WEBVTT 00:00:00.060 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.810 under a Creative Commons license. 00:00:03.810 --> 00:00:06.540 Your support will help MIT OpenCourseWare continue 00:00:06.540 --> 00:00:10.130 to offer high quality educational resources for free. 00:00:10.130 --> 00:00:12.690 To make a donation, or to view additional materials 00:00:12.690 --> 00:00:16.315 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.315 --> 00:00:16.940 at ocw.mit.edu. 00:00:21.880 --> 00:00:23.510 PROFESSOR: So let's get started. 00:00:23.510 --> 00:00:28.160 So I'm going to lecture today, Professor Zweback's away. 00:00:28.160 --> 00:00:30.180 And I just wanted to say a couple of things, 00:00:30.180 --> 00:00:32.140 just in case you haven't noticed. 00:00:32.140 --> 00:00:34.950 We posted the solutions for P-set 11. 00:00:34.950 --> 00:00:38.410 And then also later in the week, we'll 00:00:38.410 --> 00:00:40.920 post the solutions for the extra problems that came along 00:00:40.920 --> 00:00:44.330 with P-set 11, so you can look at those. 00:00:44.330 --> 00:00:47.220 And also, there's two past exams with solutions 00:00:47.220 --> 00:00:48.220 also on the website now. 00:00:48.220 --> 00:00:51.110 So you can start going through those. 00:00:51.110 --> 00:00:54.020 And also, there's a formula sheet there. 00:00:54.020 --> 00:00:55.900 And if you've got suggestions for things 00:00:55.900 --> 00:00:57.983 that you think should be on there that aren't, let 00:00:57.983 --> 00:01:01.810 us know and they probably can be put on there. 00:01:01.810 --> 00:01:03.477 So I want to turn back to what we 00:01:03.477 --> 00:01:05.810 were doing at the end of last lecture, which was talking 00:01:05.810 --> 00:01:07.850 about the spin-orbit coupling. 00:01:07.850 --> 00:01:10.240 And so this is a contribution to our Hamiltonian 00:01:10.240 --> 00:01:12.310 that looks like spin of the electron 00:01:12.310 --> 00:01:14.930 dotted into the angular momentum that the electron has 00:01:14.930 --> 00:01:19.310 around the proton in the hydrogen atom. 00:01:19.310 --> 00:01:22.320 And so because of this term we had 00:01:22.320 --> 00:01:27.600 to change the complete set of commuting observables 00:01:27.600 --> 00:01:30.220 that we wanted to talk about. 00:01:30.220 --> 00:01:32.400 So we have now this full Hamiltonian 00:01:32.400 --> 00:01:36.570 that includes this piece that has the Se dot L term in it. 00:01:36.570 --> 00:01:39.870 We have L squared, we have the spin squared. 00:01:39.870 --> 00:01:43.480 But because of this piece, Lz, which was previously 00:01:43.480 --> 00:01:46.519 one of the quantum numbers we used to classify things by, 00:01:46.519 --> 00:01:48.310 that doesn't commute with this term, right? 00:01:48.310 --> 00:01:53.600 So here we have to throw that one away. 00:01:53.600 --> 00:01:55.230 Similarly, we have this throw away 00:01:55.230 --> 00:01:58.210 the z component of the electron spin. 00:01:58.210 --> 00:02:00.800 That doesn't commute with this either. 00:02:00.800 --> 00:02:03.750 And what we replace those by is actually 00:02:03.750 --> 00:02:08.169 the J squared and the Z component of J. 00:02:08.169 --> 00:02:13.582 So J is the vector sum of the angular momentum 00:02:13.582 --> 00:02:14.790 and the spin of the electron. 00:02:17.367 --> 00:02:18.575 And this is very interesting. 00:02:21.290 --> 00:02:24.230 This term does something interesting. 00:02:24.230 --> 00:02:27.870 So if we look at-- let me go up here. 00:02:31.720 --> 00:02:35.469 If we remember the hydrogen states 00:02:35.469 --> 00:02:37.010 when we don't have this term, there's 00:02:37.010 --> 00:02:41.950 a state that has n equals 2 and l equals 1. 00:02:41.950 --> 00:02:45.595 And you can think of that as three states. 00:02:48.660 --> 00:02:52.360 And then we've got to tensor that with the spin, 00:02:52.360 --> 00:02:55.400 so the spin of the electron could be spin up or spin down. 00:02:55.400 --> 00:02:59.630 So there's a spin a half, so this is two states. 00:03:04.280 --> 00:03:06.210 And so you've got a total of six states 00:03:06.210 --> 00:03:07.790 you're going to talk about. 00:03:07.790 --> 00:03:10.237 And now what we have to do is classify these 00:03:10.237 --> 00:03:12.320 according to the quantum numbers that are actually 00:03:12.320 --> 00:03:13.360 preserved by the system. 00:03:13.360 --> 00:03:17.630 So we can't use Lz or Sz. 00:03:17.630 --> 00:03:21.330 We have to use J squared and Jz. 00:03:21.330 --> 00:03:29.150 So we've got a J equals 3/2 multiplet-- 00:03:29.150 --> 00:03:35.680 and that's four states-- plus a J equals 1/2. 00:03:42.442 --> 00:03:44.400 And you can see the number of states works out. 00:03:44.400 --> 00:03:48.180 We've got 3 times 2 is equal to 4 plus 2. 00:03:48.180 --> 00:03:56.150 And so this L dot S term takes these original six states, 00:03:56.150 --> 00:04:00.190 which without this interaction degenerate, 00:04:00.190 --> 00:04:03.770 and it splits them into the four states 00:04:03.770 --> 00:04:10.340 up here, and then two states down here. 00:04:10.340 --> 00:04:15.050 The J equals 1/2, J equals 3/2. 00:04:15.050 --> 00:04:17.470 And we also worked out the splittings. 00:04:20.540 --> 00:04:26.620 If I do this, this is plus h bar squared over 2. 00:04:26.620 --> 00:04:30.710 And this is minus h bar squared. 00:04:30.710 --> 00:04:33.340 So this gives you a splitting. 00:04:33.340 --> 00:04:37.630 Now this is not the only thing that happens in hydrogen, 00:04:37.630 --> 00:04:40.680 because you probably all know that the proton itself 00:04:40.680 --> 00:04:41.650 has spin. 00:04:41.650 --> 00:04:43.580 The proton has a spin 1/2 particle, just 00:04:43.580 --> 00:04:45.190 like the electron. 00:04:45.190 --> 00:04:46.890 It's even more complicated because it's 00:04:46.890 --> 00:04:49.760 a composite object. 00:04:49.760 --> 00:04:56.010 But that leads to additional splittings in hydrogen. 00:04:56.010 --> 00:04:58.740 And so these ones, this one here is 00:04:58.740 --> 00:05:00.609 called the defined structure. 00:05:00.609 --> 00:05:02.900 Or we can also talk about the type hyperfine structure. 00:05:11.800 --> 00:05:14.180 So this is going to be a small effect on top of this one. 00:05:23.590 --> 00:05:25.180 So we have the proton that's spin 1/2, 00:05:25.180 --> 00:05:26.555 we have the electron spin a half, 00:05:26.555 --> 00:05:30.350 and then we have the relative orbital angular momentum. 00:05:30.350 --> 00:05:39.070 And so the total angular momentum, 00:05:39.070 --> 00:05:46.640 which is J, which is going to be the sum of L plus the spin 00:05:46.640 --> 00:05:51.480 of the electron plus the spin of the proton, this is conserved. 00:05:54.630 --> 00:05:56.510 And the thing we were talking about here 00:05:56.510 --> 00:05:59.090 is actually not conserved. 00:05:59.090 --> 00:06:01.250 So once you worry about the spin of the proton 00:06:01.250 --> 00:06:03.370 you've got to look at the total angular momentum. 00:06:03.370 --> 00:06:06.930 And that's what will be conserved. 00:06:06.930 --> 00:06:11.200 And so our complete set of commuting observables 00:06:11.200 --> 00:06:15.810 is going to be a four Hamiltonian, which we'll get to 00:06:15.810 --> 00:06:23.000 in a moment, L squared the spin squareds 00:06:23.000 --> 00:06:27.800 of the proton and the electron, and then J squared, 00:06:27.800 --> 00:06:31.850 and finally Jz is the things that we're 00:06:31.850 --> 00:06:33.666 going to end up classifying states by. 00:06:38.250 --> 00:06:40.600 So we originally thought about these two 00:06:40.600 --> 00:06:43.790 here, and did a coupling between those. 00:06:43.790 --> 00:06:45.970 It's pretty natural to assume that there maybe 00:06:45.970 --> 00:06:48.460 couplings between the angular momentum and the spin 00:06:48.460 --> 00:06:50.895 of the proton, which there are. 00:06:50.895 --> 00:06:53.270 But also there's going to be a coupling between the spins 00:06:53.270 --> 00:06:54.692 of the electron and the proton. 00:06:54.692 --> 00:06:57.150 And that's the one we're going to talk about at the moment. 00:06:57.150 --> 00:07:03.200 The other one is there but we won't go over it in any detail. 00:07:03.200 --> 00:07:08.770 So the proton and the electron both spin 1/2 particles, 00:07:08.770 --> 00:07:19.970 and they both have magnetic dipole moments, 00:07:19.970 --> 00:07:22.610 which are proportional to their spin. 00:07:22.610 --> 00:07:25.090 And so it's really a coupling between these moments that 00:07:25.090 --> 00:07:29.400 tells us what the effect of this interaction is going to be. 00:07:29.400 --> 00:07:32.880 So we have the mu of the electron 00:07:32.880 --> 00:07:38.790 is equal to e over me-- minus me-- 00:07:38.790 --> 00:07:41.900 times the spin of the electron. 00:07:41.900 --> 00:07:48.940 And mu of the proton is, let me just write it as gp. 00:07:59.290 --> 00:08:07.550 And gp happens to have the value of about 5.6. 00:08:07.550 --> 00:08:09.410 And this is actually kind of interesting. 00:08:09.410 --> 00:08:11.320 So if you look at the formula up here, 00:08:11.320 --> 00:08:15.990 really I could have written this as a g over 2, with g being 2. 