1 00:00:01,080 --> 00:00:03,510 PROFESSOR: All right, so our next thing will be 2 00:00:03,510 --> 00:00:05,500 getting into the electron spin. 3 00:00:05,500 --> 00:00:09,720 And I want to discuss of two things, a little bit 4 00:00:09,720 --> 00:00:12,280 of the Dirac equation and the Pauli 5 00:00:12,280 --> 00:00:15,150 equation for the electron. 6 00:00:15,150 --> 00:00:18,660 We need to understand electron a little better 7 00:00:18,660 --> 00:00:22,770 and understand the perturbations, 8 00:00:22,770 --> 00:00:25,560 the relativistic corrections so we'll 9 00:00:25,560 --> 00:00:29,120 consider the Pauli equation. 10 00:00:33,280 --> 00:00:38,280 So the Pauli equation is what we have to do now. 11 00:00:38,280 --> 00:00:41,490 So what is the Pauli equation? 12 00:00:41,490 --> 00:00:44,910 It is the first attempt there was 13 00:00:44,910 --> 00:00:48,450 to figure out how the Schrodinger equation should 14 00:00:48,450 --> 00:00:51,580 be tailored for an electron. 15 00:00:51,580 --> 00:00:55,500 So we can also think of the Pauli equation 16 00:00:55,500 --> 00:00:59,010 as a baby version of the Dirac equation, which 17 00:00:59,010 --> 00:01:02,230 is the complete equation for the electron. 18 00:01:02,230 --> 00:01:04,800 So it all begins at the motivations 19 00:01:04,800 --> 00:01:19,630 by considering what is the magnetic moment of an electron? 20 00:01:25,840 --> 00:01:28,570 So it's a question that you've been addressing 21 00:01:28,570 --> 00:01:33,650 in this course for a while, from Stern-Gerlach experiment 22 00:01:33,650 --> 00:01:37,270 and how magnetic fields interact with electron. 23 00:01:37,270 --> 00:01:49,080 So classically and in Gaussian units, 24 00:01:49,080 --> 00:01:55,505 if you have a current loop and an area a, 25 00:01:55,505 --> 00:02:04,460 the magnetic moment is equal to i times the area 26 00:02:04,460 --> 00:02:08,410 vector divided by c. 27 00:02:11,690 --> 00:02:17,275 You can use that to imagine an object that is rotating. 28 00:02:17,275 --> 00:02:22,190 And as the object rotates, if its charged, 29 00:02:22,190 --> 00:02:26,000 it generates a magnetic moment, and the magnetic moment 30 00:02:26,000 --> 00:02:28,740 depends on the amount of rotation of the orbit. 31 00:02:28,740 --> 00:02:32,060 So it turns out that, with a little classical argument, 32 00:02:32,060 --> 00:02:37,010 you can derive that the magnetic moment is related 33 00:02:37,010 --> 00:02:48,150 to the angular momentum by a relation of the form L, 34 00:02:48,150 --> 00:02:52,320 so the charge of the object, the mass of the object c, 35 00:02:52,320 --> 00:02:53,730 and the angular momentum. 36 00:02:53,730 --> 00:02:56,730 And that's perfectly correct classically. 37 00:02:56,730 --> 00:02:59,850 So people thought that quantum mechanically 38 00:02:59,850 --> 00:03:07,680 they expect that for an electron or for an elementary particle, 39 00:03:07,680 --> 00:03:15,280 you would get q times 2 mc times the spin of the particle. 40 00:03:15,280 --> 00:03:20,070 So this would be the intrinsic magnetic moment of an electron 41 00:03:20,070 --> 00:03:29,140 that it seems to have times a fudge factor, g 42 00:03:29,140 --> 00:03:30,610 that you would have to measure. 43 00:03:30,610 --> 00:03:33,640 Because after all, the world is not classical. 44 00:03:33,640 --> 00:03:38,050 There's no reason why a classical relation like that 45 00:03:38,050 --> 00:03:42,610 would give you the right exact value of the magnetic moment 46 00:03:42,610 --> 00:03:44,200 of the electron. 