, 00:26:44.570 --> 00:26:45.590 because it's unit steps. 00:26:45.590 --> 00:26:47.440 Cool? 00:26:47.440 --> 00:26:51.690 So for example, if there were two states, so N is 2, 00:26:51.690 --> 00:26:54.530 2l is equal to-- 00:26:54.530 --> 00:26:58.575 well, that's 1, which is 2 minus 1, 00:26:58.575 --> 00:27:00.492 the number of states minus 1, also known as 1. 00:27:00.492 --> 00:27:02.700 So l is 1/2. 00:27:02.700 --> 00:27:04.420 And more generally we find that l 00:27:04.420 --> 00:27:06.107 must be, in order for this process 00:27:06.107 --> 00:27:07.690 to make sense, in order for the m plus 00:27:07.690 --> 00:27:11.380 and the m minus to match up, we need 00:27:11.380 --> 00:27:17.860 that l is of the form an integer N sub l minus 1 upon 2. 00:27:17.860 --> 00:27:20.670 So this tells us that where Nl is an integer-- 00:27:20.670 --> 00:27:23.780 Nl is just the number of states in this tower, 00:27:23.780 --> 00:27:25.580 and it's a strictly positive integer. 00:27:25.580 --> 00:27:27.122 If you have zero states in the tower, 00:27:27.122 --> 00:27:29.440 then that's not very interesting. 00:27:29.440 --> 00:27:40.500 So this tells us that l is an integer a half integer, 00:27:40.500 --> 00:27:42.810 but nothing else. 00:27:42.810 --> 00:27:43.310 Cool? 00:27:49.106 --> 00:27:52.940 In particular, if it's a half integer, 00:27:52.940 --> 00:27:59.837 that means 1/2, 3/2, 5/2, if it's an integer or it can be 0. 00:27:59.837 --> 00:28:01.420 There's nothing wrong with it being 0. 00:28:01.420 --> 00:28:03.376 But then 1 on. 00:28:07.750 --> 00:28:12.010 So that means we can plot our system in the following way. 00:28:12.010 --> 00:28:22.320 If we have l equals 0, how many states do we have? 00:28:22.320 --> 00:28:24.690 If little l is 0, how many states are? 00:28:24.690 --> 00:28:28.990 What's the largest value of Lz? 00:28:28.990 --> 00:28:32.230 What's the largest allowed value of Lz or of m 00:28:32.230 --> 00:28:34.160 if little l is equal to 0? 00:28:34.160 --> 00:28:34.660 AUDIENCE: 0. 00:28:34.660 --> 00:28:35.202 PROFESSOR: 0. 00:28:35.202 --> 00:28:37.350 Because it goes from plus 0 to minus 0. 00:28:37.350 --> 00:28:38.760 So that's pretty much 0. 00:28:38.760 --> 00:28:43.930 So we have a single state with m equals 0 and l equals 0. 00:28:43.930 --> 00:28:46.470 If l is equal to-- what's the next possible value?-- 00:28:46.470 --> 00:28:52.130 1/2, then m can be either 1/2 or we lower it 00:28:52.130 --> 00:28:53.740 by 1, which is minus 1/2. 00:28:53.740 --> 00:28:57.460 So there are two states, m equals 00:28:57.460 --> 00:28:59.400 1/2 and m equals minus 1/2. 00:29:02.030 --> 00:29:03.040 So there's one power. 00:29:03.040 --> 00:29:04.040 It's a very short tower. 00:29:04.040 --> 00:29:07.400 This is the shortest possible tower. 00:29:07.400 --> 00:29:11.670 Then we also have the state l equals 1, which has m equals 0. 00:29:11.670 --> 00:29:15.050 It has m equals 1. 00:29:15.050 --> 00:29:18.120 And it has m equals minus 1. 00:29:18.120 --> 00:29:20.900 And that's it. 00:29:20.900 --> 00:29:22.030 And so on and so forth. 00:29:25.570 --> 00:29:33.190 Four states for l equals 3/2, with states 3/2, 1/2, this is m 00:29:33.190 --> 00:29:38.710 equals 1/2, m equals minus 1/2, and minus 3/2. 00:29:38.710 --> 00:29:41.390 The values of m span from minus l to l with integer steps. 00:29:46.480 --> 00:29:51.260 And this is possible for all values of l which are 00:29:51.260 --> 00:30:02.920 of the form all integer or half integer l's. 00:30:02.920 --> 00:30:05.040 So for every different value of the total angular 00:30:05.040 --> 00:30:06.390 momentum, the total amount of angular momentum 00:30:06.390 --> 00:30:08.300 you have, you have a tower of states 00:30:08.300 --> 00:30:10.660 labeled by Lz's eigenvalue in this fashion. 00:30:13.960 --> 00:30:14.460 Questions? 00:30:17.508 --> 00:30:21.510 Does anyone notice anything troubling about these? 00:30:21.510 --> 00:30:23.458 Something physically a little discomfiting 00:30:23.458 --> 00:30:24.250 about any of these? 00:30:27.262 --> 00:30:29.470 What does it mean to say I'm in the state l equals 0, 00:30:29.470 --> 00:30:29.970 m equals 0? 00:30:29.970 --> 00:30:33.050 What is that telling you? 00:30:33.050 --> 00:30:34.060 Zero angular momentum. 00:30:34.060 --> 00:30:36.540 What's the expectation value of l squared? 00:30:36.540 --> 00:30:37.040 Zero. 00:30:37.040 --> 00:30:39.400 The expectation value of Lz? 00:30:39.400 --> 00:30:44.050 And by rotational invariance, the expected value of Lx or Ly? 00:30:44.050 --> 00:30:45.500 That thing is not rotating. 00:30:45.500 --> 00:30:47.580 There's no angular momentum. 00:30:47.580 --> 00:30:49.990 So the no angular momentum state is one with l equals 0, 00:30:49.990 --> 00:30:50.490 m equals 0. 00:30:50.490 --> 00:30:52.840 So this is not spinning. 00:30:52.840 --> 00:30:54.590 What about this guy? 00:30:54.590 --> 00:30:56.900 L equals 1, m equals 0. 00:30:56.900 --> 00:31:00.300 Does this thing carry angular momentum? 00:31:00.300 --> 00:31:01.965 Yeah, absolutely. 00:31:01.965 --> 00:31:03.840 So it just doesn't carry any angular momentum 00:31:03.840 --> 00:31:04.680 in the z direction. 00:31:04.680 --> 00:31:06.660 But its total angular momentum on average, 00:31:06.660 --> 00:31:09.180 its expected total angular momentum 00:31:09.180 --> 00:31:13.940 is ll plus 1, which is 2, times h bar squared. 00:31:13.940 --> 00:31:18.295 So if you measure Lx and Ly what do you expect to get? 00:31:18.295 --> 00:31:20.170 Well, if you measure Lx squared or Ly squared 00:31:20.170 --> 00:31:22.087 you probably expect to get something non-zero. 00:31:22.087 --> 00:31:24.260 We'll come back to that in just a second. 00:31:24.260 --> 00:31:28.073 And what about the state l equals 1, m equals 1? 00:31:28.073 --> 00:31:29.490 Your angular momentum-- you've got 00:31:29.490 --> 00:31:30.850 as much angular momentum in the z direction as you 00:31:30.850 --> 00:31:31.970 possibly can. 00:31:31.970 --> 00:31:33.890 So that definitely carries angular momentum. 00:31:33.890 --> 00:31:35.580 So there's a state that has no angular 00:31:35.580 --> 00:31:36.860 momentum in the z direction. 00:31:36.860 --> 00:31:38.277 And there's a state that has some, 00:31:38.277 --> 00:31:40.210 and there's a state that has less. 00:31:40.210 --> 00:31:41.190 That make sense. 00:31:41.190 --> 00:31:42.628 Yeah? 00:31:42.628 --> 00:31:45.048 AUDIENCE: If m equals 1 and l equals 1, 00:31:45.048 --> 00:31:47.952 that means that the angular momentum for the z direction 00:31:47.952 --> 00:31:50.173 is the total angular momentum? 00:31:50.173 --> 00:31:50.881 PROFESSOR: Is it? 00:31:50.881 --> 00:31:54.180 AUDIENCE: And then Lx and Ly was 0? 00:31:54.180 --> 00:31:56.998 PROFESSOR: OK, that's an excellent question. 00:31:56.998 --> 00:31:58.540 Let me answer that now, and then I'll 00:31:58.540 --> 00:32:00.470 come back to the point I wanted to make. 00:32:00.470 --> 00:32:00.730 Hold on a second. 00:32:00.730 --> 00:32:02.105 Let me just answer that question. 00:32:05.610 --> 00:32:06.270 I'll work here. 00:32:11.390 --> 00:32:14.120 So here's the crucial thing. 00:32:14.120 --> 00:32:16.860 Even in this state, so you were asking about the state l 00:32:16.860 --> 00:32:18.110 equals 1, m equals 1. 00:32:18.110 --> 00:32:21.170 And the question that was asked, a very good question, is look, 00:32:21.170 --> 00:32:23.930 does that mean all the angular momentum is in the z direction, 00:32:23.930 --> 00:32:26.247 and Lx and Ly are 0? 00:32:26.247 --> 00:32:27.830 But let me just ask this more broadly. 00:32:27.830 --> 00:32:29.830 Suppose we have a state with angular momentum l, 00:32:29.830 --> 00:32:31.840 and m is equal to l. 00:32:31.840 --> 00:32:33.590 The most angular momentum you can possibly 00:32:33.590 --> 00:32:35.215 have in the z direction, same question. 00:32:35.215 --> 00:32:38.025 Is all the angular momentum in the z direction? 00:32:38.025 --> 00:32:39.650 And what I want to emphasize you is no, 00:32:39.650 --> 00:32:41.040 that's absolutely not the case. 00:32:41.040 --> 00:32:42.730 So two arguments for that. 00:32:42.730 --> 00:32:45.950 The first is, suppose it's true that Lx and Ly are identically 00:32:45.950 --> 00:32:48.512 0. 00:32:48.512 --> 00:32:50.220 Can that satisfy the uncertainty relation 00:32:50.220 --> 00:32:51.430 due to those commutators? 00:32:51.430 --> 00:32:51.930 No. 00:32:51.930 --> 00:32:53.670 There must be uncertainty in Lx and Ly, 00:32:53.670 --> 00:32:57.540 because Lz has a non-zero expectation value. 00:32:57.540 --> 00:32:59.873 So it can't be that Lx squared and Ly squared 00:32:59.873 --> 00:33:01.040 have zero expectation value. 00:33:01.040 --> 00:33:03.100 But let's be more precise about this. 00:33:03.100 --> 00:33:05.440 The expectation value of L squared is easy to calculate. 00:33:05.440 --> 00:33:09.050 It's h bar squared l, plus 1. 00:33:09.050 --> 00:33:10.570 Because we're in the state Ylm. 00:33:10.570 --> 00:33:15.660 This is the state Y l sub l, or ll. 00:33:15.660 --> 00:33:21.490 The expectation of Lz squared is equal to h bar squared, 00:33:21.490 --> 00:33:22.492 m squared. 00:33:22.492 --> 00:33:23.950 And m squared now, m is equal to l. 00:33:23.950 --> 00:33:25.880 So h bar squared, l squared. 00:33:25.880 --> 00:33:26.425 Aha. 00:33:26.425 --> 00:33:27.800 So the expected value of l square 00:33:27.800 --> 00:33:29.425 is not the same as Lx squared, but this 00:33:29.425 --> 00:33:33.510 is equal to the expected value of Lx squared plus Ly squared 00:33:33.510 --> 00:33:34.340 plus Lz squared. 00:33:41.790 --> 00:33:44.260 Therefore the expected value of Lx 00:33:44.260 --> 00:33:49.160 squared plus the expected value of Ly squared 00:33:49.160 --> 00:33:51.498 is equal to the difference between this and this. 00:33:51.498 --> 00:33:54.040 We just subtract this off h bar squared ll plus 1 minus h bar 00:33:54.040 --> 00:33:57.000 squared, l squared, h bar squared l. 00:33:57.000 --> 00:33:58.900 And by symmetry you don't expect the symmetry 00:33:58.900 --> 00:34:00.932 to be broken between Lx and Ly. 00:34:00.932 --> 00:34:02.390 You can actually do the calculation 00:34:02.390 --> 00:34:03.450 and not just be glib about it. 00:34:03.450 --> 00:34:05.408 But both arguments give you the correct answer. 