1 00:00:00,060 --> 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 --> 00:00:10,110 to offer high quality educational resources for free. 5 00:00:10,110 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,590 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,590 --> 00:00:17,251 at ocw.mit.edu. 8 00:00:27,307 --> 00:00:28,265 PROFESSOR: Hi everyone. 9 00:00:32,650 --> 00:00:34,000 I'm in a very good mood today. 10 00:00:38,260 --> 00:00:42,370 It's nothing to do with the class, but I'm having a baby. 11 00:00:42,370 --> 00:00:46,370 [APPLAUSE] 12 00:00:50,314 --> 00:00:52,660 PROFESSOR: So that's kind of exciting. 13 00:00:52,660 --> 00:00:55,500 So if I just started giggling, you'll know why. 14 00:00:55,500 --> 00:00:58,650 And in six months if I am just weeping and on the ground, 15 00:00:58,650 --> 00:00:59,550 you'll also know why. 16 00:01:02,320 --> 00:01:04,330 So today we're going to do two things. 17 00:01:04,330 --> 00:01:07,590 The first is I'm going to give you-- well, 18 00:01:07,590 --> 00:01:11,210 the first is review a little bit of practice 19 00:01:11,210 --> 00:01:13,180 for the exam we're going to have on Thursday. 20 00:01:13,180 --> 00:01:15,346 So let me tell you a little bit more about the exam. 21 00:01:15,346 --> 00:01:18,210 The exam, by the way, has been rescheduled to be in the 6120, 22 00:01:18,210 --> 00:01:19,550 not in Walker Gym. 23 00:01:19,550 --> 00:01:22,190 So it's going to be the usual place, we're not moving. 24 00:01:22,190 --> 00:01:25,760 And the reason is I'm changing the format on this exam, 25 00:01:25,760 --> 00:01:28,160 in part to make it a little less of a burden to everyone. 26 00:01:28,160 --> 00:01:30,035 But also in part because I've been struggling 27 00:01:30,035 --> 00:01:33,140 with the question of how to make the exam most useful. 28 00:01:33,140 --> 00:01:35,160 The purpose of an exam like this is not 29 00:01:35,160 --> 00:01:37,470 to get grades for you guys, although that's 30 00:01:37,470 --> 00:01:38,910 an incidental byproduct. 31 00:01:38,910 --> 00:01:41,680 The purpose is to give you some feedback on how you're 32 00:01:41,680 --> 00:01:45,180 doing, how your command of the material has evolved. 33 00:01:45,180 --> 00:01:50,010 And also to help you learn some of the things 34 00:01:50,010 --> 00:01:51,820 that you might not have mastered. 35 00:01:51,820 --> 00:01:54,065 So the way the exam is going to be structured 36 00:01:54,065 --> 00:01:57,310 is going about 15 minutes of short answer questions-- 37 00:01:57,310 --> 00:01:59,890 a couple of very short computations but mostly short 38 00:01:59,890 --> 00:02:01,297 answer questions-- on paper. 39 00:02:01,297 --> 00:02:02,880 You'll hand those back, and then we'll 40 00:02:02,880 --> 00:02:06,340 go over those questions in class afterwards. 41 00:02:06,340 --> 00:02:08,630 So it's a relatively low pressure exam 42 00:02:08,630 --> 00:02:10,799 and it's mostly conceptual. 43 00:02:10,799 --> 00:02:13,340 It will cover everything we've done through this last problem 44 00:02:13,340 --> 00:02:16,490 set to the degree that we get to the lecture part of today 45 00:02:16,490 --> 00:02:18,850 after questions. 46 00:02:18,850 --> 00:02:21,460 Today's lecture will not be directly covered, 47 00:02:21,460 --> 00:02:24,935 however it will be fair game for the next midterm-- which 48 00:02:24,935 --> 00:02:26,560 will be more of a traditional midterm-- 49 00:02:26,560 --> 00:02:28,840 and that's coming in April. 50 00:02:28,840 --> 00:02:30,912 So the structure of today is, I'm 51 00:02:30,912 --> 00:02:33,120 going to give you a whole bunch of clicker questions. 52 00:02:33,120 --> 00:02:35,309 So make sure you've got your clickers out. 53 00:02:35,309 --> 00:02:36,850 And those clicker questions are going 54 00:02:36,850 --> 00:02:40,540 to give you a sense for the level and scope of the exam. 55 00:02:40,540 --> 00:02:43,190 The exam will be a little harder than the clicker questions, 56 00:02:43,190 --> 00:02:44,420 but not a whole lot. 57 00:02:44,420 --> 00:02:45,920 And the difference is just that it's 58 00:02:45,920 --> 00:02:49,199 going to be on paper in front of you instead of clickered. 59 00:02:49,199 --> 00:02:50,990 And thus, that gives you a little more room 60 00:02:50,990 --> 00:02:57,120 to do calculations on a piece of paper, short calculations. 61 00:02:57,120 --> 00:03:02,860 And then after that we'll come on to-- in some sense a review, 62 00:03:02,860 --> 00:03:06,370 but also an introduction to the Dirac Bra-Ket notation 63 00:03:06,370 --> 00:03:08,250 that many of our textbooks use, but that we 64 00:03:08,250 --> 00:03:10,740 haven't introduced in lectures so far. 65 00:03:10,740 --> 00:03:12,650 Any questions before we get started? 66 00:03:12,650 --> 00:03:15,050 AUDIENCE: What channel are we? 67 00:03:15,050 --> 00:03:17,060 PROFESSOR: 41. 68 00:03:17,060 --> 00:03:19,478 Other questions? 69 00:03:19,478 --> 00:03:20,761 AUDIENCE: So no practice exam? 70 00:03:20,761 --> 00:03:22,510 PROFESSOR: I think there will probably not 71 00:03:22,510 --> 00:03:24,680 be a practice exam because of the shift in format. 72 00:03:24,680 --> 00:03:28,540 Today will basically be your practice exam. 73 00:03:28,540 --> 00:03:32,550 If I can get my work together and get 74 00:03:32,550 --> 00:03:38,280 you guys a practice exam of the right format, then maybe. 75 00:03:38,280 --> 00:03:41,240 But it wouldn't be all that useful is the problem. 76 00:03:41,240 --> 00:03:43,423 So watch what happens today. 77 00:03:43,423 --> 00:03:45,355 AUDIENCE: Will you post the clicker questions 78 00:03:45,355 --> 00:03:48,682 from today and the last time on the website? 79 00:03:48,682 --> 00:03:50,640 PROFESSOR: I will post the questions from today 80 00:03:50,640 --> 00:03:51,944 on the website, yes. 81 00:03:51,944 --> 00:03:53,735 AUDIENCE: So it's 50 minutes, short answer. 82 00:03:53,735 --> 00:03:55,675 Is there a certain number of questions 83 00:03:55,675 --> 00:03:57,370 that's going to be on it? 84 00:03:57,370 --> 00:03:59,120 PROFESSOR: I'm not going to tell you that. 85 00:03:59,120 --> 00:04:01,510 But it's not going to be a time trial. 86 00:04:01,510 --> 00:04:06,154 You're not going to be racing to get the, you know. 87 00:04:06,154 --> 00:04:09,439 AUDIENCE: Will it still be worth as much as an exam? 88 00:04:09,439 --> 00:04:10,230 PROFESSOR: It will. 89 00:04:10,230 --> 00:04:12,770 Because to my mind, part of the reason 90 00:04:12,770 --> 00:04:14,900 to design the exam this way is that it's 91 00:04:14,900 --> 00:04:17,250 testing your conceptual understanding, which 92 00:04:17,250 --> 00:04:19,769 is the important thing. 93 00:04:19,769 --> 00:04:20,930 So it will be. 94 00:04:23,470 --> 00:04:25,880 Other questions? 95 00:04:25,880 --> 00:04:26,380 All right. 96 00:04:26,380 --> 00:04:29,450 So let's get started with the clicker part of today. 97 00:04:33,950 --> 00:04:36,743 So we're on channel 41. 98 00:04:44,120 --> 00:04:46,610 So consider this eigenvalue equation, 99 00:04:46,610 --> 00:04:50,030 two derivatives on f-- it's constant-- times f of x. 100 00:04:52,880 --> 00:04:54,380 How many of these are eigenfunctions 101 00:04:54,380 --> 00:04:55,796 with the corresponding eigenvalue? 102 00:05:32,308 --> 00:05:34,230 You've got about 10 seconds. 103 00:05:34,230 --> 00:05:35,870 So go ahead and put in your clicks. 104 00:05:44,960 --> 00:05:49,610 And so your-- whoops, oh sorry. 105 00:05:49,610 --> 00:05:51,920 That just cleared your responses, unfortunately. 106 00:05:51,920 --> 00:05:53,722 Don't worry, they're saved in memory. 107 00:05:53,722 --> 00:05:55,180 It just cleared them off my screen. 108 00:05:55,180 --> 00:05:58,380 So the responses-- here we go. 109 00:05:58,380 --> 00:06:00,686 Oops, that didn't work. 110 00:06:00,686 --> 00:06:04,880 Wow, it just totally disappeared out of that app. 111 00:06:04,880 --> 00:06:06,200 Wow, that's so weird. 112 00:06:11,890 --> 00:06:13,180 Oh no, that didn't work. 113 00:06:13,180 --> 00:06:14,510 Let's try this. 114 00:06:14,510 --> 00:06:17,850 Where did it go? 115 00:06:17,850 --> 00:06:20,110 Ah, there it is. 116 00:06:20,110 --> 00:06:21,705 Whoa, this is all very confusing. 117 00:06:24,460 --> 00:06:30,660 So that was your response, B and C, 63% and 38%. 118 00:06:30,660 --> 00:06:33,580 So let's go back to this. 119 00:06:33,580 --> 00:06:35,305 Quickly discuss this with your neighbor. 120 00:07:03,200 --> 00:07:05,400 And now go ahead and click in your new answers 121 00:07:05,400 --> 00:07:12,170 if you-- you can keep talking, that's fine. 122 00:07:18,840 --> 00:07:19,725 Another five seconds. 123 00:07:22,890 --> 00:07:24,850 Awesome. 124 00:07:24,850 --> 00:07:25,990 OK, that's it. 125 00:07:25,990 --> 00:07:31,330 So the answer is B. And 99% of you all got that. 126 00:07:31,330 --> 00:07:33,310 I suspect someone didn't click. 127 00:07:33,310 --> 00:07:35,910 OK good, next question. 128 00:07:35,910 --> 00:07:38,000 Psi I and Psi II are two solutions 129 00:07:38,000 --> 00:07:39,830 of the Schrodinger equation. 130 00:07:39,830 --> 00:07:42,857 Is the sum of the two of them with coefficients A and B 131 00:07:42,857 --> 00:07:45,820 also a solution of the Schrodinger equation? 132 00:07:45,820 --> 00:07:48,140 Oh, and I forgot to start the clicky thingy. 133 00:07:48,140 --> 00:07:50,949 So click now. 134 00:07:50,949 --> 00:07:53,910 AUDIENCE: Did you mean to say B Psi 2 on that thing? 135 00:07:53,910 --> 00:07:56,290 PROFESSOR: Oh yeah, it is supposed to say B Psi 2. 136 00:07:56,290 --> 00:07:57,810 It says B Psi 2 on the next-- sorry, 137 00:07:57,810 --> 00:08:00,340 it's A Psi I plus B Psi I. It should 138 00:08:00,340 --> 00:08:02,460 say I Psi I plus B Psi II. 139 00:08:02,460 --> 00:08:04,032 Thank you. 140 00:08:04,032 --> 00:08:06,032 AUDIENCE: Does it mean that the same Schrodinger 141 00:08:06,032 --> 00:08:07,480 equation with the same potential? 142 00:08:07,480 --> 00:08:09,188 PROFESSOR: Yeah, with the same potential. 143 00:08:09,188 --> 00:08:10,486 Yeah, Yeah. 144 00:08:10,486 --> 00:08:16,230 [LAUGHTER] 145 00:08:16,230 --> 00:08:18,170 PROFESSOR: Wow, Yeah. 146 00:08:18,170 --> 00:08:20,170 So you know Einstein said, God didn't play dice. 147 00:08:20,170 --> 00:08:22,032 And let me paraphrase that as, God doesn't 148 00:08:22,032 --> 00:08:23,490 mess with you in clicker questions. 149 00:08:27,010 --> 00:08:29,530 And you guys have effectively universally that the answer 150 00:08:29,530 --> 00:08:33,950 is A, yes, superposition principle. 151 00:08:33,950 --> 00:08:40,750 OK next question is going to be for answer-- here we go. 152 00:08:40,750 --> 00:08:44,670 Consider an infinite square well with width a. 153 00:08:44,670 --> 00:08:48,500 And compared to the infinite square well with A, 154 00:08:48,500 --> 00:08:51,610 the ground state of a finite well 155 00:08:51,610 --> 00:08:54,905 is lower, higher, same energy, or undetermined. 