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 --> 00:00:10.120 to offer high-quality educational resources for free. 00:00:10.120 --> 00:00:12.700 To make a donation, or to view additional materials 00:00:12.700 --> 00:00:16.603 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.603 --> 00:00:17.565 at ocw.mit.edu. 00:00:21.885 --> 00:00:27.260 PROFESSOR: OK so we're going to do this thing of the hydrogen 00:00:27.260 --> 00:00:31.350 atom and the algebraic solution. 00:00:31.350 --> 00:00:38.930 And I think it's not that long stuff 00:00:38.930 --> 00:00:41.140 so we can take it easy as we go along. 00:00:41.140 --> 00:00:47.480 I want to remind you of a couple of facts that will play a role. 00:00:47.480 --> 00:00:51.290 One result that is very general about the addition of angular 00:00:51.290 --> 00:00:55.850 momentum that you should of course know 00:00:55.850 --> 00:01:02.530 is that if you have a j1 times j2. 00:01:02.530 --> 00:01:05.500 What does this mean? 00:01:05.500 --> 00:01:15.650 You have some states of-- first angular momentum J1 00:01:15.650 --> 00:01:20.270 that so you have a whole multiplet with J1 00:01:20.270 --> 00:01:22.060 equals little j1. 00:01:22.060 --> 00:01:28.260 Which means the states in that multiplet have J1 squared, 00:01:28.260 --> 00:01:30.160 giving you h squared. 00:01:30.160 --> 00:01:33.370 Little j1 times little j1 plus 1. 00:01:33.370 --> 00:01:36.430 That's having a j1 multiplet. 00:01:36.430 --> 00:01:38.050 You have a j2 multiplet. 00:01:42.330 --> 00:01:46.160 And these are two independent commuting angular momenta 00:01:46.160 --> 00:01:48.440 acting on different degrees of freedom 00:01:48.440 --> 00:01:53.020 of the same particle or different particles. 00:01:53.020 --> 00:01:56.210 And what you've learned is that this 00:01:56.210 --> 00:02:00.570 can be written as a sum of representations. 00:02:00.570 --> 00:02:04.580 As a direct sum of angular momenta, 00:02:04.580 --> 00:02:17.114 which goes from j1 plus j2 plus j1 plus j2 minus 1 00:02:17.114 --> 00:02:25.490 all the way up to the representation 00:02:25.490 --> 00:02:29.390 with j1 minus j2. 00:02:29.390 --> 00:02:34.750 And these are all representations or multiplets 00:02:34.750 --> 00:02:37.070 that live in the tense or product, 00:02:37.070 --> 00:02:42.625 but they are multiplets of J equals j1 plus j2. 00:02:52.600 --> 00:02:58.140 These states here can be reorganized 00:02:58.140 --> 00:02:59.970 into these multiplets, and that's 00:02:59.970 --> 00:03:03.120 our main result for the addition of angular momentum. 00:03:03.120 --> 00:03:06.435 Mathematically, this formula summarizes it all. 00:03:09.950 --> 00:03:12.390 These states, when you write them 00:03:12.390 --> 00:03:15.520 as a basis here-- you take a basis state here 00:03:15.520 --> 00:03:20.540 times a basis state here-- these are called the coupled bases. 00:03:20.540 --> 00:03:23.990 And then you reorganize, you form linear combinations 00:03:23.990 --> 00:03:26.220 that you have been playing with, and then 00:03:26.220 --> 00:03:28.500 they get reorganized into these states. 00:03:28.500 --> 00:03:30.610 So these are called the coupled bases 00:03:30.610 --> 00:03:32.720 in which we're talking about states 00:03:32.720 --> 00:03:35.660 of the sum of angular momentum. 00:03:35.660 --> 00:03:39.500 So that's one fact we've learned about. 00:03:39.500 --> 00:03:42.810 Now as far as hydrogen is concerned, 00:03:42.810 --> 00:03:47.490 we're going to try today to understand the spectrum. 00:03:47.490 --> 00:03:50.830 And for that let me remind you what the spectrum was. 00:03:50.830 --> 00:03:58.750 The way we organized it was with an L versus energy levels. 00:03:58.750 --> 00:04:04.770 And we would put an L equals 0 state here-- well maybe-- 00:04:04.770 --> 00:04:08.259 there's color, so why not using color. 00:04:08.259 --> 00:04:09.175 Let's see if it works. 00:04:16.715 --> 00:04:18.100 Yeah, it's OK. 00:04:18.100 --> 00:04:21.050 L equals 0. 00:04:21.050 --> 00:04:24.280 And this was called n equals 1. 00:04:24.280 --> 00:04:31.020 There's an n equals 2 that has an L equals 0 state 00:04:31.020 --> 00:04:32.915 and an L equals 1 state. 00:04:36.600 --> 00:04:42.650 There's an n equals 3 state, set of states 00:04:42.650 --> 00:04:47.800 that come within L equals 0 and L equals 1 and an L equals 2. 00:04:51.520 --> 00:04:54.980 And it just goes on and on. 00:04:54.980 --> 00:05:02.910 With the energy levels En equals minus e squared over 2a0. 00:05:02.910 --> 00:05:09.250 That combination is familiar for energy, Bohr radius, charge 00:05:09.250 --> 00:05:13.470 of electron, with a 1 over n squared. 00:05:13.470 --> 00:05:21.280 And the fact is that for any level, for each n, 00:05:21.280 --> 00:05:27.625 L goes from 0, 1, 2, up to n minus 1. 00:05:30.780 --> 00:05:39.785 And for each n there's a total of n squared states. 00:05:44.050 --> 00:05:50.310 And you see it here, you have n equals 2, n equals 1, one 00:05:50.310 --> 00:05:50.930 state. 00:05:50.930 --> 00:05:54.660 n equals 2, you have L equals 0, one state. 00:05:54.660 --> 00:05:56.530 L equals 1 is 3 states. 00:05:56.530 --> 00:05:58.480 So it's 4. 00:05:58.480 --> 00:06:01.080 Here we'll have 4 plus 5. 00:06:01.080 --> 00:06:01.960 So 9. 00:06:01.960 --> 00:06:04.990 And maybe you can do it, it's a famous thing, 00:06:04.990 --> 00:06:09.470 there's n squared states at every level. 00:06:09.470 --> 00:06:12.990 So this pattern that of course continues 00:06:12.990 --> 00:06:16.380 and-- it's a little difficult to do 00:06:16.380 --> 00:06:19.150 a nice diagram of the hydrogen atom in scale 00:06:19.150 --> 00:06:22.050 because it's all pushed towards the zero energy 00:06:22.050 --> 00:06:25.520 with 1 over n squared, but that's how it goes. 00:06:25.520 --> 00:06:30.165 For n equals 4, you have 1, 2, 3, 4 for example. 00:06:33.510 --> 00:06:35.383 And this is what we want to understand. 00:06:41.390 --> 00:06:53.060 So in order to do that, let's return to this Hamiltonian, 00:06:53.060 --> 00:06:59.690 which is p squared over 2m minus e squared over r. 00:06:59.690 --> 00:07:04.486 And to the Runge-Lenz vector that we talked about in lecture 00:07:04.486 --> 00:07:07.340 and you've been playing with. 00:07:07.340 --> 00:07:13.170 So this Runge-Lenz vector, r, is defined 00:07:13.170 --> 00:07:26.830 to be 1 over 2me squared p cross L minus L cross p minus r 00:07:26.830 --> 00:07:29.470 over r. 00:07:29.470 --> 00:07:30.860 And it has no units. 00:07:38.000 --> 00:07:40.290 It's a vector that you've learned 00:07:40.290 --> 00:07:44.150 has interpretation of a constant vector that 00:07:44.150 --> 00:07:48.520 points in this direction, r. 00:07:48.520 --> 00:07:52.010 And it just stays fixed wherever the particle is going. 00:07:52.010 --> 00:07:54.570 Classically this is a constant vector 00:07:54.570 --> 00:07:58.820 that points in the direction of the major axis of the ellipse. 00:08:01.770 --> 00:08:06.080 With respect to this vector, this vector is Hermitian. 00:08:06.080 --> 00:08:10.320 And you may recall that when we did the classical vector, 00:08:10.320 --> 00:08:16.660 you had just p cross L and no 2 here. 00:08:16.660 --> 00:08:19.550 There are now two terms here. 00:08:19.550 --> 00:08:23.040 And they are necessary because we want to have a Hermitian 00:08:23.040 --> 00:08:25.790 operator, and this is the simplest 00:08:25.790 --> 00:08:29.790 way to construct the Hermitian operator, r. 00:08:29.790 --> 00:08:35.510 And the way is that you add to this this term, 00:08:35.510 --> 00:08:40.580 that if L and p commuted as they do in classical mechanics, 00:08:40.580 --> 00:08:43.240 that the term is identical to this. 00:08:43.240 --> 00:08:45.530 And you get back to the conventional thing 00:08:45.530 --> 00:08:47.920 that you had in classical mechanics. 