1 00:00:01,350 --> 00:00:04,200 PROFESSOR: So we continue today our study 2 00:00:04,200 --> 00:00:08,220 of electromagnetic fields, and quantum mechanics, 3 00:00:08,220 --> 00:00:11,980 and particles in those electromagnetic fields. 4 00:00:11,980 --> 00:00:17,430 So last time, we described what we 5 00:00:17,430 --> 00:00:20,970 must do in order to couple a particle 6 00:00:20,970 --> 00:00:22,570 to an electromagnetic field. 7 00:00:22,570 --> 00:00:25,600 And the rule was oddly simple. 8 00:00:25,600 --> 00:00:27,900 You get the Schrodinger equation, 9 00:00:27,900 --> 00:00:34,400 and the Hamiltonian is now changed into this form. 10 00:00:34,400 --> 00:00:36,860 If you had a free particle, you're 11 00:00:36,860 --> 00:00:39,950 accustomed to have p squared over 2m. 12 00:00:39,950 --> 00:00:43,880 That has changed, and you have a new term, with p, 13 00:00:43,880 --> 00:00:46,860 minus q over c, A squared. 14 00:00:46,860 --> 00:00:47,734 Thank you. 15 00:00:50,520 --> 00:00:56,650 So now our task is to understand how this is compatible, 16 00:00:56,650 --> 00:00:59,350 and what are the implications of these changes. 17 00:00:59,350 --> 00:01:01,870 They're pretty significant changes. 18 00:01:05,349 --> 00:01:08,725 This expression, p, minus q over c, 19 00:01:08,725 --> 00:01:11,710 A, is what used to be just p. 20 00:01:11,710 --> 00:01:14,230 And that used to be just the kinetic energy. 21 00:01:14,230 --> 00:01:20,350 And now it looks a little more strange, in fact. 22 00:01:20,350 --> 00:01:24,310 So in terms of electromagnetic potentials, 23 00:01:24,310 --> 00:01:27,790 we know that the potentials are not unique. 24 00:01:27,790 --> 00:01:30,190 There are this gauge transformations 25 00:01:30,190 --> 00:01:34,060 that establish that these are completely equivalent 26 00:01:34,060 --> 00:01:37,480 potentials-- so two potentials, A and phi, 27 00:01:37,480 --> 00:01:41,950 and A prime and phi prime, are physically equivalent. 28 00:01:41,950 --> 00:01:45,530 They define the same electromagnetic field 29 00:01:45,530 --> 00:01:49,780 if they are related in this way. 30 00:01:49,780 --> 00:01:52,510 We then mention that therefore you 31 00:01:52,510 --> 00:01:55,450 have an issue with the Schrodinger equation. 32 00:01:55,450 --> 00:01:59,500 You would want the physics to remain invariant 33 00:01:59,500 --> 00:02:02,780 in their gauge transformations. 34 00:02:02,780 --> 00:02:05,410 So you could write the Schrodinger equation 35 00:02:05,410 --> 00:02:09,820 for the new gauge potentials, with A prime and phi prime. 36 00:02:09,820 --> 00:02:13,240 And you could compare with the Schrodinger equation 37 00:02:13,240 --> 00:02:18,010 with the old potentials, A and phi. 38 00:02:18,010 --> 00:02:21,970 And you could ask, OK, is it solved by the same wave 39 00:02:21,970 --> 00:02:22,790 function? 40 00:02:22,790 --> 00:02:24,860 And the answer is no. 41 00:02:24,860 --> 00:02:27,560 The wave function must change. 42 00:02:27,560 --> 00:02:32,020 But there is a way to get one way function from the other. 43 00:02:32,020 --> 00:02:33,670 Here is the formula. 44 00:02:33,670 --> 00:02:40,860 Psi prime and psi are related by this factor, this function, U, 45 00:02:40,860 --> 00:02:47,890 which involves exponential of the gauge parameter, multiplied 46 00:02:47,890 --> 00:02:51,400 by a couple of physical constants, 47 00:02:51,400 --> 00:02:54,590 including the charge, q, of the particle. 