156 00:09:04,490 --> 00:09:07,350 You've got 10 seconds, so continue thinking through this. 157 00:09:11,370 --> 00:09:11,965 5 seconds. 158 00:09:15,040 --> 00:09:17,912 OK, and you mostly got it, but have 159 00:09:17,912 --> 00:09:19,620 a quick chat with the person next to you. 160 00:10:00,370 --> 00:10:02,015 All right, let's try again. 161 00:10:08,170 --> 00:10:09,125 Such a good technique. 162 00:10:12,980 --> 00:10:15,230 All right, another five seconds to put in your answer. 163 00:10:15,230 --> 00:10:15,729 4-3-2-1. 164 00:10:20,420 --> 00:10:23,797 And you virtually all got it right, a lower energy. 165 00:10:23,797 --> 00:10:25,630 And let's just think about this intuitively. 166 00:10:25,630 --> 00:10:28,867 Intuitively, the gradient of the potential is the force, right? 167 00:10:28,867 --> 00:10:30,700 So in the second case, you've got less force 168 00:10:30,700 --> 00:10:32,760 cramming the article inside the box 169 00:10:32,760 --> 00:10:34,980 so it's being squeezed less tightly. 170 00:10:34,980 --> 00:10:36,759 More physically, you see that there 171 00:10:36,759 --> 00:10:38,300 is an evanescent tail on the outside. 172 00:10:38,300 --> 00:10:39,650 What that tells you is the wave function 173 00:10:39,650 --> 00:10:41,510 didn't have to go to zero at the ends. 174 00:10:41,510 --> 00:10:45,129 It just had to get small and latch onto a dying exponential. 175 00:10:45,129 --> 00:10:47,420 That's from the qualitative analysis of wave functions. 176 00:10:47,420 --> 00:10:49,040 But meanwhile, what that tells you 177 00:10:49,040 --> 00:10:53,350 is, it has to curve less inside in order-- it doesn't have 178 00:10:53,350 --> 00:10:55,726 to get to zero, it just has to get to a small value 179 00:10:55,726 --> 00:10:57,600 where it matches to the decaying exponential. 180 00:10:57,600 --> 00:11:01,440 So if the curvature is less, then the energy is less. 181 00:11:01,440 --> 00:11:02,610 Cool? 182 00:11:02,610 --> 00:11:07,990 OK, next question. 183 00:11:07,990 --> 00:11:11,601 So any questions on that before I? 184 00:11:11,601 --> 00:11:12,100 Good. 185 00:11:16,440 --> 00:11:19,480 Time zero wave function infinite well with a 186 00:11:19,480 --> 00:11:23,530 is this, sine squared with a normalization. 187 00:11:23,530 --> 00:11:25,660 What's the wave function at a subsequent time t? 188 00:11:35,690 --> 00:11:37,360 I will remind you that you have solved 189 00:11:37,360 --> 00:11:39,200 the problem of the infinite square well, 190 00:11:39,200 --> 00:11:41,940 and you know what the eigenenergies 191 00:11:41,940 --> 00:11:46,040 and eigenfunctions are of the energy 192 00:11:46,040 --> 00:11:49,002 operator in the infinite square well. 193 00:11:49,002 --> 00:11:50,460 So remember back to what those are. 194 00:11:56,040 --> 00:11:57,830 All right, and you have five seconds. 195 00:12:03,130 --> 00:12:06,780 OK, we are at about 50-50 correct. 196 00:12:06,780 --> 00:12:08,280 So chat with the person next to you. 197 00:13:01,406 --> 00:13:03,910 All right. 198 00:13:03,910 --> 00:13:09,245 And now, any moment, go ahead and vote again. 199 00:13:18,870 --> 00:13:21,500 Good, five more seconds and then put in your final vote. 200 00:13:24,220 --> 00:13:26,690 OK, that's it for now. 201 00:13:26,690 --> 00:13:28,582 So what's the answer? 202 00:13:28,582 --> 00:13:31,040 D is the answer, but a lot of people still had some doubts. 203 00:13:31,040 --> 00:13:33,410 So who wants to give an explanation for why it's D? 204 00:13:39,360 --> 00:13:42,696 AUDIENCE: So sine squared is not an eigenfunction. 205 00:13:42,696 --> 00:13:44,624 PROFESSOR: Fantastic. 206 00:13:44,624 --> 00:13:47,034 AUDIENCE: So in some way it hast to be 207 00:13:47,034 --> 00:13:49,540 a summation of eigenfunctions. 208 00:13:49,540 --> 00:13:53,010 So not even having to know what the eigenfunctions are, 209 00:13:53,010 --> 00:13:54,420 there's only one summation in it. 210 00:13:57,720 --> 00:13:59,055 PROFESSOR: Excellent, excellent. 211 00:13:59,055 --> 00:14:00,790 AUDIENCE: And if you do know what the eigenfunctions are, 212 00:14:00,790 --> 00:14:01,460 now you know that [INAUDIBLE]. 213 00:14:01,460 --> 00:14:02,830 PROFESSOR: Brilliant, so I'm going to restate that. 214 00:14:02,830 --> 00:14:04,660 That was exactly correct in every step. 215 00:14:04,660 --> 00:14:08,100 So the first thing is, that wave function at time 0, 216 00:14:08,100 --> 00:14:11,160 sine squared of x, is not an eigenfunction of the energy 217 00:14:11,160 --> 00:14:12,160 operator for the system. 218 00:14:12,160 --> 00:14:14,539 In fact, we've computed the eigenfunctions for the energy 219 00:14:14,539 --> 00:14:16,080 operator in the infinite square well, 220 00:14:16,080 --> 00:14:20,340 and they're sines where they get a zero at the ends. 221 00:14:20,340 --> 00:14:23,380 On the other hand, any function satisfying the boundary 222 00:14:23,380 --> 00:14:25,910 conditions-- normalizable, hits zero at the boundaries-- 223 00:14:25,910 --> 00:14:29,740 is a superposition of energy eigenfunctions, 224 00:14:29,740 --> 00:14:32,350 and we can use that to determine the time evolution we take 225 00:14:32,350 --> 00:14:35,520 that superposition and add on a phase-- e 226 00:14:35,520 --> 00:14:39,480 to the minus i et upon h bar for each of the energy 227 00:14:39,480 --> 00:14:40,330 eigenfunctions. 228 00:14:40,330 --> 00:14:42,530 And in part D we express that wave 229 00:14:42,530 --> 00:14:46,850 function as a superposition with the coefficient cn determined 230 00:14:46,850 --> 00:14:50,830 from the overlap of our original function and the energy 231 00:14:50,830 --> 00:14:51,960 eigenfunctions. 232 00:14:51,960 --> 00:14:53,070 Everyone cool with that? 233 00:14:53,070 --> 00:14:55,390 So this is literally just a transcription of one 234 00:14:55,390 --> 00:14:57,670 of our postulates. 235 00:14:57,670 --> 00:14:58,620 OK, questions? 236 00:15:01,440 --> 00:15:05,180 If you have any questions at all, ask them. 237 00:15:05,180 --> 00:15:07,410 This is the time to ask them. 238 00:15:07,410 --> 00:15:08,750 Yeah. 239 00:15:08,750 --> 00:15:11,759 AUDIENCE: So the way get cn [INAUDIBLE]? 240 00:15:11,759 --> 00:15:12,550 PROFESSOR: Exactly. 241 00:15:12,550 --> 00:15:15,360 So the way you get cn is by saying, 242 00:15:15,360 --> 00:15:17,530 look I have my wave function, psi 243 00:15:17,530 --> 00:15:23,780 of x is equal to sum over n cn phi n of z, 244 00:15:23,780 --> 00:15:26,360 where these phi n's energy eigenfunctions. 245 00:15:26,360 --> 00:15:31,320 E phi n is equal to en phi m of x. 246 00:15:34,600 --> 00:15:38,130 And we also know that the integral of phi n's 247 00:15:38,130 --> 00:15:42,734 complex conjugate phi m is equal to delta mn. 248 00:15:42,734 --> 00:15:44,150 This is the statement that they're 249 00:15:44,150 --> 00:15:47,940 orthogonal and properly normalized. 250 00:15:47,940 --> 00:15:54,770 We also write this as equal to phi n phi m. 251 00:15:54,770 --> 00:15:58,370 And we can use this to determine the cn is 252 00:15:58,370 --> 00:16:04,260 equal to the inner product of phi m with our wave function. 253 00:16:04,260 --> 00:16:10,040 This is equal to the integral dx phi star phi n star 254 00:16:10,040 --> 00:16:15,320 psi, which is equal to the integral dx phi n star 255 00:16:15,320 --> 00:16:19,990 sum over m of cm phi m. 256 00:16:19,990 --> 00:16:22,510 But this sum over m of cm can be pulled out 257 00:16:22,510 --> 00:16:25,080 because this is just an integral over a sum of terms, which 258 00:16:25,080 --> 00:16:29,900 is the same as the sum over m cm integral phi 259 00:16:29,900 --> 00:16:34,040 m-- complex conjugate-- phi m. 260 00:16:34,040 --> 00:16:37,584 And that's delta mn, which is 0 unless n is equal to m, 261 00:16:37,584 --> 00:16:39,500 because these guys are orthogonal and properly 262 00:16:39,500 --> 00:16:40,250 normalized. 263 00:16:40,250 --> 00:16:42,246 So this is zero unless n is equal to m. 264 00:16:42,246 --> 00:16:44,120 So in the sum, the only term that contributes 265 00:16:44,120 --> 00:16:46,860 is when m is equal to n, this is equal to cn. 266 00:16:46,860 --> 00:16:49,630 So cn is given by the overlap of our wave function 267 00:16:49,630 --> 00:16:51,600 with the corresponding eigenfunction. 268 00:16:51,600 --> 00:16:54,470 This allows us to take our function, a known function-- 269 00:16:54,470 --> 00:16:58,870 for example, sine squared-- and express the coefficients 270 00:16:58,870 --> 00:17:01,730 in terms of an overlap of our wave function sine squared 271 00:17:01,730 --> 00:17:04,530 with the wave functions sine. 272 00:17:04,530 --> 00:17:06,670 And that's exactly the expression you see below, 273 00:17:06,670 --> 00:17:08,572 cn is equal to the integral of sine 274 00:17:08,572 --> 00:17:10,030 squared-- our wave function-- times 275 00:17:10,030 --> 00:17:11,696 sine, which is the energy eigenfunction. 276 00:17:14,703 --> 00:17:15,369 Other questions? 277 00:17:18,050 --> 00:17:18,910 OK, great. 278 00:17:22,819 --> 00:17:25,220 Next problem. 279 00:17:25,220 --> 00:17:26,295 Come to me computer. 280 00:17:33,535 --> 00:17:35,160 Why can they just be written in Python. 281 00:17:35,160 --> 00:17:38,465 OK, good. 282 00:17:38,465 --> 00:17:41,900 the eigenstates phi n-- which we usually call psi n, 283 00:17:41,900 --> 00:17:44,280 but there it is-- form an orthonormal set. 284 00:17:44,280 --> 00:17:48,380 Meaning integral of phi m star phi n is delta mn. 285 00:17:48,380 --> 00:17:53,686 What is the value of integral psi m against the sum cn phi n? 286 00:17:57,220 --> 00:17:59,904 You've got five seconds. 287 00:17:59,904 --> 00:18:01,320 I'll give you a little extra time, 288 00:18:01,320 --> 00:18:02,695 because people are clicking away. 289 00:18:07,954 --> 00:18:10,370 OK you now have seven seconds, because time is non-linear. 290 00:18:16,460 --> 00:18:21,890 OK, so quickly discuss, because there's still 291 00:18:21,890 --> 00:18:22,920 some ambiguity here. 292 00:18:37,650 --> 00:18:40,600 All right, you have 10 seconds to modify your clicks. 293 00:18:40,600 --> 00:18:41,100 Click. 294 00:18:44,390 --> 00:18:51,350 All right, yes and the answer is, C, excellent. 295 00:18:51,350 --> 00:18:51,850 Right? 296 00:18:51,850 --> 00:18:53,440 Because in fact, we just did that. 297 00:18:56,510 --> 00:18:58,070 That won't happen on the actual exam. 298 00:18:58,070 --> 00:19:00,150 So here's the next question. 