00:34:05.408 --> 00:34:08.110 The expectation value of Lx squared 00:34:08.110 --> 00:34:11.960 is equal to 1/2 h bar squared l. 00:34:11.960 --> 00:34:16.690 And ditto for y, in this state. 00:34:16.690 --> 00:34:17.667 So notice two things. 00:34:17.667 --> 00:34:20.000 First off is we make the total angular momentum little l 00:34:20.000 --> 00:34:21.719 large. 00:34:21.719 --> 00:34:24.460 The amount by which we fail to have all the angular 00:34:24.460 --> 00:34:26.860 momentum in the z direction is getting larger and larger. 00:34:26.860 --> 00:34:30.410 We're increasing the crappiness of putting all the angular 00:34:30.410 --> 00:34:32.010 momentum in the z direction. 00:34:32.010 --> 00:34:37.080 However, as a ratio of the total angular momentum divided by L 00:34:37.080 --> 00:34:40.630 squared, so this divided by L squared, 00:34:40.630 --> 00:34:43.280 and this is h bar squared, l, l plus 1, 00:34:43.280 --> 00:34:45.690 and in particular this is l squared plus l, 00:34:45.690 --> 00:34:48.578 so if we took the ratio, the rational mismatch 00:34:48.578 --> 00:34:49.870 is getting smaller and smaller. 00:34:49.870 --> 00:34:50.449 And that's good. 00:34:50.449 --> 00:34:52.280 Because as we go to very large angular momentums 00:34:52.280 --> 00:34:53.719 where things should start getting classical 00:34:53.719 --> 00:34:56.250 in some sense, we should get back the familiar intuition 00:34:56.250 --> 00:34:59.290 that you can put all the angular momentum in the z direction. 00:34:59.290 --> 00:35:00.228 Yeah? 00:35:00.228 --> 00:35:02.520 AUDIENCE: Why are we imposing the [INAUDIBLE] condition 00:35:02.520 --> 00:35:05.440 that the expectation values of Lx and Ly should be identical? 00:35:05.440 --> 00:35:05.600 PROFESSOR: Excellent. 00:35:05.600 --> 00:35:07.030 That's why I was saying this is a glib argument. 00:35:07.030 --> 00:35:08.190 You don't have to impose-- 00:35:08.190 --> 00:35:10.130 that needs to be done a little bit more delicately. 00:35:10.130 --> 00:35:11.400 But we can just directly compute this. 00:35:11.400 --> 00:35:12.830 And you do so on your problem set. 00:35:15.610 --> 00:35:16.140 Yeah? 00:35:16.140 --> 00:35:18.250 AUDIENCE: Why doesn't the existence 00:35:18.250 --> 00:35:21.770 of the l equals 0 eigenfunction where the angular 00:35:21.770 --> 00:35:26.095 momentum the L squared is definitely 0 00:35:26.095 --> 00:35:27.870 violate the uncertainty principle? 00:35:27.870 --> 00:35:28.250 PROFESSOR: Awesome. 00:35:28.250 --> 00:35:30.250 On your problem set you're going to answer that, 00:35:30.250 --> 00:35:33.030 but let me give you a quick preview. 00:35:33.030 --> 00:35:34.520 This is such a great response. 00:35:34.520 --> 00:35:37.140 Just Let me give you a quick preview. 00:35:37.140 --> 00:35:38.420 So from that Lx-- 00:35:38.420 --> 00:35:40.090 OK, this is a really fun question. 00:35:40.090 --> 00:35:43.060 Let me go into it in some detail. 00:35:43.060 --> 00:35:44.610 Wow, I'm going fast today. 00:35:44.610 --> 00:35:46.320 Am I going way too fast? 00:35:46.320 --> 00:35:47.550 No? 00:35:47.550 --> 00:35:48.110 A little? 00:35:48.110 --> 00:35:49.210 Little too fast? 00:35:49.210 --> 00:35:50.960 OK, ask me more questions to slow me down. 00:35:50.960 --> 00:35:51.520 I'm excited. 00:35:51.520 --> 00:35:52.978 I didn't get much sleep last night. 00:35:57.090 --> 00:35:58.840 One of the great joys of being a physicist 00:35:58.840 --> 00:36:01.280 is working with other physicists. 00:36:01.280 --> 00:36:03.210 So yesterday one of my very good friends 00:36:03.210 --> 00:36:06.870 and a collaborator I really delight in talking with 00:36:06.870 --> 00:36:07.790 came to visit. 00:36:07.790 --> 00:36:09.210 And we had a late night dinner. 00:36:09.210 --> 00:36:11.835 And this led to me late at night, not doing my work, 00:36:11.835 --> 00:36:14.210 but reading papers about what our conversation was about. 00:36:14.210 --> 00:36:17.552 And only then at the very end, when I was just about to die 00:36:17.552 --> 00:36:19.760 did I write the response to the reviewer on the paper 00:36:19.760 --> 00:36:22.135 that I was supposed to be doing by last night's deadline. 00:36:22.135 --> 00:36:24.750 So I'm kind of tired, but I'm in a really good mood. 00:36:28.610 --> 00:36:29.610 It's a really good job. 00:36:34.580 --> 00:36:36.330 Now I've totally lost my train of thought. 00:36:36.330 --> 00:36:37.992 What was the question again? 00:36:37.992 --> 00:36:39.760 AUDIENCE: The existence of the l equals 00:36:39.760 --> 00:36:43.250 0 state where the total angular momentum is definitely-- 00:36:43.250 --> 00:36:45.000 PROFESSOR: Excellent, and the uncertainty. 00:36:45.000 --> 00:36:47.910 So the question is, why don't we violate the uncertainty when 00:36:47.910 --> 00:36:49.530 we know that L equals-- where am I; 00:36:49.530 --> 00:36:53.608 I just covered it-- when we know that L equals 0 and m equals 0, 00:36:53.608 --> 00:36:55.150 doesn't that destroy our uncertainty? 00:36:55.150 --> 00:36:57.233 Because we know that the angular momentum Lz is 0. 00:36:57.233 --> 00:36:58.442 Angular momentum for Lx is 0. 00:36:58.442 --> 00:36:59.665 Angular momentum for Ly is 0. 00:36:59.665 --> 00:37:00.902 All of them vanish. 00:37:00.902 --> 00:37:02.860 Doesn't that violate the uncertainty principle? 00:37:02.860 --> 00:37:04.980 So let's remind ourselves what the form of that uncertainty 00:37:04.980 --> 00:37:05.750 relation is. 00:37:05.750 --> 00:37:08.680 The form of the uncertainty relation following from Lx Ly 00:37:08.680 --> 00:37:10.200 is i h bar Lz. 00:37:10.200 --> 00:37:13.260 Recall the general statement. 00:37:13.260 --> 00:37:16.710 The uncertainty in A times the uncertainty in B, squared, 00:37:16.710 --> 00:37:18.700 squared, is equal to-- 00:37:18.700 --> 00:37:23.430 let me just write it as h bar upon 2-- 00:37:23.430 --> 00:37:25.290 sorry, 1/2. 00:37:25.290 --> 00:37:27.230 The absolute value of the expectation value 00:37:27.230 --> 00:37:33.550 of the commutator, A with B. Good lord. 00:37:33.550 --> 00:37:35.230 Dimensionally, does this work? 00:37:35.230 --> 00:37:36.090 Yes. 00:37:36.090 --> 00:37:36.590 OK, good. 00:37:36.590 --> 00:37:39.570 Because units of A, units of B. Unites of A, units of B. 00:37:39.570 --> 00:37:41.602 Triumph. 00:37:41.602 --> 00:37:43.810 And these are going to be quantum-mechanically small, 00:37:43.810 --> 00:37:45.660 because commutators have h bars. 00:37:45.660 --> 00:37:47.120 And commutators have h bars why? 00:37:49.850 --> 00:37:51.020 Not because God hates us. 00:37:51.020 --> 00:37:52.460 Why do commutators have h bars? 00:37:55.810 --> 00:37:58.950 What happens classically? 00:37:58.950 --> 00:38:01.670 In classical mechanics, do things commute? 00:38:01.670 --> 00:38:02.250 Yes. 00:38:02.250 --> 00:38:04.470 Why are the h bars and commutators 00:38:04.470 --> 00:38:06.020 physical observables? 00:38:06.020 --> 00:38:08.133 Because we exist. 00:38:08.133 --> 00:38:09.550 Because there's a classical limit. 00:38:09.550 --> 00:38:11.630 OK so this is going to make quantum-mechanically small, 00:38:11.630 --> 00:38:13.505 so we expect the uncertainty relation to also 00:38:13.505 --> 00:38:15.398 be quantum-mechanically small. 00:38:15.398 --> 00:38:16.440 Just important intuition. 00:38:16.440 --> 00:38:20.730 So let's look at the specific example of Lx, Ly, and Lz. 00:38:20.730 --> 00:38:23.290 So the uncertainty in Lx, in any particular-- 00:38:23.290 --> 00:38:24.550 remember that this is defined as the uncertainty 00:38:24.550 --> 00:38:26.770 in a particular state psi, in a particular state psi. 00:38:26.770 --> 00:38:29.145 And this expectation value is taken in a particular state 00:38:29.145 --> 00:38:32.010 psi, that same state. 00:38:32.010 --> 00:38:37.470 So the uncertainty of Lx, in some state Ylm, 00:38:37.470 --> 00:38:41.610 times the uncertainty of Ly in that same state Ylm 00:38:41.610 --> 00:38:47.171 should be greater than or equal to 1/2 00:38:47.171 --> 00:38:48.963 the absolute value of the expectation value 00:38:48.963 --> 00:38:50.330 of the commutator of Lx and Ly. 00:38:50.330 --> 00:38:53.390 But the commutator of Lx and Ly is i h bar Lz. 00:38:53.390 --> 00:38:56.140 And i, when we pull it through this absolute value, 00:38:56.140 --> 00:38:58.710 is going to give me just 1. h bar is going to give me h bar. 00:38:58.710 --> 00:39:04.330 So h bar upon 2, expectation value of Lz, absolute value. 00:39:04.330 --> 00:39:06.420 Yeah? 00:39:06.420 --> 00:39:08.480 Can Lx and Ly have zero uncertainty? 00:39:16.340 --> 00:39:18.450 When? 00:39:18.450 --> 00:39:21.197 Expectation value of Lz is 0. 00:39:21.197 --> 00:39:22.030 So that sounds good. 00:39:22.030 --> 00:39:27.080 It sounds like if Lz has expectation value of 0, 00:39:27.080 --> 00:39:29.590 then we can have Lx and Ly, definite. 00:39:29.590 --> 00:39:31.430 But that's bad. 00:39:31.430 --> 00:39:32.700 Really? 00:39:32.700 --> 00:39:34.460 Really, can we do that? 00:39:34.460 --> 00:39:35.790 Why not? 00:39:35.790 --> 00:39:37.690 AUDIENCE: [INAUDIBLE] 00:39:37.690 --> 00:39:40.065 [LAUGHTER] 00:39:43.532 --> 00:39:44.740 PROFESSOR: Bless you, my son. 00:39:51.060 --> 00:39:54.520 Can we have Lx and Ly both take definite values, just 00:39:54.520 --> 00:39:57.050 because Lz? 00:39:57.050 --> 00:39:59.210 Why? 00:39:59.210 --> 00:40:03.070 What else do we have to satisfy? 00:40:03.070 --> 00:40:05.750 What other uncertainty relations must we satisfy? 00:40:08.390 --> 00:40:10.820 There are two more. 00:40:10.820 --> 00:40:15.230 And I invite you to go look at what those two more are 00:40:15.230 --> 00:40:19.520 and deduce that this is only possible if Lx squared, 00:40:19.520 --> 00:40:23.110 Ly squared, and Lz squared all have zero expectation value. 