1 00:00:00,070 --> 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 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,056 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,056 --> 00:00:17,205 at ocs.mit.edu. 8 00:00:21,240 --> 00:00:23,770 PROFESSOR: Today we have plenty to do. 9 00:00:23,770 --> 00:00:29,780 We really begin in all generality 10 00:00:29,780 --> 00:00:32,580 the addition of angular momentum. 11 00:00:32,580 --> 00:00:37,970 But we will do it in the set up of a physical problem. 12 00:00:37,970 --> 00:00:40,020 The problem of computing the spin 13 00:00:40,020 --> 00:00:46,960 orbit interactions of electrons with the nucleus. 14 00:00:46,960 --> 00:00:51,940 So this is a rather interesting and complicated interaction. 15 00:00:51,940 --> 00:00:54,610 So we'll spend a little time telling you 16 00:00:54,610 --> 00:00:57,520 about the physics of this interaction. 17 00:00:57,520 --> 00:01:01,230 And then once the physics is clear, 18 00:01:01,230 --> 00:01:04,720 it will become more obvious why we 19 00:01:04,720 --> 00:01:08,980 have to do these mathematical contortions of adding angular 20 00:01:08,980 --> 00:01:13,960 momentum in order to solve this physical problem. 21 00:01:13,960 --> 00:01:17,790 So it's a sophisticated problem that requires several steps. 22 00:01:17,790 --> 00:01:22,420 The first step is something that is 23 00:01:22,420 --> 00:01:26,480 a result in perturbation theory. 24 00:01:26,480 --> 00:01:30,460 Feynman Hellman Theorem of perturbation theory. 25 00:01:30,460 --> 00:01:32,270 And that's where we begin. 26 00:01:32,270 --> 00:01:35,770 So it's called Feynman Hellman Theorem. 27 00:01:35,770 --> 00:01:41,900 It's a very simple result. 28 00:01:46,605 --> 00:01:47,105 Theorem. 29 00:01:50,580 --> 00:01:54,530 And we'll need it in order to understand 30 00:01:54,530 --> 00:01:57,990 how a small perturbation to the Hamiltonian 31 00:01:57,990 --> 00:02:00,240 changes the energy spectrum. 32 00:02:00,240 --> 00:02:12,110 So we have H of lambda be a Hamiltonian with a parameter 33 00:02:12,110 --> 00:02:13,252 in lambda. 34 00:02:17,190 --> 00:02:19,540 Lambda. 35 00:02:19,540 --> 00:02:30,640 And psi n of lambda being normalized energy 36 00:02:30,640 --> 00:02:40,550 eigenstate with energy, En of lambda. 37 00:02:44,460 --> 00:02:48,890 So that's the whole assumption of the theorem. 38 00:02:48,890 --> 00:02:51,130 We have a Hamiltonian. 39 00:02:51,130 --> 00:02:55,080 It depends on some parameter that we're going to vary. 40 00:02:55,080 --> 00:03:01,530 And suppose we consider now an eigenstate of this Hamiltonian 41 00:03:01,530 --> 00:03:04,140 that depends on lambda, so the eigenstate also 42 00:03:04,140 --> 00:03:06,050 depends on lambda. 43 00:03:06,050 --> 00:03:09,770 And it has an energy, En of lambda. 44 00:03:09,770 --> 00:03:11,860 So the purpose of this theorem is 45 00:03:11,860 --> 00:03:15,130 to relate these various quantities. 46 00:03:15,130 --> 00:03:22,530 And the claim is that the rate of change 47 00:03:22,530 --> 00:03:26,810 of the energy with respect to lambda 48 00:03:26,810 --> 00:03:34,900 can be computed pretty much by evaluating the rate of change 49 00:03:34,900 --> 00:03:38,140 of the Hamiltonian on the relevant states. 50 00:03:50,740 --> 00:03:51,660 So that's the claim. 51 00:04:01,010 --> 00:04:05,740 And it's a pretty nice result. 52 00:04:05,740 --> 00:04:08,940 It's useful in many circumstances. 53 00:04:08,940 --> 00:04:13,630 And for us will be a way to discuss a little perturbation 54 00:04:13,630 --> 00:04:14,470 theory. 55 00:04:14,470 --> 00:04:19,240 Perturbation theory is the subject of 806 in all details. 56 00:04:19,240 --> 00:04:21,970 And it's a very sophisticated subject. 57 00:04:21,970 --> 00:04:25,970 Even today we were going to be finding that it's not 58 00:04:25,970 --> 00:04:30,140 all that easy to carry it out. 59 00:04:30,140 --> 00:04:32,100 So how does this begin? 60 00:04:32,100 --> 00:04:32,890 Well, proof. 61 00:04:36,290 --> 00:04:39,410 You begin by saying that En of lambda 62 00:04:39,410 --> 00:04:42,860 is the energy eigenstate, is nothing else 63 00:04:42,860 --> 00:04:47,200 but psi n of lambda. 64 00:04:47,200 --> 00:04:49,950 H of lambda. 65 00:04:49,950 --> 00:04:51,810 Psi n of lambda. 66 00:04:55,730 --> 00:05:01,860 And the reason is, of course, that H and psi n 67 00:05:01,860 --> 00:05:07,400 is En of lambda times psi n of lambda. 68 00:05:07,400 --> 00:05:09,385 And this is a number goes out. 69 00:05:09,385 --> 00:05:12,400 And the inner product of this things 70 00:05:12,400 --> 00:05:16,140 is 1, because the state is normalized. 71 00:05:16,140 --> 00:05:18,980 So this is a good starting point. 72 00:05:18,980 --> 00:05:24,780 And the funny thing that you see already is that, in some sense, 73 00:05:24,780 --> 00:05:27,840 you just get the middle term when 74 00:05:27,840 --> 00:05:30,290 you take the derivative with respect to lambda. 75 00:05:30,290 --> 00:05:32,320 You don't get anything from these two. 76 00:05:34,870 --> 00:05:36,540 And it's simple in fact. 77 00:05:36,540 --> 00:05:38,310 So let me just do it. 78 00:05:38,310 --> 00:05:43,880 V, En, V lambda would be the term 79 00:05:43,880 --> 00:05:49,140 that Feynman and Hellman gave. 80 00:05:49,140 --> 00:05:55,400 V, H, V lambda, psi n of lambda. 81 00:05:55,400 --> 00:05:58,470 Plus one term in which we differentiate this one. 82 00:05:58,470 --> 00:06:05,500 V, d lambda of the state psi n of lambda. 83 00:06:09,340 --> 00:06:17,130 Times H of lambda, psi n of lambda. 84 00:06:17,130 --> 00:06:21,380 Plus the other term in which you differentiate the ket. 85 00:06:21,380 --> 00:06:32,610 So psi n of lambda, H of lambda, d, d lambda of psi n of lambda. 86 00:06:38,880 --> 00:06:46,710 And the reason these terms are going to vanish 87 00:06:46,710 --> 00:06:53,020 is that you can now act with H again on psi n. 88 00:06:53,020 --> 00:06:57,620 H is supposed to be Hermitian, so it can act on the left. 89 00:06:57,620 --> 00:07:07,220 And therefore, these two terms give you En of lambda, 90 00:07:07,220 --> 00:07:19,400 times d d lambda of psi n of lambda-- psi n of lambda-- 91 00:07:19,400 --> 00:07:23,750 plus the other term, which would be psi n of lambda, 92 00:07:23,750 --> 00:07:27,395 the bra times the derivative of the ket. 93 00:07:35,450 --> 00:07:43,550 But this is nothing else than the derivative 94 00:07:43,550 --> 00:07:44,730 of the inner product. 95 00:07:55,480 --> 00:07:57,250 In the inner product to differentiate-- 96 00:07:57,250 --> 00:08:00,020 the inner product differentiates the bra, 97 00:08:00,020 --> 00:08:01,920 it differentiates the ket. 98 00:08:01,920 --> 00:08:04,280 And do it. 99 00:08:04,280 --> 00:08:08,220 And this thing is equal to 1, because it's normalized. 100 00:08:08,220 --> 00:08:10,890 So this is 0. 101 00:08:10,890 --> 00:08:12,650 End of proof. 102 00:08:12,650 --> 00:08:16,320 These two terms vanish. 103 00:08:16,320 --> 00:08:20,860 And the result holds. 104 00:08:20,860 --> 00:08:21,652 Yes? 105 00:08:21,652 --> 00:08:23,527 AUDIENCE: How do you know it stays normalized 106 00:08:23,527 --> 00:08:24,505 when you vary lambda? 107 00:08:24,505 --> 00:08:28,440 PROFESSOR: It's an assumption. 108 00:08:28,440 --> 00:08:31,770 The state is normalized for all values of n. 109 00:08:31,770 --> 00:08:34,925 So if you have a state that you've constructed, 110 00:08:34,925 --> 00:08:39,669 that is normalized, you can have this result. 111 00:08:39,669 --> 00:08:41,030 So it's an assumption. 112 00:08:41,030 --> 00:08:42,845 You have to keep the state normalized. 113 00:08:46,900 --> 00:08:50,900 Now this is a baby version of perturbation theory. 114 00:08:50,900 --> 00:08:55,320 It's a result I think that Feynman did as an undergrad. 115 00:08:55,320 --> 00:08:59,290 And as you can see, it's very simple. 116 00:08:59,290 --> 00:09:03,290 Calling it a theorem is a little too much. 117 00:09:03,290 --> 00:09:06,540 But still, the fact is that it's useful. 118 00:09:06,540 --> 00:09:10,670 And so we'll just go ahead and use it. 119 00:09:10,670 --> 00:09:14,910 Now I want to rewrite it in another way. 120 00:09:14,910 --> 00:09:19,250 So, suppose you have a Hamiltonian, H, 121 00:09:19,250 --> 00:09:24,755 which has a term H0, plus lambda, H1. 122 00:09:27,380 --> 00:09:41,770 So, the parameter lambda, H of lambda, is given in this way. 123 00:09:41,770 --> 00:09:45,180 And that's a reasonable H of lambda. 124 00:09:45,180 --> 00:09:49,860 Sometimes, this could be written as H0 plus something 125 00:09:49,860 --> 00:09:52,670 that we will call the change in the Hamiltonian. 126 00:09:52,670 --> 00:09:57,540 And we usually think of it as a small thing. 127 00:10:01,300 --> 00:10:05,190 So what do we have from this theorem? 128 00:10:08,410 --> 00:10:18,020 From this here we would have the d, En, d lambda 129 00:10:18,020 --> 00:10:29,380 is equal to psi n of lambda, H1, psi n of lambda. 130 00:10:34,100 --> 00:10:37,400 Now, we can be particularly interested 131 00:10:37,400 --> 00:10:41,250 in the evaluation of this thing at lambda equals 0. 132 00:10:41,250 --> 00:10:44,830 So what is d En of lambda? 133 00:10:44,830 --> 00:10:55,330 d lambda at lambda equals 0 would be psi n at zero, 134 00:10:55,330 --> 00:11:00,970 H1, psi n at 0. 135 00:11:00,970 --> 00:11:07,870 And therefore, you would say that the En of lambda energies 136 00:11:07,870 --> 00:11:19,790 would be the energies at 0, plus lambda, d En 137 00:11:19,790 --> 00:11:25,510 of lambda, d lambda at lambda equals 0, 138 00:11:25,510 --> 00:11:28,093 plus order lambda squared. 139 00:11:35,610 --> 00:11:38,740 I'm doing just the Taylor expansion 140 00:11:38,740 --> 00:11:46,450 of En's of lambda from lambda equals 0. 141 00:11:46,450 --> 00:11:56,480 So this thing tells you that En of lambda is equal En of 0, 142 00:11:56,480 --> 00:12:08,302 plus-- this derivative you can write it as psi n, lambda, H1, 143 00:12:08,302 --> 00:12:13,008 psi n, all at 0. 144 00:12:13,008 --> 00:12:13,980 Like that. 145 00:12:13,980 --> 00:12:16,955 Plus order lambda squared. 146 00:12:23,650 --> 00:12:28,600 So in this step, I just use the evaluation that we did here. 147 00:12:28,600 --> 00:12:32,380 I substituted that and put the lambda in. 148 00:12:32,380 --> 00:12:45,470 So that I recognize now that En of lambda is equal to En of 0, 149 00:12:45,470 --> 00:12:53,990 plus psi n of 0-- and I can write this as delta H-- psi 150 00:12:53,990 --> 00:13:00,803 n of 0, plus order delta H squared. 