299 00:19:00,150 --> 00:19:03,800 Let the Hamiltonian on a dagger Un 300 00:19:03,800 --> 00:19:05,980 equal en plus h omega-- that should 301 00:19:05,980 --> 00:19:09,720 be an h bar-- a dagger Un. 302 00:19:09,720 --> 00:19:11,480 What can you say about a dagger Un? 303 00:19:17,310 --> 00:19:18,710 Here we should probably say, what 304 00:19:18,710 --> 00:19:26,380 can one say, because it's possible-- OK, 305 00:19:26,380 --> 00:19:28,050 what can one say? 306 00:19:38,560 --> 00:19:40,464 Here the assumption-- just to say it out 307 00:19:40,464 --> 00:19:42,630 loud-- the assumption is that Un is an eigenfunction 308 00:19:42,630 --> 00:19:46,300 of the energy operator h with eigenvalue en. 309 00:19:52,171 --> 00:19:53,170 You've got five seconds. 310 00:19:58,880 --> 00:20:02,340 All right, we are at 50/50. 311 00:20:02,340 --> 00:20:04,130 So discuss amongst yourselves. 312 00:21:17,240 --> 00:21:17,905 All right. 313 00:21:21,792 --> 00:21:23,000 OK, go ahead and enter again. 314 00:21:23,000 --> 00:21:25,710 Enter your modified guesses. 315 00:21:25,710 --> 00:21:28,575 And you have 10 seconds to do so. 316 00:21:28,575 --> 00:21:29,450 This is a lot better. 317 00:21:34,350 --> 00:21:36,687 OK, this is great. 318 00:21:36,687 --> 00:21:38,770 So this is one of those really satisfying moments. 319 00:21:38,770 --> 00:21:41,990 It's improved, but there's still some real doubt here. 320 00:21:41,990 --> 00:21:45,150 So I would like to get one person to argue 321 00:21:45,150 --> 00:21:47,100 for b and one person to argue for c. 322 00:21:47,100 --> 00:21:49,250 So who wants to volunteer for each of those? 323 00:21:49,250 --> 00:21:50,440 AUDIENCE: I'll argue for c. 324 00:21:50,440 --> 00:21:52,475 PROFESSOR: Who's going to argue for b? 325 00:21:52,475 --> 00:21:54,721 Someone's got to argue for b, come on. 326 00:21:54,721 --> 00:21:55,720 AUDIENCE: You can argue. 327 00:21:55,720 --> 00:21:57,160 PROFESSOR: I'm not going to argue for b. 328 00:21:57,160 --> 00:21:58,320 I'm not going to argue for c either. 329 00:21:58,320 --> 00:21:59,385 That defeats the purpose. 330 00:21:59,385 --> 00:21:59,880 I'm the professor. 331 00:21:59,880 --> 00:22:01,213 I have to say this all the time. 332 00:22:01,213 --> 00:22:04,600 So who's going to argue for b? 333 00:22:04,600 --> 00:22:05,490 OK, you argue for b. 334 00:22:05,490 --> 00:22:08,270 Who's going to argue for c? 335 00:22:08,270 --> 00:22:11,420 All right, yeah, that worked out well. 336 00:22:11,420 --> 00:22:14,222 Great. 337 00:22:14,222 --> 00:22:14,830 Argue for b. 338 00:22:14,830 --> 00:22:16,255 You can do it. 339 00:22:16,255 --> 00:22:18,463 AUDIENCE: So I originally accidentally mis-clicked b, 340 00:22:18,463 --> 00:22:20,700 so I guess I can do this. 341 00:22:20,700 --> 00:22:23,780 So originally I didn't read the question, 342 00:22:23,780 --> 00:22:25,700 and I thought since you were acting the letter 343 00:22:25,700 --> 00:22:27,770 operator on the eigenstate that you'd 344 00:22:27,770 --> 00:22:31,450 get a proportionality constant times some eigenstate. 345 00:22:31,450 --> 00:22:33,550 So that's why it could potentially be b. 346 00:22:33,550 --> 00:22:37,360 But I disbelieve. 347 00:22:37,360 --> 00:22:40,010 PROFESSOR: It's not the best argument imaginable for b, 348 00:22:40,010 --> 00:22:41,150 but we'll take it. 349 00:22:41,150 --> 00:22:42,240 So thank you. 350 00:22:47,385 --> 00:22:49,260 When I put someone in an impossible position. 351 00:22:49,260 --> 00:22:50,427 So c. 352 00:22:50,427 --> 00:22:51,260 AUDIENCE: All right. 353 00:22:51,260 --> 00:22:52,715 I argue for c. 354 00:22:52,715 --> 00:22:55,867 So we can see just from the first line 355 00:22:55,867 --> 00:22:59,191 here that this is clearly going to be a stationary [INAUDIBLE]. 356 00:22:59,191 --> 00:23:03,120 However, we can also see that it is a distinctly different 357 00:23:03,120 --> 00:23:05,532 energy from the state that you get 358 00:23:05,532 --> 00:23:06,740 an eigenfunction [INAUDIBLE]. 359 00:23:06,740 --> 00:23:09,476 So I said c, because it was not the same state. 360 00:23:09,476 --> 00:23:10,350 PROFESSOR: Fantastic. 361 00:23:10,350 --> 00:23:11,210 That's exactly right. 362 00:23:11,210 --> 00:23:12,000 So let me walk through that. 363 00:23:12,000 --> 00:23:13,080 Let me say that allowed. 364 00:23:13,080 --> 00:23:19,030 So from the first line, it is clear that the object a Un 365 00:23:19,030 --> 00:23:21,860 is an eigenfunction of the energy operator. 366 00:23:21,860 --> 00:23:24,760 And it's got eigenvalue en plus h omega. 367 00:23:24,760 --> 00:23:25,927 So it is a stationary state. 368 00:23:25,927 --> 00:23:27,926 It's an eigenfunction of the energy [INAUDIBLE]. 369 00:23:27,926 --> 00:23:29,890 However, it is not the same one, because it 370 00:23:29,890 --> 00:23:31,610 has a different eigenvalue. 371 00:23:31,610 --> 00:23:34,640 So what it means to be the same energy eigenstate 372 00:23:34,640 --> 00:23:36,730 is it has the same eigenvalue. 373 00:23:36,730 --> 00:23:38,702 If it has a different energy eigenvalue, 374 00:23:38,702 --> 00:23:40,660 it is a different state and they're orthogonal. 375 00:23:40,660 --> 00:23:41,730 We proved this before. 376 00:23:41,730 --> 00:23:44,020 Two states with different energy are orthogonal. 377 00:23:44,020 --> 00:23:45,580 So not only are they not the same, 378 00:23:45,580 --> 00:23:48,580 they don't even have any overlap. 379 00:23:48,580 --> 00:23:50,590 So it is a stationary state, but it's not 380 00:23:50,590 --> 00:23:52,744 proportional with the state Un. 381 00:23:52,744 --> 00:23:53,660 It's an important one. 382 00:23:58,130 --> 00:24:01,020 Questions about that one before we move on to the next? 383 00:24:01,020 --> 00:24:02,520 AUDIENCE: How did you tell that it's 384 00:24:02,520 --> 00:24:03,935 proportional to the state Un? 385 00:24:03,935 --> 00:24:06,060 PROFESSOR: If it were proportional to the state Un, 386 00:24:06,060 --> 00:24:07,911 then it would be some constant times Un. 387 00:24:07,911 --> 00:24:09,910 That's what we mean by saying it's proportional. 388 00:24:09,910 --> 00:24:12,410 But then if we acted on it with the energy operator, 389 00:24:12,410 --> 00:24:15,264 what would the eigenvalue be? 390 00:24:15,264 --> 00:24:16,180 Let me say this again. 391 00:24:16,180 --> 00:24:17,972 Suppose I have a state the Un, which 392 00:24:17,972 --> 00:24:20,180 I know that if I act with the energy operator on it-- 393 00:24:20,180 --> 00:24:24,140 or this is sometimes called h Un is equal to En Un. 394 00:24:26,990 --> 00:24:30,560 If I act with E on alpha Un, where alpha's a constant, 395 00:24:30,560 --> 00:24:33,620 what is this equal to? 396 00:24:33,620 --> 00:24:36,560 Alpha EnUn. 397 00:24:36,560 --> 00:24:39,100 So the eigenvalue is the same. 398 00:24:39,100 --> 00:24:43,280 It's En, because it can divide through by [INAUDIBLE]. 399 00:24:43,280 --> 00:24:46,240 So if you have the same state-- meaning proportional to it-- 400 00:24:46,240 --> 00:24:47,740 then we'll have the same eigenvalue. 401 00:24:47,740 --> 00:24:50,430 But this manifestly has a different eigenvalue. 402 00:24:50,430 --> 00:24:51,540 Cool? 403 00:24:51,540 --> 00:24:53,527 Awesome, OK other questions? 404 00:24:53,527 --> 00:24:54,219 Yeah. 405 00:24:54,219 --> 00:24:56,510 AUDIENCE: I must have missed the-- this is [INAUDIBLE]. 406 00:24:56,510 --> 00:24:57,660 What is the [INAUDIBLE]. 407 00:24:57,660 --> 00:24:58,930 PROFESSOR: Ah, good. 408 00:24:58,930 --> 00:25:01,350 So H is another name. 409 00:25:01,350 --> 00:25:04,020 So some books use H and some books use E. 410 00:25:04,020 --> 00:25:09,300 And the reason goes back to classical mechanics. 411 00:25:09,300 --> 00:25:12,080 There's an object called the Hamiltonian, which 412 00:25:12,080 --> 00:25:13,890 is sort of like saying-- I don't know. 413 00:25:13,890 --> 00:25:17,880 It's like saying bread and [NON ENGLISH SPEECH] 414 00:25:17,880 --> 00:25:19,210 are the same thing. 415 00:25:19,210 --> 00:25:20,620 But they evoke different things. 416 00:25:20,620 --> 00:25:22,354 You say bread and you think like Dixie. 417 00:25:22,354 --> 00:25:24,020 And you say [NON ENGLISH SPEECH] and you 418 00:25:24,020 --> 00:25:29,060 think, the me of some gorgeous baguette. 419 00:25:29,060 --> 00:25:31,310 It has a different feel. 420 00:25:31,310 --> 00:25:32,410 It has a different feel. 421 00:25:32,410 --> 00:25:33,870 So this evokes different things. 422 00:25:33,870 --> 00:25:36,120 In particular, to a classical mechanic 423 00:25:36,120 --> 00:25:39,192 the Hamiltonian is the generator of time translations. 424 00:25:39,192 --> 00:25:40,650 Now, we don't usually-- that wasn't 425 00:25:40,650 --> 00:25:43,020 the classic way of thinking about the energy operator 426 00:25:43,020 --> 00:25:45,510 in, say, Newton's time. 427 00:25:45,510 --> 00:25:47,500 But it is how we think about energy now. 428 00:25:47,500 --> 00:25:49,169 So that's just terminology. 429 00:25:49,169 --> 00:25:51,210 For anyone who didn't get that, sorry about that. 430 00:25:51,210 --> 00:25:53,700 This is something that we've run across before, 431 00:25:53,700 --> 00:25:55,037 but I should emphasize. 432 00:25:55,037 --> 00:25:56,870 So when you hear people say the Hamiltonian, 433 00:25:56,870 --> 00:25:58,576 that means the energy operator. 434 00:25:58,576 --> 00:26:00,200 The reason I say energy operator rather 435 00:26:00,200 --> 00:26:01,280 than saying Hamiltonian-- which is 436 00:26:01,280 --> 00:26:03,489 the more common thing in the Lingua Franca of quantum 437 00:26:03,489 --> 00:26:05,738 mechanics-- is because I want to impress upon you guys 438 00:26:05,738 --> 00:26:06,770 that it's just energy. 439 00:26:06,770 --> 00:26:08,810 It is the energy, it's nothing else. 440 00:26:08,810 --> 00:26:10,520 It's not some strange beast. 441 00:26:10,520 --> 00:26:13,040 It's just energy. 442 00:26:13,040 --> 00:26:14,686 Other questions? 443 00:26:14,686 --> 00:26:16,617 AUDIENCE: Which one's the bread? 444 00:26:16,617 --> 00:26:17,325 PROFESSOR: Bread. 445 00:26:22,969 --> 00:26:23,635 Other questions? 446 00:26:27,420 --> 00:26:29,330 Next question. 447 00:26:29,330 --> 00:26:32,190 Which of these graphs shows the curvature of the wave function 448 00:26:32,190 --> 00:26:34,230 in a classically allowed region? 449 00:26:41,390 --> 00:26:42,630 OK, five seconds left. 450 00:26:47,410 --> 00:26:49,680 All right, fantastic. 451 00:26:49,680 --> 00:26:52,770 And the answer is a, because the curvature 452 00:26:52,770 --> 00:26:55,110 has to be towards the axis. 453 00:26:55,110 --> 00:26:56,820 In the classically allowed region, 454 00:26:56,820 --> 00:26:58,980 the wave function is sinusoidal. 455 00:26:58,980 --> 00:27:01,900 it's oscillatory. 456 00:27:01,900 --> 00:27:04,970 All right, and-- whoops. 457 00:27:04,970 --> 00:27:07,670 And the partner of that question is, 458 00:27:07,670 --> 00:27:09,670 what about in the classically disallowed region? 