00:40:23.110 --> 00:40:26.770 In fact, I think it's just a great question that I 00:40:26.770 --> 00:40:30.358 think it's on your problem set. 00:40:30.358 --> 00:40:31.650 So thank you for that question. 00:40:31.650 --> 00:40:32.460 It's a really good question. 00:40:32.460 --> 00:40:34.040 There was another question in here. 00:40:34.040 --> 00:40:35.160 Yeah? 00:40:35.160 --> 00:40:40.944 AUDIENCE: Something that you said earlier [INAUDIBLE] 00:40:40.944 --> 00:40:41.812 half integer. 00:40:41.812 --> 00:40:43.354 So were you deriving this by counting 00:40:43.354 --> 00:40:46.728 the number of equations and somehow asserting-- 00:40:46.728 --> 00:40:47.792 Why does--? 00:40:47.792 --> 00:40:48.500 PROFESSOR: Great. 00:40:48.500 --> 00:40:50.570 So the question is, wait, really? 00:40:50.570 --> 00:40:52.273 Why is L and integer a half integer. 00:40:52.273 --> 00:40:53.440 That was a little too quick. 00:40:53.440 --> 00:40:55.017 Is that roughly the right statement? 00:40:55.017 --> 00:40:56.503 AUDIENCE: [INAUDIBLE]. 00:40:56.503 --> 00:40:57.170 PROFESSOR: Good. 00:40:57.170 --> 00:40:57.660 Excellent. 00:40:57.660 --> 00:40:58.827 Let me go through the logic. 00:40:58.827 --> 00:41:01.590 So the logic goes like this. 00:41:01.590 --> 00:41:02.870 I know that the Ylm's-- 00:41:02.870 --> 00:41:06.640 if the Ylm's are eigenfunctions of L squared and Lz, 00:41:06.640 --> 00:41:09.178 and I've constructed this tower of them using the raising 00:41:09.178 --> 00:41:10.720 and lowering operators, we've already 00:41:10.720 --> 00:41:13.300 shown that the largest possible value of m is l 00:41:13.300 --> 00:41:15.470 and the least possible value is minus l. 00:41:15.470 --> 00:41:17.000 And these states must be separated 00:41:17.000 --> 00:41:19.370 by integer steps in m. 00:41:19.370 --> 00:41:19.870 OK, good. 00:41:19.870 --> 00:41:23.960 So pick a value of l, a particular tower. 00:41:23.960 --> 00:41:27.530 And let the number of states in that tower be N sub l. 00:41:27.530 --> 00:41:28.990 So there's N sub l of them. 00:41:28.990 --> 00:41:29.700 Great. 00:41:29.700 --> 00:41:34.355 And what's the distance between these guys? 00:41:34.355 --> 00:41:35.980 We haven't assumed l is an integer yet. 00:41:35.980 --> 00:41:38.610 We haven't assumed that. 00:41:38.610 --> 00:41:42.431 So this N sub l is an integer. 00:41:42.431 --> 00:41:43.848 Because it's the number of states. 00:41:43.848 --> 00:41:45.938 And the number of states can't be a pi. 00:41:50.313 --> 00:41:51.230 Now let's count that-- 00:41:51.230 --> 00:41:53.693 that so how many states are there? 00:41:53.693 --> 00:41:56.195 There are Nl. 00:41:56.195 --> 00:41:59.360 But if I count one, two, three, four, 00:41:59.360 --> 00:42:02.320 the total angular momentum down here is 2l. 00:42:04.830 --> 00:42:11.500 I had some pithy way of giving this a fancy name. 00:42:11.500 --> 00:42:13.820 But I can't remember what it was. 00:42:13.820 --> 00:42:15.670 So the length of the tower in units 00:42:15.670 --> 00:42:22.140 of h bar, the height of this tower is 2l. 00:42:22.140 --> 00:42:27.030 But the number of steps I took in here was Nl minus 1. 00:42:27.030 --> 00:42:29.550 And that number of steps is times 1. 00:42:29.550 --> 00:42:32.435 So we get that 2l is N minus 1. 00:42:32.435 --> 00:42:33.560 There's nothing fancy here. 00:42:33.560 --> 00:42:36.128 I'm just saying if L is 0 we go from here to here. 00:42:36.128 --> 00:42:37.170 There's just one element. 00:42:37.170 --> 00:42:39.400 So number states is 1, L is 0. 00:42:39.400 --> 00:42:46.288 So it's that same logic just repeated for every value of L. 00:42:46.288 --> 00:42:46.955 Other questions? 00:42:52.890 --> 00:42:59.030 Coming back to this, something on this board 00:42:59.030 --> 00:43:01.825 should cause you some serious physical discomfort. 00:43:05.525 --> 00:43:07.650 We've talked about the l equals 0 m equals 0 state. 00:43:07.650 --> 00:43:10.025 This is a state which has no angular momentum whatsoever, 00:43:10.025 --> 00:43:11.490 in any direction at all. 00:43:11.490 --> 00:43:13.867 We've talked about the l equals 1 m equals 0 state. 00:43:13.867 --> 00:43:15.950 It carries no angular momentum in the z direction, 00:43:15.950 --> 00:43:18.540 but it presumably has non-zero expectation value for L 00:43:18.540 --> 00:43:22.830 squared x and a Ly squared. 00:43:22.830 --> 00:43:23.980 These guys are also fine. 00:43:23.980 --> 00:43:26.980 What about these guys? 00:43:26.980 --> 00:43:29.920 AUDIENCE: [INAUDIBLE]. 00:43:29.920 --> 00:43:32.290 PROFESSOR: Yes! 00:43:32.290 --> 00:43:34.180 That's disconcerting. 00:43:34.180 --> 00:43:38.034 Do I have to have angular momentum in the z direction? 00:43:38.034 --> 00:43:40.022 AUDIENCE: [INAUDIBLE] 00:43:40.022 --> 00:43:42.010 [LAUGHTER] 00:43:45.000 --> 00:43:48.620 PROFESSOR: That should go on a shirt somewhere. 00:43:48.620 --> 00:43:50.420 Let me ask the question more precisely, 00:43:50.420 --> 00:43:54.080 or in a way that's a little less threatening to me. 00:43:54.080 --> 00:43:58.420 Do you have to have angular momentum in the z direction? 00:43:58.420 --> 00:43:59.630 I'm sorry, what's your name? 00:43:59.630 --> 00:44:02.587 Does David need to have angular momentum in the z direction? 00:44:02.587 --> 00:44:03.170 AUDIENCE: Yes. 00:44:03.170 --> 00:44:04.710 PROFESSOR: Does this chalk need to have angular dimension 00:44:04.710 --> 00:44:05.240 in the-- 00:44:05.240 --> 00:44:06.760 well, the chalk's-- 00:44:06.760 --> 00:44:10.163 OK, classically no. 00:44:10.163 --> 00:44:12.330 It can have some total angular momentum, which is 0, 00:44:12.330 --> 00:44:14.840 and it can be rotating not in the z direction. 00:44:14.840 --> 00:44:17.720 It can be rotating in the zx plane. 00:44:17.720 --> 00:44:19.430 If it's rotating in the zx plane, 00:44:19.430 --> 00:44:22.670 it's got total angular momentum L squared as non-zero. 00:44:22.670 --> 00:44:25.410 But its angular momentum in the z direction is 0. 00:44:25.410 --> 00:44:28.610 Its axis is exactly along the y direction. 00:44:28.610 --> 00:44:30.360 And so it's got no angular moment-- that's 00:44:30.360 --> 00:44:32.020 perfectly possible classically. 00:44:32.020 --> 00:44:34.690 And that's perfectly possible when L is an integer. 00:44:34.690 --> 00:44:36.600 Similarly when L is 2. 00:44:36.600 --> 00:44:39.200 This is the particular tower that I love the most. 00:44:39.200 --> 00:44:43.790 2, 1, 0, minus 1, minus 2. 00:44:43.790 --> 00:44:45.900 The reason I love this the most is 00:44:45.900 --> 00:44:49.300 that it's related to gravity, which is pretty awesome. 00:44:52.035 --> 00:44:53.160 That's a whole other story. 00:44:56.050 --> 00:44:59.760 I really shouldn't have said that. 00:44:59.760 --> 00:45:01.740 That's only going to confuse you. 00:45:01.740 --> 00:45:04.980 So for any integer, it's possible to have no angular 00:45:04.980 --> 00:45:06.573 momentum in the z direction. 00:45:06.573 --> 00:45:07.990 That means it's possible to rotate 00:45:07.990 --> 00:45:11.543 around the x-axis or the y-axis, orthogonal to the z-axis. 00:45:11.543 --> 00:45:12.210 That make sense? 00:45:12.210 --> 00:45:15.690 But for these half integer guys, you are inescapably spinning. 00:45:15.690 --> 00:45:18.160 There is no such thing as a state of this guy that 00:45:18.160 --> 00:45:19.880 carries no angular momentum. 00:45:19.880 --> 00:45:23.780 Anything well described by these quantum states 00:45:23.780 --> 00:45:26.830 is perpetually rotating or spinning. 00:45:26.830 --> 00:45:29.380 It carries angular momentum, we say precisely. 00:45:29.380 --> 00:45:31.797 Perpetually carries angular momentum in the z direction. 00:45:31.797 --> 00:45:34.130 Any time you measure it, it carries an angular momentum, 00:45:34.130 --> 00:45:36.920 either plus a half integer or minus a half integer. 00:45:36.920 --> 00:45:40.700 But never, ever zero. 00:45:40.700 --> 00:45:43.390 AUDIENCE: But in the classical limit where 00:45:43.390 --> 00:45:47.610 you have very large L, the m equals one half state, 00:45:47.610 --> 00:45:49.534 or the m equals minus a half state is 00:45:49.534 --> 00:45:51.784 going to get arbitrarily small compared to the angular 00:45:51.784 --> 00:45:52.284 momentum. 00:45:52.284 --> 00:45:53.880 So isn't it just like where we said, 00:45:53.880 --> 00:45:56.196 well, OK, you can never have your angular momentum 00:45:56.196 --> 00:45:58.710 only in the z direction, but we don't care? 00:45:58.710 --> 00:46:01.985 Because in the classical limit it gets arbitrarily 00:46:01.985 --> 00:46:02.610 close to there. 00:46:02.610 --> 00:46:03.920 PROFESSOR: See, one of the nice things about writing 00:46:03.920 --> 00:46:06.462 lectures like this is that you get to leave little landlines. 00:46:06.462 --> 00:46:07.970 So this is exactly one of those. 00:46:07.970 --> 00:46:09.700 Thank you for asking this question. 00:46:09.700 --> 00:46:11.470 Let me rephrase that question. 00:46:11.470 --> 00:46:14.250 Look, we all took high school chemistry. 00:46:14.250 --> 00:46:16.080 We all know about spin. 00:46:16.080 --> 00:46:17.577 The nuclei have spin. 00:46:17.577 --> 00:46:18.910 They have some angular momentum. 00:46:18.910 --> 00:46:20.327 But if you build up a lot of them, 00:46:20.327 --> 00:46:21.830 you build up a piece of chalk, look, 00:46:21.830 --> 00:46:23.560 as we said before, while it's true 00:46:23.560 --> 00:46:26.385 that there's some mismatch in the angular 00:46:26.385 --> 00:46:28.510 momentum in the z direction for some large L state, 00:46:28.510 --> 00:46:29.802 it's not only angular momentum. 00:46:29.802 --> 00:46:32.025 Some is in Lz, Lx, and Ly as well. 00:46:32.025 --> 00:46:34.700 It's preposterously small for a macroscopic object 00:46:34.700 --> 00:46:36.200 where L is macroscopic. 00:46:36.200 --> 00:46:38.030 It's the angular momentum in Planck units, 00:46:38.030 --> 00:46:41.010 in units of the Planck constant. 00:46:41.010 --> 00:46:42.220 10 to the 26th-- 00:46:42.220 --> 00:46:43.960 something huge. 00:46:43.960 --> 00:46:45.410 Why would we even notice? 00:46:45.410 --> 00:46:46.700 But here's the real statement. 