1 00:00:00,060 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation, or to view additional materials 6 00:00:12,700 --> 00:00:16,603 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,603 --> 00:00:17,565 at ocw.mit.edu. 8 00:00:21,885 --> 00:00:27,260 PROFESSOR: OK so we're going to do this thing of the hydrogen 9 00:00:27,260 --> 00:00:31,350 atom and the algebraic solution. 10 00:00:31,350 --> 00:00:38,930 And I think it's not that long stuff 11 00:00:38,930 --> 00:00:41,140 so we can take it easy as we go along. 12 00:00:41,140 --> 00:00:47,480 I want to remind you of a couple of facts that will play a role. 13 00:00:47,480 --> 00:00:51,290 One result that is very general about the addition of angular 14 00:00:51,290 --> 00:00:55,850 momentum that you should of course know 15 00:00:55,850 --> 00:01:02,530 is that if you have a j1 times j2. 16 00:01:02,530 --> 00:01:05,500 What does this mean? 17 00:01:05,500 --> 00:01:15,650 You have some states of-- first angular momentum J1 18 00:01:15,650 --> 00:01:20,270 that so you have a whole multiplet with J1 19 00:01:20,270 --> 00:01:22,060 equals little j1. 20 00:01:22,060 --> 00:01:28,260 Which means the states in that multiplet have J1 squared, 21 00:01:28,260 --> 00:01:30,160 giving you h squared. 22 00:01:30,160 --> 00:01:33,370 Little j1 times little j1 plus 1. 23 00:01:33,370 --> 00:01:36,430 That's having a j1 multiplet. 24 00:01:36,430 --> 00:01:38,050 You have a j2 multiplet. 25 00:01:42,330 --> 00:01:46,160 And these are two independent commuting angular momenta 26 00:01:46,160 --> 00:01:48,440 acting on different degrees of freedom 27 00:01:48,440 --> 00:01:53,020 of the same particle or different particles. 28 00:01:53,020 --> 00:01:56,210 And what you've learned is that this 29 00:01:56,210 --> 00:02:00,570 can be written as a sum of representations. 30 00:02:00,570 --> 00:02:04,580 As a direct sum of angular momenta, 31 00:02:04,580 --> 00:02:17,114 which goes from j1 plus j2 plus j1 plus j2 minus 1 32 00:02:17,114 --> 00:02:25,490 all the way up to the representation 33 00:02:25,490 --> 00:02:29,390 with j1 minus j2. 34 00:02:29,390 --> 00:02:34,750 And these are all representations or multiplets 35 00:02:34,750 --> 00:02:37,070 that live in the tense or product, 36 00:02:37,070 --> 00:02:42,625 but they are multiplets of J equals j1 plus j2. 37 00:02:52,600 --> 00:02:58,140 These states here can be reorganized 38 00:02:58,140 --> 00:02:59,970 into these multiplets, and that's 39 00:02:59,970 --> 00:03:03,120 our main result for the addition of angular momentum. 40 00:03:03,120 --> 00:03:06,435 Mathematically, this formula summarizes it all. 41 00:03:09,950 --> 00:03:12,390 These states, when you write them 42 00:03:12,390 --> 00:03:15,520 as a basis here-- you take a basis state here 43 00:03:15,520 --> 00:03:20,540 times a basis state here-- these are called the coupled bases. 44 00:03:20,540 --> 00:03:23,990 And then you reorganize, you form linear combinations 45 00:03:23,990 --> 00:03:26,220 that you have been playing with, and then 46 00:03:26,220 --> 00:03:28,500 they get reorganized into these states. 47 00:03:28,500 --> 00:03:30,610 So these are called the coupled bases 48 00:03:30,610 --> 00:03:32,720 in which we're talking about states 49 00:03:32,720 --> 00:03:35,660 of the sum of angular momentum. 50 00:03:35,660 --> 00:03:39,500 So that's one fact we've learned about. 51 00:03:39,500 --> 00:03:42,810 Now as far as hydrogen is concerned, 52 00:03:42,810 --> 00:03:47,490 we're going to try today to understand the spectrum. 53 00:03:47,490 --> 00:03:50,830 And for that let me remind you what the spectrum was. 54 00:03:50,830 --> 00:03:58,750 The way we organized it was with an L versus energy levels. 55 00:03:58,750 --> 00:04:04,770 And we would put an L equals 0 state here-- well maybe-- 56 00:04:04,770 --> 00:04:08,259 there's color, so why not using color. 57 00:04:08,259 --> 00:04:09,175 Let's see if it works. 58 00:04:16,715 --> 00:04:18,100 Yeah, it's OK. 59 00:04:18,100 --> 00:04:21,050 L equals 0. 60 00:04:21,050 --> 00:04:24,280 And this was called n equals 1. 61 00:04:24,280 --> 00:04:31,020 There's an n equals 2 that has an L equals 0 state 62 00:04:31,020 --> 00:04:32,915 and an L equals 1 state. 63 00:04:36,600 --> 00:04:42,650 There's an n equals 3 state, set of states 64 00:04:42,650 --> 00:04:47,800 that come within L equals 0 and L equals 1 and an L equals 2. 65 00:04:51,520 --> 00:04:54,980 And it just goes on and on. 66 00:04:54,980 --> 00:05:02,910 With the energy levels En equals minus e squared over 2a0. 67 00:05:02,910 --> 00:05:09,250 That combination is familiar for energy, Bohr radius, charge 68 00:05:09,250 --> 00:05:13,470 of electron, with a 1 over n squared. 69 00:05:13,470 --> 00:05:21,280 And the fact is that for any level, for each n, 70 00:05:21,280 --> 00:05:27,625 L goes from 0, 1, 2, up to n minus 1. 71 00:05:30,780 --> 00:05:39,785 And for each n there's a total of n squared states. 72 00:05:44,050 --> 00:05:50,310 And you see it here, you have n equals 2, n equals 1, one 73 00:05:50,310 --> 00:05:50,930 state. 74 00:05:50,930 --> 00:05:54,660 n equals 2, you have L equals 0, one state. 75 00:05:54,660 --> 00:05:56,530 L equals 1 is 3 states. 76 00:05:56,530 --> 00:05:58,480 So it's 4. 77 00:05:58,480 --> 00:06:01,080 Here we'll have 4 plus 5. 78 00:06:01,080 --> 00:06:01,960 So 9. 79 00:06:01,960 --> 00:06:04,990 And maybe you can do it, it's a famous thing, 80 00:06:04,990 --> 00:06:09,470 there's n squared states at every level. 81 00:06:09,470 --> 00:06:12,990 So this pattern that of course continues 82 00:06:12,990 --> 00:06:16,380 and-- it's a little difficult to do 83 00:06:16,380 --> 00:06:19,150 a nice diagram of the hydrogen atom in scale 84 00:06:19,150 --> 00:06:22,050 because it's all pushed towards the zero energy 85 00:06:22,050 --> 00:06:25,520 with 1 over n squared, but that's how it goes. 86 00:06:25,520 --> 00:06:30,165 For n equals 4, you have 1, 2, 3, 4 for example. 87 00:06:33,510 --> 00:06:35,383 And this is what we want to understand. 88 00:06:41,390 --> 00:06:53,060 So in order to do that, let's return to this Hamiltonian, 89 00:06:53,060 --> 00:06:59,690 which is p squared over 2m minus e squared over r. 90 00:06:59,690 --> 00:07:04,486 And to the Runge-Lenz vector that we talked about in lecture 91 00:07:04,486 --> 00:07:07,340 and you've been playing with. 92 00:07:07,340 --> 00:07:13,170 So this Runge-Lenz vector, r, is defined 93 00:07:13,170 --> 00:07:26,830 to be 1 over 2me squared p cross L minus L cross p minus r 94 00:07:26,830 --> 00:07:29,470 over r. 95 00:07:29,470 --> 00:07:30,860 And it has no units. 96 00:07:38,000 --> 00:07:40,290 It's a vector that you've learned 97 00:07:40,290 --> 00:07:44,150 has interpretation of a constant vector that 98 00:07:44,150 --> 00:07:48,520 points in this direction, r. 99 00:07:48,520 --> 00:07:52,010 And it just stays fixed wherever the particle is going. 100 00:07:52,010 --> 00:07:54,570 Classically this is a constant vector 101 00:07:54,570 --> 00:07:58,820 that points in the direction of the major axis of the ellipse. 102 00:08:01,770 --> 00:08:06,080 With respect to this vector, this vector is Hermitian. 103 00:08:06,080 --> 00:08:10,320 And you may recall that when we did the classical vector, 104 00:08:10,320 --> 00:08:16,660 you had just p cross L and no 2 here. 105 00:08:16,660 --> 00:08:19,550 There are now two terms here. 106 00:08:19,550 --> 00:08:23,040 And they are necessary because we want to have a Hermitian 107 00:08:23,040 --> 00:08:25,790 operator, and this is the simplest 108 00:08:25,790 --> 00:08:29,790 way to construct the Hermitian operator, r. 109 00:08:29,790 --> 00:08:35,510 And the way is that you add to this this term, 110 00:08:35,510 --> 00:08:40,580 that if L and p commuted as they do in classical mechanics, 111 00:08:40,580 --> 00:08:43,240 that the term is identical to this. 112 00:08:43,240 --> 00:08:45,530 And you get back to the conventional thing 113 00:08:45,530 --> 00:08:47,920 that you had in classical mechanics. 114 00:08:47,920 --> 00:08:51,130 But in quantum mechanics, of course, they don't commute, 115 00:08:51,130 --> 00:08:54,110 so it's a little bit different. 116 00:08:54,110 --> 00:08:59,580 And moreover this thing, r, is Hermitian. 