459 00:27:17,940 --> 00:27:18,596 Five seconds. 460 00:27:22,680 --> 00:27:25,970 OK, awesome, and basically everyone got this. 461 00:27:25,970 --> 00:27:27,501 B, great. 462 00:27:27,501 --> 00:27:28,000 Questions? 463 00:27:32,652 --> 00:27:34,548 Yeah, sorry. 464 00:27:34,548 --> 00:27:37,422 AUDIENCE: [INAUDIBLE]. 465 00:27:37,422 --> 00:27:38,477 PROFESSOR: No. 466 00:27:38,477 --> 00:27:39,810 No, that's a very good question. 467 00:27:39,810 --> 00:27:42,185 So C and D, let me be a little bit more precise about it. 468 00:27:42,185 --> 00:27:45,051 It's a very reasonable question, so let me give you an answer. 469 00:27:45,051 --> 00:27:46,550 Suppose you saw a wave function that 470 00:27:46,550 --> 00:27:48,060 had the curvature structure of C. 471 00:27:48,060 --> 00:27:50,820 It was curved oscillatory over there 472 00:27:50,820 --> 00:27:54,670 and curved exponential over there on the right. 473 00:27:54,670 --> 00:27:57,860 What would you say about the potential relative 474 00:27:57,860 --> 00:27:58,510 to the energy? 475 00:28:02,550 --> 00:28:04,420 Say again? 476 00:28:04,420 --> 00:28:05,080 Yeah, good. 477 00:28:05,080 --> 00:28:07,720 So on one side we're allowed, and on the other side 478 00:28:07,720 --> 00:28:08,444 we're disallowed. 479 00:28:08,444 --> 00:28:09,860 So if you saw a wave function that 480 00:28:09,860 --> 00:28:11,440 had that curvature structure, you'd 481 00:28:11,440 --> 00:28:14,132 immediately know that it was an allowed region on the left. 482 00:28:14,132 --> 00:28:16,340 But then on the right, it was classically disallowed, 483 00:28:16,340 --> 00:28:18,500 because it had curvature away from the axis. 484 00:28:18,500 --> 00:28:19,196 That cool? 485 00:28:19,196 --> 00:28:21,320 So if you saw a wave function-- so for example, you 486 00:28:21,320 --> 00:28:24,570 might say look, you never get a wave function that does this. 487 00:28:24,570 --> 00:28:25,970 And that's not true. 488 00:28:29,140 --> 00:28:30,110 You could do that. 489 00:28:30,110 --> 00:28:31,950 So this would be classically disallowed, 490 00:28:31,950 --> 00:28:33,214 because it's curvature down. 491 00:28:33,214 --> 00:28:34,630 This would be classically allowed. 492 00:28:34,630 --> 00:28:37,252 So the potential, for example, if this is the energy, 493 00:28:37,252 --> 00:28:38,210 the potential could be. 494 00:28:44,760 --> 00:28:48,592 So a potential like this could lead to a function like this. 495 00:28:48,592 --> 00:28:49,800 So it's a very good question. 496 00:28:49,800 --> 00:28:51,060 I'm sorry, I should have seen what 497 00:28:51,060 --> 00:28:52,434 you were asking more immediately. 498 00:28:54,909 --> 00:28:56,450 Other questions following up on that? 499 00:29:01,470 --> 00:29:03,354 No one ever asked that question before. 500 00:29:03,354 --> 00:29:05,770 And I just want to emphasize that you guys should just ask 501 00:29:05,770 --> 00:29:07,640 questions when you don't-- when you've confused about 502 00:29:07,640 --> 00:29:09,170 something, someone else is going to be confused about it. 503 00:29:09,170 --> 00:29:10,820 I wouldn't even necessarily know that the question 504 00:29:10,820 --> 00:29:11,861 is a meaningful question. 505 00:29:11,861 --> 00:29:13,760 I get questions I've never heard before all 506 00:29:13,760 --> 00:29:14,986 the time in this class. 507 00:29:14,986 --> 00:29:16,110 Just don't hesitate to ask. 508 00:29:16,110 --> 00:29:17,370 It's always a good idea. 509 00:29:17,370 --> 00:29:19,840 OK, next one. 510 00:29:19,840 --> 00:29:22,580 Wave function psi has been expressed 511 00:29:22,580 --> 00:29:25,506 as a sum over energy eigenfunction Un. 512 00:29:25,506 --> 00:29:27,130 Compared to the original wave function, 513 00:29:27,130 --> 00:29:28,870 is a function of x a set of coefficients, 514 00:29:28,870 --> 00:29:31,607 C1 dot dot dot contains more or less, same information? 515 00:29:31,607 --> 00:29:32,690 Or it can't be determined? 516 00:29:32,690 --> 00:29:33,190 Or depends? 517 00:29:36,170 --> 00:29:36,750 Five seconds. 518 00:29:41,880 --> 00:29:42,880 And the answer is. 519 00:29:42,880 --> 00:29:43,704 AUDIENCE: C. 520 00:29:43,704 --> 00:29:44,870 PROFESSOR: Same information. 521 00:29:44,870 --> 00:29:45,495 Good questions. 522 00:29:49,425 --> 00:29:50,800 Some of these you've seen before. 523 00:29:53,638 --> 00:29:58,010 f and g are wave functions and c is a constant. 524 00:29:58,010 --> 00:30:01,530 Then the inner product cf with g is equal to? 525 00:30:01,530 --> 00:30:03,595 Click away, 10 seconds. 526 00:30:11,150 --> 00:30:11,680 All right. 527 00:30:17,560 --> 00:30:19,870 OK, you've got about five seconds. 528 00:30:19,870 --> 00:30:22,840 Still a lot of fluctuations here. 529 00:30:22,840 --> 00:30:25,770 Wow, I'm surprised by this one. 530 00:30:25,770 --> 00:30:26,404 Yeah, question. 531 00:30:26,404 --> 00:30:28,570 AUDIENCE: Sorry, quick question on the previous one. 532 00:30:28,570 --> 00:30:29,430 PROFESSOR: Oh, let's come back to the previous one 533 00:30:29,430 --> 00:30:30,500 after we do this one. 534 00:30:30,500 --> 00:30:31,700 Thanks. 535 00:30:31,700 --> 00:30:33,850 But remind me too after we move on. 536 00:30:33,850 --> 00:30:37,650 So discuss, because about 1/3 of you got this one wrong. 537 00:30:37,650 --> 00:30:38,175 So discuss. 538 00:31:13,280 --> 00:31:16,871 So go ahead and put in your answers again. 539 00:31:16,871 --> 00:31:17,370 Good. 540 00:31:24,137 --> 00:31:25,470 All right, another five seconds. 541 00:31:30,610 --> 00:31:33,032 OK, and the answer is. 542 00:31:33,032 --> 00:31:34,740 Good, get that complex conjugation right. 543 00:31:34,740 --> 00:31:35,740 Little things like that can cause 544 00:31:35,740 --> 00:31:36,860 you an infinite amount of trouble. 545 00:31:36,860 --> 00:31:39,276 Especially as you'll notice on the problem set when you're 546 00:31:39,276 --> 00:31:42,037 computing time dependence of expectation values, 547 00:31:42,037 --> 00:31:43,620 getting that complex conjugation right 548 00:31:43,620 --> 00:31:46,250 is essential, because you need to see interference terms. 549 00:31:46,250 --> 00:31:47,380 And without the complex conjugation 550 00:31:47,380 --> 00:31:49,296 you will not get the right interference terms. 551 00:31:51,064 --> 00:31:52,980 Oh yeah, a question from the previous problem. 552 00:31:52,980 --> 00:31:54,316 Thank you. 553 00:31:54,316 --> 00:31:59,146 AUDIENCE: So since the summation with the constants, 554 00:31:59,146 --> 00:32:01,319 doesn't that technically give you 555 00:32:01,319 --> 00:32:05,440 the probability of each individual eigenstate? 556 00:32:05,440 --> 00:32:07,952 PROFESSOR: The norm squared gives you the probabilities. 557 00:32:07,952 --> 00:32:09,410 But if you know the coefficients--- 558 00:32:09,410 --> 00:32:10,380 so this is a good question. 559 00:32:10,380 --> 00:32:11,713 So let me rephrase the question. 560 00:32:11,713 --> 00:32:16,230 The question is, in the previous question we said, 561 00:32:16,230 --> 00:32:19,280 our wave function phi of x is some function which 562 00:32:19,280 --> 00:32:20,257 we could draw. 563 00:32:20,257 --> 00:32:21,840 So I could represent this in two ways. 564 00:32:21,840 --> 00:32:24,160 I could either draw it, or I could represent 565 00:32:24,160 --> 00:32:27,635 it as equal the sum over n of cn phi 566 00:32:27,635 --> 00:32:30,620 n where phi n are the known energy functions. 567 00:32:30,620 --> 00:32:34,109 And the question is, did the cn contain the same information? 568 00:32:34,109 --> 00:32:35,150 So that was the question. 569 00:32:35,150 --> 00:32:37,030 And the point that's being asked about here 570 00:32:37,030 --> 00:32:38,210 is a very good question. 571 00:32:38,210 --> 00:32:41,720 The question is, look, we know what these cn's mean. 572 00:32:41,720 --> 00:32:43,940 What the cn's mean is the probability 573 00:32:43,940 --> 00:32:45,575 that you measure the energy-- given 574 00:32:45,575 --> 00:32:47,950 that the state of psi, the probability [INAUDIBLE] energy 575 00:32:47,950 --> 00:32:51,920 En is equal to norm cn squared. 576 00:32:51,920 --> 00:32:54,147 And so the cn's tell you about probabilities. 577 00:32:54,147 --> 00:32:55,730 They don't tell you what the state is. 578 00:32:55,730 --> 00:32:56,860 They just tell you what a probability 579 00:32:56,860 --> 00:32:58,150 is that you'll measure a particular energy. 580 00:32:58,150 --> 00:32:59,180 Is that right? 581 00:32:59,180 --> 00:33:02,442 So how can it contain the same information? 582 00:33:02,442 --> 00:33:04,650 And there are two important things to say about this. 583 00:33:04,650 --> 00:33:07,190 First off, this is of course true. 584 00:33:07,190 --> 00:33:08,871 And this does give you the probability. 585 00:33:08,871 --> 00:33:10,870 And the probability comes from the norm squared. 586 00:33:10,870 --> 00:33:14,430 So the phase of cn doesn't particularly matter for that. 587 00:33:14,430 --> 00:33:17,800 However the claim that I can expand this function 588 00:33:17,800 --> 00:33:21,140 in a basis of the energy eigenfunctions, 589 00:33:21,140 --> 00:33:23,740 in a basis of other functions, is a statement 590 00:33:23,740 --> 00:33:27,130 that if know these cn's, then I take c1 multiplied by phi 1, 591 00:33:27,130 --> 00:33:29,810 c2 multiplied by phi 2, add them all back up, 592 00:33:29,810 --> 00:33:33,490 and I get nothing other than this original function. 593 00:33:33,490 --> 00:33:35,690 So if I know the cn's, I can just 594 00:33:35,690 --> 00:33:39,780 construct that original function explicitly. 595 00:33:39,780 --> 00:33:43,130 So this can't contain any more or any less 596 00:33:43,130 --> 00:33:45,230 information than the list of cn's, as long 597 00:33:45,230 --> 00:33:47,690 as I know what basis I'm talking about. 598 00:33:47,690 --> 00:33:49,750 So here's the disturbing thing about that. 599 00:33:49,750 --> 00:33:52,960 When we say the wave function is the state of the system-- 600 00:33:52,960 --> 00:33:56,290 it contains all knowledge you could possibly have access to. 601 00:33:56,290 --> 00:33:58,600 And not just knowledge that you could have access to, 602 00:33:58,600 --> 00:34:00,877 it contains all of the information of the state. 603 00:34:00,877 --> 00:34:02,710 And then we look at this and say, well look, 604 00:34:02,710 --> 00:34:03,970 it doesn't give us something deterministic. 605 00:34:03,970 --> 00:34:05,270 It gives us something probabilistic. 606 00:34:05,270 --> 00:34:06,780 It tells us the probability distribution 607 00:34:06,780 --> 00:34:08,440 for measuring the various different energies. 608 00:34:08,440 --> 00:34:09,719 What does that immediately tell you 609 00:34:09,719 --> 00:34:11,260 since they're equivalent information? 