1 00:00:01,127 --> 00:00:01,710 PROFESSOR: OK. 2 00:00:01,710 --> 00:00:05,740 Let's turn then to the Dirac equation 3 00:00:05,740 --> 00:00:11,550 and motivated, basically. 4 00:00:17,890 --> 00:00:21,250 Through the Dirac equation is the simplest way 5 00:00:21,250 --> 00:00:25,930 to do perturbation theory for the hydrogen atom. 6 00:00:25,930 --> 00:00:30,700 It helps us derive what we should think of perturbation. 7 00:00:30,700 --> 00:00:35,551 So I'll discuss that quickly. 8 00:00:35,551 --> 00:00:40,530 And there may be some of that done in recitation. 9 00:00:40,530 --> 00:00:43,140 So we're going to discuss now the Dirac equation. 10 00:00:49,650 --> 00:00:53,810 So the Dirac equation begins with the observation 11 00:00:53,810 --> 00:00:58,370 that we have E squared minus p squared 12 00:00:58,370 --> 00:01:03,950 c squared for a free particle is m squared c to the fourth. 13 00:01:03,950 --> 00:01:08,960 This dispersion relation that rates E and p for any particle. 14 00:01:08,960 --> 00:01:13,550 So if you wanted to describe the dynamics 15 00:01:13,550 --> 00:01:17,780 of a relativistic particle, you could say, look, 16 00:01:17,780 --> 00:01:24,470 the energy is the square root of p squared c squared 17 00:01:24,470 --> 00:01:27,800 plus m squared c to the fourth. 18 00:01:27,800 --> 00:01:32,960 Therefore, I should take the Hamiltonian to be that thing. 19 00:01:32,960 --> 00:01:37,610 I should take H' to be that. 20 00:01:37,610 --> 00:01:40,760 And work with a Schrodinger equation 21 00:01:40,760 --> 00:01:43,850 that has this H with the square root. 22 00:01:43,850 --> 00:01:46,880 Nobody does that, of course. 23 00:01:46,880 --> 00:01:50,420 But you can do a little bit with it. 24 00:01:50,420 --> 00:01:57,830 If you notice that this is equal to mc squared square root of 1 25 00:01:57,830 --> 00:02:02,415 plus p squared over m squared c squared. 26 00:02:05,170 --> 00:02:07,540 And then expand. 27 00:02:07,540 --> 00:02:12,910 In small velocities where this is less than 1, 28 00:02:12,910 --> 00:02:17,440 you get 1 plus 1/2 p squared over m 29 00:02:17,440 --> 00:02:22,480 squared c squared minus 1/8 of this term squared. 30 00:02:22,480 --> 00:02:29,650 So p squared p squared over m to the fourth c to the fourth. 31 00:02:29,650 --> 00:02:31,640 And then what do you get? 32 00:02:31,640 --> 00:02:38,710 mc squared plus 1/2 2m p squared. 33 00:02:38,710 --> 00:02:39,880 That's very nice. 34 00:02:39,880 --> 00:02:42,280 Apart from the rest energy, you get 35 00:02:42,280 --> 00:02:46,430 the p squared over 2m that you would expect. 36 00:02:46,430 --> 00:02:51,270 And the next correction is minus 18 p 37 00:02:51,270 --> 00:02:59,930 squared p squared over m cube c squared plus dot dot dot. 38 00:02:59,930 --> 00:03:03,210 You know, that's one way you could treat the Hamiltonian. 39 00:03:03,210 --> 00:03:05,930 But it's very complicated. 40 00:03:05,930 --> 00:03:09,620 Don't treat the Hamiltonian too seriously 41 00:03:09,620 --> 00:03:12,350 because the Schrodinger equation would 42 00:03:12,350 --> 00:03:15,460 have infinite number of spatial derivatives. 43 00:03:15,460 --> 00:03:18,710 p to the fourth, p to the sixth, p to the eighth. 44 00:03:18,710 --> 00:03:21,890 Never going to get that understood. 45 00:03:21,890 --> 00:03:24,530 But in terms of perturbation theory, 46 00:03:24,530 --> 00:03:30,050 we can think of p squared over 2m, your original Hamiltonian. 47 00:03:30,050 --> 00:03:35,290 And here is a relativistic correction that is small. 