WEBVTT 00:00:00.070 --> 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 --> 00:00:10.120 to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.056 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.056 --> 00:00:17.205 at ocs.mit.edu. 00:00:21.240 --> 00:00:23.770 PROFESSOR: Today we have plenty to do. 00:00:23.770 --> 00:00:29.780 We really begin in all generality 00:00:29.780 --> 00:00:32.580 the addition of angular momentum. 00:00:32.580 --> 00:00:37.970 But we will do it in the set up of a physical problem. 00:00:37.970 --> 00:00:40.020 The problem of computing the spin 00:00:40.020 --> 00:00:46.960 orbit interactions of electrons with the nucleus. 00:00:46.960 --> 00:00:51.940 So this is a rather interesting and complicated interaction. 00:00:51.940 --> 00:00:54.610 So we'll spend a little time telling you 00:00:54.610 --> 00:00:57.520 about the physics of this interaction. 00:00:57.520 --> 00:01:01.230 And then once the physics is clear, 00:01:01.230 --> 00:01:04.720 it will become more obvious why we 00:01:04.720 --> 00:01:08.980 have to do these mathematical contortions of adding angular 00:01:08.980 --> 00:01:13.960 momentum in order to solve this physical problem. 00:01:13.960 --> 00:01:17.790 So it's a sophisticated problem that requires several steps. 00:01:17.790 --> 00:01:22.420 The first step is something that is 00:01:22.420 --> 00:01:26.480 a result in perturbation theory. 00:01:26.480 --> 00:01:30.460 Feynman Hellman Theorem of perturbation theory. 00:01:30.460 --> 00:01:32.270 And that's where we begin. 00:01:32.270 --> 00:01:35.770 So it's called Feynman Hellman Theorem. 00:01:35.770 --> 00:01:41.900 It's a very simple result. 00:01:46.605 --> 00:01:47.105 Theorem. 00:01:50.580 --> 00:01:54.530 And we'll need it in order to understand 00:01:54.530 --> 00:01:57.990 how a small perturbation to the Hamiltonian 00:01:57.990 --> 00:02:00.240 changes the energy spectrum. 00:02:00.240 --> 00:02:12.110 So we have H of lambda be a Hamiltonian with a parameter 00:02:12.110 --> 00:02:13.252 in lambda. 00:02:17.190 --> 00:02:19.540 Lambda. 00:02:19.540 --> 00:02:30.640 And psi n of lambda being normalized energy 00:02:30.640 --> 00:02:40.550 eigenstate with energy, En of lambda. 00:02:44.460 --> 00:02:48.890 So that's the whole assumption of the theorem. 00:02:48.890 --> 00:02:51.130 We have a Hamiltonian. 00:02:51.130 --> 00:02:55.080 It depends on some parameter that we're going to vary. 00:02:55.080 --> 00:03:01.530 And suppose we consider now an eigenstate of this Hamiltonian 00:03:01.530 --> 00:03:04.140 that depends on lambda, so the eigenstate also 00:03:04.140 --> 00:03:06.050 depends on lambda. 00:03:06.050 --> 00:03:09.770 And it has an energy, En of lambda. 00:03:09.770 --> 00:03:11.860 So the purpose of this theorem is 00:03:11.860 --> 00:03:15.130 to relate these various quantities. 00:03:15.130 --> 00:03:22.530 And the claim is that the rate of change 00:03:22.530 --> 00:03:26.810 of the energy with respect to lambda 00:03:26.810 --> 00:03:34.900 can be computed pretty much by evaluating the rate of change 00:03:34.900 --> 00:03:38.140 of the Hamiltonian on the relevant states. 00:03:50.740 --> 00:03:51.660 So that's the claim. 00:04:01.010 --> 00:04:05.740 And it's a pretty nice result. 00:04:05.740 --> 00:04:08.940 It's useful in many circumstances. 00:04:08.940 --> 00:04:13.630 And for us will be a way to discuss a little perturbation 00:04:13.630 --> 00:04:14.470 theory. 00:04:14.470 --> 00:04:19.240 Perturbation theory is the subject of 806 in all details. 00:04:19.240 --> 00:04:21.970 And it's a very sophisticated subject. 00:04:21.970 --> 00:04:25.970 Even today we were going to be finding that it's not 00:04:25.970 --> 00:04:30.140 all that easy to carry it out. 00:04:30.140 --> 00:04:32.100 So how does this begin? 00:04:32.100 --> 00:04:32.890 Well, proof. 00:04:36.290 --> 00:04:39.410 You begin by saying that En of lambda 00:04:39.410 --> 00:04:42.860 is the energy eigenstate, is nothing else 00:04:42.860 --> 00:04:47.200 but psi n of lambda. 00:04:47.200 --> 00:04:49.950 H of lambda. 00:04:49.950 --> 00:04:51.810 Psi n of lambda. 00:04:55.730 --> 00:05:01.860 And the reason is, of course, that H and psi n 00:05:01.860 --> 00:05:07.400 is En of lambda times psi n of lambda. 00:05:07.400 --> 00:05:09.385 And this is a number goes out. 00:05:09.385 --> 00:05:12.400 And the inner product of this things 00:05:12.400 --> 00:05:16.140 is 1, because the state is normalized. 00:05:16.140 --> 00:05:18.980 So this is a good starting point. 00:05:18.980 --> 00:05:24.780 And the funny thing that you see already is that, in some sense, 00:05:24.780 --> 00:05:27.840 you just get the middle term when 00:05:27.840 --> 00:05:30.290 you take the derivative with respect to lambda. 00:05:30.290 --> 00:05:32.320 You don't get anything from these two. 00:05:34.870 --> 00:05:36.540 And it's simple in fact. 00:05:36.540 --> 00:05:38.310 So let me just do it. 00:05:38.310 --> 00:05:43.880 V, En, V lambda would be the term 00:05:43.880 --> 00:05:49.140 that Feynman and Hellman gave. 00:05:49.140 --> 00:05:55.400 V, H, V lambda, psi n of lambda. 00:05:55.400 --> 00:05:58.470 Plus one term in which we differentiate this one. 00:05:58.470 --> 00:06:05.500 V, d lambda of the state psi n of lambda. 00:06:09.340 --> 00:06:17.130 Times H of lambda, psi n of lambda. 00:06:17.130 --> 00:06:21.380 Plus the other term in which you differentiate the ket. 00:06:21.380 --> 00:06:32.610 So psi n of lambda, H of lambda, d, d lambda of psi n of lambda. 00:06:38.880 --> 00:06:46.710 And the reason these terms are going to vanish 00:06:46.710 --> 00:06:53.020 is that you can now act with H again on psi n. 00:06:53.020 --> 00:06:57.620 H is supposed to be Hermitian, so it can act on the left. 00:06:57.620 --> 00:07:07.220 And therefore, these two terms give you En of lambda, 00:07:07.220 --> 00:07:19.400 times d d lambda of psi n of lambda-- psi n of lambda-- 00:07:19.400 --> 00:07:23.750 plus the other term, which would be psi n of lambda, 00:07:23.750 --> 00:07:27.395 the bra times the derivative of the ket. 00:07:35.450 --> 00:07:43.550 But this is nothing else than the derivative 00:07:43.550 --> 00:07:44.730 of the inner product. 00:07:55.480 --> 00:07:57.250 In the inner product to differentiate-- 00:07:57.250 --> 00:08:00.020 the inner product differentiates the bra, 00:08:00.020 --> 00:08:01.920 it differentiates the ket. 00:08:01.920 --> 00:08:04.280 And do it. 00:08:04.280 --> 00:08:08.220 And this thing is equal to 1, because it's normalized. 00:08:08.220 --> 00:08:10.890 So this is 0. 00:08:10.890 --> 00:08:12.650 End of proof. 00:08:12.650 --> 00:08:16.320 These two terms vanish. 00:08:16.320 --> 00:08:20.860 And the result holds. 00:08:20.860 --> 00:08:21.652 Yes? 00:08:21.652 --> 00:08:23.527 AUDIENCE: How do you know it stays normalized 00:08:23.527 --> 00:08:24.505 when you vary lambda? 00:08:24.505 --> 00:08:28.440 PROFESSOR: It's an assumption. 00:08:28.440 --> 00:08:31.770 The state is normalized for all values of n. 00:08:31.770 --> 00:08:34.925 So if you have a state that you've constructed, 00:08:34.925 --> 00:08:39.669 that is normalized, you can have this result. 00:08:39.669 --> 00:08:41.030 So it's an assumption. 00:08:41.030 --> 00:08:42.845 You have to keep the state normalized. 00:08:46.900 --> 00:08:50.900 Now this is a baby version of perturbation theory. 00:08:50.900 --> 00:08:55.320 It's a result I think that Feynman did as an undergrad. 00:08:55.320 --> 00:08:59.290 And as you can see, it's very simple. 00:08:59.290 --> 00:09:03.290 Calling it a theorem is a little too much. 00:09:03.290 --> 00:09:06.540 But still, the fact is that it's useful. 00:09:06.540 --> 00:09:10.670 And so we'll just go ahead and use it. 00:09:10.670 --> 00:09:14.910 Now I want to rewrite it in another way. 00:09:14.910 --> 00:09:19.250 So, suppose you have a Hamiltonian, H, 00:09:19.250 --> 00:09:24.755 which has a term H0, plus lambda, H1. 00:09:27.380 --> 00:09:41.770 So, the parameter lambda, H of lambda, is given in this way. 00:09:41.770 --> 00:09:45.180 And that's a reasonable H of lambda. 00:09:45.180 --> 00:09:49.860 Sometimes, this could be written as H0 plus something 00:09:49.860 --> 00:09:52.670 that we will call the change in the Hamiltonian. 00:09:52.670 --> 00:09:57.540 And we usually think of it as a small thing. 00:10:01.300 --> 00:10:05.190 So what do we have from this theorem? 00:10:08.410 --> 00:10:18.020 From this here we would have the d, En, d lambda 00:10:18.020 --> 00:10:29.380 is equal to psi n of lambda, H1, psi n of lambda. 00:10:34.100 --> 00:10:37.400 Now, we can be particularly interested 00:10:37.400 --> 00:10:41.250 in the evaluation of this thing at lambda equals 0. 00:10:41.250 --> 00:10:44.830 So what is d En of lambda? 00:10:44.830 --> 00:10:55.330 d lambda at lambda equals 0 would be psi n at zero, 00:10:55.330 --> 00:11:00.970 H1, psi n at 0. 00:11:00.970 --> 00:11:07.870 And therefore, you would say that the En of lambda energies 00:11:07.870 --> 00:11:19.790 would be the energies at 0, plus lambda, d En 00:11:19.790 --> 00:11:25.510 of lambda, d lambda at lambda equals 0, 00:11:25.510 --> 00:11:28.093 plus order lambda squared. 00:11:35.610 --> 00:11:38.740 I'm doing just the Taylor expansion 00:11:38.740 --> 00:11:46.450 of En's of lambda from lambda equals 0. 00:11:46.450 --> 00:11:56.480 So this thing tells you that En of lambda is equal En of 0, 00:11:56.480 --> 00:12:08.302 plus-- this derivative you can write it as psi n, lambda, H1, 00:12:08.302 --> 00:12:13.008 psi n, all at 0. 00:12:13.008 --> 00:12:13.980 Like that. 00:12:13.980 --> 00:12:16.955 Plus order lambda squared. 