1 00:00:00,070 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,800 under a Creative Commons License. 3 00:00:03,800 --> 00:00:06,530 Your support will help MIT OpenCourseWare continue 4 00:00:06,530 --> 00:00:10,130 to offer high quality educational resources for free. 5 00:00:10,130 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:16,120 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,120 --> 00:00:17,090 at ocw.mit.edu. 8 00:00:23,200 --> 00:00:26,000 PROFESSOR: OK, so, we're going to pick up 9 00:00:26,000 --> 00:00:28,564 on our study of periodic potentials 10 00:00:28,564 --> 00:00:29,980 and our search for the explanation 11 00:00:29,980 --> 00:00:32,280 of the physics of solids. 12 00:00:32,280 --> 00:00:35,950 And I want to quickly remind you of the logic 13 00:00:35,950 --> 00:00:37,390 we went through last time. 14 00:00:37,390 --> 00:00:40,470 This is one of the more subtle bits of logic in the semester, 15 00:00:40,470 --> 00:00:42,884 so I'm gonna do it again, a little faster. 16 00:00:42,884 --> 00:00:45,050 I'm mostly gonna do it in a slightly different form. 17 00:00:45,050 --> 00:00:48,504 I want to think for a moment about just a free particle, 18 00:00:48,504 --> 00:00:49,920 but I want to use all the language 19 00:00:49,920 --> 00:00:52,710 and formalism we used for the periodic potential last time-- 20 00:00:52,710 --> 00:00:55,430 translation operators, the block wave functions-- I want 21 00:00:55,430 --> 00:00:57,360 to use all of that for just a free particle. 22 00:00:57,360 --> 00:00:58,901 So, first let's just remind ourselves 23 00:00:58,901 --> 00:01:01,280 how the free particle story goes. 24 00:01:01,280 --> 00:01:09,490 So, for a free particle, by which 25 00:01:09,490 --> 00:01:11,440 I mean the energy operator is p squared 26 00:01:11,440 --> 00:01:15,690 upon 2m, and no potential, plus 0. 27 00:01:15,690 --> 00:01:18,977 We have a very important consequence of being free-- 28 00:01:18,977 --> 00:01:21,310 that the energy can be written as just p squared-- which 29 00:01:21,310 --> 00:01:23,680 is that the energy commutes with the momentum operator. 30 00:01:23,680 --> 00:01:25,440 And what this tells us is that we 31 00:01:25,440 --> 00:01:28,050 can find a basis of eigenfunctions which 32 00:01:28,050 --> 00:01:30,580 are common eigenfunctions of E and p, right? 33 00:01:30,580 --> 00:01:32,330 We can expand every wave function 34 00:01:32,330 --> 00:01:34,980 at any arbitrary-- fantastic. 35 00:01:34,980 --> 00:01:35,690 Thank you, sir. 36 00:01:38,822 --> 00:01:40,470 Oh! 37 00:01:40,470 --> 00:01:41,170 Brilliant! 38 00:01:41,170 --> 00:01:44,551 GUEST: Plan A, Plan B. 39 00:01:44,551 --> 00:01:45,550 PROFESSOR: I don't know. 40 00:01:45,550 --> 00:01:46,760 That's an awful huge straw. 41 00:01:49,450 --> 00:01:51,760 Pay no attention to the device beneath the table. 42 00:01:51,760 --> 00:01:57,660 So, 'cause you know, it's quantum mechanics, right? 43 00:01:57,660 --> 00:02:00,220 Bowl of water-- 44 00:02:00,220 --> 00:02:01,780 So, the energy and momentum commute. 45 00:02:01,780 --> 00:02:05,450 And as a consequence, we can find a basis of eigenfunctions 46 00:02:05,450 --> 00:02:12,460 exists which are common eigenfunction of E, E on phi E, 47 00:02:12,460 --> 00:02:14,840 and I'll label k, by the eigenvalue of k, 48 00:02:14,840 --> 00:02:19,490 is equal to E phi Ek. 49 00:02:19,490 --> 00:02:28,527 And p on phi Ek is equal to h bar k phi Ek. 50 00:02:28,527 --> 00:02:30,485 You'll notice, of course, that I don't label it 51 00:02:30,485 --> 00:02:31,510 by the eigenvalue, but I label it 52 00:02:31,510 --> 00:02:33,700 by something that's determined by the label, the eigenvalue's 53 00:02:33,700 --> 00:02:35,160 determined by the labels, h bar k. 54 00:02:35,160 --> 00:02:38,150 Now, here's the thing, can we allow E and k 55 00:02:38,150 --> 00:02:42,100 to be any values we want independently? 56 00:02:42,100 --> 00:02:45,290 Can I make E, 3, and k, 47? 57 00:02:45,290 --> 00:02:46,650 No. 58 00:02:46,650 --> 00:02:48,280 There's a consistency relation, which 59 00:02:48,280 --> 00:02:50,451 involves satisfying the energy eigenvalue equation. 60 00:02:50,451 --> 00:02:52,700 We actually have to find solutions of these equations. 61 00:02:52,700 --> 00:02:55,120 And once we find the solutions are those equations, 62 00:02:55,120 --> 00:02:57,270 we'll find relations between k and E. 63 00:02:57,270 --> 00:02:59,000 And in particular, for a free particle, 64 00:02:59,000 --> 00:03:02,210 we find that the relation that we need is E is equal to-- 65 00:03:02,210 --> 00:03:05,200 and I will write E sub k-- is equal to h bar squared 66 00:03:05,200 --> 00:03:06,560 k squared upon 2m. 67 00:03:06,560 --> 00:03:11,040 This is also known as omega is equal to h 68 00:03:11,040 --> 00:03:13,500 bar k squared upon 2m. 69 00:03:13,500 --> 00:03:15,630 Just to remind you, the [INAUDIBLE] E 70 00:03:15,630 --> 00:03:19,660 is h bar omega, omega sub k. 71 00:03:19,660 --> 00:03:21,680 That's not the thing I wanted to look at. 72 00:03:21,680 --> 00:03:23,630 This is. 73 00:03:23,630 --> 00:03:26,340 And when we work this way, we find 74 00:03:26,340 --> 00:03:32,470 that the energy eigenvalues, E, as a function of k, 75 00:03:32,470 --> 00:03:33,650 take a determined form. 76 00:03:33,650 --> 00:03:36,066 And that determined form is quite simple, it's a parabola. 77 00:03:38,630 --> 00:03:40,750 Everyone cool with that? 78 00:03:40,750 --> 00:03:45,370 So, if you know k, you know E. But if you know E, 79 00:03:45,370 --> 00:03:47,260 it turns out it's doubly degenerate. 80 00:03:47,260 --> 00:03:49,520 So, if you know E, you're one of two k's. 81 00:03:49,520 --> 00:03:53,170 You could be either the ikx-- oh, sorry, 82 00:03:53,170 --> 00:03:57,720 and I needed to say-- that the function, so, from solving 83 00:03:57,720 --> 00:04:02,520 the equation, gave us that the energy eigenfunctions, 84 00:04:02,520 --> 00:04:07,600 phi Ek are equal to some normalization 1 over 2 85 00:04:07,600 --> 00:04:10,165 pi as conventional, e to the ikx. 86 00:04:12,880 --> 00:04:13,760 OK. 87 00:04:13,760 --> 00:04:15,920 So, if I tell you the energy, you 88 00:04:15,920 --> 00:04:20,004 don't know which momentum state I'm talking about. 89 00:04:20,004 --> 00:04:22,170 But if I tell you the momentum, you know the energy. 90 00:04:22,170 --> 00:04:23,503 So, there's a little degeneracy. 91 00:04:23,503 --> 00:04:26,240 That degeneracy is coming from the parity symmetry, 92 00:04:26,240 --> 00:04:28,580 the system, right? 93 00:04:28,580 --> 00:04:30,080 And note that what we've done here, 94 00:04:30,080 --> 00:04:32,080 did we have to write the wave functions in terms 95 00:04:32,080 --> 00:04:34,140 of these, the ikxes. 96 00:04:34,140 --> 00:04:35,677 Is there another basis we could've 97 00:04:35,677 --> 00:04:37,010 used for the energy eigenstates? 98 00:04:39,911 --> 00:04:40,410 Yeah. 99 00:04:40,410 --> 00:04:41,600 Because these are degenerate, we could've 100 00:04:41,600 --> 00:04:43,724 taken any linear combinations of E to the ikx and E 101 00:04:43,724 --> 00:04:45,280 to the minus ikx, and we could have 102 00:04:45,280 --> 00:04:47,005 got, for example, sine and cosine. 103 00:04:47,005 --> 00:04:48,380 That would have had the advantage 104 00:04:48,380 --> 00:04:50,390 that the energy eigenfunctions would have been pure real, 105 00:04:50,390 --> 00:04:51,889 which is one we proved was possible. 106 00:04:51,889 --> 00:04:54,580 You can always choose pure real energy eigenfunctions. 107 00:04:54,580 --> 00:04:56,480 However, it would have had the disadvantage that these states 108 00:04:56,480 --> 00:04:58,021 would not-- sine and cosine-- are not 109 00:04:58,021 --> 00:05:00,330 eigenstates of momentum, right? 110 00:05:00,330 --> 00:05:02,940 It's convenient to work with common eigenstates of momentum 111 00:05:02,940 --> 00:05:06,140 and energy, so we give up on having a real wave function 112 00:05:06,140 --> 00:05:10,350 and that's the cost of finding shared eigenfunctions of energy 113 00:05:10,350 --> 00:05:11,375 and momentum. 114 00:05:11,375 --> 00:05:12,830 Everyone cool? 115 00:05:12,830 --> 00:05:13,791 Any questions? 116 00:05:13,791 --> 00:05:14,290 OK. 117 00:05:14,290 --> 00:05:17,062 So, this is the usual story for the free particle. 118 00:05:17,062 --> 00:05:19,270 But we learn a couple of things about it immediately. 119 00:05:19,270 --> 00:05:22,310 Just by eyeballing the wave function 120 00:05:22,310 --> 00:05:26,310 we see some things to note. 121 00:05:26,310 --> 00:05:30,760 Note, all states-- all eigenfunctions-- phi sub 122 00:05:30,760 --> 00:05:35,914 E, which are of the form E to the ikxx-- 123 00:05:35,914 --> 00:05:37,830 and I'm gonna add the time dependence in now-- 124 00:05:37,830 --> 00:05:40,935 minus omega sum kt, are extended. 125 00:05:46,240 --> 00:05:48,652 They have equal probability density everywhere, 126 00:05:48,652 --> 00:05:50,360 and the probability distribution does not 127 00:05:50,360 --> 00:05:52,324 fall off to 0 as we go far away. 128 00:05:52,324 --> 00:05:53,990 And so, what this tells us we have to do 129 00:05:53,990 --> 00:05:56,620 is we have to build-- if we want to study real localized 130 00:05:56,620 --> 00:05:59,760 states that are normalizable-- we have to build 131 00:05:59,760 --> 00:06:09,610 wave packets of the form psi is equal to integral dk sum 132 00:06:09,610 --> 00:06:18,110 overall momentum modes f of k E to the ikx minus omega t sub k. 133 00:06:18,110 --> 00:06:20,630 Where this f we will generally take-- although we don't have 134 00:06:20,630 --> 00:06:26,680 to take, to be a real function-- but, is peaked at some k0. 135 00:06:28,955 --> 00:06:31,580 So, this is what we did when we studied the evolution of a wave 136 00:06:31,580 --> 00:06:34,120 packet of this form. 137 00:06:34,120 --> 00:06:36,875 But in particular, such wave packets 138 00:06:36,875 --> 00:06:38,250 have a couple of nice properties. 139 00:06:38,250 --> 00:06:39,849 The first property we've studied. 140 00:06:39,849 --> 00:06:41,390 And if you're not totally comfortable 141 00:06:41,390 --> 00:06:42,610 with the idea of group velocity, you really 142 00:06:42,610 --> 00:06:43,545 need to go over it again. 143 00:06:43,545 --> 00:06:45,711 It's be an excellent thing to show up in recitation, 144 00:06:45,711 --> 00:06:47,080 for example, stationary phase. 145 00:06:47,080 --> 00:06:49,090 So, the group velocity of such a wave packet-- 146 00:06:49,090 --> 00:06:52,080 as long as it's of this form and peaked around a single momentum 147 00:06:52,080 --> 00:07:00,970 value, k0-- the group velocity is d omega dk evaluated at k0, 148 00:07:00,970 --> 00:07:03,830 at the peak value of the distribution. 149 00:07:03,830 --> 00:07:05,750 And note, I wanna just quickly note 150 00:07:05,750 --> 00:07:08,960 that this is equal for the free particle. 151 00:07:08,960 --> 00:07:10,976 Omega is h bar k squared upon 2m, 152 00:07:10,976 --> 00:07:13,100 take a derivative with respect to k, the 2 cancels. 153 00:07:13,100 --> 00:07:14,465 We get h bar k upon m. 154 00:07:18,280 --> 00:07:20,500 Which notably, is equal to the expectation 155 00:07:20,500 --> 00:07:24,730 value of the momentum over the mass. 156 00:07:24,730 --> 00:07:25,860 Everyone cool with that? 157 00:07:25,860 --> 00:07:27,770 So, this gives us a nice way to, by the way, 158 00:07:27,770 --> 00:07:30,370 if you didn't know the mass, but you could measure the system, 159 00:07:30,370 --> 00:07:31,950 for example, you can compute on average the momentum, 160 00:07:31,950 --> 00:07:33,616 and you could compute the group velocity 161 00:07:33,616 --> 00:07:35,170 by watching wave packets move, this 162 00:07:35,170 --> 00:07:37,020 is a nifty way to compute the mass. 163 00:07:37,020 --> 00:07:39,610 The mass can be given by, well, put the mass up here, 164 00:07:39,610 --> 00:07:42,090 and divide by the group velocity. 165 00:07:42,090 --> 00:07:46,390 Measure the momentum and divide by the group velocity. 166 00:07:46,390 --> 00:07:47,970 OK, so, I can take this, and notice, 167 00:07:47,970 --> 00:07:50,950 that this just exactly gives me the mass. 168 00:07:50,950 --> 00:07:52,780 So, this is a way to, given a system 169 00:07:52,780 --> 00:07:55,760 with a momentum and a group velocity, 170 00:07:55,760 --> 00:07:57,750 I can compute an effective mass. 171 00:07:57,750 --> 00:08:00,761 Which for the free particle, is just the mass. 172 00:08:00,761 --> 00:08:01,260 Cool? 173 00:08:01,260 --> 00:08:02,922 Just a side observation. 174 00:08:02,922 --> 00:08:04,630 But this is all a consequence of the fact 175 00:08:04,630 --> 00:08:07,240 that we have wave packets of this form. 176 00:08:07,240 --> 00:08:09,150 Their plane waves, or at least things that 177 00:08:09,150 --> 00:08:12,090 have a phase, times what could be a real-- though it could 178 00:08:12,090 --> 00:08:14,790 but an overall phase on the whole thing-- a real function, 179 00:08:14,790 --> 00:08:17,370 which is peaked at a particular value of k0. 180 00:08:17,370 --> 00:08:20,170 Usual wave packet story. 181 00:08:20,170 --> 00:08:22,651 Any questions? 182 00:08:22,651 --> 00:08:23,150 Good. 183 00:08:23,150 --> 00:08:24,270 So, this should all be pretty familiar. 184 00:08:24,270 --> 00:08:25,824 Now, here's the thing, when we talk 185 00:08:25,824 --> 00:08:27,240 about a periodic potential, what I 186 00:08:27,240 --> 00:08:28,656 want to do next is, I want to talk 187 00:08:28,656 --> 00:08:29,800 about a periodic potential. 188 00:08:29,800 --> 00:08:33,190 So, now, we have a system that has maybe steps, 189 00:08:33,190 --> 00:08:38,304 and it's periodic, with period l, OK? 190 00:08:38,304 --> 00:08:38,970 And then, ditto. 191 00:08:42,030 --> 00:08:43,987 It's gonna be periodic. 192 00:08:43,987 --> 00:08:45,820 And I want to run through the same analysis. 193 00:08:45,820 --> 00:08:47,528 I want to find the energy eigenfunctions. 194 00:08:47,528 --> 00:08:49,721 I want to organize them in a nice way. 195 00:08:49,721 --> 00:08:51,720 And in particular, I want to know the following, 196 00:08:51,720 --> 00:08:57,320 if this is equal to v of x, and v of x, 197 00:08:57,320 --> 00:08:59,670 and I call the energy operator E sub 198 00:08:59,670 --> 00:09:06,570 g is equal to p squared upon 2m plus a coefficient, g, 199 00:09:06,570 --> 00:09:09,482 times-- let me call it g sub 0, just 200 00:09:09,482 --> 00:09:12,090 to make it unambiguous-- times v of x. 201 00:09:12,090 --> 00:09:16,130 Notice that when g0 is 0, this is just the free particle. 202 00:09:16,130 --> 00:09:17,860 I added a periodic potential times 0. 203 00:09:17,860 --> 00:09:21,330 That is stupid, but doable. 204 00:09:21,330 --> 00:09:24,570 But as I slowly turn on g0, what's going to happen? 205 00:09:24,570 --> 00:09:26,900 The energy eigenfunctions are going to slowly change. 206 00:09:26,900 --> 00:09:29,280 And the energy eigenvalues are going to slowly change. 207 00:09:29,280 --> 00:09:31,821 And the actual system we care about has some actual value of, 208 00:09:31,821 --> 00:09:33,380 you know, like 7, or e, or whatever. 209 00:09:33,380 --> 00:09:35,680 But this is a way to connect the free particle 210 00:09:35,680 --> 00:09:39,194 to the particle in a periodic potential. 211 00:09:39,194 --> 00:09:41,610 You know, and if you just turn on a little weak potential, 212 00:09:41,610 --> 00:09:43,609 you don't expect the system to radically change. 213 00:09:43,609 --> 00:09:46,292 So, one way to organize the question of periodic potentials 214 00:09:46,292 --> 00:09:47,750 is, what happens as you slowly turn 215 00:09:47,750 --> 00:09:50,170 on this periodic potential? 216 00:09:50,170 --> 00:09:52,290 But here's the first thing I want to notice, 217 00:09:52,290 --> 00:09:53,748 the first thing I want to notice is 218 00:09:53,748 --> 00:09:56,620 that it's no longer true that the energy commutes 219 00:09:56,620 --> 00:09:57,370 with the momentum. 220 00:10:01,910 --> 00:10:04,050 So, it would be silly to try to organize our wave 221 00:10:04,050 --> 00:10:06,620 functions in terms of energy eigenfunctions, which are also 222 00:10:06,620 --> 00:10:07,840 momentum eigenfunctions. 223 00:10:07,840 --> 00:10:09,589 Because that will only be a sensible thing 224 00:10:09,589 --> 00:10:11,400 to do when this interaction term-- 225 00:10:11,400 --> 00:10:13,490 when this potential-- is 0, right? 226 00:10:13,490 --> 00:10:15,940 So, instead, I want to organize them 227 00:10:15,940 --> 00:10:19,110 in terms of energy eigenvalues and any other operator that 228 00:10:19,110 --> 00:10:21,560 commutes with the energy eigenfunction. 229 00:10:21,560 --> 00:10:23,420 In particular, this is not invariant 230 00:10:23,420 --> 00:10:25,090 under arbitrarily translations, which 231 00:10:25,090 --> 00:10:28,320 it would have to be to be invariant to conserve momentum. 232 00:10:28,320 --> 00:10:31,760 But it is invariant under translations by L. 233 00:10:31,760 --> 00:10:35,970 And as a consequence, we have that the energy operator 234 00:10:35,970 --> 00:10:40,130 commutes with translations by L. Where translations by L. 235 00:10:40,130 --> 00:10:42,890 Takes a function and shifts it over by L. 236 00:10:42,890 --> 00:10:45,401 And so, because translations by L as we showed in a problem 237 00:10:45,401 --> 00:10:47,817 set long ago commutes with p squared, because this is just 238 00:10:47,817 --> 00:10:50,720 an exponential of Lp, with coefficients, with an i 239 00:10:50,720 --> 00:10:51,580 and an h bar. 240 00:10:51,580 --> 00:10:54,940 And because the potential is invariant under shifting by l, 241 00:10:54,940 --> 00:10:57,170 and so it commutes with TL, this is 0. 242 00:10:57,170 --> 00:11:00,180 So, what this tells us is we can find common eigenfunctions. 243 00:11:06,920 --> 00:11:13,760 In fact, a common eigenbasis of E and TL. 244 00:11:13,760 --> 00:11:15,640 And here's the thing that's nice about this, 245 00:11:15,640 --> 00:11:18,015 I could've made this argument whether the potential was 0 246 00:11:18,015 --> 00:11:18,830 or not. 247 00:11:18,830 --> 00:11:20,480 For the free particle, it's also true 248 00:11:20,480 --> 00:11:23,270 that the system commutes with translations by l. 249 00:11:23,270 --> 00:11:24,651 It's free, there is no potential. 