610 00:34:11,260 --> 00:34:13,559 This guy tells you nothing more than the probabilities. 611 00:34:13,559 --> 00:34:15,600 It doesn't tell you what energies you'll measure, 612 00:34:15,600 --> 00:34:17,550 it tells you the probabilities with which 613 00:34:17,550 --> 00:34:20,670 you will measure energies. 614 00:34:20,670 --> 00:34:22,360 So they're the same thing. 615 00:34:22,360 --> 00:34:23,730 It's a very good question. 616 00:34:23,730 --> 00:34:26,034 And I think the way to take the sort of discomfort-- 617 00:34:26,034 --> 00:34:27,409 I don't know about you-- but that 618 00:34:27,409 --> 00:34:29,190 causes me is just to read that as saying, look, 619 00:34:29,190 --> 00:34:31,579 the wave function is giving you probabilistic information 620 00:34:31,579 --> 00:34:33,118 about measurements. 621 00:34:33,118 --> 00:34:34,114 AUDIENCE: I get that. 622 00:34:34,114 --> 00:34:38,596 It's just that if you have a wave function written 623 00:34:38,596 --> 00:34:42,082 as that or a wave function written as the [INAUDIBLE], 624 00:34:42,082 --> 00:34:45,440 I just see-- since it's written that way, 625 00:34:45,440 --> 00:34:47,890 then you know what all the cn's are 626 00:34:47,890 --> 00:34:50,340 and what all the possible eigenstates are. 627 00:34:50,340 --> 00:34:52,790 And just [INAUDIBLE]. 628 00:34:52,790 --> 00:34:55,240 I understand they both contain the same information, 629 00:34:55,240 --> 00:34:56,710 but just written as each. 630 00:34:56,710 --> 00:34:57,220 PROFESSOR: Excellent, excellent. 631 00:34:57,220 --> 00:34:58,070 So let me turn this around. 632 00:34:58,070 --> 00:34:59,020 Let me phrase this slightly different. 633 00:34:59,020 --> 00:35:00,895 Tell me if this is getting at the same point. 634 00:35:00,895 --> 00:35:03,043 If I gave you a list of numbers, 1, 14, 12, 635 00:35:03,043 --> 00:35:06,600 does that tell you what function I'm talking about? 636 00:35:06,600 --> 00:35:07,170 No. 637 00:35:07,170 --> 00:35:09,910 I have to tell you that they're the expansion coefficients 638 00:35:09,910 --> 00:35:12,800 in, say, sine waves of a particular wavelength. 639 00:35:12,800 --> 00:35:15,300 Or energy eigenfunctions of some particular energy operator. 640 00:35:15,300 --> 00:35:16,270 If I just give you a list of numbers, 641 00:35:16,270 --> 00:35:17,450 it doesn't tell you what function it is. 642 00:35:17,450 --> 00:35:18,600 I have to give you a list of numbers, 643 00:35:18,600 --> 00:35:21,100 and I have to tell you what basis I'm talking about. 644 00:35:21,100 --> 00:35:22,620 If you have the wave function, you 645 00:35:22,620 --> 00:35:24,745 don't need to know what basis you're talking about. 646 00:35:24,745 --> 00:35:26,620 The wave function is basis independent. 647 00:35:26,620 --> 00:35:28,780 It's just the function. 648 00:35:28,780 --> 00:35:31,260 In a few minutes, I'm going to come back and subvert 649 00:35:31,260 --> 00:35:33,809 what I just said. 650 00:35:33,809 --> 00:35:36,100 So the wave function is just what the wave function is. 651 00:35:36,100 --> 00:35:37,920 And if I give you a list of numbers, 652 00:35:37,920 --> 00:35:40,740 they're only equivalent when you also know the wave function. 653 00:35:40,740 --> 00:35:42,150 So in order for these two things to be equivalent, 654 00:35:42,150 --> 00:35:44,567 it's important that you know what bases are talking about. 655 00:35:44,567 --> 00:35:46,149 In particular, it's important that you 656 00:35:46,149 --> 00:35:47,550 know the energy eigenfunctions. 657 00:35:47,550 --> 00:35:52,350 Is that getting at the distinction you're making? 658 00:35:52,350 --> 00:35:53,350 OK, think more about it. 659 00:35:53,350 --> 00:35:57,094 And when the question becomes more sharp, ask. 660 00:35:57,094 --> 00:35:57,760 Other questions? 661 00:36:01,472 --> 00:36:04,470 AUDIENCE: I think you kind of covered what my question was. 662 00:36:04,470 --> 00:36:08,700 But for the future, if the question is phrased such 663 00:36:08,700 --> 00:36:12,130 that if you know the cn's, you know all the information, 664 00:36:12,130 --> 00:36:14,580 are we supposed to just infer that if we know the cn, 665 00:36:14,580 --> 00:36:16,060 we also know [INAUDIBLE]? 666 00:36:16,060 --> 00:36:18,330 PROFESSOR: Yeah, usually. 667 00:36:18,330 --> 00:36:22,300 I try on problem sets in exams to make that distinction clear, 668 00:36:22,300 --> 00:36:25,234 but that will generally be assumed. 669 00:36:25,234 --> 00:36:25,900 Other questions? 670 00:36:29,169 --> 00:36:30,460 I'm going to come back to this. 671 00:36:30,460 --> 00:36:31,918 I'm going to come back to this when 672 00:36:31,918 --> 00:36:35,030 we finish the clicker questions. 673 00:36:35,030 --> 00:36:36,284 Next question. 674 00:36:36,284 --> 00:36:37,700 In the simple harmonic oscillator, 675 00:36:37,700 --> 00:36:40,800 the eigenvalues-- as we've described twice-- 676 00:36:40,800 --> 00:36:45,705 our En is h bar omega times n plus 1/2 for an integer n, 677 00:36:45,705 --> 00:36:47,330 and a measurement of energy will always 678 00:36:47,330 --> 00:36:48,630 observe one of these values. 679 00:36:48,630 --> 00:36:50,890 That's from postulate four. 680 00:36:50,890 --> 00:36:52,280 Three? 681 00:36:52,280 --> 00:36:54,310 What can we say about the expectation value 682 00:36:54,310 --> 00:36:56,515 of the energy in some arbitrary state? 683 00:37:04,950 --> 00:37:07,250 This one isn't tricky, but it's not trivial. 684 00:37:07,250 --> 00:37:08,450 So think carefully about it. 685 00:37:20,610 --> 00:37:22,400 You have five seconds. 686 00:37:22,400 --> 00:37:24,820 Enter your best guess. 687 00:37:29,690 --> 00:37:33,840 And you are at complete chance. 688 00:37:33,840 --> 00:37:36,830 I think we've just put you through a hardness box. 689 00:37:36,830 --> 00:37:38,660 So discuss. 690 00:38:46,240 --> 00:38:53,873 OK go ahead and start putting in your modified response. 691 00:38:58,970 --> 00:39:01,405 You have another five seconds or so. 692 00:39:05,310 --> 00:39:07,190 OK, click. 693 00:39:07,190 --> 00:39:10,520 OK, there's still a lot of disagreement here. 694 00:39:10,520 --> 00:39:16,410 So let's see. 695 00:39:16,410 --> 00:39:18,333 So who would like to argue for C? 696 00:39:22,420 --> 00:39:23,670 Who would like to argue for D? 697 00:39:27,880 --> 00:39:29,510 Go ahead. 698 00:39:29,510 --> 00:39:33,200 AUDIENCE: OK, so I'm going to argue for D. 699 00:39:33,200 --> 00:39:39,860 And against C. So C suggests that we would measure 700 00:39:39,860 --> 00:39:42,680 the value of the [INAUDIBLE] only for eigenstate that it 701 00:39:42,680 --> 00:39:44,720 addressed. 702 00:39:44,720 --> 00:39:51,940 But even if that changed, the stationary state 703 00:39:51,940 --> 00:39:54,790 consists of one and one eigenstate. 704 00:39:57,680 --> 00:40:03,864 So if we were to find the value, it 705 00:40:03,864 --> 00:40:08,950 can still be a combination of 1 and [INAUDIBLE]. 706 00:40:08,950 --> 00:40:15,297 And it doesn't have to fall to a certain eigenvalue. 707 00:40:15,297 --> 00:40:16,630 PROFESSOR: Excellent, excellent. 708 00:40:16,630 --> 00:40:19,160 So let me restate that. 709 00:40:19,160 --> 00:40:23,630 So the statement-- if I'm following your logic. 710 00:40:23,630 --> 00:40:24,290 Thank you. 711 00:40:24,290 --> 00:40:25,820 So at the statement if I'm following your logic 712 00:40:25,820 --> 00:40:26,861 goes something like this. 713 00:40:26,861 --> 00:40:28,620 Look, if we had the wave function which 714 00:40:28,620 --> 00:40:31,840 was phi sub n times some phase, some ridiculous phase. 715 00:40:31,840 --> 00:40:32,930 Let's say the phase was 1. 716 00:40:32,930 --> 00:40:34,420 If we had the wave function as phi sub n, 717 00:40:34,420 --> 00:40:36,740 what will the expectation value of the energy operator 718 00:40:36,740 --> 00:40:37,900 be in this state? 719 00:40:40,770 --> 00:40:41,450 En, right? 720 00:40:41,450 --> 00:40:43,160 Because it's the energy. 721 00:40:43,160 --> 00:40:46,770 It's a sum of all possible n of En times the probability 722 00:40:46,770 --> 00:40:51,820 that we measure En, by definition, 723 00:40:51,820 --> 00:40:55,860 but this is equal to sum over n of En times 724 00:40:55,860 --> 00:40:58,540 the expansion coefficient of our wave function in the energy 725 00:40:58,540 --> 00:41:01,270 eigenstate basis cn squared. 726 00:41:01,270 --> 00:41:04,011 Now, what's the expansion coefficient 727 00:41:04,011 --> 00:41:05,260 in our energy eigenbasis here? 728 00:41:05,260 --> 00:41:09,040 It's 1 for a particular value of n, and 0 for everything else. 729 00:41:09,040 --> 00:41:10,330 So what will our answer be? 730 00:41:10,330 --> 00:41:12,184 It will be 0 for everything except 1 731 00:41:12,184 --> 00:41:14,350 for a particular value En corresponding to little m. 732 00:41:14,350 --> 00:41:18,080 I should really call this m, so that dummy variables that we're 733 00:41:18,080 --> 00:41:19,580 summing over don't matter. 734 00:41:19,580 --> 00:41:26,250 So for this particular case the expectation value of e 735 00:41:26,250 --> 00:41:27,130 is equal to En. 736 00:41:27,130 --> 00:41:28,130 Everyone cool with that? 737 00:41:28,130 --> 00:41:30,150 So it certainly can be En. 738 00:41:30,150 --> 00:41:33,160 But is this the only way to get En? 739 00:41:33,160 --> 00:41:33,660 No. 740 00:41:33,660 --> 00:41:42,760 So for example, suppose we have alpha phi 1 plus beta phi 3. 741 00:41:42,760 --> 00:41:44,427 What will the answer be? 742 00:41:44,427 --> 00:41:46,510 Now first off, in order to make this normalizable, 743 00:41:46,510 --> 00:41:47,593 what does beta have to be? 744 00:41:47,593 --> 00:41:52,960 It needs to be square of 1 minus alpha norm squared. 745 00:41:52,960 --> 00:41:56,800 Just so that they sum to 1 when norm squared. 746 00:41:56,800 --> 00:41:59,370 So what I want to plot now the expectation 747 00:41:59,370 --> 00:42:04,420 value of the energy as a function of alpha. 748 00:42:04,420 --> 00:42:07,140 When alpha is 0, what is the expectation value 749 00:42:07,140 --> 00:42:10,440 of the energy? 750 00:42:10,440 --> 00:42:12,060 E3. 751 00:42:12,060 --> 00:42:15,700 When alpha is 1-- which it can't be anything greater than 1-- 752 00:42:15,700 --> 00:42:19,150 when alpha is 1, what is the expectation value 753 00:42:19,150 --> 00:42:20,900 of the energy? 754 00:42:20,900 --> 00:42:21,910 E1. 755 00:42:21,910 --> 00:42:25,050 And how does the expectation value of the energy 756 00:42:25,050 --> 00:42:28,380 vary between these two as a function of alpha? 757 00:42:28,380 --> 00:42:30,830 Well, it's smooth. 758 00:42:30,830 --> 00:42:33,700 In particular, it's quadratic because it 759 00:42:33,700 --> 00:42:36,080 goes like the square of these coefficients. 