00:12:23.650 --> 00:12:28.600 So in this step, I just use the evaluation that we did here. 00:12:28.600 --> 00:12:32.380 I substituted that and put the lambda in. 00:12:32.380 --> 00:12:45.470 So that I recognize now that En of lambda is equal to En of 0, 00:12:45.470 --> 00:12:53.990 plus psi n of 0-- and I can write this as delta H-- psi 00:12:53.990 --> 00:13:00.803 n of 0, plus order delta H squared. 00:13:09.260 --> 00:13:12.780 It's nice to write it this way, because you appreciate 00:13:12.780 --> 00:13:15.540 more the power of the theorem. 00:13:15.540 --> 00:13:20.640 The theorem here doesn't assume which value of lambda you have. 00:13:20.640 --> 00:13:23.070 And you have to have normalized eigenstates. 00:13:23.070 --> 00:13:26.320 And you wonder what is it helping you with, 00:13:26.320 --> 00:13:30.820 if finding the states for every value of lambda is complicated. 00:13:30.820 --> 00:13:37.160 Well, it certainly helps you to figure out 00:13:37.160 --> 00:13:42.960 how the energy of the state varies by a simple calculation. 00:13:42.960 --> 00:13:46.725 Suppose you know the states of the simple Hamiltonian. 00:13:49.980 --> 00:13:55.630 Those are the psi's, n, 0. 00:13:55.630 --> 00:13:59.870 So if you have the psi n 0 over here, 00:13:59.870 --> 00:14:03.120 you can do the following step. 00:14:03.120 --> 00:14:07.930 If you want to figure out how it's energy has varied, 00:14:07.930 --> 00:14:14.260 use this formula in which you compute the expectation 00:14:14.260 --> 00:14:20.370 value of the change in the Hamiltonian on that state. 00:14:20.370 --> 00:14:26.310 And that is the first correction to the energy of the state. 00:14:26.310 --> 00:14:29.010 So you have this state. 00:14:29.010 --> 00:14:31.710 You compute the expectation value 00:14:31.710 --> 00:14:34.880 of the extra piece in the Hamiltonian. 00:14:34.880 --> 00:14:39.270 And that's the correction to the energy. 00:14:39.270 --> 00:14:41.300 It's a little more complicated of course 00:14:41.300 --> 00:14:43.600 to compute the correction to the state. 00:14:43.600 --> 00:14:47.570 But that's a subject of perturbation theory. 00:14:47.570 --> 00:14:51.790 And that's not what we care about right now. 00:14:51.790 --> 00:14:57.410 So the reason we're doing this is because actually whatever 00:14:57.410 --> 00:15:01.010 we're going to have with spin orbit coupling 00:15:01.010 --> 00:15:04.100 represents an addition to the hydrogen 00:15:04.100 --> 00:15:06.680 Hamiltonian of a new term. 00:15:06.680 --> 00:15:10.060 Therefore, you want to know what happens to the energy levels. 00:15:10.060 --> 00:15:11.970 And the best thing to think about them 00:15:11.970 --> 00:15:15.360 is to-- if you know the energy levels of this one, 00:15:15.360 --> 00:15:17.980 well, a formula of this type can let 00:15:17.980 --> 00:15:20.082 you know what happens to the energy levels 00:15:20.082 --> 00:15:21.040 after the perturbation. 00:15:23.850 --> 00:15:25.790 There will be an extra complication 00:15:25.790 --> 00:15:28.500 in that the energy levels that we're going to deal with 00:15:28.500 --> 00:15:29.930 are going to be degenerate. 00:15:29.930 --> 00:15:32.980 But let's wait for that complication until it appears. 00:15:32.980 --> 00:15:34.650 So any questions? 00:15:37.960 --> 00:15:38.870 Yes? 00:15:38.870 --> 00:15:40.820 AUDIENCE: So I would imagine that this 00:15:40.820 --> 00:15:42.704 would work just as well for time. 00:15:42.704 --> 00:15:45.974 Because time [INAUDIBLE] a parameter in quantum mechanics. 00:15:45.974 --> 00:15:48.404 So [INAUDIBLE] 00:15:48.404 --> 00:15:50.830 PROFESSOR: Time dependent perturbation theory 00:15:50.830 --> 00:15:52.930 is a bit more complicated. 00:15:52.930 --> 00:15:56.860 I'd rather not get into it now. 00:15:56.860 --> 00:16:01.450 So let's leave it here, in which we don't have time. 00:16:01.450 --> 00:16:03.280 And the Schrodinger equation is something 00:16:03.280 --> 00:16:07.617 like H psi equal [INAUDIBLE] psi, that's all we care. 00:16:07.617 --> 00:16:11.250 And leave it for that moment. 00:16:11.250 --> 00:16:11.933 Other questions? 00:16:21.496 --> 00:16:22.460 OK. 00:16:22.460 --> 00:16:28.530 So let's proceed with addition of angular momentum. 00:16:28.530 --> 00:16:34.160 So first, let me give you the fundamental result 00:16:34.160 --> 00:16:36.010 of addition of angular momentum. 00:16:36.010 --> 00:16:39.880 It's a little abstract, but it's what we really 00:16:39.880 --> 00:16:41.725 mean by addition of angular momentum. 00:16:45.197 --> 00:16:48.763 Of angular momentum. 00:16:51.520 --> 00:16:53.960 And the main result is the following. 00:16:53.960 --> 00:17:00.450 Suppose you have a set of operators, J, i, 00:17:00.450 --> 00:17:04.929 1, that have the algebra of angular momentum. 00:17:08.200 --> 00:17:11.250 Of angular momentum. 00:17:11.250 --> 00:17:21.609 Which is to say Ji1, JJ1, is equal 00:17:21.609 --> 00:17:25.712 i, h bar, epsilon iJK, JK1. 00:17:29.870 --> 00:17:34.180 And this algebra is realized on some state space. 00:17:34.180 --> 00:17:41.330 On some vector space, V1. 00:17:41.330 --> 00:17:44.760 And suppose you have another operator, 00:17:44.760 --> 00:17:48.180 J-- set of operators actually. 00:17:48.180 --> 00:17:51.395 Ji2, which have the algebra of angular momentum. 00:17:51.395 --> 00:17:53.470 I will not write that. 00:17:53.470 --> 00:17:56.280 On some V2. 00:17:59.530 --> 00:18:00.640 OK. 00:18:00.640 --> 00:18:03.340 Angular momentum, some sets of states. 00:18:03.340 --> 00:18:07.570 Angular momentum on some other set of states. 00:18:07.570 --> 00:18:10.090 Here comes the thing. 00:18:10.090 --> 00:18:13.620 There is a new angular momentum, which 00:18:13.620 --> 00:18:24.970 is the sum Ji defined as Ji1, added with Ji2. 00:18:24.970 --> 00:18:30.560 Now, soon enough you will just write Ji1, plus Ji2. 00:18:30.560 --> 00:18:34.590 But let me be a little more careful now. 00:18:34.590 --> 00:18:45.290 This sum is Ji1, plus 1, tensor Ji2. 00:18:45.290 --> 00:18:48.080 So i is the same index. 00:18:48.080 --> 00:18:53.230 But here, we're having this operator 00:18:53.230 --> 00:18:57.140 that we're being defined that we call it the sum. 00:18:57.140 --> 00:19:01.730 Now how do you sum two operators that act in different spaces? 00:19:01.730 --> 00:19:04.680 Well, the only thing that you can actually do 00:19:04.680 --> 00:19:07.120 is sum them in the tensor product. 00:19:07.120 --> 00:19:19.136 So the claim is that this is an angular momentum in V1 tensor 00:19:19.136 --> 00:19:19.635 V2. 00:19:25.810 --> 00:19:28.190 That is an operator. 00:19:28.190 --> 00:19:29.690 You see, you have to sum them. 00:19:29.690 --> 00:19:34.490 So you have to create a space where both can act, 00:19:34.490 --> 00:19:36.180 and you can sum them. 00:19:36.180 --> 00:19:40.320 You cannot sum a thing, an operator that acts on one 00:19:40.320 --> 00:19:43.420 vector space to an operator that acts on another vector space. 00:19:43.420 --> 00:19:47.850 You have to create one vector space where both act. 00:19:47.850 --> 00:19:51.250 And then you can define the sum of the operators. 00:19:51.250 --> 00:19:53.750 Sum of operators is a simple thing. 00:19:53.750 --> 00:19:56.460 So you form the tensor product. 00:19:56.460 --> 00:20:00.680 In here, this operator gets upgraded 00:20:00.680 --> 00:20:05.390 in this way, in which in the tensor product it has a 1 00:20:05.390 --> 00:20:06.970 for the second input. 00:20:06.970 --> 00:20:09.290 This one gets upgrade to this way. 00:20:09.290 --> 00:20:10.250 And this is the sum. 00:20:12.950 --> 00:20:17.270 So this is a claim-- this is a definition. 00:20:17.270 --> 00:20:18.640 And this is a claim. 00:20:18.640 --> 00:20:21.430 So this has to be proven. 00:20:21.430 --> 00:20:23.370 So let me prove it. 00:20:26.140 --> 00:20:28.910 Ji, JJ. 00:20:28.910 --> 00:20:30.730 I compute this commutator. 00:20:30.730 --> 00:20:32.930 So I don't have to do the following. 00:20:32.930 --> 00:20:45.530 I have to do Ji1, tensor 1, plus 1 tensor Ji2. 00:20:45.530 --> 00:20:56.620 And then the JJ would be JJ1, tensor 1, plus 1, tensor JJ2. 00:20:59.670 --> 00:21:03.320 Have to compute this commutator. 00:21:03.320 --> 00:21:06.880 Now, an important fact about this 00:21:06.880 --> 00:21:09.320 result that I'm not trying to generalize, 00:21:09.320 --> 00:21:12.700 if you had put a minus here, it wouldn't work out. 00:21:12.700 --> 00:21:15.480 If you would have put a 2 here, it wouldn't work out. 00:21:15.480 --> 00:21:19.130 If you would have put a 1/2 here, it won't work out. 00:21:19.130 --> 00:21:22.060 This is pretty much the only way you 00:21:22.060 --> 00:21:26.370 can have two angular momenta, and create a third angular 00:21:26.370 --> 00:21:27.620 momentum. 00:21:27.620 --> 00:21:29.240 So look at this. 00:21:32.730 --> 00:21:34.820 It looks like we're going to have to work hard, 00:21:34.820 --> 00:21:38.490 but that's not true. 00:21:38.490 --> 00:21:40.650 Consider this commutator. 00:21:40.650 --> 00:21:42.970 The commutator of this term with this term. 00:21:45.510 --> 00:21:47.430 That's 0 actually. 00:21:47.430 --> 00:21:51.950 Because if you multiply them in this order, this times that, 00:21:51.950 --> 00:21:57.080 you get Ji1 times Ji2, because the ones do nothing. 00:21:57.080 --> 00:21:59.710 You multiply them in the reverse order, 00:21:59.710 --> 00:22:03.860 you get again, Ji1 times Ji2. 00:22:03.860 --> 00:22:08.895 This is to say that the operators that originally lived 00:22:08.895 --> 00:22:13.510 in the different vector spaces commute. 00:22:13.510 --> 00:22:15.260 Yes? 00:22:15.260 --> 00:22:19.320 AUDIENCE: Since the cross terms between those two 00:22:19.320 --> 00:22:22.900 are 0-- like you just said, the cross terms are 0. 00:22:22.900 --> 00:22:27.600 And if you put a minus sign in there, it will cancel. 