00:08:47.920 --> 00:08:51.130 But in quantum mechanics, of course, they don't commute, 00:08:51.130 --> 00:08:54.110 so it's a little bit different. 00:08:54.110 --> 00:08:59.580 And moreover this thing, r, is Hermitian. 00:08:59.580 --> 00:09:02.780 L and p are Hermitian but when you take the Hermitian 00:09:02.780 --> 00:09:08.110 conjugate, L goes to the other side of p. 00:09:08.110 --> 00:09:11.050 And since they don't commute, that's not the same thing. 00:09:11.050 --> 00:09:15.100 So actually the Hermitian conjugate of this term is this. 00:09:15.100 --> 00:09:18.100 There's an extra minus sign in hermiticity 00:09:18.100 --> 00:09:20.500 when you have a cross product. 00:09:20.500 --> 00:09:22.500 So this is the Hermitian conjugate of this, 00:09:22.500 --> 00:09:25.120 this is the Hermitian conjugate of this second term, 00:09:25.120 --> 00:09:29.300 here's the first and therefore this is actually 00:09:29.300 --> 00:09:31.010 a Hermitian operator. 00:09:31.010 --> 00:09:32.120 And you can work with it. 00:09:34.680 --> 00:09:39.480 Moreover, in the case of classical mechanics, 00:09:39.480 --> 00:09:42.190 it was conserved. 00:09:42.190 --> 00:09:45.240 In the case of quantum mechanics this statement 00:09:45.240 --> 00:09:47.350 of conservation quantum mechanics 00:09:47.350 --> 00:09:53.610 is something that in one of the exercises that you were asked 00:09:53.610 --> 00:09:57.460 to try to do this computation so these computations 00:09:57.460 --> 00:10:00.490 are challenging. 00:10:00.490 --> 00:10:04.280 They're not all that trivial and are good exercises. 00:10:04.280 --> 00:10:07.670 So this is one of them. 00:10:07.670 --> 00:10:09.400 This is practice. 00:10:19.570 --> 00:10:21.940 OK this is the vector r. 00:10:21.940 --> 00:10:23.140 What about it? 00:10:23.140 --> 00:10:27.900 A few more things about it that are interesting. 00:10:27.900 --> 00:10:31.840 Because of the hermiticity condition 00:10:31.840 --> 00:10:35.840 or in-- a way you can check this directly 00:10:35.840 --> 00:10:39.710 in fact was one of the exercises for you to do, 00:10:39.710 --> 00:10:42.770 was p cross L-- you did it long time ago, 00:10:42.770 --> 00:10:50.320 I think-- is equal to minus L cross b plus 2ih bar p. 00:10:52.833 --> 00:10:53.666 This is an identity. 00:11:00.820 --> 00:11:12.100 And this identity helps you write this kind of term 00:11:12.100 --> 00:11:16.120 in a way in which you have just one order of products 00:11:16.120 --> 00:11:18.400 and a little extra term, rather than 00:11:18.400 --> 00:11:21.290 having two complicated terms. 00:11:21.290 --> 00:11:30.100 So the r can be written as 1 over me 00:11:30.100 --> 00:11:45.650 squared alone, p cross L minus ihp minus r over r. 00:11:45.650 --> 00:11:46.243 For example. 00:11:48.830 --> 00:11:58.410 By writing this term as another p cross L minus that thing 00:11:58.410 --> 00:12:01.550 gives you that expression for r. 00:12:01.550 --> 00:12:03.630 You have an alternative expression 00:12:03.630 --> 00:12:05.720 in which you solve for the other one. 00:12:05.720 --> 00:12:14.820 So it's 1 over me squared minus L cross p plus ih bar p. 00:12:21.160 --> 00:12:27.460 Now, r-- we need to understand r better. 00:12:27.460 --> 00:12:31.930 That's really the challenge of this whole derivation. 00:12:31.930 --> 00:12:35.250 So we have one thing that is conserved. 00:12:35.250 --> 00:12:37.440 Angular momentum is conserved. 00:12:37.440 --> 00:12:40.380 It commutes with the Hamiltonian. 00:12:40.380 --> 00:12:44.530 We have another thing that is conserved, this r. 00:12:44.530 --> 00:12:47.200 But we have to understand better what it is. 00:12:47.200 --> 00:12:52.560 So one thing that you can ask is, well, r is conserved, 00:12:52.560 --> 00:12:56.320 so r squared is conserved as well. 00:12:56.320 --> 00:13:01.260 So r squared, if I can simplify it-- if I can do the algebra 00:13:01.260 --> 00:13:06.650 and simplify it-- it should not be that complicated. 00:13:06.650 --> 00:13:11.810 So again a practice problem was given to do that computation. 00:13:11.810 --> 00:13:17.840 And I think these forms are useful for that, to work less. 00:13:17.840 --> 00:13:20.520 And the computation gives a very nice result, 00:13:20.520 --> 00:13:29.340 where r squared is equal to 1 plus 2 H over me 00:13:29.340 --> 00:13:35.690 to the fourth L squared plus h bar squared. 00:13:35.690 --> 00:13:40.160 Kind of a strange result if you think about it. 00:13:40.160 --> 00:13:42.380 People that want to do this classically 00:13:42.380 --> 00:13:44.855 first would find that there's no h squared. 00:13:48.160 --> 00:13:53.040 And here, this h, is that whole h that we have here. 00:13:53.040 --> 00:13:55.430 It's a complicated thing. 00:13:55.430 --> 00:14:00.360 So this right hand side is quite substantial. 00:14:00.360 --> 00:14:03.580 You don't have to worry that h is in this side 00:14:03.580 --> 00:14:05.670 or whether it's on the other side 00:14:05.670 --> 00:14:10.480 because h commutes with L. L is conserved. 00:14:10.480 --> 00:14:13.550 So h appears like that. 00:14:13.550 --> 00:14:17.376 And this, again, is the result of another computation. 00:14:21.570 --> 00:14:23.120 So we've learned something. 00:14:23.120 --> 00:14:26.630 r-- oh, I'm sorry, this is r squared. 00:14:26.630 --> 00:14:29.070 Apologies. 00:14:29.070 --> 00:14:30.590 r is conserved. 00:14:30.590 --> 00:14:34.630 r squared must be conserved because if h commutes with r 00:14:34.630 --> 00:14:37.310 it commutes with r squared as well. 00:14:37.310 --> 00:14:40.050 And therefore whatever you see on the right hand side, 00:14:40.050 --> 00:14:41.810 the whole thing must be conserved. 00:14:41.810 --> 00:14:44.660 And h is conserved, of course. 00:14:44.660 --> 00:14:49.320 And L squared is conserved. 00:14:49.320 --> 00:14:55.220 Now we need one more property of a relation-- you see, 00:14:55.220 --> 00:14:57.540 you have to do these things. 00:14:57.540 --> 00:15:02.470 Even if you probably don't have an inspiration at this moment 00:15:02.470 --> 00:15:04.610 how you're going to try to understand this, 00:15:04.610 --> 00:15:06.935 there are things that just curiosity 00:15:06.935 --> 00:15:09.370 should tell that you should do. 00:15:09.370 --> 00:15:12.270 We have L, we do L squared. 00:15:12.270 --> 00:15:14.560 It's an important operator. 00:15:14.560 --> 00:15:15.060 OK. 00:15:15.060 --> 00:15:19.920 We had r, we did r squared, which is an important operator. 00:15:19.920 --> 00:15:23.396 But one thing we can do is L dot r. 00:15:23.396 --> 00:15:26.030 It's a good question what L dot r is. 00:15:28.850 --> 00:15:31.430 So what is L dot r? 00:15:39.760 --> 00:15:47.560 So-- or r dot L. What is it? 00:15:47.560 --> 00:15:53.830 Well a few things that are important to note 00:15:53.830 --> 00:15:58.902 are that you did show before that you 00:15:58.902 --> 00:16:07.490 know that r dot L, little r dot L, is 0. 00:16:07.490 --> 00:16:14.220 And little p dot L is 0. 00:16:14.220 --> 00:16:18.500 These are obvious classically, because L 00:16:18.500 --> 00:16:21.210 is perpendicular to both r and p. 00:16:21.210 --> 00:16:26.480 But quantum mechanically they take a little more work. 00:16:26.480 --> 00:16:30.170 They're not complicated, but you've shown those two. 00:16:30.170 --> 00:16:38.660 So if you have r dot L, you would have, 00:16:38.660 --> 00:16:42.550 for example, here-- r dot L, you would 00:16:42.550 --> 00:16:50.260 have to do and think of this whole r 00:16:50.260 --> 00:16:52.370 and put an L on the right. 00:16:52.370 --> 00:16:59.660 Well this little r dotted with the L on the right would be 0. 00:16:59.660 --> 00:17:06.569 That p dotted with L on the right would be 0. 