117 00:08:59,580 --> 00:09:02,780 L and p are Hermitian but when you take the Hermitian 118 00:09:02,780 --> 00:09:08,110 conjugate, L goes to the other side of p. 119 00:09:08,110 --> 00:09:11,050 And since they don't commute, that's not the same thing. 120 00:09:11,050 --> 00:09:15,100 So actually the Hermitian conjugate of this term is this. 121 00:09:15,100 --> 00:09:18,100 There's an extra minus sign in hermiticity 122 00:09:18,100 --> 00:09:20,500 when you have a cross product. 123 00:09:20,500 --> 00:09:22,500 So this is the Hermitian conjugate of this, 124 00:09:22,500 --> 00:09:25,120 this is the Hermitian conjugate of this second term, 125 00:09:25,120 --> 00:09:29,300 here's the first and therefore this is actually 126 00:09:29,300 --> 00:09:31,010 a Hermitian operator. 127 00:09:31,010 --> 00:09:32,120 And you can work with it. 128 00:09:34,680 --> 00:09:39,480 Moreover, in the case of classical mechanics, 129 00:09:39,480 --> 00:09:42,190 it was conserved. 130 00:09:42,190 --> 00:09:45,240 In the case of quantum mechanics this statement 131 00:09:45,240 --> 00:09:47,350 of conservation quantum mechanics 132 00:09:47,350 --> 00:09:53,610 is something that in one of the exercises that you were asked 133 00:09:53,610 --> 00:09:57,460 to try to do this computation so these computations 134 00:09:57,460 --> 00:10:00,490 are challenging. 135 00:10:00,490 --> 00:10:04,280 They're not all that trivial and are good exercises. 136 00:10:04,280 --> 00:10:07,670 So this is one of them. 137 00:10:07,670 --> 00:10:09,400 This is practice. 138 00:10:19,570 --> 00:10:21,940 OK this is the vector r. 139 00:10:21,940 --> 00:10:23,140 What about it? 140 00:10:23,140 --> 00:10:27,900 A few more things about it that are interesting. 141 00:10:27,900 --> 00:10:31,840 Because of the hermiticity condition 142 00:10:31,840 --> 00:10:35,840 or in-- a way you can check this directly 143 00:10:35,840 --> 00:10:39,710 in fact was one of the exercises for you to do, 144 00:10:39,710 --> 00:10:42,770 was p cross L-- you did it long time ago, 145 00:10:42,770 --> 00:10:50,320 I think-- is equal to minus L cross b plus 2ih bar p. 146 00:10:52,833 --> 00:10:53,666 This is an identity. 147 00:11:00,820 --> 00:11:12,100 And this identity helps you write this kind of term 148 00:11:12,100 --> 00:11:16,120 in a way in which you have just one order of products 149 00:11:16,120 --> 00:11:18,400 and a little extra term, rather than 150 00:11:18,400 --> 00:11:21,290 having two complicated terms. 151 00:11:21,290 --> 00:11:30,100 So the r can be written as 1 over me 152 00:11:30,100 --> 00:11:45,650 squared alone, p cross L minus ihp minus r over r. 153 00:11:45,650 --> 00:11:46,243 For example. 154 00:11:48,830 --> 00:11:58,410 By writing this term as another p cross L minus that thing 155 00:11:58,410 --> 00:12:01,550 gives you that expression for r. 156 00:12:01,550 --> 00:12:03,630 You have an alternative expression 157 00:12:03,630 --> 00:12:05,720 in which you solve for the other one. 158 00:12:05,720 --> 00:12:14,820 So it's 1 over me squared minus L cross p plus ih bar p. 159 00:12:21,160 --> 00:12:27,460 Now, r-- we need to understand r better. 160 00:12:27,460 --> 00:12:31,930 That's really the challenge of this whole derivation. 161 00:12:31,930 --> 00:12:35,250 So we have one thing that is conserved. 162 00:12:35,250 --> 00:12:37,440 Angular momentum is conserved. 163 00:12:37,440 --> 00:12:40,380 It commutes with the Hamiltonian. 164 00:12:40,380 --> 00:12:44,530 We have another thing that is conserved, this r. 165 00:12:44,530 --> 00:12:47,200 But we have to understand better what it is. 166 00:12:47,200 --> 00:12:52,560 So one thing that you can ask is, well, r is conserved, 167 00:12:52,560 --> 00:12:56,320 so r squared is conserved as well. 168 00:12:56,320 --> 00:13:01,260 So r squared, if I can simplify it-- if I can do the algebra 169 00:13:01,260 --> 00:13:06,650 and simplify it-- it should not be that complicated. 170 00:13:06,650 --> 00:13:11,810 So again a practice problem was given to do that computation. 171 00:13:11,810 --> 00:13:17,840 And I think these forms are useful for that, to work less. 172 00:13:17,840 --> 00:13:20,520 And the computation gives a very nice result, 173 00:13:20,520 --> 00:13:29,340 where r squared is equal to 1 plus 2 H over me 174 00:13:29,340 --> 00:13:35,690 to the fourth L squared plus h bar squared. 175 00:13:35,690 --> 00:13:40,160 Kind of a strange result if you think about it. 176 00:13:40,160 --> 00:13:42,380 People that want to do this classically 177 00:13:42,380 --> 00:13:44,855 first would find that there's no h squared. 178 00:13:48,160 --> 00:13:53,040 And here, this h, is that whole h that we have here. 179 00:13:53,040 --> 00:13:55,430 It's a complicated thing. 180 00:13:55,430 --> 00:14:00,360 So this right hand side is quite substantial. 181 00:14:00,360 --> 00:14:03,580 You don't have to worry that h is in this side 182 00:14:03,580 --> 00:14:05,670 or whether it's on the other side 183 00:14:05,670 --> 00:14:10,480 because h commutes with L. L is conserved. 184 00:14:10,480 --> 00:14:13,550 So h appears like that. 185 00:14:13,550 --> 00:14:17,376 And this, again, is the result of another computation. 186 00:14:21,570 --> 00:14:23,120 So we've learned something. 187 00:14:23,120 --> 00:14:26,630 r-- oh, I'm sorry, this is r squared. 188 00:14:26,630 --> 00:14:29,070 Apologies. 189 00:14:29,070 --> 00:14:30,590 r is conserved. 190 00:14:30,590 --> 00:14:34,630 r squared must be conserved because if h commutes with r 191 00:14:34,630 --> 00:14:37,310 it commutes with r squared as well. 192 00:14:37,310 --> 00:14:40,050 And therefore whatever you see on the right hand side, 193 00:14:40,050 --> 00:14:41,810 the whole thing must be conserved. 194 00:14:41,810 --> 00:14:44,660 And h is conserved, of course. 195 00:14:44,660 --> 00:14:49,320 And L squared is conserved. 196 00:14:49,320 --> 00:14:55,220 Now we need one more property of a relation-- you see, 197 00:14:55,220 --> 00:14:57,540 you have to do these things. 198 00:14:57,540 --> 00:15:02,470 Even if you probably don't have an inspiration at this moment 199 00:15:02,470 --> 00:15:04,610 how you're going to try to understand this, 200 00:15:04,610 --> 00:15:06,935 there are things that just curiosity 201 00:15:06,935 --> 00:15:09,370 should tell that you should do. 202 00:15:09,370 --> 00:15:12,270 We have L, we do L squared. 203 00:15:12,270 --> 00:15:14,560 It's an important operator. 204 00:15:14,560 --> 00:15:15,060 OK. 205 00:15:15,060 --> 00:15:19,920 We had r, we did r squared, which is an important operator. 206 00:15:19,920 --> 00:15:23,396 But one thing we can do is L dot r. 207 00:15:23,396 --> 00:15:26,030 It's a good question what L dot r is. 208 00:15:28,850 --> 00:15:31,430 So what is L dot r? 209 00:15:39,760 --> 00:15:47,560 So-- or r dot L. What is it? 210 00:15:47,560 --> 00:15:53,830 Well a few things that are important to note 211 00:15:53,830 --> 00:15:58,902 are that you did show before that you 212 00:15:58,902 --> 00:16:07,490 know that r dot L, little r dot L, is 0. 213 00:16:07,490 --> 00:16:14,220 And little p dot L is 0. 214 00:16:14,220 --> 00:16:18,500 These are obvious classically, because L 215 00:16:18,500 --> 00:16:21,210 is perpendicular to both r and p. 216 00:16:21,210 --> 00:16:26,480 But quantum mechanically they take a little more work. 217 00:16:26,480 --> 00:16:30,170 They're not complicated, but you've shown those two. 218 00:16:30,170 --> 00:16:38,660 So if you have r dot L, you would have, 219 00:16:38,660 --> 00:16:42,550 for example, here-- r dot L, you would 220 00:16:42,550 --> 00:16:50,260 have to do and think of this whole r 221 00:16:50,260 --> 00:16:52,370 and put an L on the right. 222 00:16:52,370 --> 00:16:59,660 Well this little r dotted with the L on the right would be 0. 223 00:16:59,660 --> 00:17:06,569 That p dotted with L on the right would be 0. 224 00:17:06,569 --> 00:17:11,530 And we're almost there, but p cross L dot L, 225 00:17:11,530 --> 00:17:14,460 well, what is that? 226 00:17:14,460 --> 00:17:16,490 Let me talk about it here. 227 00:17:16,490 --> 00:17:24,130 P cross L dot L-- so this is part 228 00:17:24,130 --> 00:17:27,750 of the computation of this r dot L. We've already 229 00:17:27,750 --> 00:17:29,510 seen this term will give nothing, 230 00:17:29,510 --> 00:17:30,990 this term will give nothing. 