250 00:11:24,651 --> 00:11:26,150 It sort of silly, because I could've 251 00:11:26,150 --> 00:11:27,340 just used the momentum. 252 00:11:27,340 --> 00:11:29,180 But let's see what happens if we use 253 00:11:29,180 --> 00:11:30,400 the translate by l operator. 254 00:11:30,400 --> 00:11:31,821 Everyone cool with that? 255 00:11:31,821 --> 00:11:33,570 So, we want to find common eigenfunctions. 256 00:11:36,570 --> 00:11:38,570 So, as you guys showed on a problem set, 257 00:11:38,570 --> 00:11:40,760 the eigenfunctions of TL-- so, in order to do this, 258 00:11:40,760 --> 00:11:42,260 what are the eigenfunctions of TL?-- 259 00:11:42,260 --> 00:11:45,420 you showed that the eigenfunctions of TL must take 260 00:11:45,420 --> 00:11:54,760 the form-- sorry-- phi sub q of x is equal to E to the iqx 261 00:11:54,760 --> 00:12:00,800 times u of x, for some q where q is-- where u-- 262 00:12:00,800 --> 00:12:04,982 is a real periodic-- I don't really need the real, 263 00:12:04,982 --> 00:12:06,440 but it's going to simplify my life, 264 00:12:06,440 --> 00:12:10,190 so I'll take it to be real-- periodic function. 265 00:12:13,297 --> 00:12:14,880 Well, let's just drop real altogether, 266 00:12:14,880 --> 00:12:17,120 because we don't need that. 267 00:12:17,120 --> 00:12:22,440 We'll use a periodic function, and phi of q is this phase. 268 00:12:22,440 --> 00:12:23,890 It times the periodic function. 269 00:12:23,890 --> 00:12:25,780 So in particular, phi is not periodic. 270 00:12:25,780 --> 00:12:29,890 And as a consequence, TL on phi q is equal to E 271 00:12:29,890 --> 00:12:34,480 to the iqL phi q, because this guy shifts, 272 00:12:34,480 --> 00:12:37,569 and this is invariant periodic. 273 00:12:37,569 --> 00:12:39,110 So, this is the form-- general form-- 274 00:12:39,110 --> 00:12:40,776 of the eigenfunctions of translate by l. 275 00:12:44,650 --> 00:12:49,462 And periodic means that u of x plus l is equal to u of x, 276 00:12:49,462 --> 00:12:51,330 just be to explicit. 277 00:12:51,330 --> 00:12:53,050 OK. 278 00:12:53,050 --> 00:13:00,810 So, what this tells us is that the energy eigenfunctions can 279 00:13:00,810 --> 00:13:11,130 be put in a form-- can be organized-- as phi sub Eq, 280 00:13:11,130 --> 00:13:16,240 such that E acting on phi sub Eq is equal to E phi sub Eq. 281 00:13:22,290 --> 00:13:24,150 Just like this line, phi sub Eq. 282 00:13:24,150 --> 00:13:31,585 And TL phi Eq is equal to E to the iqL phi Eq. 283 00:13:31,585 --> 00:13:32,960 And it's just saying then we have 284 00:13:32,960 --> 00:13:35,180 common eigenfunctions, yeah, with energy E 285 00:13:35,180 --> 00:13:36,632 and TL eigenvalue, q. 286 00:13:36,632 --> 00:13:37,548 AUDIENCE: [INAUDIBLE]? 287 00:13:41,380 --> 00:13:44,020 PROFESSOR: No, because under translation by l, 288 00:13:44,020 --> 00:13:45,590 this x goes to x plus l. 289 00:13:45,590 --> 00:13:47,325 Oh, I've slightly changed notation 290 00:13:47,325 --> 00:13:48,450 from earlier in the course. 291 00:13:48,450 --> 00:13:48,991 You're right. 292 00:13:48,991 --> 00:13:49,960 I've slightly switched. 293 00:13:49,960 --> 00:13:51,370 So, ah! 294 00:13:51,370 --> 00:13:51,870 Good. 295 00:13:51,870 --> 00:13:52,790 So, yes. 296 00:13:52,790 --> 00:13:54,130 So, thank you. 297 00:13:54,130 --> 00:13:55,490 So, let's fix notation. 298 00:13:55,490 --> 00:14:01,040 TL takes f of x to f of x minus l, 299 00:14:01,040 --> 00:14:02,630 is what we've used previously, right? 300 00:14:02,630 --> 00:14:04,880 But I'm going to just switch notation, and call this x 301 00:14:04,880 --> 00:14:05,420 plus l. 302 00:14:05,420 --> 00:14:07,860 So, for all the notes in here, this 303 00:14:07,860 --> 00:14:11,959 is x plus L. Yeah, it doesn't really make a difference, 304 00:14:11,959 --> 00:14:13,250 you just have to be consistent. 305 00:14:13,250 --> 00:14:15,420 And you're right, I switched between lecture-- 306 00:14:15,420 --> 00:14:17,990 what was it?-- eight, and this one. 307 00:14:17,990 --> 00:14:20,050 Sorry about that. 308 00:14:20,050 --> 00:14:20,870 Yeah, good catch. 309 00:14:20,870 --> 00:14:23,240 Thank you for pointing that out. 310 00:14:23,240 --> 00:14:26,000 So, in which case, this shifts by x to x plus l. 311 00:14:26,000 --> 00:14:27,370 And we get E to the iqL. 312 00:14:27,370 --> 00:14:27,910 Good catch. 313 00:14:30,870 --> 00:14:31,920 Good. 314 00:14:31,920 --> 00:14:33,670 So, again, we have this property that we 315 00:14:33,670 --> 00:14:34,961 can find common eigenfunctions. 316 00:14:34,961 --> 00:14:39,350 However, can E and q be specified independently? 317 00:14:39,350 --> 00:14:41,350 We know on general grounds that it's 318 00:14:41,350 --> 00:14:42,900 possible to find eigenfunctions which 319 00:14:42,900 --> 00:14:45,377 are common eigenfunctions, because these guys commute-- 320 00:14:45,377 --> 00:14:47,710 E and TL commute-- so we can find common eigenfunctions. 321 00:14:47,710 --> 00:14:48,270 And here they are. 322 00:14:48,270 --> 00:14:50,811 And, meanwhile, we know that the form of these eigenfunctions 323 00:14:50,811 --> 00:14:51,830 is of this form. 324 00:14:51,830 --> 00:14:53,430 Because any eigenfunction of TL is of 325 00:14:53,430 --> 00:14:57,047 this form, a phase times a periodic functions. 326 00:14:57,047 --> 00:14:58,880 Does that, by itself, tell us a relationship 327 00:14:58,880 --> 00:15:00,600 between the eigenvalues E and q? 328 00:15:03,600 --> 00:15:04,420 No, right? 329 00:15:04,420 --> 00:15:06,190 Just like E and k were independent, 330 00:15:06,190 --> 00:15:10,240 until we solved the actual eigenvalue equations, right? 331 00:15:10,240 --> 00:15:13,580 And it required things like regularity, 332 00:15:13,580 --> 00:15:15,540 and no similarities. 333 00:15:15,540 --> 00:15:17,570 So, what we have to do at this point, 334 00:15:17,570 --> 00:15:23,580 is solve the differential equations for continuity, 335 00:15:23,580 --> 00:15:27,087 periodicity, and solving the energy eigenvalue equations. 336 00:15:27,087 --> 00:15:28,670 And when we solve the equations, we'll 337 00:15:28,670 --> 00:15:36,650 get a relationship between E and q. 338 00:15:40,960 --> 00:15:43,810 And in particular, if we do that, 339 00:15:43,810 --> 00:15:52,420 here-- do I want to do it this way? 340 00:15:52,420 --> 00:15:54,830 Yes. 341 00:15:54,830 --> 00:16:01,290 So, in particular for-- write this in the following 342 00:16:01,290 --> 00:16:08,390 way-- for g equals 0-- g0 equls 0, which is a free particle-- 343 00:16:08,390 --> 00:16:16,230 we find E is equal to h bar squared 344 00:16:16,230 --> 00:16:21,270 q squared E sub q is equal to h bar squared q squared upon 2m. 345 00:16:21,270 --> 00:16:26,880 And yeah, and the wave functions are of the form phi 346 00:16:26,880 --> 00:16:32,880 sub qE are equal to some normalization, 1 over 2 pi, 347 00:16:32,880 --> 00:16:36,400 times E to the iqx. 348 00:16:36,400 --> 00:16:37,940 And the coefficient function, u, is 349 00:16:37,940 --> 00:16:41,400 just constant, which is nice. 350 00:16:41,400 --> 00:16:42,210 So, times constant. 351 00:16:46,180 --> 00:16:48,810 And so, this is reassuring because it's exactly 352 00:16:48,810 --> 00:16:50,550 the same form of the wave functions, 353 00:16:50,550 --> 00:16:54,602 with exactly the same form of the energy. 354 00:16:54,602 --> 00:16:56,310 But where here, the label-- the value-- q 355 00:16:56,310 --> 00:17:03,620 is representing the possible value of-- it's 356 00:17:03,620 --> 00:17:05,880 controlling the eigenvalue of TL. 357 00:17:05,880 --> 00:17:07,547 So, there was a question? 358 00:17:07,547 --> 00:17:08,130 AUDIENCE: Yes. 359 00:17:08,130 --> 00:17:09,079 PROFESSOR: Yes? 360 00:17:09,079 --> 00:17:12,452 AUDIENCE: So, for the translation operator-- 361 00:17:12,452 --> 00:17:13,416 PROFESSOR: Yes? 362 00:17:13,416 --> 00:17:15,826 AUDIENCE: --that E iqx equal to x 363 00:17:15,826 --> 00:17:20,342 will work for any complex q in any u with period [INAUDIBLE]? 364 00:17:20,342 --> 00:17:21,800 Or any periodic u with [INAUDIBLE]? 365 00:17:21,800 --> 00:17:23,970 PROFESSOR: Yeah, although, if q is complex, 366 00:17:23,970 --> 00:17:26,619 then this is badly non-normalizable. 367 00:17:26,619 --> 00:17:28,792 Because if q has an imaginary part, 368 00:17:28,792 --> 00:17:30,250 then that function is gonna diverge 369 00:17:30,250 --> 00:17:31,500 in one direction or the other. 370 00:17:31,500 --> 00:17:33,090 So, q had better be real. 371 00:17:33,090 --> 00:17:34,370 But, that's correct. 372 00:17:34,370 --> 00:17:36,160 This works for-- this expression-- 373 00:17:36,160 --> 00:17:39,070 A function of this form, a periodic function-- periodic, 374 00:17:39,070 --> 00:17:41,660 with period L-- times an E to the iqx, 375 00:17:41,660 --> 00:17:44,420 for any real value q, and any periodic u, 376 00:17:44,420 --> 00:17:47,898 will be an eigenfunction of TL, translation. 377 00:17:47,898 --> 00:17:50,007 AUDIENCE: [INAUDIBLE]? 378 00:17:50,007 --> 00:17:50,590 PROFESSOR: No. 379 00:17:50,590 --> 00:17:52,560 It's not just non-hermitian. 380 00:17:52,560 --> 00:17:54,200 It's, in fact, unitary. 381 00:17:54,200 --> 00:17:55,950 I mean, what exactly is your question? 382 00:17:55,950 --> 00:17:57,116 Sorry, I'm misunderstanding. 383 00:17:57,116 --> 00:18:00,366 AUDIENCE: So, I mean, just the translation operator 384 00:18:00,366 --> 00:18:05,400 has a really, really large number of eigenfunctions? 385 00:18:05,400 --> 00:18:07,010 PROFESSOR: Indeed, it does. 386 00:18:07,010 --> 00:18:10,370 And that number is the same as the number 387 00:18:10,370 --> 00:18:13,420 of momentum eigenfunctions. 388 00:18:13,420 --> 00:18:16,160 Because they can be both used as a basis. 389 00:18:16,160 --> 00:18:20,290 They're both operators whose eigenfunctions form a basis. 390 00:18:20,290 --> 00:18:22,024 That's exactly right. 391 00:18:22,024 --> 00:18:23,440 So, I think the difficult thing is 392 00:18:23,440 --> 00:18:26,030 to think that-- we have a tendency 393 00:18:26,030 --> 00:18:28,192 to think that there aren't very many momentum 394 00:18:28,192 --> 00:18:29,150 functions, for example. 395 00:18:29,150 --> 00:18:30,900 But there's enough momentum eigenfunctions 396 00:18:30,900 --> 00:18:31,950 to be complete basis. 397 00:18:31,950 --> 00:18:39,424 So, if this seems like a vastly larger set, then yeah. 398 00:18:39,424 --> 00:18:40,420 Indeed. 399 00:18:40,420 --> 00:18:42,040 OK. 400 00:18:42,040 --> 00:18:45,821 So, when we do our analysis for the free particle of this form, 401 00:18:45,821 --> 00:18:47,070 we get exactly the same story. 402 00:18:47,070 --> 00:18:49,486 And I want to point out a couple of consequences for this. 403 00:18:49,486 --> 00:18:52,090 Working with this TL operation-- or working with TL 404 00:18:52,090 --> 00:18:54,880 eigenfunctions--- we have exactly the same properties. 405 00:18:54,880 --> 00:18:56,670 Our eigenfunctions are, again, extended. 406 00:18:56,670 --> 00:18:57,917 We have to use wave packets. 407 00:18:57,917 --> 00:19:00,000 And those wave packets move with a group velocity, 408 00:19:00,000 --> 00:19:02,970 which is given by d omega dk. 409 00:19:02,970 --> 00:19:07,890 On the other hand, here, when we write super positions, 410 00:19:07,890 --> 00:19:10,223 our general wave function-- our general wave pack, which 411 00:19:10,223 --> 00:19:12,480 I'm gonna write here-- psi, is gonna 412 00:19:12,480 --> 00:19:18,834 be of the form integral dq of E to the iqx 413 00:19:18,834 --> 00:19:26,751 u of x times some f of x-- or sorry, f of q-- u sub q. 414 00:19:29,580 --> 00:19:34,250 So, as long as this is a periodic function, then 415 00:19:34,250 --> 00:19:38,742 the-- oh, and I should say minus omega sub qt, 416 00:19:38,742 --> 00:19:40,950 to get the time dependence in there-- as long as this 417 00:19:40,950 --> 00:19:43,110 is a nice periodic function, especially for this case 418 00:19:43,110 --> 00:19:45,568 was, when it's real, and this is a sharply peaked function, 419 00:19:45,568 --> 00:19:47,960 which is sharply peaked around some q0, 420 00:19:47,960 --> 00:19:55,230 then we will find that the group velocity is, again, the omega, 421 00:19:55,230 --> 00:20:00,149 now dq, evaluated at the peak value, q0. 422 00:20:00,149 --> 00:20:01,690 And that's just a general consequence 423 00:20:01,690 --> 00:20:04,880 of this form of our wave packet. 424 00:20:04,880 --> 00:20:08,120 As I encourage the recitation searchers 425 00:20:08,120 --> 00:20:14,290 to review by stationary phase in recitation. 426 00:20:14,290 --> 00:20:18,000 And secondly, again, we can find this nice notion 427 00:20:18,000 --> 00:20:20,230 of a mass, which is the expectation 428 00:20:20,230 --> 00:20:24,922 value of the momentum divided by the group velocity. 429 00:20:24,922 --> 00:20:26,880 And, again, this, for our simple free particle, 430 00:20:26,880 --> 00:20:28,796 this is just a nice way of measuring the mass, 431 00:20:28,796 --> 00:20:35,350 if you happen to want to do so for your free particle. 432 00:20:35,350 --> 00:20:36,860 No one told you the mass. 433 00:20:36,860 --> 00:20:38,660 Everyone cool with that? 434 00:20:38,660 --> 00:20:39,860 OK. 435 00:20:39,860 --> 00:20:41,952 Questions? 436 00:20:41,952 --> 00:20:42,452 Yeah? 437 00:20:42,452 --> 00:20:46,120 AUDIENCE: Is that still the momentum when we have g0 438 00:20:46,120 --> 00:20:47,060 is not equal to 0? 439 00:20:47,060 --> 00:20:48,310 PROFESSOR: Excellent question! 440 00:20:48,310 --> 00:20:51,071 So, now, let's study the case with g0 not equal to 0. 441 00:20:51,071 --> 00:20:51,570 OK? 442 00:20:51,570 --> 00:20:54,105 So, so far we looked at for g0 no equal to-- or g not equal 443 00:20:54,105 --> 00:20:55,820 to 0-- that was just this line. 444 00:20:55,820 --> 00:20:57,990 But everything else, was for general case 445 00:20:57,990 --> 00:21:00,020 of studying common eigenfunctions of E and TL. 446 00:21:00,020 --> 00:21:02,210 So, now, let's study the case for 447 00:21:02,210 --> 00:21:04,890 an actual periodic potential. 448 00:21:04,890 --> 00:21:11,440 So, in particular, repeat for g0 not equal to 0. 449 00:21:11,440 --> 00:21:13,420 And this is the analysis we did last time. 450 00:21:13,420 --> 00:21:15,059 We did exactly this analysis. 451 00:21:15,059 --> 00:21:16,850 And we looked for the common eigenfunctions 452 00:21:16,850 --> 00:21:19,679 of E and translate by L, right? 453 00:21:19,679 --> 00:21:21,470 And we found that the common eigenfunctions 454 00:21:21,470 --> 00:21:28,580 took the form Eq is equal to E to the iq of x 455 00:21:28,580 --> 00:21:30,330 times some u of x. 456 00:21:33,180 --> 00:21:35,596 And now, we need to find what's the relationship between E 457 00:21:35,596 --> 00:21:36,096 and q? 458 00:21:36,096 --> 00:21:37,739 Are any E and any q allowed? 459 00:21:37,739 --> 00:21:39,280 And do those give you eigenfunctions? 460 00:21:39,280 --> 00:21:40,752 From the free particle, we expect 461 00:21:40,752 --> 00:21:42,210 that that's certainly not the case. 462 00:21:42,210 --> 00:21:43,130 On the other hand, we have a guess 463 00:21:43,130 --> 00:21:44,710 as to what it should look like. 464 00:21:44,710 --> 00:21:47,620 The energy eigenfunction for-- let me, actually, 465 00:21:47,620 --> 00:21:56,650 write this here-- the energy eigenfunctions and energy 466 00:21:56,650 --> 00:22:02,520 eigenvalues, more importantly, of the free particle organized 467 00:22:02,520 --> 00:22:05,880 in terms of q, we just immediately 468 00:22:05,880 --> 00:22:09,310 see the energy is just h bar squared q squared upon 2m. 469 00:22:09,310 --> 00:22:13,644 So, the potential-- or sorry, the energy as a function of q-- 470 00:22:13,644 --> 00:22:14,560 is, again, a parabola. 471 00:22:17,900 --> 00:22:20,200 OK, so, here's q, and here's the energy. 472 00:22:20,200 --> 00:22:22,540 And this is for a free particle, right? 473 00:22:22,540 --> 00:22:24,320 Free particle. 474 00:22:24,320 --> 00:22:27,877 E equals h bar squared q squared upon 2m. 475 00:22:27,877 --> 00:22:29,710 Now, I want to ask what happens as we slowly 476 00:22:29,710 --> 00:22:31,160 turn on the potential? 477 00:22:31,160 --> 00:22:34,280 As we take g0 not equal 0. 478 00:22:34,280 --> 00:22:35,836 So, we did this last time. 479 00:22:35,836 --> 00:22:36,960 We turned on the potential. 480 00:22:36,960 --> 00:22:39,110 We turned on the a potential, which 481 00:22:39,110 --> 00:22:40,519 is, in fact, delta functions. 482 00:22:40,519 --> 00:22:42,810 So, now , in order to solve the differential equations, 483 00:22:42,810 --> 00:22:44,875 we have to actually know what the potential is. 484 00:22:44,875 --> 00:22:46,340 There's no getting around it. 485 00:22:46,340 --> 00:22:48,090 And so, we took a bunch of delta functions 486 00:22:48,090 --> 00:22:53,030 with dimensionless strength g0, and spaced by L. 487 00:22:53,030 --> 00:22:55,460 And what we found was an equation relating 488 00:22:55,460 --> 00:22:56,920 E and L, which took the following 489 00:22:56,920 --> 00:23:01,550 form-- and for simplicity, I'm gonna write E is equal to, this 490 00:23:01,550 --> 00:23:03,780 has nothing to do with momentum, although it looks 491 00:23:03,780 --> 00:23:05,821 like momentum, I'm just going to write E formally 492 00:23:05,821 --> 00:23:10,430 as h bar squared Scripty k squared upon 2m-- so, k, 493 00:23:10,430 --> 00:23:12,442 here, should just be understood as root 2m 494 00:23:12,442 --> 00:23:14,900 over h bar squared, which I hate writing out over, and over 495 00:23:14,900 --> 00:23:15,460 again. 496 00:23:15,460 --> 00:23:17,940 So, let me, actually, write that the other way around. 497 00:23:17,940 --> 00:23:24,260 Scripty k is equal to the square root of 2mE upon h bar squared. 498 00:23:24,260 --> 00:23:32,270 So, when you see k think E. When you see lions, think lions. 499 00:23:32,270 --> 00:23:33,090 OK. 500 00:23:33,090 --> 00:23:35,625 So, we found that the condition-- 501 00:23:35,625 --> 00:23:37,750 in order that we solve the differential equations-- 502 00:23:37,750 --> 00:23:41,520 the condition, in order that E and q label energy eigenstates 503 00:23:41,520 --> 00:23:43,270 which actually are energy eigenfunctions-- 504 00:23:43,270 --> 00:23:45,367 common energy eigenfunctions-- of the energy 505 00:23:45,367 --> 00:23:47,700 and the translation by operator, we found that there was 506 00:23:47,700 --> 00:23:49,450 a relation, which was cosine of qL-- 507 00:23:49,450 --> 00:23:50,520 AUDIENCE: [SNEEZE} 508 00:23:50,520 --> 00:23:56,230 PROFESSOR: --is equal to-- Bless you-- cosine of kL plus g0 509 00:23:56,230 --> 00:24:05,070 upon 2kL sine kL, right? 