00:08:15.990 --> 00:08:21.259 So for the electron, the g factor is very close to 2. 00:08:21.259 --> 00:08:23.050 This is because the electron is essentially 00:08:23.050 --> 00:08:26.120 a fundamental particle, with no substructure. 00:08:26.120 --> 00:08:28.940 But the proton, which is made up of quarks and gluons 00:08:28.940 --> 00:08:32.770 flying around inside some region, has a lot of structure. 00:08:32.770 --> 00:08:42.190 And so this is really indicative of it 00:08:42.190 --> 00:08:43.450 being a composite particle. 00:08:49.640 --> 00:08:52.410 Because a fundamental spin 1/2 particle 00:08:52.410 --> 00:08:54.040 should have this g being 2. 00:08:57.260 --> 00:09:00.420 So we've got these two dipole moments. 00:09:00.420 --> 00:09:01.970 And one way to think about this is 00:09:01.970 --> 00:09:04.360 you've got this dipole of the proton. 00:09:04.360 --> 00:09:06.770 We're going to think about the proton having 00:09:06.770 --> 00:09:11.160 a little dipole charge-- sorry, dipole magnetic moment-- 00:09:11.160 --> 00:09:13.670 and this produces a magnetic field. 00:09:13.670 --> 00:09:16.560 And the electron is sitting in that magnetic field. 00:09:16.560 --> 00:09:20.290 And its spin can couple to the field. 00:09:20.290 --> 00:09:23.851 So we're going to have a Hamiltonian, 00:09:23.851 --> 00:09:26.040 a hyperfine Hamiltonian, that looks 00:09:26.040 --> 00:09:29.530 like minus mu of the electron dotted 00:09:29.530 --> 00:09:37.460 into a magnetic field produced by the proton, which is going 00:09:37.460 --> 00:09:40.840 to depend on r, on where the electron is. 00:09:44.470 --> 00:09:46.850 And you can simplify this as just 00:09:46.850 --> 00:09:50.930 e over m spin of the electron dotted 00:09:50.930 --> 00:09:52.600 into this B of the proton. 00:09:55.980 --> 00:09:58.890 So we need to know what this dipole field is. 00:09:58.890 --> 00:10:03.730 And for that you really have to go back to electromagnetism. 00:10:03.730 --> 00:10:06.780 And you've probably seen this before. 00:10:06.780 --> 00:10:10.640 But let me just write it down, and we won't derive it here. 00:10:13.320 --> 00:10:16.240 But let's go down here. 00:10:24.310 --> 00:10:26.030 This has a kind of complicated form. 00:10:48.956 --> 00:10:50.580 So there's this piece, and then there's 00:10:50.580 --> 00:11:03.330 another piece that looks like 8 pi over 3c squared mu p times 00:11:03.330 --> 00:11:05.600 the delta function at the origin. 00:11:05.600 --> 00:11:10.070 And so you think about the dipole field 00:11:10.070 --> 00:11:15.010 arising from a spinning charge distribution here. 00:11:15.010 --> 00:11:18.010 So we've got a magnetic dipole moment pointing up. 00:11:18.010 --> 00:11:22.710 This produces a field like this, a dipole type 00:11:22.710 --> 00:11:24.930 field going this way. 00:11:24.930 --> 00:11:26.630 So this is our B. 00:11:26.630 --> 00:11:30.430 And then inside here, you should really 00:11:30.430 --> 00:11:34.160 think of taking the limit as this thing goes to 0 size. 00:11:34.160 --> 00:11:37.710 And so in order to get the right field in the middle, 00:11:37.710 --> 00:11:39.153 you need to have this term here. 00:11:57.300 --> 00:12:00.007 And so if you want to see this being derived 00:12:00.007 --> 00:12:01.090 you can look in Griffiths. 00:12:01.090 --> 00:12:03.170 That does the derivation of this. 00:12:03.170 --> 00:12:05.320 But we will skip that. 00:12:10.222 --> 00:12:11.180 So we've got the field. 00:12:11.180 --> 00:12:13.160 And now we can put it into our Hamiltonian. 00:12:17.560 --> 00:12:19.010 So it's mu e. 00:12:26.430 --> 00:12:30.670 So I could replace my mu's with the spins. 00:12:30.670 --> 00:12:32.270 So I get some factor out the front 00:12:32.270 --> 00:12:42.870 that looks like ge squared over 2 Me Mp c squared. 00:12:42.870 --> 00:13:08.170 And then I get 1 over r cubed plus-- 00:13:23.090 --> 00:13:26.335 So just plugging those in we get this Hamiltonian here. 00:13:29.820 --> 00:13:31.650 And let me just simplify a little bit. 00:13:31.650 --> 00:13:35.520 Let's just call this thing q. 00:13:35.520 --> 00:13:40.150 And so this Hamiltonian is going to be given by q. 00:13:40.150 --> 00:13:45.810 And I can write it as the i-th component of the electron spin, 00:13:45.810 --> 00:13:49.490 the j-th component of the proton spin, 00:13:49.490 --> 00:14:17.850 dotted into r hat i hat j minus-- So just taking 00:14:17.850 --> 00:14:21.210 the common factors of the spins components out the front. 00:14:24.280 --> 00:14:26.319 So if we've got this, we want to ask 00:14:26.319 --> 00:14:28.360 what it's going to do to the energy of the ground 00:14:28.360 --> 00:14:30.320 state of hydrogen. 00:14:30.320 --> 00:14:34.280 So we're going to take matrix elements of this 00:14:34.280 --> 00:14:37.830 between the hydrogen wave functions. 00:14:37.830 --> 00:14:40.767 So does anyone have questions so far? 00:14:40.767 --> 00:14:41.267 Yes. 00:14:41.267 --> 00:14:43.022 AUDIENCE: Can you use r as a [INAUDIBLE]? 00:14:43.022 --> 00:14:43.980 PROFESSOR: Right right. 00:14:43.980 --> 00:14:47.310 So these are unit vectors in the r direction. 00:14:47.310 --> 00:14:52.220 And this r is the length of the vector, r vector. 00:14:52.220 --> 00:14:54.160 The usual thing. 00:14:54.160 --> 00:14:57.490 So what we're going to try and evaluate 00:14:57.490 --> 00:15:02.815 is the expectation value. 00:15:16.430 --> 00:15:18.220 So we're going to do this. 00:15:18.220 --> 00:15:21.880 Because going back to the start of last lecture, 00:15:21.880 --> 00:15:23.702 this is going to be a small correction. 00:15:23.702 --> 00:15:25.910 And so we can work out its contribution to the energy 00:15:25.910 --> 00:15:28.300 by using the original wave functions, 00:15:28.300 --> 00:15:30.560 but just calculating its matrix elements. 00:15:30.560 --> 00:15:43.320 So we're going to calculate-- and let me just 00:15:43.320 --> 00:15:44.650 give this a name. 00:15:44.650 --> 00:15:45.370 This can be-- 00:15:55.560 --> 00:16:26.220 So this is q, and the ground state 00:16:26.220 --> 00:16:27.580 has no angular dependents. 00:16:27.580 --> 00:16:29.320 So in fact, for the ground state, 00:16:29.320 --> 00:16:32.891 I can just write this is a function of r squared. 00:16:32.891 --> 00:16:35.250 For overtly excited states I can't do that. 00:16:35.250 --> 00:16:37.589 But for the ground state that works. 00:16:37.589 --> 00:16:39.630 And then we have, so we've got the wave function. 00:16:39.630 --> 00:16:44.170 And then in between them we have to put this stuff over here. 00:16:44.170 --> 00:16:45.440 So let's put the there. 00:17:05.790 --> 00:17:09.528 So one of these terms is very easy to evaluate. 00:17:09.528 --> 00:17:11.069 With this [INAUDIBLE] function I just 00:17:11.069 --> 00:17:14.339 get the wave function at the origin. 00:17:14.339 --> 00:17:16.960 And the second term is actually also 00:17:16.960 --> 00:17:18.855 relatively easy to evaluate. 00:17:29.348 --> 00:17:40.780 Who can tell me what this integral over all three 00:17:40.780 --> 00:17:44.030 directions of just one direction? 00:17:44.030 --> 00:17:45.200 What's that? 00:17:45.200 --> 00:17:46.100 AUDIENCE: 0. 00:17:46.100 --> 00:17:48.130 PROFESSOR: 0. 00:17:48.130 --> 00:17:51.380 And you can argue that by just asking, well what can it be? 00:17:51.380 --> 00:17:54.525 It's got to carry an index, because there's 00:17:54.525 --> 00:17:56.494 an index on this side of the equation. 00:17:56.494 --> 00:17:58.660 And there's no other vectors around in this problem. 00:17:58.660 --> 00:18:02.440 So the only thing it can be is 0. 00:18:02.440 --> 00:18:11.460 So if I do integral d3r of ri rj, what can that be? 00:18:15.220 --> 00:18:15.720 Sorry? 00:18:15.720 --> 00:18:17.130 AUDIENCE: 1. 00:18:17.130 --> 00:18:18.380 PROFESSOR: 1? 00:18:18.380 --> 00:18:19.110 No. 00:18:19.110 --> 00:18:21.530 So it's got two indices. 00:18:21.530 --> 00:18:23.280 So the thing on this side of the equation 00:18:23.280 --> 00:18:25.962 also has to have two indices. 00:18:25.962 --> 00:18:26.940 AUDIENCE: Delta ij? 00:18:26.940 --> 00:18:28.630 PROFESSOR: Delta ij, very good. 00:18:28.630 --> 00:18:32.300 So the only thing that can carry two indices is delta ij. 00:18:32.300 --> 00:18:34.090 And then there might be some number here. 00:18:41.980 --> 00:18:48.010 And it actually turns out that you 00:18:48.010 --> 00:18:49.720 can do an even more complicated integral. 00:18:49.720 --> 00:18:58.370 We can look at integral d3r of ri rj sum f of r squared. 00:18:58.370 --> 00:19:00.150 And that is also just some number, 00:19:00.150 --> 00:19:05.550 which depends on what f is, times delta ij. 00:19:05.550 --> 00:19:12.160 And if you go along these lines and actually look at this, 00:19:12.160 --> 00:19:14.370 the difference between the integral of this piece 00:19:14.370 --> 00:19:18.400 and the integral of this piece is actually a factor of 1/3. 00:19:18.400 --> 00:19:24.760 And so this actually integrates to 0. 