48 00:02:54,590 --> 00:02:58,480 Now we're going to use the notation, q, all the time. 49 00:02:58,480 --> 00:03:01,540 Many books put E in there. 50 00:03:01,540 --> 00:03:03,640 And that's the charge of the electron. 51 00:03:03,640 --> 00:03:08,200 And then you always wonder, is that E, or minus E, 52 00:03:08,200 --> 00:03:10,300 or which sign is it. 53 00:03:10,300 --> 00:03:14,890 So here, q is the charge of the particle. 54 00:03:14,890 --> 00:03:18,010 If you have an electron, q is minus E. 55 00:03:18,010 --> 00:03:22,810 But let's leave it q open there, so you can work with arbitrary 56 00:03:22,810 --> 00:03:29,120 charges in any circumstance. 57 00:03:29,120 --> 00:03:32,500 So this Hamiltonian is what we have to understand. 58 00:03:32,500 --> 00:03:36,040 They want to make a couple more comments 59 00:03:36,040 --> 00:03:38,890 about what this Hamiltonian is. 60 00:03:38,890 --> 00:03:42,580 And we must think a little about the gauge invariance as well. 61 00:03:49,360 --> 00:03:50,570 So there it is. 62 00:03:50,570 --> 00:03:54,560 A Hamiltonian contains this term squared. 63 00:03:54,560 --> 00:03:57,280 So let's just ask ourselves, what 64 00:03:57,280 --> 00:04:00,860 does that term really imply. 65 00:04:00,860 --> 00:04:04,960 Well, there's 1 over 2m. 66 00:04:04,960 --> 00:04:09,310 So for the Hamiltonian, 1 over 2m. 67 00:04:09,310 --> 00:04:11,200 And we have this factor squared. 68 00:04:11,200 --> 00:04:15,550 So following the careful things we're 69 00:04:15,550 --> 00:04:17,680 accustomed to do in quantum mechanics, 70 00:04:17,680 --> 00:04:30,010 I would put P squared here, minus q over c, P dot A, 71 00:04:30,010 --> 00:04:41,920 minus q over c, A dot P, plus q squared over c squared, 72 00:04:41,920 --> 00:04:50,090 A squared, plus q phi. 73 00:04:50,090 --> 00:04:52,010 That's how the Hamiltonian looks. 74 00:04:56,730 --> 00:04:57,855 But you have to be careful. 75 00:05:01,110 --> 00:05:04,290 Here, this term we kind of understand what it is. 76 00:05:04,290 --> 00:05:07,530 It's vector potential squared-- 77 00:05:07,530 --> 00:05:08,790 no problem. 78 00:05:08,790 --> 00:05:10,740 Here is p squared. 79 00:05:10,740 --> 00:05:12,070 We're accustomed to that. 80 00:05:12,070 --> 00:05:14,090 That's a Lapacian. 81 00:05:14,090 --> 00:05:17,850 We have two terms that could be a little strange. 82 00:05:17,850 --> 00:05:22,200 A dot P, probably is all clear. 83 00:05:22,200 --> 00:05:28,500 A is dotted with a vector momentum, which is an operator. 84 00:05:28,500 --> 00:05:29,770 It's the gradient. 85 00:05:29,770 --> 00:05:33,510 So that's going to differentiate whatever is to the right. 86 00:05:33,510 --> 00:05:36,740 The question is, what this P dot A? 87 00:05:36,740 --> 00:05:42,750 And P dot A must be thought in the operator way, 88 00:05:42,750 --> 00:05:49,320 that P acts on everything to the right, including A. 89 00:05:49,320 --> 00:05:53,880 So if this Hamiltonian is to act in a wave function, that 90 00:05:53,880 --> 00:05:57,080 doesn't mean that the P operator just acts on A. 91 00:05:57,080 --> 00:06:01,770 It acts on A and everything to the right. 92 00:06:01,770 --> 00:06:10,200 So P dot A, if you think of it as a derivative, 93 00:06:10,200 --> 00:06:29,650 is h bar over i, divergence of A, plus A dot P. You see, 94 00:06:29,650 --> 00:06:36,950 this P is h bar over i gradient, but it's acting on everything. 