510 00:24:05,070 --> 00:24:06,830 So, first off, let's make sure that this 511 00:24:06,830 --> 00:24:10,077 agrees with what we found when g0 was equal to 0. 512 00:24:10,077 --> 00:24:11,660 When g0 was equal to 0 we found that E 513 00:24:11,660 --> 00:24:15,801 was equal to h bar squared q squared for any q. 514 00:24:15,801 --> 00:24:18,050 And, well, when g0 is equal to 0, this term goes away. 515 00:24:18,050 --> 00:24:20,580 And we get cosine of qL is equal to cosine of kL. 516 00:24:20,580 --> 00:24:25,060 And an example of a solution for that is qL is equal to kL. 517 00:24:25,060 --> 00:24:26,700 Or q is equal to k. 518 00:24:26,700 --> 00:24:29,380 In which case, E is equal to h bar squared k squared upon 2m, 519 00:24:29,380 --> 00:24:31,150 by plugging in that expression. 520 00:24:31,150 --> 00:24:31,690 Cool? 521 00:24:31,690 --> 00:24:33,559 So, this reproduces the free particle result 522 00:24:33,559 --> 00:24:34,350 when we need it to. 523 00:24:36,920 --> 00:24:40,240 But in general, it's not the same. 524 00:24:40,240 --> 00:24:43,510 So, what does it look like? 525 00:24:43,510 --> 00:24:45,620 So, let's take a look at that. 526 00:24:45,620 --> 00:24:53,250 So, here, first off, what I'm plotting here 527 00:24:53,250 --> 00:24:55,650 is the plot that we draw on the board 528 00:24:55,650 --> 00:24:57,900 last time, only here I'm plotting it here horizontally 529 00:24:57,900 --> 00:25:03,400 is the energy, E, and vertically is cosine of qL. 530 00:25:03,400 --> 00:25:05,210 Now, for q a real number, cosine of qL 531 00:25:05,210 --> 00:25:09,620 goes between 1 and negative 1. 532 00:25:09,620 --> 00:25:11,870 And so, any value in here corresponds 533 00:25:11,870 --> 00:25:14,385 to a valid value of cosine-- or of q-- sorry. 534 00:25:14,385 --> 00:25:17,890 So, any value, any point, between 1 and minus 1 535 00:25:17,890 --> 00:25:19,820 is an allowable value of q. 536 00:25:19,820 --> 00:25:22,380 So, for example, this point, right here, 537 00:25:22,380 --> 00:25:25,940 corresponds to this value is q, OK? 538 00:25:25,940 --> 00:25:27,880 Just horizontal lines. 539 00:25:27,880 --> 00:25:29,780 Meanwhile, this curve is the right hand side 540 00:25:29,780 --> 00:25:30,800 of that expression. 541 00:25:30,800 --> 00:25:34,360 It's cosine of kL plus-- or sorry-- in this case, 542 00:25:34,360 --> 00:25:36,730 it's cosine of root 2mE upon h bar squared 543 00:25:36,730 --> 00:25:40,630 L. This right-hand side plugged in here, 544 00:25:40,630 --> 00:25:42,480 and I'm plotting it as a function of energy. 545 00:25:42,480 --> 00:25:47,690 Just to be explicit, so this is energy-- or energy-- and the q. 546 00:25:47,690 --> 00:25:49,070 And here's that curve. 547 00:25:49,070 --> 00:25:53,350 So, for g equals 0-- which is what I've set it to here-- g 548 00:25:53,350 --> 00:25:55,490 is equal to 0-- that didn't work, 549 00:25:55,490 --> 00:25:57,110 there we go-- when g is equal to 0, 550 00:25:57,110 --> 00:25:59,490 we see that it goes between 0, and this is just 551 00:25:59,490 --> 00:26:01,629 cosine of root E. And there's cosine of root E. 552 00:26:01,629 --> 00:26:03,670 It gets spread out because it's square root of E, 553 00:26:03,670 --> 00:26:07,040 rather than E. Everyone cool with that? 554 00:26:07,040 --> 00:26:11,328 And so, here's my question, what pairs of E and q 555 00:26:11,328 --> 00:26:13,161 correspond to allowed energy eigenfunctions? 556 00:26:15,501 --> 00:26:17,250 How do you look at this diagram and decide 557 00:26:17,250 --> 00:26:21,220 which values of E and q are good energy eigenfunctions? 558 00:26:21,220 --> 00:26:23,428 AUDIENCE: Well, the fact that the boundary conditions 559 00:26:23,428 --> 00:26:28,170 of the q is such that cosine of qL is between negative 1 and 1? 560 00:26:28,170 --> 00:26:29,080 PROFESSOR: Good 561 00:26:29,080 --> 00:26:30,880 AUDIENCE: Or all values are allowed? 562 00:26:30,880 --> 00:26:31,240 PROFESSOR: Awesome. 563 00:26:31,240 --> 00:26:32,010 So, let me be explicit. 564 00:26:32,010 --> 00:26:32,540 So, how would I find? 565 00:26:32,540 --> 00:26:33,155 That's exactly right. 566 00:26:33,155 --> 00:26:35,280 Let me rephrase that and be a little more explicit. 567 00:26:35,280 --> 00:26:38,514 So, what we need is we need a solution of cosine qL 568 00:26:38,514 --> 00:26:40,430 is equal to the right hand side, in this case, 569 00:26:40,430 --> 00:26:43,592 cosine of root energy, L. So, that 570 00:26:43,592 --> 00:26:45,800 means we need to find a horizontal line corresponding 571 00:26:45,800 --> 00:26:47,883 to a particular value of q-- or a particular value 572 00:26:47,883 --> 00:26:51,680 of cosine qL-- and a point on the curve, cosine 573 00:26:51,680 --> 00:26:54,640 of root EL, which is this guy. 574 00:26:54,640 --> 00:26:57,160 So, at any point where we are inside these two 575 00:26:57,160 --> 00:26:59,760 extreme values, and this E, will give us a value. 576 00:26:59,760 --> 00:27:01,720 So, for example, for this point-- ah, 577 00:27:01,720 --> 00:27:06,000 I wish I could draw-- for this point, that corresponds 578 00:27:06,000 --> 00:27:09,110 to a particular value of q, right? 579 00:27:09,110 --> 00:27:11,360 Where the cosine of q is equal to that vertical value. 580 00:27:11,360 --> 00:27:14,060 And it corresponds to a particular value of E. 581 00:27:14,060 --> 00:27:16,785 So, each point, here, corresponds to a value of E-- 582 00:27:16,785 --> 00:27:18,890 an allowed value of E-- on the horizontal, 583 00:27:18,890 --> 00:27:22,005 and an allowed value of q, such that cosine qL 584 00:27:22,005 --> 00:27:24,120 is the vertical value. 585 00:27:24,120 --> 00:27:25,280 Everyone cool with that? 586 00:27:25,280 --> 00:27:27,215 So, what val-- Yeah? 587 00:27:27,215 --> 00:27:31,015 AUDIENCE: [INAUDIBLE] p degeneracy for values of q. 588 00:27:31,015 --> 00:27:32,677 Like for a specific q, we're going 589 00:27:32,677 --> 00:27:36,215 to find a lot of [INAUDIBLE]? 590 00:27:36,215 --> 00:27:38,781 PROFESSOR: For specific values of q, why does it-- 591 00:27:38,781 --> 00:27:40,689 AUDIENCE: [INAUDIBLE] horizontal line. 592 00:27:40,689 --> 00:27:42,572 You'll have a lot of intersections, right? 593 00:27:42,572 --> 00:27:44,030 PROFESSOR: Yeah, that's, excellent. 594 00:27:44,030 --> 00:27:44,240 OK. 595 00:27:44,240 --> 00:27:44,940 So, good. 596 00:27:44,940 --> 00:27:47,144 So, there are two ways of answering that. 597 00:27:47,144 --> 00:27:48,310 That's a very good question. 598 00:27:48,310 --> 00:27:49,770 So, the question is, look, this is 599 00:27:49,770 --> 00:27:51,160 slightly funny state of affairs. 600 00:27:51,160 --> 00:27:53,880 If we plot-- if we take a horizontal line-- corresponding 601 00:27:53,880 --> 00:27:56,540 to a particular value q, it hits at various different values 602 00:27:56,540 --> 00:27:57,900 of the energy. 603 00:27:57,900 --> 00:27:58,817 Excellent observation. 604 00:27:58,817 --> 00:28:00,400 So, let me make two points about that. 605 00:28:00,400 --> 00:28:05,970 The first point is that when we defined q, how did we define q? 606 00:28:05,970 --> 00:28:11,920 We said the translate be L on phi Eq was equal to E 607 00:28:11,920 --> 00:28:15,690 to the iqL phi eq. 608 00:28:15,690 --> 00:28:23,160 But now, suppose we take q to q plus 2pi upon L, OK? 609 00:28:23,160 --> 00:28:25,710 OK, so if we take q to q plus 2pi upon L, 610 00:28:25,710 --> 00:28:27,980 we don't do anything to the eigenvalue. 611 00:28:27,980 --> 00:28:29,980 So, that's not enough to specify which 612 00:28:29,980 --> 00:28:31,714 function we're talking about. 613 00:28:31,714 --> 00:28:33,130 Because we changed the value of q, 614 00:28:33,130 --> 00:28:33,900 and it doesn't change the eigenvalue. 615 00:28:33,900 --> 00:28:35,691 So, you can't just focus on the eigenvalue. 616 00:28:35,691 --> 00:28:38,240 There are two ways of thinking about that. 617 00:28:38,240 --> 00:28:41,470 One is, look, since q and q plus 2 pi upon L 618 00:28:41,470 --> 00:28:45,420 give you the same eigenvalue of TL, 619 00:28:45,420 --> 00:28:48,250 we should just think of them as equivalent. 620 00:28:48,250 --> 00:28:50,480 So, one, we should just-- look, any time you have q, 621 00:28:50,480 --> 00:28:56,449 and you add plus 2pi upon L, you just think of this as q. 622 00:28:56,449 --> 00:28:57,240 It's not different. 623 00:28:57,240 --> 00:29:00,230 There's just, you know, q now is valued periodic. 624 00:29:00,230 --> 00:29:02,090 If you shift q by 2 pi, you're actually back 625 00:29:02,090 --> 00:29:02,839 at the same point. 626 00:29:02,839 --> 00:29:05,650 So, q is now a periodic variable. 627 00:29:05,650 --> 00:29:07,395 And but, that's not enough. 628 00:29:07,395 --> 00:29:09,520 Different values of these q correspond to different 629 00:29:09,520 --> 00:29:11,580 values-- the same eigenvalue. 630 00:29:11,580 --> 00:29:13,890 So, you're getting multiple solutions as a consequence. 631 00:29:13,890 --> 00:29:16,590 But there's another way to think about this. 632 00:29:16,590 --> 00:29:18,990 Which is just in terms of this graph. 633 00:29:18,990 --> 00:29:25,560 In terms of this graph, if what we're specifying 634 00:29:25,560 --> 00:29:30,490 is cosine of qL, but we're not specifying the value for q, 635 00:29:30,490 --> 00:29:31,140 yeah? 636 00:29:31,140 --> 00:29:42,300 So, if we instead plotted this as a function of q, 637 00:29:42,300 --> 00:29:50,610 without assuming q is equivalent to 2pi q plus 2pi 638 00:29:50,610 --> 00:29:53,210 upon L-- which we can do, we're perfectly entitled 639 00:29:53,210 --> 00:29:54,490 to do that-- what do we get? 640 00:29:59,720 --> 00:30:01,960 What do you think you get? 641 00:30:01,960 --> 00:30:04,000 For the free particle, we know exactly what 642 00:30:04,000 --> 00:30:07,320 we get for the free particle. 643 00:30:07,320 --> 00:30:07,900 The parabola. 644 00:30:07,900 --> 00:30:10,275 And if you just take this, and I'm going to give you guys 645 00:30:10,275 --> 00:30:12,066 the Mathematica files for all these things. 646 00:30:12,066 --> 00:30:15,310 Yeah, if you just take this, and open it up, 647 00:30:15,310 --> 00:30:17,310 instead of plotting it in terms of cosine of qL, 648 00:30:17,310 --> 00:30:20,150 plot it in terms of q, that's exactly where you get. 649 00:30:20,150 --> 00:30:22,144 The satisfying thing. 650 00:30:22,144 --> 00:30:24,310 And I'm going to come back to that in just a minute, 651 00:30:24,310 --> 00:30:25,685 but it was a really good question 652 00:30:25,685 --> 00:30:27,870 and we'll come back to it. 653 00:30:27,870 --> 00:30:30,050 Other questions? 654 00:30:30,050 --> 00:30:35,380 OK, now, let's ask what happens as we turn on g0. 655 00:30:35,380 --> 00:30:37,730 As we turn on g0, the solutions to this equation 656 00:30:37,730 --> 00:30:39,355 are going to be different, because it's 657 00:30:39,355 --> 00:30:40,830 going to be a difference equation. 658 00:30:40,830 --> 00:30:41,410 So, let's do it. 659 00:30:41,410 --> 00:30:42,035 So, here it is. 660 00:30:42,035 --> 00:30:43,930 We have g0, we can tune it. 661 00:30:43,930 --> 00:30:46,380 And I'm going to slowly turn on g0. 662 00:30:46,380 --> 00:30:48,020 And watch what happens to these curves. 663 00:30:50,914 --> 00:30:52,330 So, now we see that the right hand 664 00:30:52,330 --> 00:30:57,480 side of the yellowish curve is now exceeding 1 and minus 1 665 00:30:57,480 --> 00:31:00,040 in various places, right? 666 00:31:00,040 --> 00:31:02,530 And why is it doing that? 667 00:31:02,530 --> 00:31:05,476 Let's get some reasonable down here. 668 00:31:05,476 --> 00:31:06,600 So, why is it doing that's? 669 00:31:06,600 --> 00:31:11,300 It's doing that because this can now be greater than 1. 670 00:31:11,300 --> 00:31:13,840 Cosine is bounded between 1 and minus 1, but this is not. 671 00:31:16,680 --> 00:31:18,180 And so, we overshoot in some places. 672 00:31:18,180 --> 00:31:19,480 And what does this tell us? 673 00:31:19,480 --> 00:31:21,620 What does this really telling us? 674 00:31:21,620 --> 00:31:24,470 What its really telling us is that-- there we 675 00:31:24,470 --> 00:31:27,536 go-- how do we find allowed values of E and q 676 00:31:27,536 --> 00:31:29,660 such that we have a good energy eigenfunction which 677 00:31:29,660 --> 00:31:32,830 is simultaneously and eigenfunction of E and of TL? 678 00:31:32,830 --> 00:31:35,880 We need to find points where a horizontal line between 1 679 00:31:35,880 --> 00:31:39,010 and minus 1 intersects the right hand side curve. 680 00:31:39,010 --> 00:31:42,320 But for the energies between here-- 681 00:31:42,320 --> 00:31:44,700 this value of the energy, and this value of the energy--- 682 00:31:44,700 --> 00:31:47,390 there is no such point. 683 00:31:47,390 --> 00:31:48,802 The energies in here-- any energy 684 00:31:48,802 --> 00:31:51,010 in here-- in order for those lines to intersect, line 685 00:31:51,010 --> 00:31:54,350 must be greater than 1, or less than minus 1 down here, 686 00:31:54,350 --> 00:31:56,480 or here. 687 00:31:56,480 --> 00:31:59,404 So, there's no allowed value of q, 688 00:31:59,404 --> 00:32:01,820 such that there's a solution with energy between these two 689 00:32:01,820 --> 00:32:03,220 points. 690 00:32:03,220 --> 00:32:05,310 Everyone cool with that? 691 00:32:05,310 --> 00:32:11,290 So, that's telling us that there are values of energy where 692 00:32:11,290 --> 00:32:14,850 there are no energy eigenvalues with that corresponding energy. 693 00:32:14,850 --> 00:32:16,450 Because there simply aren't solutions 694 00:32:16,450 --> 00:32:21,440 of this equation for any value of q with that energy. 695 00:32:21,440 --> 00:32:23,350 On the other hand, when the right hand side 696 00:32:23,350 --> 00:32:27,430 is between 1 and minus 1, we have allowed values of energy. 697 00:32:27,430 --> 00:32:29,327 We have a value of q, and a value of E 698 00:32:29,327 --> 00:32:31,410 that correspond to each other, and they correspond 699 00:32:31,410 --> 00:32:34,860 to a solution of the energy eigenvalue equation and the TL 700 00:32:34,860 --> 00:32:38,340 eigenvalue equation. 701 00:32:38,340 --> 00:32:43,436 So, let's see what that looks like in this presentation. 702 00:32:43,436 --> 00:32:47,500 Oh, before I do, I want to say one other thing about this. 703 00:32:47,500 --> 00:32:52,150 Let's take this plot, and let's use this periodicity. 704 00:32:52,150 --> 00:32:54,970 So, this periodicity is going to say that-- So, here's 705 00:32:54,970 --> 00:32:59,475 pi upon L, here's minus pi upon L-- let's see, yeah, 706 00:32:59,475 --> 00:33:06,450 good-- there's minus pi upon L, there's 2pi upon L, 707 00:33:06,450 --> 00:33:09,060 and 2pi upon L. If these guys are periodic, 708 00:33:09,060 --> 00:33:11,180 than I could've just said, look, at this point-- 709 00:33:11,180 --> 00:33:14,832 this is pi upon L, and minus pi upon L-- 710 00:33:14,832 --> 00:33:17,040 I could've said this point is the same as this point. 711 00:33:17,040 --> 00:33:19,081 So, I could've taken this whole bit of the curve, 712 00:33:19,081 --> 00:33:21,290 and I could've moved it over here. 713 00:33:21,290 --> 00:33:22,520 Everyone cool with that? 714 00:33:22,520 --> 00:33:25,100 Because this value of q, is the same as this value, q. 715 00:33:25,100 --> 00:33:28,329 This value of q, is the same as this value of q, right? 716 00:33:28,329 --> 00:33:30,870 So, I could have written this, and if we just take this over, 717 00:33:30,870 --> 00:33:33,030 you can see it's, you know, symmetric. 718 00:33:33,030 --> 00:33:41,850 As a consequence, we get-- And so, we 719 00:33:41,850 --> 00:33:46,610 can write-- we can draw-- this entire parabola folded up 720 00:33:46,610 --> 00:33:49,640 into one region, using the periodicity with q as 2pi 721 00:33:49,640 --> 00:33:53,020 upon L. And when you fold it up into one region, 722 00:33:53,020 --> 00:33:55,122 just using q as periodic with period 2pi over L, 723 00:33:55,122 --> 00:33:57,080 and now noting that there are several solutions 724 00:33:57,080 --> 00:33:59,860 for every value q-- corresponding to how far out 725 00:33:59,860 --> 00:34:01,374 on the parabola you went-- now we 726 00:34:01,374 --> 00:34:03,790 have a much simpler way of presenting this, where we don't 727 00:34:03,790 --> 00:34:05,400 need to see the whole parabola. 728 00:34:05,400 --> 00:34:06,980 This is a real advantage. 729 00:34:06,980 --> 00:34:10,310 When we plot things in this way, the structure 730 00:34:10,310 --> 00:34:12,530 of the energy bands and energy gaps 731 00:34:12,530 --> 00:34:15,100 becomes considerably more simple. 732 00:34:15,100 --> 00:34:16,420 So, let me do that. 733 00:34:16,420 --> 00:34:19,600 First let me set the g to 0. 734 00:34:19,600 --> 00:34:22,493 OK, so, here is our free particle. 735 00:34:22,493 --> 00:34:23,659 And let's look at that plot. 736 00:34:23,659 --> 00:34:26,659 So, here's the plot that I was just describing. 737 00:34:26,659 --> 00:34:31,120 What you see here, is the parabola. 738 00:34:31,120 --> 00:34:33,310 And I've use a periodicity to box it up 739 00:34:33,310 --> 00:34:36,280 into one fundamental period between pi and minus pi 740 00:34:36,280 --> 00:34:38,020 upon L, OK? 741 00:34:38,020 --> 00:34:39,980 And so, the blue line corresponds 742 00:34:39,980 --> 00:34:42,155 to the energy as a function of q-- 743 00:34:42,155 --> 00:34:43,780 the allowed energy as a function of q-- 744 00:34:43,780 --> 00:34:45,540 and what the green background represents 745 00:34:45,540 --> 00:34:47,880 is I'm going to put the horizontal section in green 746 00:34:47,880 --> 00:34:50,139 anytime there's an allowed energy with that. 747 00:34:50,139 --> 00:34:51,954 So, energy, here, is vertical, and q 748 00:34:51,954 --> 00:34:54,370 is on the horizontal, going between minus pi over L and pi 749 00:34:54,370 --> 00:34:56,294 over L. Yeah? 750 00:34:56,294 --> 00:34:58,679 AUDIENCE: I might be getting this just out, 751 00:34:58,679 --> 00:35:04,950 but is there a reason that as the energy increases the bands 752 00:35:04,950 --> 00:35:06,270 get broader and broader? 