00:19:27.410 --> 00:19:29.570 So when I integrate over this one, 00:19:29.570 --> 00:19:32.090 I get something times delta ij. 00:19:32.090 --> 00:19:33.610 And that something is actually 1/3. 00:19:37.240 --> 00:19:41.310 And so this term and this term cancel in the integral. 00:19:41.310 --> 00:19:43.950 And so you just get the delta function contributions. 00:19:43.950 --> 00:19:51.540 So you get some number times Sei S delta ij. 00:19:51.540 --> 00:19:54.365 So it becomes Se dotted into Sp. 00:20:02.520 --> 00:20:04.590 8 pi over 3. 00:20:04.590 --> 00:20:09.090 And then it's psi 100 at the origin. 00:20:14.710 --> 00:20:18.900 So this we know, we've already computed these radial wave 00:20:18.900 --> 00:20:22.260 functions, and saw at the origin this one 00:20:22.260 --> 00:20:28.220 is actually 1 over pi times the Bohr constant. 00:20:30.910 --> 00:20:35.270 And if you plug-in what Q is, and what the Bohr constant is, 00:20:35.270 --> 00:20:38.540 you can just find out that this whole thing ends up 00:20:38.540 --> 00:21:01.570 looking like 4/3 this gp and this 00:21:01.570 --> 00:21:07.100 we can call delta e hyperfine. 00:21:07.100 --> 00:21:09.010 So you end up with a very simple thing. 00:21:09.010 --> 00:21:11.390 And it's just proportional to the dot product of the two 00:21:11.390 --> 00:21:11.890 spins. 00:21:20.440 --> 00:21:26.500 So you've seen, essentially, you saw this term in your homework. 00:21:26.500 --> 00:21:31.100 So we just assume that this thing here came out of nowhere 00:21:31.100 --> 00:21:33.850 and was just some number times Se dot Sp, 00:21:33.850 --> 00:21:36.740 and this was a contribution to your Hamiltonian. 00:21:36.740 --> 00:21:40.420 But now we actually know where that comes from. 00:21:40.420 --> 00:21:48.250 And interestingly, this thing here, this whole thing, it's 00:21:48.250 --> 00:21:52.760 still an operator because it's got these spins in it. 00:21:56.530 --> 00:22:01.580 And that's-- put a star next to that because it's important. 00:22:01.580 --> 00:22:07.940 So now we need to ask, well what are the real states of hydrogen 00:22:07.940 --> 00:22:09.784 so they're where we've got two spins? 00:22:09.784 --> 00:22:11.700 The spin of the proton, they could be aligned, 00:22:11.700 --> 00:22:13.460 or they could be anti-aligned. 00:22:16.300 --> 00:22:17.000 Oh, sorry. 00:22:17.000 --> 00:22:18.770 We have a question up there. 00:22:18.770 --> 00:22:22.040 AUDIENCE: Is that np over np, or mu u? 00:22:22.040 --> 00:22:23.490 PROFESSOR: No, me, mass. 00:22:23.490 --> 00:22:25.364 Mass of the electron over mass of the proton. 00:22:30.350 --> 00:22:34.660 So you have to remember that the spins of the proton 00:22:34.660 --> 00:22:37.335 and the electron to can parallel or they can be anti-parallel. 00:22:37.335 --> 00:22:40.090 Or they can be both down. 00:22:40.090 --> 00:22:43.780 And so we have to go back and work out-- we 00:22:43.780 --> 00:22:47.340 have to realize that because of these terms 00:22:47.340 --> 00:22:53.290 the z components of those spins are not good quantum numbers. 00:22:53.290 --> 00:22:55.400 The only z component that appears in our list 00:22:55.400 --> 00:22:59.375 is Jz, so the total z component of angular momentum. 00:23:01.910 --> 00:23:06.620 So we need to go back and do what you-- you probably 00:23:06.620 --> 00:23:08.440 have done this to the P-set. 00:23:08.440 --> 00:23:10.080 But let's just do it very quickly. 00:23:10.080 --> 00:23:17.310 We'll take those two spin 1/2 things 00:23:17.310 --> 00:23:23.020 and so let's make this J1 and this is J2. 00:23:23.020 --> 00:23:32.970 And we're going to have J. 00:23:32.970 --> 00:23:35.520 So if I've got these two spins I can make various things. 00:23:35.520 --> 00:23:45.770 I can write down-- And if I've done this than 00:23:45.770 --> 00:23:49.405 I should also write that the m, the m quantum number that 00:23:49.405 --> 00:23:50.780 goes with the J quantum number is 00:23:50.780 --> 00:23:54.650 going to be equal to m1 plus m2. 00:23:54.650 --> 00:23:57.780 So this state here, because both of the spins are pointing up, 00:23:57.780 --> 00:23:59.490 this is an m equals 1 state. 00:24:02.210 --> 00:24:06.650 And then we can also have something like 1/2, 1/2. 00:24:22.620 --> 00:24:24.090 You could have these two states. 00:24:24.090 --> 00:24:27.740 So they both have m equals 0. 00:24:27.740 --> 00:24:38.470 And then there's an m equals minus 1, which is 1/2 this guy. 00:24:41.110 --> 00:24:45.500 So since this has m equals 1, and 1/2 cross 00:24:45.500 --> 00:24:51.770 1/2 is going to give us a spin 0 multiplet and a spin 1 00:24:51.770 --> 00:24:56.180 multiplet, because it's got m equals 1, 00:24:56.180 --> 00:24:58.950 this has to be J equals 1 as well. 00:24:58.950 --> 00:25:01.190 And this one has to be J equals 1. 00:25:01.190 --> 00:25:03.970 But the two states in the middle, 00:25:03.970 --> 00:25:05.830 we don't know what those are. 00:25:08.750 --> 00:25:10.300 There's going to be a J equals 1, 00:25:10.300 --> 00:25:12.110 m equals 0 state, which is going to be 00:25:12.110 --> 00:25:15.350 some linear combination of these two. 00:25:15.350 --> 00:25:20.955 So let's just go over here. 00:25:20.955 --> 00:25:22.060 We don't need any of this. 00:25:27.710 --> 00:25:31.290 And we need to work out what the linear combination is. 00:25:31.290 --> 00:25:34.740 So something to remember is this. 00:25:34.740 --> 00:25:39.451 The J plus or minus acting on Jm is this funny square root 00:25:39.451 --> 00:25:39.950 thing. 00:25:52.111 --> 00:25:54.110 So these are the raising and lowering operators. 00:25:54.110 --> 00:25:57.346 They take us from one state to the one with a different m 00:25:57.346 --> 00:25:57.845 value. 00:26:00.410 --> 00:26:03.130 And so we can use that to start with. 00:26:03.130 --> 00:26:05.925 We could basically take J minus on our state. 00:26:10.390 --> 00:26:12.960 And according to this formula, this 00:26:12.960 --> 00:26:23.160 will give us the square root of 1 times 2 minus m is 1. 00:26:32.230 --> 00:26:36.030 And this should be 0, right? 00:26:36.030 --> 00:26:38.490 I think I've got this sign up the wrong way. 00:26:38.490 --> 00:26:40.460 I think this is minus plus. 00:26:44.470 --> 00:26:46.950 No, sorry, that's right. 00:26:46.950 --> 00:26:49.980 It should be-- I'm doing the J minus 00:26:49.980 --> 00:26:57.300 so I have 1 times 1 minus-- yeah, right, so it's this. 00:26:57.300 --> 00:27:03.370 So this is square root 2 times Jm minus 1. 00:27:03.370 --> 00:27:10.150 But we also know that J minus a is equal to J1 minus plus J2 00:27:10.150 --> 00:27:17.310 minus because J is just the vector sum of the two J's. 00:27:17.310 --> 00:27:26.230 So we can ask what J minus on the state is. 00:27:26.230 --> 00:27:30.440 But this state we can write in terms of the tensor product. 00:27:38.612 --> 00:27:45.220 So this is equal to J1 minus 1. 00:28:09.030 --> 00:28:11.140 If we use this formula for lowering something 00:28:11.140 --> 00:28:18.100 with spin 1/2 we get 1/2 times 3/2 minus 1/2 times minus 1/2, 00:28:18.100 --> 00:28:21.360 which is actually 1 under that square root. 00:28:21.360 --> 00:28:28.740 And so this actually equals 1/2 minus 1/2 tensor 1/2, 1/2. 00:28:41.300 --> 00:28:45.520 So these two things are equal. 00:28:45.520 --> 00:28:50.220 And so that tells us, in fact, that the 1, 0 state, 00:28:50.220 --> 00:28:54.331 which is what's over here-- oh, sorry. 00:28:54.331 --> 00:28:55.350 Oh, why did I do that? 00:28:55.350 --> 00:28:59.610 This should be J equals 1 and M equals 00:28:59.610 --> 00:29:03.600 what it was, minus 1, this. 00:29:03.600 --> 00:29:06.830 So the 1, 0 state, if we bring that 1 over root 2 00:29:06.830 --> 00:29:26.860 on the other side is this combination. 00:29:26.860 --> 00:29:28.995 So it's one linear combination of those two pieces. 00:29:32.600 --> 00:29:34.510 We also want the other one. 00:29:34.510 --> 00:29:36.990 So we've got three of our states. 00:29:36.990 --> 00:29:38.540 The fourth state is then, of course, 00:29:38.540 --> 00:29:44.480 the other linear combination of the two states over there. 00:29:44.480 --> 00:29:49.050 And so that's going to be our J equals 0, M equals 0 state. 00:29:53.660 --> 00:30:01.070 So this state is going to be orthogonal to the one we've 00:30:01.070 --> 00:30:02.450 just written here. 00:30:02.450 --> 00:30:04.480 And so this is pretty easy to work out. 00:30:04.480 --> 00:30:06.000 Since there's only two terms, all we 00:30:06.000 --> 00:30:07.840 do is change the sign of one of them 00:30:07.840 --> 00:30:11.705 and it becomes orthogonal, because these states here 00:30:11.705 --> 00:30:12.560 are normalized. 00:30:15.660 --> 00:30:20.602 So this becomes 1/2, 1/2 tensor-- 00:30:20.602 --> 00:30:24.070 let me just write it in the same way that-- 00:30:24.070 --> 00:30:28.125 1/2 minus 1/2 minus-- 00:30:38.150 --> 00:30:42.410 And so our four states, so we can condense our notation 00:30:42.410 --> 00:30:49.340 so we can say that this state we can just label as this. 00:30:49.340 --> 00:30:57.710 And we can just label as a down arrow. 00:30:57.710 --> 00:31:11.240 And then something like we can label as just up down, just 00:31:11.240 --> 00:31:14.140 to make everything compact. 