95 00:06:36,950 --> 00:06:39,730 So you could put up psi to the right of here, 96 00:06:39,730 --> 00:06:43,010 and it would act on everything there. 97 00:06:43,010 --> 00:06:47,200 So maybe in order to justify this equation, 98 00:06:47,200 --> 00:06:54,820 you should just write h bar over i gradient acting on A times 99 00:06:54,820 --> 00:06:56,800 a wave function, psi. 100 00:06:56,800 --> 00:07:00,470 And go through this and see that at the end of the day, 101 00:07:00,470 --> 00:07:05,800 this P dot with A means act on A, 102 00:07:05,800 --> 00:07:08,720 but then you still have to act on everything to the right. 103 00:07:08,720 --> 00:07:12,880 So that's this term here. 104 00:07:12,880 --> 00:07:18,000 So this term is pretty important. 105 00:07:18,000 --> 00:07:22,290 Sometimes your A satisfies what is called the Coulomb gauge, 106 00:07:22,290 --> 00:07:25,560 in which delta of A is zero, but in general it doesn't. 107 00:07:25,560 --> 00:07:29,490 So this helps clarify what the Hamiltonian really is. 108 00:07:33,150 --> 00:07:35,430 It would be a mistake to say that these two 109 00:07:35,430 --> 00:07:36,760 terms are the same. 110 00:07:36,760 --> 00:07:40,700 And it would be also a mistake to say that this thing is just 111 00:07:40,700 --> 00:07:44,090 the divergence of A. Both are wrong things. 112 00:07:44,090 --> 00:07:54,390 So the whole Hamiltonian, now, is P squared over 2m. 113 00:07:54,390 --> 00:07:57,270 Then you have twice-- 114 00:07:57,270 --> 00:07:59,430 well, let's have this term. 115 00:07:59,430 --> 00:08:05,490 So you have plus, the i goes to the numerator, 116 00:08:05,490 --> 00:08:14,280 as i h bar q, over 2m c, divergence of A. 117 00:08:14,280 --> 00:08:17,880 And that's just the divergence of A, the function. 118 00:08:17,880 --> 00:08:20,550 It's not any more a differential operator. 119 00:08:20,550 --> 00:08:22,740 It's just acted on it. 120 00:08:22,740 --> 00:08:25,290 Then you have this term twice. 121 00:08:25,290 --> 00:08:36,809 So minus q over mc, A dot P, plus this term, q squared 122 00:08:36,809 --> 00:08:43,279 over 2mc squared, A squared, plus q phi. 123 00:08:46,220 --> 00:08:48,290 So that's the Hamiltonian. 124 00:08:48,290 --> 00:08:50,990 Let me just make sure I got it right-- 125 00:08:50,990 --> 00:08:57,485 P squared over 2m, plus [INAUDIBLE],, minus qAP-- 126 00:08:57,485 --> 00:09:00,270 OK, so that's right. 127 00:09:00,270 --> 00:09:02,420 OK, so this is our Hamiltonian. 128 00:09:02,420 --> 00:09:08,450 If you just want to write it very explicitly. 129 00:09:08,450 --> 00:09:11,660 It's not generally all that useful to have 130 00:09:11,660 --> 00:09:12,600 the explicit form. 131 00:09:12,600 --> 00:09:15,560 But in some examples-- there will be at least one example 132 00:09:15,560 --> 00:09:18,480 where it is nice to know. 133 00:09:18,480 --> 00:09:23,560 And in fact, the H formula on the top of the blackboard 134 00:09:23,560 --> 00:09:28,100 hides, in a nice way, a little bit the complexity 135 00:09:28,100 --> 00:09:29,730 of this whole coupling. 136 00:09:29,730 --> 00:09:30,230 Yes. 137 00:09:30,230 --> 00:09:32,530 AUDIENCE: Is this for scalar particles? 138 00:09:32,530 --> 00:09:36,010 PROFESSOR: Yes, this is for a scalar particle. 139 00:09:36,010 --> 00:09:42,490 So we're taking a particle, at this moment, without spin. 140 00:09:42,490 --> 00:09:46,160 For spin particles, there would be a little extra term 141 00:09:46,160 --> 00:09:47,060 sometimes. 