753 00:35:06,270 --> 00:35:07,425 Is that just an artifact of the math, 754 00:35:07,425 --> 00:35:08,900 or is there a good physical reason for that? 755 00:35:08,900 --> 00:35:10,191 PROFESSOR: Yeah, there're both. 756 00:35:10,191 --> 00:35:15,700 So, one way to think about it, so just purely mathematically, 757 00:35:15,700 --> 00:35:19,510 this is saying that cosine of qL is cosine of root EL. 758 00:35:19,510 --> 00:35:22,270 So, as we make the energy larger, and larger, 759 00:35:22,270 --> 00:35:23,780 some small variation in q is going 760 00:35:23,780 --> 00:35:26,550 to correspond to a quadratic variation in E. 761 00:35:26,550 --> 00:35:28,330 And that's what's making it stretched out. 762 00:35:28,330 --> 00:35:31,560 If we plotted this as a function of kL, instead of E, 763 00:35:31,560 --> 00:35:35,390 then it would have looked much more constant period, OK? 764 00:35:35,390 --> 00:35:37,120 So, that's the mathematical answer. 765 00:35:37,120 --> 00:35:40,487 The physical answer is this, if we have some potential, 766 00:35:40,487 --> 00:35:42,570 if I put you in a finite well-- So, there you are. 767 00:35:42,570 --> 00:35:43,234 You're in a finite well. 768 00:35:43,234 --> 00:35:45,317 And you look at the ceiling, you're like, damn it! 769 00:35:45,317 --> 00:35:47,130 And so, that's frustrating, right? 770 00:35:47,130 --> 00:35:49,350 And it's deeply frustrating if it's a deep well, 771 00:35:49,350 --> 00:35:51,724 but if it's a shallow well, it's not all that big a deal, 772 00:35:51,724 --> 00:35:53,560 you climb out. 773 00:35:53,560 --> 00:35:56,170 Being in a deep potential-- being at low energy, 774 00:35:56,170 --> 00:35:58,000 compared to the potential-- means 775 00:35:58,000 --> 00:35:59,640 that you're tightly bound, right? 776 00:35:59,640 --> 00:36:01,350 Takes a huge amount of energy to get out. 777 00:36:01,350 --> 00:36:02,810 And as a consequence, what you saw in your problem 778 00:36:02,810 --> 00:36:04,396 set is that the bands are quite thin. 779 00:36:04,396 --> 00:36:06,270 But as you get closer, and closer to the top, 780 00:36:06,270 --> 00:36:08,650 or as you go to higher energy, the thickness of that band 781 00:36:08,650 --> 00:36:10,710 grows as a measure of the fact that you're 782 00:36:10,710 --> 00:36:13,920 less tightly constrained by the potential. 783 00:36:13,920 --> 00:36:15,340 OK? 784 00:36:15,340 --> 00:36:15,840 Good. 785 00:36:15,840 --> 00:36:17,256 So, that's the physical intuition. 786 00:36:17,256 --> 00:36:18,490 Did that make sense? 787 00:36:18,490 --> 00:36:19,460 Good. 788 00:36:19,460 --> 00:36:21,010 OK, so, yeah? 789 00:36:21,010 --> 00:36:24,930 AUDIENCE: [INAUDIBLE] g0 different from 0, 790 00:36:24,930 --> 00:36:27,912 do you ever get to the point where the solution is always 791 00:36:27,912 --> 00:36:30,804 inside the [INAUDIBLE]. 792 00:36:30,804 --> 00:36:32,732 So, like the bumps [INAUDIBLE]. 793 00:36:38,516 --> 00:36:39,470 PROFESSOR: Ah! 794 00:36:39,470 --> 00:36:39,970 Good. 795 00:36:39,970 --> 00:36:40,470 Let's look. 796 00:36:40,470 --> 00:36:41,760 Let's look. 797 00:36:41,760 --> 00:36:43,190 So, let's answer that by looking. 798 00:36:43,190 --> 00:36:45,961 So, the question is, basically, look, as you crank up g0, 799 00:36:45,961 --> 00:36:47,460 do you ever lose the bands entirely, 800 00:36:47,460 --> 00:36:48,940 or do the bands just disappear? 801 00:36:48,940 --> 00:36:50,114 Is that the question? 802 00:36:50,114 --> 00:36:51,780 AUDIENCE: As you get to higher energies. 803 00:36:51,780 --> 00:36:53,070 PROFESSOR: Yeah, as you go to higher energies. 804 00:36:53,070 --> 00:36:53,570 Good. 805 00:36:53,570 --> 00:36:54,760 So, we can answer that. 806 00:36:54,760 --> 00:36:56,142 Let me do that in two ways. 807 00:36:56,142 --> 00:36:57,600 So, the first thing I want to do is 808 00:36:57,600 --> 00:36:59,760 I want to ask the first of that question, which 809 00:36:59,760 --> 00:37:01,594 is what happens as we turn on the potential? 810 00:37:01,594 --> 00:37:04,093 Here we have the free particle, let's turn on the potential. 811 00:37:04,093 --> 00:37:05,529 We know what happens down here, I 812 00:37:05,529 --> 00:37:08,070 want to ask what happens to the picture of the allowed values 813 00:37:08,070 --> 00:37:09,650 of q and the allowed value-- sorry, 814 00:37:09,650 --> 00:37:12,191 the allowed values of energy-- and their corresponding values 815 00:37:12,191 --> 00:37:12,720 of q. 816 00:37:12,720 --> 00:37:13,553 So, here, it's easy. 817 00:37:13,553 --> 00:37:15,180 This particular point on the blue line 818 00:37:15,180 --> 00:37:19,602 corresponds to a value of the energy, and a value of q. 819 00:37:19,602 --> 00:37:21,060 What happens to those allowed pairs 820 00:37:21,060 --> 00:37:23,375 as we increase the potential? 821 00:37:23,375 --> 00:37:24,750 And as we increase the potential, 822 00:37:24,750 --> 00:37:28,580 you can see on the left that the right hand 823 00:37:28,580 --> 00:37:30,340 side is exceeding 1 and minus 1. 824 00:37:30,340 --> 00:37:31,810 And correspondingly, on the right, 825 00:37:31,810 --> 00:37:35,630 here the red dot is what was the free particle-- parabola-- 826 00:37:35,630 --> 00:37:37,890 and the blue line is the actual solutions. 827 00:37:37,890 --> 00:37:42,360 What you find is when you turn on the potential, 828 00:37:42,360 --> 00:37:44,900 the actual solution moves away from the free particle 829 00:37:44,900 --> 00:37:46,500 line in a very particular way. 830 00:37:46,500 --> 00:37:49,980 In particular, for this value of energy-- around 100-- 831 00:37:49,980 --> 00:37:52,192 there are absolutely no allowed energy eigenstates. 832 00:37:52,192 --> 00:37:53,150 There are no solutions. 833 00:37:53,150 --> 00:37:55,524 And so as correspondingly, that area is not shaded green. 834 00:37:55,524 --> 00:37:58,720 There are only solutions where you have this blue line. 835 00:37:58,720 --> 00:38:01,150 Everyone see that? 836 00:38:01,150 --> 00:38:02,190 So, we get these gaps. 837 00:38:02,190 --> 00:38:04,860 And we get bands of continuously allowed energies. 838 00:38:04,860 --> 00:38:08,630 And then gaps between those allowed bands. 839 00:38:08,630 --> 00:38:12,930 And I'm going to post this-- the guy-- to play with on Stellar. 840 00:38:12,930 --> 00:38:14,940 So, now, let's go up to very, very large values 841 00:38:14,940 --> 00:38:15,902 of the interaction. 842 00:38:15,902 --> 00:38:17,360 OK, and let's see what happens when 843 00:38:17,360 --> 00:38:19,437 you go to large of interaction. 844 00:38:19,437 --> 00:38:21,770 Well, one thing that happens it that Mathematica sort of 845 00:38:21,770 --> 00:38:24,000 panics down here, so it lost an entire band, which 846 00:38:24,000 --> 00:38:25,034 shouldn't happen. 847 00:38:25,034 --> 00:38:26,200 Let's try this, there we go. 848 00:38:26,200 --> 00:38:32,670 That's better There's a wonderful book 849 00:38:32,670 --> 00:38:35,580 on spectral methods in solving differential equations 850 00:38:35,580 --> 00:38:37,980 by a guy named Boyd. 851 00:38:37,980 --> 00:38:41,450 And he's pithy, if somewhat degenerate, 852 00:38:41,450 --> 00:38:45,520 and he says at one point, the definition of an idiot 853 00:38:45,520 --> 00:38:48,570 is-- or he says, idiot definition, 854 00:38:48,570 --> 00:38:51,470 someone who doesn't reproduce their numerics twice 855 00:38:51,470 --> 00:38:52,910 with different parameter values. 856 00:38:52,910 --> 00:38:55,560 And so, we could have just lost this band and written a paper, 857 00:38:55,560 --> 00:38:56,490 oh, band's disappear! 858 00:38:56,490 --> 00:38:57,747 This is great! 859 00:38:57,747 --> 00:38:59,830 Never believe Mathematica until you've checked it. 860 00:38:59,830 --> 00:39:03,407 So, anyway, here you see a band, and another band, and another. 861 00:39:03,407 --> 00:39:05,490 But what's happened is the bands become very thin. 862 00:39:05,490 --> 00:39:07,060 But that makes sense from the earlier intuition. 863 00:39:07,060 --> 00:39:08,768 We're making the potential much stronger. 864 00:39:08,768 --> 00:39:11,430 We're more tightly binding particles into wells. 865 00:39:11,430 --> 00:39:14,085 And as a consequence, the bands are very tightly restricted. 866 00:39:14,085 --> 00:39:15,460 And that fits very much with what 867 00:39:15,460 --> 00:39:17,589 we saw on your phet simulations. 868 00:39:17,589 --> 00:39:19,380 OK, but-- hold on for one sec-- coming back 869 00:39:19,380 --> 00:39:21,916 to the question over here-- yours-- coming back 870 00:39:21,916 --> 00:39:24,540 to that question, if you look at higher and higher energy, what 871 00:39:24,540 --> 00:39:27,390 you're going to find is that the bands get wider and wider, 872 00:39:27,390 --> 00:39:30,367 and the gaps get thinner and thinner, 873 00:39:30,367 --> 00:39:32,700 as you go to high-- well, it's a little more complicated 874 00:39:32,700 --> 00:39:34,825 than that, it depends on exactly what you're doing, 875 00:39:34,825 --> 00:39:37,130 but-- the bands get wider, the gaps get wider, 876 00:39:37,130 --> 00:39:40,299 but both of them don't get wider at the same rate. 877 00:39:40,299 --> 00:39:41,840 So, this is actually something you'll 878 00:39:41,840 --> 00:39:43,070 look at on your problem set. 879 00:39:43,070 --> 00:39:46,550 But the important thing you will find 880 00:39:46,550 --> 00:39:48,950 is that the bands never disappear. 881 00:39:48,950 --> 00:39:52,400 Like a band in 1D doesn't just close and disappear altogether, 882 00:39:52,400 --> 00:39:54,201 and the bands don't overlap. 883 00:39:54,201 --> 00:39:55,700 And here's an easy way to understand 884 00:39:55,700 --> 00:39:57,640 why the bands don't overlap. 885 00:39:57,640 --> 00:39:59,994 Let's look at what these states are. 886 00:39:59,994 --> 00:40:01,910 So in particular, I want to look at the states 887 00:40:01,910 --> 00:40:04,830 at the bottom of each loud energy band. 888 00:40:04,830 --> 00:40:07,890 So, let's go back down to small value of g0. 889 00:40:07,890 --> 00:40:09,930 So, we have these big thick bands 890 00:40:09,930 --> 00:40:11,447 with a weak little potential. 891 00:40:11,447 --> 00:40:13,280 We have big, thick bands, little, tiny gaps, 892 00:40:13,280 --> 00:40:15,696 and I want to look at the states at the bottom of the gap. 893 00:40:15,696 --> 00:40:17,830 Let's make it maybe a little bit bigger. 894 00:40:17,830 --> 00:40:20,080 So, we'll look at the states at the bottom of the gap. 895 00:40:20,080 --> 00:40:22,204 And there's an optional problem in your problem set 896 00:40:22,204 --> 00:40:23,960 that works through the mechanics of this. 897 00:40:23,960 --> 00:40:26,340 But I want to look through it in some detail. 898 00:40:26,340 --> 00:40:34,345 So, what we're plotting here-- Did that work? 899 00:40:41,021 --> 00:40:41,520 Crap. 900 00:40:49,170 --> 00:40:54,230 Unfortunately, this is-- OK, good. 901 00:40:54,230 --> 00:40:56,610 So, here what we have is the ground state 902 00:40:56,610 --> 00:40:58,240 of the lowest band at zero interaction. 903 00:40:58,240 --> 00:41:00,100 So, here, is g0. 904 00:41:00,100 --> 00:41:02,750 And it's 0.001, it's basically 0. 905 00:41:02,750 --> 00:41:04,900 As I turn on the delta function potentials, 906 00:41:04,900 --> 00:41:07,233 what's going to happen to the lowest energy state, which 907 00:41:07,233 --> 00:41:09,400 is just the constant, has momentum 0. 908 00:41:09,400 --> 00:41:11,140 What's going to happen to this guy? 909 00:41:11,140 --> 00:41:14,830 Well, we saw from this guy, that the lowest energy state 910 00:41:14,830 --> 00:41:16,925 got a little bit of a gap there. 911 00:41:16,925 --> 00:41:18,730 There is no state at 0 energy anymore. 912 00:41:18,730 --> 00:41:21,296 There's now a state only at slightly positive energy. 913 00:41:21,296 --> 00:41:23,920 A wave from what would have been the zero energy, zero momentum 914 00:41:23,920 --> 00:41:24,820 state. 915 00:41:24,820 --> 00:41:26,147 Everyone see that? 916 00:41:26,147 --> 00:41:28,230 There's this little gap between 0, and the bottom. 917 00:41:28,230 --> 00:41:29,816 And if we make the interaction bigger, 918 00:41:29,816 --> 00:41:31,440 we'll see that a little more obviously. 919 00:41:31,440 --> 00:41:33,670 Here's the red is the zero energy, 920 00:41:33,670 --> 00:41:38,220 and it goes up to finite energy. 921 00:41:38,220 --> 00:41:41,470 But has that lowest state developed any nodes? 922 00:41:45,050 --> 00:41:48,150 Does the lowest state in a 1D potential have nodes? 923 00:41:48,150 --> 00:41:48,659 No, right? 924 00:41:48,659 --> 00:41:49,450 There's nodes here. 925 00:41:49,450 --> 00:41:50,940 0 nodes, then one, then two. 926 00:41:50,940 --> 00:41:52,350 And do they ever switch orders? 927 00:41:52,350 --> 00:41:54,080 Does it ever go zero, two, one? 928 00:41:54,080 --> 00:41:54,580 No. 929 00:41:54,580 --> 00:41:55,180 There's the node theory. 930 00:41:55,180 --> 00:41:56,400 So, what happens to these wave functions 931 00:41:56,400 --> 00:41:57,983 is all the wave functions continuously 932 00:41:57,983 --> 00:41:59,550 change as we turn on the potential, 933 00:41:59,550 --> 00:42:00,800 but they don't switch order. 934 00:42:00,800 --> 00:42:03,900 Their ordering is completely fixed by the number of nodes. 935 00:42:03,900 --> 00:42:07,679 So, let's watch what happens to the actual wave functions. 936 00:42:07,679 --> 00:42:09,220 So, here's that lowest state, and I'm 937 00:42:09,220 --> 00:42:10,030 going to turn on the potential. 938 00:42:10,030 --> 00:42:11,320 What's going to happen? 939 00:42:11,320 --> 00:42:12,750 Well, we're going to see the effect of the delta function. 940 00:42:12,750 --> 00:42:15,910 And there, you see the effect of the delta function turning on. 941 00:42:15,910 --> 00:42:18,224 And as we crank up the potential, 942 00:42:18,224 --> 00:42:19,890 you see that the wave function generates 943 00:42:19,890 --> 00:42:22,340 a kink where there is a delta function. 944 00:42:22,340 --> 00:42:23,800 And that state has zero nodes. 945 00:42:23,800 --> 00:42:24,930 That's the ground state. 946 00:42:24,930 --> 00:42:26,530 Everyone cool with that? 947 00:42:26,530 --> 00:42:27,200 OK. 948 00:42:27,200 --> 00:42:29,770 Similarly, let's go to 0. 949 00:42:29,770 --> 00:42:30,910 Whoops. 950 00:42:30,910 --> 00:42:32,790 Oh, shoot. 951 00:42:32,790 --> 00:42:36,020 Don't give me infinite expressions. 952 00:42:36,020 --> 00:42:37,520 You're not allowed to divide by 0. 953 00:42:37,520 --> 00:42:38,190 Oh, shoot! 954 00:42:42,800 --> 00:42:47,274 OK, here's the second excited state, this guy. 955 00:42:47,274 --> 00:42:49,690 And let's look at what happens as we turn up the potential 956 00:42:49,690 --> 00:42:50,192 now. 957 00:42:50,192 --> 00:42:51,150 Exactly the same thing. 958 00:42:51,150 --> 00:42:52,025 We see a little kink. 959 00:42:52,025 --> 00:42:53,850 We don't get new nodes. 960 00:42:53,850 --> 00:42:55,860 But what we do get are kinks at the potential. 961 00:42:55,860 --> 00:42:58,401 This is for the special case of the delta function potential, 962 00:42:58,401 --> 00:43:00,720 but it's illustrative of everything else. 963 00:43:00,720 --> 00:43:03,075 So, going back to the bands, what's going on here 964 00:43:03,075 --> 00:43:04,950 is that these states are getting compressed-- 965 00:43:04,950 --> 00:43:06,450 they're getting pushed up together-- 966 00:43:06,450 --> 00:43:08,850 but they're not getting swapped in their order. 967 00:43:08,850 --> 00:43:10,066 OK? 968 00:43:10,066 --> 00:43:12,440 And this was in the service of a question that was asked. 969 00:43:12,440 --> 00:43:13,564 What was the last question? 970 00:43:17,560 --> 00:43:19,535 AUDIENCE: [INAUDIBLE]. 971 00:43:19,535 --> 00:43:21,160 PROFESSOR: OK, I don't remember exactly 972 00:43:21,160 --> 00:43:23,201 which questions I was answering, I'm sorry, but-- 973 00:43:23,201 --> 00:43:24,172 AUDIENCE: [INAUDIBLE]. 974 00:43:24,172 --> 00:43:24,880 PROFESSOR: Sorry? 975 00:43:24,880 --> 00:43:26,621 AUDIENCE: I think it was if you lose the bands. 976 00:43:26,621 --> 00:43:27,040 PROFESSOR: Ah, yeah. 977 00:43:27,040 --> 00:43:27,550 If you lose the bands. 978 00:43:27,550 --> 00:43:30,050 So, you know by the node theorem that you never lose states. 979 00:43:30,050 --> 00:43:32,326 States with three nodes can't disappear. 980 00:43:32,326 --> 00:43:33,700 That same with three nodes always 981 00:43:33,700 --> 00:43:34,760 has to be between the state with two-- 982 00:43:34,760 --> 00:43:36,634 thank you-- between the state with two nodes, 983 00:43:36,634 --> 00:43:37,970 and the state with four nodes. 984 00:43:37,970 --> 00:43:40,059 And they can't just disappear. 985 00:43:40,059 --> 00:43:41,350 So, those gaps never disappear. 986 00:43:41,350 --> 00:43:42,670 And they never overlap, because they never 987 00:43:42,670 --> 00:43:44,600 cross each other, again, by the node theorem. 988 00:43:44,600 --> 00:43:46,080 All you get is things separating into bands 989 00:43:46,080 --> 00:43:47,910 and squishing together, or spreading back apart 990 00:43:47,910 --> 00:43:49,215 and becoming the free particle. 991 00:43:49,215 --> 00:43:51,394 AUDIENCE: [INAUDIBLE]. 992 00:43:51,394 --> 00:43:52,060 PROFESSOR: Yeah? 993 00:43:52,060 --> 00:43:56,052 AUDIENCE: [INAUDIBLE] strange periodic potential and you'll 994 00:43:56,052 --> 00:43:57,410 get the [INAUDIBLE]. 995 00:43:57,410 --> 00:43:58,160 PROFESSOR: Indeed. 996 00:43:58,160 --> 00:43:58,660 Indeed. 997 00:43:58,660 --> 00:43:59,910 So, that's exactly right. 998 00:43:59,910 --> 00:44:02,210 So, the observation is, look, this was for the delta function 999 00:44:02,210 --> 00:44:04,010 potential, but we didn't have to use delta function barriers. 1000 00:44:04,010 --> 00:44:05,210 We could have use little square barriers, 1001 00:44:05,210 --> 00:44:06,790 I could have used little, you know, goat-shaped barriers. 1002 00:44:06,790 --> 00:44:07,914 I could have used whatever. 1003 00:44:07,914 --> 00:44:09,502 And there's going to be some answer. 1004 00:44:09,502 --> 00:44:10,710 And how's it going to change? 1005 00:44:10,710 --> 00:44:13,760 Well, the way it's going to change is where the bands are, 1006 00:44:13,760 --> 00:44:17,511 how wide they are, and how wide the gaps are. 1007 00:44:17,511 --> 00:44:20,010 But the rest of the story goes over in exactly the same way. 1008 00:44:20,010 --> 00:44:21,718 The energy eigenfunctions take the form E 1009 00:44:21,718 --> 00:44:24,240 to the iqx times the periodic function. 