00:46:46.700 --> 00:46:48.230 The statement isn't just that this 00:46:48.230 --> 00:46:49.837 is true of macroscopic objects. 00:46:49.837 --> 00:46:51.420 But imagine you take a small particle. 00:46:51.420 --> 00:46:54.500 Imagine you take a single atom. 00:46:54.500 --> 00:46:56.690 We're deep in the quantum mechanical regime. 00:46:56.690 --> 00:46:59.220 We're not in the classical regime. 00:46:59.220 --> 00:47:01.540 We take that single atom. 00:47:01.540 --> 00:47:03.280 And if it carries angular momentum 00:47:03.280 --> 00:47:05.240 and it's described by L equals 1/2 state, 00:47:05.240 --> 00:47:09.000 that atom will never, ever, ever be measured to have its angular 00:47:09.000 --> 00:47:13.600 momentum in the z direction, or indeed any direction, be 0. 00:47:13.600 --> 00:47:15.278 You will never, in any direction, 00:47:15.278 --> 00:47:16.820 measure its angular momentum to be 0. 00:47:16.820 --> 00:47:20.110 That atom perpetually carries angular momentum. 00:47:20.110 --> 00:47:22.010 And that is weird. 00:47:22.010 --> 00:47:23.798 OK, maybe you don't find it weird. 00:47:23.798 --> 00:47:24.340 This is good. 00:47:24.340 --> 00:47:26.673 You've grown up in an era when that's not a weird thing. 00:47:26.673 --> 00:47:29.230 But I find this deeply disconcerting. 00:47:29.230 --> 00:47:32.680 And you might say, look, we never actually measure an atom. 00:47:32.680 --> 00:47:33.640 But we do. 00:47:33.640 --> 00:47:34.803 We do all the time. 00:47:34.803 --> 00:47:36.970 Because we measure things like the spectre of light, 00:47:36.970 --> 00:47:40.960 as we'll study when we study atoms in a week, a couple 00:47:40.960 --> 00:47:42.580 weeks because of the exam-- 00:47:42.580 --> 00:47:44.650 sorry guys, there has to be an exam-- 00:47:44.650 --> 00:47:46.400 as we will find when we study atoms 00:47:46.400 --> 00:47:50.210 in more detail, or indeed, at all, 00:47:50.210 --> 00:47:52.140 we'll be sensitive to the angular 00:47:52.140 --> 00:47:54.780 momentum of the constituents of the atom. 00:47:54.780 --> 00:47:56.190 And we'll see different spectra. 00:47:56.190 --> 00:48:01.730 So it's an observable property when you shine light on gases. 00:48:01.730 --> 00:48:05.380 This is something we can really observe. 00:48:05.380 --> 00:48:09.390 So what this suggests is one of two things. 00:48:09.390 --> 00:48:13.005 Either these are just crazy and ridiculous 00:48:13.005 --> 00:48:15.305 and we should ignore them, or there's 00:48:15.305 --> 00:48:16.930 something interesting and intrinsically 00:48:16.930 --> 00:48:19.580 quantum-mechanical about them that's not so familiar. 00:48:19.580 --> 00:48:21.038 And the answer is going to turn out 00:48:21.038 --> 00:48:24.500 to be the second, the latter of those. 00:48:24.500 --> 00:48:25.030 The ladder? 00:48:28.210 --> 00:48:31.120 That was not intentional. 00:48:31.120 --> 00:48:32.300 Maybe it was subconscious. 00:48:32.300 --> 00:48:35.300 OK, so I want to think about some more consequences 00:48:35.300 --> 00:48:37.630 of the structure of the Ylm's and the eigenvalues, 00:48:37.630 --> 00:48:39.698 in particular of this tower structure. 00:48:39.698 --> 00:48:40.990 I want to understand some more. 00:48:40.990 --> 00:48:43.925 What other physics can we extract from this story? 00:48:48.540 --> 00:48:50.290 First, a very useful thing is just 00:48:50.290 --> 00:48:52.370 to get a picture of these guys in your head. 00:48:52.370 --> 00:48:55.940 Let's draw the angular momentum eigenfunctions. 00:48:58.232 --> 00:48:59.190 So what does that mean? 00:48:59.190 --> 00:49:02.350 Well first, when we talked about the eigenfunctions of momentum, 00:49:02.350 --> 00:49:04.730 linear momentum in one dimension, 00:49:04.730 --> 00:49:06.740 we immediately went to the wave function. 00:49:06.740 --> 00:49:09.260 We talked about how the amplitude 00:49:09.260 --> 00:49:11.450 to beat a particular spot varied in space. 00:49:11.450 --> 00:49:13.820 And the amplitude was just e to the iKX. 00:49:13.820 --> 00:49:15.660 So the amplitude was an oscillating-- 00:49:15.660 --> 00:49:17.010 the phase rotated. 00:49:17.010 --> 00:49:19.060 And the absolute value of the probability density 00:49:19.060 --> 00:49:19.940 was completely constant. 00:49:19.940 --> 00:49:21.040 Everybody cool with that? 00:49:21.040 --> 00:49:22.537 That was the 1D plane wave. 00:49:22.537 --> 00:49:25.120 So the variables there, we had an angular momentum eigenstate. 00:49:25.120 --> 00:49:27.913 And that's a function of the position. 00:49:27.913 --> 00:49:29.580 Or, sorry, a linear momentum eigenstate. 00:49:29.580 --> 00:49:30.340 And that's a function of the position. 00:49:30.340 --> 00:49:32.110 Angular momentum eigenstates are going 00:49:32.110 --> 00:49:35.920 to be functions of angular position. 00:49:35.920 --> 00:49:39.770 So I want to know what these wave functions look like, not 00:49:39.770 --> 00:49:40.658 just the eigenvalues. 00:49:40.658 --> 00:49:42.950 But I want to know what is the wave function associated 00:49:42.950 --> 00:49:44.980 to eigenvalues little l and little m 00:49:44.980 --> 00:49:48.910 look like, y sub lm of the angles theta and phi. 00:49:48.910 --> 00:49:52.118 What do these guys look like? 00:49:52.118 --> 00:49:52.910 How do we get them? 00:49:54.358 --> 00:49:56.650 This is going to be your goal for the next two minutes. 00:49:59.160 --> 00:50:03.970 So the first thing to notice is that we 00:50:03.970 --> 00:50:06.340 know what the form of the eigenvalues and eigenfunctions 00:50:06.340 --> 00:50:06.840 are. 00:50:06.840 --> 00:50:09.610 If we act with Lz we get h bar m back. 00:50:09.610 --> 00:50:13.510 If we act with L squared we get h bar squared ll plus 1 back. 00:50:13.510 --> 00:50:16.910 But we also have other expressions for Lz and L 00:50:16.910 --> 00:50:17.410 squared. 00:50:17.410 --> 00:50:19.210 In particular-- I wrote them down last time 00:50:19.210 --> 00:50:20.430 in spherical coordinates. 00:50:20.430 --> 00:50:23.140 So I'm going to working in the spherical coordinates where 00:50:23.140 --> 00:50:26.440 the declination from the vertical is an angle theta, 00:50:26.440 --> 00:50:31.510 and the angle around the equator is an angle phi. 00:50:31.510 --> 00:50:34.523 And theta equals 0 is going to be up in the z direction. 00:50:34.523 --> 00:50:35.940 It's just a choice of coordinates. 00:50:35.940 --> 00:50:37.942 There's nothing deep. 00:50:37.942 --> 00:50:39.150 There's nothing even shallow. 00:50:39.150 --> 00:50:40.653 It's just definitions. 00:50:40.653 --> 00:50:42.820 So we're going to work in the spherical coordinates. 00:50:42.820 --> 00:50:44.620 And in spherical coordinates we observe 00:50:44.620 --> 00:50:46.370 that this angular momentum, just following 00:50:46.370 --> 00:50:49.850 the definition from r cross p, takes 00:50:49.850 --> 00:50:51.040 a particularly simple form. 00:50:54.580 --> 00:50:55.790 That's a typo. 00:50:55.790 --> 00:50:59.310 h bar upon i, dd phi. 00:50:59.310 --> 00:51:00.810 And instead of writing L squared I'm 00:51:00.810 --> 00:51:02.980 going to write L plus minus, because it's shorter 00:51:02.980 --> 00:51:05.397 and also because it's going to turn out to be more useful. 00:51:05.397 --> 00:51:08.260 So this takes the form h bar, e to the plus minus i 00:51:08.260 --> 00:51:19.060 phi, d theta, plus or minus cotangent of theta, d phi. 00:51:23.652 --> 00:51:25.110 So these are the expressions for Lz 00:51:25.110 --> 00:51:27.943 and L plus minus in spherical coordinates, 00:51:27.943 --> 00:51:29.235 in these spherical coordinates. 00:51:34.900 --> 00:51:36.900 I want to know how Ylm depends on theta and phi. 00:51:36.900 --> 00:51:38.358 And it's clearly going to be easier 00:51:38.358 --> 00:51:40.660 to ask about the Lz eigenequation. 00:51:40.660 --> 00:51:41.960 So let's look at that. 00:51:41.960 --> 00:51:48.530 Lz on Ylm gives me h-- 00:51:48.530 --> 00:51:49.040 Oh yeah? 00:51:49.040 --> 00:51:51.255 AUDIENCE: Is it h or h bar? 00:51:51.255 --> 00:51:52.130 PROFESSOR: Oh, Jesus. 00:51:56.140 --> 00:51:59.580 When I write letters by hand, which 00:51:59.580 --> 00:52:03.610 is basically when I write to my mom, all my h's are crossed. 00:52:03.610 --> 00:52:04.283 I can't help it. 00:52:04.283 --> 00:52:05.450 So this is like the inverse. 00:52:10.550 --> 00:52:12.550 It's been a long time since I made that mistake. 00:52:12.550 --> 00:52:14.120 It's usually the other. 00:52:14.120 --> 00:52:22.122 So Lz acting on Ylm gives me h bar m, acting on Ylm. 00:52:22.122 --> 00:52:23.830 But that's what I get when I act with Lz, 00:52:23.830 --> 00:52:25.955 so let's just write out the differential equation . 00:52:25.955 --> 00:52:29.150 So h bar m Ylm where m is an integer, or a half integer, 00:52:29.150 --> 00:52:31.660 depending on whether l is an integer or half integer. 00:52:31.660 --> 00:52:37.000 h bar m Ylm is equal to h bar upin i, d phi of Ylm. 00:52:39.360 --> 00:52:41.610 Using the awesome power of division and multiplication 00:52:41.610 --> 00:52:43.860 I will divide both sides by h bar. 00:52:43.860 --> 00:52:47.730 And I will multiply both sides by i. 00:52:47.730 --> 00:52:53.707 And we now have the equation for the eigenfunctions of Lz, 00:52:53.707 --> 00:52:55.290 which we actually worked on last time. 00:52:55.290 --> 00:52:57.410 And we can solve this very simply. 00:52:57.410 --> 00:52:59.973 This says that, remember, Ylm is a function of theta and phi. 00:52:59.973 --> 00:53:01.890 Here we're only looking at the phi dependence, 00:53:01.890 --> 00:53:04.015 because that's all that showed up in this equation. 00:53:04.015 --> 00:53:06.820 So this tells us that the eigenfunctions Ylm are 00:53:06.820 --> 00:53:12.080 of the form of theta and phi, are of the form e to the im 00:53:12.080 --> 00:53:17.540 phi times some remaining dependence 00:53:17.540 --> 00:53:21.080 on theta, which I'll write as p of theta. 00:53:21.080 --> 00:53:24.510 And that p could depend on l and m. 00:53:24.510 --> 00:53:25.010 Cool? 00:53:34.820 --> 00:53:36.980 Already, before we even ask about that dependence 00:53:36.980 --> 00:53:41.230 on theta, the p dependence, we learn something pretty awesome. 