00:17:06.569 --> 00:17:11.530 And we're almost there, but p cross L dot L, 00:17:11.530 --> 00:17:14.460 well, what is that? 00:17:14.460 --> 00:17:16.490 Let me talk about it here. 00:17:16.490 --> 00:17:24.130 P cross L dot L-- so this is part 00:17:24.130 --> 00:17:27.750 of the computation of this r dot L. We've already 00:17:27.750 --> 00:17:29.510 seen this term will give nothing, 00:17:29.510 --> 00:17:30.990 this term will give nothing. 00:17:30.990 --> 00:17:34.180 But this term could give something. 00:17:34.180 --> 00:17:37.360 So when you face something like that, 00:17:37.360 --> 00:17:38.900 maybe you say, well, I don't know 00:17:38.900 --> 00:17:41.610 any identities I should be using here. 00:17:44.320 --> 00:17:48.510 So you just do it. 00:17:48.510 --> 00:17:51.180 Then you say, this is i-th component 00:17:51.180 --> 00:17:53.460 of this vector times the i-th of that. 00:17:53.460 --> 00:17:55.070 So it's epsilon ijkpjLkLi. 00:18:08.170 --> 00:18:12.960 And then you say look this looks a little-- you could 00:18:12.960 --> 00:18:16.850 say many things that are wrong and get the right answer. 00:18:16.850 --> 00:18:21.685 So you could say, oh, ki symmetric 00:18:21.685 --> 00:18:23.560 and ki anti-symmetric. 00:18:23.560 --> 00:18:27.000 But that's wrong, because these k and i are not 00:18:27.000 --> 00:18:32.280 symmetric really because these operators don't commute. 00:18:32.280 --> 00:18:39.730 So the answer will be zero, but for a more complicated reason. 00:18:39.730 --> 00:18:41.600 So what do you have in here? 00:18:41.600 --> 00:18:42.930 ki. 00:18:42.930 --> 00:18:47.798 Let's move the i to the end of the epsilon, so jkipjLkLi. 00:18:53.980 --> 00:19:02.390 And now you see this part is pj L cross cross L j. 00:19:10.110 --> 00:19:12.740 Is the cross product of this. 00:19:12.740 --> 00:19:14.870 But what is L cross L? 00:19:14.870 --> 00:19:16.340 You probably remember. 00:19:16.340 --> 00:19:24.010 This is ih bar L L cross L, that's 00:19:24.010 --> 00:19:28.240 the computation in relation of angular momentum. 00:19:28.240 --> 00:19:31.680 In case you kind of don't remember 00:19:31.680 --> 00:19:35.120 it was ih bar L. Like that. 00:19:35.120 --> 00:19:40.190 So now p dot L is anyway 0. 00:19:40.190 --> 00:19:43.160 So this is 0. 00:19:43.160 --> 00:19:48.485 So it's kind of-- it's a little delicate 00:19:48.485 --> 00:19:50.080 to do these computations. 00:19:50.080 --> 00:19:54.830 But so since that term is zero, this thing is zero. 00:20:01.560 --> 00:20:05.170 Now you may as well-- r dot L is 0. 00:20:05.170 --> 00:20:09.610 Is L dot r also 0? 00:20:09.610 --> 00:20:12.150 It's not all that obvious you can even do that. 00:20:12.150 --> 00:20:15.730 Well in a second we'll see that that's true as well. 00:20:15.730 --> 00:20:20.840 L dot r and r dot L, capital R, are 0. 00:20:33.780 --> 00:20:37.000 Let's remember-- let's continue-- let's see, 00:20:37.000 --> 00:20:41.510 I wanted to number some of these equations. 00:20:41.510 --> 00:20:45.090 We're going to need them. 00:20:45.090 --> 00:20:46.850 So this will be equation one. 00:20:53.390 --> 00:20:57.030 This will be equation two, it's an important one. 00:21:05.060 --> 00:21:11.700 Now what was-- let me remind you of a notation 00:21:11.700 --> 00:21:15.260 we also had about vectors and their rotations. 00:21:15.260 --> 00:21:19.015 Vector under rotations. 00:21:26.410 --> 00:21:32.020 So what was a vector, Vi, under rotations 00:21:32.020 --> 00:21:39.380 was something that you had LiVj equals ih bar epsilon ijkvk. 00:21:46.250 --> 00:21:51.760 So there is a way to write this with cross products that 00:21:51.760 --> 00:21:53.500 is useful in some cases. 00:21:53.500 --> 00:21:58.270 So I will do it. 00:21:58.270 --> 00:22:00.400 You probably have seen that in the notes, 00:22:00.400 --> 00:22:02.090 but let me remind you. 00:22:02.090 --> 00:22:06.370 Consider this product, L cross V plus V 00:22:06.370 --> 00:22:11.230 cross L and the i-th component of it. 00:22:14.260 --> 00:22:17.440 i-th component of this product. 00:22:17.440 --> 00:22:25.737 So this is epsilon ijk and you have LjVk plus VjLk. 00:22:34.130 --> 00:22:42.160 Now in this term you can do something nice. 00:22:42.160 --> 00:22:45.160 If you think of it like expanded out, 00:22:45.160 --> 00:22:48.580 you have the second term has epsilon ijkVjk. 00:22:51.490 --> 00:22:54.730 Change j for k. 00:22:54.730 --> 00:22:57.286 If you change j for k, this will be VkLj. 00:22:59.960 --> 00:23:02.320 And these would have the opposite order. 00:23:02.320 --> 00:23:06.460 But this order can be changed up to the cost of a minus sign. 00:23:06.460 --> 00:23:14.832 So I claim this is ijk-- first term is the same-- minus VkLj. 00:23:19.160 --> 00:23:23.920 So in the second term, for this term alone, we've 00:23:23.920 --> 00:23:26.960 done for this term, multiplied with this of course, 00:23:26.960 --> 00:23:29.560 we've done j and k relabeling. 00:23:39.160 --> 00:23:46.320 But this is nothing else than the commutator of L with V. 00:23:46.320 --> 00:23:47.790 So this is epsilon ijkLjVk. 00:23:57.600 --> 00:24:05.560 That's epsilon ijk, and this is epsilon jkp or LVL. 00:24:13.160 --> 00:24:17.380 Now, 2 epsilons with 2 common indices 00:24:17.380 --> 00:24:20.410 is something that it simple. 00:24:20.410 --> 00:24:24.150 It's a commutator dealt on the other indices. 00:24:24.150 --> 00:24:28.740 Now it's better if they are sort of all aligned in the same way, 00:24:28.740 --> 00:24:33.140 but they kind of are because this L, without paying a price, 00:24:33.140 --> 00:24:34.830 can be put as the first index. 00:24:34.830 --> 00:24:38.080 So you have jk as the second and third 00:24:38.080 --> 00:24:40.290 and-- jk as the second and third-- 00:24:40.290 --> 00:24:44.230 once L has been moved to the first position. 00:24:44.230 --> 00:24:52.720 So this thing is 2 times delta iL. 00:24:52.720 --> 00:24:57.660 And there's an h bar, ih bar I forgot here. 00:24:57.660 --> 00:24:59.070 ih bar. 00:24:59.070 --> 00:25:06.410 2 delta ik ih bar iL ih bar VL. 00:25:06.410 --> 00:25:10.070 So this is 2 ih bar Vi. 00:25:12.600 --> 00:25:16.610 So this whole thing the i-th component of this thing, 00:25:16.610 --> 00:25:19.720 using this commutation relation is this. 00:25:19.720 --> 00:25:23.060 So what we've learned is that L cross 00:25:23.060 --> 00:25:33.820 V plus V cross L you see go to 2 ih bar V. 00:25:33.820 --> 00:25:42.430 And that's a statement as a vector relation of the fact 00:25:42.430 --> 00:25:46.160 that V is a vector in the rotations. 00:25:46.160 --> 00:25:52.830 So for V to be a vector in the rotations means this. 00:25:52.830 --> 00:25:57.200 And if you wish, it means this thing as well. 00:25:57.200 --> 00:26:00.470 It's just another thing of what it means. 00:26:00.470 --> 00:26:05.280 Now R is a vector in the rotations. 00:26:05.280 --> 00:26:08.980 This capital R. Why? 00:26:08.980 --> 00:26:15.850 You've shown that if you have a vector in the rotations 00:26:15.850 --> 00:26:18.670 and you multiply it by another vector in the rotations 00:26:18.670 --> 00:26:20.510 under the cross product, it's still 00:26:20.510 --> 00:26:22.040 a vector in the rotations. 00:26:22.040 --> 00:26:24.810 So this is a vector in the rotations, this is, 00:26:24.810 --> 00:26:27.850 and this is a vector in the rotations. 00:26:27.850 --> 00:26:29.620 R is a vector in the rotations. 00:26:29.620 --> 00:26:45.630 So this capital R is a vector on the rotations, 00:26:45.630 --> 00:26:48.110 which means two things. 00:26:48.110 --> 00:26:52.890 It means it satisfies this kind of equation. 00:26:52.890 --> 00:27:19.910 So r cross-- or L cross R plus R cross L is equal to ih bar R. 00:27:19.910 --> 00:27:23.570 So R is a vector in the rotation. 00:27:23.570 --> 00:27:27.330 It's a fact beyond doubt. 00:27:27.330 --> 00:27:31.030 And that means that we now know the commutation relations 00:27:31.030 --> 00:27:35.060 between L and R. So we're starting to put together 00:27:35.060 --> 00:27:39.160 this picture in which we get familiar with R 00:27:39.160 --> 00:27:41.950 and the commutators that are possible. 