47 00:03:44,200 --> 00:03:50,170 As it turns out, this g happens to be equal to 2. 48 00:03:56,980 --> 00:03:58,730 And we get the following. 49 00:03:58,730 --> 00:04:08,380 So for electrons g happens to be equal to 2. 50 00:04:08,380 --> 00:04:14,820 So mu is equal to 2 times minus e-- the charge of the electron 51 00:04:14,820 --> 00:04:19,100 is minus 2 mc, and the spin of the electron 52 00:04:19,100 --> 00:04:25,100 is h bar over 2 times sigma, the Pauli matrices. 53 00:04:25,100 --> 00:04:39,750 So this thing is minus eh bar over 2 mc sigma 54 00:04:39,750 --> 00:04:40,570 mu of the electron. 55 00:04:44,430 --> 00:04:54,620 So if you have a magnetic field, a B external, 56 00:04:54,620 --> 00:04:59,000 the Hamiltonian that includes the energetics 57 00:04:59,000 --> 00:05:00,980 of the magnetic field interacting 58 00:05:00,980 --> 00:05:08,320 with the dipole moment, it's always minus mu dot b. 59 00:05:08,320 --> 00:05:19,110 So in this case it would be eh bar over 2 mc sigma dot b. 60 00:05:19,110 --> 00:05:21,900 That's the Hamiltonian for an electron 61 00:05:21,900 --> 00:05:26,550 in a magnetic field, something you've used many, many times. 62 00:05:26,550 --> 00:05:31,530 The most unpleasant part of this thing is this g equal 2. 63 00:05:31,530 --> 00:05:34,860 Why did it turn out to be g equal 2? 64 00:05:34,860 --> 00:05:37,920 People knew it was g equal 2. 65 00:05:37,920 --> 00:05:39,600 But why? 66 00:05:39,600 --> 00:05:43,230 And here we're going to see the beginning of an explanation. 67 00:05:43,230 --> 00:05:45,900 In fact, it comes very close to an explanation of that 68 00:05:45,900 --> 00:05:49,290 through the Pauli equation. 69 00:05:49,290 --> 00:05:53,190 The Pauli equation is a nice equation you can write, 70 00:05:53,190 --> 00:05:55,260 and you can motivate. 71 00:05:55,260 --> 00:05:56,850 And it predicts. 72 00:05:56,850 --> 00:05:59,790 Once you admit the Pauli equation, 73 00:05:59,790 --> 00:06:02,350 it predicts that g would be equal to 2. 74 00:06:04,990 --> 00:06:08,320 And that, of course, happens also for the Dirac equation. 75 00:06:08,320 --> 00:06:12,340 And Dirac noticed that, and he was very happy about it. 76 00:06:15,130 --> 00:06:21,190 So it's quite remarkable how these things showed up, 77 00:06:21,190 --> 00:06:27,450 and the G equal to 2 was perfectly natural. 78 00:06:27,450 --> 00:06:33,690 So let's see this Pauli equation first. 79 00:06:33,690 --> 00:06:34,430 What is it? 80 00:06:38,870 --> 00:06:43,600 Well, when you do the Schrodinger equation for a wave 81 00:06:43,600 --> 00:06:47,260 function, you say p squared over 2m-- 82 00:06:47,260 --> 00:06:50,850 say a free particle, and you would 83 00:06:50,850 --> 00:06:57,170 put the wave function is equal to energies times the wave 84 00:06:57,170 --> 00:06:59,510 function. 85 00:06:59,510 --> 00:07:03,440 h on the wave function is equal to energy terms the wave 86 00:07:03,440 --> 00:07:04,820 function. 87 00:07:04,820 --> 00:07:11,000 So for this electron, you already 88 00:07:11,000 --> 00:07:14,990 know it's a two-dimensional Hilbert space, the state 89 00:07:14,990 --> 00:07:16,610 space of the up and down. 90 00:07:16,610 --> 00:07:21,230 So it's convenient to change this and to say, you know what? 91 00:07:21,230 --> 00:07:23,450 I'm going to put here something I'll call a spinor. 92 00:07:26,130 --> 00:07:29,110 Chi is a Pauli. 