00:53:41.230 --> 00:53:43.880 Look at this wave function. 00:53:43.880 --> 00:53:48.450 Whatever else we know, its dependence on phi 00:53:48.450 --> 00:53:50.500 is e to the im phi. 00:53:50.500 --> 00:53:52.260 Now, remember what phi is. 00:53:52.260 --> 00:53:54.670 Phi is the angle around the equator. 00:53:54.670 --> 00:53:57.180 So it goes from 0 to 2 pi. 00:53:57.180 --> 00:53:59.790 And when it comes back to 2 pi it's the same point. 00:53:59.790 --> 00:54:01.250 Phi equals 0 and phi equals pi are 00:54:01.250 --> 00:54:03.120 two names for the same point. 00:54:03.120 --> 00:54:03.620 Yes? 00:54:06.750 --> 00:54:08.520 But that should worry you. 00:54:08.520 --> 00:54:11.540 Because note that as a consequence of this, 00:54:11.540 --> 00:54:18.630 Ylm of theta 0 is equal to, well, whatever it is. 00:54:21.600 --> 00:54:31.120 Sorry, theta of 2 pi is equal to e to the im 2 pi times Plm 00:54:31.120 --> 00:54:31.620 to theta. 00:54:35.370 --> 00:54:36.720 But this is equal to-- 00:54:36.720 --> 00:54:39.230 oh, now we're in trouble. 00:54:39.230 --> 00:54:45.710 If m is an integer, this is equal to e to the i integer 2 00:54:45.710 --> 00:54:50.240 pi 1. 00:54:50.240 --> 00:54:51.090 Yes, OK. 00:54:51.090 --> 00:54:52.715 You're supposed to cheer at that point. 00:54:52.715 --> 00:54:54.548 It's like the coolest identity in the world. 00:54:54.548 --> 00:54:58.040 So e to the i 2 pi, that's one. 00:54:58.040 --> 00:55:05.400 So if m is an integer then this is just 00:55:05.400 --> 00:55:10.180 Plm of theta, which is also what we get by putting in phi 00:55:10.180 --> 00:55:11.210 equals 0. 00:55:11.210 --> 00:55:17.930 So this is equal to Ylm of theta, comma, 0. 00:55:17.930 --> 00:55:21.350 But if m is a half integer then e to the i half 00:55:21.350 --> 00:55:26.010 integer times 2 pi is minus 1. 00:55:26.010 --> 00:55:31.650 And so that gives us minus Ylm of theta and 0, 00:55:31.650 --> 00:55:34.010 if m is a half integer. 00:55:38.330 --> 00:55:43.730 So let me say that again, in the same words, actually. 00:55:43.730 --> 00:55:47.760 But let me just say it again with different emphasis. 00:55:47.760 --> 00:55:53.180 What this tells us is that Ylm at 0, 00:55:53.180 --> 00:55:55.805 as function of the coordinates theta and phi, Ylm-- 00:55:58.390 --> 00:56:02.240 so let's take m as an integer. 00:56:02.240 --> 00:56:11.530 Ylm at theta and 0 is equal to Ylm at theta and 0. 00:56:11.530 --> 00:56:12.630 That's good. 00:56:12.630 --> 00:56:20.310 But if Y is a half integer, then Ylm at theta and 2 pi, 00:56:20.310 --> 00:56:23.760 which is the same point as Ylm at theta and 0, 00:56:23.760 --> 00:56:30.000 is equal to minus Ylm at theta and 0. 00:56:30.000 --> 00:56:31.660 That's less good. 00:56:31.660 --> 00:56:34.494 What must be true of Ylm, of theta and 0? 00:56:34.494 --> 00:56:36.150 0. 00:56:36.150 --> 00:56:38.470 And was there anything special about the point 0? 00:56:38.470 --> 00:56:38.970 No. 00:56:38.970 --> 00:56:40.345 I could have just taken any point 00:56:40.345 --> 00:56:41.690 and rotated it around by pi. 00:56:41.690 --> 00:56:47.610 So this tells us that Ylm of theta and phi 00:56:47.610 --> 00:56:51.990 is identically equal to 0 if m is a half integer. 00:56:57.410 --> 00:56:58.700 Huh. 00:56:58.700 --> 00:57:00.387 That's bad. 00:57:00.387 --> 00:57:01.970 Because what's the probability density 00:57:01.970 --> 00:57:05.030 of being found at any particular angular position? 00:57:05.030 --> 00:57:06.390 0. 00:57:06.390 --> 00:57:07.814 Can you normalize that state? 00:57:07.814 --> 00:57:08.603 No. 00:57:08.603 --> 00:57:10.520 That is not a state that describes a particle. 00:57:10.520 --> 00:57:12.895 That is a state that describes the absence of a particle. 00:57:12.895 --> 00:57:14.400 That is not what we want. 00:57:14.400 --> 00:57:20.060 So these states cannot describe these values of l and m, 00:57:20.060 --> 00:57:22.480 which seem like perfectly reasonable values of l and m, 00:57:22.480 --> 00:57:25.740 perfectly reasonable eigenvalues of L squared and lm. 00:57:29.930 --> 00:57:35.570 They cannot be used to label wave functions of physical 00:57:35.570 --> 00:57:38.670 states corresponding to wave functions on a sphere. 00:57:38.670 --> 00:57:40.070 You can't do it. 00:57:40.070 --> 00:57:42.560 Because if you try, you find that those wave functions 00:57:42.560 --> 00:57:44.750 identically vanish. 00:57:44.750 --> 00:57:45.840 OK? 00:57:45.840 --> 00:57:47.310 So these cannot be used. 00:57:47.310 --> 00:57:56.755 These do not describe wave functions of a particle 00:57:56.755 --> 00:57:57.630 in quantum mechanics. 00:57:57.630 --> 00:57:58.463 They cannot be used. 00:57:58.463 --> 00:58:01.598 Those values, those towers cannot be used to describe 00:58:01.598 --> 00:58:03.140 particles moving in three dimensions. 00:58:06.950 --> 00:58:07.830 Questions about that? 00:58:07.830 --> 00:58:11.550 This is a slightly subtle argument. 00:58:11.550 --> 00:58:12.050 Yeah? 00:58:12.050 --> 00:58:13.800 AUDIENCE: You said something earlier about 00:58:13.800 --> 00:58:17.484 how atoms would never have any zero angular momentum. 00:58:17.484 --> 00:58:20.760 And so the ones that have no zero angular momentum, 00:58:20.760 --> 00:58:22.177 we just said they're not possible. 00:58:22.177 --> 00:58:23.170 So [INAUDIBLE]? 00:58:23.170 --> 00:58:24.420 PROFESSOR: Excellent question. 00:58:24.420 --> 00:58:26.295 I said it slightly diff-- so the question is, 00:58:26.295 --> 00:58:28.555 look, earlier you were saying, yeah, yeah, yeah. 00:58:28.555 --> 00:58:31.180 There are atoms in the world and they have half integer angular 00:58:31.180 --> 00:58:31.690 momentum. 00:58:31.690 --> 00:58:33.065 And you can shine a light on them 00:58:33.065 --> 00:58:35.620 and you can tell and stuff. 00:58:35.620 --> 00:58:38.490 But you just said these can't exist. 00:58:38.490 --> 00:58:41.143 So how can those two things both be true? 00:58:41.143 --> 00:58:42.310 Thank you for this question. 00:58:42.310 --> 00:58:43.393 It's a very good question. 00:58:43.393 --> 00:58:45.490 I actually said a slightly different thing. 00:58:45.490 --> 00:58:49.660 What I said was, states where angular momentum lm are half 00:58:49.660 --> 00:58:53.070 integers cannot be described by a wave function 00:58:53.070 --> 00:58:54.500 of the coordinates. 00:58:54.500 --> 00:58:56.790 We're going to need some different description. 00:58:56.790 --> 00:58:58.420 And in particular, we're going to need a different description 00:58:58.420 --> 00:58:59.440 that does what? 00:58:59.440 --> 00:59:06.650 Well, as we take phi from 0 to 2 pi, as we rotate the system, 00:59:06.650 --> 00:59:08.750 we're going to pick up a minus sign. 00:59:08.750 --> 00:59:11.020 So in order to describe an object with lm 00:59:11.020 --> 00:59:13.710 being a half integer, we can't use the wave function. 00:59:13.710 --> 00:59:17.125 We need something that is allowed to be doubly valued. 00:59:17.125 --> 00:59:19.500 And in particular, we need something that behaves nicely. 00:59:19.500 --> 00:59:21.830 When you rotate around by 2 pi, we 00:59:21.830 --> 00:59:25.330 need to come back to a minus sign, not itself. 00:59:25.330 --> 00:59:29.910 So at some object that's not a function, 00:59:29.910 --> 00:59:31.420 it's called a spinner. 00:59:31.420 --> 00:59:35.080 So we'll talk about it later. 00:59:35.080 --> 00:59:36.880 We need some object that does that. 00:59:36.880 --> 00:59:39.347 So there's this classic demonstration 00:59:39.347 --> 00:59:41.680 at this point, which is supposed to be done in a quantum 00:59:41.680 --> 00:59:42.675 mechanics class. 00:59:42.675 --> 00:59:44.050 So at this point the lecturist is 00:59:44.050 --> 00:59:45.650 obliged to do the following thing. 00:59:45.650 --> 00:59:48.700 You say, blah, blah, blah, if things rotate by 2 pi 00:59:48.700 --> 00:59:50.480 they have to come back to themselves. 00:59:50.480 --> 00:59:52.790 And then you do this. 00:59:52.790 --> 00:59:59.170 I'm going to rotate my hand like a record by 2 pi. 00:59:59.170 --> 01:00:00.760 And it will not come back to itself. 01:00:00.760 --> 01:00:01.260 OK? 01:00:04.250 --> 01:00:07.080 And it quite uncomfortably has not come back to itself. 01:00:07.080 --> 01:00:08.970 But I can do a further rotation to show you 01:00:08.970 --> 01:00:09.928 that it's a minus sign. 01:00:09.928 --> 01:00:11.640 I can do a further the rotation by 2 pi 01:00:11.640 --> 01:00:12.973 and have it come back to itself. 01:00:15.390 --> 01:00:16.970 And I kept the axis vertical, yeah? 01:00:16.970 --> 01:00:19.512 OK, so at this point you're all supposed to go like, oh, yes, 01:00:19.512 --> 01:00:22.070 uh-huh, mmm. 01:00:22.070 --> 01:00:28.460 So now that we've got that out of the way, I have an arm. 01:00:28.460 --> 01:00:30.687 So the story is a little more complicated than that. 01:00:30.687 --> 01:00:32.270 This is actually a fair demonstration, 01:00:32.270 --> 01:00:33.645 but it's a slightly subtle story. 01:00:33.645 --> 01:00:36.145 If you want to understand it, ask your recitation instructor 01:00:36.145 --> 01:00:37.383 or come to my office hours. 01:00:37.383 --> 01:00:40.100 AUDIENCE: What about USB sticks? 01:00:40.100 --> 01:00:41.342 PROFESSOR: USB sticks? 01:00:41.342 --> 01:00:43.800 AUDIENCE: You insert them here and they don't go and insert 01:00:43.800 --> 01:00:45.045 the other way. 01:00:45.045 --> 01:00:46.905 [LAUGHTER] 01:00:48.805 --> 01:00:50.180 PROFESSOR: What about USB sticks? 