00:22:27.600 --> 00:22:29.494 But when you do the multiplications 00:22:29.494 --> 00:22:33.318 with the second ones, why can't you put a minus sign in there? 00:22:33.318 --> 00:22:34.170 [INAUDIBLE] 00:22:34.170 --> 00:22:35.420 PROFESSOR: In the whole thing? 00:22:35.420 --> 00:22:37.790 In this definition, a minus sign? 00:22:37.790 --> 00:22:38.720 AUDIENCE: Yeah. 00:22:38.720 --> 00:22:41.900 PROFESSOR: Well, here if I put a minus-- 00:22:41.900 --> 00:22:45.270 it's like I'm going to prove that this works. 00:22:45.270 --> 00:22:48.860 So if-- I'm going to get an angular momentum. 00:22:48.860 --> 00:22:51.750 If I put a minus sign to angular momentum, 00:22:51.750 --> 00:22:54.420 I ruin the algebra here. 00:22:54.420 --> 00:22:56.810 I put a minus minus, it cancels. 00:22:56.810 --> 00:23:00.010 But then I get a minus sign here. 00:23:00.010 --> 00:23:03.940 So I cannot really even change a sign. 00:23:03.940 --> 00:23:09.690 So any way, these are operators acting on different spaces. 00:23:09.690 --> 00:23:11.320 They commute. 00:23:11.320 --> 00:23:13.375 It's clear they commute. 00:23:13.375 --> 00:23:15.700 You just multiply them, and see that. 00:23:15.700 --> 00:23:18.860 These one's commute as well. 00:23:18.860 --> 00:23:23.730 The only ones that don't commute are this with this. 00:23:23.730 --> 00:23:25.310 And that with that. 00:23:25.310 --> 00:23:27.520 So let me just write them. 00:23:27.520 --> 00:23:38.430 Ji1, tensor 1, with JJ1, tensor 1. 00:23:38.430 --> 00:23:50.347 Plus this one, 1 tensor Ji2, 1 tensor JJ2. 00:23:58.730 --> 00:24:01.870 OK, next step is to realize that actually 00:24:01.870 --> 00:24:07.280 the 1 is a spectator here. 00:24:07.280 --> 00:24:10.180 Therefore, this commutator is nothing 00:24:10.180 --> 00:24:19.130 but the commutator Ji1 with JJ1, tensor 1. 00:24:23.370 --> 00:24:24.510 You can do it. 00:24:24.510 --> 00:24:27.530 If you prefer to write it, write it. 00:24:27.530 --> 00:24:32.930 This product is Ji times JJ, tensor 1. 00:24:32.930 --> 00:24:37.240 And the other product is JJ, Ji, tensor 1. 00:24:37.240 --> 00:24:41.040 So the tensor 1 factors out. 00:24:41.040 --> 00:24:44.980 Here the tensor 1 also factors out. 00:24:44.980 --> 00:24:50.470 And you get an honest commutator, Ji2, JJ2. 00:24:53.300 --> 00:24:57.290 So one last step. 00:24:57.290 --> 00:25:01.920 This is i, h bar, epsilon, iJK. 00:25:01.920 --> 00:25:04.750 I'll put a big parentheses. 00:25:04.750 --> 00:25:12.990 JK1, tensor 1, for the first one. 00:25:12.990 --> 00:25:17.530 Because J1 forms an angular momentum algebra. 00:25:17.530 --> 00:25:21.750 And here, 1 tensor JK2. 00:25:30.670 --> 00:25:36.280 And this thing is i, h bar, epsilon, iJK. 00:25:36.280 --> 00:25:41.880 The total angular momentum, K. And you've 00:25:41.880 --> 00:25:43.990 shown the algebra works out. 00:25:48.370 --> 00:25:52.620 Now most people after a little practice, 00:25:52.620 --> 00:26:02.890 they just say, oh, Ji is J1 plus J2, J1 plus J2. 00:26:02.890 --> 00:26:04.910 J1 and J2 don't commute. 00:26:04.910 --> 00:26:07.870 J2 and J1-- I'm sorry. 00:26:07.870 --> 00:26:09.390 J1 and J2 commute. 00:26:09.390 --> 00:26:11.110 J2 and J1 commute. 00:26:11.110 --> 00:26:14.470 Therefore you get this 2, like that. 00:26:14.470 --> 00:26:18.760 And this gives you-- J1 and J1 gives you J1. 00:26:18.760 --> 00:26:22.230 J2 and J2 gives you J2, so the sum works out. 00:26:22.230 --> 00:26:25.530 So most people after a little practice 00:26:25.530 --> 00:26:28.100 just don't put all these tensor things. 00:26:28.100 --> 00:26:32.320 But at the beginning it's nice to just make sure 00:26:32.320 --> 00:26:36.700 that you understand what these tensor things do. 00:26:36.700 --> 00:26:37.280 All right. 00:26:37.280 --> 00:26:40.700 So that's our main theorem-- that you 00:26:40.700 --> 00:26:45.180 start with one angular momentum on a state space. 00:26:45.180 --> 00:26:49.090 Another angular momentum that has nothing to do perhaps 00:26:49.090 --> 00:26:53.170 with the first on another vector space. 00:26:53.170 --> 00:26:58.390 And on the tensor product you have another angular momentum, 00:26:58.390 --> 00:26:59.420 which is the sum. 00:27:01.940 --> 00:27:02.670 All right. 00:27:02.670 --> 00:27:05.590 So now, we do spin orbit coupling 00:27:05.590 --> 00:27:08.660 to try to apply these ideas. 00:27:08.660 --> 00:27:25.060 So for spin orbit coupling, we will consider the hydrogen atom 00:27:25.060 --> 00:27:28.000 coupling. 00:27:28.000 --> 00:27:36.610 And the new term in the Hamiltonian, mu dot B. 00:27:36.610 --> 00:27:41.390 The kind of term that we've done so much in this semester. 00:27:41.390 --> 00:27:43.880 We've looked over magnetic ones. 00:27:43.880 --> 00:27:46.700 So which magnetic moment at which B? 00:27:46.700 --> 00:27:50.050 There was no B in the hydrogen atom. 00:27:50.050 --> 00:27:54.850 Well, there's no B to begin with. 00:27:54.850 --> 00:27:58.980 But here is one where you can think there is a B. First, 00:27:58.980 --> 00:28:06.280 this will be the electron dipole moment. 00:28:06.280 --> 00:28:07.790 Magnetic dipole moment. 00:28:07.790 --> 00:28:11.350 So we have a formula for it. 00:28:11.350 --> 00:28:17.370 The formula for it is the mu of the electron is minus E over m, 00:28:17.370 --> 00:28:19.960 times the spin of the electron. 00:28:19.960 --> 00:28:25.090 And I actually will use a little different formula 00:28:25.090 --> 00:28:28.130 that is valued in Gaussian units. 00:28:28.130 --> 00:28:35.550 ge over mC, S, in Gaussian units. 00:28:38.590 --> 00:28:42.070 And g is the g factor of the electron, which is 2. 00:28:42.070 --> 00:28:42.770 I'm sorry. 00:28:42.770 --> 00:28:44.850 There's a 2 here. 00:28:44.850 --> 00:28:45.440 OK. 00:28:45.440 --> 00:28:46.990 So look what I've written. 00:28:46.990 --> 00:28:49.540 I don't want to distract you with this too much. 00:28:49.540 --> 00:28:53.270 But you know that the magnetic dipole of the electron 00:28:53.270 --> 00:28:55.310 is given by this quantity. 00:28:55.310 --> 00:28:57.930 Now, you could put a 2 up, and a 2 down. 00:28:57.930 --> 00:29:02.120 And that's why people actually classically 00:29:02.120 --> 00:29:04.410 there seems to be a 2 down. 00:29:04.410 --> 00:29:08.220 But there's a 2 up, because it's an effect of the electron. 00:29:08.220 --> 00:29:09.640 And you have this formula. 00:29:09.640 --> 00:29:11.650 The only thing I've added in that formula 00:29:11.650 --> 00:29:16.390 is a factor of C that is because of Gaussian units. 00:29:16.390 --> 00:29:20.250 And it allows you to estimate terms a little more easily. 00:29:20.250 --> 00:29:22.610 So that's the mu of the electron. 00:29:22.610 --> 00:29:28.600 But the electron apparently would feel no magnetic field. 00:29:28.600 --> 00:29:31.320 You didn't put an external magnetic field. 00:29:31.320 --> 00:29:37.070 Except that here you go in this way 00:29:37.070 --> 00:29:40.840 of thinking-- you think suppose you are the electron. 00:29:40.840 --> 00:29:45.340 You see a proton, which is a nucleus going around you. 00:29:45.340 --> 00:29:49.750 And a proton going around you is a current going around you. 00:29:49.750 --> 00:29:52.280 It generates a magnetic field. 00:29:52.280 --> 00:29:54.840 And therefore, you see a magnetic field 00:29:54.840 --> 00:30:00.260 created by the proton going around you. 00:30:00.260 --> 00:30:02.400 So there is a magnetic field. 00:30:02.400 --> 00:30:04.720 And there's a magnetic field experienced 00:30:04.720 --> 00:30:11.440 by the electron-- felt by electron. 00:30:14.300 --> 00:30:19.570 So you can think of this, the electron. 00:30:19.570 --> 00:30:22.100 Here is the proton with the plus charge, 00:30:22.100 --> 00:30:25.190 and here's the electron. 00:30:25.190 --> 00:30:29.060 And the electron is going around the proton. 00:30:29.060 --> 00:30:31.730 Now, from the viewpoint of the electron, 00:30:31.730 --> 00:30:35.690 the proton is going around him. 00:30:35.690 --> 00:30:38.000 So here is the proton. 00:30:38.000 --> 00:30:40.330 Here is the electron going like that. 00:30:40.330 --> 00:30:42.040 From the viewpoint of the electron, 00:30:42.040 --> 00:30:44.760 the proton is going like this. 00:30:44.760 --> 00:30:48.350 Also, from the viewpoint of the electron, 00:30:48.350 --> 00:30:51.335 the proton would be going in this direction 00:30:51.335 --> 00:30:54.150 and creating a magnetic field up. 00:30:59.990 --> 00:31:06.570 And the magnetic field up corresponds actually 00:31:06.570 --> 00:31:11.775 to the idea that the angular momentum of the electron 00:31:11.775 --> 00:31:16.970 is also up-- L of the electron is also up. 00:31:16.970 --> 00:31:20.870 So the whole point of this thing is 00:31:20.870 --> 00:31:25.670 that somehow this magnetic field is proportional to the angular 00:31:25.670 --> 00:31:27.100 momentum. 00:31:27.100 --> 00:31:29.900 And then, L will come here. 00:31:29.900 --> 00:31:31.770 And here, you have S. So you have 00:31:31.770 --> 00:31:39.440 L dot S. That's the fine structure coupling. 00:31:39.440 --> 00:31:45.300 Now let me do a little of this so that we just 00:31:45.300 --> 00:31:49.080 get a bit more feeling, although it's unfortunately 00:31:49.080 --> 00:31:53.000 a somewhat frustrating exercise. 00:31:53.000 --> 00:31:56.790 So let me tell you what's going on. 00:31:56.790 --> 00:32:00.836 So consider the electron. 00:32:00.836 --> 00:32:05.600 At some point, look at it and draw a plane. 00:32:05.600 --> 00:32:09.000 So the electron-- let's assume it's going down. 00:32:09.000 --> 00:32:10.090 Here is the proton. 00:32:10.090 --> 00:32:11.670 It's going around in circles. 