231 00:17:30,990 --> 00:17:34,180 But this term could give something. 232 00:17:34,180 --> 00:17:37,360 So when you face something like that, 233 00:17:37,360 --> 00:17:38,900 maybe you say, well, I don't know 234 00:17:38,900 --> 00:17:41,610 any identities I should be using here. 235 00:17:44,320 --> 00:17:48,510 So you just do it. 236 00:17:48,510 --> 00:17:51,180 Then you say, this is i-th component 237 00:17:51,180 --> 00:17:53,460 of this vector times the i-th of that. 238 00:17:53,460 --> 00:17:55,070 So it's epsilon ijkpjLkLi. 239 00:18:08,170 --> 00:18:12,960 And then you say look this looks a little-- you could 240 00:18:12,960 --> 00:18:16,850 say many things that are wrong and get the right answer. 241 00:18:16,850 --> 00:18:21,685 So you could say, oh, ki symmetric 242 00:18:21,685 --> 00:18:23,560 and ki anti-symmetric. 243 00:18:23,560 --> 00:18:27,000 But that's wrong, because these k and i are not 244 00:18:27,000 --> 00:18:32,280 symmetric really because these operators don't commute. 245 00:18:32,280 --> 00:18:39,730 So the answer will be zero, but for a more complicated reason. 246 00:18:39,730 --> 00:18:41,600 So what do you have in here? 247 00:18:41,600 --> 00:18:42,930 ki. 248 00:18:42,930 --> 00:18:47,798 Let's move the i to the end of the epsilon, so jkipjLkLi. 249 00:18:53,980 --> 00:19:02,390 And now you see this part is pj L cross cross L j. 250 00:19:10,110 --> 00:19:12,740 Is the cross product of this. 251 00:19:12,740 --> 00:19:14,870 But what is L cross L? 252 00:19:14,870 --> 00:19:16,340 You probably remember. 253 00:19:16,340 --> 00:19:24,010 This is ih bar L L cross L, that's 254 00:19:24,010 --> 00:19:28,240 the computation in relation of angular momentum. 255 00:19:28,240 --> 00:19:31,680 In case you kind of don't remember 256 00:19:31,680 --> 00:19:35,120 it was ih bar L. Like that. 257 00:19:35,120 --> 00:19:40,190 So now p dot L is anyway 0. 258 00:19:40,190 --> 00:19:43,160 So this is 0. 259 00:19:43,160 --> 00:19:48,485 So it's kind of-- it's a little delicate 260 00:19:48,485 --> 00:19:50,080 to do these computations. 261 00:19:50,080 --> 00:19:54,830 But so since that term is zero, this thing is zero. 262 00:20:01,560 --> 00:20:05,170 Now you may as well-- r dot L is 0. 263 00:20:05,170 --> 00:20:09,610 Is L dot r also 0? 264 00:20:09,610 --> 00:20:12,150 It's not all that obvious you can even do that. 265 00:20:12,150 --> 00:20:15,730 Well in a second we'll see that that's true as well. 266 00:20:15,730 --> 00:20:20,840 L dot r and r dot L, capital R, are 0. 267 00:20:33,780 --> 00:20:37,000 Let's remember-- let's continue-- let's see, 268 00:20:37,000 --> 00:20:41,510 I wanted to number some of these equations. 269 00:20:41,510 --> 00:20:45,090 We're going to need them. 270 00:20:45,090 --> 00:20:46,850 So this will be equation one. 271 00:20:53,390 --> 00:20:57,030 This will be equation two, it's an important one. 272 00:21:05,060 --> 00:21:11,700 Now what was-- let me remind you of a notation 273 00:21:11,700 --> 00:21:15,260 we also had about vectors and their rotations. 274 00:21:15,260 --> 00:21:19,015 Vector under rotations. 275 00:21:26,410 --> 00:21:32,020 So what was a vector, Vi, under rotations 276 00:21:32,020 --> 00:21:39,380 was something that you had LiVj equals ih bar epsilon ijkvk. 277 00:21:46,250 --> 00:21:51,760 So there is a way to write this with cross products that 278 00:21:51,760 --> 00:21:53,500 is useful in some cases. 279 00:21:53,500 --> 00:21:58,270 So I will do it. 280 00:21:58,270 --> 00:22:00,400 You probably have seen that in the notes, 281 00:22:00,400 --> 00:22:02,090 but let me remind you. 282 00:22:02,090 --> 00:22:06,370 Consider this product, L cross V plus V 283 00:22:06,370 --> 00:22:11,230 cross L and the i-th component of it. 284 00:22:14,260 --> 00:22:17,440 i-th component of this product. 285 00:22:17,440 --> 00:22:25,737 So this is epsilon ijk and you have LjVk plus VjLk. 286 00:22:34,130 --> 00:22:42,160 Now in this term you can do something nice. 287 00:22:42,160 --> 00:22:45,160 If you think of it like expanded out, 288 00:22:45,160 --> 00:22:48,580 you have the second term has epsilon ijkVjk. 289 00:22:51,490 --> 00:22:54,730 Change j for k. 290 00:22:54,730 --> 00:22:57,286 If you change j for k, this will be VkLj. 291 00:22:59,960 --> 00:23:02,320 And these would have the opposite order. 292 00:23:02,320 --> 00:23:06,460 But this order can be changed up to the cost of a minus sign. 293 00:23:06,460 --> 00:23:14,832 So I claim this is ijk-- first term is the same-- minus VkLj. 294 00:23:19,160 --> 00:23:23,920 So in the second term, for this term alone, we've 295 00:23:23,920 --> 00:23:26,960 done for this term, multiplied with this of course, 296 00:23:26,960 --> 00:23:29,560 we've done j and k relabeling. 297 00:23:39,160 --> 00:23:46,320 But this is nothing else than the commutator of L with V. 298 00:23:46,320 --> 00:23:47,790 So this is epsilon ijkLjVk. 299 00:23:57,600 --> 00:24:05,560 That's epsilon ijk, and this is epsilon jkp or LVL. 300 00:24:13,160 --> 00:24:17,380 Now, 2 epsilons with 2 common indices 301 00:24:17,380 --> 00:24:20,410 is something that it simple. 302 00:24:20,410 --> 00:24:24,150 It's a commutator dealt on the other indices. 303 00:24:24,150 --> 00:24:28,740 Now it's better if they are sort of all aligned in the same way, 304 00:24:28,740 --> 00:24:33,140 but they kind of are because this L, without paying a price, 305 00:24:33,140 --> 00:24:34,830 can be put as the first index. 306 00:24:34,830 --> 00:24:38,080 So you have jk as the second and third 307 00:24:38,080 --> 00:24:40,290 and-- jk as the second and third-- 308 00:24:40,290 --> 00:24:44,230 once L has been moved to the first position. 309 00:24:44,230 --> 00:24:52,720 So this thing is 2 times delta iL. 310 00:24:52,720 --> 00:24:57,660 And there's an h bar, ih bar I forgot here. 311 00:24:57,660 --> 00:24:59,070 ih bar. 312 00:24:59,070 --> 00:25:06,410 2 delta ik ih bar iL ih bar VL. 313 00:25:06,410 --> 00:25:10,070 So this is 2 ih bar Vi. 314 00:25:12,600 --> 00:25:16,610 So this whole thing the i-th component of this thing, 315 00:25:16,610 --> 00:25:19,720 using this commutation relation is this. 316 00:25:19,720 --> 00:25:23,060 So what we've learned is that L cross 317 00:25:23,060 --> 00:25:33,820 V plus V cross L you see go to 2 ih bar V. 318 00:25:33,820 --> 00:25:42,430 And that's a statement as a vector relation of the fact 319 00:25:42,430 --> 00:25:46,160 that V is a vector in the rotations. 320 00:25:46,160 --> 00:25:52,830 So for V to be a vector in the rotations means this. 321 00:25:52,830 --> 00:25:57,200 And if you wish, it means this thing as well. 322 00:25:57,200 --> 00:26:00,470 It's just another thing of what it means. 323 00:26:00,470 --> 00:26:05,280 Now R is a vector in the rotations. 324 00:26:05,280 --> 00:26:08,980 This capital R. Why? 325 00:26:08,980 --> 00:26:15,850 You've shown that if you have a vector in the rotations 326 00:26:15,850 --> 00:26:18,670 and you multiply it by another vector in the rotations 327 00:26:18,670 --> 00:26:20,510 under the cross product, it's still 328 00:26:20,510 --> 00:26:22,040 a vector in the rotations. 329 00:26:22,040 --> 00:26:24,810 So this is a vector in the rotations, this is, 330 00:26:24,810 --> 00:26:27,850 and this is a vector in the rotations. 331 00:26:27,850 --> 00:26:29,620 R is a vector in the rotations. 332 00:26:29,620 --> 00:26:45,630 So this capital R is a vector on the rotations, 333 00:26:45,630 --> 00:26:48,110 which means two things. 334 00:26:48,110 --> 00:26:52,890 It means it satisfies this kind of equation. 335 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. 336 00:27:19,910 --> 00:27:23,570 So R is a vector in the rotation. 337 00:27:23,570 --> 00:27:27,330 It's a fact beyond doubt. 338 00:27:27,330 --> 00:27:31,030 And that means that we now know the commutation relations 339 00:27:31,030 --> 00:27:35,060 between L and R. So we're starting to put together 340 00:27:35,060 --> 00:27:39,160 this picture in which we get familiar with R 341 00:27:39,160 --> 00:27:41,950 and the commutators that are possible. 