00:31:14.140 --> 00:31:16.230 You just have to remember that this is referring 00:31:16.230 --> 00:31:17.605 to the first spin, this is always 00:31:17.605 --> 00:31:19.760 referring to the second spin. 00:31:19.760 --> 00:31:21.500 And so then we can write our multiplets. 00:31:24.105 --> 00:31:25.840 J equals 1 has three states. 00:31:25.840 --> 00:31:28.720 It has up, up. 00:31:28.720 --> 00:31:37.540 It has up, down plus down, up. 00:31:37.540 --> 00:31:41.200 And it has down, down. 00:31:41.200 --> 00:31:44.830 So those are our three states that have J equals 1. 00:31:44.830 --> 00:31:48.320 And then we have J equals 0, which 00:31:48.320 --> 00:31:57.861 just has 1 over square root 2 up, down minus down, up. 00:32:00.950 --> 00:32:03.050 And so the two spins in our hydrogen 00:32:03.050 --> 00:32:05.270 atom, the spin of the proton and the electron, 00:32:05.270 --> 00:32:10.020 can combine to be a J equals 1 or a J equals 0 system. 00:32:10.020 --> 00:32:12.600 And since we're talking about the ground state of hydrogen, 00:32:12.600 --> 00:32:15.680 it has 0 angular momentum. 00:32:15.680 --> 00:32:20.610 And so I'm really just talking about J total, here. 00:32:26.360 --> 00:32:31.490 So if we now have this Hamiltonian, which is still 00:32:31.490 --> 00:32:33.540 an operator in spin-- we've dealt 00:32:33.540 --> 00:32:36.860 with the spacial dependence of the wave functions, 00:32:36.860 --> 00:32:39.210 but it's still an operator in spin-- 00:32:39.210 --> 00:32:42.090 we can now evaluate this. 00:32:42.090 --> 00:32:51.840 So we can take its expectation value in either the J equals 1 00:32:51.840 --> 00:32:53.940 multiplet or the J equals 0 multiplet. 00:32:56.880 --> 00:32:59.380 So let's just write it out again. 00:32:59.380 --> 00:33:04.610 So we have h hyperfine 1, 0, 0. 00:33:04.610 --> 00:33:12.039 This is equal to some delta E HF spin of the electron dotted 00:33:12.039 --> 00:33:13.205 into the spin of the proton. 00:33:16.070 --> 00:33:20.260 We can rewrite this using something we did last time. 00:33:24.540 --> 00:33:31.520 We can write this as J squared minus Se squared 00:33:31.520 --> 00:33:36.730 minus Sp squared, with the 1/2 out the front. 00:33:36.730 --> 00:33:42.790 So here, because l equals 0 because we're in the ground 00:33:42.790 --> 00:33:48.634 state, then J equals Se plus Sp. 00:33:48.634 --> 00:33:51.030 And so J squared is going to give us 00:33:51.030 --> 00:33:55.860 Se squared, Sp squared, and then the dot product. 00:33:55.860 --> 00:33:58.910 So great. 00:33:58.910 --> 00:34:01.650 So what is this, the spin squared of the electron? 00:34:07.530 --> 00:34:11.880 What's the eigenvalue of J squared, always? 00:34:11.880 --> 00:34:16.170 J, J plus 1 times h bar squared. 00:34:16.170 --> 00:34:18.376 And what is J for the electron? 00:34:18.376 --> 00:34:19.079 1/2. 00:34:19.079 --> 00:34:21.302 And what about the proton? 00:34:21.302 --> 00:34:22.218 AUDIENCE: 1/2 as well. 00:34:22.218 --> 00:34:22.843 PROFESSOR: 1/2. 00:34:22.843 --> 00:34:25.949 So we've got 1/2 times 1/2 plus 1, so 3/2. 00:34:25.949 --> 00:34:28.330 So this gives us minus 3/4. 00:34:28.330 --> 00:34:30.790 This gives us minus 3/4. 00:34:30.790 --> 00:34:35.399 So this just looks like delta e HF 00:34:35.399 --> 00:34:44.989 over 2 J squared minus 3/2 h bar squared. 00:34:44.989 --> 00:34:45.489 OK? 00:34:48.179 --> 00:34:49.420 Anyone lost doing that? 00:34:49.420 --> 00:34:50.587 Or is that OK? 00:34:50.587 --> 00:34:51.900 AUDIENCE: [INAUDIBLE] 00:34:51.900 --> 00:34:52.980 PROFESSOR: Yep. 00:34:52.980 --> 00:34:58.150 AUDIENCE: So, when you define delta e [INAUDIBLE] 00:34:58.150 --> 00:35:00.564 over there, that exudes energy? 00:35:00.564 --> 00:35:01.730 PROFESSOR: Oh, you're right. 00:35:01.730 --> 00:35:02.840 You're very right. 00:35:02.840 --> 00:35:03.471 I've messed up. 00:35:03.471 --> 00:35:03.970 I've-- 00:35:03.970 --> 00:35:06.470 AUDIENCE: [INAUDIBLE] 00:35:06.470 --> 00:35:08.750 PROFESSOR: Let me see. 00:35:08.750 --> 00:35:15.400 Yeah, really I have an h bar squared here. 00:35:18.640 --> 00:35:21.460 I think I should have had an h bar squared over there as well. 00:35:25.790 --> 00:35:27.545 Yeah. 00:35:27.545 --> 00:35:32.180 That should be over h bar squared here. 00:35:32.180 --> 00:35:35.110 Thank you. 00:35:35.110 --> 00:35:36.396 OK so-- 00:35:36.396 --> 00:35:38.622 AUDIENCE: [INAUDIBLE] 00:35:38.622 --> 00:35:39.330 PROFESSOR: Sorry? 00:35:39.330 --> 00:35:43.016 AUDIENCE: When does it get [INAUDIBLE] 00:35:43.016 --> 00:35:44.640 PROFESSOR: That was just in the algebra 00:35:44.640 --> 00:35:48.820 going from this expression, writing it in terms of alpha, 00:35:48.820 --> 00:35:49.560 things like that. 00:35:49.560 --> 00:35:53.850 So it's just some algebra. 00:35:53.850 --> 00:35:56.585 OK, anything else? 00:35:56.585 --> 00:35:57.085 No? 00:35:57.085 --> 00:35:57.585 Good. 00:35:57.585 --> 00:36:02.330 OK so now we can easily evaluate these things. 00:36:02.330 --> 00:36:17.040 We can now take J equals 1 and some M-- 00:36:17.040 --> 00:36:22.650 and this is for M equals all three states here-- and just 00:36:22.650 --> 00:36:24.310 evaluate this. 00:36:24.310 --> 00:36:26.250 And all that means is we have this J squared 00:36:26.250 --> 00:36:29.170 operator acting on this state here. 00:36:29.170 --> 00:36:32.680 And this gives us h bar squared 1 times 1 plus 1, 00:36:32.680 --> 00:36:34.730 or 2h bar squared. 00:36:34.730 --> 00:36:42.390 So this will give us delta e hyperfine over 2. 00:36:42.390 --> 00:36:45.010 And then we've got, let's pull the-- sorry there's still 00:36:45.010 --> 00:36:51.290 and h bar squared here, and an h bar squared there. 00:36:51.290 --> 00:36:54.130 But now we can evaluate. 00:36:54.130 --> 00:36:56.060 The h bar squared here cancels that one, 00:36:56.060 --> 00:37:02.750 and we get a 1 times a 1 plus 1 minus 3/2. 00:37:02.750 --> 00:37:05.730 And that's just one quarter, which is-- 00:37:12.430 --> 00:37:27.050 And similarly we can take the J equals 0 state, 00:37:27.050 --> 00:37:33.880 and this one gives us delta e hyperfine over 2. 00:37:33.880 --> 00:37:38.515 And then it's 0 time 1 minus 3/2. 00:37:38.515 --> 00:37:43.180 And so that equals minus 3/4 EHF. 00:37:48.206 --> 00:37:49.580 So what we're doing is evaluating 00:37:49.580 --> 00:37:51.151 these in these particular J states. 00:37:51.151 --> 00:37:53.400 And now we end up with something that's just a number. 00:37:53.400 --> 00:37:54.690 It's no longer an operator. 00:37:54.690 --> 00:37:56.650 It's an energy that we can measure. 00:37:56.650 --> 00:37:58.400 Yeah? 00:37:58.400 --> 00:38:01.400 AUDIENCE: So, this expectation value 00:38:01.400 --> 00:38:04.817 h hyperfine 1, 0, 0, is still an operator. 00:38:04.817 --> 00:38:06.900 Is that because we only took the expectation value 00:38:06.900 --> 00:38:08.900 over the angular [INAUDIBLE] 00:38:08.900 --> 00:38:12.060 PROFESSOR: We took over the spacial wave function. 00:38:12.060 --> 00:38:13.780 We did the r integral, right? 00:38:13.780 --> 00:38:14.480 But we didn't-- 00:38:14.480 --> 00:38:15.355 AUDIENCE: [INAUDIBLE] 00:38:15.355 --> 00:38:16.831 PROFESSOR: Right, right. 00:38:16.831 --> 00:38:17.330 Yeah. 00:38:20.100 --> 00:38:22.935 So this is actually a really important system. 00:38:28.400 --> 00:38:35.880 So let's just draw the energy level diagram here. 00:38:35.880 --> 00:38:37.800 And here we have four states. 00:38:37.800 --> 00:38:41.860 We have the spin 1/2 times spin 1/2. 00:38:41.860 --> 00:38:43.230 So 2 times 2 states. 00:38:43.230 --> 00:38:45.210 So we get a triplet and a singlet. 00:38:45.210 --> 00:38:47.040 And what this hyperfine splitting does 00:38:47.040 --> 00:38:53.070 is take those four states and split the triplet up here, 00:38:53.070 --> 00:38:56.270 and split the singlet down here. 00:38:56.270 --> 00:38:59.792 And this gap we can see is-- oops, 00:38:59.792 --> 00:39:03.380 so this should be a delta HF. 00:39:03.380 --> 00:39:08.150 So this gap is delta E HF. 00:39:08.150 --> 00:39:10.245 So it's 1/4 and minus 3/4. 00:39:13.150 --> 00:39:19.070 And if you plug numbers in, delta E HF, 00:39:19.070 --> 00:39:26.330 this actually ends up being 5.9 times 10 to the minus 6 00:39:26.330 --> 00:39:32.620 electron volts, which is a pretty small scale. 00:39:32.620 --> 00:39:35.590 So you should be comparing that to the binding energy 00:39:35.590 --> 00:39:39.550 of the ground state of hydrogen of 13.6 electron volts. 00:39:39.550 --> 00:39:41.750 So this is a very small effect. 00:39:41.750 --> 00:39:44.890 And you can really think about the relative size. 00:39:44.890 --> 00:39:52.550 So the Bohr energy, so that 13.6, formally this 00:39:52.550 --> 00:39:58.060 goes like, alpha squared times Me c squared. 00:40:01.060 --> 00:40:06.200 Then last time we talked about the S coupling, so the spin 00:40:06.200 --> 00:40:10.625 orbit, or fine structure. 00:40:14.120 --> 00:40:19.162 And so this one we found went like alpha to the fourth Me C 00:40:19.162 --> 00:40:20.660 squared. 