00:32:11.670 --> 00:32:14.660 So here, it's going down. 00:32:14.660 --> 00:32:16.280 The electron is going down. 00:32:16.280 --> 00:32:20.900 Electron, its velocity of the electron is going down. 00:32:20.900 --> 00:32:24.380 The proton is over here. 00:32:24.380 --> 00:32:28.740 And the electron is going around like that. 00:32:28.740 --> 00:32:32.350 The proton would produce an electric field of this form. 00:32:35.770 --> 00:32:42.080 Now, in relativity, the electric and magnetic fields 00:32:42.080 --> 00:32:44.920 seen by different observers are different. 00:32:44.920 --> 00:32:48.210 So there is this electric field that we see. 00:32:48.210 --> 00:32:52.260 We sit here, and we see in our rest frame 00:32:52.260 --> 00:32:56.330 this proton creates an electric field. 00:32:56.330 --> 00:33:00.760 And then, from the viewpoint of the electron, 00:33:00.760 --> 00:33:02.280 the electron is moving. 00:33:02.280 --> 00:33:04.840 And there is an electric field. 00:33:04.840 --> 00:33:09.020 But whenever you are moving inside an electric field, 00:33:09.020 --> 00:33:12.560 you also see a magnetic field generated by the motion, 00:33:12.560 --> 00:33:14.910 by relativistic effects. 00:33:14.910 --> 00:33:20.010 The magnetic field that you see is roughly 00:33:20.010 --> 00:33:27.560 given to first order in relativity by V cross E over c. 00:33:30.350 --> 00:33:37.120 So V cross E, VE V cross E over c 00:33:37.120 --> 00:33:39.900 up-- change sign because of this. 00:33:39.900 --> 00:33:42.740 And the magnetic field consistently, 00:33:42.740 --> 00:33:45.860 as we would expect, goes in this direction. 00:33:45.860 --> 00:33:48.210 So it's consistent with the picture 00:33:48.210 --> 00:33:51.840 that we developed that if you were the electron, the proton, 00:33:51.840 --> 00:33:53.940 would be going around in circles like that 00:33:53.940 --> 00:33:55.790 and the magnetic field would be up. 00:33:58.430 --> 00:34:07.730 Now here I can change the sign by doing E cross V over c. 00:34:07.730 --> 00:34:11.783 So this is the magnetic field seen by the electron. 00:34:23.389 --> 00:34:30.010 OK, so we need a little more work on that magnetic field 00:34:30.010 --> 00:34:33.610 by calculating the electric field. 00:34:33.610 --> 00:34:36.219 Now, what is the electric field? 00:34:36.219 --> 00:34:39.865 Well, the scalar potential for the hydrogen atom, 00:34:39.865 --> 00:34:44.670 we write it as minus e squared over r. 00:34:44.670 --> 00:34:47.159 It's actually not quite the scalar potential. 00:34:47.159 --> 00:34:49.980 But it is the potential energy. 00:34:49.980 --> 00:34:55.440 It has one factor of e more than what the scalar potential is. 00:34:55.440 --> 00:34:57.940 Remember, the scalar potential in electromagnetism 00:34:57.940 --> 00:35:00.260 is charge divided by r. 00:35:00.260 --> 00:35:04.000 So it has one factor of e more. 00:35:04.000 --> 00:35:07.100 What is the derivative of this potential? 00:35:07.100 --> 00:35:12.370 With respect to r, it's e squared over r squared. 00:35:12.370 --> 00:35:16.990 So the electric field goes like e over r squared. 00:35:16.990 --> 00:35:29.410 So the electric field is equal to dV dr divided by e. 00:35:32.220 --> 00:35:34.500 That's the magnitude of the electric field. 00:35:34.500 --> 00:35:40.150 And its direction is radial from the viewpoint of the proton. 00:35:40.150 --> 00:35:42.935 The electric field is here. 00:35:47.790 --> 00:35:51.920 So this can be written as r vector divided by r. 00:35:58.880 --> 00:36:07.000 Therefore, the magnetic field will-- [INAUDIBLE] this. 00:36:07.000 --> 00:36:10.320 The magnetic field now can be calculated. 00:36:10.320 --> 00:36:16.400 And we'll see what we claimed was 00:36:16.400 --> 00:36:18.930 the relation with angular momentum. 00:36:18.930 --> 00:36:25.850 Because B prime is now E cross V. 00:36:25.850 --> 00:36:37.170 So you have 1 over ec 1 over r dV dr. 00:36:37.170 --> 00:36:38.960 I've taken care of this. 00:36:38.960 --> 00:36:44.770 And now I just have r cross V. Well, 00:36:44.770 --> 00:36:51.240 r cross V is your angular momentum if you had p here. 00:36:51.240 --> 00:36:56.970 So we borrow a factor of the mass of the electron, 00:36:56.970 --> 00:37:06.570 ecm 1 over r dV dr L, L of the electron. 00:37:11.710 --> 00:37:16.380 p equals mv. 00:37:16.380 --> 00:37:22.180 So we have a nice formula for B. And then, we 00:37:22.180 --> 00:37:28.460 can go and calculate delta H. Delta H would then 00:37:28.460 --> 00:37:45.380 be minus mu dot B. And that would be ge over 2mc spin 00:37:45.380 --> 00:38:01.670 dot L-- mu was given here-- S dot L 1 over r dV dr. 00:38:01.670 --> 00:38:07.150 And that is the split spin orbit interaction. 00:38:07.150 --> 00:38:10.150 Now, the downside of this derivation 00:38:10.150 --> 00:38:13.015 is that it has a relativistic error. 00:38:16.740 --> 00:38:20.370 There's a phenomenon called Thomas precession 00:38:20.370 --> 00:38:23.870 that affects this result. 00:38:23.870 --> 00:38:26.330 We didn't waste our time. 00:38:26.330 --> 00:38:32.250 The true result is that you must subtract from this g 1. 00:38:32.250 --> 00:38:37.730 So g must really be replaced by g minus 1. 00:38:37.730 --> 00:38:41.040 Since g is approximately 2 for the electron, 00:38:41.040 --> 00:38:46.080 the true result is really 1/2 of this thing. 00:38:46.080 --> 00:38:49.760 So this should not be in parentheses, 00:38:49.760 --> 00:38:54.200 but true result is this. 00:38:54.200 --> 00:38:59.770 And the mistake that is done in calculating this spin orbit 00:38:59.770 --> 00:39:04.970 coupling is that this spin orbit coupling 00:39:04.970 --> 00:39:08.510 affects precession rates. 00:39:08.510 --> 00:39:12.260 All these interactions of magnetic dipoles 00:39:12.260 --> 00:39:15.540 with magnetic fields affect precession rates. 00:39:15.540 --> 00:39:18.020 And you have to be a little more careful here 00:39:18.020 --> 00:39:24.200 that the system where you've worked, the electron rest frame 00:39:24.200 --> 00:39:26.370 is not quite an inertial system. 00:39:26.370 --> 00:39:29.330 Because it's doing circular motion. 00:39:29.330 --> 00:39:33.420 So there's an extra correction that has to be done. 00:39:33.420 --> 00:39:37.350 Thomas precession or Thomas correction it's called. 00:39:37.350 --> 00:39:41.170 And it would be a detour of about one hour 00:39:41.170 --> 00:39:43.900 in special relativity to do it right. 00:39:43.900 --> 00:39:47.676 So Griffiths doesn't do it. 00:39:47.676 --> 00:39:51.010 I don't think Shankar does it. 00:39:51.010 --> 00:39:53.280 Pretty much graduate books do it. 00:39:56.088 --> 00:39:59.990 So we will not try to do better. 00:39:59.990 --> 00:40:02.550 I mentioned that fact that this really 00:40:02.550 --> 00:40:06.260 should be reduced to one half of its value. 00:40:06.260 --> 00:40:08.780 And it's an interesting system to analyze. 00:40:08.780 --> 00:40:15.170 So Thomas precession is that relativistic correction 00:40:15.170 --> 00:40:18.540 to precession rates when the object that is precessing 00:40:18.540 --> 00:40:21.820 is in an accelerated frame. 00:40:21.820 --> 00:40:24.670 And any rotating frame is accelerated. 00:40:24.670 --> 00:40:31.200 So this result needs correction. 00:40:31.200 --> 00:40:36.450 OK, but let's take this result as it is-- instead of g, 00:40:36.450 --> 00:40:37.640 g minus 1. 00:40:37.640 --> 00:40:39.810 Let's not worry too much about it. 00:40:39.810 --> 00:40:44.686 And let's just estimate how big this effect is. 00:40:44.686 --> 00:40:50.470 It's the last thing I want to do as a way of motivating 00:40:50.470 --> 00:40:51.460 this subject. 00:40:51.460 --> 00:40:53.850 So delta H is this. 00:40:53.850 --> 00:40:55.770 Let's estimate it. 00:40:55.770 --> 00:41:03.090 Now for estimates, a couple of things are useful to remember, 00:41:03.090 --> 00:41:08.640 that Bohr radius is h squared over me squared. 00:41:08.640 --> 00:41:10.740 We did that last time. 00:41:10.740 --> 00:41:13.680 And there's this constant that is very useful, 00:41:13.680 --> 00:41:18.620 the fine structure constant, which is e squared over hc. 00:41:18.620 --> 00:41:21.885 And it's about 1 over 137. 00:41:21.885 --> 00:41:27.090 And it helps you estimate all kinds of things. 00:41:27.090 --> 00:41:31.580 Because it's a rather complicated number to evaluate, 00:41:31.580 --> 00:41:36.450 you need all kinds of units and things like that. 00:41:36.450 --> 00:41:43.180 So the charge of the electron divided by hc being 1 over 137 00:41:43.180 --> 00:41:46.680 is quite nice. 00:41:46.680 --> 00:41:53.050 So let's estimate delta H. Well, g 00:41:53.050 --> 00:41:58.160 we won't worry-- 2, 1, doesn't matter. 00:41:58.160 --> 00:42:07.530 e mc-- so far, that is kind of simple. 00:42:07.530 --> 00:42:13.036 Then we have S dot L. Well, how do I estimate S dot L? 00:42:13.036 --> 00:42:15.740 I don't do too much. 00:42:15.740 --> 00:42:18.850 S spin is multiples of h bar. 00:42:18.850 --> 00:42:23.960 L for an atomic state will be 1, 2, 3, so multiples of h bar. 00:42:23.960 --> 00:42:29.430 So h bar squared, that's it for S dot L. 00:42:29.430 --> 00:42:33.500 1 over r is 1 over r. 00:42:33.500 --> 00:42:37.570 dV dr is e squared over r squared. 00:42:37.570 --> 00:42:39.280 And that's it. 00:42:39.280 --> 00:42:43.760 But here, instead of r, I should put the typical length 00:42:43.760 --> 00:42:46.950 of the hydrogen atom, which is a0. 00:42:46.950 --> 00:42:48.210 So what do I get? 00:42:53.230 --> 00:42:57.300 I'm sorry, I made a mistake here. 00:42:57.300 --> 00:42:59.407 AUDIENCE: Yeah, it's up there. 00:42:59.407 --> 00:43:03.079 PROFESSOR: Oh, I made a mistake here 00:43:03.079 --> 00:43:09.910 in that I didn't put this factor, 1 over ecm. 00:43:09.910 --> 00:43:11.980 So the e cancels. 00:43:11.980 --> 00:43:15.920 And this is the result here-- g over 2m 00:43:15.920 --> 00:43:39.900 squared c squared S dot L 1 over r dV dr. So let me start again. 