48 00:03:35,290 --> 00:03:44,430 But Dirac was puzzled by that square root. 49 00:03:44,430 --> 00:03:48,180 He basically looked at it and I think 50 00:03:48,180 --> 00:03:52,950 he had sort of mathematical inspiration in here. 51 00:03:52,950 --> 00:03:55,440 He looked at this equation and that square root 52 00:03:55,440 --> 00:03:57,990 and he said, how much would I wish 53 00:03:57,990 --> 00:04:01,270 I could take that square root. 54 00:04:01,270 --> 00:04:03,910 What can I do to take that square root? 55 00:04:03,910 --> 00:04:08,610 So he said, well, I could take that square root. 56 00:04:08,610 --> 00:04:14,460 This has p squared so c squared plus m squared c to the fourth. 57 00:04:14,460 --> 00:04:17,700 I could take that square root if that would 58 00:04:17,700 --> 00:04:20,769 be equal to something squared. 59 00:04:20,769 --> 00:04:24,690 Which it's not because it's missing the cross product 60 00:04:24,690 --> 00:04:26,040 of the right amount. 61 00:04:26,040 --> 00:04:28,900 And it's not there. 62 00:04:28,900 --> 00:04:31,710 But if it would be there, the cross product, 63 00:04:31,710 --> 00:04:35,610 it would be a linear function of p and mc squared. 64 00:04:35,610 --> 00:04:41,010 So he says, OK, let me put a linear function of p. 65 00:04:41,010 --> 00:04:45,840 And the most general linear function of p is obtained, 66 00:04:45,840 --> 00:04:48,720 I want the c squared, so c, by doing 67 00:04:48,720 --> 00:04:54,640 the dot product of a vector with p, a constant vector. 68 00:04:54,640 --> 00:04:58,350 A constant vector times p is the most general linear function 69 00:04:58,350 --> 00:04:59,580 of p. 70 00:04:59,580 --> 00:05:08,990 And here I'll output another constant, mc squared. 71 00:05:08,990 --> 00:05:12,120 And I hope I can take the square root 72 00:05:12,120 --> 00:05:14,970 because there will exist some constants 73 00:05:14,970 --> 00:05:16,300 and it somehow will work. 74 00:05:16,300 --> 00:05:19,390 As of now, it cannot work because it's always across 75 00:05:19,390 --> 00:05:19,890 product. 76 00:05:19,890 --> 00:05:22,930 But just follow these dots. 77 00:05:22,930 --> 00:05:25,290 And then alpha dot p. 78 00:05:25,290 --> 00:05:31,680 This is equal to c alpha 1 p1 plus c alpha 2 p2 79 00:05:31,680 --> 00:05:37,110 plus c alpha 3 p3 plus beta mc squared. 80 00:05:37,110 --> 00:05:40,470 So let's list all the things that should 81 00:05:40,470 --> 00:05:43,380 happen for this to work out. 82 00:05:43,380 --> 00:05:46,496 Well, this square includes the square of this, 83 00:05:46,496 --> 00:05:48,120 the square of this, the square of this, 84 00:05:48,120 --> 00:05:49,078 and the square of this. 85 00:05:49,078 --> 00:05:50,530 And those are what we want. 86 00:05:50,530 --> 00:05:54,900 So we should have that alpha 1 squared 87 00:05:54,900 --> 00:05:57,990 is equal to alpha 2 squared is equal to alpha 88 00:05:57,990 --> 00:06:03,990 3 squared is equal to beta squared is equal to 1. 89 00:06:03,990 --> 00:06:07,260 If that happens, well, this term will give you 90 00:06:07,260 --> 00:06:10,320 c squared p1 squared, c squared p2 squared, 91 00:06:10,320 --> 00:06:14,550 c squared p3 squared, and m squared c to the fourth, 92 00:06:14,550 --> 00:06:17,100 and I get everything to work. 93 00:06:17,100 --> 00:06:19,770 But the cross products must also work. 94 00:06:19,770 --> 00:06:22,590 So for example, the cross product of p1 with p2 95 00:06:22,590 --> 00:06:24,940 should vanish. 96 00:06:24,940 --> 00:06:28,660 And that, well, if you ride these two factors like 97 00:06:28,660 --> 00:06:31,840 that here you would say, oh, what they need now 98 00:06:31,840 --> 00:06:39,220 is that alpha i times alpha j plus alpha j times alpha i 99 00:06:39,220 --> 00:06:42,890 should be 0 for all i different from j. 100 00:06:42,890 --> 00:06:47,260 So for example, alpha 1, alpha 2 plus alpha 2 alpha 1, 101 00:06:47,260 --> 00:06:49,120 when you do the product, should vanish. 102 00:06:49,120 --> 00:06:52,490 And I kept the order of these things there. 103 00:06:52,490 --> 00:06:54,610 When I'm starting to think that maybe 104 00:06:54,610 --> 00:06:56,830 numbers is not going to work. 105 00:06:56,830 --> 00:06:58,600 So i difference from j. 106 00:06:58,600 --> 00:07:02,050 This must happen too for this to be equal. 