93 00:07:29,110 --> 00:07:35,950 It has two things, a chi 1 and a chi 2, and it's a Pauli spinor. 94 00:07:40,790 --> 00:07:44,250 And this is reasonable so far. 95 00:07:44,250 --> 00:07:50,470 The spin is defined by 2 degrees of freedom, 2 basis is vectors. 96 00:07:50,470 --> 00:07:53,630 So you've done wave functions for spin. 97 00:07:53,630 --> 00:07:56,900 And the wave functions for spin have these things, 98 00:07:56,900 --> 00:08:00,680 and each themself can be a function of position. 99 00:08:00,680 --> 00:08:03,740 So that's perfectly reasonable. 100 00:08:03,740 --> 00:08:11,270 I don't think anyone of you is very impressed by this so far. 101 00:08:11,270 --> 00:08:15,380 But here comes the funny thing. 102 00:08:15,380 --> 00:08:24,440 In a sense here, there is a 2 by 2 identity matrix sitting here. 103 00:08:24,440 --> 00:08:30,290 So Pauli observes the following identity. 104 00:08:30,290 --> 00:08:37,559 If you have sigma dot 8 and sigma dot b, 105 00:08:37,559 --> 00:08:50,910 it's equal to a dot b times 1 plus i sigma plot a cross b. 106 00:08:50,910 --> 00:08:55,330 Look, these are two vectors, a and b. 107 00:08:55,330 --> 00:08:59,520 And if you multiply half this product-- first, 108 00:08:59,520 --> 00:09:00,900 this is a dot product. 109 00:09:00,900 --> 00:09:06,450 It means sigma 1 times a1 sigma 2 times a2 sigma 3 times a3. 110 00:09:06,450 --> 00:09:08,010 So this is a dot product. 111 00:09:08,010 --> 00:09:10,770 This product here is a matrix product 112 00:09:10,770 --> 00:09:12,930 because this is already a matrix, 113 00:09:12,930 --> 00:09:16,590 this is already a matrix, so this is a matrix product. 114 00:09:16,590 --> 00:09:18,630 And this is the result. This comes 115 00:09:18,630 --> 00:09:22,740 from properties of the Pauli matrices you already know. 116 00:09:22,740 --> 00:09:26,070 In particular, their commutator determines 117 00:09:26,070 --> 00:09:29,140 this piece and the anticommutator 118 00:09:29,140 --> 00:09:30,590 determines this piece. 119 00:09:30,590 --> 00:09:34,960 So here, if you have, for example, 120 00:09:34,960 --> 00:09:40,900 sigma dot p times sigma dot p-- 121 00:09:40,900 --> 00:09:44,620 see, these two vectors are the same in this case-- 122 00:09:44,620 --> 00:09:51,380 you'll get just p squared times 1 plus 0. 123 00:09:51,380 --> 00:09:57,110 Because p times P, those are two vector operators if you wish, 124 00:09:57,110 --> 00:10:00,310 but still they commute. 125 00:10:00,310 --> 00:10:04,240 So this is equal to 0. 126 00:10:04,240 --> 00:10:08,920 So your p squared can be replaced by sigma dot p sigma 127 00:10:08,920 --> 00:10:09,580 dot p. 128 00:10:09,580 --> 00:10:15,310 So h, the Hamiltonian-- that is p squared over 2 m times 129 00:10:15,310 --> 00:10:18,310 the identity matrix-- 130 00:10:18,310 --> 00:10:21,890 can be perhaps better thought as sigma 131 00:10:21,890 --> 00:10:29,680 dot p times sigma dot p divided by 2m. 132 00:10:35,990 --> 00:10:38,310 So this is the first step. 133 00:10:38,310 --> 00:10:41,190 You haven't done really much, but you've 134 00:10:41,190 --> 00:10:44,690 rewritten the Hamiltonian with a unit matrix 135 00:10:44,690 --> 00:10:50,100 here perhaps in a somewhat provocative way. 136 00:10:50,100 --> 00:10:54,240 In quantum mechanics, when we couple to electromagnetism, 137 00:10:54,240 --> 00:10:58,290 there are simple changes we have to do. 138 00:10:58,290 --> 00:11:04,720 And we will study that in detail in about three weeks, 139 00:11:04,720 --> 00:11:07,860 but today I will just bench on what 140 00:11:07,860 --> 00:11:13,320 you're supposed to do in order to couple to electromagnetism. 