00:43:39.900 --> 00:43:49.680 1 over m squared c squared h bar squared 1 over r dV dr-- 00:43:49.680 --> 00:43:51.550 that much I got right. 00:43:51.550 --> 00:43:58.810 So this is roughly 1 over [INAUDIBLE] 00:43:58.810 --> 00:44:02.220 of the electron squared c squared e 00:44:02.220 --> 00:44:08.780 squared over a0 cubed h squared-- still quite 00:44:08.780 --> 00:44:12.610 messy, but not that terrible. 00:44:12.610 --> 00:44:16.700 So in order to get an idea of how big this is, 00:44:16.700 --> 00:44:19.720 the ground state energy of the hydrogen atom 00:44:19.720 --> 00:44:23.400 was e squared over 2a0. 00:44:23.400 --> 00:44:30.610 So let's divide delta H over the ground state energy. 00:44:30.610 --> 00:44:32.220 And that's how much? 00:44:32.220 --> 00:44:37.850 Well, we have all this quantity, 1 over m 00:44:37.850 --> 00:44:44.490 squared c squared e squared a0 cubed h squared. 00:44:44.490 --> 00:44:53.110 And now, we must divide by e squared over a0 like this. 00:44:59.160 --> 00:45:02.370 Well, the e squareds cancel. 00:45:02.370 --> 00:45:11.800 And we get h squared over m squared c squared a0 squared. 00:45:11.800 --> 00:45:13.950 You need to know what a0 is. 00:45:13.950 --> 00:45:16.640 Let's just boil it down to the simplest thing, 00:45:16.640 --> 00:45:20.760 so h squared m squared c squared. 00:45:20.760 --> 00:45:26.120 a0 squared would be h to the fourth m squared e 00:45:26.120 --> 00:45:28.540 to the fourth. 00:45:28.540 --> 00:45:33.840 So this is actually e to the fourth over h 00:45:33.840 --> 00:45:41.200 squared c squared, or e squared over hc squared, which is alpha 00:45:41.200 --> 00:45:41.700 squared. 00:45:41.700 --> 00:45:45.510 Whew-- lots of work to get something very nice. 00:45:48.030 --> 00:45:56.910 The ratio of the spin orbit coupling to the ground state 00:45:56.910 --> 00:46:00.850 energy is 1 over alpha squared. 00:46:00.850 --> 00:46:05.310 It's alpha squared, which is 1 over 137 squared. 00:46:05.310 --> 00:46:08.040 So it's a pretty small thing. 00:46:08.040 --> 00:46:14.250 It's about 1 over 19,000. 00:46:14.250 --> 00:46:19.850 So when this is called fine structure of the hydrogen atom, 00:46:19.850 --> 00:46:24.300 it means that it's in the level in your page 00:46:24.300 --> 00:46:30.040 that you use a few inches to plot the 13.6 electron 00:46:30.040 --> 00:46:35.450 volts-- well, you're talking about 20,000 times smaller, 00:46:35.450 --> 00:46:38.820 something that you don't see. 00:46:38.820 --> 00:46:41.410 But of course, it's a pretty important thing. 00:46:41.410 --> 00:46:48.960 So all in all, in the conventions of-- this 00:46:48.960 --> 00:46:53.900 is done in Gaussian units. 00:46:53.900 --> 00:47:01.140 In SI units, which is what Griffiths uses, 00:47:01.140 --> 00:47:08.560 delta H is e squared over 8 pi epsilon 0 1 over m 00:47:08.560 --> 00:47:17.450 squared c squared r cubed S dot L. That's for reference. 00:47:17.450 --> 00:47:18.275 This is Griffiths. 00:47:22.300 --> 00:47:24.310 But this is correct as well. 00:47:24.310 --> 00:47:27.440 This is the correct value. 00:47:27.440 --> 00:47:30.960 This is the correct value already taking 00:47:30.960 --> 00:47:33.340 into account the relativistic correction. 00:47:33.340 --> 00:47:37.270 So here, you're supposed to let g go to g minus 1. 00:47:37.270 --> 00:47:42.490 So you can put the 1 there, and it's pretty accurate. 00:47:42.490 --> 00:47:45.590 All right, so what is the physics question 00:47:45.590 --> 00:47:49.500 we want to answer with this spin orbit coupling? 00:47:49.500 --> 00:47:54.640 So here it comes. 00:47:54.640 --> 00:47:58.920 You have the hydrogen atom spectrum. 00:47:58.920 --> 00:48:01.460 And that spectrum you know. 00:48:01.460 --> 00:48:05.050 At L equals 0, you have one state here. 00:48:05.050 --> 00:48:09.890 Then, that's n equals 1, n equals 2. 00:48:12.470 --> 00:48:16.580 You have one state here and one state here at L equals 1. 00:48:16.580 --> 00:48:21.260 Then n equals 3, they start getting very close together. 00:48:21.260 --> 00:48:25.980 n equals 4 is like that. 00:48:25.980 --> 00:48:31.370 Let's consider if you want to have spin orbit coupling, 00:48:31.370 --> 00:48:35.880 we must have angular momentum. 00:48:35.880 --> 00:48:42.330 And that's L. And therefore, let's consider this state here. 00:48:42.330 --> 00:48:47.420 l equals 1, n equals 1-- n equals 2, I'm sorry. 00:48:51.410 --> 00:48:56.520 What happens to those states, is the question. 00:48:56.520 --> 00:48:59.110 First, how many states do you have there 00:48:59.110 --> 00:49:02.120 and how should you think of them? 00:49:02.120 --> 00:49:06.430 Well actually, we know that an l equals 1 corresponds 00:49:06.430 --> 00:49:07.970 to three states. 00:49:07.970 --> 00:49:13.820 So you'd have lm with l equals 1. 00:49:13.820 --> 00:49:18.310 And then m can be 1, 0, or minus 1. 00:49:18.310 --> 00:49:20.730 So you have three states. 00:49:20.730 --> 00:49:23.840 But there's not really three states. 00:49:23.840 --> 00:49:26.420 Because the electron can have spin. 00:49:26.420 --> 00:49:31.770 So here it is, a tensor product that appears in your face 00:49:31.770 --> 00:49:36.150 because there is more than angular 00:49:36.150 --> 00:49:37.430 momentum to the electron. 00:49:37.430 --> 00:49:38.630 There's spin. 00:49:38.630 --> 00:49:42.130 And it's a totally different vector space, the same particle 00:49:42.130 --> 00:49:45.780 but another vector space, the spin space. 00:49:45.780 --> 00:49:50.140 So here, you have the possible spins of the electron. 00:49:50.140 --> 00:49:53.130 So that's another angular momentum. 00:49:53.130 --> 00:49:57.525 And well, you could have the plus/minus states, for example. 00:50:00.730 --> 00:50:05.290 So you have three states here and two states here. 00:50:05.290 --> 00:50:16.440 So this is really six states, so six states 00:50:16.440 --> 00:50:20.700 whose fate we would like to understand 00:50:20.700 --> 00:50:23.355 due to this spin orbit coupling. 00:50:27.600 --> 00:50:35.610 So to use the language of angular momentum, 00:50:35.610 --> 00:50:37.740 instead of writing plus/minus, you 00:50:37.740 --> 00:50:49.990 could write Smz, if you will-- ms I will call, spin of s. 00:50:49.990 --> 00:50:59.780 You have here spin of 1/2 and states 1/2 or minus 1/2. 00:50:59.780 --> 00:51:01.040 This is the up. 00:51:01.040 --> 00:51:07.040 When the z component of the spin that we always call m-- m now 00:51:07.040 --> 00:51:09.760 corresponds to the z component of angular momentum. 00:51:09.760 --> 00:51:12.846 So in general, even for spin, we use m. 00:51:12.846 --> 00:51:16.410 And we have that our two spin states of the electron 00:51:16.410 --> 00:51:21.880 are spin 1/2 particle with plus spin in the z direction, 00:51:21.880 --> 00:51:26.550 spin 1/2 particle with minus spin in the z direction. 00:51:26.550 --> 00:51:29.380 We usually never put this 1/2 here. 00:51:29.380 --> 00:51:32.700 But now you have here really three states-- 00:51:32.700 --> 00:51:39.950 1, 1, 1, 0, 1, minus 1, the first telling you 00:51:39.950 --> 00:51:42.810 about the total angular momentum. 00:51:42.810 --> 00:51:45.480 Here, the total spin is 1/2. 00:51:45.480 --> 00:51:47.860 But it happens to be either up or down. 00:51:47.860 --> 00:51:50.550 Here, the total angular momentum is 1. 00:51:50.550 --> 00:51:56.010 But it happens to be plus 1, 0, or minus 1 here. 00:51:56.010 --> 00:51:58.830 So these are our six states. 00:51:58.830 --> 00:52:01.600 You can combine this with this, this with that, this with this, 00:52:01.600 --> 00:52:02.320 this with that. 00:52:02.320 --> 00:52:03.870 You make all the products. 00:52:03.870 --> 00:52:06.610 And these are the six states of the hydrogen 00:52:06.610 --> 00:52:08.090 atom at this level. 00:52:08.090 --> 00:52:13.100 And we wish to know what happens to them. 00:52:13.100 --> 00:52:17.910 Now, this correction is small. 00:52:17.910 --> 00:52:22.750 So it fits our understanding of the perturbation theory 00:52:22.750 --> 00:52:25.310 of Feynman-Hellman in which we try 00:52:25.310 --> 00:52:29.660 to find the corrections to these things. 00:52:29.660 --> 00:52:35.130 Our difficulty now is a little serious, however. 00:52:35.130 --> 00:52:37.290 It's the fact that Feynman-Hellman 00:52:37.290 --> 00:52:39.870 assumed that you had a state. 00:52:39.870 --> 00:52:43.600 And it was an eigenstate of the corrected Hamiltonian 00:52:43.600 --> 00:52:46.930 as you moved along. 00:52:46.930 --> 00:52:51.220 And then, you could compute how its energy changes. 00:52:51.220 --> 00:52:56.260 Here, unfortunately, we have a much more difficult situation. 00:52:56.260 --> 00:52:59.580 These six states that I'm not listing yet, 00:52:59.580 --> 00:53:07.170 but I will list very soon, are not obviously eigenstates 00:53:07.170 --> 00:53:12.340 of delta H. In fact, they are not eigenstates of delta H. 00:53:12.340 --> 00:53:15.790 They're degenerate states, six degenerate states, 00:53:15.790 --> 00:53:19.440 that are not eigenstates of delta H. Therefore, 00:53:19.440 --> 00:53:24.360 I cannot use the Feynman-Hellman theorem until I find what are 00:53:24.360 --> 00:53:28.499 the combinations that are eigenstates of this 00:53:28.499 --> 00:53:29.040 perturbation. 00:53:32.680 --> 00:53:35.380 So we are a little bit in trouble. 00:53:35.380 --> 00:53:43.010 Because we have a perturbation for which these product 00:53:43.010 --> 00:53:52.110 states-- we call them uncoupled bases-- are not eigenstates. 00:53:52.110 --> 00:53:56.850 Now, we've written this operator a little naively. 00:53:56.850 --> 00:54:01.270 What does this operator really mean, S dot L? 00:54:06.150 --> 00:54:15.030 In our tensor products, it means S1 tensor L1. 00:54:15.030 --> 00:54:19.180 Actually, I'll use L dot S. I'll always 00:54:19.180 --> 00:54:24.740 put the L information first and the S information afterward. 