142 00:09:49,570 --> 00:09:57,280 So let's say a couple more words about this thing, 143 00:09:57,280 --> 00:09:59,320 in particular the gauge invariance. 144 00:09:59,320 --> 00:10:04,450 So we now have seen what the Hamiltonian is. 145 00:10:04,450 --> 00:10:07,790 Let's think of the gauge invariance. 146 00:10:07,790 --> 00:10:12,560 How could you establish this gauge invariance? 147 00:10:12,560 --> 00:10:17,830 So an identity that is very useful 148 00:10:17,830 --> 00:10:20,410 takes the following form. 149 00:10:20,410 --> 00:10:27,430 This differential operator-- h bar over i gradient, the p, 150 00:10:27,430 --> 00:10:38,310 minus q over c, A prime, times U-- 151 00:10:38,310 --> 00:10:42,270 and you could put the psi here if you wish-- 152 00:10:42,270 --> 00:10:49,290 is in fact equal to U, h bar over i gradient, 153 00:10:49,290 --> 00:10:55,020 minus q over c, A, psi. 154 00:11:03,930 --> 00:11:08,345 So I call this a very remarkable identity. 155 00:11:14,180 --> 00:11:18,200 And look what's happening here. 156 00:11:18,200 --> 00:11:25,850 This is useful for this kind of equation. 157 00:11:25,850 --> 00:11:30,220 If you have a psi prime which is U psi, 158 00:11:30,220 --> 00:11:38,330 the factor U interacts very nicely with this operator. 159 00:11:38,330 --> 00:11:44,200 In fact, as you move U from the right to the left, 160 00:11:44,200 --> 00:11:49,675 the gauge potential goes from A prime to A. 161 00:11:49,675 --> 00:11:53,960 A rather nice thing about this derivative-- it's 162 00:11:53,960 --> 00:11:57,680 as if this derivative had a very special symmetry 163 00:11:57,680 --> 00:12:03,080 property that U can be moved across, 164 00:12:03,080 --> 00:12:05,510 almost as if U was a constant. 165 00:12:05,510 --> 00:12:07,190 But of course it's not. 166 00:12:07,190 --> 00:12:09,410 But when you move it across, the only effect 167 00:12:09,410 --> 00:12:13,490 is to change the A prime to A. So a gauge 168 00:12:13,490 --> 00:12:17,222 transformation on the A. 169 00:12:17,222 --> 00:12:24,090 That is the reason this derivative, equipped 170 00:12:24,090 --> 00:12:28,110 with an extra term, is sometimes called a gauge covariant 171 00:12:28,110 --> 00:12:29,340 derivative. 172 00:12:29,340 --> 00:12:33,660 That is, it transforms nicely under gauge transformations. 173 00:12:33,660 --> 00:12:36,840 It does a nice job. 174 00:12:36,840 --> 00:12:40,260 I write this equation because, in part of the exercises, 175 00:12:40,260 --> 00:12:45,990 you will be asked to show that this statement about gauge 176 00:12:45,990 --> 00:12:50,790 transformations is correct, that if this equation-- top-- holds, 177 00:12:50,790 --> 00:12:55,680 the bottom equation holds with the replacements indicated 178 00:12:55,680 --> 00:12:56,940 here. 179 00:12:56,940 --> 00:13:02,490 And this identity makes the task of proving the equation-- 180 00:13:02,490 --> 00:13:04,950 the gauge invariance-- very simple. 181 00:13:04,950 --> 00:13:07,500 Because you can imagine this psi prime-- 182 00:13:07,500 --> 00:13:11,700 put U psi-- and then getting the U out, 183 00:13:11,700 --> 00:13:15,000 so the psi prime just will become psi, 184 00:13:15,000 --> 00:13:18,210 and the U going through these factors 185 00:13:18,210 --> 00:13:20,860 and simplifying very nicely. 