342 00:27:41,950 --> 00:27:45,470 So I can summarize it here. 343 00:27:55,170 --> 00:28:03,900 L dot R LiRj is ih bar epsilon ijkRk. 344 00:28:08,410 --> 00:28:12,900 That's the same statement as this one but in components. 345 00:28:12,900 --> 00:28:23,140 And now you see why R dot L is equal to L dot R. 346 00:28:23,140 --> 00:28:29,150 Because actually if you put the same two indices here, i and i, 347 00:28:29,150 --> 00:28:30,130 you get zero. 348 00:28:30,130 --> 00:28:39,410 So when you have R dot L you have R1L1 plus R2L2 plus R3L3. 349 00:28:39,410 --> 00:28:41,700 And each of these two commute when 350 00:28:41,700 --> 00:28:43,740 the two indices are the same. 351 00:28:43,740 --> 00:28:45,090 Because of the epsilon. 352 00:28:45,090 --> 00:28:47,640 So R dot L is 0. 353 00:28:47,640 --> 00:28:54,710 And now you also appreciate that L dot R is also 0, too. 354 00:29:00,226 --> 00:29:00,725 OK. 355 00:29:03,900 --> 00:29:09,440 Now comes, in a sense, the most difficult of all calculations. 356 00:29:09,440 --> 00:29:11,840 Even if this seemed a little easy. 357 00:29:11,840 --> 00:29:15,680 But you can get quite far with it. 358 00:29:15,680 --> 00:29:18,800 So what do you do with Ls? 359 00:29:18,800 --> 00:29:22,550 You computed L commutators and you 360 00:29:22,550 --> 00:29:24,750 got the algebra of angular momentum. 361 00:29:28,340 --> 00:29:29,810 Over here. 362 00:29:29,810 --> 00:29:32,440 This is the algebra for angular momentum. 363 00:29:32,440 --> 00:29:36,610 And this kind of nontrivial calculation, 364 00:29:36,610 --> 00:29:38,560 you did it by building results. 365 00:29:38,560 --> 00:29:41,510 You knew how R was a vector in the rotation 366 00:29:41,510 --> 00:29:43,430 or how p was a vector in the rotation. 367 00:29:43,430 --> 00:29:48,170 You multiplied the two of them, and it was not so difficult. 368 00:29:48,170 --> 00:29:54,070 But the calculation that you really need to do now 369 00:29:54,070 --> 00:30:01,540 is the calculation of the commutator say of Ri with Rj. 370 00:30:01,540 --> 00:30:04,340 And that looks like a little bit of a nightmare. 371 00:30:04,340 --> 00:30:10,420 You have to commute this whole thing with itself. 372 00:30:10,420 --> 00:30:13,750 Lots of p's, L's, R's. 373 00:30:13,750 --> 00:30:16,790 1 over R's, those don't commute with p. 374 00:30:16,790 --> 00:30:18,870 You remember that. 375 00:30:18,870 --> 00:30:21,550 So this kind of calculation done by brute force. 376 00:30:21,550 --> 00:30:25,120 You're talking a day, probably. 377 00:30:25,120 --> 00:30:26,690 I think so. 378 00:30:26,690 --> 00:30:29,500 And probably it becomes a mess, but. 379 00:30:29,500 --> 00:30:32,160 You'll find a little trick helps to organize it better. 380 00:30:32,160 --> 00:30:34,380 It's less of a mess, but still you 381 00:30:34,380 --> 00:30:38,200 don't get it and-- try several times. 382 00:30:38,200 --> 00:30:42,320 So what we're going to do is try to think 383 00:30:42,320 --> 00:30:46,550 of what the answer could be by some arguments. 384 00:30:46,550 --> 00:30:50,210 And then once we know what the answer can be, 385 00:30:50,210 --> 00:30:53,020 there's still one calculation to be done. 386 00:30:53,020 --> 00:30:55,780 That I will probably put in the notes, 387 00:30:55,780 --> 00:30:58,050 but it's not a difficult one. 388 00:30:58,050 --> 00:31:01,420 And the answer just pops out. 389 00:31:01,420 --> 00:31:10,220 So the question is what is R cross R. R cross R is really 390 00:31:10,220 --> 00:31:14,040 what we have when we have this commutator. 391 00:31:14,040 --> 00:31:19,650 So we need to know what R cross R is, just like L cross L. 392 00:31:19,650 --> 00:31:25,420 Now R is not likely to be an angular momentum. 393 00:31:25,420 --> 00:31:27,880 It's a vector but it's not an angular momentum. 394 00:31:27,880 --> 00:31:30,610 Has nothing to do with it. 395 00:31:30,610 --> 00:31:32,270 It's more complicated. 396 00:31:32,270 --> 00:31:36,140 So what is R cross R quantum-mechanically? 397 00:31:36,140 --> 00:31:40,200 Classically, of course, it would be zero. 398 00:31:40,200 --> 00:31:46,230 So first thing is you think of what this should be. 399 00:31:50,330 --> 00:31:56,720 We have a vector, because the cross product of two vectors. 400 00:31:56,720 --> 00:32:01,900 Now I want to emphasize one other thing, 401 00:32:01,900 --> 00:32:08,870 that it should be this thing-- R cross R-- is 402 00:32:08,870 --> 00:32:10,330 tantamount to this thing. 403 00:32:10,330 --> 00:32:11,590 What is this thing? 404 00:32:14,450 --> 00:32:17,660 It should be actually proportional 405 00:32:17,660 --> 00:32:19,675 to some conserved quantity. 406 00:32:22,510 --> 00:32:26,640 And the reason is quite interesting. 407 00:32:26,640 --> 00:32:29,545 So this is a small aside here. 408 00:32:33,780 --> 00:32:36,870 If some operator is conserved, it 409 00:32:36,870 --> 00:32:38,440 commutes with the Hamiltonian. 410 00:32:38,440 --> 00:32:52,595 Say if S1 and S2 are symmetries, that means that S1 with h 411 00:32:52,595 --> 00:32:59,506 is equal to S2 with h is equal to zero. 412 00:32:59,506 --> 00:33:05,580 Then the claim is that the commutator of this S1 and S2 413 00:33:05,580 --> 00:33:12,255 claim S1 commutator with S2 is also a symmetry. 414 00:33:17,140 --> 00:33:21,120 So the reason is because commutator 415 00:33:21,120 --> 00:33:29,425 of S1 S2 commutator with h is equal actually to zero. 416 00:33:29,425 --> 00:33:32,500 And why would it be equal to zero? 417 00:33:32,500 --> 00:33:34,840 It's because of the so-called Jacobi 418 00:33:34,840 --> 00:33:37,580 identity for commutators. 419 00:33:37,580 --> 00:33:41,930 You'll remember when you have three things like that, 420 00:33:41,930 --> 00:33:46,355 this term is equal to 1-- this term plus 1, 421 00:33:46,355 --> 00:33:48,050 in which you cycle them. 422 00:33:48,050 --> 00:33:51,390 And plus another one where you cycle them again 423 00:33:51,390 --> 00:33:53,230 is equal to zero. 424 00:33:53,230 --> 00:33:55,600 That's a Jacobi identity. 425 00:33:55,600 --> 00:33:59,460 And in those cyclings you get an h with S2, 426 00:33:59,460 --> 00:34:01,710 for example, that is zero. 427 00:34:01,710 --> 00:34:04,090 And then an h with S1, which is zero. 428 00:34:04,090 --> 00:34:08,310 So you use these things here and you prove that. 429 00:34:08,310 --> 00:34:09,959 So I write here, by Jacobi. 430 00:34:14,250 --> 00:34:17,520 So if you have a conserved-- this 431 00:34:17,520 --> 00:34:20,630 is the great thing about conserved quantities, 432 00:34:20,630 --> 00:34:24,050 if you have one conserved quantity, it's OK. 433 00:34:24,050 --> 00:34:26,300 But if you have two, you're in business. 434 00:34:26,300 --> 00:34:29,985 Because you can then take the commutator of these two and you 435 00:34:29,985 --> 00:34:31,400 get another conserved quantity. 436 00:34:31,400 --> 00:34:34,830 And then more commutators and you keep taking commutators 437 00:34:34,830 --> 00:34:39,420 and if you're lucky you get all of the conserved quantities. 438 00:34:39,420 --> 00:34:45,480 So here R cross R refers to this commutator. 439 00:34:45,480 --> 00:34:48,670 So whatever is on the right should be a vector 440 00:34:48,670 --> 00:34:50,375 and should be conserved. 441 00:34:57,460 --> 00:35:03,270 And what are our conserved vectors? 442 00:35:03,270 --> 00:35:11,210 Well our conserved vectors-- candidates here-- 443 00:35:11,210 --> 00:35:25,130 are L, R itself, and L cross R. That's pretty much it. 444 00:35:25,130 --> 00:35:30,330 L and R are our only conserved things, so it better be that. 445 00:35:35,450 --> 00:35:39,370 Still this is far too much. 446 00:35:39,370 --> 00:35:41,330 So there could be a term proportional 447 00:35:41,330 --> 00:35:46,650 to L, a term proportional to R, a term proportional to L dot R. 448 00:35:46,650 --> 00:35:51,940 So this kind of analysis is based by something 449 00:35:51,940 --> 00:35:53,760 that Julian Schwinger did. 450 00:35:53,760 --> 00:35:57,090 This same guy that actually did quantum 451 00:35:57,090 --> 00:36:01,940 electrodynamics along with Feynman and Tomonaga. 452 00:36:01,940 --> 00:36:04,360 And he's the one of those who invented 453 00:36:04,360 --> 00:36:07,870 the trick of using three-dimensional angular 454 00:36:07,870 --> 00:36:10,770 momentum for the two-dimensional oscillator. 