151 00:13:09,260 --> 00:13:12,780 It's nice to write it this way, because you appreciate 152 00:13:12,780 --> 00:13:15,540 more the power of the theorem. 153 00:13:15,540 --> 00:13:20,640 The theorem here doesn't assume which value of lambda you have. 154 00:13:20,640 --> 00:13:23,070 And you have to have normalized eigenstates. 155 00:13:23,070 --> 00:13:26,320 And you wonder what is it helping you with, 156 00:13:26,320 --> 00:13:30,820 if finding the states for every value of lambda is complicated. 157 00:13:30,820 --> 00:13:37,160 Well, it certainly helps you to figure out 158 00:13:37,160 --> 00:13:42,960 how the energy of the state varies by a simple calculation. 159 00:13:42,960 --> 00:13:46,725 Suppose you know the states of the simple Hamiltonian. 160 00:13:49,980 --> 00:13:55,630 Those are the psi's, n, 0. 161 00:13:55,630 --> 00:13:59,870 So if you have the psi n 0 over here, 162 00:13:59,870 --> 00:14:03,120 you can do the following step. 163 00:14:03,120 --> 00:14:07,930 If you want to figure out how it's energy has varied, 164 00:14:07,930 --> 00:14:14,260 use this formula in which you compute the expectation 165 00:14:14,260 --> 00:14:20,370 value of the change in the Hamiltonian on that state. 166 00:14:20,370 --> 00:14:26,310 And that is the first correction to the energy of the state. 167 00:14:26,310 --> 00:14:29,010 So you have this state. 168 00:14:29,010 --> 00:14:31,710 You compute the expectation value 169 00:14:31,710 --> 00:14:34,880 of the extra piece in the Hamiltonian. 170 00:14:34,880 --> 00:14:39,270 And that's the correction to the energy. 171 00:14:39,270 --> 00:14:41,300 It's a little more complicated of course 172 00:14:41,300 --> 00:14:43,600 to compute the correction to the state. 173 00:14:43,600 --> 00:14:47,570 But that's a subject of perturbation theory. 174 00:14:47,570 --> 00:14:51,790 And that's not what we care about right now. 175 00:14:51,790 --> 00:14:57,410 So the reason we're doing this is because actually whatever 176 00:14:57,410 --> 00:15:01,010 we're going to have with spin orbit coupling 177 00:15:01,010 --> 00:15:04,100 represents an addition to the hydrogen 178 00:15:04,100 --> 00:15:06,680 Hamiltonian of a new term. 179 00:15:06,680 --> 00:15:10,060 Therefore, you want to know what happens to the energy levels. 180 00:15:10,060 --> 00:15:11,970 And the best thing to think about them 181 00:15:11,970 --> 00:15:15,360 is to-- if you know the energy levels of this one, 182 00:15:15,360 --> 00:15:17,980 well, a formula of this type can let 183 00:15:17,980 --> 00:15:20,082 you know what happens to the energy levels 184 00:15:20,082 --> 00:15:21,040 after the perturbation. 185 00:15:23,850 --> 00:15:25,790 There will be an extra complication 186 00:15:25,790 --> 00:15:28,500 in that the energy levels that we're going to deal with 187 00:15:28,500 --> 00:15:29,930 are going to be degenerate. 188 00:15:29,930 --> 00:15:32,980 But let's wait for that complication until it appears. 189 00:15:32,980 --> 00:15:34,650 So any questions? 190 00:15:37,960 --> 00:15:38,870 Yes? 191 00:15:38,870 --> 00:15:40,820 AUDIENCE: So I would imagine that this 192 00:15:40,820 --> 00:15:42,704 would work just as well for time. 193 00:15:42,704 --> 00:15:45,974 Because time [INAUDIBLE] a parameter in quantum mechanics. 194 00:15:45,974 --> 00:15:48,404 So [INAUDIBLE] 195 00:15:48,404 --> 00:15:50,830 PROFESSOR: Time dependent perturbation theory 196 00:15:50,830 --> 00:15:52,930 is a bit more complicated. 197 00:15:52,930 --> 00:15:56,860 I'd rather not get into it now. 198 00:15:56,860 --> 00:16:01,450 So let's leave it here, in which we don't have time. 199 00:16:01,450 --> 00:16:03,280 And the Schrodinger equation is something 200 00:16:03,280 --> 00:16:07,617 like H psi equal [INAUDIBLE] psi, that's all we care. 201 00:16:07,617 --> 00:16:11,250 And leave it for that moment. 202 00:16:11,250 --> 00:16:11,933 Other questions? 203 00:16:21,496 --> 00:16:22,460 OK. 204 00:16:22,460 --> 00:16:28,530 So let's proceed with addition of angular momentum. 205 00:16:28,530 --> 00:16:34,160 So first, let me give you the fundamental result 206 00:16:34,160 --> 00:16:36,010 of addition of angular momentum. 207 00:16:36,010 --> 00:16:39,880 It's a little abstract, but it's what we really 208 00:16:39,880 --> 00:16:41,725 mean by addition of angular momentum. 209 00:16:45,197 --> 00:16:48,763 Of angular momentum. 210 00:16:51,520 --> 00:16:53,960 And the main result is the following. 211 00:16:53,960 --> 00:17:00,450 Suppose you have a set of operators, J, i, 212 00:17:00,450 --> 00:17:04,929 1, that have the algebra of angular momentum. 213 00:17:08,200 --> 00:17:11,250 Of angular momentum. 214 00:17:11,250 --> 00:17:21,609 Which is to say Ji1, JJ1, is equal 215 00:17:21,609 --> 00:17:25,712 i, h bar, epsilon iJK, JK1. 216 00:17:29,870 --> 00:17:34,180 And this algebra is realized on some state space. 217 00:17:34,180 --> 00:17:41,330 On some vector space, V1. 218 00:17:41,330 --> 00:17:44,760 And suppose you have another operator, 219 00:17:44,760 --> 00:17:48,180 J-- set of operators actually. 220 00:17:48,180 --> 00:17:51,395 Ji2, which have the algebra of angular momentum. 221 00:17:51,395 --> 00:17:53,470 I will not write that. 222 00:17:53,470 --> 00:17:56,280 On some V2. 223 00:17:59,530 --> 00:18:00,640 OK. 224 00:18:00,640 --> 00:18:03,340 Angular momentum, some sets of states. 225 00:18:03,340 --> 00:18:07,570 Angular momentum on some other set of states. 226 00:18:07,570 --> 00:18:10,090 Here comes the thing. 227 00:18:10,090 --> 00:18:13,620 There is a new angular momentum, which 228 00:18:13,620 --> 00:18:24,970 is the sum Ji defined as Ji1, added with Ji2. 229 00:18:24,970 --> 00:18:30,560 Now, soon enough you will just write Ji1, plus Ji2. 230 00:18:30,560 --> 00:18:34,590 But let me be a little more careful now. 231 00:18:34,590 --> 00:18:45,290 This sum is Ji1, plus 1, tensor Ji2. 232 00:18:45,290 --> 00:18:48,080 So i is the same index. 233 00:18:48,080 --> 00:18:53,230 But here, we're having this operator 234 00:18:53,230 --> 00:18:57,140 that we're being defined that we call it the sum. 235 00:18:57,140 --> 00:19:01,730 Now how do you sum two operators that act in different spaces? 236 00:19:01,730 --> 00:19:04,680 Well, the only thing that you can actually do 237 00:19:04,680 --> 00:19:07,120 is sum them in the tensor product. 238 00:19:07,120 --> 00:19:19,136 So the claim is that this is an angular momentum in V1 tensor 239 00:19:19,136 --> 00:19:19,635 V2. 240 00:19:25,810 --> 00:19:28,190 That is an operator. 241 00:19:28,190 --> 00:19:29,690 You see, you have to sum them. 242 00:19:29,690 --> 00:19:34,490 So you have to create a space where both can act, 243 00:19:34,490 --> 00:19:36,180 and you can sum them. 244 00:19:36,180 --> 00:19:40,320 You cannot sum a thing, an operator that acts on one 245 00:19:40,320 --> 00:19:43,420 vector space to an operator that acts on another vector space. 246 00:19:43,420 --> 00:19:47,850 You have to create one vector space where both act. 247 00:19:47,850 --> 00:19:51,250 And then you can define the sum of the operators. 248 00:19:51,250 --> 00:19:53,750 Sum of operators is a simple thing. 249 00:19:53,750 --> 00:19:56,460 So you form the tensor product. 250 00:19:56,460 --> 00:20:00,680 In here, this operator gets upgraded 251 00:20:00,680 --> 00:20:05,390 in this way, in which in the tensor product it has a 1 252 00:20:05,390 --> 00:20:06,970 for the second input. 253 00:20:06,970 --> 00:20:09,290 This one gets upgrade to this way. 254 00:20:09,290 --> 00:20:10,250 And this is the sum. 255 00:20:12,950 --> 00:20:17,270 So this is a claim-- this is a definition. 256 00:20:17,270 --> 00:20:18,640 And this is a claim. 257 00:20:18,640 --> 00:20:21,430 So this has to be proven. 258 00:20:21,430 --> 00:20:23,370 So let me prove it. 259 00:20:26,140 --> 00:20:28,910 Ji, JJ. 260 00:20:28,910 --> 00:20:30,730 I compute this commutator. 261 00:20:30,730 --> 00:20:32,930 So I don't have to do the following. 262 00:20:32,930 --> 00:20:45,530 I have to do Ji1, tensor 1, plus 1 tensor Ji2. 263 00:20:45,530 --> 00:20:56,620 And then the JJ would be JJ1, tensor 1, plus 1, tensor JJ2. 264 00:20:59,670 --> 00:21:03,320 Have to compute this commutator. 265 00:21:03,320 --> 00:21:06,880 Now, an important fact about this 266 00:21:06,880 --> 00:21:09,320 result that I'm not trying to generalize, 267 00:21:09,320 --> 00:21:12,700 if you had put a minus here, it wouldn't work out. 268 00:21:12,700 --> 00:21:15,480 If you would have put a 2 here, it wouldn't work out. 269 00:21:15,480 --> 00:21:19,130 If you would have put a 1/2 here, it won't work out. 270 00:21:19,130 --> 00:21:22,060 This is pretty much the only way you 271 00:21:22,060 --> 00:21:26,370 can have two angular momenta, and create a third angular 272 00:21:26,370 --> 00:21:27,620 momentum. 273 00:21:27,620 --> 00:21:29,240 So look at this. 274 00:21:32,730 --> 00:21:34,820 It looks like we're going to have to work hard, 275 00:21:34,820 --> 00:21:38,490 but that's not true. 276 00:21:38,490 --> 00:21:40,650 Consider this commutator. 277 00:21:40,650 --> 00:21:42,970 The commutator of this term with this term. 278 00:21:45,510 --> 00:21:47,430 That's 0 actually. 279 00:21:47,430 --> 00:21:51,950 Because if you multiply them in this order, this times that, 280 00:21:51,950 --> 00:21:57,080 you get Ji1 times Ji2, because the ones do nothing. 281 00:21:57,080 --> 00:21:59,710 You multiply them in the reverse order, 282 00:21:59,710 --> 00:22:03,860 you get again, Ji1 times Ji2. 283 00:22:03,860 --> 00:22:08,895 This is to say that the operators that originally lived 284 00:22:08,895 --> 00:22:13,510 in the different vector spaces commute. 285 00:22:13,510 --> 00:22:15,260 Yes? 286 00:22:15,260 --> 00:22:19,320 AUDIENCE: Since the cross terms between those two 287 00:22:19,320 --> 00:22:22,900 are 0-- like you just said, the cross terms are 0. 288 00:22:22,900 --> 00:22:27,600 And if you put a minus sign in there, it will cancel. 289 00:22:27,600 --> 00:22:29,494 But when you do the multiplications 290 00:22:29,494 --> 00:22:33,318 with the second ones, why can't you put a minus sign in there? 