1010 00:44:24,240 --> 00:44:24,744 Cool? 1011 00:44:24,744 --> 00:44:26,160 OK, and on your problem set, where 1012 00:44:26,160 --> 00:44:27,860 you're gonna show-- one of the things you're gonna show-- 1013 00:44:27,860 --> 00:44:29,939 and I love this result, because it encapsulates 1014 00:44:29,939 --> 00:44:31,480 so much physics, you're going to show 1015 00:44:31,480 --> 00:44:34,650 that if you have a potential-- a single potential barrier-- 1016 00:44:34,650 --> 00:44:36,414 and you know the reflection amplitude, 1017 00:44:36,414 --> 00:44:38,580 and the transmission amplitude-- or really, you just 1018 00:44:38,580 --> 00:44:40,530 need to know one-- if you know the reflection 1019 00:44:40,530 --> 00:44:42,740 amplitude-- if it's parity symmetric-- then 1020 00:44:42,740 --> 00:44:45,290 you can write down the band structure entirely 1021 00:44:45,290 --> 00:44:47,340 in terms of-- you can write down an equation 1022 00:44:47,340 --> 00:44:48,890 for the allowed energies as a function of q-- 1023 00:44:48,890 --> 00:44:50,350 entirely in terms of the reflection 1024 00:44:50,350 --> 00:44:52,690 amplitude and the phase shift. 1025 00:44:52,690 --> 00:44:53,920 Which is an amazing fact. 1026 00:44:53,920 --> 00:44:56,240 So, you know about reflection off of one barrier, 1027 00:44:56,240 --> 00:44:58,790 you make an infinite lattice of those barriers, 1028 00:44:58,790 --> 00:45:01,345 and you deduce the structure of the allowed energy bands 1029 00:45:01,345 --> 00:45:03,141 and gaps, which is cool. 1030 00:45:03,141 --> 00:45:05,390 So, the scattering information-- as I kept promising-- 1031 00:45:05,390 --> 00:45:09,135 contains a huge amount of the physics of your system. 1032 00:45:09,135 --> 00:45:11,100 So, we saw that it contains bound energies, 1033 00:45:11,100 --> 00:45:14,370 but it also contains these band gaps. 1034 00:45:14,370 --> 00:45:15,170 OK, questions. 1035 00:45:15,170 --> 00:45:15,919 Yeah? 1036 00:45:15,919 --> 00:45:19,202 AUDIENCE: [INAUDIBLE] If we make the g0 go to infinity, 1037 00:45:19,202 --> 00:45:22,020 would that approach a single [INAUDIBLE]? 1038 00:45:22,020 --> 00:45:23,160 PROFESSOR: Yeah, exactly. 1039 00:45:23,160 --> 00:45:23,660 Excellent. 1040 00:45:29,334 --> 00:45:31,750 So, the question was, what happens as g0 goes to infinity, 1041 00:45:31,750 --> 00:45:33,200 is it like getting single wells? 1042 00:45:33,200 --> 00:45:33,820 So, let's look at that. 1043 00:45:33,820 --> 00:45:34,750 It's a very good observation. 1044 00:45:34,750 --> 00:45:36,060 So, here's a band between-- I really 1045 00:45:36,060 --> 00:45:37,420 should put a line here-- but between here, 1046 00:45:37,420 --> 00:45:38,850 and here, we have this band. 1047 00:45:38,850 --> 00:45:42,580 And that corresponds to between here, and here. 1048 00:45:42,580 --> 00:45:47,022 As we make g0 stronger, the derivation from simple cosine 1049 00:45:47,022 --> 00:45:47,980 gets larger and larger. 1050 00:45:47,980 --> 00:45:49,521 The amplitude gets larger and larger, 1051 00:45:49,521 --> 00:45:51,800 and those bands get steeper, and steeper, and steeper. 1052 00:45:51,800 --> 00:45:53,940 And the allowed energy bands get thinner, and thinner, 1053 00:45:53,940 --> 00:45:54,400 and thinner. 1054 00:45:54,400 --> 00:45:56,400 And, in fact, they get so thin, that Mathematica 1055 00:45:56,400 --> 00:45:59,390 loses track of some of them, which is the annoyance. 1056 00:45:59,390 --> 00:46:01,750 So, here we have very, very thin bands. 1057 00:46:01,750 --> 00:46:04,340 What happens is we take g0 very, very large. 1058 00:46:04,340 --> 00:46:06,310 So, let's make it 400. 1059 00:46:06,310 --> 00:46:08,310 And, you can see, Mathematica doesn't like that. 1060 00:46:08,310 --> 00:46:09,434 It's totally lost the band. 1061 00:46:09,434 --> 00:46:11,970 So, let's try something a little less extreme. 1062 00:46:11,970 --> 00:46:14,910 Oh, no, still too big for Mathematica. 1063 00:46:14,910 --> 00:46:16,715 Eight, you can do it, little buddy. 1064 00:46:16,715 --> 00:46:17,570 Ah, there we go! 1065 00:46:17,570 --> 00:46:18,700 We can keep tack of one. 1066 00:46:18,700 --> 00:46:19,960 So, what's happening is these bands 1067 00:46:19,960 --> 00:46:20,900 are getting extremely thin. 1068 00:46:20,900 --> 00:46:22,540 And what's happening, you're erecting 1069 00:46:22,540 --> 00:46:26,040 infinitely high barriers between each period in the lattice. 1070 00:46:26,040 --> 00:46:28,105 And if you have infinitely high barriers, 1071 00:46:28,105 --> 00:46:29,480 what are the allowed eigenvalues? 1072 00:46:32,050 --> 00:46:34,390 It's the same as the energy eigenvalues of the infinite 1073 00:46:34,390 --> 00:46:37,680 well, because it has to be 0 at those infinite barriers, right? 1074 00:46:37,680 --> 00:46:40,200 So, all of those states-- all of the states in the band-- 1075 00:46:40,200 --> 00:46:42,491 become degenerate, and have the same energy, the energy 1076 00:46:42,491 --> 00:46:44,990 of the periodic well, or of the infinite well. 1077 00:46:44,990 --> 00:46:46,360 Very good question. 1078 00:46:46,360 --> 00:46:47,090 OK. 1079 00:46:47,090 --> 00:46:51,630 So, let's let Mathematica relax a little bit again. 1080 00:46:51,630 --> 00:46:52,160 Well done. 1081 00:46:52,160 --> 00:46:53,360 OK. 1082 00:46:53,360 --> 00:46:56,240 And so, I want to quickly ask why do we 1083 00:46:56,240 --> 00:46:57,540 have bands in the first place? 1084 00:46:57,540 --> 00:46:59,807 Why do we have gaps? 1085 00:46:59,807 --> 00:47:00,640 So, what's going on? 1086 00:47:00,640 --> 00:47:02,264 Why do we have gaps in the first place? 1087 00:47:07,088 --> 00:47:09,050 AUDIENCE: Because our math told us we had to? 1088 00:47:09,050 --> 00:47:09,660 PROFESSOR: OK. 1089 00:47:09,660 --> 00:47:10,159 Good. 1090 00:47:10,159 --> 00:47:12,060 This is an excellent answer. 1091 00:47:12,060 --> 00:47:13,310 Not the one I was looking for. 1092 00:47:13,310 --> 00:47:15,726 The answer is a good answer because we did the calculation 1093 00:47:15,726 --> 00:47:18,020 and that's what the calculation tells us. 1094 00:47:18,020 --> 00:47:21,900 And that is a completely valid answer for a theorist, right? 1095 00:47:21,900 --> 00:47:24,005 At this stage of writing down what your theory is 1096 00:47:24,005 --> 00:47:24,710 and making a prediction. 1097 00:47:24,710 --> 00:47:27,300 But before you can write that paper, publish it, and say, 1098 00:47:27,300 --> 00:47:27,800 aha! 1099 00:47:27,800 --> 00:47:28,890 I've discovered bands! 1100 00:47:28,890 --> 00:47:31,485 You must have an explanation to the following question, why? 1101 00:47:31,485 --> 00:47:33,360 How do you know your calculation was correct, 1102 00:47:33,360 --> 00:47:34,430 before you've done the experiment? 1103 00:47:34,430 --> 00:47:35,850 So, one answer is, of course, you do the experiment, 1104 00:47:35,850 --> 00:47:36,510 and it fits like a champ. 1105 00:47:36,510 --> 00:47:38,385 But that's not a very illuminating experiment 1106 00:47:38,385 --> 00:47:41,104 as you know you're going to find departures from your theory, 1107 00:47:41,104 --> 00:47:42,520 you want to improve the situation. 1108 00:47:42,520 --> 00:47:45,330 You want some intuition about why this works. 1109 00:47:45,330 --> 00:47:47,880 And you actually all know the physics behind this. 1110 00:47:47,880 --> 00:47:50,680 And we did it on the first problem set. 1111 00:47:50,680 --> 00:47:55,170 Remember the experiment that we studied on a problem set, 1112 00:47:55,170 --> 00:47:58,270 the Davis and Germer Experiment, where you send plane waves 1113 00:47:58,270 --> 00:48:01,430 onto a crystal, and due to interference effects, 1114 00:48:01,430 --> 00:48:06,120 the transmission amplitude changes-- the transition 1115 00:48:06,120 --> 00:48:08,250 probability changes-- as a function of the angle. 1116 00:48:08,250 --> 00:48:10,750 And the reason it changes as a function of the angle was you 1117 00:48:10,750 --> 00:48:11,990 had a crystal plane, right? 1118 00:48:11,990 --> 00:48:15,860 So, atom, atom, atom, atom, atom, atom, atom, atom, 1119 00:48:15,860 --> 00:48:19,100 and a plane wave sent in could scatter off 1120 00:48:19,100 --> 00:48:21,730 of any of these scatterers, right? 1121 00:48:21,730 --> 00:48:24,240 Or it could transmit and scatter off a further one down. 1122 00:48:24,240 --> 00:48:26,530 And if you look at the scattering 1123 00:48:26,530 --> 00:48:30,550 off different scatterers, if the effect of those scatterings-- 1124 00:48:30,550 --> 00:48:33,570 so one down, and then one much deeper down-- if those 1125 00:48:33,570 --> 00:48:36,530 interfere destructively, then that's 1126 00:48:36,530 --> 00:48:38,260 going to suppress the reflection. 1127 00:48:38,260 --> 00:48:40,130 And if they interfere constructively, 1128 00:48:40,130 --> 00:48:43,381 it's gonna enhance the reflection. 1129 00:48:43,381 --> 00:48:45,880 So, usually we think about that Davis and Germer Experiment, 1130 00:48:45,880 --> 00:48:48,296 we think about it as a function of the angle of incidence, 1131 00:48:48,296 --> 00:48:50,545 either straight at, or as a function of angle. 1132 00:48:50,545 --> 00:48:53,250 And we ask, how does the transmission 1133 00:48:53,250 --> 00:48:55,040 depend on those angles? 1134 00:48:55,040 --> 00:48:56,650 But there's another way to phrase 1135 00:48:56,650 --> 00:48:58,940 that question, you can ask, look, on average, 1136 00:48:58,940 --> 00:49:00,740 how deep does that wave go? 1137 00:49:00,740 --> 00:49:02,600 Does the wave propagate through the crystal? 1138 00:49:02,600 --> 00:49:04,640 Or if it doesn't propagate through the crystal, 1139 00:49:04,640 --> 00:49:08,080 it's gonna exponentially decay with some wavelength, 1140 00:49:08,080 --> 00:49:11,830 with some decay-length. 1141 00:49:11,830 --> 00:49:18,612 And what you find is that whether the wave propagates in, 1142 00:49:18,612 --> 00:49:20,070 or whether it decays exponentially, 1143 00:49:20,070 --> 00:49:21,990 depends on the energy and the lattice 1144 00:49:21,990 --> 00:49:24,870 spacing of that crystal. 1145 00:49:24,870 --> 00:49:26,720 And this is exactly the same thing, 1146 00:49:26,720 --> 00:49:28,534 only we have a crystal in both directions. 1147 00:49:28,534 --> 00:49:30,950 We don't have empty air and we're sending in a plane wave. 1148 00:49:30,950 --> 00:49:33,224 Now we have a plane wave, can it propagate through? 1149 00:49:33,224 --> 00:49:34,890 Well, for some energies, it'll propagate 1150 00:49:34,890 --> 00:49:37,411 with continuous energy, or it'll propagate continuously. 1151 00:49:37,411 --> 00:49:39,660 And for some energies, it will damp out exponentially. 1152 00:49:39,660 --> 00:49:40,700 But if it damps out exponentially, 1153 00:49:40,700 --> 00:49:42,840 and it damps out exponentially in the other direction too, 1154 00:49:42,840 --> 00:49:44,464 then you don't get any solution at all. 1155 00:49:44,464 --> 00:49:45,940 It just completely dies. 1156 00:49:45,940 --> 00:49:48,340 So, what's really going on here is 1157 00:49:48,340 --> 00:49:51,940 the fact that all of these scatterers in this lattice 1158 00:49:51,940 --> 00:49:55,554 are scattering centers. 1159 00:49:55,554 --> 00:49:57,970 And if you send a little wave packet in-- which you should 1160 00:49:57,970 --> 00:50:00,380 think of as a superposition of energy eigenstates-- 1161 00:50:00,380 --> 00:50:02,131 if you send a wave packet in, the probably 1162 00:50:02,131 --> 00:50:03,546 to get-- the probability amplitude 1163 00:50:03,546 --> 00:50:05,970 to get from here to here-- is the probability amplitude 1164 00:50:05,970 --> 00:50:09,350 to go from here to here, plus to scatter through-- 1165 00:50:09,350 --> 00:50:12,200 to transmit through and reflect back-- plus to transmit, 1166 00:50:12,200 --> 00:50:15,890 transmit, reflect, reflect-- or sorry, transmit, transmit, 1167 00:50:15,890 --> 00:50:18,840 reflect, transmit-- plus all the other crazy things you 1168 00:50:18,840 --> 00:50:19,340 could do. 1169 00:50:19,340 --> 00:50:23,334 And you have to sum up over all those terms. 1170 00:50:23,334 --> 00:50:25,000 You have to sum the effect of how do you 1171 00:50:25,000 --> 00:50:26,042 get from here to here? 1172 00:50:26,042 --> 00:50:27,500 Every possible way to get from here 1173 00:50:27,500 --> 00:50:29,250 to there, a la the two slit experiment. 1174 00:50:29,250 --> 00:50:30,935 Sum of over all possible paths where 1175 00:50:30,935 --> 00:50:32,730 you bounce around scattering off each of these scattering 1176 00:50:32,730 --> 00:50:33,360 centers. 1177 00:50:33,360 --> 00:50:36,220 And that's why the band structure's 1178 00:50:36,220 --> 00:50:38,179 encoded in the reflection amplitudes 1179 00:50:38,179 --> 00:50:39,720 from an individual scattering center. 1180 00:50:39,720 --> 00:50:41,210 Because now, it's just a common [INAUDIBLE] problems 1181 00:50:41,210 --> 00:50:43,418 of how many different paths, and summing up the phase 1182 00:50:43,418 --> 00:50:44,570 that you get. 1183 00:50:44,570 --> 00:50:45,070 OK? 1184 00:50:45,070 --> 00:50:51,370 So, it's just about coherent and incoherent scattering. 1185 00:50:51,370 --> 00:50:52,588 Yeah? 1186 00:50:52,588 --> 00:50:54,070 AUDIENCE: [INAUDIBLE] crystal and I have an electron. 1187 00:50:54,070 --> 00:50:54,564 PROFESSOR: Yeah. 1188 00:50:54,564 --> 00:50:56,144 AUDIENCE: If I sent the electron through the crystal in just 1189 00:50:56,144 --> 00:50:57,894 the right way, with just the right energy, 1190 00:50:57,894 --> 00:51:00,690 won't it just go through in some states, without seeing it? 1191 00:51:00,690 --> 00:51:01,160 PROFESSOR: Excellent. 1192 00:51:01,160 --> 00:51:01,660 Excellent. 1193 00:51:01,660 --> 00:51:03,952 So, the question is under what conditions can you 1194 00:51:03,952 --> 00:51:05,910 send an electron and it'll just cruise right on 1195 00:51:05,910 --> 00:51:07,180 through the potential? 1196 00:51:07,180 --> 00:51:08,520 So, let's answer that question. 1197 00:51:08,520 --> 00:51:10,260 That's exactly the right question. 1198 00:51:10,260 --> 00:51:15,140 So, let's pick up from here. 1199 00:51:15,140 --> 00:51:17,300 And I want to ask, what have we learned? 1200 00:51:17,300 --> 00:51:21,490 So, what are the lessons for g0 equal to 0? 1201 00:51:24,420 --> 00:51:30,410 So, the first lesson is that the energy eigenvalues, 1202 00:51:30,410 --> 00:51:41,440 E eigenvalues, are restricted to lie 1203 00:51:41,440 --> 00:51:46,500 within a band-- this bands-- and the bands 1204 00:51:46,500 --> 00:51:47,460 are separated by gaps. 1205 00:51:50,220 --> 00:51:53,550 Two, the energy eigenstates-- so this 1206 00:51:53,550 --> 00:52:01,720 was eigenvalues-- the energy eigenstates are all extended. 1207 00:52:04,290 --> 00:52:09,120 And we didn't even need to find the full solution 1208 00:52:09,120 --> 00:52:10,370 of this equation to know that. 1209 00:52:10,370 --> 00:52:13,150 All we needed to know is that it was a solution. 1210 00:52:13,150 --> 00:52:15,460 That the energy eigenfunction are eigenfunctions 1211 00:52:15,460 --> 00:52:17,822 of the translation operator as well. 1212 00:52:17,822 --> 00:52:19,780 And that tells us the probability of that form. 1213 00:52:19,780 --> 00:52:21,670 And that tells us what the probability distribution is, 1214 00:52:21,670 --> 00:52:23,750 which is the norm squared of that guy, is independent, 1215 00:52:23,750 --> 00:52:24,333 it's periodic. 1216 00:52:24,333 --> 00:52:26,180 And if it's periodic, it can't fall off 1217 00:52:26,180 --> 00:52:28,794 exponentially one direction or the other. 1218 00:52:28,794 --> 00:52:30,710 So, we've seen that the eigenfunctions are all 1219 00:52:30,710 --> 00:52:33,450 extended, and so as a consequence, 1220 00:52:33,450 --> 00:52:35,580 we have to build wave packets. 1221 00:52:35,580 --> 00:52:39,785 So, therefore, we must build wave packets. 1222 00:52:43,410 --> 00:52:47,960 And, again, the form of the wave packets, psi 1223 00:52:47,960 --> 00:52:54,120 is equal to the integral dq f of q-- sorry, 1224 00:52:54,120 --> 00:52:58,440 I'll say fq0-- sharply peaked around the point q0 of q. 1225 00:52:58,440 --> 00:53:01,420 And then the wave function's E to the iqz 1226 00:53:01,420 --> 00:53:08,680 minus omega t times ru of x sub q. 1227 00:53:08,680 --> 00:53:10,640 OK. 1228 00:53:10,640 --> 00:53:12,170 So, we have to build wave packets. 1229 00:53:12,170 --> 00:53:14,450 So, three, this tells us that those wave packets 1230 00:53:14,450 --> 00:53:16,720 move with group velocity, by the same logic 1231 00:53:16,720 --> 00:53:18,510 as for the free particle, V group is 1232 00:53:18,510 --> 00:53:24,247 equal to d omega dk-- or dq-- evaluated at q0. 1233 00:53:24,247 --> 00:53:26,580 OK, where q0 is where our wave packet is sharply peaked. 1234 00:53:30,220 --> 00:53:35,850 So, that's already telling you something really cool. 1235 00:53:35,850 --> 00:53:38,880 Look at this lowest band. 1236 00:53:38,880 --> 00:53:40,920 Let's make this a little more reasonable. 1237 00:53:40,920 --> 00:53:42,456 So, let's look at this lowest band. 1238 00:53:42,456 --> 00:53:46,596 And in particular, I'm going to make this big. 1239 00:53:46,596 --> 00:53:48,470 So, focus in on the lowest energy band, here. 1240 00:53:48,470 --> 00:53:49,710 So, there's the lowest energy band. 1241 00:53:49,710 --> 00:53:51,420 It goes from the top of the band to the bottom of the band. 1242 00:53:51,420 --> 00:53:53,180 It's just a little bit away from the free particle, 1243 00:53:53,180 --> 00:53:54,721 because we're looking at a relatively 1244 00:53:54,721 --> 00:53:57,070 weakly, perturbed system. 1245 00:53:57,070 --> 00:53:58,442 I chose a small value of q0. 1246 00:53:58,442 --> 00:54:00,525 Actually, let's make it a little more exaggerated. 1247 00:54:08,800 --> 00:54:11,040 OK, so there's the free particle curve, the red one, 1248 00:54:11,040 --> 00:54:12,540 and the blue one is our actual band. 1249 00:54:12,540 --> 00:54:14,991 So, that's the energy as a function of q. 1250 00:54:14,991 --> 00:54:18,587 And let's make it even bigger. 1251 00:54:18,587 --> 00:54:19,420 This one goes to 11. 1252 00:54:22,576 --> 00:54:23,290 Let's see. 1253 00:54:25,820 --> 00:54:26,350 OK. 1254 00:54:26,350 --> 00:54:27,980 And so what you can see is the following, it looks, 1255 00:54:27,980 --> 00:54:30,313 again, it's looks sort of like a parabola at the bottom. 