455 00:36:10,770 --> 00:36:13,860 And had lots of bags of tricks. 456 00:36:13,860 --> 00:36:16,920 So actually this whole discussion of the hydrogen 457 00:36:16,920 --> 00:36:22,200 atom-- most books just say, well, these calculations are 458 00:36:22,200 --> 00:36:22,810 hopeless. 459 00:36:22,810 --> 00:36:25,330 Let me give you the answers. 460 00:36:25,330 --> 00:36:28,780 Schwinger, on the other hand, in his book on quantum mechanics-- 461 00:36:28,780 --> 00:36:33,940 which is kind of interesting but very idiosyncratic-- 462 00:36:33,940 --> 00:36:37,910 finds a trick to do every calculation. 463 00:36:37,910 --> 00:36:41,230 So you never get into a big mess. 464 00:36:41,230 --> 00:36:44,980 He's absolutely elegant and keeps pulling tricks 465 00:36:44,980 --> 00:36:45,700 from the bag. 466 00:36:45,700 --> 00:36:49,830 So this is one of those tricks. 467 00:36:49,830 --> 00:36:54,180 Basically he goes through the following analysis now 468 00:36:54,180 --> 00:36:59,490 and says, look, suppose I have the vector R 469 00:36:59,490 --> 00:37:01,960 and I do a parity transformation. 470 00:37:01,960 --> 00:37:05,490 I change it for minus R. 471 00:37:05,490 --> 00:37:10,010 What happens under those circumstances? 472 00:37:10,010 --> 00:37:15,950 Well the momentum is the rate of change of R, 473 00:37:15,950 --> 00:37:18,260 should also change sign. 474 00:37:18,260 --> 00:37:20,840 Quantum mechanically this is consistent, 475 00:37:20,840 --> 00:37:25,590 because a commutation between R and p should give you h bar. 476 00:37:25,590 --> 00:37:31,810 And if R changes, p should change sign. 477 00:37:31,810 --> 00:37:37,530 But now when you do this, L, which is R cross p, 478 00:37:37,530 --> 00:37:47,960 just goes to L. And R, however, changes sign 479 00:37:47,960 --> 00:37:53,320 because L doesn't change sign but p does and R does. 480 00:37:53,320 --> 00:37:57,330 So under these changes-- so this is the originator, 481 00:37:57,330 --> 00:38:00,140 the troublemaker and then everybody else 482 00:38:00,140 --> 00:38:04,875 follows-- R also changes sign. 483 00:38:11,200 --> 00:38:16,100 So this is extremely powerful because if you imagine 484 00:38:16,100 --> 00:38:18,840 this being equal to something, well it 485 00:38:18,840 --> 00:38:21,980 should be consistent with the symmetries. 486 00:38:21,980 --> 00:38:28,470 So as I change R to minus R, capital R changes sign 487 00:38:28,470 --> 00:38:32,550 but the left hand side doesn't change sign. 488 00:38:32,550 --> 00:38:36,210 Therefore the right hand side should not change sign. 489 00:38:36,210 --> 00:38:42,080 And R changes sign and L cross R changes sign. 490 00:38:42,080 --> 00:38:45,590 So computation kind of finished because the only thing 491 00:38:45,590 --> 00:38:53,220 you can get on the right is L. 492 00:38:53,220 --> 00:38:55,520 This is the kind of thing that you do 493 00:38:55,520 --> 00:38:57,910 and probably if you were writing a paper on 494 00:38:57,910 --> 00:39:01,400 that you would anyway do the calculation. 495 00:39:01,400 --> 00:39:05,420 The silly way, the- the right way. 496 00:39:05,420 --> 00:39:08,880 But this is quite save of times. 497 00:39:08,880 --> 00:39:11,300 So actually what you have learned 498 00:39:11,300 --> 00:39:24,940 is that R cross R is equal to some scalar conserved quantity, 499 00:39:24,940 --> 00:39:28,240 which is something that is conserved that could 500 00:39:28,240 --> 00:39:31,020 be like an h, for example, here. 501 00:39:31,020 --> 00:39:33,130 But it's a scalar. 502 00:39:33,130 --> 00:39:37,480 And, L. 503 00:39:37,480 --> 00:39:41,290 Well once you know that much, it doesn't take much work 504 00:39:41,290 --> 00:39:43,810 to do this and to calculate what it is. 505 00:39:43,810 --> 00:39:47,670 But I will skip that calculation. 506 00:39:47,670 --> 00:39:51,440 This is the sort of thoughtful part of it. 507 00:39:51,440 --> 00:39:59,820 And R cross R turns out to be ih bar minus 2 h again. 508 00:39:59,820 --> 00:40:04,060 h shows up in several places, like here, 509 00:40:04,060 --> 00:40:08,920 so it tends-- it has a tendency to show up. 510 00:40:08,920 --> 00:40:16,210 me to the fourth L. 511 00:40:16,210 --> 00:40:24,640 So this is our equation for-- and in a sense, 512 00:40:24,640 --> 00:40:29,020 all the hard work has been done. 513 00:40:29,020 --> 00:40:32,940 Because now you have a complete understanding of these two 514 00:40:32,940 --> 00:40:36,470 vectors, L and R. You know what L squared is, 515 00:40:36,470 --> 00:40:39,235 what R squared is, what L dot R is. 516 00:40:39,235 --> 00:40:41,115 And you know all the commutators, 517 00:40:41,115 --> 00:40:46,600 you know the commutation of L with L, L with R, and R with R. 518 00:40:46,600 --> 00:40:50,140 You've done all the algebraic work. 519 00:40:50,140 --> 00:40:53,360 And the question is, how do we proceed from now 520 00:40:53,360 --> 00:40:57,960 to solve the hydrogen atom. 521 00:40:57,960 --> 00:41:02,610 So the way we proceed is kind of interesting. 522 00:41:02,610 --> 00:41:07,270 We're going to try to build from this L that 523 00:41:07,270 --> 00:41:09,530 is an angular momentum. 524 00:41:09,530 --> 00:41:14,890 And this R that is not an angular momentum. 525 00:41:14,890 --> 00:41:18,300 Two sets of angular momenta. 526 00:41:18,300 --> 00:41:19,620 You have two vectors. 527 00:41:19,620 --> 00:41:23,730 So somehow we want to try to combine them in such a way 528 00:41:23,730 --> 00:41:27,500 that we can invent two angular momenta. 529 00:41:27,500 --> 00:41:30,070 Just like the angular momentum in 530 00:41:30,070 --> 00:41:32,810 the two-dimensional harmonic oscillator. 531 00:41:32,810 --> 00:41:37,860 It was not directly through angular momentum, 532 00:41:37,860 --> 00:41:40,450 but was mathematical angular momentum. 533 00:41:40,450 --> 00:41:43,910 These two angular momenta we're going to build, one of them 534 00:41:43,910 --> 00:41:46,870 is going to be recognizable. 535 00:41:46,870 --> 00:41:49,130 The other one is going to be a little unfamiliar. 536 00:41:51,910 --> 00:41:55,540 But now I have to do something that-- it 537 00:41:55,540 --> 00:42:00,380 may sound a little unusual, but is 538 00:42:00,380 --> 00:42:03,570 necessary to simplify our life. 539 00:42:03,570 --> 00:42:07,120 I want to say some words that will 540 00:42:07,120 --> 00:42:13,640 allow me to think of this h here as a number. 541 00:42:13,640 --> 00:42:19,280 And would allow me to think of this h as a number. 542 00:42:19,280 --> 00:42:22,150 So here's what we're going to say. 543 00:42:22,150 --> 00:42:25,025 It's an assumption-- it's no assumption, 544 00:42:25,025 --> 00:42:27,180 but it sounds like an assumption. 545 00:42:27,180 --> 00:42:29,330 But there's no assumption whatsoever. 546 00:42:29,330 --> 00:42:33,880 We say the following: this hydrogen atom 547 00:42:33,880 --> 00:42:36,990 is going to have some states. 548 00:42:36,990 --> 00:42:44,590 So let's assume there is one state, and it has some energy. 549 00:42:44,590 --> 00:42:48,180 If I have that state with some energy, well, 550 00:42:48,180 --> 00:42:50,360 that would be the end of the story. 551 00:42:50,360 --> 00:42:53,570 But in fact, the thing that they want 552 00:42:53,570 --> 00:42:57,200 to allow the possibility for is that at that the energy there 553 00:42:57,200 --> 00:42:59,870 are more states. 554 00:42:59,870 --> 00:43:04,070 One state would be OK, maybe sometimes it happens. 555 00:43:04,070 --> 00:43:08,250 But in general there are more states at that energy. 556 00:43:08,250 --> 00:43:13,370 So I don't-- I'm not making any physical assumption to state 557 00:43:13,370 --> 00:43:18,040 that there is a subspace of degenerate states. 558 00:43:18,040 --> 00:43:22,530 And in that subspace of degenerate states, 559 00:43:22,530 --> 00:43:24,820 there may be just one state, there are two states, 560 00:43:24,820 --> 00:43:26,278 there are three states, but there's 561 00:43:26,278 --> 00:43:31,740 subspace of degenerate states that have some energy. 562 00:43:31,740 --> 00:43:35,750 And I'm going to work in that subspace. 563 00:43:35,750 --> 00:43:37,830 And all the operators that I have 564 00:43:37,830 --> 00:43:40,500 are going to be acting in that subspace. 