291 00:22:33,318 --> 00:22:34,170 [INAUDIBLE] 292 00:22:34,170 --> 00:22:35,420 PROFESSOR: In the whole thing? 293 00:22:35,420 --> 00:22:37,790 In this definition, a minus sign? 294 00:22:37,790 --> 00:22:38,720 AUDIENCE: Yeah. 295 00:22:38,720 --> 00:22:41,900 PROFESSOR: Well, here if I put a minus-- 296 00:22:41,900 --> 00:22:45,270 it's like I'm going to prove that this works. 297 00:22:45,270 --> 00:22:48,860 So if-- I'm going to get an angular momentum. 298 00:22:48,860 --> 00:22:51,750 If I put a minus sign to angular momentum, 299 00:22:51,750 --> 00:22:54,420 I ruin the algebra here. 300 00:22:54,420 --> 00:22:56,810 I put a minus minus, it cancels. 301 00:22:56,810 --> 00:23:00,010 But then I get a minus sign here. 302 00:23:00,010 --> 00:23:03,940 So I cannot really even change a sign. 303 00:23:03,940 --> 00:23:09,690 So any way, these are operators acting on different spaces. 304 00:23:09,690 --> 00:23:11,320 They commute. 305 00:23:11,320 --> 00:23:13,375 It's clear they commute. 306 00:23:13,375 --> 00:23:15,700 You just multiply them, and see that. 307 00:23:15,700 --> 00:23:18,860 These one's commute as well. 308 00:23:18,860 --> 00:23:23,730 The only ones that don't commute are this with this. 309 00:23:23,730 --> 00:23:25,310 And that with that. 310 00:23:25,310 --> 00:23:27,520 So let me just write them. 311 00:23:27,520 --> 00:23:38,430 Ji1, tensor 1, with JJ1, tensor 1. 312 00:23:38,430 --> 00:23:50,347 Plus this one, 1 tensor Ji2, 1 tensor JJ2. 313 00:23:58,730 --> 00:24:01,870 OK, next step is to realize that actually 314 00:24:01,870 --> 00:24:07,280 the 1 is a spectator here. 315 00:24:07,280 --> 00:24:10,180 Therefore, this commutator is nothing 316 00:24:10,180 --> 00:24:19,130 but the commutator Ji1 with JJ1, tensor 1. 317 00:24:23,370 --> 00:24:24,510 You can do it. 318 00:24:24,510 --> 00:24:27,530 If you prefer to write it, write it. 319 00:24:27,530 --> 00:24:32,930 This product is Ji times JJ, tensor 1. 320 00:24:32,930 --> 00:24:37,240 And the other product is JJ, Ji, tensor 1. 321 00:24:37,240 --> 00:24:41,040 So the tensor 1 factors out. 322 00:24:41,040 --> 00:24:44,980 Here the tensor 1 also factors out. 323 00:24:44,980 --> 00:24:50,470 And you get an honest commutator, Ji2, JJ2. 324 00:24:53,300 --> 00:24:57,290 So one last step. 325 00:24:57,290 --> 00:25:01,920 This is i, h bar, epsilon, iJK. 326 00:25:01,920 --> 00:25:04,750 I'll put a big parentheses. 327 00:25:04,750 --> 00:25:12,990 JK1, tensor 1, for the first one. 328 00:25:12,990 --> 00:25:17,530 Because J1 forms an angular momentum algebra. 329 00:25:17,530 --> 00:25:21,750 And here, 1 tensor JK2. 330 00:25:30,670 --> 00:25:36,280 And this thing is i, h bar, epsilon, iJK. 331 00:25:36,280 --> 00:25:41,880 The total angular momentum, K. And you've 332 00:25:41,880 --> 00:25:43,990 shown the algebra works out. 333 00:25:48,370 --> 00:25:52,620 Now most people after a little practice, 334 00:25:52,620 --> 00:26:02,890 they just say, oh, Ji is J1 plus J2, J1 plus J2. 335 00:26:02,890 --> 00:26:04,910 J1 and J2 don't commute. 336 00:26:04,910 --> 00:26:07,870 J2 and J1-- I'm sorry. 337 00:26:07,870 --> 00:26:09,390 J1 and J2 commute. 338 00:26:09,390 --> 00:26:11,110 J2 and J1 commute. 339 00:26:11,110 --> 00:26:14,470 Therefore you get this 2, like that. 340 00:26:14,470 --> 00:26:18,760 And this gives you-- J1 and J1 gives you J1. 341 00:26:18,760 --> 00:26:22,230 J2 and J2 gives you J2, so the sum works out. 342 00:26:22,230 --> 00:26:25,530 So most people after a little practice 343 00:26:25,530 --> 00:26:28,100 just don't put all these tensor things. 344 00:26:28,100 --> 00:26:32,320 But at the beginning it's nice to just make sure 345 00:26:32,320 --> 00:26:36,700 that you understand what these tensor things do. 346 00:26:36,700 --> 00:26:37,280 All right. 347 00:26:37,280 --> 00:26:40,700 So that's our main theorem-- that you 348 00:26:40,700 --> 00:26:45,180 start with one angular momentum on a state space. 349 00:26:45,180 --> 00:26:49,090 Another angular momentum that has nothing to do perhaps 350 00:26:49,090 --> 00:26:53,170 with the first on another vector space. 351 00:26:53,170 --> 00:26:58,390 And on the tensor product you have another angular momentum, 352 00:26:58,390 --> 00:26:59,420 which is the sum. 353 00:27:01,940 --> 00:27:02,670 All right. 354 00:27:02,670 --> 00:27:05,590 So now, we do spin orbit coupling 355 00:27:05,590 --> 00:27:08,660 to try to apply these ideas. 356 00:27:08,660 --> 00:27:25,060 So for spin orbit coupling, we will consider the hydrogen atom 357 00:27:25,060 --> 00:27:28,000 coupling. 358 00:27:28,000 --> 00:27:36,610 And the new term in the Hamiltonian, mu dot B. 359 00:27:36,610 --> 00:27:41,390 The kind of term that we've done so much in this semester. 360 00:27:41,390 --> 00:27:43,880 We've looked over magnetic ones. 361 00:27:43,880 --> 00:27:46,700 So which magnetic moment at which B? 362 00:27:46,700 --> 00:27:50,050 There was no B in the hydrogen atom. 363 00:27:50,050 --> 00:27:54,850 Well, there's no B to begin with. 364 00:27:54,850 --> 00:27:58,980 But here is one where you can think there is a B. First, 365 00:27:58,980 --> 00:28:06,280 this will be the electron dipole moment. 366 00:28:06,280 --> 00:28:07,790 Magnetic dipole moment. 367 00:28:07,790 --> 00:28:11,350 So we have a formula for it. 368 00:28:11,350 --> 00:28:17,370 The formula for it is the mu of the electron is minus E over m, 369 00:28:17,370 --> 00:28:19,960 times the spin of the electron. 370 00:28:19,960 --> 00:28:25,090 And I actually will use a little different formula 371 00:28:25,090 --> 00:28:28,130 that is valued in Gaussian units. 372 00:28:28,130 --> 00:28:35,550 ge over mC, S, in Gaussian units. 373 00:28:38,590 --> 00:28:42,070 And g is the g factor of the electron, which is 2. 374 00:28:42,070 --> 00:28:42,770 I'm sorry. 375 00:28:42,770 --> 00:28:44,850 There's a 2 here. 376 00:28:44,850 --> 00:28:45,440 OK. 377 00:28:45,440 --> 00:28:46,990 So look what I've written. 378 00:28:46,990 --> 00:28:49,540 I don't want to distract you with this too much. 379 00:28:49,540 --> 00:28:53,270 But you know that the magnetic dipole of the electron 380 00:28:53,270 --> 00:28:55,310 is given by this quantity. 381 00:28:55,310 --> 00:28:57,930 Now, you could put a 2 up, and a 2 down. 382 00:28:57,930 --> 00:29:02,120 And that's why people actually classically 383 00:29:02,120 --> 00:29:04,410 there seems to be a 2 down. 384 00:29:04,410 --> 00:29:08,220 But there's a 2 up, because it's an effect of the electron. 385 00:29:08,220 --> 00:29:09,640 And you have this formula. 386 00:29:09,640 --> 00:29:11,650 The only thing I've added in that formula 387 00:29:11,650 --> 00:29:16,390 is a factor of C that is because of Gaussian units. 388 00:29:16,390 --> 00:29:20,250 And it allows you to estimate terms a little more easily. 389 00:29:20,250 --> 00:29:22,610 So that's the mu of the electron. 390 00:29:22,610 --> 00:29:28,600 But the electron apparently would feel no magnetic field. 391 00:29:28,600 --> 00:29:31,320 You didn't put an external magnetic field. 392 00:29:31,320 --> 00:29:37,070 Except that here you go in this way 393 00:29:37,070 --> 00:29:40,840 of thinking-- you think suppose you are the electron. 394 00:29:40,840 --> 00:29:45,340 You see a proton, which is a nucleus going around you. 395 00:29:45,340 --> 00:29:49,750 And a proton going around you is a current going around you. 396 00:29:49,750 --> 00:29:52,280 It generates a magnetic field. 397 00:29:52,280 --> 00:29:54,840 And therefore, you see a magnetic field 398 00:29:54,840 --> 00:30:00,260 created by the proton going around you. 399 00:30:00,260 --> 00:30:02,400 So there is a magnetic field. 400 00:30:02,400 --> 00:30:04,720 And there's a magnetic field experienced 401 00:30:04,720 --> 00:30:11,440 by the electron-- felt by electron. 402 00:30:14,300 --> 00:30:19,570 So you can think of this, the electron. 403 00:30:19,570 --> 00:30:22,100 Here is the proton with the plus charge, 404 00:30:22,100 --> 00:30:25,190 and here's the electron. 405 00:30:25,190 --> 00:30:29,060 And the electron is going around the proton. 406 00:30:29,060 --> 00:30:31,730 Now, from the viewpoint of the electron, 407 00:30:31,730 --> 00:30:35,690 the proton is going around him. 408 00:30:35,690 --> 00:30:38,000 So here is the proton. 409 00:30:38,000 --> 00:30:40,330 Here is the electron going like that. 410 00:30:40,330 --> 00:30:42,040 From the viewpoint of the electron, 411 00:30:42,040 --> 00:30:44,760 the proton is going like this. 412 00:30:44,760 --> 00:30:48,350 Also, from the viewpoint of the electron, 413 00:30:48,350 --> 00:30:51,335 the proton would be going in this direction 414 00:30:51,335 --> 00:30:54,150 and creating a magnetic field up. 415 00:30:59,990 --> 00:31:06,570 And the magnetic field up corresponds actually 416 00:31:06,570 --> 00:31:11,775 to the idea that the angular momentum of the electron 417 00:31:11,775 --> 00:31:16,970 is also up-- L of the electron is also up. 418 00:31:16,970 --> 00:31:20,870 So the whole point of this thing is 419 00:31:20,870 --> 00:31:25,670 that somehow this magnetic field is proportional to the angular 420 00:31:25,670 --> 00:31:27,100 momentum. 421 00:31:27,100 --> 00:31:29,900 And then, L will come here. 422 00:31:29,900 --> 00:31:31,770 And here, you have S. So you have 423 00:31:31,770 --> 00:31:39,440 L dot S. That's the fine structure coupling. 424 00:31:39,440 --> 00:31:45,300 Now let me do a little of this so that we just 425 00:31:45,300 --> 00:31:49,080 get a bit more feeling, although it's unfortunately 426 00:31:49,080 --> 00:31:53,000 a somewhat frustrating exercise. 427 00:31:53,000 --> 00:31:56,790 So let me tell you what's going on. 428 00:31:56,790 --> 00:32:00,836 So consider the electron. 429 00:32:00,836 --> 00:32:05,600 At some point, look at it and draw a plane. 430 00:32:05,600 --> 00:32:09,000 So the electron-- let's assume it's going down. 431 00:32:09,000 --> 00:32:10,090 Here is the proton. 432 00:32:10,090 --> 00:32:11,670 It's going around in circles. 433 00:32:11,670 --> 00:32:14,660 So here, it's going down. 434 00:32:14,660 --> 00:32:16,280 The electron is going down. 435 00:32:16,280 --> 00:32:20,900 Electron, its velocity of the electron is going down. 436 00:32:20,900 --> 00:32:24,380 The proton is over here. 437 00:32:24,380 --> 00:32:28,740 And the electron is going around like that. 438 00:32:28,740 --> 00:32:32,350 The proton would produce an electric field of this form. 439 00:32:35,770 --> 00:32:42,080 Now, in relativity, the electric and magnetic fields 440 00:32:42,080 --> 00:32:44,920 seen by different observers are different. 