00:54:24.740 --> 00:54:29.850 So L dot S is clearly an operator that 00:54:29.850 --> 00:54:33.910 must be thought to act on the tensor product. 00:54:33.910 --> 00:54:35.830 Because both have to act. 00:54:35.830 --> 00:54:37.990 S has to act and L has to act. 00:54:37.990 --> 00:54:39.800 So it only lives in the tensor product. 00:54:39.800 --> 00:54:41.400 So what does it mean? 00:54:41.400 --> 00:54:55.960 It means this-- S2 L2 plus S3 L3, or sum over i Si tensor Li. 00:54:55.960 --> 00:54:59.860 So this is the kind of thing that you need to understand-- 00:54:59.860 --> 00:55:05.080 how do you find for this operator's eigenstates here? 00:55:07.940 --> 00:55:16.400 So that is our difficulty. 00:55:16.400 --> 00:55:19.910 And that's what we have to solve. 00:55:19.910 --> 00:55:23.770 We're going to solve it in the next half hour. 00:55:23.770 --> 00:55:31.140 So it's a complicated operator, L dot S. But on the other hand, 00:55:31.140 --> 00:55:34.780 we have to use our ideas that we've learned already 00:55:34.780 --> 00:55:40.090 about summing angular momenta. 00:55:40.090 --> 00:55:49.310 What if I define J to be L plus S, 00:55:49.310 --> 00:55:58.545 which really means L tensor 1 plus 1 tensor S? 00:56:03.940 --> 00:56:09.130 So this is what I really mean by this operator. 00:56:12.170 --> 00:56:17.440 J, as we've demonstrated, will be an angular momentum, 00:56:17.440 --> 00:56:20.510 because this satisfies the algebra of angular momentum 00:56:20.510 --> 00:56:23.820 and this satisfies the algebra of angular momentum. 00:56:23.820 --> 00:56:28.280 So this thing satisfies the algebra of angular momentum. 00:56:28.280 --> 00:56:32.310 And why do we look at that term? 00:56:32.310 --> 00:56:36.340 Because of the following reason. 00:56:36.340 --> 00:56:39.590 We can square it-- JiJi. 00:56:43.930 --> 00:56:48.870 Now we would have to square this thing. 00:56:48.870 --> 00:56:50.510 How do you square this thing? 00:56:50.510 --> 00:56:52.590 Well, there's two ways. 00:56:52.590 --> 00:56:56.060 Naively-- L squared plus L squared plus 2L 00:56:56.060 --> 00:56:59.285 dot S-- basically correct. 00:56:59.285 --> 00:57:01.750 But you can do it a little more slowly. 00:57:01.750 --> 00:57:07.120 If you square this term, you get L squared tensor 1. 00:57:07.120 --> 00:57:14.000 If you square this term, you get 1 tensor S squared. 00:57:14.000 --> 00:57:16.210 But when you do the mixed products, 00:57:16.210 --> 00:57:20.070 you just must take the i's here and the i's here 00:57:20.070 --> 00:57:21.250 and multiply them. 00:57:21.250 --> 00:57:28.530 So actually, you do get two i's, the sum over i Li tensor Si. 00:57:35.710 --> 00:57:37.320 This is sum over i. 00:57:37.320 --> 00:57:38.500 This is J squared. 00:57:42.500 --> 00:57:50.420 So basically, what I'm saying is that J squared naively 00:57:50.420 --> 00:57:56.100 is L squared plus S squared plus our interaction 00:57:56.100 --> 00:58:01.425 2L dot S defined property. 00:58:04.110 --> 00:58:16.900 So L dot S is equal to 1/2 of J squared minus L squared 00:58:16.900 --> 00:58:19.690 minus S squared. 00:58:19.690 --> 00:58:26.020 And that tells you all kinds of interesting things about L dot 00:58:26.020 --> 00:58:28.470 S. 00:58:28.470 --> 00:58:34.180 Basically, we can trade L dot S for J squared, 00:58:34.180 --> 00:58:36.490 L squared, and S squared. 00:58:36.490 --> 00:58:39.590 L squared is very simple, and S squared 00:58:39.590 --> 00:58:42.190 is extremely simple as well. 00:58:42.190 --> 00:58:46.520 Remember, L squared commutes with any Li. 00:58:46.520 --> 00:58:51.340 So L squared with any Li is equal to 0. 00:58:51.340 --> 00:58:55.740 S squared with any Si is equal to 0. 00:58:55.740 --> 00:58:58.180 And Li's and Si's commute. 00:58:58.180 --> 00:58:59.780 They live in different worlds. 00:58:59.780 --> 00:59:03.280 So L squared and Si's commute. 00:59:03.280 --> 00:59:06.720 S squareds and Li's commute. 00:59:06.720 --> 00:59:11.230 These things are pretty nice and simple. 00:59:11.230 --> 00:59:15.600 So let's think now of our Hamiltonian 00:59:15.600 --> 00:59:22.260 and what is happening to it. 00:59:22.260 --> 00:59:31.260 Whenever we had the hydrogen atom, 00:59:31.260 --> 00:59:39.410 we had a set of commuting observables H, L squared, 00:59:39.410 --> 00:59:40.120 and Lz. 00:59:44.390 --> 00:59:48.470 It's a complete set of commuting observables. 00:59:48.470 --> 00:59:53.020 Now, in the hydrogen atom, you could add to it 00:59:53.020 --> 00:59:57.620 S squared and Sz. 00:59:57.620 --> 01:00:00.160 We didn't talk about spin at the beginning, 01:00:00.160 --> 01:00:02.760 because we just considered a particle going 01:00:02.760 --> 01:00:04.520 around the hydrogen atom. 01:00:04.520 --> 01:00:09.330 But if you have spin, the hydrogen atom Hamiltonian, 01:00:09.330 --> 01:00:13.820 the original one, doesn't involve spin in any way. 01:00:13.820 --> 01:00:16.890 So certainly, Hamiltonian commutes with spin, 01:00:16.890 --> 01:00:18.740 with spin z. 01:00:18.740 --> 01:00:21.880 L and S don't talk, so this is the complete set 01:00:21.880 --> 01:00:24.870 of commuting observables. 01:00:24.870 --> 01:00:27.215 But what happens to this list? 01:00:27.215 --> 01:00:35.210 This is our problem for H0, the hydrogen atom, 01:00:35.210 --> 01:00:43.450 plus delta H that has the S dot L. 01:00:43.450 --> 01:00:49.100 Well, what are complete set of commuting observables? 01:00:49.100 --> 01:00:51.550 This is a very important question. 01:00:51.550 --> 01:00:53.560 Because this is what tells you how 01:00:53.560 --> 01:00:56.050 you're going to try to organize the spectrum. 01:00:56.050 --> 01:01:01.880 So we could have H, the total, H total. 01:01:05.800 --> 01:01:08.960 And what else? 01:01:08.960 --> 01:01:14.630 Well, can I still have L squared here? 01:01:18.450 --> 01:01:23.135 Can I include L squared and say it commutes with the total H? 01:01:27.229 --> 01:01:30.170 A little worrisome, but actually, 01:01:30.170 --> 01:01:35.500 you know that L squared commutes with the original Hamiltonian. 01:01:35.500 --> 01:01:38.260 Now, the question is whether L squared commutes 01:01:38.260 --> 01:01:40.320 with this extra piece. 01:01:40.320 --> 01:01:44.110 Well, but L squared commutes with any Li. 01:01:44.110 --> 01:01:47.730 And it doesn't even talk to S. So L squared is safe. 01:01:47.730 --> 01:01:50.400 L squared we can keep. 01:01:50.400 --> 01:01:55.660 OK, S squared-- can we keep S squared? 01:01:55.660 --> 01:01:57.480 Well, S squared was here. 01:01:57.480 --> 01:02:01.420 So it commuted with the Hamiltonian, and that was good. 01:02:01.420 --> 01:02:06.000 S squared commutes with any Si, and it doesn't talk to L. 01:02:06.000 --> 01:02:07.730 So S squared can stay. 01:02:11.040 --> 01:02:13.550 But that's not good enough. 01:02:13.550 --> 01:02:16.950 We won't be able to solve the problem with this still. 01:02:16.950 --> 01:02:17.790 We need more. 01:02:21.100 --> 01:02:22.166 How about Lz? 01:02:22.166 --> 01:02:24.885 It was here, so let's try our luck. 01:02:29.010 --> 01:02:32.340 Any opinions on Lz-- can we keep it or not? 01:02:37.840 --> 01:02:38.838 Yes. 01:02:38.838 --> 01:02:40.182 AUDIENCE: I don't think so. 01:02:40.182 --> 01:02:44.117 Because in the J term, we have Lx's and Ly's, which 01:02:44.117 --> 01:02:45.340 don't commute with Lz. 01:02:45.340 --> 01:02:47.980 PROFESSOR: Right, it can't be kept. 01:02:47.980 --> 01:02:57.540 Here, this term has SxLx plus SyLy plus SzLz. 01:02:57.540 --> 01:03:02.370 And Lz doesn't commute with this one. 01:03:02.370 --> 01:03:04.995 So no, you can't keep Lz-- no good. 01:03:09.680 --> 01:03:14.496 On the other hand, let's think about J squared. 01:03:17.640 --> 01:03:22.670 J squared is here. 01:03:22.670 --> 01:03:26.685 And J squared commutes with L squared and with S squared. 01:03:29.690 --> 01:03:36.600 J squared, therefore, is-- well, let me say it this way. 01:03:36.600 --> 01:03:42.460 Here is L dot S, which is our extra interaction. 01:03:42.460 --> 01:03:44.920 Here we have this thing. 01:03:44.920 --> 01:03:49.700 I would like to say on behalf of J squared 01:03:49.700 --> 01:03:56.200 that we can include it here, J squared, 01:03:56.200 --> 01:04:00.840 because J squared is really pretty much the same 01:04:00.840 --> 01:04:05.860 as L dot S up to this L squared and S squared. 01:04:05.860 --> 01:04:09.640 But J squared commutes with L squared and S squared. 01:04:09.640 --> 01:04:11.400 I should probably write it there. 01:04:15.120 --> 01:04:22.350 J squared commutes with L squared. 01:04:22.350 --> 01:04:29.010 And J squared communicates with S squared that we have here. 01:04:29.010 --> 01:04:38.300 And moreover, we have over here that J squared therefore 01:04:38.300 --> 01:04:41.960 will commute, or it's pretty much the same, 01:04:41.960 --> 01:04:45.600 as L dot S. J squared with L dot S would 01:04:45.600 --> 01:04:50.810 be J squared times this thing, which is 0. 01:04:50.810 --> 01:04:55.840 So J squared commutes with this term. 01:04:55.840 --> 01:04:59.210 And it commutes with the Hamiltonian, 01:04:59.210 --> 01:05:02.410 your original hydrogen Hamiltonian. 01:05:02.410 --> 01:05:06.240 So J squared can be added here. 01:05:09.670 --> 01:05:13.980 J square is a good operator to have. 01:05:13.980 --> 01:05:19.110 And now we can get one more kind of free from here. 01:05:19.110 --> 01:05:21.290 It's Jz. 01:05:21.290 --> 01:05:23.280 Z 01:05:23.280 --> 01:05:29.630 Because Jz commutes with J squared. 01:05:29.630 --> 01:05:32.140 Jz commutes with these things. 01:05:32.140 --> 01:05:38.090 And Jz, which is a symmetry of the original Hamiltonian, 01:05:38.090 --> 01:05:44.342 also commutes with our new interaction, the L dot S, 01:05:44.342 --> 01:05:47.770 which is proportional to J squared. 