107 00:07:02,050 --> 00:07:07,300 And moreover for all i alpha i beta. 108 00:07:07,300 --> 00:07:09,990 There shouldn't be cross products between these ones 109 00:07:09,990 --> 00:07:11,830 and this term either. 110 00:07:11,830 --> 00:07:17,560 Plus beta alpha i is equal to 0. 111 00:07:17,560 --> 00:07:20,690 And that's all that should happen. 112 00:07:20,690 --> 00:07:23,180 If that happens, we can take the square root. 113 00:07:27,290 --> 00:07:31,370 But if these things are numbers, well, this 114 00:07:31,370 --> 00:07:35,160 could be phases or 1 or minus 1. 115 00:07:35,160 --> 00:07:37,730 And that's never going to work. 116 00:07:37,730 --> 00:07:39,560 This is twice this product. 117 00:07:39,560 --> 00:07:42,620 So [INAUDIBLE] OK, they're matrices. 118 00:07:42,620 --> 00:07:45,470 But that's not bad, if they're matrices, 119 00:07:45,470 --> 00:07:49,420 we'll have like spinors, like Pauli was doing. 120 00:07:49,420 --> 00:07:50,750 And we'll work on spinors. 121 00:07:50,750 --> 00:07:54,710 But it turns out that this one, the simplest solution 122 00:07:54,710 --> 00:07:58,610 is something with 4 by 4 matrices. 123 00:07:58,610 --> 00:08:10,250 So the solution is alpha is 0 sigma sigma 0. 124 00:08:10,250 --> 00:08:17,490 And beta is 1 minus 100. 125 00:08:17,490 --> 00:08:22,370 This is the simplest solution, 4 by 4 matrices. 126 00:08:22,370 --> 00:08:34,890 So that Dirac Hamiltonian, H Dirac, is equal to the energy. 127 00:08:34,890 --> 00:08:38,240 And the energy was the square root of this. 128 00:08:38,240 --> 00:08:39,760 So it's just this thing. 129 00:08:39,760 --> 00:08:47,095 So it's c alpha dot p plus b mc squared. 130 00:08:50,220 --> 00:08:51,900 That's our Dirac Hamiltonian. 131 00:08:59,250 --> 00:09:05,100 But we need to deal with spinors that now are four dimensional. 132 00:09:05,100 --> 00:09:07,950 These matrices, alpha and beta, are 133 00:09:07,950 --> 00:09:12,990 four dimensional because the segments that are inside 134 00:09:12,990 --> 00:09:14,990 are two dimensional. 135 00:09:14,990 --> 00:09:16,860 And these are 2 by 2 matrices. 136 00:09:20,520 --> 00:09:29,790 So when you write the Dirac equation as i H bar d psi dt 137 00:09:29,790 --> 00:09:39,340 equal H Dirac psi, psi is a four component thing, a chi 138 00:09:39,340 --> 00:09:41,252 and a phi. 139 00:09:41,252 --> 00:09:44,070 And what people discover is that chi 140 00:09:44,070 --> 00:09:49,330 behaves like the Pauli spinor. 141 00:09:49,330 --> 00:09:54,880 And this one is a small thing, a small correction. 142 00:09:54,880 --> 00:10:03,670 So there is a whole analysis of this system in which, in order 143 00:10:03,670 --> 00:10:07,900 to put the magnetic field, you can put in here 144 00:10:07,900 --> 00:10:16,550 for the Dirac Hamiltonian, H Dirac, 145 00:10:16,550 --> 00:10:23,465 coupled to electromagnetism has a c alpha p plus e over 146 00:10:23,465 --> 00:10:30,970 ca plus beta mc squared. 147 00:10:30,970 --> 00:10:34,290 And you still have to add the potential, V 148 00:10:34,290 --> 00:10:39,720 of r, that comes from [INAUDIBLE] minus e squared 149 00:10:39,720 --> 00:10:40,530 over r. 150 00:10:44,930 --> 00:10:50,060 It's the value of the electron charge 151 00:10:50,060 --> 00:10:54,280 that is the potential of r. 152 00:10:54,280 --> 00:10:59,390 The scalar potential phi of r. 153 00:10:59,390 --> 00:11:02,380 So this is the Dirac Hamiltonian. 154 00:11:02,380 --> 00:11:04,250 And you have these things. 155 00:11:04,250 --> 00:11:06,950 And this is something you can read 156 00:11:06,950 --> 00:11:09,530 if you're interested in [? Shankar. ?] There 157 00:11:09,530 --> 00:11:18,390 is a derivation of the Pauli equation for the electron. 