441 00:32:44,920 --> 00:32:48,210 So there is this electric field that we see. 442 00:32:48,210 --> 00:32:52,260 We sit here, and we see in our rest frame 443 00:32:52,260 --> 00:32:56,330 this proton creates an electric field. 444 00:32:56,330 --> 00:33:00,760 And then, from the viewpoint of the electron, 445 00:33:00,760 --> 00:33:02,280 the electron is moving. 446 00:33:02,280 --> 00:33:04,840 And there is an electric field. 447 00:33:04,840 --> 00:33:09,020 But whenever you are moving inside an electric field, 448 00:33:09,020 --> 00:33:12,560 you also see a magnetic field generated by the motion, 449 00:33:12,560 --> 00:33:14,910 by relativistic effects. 450 00:33:14,910 --> 00:33:20,010 The magnetic field that you see is roughly 451 00:33:20,010 --> 00:33:27,560 given to first order in relativity by V cross E over c. 452 00:33:30,350 --> 00:33:37,120 So V cross E, VE V cross E over c 453 00:33:37,120 --> 00:33:39,900 up-- change sign because of this. 454 00:33:39,900 --> 00:33:42,740 And the magnetic field consistently, 455 00:33:42,740 --> 00:33:45,860 as we would expect, goes in this direction. 456 00:33:45,860 --> 00:33:48,210 So it's consistent with the picture 457 00:33:48,210 --> 00:33:51,840 that we developed that if you were the electron, the proton, 458 00:33:51,840 --> 00:33:53,940 would be going around in circles like that 459 00:33:53,940 --> 00:33:55,790 and the magnetic field would be up. 460 00:33:58,430 --> 00:34:07,730 Now here I can change the sign by doing E cross V over c. 461 00:34:07,730 --> 00:34:11,783 So this is the magnetic field seen by the electron. 462 00:34:23,389 --> 00:34:30,010 OK, so we need a little more work on that magnetic field 463 00:34:30,010 --> 00:34:33,610 by calculating the electric field. 464 00:34:33,610 --> 00:34:36,219 Now, what is the electric field? 465 00:34:36,219 --> 00:34:39,865 Well, the scalar potential for the hydrogen atom, 466 00:34:39,865 --> 00:34:44,670 we write it as minus e squared over r. 467 00:34:44,670 --> 00:34:47,159 It's actually not quite the scalar potential. 468 00:34:47,159 --> 00:34:49,980 But it is the potential energy. 469 00:34:49,980 --> 00:34:55,440 It has one factor of e more than what the scalar potential is. 470 00:34:55,440 --> 00:34:57,940 Remember, the scalar potential in electromagnetism 471 00:34:57,940 --> 00:35:00,260 is charge divided by r. 472 00:35:00,260 --> 00:35:04,000 So it has one factor of e more. 473 00:35:04,000 --> 00:35:07,100 What is the derivative of this potential? 474 00:35:07,100 --> 00:35:12,370 With respect to r, it's e squared over r squared. 475 00:35:12,370 --> 00:35:16,990 So the electric field goes like e over r squared. 476 00:35:16,990 --> 00:35:29,410 So the electric field is equal to dV dr divided by e. 477 00:35:32,220 --> 00:35:34,500 That's the magnitude of the electric field. 478 00:35:34,500 --> 00:35:40,150 And its direction is radial from the viewpoint of the proton. 479 00:35:40,150 --> 00:35:42,935 The electric field is here. 480 00:35:47,790 --> 00:35:51,920 So this can be written as r vector divided by r. 481 00:35:58,880 --> 00:36:07,000 Therefore, the magnetic field will-- [INAUDIBLE] this. 482 00:36:07,000 --> 00:36:10,320 The magnetic field now can be calculated. 483 00:36:10,320 --> 00:36:16,400 And we'll see what we claimed was 484 00:36:16,400 --> 00:36:18,930 the relation with angular momentum. 485 00:36:18,930 --> 00:36:25,850 Because B prime is now E cross V. 486 00:36:25,850 --> 00:36:37,170 So you have 1 over ec 1 over r dV dr. 487 00:36:37,170 --> 00:36:38,960 I've taken care of this. 488 00:36:38,960 --> 00:36:44,770 And now I just have r cross V. Well, 489 00:36:44,770 --> 00:36:51,240 r cross V is your angular momentum if you had p here. 490 00:36:51,240 --> 00:36:56,970 So we borrow a factor of the mass of the electron, 491 00:36:56,970 --> 00:37:06,570 ecm 1 over r dV dr L, L of the electron. 492 00:37:11,710 --> 00:37:16,380 p equals mv. 493 00:37:16,380 --> 00:37:22,180 So we have a nice formula for B. And then, we 494 00:37:22,180 --> 00:37:28,460 can go and calculate delta H. Delta H would then 495 00:37:28,460 --> 00:37:45,380 be minus mu dot B. And that would be ge over 2mc spin 496 00:37:45,380 --> 00:38:01,670 dot L-- mu was given here-- S dot L 1 over r dV dr. 497 00:38:01,670 --> 00:38:07,150 And that is the split spin orbit interaction. 498 00:38:07,150 --> 00:38:10,150 Now, the downside of this derivation 499 00:38:10,150 --> 00:38:13,015 is that it has a relativistic error. 500 00:38:16,740 --> 00:38:20,370 There's a phenomenon called Thomas precession 501 00:38:20,370 --> 00:38:23,870 that affects this result. 502 00:38:23,870 --> 00:38:26,330 We didn't waste our time. 503 00:38:26,330 --> 00:38:32,250 The true result is that you must subtract from this g 1. 504 00:38:32,250 --> 00:38:37,730 So g must really be replaced by g minus 1. 505 00:38:37,730 --> 00:38:41,040 Since g is approximately 2 for the electron, 506 00:38:41,040 --> 00:38:46,080 the true result is really 1/2 of this thing. 507 00:38:46,080 --> 00:38:49,760 So this should not be in parentheses, 508 00:38:49,760 --> 00:38:54,200 but true result is this. 509 00:38:54,200 --> 00:38:59,770 And the mistake that is done in calculating this spin orbit 510 00:38:59,770 --> 00:39:04,970 coupling is that this spin orbit coupling 511 00:39:04,970 --> 00:39:08,510 affects precession rates. 512 00:39:08,510 --> 00:39:12,260 All these interactions of magnetic dipoles 513 00:39:12,260 --> 00:39:15,540 with magnetic fields affect precession rates. 514 00:39:15,540 --> 00:39:18,020 And you have to be a little more careful here 515 00:39:18,020 --> 00:39:24,200 that the system where you've worked, the electron rest frame 516 00:39:24,200 --> 00:39:26,370 is not quite an inertial system. 517 00:39:26,370 --> 00:39:29,330 Because it's doing circular motion. 518 00:39:29,330 --> 00:39:33,420 So there's an extra correction that has to be done. 519 00:39:33,420 --> 00:39:37,350 Thomas precession or Thomas correction it's called. 520 00:39:37,350 --> 00:39:41,170 And it would be a detour of about one hour 521 00:39:41,170 --> 00:39:43,900 in special relativity to do it right. 522 00:39:43,900 --> 00:39:47,676 So Griffiths doesn't do it. 523 00:39:47,676 --> 00:39:51,010 I don't think Shankar does it. 524 00:39:51,010 --> 00:39:53,280 Pretty much graduate books do it. 525 00:39:56,088 --> 00:39:59,990 So we will not try to do better. 526 00:39:59,990 --> 00:40:02,550 I mentioned that fact that this really 527 00:40:02,550 --> 00:40:06,260 should be reduced to one half of its value. 528 00:40:06,260 --> 00:40:08,780 And it's an interesting system to analyze. 529 00:40:08,780 --> 00:40:15,170 So Thomas precession is that relativistic correction 530 00:40:15,170 --> 00:40:18,540 to precession rates when the object that is precessing 531 00:40:18,540 --> 00:40:21,820 is in an accelerated frame. 532 00:40:21,820 --> 00:40:24,670 And any rotating frame is accelerated. 533 00:40:24,670 --> 00:40:31,200 So this result needs correction. 534 00:40:31,200 --> 00:40:36,450 OK, but let's take this result as it is-- instead of g, 535 00:40:36,450 --> 00:40:37,640 g minus 1. 536 00:40:37,640 --> 00:40:39,810 Let's not worry too much about it. 537 00:40:39,810 --> 00:40:44,686 And let's just estimate how big this effect is. 538 00:40:44,686 --> 00:40:50,470 It's the last thing I want to do as a way of motivating 539 00:40:50,470 --> 00:40:51,460 this subject. 540 00:40:51,460 --> 00:40:53,850 So delta H is this. 541 00:40:53,850 --> 00:40:55,770 Let's estimate it. 542 00:40:55,770 --> 00:41:03,090 Now for estimates, a couple of things are useful to remember, 543 00:41:03,090 --> 00:41:08,640 that Bohr radius is h squared over me squared. 544 00:41:08,640 --> 00:41:10,740 We did that last time. 545 00:41:10,740 --> 00:41:13,680 And there's this constant that is very useful, 546 00:41:13,680 --> 00:41:18,620 the fine structure constant, which is e squared over hc. 547 00:41:18,620 --> 00:41:21,885 And it's about 1 over 137. 548 00:41:21,885 --> 00:41:27,090 And it helps you estimate all kinds of things. 549 00:41:27,090 --> 00:41:31,580 Because it's a rather complicated number to evaluate, 550 00:41:31,580 --> 00:41:36,450 you need all kinds of units and things like that. 551 00:41:36,450 --> 00:41:43,180 So the charge of the electron divided by hc being 1 over 137 552 00:41:43,180 --> 00:41:46,680 is quite nice. 553 00:41:46,680 --> 00:41:53,050 So let's estimate delta H. Well, g 554 00:41:53,050 --> 00:41:58,160 we won't worry-- 2, 1, doesn't matter. 555 00:41:58,160 --> 00:42:07,530 e mc-- so far, that is kind of simple. 556 00:42:07,530 --> 00:42:13,036 Then we have S dot L. Well, how do I estimate S dot L? 557 00:42:13,036 --> 00:42:15,740 I don't do too much. 558 00:42:15,740 --> 00:42:18,850 S spin is multiples of h bar. 559 00:42:18,850 --> 00:42:23,960 L for an atomic state will be 1, 2, 3, so multiples of h bar. 560 00:42:23,960 --> 00:42:29,430 So h bar squared, that's it for S dot L. 561 00:42:29,430 --> 00:42:33,500 1 over r is 1 over r. 562 00:42:33,500 --> 00:42:37,570 dV dr is e squared over r squared. 563 00:42:37,570 --> 00:42:39,280 And that's it. 564 00:42:39,280 --> 00:42:43,760 But here, instead of r, I should put the typical length 565 00:42:43,760 --> 00:42:46,950 of the hydrogen atom, which is a0. 566 00:42:46,950 --> 00:42:48,210 So what do I get? 567 00:42:53,230 --> 00:42:57,300 I'm sorry, I made a mistake here. 568 00:42:57,300 --> 00:42:59,407 AUDIENCE: Yeah, it's up there. 569 00:42:59,407 --> 00:43:03,079 PROFESSOR: Oh, I made a mistake here 570 00:43:03,079 --> 00:43:09,910 in that I didn't put this factor, 1 over ecm. 571 00:43:09,910 --> 00:43:11,980 So the e cancels. 572 00:43:11,980 --> 00:43:15,920 And this is the result here-- g over 2m 573 00:43:15,920 --> 00:43:39,900 squared c squared S dot L 1 over r dV dr. So let me start again. 574 00:43:39,900 --> 00:43:49,680 1 over m squared c squared h bar squared 1 over r dV dr-- 575 00:43:49,680 --> 00:43:51,550 that much I got right. 