00:40:20.660 --> 00:40:23.180 So smaller than the binding energy 00:40:23.180 --> 00:40:28.070 by a factor of 1 over 137 squared, or about 20,000. 00:40:28.070 --> 00:40:29.800 And then this one that we're talking 00:40:29.800 --> 00:40:39.210 about here, the hyperfine, this, if you look over here, 00:40:39.210 --> 00:40:42.660 this is going like alpha to the fourth Me C 00:40:42.660 --> 00:40:46.780 squared times an additional factor of Me over Mp. 00:40:49.430 --> 00:40:51.820 And the mass of the proton is about 2,000 times 00:40:51.820 --> 00:40:53.240 the mass of the electron. 00:40:53.240 --> 00:40:55.710 And so this again is-- oh, sorry. 00:40:55.710 --> 00:40:58.320 This is alpha to the fourth. 00:40:58.320 --> 00:41:03.210 So this is suppressed by about another factor of 2,000. 00:41:03.210 --> 00:41:05.270 You can go further. 00:41:05.270 --> 00:41:07.440 There are further corrections to this 00:41:07.440 --> 00:41:10.620 in something called the Lamb shift, which 00:41:10.620 --> 00:41:12.740 we won't say anything else about. 00:41:12.740 --> 00:41:15.700 This goes like alpha to the fifth Me C squared. 00:41:15.700 --> 00:41:18.260 And there's a whole host of higher order corrections. 00:41:18.260 --> 00:41:20.520 People actually calculate these energy levels 00:41:20.520 --> 00:41:23.530 to very high precision. 00:41:23.530 --> 00:41:25.650 But we won't do any more. 00:41:25.650 --> 00:41:33.850 So this transition here is actually 00:41:33.850 --> 00:41:36.200 astrophysically extremely important. 00:41:36.200 --> 00:41:39.170 So if we think about something sitting in the state here, 00:41:39.170 --> 00:41:43.890 it can decay down to the ground state by emitting a photon. 00:41:43.890 --> 00:41:51.765 So we can decay from J equals 1 to J equals 0 by a photon. 00:41:54.500 --> 00:42:01.030 And that photon will have a wavelength 00:42:01.030 --> 00:42:04.830 that corresponds exactly to this energy difference. 00:42:04.830 --> 00:42:07.230 And so that wavelength is going to be, 00:42:07.230 --> 00:42:10.970 we can write it as c over the frequency, 00:42:10.970 --> 00:42:18.040 or hc-- oh, hc not h bar c-- hc over this delta e hyperfine. 00:42:18.040 --> 00:42:21.510 If you plug numbers into this you 00:42:21.510 --> 00:42:29.660 find out that this is approximately 21.1 centimeters. 00:42:29.660 --> 00:42:38.540 And the frequency is 1,420 megahertz. 00:42:38.540 --> 00:42:40.880 And so right in the middle-- well, 00:42:40.880 --> 00:42:43.740 at the end-- of the FM band in radio. 00:42:43.740 --> 00:42:46.400 So theses are radio waves. 00:42:46.400 --> 00:42:52.750 So the size of this wavelength is firstly important 00:42:52.750 --> 00:42:55.690 because it's large compared to the size of dust 00:42:55.690 --> 00:42:56.910 in the universe. 00:42:56.910 --> 00:42:58.640 So dust is little stuff. 00:42:58.640 --> 00:43:02.230 So this is essentially goes straight through dust. 00:43:02.230 --> 00:43:04.860 So these photons will go straight through dust. 00:43:04.860 --> 00:43:10.910 The other important thing is that you probably 00:43:10.910 --> 00:43:14.170 know that there's a cosmic microwave background 00:43:14.170 --> 00:43:17.490 radiation in the universe, that's essentially very 00:43:17.490 --> 00:43:18.710 close to constant everywhere. 00:43:28.181 --> 00:43:29.680 And so we have, essentially, we have 00:43:29.680 --> 00:43:36.070 a temperature of 2.7 Kelvin. 00:43:36.070 --> 00:43:43.760 That corresponds to photons with an energy kT, which 00:43:43.760 --> 00:43:48.790 is about 0.2 times 10 to the minus 3 electron volts. 00:43:48.790 --> 00:43:51.830 So milli electron volts. 00:43:51.830 --> 00:43:56.550 But if you compare this number to what we have here, 00:43:56.550 --> 00:44:00.250 this cosmic background microwave radiation 00:44:00.250 --> 00:44:03.570 can excite hydrogen from here up to here. 00:44:03.570 --> 00:44:05.555 There's enough energy for one of those photons 00:44:05.555 --> 00:44:07.380 to come along, knock the hydrogen atom, 00:44:07.380 --> 00:44:09.270 and excite it up to here. 00:44:09.270 --> 00:44:11.940 And then it will decay and will emit 00:44:11.940 --> 00:44:18.110 this beautiful 21 centimeter line, 00:44:18.110 --> 00:44:19.680 which will go through all the dust. 00:44:19.680 --> 00:44:22.721 And so we can actually see the universe in this 21 centimeter 00:44:22.721 --> 00:44:23.220 line. 00:44:27.270 --> 00:44:31.490 Even more remarkable is, we can't calculate this 00:44:31.490 --> 00:44:34.360 at the moment, but you can show that the lifetime 00:44:34.360 --> 00:44:42.470 for this transition to happen is about 10 to the 7 years. 00:44:46.380 --> 00:44:49.740 So we can never measure that in a lab. 00:44:49.740 --> 00:44:53.985 But because these hydrogen atoms can 00:44:53.985 --> 00:44:55.360 be wandering around the universe, 00:44:55.360 --> 00:44:58.870 not interacting for that long, then they can emit. 00:44:58.870 --> 00:45:01.620 And so we can see that. 00:45:01.620 --> 00:45:05.120 This was first observed in about 1951, 00:45:05.120 --> 00:45:07.340 and is the first way that we actually 00:45:07.340 --> 00:45:12.130 saw that the galaxy had spiral shaped arms. 00:45:12.130 --> 00:45:13.700 So it's pretty important. 00:45:13.700 --> 00:45:16.120 And another nice thing about this 00:45:16.120 --> 00:45:19.020 is if you think about another galaxy-- 00:45:19.020 --> 00:45:21.570 so let me just draw another galaxy, a spiral 00:45:21.570 --> 00:45:25.580 galaxy somewhere else, like this. 00:45:25.580 --> 00:45:29.750 Let's have us over here looking at this galaxy from side on. 00:45:29.750 --> 00:45:31.660 This galaxy is rotating. 00:45:31.660 --> 00:45:34.560 So this one's moving this way, this one to moving this way. 00:45:34.560 --> 00:45:36.870 There's hydrogen over here and over here. 00:45:36.870 --> 00:45:39.170 And so we get these photons coming over here, 00:45:39.170 --> 00:45:41.840 and photons coming to us from there. 00:45:41.840 --> 00:45:43.933 But what's going to be different about these? 00:45:43.933 --> 00:45:44.920 AUDIENCE: [INAUDIBLE] 00:45:44.920 --> 00:45:45.810 PROFESSOR: Their rate shifted, right? 00:45:45.810 --> 00:45:46.770 The Doppler shifted. 00:45:46.770 --> 00:45:51.502 So this is my 21 centimeter photon. 00:45:51.502 --> 00:45:52.710 But they get Doppler shifted. 00:45:52.710 --> 00:45:54.500 And so we can measure the difference 00:45:54.500 --> 00:45:57.190 in the frequencies of those. 00:45:57.190 --> 00:45:58.725 What does that tell us? 00:45:58.725 --> 00:46:00.210 AUDIENCE: [INAUDIBLE] 00:46:00.210 --> 00:46:03.710 PROFESSOR: How fast this galaxy is spinning, right? 00:46:03.710 --> 00:46:05.090 And so one very interesting thing 00:46:05.090 --> 00:46:08.050 you find from that is if you look at the galaxy 00:46:08.050 --> 00:46:09.820 and count how many stars are in it, 00:46:09.820 --> 00:46:13.130 and essentially work out how massive 00:46:13.130 --> 00:46:15.910 that galaxy is, the speed of rotation 00:46:15.910 --> 00:46:19.750 here is actually-- that you measure from these hydrogen 00:46:19.750 --> 00:46:22.480 lines-- is that it's actually faster 00:46:22.480 --> 00:46:26.740 than the escape velocity of the matter. 00:46:26.740 --> 00:46:30.930 And so if all that was there was the visible matter, then 00:46:30.930 --> 00:46:32.630 the thing would just fall apart. 00:46:32.630 --> 00:46:34.421 And so this actually tells you that there's 00:46:34.421 --> 00:46:37.650 dark matter that doesn't interact 00:46:37.650 --> 00:46:43.860 with visible light, that's kind of all over here. 00:46:43.860 --> 00:46:45.740 So that's kind of a pretty interesting thing. 00:46:48.580 --> 00:46:53.450 So that, I think, yeah. 00:46:53.450 --> 00:46:55.074 So any questions about that? 00:46:55.074 --> 00:46:56.740 We're going to move on to another topic. 00:46:59.770 --> 00:47:00.270 Yep? 00:47:00.270 --> 00:47:01.853 AUDIENCE: You said earlier [INAUDIBLE] 00:47:01.853 --> 00:47:02.820 it's 10 to the 7 years. 00:47:02.820 --> 00:47:05.907 Does that mean it takes an average of 10 00:47:05.907 --> 00:47:09.792 to the 7 years for the cosmic microwave background energy 00:47:09.792 --> 00:47:13.450 to shift back [INAUDIBLE] 00:47:13.450 --> 00:47:15.600 PROFESSOR: No, it's really, if I just 00:47:15.600 --> 00:47:18.487 took hydrogen in this state, and took a sample of it, 00:47:18.487 --> 00:47:20.320 that's how long it would take for half of it 00:47:20.320 --> 00:47:23.240 to have gone and made the decay. 00:47:23.240 --> 00:47:25.125 So it can happen much faster. 00:47:28.250 --> 00:47:30.090 And there's a lot of it in the universe. 00:47:30.090 --> 00:47:33.730 So there's many more than 10 to the 7 atoms 00:47:33.730 --> 00:47:34.900 of hydrogen in the universe. 00:47:34.900 --> 00:47:37.800 So we see more than one of these things every year. 00:47:37.800 --> 00:47:40.870 So if you were just looking at one of them 00:47:40.870 --> 00:47:42.794 you would have to wait a long time. 00:47:42.794 --> 00:47:45.335 AUDIENCE: But the thing about the cosmic microwave background 00:47:45.335 --> 00:47:49.220 is to go from [INAUDIBLE] from 0 to 0 up to get [INAUDIBLE] 00:47:49.220 --> 00:47:51.282 PROFESSOR: Right so I mean the energy is large 00:47:51.282 --> 00:47:51.990 compared to that. 00:47:51.990 --> 00:47:54.709 So it will typically knock you up into an even higher state. 