01:00:50.180 --> 01:00:51.410 You insert them this way, they don't work. 01:00:51.410 --> 01:00:53.160 You insert them this way, they don't work. 01:00:53.160 --> 01:00:55.450 But if you do it again, then they do. 01:00:55.450 --> 01:00:57.410 [LAUGHTER] 01:00:59.860 --> 01:01:01.820 [APPLAUSE] 01:01:02.707 --> 01:01:03.290 PROFESSOR: OK. 01:01:07.220 --> 01:01:10.200 That's pretty good. 01:01:10.200 --> 01:01:13.245 So for the moment, as long as we want to describe our system 01:01:13.245 --> 01:01:14.620 with a wave function of position, 01:01:14.620 --> 01:01:16.870 which means we're thinking about where will we find it 01:01:16.870 --> 01:01:19.230 as a function of angle, we cannot use the half integer l 01:01:19.230 --> 01:01:20.932 or m. 01:01:20.932 --> 01:01:23.015 So if we can't use the half interger l or m, fine. 01:01:23.015 --> 01:01:24.230 We'll just throw them out for the moment 01:01:24.230 --> 01:01:26.050 and we'll use the integer l and m. 01:01:26.050 --> 01:01:28.860 And let's keep going. 01:01:28.860 --> 01:01:30.500 What we need to determine now is the P. 01:01:30.500 --> 01:01:32.000 We've determined the phi dependence, 01:01:32.000 --> 01:01:33.430 but we need the theta dependence. 01:01:33.430 --> 01:01:36.940 We can get the theta dependence in a sneaky fashion. 01:01:36.940 --> 01:01:39.470 Remember the harmonic oscillator in 1D. 01:01:39.470 --> 01:01:41.720 When we wanted to find the ground state wave function, 01:01:41.720 --> 01:01:44.230 we could either solve the energy eigenvalue equation, which 01:01:44.230 --> 01:01:47.380 is a second order differential equation and kind of horrible, 01:01:47.380 --> 01:01:50.450 or we could solve the ground state equation that 01:01:50.450 --> 01:01:53.020 said that the ground state is annihilated by the annihilation 01:01:53.020 --> 01:01:55.458 operator, which is a first order difference equation 01:01:55.458 --> 01:01:56.500 and much easier to solve. 01:01:56.500 --> 01:01:57.140 Yeah? 01:01:57.140 --> 01:01:58.140 Let's do the same thing. 01:01:58.140 --> 01:01:59.620 We have an annihilation condition. 01:01:59.620 --> 01:02:04.047 If we have the top state, L plus on Y ll is equal to 0. 01:02:04.047 --> 01:02:06.130 But L plus is a first order differential operator. 01:02:06.130 --> 01:02:07.370 This is going to be easier. 01:02:07.370 --> 01:02:09.490 So we need to find a solution to this equation. 01:02:09.490 --> 01:02:11.327 And do I want to go through this? 01:02:11.327 --> 01:02:11.910 Yeah, why not. 01:02:11.910 --> 01:02:13.440 OK, so I want this to be equal to 0. 01:02:13.440 --> 01:02:16.340 But L plus Yll is equal to-- well, 01:02:16.340 --> 01:02:25.460 it's h bar, e to the plus i phi, d theta, plus cotangent theta, 01:02:25.460 --> 01:02:28.560 d phi on Yll. 01:02:28.560 --> 01:02:36.390 And Yll is e to the i l phi times Plm of theta. 01:02:36.390 --> 01:02:37.720 Cool? 01:02:37.720 --> 01:02:41.760 So dd phi on e to the il phi, there's no phi dependence here. 01:02:41.760 --> 01:02:44.970 It's just going to give us a vector of il. 01:02:44.970 --> 01:02:46.336 Did I get the-- 01:02:46.336 --> 01:02:47.210 yeah. 01:02:47.210 --> 01:02:52.030 So that's going to give us a plus il and no dd phi. 01:02:58.860 --> 01:03:01.500 And this e to the il phi we can pull out. 01:03:01.500 --> 01:03:02.840 But this has to be equal to 0. 01:03:02.840 --> 01:03:10.670 So this says that 0 is equal to d theta plus il cotangent theta 01:03:10.670 --> 01:03:13.090 P lm of theta. 01:03:15.843 --> 01:03:17.510 And this is actually much better than it 01:03:17.510 --> 01:03:18.802 seems for the following reason. 01:03:18.802 --> 01:03:21.420 dd theta-- find the dd theta. 01:03:21.420 --> 01:03:25.350 Cotangent of theta is cosine over sine. 01:03:25.350 --> 01:03:28.030 That's what you get if you take the derivative of lots 01:03:28.030 --> 01:03:30.270 of sine functions-- 01:03:30.270 --> 01:03:32.210 sine to the l, say. 01:03:32.210 --> 01:03:32.960 Take a derivative. 01:03:32.960 --> 01:03:35.418 You lose a power of sine and you pick up a power of cosine. 01:03:35.418 --> 01:03:37.840 So multiplying by cotangent gets rid of a power of sign 01:03:37.840 --> 01:03:43.450 and gives you a power of cosine, which is a derivative of sine. 01:03:43.450 --> 01:03:45.830 So Plm, noticing the l-- 01:03:45.830 --> 01:03:49.400 and I screwed up an i somewhere. 01:03:49.400 --> 01:03:52.330 I think I wanted an i up here. 01:03:52.330 --> 01:03:53.980 Let's see. 01:03:53.980 --> 01:03:54.480 i. 01:04:06.290 --> 01:04:06.790 Sorry. 01:04:10.280 --> 01:04:12.490 Yes, I want an i cotangent. 01:04:12.490 --> 01:04:14.280 OK, that's much better. 01:04:14.280 --> 01:04:16.510 So i cotangent-- good, good. 01:04:16.510 --> 01:04:20.970 And then the i squareds give me a minus l. 01:04:20.970 --> 01:04:25.260 And this tells us that Plm, so if this is sine to the l, 01:04:25.260 --> 01:04:29.690 then d theta gives us an l sine to the l minus 1 cosine, which 01:04:29.690 --> 01:04:31.777 is what I get by taking sine to the l 01:04:31.777 --> 01:04:33.610 and multiplying by cosine, dividing by sine, 01:04:33.610 --> 01:04:35.200 and multplying by l. 01:04:35.200 --> 01:04:38.120 So this gives me Pll. 01:04:38.120 --> 01:04:40.847 This is for the particular state ll. 01:04:40.847 --> 01:04:43.180 We're looking at the top state and we're annhilating it. 01:04:43.180 --> 01:04:47.570 So Pll is equal to some coefficient, 01:04:47.570 --> 01:04:52.150 so I'll just say proportional to sine to the l of theta. 01:04:55.450 --> 01:05:02.710 So this tells us that Yll is equal to some normalization, 01:05:02.710 --> 01:05:06.730 sub ll, just some number, times from the phi dependence 01:05:06.730 --> 01:05:10.400 e to the il phi, and from the theta dependence, 01:05:10.400 --> 01:05:12.640 sine to the l of theta. 01:05:16.580 --> 01:05:19.570 So this is the form. 01:05:19.570 --> 01:05:20.846 Sorry, go ahead. 01:05:20.846 --> 01:05:22.980 AUDIENCE: What's the symbol there? 01:05:22.980 --> 01:05:25.250 PROFESSOR: Oh, this twisted horrible thing? 01:05:25.250 --> 01:05:26.610 It's proportional to. 01:05:26.610 --> 01:05:29.930 But it was a long night. 01:05:33.350 --> 01:05:36.160 So now we have the wave function explicitly as a function phi 01:05:36.160 --> 01:05:38.870 and as a function of theta completely understood 01:05:38.870 --> 01:05:41.640 for the top state in any tower. 01:05:41.640 --> 01:05:43.740 This is for any L. The top state in any tower 01:05:43.740 --> 01:05:45.070 is e to the il phi. 01:05:45.070 --> 01:05:46.050 Does that make sense? 01:05:46.050 --> 01:05:48.495 Well Lz is h bar upon id phi. 01:05:48.495 --> 01:05:51.760 So that gives us h bar as m as l. 01:05:51.760 --> 01:05:52.400 So that's good. 01:05:52.400 --> 01:05:53.340 That's the top state. 01:05:53.340 --> 01:05:55.007 And from the sine theta we just checked. 01:05:55.007 --> 01:05:58.490 We constructed that this indeed has the-- 01:05:58.490 --> 01:06:00.860 well, if you then check you will find that the L squared 01:06:00.860 --> 01:06:02.818 eigenvalue, which you'll do on the problem set, 01:06:02.818 --> 01:06:05.990 the l squared eigenvalue is h bar squared ll plus 1. 01:06:05.990 --> 01:06:07.720 AUDIENCE: Quick question. 01:06:07.720 --> 01:06:10.530 The expression we have for the L plus minus operators, 01:06:10.530 --> 01:06:12.970 how did we construct the expressions for Lx and Ly? 01:06:12.970 --> 01:06:14.130 PROFESSOR: Good. 01:06:14.130 --> 01:06:15.800 It's much easier than you think. 01:06:15.800 --> 01:06:18.380 So Lx is equal to-- 01:06:18.380 --> 01:06:19.500 L is r cross b, right? 01:06:19.500 --> 01:06:22.460 So this is going to be yPz minus zPy. 01:06:28.200 --> 01:06:37.391 And this is equal to h bar upon i, ydz minus zdy. 01:06:37.391 --> 01:06:40.010 But you know what y is in spherical coordinates. 01:06:40.010 --> 01:06:40.990 And you know what derivitive with respect to z 01:06:40.990 --> 01:06:41.750 is in spherical coordinates. 01:06:41.750 --> 01:06:43.810 Because you know what z is, and you know the chain rule. 01:06:43.810 --> 01:06:45.940 So taking this and just plugging in explicit expression 01:06:45.940 --> 01:06:48.107 for the change of variables to spherical coordinates 01:06:48.107 --> 01:06:49.550 takes care of it. 01:06:49.550 --> 01:06:50.050 Yeah? 01:06:50.050 --> 01:06:51.855 AUDIENCE: What does the superscript of the sine 01:06:51.855 --> 01:06:52.465 indicate? 01:06:52.465 --> 01:06:53.960 Is that sine to the power of l? 01:06:53.960 --> 01:06:54.668 PROFESSOR: Sorry. 01:06:54.668 --> 01:06:56.090 This is bad notation. 01:06:56.090 --> 01:06:59.410 It's not bad notation, it's just not familiar notation. 01:06:59.410 --> 01:07:02.075 It's notation that is used throughout theoretical physics. 01:07:02.075 --> 01:07:04.310 It means this, sine theta to the l. 01:07:04.310 --> 01:07:08.090 The lth power of sine. 01:07:08.090 --> 01:07:10.250 For typesetting reasons we often put the power 01:07:10.250 --> 01:07:12.130 before the argument. 01:07:12.130 --> 01:07:13.630 Yeah, no, it's a very good question. 01:07:13.630 --> 01:07:15.505 Thank you for asking, because it was unclear. 01:07:15.505 --> 01:07:16.610 I appreciate that. 01:07:16.610 --> 01:07:19.122 Other questions? 01:07:19.122 --> 01:07:22.220 AUDIENCE: [INAUDIBLE] L hat [INAUDIBLE]?? 01:07:22.220 --> 01:07:24.860 PROFESSOR: How did we come up with the L hat plus minus? 01:07:24.860 --> 01:07:27.030 That was from this. 01:07:27.030 --> 01:07:29.645 So we know what the components of the angular momentum 01:07:29.645 --> 01:07:31.600 are in Cartesian coordinates. 01:07:31.600 --> 01:07:33.910 And you know how, because it's coordinates, 01:07:33.910 --> 01:07:36.640 to change variables from Cartesian to spherical. 01:07:36.640 --> 01:07:38.290 So you just plug this in for Lx. 01:07:38.290 --> 01:07:41.200 But L plus is Lx plus ioy, and so you just 01:07:41.200 --> 01:07:43.130 take these guys in spherical form 01:07:43.130 --> 01:07:44.880 and add them together with the relative i. 01:07:44.880 --> 01:07:48.008 And that gives you that expression. 01:07:48.008 --> 01:07:49.966 AUDIENCE: Maybe I missed this, but can you just 01:07:49.966 --> 01:07:53.173 explain the distinction between Y sub ll and Y sub lm? 01:07:53.173 --> 01:07:54.340 PROFESSOR: Yeah, absolutely. 01:07:54.340 --> 01:08:01.756 So Y sub ll, it means Y sub lm where m is equal to l. 01:08:01.756 --> 01:08:02.464 AUDIENCE: Oh, OK. 01:08:02.464 --> 01:08:04.780 It was just the generic [INAUDIBLE].. 