576 00:43:51,550 --> 00:43:58,810 So this is roughly 1 over [INAUDIBLE] 577 00:43:58,810 --> 00:44:02,220 of the electron squared c squared e 578 00:44:02,220 --> 00:44:08,780 squared over a0 cubed h squared-- still quite 579 00:44:08,780 --> 00:44:12,610 messy, but not that terrible. 580 00:44:12,610 --> 00:44:16,700 So in order to get an idea of how big this is, 581 00:44:16,700 --> 00:44:19,720 the ground state energy of the hydrogen atom 582 00:44:19,720 --> 00:44:23,400 was e squared over 2a0. 583 00:44:23,400 --> 00:44:30,610 So let's divide delta H over the ground state energy. 584 00:44:30,610 --> 00:44:32,220 And that's how much? 585 00:44:32,220 --> 00:44:37,850 Well, we have all this quantity, 1 over m 586 00:44:37,850 --> 00:44:44,490 squared c squared e squared a0 cubed h squared. 587 00:44:44,490 --> 00:44:53,110 And now, we must divide by e squared over a0 like this. 588 00:44:59,160 --> 00:45:02,370 Well, the e squareds cancel. 589 00:45:02,370 --> 00:45:11,800 And we get h squared over m squared c squared a0 squared. 590 00:45:11,800 --> 00:45:13,950 You need to know what a0 is. 591 00:45:13,950 --> 00:45:16,640 Let's just boil it down to the simplest thing, 592 00:45:16,640 --> 00:45:20,760 so h squared m squared c squared. 593 00:45:20,760 --> 00:45:26,120 a0 squared would be h to the fourth m squared e 594 00:45:26,120 --> 00:45:28,540 to the fourth. 595 00:45:28,540 --> 00:45:33,840 So this is actually e to the fourth over h 596 00:45:33,840 --> 00:45:41,200 squared c squared, or e squared over hc squared, which is alpha 597 00:45:41,200 --> 00:45:41,700 squared. 598 00:45:41,700 --> 00:45:45,510 Whew-- lots of work to get something very nice. 599 00:45:48,030 --> 00:45:56,910 The ratio of the spin orbit coupling to the ground state 600 00:45:56,910 --> 00:46:00,850 energy is 1 over alpha squared. 601 00:46:00,850 --> 00:46:05,310 It's alpha squared, which is 1 over 137 squared. 602 00:46:05,310 --> 00:46:08,040 So it's a pretty small thing. 603 00:46:08,040 --> 00:46:14,250 It's about 1 over 19,000. 604 00:46:14,250 --> 00:46:19,850 So when this is called fine structure of the hydrogen atom, 605 00:46:19,850 --> 00:46:24,300 it means that it's in the level in your page 606 00:46:24,300 --> 00:46:30,040 that you use a few inches to plot the 13.6 electron 607 00:46:30,040 --> 00:46:35,450 volts-- well, you're talking about 20,000 times smaller, 608 00:46:35,450 --> 00:46:38,820 something that you don't see. 609 00:46:38,820 --> 00:46:41,410 But of course, it's a pretty important thing. 610 00:46:41,410 --> 00:46:48,960 So all in all, in the conventions of-- this 611 00:46:48,960 --> 00:46:53,900 is done in Gaussian units. 612 00:46:53,900 --> 00:47:01,140 In SI units, which is what Griffiths uses, 613 00:47:01,140 --> 00:47:08,560 delta H is e squared over 8 pi epsilon 0 1 over m 614 00:47:08,560 --> 00:47:17,450 squared c squared r cubed S dot L. That's for reference. 615 00:47:17,450 --> 00:47:18,275 This is Griffiths. 616 00:47:22,300 --> 00:47:24,310 But this is correct as well. 617 00:47:24,310 --> 00:47:27,440 This is the correct value. 618 00:47:27,440 --> 00:47:30,960 This is the correct value already taking 619 00:47:30,960 --> 00:47:33,340 into account the relativistic correction. 620 00:47:33,340 --> 00:47:37,270 So here, you're supposed to let g go to g minus 1. 621 00:47:37,270 --> 00:47:42,490 So you can put the 1 there, and it's pretty accurate. 622 00:47:42,490 --> 00:47:45,590 All right, so what is the physics question 623 00:47:45,590 --> 00:47:49,500 we want to answer with this spin orbit coupling? 624 00:47:49,500 --> 00:47:54,640 So here it comes. 625 00:47:54,640 --> 00:47:58,920 You have the hydrogen atom spectrum. 626 00:47:58,920 --> 00:48:01,460 And that spectrum you know. 627 00:48:01,460 --> 00:48:05,050 At L equals 0, you have one state here. 628 00:48:05,050 --> 00:48:09,890 Then, that's n equals 1, n equals 2. 629 00:48:12,470 --> 00:48:16,580 You have one state here and one state here at L equals 1. 630 00:48:16,580 --> 00:48:21,260 Then n equals 3, they start getting very close together. 631 00:48:21,260 --> 00:48:25,980 n equals 4 is like that. 632 00:48:25,980 --> 00:48:31,370 Let's consider if you want to have spin orbit coupling, 633 00:48:31,370 --> 00:48:35,880 we must have angular momentum. 634 00:48:35,880 --> 00:48:42,330 And that's L. And therefore, let's consider this state here. 635 00:48:42,330 --> 00:48:47,420 l equals 1, n equals 1-- n equals 2, I'm sorry. 636 00:48:51,410 --> 00:48:56,520 What happens to those states, is the question. 637 00:48:56,520 --> 00:48:59,110 First, how many states do you have there 638 00:48:59,110 --> 00:49:02,120 and how should you think of them? 639 00:49:02,120 --> 00:49:06,430 Well actually, we know that an l equals 1 corresponds 640 00:49:06,430 --> 00:49:07,970 to three states. 641 00:49:07,970 --> 00:49:13,820 So you'd have lm with l equals 1. 642 00:49:13,820 --> 00:49:18,310 And then m can be 1, 0, or minus 1. 643 00:49:18,310 --> 00:49:20,730 So you have three states. 644 00:49:20,730 --> 00:49:23,840 But there's not really three states. 645 00:49:23,840 --> 00:49:26,420 Because the electron can have spin. 646 00:49:26,420 --> 00:49:31,770 So here it is, a tensor product that appears in your face 647 00:49:31,770 --> 00:49:36,150 because there is more than angular 648 00:49:36,150 --> 00:49:37,430 momentum to the electron. 649 00:49:37,430 --> 00:49:38,630 There's spin. 650 00:49:38,630 --> 00:49:42,130 And it's a totally different vector space, the same particle 651 00:49:42,130 --> 00:49:45,780 but another vector space, the spin space. 652 00:49:45,780 --> 00:49:50,140 So here, you have the possible spins of the electron. 653 00:49:50,140 --> 00:49:53,130 So that's another angular momentum. 654 00:49:53,130 --> 00:49:57,525 And well, you could have the plus/minus states, for example. 655 00:50:00,730 --> 00:50:05,290 So you have three states here and two states here. 656 00:50:05,290 --> 00:50:16,440 So this is really six states, so six states 657 00:50:16,440 --> 00:50:20,700 whose fate we would like to understand 658 00:50:20,700 --> 00:50:23,355 due to this spin orbit coupling. 659 00:50:27,600 --> 00:50:35,610 So to use the language of angular momentum, 660 00:50:35,610 --> 00:50:37,740 instead of writing plus/minus, you 661 00:50:37,740 --> 00:50:49,990 could write Smz, if you will-- ms I will call, spin of s. 662 00:50:49,990 --> 00:50:59,780 You have here spin of 1/2 and states 1/2 or minus 1/2. 663 00:50:59,780 --> 00:51:01,040 This is the up. 664 00:51:01,040 --> 00:51:07,040 When the z component of the spin that we always call m-- m now 665 00:51:07,040 --> 00:51:09,760 corresponds to the z component of angular momentum. 666 00:51:09,760 --> 00:51:12,846 So in general, even for spin, we use m. 667 00:51:12,846 --> 00:51:16,410 And we have that our two spin states of the electron 668 00:51:16,410 --> 00:51:21,880 are spin 1/2 particle with plus spin in the z direction, 669 00:51:21,880 --> 00:51:26,550 spin 1/2 particle with minus spin in the z direction. 670 00:51:26,550 --> 00:51:29,380 We usually never put this 1/2 here. 671 00:51:29,380 --> 00:51:32,700 But now you have here really three states-- 672 00:51:32,700 --> 00:51:39,950 1, 1, 1, 0, 1, minus 1, the first telling you 673 00:51:39,950 --> 00:51:42,810 about the total angular momentum. 674 00:51:42,810 --> 00:51:45,480 Here, the total spin is 1/2. 675 00:51:45,480 --> 00:51:47,860 But it happens to be either up or down. 676 00:51:47,860 --> 00:51:50,550 Here, the total angular momentum is 1. 677 00:51:50,550 --> 00:51:56,010 But it happens to be plus 1, 0, or minus 1 here. 678 00:51:56,010 --> 00:51:58,830 So these are our six states. 679 00:51:58,830 --> 00:52:01,600 You can combine this with this, this with that, this with this, 680 00:52:01,600 --> 00:52:02,320 this with that. 681 00:52:02,320 --> 00:52:03,870 You make all the products. 682 00:52:03,870 --> 00:52:06,610 And these are the six states of the hydrogen 683 00:52:06,610 --> 00:52:08,090 atom at this level. 684 00:52:08,090 --> 00:52:13,100 And we wish to know what happens to them. 685 00:52:13,100 --> 00:52:17,910 Now, this correction is small. 686 00:52:17,910 --> 00:52:22,750 So it fits our understanding of the perturbation theory 687 00:52:22,750 --> 00:52:25,310 of Feynman-Hellman in which we try 688 00:52:25,310 --> 00:52:29,660 to find the corrections to these things. 689 00:52:29,660 --> 00:52:35,130 Our difficulty now is a little serious, however. 690 00:52:35,130 --> 00:52:37,290 It's the fact that Feynman-Hellman 691 00:52:37,290 --> 00:52:39,870 assumed that you had a state. 692 00:52:39,870 --> 00:52:43,600 And it was an eigenstate of the corrected Hamiltonian 693 00:52:43,600 --> 00:52:46,930 as you moved along. 694 00:52:46,930 --> 00:52:51,220 And then, you could compute how its energy changes. 695 00:52:51,220 --> 00:52:56,260 Here, unfortunately, we have a much more difficult situation. 696 00:52:56,260 --> 00:52:59,580 These six states that I'm not listing yet, 697 00:52:59,580 --> 00:53:07,170 but I will list very soon, are not obviously eigenstates 698 00:53:07,170 --> 00:53:12,340 of delta H. In fact, they are not eigenstates of delta H. 699 00:53:12,340 --> 00:53:15,790 They're degenerate states, six degenerate states, 700 00:53:15,790 --> 00:53:19,440 that are not eigenstates of delta H. Therefore, 701 00:53:19,440 --> 00:53:24,360 I cannot use the Feynman-Hellman theorem until I find what are 702 00:53:24,360 --> 00:53:28,499 the combinations that are eigenstates of this 703 00:53:28,499 --> 00:53:29,040 perturbation. 704 00:53:32,680 --> 00:53:35,380 So we are a little bit in trouble. 705 00:53:35,380 --> 00:53:43,010 Because we have a perturbation for which these product 706 00:53:43,010 --> 00:53:52,110 states-- we call them uncoupled bases-- are not eigenstates. 707 00:53:52,110 --> 00:53:56,850 Now, we've written this operator a little naively. 708 00:53:56,850 --> 00:54:01,270 What does this operator really mean, S dot L? 709 00:54:06,150 --> 00:54:15,030 In our tensor products, it means S1 tensor L1. 710 00:54:15,030 --> 00:54:19,180 Actually, I'll use L dot S. I'll always 711 00:54:19,180 --> 00:54:24,740 put the L information first and the S information afterward. 