00:27:41.950 --> 00:27:45.470 So I can summarize it here. 00:27:55.170 --> 00:28:03.900 L dot R LiRj is ih bar epsilon ijkRk. 00:28:08.410 --> 00:28:12.900 That's the same statement as this one but in components. 00:28:12.900 --> 00:28:23.140 And now you see why R dot L is equal to L dot R. 00:28:23.140 --> 00:28:29.150 Because actually if you put the same two indices here, i and i, 00:28:29.150 --> 00:28:30.130 you get zero. 00:28:30.130 --> 00:28:39.410 So when you have R dot L you have R1L1 plus R2L2 plus R3L3. 00:28:39.410 --> 00:28:41.700 And each of these two commute when 00:28:41.700 --> 00:28:43.740 the two indices are the same. 00:28:43.740 --> 00:28:45.090 Because of the epsilon. 00:28:45.090 --> 00:28:47.640 So R dot L is 0. 00:28:47.640 --> 00:28:54.710 And now you also appreciate that L dot R is also 0, too. 00:29:00.226 --> 00:29:00.725 OK. 00:29:03.900 --> 00:29:09.440 Now comes, in a sense, the most difficult of all calculations. 00:29:09.440 --> 00:29:11.840 Even if this seemed a little easy. 00:29:11.840 --> 00:29:15.680 But you can get quite far with it. 00:29:15.680 --> 00:29:18.800 So what do you do with Ls? 00:29:18.800 --> 00:29:22.550 You computed L commutators and you 00:29:22.550 --> 00:29:24.750 got the algebra of angular momentum. 00:29:28.340 --> 00:29:29.810 Over here. 00:29:29.810 --> 00:29:32.440 This is the algebra for angular momentum. 00:29:32.440 --> 00:29:36.610 And this kind of nontrivial calculation, 00:29:36.610 --> 00:29:38.560 you did it by building results. 00:29:38.560 --> 00:29:41.510 You knew how R was a vector in the rotation 00:29:41.510 --> 00:29:43.430 or how p was a vector in the rotation. 00:29:43.430 --> 00:29:48.170 You multiplied the two of them, and it was not so difficult. 00:29:48.170 --> 00:29:54.070 But the calculation that you really need to do now 00:29:54.070 --> 00:30:01.540 is the calculation of the commutator say of Ri with Rj. 00:30:01.540 --> 00:30:04.340 And that looks like a little bit of a nightmare. 00:30:04.340 --> 00:30:10.420 You have to commute this whole thing with itself. 00:30:10.420 --> 00:30:13.750 Lots of p's, L's, R's. 00:30:13.750 --> 00:30:16.790 1 over R's, those don't commute with p. 00:30:16.790 --> 00:30:18.870 You remember that. 00:30:18.870 --> 00:30:21.550 So this kind of calculation done by brute force. 00:30:21.550 --> 00:30:25.120 You're talking a day, probably. 00:30:25.120 --> 00:30:26.690 I think so. 00:30:26.690 --> 00:30:29.500 And probably it becomes a mess, but. 00:30:29.500 --> 00:30:32.160 You'll find a little trick helps to organize it better. 00:30:32.160 --> 00:30:34.380 It's less of a mess, but still you 00:30:34.380 --> 00:30:38.200 don't get it and-- try several times. 00:30:38.200 --> 00:30:42.320 So what we're going to do is try to think 00:30:42.320 --> 00:30:46.550 of what the answer could be by some arguments. 00:30:46.550 --> 00:30:50.210 And then once we know what the answer can be, 00:30:50.210 --> 00:30:53.020 there's still one calculation to be done. 00:30:53.020 --> 00:30:55.780 That I will probably put in the notes, 00:30:55.780 --> 00:30:58.050 but it's not a difficult one. 00:30:58.050 --> 00:31:01.420 And the answer just pops out. 00:31:01.420 --> 00:31:10.220 So the question is what is R cross R. R cross R is really 00:31:10.220 --> 00:31:14.040 what we have when we have this commutator. 00:31:14.040 --> 00:31:19.650 So we need to know what R cross R is, just like L cross L. 00:31:19.650 --> 00:31:25.420 Now R is not likely to be an angular momentum. 00:31:25.420 --> 00:31:27.880 It's a vector but it's not an angular momentum. 00:31:27.880 --> 00:31:30.610 Has nothing to do with it. 00:31:30.610 --> 00:31:32.270 It's more complicated. 00:31:32.270 --> 00:31:36.140 So what is R cross R quantum-mechanically? 00:31:36.140 --> 00:31:40.200 Classically, of course, it would be zero. 00:31:40.200 --> 00:31:46.230 So first thing is you think of what this should be. 00:31:50.330 --> 00:31:56.720 We have a vector, because the cross product of two vectors. 00:31:56.720 --> 00:32:01.900 Now I want to emphasize one other thing, 00:32:01.900 --> 00:32:08.870 that it should be this thing-- R cross R-- is 00:32:08.870 --> 00:32:10.330 tantamount to this thing. 00:32:10.330 --> 00:32:11.590 What is this thing? 00:32:14.450 --> 00:32:17.660 It should be actually proportional 00:32:17.660 --> 00:32:19.675 to some conserved quantity. 00:32:22.510 --> 00:32:26.640 And the reason is quite interesting. 00:32:26.640 --> 00:32:29.545 So this is a small aside here. 00:32:33.780 --> 00:32:36.870 If some operator is conserved, it 00:32:36.870 --> 00:32:38.440 commutes with the Hamiltonian. 00:32:38.440 --> 00:32:52.595 Say if S1 and S2 are symmetries, that means that S1 with h 00:32:52.595 --> 00:32:59.506 is equal to S2 with h is equal to zero. 00:32:59.506 --> 00:33:05.580 Then the claim is that the commutator of this S1 and S2 00:33:05.580 --> 00:33:12.255 claim S1 commutator with S2 is also a symmetry. 00:33:17.140 --> 00:33:21.120 So the reason is because commutator 00:33:21.120 --> 00:33:29.425 of S1 S2 commutator with h is equal actually to zero. 00:33:29.425 --> 00:33:32.500 And why would it be equal to zero? 00:33:32.500 --> 00:33:34.840 It's because of the so-called Jacobi 00:33:34.840 --> 00:33:37.580 identity for commutators. 00:33:37.580 --> 00:33:41.930 You'll remember when you have three things like that, 00:33:41.930 --> 00:33:46.355 this term is equal to 1-- this term plus 1, 00:33:46.355 --> 00:33:48.050 in which you cycle them. 00:33:48.050 --> 00:33:51.390 And plus another one where you cycle them again 00:33:51.390 --> 00:33:53.230 is equal to zero. 00:33:53.230 --> 00:33:55.600 That's a Jacobi identity. 00:33:55.600 --> 00:33:59.460 And in those cyclings you get an h with S2, 00:33:59.460 --> 00:34:01.710 for example, that is zero. 00:34:01.710 --> 00:34:04.090 And then an h with S1, which is zero. 00:34:04.090 --> 00:34:08.310 So you use these things here and you prove that. 00:34:08.310 --> 00:34:09.959 So I write here, by Jacobi. 00:34:14.250 --> 00:34:17.520 So if you have a conserved-- this 00:34:17.520 --> 00:34:20.630 is the great thing about conserved quantities, 00:34:20.630 --> 00:34:24.050 if you have one conserved quantity, it's OK. 00:34:24.050 --> 00:34:26.300 But if you have two, you're in business. 00:34:26.300 --> 00:34:29.985 Because you can then take the commutator of these two and you 00:34:29.985 --> 00:34:31.400 get another conserved quantity. 00:34:31.400 --> 00:34:34.830 And then more commutators and you keep taking commutators 00:34:34.830 --> 00:34:39.420 and if you're lucky you get all of the conserved quantities. 00:34:39.420 --> 00:34:45.480 So here R cross R refers to this commutator. 00:34:45.480 --> 00:34:48.670 So whatever is on the right should be a vector 00:34:48.670 --> 00:34:50.375 and should be conserved. 00:34:57.460 --> 00:35:03.270 And what are our conserved vectors? 00:35:03.270 --> 00:35:11.210 Well our conserved vectors-- candidates here-- 00:35:11.210 --> 00:35:25.130 are L, R itself, and L cross R. That's pretty much it. 00:35:25.130 --> 00:35:30.330 L and R are our only conserved things, so it better be that. 00:35:35.450 --> 00:35:39.370 Still this is far too much. 00:35:39.370 --> 00:35:41.330 So there could be a term proportional 00:35:41.330 --> 00:35:46.650 to L, a term proportional to R, a term proportional to L dot R. 00:35:46.650 --> 00:35:51.940 So this kind of analysis is based by something 00:35:51.940 --> 00:35:53.760 that Julian Schwinger did. 00:35:53.760 --> 00:35:57.090 This same guy that actually did quantum 00:35:57.090 --> 00:36:01.940 electrodynamics along with Feynman and Tomonaga. 