186 00:13:20,860 --> 00:13:26,160 So this is a simple equation to show. 187 00:13:26,160 --> 00:13:30,780 For example, I can take the left-hand side 188 00:13:30,780 --> 00:13:33,040 and see what happens. 189 00:13:33,040 --> 00:13:36,960 So the first term, you have a gradient 190 00:13:36,960 --> 00:13:39,970 acting on the product of two functions. 191 00:13:39,970 --> 00:13:43,020 And that's just the gradient acting on each. 192 00:13:43,020 --> 00:13:53,070 So h over i, gradient of U, times psi, plus U, 193 00:13:53,070 --> 00:13:57,690 h over i, gradient of psi. 194 00:13:57,690 --> 00:13:59,060 That's the first term. 195 00:14:01,710 --> 00:14:11,160 The second term is minus q over c, A, times U psi-- 196 00:14:11,160 --> 00:14:16,290 because A prime is A plus gradient of lambda. 197 00:14:16,290 --> 00:14:20,820 So you're going to have minus q over c, 198 00:14:20,820 --> 00:14:25,470 gradient of lambda, U psi. 199 00:14:28,620 --> 00:14:30,975 So I wrote the first line. 200 00:14:34,270 --> 00:14:39,580 Now we can take the gradient of U. U is here. 201 00:14:39,580 --> 00:14:42,940 When we take the gradient of U, what do we get? 202 00:14:42,940 --> 00:14:47,590 h bar over i, now gradient of U-- 203 00:14:47,590 --> 00:14:49,200 you differentiate the exponentials, 204 00:14:49,200 --> 00:14:53,150 so you take the gradient of what is in the exponent. 205 00:14:53,150 --> 00:15:00,570 So it's iq over h bar c, gradient of lambda, 206 00:15:00,570 --> 00:15:04,270 times U itself, because you're differentiating 207 00:15:04,270 --> 00:15:07,285 an exponential, times psi. 208 00:15:10,820 --> 00:15:15,720 And then you have all these other terms. 209 00:15:15,720 --> 00:15:19,730 I'll couple this term. 210 00:15:22,660 --> 00:15:33,430 These two terms here are plus U h bar over i, gradient of psi, 211 00:15:33,430 --> 00:15:36,476 minus q over c. 212 00:15:36,476 --> 00:15:43,050 U and A commute, because A is a function of x and t, and U 213 00:15:43,050 --> 00:15:47,140 is a function of lambda, which is also a function of x and t. 214 00:15:47,140 --> 00:15:49,090 So there's no momentum here. 215 00:15:49,090 --> 00:15:53,950 These two things commute, so U can be moved to the left-- 216 00:15:53,950 --> 00:15:59,780 A psi minus this term. 217 00:16:04,550 --> 00:16:06,960 And if we've done the arithmetic right, 218 00:16:06,960 --> 00:16:11,690 the first and last term should cancel, and they do. 219 00:16:11,690 --> 00:16:14,320 That i cancels, the h bar cancels, 220 00:16:14,320 --> 00:16:18,500 there's q over c times that, and there's minus that term, 221 00:16:18,500 --> 00:16:21,020 so these two cancel. 222 00:16:21,020 --> 00:16:24,110 And this is precisely the right-hand side. 223 00:16:29,840 --> 00:16:32,435 So this covariant derivative is very nice. 224 00:16:35,470 --> 00:16:38,210 There's something about the Schrodinger equation of course 225 00:16:38,210 --> 00:16:44,090 that, in a sense, it's all made of covariant derivatives. 226 00:16:44,090 --> 00:16:48,080 Let's look at that. 227 00:16:48,080 --> 00:16:51,590 So any version of the Schrodinger equation-- 228 00:16:51,590 --> 00:16:54,760 here is the typical version of the Schrodinger equation. 229 00:16:54,760 --> 00:17:00,550 So recall that the vector potential in general 230 00:17:00,550 --> 00:17:03,610 can be thought of as a four-vector. 231 00:17:03,610 --> 00:17:06,609 You may or may not have seen this in the literal dynamics. 232 00:17:06,609 --> 00:17:11,260 But it's just like time and x form a four-vector. 