712 00:54:24,740 --> 00:54:29,850 So L dot S is clearly an operator that 713 00:54:29,850 --> 00:54:33,910 must be thought to act on the tensor product. 714 00:54:33,910 --> 00:54:35,830 Because both have to act. 715 00:54:35,830 --> 00:54:37,990 S has to act and L has to act. 716 00:54:37,990 --> 00:54:39,800 So it only lives in the tensor product. 717 00:54:39,800 --> 00:54:41,400 So what does it mean? 718 00:54:41,400 --> 00:54:55,960 It means this-- S2 L2 plus S3 L3, or sum over i Si tensor Li. 719 00:54:55,960 --> 00:54:59,860 So this is the kind of thing that you need to understand-- 720 00:54:59,860 --> 00:55:05,080 how do you find for this operator's eigenstates here? 721 00:55:07,940 --> 00:55:16,400 So that is our difficulty. 722 00:55:16,400 --> 00:55:19,910 And that's what we have to solve. 723 00:55:19,910 --> 00:55:23,770 We're going to solve it in the next half hour. 724 00:55:23,770 --> 00:55:31,140 So it's a complicated operator, L dot S. But on the other hand, 725 00:55:31,140 --> 00:55:34,780 we have to use our ideas that we've learned already 726 00:55:34,780 --> 00:55:40,090 about summing angular momenta. 727 00:55:40,090 --> 00:55:49,310 What if I define J to be L plus S, 728 00:55:49,310 --> 00:55:58,545 which really means L tensor 1 plus 1 tensor S? 729 00:56:03,940 --> 00:56:09,130 So this is what I really mean by this operator. 730 00:56:12,170 --> 00:56:17,440 J, as we've demonstrated, will be an angular momentum, 731 00:56:17,440 --> 00:56:20,510 because this satisfies the algebra of angular momentum 732 00:56:20,510 --> 00:56:23,820 and this satisfies the algebra of angular momentum. 733 00:56:23,820 --> 00:56:28,280 So this thing satisfies the algebra of angular momentum. 734 00:56:28,280 --> 00:56:32,310 And why do we look at that term? 735 00:56:32,310 --> 00:56:36,340 Because of the following reason. 736 00:56:36,340 --> 00:56:39,590 We can square it-- JiJi. 737 00:56:43,930 --> 00:56:48,870 Now we would have to square this thing. 738 00:56:48,870 --> 00:56:50,510 How do you square this thing? 739 00:56:50,510 --> 00:56:52,590 Well, there's two ways. 740 00:56:52,590 --> 00:56:56,060 Naively-- L squared plus L squared plus 2L 741 00:56:56,060 --> 00:56:59,285 dot S-- basically correct. 742 00:56:59,285 --> 00:57:01,750 But you can do it a little more slowly. 743 00:57:01,750 --> 00:57:07,120 If you square this term, you get L squared tensor 1. 744 00:57:07,120 --> 00:57:14,000 If you square this term, you get 1 tensor S squared. 745 00:57:14,000 --> 00:57:16,210 But when you do the mixed products, 746 00:57:16,210 --> 00:57:20,070 you just must take the i's here and the i's here 747 00:57:20,070 --> 00:57:21,250 and multiply them. 748 00:57:21,250 --> 00:57:28,530 So actually, you do get two i's, the sum over i Li tensor Si. 749 00:57:35,710 --> 00:57:37,320 This is sum over i. 750 00:57:37,320 --> 00:57:38,500 This is J squared. 751 00:57:42,500 --> 00:57:50,420 So basically, what I'm saying is that J squared naively 752 00:57:50,420 --> 00:57:56,100 is L squared plus S squared plus our interaction 753 00:57:56,100 --> 00:58:01,425 2L dot S defined property. 754 00:58:04,110 --> 00:58:16,900 So L dot S is equal to 1/2 of J squared minus L squared 755 00:58:16,900 --> 00:58:19,690 minus S squared. 756 00:58:19,690 --> 00:58:26,020 And that tells you all kinds of interesting things about L dot 757 00:58:26,020 --> 00:58:28,470 S. 758 00:58:28,470 --> 00:58:34,180 Basically, we can trade L dot S for J squared, 759 00:58:34,180 --> 00:58:36,490 L squared, and S squared. 760 00:58:36,490 --> 00:58:39,590 L squared is very simple, and S squared 761 00:58:39,590 --> 00:58:42,190 is extremely simple as well. 762 00:58:42,190 --> 00:58:46,520 Remember, L squared commutes with any Li. 763 00:58:46,520 --> 00:58:51,340 So L squared with any Li is equal to 0. 764 00:58:51,340 --> 00:58:55,740 S squared with any Si is equal to 0. 765 00:58:55,740 --> 00:58:58,180 And Li's and Si's commute. 766 00:58:58,180 --> 00:58:59,780 They live in different worlds. 767 00:58:59,780 --> 00:59:03,280 So L squared and Si's commute. 768 00:59:03,280 --> 00:59:06,720 S squareds and Li's commute. 769 00:59:06,720 --> 00:59:11,230 These things are pretty nice and simple. 770 00:59:11,230 --> 00:59:15,600 So let's think now of our Hamiltonian 771 00:59:15,600 --> 00:59:22,260 and what is happening to it. 772 00:59:22,260 --> 00:59:31,260 Whenever we had the hydrogen atom, 773 00:59:31,260 --> 00:59:39,410 we had a set of commuting observables H, L squared, 774 00:59:39,410 --> 00:59:40,120 and Lz. 775 00:59:44,390 --> 00:59:48,470 It's a complete set of commuting observables. 776 00:59:48,470 --> 00:59:53,020 Now, in the hydrogen atom, you could add to it 777 00:59:53,020 --> 00:59:57,620 S squared and Sz. 778 00:59:57,620 --> 01:00:00,160 We didn't talk about spin at the beginning, 779 01:00:00,160 --> 01:00:02,760 because we just considered a particle going 780 01:00:02,760 --> 01:00:04,520 around the hydrogen atom. 781 01:00:04,520 --> 01:00:09,330 But if you have spin, the hydrogen atom Hamiltonian, 782 01:00:09,330 --> 01:00:13,820 the original one, doesn't involve spin in any way. 783 01:00:13,820 --> 01:00:16,890 So certainly, Hamiltonian commutes with spin, 784 01:00:16,890 --> 01:00:18,740 with spin z. 785 01:00:18,740 --> 01:00:21,880 L and S don't talk, so this is the complete set 786 01:00:21,880 --> 01:00:24,870 of commuting observables. 787 01:00:24,870 --> 01:00:27,215 But what happens to this list? 788 01:00:27,215 --> 01:00:35,210 This is our problem for H0, the hydrogen atom, 789 01:00:35,210 --> 01:00:43,450 plus delta H that has the S dot L. 790 01:00:43,450 --> 01:00:49,100 Well, what are complete set of commuting observables? 791 01:00:49,100 --> 01:00:51,550 This is a very important question. 792 01:00:51,550 --> 01:00:53,560 Because this is what tells you how 793 01:00:53,560 --> 01:00:56,050 you're going to try to organize the spectrum. 794 01:00:56,050 --> 01:01:01,880 So we could have H, the total, H total. 795 01:01:05,800 --> 01:01:08,960 And what else? 796 01:01:08,960 --> 01:01:14,630 Well, can I still have L squared here? 797 01:01:18,450 --> 01:01:23,135 Can I include L squared and say it commutes with the total H? 798 01:01:27,229 --> 01:01:30,170 A little worrisome, but actually, 799 01:01:30,170 --> 01:01:35,500 you know that L squared commutes with the original Hamiltonian. 800 01:01:35,500 --> 01:01:38,260 Now, the question is whether L squared commutes 801 01:01:38,260 --> 01:01:40,320 with this extra piece. 802 01:01:40,320 --> 01:01:44,110 Well, but L squared commutes with any Li. 803 01:01:44,110 --> 01:01:47,730 And it doesn't even talk to S. So L squared is safe. 804 01:01:47,730 --> 01:01:50,400 L squared we can keep. 805 01:01:50,400 --> 01:01:55,660 OK, S squared-- can we keep S squared? 806 01:01:55,660 --> 01:01:57,480 Well, S squared was here. 807 01:01:57,480 --> 01:02:01,420 So it commuted with the Hamiltonian, and that was good. 808 01:02:01,420 --> 01:02:06,000 S squared commutes with any Si, and it doesn't talk to L. 809 01:02:06,000 --> 01:02:07,730 So S squared can stay. 810 01:02:11,040 --> 01:02:13,550 But that's not good enough. 811 01:02:13,550 --> 01:02:16,950 We won't be able to solve the problem with this still. 812 01:02:16,950 --> 01:02:17,790 We need more. 813 01:02:21,100 --> 01:02:22,166 How about Lz? 814 01:02:22,166 --> 01:02:24,885 It was here, so let's try our luck. 815 01:02:29,010 --> 01:02:32,340 Any opinions on Lz-- can we keep it or not? 816 01:02:37,840 --> 01:02:38,838 Yes. 817 01:02:38,838 --> 01:02:40,182 AUDIENCE: I don't think so. 818 01:02:40,182 --> 01:02:44,117 Because in the J term, we have Lx's and Ly's, which 819 01:02:44,117 --> 01:02:45,340 don't commute with Lz. 820 01:02:45,340 --> 01:02:47,980 PROFESSOR: Right, it can't be kept. 821 01:02:47,980 --> 01:02:57,540 Here, this term has SxLx plus SyLy plus SzLz. 822 01:02:57,540 --> 01:03:02,370 And Lz doesn't commute with this one. 823 01:03:02,370 --> 01:03:04,995 So no, you can't keep Lz-- no good. 824 01:03:09,680 --> 01:03:14,496 On the other hand, let's think about J squared. 825 01:03:17,640 --> 01:03:22,670 J squared is here. 826 01:03:22,670 --> 01:03:26,685 And J squared commutes with L squared and with S squared. 827 01:03:29,690 --> 01:03:36,600 J squared, therefore, is-- well, let me say it this way. 828 01:03:36,600 --> 01:03:42,460 Here is L dot S, which is our extra interaction. 829 01:03:42,460 --> 01:03:44,920 Here we have this thing. 830 01:03:44,920 --> 01:03:49,700 I would like to say on behalf of J squared 831 01:03:49,700 --> 01:03:56,200 that we can include it here, J squared, 832 01:03:56,200 --> 01:04:00,840 because J squared is really pretty much the same 833 01:04:00,840 --> 01:04:05,860 as L dot S up to this L squared and S squared. 834 01:04:05,860 --> 01:04:09,640 But J squared commutes with L squared and S squared. 835 01:04:09,640 --> 01:04:11,400 I should probably write it there. 836 01:04:15,120 --> 01:04:22,350 J squared commutes with L squared. 837 01:04:22,350 --> 01:04:29,010 And J squared communicates with S squared that we have here. 838 01:04:29,010 --> 01:04:38,300 And moreover, we have over here that J squared therefore 839 01:04:38,300 --> 01:04:41,960 will commute, or it's pretty much the same, 840 01:04:41,960 --> 01:04:45,600 as L dot S. J squared with L dot S would 841 01:04:45,600 --> 01:04:50,810 be J squared times this thing, which is 0. 842 01:04:50,810 --> 01:04:55,840 So J squared commutes with this term. 843 01:04:55,840 --> 01:04:59,210 And it commutes with the Hamiltonian, 844 01:04:59,210 --> 01:05:02,410 your original hydrogen Hamiltonian. 845 01:05:02,410 --> 01:05:06,240 So J squared can be added here. 846 01:05:09,670 --> 01:05:13,980 J square is a good operator to have. 847 01:05:13,980 --> 01:05:19,110 And now we can get one more kind of free from here. 848 01:05:19,110 --> 01:05:21,290 It's Jz. 849 01:05:21,290 --> 01:05:23,280 Z 850 01:05:23,280 --> 01:05:29,630 Because Jz commutes with J squared. 851 01:05:29,630 --> 01:05:32,140 Jz commutes with these things. 852 01:05:32,140 --> 01:05:38,090 And Jz, which is a symmetry of the original Hamiltonian, 853 01:05:38,090 --> 01:05:44,342 also commutes with our new interaction, the L dot S, 854 01:05:44,342 --> 01:05:47,770 which is proportional to J squared. 