00:36:01.940 --> 00:36:04.360 And he's the one of those who invented 00:36:04.360 --> 00:36:07.870 the trick of using three-dimensional angular 00:36:07.870 --> 00:36:10.770 momentum for the two-dimensional oscillator. 00:36:10.770 --> 00:36:13.860 And had lots of bags of tricks. 00:36:13.860 --> 00:36:16.920 So actually this whole discussion of the hydrogen 00:36:16.920 --> 00:36:22.200 atom-- most books just say, well, these calculations are 00:36:22.200 --> 00:36:22.810 hopeless. 00:36:22.810 --> 00:36:25.330 Let me give you the answers. 00:36:25.330 --> 00:36:28.780 Schwinger, on the other hand, in his book on quantum mechanics-- 00:36:28.780 --> 00:36:33.940 which is kind of interesting but very idiosyncratic-- 00:36:33.940 --> 00:36:37.910 finds a trick to do every calculation. 00:36:37.910 --> 00:36:41.230 So you never get into a big mess. 00:36:41.230 --> 00:36:44.980 He's absolutely elegant and keeps pulling tricks 00:36:44.980 --> 00:36:45.700 from the bag. 00:36:45.700 --> 00:36:49.830 So this is one of those tricks. 00:36:49.830 --> 00:36:54.180 Basically he goes through the following analysis now 00:36:54.180 --> 00:36:59.490 and says, look, suppose I have the vector R 00:36:59.490 --> 00:37:01.960 and I do a parity transformation. 00:37:01.960 --> 00:37:05.490 I change it for minus R. 00:37:05.490 --> 00:37:10.010 What happens under those circumstances? 00:37:10.010 --> 00:37:15.950 Well the momentum is the rate of change of R, 00:37:15.950 --> 00:37:18.260 should also change sign. 00:37:18.260 --> 00:37:20.840 Quantum mechanically this is consistent, 00:37:20.840 --> 00:37:25.590 because a commutation between R and p should give you h bar. 00:37:25.590 --> 00:37:31.810 And if R changes, p should change sign. 00:37:31.810 --> 00:37:37.530 But now when you do this, L, which is R cross p, 00:37:37.530 --> 00:37:47.960 just goes to L. And R, however, changes sign 00:37:47.960 --> 00:37:53.320 because L doesn't change sign but p does and R does. 00:37:53.320 --> 00:37:57.330 So under these changes-- so this is the originator, 00:37:57.330 --> 00:38:00.140 the troublemaker and then everybody else 00:38:00.140 --> 00:38:04.875 follows-- R also changes sign. 00:38:11.200 --> 00:38:16.100 So this is extremely powerful because if you imagine 00:38:16.100 --> 00:38:18.840 this being equal to something, well it 00:38:18.840 --> 00:38:21.980 should be consistent with the symmetries. 00:38:21.980 --> 00:38:28.470 So as I change R to minus R, capital R changes sign 00:38:28.470 --> 00:38:32.550 but the left hand side doesn't change sign. 00:38:32.550 --> 00:38:36.210 Therefore the right hand side should not change sign. 00:38:36.210 --> 00:38:42.080 And R changes sign and L cross R changes sign. 00:38:42.080 --> 00:38:45.590 So computation kind of finished because the only thing 00:38:45.590 --> 00:38:53.220 you can get on the right is L. 00:38:53.220 --> 00:38:55.520 This is the kind of thing that you do 00:38:55.520 --> 00:38:57.910 and probably if you were writing a paper on 00:38:57.910 --> 00:39:01.400 that you would anyway do the calculation. 00:39:01.400 --> 00:39:05.420 The silly way, the- the right way. 00:39:05.420 --> 00:39:08.880 But this is quite save of times. 00:39:08.880 --> 00:39:11.300 So actually what you have learned 00:39:11.300 --> 00:39:24.940 is that R cross R is equal to some scalar conserved quantity, 00:39:24.940 --> 00:39:28.240 which is something that is conserved that could 00:39:28.240 --> 00:39:31.020 be like an h, for example, here. 00:39:31.020 --> 00:39:33.130 But it's a scalar. 00:39:33.130 --> 00:39:37.480 And, L. 00:39:37.480 --> 00:39:41.290 Well once you know that much, it doesn't take much work 00:39:41.290 --> 00:39:43.810 to do this and to calculate what it is. 00:39:43.810 --> 00:39:47.670 But I will skip that calculation. 00:39:47.670 --> 00:39:51.440 This is the sort of thoughtful part of it. 00:39:51.440 --> 00:39:59.820 And R cross R turns out to be ih bar minus 2 h again. 00:39:59.820 --> 00:40:04.060 h shows up in several places, like here, 00:40:04.060 --> 00:40:08.920 so it tends-- it has a tendency to show up. 00:40:08.920 --> 00:40:16.210 me to the fourth L. 00:40:16.210 --> 00:40:24.640 So this is our equation for-- and in a sense, 00:40:24.640 --> 00:40:29.020 all the hard work has been done. 00:40:29.020 --> 00:40:32.940 Because now you have a complete understanding of these two 00:40:32.940 --> 00:40:36.470 vectors, L and R. You know what L squared is, 00:40:36.470 --> 00:40:39.235 what R squared is, what L dot R is. 00:40:39.235 --> 00:40:41.115 And you know all the commutators, 00:40:41.115 --> 00:40:46.600 you know the commutation of L with L, L with R, and R with R. 00:40:46.600 --> 00:40:50.140 You've done all the algebraic work. 00:40:50.140 --> 00:40:53.360 And the question is, how do we proceed from now 00:40:53.360 --> 00:40:57.960 to solve the hydrogen atom. 00:40:57.960 --> 00:41:02.610 So the way we proceed is kind of interesting. 00:41:02.610 --> 00:41:07.270 We're going to try to build from this L that 00:41:07.270 --> 00:41:09.530 is an angular momentum. 00:41:09.530 --> 00:41:14.890 And this R that is not an angular momentum. 00:41:14.890 --> 00:41:18.300 Two sets of angular momenta. 00:41:18.300 --> 00:41:19.620 You have two vectors. 00:41:19.620 --> 00:41:23.730 So somehow we want to try to combine them in such a way 00:41:23.730 --> 00:41:27.500 that we can invent two angular momenta. 00:41:27.500 --> 00:41:30.070 Just like the angular momentum in 00:41:30.070 --> 00:41:32.810 the two-dimensional harmonic oscillator. 00:41:32.810 --> 00:41:37.860 It was not directly through angular momentum, 00:41:37.860 --> 00:41:40.450 but was mathematical angular momentum. 00:41:40.450 --> 00:41:43.910 These two angular momenta we're going to build, one of them 00:41:43.910 --> 00:41:46.870 is going to be recognizable. 00:41:46.870 --> 00:41:49.130 The other one is going to be a little unfamiliar. 00:41:51.910 --> 00:41:55.540 But now I have to do something that-- it 00:41:55.540 --> 00:42:00.380 may sound a little unusual, but is 00:42:00.380 --> 00:42:03.570 necessary to simplify our life. 00:42:03.570 --> 00:42:07.120 I want to say some words that will 00:42:07.120 --> 00:42:13.640 allow me to think of this h here as a number. 00:42:13.640 --> 00:42:19.280 And would allow me to think of this h as a number. 00:42:19.280 --> 00:42:22.150 So here's what we're going to say. 00:42:22.150 --> 00:42:25.025 It's an assumption-- it's no assumption, 00:42:25.025 --> 00:42:27.180 but it sounds like an assumption. 00:42:27.180 --> 00:42:29.330 But there's no assumption whatsoever. 00:42:29.330 --> 00:42:33.880 We say the following: this hydrogen atom 00:42:33.880 --> 00:42:36.990 is going to have some states. 00:42:36.990 --> 00:42:44.590 So let's assume there is one state, and it has some energy. 00:42:44.590 --> 00:42:48.180 If I have that state with some energy, well, 00:42:48.180 --> 00:42:50.360 that would be the end of the story. 00:42:50.360 --> 00:42:53.570 But in fact, the thing that they want 00:42:53.570 --> 00:42:57.200 to allow the possibility for is that at that the energy there 00:42:57.200 --> 00:42:59.870 are more states. 00:42:59.870 --> 00:43:04.070 One state would be OK, maybe sometimes it happens. 00:43:04.070 --> 00:43:08.250 But in general there are more states at that energy. 00:43:08.250 --> 00:43:13.370 So I don't-- I'm not making any physical assumption to state 00:43:13.370 --> 00:43:18.040 that there is a subspace of degenerate states. 00:43:18.040 --> 00:43:22.530 And in that subspace of degenerate states, 00:43:22.530 --> 00:43:24.820 there may be just one state, there are two states, 00:43:24.820 --> 00:43:26.278 there are three states, but there's 00:43:26.278 --> 00:43:31.740 subspace of degenerate states that have some energy. 00:43:31.740 --> 00:43:35.750 And I'm going to work in that subspace. 