760 00:42:36,080 --> 00:42:38,090 So it's something smooth that does this and has 761 00:42:38,090 --> 00:42:40,680 slope 0 at the end. 762 00:42:40,680 --> 00:42:44,750 And somewhere in between, it will hit E2. 763 00:42:44,750 --> 00:42:47,070 But not only is this not equal to E2, 764 00:42:47,070 --> 00:42:52,050 it doesn't even contain E2 as an element of the superposition. 765 00:42:52,050 --> 00:42:58,380 So this is a counter-example to C. This is an important point. 766 00:42:58,380 --> 00:43:01,640 Now if I know that at some moment in time, 767 00:43:01,640 --> 00:43:04,450 the energy expectation value is this value, 768 00:43:04,450 --> 00:43:06,610 corresponding let's say to some particular alpha, 769 00:43:06,610 --> 00:43:09,120 and I let the system evolve in time, 770 00:43:09,120 --> 00:43:11,960 how does that expectation value evolve in time? 771 00:43:17,390 --> 00:43:23,530 How does the expectation value of the energy change over time? 772 00:43:23,530 --> 00:43:25,000 It doesn't. 773 00:43:25,000 --> 00:43:28,177 The expectation value of energy is always time independent. 774 00:43:30,740 --> 00:43:34,880 Does the expectation value of position change over time? 775 00:43:34,880 --> 00:43:36,250 In general, yes. 776 00:43:36,250 --> 00:43:39,300 When does it not change over time? 777 00:43:39,300 --> 00:43:41,040 In a stationary state. 778 00:43:41,040 --> 00:43:43,129 So working through this is what's 779 00:43:43,129 --> 00:43:44,670 on your previous problem set, as well 780 00:43:44,670 --> 00:43:47,292 as on your current problem set. 781 00:43:47,292 --> 00:43:49,250 AUDIENCE: So is this anything like conservation 782 00:43:49,250 --> 00:43:52,340 of energy, where the expectation value of energy 783 00:43:52,340 --> 00:43:54,540 needs to stay the same if there's nothing going on? 784 00:43:54,540 --> 00:43:56,820 PROFESSOR: Excellent, it sort of sounds like that. 785 00:43:56,820 --> 00:43:58,529 We'll make a sharp version of that later. 786 00:43:58,529 --> 00:44:00,903 So you should have that tinkling in the back of your head 787 00:44:00,903 --> 00:44:03,400 as like, oh look, we have a time independent potential, 788 00:44:03,400 --> 00:44:04,780 energy is conserved. 789 00:44:04,780 --> 00:44:07,617 That sounds kind of like the expectation value is constant. 790 00:44:07,617 --> 00:44:09,200 But we need to make that more precise. 791 00:44:09,200 --> 00:44:10,950 So we'll find a precise version of that statement later 792 00:44:10,950 --> 00:44:11,950 in the semester. 793 00:44:11,950 --> 00:44:14,450 But it's a very good intuition to have. 794 00:44:14,450 --> 00:44:16,730 Other questions? 795 00:44:16,730 --> 00:44:22,059 OK, so the answer is D. And you guys were about 4/5 796 00:44:22,059 --> 00:44:22,850 on that at the end. 797 00:44:25,453 --> 00:44:27,280 Oops, that was not right. 798 00:44:30,860 --> 00:44:31,940 OK, next one. 799 00:44:31,940 --> 00:44:34,790 This is a three-step question. 800 00:44:34,790 --> 00:44:37,750 We have observable A and B with eigenstates psi 1 and psi 2, 801 00:44:37,750 --> 00:44:40,700 and phi 1 and phi 2 of A and B, and eigenvalues 802 00:44:40,700 --> 00:44:42,802 of a1, a2, and b1, b2. 803 00:44:42,802 --> 00:44:46,590 And the eigenstate's related in this linear fashion. 804 00:44:46,590 --> 00:44:51,390 I measure observable A and get the value of a1. 805 00:44:51,390 --> 00:44:54,257 What's the state immediately after that measurement? 806 00:44:54,257 --> 00:44:55,340 What is the wave function? 807 00:45:06,250 --> 00:45:06,835 Five seconds. 808 00:45:11,980 --> 00:45:13,270 2-1. 809 00:45:13,270 --> 00:45:14,620 Click away. 810 00:45:14,620 --> 00:45:18,030 So there's pretty strong agreement among you all 811 00:45:18,030 --> 00:45:22,156 that the answer is A, and that is indeed the answer. 812 00:45:22,156 --> 00:45:22,655 Questions? 813 00:45:27,550 --> 00:45:31,614 So a slightly less obvious one. 814 00:45:31,614 --> 00:45:33,280 Immediately after this measurement of A, 815 00:45:33,280 --> 00:45:34,920 observable B is measured. 816 00:45:34,920 --> 00:45:36,770 What's the probability the b1 is found? 817 00:45:54,980 --> 00:45:57,230 Wow, there's some amazing initial transience there. 818 00:46:01,780 --> 00:46:03,612 I could use the first couple of seconds 819 00:46:03,612 --> 00:46:04,820 as a random number generator. 820 00:46:08,080 --> 00:46:10,000 OK, that's it. 821 00:46:10,000 --> 00:46:12,110 And there's actually a third of you 822 00:46:12,110 --> 00:46:14,420 disagree with the correct answer. 823 00:46:14,420 --> 00:46:16,920 So I'm going to invite you to chat with each other. 824 00:46:45,515 --> 00:46:46,890 Go ahead and update your answers. 825 00:46:54,620 --> 00:46:55,786 OK, five more seconds. 826 00:46:59,720 --> 00:47:00,820 One second. 827 00:47:00,820 --> 00:47:01,560 OK, nice. 828 00:47:01,560 --> 00:47:02,060 Excellent. 829 00:47:02,060 --> 00:47:06,630 You guys went from about 65% to 90%, so that was great. 830 00:47:06,630 --> 00:47:09,530 The answer is E. It's the norm squared 831 00:47:09,530 --> 00:47:14,250 of the coefficient of eigenfunction corresponding 832 00:47:14,250 --> 00:47:17,096 to b1 in the original state, which was psi 1. 833 00:47:19,960 --> 00:47:23,580 Now imagine, on the other hand, that the grad student doing 834 00:47:23,580 --> 00:47:25,470 the measurement failed to in fact measure 835 00:47:25,470 --> 00:47:29,230 the observable B-- it was a bad day-- but instead measured A 836 00:47:29,230 --> 00:47:29,790 again. 837 00:47:29,790 --> 00:47:31,590 What's the probability that the second measurement 838 00:47:31,590 --> 00:47:32,395 will yield a1? 839 00:47:36,710 --> 00:47:40,120 Trust me on this one, there's so many ways. 840 00:47:40,120 --> 00:47:44,508 They read the package wrong, they did the wrong measurement. 841 00:47:44,508 --> 00:47:46,597 AUDIENCE: [INAUDIBLE]. 842 00:47:46,597 --> 00:47:48,680 PROFESSOR: Sorry, thank you, thank you, thank you. 843 00:47:48,680 --> 00:47:49,530 Very good question. 844 00:47:49,530 --> 00:47:53,670 He accidentally measured B instead of A. 845 00:47:53,670 --> 00:47:54,900 AUDIENCE: [INAUDIBLE]. 846 00:47:54,900 --> 00:47:58,400 PROFESSOR: I mean A instead of B. A instead of B. See 847 00:47:58,400 --> 00:48:00,200 how easy it is to make that mistake? 848 00:48:00,200 --> 00:48:04,700 He accidentally measured A instead of B. 849 00:48:04,700 --> 00:48:07,810 I wish that was intentional. 850 00:48:07,810 --> 00:48:10,950 OK, and so and we got almost total unanimity. 851 00:48:10,950 --> 00:48:14,584 The answer is B. Yes, question. 852 00:48:14,584 --> 00:48:17,857 AUDIENCE: I was just going to ask how quickly after. 853 00:48:17,857 --> 00:48:18,940 PROFESSOR: Great question. 854 00:48:18,940 --> 00:48:20,481 Let's just assume it's instantaneous. 855 00:48:22,470 --> 00:48:24,020 Maybe sloppy, but he's very quick. 856 00:48:29,200 --> 00:48:31,090 OK, next question. 857 00:48:31,090 --> 00:48:34,170 A system is in a state which is a linear superposition of n 858 00:48:34,170 --> 00:48:37,962 equals 1 and n equals 2 energy eigenstates, blah. 859 00:48:37,962 --> 00:48:40,170 What's the probability that measurement of the energy 860 00:48:40,170 --> 00:48:42,516 will yield the first energy eigenvalue? 861 00:48:47,480 --> 00:48:49,400 Another five seconds, 2-3-4-5. 862 00:48:52,160 --> 00:48:57,290 Great, you guys totally agree that the answer is C. Great. 863 00:48:57,290 --> 00:49:00,094 That's it for the clicker questions for today. 864 00:49:00,094 --> 00:49:01,760 So here's the thing I want to emphasize. 865 00:49:01,760 --> 00:49:04,500 The first is that these were totally conceptual. 866 00:49:04,500 --> 00:49:06,820 They did not involve any computations. 867 00:49:06,820 --> 00:49:09,550 And yet, they're not trivial. 868 00:49:09,550 --> 00:49:14,200 Some of them got 50% consistently across the class. 869 00:49:14,200 --> 00:49:18,570 So it's not just the calculations that are hard. 870 00:49:18,570 --> 00:49:21,710 Just thinking through the basic premises, the basic postulates, 871 00:49:21,710 --> 00:49:23,252 is really essential at this point. 872 00:49:23,252 --> 00:49:24,960 It's more important than the computation. 873 00:49:24,960 --> 00:49:25,812 Yeah. 874 00:49:25,812 --> 00:49:28,678 AUDIENCE: Are you going to post this? 875 00:49:28,678 --> 00:49:29,886 PROFESSOR: I will post these. 876 00:49:29,886 --> 00:49:31,100 I will post these. 877 00:49:31,100 --> 00:49:33,910 Any questions remaining after this-- oh, shoot! 878 00:49:37,710 --> 00:49:41,260 I missed out on a whole opportunity. 879 00:49:41,260 --> 00:49:42,710 I'll do it next time. 880 00:49:42,710 --> 00:49:43,440 Poo. 881 00:49:43,440 --> 00:49:45,874 Any other questions before we move on to lecture part? 882 00:49:45,874 --> 00:49:46,790 AUDIENCE: [INAUDIBLE]. 883 00:49:46,790 --> 00:49:49,400 PROFESSOR: Oh, you totally have to wait. 884 00:49:49,400 --> 00:49:50,800 Yeah, you totally have to wait. 885 00:49:50,800 --> 00:49:52,010 Any other questions? 886 00:49:52,010 --> 00:49:53,974 Yeah. 887 00:49:53,974 --> 00:49:56,015 AUDIENCE: Do you ever have a [INAUDIBLE] function 888 00:49:56,015 --> 00:49:56,720 that's odd? 889 00:49:56,720 --> 00:49:59,776 Or just continues down [INAUDIBLE]. 890 00:49:59,776 --> 00:50:01,650 PROFESSOR: Well, that's a very good question. 891 00:50:01,650 --> 00:50:03,191 So the question is, can you ever have 892 00:50:03,191 --> 00:50:05,230 a potential function that is odd? 893 00:50:05,230 --> 00:50:08,006 So for example-- you're actually asking two questions, 894 00:50:08,006 --> 00:50:09,130 so let me disentangle them. 895 00:50:09,130 --> 00:50:14,200 So one question is, can I have an a potential function 896 00:50:14,200 --> 00:50:17,760 which is odd and linear? 897 00:50:17,760 --> 00:50:19,870 So it just ramps all the way down. 898 00:50:19,870 --> 00:50:23,230 So does that seem physical? 899 00:50:23,230 --> 00:50:25,347 Is the energy bounded from below? 900 00:50:25,347 --> 00:50:27,180 Well, that's a not a terribly good criterion 901 00:50:27,180 --> 00:50:30,580 because it's not bounded from below for the Coulomb 902 00:50:30,580 --> 00:50:32,532 potential either. 903 00:50:32,532 --> 00:50:34,240 But in the case of the Coulomb potential, 904 00:50:34,240 --> 00:50:37,510 it's not bounded from below in a little tiny region. 905 00:50:37,510 --> 00:50:39,932 Not in a huge swath. 906 00:50:39,932 --> 00:50:42,140 So we're going to need a better definition than this, 907 00:50:42,140 --> 00:50:44,060 but the short answer to this one is no. 908 00:50:44,060 --> 00:50:46,880 This is not good. 909 00:50:46,880 --> 00:50:49,330 On the other hand, it's not the oddness that's bad. 910 00:50:49,330 --> 00:50:51,230 It's that the energy is continuing 911 00:50:51,230 --> 00:50:53,640 to be unbounded from below. 