1256 00:54:30,313 --> 00:54:32,360 And at the top, it curves over, and, actually, 1257 00:54:32,360 --> 00:54:33,170 goes to zero derivative. 1258 00:54:33,170 --> 00:54:35,128 And it has to, because it's got to be periodic. 1259 00:54:35,128 --> 00:54:38,119 Because q is periodic with period 2pi over L. So, 1260 00:54:38,119 --> 00:54:39,910 what does that tell you about the velocity? 1261 00:54:39,910 --> 00:54:41,380 Well, let's look at the velocity. 1262 00:54:41,380 --> 00:54:46,610 V group velocity, of wave packet with-- whoops, 1263 00:54:46,610 --> 00:54:55,220 say, here's 0-- as a function of q between minus 2pi over L, 1264 00:54:55,220 --> 00:54:58,275 and 2pi over L. 1265 00:54:58,275 --> 00:54:59,400 What is the group velocity? 1266 00:54:59,400 --> 00:55:00,358 And here's what I mean. 1267 00:55:00,358 --> 00:55:02,730 What I mean is let's build a wave packet which 1268 00:55:02,730 --> 00:55:04,930 has a reasonably well-localized momentum, 1269 00:55:04,930 --> 00:55:05,930 say, at this value of q. 1270 00:55:05,930 --> 00:55:09,260 So, I built a little wave packet it has a peak here, 1271 00:55:09,260 --> 00:55:11,500 and what does that wave function do? 1272 00:55:11,500 --> 00:55:12,850 Well, it has a group velocity. 1273 00:55:12,850 --> 00:55:15,550 And the group velocity is given by d omega dq. 1274 00:55:15,550 --> 00:55:18,010 And now, we can't solve that equation analytically, 1275 00:55:18,010 --> 00:55:19,051 because it's [INAUDIBLE]. 1276 00:55:19,051 --> 00:55:21,400 But we can see, just by eyeball, what this does. 1277 00:55:21,400 --> 00:55:24,225 When q is equal to 0, what's the group velocity? 1278 00:55:30,371 --> 00:55:30,870 Yeah. 1279 00:55:30,870 --> 00:55:32,085 It's the slope, right? 1280 00:55:32,085 --> 00:55:32,640 D omega dq. 1281 00:55:32,640 --> 00:55:35,090 So, what's d omega dq at the origin? 1282 00:55:35,090 --> 00:55:36,890 0, right? 'Cause it's just horizontal. 1283 00:55:36,890 --> 00:55:38,490 So, it's 0. 1284 00:55:38,490 --> 00:55:39,280 Good. 1285 00:55:39,280 --> 00:55:42,630 What about a little bit to the right of 0? 1286 00:55:42,630 --> 00:55:45,040 What about if I give it a small positive q-- wave 1287 00:55:45,040 --> 00:55:47,170 packet localized with a small positive q? 1288 00:55:47,170 --> 00:55:49,550 It's got a little bit of a slope now, right? 1289 00:55:49,550 --> 00:55:51,420 Little bit of a positive slope. 1290 00:55:51,420 --> 00:55:52,950 So, it increases. 1291 00:55:52,950 --> 00:55:55,150 But then, it reaches a maximum slope, halfway 1292 00:55:55,150 --> 00:56:00,770 through the domain. 1293 00:56:00,770 --> 00:56:02,952 So, here, it hits a maximum value, 1294 00:56:02,952 --> 00:56:04,410 and then the slope comes back down. 1295 00:56:06,816 --> 00:56:08,440 And if we'd went the other way, if we'd 1296 00:56:08,440 --> 00:56:12,250 made q a little bit negative, the slope becomes negative. 1297 00:56:12,250 --> 00:56:15,400 And that's the velocity. 1298 00:56:15,400 --> 00:56:17,360 So, that's a slightly strange thing. 1299 00:56:17,360 --> 00:56:20,170 As we increase q, it increases the velocity. 1300 00:56:20,170 --> 00:56:22,170 Increasing q increases the velocity for a while. 1301 00:56:22,170 --> 00:56:23,750 That's just like k and momentum. 1302 00:56:23,750 --> 00:56:24,947 Momentum is h bar k. 1303 00:56:24,947 --> 00:56:26,530 So, that would be increasing the group 1304 00:56:26,530 --> 00:56:27,870 velocity for a free particle. 1305 00:56:27,870 --> 00:56:30,400 But now, as we increase the q to a reasonably large value, 1306 00:56:30,400 --> 00:56:38,400 to 2pi upon L-- oh, sorry, this is pi upon L, and pi upon L-- 1307 00:56:38,400 --> 00:56:43,200 so, if we do this to pi upon 2L, or roughly, to the middle, 1308 00:56:43,200 --> 00:56:44,580 we get to a maximum value. 1309 00:56:44,580 --> 00:56:48,130 And if we increase q further, what happens? 1310 00:56:48,130 --> 00:56:49,867 The velocity goes down, right? 1311 00:56:49,867 --> 00:56:51,700 If we increase q further, the velocity goes. 1312 00:56:51,700 --> 00:56:52,410 This is strange. 1313 00:56:52,410 --> 00:56:54,531 For a free particle, if you had momentum h bar k, 1314 00:56:54,531 --> 00:56:56,780 and you increased k, what happens to the expectation-? 1315 00:56:56,780 --> 00:56:59,738 What happens to the group velocity? 1316 00:56:59,738 --> 00:57:01,237 If you increase the momentum, what 1317 00:57:01,237 --> 00:57:03,320 happens to the group velocity for a free particle? 1318 00:57:03,320 --> 00:57:04,030 AUDIENCE: [MURMURS] 1319 00:57:04,030 --> 00:57:05,613 PROFESSOR: It's gonna increase, right? 1320 00:57:05,613 --> 00:57:08,410 But not for a particle in the periodic potential. 1321 00:57:08,410 --> 00:57:10,740 The wave packet, as you increase the crystal momentum, 1322 00:57:10,740 --> 00:57:15,430 q, past this maximum, as you increase the crystal momentum, 1323 00:57:15,430 --> 00:57:17,560 you actually decrease the velocity. 1324 00:57:17,560 --> 00:57:19,190 And, in fact, if you keep increasing q, 1325 00:57:19,190 --> 00:57:21,090 eventually the velocity goes to 0. 1326 00:57:21,090 --> 00:57:22,630 And you increase q a little more, 1327 00:57:22,630 --> 00:57:25,190 and it actually goes negative. 1328 00:57:25,190 --> 00:57:26,674 So, this is a funny thing, it's not 1329 00:57:26,674 --> 00:57:28,840 behaving like normally you'd think of as a momentum. 1330 00:57:28,840 --> 00:57:32,340 And, indeed, is there momentum conservation in this system? 1331 00:57:32,340 --> 00:57:32,840 No. 1332 00:57:32,840 --> 00:57:34,048 We have a periodic potential. 1333 00:57:34,048 --> 00:57:35,380 There are forces at work. 1334 00:57:35,380 --> 00:57:37,950 Is momentum conserved? 1335 00:57:37,950 --> 00:57:38,480 No. 1336 00:57:38,480 --> 00:57:40,190 This is not a system that conserves a momentum. 1337 00:57:40,190 --> 00:57:42,106 Momentum can be exchanged between the particle 1338 00:57:42,106 --> 00:57:43,910 and the potential. 1339 00:57:43,910 --> 00:57:46,010 Q is a good quantity. 1340 00:57:46,010 --> 00:57:48,210 It's the eigenfunction of TL, and that does commute 1341 00:57:48,210 --> 00:57:49,300 with the energy operator. 1342 00:57:49,300 --> 00:57:51,008 So, it's a perfectly reasonable quantity. 1343 00:57:51,008 --> 00:57:54,770 But the momentum itself, p, is not conserved. 1344 00:57:54,770 --> 00:57:56,920 So, this crystal momentum, q, is not 1345 00:57:56,920 --> 00:57:59,090 the momentum, for all these reasons. 1346 00:57:59,090 --> 00:58:00,590 But it plays a lot of the same role, 1347 00:58:00,590 --> 00:58:03,291 especially near the bottom of the band. 1348 00:58:03,291 --> 00:58:05,599 Near the top of the band, it's a little bit funnier, 1349 00:58:05,599 --> 00:58:07,890 because if you increase q, you get a negative velocity. 1350 00:58:07,890 --> 00:58:10,280 If you decrease q, you get a positive velocity. 1351 00:58:10,280 --> 00:58:13,230 It's exactly the opposite of what you'd normally expect. 1352 00:58:13,230 --> 00:58:17,520 But let's hold on to that thought for a second. 1353 00:58:17,520 --> 00:58:20,120 So, there's another sense in which the crystal momentum 1354 00:58:20,120 --> 00:58:21,070 is like a momentum. 1355 00:58:21,070 --> 00:58:23,804 And you're gonna show this on your problem set. 1356 00:58:23,804 --> 00:58:26,220 But it's very important in giving you some understanding-- 1357 00:58:26,220 --> 00:58:28,160 some intuition-- about what the crystal momentum is. 1358 00:58:28,160 --> 00:58:29,470 On your problem set, you'll show the following. 1359 00:58:29,470 --> 00:58:31,114 Suppose I induce a constant force. 1360 00:58:31,114 --> 00:58:32,530 For example, imagine this particle 1361 00:58:32,530 --> 00:58:34,113 in my periodic potential had a charge, 1362 00:58:34,113 --> 00:58:36,490 and I turn on a constant, uniform electric field. 1363 00:58:36,490 --> 00:58:39,020 Then that particle experiences a constant, uniform driving 1364 00:58:39,020 --> 00:58:42,301 force, which I can write as a linearly increasing potential, 1365 00:58:42,301 --> 00:58:44,534 right? 1366 00:58:44,534 --> 00:58:45,950 So, what you showed on the problem 1367 00:58:45,950 --> 00:58:48,600 set is that the force-- the expectation value 1368 00:58:48,600 --> 00:58:52,310 of the force-- is equal to-- for a wave packet-- sharply 1369 00:58:52,310 --> 00:58:56,410 localized around some value, q0, the force, 1370 00:58:56,410 --> 00:58:58,980 it gives the time rate of change, d dt, 1371 00:58:58,980 --> 00:59:02,624 of the expectation value of q. 1372 00:59:02,624 --> 00:59:04,290 So, if you turn on a force, what happens 1373 00:59:04,290 --> 00:59:06,840 is q increases linearly. 1374 00:59:06,840 --> 00:59:07,450 Thank you. 1375 00:59:10,420 --> 00:59:12,650 It's so hard. 1376 00:59:12,650 --> 00:59:14,020 It's just so obviously 1. 1377 00:59:17,424 --> 00:59:18,840 So, if the force is constant, then 1378 00:59:18,840 --> 00:59:21,810 h bar q increases constantly. 1379 00:59:21,810 --> 00:59:25,500 So, q just increases linearly in time, yeah? 1380 00:59:25,500 --> 00:59:26,765 But that's funny. 1381 00:59:26,765 --> 00:59:28,390 Well, we'll come back to that, just how 1382 00:59:28,390 --> 00:59:32,133 funny that is in just a-- yeah? 1383 00:59:32,133 --> 00:59:33,119 Question? 1384 00:59:33,119 --> 00:59:34,221 No, OK. 1385 00:59:34,221 --> 00:59:35,970 Well, let's think about how funny that it. 1386 00:59:35,970 --> 00:59:38,860 Let's tackle this, right now. 1387 00:59:38,860 --> 00:59:42,620 So, imagine we start with a wave packet, which 1388 00:59:42,620 --> 00:59:44,760 is well localized around zero crystal momentum, 1389 00:59:44,760 --> 00:59:45,634 around q equals 0. 1390 00:59:45,634 --> 00:59:47,550 Then what is that wave function doing in time? 1391 00:59:47,550 --> 00:59:48,841 How does that wave packet move? 1392 00:59:53,090 --> 00:59:55,240 What's its group velocity? 1393 00:59:55,240 --> 00:59:55,740 0. 1394 00:59:55,740 --> 00:59:57,000 So, it's just sitting there. 1395 00:59:57,000 --> 00:59:59,145 It's just little wave packets slowly dispersing, 1396 00:59:59,145 --> 01:00:00,270 quantum mechanically, yeah? 1397 01:00:00,270 --> 01:00:02,710 But if we don't make it too tightly constrained, and p 1398 01:00:02,710 --> 01:00:03,535 and 2 too tightly constrained. 1399 01:00:03,535 --> 01:00:05,370 And the dispersion could be made quite slow. 1400 01:00:05,370 --> 01:00:06,828 So, it's just a little wave packet, 1401 01:00:06,828 --> 01:00:08,930 localized around a particular position. 1402 01:00:08,930 --> 01:00:10,490 And localized around zero momentum, 1403 01:00:10,490 --> 01:00:11,850 and it's just sitting there. 1404 01:00:11,850 --> 01:00:12,550 Cool? 1405 01:00:12,550 --> 01:00:14,450 Now, let's apply a force. 1406 01:00:14,450 --> 01:00:15,710 What happens? 1407 01:00:15,710 --> 01:00:19,000 As we apply a force, a constant, force with a positive sign. 1408 01:00:19,000 --> 01:00:19,950 Q is gonna increase. 1409 01:00:19,950 --> 01:00:21,980 So, the central value-- I should say, yeah-- so, 1410 01:00:21,980 --> 01:00:23,480 the central value of our wave packet 1411 01:00:23,480 --> 01:00:26,160 is gonna go from 0 to something slightly positive. 1412 01:00:26,160 --> 01:00:27,160 Because q is increasing. 1413 01:00:27,160 --> 01:00:28,785 What that means is that the velocity is 1414 01:00:28,785 --> 01:00:29,712 increasing at first. 1415 01:00:29,712 --> 01:00:31,170 So, the velocity is gonna increase, 1416 01:00:31,170 --> 01:00:32,711 and in equal units of time it's gonna 1417 01:00:32,711 --> 01:00:34,280 march linearly along this way. 1418 01:00:34,280 --> 01:00:36,710 So, the particle is going faster, and faster, and faster, 1419 01:00:36,710 --> 01:00:37,426 and faster. 1420 01:00:37,426 --> 01:00:38,300 And that makes sense. 1421 01:00:38,300 --> 01:00:40,675 You're driving the system by putting on a constant force. 1422 01:00:40,675 --> 01:00:42,574 Of course it accelerates. 1423 01:00:42,574 --> 01:00:43,990 The velocity accelerates linearly, 1424 01:00:43,990 --> 01:00:46,410 so the position accelerates quadratically. 1425 01:00:46,410 --> 01:00:49,380 So, let's plot the position-- expectation 1426 01:00:49,380 --> 01:00:51,840 value of the position-- as a function of time. 1427 01:00:51,840 --> 01:00:53,300 At zero time-- it's at the origin-- 1428 01:00:53,300 --> 01:00:55,780 it's at some trivial position, x0. 1429 01:00:55,780 --> 01:00:58,880 As we increase time, velocity's increasing linearly for awhile, 1430 01:00:58,880 --> 01:01:00,980 and the position is gonna increase quadratically. 1431 01:01:00,980 --> 01:01:02,650 Everyone cool with that? 1432 01:01:02,650 --> 01:01:05,369 But eventually, at some point in time, the velocity-- or the q-- 1433 01:01:05,369 --> 01:01:06,910 is gonna get to the point where we're 1434 01:01:06,910 --> 01:01:08,980 at the maximum allowed velocity. 1435 01:01:08,980 --> 01:01:13,030 And at that point, what is the curve for x gonna look like? 1436 01:01:13,030 --> 01:01:15,140 It's gonna have an inflection point. 1437 01:01:15,140 --> 01:01:18,870 And as we increase-- as we wait longer-- q 1438 01:01:18,870 --> 01:01:20,790 is gonna continue linearly increasing, which 1439 01:01:20,790 --> 01:01:23,740 means the velocity is going to slow down. 1440 01:01:23,740 --> 01:01:26,079 So, the thing slows down, so its slope decreases, 1441 01:01:26,079 --> 01:01:27,870 until finally it's got zero velocity again. 1442 01:01:27,870 --> 01:01:30,870 There we go-- zero velocity-- dx dt. 1443 01:01:30,870 --> 01:01:32,260 And then what happens? 1444 01:01:32,260 --> 01:01:33,320 We keep forcing it. 1445 01:01:33,320 --> 01:01:34,760 It's a constant driving force. 1446 01:01:34,760 --> 01:01:37,800 But this is this, by periodicity of q, 1447 01:01:37,800 --> 01:01:40,580 and the thing is, we increase q further linearly in time. 1448 01:01:40,580 --> 01:01:44,180 As we increased q further, the velocity becomes negative. 1449 01:01:44,180 --> 01:01:46,220 So, instead of continuing to accelerate-- 1450 01:01:46,220 --> 01:01:49,740 instead of continuing to move-- in a positive x direction, 1451 01:01:49,740 --> 01:01:52,750 it starts going backwards! 1452 01:01:52,750 --> 01:01:53,440 That's weird. 1453 01:01:53,440 --> 01:01:55,148 You're putting a force in this direction, 1454 01:01:55,148 --> 01:01:58,120 and it's accelerating in that direction. 1455 01:01:58,120 --> 01:01:59,675 That's a strange effect. 1456 01:01:59,675 --> 01:02:01,050 And as we continue, eventually we 1457 01:02:01,050 --> 01:02:02,390 get to maximum negative velocity. 1458 01:02:02,390 --> 01:02:03,530 Again, an inflection point. 1459 01:02:07,190 --> 01:02:08,565 And, at which point, the velocity 1460 01:02:08,565 --> 01:02:11,106 starts getting less, and less negative, or closer, and closer 1461 01:02:11,106 --> 01:02:11,640 to 0. 1462 01:02:11,640 --> 01:02:13,100 And we return. 1463 01:02:13,100 --> 01:02:15,000 And so, in some period, capital T, 1464 01:02:15,000 --> 01:02:17,850 which is determined by how big this constant driving force is, 1465 01:02:17,850 --> 01:02:21,270 the particle returns to its original position. 1466 01:02:21,270 --> 01:02:23,841 Everyone see that? 1467 01:02:23,841 --> 01:02:24,340 Yeah? 1468 01:02:24,340 --> 01:02:26,320 AUDIENCE: Is it starting at q equals 0? 1469 01:02:26,320 --> 01:02:27,600 PROFESSOR: Well, yeah. 1470 01:02:27,600 --> 01:02:28,270 At q0 equals 0. 1471 01:02:28,270 --> 01:02:28,769 Exactly. 1472 01:02:28,769 --> 01:02:34,680 So, at first point, so this is q equals 0, this is-- sorry, 1473 01:02:34,680 --> 01:02:36,730 q equals 0-- is the initial point. 1474 01:02:36,730 --> 01:02:39,790 And x is equal to x0. 1475 01:02:39,790 --> 01:02:41,820 So, under this process, the particle 1476 01:02:41,820 --> 01:02:45,640 goes through an oscillation, called a Bloch Oscillation. 1477 01:02:45,640 --> 01:02:47,790 It's named after the guy who invented the wave 1478 01:02:47,790 --> 01:02:49,335 functions, the Bloch Wave Functions. 1479 01:02:49,335 --> 01:02:51,590 And this is a deeply weird thing. 1480 01:02:51,590 --> 01:02:55,070 Normally, you think, well, look, if I take a charged particle, 1481 01:02:55,070 --> 01:02:57,240 and I just put it in an empty space, or in a box, 1482 01:02:57,240 --> 01:02:58,350 or whatever, and I turn on an electric field, 1483 01:02:58,350 --> 01:02:59,600 what happens to it? 1484 01:02:59,600 --> 01:03:00,630 It accelerates. 1485 01:03:00,630 --> 01:03:03,650 But if you put that charged particle in a periodic lattice, 1486 01:03:03,650 --> 01:03:10,000 like copper-- idealized copper, here, perfect uniform lattice-- 1487 01:03:10,000 --> 01:03:11,660 and you turn on an electric field-- 1488 01:03:11,660 --> 01:03:14,300 or you put a capacitor plate across the thing-- 1489 01:03:14,300 --> 01:03:16,499 that electron, at first, wants to behave 1490 01:03:16,499 --> 01:03:17,790 like an electron in free space. 1491 01:03:17,790 --> 01:03:19,060 It wants to accelerate. 1492 01:03:19,060 --> 01:03:21,630 But then it finds out it's in a lattice. 1493 01:03:21,630 --> 01:03:23,770 And it goes backwards. 1494 01:03:23,770 --> 01:03:26,120 And it just oscillates back and forth, [INAUDIBLE]-like, 1495 01:03:26,120 --> 01:03:26,721 you know? 1496 01:03:26,721 --> 01:03:28,220 I really want to get there, oh crap. 1497 01:03:28,220 --> 01:03:28,925 I really want to get there. 1498 01:03:28,925 --> 01:03:30,440 Oh no, I'm in a periodic potential. 1499 01:03:30,440 --> 01:03:31,880 And it just oscillates back and forth. 1500 01:03:31,880 --> 01:03:33,530 Is there any conductions in this system 1501 01:03:33,530 --> 01:03:36,880 due to the electromagnetic field that we've induced? 1502 01:03:36,880 --> 01:03:38,040 None. 1503 01:03:38,040 --> 01:03:40,360 That is a strange little beasty. 1504 01:03:40,360 --> 01:03:41,870 That is a strange, strange property. 1505 01:03:41,870 --> 01:03:44,240 And yet, copper conducts. 1506 01:03:44,240 --> 01:03:45,740 Copper, which is a uniform lattice 1507 01:03:45,740 --> 01:03:48,570 of ions-- of potentials-- to which 1508 01:03:48,570 --> 01:03:51,820 electrons are stuck in bands. 1509 01:03:51,820 --> 01:03:52,900 Copper conducts! 1510 01:03:52,900 --> 01:03:55,040 The electrons don't do this Bloch Oscillation. 1511 01:03:55,040 --> 01:03:55,670 How come? 1512 01:04:00,010 --> 01:04:00,510 Yeah. 1513 01:04:00,510 --> 01:04:01,130 It's not perfect. 