00:43:35.750 --> 00:43:37.830 And all the operators that I have 00:43:37.830 --> 00:43:40.500 are going to be acting in that subspace. 00:43:40.500 --> 00:43:42.790 And I'm going to analyze subspace 00:43:42.790 --> 00:43:45.840 by subspace of different energies. 00:43:45.840 --> 00:43:48.320 So we're going to work with one subspace 00:43:48.320 --> 00:43:49.480 of degenerate energies. 00:43:49.480 --> 00:43:54.750 And if I have, for example, the operator R squared 00:43:54.750 --> 00:43:58.580 acting on any state of that subspace, 00:43:58.580 --> 00:44:01.390 since h commutes with L squared, h 00:44:01.390 --> 00:44:04.740 can go here, acts on this thing, becomes a number. 00:44:04.740 --> 00:44:07.460 So you might as well put a number here. 00:44:07.460 --> 00:44:10.440 You might as well put a number here as well. 00:44:13.272 --> 00:44:15.370 It has to be stated like that. 00:44:15.370 --> 00:44:15.870 Carefully. 00:44:15.870 --> 00:44:18.450 We're going to work on a degenerate subspace 00:44:18.450 --> 00:44:19.550 of some energy. 00:44:19.550 --> 00:44:24.050 But then we can treat the h as a number. 00:44:24.050 --> 00:44:25.800 So let me say it here. 00:44:25.800 --> 00:44:38.220 We'll work in a degenerate subspace 00:44:38.220 --> 00:44:50.080 with eigenvalues of h equal to h prime, for h prime. 00:44:50.080 --> 00:44:56.100 Now I want to write some numbers here to simplify my algebra. 00:44:56.100 --> 00:44:59.030 So without loss of generality we put 00:44:59.030 --> 00:45:08.185 what this dimensionless-- this is dimensionless. 00:45:11.370 --> 00:45:13.010 I'm sorry, this is not dimensionless. 00:45:13.010 --> 00:45:16.770 This one has units of energy. 00:45:16.770 --> 00:45:19.660 This is roughly the right energy, 00:45:19.660 --> 00:45:24.050 with this one would be the right energy for the ground state. 00:45:24.050 --> 00:45:26.530 Now we don't know the energies and this 00:45:26.530 --> 00:45:28.850 is going to give us the energies as well. 00:45:28.850 --> 00:45:31.810 So without solving the differential equation, 00:45:31.810 --> 00:45:33.580 we're going to get the energies. 00:45:33.580 --> 00:45:36.680 So if I say, well that's the energies you would say, 00:45:36.680 --> 00:45:38.830 come on, you're cheating. 00:45:38.830 --> 00:45:44.190 So I'll put one over nu squared where nu can be anything. 00:45:44.190 --> 00:45:45.045 Nu is real. 00:45:47.790 --> 00:45:50.540 And that's just a way to write things 00:45:50.540 --> 00:45:52.610 in order to simplify the algebra. 00:45:52.610 --> 00:45:55.190 I don't know what nu is. 00:45:55.190 --> 00:45:58.180 How you say-- you don't know, but you have this in mind 00:45:58.180 --> 00:46:00.450 and it's going to be an integer, sure. 00:46:00.450 --> 00:46:03.030 That's what good notation is all about. 00:46:03.030 --> 00:46:06.700 You write things and then, you know, it's nu. 00:46:06.700 --> 00:46:10.280 You don't call it N. Because you don't know it's an integer. 00:46:10.280 --> 00:46:14.900 You call it nu, and you proceed. 00:46:14.900 --> 00:46:18.980 So once you have called it nu, you 00:46:18.980 --> 00:46:22.440 see here that, well, that's what we call h really. 00:46:22.440 --> 00:46:26.240 h will be-- this h prime is kind of not necessary. 00:46:26.240 --> 00:46:30.950 This is what-- where h becomes in every formula. 00:46:30.950 --> 00:46:38.110 So from here you get that minus 2h over me to the fourth 00:46:38.110 --> 00:46:41.490 is 1 over h squared nu squared. 00:46:45.060 --> 00:46:47.970 I have a minus here, I'm sorry. 00:46:47.970 --> 00:46:53.340 2h minus me to the fourth down is h squared nu squared. 00:46:53.340 --> 00:46:58.190 So we can substitute that in our nice formulas 00:46:58.190 --> 00:47:04.790 that hme to the fourth so our formulas four and five 00:47:04.790 --> 00:47:08.745 have become-- I'm going to use this blackboard. 00:47:13.370 --> 00:47:18.010 Any blackboard where I don't have a formula boxed 00:47:18.010 --> 00:47:18.880 can be erased. 00:47:18.880 --> 00:47:20.550 So I will continue here. 00:47:28.350 --> 00:47:31.720 And so what do we have? 00:47:31.720 --> 00:47:40.240 R cross R, from that formula, well this thing 00:47:40.240 --> 00:47:43.650 is over there minus 2h over me to the fourth, 00:47:43.650 --> 00:47:45.090 you substitute it in here. 00:47:45.090 --> 00:47:49.680 So it's i over h bar, one over nu 00:47:49.680 --> 00:47:53.700 squared L. Doesn't look that bad. 00:47:53.700 --> 00:48:05.260 And, R squared is equal to 1 minus 1 over h bar nu squared. 00:48:05.260 --> 00:48:06.760 Like this. 00:48:06.760 --> 00:48:08.745 L squared plus h squared. 00:48:12.920 --> 00:48:16.204 2h, that's minus h squared nu squared. 00:48:16.204 --> 00:48:18.180 Yeah. 00:48:18.180 --> 00:48:20.650 So these are nice formulas. 00:48:20.650 --> 00:48:24.570 These are already quite clean. 00:48:24.570 --> 00:48:27.205 We'll call them five, equation five. 00:48:29.850 --> 00:48:37.330 I still want to rewrite them in a way that perhaps 00:48:37.330 --> 00:48:41.380 is a little more understandable or suggestive. 00:48:41.380 --> 00:48:45.230 I will put an h bar nu together with each R. 00:48:45.230 --> 00:48:55.260 So h nu R cross h nu R is equal to ih 00:48:55.260 --> 00:49:01.720 bar L. Makes it look nice. 00:49:01.720 --> 00:49:06.440 Then for this one you'll put h squared 00:49:06.440 --> 00:49:16.150 nu squared R squared is equal to h squared nu squared minus 1 00:49:16.150 --> 00:49:19.420 minus L squared. 00:49:19.420 --> 00:49:22.190 It's sort of trivial algebra. 00:49:22.190 --> 00:49:25.980 You multiply by h squared nu squared, you get this. 00:49:25.980 --> 00:49:31.260 You get h squared nu squared minus L squared because it's 00:49:31.260 --> 00:49:35.010 all multiplied minus h squared. 00:49:35.010 --> 00:49:41.370 So these two equations, five, have become six. 00:49:41.370 --> 00:49:45.810 So five and six are really the same equations. 00:49:45.810 --> 00:49:49.760 Nothing much has been done. 00:49:49.760 --> 00:49:55.080 And if you wish, in terms of commutators 00:49:55.080 --> 00:50:05.590 this equation says that the commutator h nu Ri with h nu Rj 00:50:05.590 --> 00:50:09.640 is equal to ih bar epsilon ijkLk. 00:50:14.030 --> 00:50:22.250 H cross this thing h nu R, h nu R cross equal iHL in components 00:50:22.250 --> 00:50:23.070 means this. 00:50:25.660 --> 00:50:28.770 That is not totally obvious. 00:50:28.770 --> 00:50:32.850 It requires a small computation, but is the same computation 00:50:32.850 --> 00:50:35.820 that shows that this thing is really 00:50:35.820 --> 00:50:42.420 LiLj equal ih bar epsilon ijkLk. 00:50:42.420 --> 00:50:45.850 In which these L's have now become R's. 00:50:48.660 --> 00:50:54.430 OK so, we've cleaned up everything. 00:50:58.970 --> 00:51:01.550 We've made great progress even though at this moment 00:51:01.550 --> 00:51:04.745 it still looks like we haven't solved the problem at all. 00:51:04.745 --> 00:51:07.320 But we're very close. 00:51:07.320 --> 00:51:12.510 So are there any questions about what we've done so far? 00:51:12.510 --> 00:51:17.350 Have I lost you in the algebra, or any goals here? 00:51:17.350 --> 00:51:18.100 Yes. 00:51:18.100 --> 00:51:21.490 AUDIENCE: Why is R cross R not a commutation? 00:51:21.490 --> 00:51:25.440 Why would we expect that to not be a commutation? 00:51:25.440 --> 00:51:30.480 PROFESSOR: In general, it's the same thing as here. 00:51:30.480 --> 00:51:33.590 L cross L is this. 00:51:33.590 --> 00:51:38.380 The commutator of two Hermitian operators is anti-Hermitian. 00:51:38.380 --> 00:51:41.600 So there's always an i over there. 00:51:44.430 --> 00:51:45.240 Other questions? 00:51:48.232 --> 00:51:49.690 It's good, you have to-- you should 00:51:49.690 --> 00:51:52.386 worry about those things. 00:51:52.386 --> 00:51:56.020 Are the units right, or the right number of i's 00:51:56.020 --> 00:51:57.550 on the right hand side. 00:51:57.550 --> 00:52:01.150 That's a great way to catch mistakes. 