00:47:54.709 --> 00:47:56.250 And then you will kind of decay down. 00:47:56.250 --> 00:47:59.000 But then this last decay is-- because this lifetime 00:47:59.000 --> 00:48:02.500 is very long, the width of this line is also very, very narrow. 00:48:06.240 --> 00:48:10.510 So now let's talk more about adding angular momenta. 00:48:13.235 --> 00:48:14.855 Oh, maybe I should have left that up. 00:48:14.855 --> 00:48:15.355 Too late. 00:48:30.860 --> 00:48:35.060 So we're going to do this in a more general sense. 00:48:35.060 --> 00:48:38.050 So we're going to take J1, some spin 00:48:38.050 --> 00:48:51.520 J1 that has states J1, M1, with M1 equals minus J up to J. 00:48:51.520 --> 00:48:53.680 And so we're sort of talking about something 00:48:53.680 --> 00:48:56.600 like the electron in the hydrogen atom. 00:48:56.600 --> 00:48:59.967 And so that's not in any particular orbital angular 00:48:59.967 --> 00:49:00.466 momentum. 00:49:03.090 --> 00:49:05.450 So we can talk about that the Hilbert space 00:49:05.450 --> 00:49:06.600 that this thing lives in. 00:49:10.150 --> 00:49:15.710 So we can think about particle 1 with angular momentum J1. 00:49:15.710 --> 00:49:20.430 And this is basically spanned by the states 00:49:20.430 --> 00:49:25.940 J1, M1 of these [INAUDIBLE]. 00:49:25.940 --> 00:49:32.640 We can take another system with another J2, 00:49:32.640 --> 00:49:41.198 and this is going to have states J2, M2, with M2-- that 00:49:41.198 --> 00:49:44.800 should be J1's there. 00:49:44.800 --> 00:49:47.600 And that would similarly talk about some Hilbert space 00:49:47.600 --> 00:49:50.110 of some fixed angular momentum. 00:49:50.110 --> 00:49:54.110 If we want to talk about the electron in a hydrogen 00:49:54.110 --> 00:49:56.690 atom, where it doesn't have a fixed angular momentum, what we 00:49:56.690 --> 00:50:00.140 really want to talk about is the Hilbert space H1, 00:50:00.140 --> 00:50:03.470 which is the sum over J1 of these Hilbert spaces. 00:50:06.680 --> 00:50:09.870 And so this is talking about-- this Hilbert space 00:50:09.870 --> 00:50:12.610 contains every state the electron 00:50:12.610 --> 00:50:14.920 can have in a hydrogen atom. 00:50:14.920 --> 00:50:18.350 It can have all the different angular momenta. 00:50:18.350 --> 00:50:21.430 And similarly we could do that for J2. 00:50:21.430 --> 00:50:29.850 We can define J, which is J1 plus J2, as you might guess. 00:50:29.850 --> 00:50:35.440 And really this you should think of as J1 tensor 00:50:35.440 --> 00:50:45.290 the identity plus the identity tensor J2, where this one is 00:50:45.290 --> 00:50:47.370 acting on things in this Hilbert space, 00:50:47.370 --> 00:50:50.170 and the 1 here is acting on things in this Hilbert space. 00:50:50.170 --> 00:50:55.830 And similarly there's an H2, J2 that 00:50:55.830 --> 00:50:59.090 goes along with these guys. 00:50:59.090 --> 00:51:01.690 And so this operator, this big J, 00:51:01.690 --> 00:51:10.660 is something that acts on vectors 00:51:10.660 --> 00:51:21.835 in things in this tensor product space. 00:51:24.730 --> 00:51:28.654 Actually I should label this with a J1. 00:51:28.654 --> 00:51:30.880 It also acts on things in the full space, 00:51:30.880 --> 00:51:33.070 but we can talk just about that one. 00:51:45.610 --> 00:51:48.140 So now we might want to construct 00:51:48.140 --> 00:51:49.330 a basis for this space. 00:51:52.320 --> 00:51:57.710 And we conversely construct an uncoupled basis 00:51:57.710 --> 00:52:01.230 which is just take the basis elements of each of the spaces 00:52:01.230 --> 00:52:03.700 and multiply them. 00:52:03.700 --> 00:52:10.000 So we would have J1, J2, M1, M2. 00:52:17.670 --> 00:52:20.240 We'd have the states here. 00:52:20.240 --> 00:52:24.285 And if we just ask what our various-- 00:52:39.430 --> 00:52:43.230 J1 just gives us h bar squared J, J plus 1. 00:52:48.020 --> 00:53:01.560 And this one gives us h bar M1 h bar squared times our state. 00:53:08.690 --> 00:53:11.420 And so we can think about all of these. 00:53:11.420 --> 00:53:13.347 And this is what we label our state with. 00:53:13.347 --> 00:53:15.180 And that's because these form a complete set 00:53:15.180 --> 00:53:16.221 of commuting observables. 00:53:18.720 --> 00:53:20.820 And we'll just call this the A set. 00:53:33.500 --> 00:53:38.090 We can also talk about our operator J 00:53:38.090 --> 00:53:39.620 and use that to define our basis. 00:53:47.420 --> 00:53:49.690 And let's just be a little explicit 00:53:49.690 --> 00:53:51.760 about what J squared is going to be. 00:53:51.760 --> 00:54:01.620 So this is J1 tensor identity plus 1 tensor J2. 00:54:06.004 --> 00:54:07.590 And the same thing here. 00:54:14.500 --> 00:54:19.530 If you expand this out you get J1 squared tensor identity 00:54:19.530 --> 00:54:26.380 plus 1 tensor J2 squared plus the dot product, which 00:54:26.380 --> 00:54:34.020 we can write as the sum of J1k tensor J2k. 00:54:37.880 --> 00:54:42.030 And because of this piece here, J squared 00:54:42.030 --> 00:54:45.857 doesn't commute with J1z, for example. 00:54:52.330 --> 00:54:55.410 So we can't add this operator to our list 00:54:55.410 --> 00:54:56.600 of operators over there. 00:55:00.420 --> 00:55:07.262 And similarly J2z J squared is not equal to 0. 00:55:07.262 --> 00:55:08.970 So if we want to talk about this operator 00:55:08.970 --> 00:55:11.740 we have to throw both of those away. 00:55:11.740 --> 00:55:22.290 But there is an operator total Jz that commutes with J 00:55:22.290 --> 00:55:23.060 squared. 00:55:23.060 --> 00:55:27.320 And it also commutes with J1 squared and J2 squared. 00:55:27.320 --> 00:55:32.470 And so we can have another complete set 00:55:32.470 --> 00:55:35.110 of commuting observables B that's 00:55:35.110 --> 00:55:43.490 equal to J1 squared, J2 squared, J squared, and Jz. 00:55:50.960 --> 00:55:58.630 And so if there are observables then the natural basis 00:55:58.630 --> 00:56:01.540 is to label them by the eigenvalues here. 00:56:01.540 --> 00:56:20.230 So we're going to have a J1, a J2, then a J and an M. 00:56:20.230 --> 00:56:23.050 And so this is the coupled basis. 00:56:23.050 --> 00:56:27.550 Now both of these bases are equally good. 00:56:27.550 --> 00:56:30.940 They both span the full space. 00:56:30.940 --> 00:56:33.180 They're both orthogonal, orthonormal. 00:56:46.040 --> 00:56:47.810 And so we can actually write one basis 00:56:47.810 --> 00:56:50.110 into in terms of the other one. 00:56:50.110 --> 00:56:52.284 And that's the generic problem that we 00:56:52.284 --> 00:56:53.700 are trying to do when we're trying 00:56:53.700 --> 00:56:55.500 to write what we did over here before, 00:56:55.500 --> 00:56:58.070 when we did spin 1/2 cross spin 1/2. 00:56:58.070 --> 00:57:01.530 We're trying to write those products states 00:57:01.530 --> 00:57:03.530 in terms of the coupled basis. 00:57:21.430 --> 00:57:23.350 Well, they're both orthonormal basis. 00:57:27.590 --> 00:57:31.830 So I can expand J1, J2. 00:57:39.860 --> 00:57:44.470 Well actually, maybe I'll say one more thing first. 00:57:44.470 --> 00:57:49.850 So being orthonormal means that, for example, sum over J1, J2-- 00:58:03.375 --> 00:58:04.700 This is 1, right? 00:58:04.700 --> 00:58:08.920 You can resolve the identity in terms of these states. 00:58:08.920 --> 00:58:17.000 And this is the identity on this Hilbert space. 00:58:17.000 --> 00:58:19.100 I can also think about the identity just 00:58:19.100 --> 00:58:25.190 on this smaller Hilbert space, where the J1 and J2 are fixed. 00:58:25.190 --> 00:58:39.905 And so I can actually write it's the identity operator. 00:58:53.790 --> 00:58:56.750 So because every state in this space 00:58:56.750 --> 00:59:00.130 has J1 equal to some fixed value and J2 equal to some fixed 00:59:00.130 --> 00:59:03.090 value, then an identity in that thing 00:59:03.090 --> 00:59:07.240 is just somewhere over the M's, because they're the only things 00:59:07.240 --> 00:59:09.570 there. 00:59:09.570 --> 00:59:17.420 So using this, because I know that the state J1, J2, 00:59:17.420 --> 00:59:21.280 Jm has some fixed value of J1 and J2, 00:59:21.280 --> 00:59:23.160 I can write this as a sum. 00:59:25.860 --> 00:59:30.275 I can use this form of the identity. 00:59:45.940 --> 00:59:50.010 So I've written my coupled basis in terms of the uncoupled basis 00:59:50.010 --> 00:59:50.980 here. 00:59:50.980 --> 00:59:53.110 And these are just coefficients. 00:59:53.110 --> 01:00:02.005 These are called Clebsch-Gordan coefficients. 01:00:05.169 --> 01:00:07.710 They're just numbers like square root 2 and things like this. 01:00:07.710 --> 01:00:11.095 And I tell you how to do this decomposition. 01:00:15.690 --> 01:00:21.415 So they have various properties. 01:00:24.930 --> 01:00:26.810 Firstly, sometimes you also see them 01:00:26.810 --> 01:00:41.120 written as C of J1J2J colon M1M2M 01:00:41.120 --> 01:00:43.230 and various other notations. 