912 00:50:53,640 --> 00:50:57,590 So let me be more precise than saying it's not happy. 913 00:50:57,590 --> 00:51:00,220 What important for an observable? 914 00:51:00,220 --> 00:51:02,380 What's important for the operator 915 00:51:02,380 --> 00:51:05,840 corresponding to an observable in quantum mechanics? 916 00:51:05,840 --> 00:51:08,870 What property must that operator have? 917 00:51:08,870 --> 00:51:10,844 It has to be Hermitian. 918 00:51:10,844 --> 00:51:12,760 And [INAUDIBLE] that implies real eigenvalues. 919 00:51:12,760 --> 00:51:15,176 It implies that there's a basis of its eigenfunctions, all 920 00:51:15,176 --> 00:51:17,100 these nice things. 921 00:51:17,100 --> 00:51:20,860 What we will find is that the Coulomb potential-- 922 00:51:20,860 --> 00:51:24,300 the hydrogen potential-- gives us a nice Hermitian energy 923 00:51:24,300 --> 00:51:24,981 operator. 924 00:51:24,981 --> 00:51:26,480 We'll prove that the energy operator 925 00:51:26,480 --> 00:51:27,940 self-adjoint on a problem set. 926 00:51:27,940 --> 00:51:30,770 However, this guy will not. 927 00:51:30,770 --> 00:51:33,230 So we're going to have one that is not bounded from below. 928 00:51:33,230 --> 00:51:35,730 And so while there's a sense in which it's Hermitian, 929 00:51:35,730 --> 00:51:37,300 there's not a good sense. 930 00:51:37,300 --> 00:51:38,600 It's got problems. 931 00:51:38,600 --> 00:51:40,340 But that's a more technical detail. 932 00:51:40,340 --> 00:51:42,590 However, there's a second part of your question, which 933 00:51:42,590 --> 00:51:44,090 is you could have been asking, look, 934 00:51:44,090 --> 00:51:45,810 can you have a potential that's odd 935 00:51:45,810 --> 00:51:48,280 but let's leave aside the diverging at infinity. 936 00:51:48,280 --> 00:51:50,030 So for example, could you have a potential 937 00:51:50,030 --> 00:51:55,920 that does this so that it's odd, but it's bounded? 938 00:51:55,920 --> 00:51:56,960 And that's fine. 939 00:51:56,960 --> 00:51:58,210 Nothing wrong with that. 940 00:51:58,210 --> 00:51:59,420 So odd isn't a problem. 941 00:51:59,420 --> 00:52:01,326 The problem is certain ways of diverging. 942 00:52:05,051 --> 00:52:06,800 AUDIENCE: So what would happen in the case 943 00:52:06,800 --> 00:52:08,841 where you did have some kind of linear potential? 944 00:52:08,841 --> 00:52:12,724 You had a uniform electric field and something was moving. 945 00:52:12,724 --> 00:52:14,640 Is the problem that that potential technically 946 00:52:14,640 --> 00:52:17,520 isn't a physical potential because it doesn't go away? 947 00:52:17,520 --> 00:52:20,220 PROFESSOR: How would you ever build a linear potential, 948 00:52:20,220 --> 00:52:20,720 right? 949 00:52:20,720 --> 00:52:22,397 So the question here is, look, I know 950 00:52:22,397 --> 00:52:23,730 how to build a linear potential. 951 00:52:23,730 --> 00:52:26,110 Turn a uniform electric field, and an electric field 952 00:52:26,110 --> 00:52:28,100 is the gradient to the electrostatic potential. 953 00:52:28,100 --> 00:52:29,400 So that means that the electrostatic potential 954 00:52:29,400 --> 00:52:30,334 is linear. 955 00:52:30,334 --> 00:52:32,250 And the potential energy of a charged particle 956 00:52:32,250 --> 00:52:34,360 with charge Q in an electrostatic potential 957 00:52:34,360 --> 00:52:36,409 is q times the electrostatic potential. 958 00:52:36,409 --> 00:52:38,450 So the potential energy for that charged particle 959 00:52:38,450 --> 00:52:39,609 is a linear function. 960 00:52:39,609 --> 00:52:40,900 So how do I build such a thing? 961 00:52:40,900 --> 00:52:42,310 Well, I take two capacitor plates, 962 00:52:42,310 --> 00:52:43,370 and I dump some charge on this one, 963 00:52:43,370 --> 00:52:44,869 and the opposite charge on this guy, 964 00:52:44,869 --> 00:52:46,620 and I build up a linear potential. 965 00:52:46,620 --> 00:52:47,484 Well yeah, exactly. 966 00:52:47,484 --> 00:52:48,400 So that's the problem. 967 00:52:48,400 --> 00:52:51,450 So can I make this linear over an arbitrarily large domain? 968 00:52:51,450 --> 00:52:51,950 No. 969 00:52:51,950 --> 00:52:53,900 I need an arbitrarily large amount of charge, 970 00:52:53,900 --> 00:52:56,274 and I need to push them apart from each other arbitrarily 971 00:52:56,274 --> 00:52:59,590 far, which takes an arbitrarily large amount of work. 972 00:52:59,590 --> 00:53:01,520 That's not terribly physical, right? 973 00:53:01,520 --> 00:53:04,160 Your arms are only so big. 974 00:53:04,160 --> 00:53:06,406 So at the end and the day, we're always 975 00:53:06,406 --> 00:53:08,530 going to discover that the pathologies that show up 976 00:53:08,530 --> 00:53:10,696 in a potential like this, like the Hamiltonian's not 977 00:53:10,696 --> 00:53:12,970 self-adjoint and compactly supported 978 00:53:12,970 --> 00:53:16,240 [INAUDIBLE] continuous functions, then OK. 979 00:53:16,240 --> 00:53:19,000 That's going to always be some mathematical version of, 980 00:53:19,000 --> 00:53:20,190 your arms are only so big. 981 00:53:20,190 --> 00:53:22,990 You cannot build an infinitely large apparatus like this. 982 00:53:22,990 --> 00:53:24,550 So yes, you're exactly right. 983 00:53:24,550 --> 00:53:26,320 It is not physical. 984 00:53:26,320 --> 00:53:28,973 Is it a mathematical problem one could analyze? 985 00:53:28,973 --> 00:53:29,940 Yes. 986 00:53:29,940 --> 00:53:33,270 One could write a dissertation on the singularity structure 987 00:53:33,270 --> 00:53:34,960 of these differential equations. 988 00:53:34,960 --> 00:53:37,750 But that dissertation would never be read by a physicist. 989 00:53:41,230 --> 00:53:42,392 Other questions? 990 00:53:42,392 --> 00:53:43,356 Yeah. 991 00:53:43,356 --> 00:53:45,647 AUDIENCE: Talking about the rising and lowering number. 992 00:53:45,647 --> 00:53:48,964 Can you find the raising and lowering for any [INAUDIBLE] 993 00:53:48,964 --> 00:53:50,040 potential. 994 00:53:50,040 --> 00:53:51,290 PROFESSOR: Excellent question. 995 00:53:51,290 --> 00:53:52,250 No. 996 00:53:52,250 --> 00:53:53,725 So here's what you can do. 997 00:53:53,725 --> 00:53:55,600 Suppose I have a bunch of energy eigenstates. 998 00:53:55,600 --> 00:53:56,740 I have some potential. 999 00:53:56,740 --> 00:53:57,910 This is the potential. 1000 00:53:57,910 --> 00:53:59,387 It's a crazy potential. 1001 00:53:59,387 --> 00:54:01,220 And it has some definite energy eigenvalues. 1002 00:54:01,220 --> 00:54:04,480 As we've shown in 1D, or as I've mentioned and proved 1003 00:54:04,480 --> 00:54:09,100 in office hours, we can't find degenerate eigenfunctions 1004 00:54:09,100 --> 00:54:11,420 in a one-dimensional potential. 1005 00:54:11,420 --> 00:54:12,810 So they're not degenerate. 1006 00:54:12,810 --> 00:54:14,460 They're spaced a finite distance apart 1007 00:54:14,460 --> 00:54:18,140 for wave functions that are convergent to 0 1008 00:54:18,140 --> 00:54:21,470 and infinity for bound states. 1009 00:54:21,470 --> 00:54:23,420 So we can always construct an operator, 1010 00:54:23,420 --> 00:54:25,820 which I will simply define in this way. 1011 00:54:25,820 --> 00:54:28,679 A, and just for flourish I will put a dagger on it 1012 00:54:28,679 --> 00:54:29,720 which does the following. 1013 00:54:29,720 --> 00:54:32,240 A dagger is the defined as the operator 1014 00:54:32,240 --> 00:54:38,320 that maps phi n to phi m, n plus 1. 1015 00:54:38,320 --> 00:54:39,250 So what's the rule? 1016 00:54:39,250 --> 00:54:40,480 It takes this state to this state, 1017 00:54:40,480 --> 00:54:42,605 this state to this state, this state to this state. 1018 00:54:42,605 --> 00:54:45,574 And since we can extend-- since these are a basis, 1019 00:54:45,574 --> 00:54:47,990 then that tells us how this acts on an arbitrary function, 1020 00:54:47,990 --> 00:54:50,000 because it acts on the superposition as acting 1021 00:54:50,000 --> 00:54:51,610 on each term. 1022 00:54:51,610 --> 00:54:54,310 So we can define it in this way, but here's the question. 1023 00:54:54,310 --> 00:54:56,900 What does it mean for the raising operator 1024 00:54:56,900 --> 00:54:58,220 to be the raising operator? 1025 00:54:58,220 --> 00:55:00,530 It wasn't that it lifted to raise the operator. 1026 00:55:00,530 --> 00:55:02,920 That's not what started out the whole machinery. 1027 00:55:02,920 --> 00:55:05,220 Where the raising operator got its juice was 1028 00:55:05,220 --> 00:55:09,356 from this computational relation, a dagger with E 1029 00:55:09,356 --> 00:55:14,310 was equal to h bar omega a dagger. 1030 00:55:14,310 --> 00:55:16,910 So can we build an operator a dagger 1031 00:55:16,910 --> 00:55:20,060 that raises states and commutes with the energy 1032 00:55:20,060 --> 00:55:21,760 operator in this fashion? 1033 00:55:21,760 --> 00:55:24,810 No, that depends on the energy operator. 1034 00:55:24,810 --> 00:55:26,130 This is not a general property. 1035 00:55:26,130 --> 00:55:29,330 This is not always something we can do. 1036 00:55:29,330 --> 00:55:32,140 If we could, then that energy operator 1037 00:55:32,140 --> 00:55:35,630 would have evenly spaced energy eigenfunctions. 1038 00:55:35,630 --> 00:55:39,120 On the other hand, it's not like it's never useful. 1039 00:55:39,120 --> 00:55:40,770 So consider the following example. 1040 00:55:40,770 --> 00:55:41,510 And I think this was on your problem set 1041 00:55:41,510 --> 00:55:42,385 but I don't remember. 1042 00:55:42,385 --> 00:55:45,280 So let me just say it. 1043 00:55:45,280 --> 00:55:50,220 If we write a dagger a is equal to n, 1044 00:55:50,220 --> 00:55:52,580 then the commutation relation for this-- so the energy 1045 00:55:52,580 --> 00:55:54,340 operator for the harmonic oscillator 1046 00:55:54,340 --> 00:55:58,290 was equal to h bar omega n plus 1/2. 1047 00:55:58,290 --> 00:56:01,610 That's what we derived last time. 1048 00:56:01,610 --> 00:56:05,000 If we define a dagger a as n, then the commutator 1049 00:56:05,000 --> 00:56:08,280 of n with a dagger is a dagger. 1050 00:56:08,280 --> 00:56:11,970 And the commutator of n with a is equal to minus a. 1051 00:56:15,590 --> 00:56:26,240 Suppose the energy operator is equal to h 1052 00:56:26,240 --> 00:56:30,030 bar omega n plus n cubed. 1053 00:56:35,120 --> 00:56:37,270 Does the raising operator commute with the energy 1054 00:56:37,270 --> 00:56:39,380 operator to give you a dagger again? 1055 00:56:41,930 --> 00:56:45,310 No, because it commutes to give you an a dagger and it commutes 1056 00:56:45,310 --> 00:56:49,650 with this guy to give you, well, a constant times n 1057 00:56:49,650 --> 00:56:52,890 squared a dagger, which is a slightly funny [INAUDIBLE]. 1058 00:56:52,890 --> 00:56:55,140 So this doesn't have the same computational relations. 