233 00:17:11,260 --> 00:17:15,640 The scalar potential and the vector potential 234 00:17:15,640 --> 00:17:19,460 form a four-vector. 235 00:17:19,460 --> 00:17:26,305 So the four-vector with index mu, 0, 1, 2, 3 form this. 236 00:17:29,180 --> 00:17:32,700 And then, if you look at the Schrodinger equation, 237 00:17:32,700 --> 00:17:34,710 what do we have? 238 00:17:34,710 --> 00:17:45,000 We have i h bar, d psi, dt, minus q phi-- 239 00:17:45,000 --> 00:17:47,660 I'm bringing the last term on the right-hand side 240 00:17:47,660 --> 00:17:48,680 to the left. 241 00:17:48,680 --> 00:18:00,980 So it's minus q times phi, which is minus A0 psi is equal to 1 242 00:18:00,980 --> 00:18:10,040 over 2m, this i h bar over i gradient, 243 00:18:10,040 --> 00:18:16,080 minus q over c, A, squared, and psi. 244 00:18:19,380 --> 00:18:20,790 And now look at this. 245 00:18:23,340 --> 00:18:26,250 It can be written as follows. 246 00:18:26,250 --> 00:18:48,070 Minus c times h over i, d over d of ct, minus q over c, A0, psi, 247 00:18:48,070 --> 00:18:50,480 equal the same thing on the right-hand side. 248 00:18:50,480 --> 00:18:54,500 So I've rewritten the left-hand side 249 00:18:54,500 --> 00:19:00,250 in a slightly different way, all the terms. 250 00:19:00,250 --> 00:19:06,132 So I put an extra c, this d, dct. 251 00:19:06,132 --> 00:19:07,570 That canceled the c. 252 00:19:07,570 --> 00:19:11,750 The h bar, the i went to the denominator. 253 00:19:11,750 --> 00:19:17,140 And now this all looks like this covariant derivative. 254 00:19:17,140 --> 00:19:20,410 Look at this covariant derivative. h over i, d dx, 255 00:19:20,410 --> 00:19:27,460 minus q over c, A. And here it is, h over i, d dx0-- 256 00:19:27,460 --> 00:19:29,470 because 0 is component of the [INAUDIBLE],, 257 00:19:29,470 --> 00:19:32,110 minus q over c, A0. 258 00:19:32,110 --> 00:19:35,260 So the whole Schrodinger equation 259 00:19:35,260 --> 00:19:38,810 is built with this funny derivatives-- d 260 00:19:38,810 --> 00:19:44,510 dx minus the vector potential added in the net. 261 00:19:44,510 --> 00:19:46,820 These are the covariant derivatives. 262 00:19:46,820 --> 00:19:51,140 These are nice operators. 263 00:19:51,140 --> 00:19:54,260 You see, the operator P is always 264 00:19:54,260 --> 00:19:58,580 called the canonical momentum-- 265 00:19:58,580 --> 00:20:01,785 canonical momentum. 266 00:20:05,570 --> 00:20:11,090 And this canonical momentum is a momentum such that x with P, 267 00:20:11,090 --> 00:20:14,675 if you put the hat, is i h bar. 268 00:20:17,890 --> 00:20:27,400 But this canonical momentum, P, is not mass times velocity, 269 00:20:27,400 --> 00:20:29,380 not at all. 270 00:20:29,380 --> 00:20:34,740 This canonical momentum is a little unintuitive. 271 00:20:34,740 --> 00:20:37,830 It's the one that generates translation. 272 00:20:37,830 --> 00:20:42,150 The one that is mass times velocity is really 273 00:20:42,150 --> 00:20:46,290 this whole combination, is mass times velocity. 274 00:20:46,290 --> 00:20:51,300 Because if it's mass times velocity, this term, 1 over 2m, 275 00:20:51,300 --> 00:20:54,030 the mass squared times velocity squared, 276 00:20:54,030 --> 00:20:56,560 that gives you kinetic energy. 277 00:20:56,560 --> 00:21:02,640 So we have to be aware that the canonical momentum is not 278 00:21:02,640 --> 00:21:09,230 necessarily the simplest, most intuitive object.