565 00:43:40,500 --> 00:43:42,790 And I'm going to analyze subspace 566 00:43:42,790 --> 00:43:45,840 by subspace of different energies. 567 00:43:45,840 --> 00:43:48,320 So we're going to work with one subspace 568 00:43:48,320 --> 00:43:49,480 of degenerate energies. 569 00:43:49,480 --> 00:43:54,750 And if I have, for example, the operator R squared 570 00:43:54,750 --> 00:43:58,580 acting on any state of that subspace, 571 00:43:58,580 --> 00:44:01,390 since h commutes with L squared, h 572 00:44:01,390 --> 00:44:04,740 can go here, acts on this thing, becomes a number. 573 00:44:04,740 --> 00:44:07,460 So you might as well put a number here. 574 00:44:07,460 --> 00:44:10,440 You might as well put a number here as well. 575 00:44:13,272 --> 00:44:15,370 It has to be stated like that. 576 00:44:15,370 --> 00:44:15,870 Carefully. 577 00:44:15,870 --> 00:44:18,450 We're going to work on a degenerate subspace 578 00:44:18,450 --> 00:44:19,550 of some energy. 579 00:44:19,550 --> 00:44:24,050 But then we can treat the h as a number. 580 00:44:24,050 --> 00:44:25,800 So let me say it here. 581 00:44:25,800 --> 00:44:38,220 We'll work in a degenerate subspace 582 00:44:38,220 --> 00:44:50,080 with eigenvalues of h equal to h prime, for h prime. 583 00:44:50,080 --> 00:44:56,100 Now I want to write some numbers here to simplify my algebra. 584 00:44:56,100 --> 00:44:59,030 So without loss of generality we put 585 00:44:59,030 --> 00:45:08,185 what this dimensionless-- this is dimensionless. 586 00:45:11,370 --> 00:45:13,010 I'm sorry, this is not dimensionless. 587 00:45:13,010 --> 00:45:16,770 This one has units of energy. 588 00:45:16,770 --> 00:45:19,660 This is roughly the right energy, 589 00:45:19,660 --> 00:45:24,050 with this one would be the right energy for the ground state. 590 00:45:24,050 --> 00:45:26,530 Now we don't know the energies and this 591 00:45:26,530 --> 00:45:28,850 is going to give us the energies as well. 592 00:45:28,850 --> 00:45:31,810 So without solving the differential equation, 593 00:45:31,810 --> 00:45:33,580 we're going to get the energies. 594 00:45:33,580 --> 00:45:36,680 So if I say, well that's the energies you would say, 595 00:45:36,680 --> 00:45:38,830 come on, you're cheating. 596 00:45:38,830 --> 00:45:44,190 So I'll put one over nu squared where nu can be anything. 597 00:45:44,190 --> 00:45:45,045 Nu is real. 598 00:45:47,790 --> 00:45:50,540 And that's just a way to write things 599 00:45:50,540 --> 00:45:52,610 in order to simplify the algebra. 600 00:45:52,610 --> 00:45:55,190 I don't know what nu is. 601 00:45:55,190 --> 00:45:58,180 How you say-- you don't know, but you have this in mind 602 00:45:58,180 --> 00:46:00,450 and it's going to be an integer, sure. 603 00:46:00,450 --> 00:46:03,030 That's what good notation is all about. 604 00:46:03,030 --> 00:46:06,700 You write things and then, you know, it's nu. 605 00:46:06,700 --> 00:46:10,280 You don't call it N. Because you don't know it's an integer. 606 00:46:10,280 --> 00:46:14,900 You call it nu, and you proceed. 607 00:46:14,900 --> 00:46:18,980 So once you have called it nu, you 608 00:46:18,980 --> 00:46:22,440 see here that, well, that's what we call h really. 609 00:46:22,440 --> 00:46:26,240 h will be-- this h prime is kind of not necessary. 610 00:46:26,240 --> 00:46:30,950 This is what-- where h becomes in every formula. 611 00:46:30,950 --> 00:46:38,110 So from here you get that minus 2h over me to the fourth 612 00:46:38,110 --> 00:46:41,490 is 1 over h squared nu squared. 613 00:46:45,060 --> 00:46:47,970 I have a minus here, I'm sorry. 614 00:46:47,970 --> 00:46:53,340 2h minus me to the fourth down is h squared nu squared. 615 00:46:53,340 --> 00:46:58,190 So we can substitute that in our nice formulas 616 00:46:58,190 --> 00:47:04,790 that hme to the fourth so our formulas four and five 617 00:47:04,790 --> 00:47:08,745 have become-- I'm going to use this blackboard. 618 00:47:13,370 --> 00:47:18,010 Any blackboard where I don't have a formula boxed 619 00:47:18,010 --> 00:47:18,880 can be erased. 620 00:47:18,880 --> 00:47:20,550 So I will continue here. 621 00:47:28,350 --> 00:47:31,720 And so what do we have? 622 00:47:31,720 --> 00:47:40,240 R cross R, from that formula, well this thing 623 00:47:40,240 --> 00:47:43,650 is over there minus 2h over me to the fourth, 624 00:47:43,650 --> 00:47:45,090 you substitute it in here. 625 00:47:45,090 --> 00:47:49,680 So it's i over h bar, one over nu 626 00:47:49,680 --> 00:47:53,700 squared L. Doesn't look that bad. 627 00:47:53,700 --> 00:48:05,260 And, R squared is equal to 1 minus 1 over h bar nu squared. 628 00:48:05,260 --> 00:48:06,760 Like this. 629 00:48:06,760 --> 00:48:08,745 L squared plus h squared. 630 00:48:12,920 --> 00:48:16,204 2h, that's minus h squared nu squared. 631 00:48:16,204 --> 00:48:18,180 Yeah. 632 00:48:18,180 --> 00:48:20,650 So these are nice formulas. 633 00:48:20,650 --> 00:48:24,570 These are already quite clean. 634 00:48:24,570 --> 00:48:27,205 We'll call them five, equation five. 635 00:48:29,850 --> 00:48:37,330 I still want to rewrite them in a way that perhaps 636 00:48:37,330 --> 00:48:41,380 is a little more understandable or suggestive. 637 00:48:41,380 --> 00:48:45,230 I will put an h bar nu together with each R. 638 00:48:45,230 --> 00:48:55,260 So h nu R cross h nu R is equal to ih 639 00:48:55,260 --> 00:49:01,720 bar L. Makes it look nice. 640 00:49:01,720 --> 00:49:06,440 Then for this one you'll put h squared 641 00:49:06,440 --> 00:49:16,150 nu squared R squared is equal to h squared nu squared minus 1 642 00:49:16,150 --> 00:49:19,420 minus L squared. 643 00:49:19,420 --> 00:49:22,190 It's sort of trivial algebra. 644 00:49:22,190 --> 00:49:25,980 You multiply by h squared nu squared, you get this. 645 00:49:25,980 --> 00:49:31,260 You get h squared nu squared minus L squared because it's 646 00:49:31,260 --> 00:49:35,010 all multiplied minus h squared. 647 00:49:35,010 --> 00:49:41,370 So these two equations, five, have become six. 648 00:49:41,370 --> 00:49:45,810 So five and six are really the same equations. 649 00:49:45,810 --> 00:49:49,760 Nothing much has been done. 650 00:49:49,760 --> 00:49:55,080 And if you wish, in terms of commutators 651 00:49:55,080 --> 00:50:05,590 this equation says that the commutator h nu Ri with h nu Rj 652 00:50:05,590 --> 00:50:09,640 is equal to ih bar epsilon ijkLk. 653 00:50:14,030 --> 00:50:22,250 H cross this thing h nu R, h nu R cross equal iHL in components 654 00:50:22,250 --> 00:50:23,070 means this. 655 00:50:25,660 --> 00:50:28,770 That is not totally obvious. 656 00:50:28,770 --> 00:50:32,850 It requires a small computation, but is the same computation 657 00:50:32,850 --> 00:50:35,820 that shows that this thing is really 658 00:50:35,820 --> 00:50:42,420 LiLj equal ih bar epsilon ijkLk. 659 00:50:42,420 --> 00:50:45,850 In which these L's have now become R's. 660 00:50:48,660 --> 00:50:54,430 OK so, we've cleaned up everything. 661 00:50:58,970 --> 00:51:01,550 We've made great progress even though at this moment 662 00:51:01,550 --> 00:51:04,745 it still looks like we haven't solved the problem at all. 663 00:51:04,745 --> 00:51:07,320 But we're very close. 664 00:51:07,320 --> 00:51:12,510 So are there any questions about what we've done so far? 665 00:51:12,510 --> 00:51:17,350 Have I lost you in the algebra, or any goals here? 666 00:51:17,350 --> 00:51:18,100 Yes. 667 00:51:18,100 --> 00:51:21,490 AUDIENCE: Why is R cross R not a commutation? 668 00:51:21,490 --> 00:51:25,440 Why would we expect that to not be a commutation? 669 00:51:25,440 --> 00:51:30,480 PROFESSOR: In general, it's the same thing as here. 670 00:51:30,480 --> 00:51:33,590 L cross L is this. 671 00:51:33,590 --> 00:51:38,380 The commutator of two Hermitian operators is anti-Hermitian. 672 00:51:38,380 --> 00:51:41,600 So there's always an i over there. 673 00:51:44,430 --> 00:51:45,240 Other questions? 674 00:51:48,232 --> 00:51:49,690 It's good, you have to-- you should 675 00:51:49,690 --> 00:51:52,386 worry about those things. 676 00:51:52,386 --> 00:51:56,020 Are the units right, or the right number of i's 677 00:51:56,020 --> 00:51:57,550 on the right hand side. 678 00:51:57,550 --> 00:52:01,150 That's a great way to catch mistakes. 