01:08:04.780 --> 01:08:07.030 PROFESSOR: It's a generic eigenfunction of the angular 01:08:07.030 --> 01:08:10.600 momentum, with the angular momentum in the z direction 01:08:10.600 --> 01:08:13.140 being equal to the angular momentum in the total angular 01:08:13.140 --> 01:08:14.200 momentum. 01:08:14.200 --> 01:08:16.680 At least for these numbers. 01:08:16.680 --> 01:08:17.380 OK? 01:08:17.380 --> 01:08:19.490 Cool. 01:08:19.490 --> 01:08:22.609 Good, so now, if we know this, how do we just as a side note-- 01:08:22.609 --> 01:08:24.609 suppose we know-- well, suppose we know this? 01:08:24.609 --> 01:08:25.170 We know this. 01:08:25.170 --> 01:08:26.760 We know what the top state in the tower looks like. 01:08:26.760 --> 01:08:28.677 How do I get the next state down in the tower? 01:08:28.677 --> 01:08:30.923 how do I get Yl l minus 1? 01:08:30.923 --> 01:08:31.715 AUDIENCE: Lower it. 01:08:31.715 --> 01:08:32.548 PROFESSOR: Lower it. 01:08:32.548 --> 01:08:33.130 Exactly. 01:08:33.130 --> 01:08:35.643 So this is easy, L minus on Yll. 01:08:35.643 --> 01:08:37.560 And we have to be careful about normalization. 01:08:37.560 --> 01:08:41.262 So again, it's proportional to. 01:08:41.262 --> 01:08:41.970 But this is easy. 01:08:41.970 --> 01:08:43.260 We don't have to solve any difference equations. 01:08:43.260 --> 01:08:44.490 We just have to take derivatives. 01:08:44.490 --> 01:08:45.819 So it's just like the raising operator 01:08:45.819 --> 01:08:47.040 for a harmonic oscillator. 01:08:47.040 --> 01:08:48.787 We can raise and lower along the tower 01:08:48.787 --> 01:08:50.120 and get the right wave function. 01:08:54.920 --> 01:08:57.924 To give you some examples-- 01:08:57.924 --> 01:08:59.043 yeah, let's do that here. 01:08:59.043 --> 01:09:01.340 Let me just quickly give you a few examples 01:09:01.340 --> 01:09:05.125 of the first few spherical harmonics. 01:09:05.125 --> 01:09:06.750 Sorry, I should give these guys a name. 01:09:06.750 --> 01:09:08.939 These functions, Ylm of theta and phi, 01:09:08.939 --> 01:09:10.970 they're called the spherical harmonics. 01:09:10.970 --> 01:09:13.630 They're called these because they solve the Laplacian 01:09:13.630 --> 01:09:16.750 equation on the sphere, which is just the eigenvalue equation. 01:09:16.750 --> 01:09:19.500 L squared on them is equal to a constant times those things 01:09:19.500 --> 01:09:20.000 back. 01:09:24.240 --> 01:09:28.279 Just to tabulate a couple of examples for you concretely, 01:09:28.279 --> 01:09:31.529 consider the l equals 0 states. 01:09:31.529 --> 01:09:34.500 What are the allowed values of m for little l equals 0? 01:09:34.500 --> 01:09:35.000 0. 01:09:35.000 --> 01:09:36.710 So Y0,0 is the only state. 01:09:36.710 --> 01:09:41.020 And if you properly normalize it, it's 1 over root 4 pi. 01:09:41.020 --> 01:09:41.660 OK, good. 01:09:41.660 --> 01:09:43.460 what about l equals 1? 01:09:43.460 --> 01:09:50.520 Then we have Y1, 0 and we have Y1 minus 1. 01:09:50.520 --> 01:09:52.960 And we have Y1,1. 01:09:52.960 --> 01:09:56.370 So these guys take a particularly simple form. 01:09:56.370 --> 01:09:59.805 Root 3-- I'm not even going to worry about the coefficient. 01:09:59.805 --> 01:10:00.680 They're in the notes. 01:10:00.680 --> 01:10:03.500 You can look them up anywhere. 01:10:03.500 --> 01:10:08.110 So first off let's look at Y1, 1. 01:10:08.110 --> 01:10:09.000 So non-linear today. 01:10:09.000 --> 01:10:11.045 So Y1, 1, it's going to be some normalization. 01:10:11.045 --> 01:10:12.130 And what is the form? 01:10:15.940 --> 01:10:19.490 It's just e to the il phi sine theta to the l. l is 1. 01:10:19.490 --> 01:10:27.040 So this is some constant times e to the i phi, sine theta. 01:10:27.040 --> 01:10:27.740 Who 1, 0? 01:10:27.740 --> 01:10:32.560 Well, it's got angular momentum 0 in the z direction, in Lz. 01:10:32.560 --> 01:10:35.540 So that means how does it depend on phi? 01:10:35.540 --> 01:10:36.590 It doesn't. 01:10:36.590 --> 01:10:39.112 And you can easily see that, because when 01:10:39.112 --> 01:10:42.500 we lower we get an e to the i minus i phi. 01:10:42.500 --> 01:10:45.780 Anyway, so this gives us a constant times 01:10:45.780 --> 01:10:47.760 no e to the i phi, no phi dependents, 01:10:47.760 --> 01:10:50.380 and cosine of theta. 01:10:50.380 --> 01:10:51.990 And if you get a cosine of theta, 01:10:51.990 --> 01:10:53.760 the d theta and the cotangent d phi 01:10:53.760 --> 01:10:55.270 will give you the same thing. 01:10:55.270 --> 01:10:58.350 And Y1 minus 1 is again a constant, times e 01:10:58.350 --> 01:10:59.790 to the minus i phi. 01:10:59.790 --> 01:11:04.530 So it's got m equals minus 1 and sine theta again. 01:11:04.530 --> 01:11:07.170 Notice a pleasing parsimony here. 01:11:07.170 --> 01:11:10.795 The theta dependence is the same for plus m and minus m. 01:11:13.660 --> 01:11:14.560 So what about Y2, 2? 01:11:17.740 --> 01:11:25.970 Some constant e to the i 2 phi, sine squared theta, dot, dot, 01:11:25.970 --> 01:11:27.957 dot, Y0, 0. 01:11:27.957 --> 01:11:29.040 And here it's interesting. 01:11:29.040 --> 01:11:31.850 Here we just got one term from taking the derivative. 01:11:31.850 --> 01:11:33.190 They both give you cosine. 01:11:33.190 --> 01:11:36.780 But now there are two ways to act with the two derivatives. 01:11:36.780 --> 01:11:38.190 And this gives you a constant. 01:11:38.190 --> 01:11:39.440 Now what's the phi dependence? 01:11:39.440 --> 01:11:41.560 It's nothing, because it's got m equals 0. 01:11:41.560 --> 01:11:43.250 And so the only dependence is on theta. 01:11:43.250 --> 01:11:46.470 And we get a cos squared whoops, there's a 3-- 01:11:46.470 --> 01:11:49.150 3 cos squared theta minus 1. 01:11:49.150 --> 01:11:50.895 AUDIENCE: Do you mean Y2, 0? 01:11:50.895 --> 01:11:51.770 PROFESSOR: Oh, shoot. 01:11:51.770 --> 01:11:52.270 Thank you. 01:11:52.270 --> 01:11:53.080 Yes, I mean Y2, 0. 01:11:53.080 --> 01:11:53.980 Thank you. 01:11:53.980 --> 01:11:58.600 And then if we continue lowering to Y2 minus 2, 01:11:58.600 --> 01:12:00.100 this is equal to, again, a constant. 01:12:00.100 --> 01:12:02.350 And the it's going to be the same dependence on theta, 01:12:02.350 --> 01:12:05.890 but a different dependence of phi. e to the minus 2i phi, 01:12:05.890 --> 01:12:07.850 sine squared theta. 01:12:07.850 --> 01:12:08.350 OK? 01:12:08.350 --> 01:12:09.348 Yeah. 01:12:09.348 --> 01:12:11.340 AUDIENCE: Aren't these not normalizable? 01:12:14.325 --> 01:12:14.950 PROFESSOR: Why? 01:12:14.950 --> 01:12:16.550 AUDIENCE: Oh, never mind. 01:12:16.550 --> 01:12:17.480 PROFESSOR: Good. 01:12:17.480 --> 01:12:18.980 So let me turn that into a question. 01:12:18.980 --> 01:12:21.790 The question is, are these normalizable? 01:12:21.790 --> 01:12:23.590 Yeah, so how would we normalize them? 01:12:23.590 --> 01:12:25.320 What's the check for if they're normalizable or not? 01:12:25.320 --> 01:12:26.270 AUDIENCE: [INAUDIBLE] 01:12:26.270 --> 01:12:27.687 PROFESSOR: Yeah, we integrate them 01:12:27.687 --> 01:12:30.890 norm squared over a sphere. 01:12:30.890 --> 01:12:32.800 Not over a volume, just over a sphere. 01:12:32.800 --> 01:12:35.150 Because they're only wave functions on the sphere. 01:12:35.150 --> 01:12:36.360 We haven't dealt with the radial function. 01:12:36.360 --> 01:12:37.485 We'll deal with that later. 01:12:37.485 --> 01:12:39.403 That will come in the next lecture. 01:12:39.403 --> 01:12:40.070 Other questions? 01:12:40.070 --> 01:12:40.630 Yeah? 01:12:40.630 --> 01:12:43.020 AUDIENCE: Can you explain one more time why m equals 01:12:43.020 --> 01:12:44.915 0 doesn't have [INAUDIBLE]? 01:12:44.915 --> 01:12:46.540 PROFESSOR: Yeah, why m equals 0 doesn't 01:12:46.540 --> 01:12:49.070 have any phi dependence? 01:12:49.070 --> 01:12:51.570 If m equals 0 had phi dependence, 01:12:51.570 --> 01:12:53.990 then we know that the eigenvalue of Llz 01:12:53.990 --> 01:12:56.540 is what we get when we take a derivative with respect to phi. 01:12:56.540 --> 01:12:59.110 But if the Lz eigenvalue is 0, that 01:12:59.110 --> 01:13:01.190 means that when we act with dd phi we get 0. 01:13:01.190 --> 01:13:03.220 That means it can't depend on phi. 01:13:03.220 --> 01:13:05.950 Cool? 01:13:05.950 --> 01:13:07.280 Other questions? 01:13:07.280 --> 01:13:09.280 What I'm going to do next is I'm going show you, 01:13:09.280 --> 01:13:10.863 walk you through some of these angular 01:13:10.863 --> 01:13:14.280 momentum eigenstates, graphically on the computer. 01:13:14.280 --> 01:13:18.700 Before we do that, any questions about the calculation so far? 01:13:18.700 --> 01:13:19.200 OK. 01:13:30.620 --> 01:13:34.690 So this mathematical package I'll post on the web page. 01:13:34.690 --> 01:13:37.050 And at the moment I think it's doing the real part. 01:13:37.050 --> 01:13:38.610 So what we're looking at now is the real part. 01:13:38.610 --> 01:13:40.402 Actually, let's look at the absolute value. 01:13:43.350 --> 01:13:44.300 Good. 01:13:44.300 --> 01:13:50.490 So here we are, looking at the absolute value of-- 01:13:50.490 --> 01:13:54.180 that's not what I wanted to do. 01:13:54.180 --> 01:13:56.465 So what we're looking at in this notation 01:13:56.465 --> 01:13:57.840 is some horrible parametric plot. 01:13:57.840 --> 01:14:00.090 You don't really need to see the mathemat-- oh, shoot. 01:14:02.330 --> 01:14:02.830 Sorry. 01:14:05.605 --> 01:14:07.480 You don't need to see the code, particularly. 01:14:07.480 --> 01:14:10.240 So I'm not going to worry about it, but it will be posted. 01:14:10.240 --> 01:14:13.230 So here we're looking at the absolute value, the norm 01:14:13.230 --> 01:14:19.810 squared of the spherical harmonic Y. And the lower 01:14:19.810 --> 01:14:22.926 eigenvalue here, lower coordinate is the l. 01:14:22.926 --> 01:14:25.440 And the upper is m. 01:14:25.440 --> 01:14:29.210 So when l is 0, what we get-- 01:14:29.210 --> 01:14:32.000 here's what this plot is indicating. 01:14:32.000 --> 01:14:34.570 The distance away from the origin in a particular angular 01:14:34.570 --> 01:14:37.430 direction is the absolute value of the wave function. 