1514 01:04:01,130 --> 01:04:03,213 And it's not perfect for a whole bunch of reasons. 1515 01:04:03,213 --> 01:04:07,680 So, one reason it's not perfect is that, so, first off, 1516 01:04:07,680 --> 01:04:09,770 in a lattice of, say, copper, or whatever, 1517 01:04:09,770 --> 01:04:12,660 the ions have a finite mass. 1518 01:04:12,660 --> 01:04:15,240 So, what that means is they're not stuck in place, 1519 01:04:15,240 --> 01:04:16,030 they also wiggle. 1520 01:04:16,030 --> 01:04:18,060 When the electron moves, one of the ions move. 1521 01:04:18,060 --> 01:04:19,560 So, the whole thing you should think 1522 01:04:19,560 --> 01:04:21,320 of as a little wiggling piece of jello. 1523 01:04:21,320 --> 01:04:24,210 And as a consequence, they're not perfectly periodic. 1524 01:04:24,210 --> 01:04:25,690 This does a bunch of things. 1525 01:04:25,690 --> 01:04:27,780 One is these are no longer the exact energy eigenfunctions. 1526 01:04:27,780 --> 01:04:28,935 You now have to deal with the oscillations 1527 01:04:28,935 --> 01:04:30,480 and wiggling of the lattice. 1528 01:04:30,480 --> 01:04:31,480 But more importantly, here's something 1529 01:04:31,480 --> 01:04:33,470 that can happen-- and electron can move along, 1530 01:04:33,470 --> 01:04:36,967 and it can kick one of these ions-- it can bounce off 1531 01:04:36,967 --> 01:04:39,050 one of the ions-- and scatter some of its momentum 1532 01:04:39,050 --> 01:04:40,410 into the ion. 1533 01:04:40,410 --> 01:04:42,552 Not into the rigid lattice, but it 1534 01:04:42,552 --> 01:04:44,260 could just make one of those ions wiggle. 1535 01:04:44,260 --> 01:04:46,870 So, it's changed the structure of the lattice. 1536 01:04:46,870 --> 01:04:47,760 OK? 1537 01:04:47,760 --> 01:04:49,525 And what you can do then, is you can have an electron that 1538 01:04:49,525 --> 01:04:51,780 is accelerating, and it hits something, and it stops. 1539 01:04:51,780 --> 01:04:52,820 And it accelerates. 1540 01:04:52,820 --> 01:04:53,587 And it stops. 1541 01:04:53,587 --> 01:04:55,670 Accelerates, and it hits something, then it stops. 1542 01:04:55,670 --> 01:04:56,390 OK? 1543 01:04:56,390 --> 01:04:58,920 So, when you have lots of disorder in your system-- 1544 01:04:58,920 --> 01:05:01,000 or when you allow the electron to bounce off 1545 01:05:01,000 --> 01:05:03,780 things in the lattice-- you get this effective conductivity. 1546 01:05:03,780 --> 01:05:05,467 Things start their Bloch Oscillation, 1547 01:05:05,467 --> 01:05:07,550 then they collide, and scatter off their momentum, 1548 01:05:07,550 --> 01:05:09,140 and go back to zero momentum, but they've 1549 01:05:09,140 --> 01:05:11,348 moved over a little bit in their oscillation already. 1550 01:05:11,348 --> 01:05:12,520 They move up, and they move. 1551 01:05:12,520 --> 01:05:14,696 So, what that picture looks like is, you accelerate, 1552 01:05:14,696 --> 01:05:16,820 and then, boom, you scatter, and you fall back down 1553 01:05:16,820 --> 01:05:18,236 to zero momentum by scattering off 1554 01:05:18,236 --> 01:05:19,990 your momentum into something else. 1555 01:05:19,990 --> 01:05:22,359 Same thing, you move up, and then, now, you 1556 01:05:22,359 --> 01:05:23,400 have zero momentum again. 1557 01:05:23,400 --> 01:05:24,660 You move up, you have zero momentum again. 1558 01:05:24,660 --> 01:05:25,890 You move up, you have zero momentum. 1559 01:05:25,890 --> 01:05:27,890 You move up, you have zero momentum, again. 1560 01:05:27,890 --> 01:05:29,870 And so, you get this effective drifting 1561 01:05:29,870 --> 01:05:34,730 of the electrons, pointed out by a guy named Drude. 1562 01:05:34,730 --> 01:05:37,335 That gives you, effectively, a conductivity and a solid. 1563 01:05:37,335 --> 01:05:38,960 Now, there are a bunch of other effects 1564 01:05:38,960 --> 01:05:40,060 that are important for conductivity. 1565 01:05:40,060 --> 01:05:42,600 Normally, we think of disorder as something that, you know, 1566 01:05:42,600 --> 01:05:44,200 if you make things messy, they're 1567 01:05:44,200 --> 01:05:45,950 probably gonna transmit less well, right? 1568 01:05:45,950 --> 01:05:49,215 But disorder is absolutely essential for conductivity 1569 01:05:49,215 --> 01:05:50,590 in real solids. 1570 01:05:50,590 --> 01:05:52,600 Both for this reason, because you scatter off 1571 01:05:52,600 --> 01:05:55,260 of fluctuations, but also, when you have a chunk of copper, 1572 01:05:55,260 --> 01:05:56,485 it's not a chunk of copper. 1573 01:05:56,485 --> 01:05:58,026 It's a chunk of copper, but it's got, 1574 01:05:58,026 --> 01:06:00,067 you know, here and there, there's a little carbon 1575 01:06:00,067 --> 01:06:02,710 atom that got stuck, and maybe there's a bit of nickel, 1576 01:06:02,710 --> 01:06:05,530 and, you know, palladium, or, you know, 1577 01:06:05,530 --> 01:06:08,116 Berkelium-- pretty unlikely, but could be there-- so, 1578 01:06:08,116 --> 01:06:10,490 you have all sorts of schmutz sort of distributed around. 1579 01:06:10,490 --> 01:06:11,500 And, again, those are things that 1580 01:06:11,500 --> 01:06:13,150 make the potential not periodic. 1581 01:06:13,150 --> 01:06:16,920 And, so, will change the conservation of this q. 1582 01:06:16,920 --> 01:06:18,120 OK? 1583 01:06:18,120 --> 01:06:19,032 Yeah? 1584 01:06:19,032 --> 01:06:20,740 AUDIENCE: [INAUDIBLE] but if you actually 1585 01:06:20,740 --> 01:06:22,660 managed to put a alternating current-- 1586 01:06:22,660 --> 01:06:25,534 or an alternating voltage-- at the right frequency, 1587 01:06:25,534 --> 01:06:27,700 you could actually drive the electrons from one side 1588 01:06:27,700 --> 01:06:29,280 to the other in this fashion? 1589 01:06:29,280 --> 01:06:29,650 PROFESSOR: Yeah, so, I-- 1590 01:06:29,650 --> 01:06:30,768 AUDIENCE: --even though the conventional way 1591 01:06:30,768 --> 01:06:31,890 of thinking about alternating currents 1592 01:06:31,890 --> 01:06:33,520 is that the electrons really don't move at all. 1593 01:06:33,520 --> 01:06:35,270 PROFESSOR: That's a very good observation. 1594 01:06:35,270 --> 01:06:37,266 It's a little subtle, so come to my office hours 1595 01:06:37,266 --> 01:06:37,830 and ask me about that. 1596 01:06:37,830 --> 01:06:40,163 But there's an interesting-- For a very specific reason, 1597 01:06:40,163 --> 01:06:41,820 because it's easy and unambiguous, 1598 01:06:41,820 --> 01:06:42,670 I chose look the DC. 1599 01:06:42,670 --> 01:06:43,580 But you're right, looking at the AC, 1600 01:06:43,580 --> 01:06:45,250 is really a very entertaining example. 1601 01:06:45,250 --> 01:06:46,862 So, yeah, I'm not gonna get into it. 1602 01:06:46,862 --> 01:06:48,820 There's a cool story with parametric resonance. 1603 01:06:48,820 --> 01:06:50,200 And yeah, there's a very nice story there. 1604 01:06:50,200 --> 01:06:51,880 Come to my office hours, because that 1605 01:06:51,880 --> 01:06:53,230 is a particularly fun story. 1606 01:06:53,230 --> 01:06:55,600 OK, but I want to do one last thing on this 1607 01:06:55,600 --> 01:06:56,650 before we move on. 1608 01:06:56,650 --> 01:07:01,340 So, in this process, something very strange happened. 1609 01:07:01,340 --> 01:07:03,590 When we got to the top of the potential, what we found 1610 01:07:03,590 --> 01:07:06,270 was as we increase-- as we continue 1611 01:07:06,270 --> 01:07:11,576 constant, positive force-- the velocity decreased. 1612 01:07:11,576 --> 01:07:12,540 Yeah? 1613 01:07:12,540 --> 01:07:14,030 That is weird. 1614 01:07:14,030 --> 01:07:16,030 Normally, when I take a force, and I take 1615 01:07:16,030 --> 01:07:18,280 some object with some mass, and I take a force 1616 01:07:18,280 --> 01:07:19,570 and I apply it to the mass, and then the acceleration 1617 01:07:19,570 --> 01:07:20,570 is the force divided by the mass. 1618 01:07:20,570 --> 01:07:22,528 In particular, mass is always a positive thing, 1619 01:07:22,528 --> 01:07:25,176 so if I take my force, and I divide it by the positive mass, 1620 01:07:25,176 --> 01:07:27,550 I get an acceleration in the same direction as the force. 1621 01:07:27,550 --> 01:07:29,400 But here, we have an applied force, 1622 01:07:29,400 --> 01:07:32,520 and we get an acceleration in the opposite direction. 1623 01:07:32,520 --> 01:07:35,160 I thought our particle had a positive mass. 1624 01:07:35,160 --> 01:07:37,010 Didn't it have a positive mass? 1625 01:07:37,010 --> 01:07:38,640 We started off with a positive mass. 1626 01:07:38,640 --> 01:07:40,181 We should still have a positive mass. 1627 01:07:40,181 --> 01:07:42,330 So, what's going on here? 1628 01:07:42,330 --> 01:07:45,330 Well, we have a rule for calculating what the mass is. 1629 01:07:45,330 --> 01:07:47,340 We take the expectation value of the momentum 1630 01:07:47,340 --> 01:07:50,589 and divide by the group velocity. 1631 01:07:50,589 --> 01:07:52,630 So, let's find out what the mass is for this guy. 1632 01:07:55,210 --> 01:07:57,670 And I'll do that here. 1633 01:07:57,670 --> 01:08:01,570 So, this is a slightly surreal moment 1634 01:08:01,570 --> 01:08:02,855 with a nice little codon. 1635 01:08:02,855 --> 01:08:04,980 So, I'm going to write that expression for the mass 1636 01:08:04,980 --> 01:08:05,854 slightly differently. 1637 01:08:05,854 --> 01:08:09,407 And you're gonna show in your problem set-- PSet-- 1638 01:08:09,407 --> 01:08:11,865 you're gonna show that that definition of the mass-- M star 1639 01:08:11,865 --> 01:08:13,960 is p upon the group velocity-- leads 1640 01:08:13,960 --> 01:08:16,880 to the following expression, you can write that as-- you can use 1641 01:08:16,880 --> 01:08:21,470 that to derive-- that the mass can be written as 1 upon h bar 1642 01:08:21,470 --> 01:08:28,660 squared d squared E dq squared. 1643 01:08:28,660 --> 01:08:31,680 So, the group velocity is the first derivative 1644 01:08:31,680 --> 01:08:34,837 of the energy with respect to q. 1645 01:08:34,837 --> 01:08:36,420 The mass-- or really q over the mass-- 1646 01:08:36,420 --> 01:08:39,840 is proportional to the second derivative, the curvature. 1647 01:08:39,840 --> 01:08:40,640 Everyone see that? 1648 01:08:40,640 --> 01:08:42,431 So, this should not be an obvious equation, 1649 01:08:42,431 --> 01:08:45,779 unless you're breathtakingly good at calculus in your head. 1650 01:08:45,779 --> 01:08:47,260 It's just chain rule, but it does 1651 01:08:47,260 --> 01:08:48,340 take a little bit of careful thinking, 1652 01:08:48,340 --> 01:08:49,290 because it's a physical argument, 1653 01:08:49,290 --> 01:08:51,040 it's not a rigorous mathematical argument. 1654 01:08:51,040 --> 01:08:52,717 You're gonna do it on your problem set. 1655 01:08:52,717 --> 01:08:54,550 So, let's take this expression and let's see 1656 01:08:54,550 --> 01:08:55,710 what the effective mass is. 1657 01:08:55,710 --> 01:08:57,270 What is the mass of our particle? 1658 01:08:57,270 --> 01:08:59,600 Now, here, when I see what is the mass of our particle? 1659 01:08:59,600 --> 01:09:01,439 I mean a very precise thing. 1660 01:09:01,439 --> 01:09:03,910 What is the mass of the object moving 1661 01:09:03,910 --> 01:09:05,750 in this periodic potential? 1662 01:09:05,750 --> 01:09:06,580 That's what I mean. 1663 01:09:06,580 --> 01:09:07,680 What is the mass of this thing? 1664 01:09:07,680 --> 01:09:08,720 And we have a definition for it. 1665 01:09:08,720 --> 01:09:09,410 Here it is. 1666 01:09:09,410 --> 01:09:13,170 And let's plot this 1 over mass. 1667 01:09:13,170 --> 01:09:14,778 So, let's plot this 1 over mass. 1668 01:09:14,778 --> 01:09:16,402 Oh, that's not the way I wanna draw it. 1669 01:09:19,229 --> 01:09:19,729 Yeah. 1670 01:09:22,429 --> 01:09:24,000 Yeah, OK, good. 1671 01:09:24,000 --> 01:09:30,630 So, here-- oh, shoot-- so, let's do it this way. 1672 01:09:30,630 --> 01:09:33,170 So, I want to plot 1 over mass, in the vertical, 1673 01:09:33,170 --> 01:09:34,960 as a function of q. 1674 01:09:34,960 --> 01:09:38,290 And here is q is equal to 0, and here is 1 1675 01:09:38,290 --> 01:09:40,640 upon mass is equal to 0. 1676 01:09:40,640 --> 01:09:44,390 So, at the bottom of the band-- and this is pi upon L, 1677 01:09:44,390 --> 01:09:49,000 and minus pi upon L. At the bottom of at the band, at q 1678 01:09:49,000 --> 01:09:51,660 equals 0, what is the mass? 1679 01:09:51,660 --> 01:09:54,140 Well, here's the velocity, and there's the band. 1680 01:09:54,140 --> 01:09:56,560 So, what's the mass at the bottom of the band? 1681 01:09:59,460 --> 01:10:02,230 Tell me properties about it. 1682 01:10:02,230 --> 01:10:02,780 Is it 0? 1683 01:10:07,310 --> 01:10:07,810 Yeah. 1684 01:10:07,810 --> 01:10:09,018 There's still some curvature. 1685 01:10:09,018 --> 01:10:14,160 It's approximated by some parabola, right? 1686 01:10:14,160 --> 01:10:15,525 It's even. 1687 01:10:15,525 --> 01:10:18,920 And, in fact, you can take that equation and derive properties 1688 01:10:18,920 --> 01:10:19,420 about it. 1689 01:10:19,420 --> 01:10:20,540 But it's some curvy thing. 1690 01:10:20,540 --> 01:10:21,562 You can see that. 1691 01:10:21,562 --> 01:10:23,770 It's more obvious if we look at a higher energy band. 1692 01:10:23,770 --> 01:10:25,260 So, here's the minimum. 1693 01:10:25,260 --> 01:10:26,100 Here's what it would have been if we 1694 01:10:26,100 --> 01:10:27,430 didn't have a potential, right? 1695 01:10:27,430 --> 01:10:29,390 But here's the minimum, and it's nice and curvy, and at the top, 1696 01:10:29,390 --> 01:10:30,590 again, it's nice and curvy. 1697 01:10:30,590 --> 01:10:32,923 But it's got a second derivative there, that's non-zero. 1698 01:10:32,923 --> 01:10:34,450 Everyone agree with that? 1699 01:10:34,450 --> 01:10:39,400 Now, here's the thing I want to ask, is that second derivative 1700 01:10:39,400 --> 01:10:44,945 the same as it would have been if we had g arbitrarily small? 1701 01:10:44,945 --> 01:10:46,024 AUDIENCE: [INAUDIBLE]? 1702 01:10:46,024 --> 01:10:46,690 PROFESSOR: Yeah. 1703 01:10:46,690 --> 01:10:47,394 Not so much. 1704 01:10:47,394 --> 01:10:49,310 No, if g was much lower, the second derivative 1705 01:10:49,310 --> 01:10:51,270 would be a bit different. 1706 01:10:51,270 --> 01:10:51,940 OK. 1707 01:10:51,940 --> 01:10:53,774 So, in particular, this is some funny value. 1708 01:10:53,774 --> 01:10:55,190 And I'm just gonna give it a name. 1709 01:10:55,190 --> 01:10:56,630 It's gonna have some value at 0. 1710 01:10:56,630 --> 01:10:58,379 And, in particular, it is positive. 1711 01:10:58,379 --> 01:10:59,420 Everyone agree with that? 1712 01:11:02,600 --> 01:11:04,970 But there's a point here-- there's an inflection point-- 1713 01:11:04,970 --> 01:11:07,429 where this thing has zero second derivative. 1714 01:11:07,429 --> 01:11:09,220 And that inflection point is the same point 1715 01:11:09,220 --> 01:11:10,761 where the velocity becomes a maximum, 1716 01:11:10,761 --> 01:11:12,550 because the velocity's a first derivative. 1717 01:11:12,550 --> 01:11:14,160 So, when the velocity's at a maximum, 1718 01:11:14,160 --> 01:11:16,290 the second derivative has a 0. 1719 01:11:16,290 --> 01:11:20,140 It's the derivative of the first velocity, zero slope. 1720 01:11:20,140 --> 01:11:21,880 And so, at that inflection point, 1721 01:11:21,880 --> 01:11:25,950 the velocity is 0-- or sorry, the velocity 1722 01:11:25,950 --> 01:11:29,770 is a maximum-- and 1 upon the mass, goes to 0. 1723 01:11:29,770 --> 01:11:35,240 And then, if we increase q further, what happens? 1724 01:11:35,240 --> 01:11:38,030 1 over m star goes negative. 1725 01:11:38,030 --> 01:11:43,660 And it reaches a minimum at pi over L, come back, 1726 01:11:43,660 --> 01:11:44,790 and, again, repeats. 1727 01:11:44,790 --> 01:11:46,260 So, it's periodic. 1728 01:11:46,260 --> 01:11:47,870 I'm just artistically challenged. 1729 01:11:47,870 --> 01:11:49,971 So, how can this possibly be? 1730 01:11:49,971 --> 01:11:50,470 Right? 1731 01:11:50,470 --> 01:11:51,928 So, first off, what this is telling 1732 01:11:51,928 --> 01:11:53,887 us is that we have a point where the mass is 0. 1733 01:11:53,887 --> 01:11:56,136 It's positive around the origin, where things normally 1734 01:11:56,136 --> 01:11:57,080 behave intuitively. 1735 01:11:57,080 --> 01:11:58,930 It goes to 0-- 1 over the mass-- goes 1736 01:11:58,930 --> 01:12:01,595 to 0, which tells me that the mass, m star, 1737 01:12:01,595 --> 01:12:02,470 is going to infinity. 1738 01:12:05,636 --> 01:12:06,760 1 over the mass going to 0? 1739 01:12:06,760 --> 01:12:07,920 Mass must be going to infinity. 1740 01:12:07,920 --> 01:12:08,690 What does that mean? 1741 01:12:08,690 --> 01:12:09,500 Well, what does it mean for something 1742 01:12:09,500 --> 01:12:10,541 to be infinitely massive? 1743 01:12:10,541 --> 01:12:14,230 It mean f equals ma, if it's infinitely massive-- then let's 1744 01:12:14,230 --> 01:12:17,390 write that as acceleration is force over m star-- then 1745 01:12:17,390 --> 01:12:19,100 that tells you that if you apply a force, 1746 01:12:19,100 --> 01:12:20,450 you get no acceleration. 1747 01:12:20,450 --> 01:12:21,850 But that's exactly what we saw. 1748 01:12:21,850 --> 01:12:24,130 We get to this point, we keep applying our force, 1749 01:12:24,130 --> 01:12:25,920 and the velocity stays constant. 1750 01:12:25,920 --> 01:12:27,020 There's no acceleration. 1751 01:12:27,020 --> 01:12:27,520 Yeah? 1752 01:12:27,520 --> 01:12:29,330 AUDIENCE: [INAUDIBLE] in this equation, 1753 01:12:29,330 --> 01:12:30,770 that's an external force, right? 1754 01:12:30,770 --> 01:12:31,515 PROFESSOR: That's an external force. 1755 01:12:31,515 --> 01:12:31,790 AUDIENCE: So, only-- 1756 01:12:31,790 --> 01:12:32,180 PROFESSOR: Exactly. 1757 01:12:32,180 --> 01:12:32,420 Yeah. 1758 01:12:32,420 --> 01:12:33,336 Purely external force. 1759 01:12:33,336 --> 01:12:35,563 I put a capacitor plate across my piece of metal. 1760 01:12:35,563 --> 01:12:37,351 AUDIENCE: So, my next question would 1761 01:12:37,351 --> 01:12:39,515 be is it all that insane then? 1762 01:12:39,515 --> 01:12:41,640 Because, like, you haven't calculated the net force 1763 01:12:41,640 --> 01:12:45,196 on the object, so to speak, and so there is the potential. 1764 01:12:45,196 --> 01:12:47,130 And so, yeah, the net force is zero, sure, 1765 01:12:47,130 --> 01:12:49,255 you're applying force, but it's not going anywhere. 