158 00:11:18,390 --> 00:11:20,210 Which is the spinors that you are 159 00:11:20,210 --> 00:11:26,270 familiar with by eliminating in a recursive expansion phi. 160 00:11:26,270 --> 00:11:28,850 And it may be that Professor Metlitski 161 00:11:28,850 --> 00:11:33,070 is going to go through that in recitation as well. 162 00:11:33,070 --> 00:11:37,180 So what happens? 163 00:11:37,180 --> 00:11:44,230 What you get is that you get an H chi equal E chi. 164 00:11:44,230 --> 00:11:45,550 And H is what? 165 00:11:48,290 --> 00:11:50,990 Here is the grand prize. 166 00:11:50,990 --> 00:11:56,600 From the Dirac equation, we can rewrite the Hamiltonian 167 00:11:56,600 --> 00:12:01,820 of the hydrogen atom in a more accurate way, a more complete 168 00:12:01,820 --> 00:12:02,930 Hamiltonian. 169 00:12:02,930 --> 00:12:11,370 And it has p squared over 2m plus V of r, 170 00:12:11,370 --> 00:12:15,080 which is what we've called H0. 171 00:12:15,080 --> 00:12:15,860 It's there. 172 00:12:15,860 --> 00:12:17,150 It's the first term. 173 00:12:17,150 --> 00:12:20,470 That's what you would expect. 174 00:12:20,470 --> 00:12:24,430 Then there is that term that we anticipated. 175 00:12:24,430 --> 00:12:28,300 Some relativistic correction here. 176 00:12:28,300 --> 00:12:37,960 So it comes like minus p to the fourth over 8m cubed c squared. 177 00:12:37,960 --> 00:12:42,190 And we call this delta H relativistic. 178 00:12:44,790 --> 00:12:50,280 And then you get this term that corresponds 179 00:12:50,280 --> 00:12:53,850 to spin orbit coupling that also shows up. 180 00:12:53,850 --> 00:13:00,960 2m squared c squared 1 over r dv dr. You've studied this thing 181 00:13:00,960 --> 00:13:04,560 and you gave a heuristic explanation for it. 182 00:13:04,560 --> 00:13:06,310 That's another term. 183 00:13:06,310 --> 00:13:10,920 It's kind of difficult to get the right value 184 00:13:10,920 --> 00:13:13,170 by a simple argument. 185 00:13:13,170 --> 00:13:15,330 You always get it off by a factor of 2 186 00:13:15,330 --> 00:13:18,060 that is associated to Thomas precession. 187 00:13:18,060 --> 00:13:22,690 But here it comes out directly with the right number. 188 00:13:22,690 --> 00:13:26,220 This is called spin orbit coupling. 189 00:13:26,220 --> 00:13:29,290 Delta H of spin orbit. 190 00:13:32,020 --> 00:13:34,240 And there's one more term. 191 00:13:34,240 --> 00:13:38,440 Plus H squared to this [INAUDIBLE] 8 m squared 192 00:13:38,440 --> 00:13:45,830 c squared Laplacian of V. Which is called of this V of r. 193 00:13:45,830 --> 00:13:48,290 It's called the Darwin term. 194 00:13:48,290 --> 00:13:49,985 Delta H Darwin. 195 00:13:52,720 --> 00:13:55,610 Not the biologist. 196 00:13:55,610 --> 00:13:58,710 Some Darwin. 197 00:13:58,710 --> 00:14:02,550 So the job is set for us. 198 00:14:02,550 --> 00:14:05,520 We have to do perturbation theory. 199 00:14:05,520 --> 00:14:10,710 Now you remember that all the terms associated with H0 200 00:14:10,710 --> 00:14:16,740 were of the form energies with alpha squared mc squared. 201 00:14:16,740 --> 00:14:18,010 This was like H0. 202 00:14:21,470 --> 00:14:26,190 If you look at the units of all of these terms, 203 00:14:26,190 --> 00:14:28,770 we looked at them in the notes. 204 00:14:28,770 --> 00:14:35,250 Delta H. They all go like alpha to the fourth mc squared. 205 00:14:35,250 --> 00:14:39,720 So there is a difference of alpha squared there. 206 00:14:39,720 --> 00:14:42,750 1/19000 smaller. 207 00:14:42,750 --> 00:14:47,280 That is the fine structure of the hydrogen atom. 208 00:14:47,280 --> 00:14:52,590 And our task next time will be to compute the effect of this 209 00:14:52,590 --> 00:14:55,860 and see what happens with all the levels of the hydrogen 210 00:14:55,860 --> 00:14:56,650 atom. 211 00:14:56,650 --> 00:14:58,450 This is our [? final ?] structure. 212 00:14:58,450 --> 00:15:01,040 So we'll do that next time.