141 00:11:13,320 --> 00:11:15,000 In order to couple to electromagnetism, 142 00:11:15,000 --> 00:11:18,780 you're supposed to change p wherever you see 143 00:11:18,780 --> 00:11:23,070 by an object you can call pi. 144 00:11:23,070 --> 00:11:27,270 It's some sort of more canonical momentum 145 00:11:27,270 --> 00:11:34,080 in which it's got to p minus q over c. 146 00:11:34,080 --> 00:11:37,380 The vector potential is a function of x. 147 00:11:37,380 --> 00:11:41,100 Well, people write it like this, qp minus q ac. 148 00:11:44,560 --> 00:11:47,290 So you're supposed to do this replacement. 149 00:11:47,290 --> 00:11:50,260 When you're dealing with a particle moving 150 00:11:50,260 --> 00:11:56,080 in some electromagnetic field, that's the change you must do. 151 00:11:56,080 --> 00:11:57,890 We will study that in detail. 152 00:11:57,890 --> 00:12:01,180 But I think an obvious question at this moment 153 00:12:01,180 --> 00:12:02,210 is the following. 154 00:12:02,210 --> 00:12:04,690 You say, well, I'm in quantum mechanics, 155 00:12:04,690 --> 00:12:07,720 and I work with p and x. 156 00:12:07,720 --> 00:12:11,350 Those are my opera-- what is a? is it an operator? 157 00:12:11,350 --> 00:12:12,610 Is it a vector? 158 00:12:12,610 --> 00:12:18,190 Is it what? p is an operator, but what is a? 159 00:12:18,190 --> 00:12:20,950 You should think of this vector potential. 160 00:12:20,950 --> 00:12:23,950 In general, this vector potential 161 00:12:23,950 --> 00:12:25,790 will depend on the position. 162 00:12:25,790 --> 00:12:30,070 So if you put a magnetic field, you require a vector potential. 163 00:12:30,070 --> 00:12:32,470 It has some position dependent. 164 00:12:32,470 --> 00:12:35,680 So actually, this a here should be 165 00:12:35,680 --> 00:12:41,140 thought as a of x, the same way as when 166 00:12:41,140 --> 00:12:44,020 you have the potential that depends on x. 167 00:12:44,020 --> 00:12:47,440 In quantum mechanics, we just think of that x as an operator. 168 00:12:47,440 --> 00:12:52,270 So the a is an operator because x is an operator, 169 00:12:52,270 --> 00:12:55,850 and a has x dependents. 170 00:12:55,850 --> 00:13:09,660 So the Pauli Hamiltonian, h Pauli is nothing else but sigma 171 00:13:09,660 --> 00:13:18,760 dot pi times sigma dot pi over 2 m. 172 00:13:18,760 --> 00:13:21,790 Because we said p must be replaced by pi, 173 00:13:21,790 --> 00:13:26,090 so this is the Pauli Hamiltonian. 174 00:13:26,090 --> 00:13:34,340 And it's equal to 1 over 2m pi squared 175 00:13:34,340 --> 00:13:45,880 pi dot pi times 1 plus i sigma dotted with pi cross pi. 176 00:13:59,190 --> 00:14:04,260 OK, here is the computation we need to do. 177 00:14:04,260 --> 00:14:07,350 What is pi cross pi? 178 00:14:07,350 --> 00:14:08,655 You know what pi is. 179 00:14:08,655 --> 00:14:10,710 It's given by this thing. 180 00:14:10,710 --> 00:14:12,870 So how much is pi cross pi? 181 00:14:16,310 --> 00:14:18,820 Well, it takes a little computation. 182 00:14:18,820 --> 00:14:21,740 I'll tell you what you have to do. 183 00:14:21,740 --> 00:14:23,840 It's a very interesting computation. 184 00:14:23,840 --> 00:14:25,160 It's small. 