01:00:43.230 --> 01:00:45.631 So basically things with six indices 01:00:45.631 --> 01:00:47.130 are probably going to be these guys. 01:00:51.990 --> 01:00:54.280 So they have various properties. 01:00:58.650 --> 01:01:11.830 The first property is they vanish 01:01:11.830 --> 01:01:19.210 if M is not equal to M1 plus M2. 01:01:19.210 --> 01:01:21.185 And this is actually very easy to prove. 01:01:24.560 --> 01:01:32.570 So remember that Jz is just going to be J1z plus J2z. 01:01:35.560 --> 01:01:42.895 So as an operator I can write Jz minus J1z minus J2z. 01:01:46.100 --> 01:01:47.205 And what is that operator? 01:01:53.054 --> 01:01:54.355 It's just 0, right? 01:01:57.119 --> 01:01:58.035 This is equal to that. 01:01:58.035 --> 01:01:59.300 So this is 0. 01:01:59.300 --> 01:02:03.010 So I can put this 0 anywhere I want and I'll still get 0. 01:02:03.010 --> 01:02:09.100 So let's put this between-- so this is 0-- put it 01:02:09.100 --> 01:02:15.610 between J1, J2, Jm on this side. 01:02:15.610 --> 01:02:17.510 So a coupled state here. 01:02:17.510 --> 01:02:20.140 And on this side I'll put it between the uncoupled state, 01:02:20.140 --> 01:02:22.920 J1, J2, M1, M2. 01:02:26.870 --> 01:02:32.520 So this state is an eigenstate of Jz. 01:02:32.520 --> 01:02:36.620 And this state is an eigenstate of J1z and J2z. 01:02:36.620 --> 01:02:40.600 So I can act to the right with Jz, 01:02:40.600 --> 01:02:44.100 and act to the left with these J1z and J2z, 01:02:44.100 --> 01:02:46.595 and they have mission operators. 01:02:46.595 --> 01:02:50.830 And I know because this is 0, this whole thing is 0. 01:02:50.830 --> 01:02:53.030 So then act this one on these guys 01:02:53.030 --> 01:02:56.040 and these two back this way. 01:02:56.040 --> 01:03:00.737 And so you see that gives me h bar. 01:03:00.737 --> 01:03:02.320 And then I get this one acting on here 01:03:02.320 --> 01:03:05.925 gives me M. And J1 acting on here gives me M1. 01:03:23.285 --> 01:03:29.820 And if M is not equal to M1 plus M2, then this term isn't 0. 01:03:29.820 --> 01:03:33.997 But the whole thing is so that has to be 0. 01:03:33.997 --> 01:03:36.310 So that's QED. 01:04:01.890 --> 01:04:29.760 The second property is that-- So they only allow values of J 01:04:29.760 --> 01:04:31.660 fall in this range here. 01:04:31.660 --> 01:04:36.590 And each J occurs once. 01:04:41.730 --> 01:04:43.640 Now one way to think about this is 01:04:43.640 --> 01:04:45.800 to think of these things as vectors. 01:04:45.800 --> 01:04:51.550 So you have vector J1, and then from the point of this 01:04:51.550 --> 01:04:52.785 you can have vector J2. 01:04:52.785 --> 01:04:55.690 But it can go into an arbitrary direction. 01:04:55.690 --> 01:05:00.400 So it can go up here, or it can go like this. 01:05:00.400 --> 01:05:02.570 These are meant to be the same length. 01:05:02.570 --> 01:05:04.430 And I can come all the way down here. 01:05:04.430 --> 01:05:10.390 But I can could only sit on integer points. 01:05:10.390 --> 01:05:12.090 And so this is kind of J2. 01:05:12.090 --> 01:05:15.660 And so the length of this thing here would be the length of J1 01:05:15.660 --> 01:05:16.700 plus the length of J2. 01:05:16.700 --> 01:05:17.870 So it would be this. 01:05:17.870 --> 01:05:21.460 And then the length of up to here would be this one. 01:05:21.460 --> 01:05:25.810 And then all of the other ones are in between. 01:05:25.810 --> 01:05:29.690 But you can also just look at the multiplicities 01:05:29.690 --> 01:05:30.840 of the different states. 01:05:30.840 --> 01:05:37.510 So if we look at the uncoupled basis-- 01:05:37.510 --> 01:05:42.242 so the first state, which was J equals J1, 01:05:42.242 --> 01:05:47.820 there are two J1 plus 1 states, because it 01:05:47.820 --> 01:05:52.480 can have all of the M values from J1 down to minus J1. 01:05:52.480 --> 01:05:57.600 And the other one can have two J2 plus 1 states. 01:05:57.600 --> 01:06:00.780 So that's the total number of states that I expect to have. 01:06:00.780 --> 01:06:05.070 So now let's assume this is correct 01:06:05.070 --> 01:06:09.120 and ask what the N coupled is. 01:06:09.120 --> 01:06:18.591 So this would be the sum over J equals mod J1 minus J2 up 01:06:18.591 --> 01:06:23.900 to J1 plus J2 of 2J plus 1. 01:06:23.900 --> 01:06:29.405 And let's assume that J1 is greater than or equal to J2, 01:06:29.405 --> 01:06:34.880 just to stop writing absolute values all the time. 01:06:34.880 --> 01:06:39.860 So we can write this as the difference of two sums, 01:06:39.860 --> 01:06:47.700 J equals 0 to J1 of-- J1 plus J2-- of 2J 01:06:47.700 --> 01:06:52.130 plus 1 minus the sum of J equals 0 01:06:52.130 --> 01:07:00.650 to J equals J1 minus J2 minus 1 of 2J plus 1. 01:07:00.650 --> 01:07:07.680 And if you go through-- so this is just N,N plus 1 over 2 01:07:07.680 --> 01:07:10.070 for each of these things. 01:07:10.070 --> 01:07:30.639 You end up with, well, you end up with this. 01:07:30.639 --> 01:07:31.930 You end up with the same thing. 01:07:31.930 --> 01:07:33.705 And so this is at least consistent 01:07:33.705 --> 01:07:36.730 that the number of states that we have 01:07:36.730 --> 01:07:38.370 is consistent with choosing this. 01:07:46.060 --> 01:07:49.010 One other thing we can do is look at the top state, 01:07:49.010 --> 01:07:50.135 and just see if that works. 01:07:53.040 --> 01:07:54.855 See if that has the right properties. 01:07:59.070 --> 01:08:09.008 So because we know that the J1, J2, J equals J1 plus J2, 01:08:09.008 --> 01:08:11.240 M equals J1 plus J2. 01:08:11.240 --> 01:08:16.140 So the maximal state, the only way we can make this 01:08:16.140 --> 01:08:25.600 is to take J1, J2, M1 equals J1, M2 equals J2. 01:08:25.600 --> 01:08:29.939 Our spins are completely aligned in the total up direction. 01:08:29.939 --> 01:08:30.594 Yeah? 01:08:30.594 --> 01:08:33.010 AUDIENCE: Sir, would you be able to write a little larger? 01:08:33.010 --> 01:08:34.560 PROFESSOR: Yes, sorry. 01:08:34.560 --> 01:08:35.420 OK, yeah. 01:08:35.420 --> 01:08:36.670 That's why I like a big chalk. 01:08:36.670 --> 01:08:39.844 But we've run out of big chalk, so I'll try. 01:08:44.450 --> 01:08:50.439 So we know, also, that J squared is 01:08:50.439 --> 01:08:55.125 equal to J1 squared plus J2 squared plus the dot product. 01:09:00.779 --> 01:09:06.300 We can write that out as J1 squared plus J2 squared 01:09:06.300 --> 01:09:22.531 plus 2J1z J2z plus J1 plus J2 minus plus J1 minus J2 plus. 01:09:25.279 --> 01:09:30.569 And then we can ask what does J squared on this state give? 01:09:33.740 --> 01:09:46.039 And this is J1, J2, J1 plus J2, J1 plus J2. 01:09:48.913 --> 01:09:53.490 So J squared, so we know what that should be. 01:09:53.490 --> 01:09:58.940 That should return J1 plus J2 times J1 plus J2 plus 1 times 01:09:58.940 --> 01:10:02.140 h bar squared, because J is the good quantum number. 01:10:02.140 --> 01:10:04.260 But let's let it act on this piece. 01:10:04.260 --> 01:10:10.300 So this equals J1 squared plus J2 01:10:10.300 --> 01:10:25.140 squared plus 2J1z J2z plus J1 plus J2 minus plus J1 minus J2 01:10:25.140 --> 01:10:32.406 plus acting on J1, J2, J1, J2. 01:10:32.406 --> 01:10:35.430 So this state here. 01:10:35.430 --> 01:10:37.410 So we know how that acts. 01:10:37.410 --> 01:10:39.520 So this one gives us-- everything 01:10:39.520 --> 01:10:41.290 gives us an h bar squared. 01:10:41.290 --> 01:10:46.390 This gives us J1 J1 plus 1, for this term, 01:10:46.390 --> 01:10:54.510 plus J2 J2 plus 1 for the second term. 01:10:54.510 --> 01:10:57.970 Each of these gives us the M quantum numbers. 01:10:57.970 --> 01:10:59.450 But that's J1 J2. 01:10:59.450 --> 01:11:05.930 So this is plus 2J1, J2. 01:11:05.930 --> 01:11:08.700 And now what does this one do? 01:11:08.700 --> 01:11:10.661 J1 plus on this state. 01:11:10.661 --> 01:11:11.452 AUDIENCE: Kills it. 01:11:11.452 --> 01:11:12.577 PROFESSOR: Kills it, right. 01:11:12.577 --> 01:11:16.200 Because it's trying to raise the M component of 1, 01:11:16.200 --> 01:11:17.800 and it's already maximal. 01:11:17.800 --> 01:11:20.800 And this one, J2 plus, also kills it. 01:11:20.800 --> 01:11:25.250 So you get plus 0 plus 0 times the state. 01:11:29.460 --> 01:11:33.590 So if you rearrange all of this you actually 01:11:33.590 --> 01:11:44.300 find you can write this as J1 plus J2, J1 01:11:44.300 --> 01:11:52.030 plus J2 plus 1 times the state, which is what you want. 01:11:55.400 --> 01:12:00.980 So the J squared operator acting in the coupled basis gives-- 01:12:00.980 --> 01:12:02.700 well, acting in the uncoupled basis 01:12:02.700 --> 01:12:04.658 gives you what you expect in the coupled basis. 01:12:10.180 --> 01:12:15.505 So now I need a big blackboard. 01:12:19.310 --> 01:12:23.400 So let's do an example of multiplying two things. 01:12:23.400 --> 01:12:25.280 So let's write out a multiplet. 01:12:25.280 --> 01:12:28.440 So we're going to take J1, and we're 01:12:28.440 --> 01:12:31.950 going to have J1 bigger than J2 here. 01:12:31.950 --> 01:12:38.130 So we've got J1 J1, J1 J1 minus 1. 