00:52:01.150 --> 00:52:05.050 OK so we're there. 00:52:05.050 --> 00:52:08.360 And now it should really almost look 00:52:08.360 --> 00:52:12.490 reasonable to do what we're going to do. 00:52:12.490 --> 00:52:15.876 h nu R with h nu R gives you like L. 00:52:15.876 --> 00:52:20.980 So you have L with L, form angular momentum. 00:52:20.980 --> 00:52:25.630 L and R are vectors in their angular momentum. 00:52:25.630 --> 00:52:30.680 Now R cross R is L. And with these units, 00:52:30.680 --> 00:52:34.570 h nu R and h nu R looks like it has 00:52:34.570 --> 00:52:37.940 the units of angular momentum. 00:52:37.940 --> 00:52:43.290 So h nu R can be added to angular momentum 00:52:43.290 --> 00:52:45.840 to form more angular momentum. 00:52:45.840 --> 00:52:48.100 So that's exactly what we're going to do. 00:52:51.480 --> 00:52:54.170 So here it comes. 00:52:54.170 --> 00:52:54.820 Key step. 00:52:57.600 --> 00:53:01.580 J1-- I'm going to define two angular momenta. 00:53:01.580 --> 00:53:04.145 Well, we hope that they are angular momenta. 00:53:04.145 --> 00:53:18.707 L plus h nu R. And J2, one half L minus h nu R. These 00:53:18.707 --> 00:53:19.373 are definitions. 00:53:22.490 --> 00:53:26.340 It's just defining two operators. 00:53:26.340 --> 00:53:29.775 We hope something good happens with these operators, 00:53:29.775 --> 00:53:32.530 but at this moment you don't know. 00:53:32.530 --> 00:53:36.130 It's a good suggestion because of the units 00:53:36.130 --> 00:53:37.660 match and all that stuff. 00:53:37.660 --> 00:53:44.200 So this is going to be our definitions, seven. 00:53:44.200 --> 00:53:50.490 And from these of course follows that L, the quantity we know, 00:53:50.490 --> 00:53:53.654 is J1 plus J2. 00:53:53.654 --> 00:53:59.360 And R, or h nu R, is J1 minus J2. 00:54:03.750 --> 00:54:07.600 You solve in the other way. 00:54:07.600 --> 00:54:15.200 Now my first claim is that J1 and J2 commute. 00:54:18.520 --> 00:54:19.960 Commute with each other. 00:54:28.460 --> 00:54:32.950 So these are nice, commuting angular momenta. 00:54:32.950 --> 00:54:40.150 Now this computation has to be done-- let me-- yeah, 00:54:40.150 --> 00:54:42.420 we can do it. 00:54:42.420 --> 00:54:45.580 J1i with J2J. 00:54:49.610 --> 00:54:53.720 It's one half and one half gives you 00:54:53.720 --> 00:55:06.920 one quarter of Li plus h nu Ri with LJ minus h nu RJ. 00:55:11.260 --> 00:55:15.430 Now the question is where do I-- I 00:55:15.430 --> 00:55:17.970 think I can erase most of this blackboard. 00:55:24.570 --> 00:55:26.300 I can leave this formula. 00:55:26.300 --> 00:55:31.445 It's kind of the only very much needed one. 00:55:34.360 --> 00:55:38.990 So I'll continue with this computation here. 00:55:38.990 --> 00:55:45.650 This gives me one quarter-- and we have a big parentheses-- ih 00:55:45.650 --> 00:55:47.040 bar epsilon iJkLk. 00:55:49.700 --> 00:55:53.470 For the commutator of these two. 00:55:53.470 --> 00:56:00.680 And then you have the commutator of the cross terms. 00:56:00.680 --> 00:56:04.320 So what do they look like? 00:56:04.320 --> 00:56:19.690 They look like minus h nu Li with RJ, and minus h 00:56:19.690 --> 00:56:25.995 nu Ri with-- no. 00:56:31.780 --> 00:56:36.810 So I have minus h nu Li with RJ, and now I 00:56:36.810 --> 00:56:38.720 have a plus of this term. 00:56:38.720 --> 00:56:44.565 But I will write this as a minus h nu of LJ with Ri. 00:56:47.390 --> 00:56:50.150 Those are the two cross products. 00:56:50.150 --> 00:56:56.827 And then finally we have this thing, the h nu with h nu Rijk. 00:56:59.700 --> 00:57:11.914 So I have minus h nu squared, and you have then RiRJk. 00:57:14.720 --> 00:57:18.940 No, I'll do it this way. 00:57:18.940 --> 00:57:20.290 I'm sorry. 00:57:20.290 --> 00:57:26.180 You have minus over there, and I have this thing 00:57:26.180 --> 00:57:35.210 so it's minus ih bar epsilon iJkLk from the last two 00:57:35.210 --> 00:57:36.230 commutators. 00:57:36.230 --> 00:57:40.445 So this one you use essentially equation six. 00:57:43.260 --> 00:57:44.025 Now look. 00:57:47.780 --> 00:57:51.860 This thing and this thing cancels. 00:57:51.860 --> 00:57:56.650 And these two terms, they actually cancel as well. 00:57:56.650 --> 00:58:00.365 Because here you get an epsilon iJR. 00:58:03.020 --> 00:58:06.560 And here there's an epsilon Ji something. 00:58:06.560 --> 00:58:10.880 So these two terms actually add up to zero. 00:58:10.880 --> 00:58:12.000 And this is zero. 00:58:12.000 --> 00:58:20.050 So indeed, J1i and J2i-- 2J-- is zero. 00:58:20.050 --> 00:58:25.150 And these are commuting things. 00:58:25.150 --> 00:58:27.540 I wanted to say commuting angular momentum, 00:58:27.540 --> 00:58:30.280 but not quite yet. 00:58:30.280 --> 00:58:33.520 Haven't shown their angular momenta. 00:58:33.520 --> 00:58:38.520 So how do we show their angular momenta? 00:58:38.520 --> 00:58:43.790 We have to try it and see if they really 00:58:43.790 --> 00:58:46.660 do form an algebra of angular momentum. 00:58:46.660 --> 00:58:53.280 So again, for saving room, I'm going to erase this formula. 00:58:53.280 --> 00:58:55.470 It will reappear in lecture notes. 00:58:55.470 --> 00:58:57.760 But now it should go. 00:59:03.240 --> 00:59:06.460 So the next computation is something that I want to do. 00:59:06.460 --> 00:59:13.080 J1 cross J1 or the J2 cross J2, to see 00:59:13.080 --> 00:59:15.590 if they form angular momenta. 00:59:15.590 --> 00:59:17.940 And I want to do them simultaneously, 00:59:17.940 --> 00:59:26.370 so I will do one quarter of J1 cross 00:59:26.370 --> 00:59:40.680 J2 would be L plus minus h nu R cross L plus minus h nu R. 00:59:40.680 --> 00:59:45.680 OK that doesn't look bad at all, especially because we 00:59:45.680 --> 00:59:50.350 have all these formulas for products. 00:59:50.350 --> 00:59:55.500 So look, you have L cross L, which we know. 00:59:55.500 --> 00:59:58.720 Then you have L cross R plus R cross L 00:59:58.720 --> 01:00:01.950 that is conveniently here. 01:00:01.950 --> 01:00:08.350 And finally, you have R cross R which is here. 01:00:08.350 --> 01:00:11.860 So it's all sort of done in a way 01:00:11.860 --> 01:00:15.220 that the composition should be easy. 01:00:15.220 --> 01:00:19.560 So indeed 1 over 4 L cross L gives you 01:00:19.560 --> 01:00:26.820 an ih bar L. From L cross L. From these ones, 01:00:26.820 --> 01:00:28.910 you get plus minus with plus minus. 01:00:28.910 --> 01:00:33.780 It's always plus but you get another ihL. 01:00:33.780 --> 01:00:35.560 So you get another ihL. 01:00:39.070 --> 01:00:49.050 And then you get plus minus L cross h nu R plus h nu R 01:00:49.050 --> 01:01:00.710 cross L. So here you get one quarter of 2 ihL. 01:01:03.300 --> 01:01:08.590 And look at this formula, just put an h nu here and h nu here 01:01:08.590 --> 01:01:10.840 and an h nu here. 01:01:10.840 --> 01:01:25.420 So you get plus minus 2 ih from here and an h nu R. 01:01:25.420 --> 01:01:31.720 OK so the twos and the fours and the iH's go out and then you 01:01:31.720 --> 01:01:39.652 get ih times one half times L plus minus h nu R, 01:01:39.652 --> 01:01:46.130 which is either J1 or J2. 01:01:46.130 --> 01:01:50.810 So, very nicely, we've shown that J1 01:01:50.810 --> 01:02:02.680 cross J1 is ih bar J1 and J2 cross J2 is ih bar J2. 01:02:06.200 --> 01:02:09.640 And now finally you can say that you've 01:02:09.640 --> 01:02:16.120 discovered two independent angular momenta in the hydrogen 01:02:16.120 --> 01:02:16.840 atom. 01:02:16.840 --> 01:02:21.520 You did have an angular momentum on an R vector, 01:02:21.520 --> 01:02:24.940 and all of our work has gone into showing now 01:02:24.940 --> 01:02:26.855 that you have two angular momenta. 01:02:32.430 --> 01:02:36.150 Pretty much we're at the end of this 01:02:36.150 --> 01:02:43.620 because, after we do one more little thing, we're there. 