855 01:05:47,770 --> 01:05:53,340 So you have to go through this yourselves 856 01:05:53,340 --> 01:05:56,980 probably even a little more slowly than I've gone. 857 01:05:56,980 --> 01:05:59,660 Just check that everything that I'm 858 01:05:59,660 --> 01:06:02,480 saying about whatever commutes commutes. 859 01:06:02,480 --> 01:06:06,350 So for example, when I say that J squared commutes 860 01:06:06,350 --> 01:06:09,700 with L dot S, it's because I can put 861 01:06:09,700 --> 01:06:12,590 instead of L dot S all of this. 862 01:06:12,590 --> 01:06:15,920 And go slowly through this. 863 01:06:15,920 --> 01:06:22,030 So this is actually the complete set of committing observables. 864 01:06:22,030 --> 01:06:25,510 And it's basically saying to us, try 865 01:06:25,510 --> 01:06:31,755 to diagonalize this thing with total angular momentum. 866 01:06:34,590 --> 01:06:38,290 So it's about time to really do it. 867 01:06:38,290 --> 01:06:39,790 We haven't done it yet. 868 01:06:39,790 --> 01:06:43,930 But now the part that we have to do now, 869 01:06:43,930 --> 01:06:46,470 it's kind of a nice exercise. 870 01:06:46,470 --> 01:06:48,970 And it's fun. 871 01:06:48,970 --> 01:06:52,180 Now, there's one problem in the homework set 872 01:06:52,180 --> 01:06:55,710 that sort of uses this kind of thing. 873 01:06:55,710 --> 01:07:01,850 And I will suggest there to Will and Aram that tomorrow, 874 01:07:01,850 --> 01:07:07,120 they spend some time discussing it and helping you with it. 875 01:07:07,120 --> 01:07:10,930 The last problem in the homework set 876 01:07:10,930 --> 01:07:15,060 would've been better if you had a little more time for it 877 01:07:15,060 --> 01:07:17,930 and you had more time to digest what I'm doing today. 878 01:07:17,930 --> 01:07:21,920 But nevertheless, go to recitation, 879 01:07:21,920 --> 01:07:23,490 learn more about the problem. 880 01:07:23,490 --> 01:07:26,740 It will not be all that difficult. 881 01:07:26,740 --> 01:07:31,900 OK, so we're trying now to finally form 882 01:07:31,900 --> 01:07:33,890 another basis of states. 883 01:07:33,890 --> 01:07:36,970 We had these six states. 884 01:07:36,970 --> 01:07:39,020 And we're going to try to organize them 885 01:07:39,020 --> 01:07:45,480 in a better way-- as eigenstates of the total angular momentum L 886 01:07:45,480 --> 01:07:51,210 plus S. So I'm going to write them here in this way. 887 01:07:51,210 --> 01:07:58,830 Here is one of the states of this L equals 888 01:07:58,830 --> 01:08:04,330 1 electron, the 1, 1 coupled to the 1/2, 1/2. 889 01:08:04,330 --> 01:08:20,740 Here are two more states- 1, 0, 1/2, 1/2, 1, 1, 1/2, minus 1/2, 890 01:08:20,740 --> 01:08:27,850 so the 1, 0 with the top, the 1, 1 with the bottom. 891 01:08:27,850 --> 01:08:35,020 Here are two more states-- 1, 0 with 1/2, 892 01:08:35,020 --> 01:08:43,620 minus 1/2 and 1, minus 1 with 1/2, 1/2. 893 01:08:47,790 --> 01:08:56,100 And here is the last state-- 1, minus 1 with 1/2, minus 1. 894 01:09:02,520 --> 01:09:05,832 These are our six states. 895 01:09:05,832 --> 01:09:09,654 And I've organized them in a nice way actually. 896 01:09:12,760 --> 01:09:14,990 I've organized them in such a way 897 01:09:14,990 --> 01:09:21,750 that you can read what is the value of Jz over h bar. 898 01:09:21,750 --> 01:09:28,563 Remember, Jz is 1 over h bar Lz plus Sz. 899 01:09:34,080 --> 01:09:35,479 So what is it? 900 01:09:35,479 --> 01:09:39,760 These are, I claim, eigenstates of Jz. 901 01:09:39,760 --> 01:09:40,899 Why? 902 01:09:40,899 --> 01:09:42,569 Because let's act on them. 903 01:09:42,569 --> 01:09:45,510 Suppose I act with Jz on this state. 904 01:09:45,510 --> 01:09:47,930 The Lz comes here and says, 1. 905 01:09:47,930 --> 01:09:50,890 The Sz comes here and says, 1/2. 906 01:09:50,890 --> 01:09:56,337 So the sum of them give you Jz over h bar equal to 3/2. 907 01:10:01,080 --> 01:10:05,240 And that's why I organized these states in such a way 908 01:10:05,240 --> 01:10:10,510 that these second things add up to the same value-- 0 and 1/2, 909 01:10:10,510 --> 01:10:12,590 1 and minus 1/2. 910 01:10:12,590 --> 01:10:16,360 So if you act with Jz on this state, 911 01:10:16,360 --> 01:10:19,330 it's an eigenstate with Jz. 912 01:10:19,330 --> 01:10:22,470 Here, 0 contribution, here 1/2. 913 01:10:22,470 --> 01:10:26,700 So this is with plus 1/2. 914 01:10:26,700 --> 01:10:31,130 Here, you have 0 and minus 1/2, minus 1, and that is minus 1/2. 915 01:10:34,330 --> 01:10:36,930 And here you have minus 3/2. 916 01:10:40,990 --> 01:10:43,943 OK, questions. 917 01:10:48,980 --> 01:10:50,330 We've written the states. 918 01:10:50,330 --> 01:10:55,240 And I'm evaluating the total z component of angular momentum. 919 01:10:55,240 --> 01:10:58,470 And these two states are like that. 920 01:10:58,470 --> 01:11:02,150 So what does our theorem guarantee for us? 921 01:11:02,150 --> 01:11:06,670 Our theorem guarantees that we have-- in this tensor product, 922 01:11:06,670 --> 01:11:10,770 there is an algebra of angular momentum of the Jz operators. 923 01:11:10,770 --> 01:11:14,270 And the states have to fall into representations 924 01:11:14,270 --> 01:11:15,930 of those operators. 925 01:11:15,930 --> 01:11:19,920 So you must have angular momentum multiplets. 926 01:11:19,920 --> 01:11:23,710 So at this moment, you can figure out 927 01:11:23,710 --> 01:11:30,240 what angular momentum you're going to get for the result. 928 01:11:30,240 --> 01:11:36,770 Here we obtained a maximum Jz of 3/2. 929 01:11:36,770 --> 01:11:42,005 So we must get a J equals 3/2 multiplet. 930 01:11:46,100 --> 01:11:48,660 Because a J equaling 3/2 multiplets 931 01:11:48,660 --> 01:11:55,650 has Jz 3/2, 1/2, minus 1/2, and 0. 932 01:11:55,650 --> 01:12:01,510 So actually, this state must be the top state of the multiplet. 933 01:12:01,510 --> 01:12:05,500 This state must be the bottom state of the multiplet. 934 01:12:05,500 --> 01:12:10,160 I don't know which one is the middle state of the multiplet 935 01:12:10,160 --> 01:12:12,250 and which one is here. 936 01:12:12,250 --> 01:12:15,670 But we have four states here, four states. 937 01:12:18,560 --> 01:12:22,940 So one linear combination of these two states 938 01:12:22,940 --> 01:12:29,190 must be, then, that Jz equals 1/2 state of the multiplet. 939 01:12:29,190 --> 01:12:31,600 And one inner combination of these two states 940 01:12:31,600 --> 01:12:35,220 must be that Jz equals minus 1/2 state of the multiplet. 941 01:12:35,220 --> 01:12:36,750 Which one is it? 942 01:12:36,750 --> 01:12:38,180 I don't know. 943 01:12:38,180 --> 01:12:40,010 But we can figure it out. 944 01:12:40,010 --> 01:12:41,930 We'll figure it out in a second. 945 01:12:41,930 --> 01:12:44,640 Once you get this J 3/2 multiplet, 946 01:12:44,640 --> 01:12:48,270 there will be one linear combination here left over 947 01:12:48,270 --> 01:12:51,500 and one linear combination here left over. 948 01:12:51,500 --> 01:12:56,300 Those are two state, one with Jz plus 1/2 and one with Jz 949 01:12:56,300 --> 01:12:58,060 equals minus 1/2. 950 01:12:58,060 --> 01:13:01,740 So you also get a J equals 1/2 multiplet. 951 01:13:07,670 --> 01:13:11,290 So the whole tensor product of six 952 01:13:11,290 --> 01:13:18,260 states-- it was the tensor product of a spin 1 953 01:13:18,260 --> 01:13:21,620 with a spin 1/2. 954 01:13:21,620 --> 01:13:24,560 So we write it like this. 955 01:13:24,560 --> 01:13:32,940 The tensor product of a spin 1 with a spin 1/2 956 01:13:32,940 --> 01:13:43,660 will give you a total spin 3/2 plus total spin 957 01:13:43,660 --> 01:13:52,940 1/2-- funny formula. 958 01:13:52,940 --> 01:13:56,820 Here is the tensor product, the tensor 959 01:13:56,820 --> 01:14:01,710 product of these three states with these two states. 960 01:14:01,710 --> 01:14:08,720 This can be written as 3 times 2 is equal to 4 plus 2 961 01:14:08,720 --> 01:14:11,870 in terms of number of states. 962 01:14:11,870 --> 01:14:16,020 The tensor product of this spin 1 and spin 1/2 963 01:14:16,020 --> 01:14:21,040 gives you a spin 3/2 multiplet with four states 964 01:14:21,040 --> 01:14:23,520 and a spin 1/2 multiplet with two states. 965 01:14:27,700 --> 01:14:33,040 So how do you calculate what are the states themselves? 966 01:14:35,890 --> 01:14:38,380 So the states themselves are the following. 967 01:14:47,100 --> 01:14:51,280 All right, here I have them. 968 01:14:51,280 --> 01:14:57,560 I claim that the J equals 3/2 states, 969 01:14:57,560 --> 01:15:02,610 m equals 3/2 states, the top state of that multiplet 970 01:15:02,610 --> 01:15:11,860 can only be the state here, the 1, 1 tensor 1/2, 1/2. 971 01:15:11,860 --> 01:15:18,490 And there's no way any other state can be put on the right. 972 01:15:18,490 --> 01:15:22,020 Because there's no other state with total z component 973 01:15:22,020 --> 01:15:24,160 of angular momentum equals 3/2. 974 01:15:24,160 --> 01:15:26,490 So that must be the state. 975 01:15:26,490 --> 01:15:33,000 Similarly, the J equals 3/2, m equals 976 01:15:33,000 --> 01:15:40,980 minus 3/2 state must be the bottom one-- 1, minus 1, 1/2, 977 01:15:40,980 --> 01:15:44,220 minus 1/2. 978 01:15:44,220 --> 01:15:46,490 The one that we wish to figure out 979 01:15:46,490 --> 01:15:53,710 is the next state here, which is the J equals 3/2, m equals 1/2. 980 01:15:53,710 --> 01:15:57,200 It's a linear combination of these two. 981 01:15:57,200 --> 01:15:58,390 But which one? 982 01:16:01,060 --> 01:16:05,250 That is kind of the last thing we want to do. 983 01:16:05,250 --> 01:16:08,520 Because it will pretty much solve the rest of the problem. 984 01:16:13,880 --> 01:16:17,540 So how do we solve for this? 985 01:16:17,540 --> 01:16:25,270 Well, we had this basic relation that we 986 01:16:25,270 --> 01:16:36,470 know how to lower or raise states of angular momentum-- 987 01:16:36,470 --> 01:16:43,530 m times m plus/minus 1 J-- I should have written it 988 01:16:43,530 --> 01:16:52,510 J plus/minus Jm equals h bar square root. 989 01:16:52,510 --> 01:16:56,790 More space for everybody to see this-- J times 990 01:16:56,790 --> 01:17:02,180 J plus 1 minus m times m plus/minus 1. 991 01:17:02,180 --> 01:17:08,120 Close the square root-- Jm plus/minus 1. 992 01:17:08,120 --> 01:17:14,020 So what I should try to do is lower this state, 993 01:17:14,020 --> 01:17:19,630 try to find this state by acting with J minus. 