1766 01:12:49,255 --> 01:12:52,910 PROFESSOR: Yeah, it's not a complete answer to what's 1767 01:12:52,910 --> 01:12:55,560 going on here, but it's a very good observation. 1768 01:12:55,560 --> 01:12:58,010 So, let me rephrase that slightly. 1769 01:12:58,010 --> 01:13:00,230 So, look, I'm doing a slightly strange thing. 1770 01:13:00,230 --> 01:13:02,170 I'm applying an external force, and I'm not 1771 01:13:02,170 --> 01:13:03,100 treating it quantum mechanically, 1772 01:13:03,100 --> 01:13:04,670 I'm treating this particle quantum mechanically. 1773 01:13:04,670 --> 01:13:05,860 That's clearly stupid. 1774 01:13:05,860 --> 01:13:07,330 We should treat everything quantum mechanically, 1775 01:13:07,330 --> 01:13:08,990 and derive the results in that fashion. 1776 01:13:08,990 --> 01:13:11,489 It turns out, in this case, that the negative effective mass 1777 01:13:11,489 --> 01:13:14,380 doesn't go away if we treat the electromagnetic field quantum 1778 01:13:14,380 --> 01:13:15,010 mechanically. 1779 01:13:15,010 --> 01:13:16,801 So, you're absolutely correct, but it's not 1780 01:13:16,801 --> 01:13:19,170 enough to sort of make us comfortable with the result. 1781 01:13:19,170 --> 01:13:20,940 It doesn't change this result. 1782 01:13:20,940 --> 01:13:23,440 But you're absolutely correct that that's an important thing 1783 01:13:23,440 --> 01:13:25,600 to do if we want to be honest. 1784 01:13:25,600 --> 01:13:29,810 But this is a perfectly good approximation for our purposes. 1785 01:13:29,810 --> 01:13:33,364 So, the mass goes infinite, and then it goes a negative! 1786 01:13:33,364 --> 01:13:35,530 1 over the mass goes negative, so that's ridiculous, 1787 01:13:35,530 --> 01:13:36,630 but that's exactly what we wanted. 1788 01:13:36,630 --> 01:13:38,630 The acceleration is the force divided by the mass, 1789 01:13:38,630 --> 01:13:40,740 but we found that when we apply a force in this direction, 1790 01:13:40,740 --> 01:13:42,577 we get an acceleration in this direction. 1791 01:13:42,577 --> 01:13:44,910 That's a negative coefficient, so, in particular, here's 1792 01:13:44,910 --> 01:13:46,034 what I want to think about. 1793 01:13:46,034 --> 01:13:48,110 I want to ask the following question, this value 1794 01:13:48,110 --> 01:13:49,651 of the mass-- even around the bottom, 1795 01:13:49,651 --> 01:13:51,810 this should already be disturbing observation-- 1796 01:13:51,810 --> 01:13:53,420 because this value of the mass is not 1797 01:13:53,420 --> 01:13:55,880 the same as the mass of our original particle. 1798 01:13:55,880 --> 01:13:57,463 And the way you can see that is if you 1799 01:13:57,463 --> 01:14:00,130 go to various strong coupling, then this band shrinks down. 1800 01:14:00,130 --> 01:14:02,440 It becomes very thin, and the second derivative 1801 01:14:02,440 --> 01:14:05,400 becomes arbitrarily small, which means 1802 01:14:05,400 --> 01:14:07,652 the mass becomes arbitrarily large. 1803 01:14:07,652 --> 01:14:09,360 So, the mass is getting larger and larger 1804 01:14:09,360 --> 01:14:11,634 as we crank up the potential-- the effective mass-- 1805 01:14:11,634 --> 01:14:13,300 of whatever this thing is that's moving. 1806 01:14:13,300 --> 01:14:14,460 But this is strange. 1807 01:14:14,460 --> 01:14:16,126 The thing that's moving is our particle. 1808 01:14:16,126 --> 01:14:17,220 It has mass, m. 1809 01:14:17,220 --> 01:14:18,660 Everyone agree with that? 1810 01:14:18,660 --> 01:14:20,890 What is going on here? 1811 01:14:20,890 --> 01:14:25,670 And here, I want to do an experiment. 1812 01:14:28,109 --> 01:14:28,900 And my experiment-- 1813 01:14:28,900 --> 01:14:32,600 AUDIENCE: [MURMURING] 1814 01:14:32,600 --> 01:14:37,110 PROFESSOR: So, yeah, the seamless pong ball company, 1815 01:14:37,110 --> 01:14:40,080 in recognition of my contributions to physics 1816 01:14:40,080 --> 01:14:45,640 and $3, has given me a series of ping pong balls, with which, 1817 01:14:45,640 --> 01:14:48,390 I'm going to do the following experiment. 1818 01:14:48,390 --> 01:14:49,110 OK. 1819 01:14:49,110 --> 01:14:52,556 Here's the calculation I want to do. 1820 01:14:52,556 --> 01:14:54,430 The lesson I want you to take away from this, 1821 01:14:54,430 --> 01:14:56,430 is that you've got to be very careful what 1822 01:14:56,430 --> 01:14:58,805 you mean by, "the mass" of an object. 1823 01:15:02,204 --> 01:15:03,620 Take one of these ping pong balls, 1824 01:15:03,620 --> 01:15:05,630 I'm not gonna do the whole experiment for you, 1825 01:15:05,630 --> 01:15:06,940 but I'll tell you the set up. 1826 01:15:06,940 --> 01:15:08,360 Take a ping pong ball, and the first thing I want to do 1827 01:15:08,360 --> 01:15:09,300 is, I want to measure it's mass. 1828 01:15:09,300 --> 01:15:10,320 Well, what do you mean by measure its mass? 1829 01:15:10,320 --> 01:15:12,130 Well, I'm gonna take it, and I'm gonna put it in a vacuum, 1830 01:15:12,130 --> 01:15:14,296 you know, we could take it to France to were they do 1831 01:15:14,296 --> 01:15:16,010 the-- And take the ping pong ball-- 1832 01:15:16,010 --> 01:15:17,510 ANNOUNCEMENT: One, two, three, four. 1833 01:15:17,510 --> 01:15:21,310 One, two, three, four. 1834 01:15:21,310 --> 01:15:24,740 One, two. 1835 01:15:24,740 --> 01:15:29,150 AUDIENCE: [LAUGHTER] 1836 01:15:29,150 --> 01:15:32,774 PROFESSOR: We're just gonna turn that down. 1837 01:15:32,774 --> 01:15:33,274 OK. 1838 01:15:33,274 --> 01:15:35,238 ANNOUNCEMENT: One, two, three, four. 1839 01:15:35,238 --> 01:15:39,670 One, two, three, four. 1840 01:15:39,670 --> 01:15:41,980 PROFESSOR: Can we kill that? 1841 01:15:41,980 --> 01:15:43,155 OK, so. 1842 01:15:46,950 --> 01:15:49,730 So, we take our ping pong ball, and we put some mass over here, 1843 01:15:49,730 --> 01:15:51,620 with a known mass, and we put it on a scale, 1844 01:15:51,620 --> 01:15:54,030 and we wait until this goes to the vertical, 1845 01:15:54,030 --> 01:15:55,030 and we declare the mass. 1846 01:15:55,030 --> 01:15:56,580 And what we find is that the mass of our ping pong 1847 01:15:56,580 --> 01:15:59,079 ball-- this is, you know, not the greatest ping pong balls-- 1848 01:15:59,079 --> 01:16:01,904 but by legislation, this is supposed to be about 2.7 grams, 1849 01:16:01,904 --> 01:16:03,070 or people get in fistfights. 1850 01:16:03,070 --> 01:16:04,486 So, the mass of the ping pong ball 1851 01:16:04,486 --> 01:16:05,790 needs to be about 2.7 grams. 1852 01:16:05,790 --> 01:16:07,740 And, meanwhile, the radius of this guy 1853 01:16:07,740 --> 01:16:12,020 is about 20 millimeters, so 2 centimeters. 1854 01:16:14,984 --> 01:16:16,130 Yeah, that's about right. 1855 01:16:16,130 --> 01:16:18,932 And yeah. 1856 01:16:18,932 --> 01:16:20,390 So, this is supposed to-- morally-- 1857 01:16:20,390 --> 01:16:21,710 it's supposed to be 2 centimeters. 1858 01:16:21,710 --> 01:16:24,180 And so, if you go through this and you compute the density 1859 01:16:24,180 --> 01:16:26,260 from these guys-- so, this gives you a volume-- 1860 01:16:26,260 --> 01:16:29,262 and if you compute the density, the density is about 12.4 1861 01:16:29,262 --> 01:16:29,845 by regulation. 1862 01:16:29,845 --> 01:16:32,100 Now, I've done this experiment, and I've done this measurement. 1863 01:16:32,100 --> 01:16:33,600 And I must've used very cheap balls, 1864 01:16:33,600 --> 01:16:35,720 because I got a very, very low density. 1865 01:16:35,720 --> 01:16:39,420 I got-- oh sorry, this is the wrong way to do it-- 1866 01:16:39,420 --> 01:16:42,550 the density is equal to 1, in usual CGS units, 1867 01:16:42,550 --> 01:16:45,380 is 1 upon 12.4. 1868 01:16:45,380 --> 01:16:47,930 And I can write that as 1 upon 12.4 times the density 1869 01:16:47,930 --> 01:16:49,090 of water. 1870 01:16:49,090 --> 01:16:49,670 OK? 1871 01:16:49,670 --> 01:16:52,330 Now, when I did this experiment, I got 1 over 20. 1872 01:16:52,330 --> 01:16:54,650 So, I was probably using very cheap ping pong balls. 1873 01:16:54,650 --> 01:16:56,649 But if you look at the regulations on Wikipedia, 1874 01:16:56,649 --> 01:16:59,570 this is what you're supposed to get. 1875 01:16:59,570 --> 01:17:01,010 Let's assume these are perfect. 1876 01:17:01,010 --> 01:17:02,152 So, you do this. 1877 01:17:02,152 --> 01:17:02,860 There's the mass. 1878 01:17:02,860 --> 01:17:03,569 That is the mass. 1879 01:17:03,569 --> 01:17:04,860 That's what I mean by the mass. 1880 01:17:04,860 --> 01:17:06,536 You take a scale, you weigh the thing! 1881 01:17:06,536 --> 01:17:07,036 Right? 1882 01:17:07,036 --> 01:17:09,620 Does anyone object to my definition of the mass? 1883 01:17:09,620 --> 01:17:11,809 It's an observational, empirical process 1884 01:17:11,809 --> 01:17:12,850 for determining the mass. 1885 01:17:12,850 --> 01:17:14,380 If I take my mass, OK? 1886 01:17:14,380 --> 01:17:17,439 And now I ask the following question, if I take some water, 1887 01:17:17,439 --> 01:17:19,730 and I pull my ping pong ball, and I put it under water, 1888 01:17:19,730 --> 01:17:21,760 and I put a spring scale here, and they 1889 01:17:21,760 --> 01:17:23,840 measure the force on this ping pong 1890 01:17:23,840 --> 01:17:26,150 ball, what will the force be? 1891 01:17:26,150 --> 01:17:28,850 Well, the force will be-- and let's just call 1892 01:17:28,850 --> 01:17:30,580 this ten, because the calculation's going 1893 01:17:30,580 --> 01:17:33,400 to be horrible otherwise-- well, whatever, let's call it 12. 1894 01:17:33,400 --> 01:17:36,220 So, the force-- and another way to write this 1895 01:17:36,220 --> 01:17:39,975 is that rho water is equal to 12 rho ping pong 1896 01:17:39,975 --> 01:17:44,510 ball-- I did math there. 1897 01:17:44,510 --> 01:17:48,020 So, the force-- we know this from Archimedes-- the force 1898 01:17:48,020 --> 01:17:50,140 is a combination of the weight of the ping pong 1899 01:17:50,140 --> 01:17:55,270 ball down, so, it's going to be the density times the volume. 1900 01:17:55,270 --> 01:18:01,800 So, the volume times rho ping pong ball down, so, minus. 1901 01:18:01,800 --> 01:18:05,920 Plus the volume times rho water up, 1902 01:18:05,920 --> 01:18:08,697 because that's the weight of the water displaced, right? 1903 01:18:08,697 --> 01:18:11,280 So, your weight down, minus the weight of the water displaced. 1904 01:18:11,280 --> 01:18:12,279 Everyone cool with that? 1905 01:18:12,279 --> 01:18:12,929 Archimedes. 1906 01:18:12,929 --> 01:18:14,470 It's pretty well known at this point. 1907 01:18:14,470 --> 01:18:17,990 So, this is equal to the volume times rho 1908 01:18:17,990 --> 01:18:21,540 water minus rho ping pong ball. 1909 01:18:21,540 --> 01:18:23,250 And I can write this as the volume times 1910 01:18:23,250 --> 01:18:25,180 rho water is 12 rho ping pong balls, 1911 01:18:25,180 --> 01:18:27,510 so that's 12 minus 1 rho ping pong ball. 1912 01:18:27,510 --> 01:18:31,550 That's 11 rho ping pong ball. 1913 01:18:31,550 --> 01:18:35,350 But this is equal to 11 times the weight of the ping 1914 01:18:35,350 --> 01:18:39,450 pong ball-- oh sorry-- 11 times the mass of the ping pong ball. 1915 01:18:39,450 --> 01:18:40,700 The volume times the density. 1916 01:18:40,700 --> 01:18:42,630 So, mass ping pong ball. 1917 01:18:42,630 --> 01:18:45,160 Now, there's an important thing that I forgot to add here, 1918 01:18:45,160 --> 01:18:47,270 what did I forget to add? 1919 01:18:47,270 --> 01:18:48,000 G. Right? 1920 01:18:48,000 --> 01:18:50,726 It's the force is the mass times g. 1921 01:18:50,726 --> 01:18:54,470 OK, g, g, and g. 1922 01:18:54,470 --> 01:18:58,399 So, we get that the force is equal to 11 m ping pong ball g. 1923 01:18:58,399 --> 01:18:59,440 Everyone agree with that? 1924 01:18:59,440 --> 01:19:01,030 I haven't done anything sneaky. 1925 01:19:01,030 --> 01:19:02,920 And so, as a consequence, what this predicts 1926 01:19:02,920 --> 01:19:05,190 is that if I take the force and I divide it 1927 01:19:05,190 --> 01:19:07,100 by the mass of the ping pong ball, 1928 01:19:07,100 --> 01:19:10,040 I get 11 times the acceleration of gravity. 1929 01:19:10,040 --> 01:19:12,590 And here's what that predicts, that predicts that when I let 1930 01:19:12,590 --> 01:19:16,710 go of this thing, what should happen? 1931 01:19:16,710 --> 01:19:19,410 It should accelerate upward through the water 1932 01:19:19,410 --> 01:19:22,360 at 11 times the acceleration of gravity, which 1933 01:19:22,360 --> 01:19:25,100 is roughly 10 meters per second squared. 1934 01:19:25,100 --> 01:19:26,920 So, 110 meters per second squared. 1935 01:19:26,920 --> 01:19:29,450 Does that sound right? 1936 01:19:29,450 --> 01:19:31,600 Just intuitively, does that sound right? 1937 01:19:31,600 --> 01:19:33,640 If you pull something like a ping pong 1938 01:19:33,640 --> 01:19:35,556 ball, or a beach ball or something underwater, 1939 01:19:35,556 --> 01:19:38,250 what happens when you let go of it? 1940 01:19:38,250 --> 01:19:40,350 Well, at this point, I'm going to remind you 1941 01:19:40,350 --> 01:19:42,760 that physics is an empirical science. 1942 01:19:42,760 --> 01:19:46,014 So, let's see if we can do this. 1943 01:19:46,014 --> 01:19:47,430 OK, now, what's supposed to happen 1944 01:19:47,430 --> 01:19:51,321 is this thing is supposed to shoot out at nine times-- or 11 1945 01:19:51,321 --> 01:19:52,820 times-- the acceleration of gravity. 1946 01:19:52,820 --> 01:19:55,660 So, it just shoots through this thing much faster 1947 01:19:55,660 --> 01:19:57,220 than equivalently. 1948 01:19:57,220 --> 01:19:59,698 So which one's going to hit first? 1949 01:19:59,698 --> 01:20:01,070 AUDIENCE: [INAUDIBLE]. 1950 01:20:01,070 --> 01:20:02,020 PROFESSOR: Should it come out, or should it 1951 01:20:02,020 --> 01:20:02,895 hit the ground first? 1952 01:20:02,895 --> 01:20:03,920 AUDIENCE: [INAUDIBLE]. 1953 01:20:03,920 --> 01:20:05,420 PROFESSOR: Yeah, the ping pong ball. 1954 01:20:05,420 --> 01:20:06,734 Great, that's helpful. 1955 01:20:06,734 --> 01:20:07,640 Uh-huh. 1956 01:20:07,640 --> 01:20:09,550 So, what's gonna happen first? 1957 01:20:09,550 --> 01:20:10,670 Is the ping pong ball gonna hit the surface, 1958 01:20:10,670 --> 01:20:12,503 or is it gonna come flying out of the water? 1959 01:20:12,503 --> 01:20:15,070 This is accelerating with the acceleration of gravity. 1960 01:20:15,070 --> 01:20:17,569 And this it's accelerating up with 11 times the acceleration 1961 01:20:17,569 --> 01:20:18,820 of gravity. 1962 01:20:18,820 --> 01:20:20,200 So, let's do the experiment. 1963 01:20:20,200 --> 01:20:22,041 And this is what they pay me for. 1964 01:20:22,041 --> 01:20:22,540 Oh. 1965 01:20:22,540 --> 01:20:23,956 This would be much more satisfying 1966 01:20:23,956 --> 01:20:33,120 if we had a nice long-- how do I want to phrase this-- cylinder. 1967 01:20:33,120 --> 01:20:33,680 Oh, shoot! 1968 01:20:33,680 --> 01:20:34,180 Stop. 1969 01:20:37,320 --> 01:20:39,470 So, it's best done with a nice, long cylinder, 1970 01:20:39,470 --> 01:20:41,380 but let's do this. 1971 01:20:41,380 --> 01:20:42,210 So, which one wins? 1972 01:20:42,210 --> 01:20:43,569 AUDIENCE: [MURMURS] 1973 01:20:43,569 --> 01:20:44,860 PROFESSOR: Gravity wins, right? 1974 01:20:44,860 --> 01:20:45,720 And why? 1975 01:20:45,720 --> 01:20:47,630 Intuitively, why does this work? 1976 01:20:47,630 --> 01:20:50,139 AUDIENCE: [MURMURING] 1977 01:20:50,139 --> 01:20:51,930 PROFESSOR: I am neglecting the drag effect, 1978 01:20:51,930 --> 01:20:53,660 but I'm not really heating up the water all that much. 1979 01:20:53,660 --> 01:20:55,160 I mean, it's not like the dissipation 1980 01:20:55,160 --> 01:20:56,160 is all that significant. 1981 01:20:56,160 --> 01:20:59,090 The dissipation of the momentum, which is drag-- the friction-- 1982 01:20:59,090 --> 01:21:01,362 is not the most important thing for our purposes here. 1983 01:21:01,362 --> 01:21:02,820 That water hasn't heated up at all. 1984 01:21:02,820 --> 01:21:05,054 So, what's going on? 1985 01:21:05,054 --> 01:21:07,747 AUDIENCE: [INAUDIBLE]. 1986 01:21:07,747 --> 01:21:09,330 PROFESSOR: OK, so here's the question. 1987 01:21:09,330 --> 01:21:11,788 Has a mass of the ping pong ball, my empty little ping pong 1988 01:21:11,788 --> 01:21:13,620 ball changed? 1989 01:21:13,620 --> 01:21:16,324 So, something like that has to be true. 1990 01:21:16,324 --> 01:21:17,490 But what's the right answer? 1991 01:21:17,490 --> 01:21:21,330 The right answer is this, the thing that's moving, 1992 01:21:21,330 --> 01:21:23,970 is not the ping pong ball. 1993 01:21:23,970 --> 01:21:26,100 The ping pong ball is interacting with the fluid. 1994 01:21:26,100 --> 01:21:28,020 And as a ping pong ball accelerates, 1995 01:21:28,020 --> 01:21:30,190 it drags the fluid along with it. 1996 01:21:30,190 --> 01:21:31,810 And as it goes faster, and faster, 1997 01:21:31,810 --> 01:21:34,239 it drags more and more of the fluid along with it. 1998 01:21:34,239 --> 01:21:36,280 And as it's dragging more and more of that fluid, 1999 01:21:36,280 --> 01:21:38,241 the effective mass of the entire object 2000 01:21:38,241 --> 01:21:39,990 that's moving-- which is now the ping pong 2001 01:21:39,990 --> 01:21:44,300 ball, and all of its associated water-- is increasing. 2002 01:21:44,300 --> 01:21:47,040 And now, if you take that force, which was constant, and divide 2003 01:21:47,040 --> 01:21:49,010 by this growing effective mass, you 2004 01:21:49,010 --> 01:21:51,777 find that the acceleration rapidly falls off. 2005 01:21:51,777 --> 01:21:53,110 And you get a terminal velocity. 2006 01:21:53,110 --> 01:21:54,050 And you know this works, because if you 2007 01:21:54,050 --> 01:21:56,150 blow a bubble under water, it doesn't shoot upwards 2008 01:21:56,150 --> 01:21:57,400 with an infinite acceleration. 2009 01:21:57,400 --> 01:21:59,020 It goes glurg, glurg, glurg. 2010 01:21:59,020 --> 01:22:02,930 And this idea, this idea of the effective mass 2011 01:22:02,930 --> 01:22:05,640 of a particle changing with its state of motion, 2012 01:22:05,640 --> 01:22:08,070 is an idea called renormalization. 2013 01:22:08,070 --> 01:22:10,690 And it plays an essential role, not just in particle physics, 2014 01:22:10,690 --> 01:22:13,140 but in all of modern condensed matter physics. 2015 01:22:13,140 --> 01:22:15,080 And we'll pick up at this point next time. 2016 01:22:15,080 --> 01:22:16,630 AUDIENCE: [APPLAUSE]