679 00:52:01,150 --> 00:52:05,050 OK so we're there. 680 00:52:05,050 --> 00:52:08,360 And now it should really almost look 681 00:52:08,360 --> 00:52:12,490 reasonable to do what we're going to do. 682 00:52:12,490 --> 00:52:15,876 h nu R with h nu R gives you like L. 683 00:52:15,876 --> 00:52:20,980 So you have L with L, form angular momentum. 684 00:52:20,980 --> 00:52:25,630 L and R are vectors in their angular momentum. 685 00:52:25,630 --> 00:52:30,680 Now R cross R is L. And with these units, 686 00:52:30,680 --> 00:52:34,570 h nu R and h nu R looks like it has 687 00:52:34,570 --> 00:52:37,940 the units of angular momentum. 688 00:52:37,940 --> 00:52:43,290 So h nu R can be added to angular momentum 689 00:52:43,290 --> 00:52:45,840 to form more angular momentum. 690 00:52:45,840 --> 00:52:48,100 So that's exactly what we're going to do. 691 00:52:51,480 --> 00:52:54,170 So here it comes. 692 00:52:54,170 --> 00:52:54,820 Key step. 693 00:52:57,600 --> 00:53:01,580 J1-- I'm going to define two angular momenta. 694 00:53:01,580 --> 00:53:04,145 Well, we hope that they are angular momenta. 695 00:53:04,145 --> 00:53:18,707 L plus h nu R. And J2, one half L minus h nu R. These 696 00:53:18,707 --> 00:53:19,373 are definitions. 697 00:53:22,490 --> 00:53:26,340 It's just defining two operators. 698 00:53:26,340 --> 00:53:29,775 We hope something good happens with these operators, 699 00:53:29,775 --> 00:53:32,530 but at this moment you don't know. 700 00:53:32,530 --> 00:53:36,130 It's a good suggestion because of the units 701 00:53:36,130 --> 00:53:37,660 match and all that stuff. 702 00:53:37,660 --> 00:53:44,200 So this is going to be our definitions, seven. 703 00:53:44,200 --> 00:53:50,490 And from these of course follows that L, the quantity we know, 704 00:53:50,490 --> 00:53:53,654 is J1 plus J2. 705 00:53:53,654 --> 00:53:59,360 And R, or h nu R, is J1 minus J2. 706 00:54:03,750 --> 00:54:07,600 You solve in the other way. 707 00:54:07,600 --> 00:54:15,200 Now my first claim is that J1 and J2 commute. 708 00:54:18,520 --> 00:54:19,960 Commute with each other. 709 00:54:28,460 --> 00:54:32,950 So these are nice, commuting angular momenta. 710 00:54:32,950 --> 00:54:40,150 Now this computation has to be done-- let me-- yeah, 711 00:54:40,150 --> 00:54:42,420 we can do it. 712 00:54:42,420 --> 00:54:45,580 J1i with J2J. 713 00:54:49,610 --> 00:54:53,720 It's one half and one half gives you 714 00:54:53,720 --> 00:55:06,920 one quarter of Li plus h nu Ri with LJ minus h nu RJ. 715 00:55:11,260 --> 00:55:15,430 Now the question is where do I-- I 716 00:55:15,430 --> 00:55:17,970 think I can erase most of this blackboard. 717 00:55:24,570 --> 00:55:26,300 I can leave this formula. 718 00:55:26,300 --> 00:55:31,445 It's kind of the only very much needed one. 719 00:55:34,360 --> 00:55:38,990 So I'll continue with this computation here. 720 00:55:38,990 --> 00:55:45,650 This gives me one quarter-- and we have a big parentheses-- ih 721 00:55:45,650 --> 00:55:47,040 bar epsilon iJkLk. 722 00:55:49,700 --> 00:55:53,470 For the commutator of these two. 723 00:55:53,470 --> 00:56:00,680 And then you have the commutator of the cross terms. 724 00:56:00,680 --> 00:56:04,320 So what do they look like? 725 00:56:04,320 --> 00:56:19,690 They look like minus h nu Li with RJ, and minus h 726 00:56:19,690 --> 00:56:25,995 nu Ri with-- no. 727 00:56:31,780 --> 00:56:36,810 So I have minus h nu Li with RJ, and now I 728 00:56:36,810 --> 00:56:38,720 have a plus of this term. 729 00:56:38,720 --> 00:56:44,565 But I will write this as a minus h nu of LJ with Ri. 730 00:56:47,390 --> 00:56:50,150 Those are the two cross products. 731 00:56:50,150 --> 00:56:56,827 And then finally we have this thing, the h nu with h nu Rijk. 732 00:56:59,700 --> 00:57:11,914 So I have minus h nu squared, and you have then RiRJk. 733 00:57:14,720 --> 00:57:18,940 No, I'll do it this way. 734 00:57:18,940 --> 00:57:20,290 I'm sorry. 735 00:57:20,290 --> 00:57:26,180 You have minus over there, and I have this thing 736 00:57:26,180 --> 00:57:35,210 so it's minus ih bar epsilon iJkLk from the last two 737 00:57:35,210 --> 00:57:36,230 commutators. 738 00:57:36,230 --> 00:57:40,445 So this one you use essentially equation six. 739 00:57:43,260 --> 00:57:44,025 Now look. 740 00:57:47,780 --> 00:57:51,860 This thing and this thing cancels. 741 00:57:51,860 --> 00:57:56,650 And these two terms, they actually cancel as well. 742 00:57:56,650 --> 00:58:00,365 Because here you get an epsilon iJR. 743 00:58:03,020 --> 00:58:06,560 And here there's an epsilon Ji something. 744 00:58:06,560 --> 00:58:10,880 So these two terms actually add up to zero. 745 00:58:10,880 --> 00:58:12,000 And this is zero. 746 00:58:12,000 --> 00:58:20,050 So indeed, J1i and J2i-- 2J-- is zero. 747 00:58:20,050 --> 00:58:25,150 And these are commuting things. 748 00:58:25,150 --> 00:58:27,540 I wanted to say commuting angular momentum, 749 00:58:27,540 --> 00:58:30,280 but not quite yet. 750 00:58:30,280 --> 00:58:33,520 Haven't shown their angular momenta. 751 00:58:33,520 --> 00:58:38,520 So how do we show their angular momenta? 752 00:58:38,520 --> 00:58:43,790 We have to try it and see if they really 753 00:58:43,790 --> 00:58:46,660 do form an algebra of angular momentum. 754 00:58:46,660 --> 00:58:53,280 So again, for saving room, I'm going to erase this formula. 755 00:58:53,280 --> 00:58:55,470 It will reappear in lecture notes. 756 00:58:55,470 --> 00:58:57,760 But now it should go. 757 00:59:03,240 --> 00:59:06,460 So the next computation is something that I want to do. 758 00:59:06,460 --> 00:59:13,080 J1 cross J1 or the J2 cross J2, to see 759 00:59:13,080 --> 00:59:15,590 if they form angular momenta. 760 00:59:15,590 --> 00:59:17,940 And I want to do them simultaneously, 761 00:59:17,940 --> 00:59:26,370 so I will do one quarter of J1 cross 762 00:59:26,370 --> 00:59:40,680 J2 would be L plus minus h nu R cross L plus minus h nu R. 763 00:59:40,680 --> 00:59:45,680 OK that doesn't look bad at all, especially because we 764 00:59:45,680 --> 00:59:50,350 have all these formulas for products. 765 00:59:50,350 --> 00:59:55,500 So look, you have L cross L, which we know. 766 00:59:55,500 --> 00:59:58,720 Then you have L cross R plus R cross L 767 00:59:58,720 --> 01:00:01,950 that is conveniently here. 768 01:00:01,950 --> 01:00:08,350 And finally, you have R cross R which is here. 769 01:00:08,350 --> 01:00:11,860 So it's all sort of done in a way 770 01:00:11,860 --> 01:00:15,220 that the composition should be easy. 771 01:00:15,220 --> 01:00:19,560 So indeed 1 over 4 L cross L gives you 772 01:00:19,560 --> 01:00:26,820 an ih bar L. From L cross L. From these ones, 773 01:00:26,820 --> 01:00:28,910 you get plus minus with plus minus. 774 01:00:28,910 --> 01:00:33,780 It's always plus but you get another ihL. 775 01:00:33,780 --> 01:00:35,560 So you get another ihL. 776 01:00:39,070 --> 01:00:49,050 And then you get plus minus L cross h nu R plus h nu R 777 01:00:49,050 --> 01:01:00,710 cross L. So here you get one quarter of 2 ihL. 778 01:01:03,300 --> 01:01:08,590 And look at this formula, just put an h nu here and h nu here 779 01:01:08,590 --> 01:01:10,840 and an h nu here. 780 01:01:10,840 --> 01:01:25,420 So you get plus minus 2 ih from here and an h nu R. 781 01:01:25,420 --> 01:01:31,720 OK so the twos and the fours and the iH's go out and then you 782 01:01:31,720 --> 01:01:39,652 get ih times one half times L plus minus h nu R, 783 01:01:39,652 --> 01:01:46,130 which is either J1 or J2. 784 01:01:46,130 --> 01:01:50,810 So, very nicely, we've shown that J1 785 01:01:50,810 --> 01:02:02,680 cross J1 is ih bar J1 and J2 cross J2 is ih bar J2. 786 01:02:06,200 --> 01:02:09,640 And now finally you can say that you've 787 01:02:09,640 --> 01:02:16,120 discovered two independent angular momenta in the hydrogen 788 01:02:16,120 --> 01:02:16,840 atom. 789 01:02:16,840 --> 01:02:21,520 You did have an angular momentum on an R vector, 790 01:02:21,520 --> 01:02:24,940 and all of our work has gone into showing now 791 01:02:24,940 --> 01:02:26,855 that you have two angular momenta. 792 01:02:32,430 --> 01:02:36,150 Pretty much we're at the end of this 793 01:02:36,150 --> 01:02:43,620 because, after we do one more little thing, we're there. 