01:14:37.430 --> 01:14:39.330 So the further away from the origin 01:14:39.330 --> 01:14:42.320 the colored point you see is, the larger the absolute value. 01:14:45.720 --> 01:14:48.560 And the color here is just to indicate depth and position. 01:14:48.560 --> 01:14:50.110 It's not terribly meaningful. 01:14:50.110 --> 01:14:52.520 So here we see that we get a sphere. 01:14:52.520 --> 01:14:54.330 So the probability density or the norm 01:14:54.330 --> 01:14:57.280 squared of the wave function of the spherical harmonic 01:14:57.280 --> 01:14:57.997 is constant. 01:14:57.997 --> 01:14:58.830 So that makes sense. 01:14:58.830 --> 01:14:59.960 It's spherically symmetric. 01:14:59.960 --> 01:15:01.400 It has no angular momentum. 01:15:01.400 --> 01:15:03.400 As we start increasing the angular momentum, 01:15:03.400 --> 01:15:06.778 let's take the angular momentum l is 1, m is 0 state, 01:15:06.778 --> 01:15:08.195 now something interesting happens. 01:15:11.037 --> 01:15:12.370 The total angular momentum is 1. 01:15:12.370 --> 01:15:15.260 And we see that there are two spheres. 01:15:15.260 --> 01:15:16.940 Let me sort of rotate this. 01:15:16.940 --> 01:15:19.700 So there are two spheres, and there's the z-axis 01:15:19.700 --> 01:15:20.720 passing through them. 01:15:20.720 --> 01:15:23.430 And so the probability is much larger 01:15:23.430 --> 01:15:25.980 around this lobe on the top or the lobe on the bottom. 01:15:25.980 --> 01:15:29.070 And it's 0 on the plane. 01:15:29.070 --> 01:15:31.280 Now, that's kind of non-intuitive 01:15:31.280 --> 01:15:37.060 if you think, well, Lz is large, so why 01:15:37.060 --> 01:15:38.220 is it along the vertical? 01:15:38.220 --> 01:15:39.300 Why is that true? 01:15:39.300 --> 01:15:41.990 So here, this is the Lz equals 0 state. 01:15:41.990 --> 01:15:46.550 That means it carries no angular momentum along the z-axis. 01:15:46.550 --> 01:15:50.740 That means it's not rotating far out in the xy plane. 01:15:50.740 --> 01:15:52.950 So your probability of finding it in the xy plane 01:15:52.950 --> 01:15:53.550 is very small. 01:15:53.550 --> 01:15:55.450 Because if it was rotating in the xy plane 01:15:55.450 --> 01:15:57.367 it would carry a large angular momentum in Lz. 01:15:57.367 --> 01:15:58.110 But Lz is 0. 01:15:58.110 --> 01:16:00.560 So it can't be extended out in the xy plane. 01:16:00.560 --> 01:16:01.580 Cool? 01:16:01.580 --> 01:16:03.656 On the other hand, it can carry angular momentum 01:16:03.656 --> 01:16:05.063 in the x or the y direction. 01:16:05.063 --> 01:16:06.480 But if it carries angular momentum 01:16:06.480 --> 01:16:07.990 in the x direction for example, that 01:16:07.990 --> 01:16:09.920 means the system is rotating around the z-- 01:16:09.920 --> 01:16:10.420 sorry. 01:16:10.420 --> 01:16:12.503 If it carries angular momentum in the x direction, 01:16:12.503 --> 01:16:15.020 it's rotating in the zy plane. 01:16:15.020 --> 01:16:17.600 So there's some probability to find it out 01:16:17.600 --> 01:16:20.090 of the plane in y and z. 01:16:20.090 --> 01:16:22.290 But it can't be in the xy plane. 01:16:22.290 --> 01:16:24.690 Hence it's got to be in the lobes up above. 01:16:24.690 --> 01:16:25.710 That cool? 01:16:25.710 --> 01:16:28.230 So it's very useful to develop an intuition for this stuff 01:16:28.230 --> 01:16:30.342 if you're going to do chemistry or crystallography 01:16:30.342 --> 01:16:31.800 or any condensed manner of physics. 01:16:31.800 --> 01:16:32.717 It's just very useful. 01:16:32.717 --> 01:16:35.330 So I encourage you to play with these little applets. 01:16:35.330 --> 01:16:38.490 And I'll post this mathematics package. 01:16:38.490 --> 01:16:41.170 But let's looked at what happens now if we crank up the angular 01:16:41.170 --> 01:16:41.670 momentum. 01:16:41.670 --> 01:16:43.897 So as we crank up the l angular momentum, 01:16:43.897 --> 01:16:45.980 now we're getting this sort of lobe-y thing, which 01:16:45.980 --> 01:16:49.290 looks like some sort of '50s sci-fi apparatus. 01:16:49.290 --> 01:16:52.460 So what's going on there? 01:16:52.460 --> 01:16:54.360 This is the 2, 0 state. 01:16:54.360 --> 01:16:57.495 And the 2, 0 state has a 3 cos squared theta minus 1. 01:16:57.495 --> 01:16:58.870 But cos squared theta, that means 01:16:58.870 --> 01:17:02.910 it's got two periods as it goes from vertical to negative. 01:17:02.910 --> 01:17:04.630 And if you take that and you square it, 01:17:04.630 --> 01:17:06.670 you get exactly this. 01:17:06.670 --> 01:17:07.540 OK? 01:17:07.540 --> 01:17:10.200 So they're using the cos squared as a function of the angle 01:17:10.200 --> 01:17:12.080 of declination from vertical. 01:17:12.080 --> 01:17:16.885 And it's m equals 0, so you're saying no dependence on phi. 01:17:16.885 --> 01:17:18.260 Of course, that's a little cheap. 01:17:18.260 --> 01:17:19.970 Because the angular dependence on phi 01:17:19.970 --> 01:17:21.870 is just an overall phase. 01:17:21.870 --> 01:17:24.416 So we're not going to see it in the absolute value. 01:17:24.416 --> 01:17:25.570 Everyone agree on that? 01:17:25.570 --> 01:17:26.810 We're not going to see the absolute-- 01:17:26.810 --> 01:17:27.090 OK. 01:17:27.090 --> 01:17:28.140 So let's check that. 01:17:28.140 --> 01:17:30.880 Let's take l2 and m2. 01:17:30.880 --> 01:17:34.520 So now when l is 2 and m is 2, we just get this donut. 01:17:34.520 --> 01:17:37.470 So what that's saying is, we've got some angular momentum. 01:17:37.470 --> 01:17:37.970 l is 2. 01:17:37.970 --> 01:17:40.390 But all the angular momentum, almost all of it, anyway, 01:17:40.390 --> 01:17:41.932 is in the z direction. 01:17:41.932 --> 01:17:43.390 And is that what we're seeing here? 01:17:43.390 --> 01:17:44.070 Well, yeah. 01:17:44.070 --> 01:17:46.340 It seems like it's most likely to find the particle, 01:17:46.340 --> 01:17:49.980 the probability is greatest, out in this donut around the plane. 01:17:49.980 --> 01:17:53.890 Now, if it were Lz is equal to l, it would be flat. 01:17:53.890 --> 01:17:55.955 It would be a strictly 0 thickness pancake. 01:17:55.955 --> 01:17:59.750 But we have some uncertainty in what Lx and Ly are, 01:17:59.750 --> 01:18:01.510 which is why we got this fattened donut. 01:18:01.510 --> 01:18:02.760 Everyone cool with that? 01:18:02.760 --> 01:18:07.630 And if we crank up l and we make-- yeah, right? 01:18:07.630 --> 01:18:10.460 So you can see that you've got some complicated shapes. 01:18:10.460 --> 01:18:12.010 But as we crank up l and crank up m, 01:18:12.010 --> 01:18:13.690 we just get a thinner and thinner donut. 01:18:13.690 --> 01:18:16.065 And the fact that the donut's getting thinner and thinner 01:18:16.065 --> 01:18:18.050 is that l over L squared that we did earlier. 01:18:18.050 --> 01:18:18.590 Cool? 01:18:18.590 --> 01:18:20.210 It's still a donut, but it's getting 01:18:20.210 --> 01:18:22.660 relatively thinner and thinner, by virtue 01:18:22.660 --> 01:18:24.940 of getting wider and wider. 01:18:24.940 --> 01:18:30.060 And a last thing to show you is let's take a look now at-- 01:18:30.060 --> 01:18:32.780 in fact, let's go to the l equals 0 state. 01:18:32.780 --> 01:18:35.620 Let's take a look at the real part. 01:18:39.490 --> 01:18:42.240 So now we're looking at the real part. 01:18:42.240 --> 01:18:45.990 And nothing much changed for the Y0, 0. 01:18:45.990 --> 01:18:49.620 But for Y2, 0, well, still not much changed. 01:18:49.620 --> 01:18:52.280 For Y2, 0 let's now-- 01:18:52.280 --> 01:18:53.540 this is sort of disheartening. 01:18:53.540 --> 01:18:56.490 Nothing really has changed. 01:18:56.490 --> 01:18:59.410 Why? 01:18:59.410 --> 01:19:00.850 Because it's real, exactly. 01:19:00.850 --> 01:19:04.530 So for Y2, 0, as long as m is equal to 0, this is real. 01:19:04.530 --> 01:19:06.210 There's no phase. 01:19:06.210 --> 01:19:08.380 The phase information contains information 01:19:08.380 --> 01:19:10.710 about the Lz eigenvalue. 01:19:10.710 --> 01:19:19.540 So we can correct this by changing the angular momentum. 01:19:19.540 --> 01:19:22.113 Let's-- oh shoot, how do I do that? 01:19:22.113 --> 01:19:23.530 I can't turn it off at the moment. 01:19:23.530 --> 01:19:24.600 OK, whatever. 01:19:24.600 --> 01:19:29.930 So here we have a large M. And now 01:19:29.930 --> 01:19:31.750 we've got this very funny lobe-y structure. 01:19:31.750 --> 01:19:34.250 So this is the Y2, 2, which a minute ago looked rotationally 01:19:34.250 --> 01:19:34.873 symmetric. 01:19:34.873 --> 01:19:36.540 And now it's not rotationally symmetric. 01:19:36.540 --> 01:19:40.340 It's this lobe-y structure, where the lobes are-- remember, 01:19:40.340 --> 01:19:41.430 previously we had a donut. 01:19:41.430 --> 01:19:44.580 Now we have these lobes when we look at the real part. 01:19:44.580 --> 01:19:45.850 How does that make sense? 01:19:45.850 --> 01:19:53.050 Well, we've got an e to the i 2 phi. 01:19:53.050 --> 01:19:56.420 And if we look at the real part, that's cosine of 2 pi. 01:19:56.420 --> 01:19:59.095 And so we're getting a cosine function modulating the donut. 01:19:59.095 --> 01:20:01.470 And if we look at the real part, let's do the same thing. 01:20:01.470 --> 01:20:03.550 Let's look at the imaginary part. 01:20:06.420 --> 01:20:10.360 And the imaginary part of Y2, 0, we get nothing. 01:20:10.360 --> 01:20:13.098 That's good, because Y2, 0 was real. 01:20:13.098 --> 01:20:14.140 That would have been bad. 01:20:14.140 --> 01:20:16.140 But if we look at the imaginary part of Y2, 2, 01:20:16.140 --> 01:20:19.380 we get the corresponding lobes, the other lobes, 01:20:19.380 --> 01:20:22.473 so that cos squared plus sine squared is 1. 01:20:22.473 --> 01:20:23.140 Play with these. 01:20:23.140 --> 01:20:24.440 Develop some intuition. 01:20:24.440 --> 01:20:26.023 They're going to be very useful for us 01:20:26.023 --> 01:20:28.640 when we talk about hydrogen and the structure of solids. 01:20:28.640 --> 01:20:30.640 And I will see you next week. 01:20:30.640 --> 01:20:33.390 [APPLAUSE]