185 00:14:25,160 --> 00:14:27,260 It reminds you of this computation 186 00:14:27,260 --> 00:14:32,060 in angular momentum, l cross l. 187 00:14:32,060 --> 00:14:35,240 Maybe you've written the algebra of angular momentum 188 00:14:35,240 --> 00:14:36,110 in this language. 189 00:14:36,110 --> 00:14:39,350 It's equal to h bar l. 190 00:14:42,420 --> 00:14:46,430 That's your computation in relations of angular momentum 191 00:14:46,430 --> 00:14:47,240 written like that. 192 00:14:47,240 --> 00:14:51,730 So pi cross pi, the kth component 193 00:14:51,730 --> 00:15:00,620 is epsilon ijk pi i pi j or 1/2 epsilon 194 00:15:00,620 --> 00:15:06,005 ijk pi I commutator with pi j. 195 00:15:09,720 --> 00:15:13,680 This last step, it can be explained 196 00:15:13,680 --> 00:15:15,990 by writing out the commutator, which 197 00:15:15,990 --> 00:15:19,650 is pi i pi j minus pi j pi i. 198 00:15:19,650 --> 00:15:22,770 But with this epsilon, those two terms are the same, 199 00:15:22,770 --> 00:15:26,000 and it becomes this. 200 00:15:26,000 --> 00:15:30,400 So how much is this thing? 201 00:15:30,400 --> 00:15:33,220 I'll leave it for you to do it. 202 00:15:33,220 --> 00:15:39,410 It's simple thing. 203 00:15:39,410 --> 00:15:42,200 You have to commute these things, 204 00:15:42,200 --> 00:15:45,860 and you must think of p as derivatives. 205 00:15:45,860 --> 00:15:57,680 So this pi i pi j is i h bar q over c di a j minus dj a i. 206 00:15:57,680 --> 00:15:59,450 You can see it coming. 207 00:15:59,450 --> 00:16:05,120 You see, you have a commutator of two factors like that. 208 00:16:05,120 --> 00:16:07,340 The a with a will commute. 209 00:16:07,340 --> 00:16:09,200 The p with p will commute. 210 00:16:09,200 --> 00:16:12,639 The cross products won't, but that's just derivatives. 211 00:16:12,639 --> 00:16:14,180 So therefore, you get the derivatives 212 00:16:14,180 --> 00:16:17,390 of a antisymmetrized. 213 00:16:17,390 --> 00:16:20,030 And the derivatives of a antisymmetrized 214 00:16:20,030 --> 00:16:21,890 is the curl of a. 215 00:16:21,890 --> 00:16:24,660 And the curl of a is the magnetic field. 216 00:16:24,660 --> 00:16:27,870 So that's why this happens. 217 00:16:27,870 --> 00:16:32,210 So at the end of the day, when you'll finish this computation, 218 00:16:32,210 --> 00:16:40,700 get that pi cross pi is just i q i h bar q over c 219 00:16:40,700 --> 00:16:44,060 times the magnetic field. 220 00:16:44,060 --> 00:16:45,190 Pretty nice equation. 221 00:16:48,630 --> 00:16:52,845 So the Pauli Hamiltonian includes h Pauli. 222 00:16:55,560 --> 00:17:06,250 It includes this term, which is i sigma over 2m 223 00:17:06,250 --> 00:17:18,520 dotted with i hQ over cb plus the other terms. 224 00:17:18,520 --> 00:17:24,700 I'm just looking at this term which has the magnetic field. 225 00:17:24,700 --> 00:17:26,740 And therefore, look at this. 226 00:17:26,740 --> 00:17:30,640 i with i is minus 1, but q is minus e. 227 00:17:30,640 --> 00:17:45,020 So this is eh bar over 2 mc sigma dot b, which is here, 228 00:17:45,020 --> 00:17:48,260 which came from g equal 2. 229 00:17:48,260 --> 00:17:53,530 Nowhere I had to say there that g is equal to 2. 230 00:17:53,530 --> 00:17:56,890 It came out of this calculation. 231 00:17:56,890 --> 00:18:00,310 The Pauli Hamiltonian knew of this. 232 00:18:00,310 --> 00:18:04,120 And therefore, it's a great progress, that Pauli 233 00:18:04,120 --> 00:18:07,960 Hamiltonian, but it suggests to us 234 00:18:07,960 --> 00:18:12,290 that we can do things still even better, as Dirac did.