01:05:47.770 --> 01:05:53.340 So you have to go through this yourselves 01:05:53.340 --> 01:05:56.980 probably even a little more slowly than I've gone. 01:05:56.980 --> 01:05:59.660 Just check that everything that I'm 01:05:59.660 --> 01:06:02.480 saying about whatever commutes commutes. 01:06:02.480 --> 01:06:06.350 So for example, when I say that J squared commutes 01:06:06.350 --> 01:06:09.700 with L dot S, it's because I can put 01:06:09.700 --> 01:06:12.590 instead of L dot S all of this. 01:06:12.590 --> 01:06:15.920 And go slowly through this. 01:06:15.920 --> 01:06:22.030 So this is actually the complete set of committing observables. 01:06:22.030 --> 01:06:25.510 And it's basically saying to us, try 01:06:25.510 --> 01:06:31.755 to diagonalize this thing with total angular momentum. 01:06:34.590 --> 01:06:38.290 So it's about time to really do it. 01:06:38.290 --> 01:06:39.790 We haven't done it yet. 01:06:39.790 --> 01:06:43.930 But now the part that we have to do now, 01:06:43.930 --> 01:06:46.470 it's kind of a nice exercise. 01:06:46.470 --> 01:06:48.970 And it's fun. 01:06:48.970 --> 01:06:52.180 Now, there's one problem in the homework set 01:06:52.180 --> 01:06:55.710 that sort of uses this kind of thing. 01:06:55.710 --> 01:07:01.850 And I will suggest there to Will and Aram that tomorrow, 01:07:01.850 --> 01:07:07.120 they spend some time discussing it and helping you with it. 01:07:07.120 --> 01:07:10.930 The last problem in the homework set 01:07:10.930 --> 01:07:15.060 would've been better if you had a little more time for it 01:07:15.060 --> 01:07:17.930 and you had more time to digest what I'm doing today. 01:07:17.930 --> 01:07:21.920 But nevertheless, go to recitation, 01:07:21.920 --> 01:07:23.490 learn more about the problem. 01:07:23.490 --> 01:07:26.740 It will not be all that difficult. 01:07:26.740 --> 01:07:31.900 OK, so we're trying now to finally form 01:07:31.900 --> 01:07:33.890 another basis of states. 01:07:33.890 --> 01:07:36.970 We had these six states. 01:07:36.970 --> 01:07:39.020 And we're going to try to organize them 01:07:39.020 --> 01:07:45.480 in a better way-- as eigenstates of the total angular momentum L 01:07:45.480 --> 01:07:51.210 plus S. So I'm going to write them here in this way. 01:07:51.210 --> 01:07:58.830 Here is one of the states of this L equals 01:07:58.830 --> 01:08:04.330 1 electron, the 1, 1 coupled to the 1/2, 1/2. 01:08:04.330 --> 01:08:20.740 Here are two more states- 1, 0, 1/2, 1/2, 1, 1, 1/2, minus 1/2, 01:08:20.740 --> 01:08:27.850 so the 1, 0 with the top, the 1, 1 with the bottom. 01:08:27.850 --> 01:08:35.020 Here are two more states-- 1, 0 with 1/2, 01:08:35.020 --> 01:08:43.620 minus 1/2 and 1, minus 1 with 1/2, 1/2. 01:08:47.790 --> 01:08:56.100 And here is the last state-- 1, minus 1 with 1/2, minus 1. 01:09:02.520 --> 01:09:05.832 These are our six states. 01:09:05.832 --> 01:09:09.654 And I've organized them in a nice way actually. 01:09:12.760 --> 01:09:14.990 I've organized them in such a way 01:09:14.990 --> 01:09:21.750 that you can read what is the value of Jz over h bar. 01:09:21.750 --> 01:09:28.563 Remember, Jz is 1 over h bar Lz plus Sz. 01:09:34.080 --> 01:09:35.479 So what is it? 01:09:35.479 --> 01:09:39.760 These are, I claim, eigenstates of Jz. 01:09:39.760 --> 01:09:40.899 Why? 01:09:40.899 --> 01:09:42.569 Because let's act on them. 01:09:42.569 --> 01:09:45.510 Suppose I act with Jz on this state. 01:09:45.510 --> 01:09:47.930 The Lz comes here and says, 1. 01:09:47.930 --> 01:09:50.890 The Sz comes here and says, 1/2. 01:09:50.890 --> 01:09:56.337 So the sum of them give you Jz over h bar equal to 3/2. 01:10:01.080 --> 01:10:05.240 And that's why I organized these states in such a way 01:10:05.240 --> 01:10:10.510 that these second things add up to the same value-- 0 and 1/2, 01:10:10.510 --> 01:10:12.590 1 and minus 1/2. 01:10:12.590 --> 01:10:16.360 So if you act with Jz on this state, 01:10:16.360 --> 01:10:19.330 it's an eigenstate with Jz. 01:10:19.330 --> 01:10:22.470 Here, 0 contribution, here 1/2. 01:10:22.470 --> 01:10:26.700 So this is with plus 1/2. 01:10:26.700 --> 01:10:31.130 Here, you have 0 and minus 1/2, minus 1, and that is minus 1/2. 01:10:34.330 --> 01:10:36.930 And here you have minus 3/2. 01:10:40.990 --> 01:10:43.943 OK, questions. 01:10:48.980 --> 01:10:50.330 We've written the states. 01:10:50.330 --> 01:10:55.240 And I'm evaluating the total z component of angular momentum. 01:10:55.240 --> 01:10:58.470 And these two states are like that. 01:10:58.470 --> 01:11:02.150 So what does our theorem guarantee for us? 01:11:02.150 --> 01:11:06.670 Our theorem guarantees that we have-- in this tensor product, 01:11:06.670 --> 01:11:10.770 there is an algebra of angular momentum of the Jz operators. 01:11:10.770 --> 01:11:14.270 And the states have to fall into representations 01:11:14.270 --> 01:11:15.930 of those operators. 01:11:15.930 --> 01:11:19.920 So you must have angular momentum multiplets. 01:11:19.920 --> 01:11:23.710 So at this moment, you can figure out 01:11:23.710 --> 01:11:30.240 what angular momentum you're going to get for the result. 01:11:30.240 --> 01:11:36.770 Here we obtained a maximum Jz of 3/2. 01:11:36.770 --> 01:11:42.005 So we must get a J equals 3/2 multiplet. 01:11:46.100 --> 01:11:48.660 Because a J equaling 3/2 multiplets 01:11:48.660 --> 01:11:55.650 has Jz 3/2, 1/2, minus 1/2, and 0. 01:11:55.650 --> 01:12:01.510 So actually, this state must be the top state of the multiplet. 01:12:01.510 --> 01:12:05.500 This state must be the bottom state of the multiplet. 01:12:05.500 --> 01:12:10.160 I don't know which one is the middle state of the multiplet 01:12:10.160 --> 01:12:12.250 and which one is here. 01:12:12.250 --> 01:12:15.670 But we have four states here, four states. 01:12:18.560 --> 01:12:22.940 So one linear combination of these two states 01:12:22.940 --> 01:12:29.190 must be, then, that Jz equals 1/2 state of the multiplet. 01:12:29.190 --> 01:12:31.600 And one inner combination of these two states 01:12:31.600 --> 01:12:35.220 must be that Jz equals minus 1/2 state of the multiplet. 01:12:35.220 --> 01:12:36.750 Which one is it? 01:12:36.750 --> 01:12:38.180 I don't know. 01:12:38.180 --> 01:12:40.010 But we can figure it out. 01:12:40.010 --> 01:12:41.930 We'll figure it out in a second. 01:12:41.930 --> 01:12:44.640 Once you get this J 3/2 multiplet, 01:12:44.640 --> 01:12:48.270 there will be one linear combination here left over 01:12:48.270 --> 01:12:51.500 and one linear combination here left over. 01:12:51.500 --> 01:12:56.300 Those are two state, one with Jz plus 1/2 and one with Jz 01:12:56.300 --> 01:12:58.060 equals minus 1/2. 01:12:58.060 --> 01:13:01.740 So you also get a J equals 1/2 multiplet. 01:13:07.670 --> 01:13:11.290 So the whole tensor product of six 01:13:11.290 --> 01:13:18.260 states-- it was the tensor product of a spin 1 01:13:18.260 --> 01:13:21.620 with a spin 1/2. 01:13:21.620 --> 01:13:24.560 So we write it like this. 01:13:24.560 --> 01:13:32.940 The tensor product of a spin 1 with a spin 1/2 01:13:32.940 --> 01:13:43.660 will give you a total spin 3/2 plus total spin 01:13:43.660 --> 01:13:52.940 1/2-- funny formula. 01:13:52.940 --> 01:13:56.820 Here is the tensor product, the tensor 01:13:56.820 --> 01:14:01.710 product of these three states with these two states. 01:14:01.710 --> 01:14:08.720 This can be written as 3 times 2 is equal to 4 plus 2 01:14:08.720 --> 01:14:11.870 in terms of number of states. 01:14:11.870 --> 01:14:16.020 The tensor product of this spin 1 and spin 1/2 01:14:16.020 --> 01:14:21.040 gives you a spin 3/2 multiplet with four states 01:14:21.040 --> 01:14:23.520 and a spin 1/2 multiplet with two states. 01:14:27.700 --> 01:14:33.040 So how do you calculate what are the states themselves? 01:14:35.890 --> 01:14:38.380 So the states themselves are the following. 01:14:47.100 --> 01:14:51.280 All right, here I have them. 01:14:51.280 --> 01:14:57.560 I claim that the J equals 3/2 states, 01:14:57.560 --> 01:15:02.610 m equals 3/2 states, the top state of that multiplet 01:15:02.610 --> 01:15:11.860 can only be the state here, the 1, 1 tensor 1/2, 1/2. 01:15:11.860 --> 01:15:18.490 And there's no way any other state can be put on the right. 01:15:18.490 --> 01:15:22.020 Because there's no other state with total z component 01:15:22.020 --> 01:15:24.160 of angular momentum equals 3/2. 01:15:24.160 --> 01:15:26.490 So that must be the state. 01:15:26.490 --> 01:15:33.000 Similarly, the J equals 3/2, m equals 01:15:33.000 --> 01:15:40.980 minus 3/2 state must be the bottom one-- 1, minus 1, 1/2, 01:15:40.980 --> 01:15:44.220 minus 1/2. 01:15:44.220 --> 01:15:46.490 The one that we wish to figure out 01:15:46.490 --> 01:15:53.710 is the next state here, which is the J equals 3/2, m equals 1/2. 01:15:53.710 --> 01:15:57.200 It's a linear combination of these two. 01:15:57.200 --> 01:15:58.390 But which one? 01:16:01.060 --> 01:16:05.250 That is kind of the last thing we want to do. 01:16:05.250 --> 01:16:08.520 Because it will pretty much solve the rest of the problem. 01:16:13.880 --> 01:16:17.540 So how do we solve for this? 01:16:17.540 --> 01:16:25.270 Well, we had this basic relation that we 01:16:25.270 --> 01:16:36.470 know how to lower or raise states of angular momentum-- 01:16:36.470 --> 01:16:43.530 m times m plus/minus 1 J-- I should have written it 01:16:43.530 --> 01:16:52.510 J plus/minus Jm equals h bar square root. 01:16:52.510 --> 01:16:56.790 More space for everybody to see this-- J times 01:16:56.790 --> 01:17:02.180 J plus 1 minus m times m plus/minus 1. 01:17:02.180 --> 01:17:08.120 Close the square root-- Jm plus/minus 1. 