794 01:02:43,620 --> 01:02:45,255 So let me do it here. 795 01:02:53,987 --> 01:02:58,630 I will not need these equations anymore. 796 01:02:58,630 --> 01:03:00,810 Except this one I will need. 797 01:03:16,900 --> 01:03:19,686 So 798 01:03:19,686 --> 01:03:23,580 L dot R is zero. 799 01:03:23,580 --> 01:03:27,740 So from L dot R equals zero, this time 800 01:03:27,740 --> 01:03:37,060 you get J1 plus J2 is equal to-- no, times-- 801 01:03:37,060 --> 01:03:41,430 J1 minus J2 is equal to zero. 802 01:03:41,430 --> 01:03:45,080 Now J1 and J2 commute. 803 01:03:45,080 --> 01:03:48,800 So the cross terms vanish. 804 01:03:48,800 --> 01:03:50,410 J1 and J2 commute. 805 01:03:50,410 --> 01:03:55,920 So this implies that J1 squared is equal to J2 squared. 806 01:04:00,800 --> 01:04:10,120 Now this is a very surprising thing. 807 01:04:10,120 --> 01:04:14,530 These two angular momenta have the same length squared. 808 01:04:14,530 --> 01:04:18,370 Let's look a little more at the length squared of it. 809 01:04:18,370 --> 01:04:21,370 So let's, for example, square J1. 810 01:04:21,370 --> 01:04:33,850 Well, if I square J1, I have one fourth L squared plus h squared 811 01:04:33,850 --> 01:04:36,470 nu squared R squared. 812 01:04:36,470 --> 01:04:42,090 No L dot R term, because L dot R is 0. 813 01:04:42,090 --> 01:04:47,530 And h squared nu squared R squared is here. 814 01:04:47,530 --> 01:04:49,650 So this is good news. 815 01:04:49,650 --> 01:04:56,150 This is one fourth L squared plus h squared 816 01:04:56,150 --> 01:05:00,620 nu squared minus 1 minus L squared. 817 01:05:00,620 --> 01:05:03,720 The L squared cancels. 818 01:05:03,720 --> 01:05:09,020 And you've got that J1 equals to J2 squared. 819 01:05:09,020 --> 01:05:22,680 And it's equal to one fourth of h squared nu squared minus 1. 820 01:05:25,625 --> 01:05:26,125 OK. 821 01:05:29,050 --> 01:05:31,330 Well the problem has been solved, 822 01:05:31,330 --> 01:05:35,170 even if you don't notice at this moment. 823 01:05:35,170 --> 01:05:37,530 It's all solved. 824 01:05:37,530 --> 01:05:39,490 Why? 825 01:05:39,490 --> 01:05:42,210 You've been talking a degenerate subspace 826 01:05:42,210 --> 01:05:47,220 with angular momentum with equal energies. 827 01:05:47,220 --> 01:05:50,610 And there's two angular momenta there. 828 01:05:50,610 --> 01:05:54,480 And their squares equal to the same thing. 829 01:05:54,480 --> 01:06:00,250 So these two angular momenta, our squares 830 01:06:00,250 --> 01:06:04,800 are the same and the square is precisely what 831 01:06:04,800 --> 01:06:13,390 we call h squared J times J plus 1, where j J is quantized. 832 01:06:13,390 --> 01:06:19,520 It can be zero, one half, one, all of this. 833 01:06:19,520 --> 01:06:23,350 So here comes a quantization. 834 01:06:23,350 --> 01:06:27,480 J squared being nu squared, we didn't know what nu squared is, 835 01:06:27,480 --> 01:06:30,830 but it's now equal to these things. 836 01:06:30,830 --> 01:06:36,490 So at this moment, things have been quantized. 837 01:06:36,490 --> 01:06:41,520 And let's look into a little more detail what has happened 838 01:06:41,520 --> 01:06:44,780 and confirm that we got everything we wanted. 839 01:06:57,940 --> 01:07:00,680 So let me write that equation again here. 840 01:07:00,680 --> 01:07:07,040 J1 squared is equal J2 squared is equal to one quarter 841 01:07:07,040 --> 01:07:10,670 h squared nu squared minus 1, which 842 01:07:10,670 --> 01:07:16,110 is h squared J times J plus 1. 843 01:07:16,110 --> 01:07:22,740 So cancel the h squares and solve for nu squared. 844 01:07:22,740 --> 01:07:28,580 Nu squared would be 1 plus 4J times 845 01:07:28,580 --> 01:07:36,380 J plus 1, which is 4J squared plus 4J plus 1, 846 01:07:36,380 --> 01:07:39,660 which is 2J plus 1 squared. 847 01:07:42,810 --> 01:07:44,550 That's pretty neat. 848 01:07:44,550 --> 01:07:46,210 Why is it so neat? 849 01:07:46,210 --> 01:07:53,220 Because as J is equal to zero, all the possible values 850 01:07:53,220 --> 01:07:55,860 of angular momentum-- three halves, 851 01:07:55,860 --> 01:08:01,780 all these things-- nu, which is 2J plus 1, 852 01:08:01,780 --> 01:08:14,220 will be equal to 1, 2, 3, 4-- all the integers. 853 01:08:14,220 --> 01:08:17,292 And what was nu? 854 01:08:17,292 --> 01:08:20,609 It was the values of the energies. 855 01:08:20,609 --> 01:08:26,420 So actually you've proven the spectrum. 856 01:08:26,420 --> 01:08:31,420 Nu has come out to be either 1, 2, 3, 857 01:08:31,420 --> 01:08:35,360 but you have all representations of angular momentum. 858 01:08:35,360 --> 01:08:38,630 You have the singlet, the spin one 859 01:08:38,630 --> 01:08:40,930 half-- where are the spins here? 860 01:08:40,930 --> 01:08:42,000 Nowhere. 861 01:08:42,000 --> 01:08:44,950 There was an electron, a proton, we never 862 01:08:44,950 --> 01:08:47,240 put spin for the hydrogen atom. 863 01:08:47,240 --> 01:08:51,180 But it all shows up as these representations 864 01:08:51,180 --> 01:08:53,880 in which they come along. 865 01:08:53,880 --> 01:08:57,740 Even more is true, as we will see right away 866 01:08:57,740 --> 01:09:02,160 and confirm that everything really shows up the right way. 867 01:09:02,160 --> 01:09:06,120 So what happened now? 868 01:09:06,120 --> 01:09:11,950 We have two independent, equal angular momentum. 869 01:09:11,950 --> 01:09:16,149 So what is this degenerate subspace we were inventing? 870 01:09:16,149 --> 01:09:24,000 Is the space J, which is J1 and m1 tensor product 871 01:09:24,000 --> 01:09:28,310 with J, which is J2 but has the same value 872 01:09:28,310 --> 01:09:31,029 because the squares are the same, m2. 873 01:09:34,450 --> 01:09:39,350 So this is an uncoupled basis. 874 01:09:39,350 --> 01:09:45,000 Uncoupled basis of states in the degenerate subspace. 875 01:09:45,000 --> 01:09:50,720 And now, you know, it's all a little surreal 876 01:09:50,720 --> 01:09:55,350 because these don't look like our states at all. 877 01:09:55,350 --> 01:09:58,780 But this is the way algebraically they show up. 878 01:09:58,780 --> 01:10:05,260 We choose our value of J, we have then that nu is equal 879 01:10:05,260 --> 01:10:09,270 to this and for that value of J there's some values of m's. 880 01:10:09,270 --> 01:10:13,470 And therefore, this must be the degenerate subspace. 881 01:10:13,470 --> 01:10:17,890 So this is nothing but the tensor product 882 01:10:17,890 --> 01:10:22,220 of a J multiplet with a J multiplet. 883 01:10:25,430 --> 01:10:29,120 Where J is that integer here. 884 01:10:29,120 --> 01:10:32,400 And what is the tensor product of a J multiplet? 885 01:10:32,400 --> 01:10:35,200 First, J is for J1. 886 01:10:35,200 --> 01:10:37,487 The second J is for J2. 887 01:10:44,230 --> 01:10:46,220 So at this moment of course we're 888 01:10:46,220 --> 01:10:51,680 calling this N for the quantum number. 889 01:10:51,680 --> 01:10:53,250 But what is this thing? 890 01:10:53,250 --> 01:11:01,850 This is 2J plus 2J minus 1 plus-- all the way 891 01:11:01,850 --> 01:11:02,810 up to the singlet. 892 01:11:08,430 --> 01:11:12,620 But what are these representations of? 893 01:11:12,620 --> 01:11:17,190 Well here we have J1 and here is J2. 894 01:11:17,190 --> 01:11:19,790 These must be the ones of the sum. 895 01:11:19,790 --> 01:11:22,640 But who is the sum, L? 896 01:11:22,640 --> 01:11:27,640 So these are the L representations that you get. 897 01:11:27,640 --> 01:11:29,850 L is your angular momentum. 898 01:11:29,850 --> 01:11:32,410 L representations. 899 01:11:32,410 --> 01:11:39,520 And if 2J plus 1 is N, you got a representation 900 01:11:39,520 --> 01:11:46,170 with L equals N minus 1, because 2J plus 1 901 01:11:46,170 --> 01:11:57,190 is N, L equals N minus 2, all the way up to L equals 0. 902 01:11:57,190 --> 01:12:04,780 Therefore, you get precisely this whole structure. 903 01:12:04,780 --> 01:12:09,870 So, just in time as we get to 2 o'clock, 904 01:12:09,870 --> 01:12:13,520 we've finished the quantization of the hydrogen atom. 905 01:12:13,520 --> 01:12:16,340 We've finished 805. 906 01:12:16,340 --> 01:12:17,560 I hope you enjoyed. 907 01:12:17,560 --> 01:12:21,820 I did a lot. [INAUDIBLE] and Will did, too. 908 01:12:21,820 --> 01:12:25,440 Good luck and we'll see you soon.