994 01:17:19,630 --> 01:17:22,660 So let me try to lower the state, so 995 01:17:22,660 --> 01:17:31,170 J minus on this state, on J equals 3/2, m equals 3/2. 996 01:17:31,170 --> 01:17:39,770 I can go to that formula and write it as h bar square root. 997 01:17:39,770 --> 01:17:47,260 J is 3/2, so 3/2 times 5/2 minus m, which is 3/2, 998 01:17:47,260 --> 01:17:49,960 times m minus 1, 1/2. 999 01:17:49,960 --> 01:17:57,570 We're doing the minus-- times the state 3/2, 1/2. 1000 01:17:57,570 --> 01:18:00,030 So the state we want is here. 1001 01:18:00,030 --> 01:18:03,570 And it's obtained by doing J minus on that. 1002 01:18:03,570 --> 01:18:05,320 But we want the number here. 1003 01:18:05,320 --> 01:18:08,340 So that's why I did all these square roots. 1004 01:18:08,340 --> 01:18:16,650 And that just gives h bar square root of 3, 3/2, 1/2. 1005 01:18:16,650 --> 01:18:19,670 Well, that still doesn't calculate it for me. 1006 01:18:19,670 --> 01:18:21,510 But it comes very close. 1007 01:18:29,490 --> 01:18:33,250 So you have it there. 1008 01:18:33,250 --> 01:18:39,580 Now I want to do this but using the right hand side. 1009 01:18:39,580 --> 01:18:41,560 So look at the right hand side. 1010 01:18:41,560 --> 01:18:52,570 We want to do J minus, but on 1, 1 tensor 1/2, 1/2. 1011 01:18:52,570 --> 01:18:55,820 So I applied J minus to the left hand side. 1012 01:18:55,820 --> 01:19:01,130 Now we have to apply J minus to the right hand side. 1013 01:19:01,130 --> 01:19:19,680 But J minus is L minus plus S minus on 1, 1 tensor 1/2, 1/2. 1014 01:19:19,680 --> 01:19:22,750 When this acts, it acts on the first. 1015 01:19:22,750 --> 01:19:31,570 So you get L minus on 1, 1 tensor 1/2, 1/2. 1016 01:19:31,570 --> 01:19:38,280 And in the second term, you get plus 1, 1 tensor S 1017 01:19:38,280 --> 01:19:40,520 minus on 1/2, 1/2. 1018 01:19:44,420 --> 01:19:47,540 Now, what is L minus on 1, 1? 1019 01:19:47,540 --> 01:19:50,330 You can use the same formula. 1020 01:19:50,330 --> 01:19:51,450 It's 1, 1. 1021 01:19:51,450 --> 01:19:53,250 And it's an angular momentum. 1022 01:19:53,250 --> 01:20:00,350 So it just goes on and gives you h bar square 1023 01:20:00,350 --> 01:20:05,520 root of 1 times 2 minus 1 times 0. 1024 01:20:05,520 --> 01:20:10,720 1, 0-- it lowers it-- times 1/2, 1/2. 1025 01:20:15,080 --> 01:20:19,270 Let me go here-- plus 1, 1. 1026 01:20:19,270 --> 01:20:21,980 And what is S minus on this? 1027 01:20:21,980 --> 01:20:26,390 Use the formula with J equals 1/2. 1028 01:20:26,390 --> 01:20:37,460 So this is h bar square root of 1/2 times 3/2 minus 1/2 times 1029 01:20:37,460 --> 01:20:43,522 minus 1/2 times 1/2 minus 1/2. 1030 01:20:43,522 --> 01:20:48,620 Whew-- well not too difficult. 1031 01:20:48,620 --> 01:20:57,630 But this gives you h over square root of 2, 1, 0 tensor 1/2, 1032 01:20:57,630 --> 01:21:01,440 1/2 plus just h bar. 1033 01:21:01,440 --> 01:21:11,030 This whole thing is 1-- 1, 1 tensor 1/2, minus 1/2. 1034 01:21:11,030 --> 01:21:16,800 OK, stop a second to see what's happened. 1035 01:21:16,800 --> 01:21:18,730 We had this equality. 1036 01:21:18,730 --> 01:21:20,260 And we acted with J minus. 1037 01:21:20,260 --> 01:21:23,390 Acting on the left, it gives us a number 1038 01:21:23,390 --> 01:21:25,980 times the state we want. 1039 01:21:25,980 --> 01:21:29,950 Acting on the right, we got this. 1040 01:21:29,950 --> 01:21:36,220 So actually, equating this to that, or left hand side 1041 01:21:36,220 --> 01:21:41,270 to right hand side, we finally found the state 3/2, 1/2. 1042 01:21:41,270 --> 01:21:49,630 So the state 3/2, 1/2 is as follows. 1043 01:21:54,410 --> 01:22:06,930 3/2, 1/2 is-- you must divide by that square root. 1044 01:22:06,930 --> 01:22:09,435 So you get the square root of 3 down. 1045 01:22:09,435 --> 01:22:12,280 The h bars cancel. 1046 01:22:12,280 --> 01:22:18,290 So here it is, a very nice little formula-- 2 over 3, 1047 01:22:18,290 --> 01:22:26,450 1, 0 tensor 1/2, 1/2 plus 1 over square root of 3, 1048 01:22:26,450 --> 01:22:34,010 1, 1 tensor 1/2, minus 1/2. 1049 01:22:34,010 --> 01:22:36,730 So we have the top state of the multiplet. 1050 01:22:40,070 --> 01:22:43,910 We have the next state of the multiplet. 1051 01:22:43,910 --> 01:22:47,240 We have-- I'm sorry, the top state of the multiplet 1052 01:22:47,240 --> 01:22:49,250 was this. 1053 01:22:49,250 --> 01:22:51,830 You have the bottom state of the multiplet, 1054 01:22:51,830 --> 01:22:53,890 the middle state of the multiplet. 1055 01:22:53,890 --> 01:22:58,920 What you're missing is the bottom and the middle term. 1056 01:22:58,920 --> 01:23:02,935 And this one can be obtained in many ways. 1057 01:23:05,660 --> 01:23:08,990 One way would be to raise this state. 1058 01:23:08,990 --> 01:23:12,330 The minus 3/2 could be raised by one unit 1059 01:23:12,330 --> 01:23:14,710 and do exactly the same thing. 1060 01:23:14,710 --> 01:23:24,760 Well, the result is square root of 2 over 3, 1, 0 tensor 1/2, 1061 01:23:24,760 --> 01:23:30,550 minus 1/2 plus 1 over square root of 3. 1062 01:23:30,550 --> 01:23:35,240 That square root of 2 doesn't look right to me now. 1063 01:23:35,240 --> 01:23:37,830 I must have copied it wrong. 1064 01:23:37,830 --> 01:23:43,940 It's 1 over square root of 3-- 1 over square root of 3, 1065 01:23:43,940 --> 01:23:50,660 1, minus 1 tensor 1/2, 1/2. 1066 01:23:50,660 --> 01:23:52,365 So you've built that whole multiplet. 1067 01:23:55,420 --> 01:24:00,480 And this state, as we said, was a linear combination 1068 01:24:00,480 --> 01:24:01,996 of the two possible states. 1069 01:24:05,800 --> 01:24:09,430 This 3 minus 1/2 was a linear combination 1070 01:24:09,430 --> 01:24:11,530 of these two possible states. 1071 01:24:11,530 --> 01:24:14,690 So the other states that are left over, 1072 01:24:14,690 --> 01:24:17,785 the other linear combinations, form 1073 01:24:17,785 --> 01:24:21,090 the J equals 1/2 multiplet. 1074 01:24:21,090 --> 01:24:26,050 So basically, every state must be orthogonal to each other. 1075 01:24:26,050 --> 01:24:35,200 So the other state, the 1/2, 1/2 and the 1/2, minus 1/2 of the J 1076 01:24:35,200 --> 01:24:41,430 equals 1/2 multiplet must be this orthogonal to this. 1077 01:24:41,430 --> 01:24:44,810 And this must be orthogonal to that. 1078 01:24:44,810 --> 01:24:50,760 So those formulas are easily found by orthogonality. 1079 01:24:50,760 --> 01:24:56,760 So I'll conclude by writing them-- minus 1 1080 01:24:56,760 --> 01:25:04,010 over square root of 3, 1, 0, 1/2, 1/2 1081 01:25:04,010 --> 01:25:13,170 plus the square root of 2 over 3, 1, 1, 1/2, minus 1/2. 1082 01:25:13,170 --> 01:25:19,770 And here, you get 1 over square root of 3, 1, 0, 1/2, 1083 01:25:19,770 --> 01:25:31,340 minus 1/2 minus 2 over square root of 3, 1, minus 1 1084 01:25:31,340 --> 01:25:33,380 tensor 1/2, 1/2. 1085 01:25:36,350 --> 01:25:45,870 So lots of terms, a little hard to read-- I apologize. 1086 01:25:45,870 --> 01:25:53,120 Now, the punchline here is that you've found these states. 1087 01:25:53,120 --> 01:25:57,050 And the claim is that these are states 1088 01:25:57,050 --> 01:26:00,610 in which L dot S is diagonal. 1089 01:26:00,610 --> 01:26:04,310 And it's kind of obvious that that should be the case. 1090 01:26:04,310 --> 01:26:07,830 Because what was L dot S? 1091 01:26:07,830 --> 01:26:20,150 So one last formula-- L dot S equals 1092 01:26:20,150 --> 01:26:26,220 1/2 of J squared minus L squared minus S squared. 1093 01:26:26,220 --> 01:26:29,770 Now, in terms of eigenvalues, this 1094 01:26:29,770 --> 01:26:36,930 is 1/2 h squared J times J plus 1 minus L times L 1095 01:26:36,930 --> 01:26:42,440 plus 1 minus S times S plus 1. 1096 01:26:42,440 --> 01:26:45,160 Now, all the states that we built 1097 01:26:45,160 --> 01:26:48,070 have definite values of J squared, 1098 01:26:48,070 --> 01:26:50,450 definite values of S squared. 1099 01:26:50,450 --> 01:26:52,740 Because L was 1. 1100 01:26:52,740 --> 01:26:55,270 And S is 1/2. 1101 01:26:55,270 --> 01:27:03,400 So here you go h squared over 2 J times J plus 1 minus 1 times 1102 01:27:03,400 --> 01:27:09,450 2 is 2 minus 1/2 times 3/2 is 3/4. 1103 01:27:12,050 --> 01:27:13,630 And that's the whole story. 1104 01:27:13,630 --> 01:27:17,170 The whole story in a sense has been summarized by this. 1105 01:27:17,170 --> 01:27:22,210 We have four states with J equals 3/2 1106 01:27:22,210 --> 01:27:25,850 and two states with J equals 1/2. 1107 01:27:25,850 --> 01:27:31,190 So these six states that you have here-- 1108 01:27:31,190 --> 01:27:33,690 split because of this interaction 1109 01:27:33,690 --> 01:27:42,700 into four states that have J equal to 3/2 and two 1110 01:27:42,700 --> 01:27:47,370 states that have J equal to 1/2. 1111 01:27:47,370 --> 01:27:49,640 And you plug the numbers here. 1112 01:27:49,640 --> 01:27:51,456 And that gives you the amount of splitting. 1113 01:27:54,790 --> 01:28:02,810 So actually, this height that this goes up here 1114 01:28:02,810 --> 01:28:05,230 is h squared over 2. 1115 01:28:05,230 --> 01:28:08,030 And this is minus h squared by the time you 1116 01:28:08,030 --> 01:28:11,860 put the numbers J, 3/2, and 1/2. 1117 01:28:11,860 --> 01:28:17,220 So all our work was because the Hamiltonian at the end 1118 01:28:17,220 --> 01:28:20,360 was simple in J squared. 1119 01:28:20,360 --> 01:28:23,240 And therefore, we needed J multiplets. 1120 01:28:23,240 --> 01:28:27,900 J multiplets are the addition of angular momentum multiplets. 1121 01:28:27,900 --> 01:28:31,270 In a sense, we don't have to construct these things 1122 01:28:31,270 --> 01:28:34,910 if you don't want to calculate very explicit details. 1123 01:28:34,910 --> 01:28:37,960 Once you have that, you have everything. 1124 01:28:37,960 --> 01:28:42,620 This product of angular momentum 1, angular momentum 1/2 1125 01:28:42,620 --> 01:28:45,800 gave you total angular momentum 3/2 1126 01:28:45,800 --> 01:28:48,990 and 1/2-- four states, two states. 1127 01:28:48,990 --> 01:28:53,380 So four states split one way, two states split the other way, 1128 01:28:53,380 --> 01:28:55,210 and that's the end of the story. 1129 01:28:55,210 --> 01:28:58,340 So more of this in recitation and more of this all 1130 01:28:58,340 --> 01:29:00,580 of next week. 1131 01:29:00,580 --> 01:29:02,790 We'll see you then.