01:17:08.120 --> 01:17:14.020 So what I should try to do is lower this state, 01:17:14.020 --> 01:17:19.630 try to find this state by acting with J minus. 01:17:19.630 --> 01:17:22.660 So let me try to lower the state, so 01:17:22.660 --> 01:17:31.170 J minus on this state, on J equals 3/2, m equals 3/2. 01:17:31.170 --> 01:17:39.770 I can go to that formula and write it as h bar square root. 01:17:39.770 --> 01:17:47.260 J is 3/2, so 3/2 times 5/2 minus m, which is 3/2, 01:17:47.260 --> 01:17:49.960 times m minus 1, 1/2. 01:17:49.960 --> 01:17:57.570 We're doing the minus-- times the state 3/2, 1/2. 01:17:57.570 --> 01:18:00.030 So the state we want is here. 01:18:00.030 --> 01:18:03.570 And it's obtained by doing J minus on that. 01:18:03.570 --> 01:18:05.320 But we want the number here. 01:18:05.320 --> 01:18:08.340 So that's why I did all these square roots. 01:18:08.340 --> 01:18:16.650 And that just gives h bar square root of 3, 3/2, 1/2. 01:18:16.650 --> 01:18:19.670 Well, that still doesn't calculate it for me. 01:18:19.670 --> 01:18:21.510 But it comes very close. 01:18:29.490 --> 01:18:33.250 So you have it there. 01:18:33.250 --> 01:18:39.580 Now I want to do this but using the right hand side. 01:18:39.580 --> 01:18:41.560 So look at the right hand side. 01:18:41.560 --> 01:18:52.570 We want to do J minus, but on 1, 1 tensor 1/2, 1/2. 01:18:52.570 --> 01:18:55.820 So I applied J minus to the left hand side. 01:18:55.820 --> 01:19:01.130 Now we have to apply J minus to the right hand side. 01:19:01.130 --> 01:19:19.680 But J minus is L minus plus S minus on 1, 1 tensor 1/2, 1/2. 01:19:19.680 --> 01:19:22.750 When this acts, it acts on the first. 01:19:22.750 --> 01:19:31.570 So you get L minus on 1, 1 tensor 1/2, 1/2. 01:19:31.570 --> 01:19:38.280 And in the second term, you get plus 1, 1 tensor S 01:19:38.280 --> 01:19:40.520 minus on 1/2, 1/2. 01:19:44.420 --> 01:19:47.540 Now, what is L minus on 1, 1? 01:19:47.540 --> 01:19:50.330 You can use the same formula. 01:19:50.330 --> 01:19:51.450 It's 1, 1. 01:19:51.450 --> 01:19:53.250 And it's an angular momentum. 01:19:53.250 --> 01:20:00.350 So it just goes on and gives you h bar square 01:20:00.350 --> 01:20:05.520 root of 1 times 2 minus 1 times 0. 01:20:05.520 --> 01:20:10.720 1, 0-- it lowers it-- times 1/2, 1/2. 01:20:15.080 --> 01:20:19.270 Let me go here-- plus 1, 1. 01:20:19.270 --> 01:20:21.980 And what is S minus on this? 01:20:21.980 --> 01:20:26.390 Use the formula with J equals 1/2. 01:20:26.390 --> 01:20:37.460 So this is h bar square root of 1/2 times 3/2 minus 1/2 times 01:20:37.460 --> 01:20:43.522 minus 1/2 times 1/2 minus 1/2. 01:20:43.522 --> 01:20:48.620 Whew-- well not too difficult. 01:20:48.620 --> 01:20:57.630 But this gives you h over square root of 2, 1, 0 tensor 1/2, 01:20:57.630 --> 01:21:01.440 1/2 plus just h bar. 01:21:01.440 --> 01:21:11.030 This whole thing is 1-- 1, 1 tensor 1/2, minus 1/2. 01:21:11.030 --> 01:21:16.800 OK, stop a second to see what's happened. 01:21:16.800 --> 01:21:18.730 We had this equality. 01:21:18.730 --> 01:21:20.260 And we acted with J minus. 01:21:20.260 --> 01:21:23.390 Acting on the left, it gives us a number 01:21:23.390 --> 01:21:25.980 times the state we want. 01:21:25.980 --> 01:21:29.950 Acting on the right, we got this. 01:21:29.950 --> 01:21:36.220 So actually, equating this to that, or left hand side 01:21:36.220 --> 01:21:41.270 to right hand side, we finally found the state 3/2, 1/2. 01:21:41.270 --> 01:21:49.630 So the state 3/2, 1/2 is as follows. 01:21:54.410 --> 01:22:06.930 3/2, 1/2 is-- you must divide by that square root. 01:22:06.930 --> 01:22:09.435 So you get the square root of 3 down. 01:22:09.435 --> 01:22:12.280 The h bars cancel. 01:22:12.280 --> 01:22:18.290 So here it is, a very nice little formula-- 2 over 3, 01:22:18.290 --> 01:22:26.450 1, 0 tensor 1/2, 1/2 plus 1 over square root of 3, 01:22:26.450 --> 01:22:34.010 1, 1 tensor 1/2, minus 1/2. 01:22:34.010 --> 01:22:36.730 So we have the top state of the multiplet. 01:22:40.070 --> 01:22:43.910 We have the next state of the multiplet. 01:22:43.910 --> 01:22:47.240 We have-- I'm sorry, the top state of the multiplet 01:22:47.240 --> 01:22:49.250 was this. 01:22:49.250 --> 01:22:51.830 You have the bottom state of the multiplet, 01:22:51.830 --> 01:22:53.890 the middle state of the multiplet. 01:22:53.890 --> 01:22:58.920 What you're missing is the bottom and the middle term. 01:22:58.920 --> 01:23:02.935 And this one can be obtained in many ways. 01:23:05.660 --> 01:23:08.990 One way would be to raise this state. 01:23:08.990 --> 01:23:12.330 The minus 3/2 could be raised by one unit 01:23:12.330 --> 01:23:14.710 and do exactly the same thing. 01:23:14.710 --> 01:23:24.760 Well, the result is square root of 2 over 3, 1, 0 tensor 1/2, 01:23:24.760 --> 01:23:30.550 minus 1/2 plus 1 over square root of 3. 01:23:30.550 --> 01:23:35.240 That square root of 2 doesn't look right to me now. 01:23:35.240 --> 01:23:37.830 I must have copied it wrong. 01:23:37.830 --> 01:23:43.940 It's 1 over square root of 3-- 1 over square root of 3, 01:23:43.940 --> 01:23:50.660 1, minus 1 tensor 1/2, 1/2. 01:23:50.660 --> 01:23:52.365 So you've built that whole multiplet. 01:23:55.420 --> 01:24:00.480 And this state, as we said, was a linear combination 01:24:00.480 --> 01:24:01.996 of the two possible states. 01:24:05.800 --> 01:24:09.430 This 3 minus 1/2 was a linear combination 01:24:09.430 --> 01:24:11.530 of these two possible states. 01:24:11.530 --> 01:24:14.690 So the other states that are left over, 01:24:14.690 --> 01:24:17.785 the other linear combinations, form 01:24:17.785 --> 01:24:21.090 the J equals 1/2 multiplet. 01:24:21.090 --> 01:24:26.050 So basically, every state must be orthogonal to each other. 01:24:26.050 --> 01:24:35.200 So the other state, the 1/2, 1/2 and the 1/2, minus 1/2 of the J 01:24:35.200 --> 01:24:41.430 equals 1/2 multiplet must be this orthogonal to this. 01:24:41.430 --> 01:24:44.810 And this must be orthogonal to that. 01:24:44.810 --> 01:24:50.760 So those formulas are easily found by orthogonality. 01:24:50.760 --> 01:24:56.760 So I'll conclude by writing them-- minus 1 01:24:56.760 --> 01:25:04.010 over square root of 3, 1, 0, 1/2, 1/2 01:25:04.010 --> 01:25:13.170 plus the square root of 2 over 3, 1, 1, 1/2, minus 1/2. 01:25:13.170 --> 01:25:19.770 And here, you get 1 over square root of 3, 1, 0, 1/2, 01:25:19.770 --> 01:25:31.340 minus 1/2 minus 2 over square root of 3, 1, minus 1 01:25:31.340 --> 01:25:33.380 tensor 1/2, 1/2. 01:25:36.350 --> 01:25:45.870 So lots of terms, a little hard to read-- I apologize. 01:25:45.870 --> 01:25:53.120 Now, the punchline here is that you've found these states. 01:25:53.120 --> 01:25:57.050 And the claim is that these are states 01:25:57.050 --> 01:26:00.610 in which L dot S is diagonal. 01:26:00.610 --> 01:26:04.310 And it's kind of obvious that that should be the case. 01:26:04.310 --> 01:26:07.830 Because what was L dot S? 01:26:07.830 --> 01:26:20.150 So one last formula-- L dot S equals 01:26:20.150 --> 01:26:26.220 1/2 of J squared minus L squared minus S squared. 01:26:26.220 --> 01:26:29.770 Now, in terms of eigenvalues, this 01:26:29.770 --> 01:26:36.930 is 1/2 h squared J times J plus 1 minus L times L 01:26:36.930 --> 01:26:42.440 plus 1 minus S times S plus 1. 01:26:42.440 --> 01:26:45.160 Now, all the states that we built 01:26:45.160 --> 01:26:48.070 have definite values of J squared, 01:26:48.070 --> 01:26:50.450 definite values of S squared. 01:26:50.450 --> 01:26:52.740 Because L was 1. 01:26:52.740 --> 01:26:55.270 And S is 1/2. 01:26:55.270 --> 01:27:03.400 So here you go h squared over 2 J times J plus 1 minus 1 times 01:27:03.400 --> 01:27:09.450 2 is 2 minus 1/2 times 3/2 is 3/4. 01:27:12.050 --> 01:27:13.630 And that's the whole story. 01:27:13.630 --> 01:27:17.170 The whole story in a sense has been summarized by this. 01:27:17.170 --> 01:27:22.210 We have four states with J equals 3/2 01:27:22.210 --> 01:27:25.850 and two states with J equals 1/2. 01:27:25.850 --> 01:27:31.190 So these six states that you have here-- 01:27:31.190 --> 01:27:33.690 split because of this interaction 01:27:33.690 --> 01:27:42.700 into four states that have J equal to 3/2 and two 01:27:42.700 --> 01:27:47.370 states that have J equal to 1/2. 01:27:47.370 --> 01:27:49.640 And you plug the numbers here. 01:27:49.640 --> 01:27:51.456 And that gives you the amount of splitting. 01:27:54.790 --> 01:28:02.810 So actually, this height that this goes up here 01:28:02.810 --> 01:28:05.230 is h squared over 2. 01:28:05.230 --> 01:28:08.030 And this is minus h squared by the time you 01:28:08.030 --> 01:28:11.860 put the numbers J, 3/2, and 1/2. 01:28:11.860 --> 01:28:17.220 So all our work was because the Hamiltonian at the end 01:28:17.220 --> 01:28:20.360 was simple in J squared. 01:28:20.360 --> 01:28:23.240 And therefore, we needed J multiplets. 01:28:23.240 --> 01:28:27.900 J multiplets are the addition of angular momentum multiplets. 01:28:27.900 --> 01:28:31.270 In a sense, we don't have to construct these things 01:28:31.270 --> 01:28:34.910 if you don't want to calculate very explicit details. 01:28:34.910 --> 01:28:37.960 Once you have that, you have everything. 01:28:37.960 --> 01:28:42.620 This product of angular momentum 1, angular momentum 1/2 01:28:42.620 --> 01:28:45.800 gave you total angular momentum 3/2 01:28:45.800 --> 01:28:48.990 and 1/2-- four states, two states. 01:28:48.990 --> 01:28:53.380 So four states split one way, two states split the other way, 01:28:53.380 --> 01:28:55.210 and that's the end of the story. 01:28:55.210 --> 01:28:58.340 So more of this in recitation and more of this all 01:28:58.340 --> 01:29:00.580 of next week. 01:29:00.580 --> 01:29:02.790 We'll see you then.