01:12:44.610 --> 01:12:53.420 And then somewhere down here I've got J1 and 2J2 minus J1. 01:12:53.420 --> 01:12:58.110 And then all the way down to J1 minus J1. 01:13:02.220 --> 01:13:05.925 So this has two J1 plus 1 states. 01:13:09.960 --> 01:13:12.660 And I'm going to tensor that was another multiplet, 01:13:12.660 --> 01:13:15.350 with my J2 multiplet, which is going to be smaller. 01:13:15.350 --> 01:13:17.910 So I'm going to have J2 J2. 01:13:23.929 --> 01:13:25.220 Oh, maybe I'll put one more in. 01:13:28.570 --> 01:13:32.220 And down here we've got J2 comma minus J2. 01:13:35.270 --> 01:13:40.175 And so here we have two J2 plus 1 states. 01:13:47.160 --> 01:13:54.400 And importantly, this left hand side 01:13:54.400 --> 01:14:03.775 has two J1 minus J2 more states than the right hand side. 01:14:06.820 --> 01:14:11.830 Just counting those states that's pretty obvious. 01:14:11.830 --> 01:14:14.840 So now let's start multiplying these things, 01:14:14.840 --> 01:14:22.680 and forming states of particular values of M, the total M. 01:14:22.680 --> 01:14:29.730 So if we say we want M equals J1 plus J2 what can we do? 01:14:29.730 --> 01:14:30.750 How can we make that? 01:14:35.590 --> 01:14:38.190 So we have to take the top state in each case. 01:14:38.190 --> 01:14:41.740 Because if I take this one and I take this M value, 01:14:41.740 --> 01:14:43.209 I can't get up to this, right? 01:14:43.209 --> 01:14:44.750 So there's only one way to make this. 01:14:48.840 --> 01:14:51.370 So I'm going to draw a diagram of this. 01:14:51.370 --> 01:14:54.120 We're going to have a one state there. 01:14:54.120 --> 01:15:00.425 The next M value, J1 plus J2 minus 1, how can I make that? 01:15:03.770 --> 01:15:05.640 So I can start with this state, and I 01:15:05.640 --> 01:15:09.696 will multiply it by this one, right? 01:15:09.696 --> 01:15:11.590 Or, what else can I do? 01:15:14.178 --> 01:15:15.678 AUDIENCE: Start with the second down 01:15:15.678 --> 01:15:17.810 on the left and tensor with the top? 01:15:17.810 --> 01:15:18.810 PROFESSOR: That's right. 01:15:18.810 --> 01:15:19.986 So I take those two. 01:15:19.986 --> 01:15:20.985 So there are two states. 01:15:24.820 --> 01:15:29.320 And those two states are just two linear combinations. 01:15:29.320 --> 01:15:33.165 So let me draw two dots here, I can form two states. 01:15:36.130 --> 01:15:40.565 Keep going-- minus 2, I get three states. 01:15:44.760 --> 01:15:51.470 And let me try and draw lines here to guide this stuff. 01:15:56.500 --> 01:15:57.910 OK, I'm not going to keep going. 01:15:57.910 --> 01:16:02.660 But at some point we get-- what's 01:16:02.660 --> 01:16:04.650 the largest number of states of a given M 01:16:04.650 --> 01:16:07.536 I can make going to be? 01:16:07.536 --> 01:16:08.630 Can anyone see that? 01:16:11.840 --> 01:16:13.458 AUDIENCE: 2J2 plus 1? 01:16:13.458 --> 01:16:15.840 PROFESSOR: 2J2 plus 2, right, because I've 01:16:15.840 --> 01:16:18.820 got 2J2 plus 1 states here. 01:16:18.820 --> 01:16:21.210 And I'm taking one of these plus one of these 01:16:21.210 --> 01:16:30.510 will give me-- so down here I'll have an M equals-- what is it-- 01:16:30.510 --> 01:16:32.342 J1 minus J2. 01:16:32.342 --> 01:16:37.530 And here I have 2J2 plus 1 states. 01:16:37.530 --> 01:16:41.320 And so let me kind of draw some of these states in. 01:16:41.320 --> 01:16:42.735 And then dot, dot, dot. 01:16:42.735 --> 01:16:49.170 And then over here we end up with this guy. 01:16:49.170 --> 01:16:51.955 So if I go down to the next one, how many states? 01:16:56.630 --> 01:17:02.910 So to form those states I was taking this top state 01:17:02.910 --> 01:17:04.280 with the bottom state here. 01:17:04.280 --> 01:17:06.640 That gives me J1 minus J2, right? 01:17:06.640 --> 01:17:09.590 Or I was taking the second state here with the second to bottom 01:17:09.590 --> 01:17:11.200 state here, and so forth. 01:17:11.200 --> 01:17:14.910 And then all the way up to here. 01:17:14.910 --> 01:17:20.020 Now if I then start shifting things down in this side, 01:17:20.020 --> 01:17:22.950 but leave exactly the same things over there, 01:17:22.950 --> 01:17:25.250 then I'll lower J by 1. 01:17:25.250 --> 01:17:29.030 And I'll keep doing it until I hit the bottom. 01:17:29.030 --> 01:17:32.860 And because there's this number of states 01:17:32.860 --> 01:17:35.906 more in the right hand side than the left hand side-- hang on, 01:17:35.906 --> 01:17:37.030 let me just write this one. 01:17:43.370 --> 01:17:45.980 OK, we might need to go onto the next board. 01:17:45.980 --> 01:17:47.740 So this keeps on going until I get 01:17:47.740 --> 01:17:53.750 to M equals-- I don't remember the number-- M 01:17:53.750 --> 01:17:58.780 equals J2 minus J1. 01:17:58.780 --> 01:18:01.715 And there are 2J plus 2 plus 1 states of this. 01:18:09.880 --> 01:18:14.560 And then once I do that, then I start 01:18:14.560 --> 01:18:15.810 having fewer and fewer states. 01:18:15.810 --> 01:18:17.190 Because I've gone basically moving out 01:18:17.190 --> 01:18:18.398 the bottom of this multiplet. 01:18:23.340 --> 01:18:30.950 And so here we have, this is 2 J1 minus J2 plus 1 rows. 01:18:33.522 --> 01:18:34.730 And then I start contracting. 01:18:39.590 --> 01:18:44.700 So the next one, M equals J2 minus J1 minus 1 01:18:44.700 --> 01:18:46.035 has two J2 states. 01:18:53.730 --> 01:18:54.820 So then we can keep going. 01:18:58.090 --> 01:19:01.410 And this is meant to continue this diagram up here. 01:19:01.410 --> 01:19:03.660 So then we keep going down, down, down. 01:19:03.660 --> 01:19:11.600 And then we'd have M equals minus J1 plus 1. 01:19:11.600 --> 01:19:13.545 And how many states can I make that have that? 01:19:18.530 --> 01:19:24.460 So I need to take this one, and I could take the-- oh, 01:19:24.460 --> 01:19:25.930 where is it? 01:19:25.930 --> 01:19:27.220 Sorry, not this. 01:19:27.220 --> 01:19:28.380 This is not what I mean. 01:19:28.380 --> 01:19:32.620 Minus J1 minus J2 plus 1. 01:19:32.620 --> 01:19:34.220 That's more obvious, right? 01:19:34.220 --> 01:19:35.170 So there's two states. 01:19:39.650 --> 01:19:42.120 And so this picture has kind of-- this line 01:19:42.120 --> 01:19:43.340 starts coming in. 01:19:43.340 --> 01:19:45.140 And now I've got my two states here. 01:19:45.140 --> 01:19:47.430 I have the next one, I've got three states. 01:19:47.430 --> 01:19:53.950 And then finally, M equals minus J1 minus J2, I have one state. 01:19:53.950 --> 01:19:54.900 And so I get this. 01:19:54.900 --> 01:19:58.620 And so you actually-- oh, that was not very well drawn. 01:20:01.760 --> 01:20:07.160 So if you look how many states there are in this first column, 01:20:07.160 --> 01:20:10.460 how many is going to be there? 01:20:10.460 --> 01:20:16.050 So it goes from plus J1 plus J2 to minus J1 minus J2. 01:20:16.050 --> 01:20:23.860 So there's two J1 plus J2 plus 1 states there. 01:20:23.860 --> 01:20:30.660 And here there is two J1 plus J2 minus 1 01:20:30.660 --> 01:20:32.850 plus 1 states in this guy. 01:20:32.850 --> 01:20:36.730 And so this is a J equals J1 plus J2. 01:20:36.730 --> 01:20:42.650 This one is J equals J1 plus J2 minus 1. 01:20:42.650 --> 01:20:46.630 And if you are careful you'd find that this one here, 01:20:46.630 --> 01:20:50.170 the last one here, this has this many states in it, 01:20:50.170 --> 01:20:53.030 two J minus 1 plus 1 states. 01:20:53.030 --> 01:20:59.224 So this is a J equals J1 minus J2. 01:20:59.224 --> 01:21:02.130 And to be completely correct, we put 01:21:02.130 --> 01:21:04.530 an absolute value in case J2 is bigger than J1. 01:21:07.710 --> 01:21:11.955 So this is our full multiplet structure of this system. 01:21:11.955 --> 01:21:15.480 So all of the states in this column 01:21:15.480 --> 01:21:17.945 will transform into each other under rotations, 01:21:17.945 --> 01:21:18.820 and things like this. 01:21:18.820 --> 01:21:21.910 And same for each column, they all form separate multiplets. 01:21:26.930 --> 01:21:36.000 So just some last things before we finish. 01:21:36.000 --> 01:21:38.780 So another property of Clebsch-Gordan 01:21:38.780 --> 01:21:49.330 coefficients we can choose them to be real. 01:21:56.962 --> 01:22:05.580 They satisfy a recursion relation 01:22:05.580 --> 01:22:07.430 but don't have a nice, closed form. 01:22:20.780 --> 01:22:24.250 I think this is in Griffiths. 01:22:24.250 --> 01:22:26.390 It gives you what this recursion relation is. 01:22:26.390 --> 01:22:30.230 I think it does, at least many books do. 01:22:30.230 --> 01:22:36.515 And also, they're tabulated in lots of places. 01:22:40.930 --> 01:22:42.400 So if you need to know the values, 01:22:42.400 --> 01:22:43.680 you can just go and look them up, 01:22:43.680 --> 01:22:45.888 rather than trying to calculate them all necessarily. 01:22:49.620 --> 01:22:52.720 And I think that's all we've got time for. 01:22:52.720 --> 01:22:54.380 So are there any questions about that? 01:22:56.994 --> 01:22:58.410 Any questions about anything else? 01:23:00.980 --> 01:23:01.630 OK, great. 01:23:01.630 --> 01:23:05.190 So we will see you on Wednesday for the last lecture.