01:02:43.620 --> 01:02:45.255 So let me do it here. 01:02:53.987 --> 01:02:58.630 I will not need these equations anymore. 01:02:58.630 --> 01:03:00.810 Except this one I will need. 01:03:16.900 --> 01:03:19.686 So 01:03:19.686 --> 01:03:23.580 L dot R is zero. 01:03:23.580 --> 01:03:27.740 So from L dot R equals zero, this time 01:03:27.740 --> 01:03:37.060 you get J1 plus J2 is equal to-- no, times-- 01:03:37.060 --> 01:03:41.430 J1 minus J2 is equal to zero. 01:03:41.430 --> 01:03:45.080 Now J1 and J2 commute. 01:03:45.080 --> 01:03:48.800 So the cross terms vanish. 01:03:48.800 --> 01:03:50.410 J1 and J2 commute. 01:03:50.410 --> 01:03:55.920 So this implies that J1 squared is equal to J2 squared. 01:04:00.800 --> 01:04:10.120 Now this is a very surprising thing. 01:04:10.120 --> 01:04:14.530 These two angular momenta have the same length squared. 01:04:14.530 --> 01:04:18.370 Let's look a little more at the length squared of it. 01:04:18.370 --> 01:04:21.370 So let's, for example, square J1. 01:04:21.370 --> 01:04:33.850 Well, if I square J1, I have one fourth L squared plus h squared 01:04:33.850 --> 01:04:36.470 nu squared R squared. 01:04:36.470 --> 01:04:42.090 No L dot R term, because L dot R is 0. 01:04:42.090 --> 01:04:47.530 And h squared nu squared R squared is here. 01:04:47.530 --> 01:04:49.650 So this is good news. 01:04:49.650 --> 01:04:56.150 This is one fourth L squared plus h squared 01:04:56.150 --> 01:05:00.620 nu squared minus 1 minus L squared. 01:05:00.620 --> 01:05:03.720 The L squared cancels. 01:05:03.720 --> 01:05:09.020 And you've got that J1 equals to J2 squared. 01:05:09.020 --> 01:05:22.680 And it's equal to one fourth of h squared nu squared minus 1. 01:05:25.625 --> 01:05:26.125 OK. 01:05:29.050 --> 01:05:31.330 Well the problem has been solved, 01:05:31.330 --> 01:05:35.170 even if you don't notice at this moment. 01:05:35.170 --> 01:05:37.530 It's all solved. 01:05:37.530 --> 01:05:39.490 Why? 01:05:39.490 --> 01:05:42.210 You've been talking a degenerate subspace 01:05:42.210 --> 01:05:47.220 with angular momentum with equal energies. 01:05:47.220 --> 01:05:50.610 And there's two angular momenta there. 01:05:50.610 --> 01:05:54.480 And their squares equal to the same thing. 01:05:54.480 --> 01:06:00.250 So these two angular momenta, our squares 01:06:00.250 --> 01:06:04.800 are the same and the square is precisely what 01:06:04.800 --> 01:06:13.390 we call h squared J times J plus 1, where j J is quantized. 01:06:13.390 --> 01:06:19.520 It can be zero, one half, one, all of this. 01:06:19.520 --> 01:06:23.350 So here comes a quantization. 01:06:23.350 --> 01:06:27.480 J squared being nu squared, we didn't know what nu squared is, 01:06:27.480 --> 01:06:30.830 but it's now equal to these things. 01:06:30.830 --> 01:06:36.490 So at this moment, things have been quantized. 01:06:36.490 --> 01:06:41.520 And let's look into a little more detail what has happened 01:06:41.520 --> 01:06:44.780 and confirm that we got everything we wanted. 01:06:57.940 --> 01:07:00.680 So let me write that equation again here. 01:07:00.680 --> 01:07:07.040 J1 squared is equal J2 squared is equal to one quarter 01:07:07.040 --> 01:07:10.670 h squared nu squared minus 1, which 01:07:10.670 --> 01:07:16.110 is h squared J times J plus 1. 01:07:16.110 --> 01:07:22.740 So cancel the h squares and solve for nu squared. 01:07:22.740 --> 01:07:28.580 Nu squared would be 1 plus 4J times 01:07:28.580 --> 01:07:36.380 J plus 1, which is 4J squared plus 4J plus 1, 01:07:36.380 --> 01:07:39.660 which is 2J plus 1 squared. 01:07:42.810 --> 01:07:44.550 That's pretty neat. 01:07:44.550 --> 01:07:46.210 Why is it so neat? 01:07:46.210 --> 01:07:53.220 Because as J is equal to zero, all the possible values 01:07:53.220 --> 01:07:55.860 of angular momentum-- three halves, 01:07:55.860 --> 01:08:01.780 all these things-- nu, which is 2J plus 1, 01:08:01.780 --> 01:08:14.220 will be equal to 1, 2, 3, 4-- all the integers. 01:08:14.220 --> 01:08:17.292 And what was nu? 01:08:17.292 --> 01:08:20.609 It was the values of the energies. 01:08:20.609 --> 01:08:26.420 So actually you've proven the spectrum. 01:08:26.420 --> 01:08:31.420 Nu has come out to be either 1, 2, 3, 01:08:31.420 --> 01:08:35.360 but you have all representations of angular momentum. 01:08:35.360 --> 01:08:38.630 You have the singlet, the spin one 01:08:38.630 --> 01:08:40.930 half-- where are the spins here? 01:08:40.930 --> 01:08:42.000 Nowhere. 01:08:42.000 --> 01:08:44.950 There was an electron, a proton, we never 01:08:44.950 --> 01:08:47.240 put spin for the hydrogen atom. 01:08:47.240 --> 01:08:51.180 But it all shows up as these representations 01:08:51.180 --> 01:08:53.880 in which they come along. 01:08:53.880 --> 01:08:57.740 Even more is true, as we will see right away 01:08:57.740 --> 01:09:02.160 and confirm that everything really shows up the right way. 01:09:02.160 --> 01:09:06.120 So what happened now? 01:09:06.120 --> 01:09:11.950 We have two independent, equal angular momentum. 01:09:11.950 --> 01:09:16.149 So what is this degenerate subspace we were inventing? 01:09:16.149 --> 01:09:24.000 Is the space J, which is J1 and m1 tensor product 01:09:24.000 --> 01:09:28.310 with J, which is J2 but has the same value 01:09:28.310 --> 01:09:31.029 because the squares are the same, m2. 01:09:34.450 --> 01:09:39.350 So this is an uncoupled basis. 01:09:39.350 --> 01:09:45.000 Uncoupled basis of states in the degenerate subspace. 01:09:45.000 --> 01:09:50.720 And now, you know, it's all a little surreal 01:09:50.720 --> 01:09:55.350 because these don't look like our states at all. 01:09:55.350 --> 01:09:58.780 But this is the way algebraically they show up. 01:09:58.780 --> 01:10:05.260 We choose our value of J, we have then that nu is equal 01:10:05.260 --> 01:10:09.270 to this and for that value of J there's some values of m's. 01:10:09.270 --> 01:10:13.470 And therefore, this must be the degenerate subspace. 01:10:13.470 --> 01:10:17.890 So this is nothing but the tensor product 01:10:17.890 --> 01:10:22.220 of a J multiplet with a J multiplet. 01:10:25.430 --> 01:10:29.120 Where J is that integer here. 01:10:29.120 --> 01:10:32.400 And what is the tensor product of a J multiplet? 01:10:32.400 --> 01:10:35.200 First, J is for J1. 01:10:35.200 --> 01:10:37.487 The second J is for J2. 01:10:44.230 --> 01:10:46.220 So at this moment of course we're 01:10:46.220 --> 01:10:51.680 calling this N for the quantum number. 01:10:51.680 --> 01:10:53.250 But what is this thing? 01:10:53.250 --> 01:11:01.850 This is 2J plus 2J minus 1 plus-- all the way 01:11:01.850 --> 01:11:02.810 up to the singlet. 01:11:08.430 --> 01:11:12.620 But what are these representations of? 01:11:12.620 --> 01:11:17.190 Well here we have J1 and here is J2. 01:11:17.190 --> 01:11:19.790 These must be the ones of the sum. 01:11:19.790 --> 01:11:22.640 But who is the sum, L? 01:11:22.640 --> 01:11:27.640 So these are the L representations that you get. 01:11:27.640 --> 01:11:29.850 L is your angular momentum. 01:11:29.850 --> 01:11:32.410 L representations. 01:11:32.410 --> 01:11:39.520 And if 2J plus 1 is N, you got a representation 01:11:39.520 --> 01:11:46.170 with L equals N minus 1, because 2J plus 1 01:11:46.170 --> 01:11:57.190 is N, L equals N minus 2, all the way up to L equals 0. 01:11:57.190 --> 01:12:04.780 Therefore, you get precisely this whole structure. 01:12:04.780 --> 01:12:09.870 So, just in time as we get to 2 o'clock, 01:12:09.870 --> 01:12:13.520 we've finished the quantization of the hydrogen atom. 01:12:13.520 --> 01:12:16.340 We've finished 805. 01:12:16.340 --> 01:12:17.560 I hope you enjoyed. 01:12:17.560 --> 01:12:21.820 I did a lot. [INAUDIBLE] and Will did, too. 01:12:21.820 --> 01:12:25.440 Good luck and we'll see you soon.