1059 00:56:55,140 --> 00:56:57,790 On the other hand, we know how to build the functions of the n 1060 00:56:57,790 --> 00:57:00,220 operator because they live in a tower by this algebra. 1061 00:57:00,220 --> 00:57:02,730 By this commutator relation, they live in a tower. 1062 00:57:02,730 --> 00:57:05,032 So I can build the eigenfunctions of n. 1063 00:57:05,032 --> 00:57:07,240 And then I can take those eigenfunctions of n and act 1064 00:57:07,240 --> 00:57:09,115 on them with e and discover that they're also 1065 00:57:09,115 --> 00:57:10,000 eigenfunctions of e. 1066 00:57:10,000 --> 00:57:12,140 Because e acting on them is just n, and n cubed 1067 00:57:12,140 --> 00:57:14,640 acting on the eigenfunctions of n. 1068 00:57:14,640 --> 00:57:16,977 So the states with definite n are still 1069 00:57:16,977 --> 00:57:18,560 eigenfunctions of the energy operator. 1070 00:57:18,560 --> 00:57:20,854 But they're not evenly spaced. 1071 00:57:20,854 --> 00:57:22,270 Now can every potential be written 1072 00:57:22,270 --> 00:57:23,660 in terms of this n operator. 1073 00:57:23,660 --> 00:57:25,250 So I leave that you as an exercise. 1074 00:57:25,250 --> 00:57:27,840 The answer is no. 1075 00:57:27,840 --> 00:57:28,400 Try it. 1076 00:57:28,400 --> 00:57:30,929 AUDIENCE: [INAUDIBLE]. 1077 00:57:30,929 --> 00:57:31,970 PROFESSOR: Yeah, exactly. 1078 00:57:31,970 --> 00:57:33,886 So there's always something you could formally 1079 00:57:33,886 --> 00:57:35,040 define in this fashion. 1080 00:57:35,040 --> 00:57:37,373 It's not always useful to you, because it doesn't always 1081 00:57:37,373 --> 00:57:39,362 commute with the energy operator in a nice way. 1082 00:57:39,362 --> 00:57:41,070 So then you can ask, are there properties 1083 00:57:41,070 --> 00:57:41,690 that aren't nice about this? 1084 00:57:41,690 --> 00:57:44,023 Well, there's something called the supersymmetric method 1085 00:57:44,023 --> 00:57:46,050 in 1D quantum mechanics. 1086 00:57:46,050 --> 00:57:47,610 But that's sort of beyond the scope. 1087 00:57:47,610 --> 00:57:51,910 Come to my office hours and ask me this question again. 1088 00:57:51,910 --> 00:57:56,740 This is a miracle of the harmonic oscillator. 1089 00:57:56,740 --> 00:57:58,280 Other questions? 1090 00:57:58,280 --> 00:58:00,310 AUDIENCE: So to construct the [INAUDIBLE] 1091 00:58:00,310 --> 00:58:03,390 we use a condition that a phi 0 is 0? 1092 00:58:03,390 --> 00:58:04,460 PROFESSOR: Yes. 1093 00:58:04,460 --> 00:58:05,760 AUDIENCE: So how did we come up with that? 1094 00:58:05,760 --> 00:58:07,810 Did we demand that on physical grounds that it had to be 0? 1095 00:58:07,810 --> 00:58:08,768 PROFESSOR: Yes, we did. 1096 00:58:08,768 --> 00:58:09,730 So excellent question. 1097 00:58:09,730 --> 00:58:13,140 So the question is this. 1098 00:58:13,140 --> 00:58:15,030 When we constructed using the operator method 1099 00:58:15,030 --> 00:58:17,488 for the harmonic oscillator, when we constructed the ground 1100 00:58:17,488 --> 00:58:19,867 state, we said the ground state, phi 0, 1101 00:58:19,867 --> 00:58:21,950 is that state which is annihilated by the learning 1102 00:58:21,950 --> 00:58:22,610 operator. 1103 00:58:22,610 --> 00:58:23,680 A phi 0 is equal to 0. 1104 00:58:23,680 --> 00:58:25,391 So where did that come from? 1105 00:58:25,391 --> 00:58:27,140 Were we forcing that for physical reasons? 1106 00:58:27,140 --> 00:58:28,181 For mathematical reasons? 1107 00:58:28,181 --> 00:58:29,330 What was the reason? 1108 00:58:29,330 --> 00:58:34,680 So this was very important, so let's go through it slowly. 1109 00:58:34,680 --> 00:58:37,270 So remember the step that came immediately before this step. 1110 00:58:37,270 --> 00:58:38,811 Immediately before this step we said, 1111 00:58:38,811 --> 00:58:41,540 look the expectation value of energy 1112 00:58:41,540 --> 00:58:44,010 can always be written as a strictly positive thing. 1113 00:58:44,010 --> 00:58:46,110 It's equal to the integral and you 1114 00:58:46,110 --> 00:58:47,620 should derive this for yourself. 1115 00:58:47,620 --> 00:58:49,090 I didn't derive it, but this is a good thing 1116 00:58:49,090 --> 00:58:50,048 to derive for yourself. 1117 00:58:50,048 --> 00:58:51,540 It takes only a couple of steps. 1118 00:58:51,540 --> 00:58:54,420 Integral overall momentum of psi tilde 1119 00:58:54,420 --> 00:58:56,960 of p-- the Fourier transformer norm squared. 1120 00:58:56,960 --> 00:58:59,474 This is the probability that you have given the state psi. 1121 00:58:59,474 --> 00:59:01,140 This is the probability density that you 1122 00:59:01,140 --> 00:59:05,130 have momentum p, p squared upon 2 m. 1123 00:59:05,130 --> 00:59:09,900 Plus the integral dx psi of x norm 1124 00:59:09,900 --> 00:59:14,285 squared m omega squared upon 2 x squared. 1125 00:59:14,285 --> 00:59:16,720 This is the probability that you're at position 1126 00:59:16,720 --> 00:59:18,180 x, given the wave function x. 1127 00:59:18,180 --> 00:59:20,850 And that's the value of the potential. 1128 00:59:20,850 --> 00:59:24,160 Now this is a strictly positive quantity, as is this. 1129 00:59:24,160 --> 00:59:26,060 Strictly positive, as is this. 1130 00:59:26,060 --> 00:59:26,980 This is a plus. 1131 00:59:26,980 --> 00:59:30,140 This quantity must always be greater than or equal to 0. 1132 00:59:30,140 --> 00:59:31,190 Correct? 1133 00:59:31,190 --> 00:59:33,490 However, we derive from the operator relations 1134 00:59:33,490 --> 00:59:38,820 that if we take away function phi sub e 1135 00:59:38,820 --> 00:59:41,120 and we act on it with the lowering operator a, 1136 00:59:41,120 --> 00:59:45,230 this defines a new state, which up to some normalization 1137 00:59:45,230 --> 00:59:48,720 is an eigenfunction of the energy operator with energy 1138 00:59:48,720 --> 00:59:52,400 e minus h bar omega. 1139 00:59:52,400 --> 00:59:55,950 And we show this by showing that the energy eigenfunction acting 1140 00:59:55,950 --> 01:00:01,654 on this gives exactly this coefficient times this back. 1141 01:00:01,654 --> 01:00:02,570 It's an eigenfunction. 1142 01:00:02,570 --> 01:00:04,420 We just showed explicitly that it's an eigenfunction. 1143 01:00:04,420 --> 01:00:06,711 So this suggest that if you have a state with energy e, 1144 01:00:06,711 --> 01:00:09,350 you could also build a state with energy e minus h bar 1145 01:00:09,350 --> 01:00:10,960 omega. 1146 01:00:10,960 --> 01:00:14,330 And thus we can repeat e minus 2 h bar omega, 1147 01:00:14,330 --> 01:00:16,230 and this turtles all the way down. 1148 01:00:16,230 --> 01:00:18,630 The problem is, for any finite value of e, 1149 01:00:18,630 --> 01:00:20,480 eventually this tower will get negative. 1150 01:00:20,480 --> 01:00:23,040 Here's 0. 1151 01:00:23,040 --> 01:00:26,156 But the energy expectation value cannot be less than 0. 1152 01:00:26,156 --> 01:00:27,780 It's got to be strictly greater than 0. 1153 01:00:30,730 --> 01:00:33,260 But if we were in this energy eigenstate 1154 01:00:33,260 --> 01:00:35,260 and we measured the expectation value of energy, 1155 01:00:35,260 --> 01:00:39,560 this is the value we would get, which would be negative. 1156 01:00:39,560 --> 01:00:42,630 Whoops, that's a 2. 1157 01:00:42,630 --> 01:00:46,680 So something is amiss. 1158 01:00:46,680 --> 01:00:48,460 How can this possibly make sense? 1159 01:00:48,460 --> 01:00:51,600 And the answer up with was, well look, 1160 01:00:51,600 --> 01:00:54,570 it's inescapable that acting with A n phi 1161 01:00:54,570 --> 01:00:57,140 gives us something that's an eigenfunction of the energy. 1162 01:00:57,140 --> 01:01:00,320 However, it's possible that this function 1163 01:01:00,320 --> 01:01:03,330 happens to be the 0 function. 1164 01:01:03,330 --> 01:01:07,140 Not 0 energy, just 0, not a function. 1165 01:01:07,140 --> 01:01:09,280 Is zero normalizable? 1166 01:01:09,280 --> 01:01:11,430 No, the integral of 0 squared is 0. 1167 01:01:11,430 --> 01:01:16,000 No coefficient times that is a finite number, is 1. 1168 01:01:16,000 --> 01:01:20,100 So what must happen is it must be true that at some point 1169 01:01:20,100 --> 01:01:21,520 we can't lower anymore. 1170 01:01:21,520 --> 01:01:24,940 But we don't always act with a, so we always try to lower. 1171 01:01:24,940 --> 01:01:28,470 The only out is if when we lower one particular guy, which 1172 01:01:28,470 --> 01:01:32,600 I will call phi 0, the only out is if we try to lower phi 0, 1173 01:01:32,600 --> 01:01:35,236 we don't get another state-- which we see we did here. 1174 01:01:35,236 --> 01:01:36,360 We don't get another state. 1175 01:01:36,360 --> 01:01:37,940 Instead, we just get 0. 1176 01:01:37,940 --> 01:01:40,240 We get no state. 1177 01:01:40,240 --> 01:01:41,120 Not energy 0. 1178 01:01:41,120 --> 01:01:44,060 0, the function is 0, identically vanishing. 1179 01:01:44,060 --> 01:01:46,585 This does not describe the configuration of [INAUDIBLE]. 1180 01:01:46,585 --> 01:01:49,240 The probability of it being anywhere is 0. 1181 01:01:49,240 --> 01:01:52,260 That's bad. 1182 01:01:52,260 --> 01:01:55,750 So in order for the tower to end, 1183 01:01:55,750 --> 01:01:58,180 in order for the energy to be bounded from below 1184 01:01:58,180 --> 01:02:01,360 as the potential is, we need that there's 1185 01:02:01,360 --> 01:02:03,860 a lowest state in the tower. 1186 01:02:03,860 --> 01:02:05,200 Does that answer your question? 1187 01:02:05,200 --> 01:02:05,990 AUDIENCE: Yes. 1188 01:02:05,990 --> 01:02:07,348 PROFESSOR: Great. 1189 01:02:07,348 --> 01:02:08,571 Question? 1190 01:02:08,571 --> 01:02:10,862 AUDIENCE: Are the energy eigenfunctions always strictly 1191 01:02:10,862 --> 01:02:12,230 real? 1192 01:02:12,230 --> 01:02:13,480 PROFESSOR: Very good question. 1193 01:02:13,480 --> 01:02:14,710 You're going to show this on a problem set 1194 01:02:14,710 --> 01:02:15,330 if you haven't already. 1195 01:02:15,330 --> 01:02:17,020 I thought you did, but maybe not. 1196 01:02:17,020 --> 01:02:25,650 The energy eigenfunctions for potentials 1197 01:02:25,650 --> 01:02:29,230 with only bound states can always 1198 01:02:29,230 --> 01:02:32,277 be expressed as purely real or purely imaginary. 1199 01:02:32,277 --> 01:02:33,860 You can always decompose them in terms 1200 01:02:33,860 --> 01:02:35,410 of purely real and purely imaginary. 1201 01:02:35,410 --> 01:02:38,050 Proving that will be a problem on one of your problem sets. 1202 01:02:38,050 --> 01:02:40,383 But for the moment, let me just say, yes, you can always 1203 01:02:40,383 --> 01:02:42,470 show that the energy eigenfunctions 1204 01:02:42,470 --> 01:02:47,974 for 1 D potential can always be expressed as purely real. 1205 01:02:47,974 --> 01:02:49,390 I need that bound state condition, 1206 01:02:49,390 --> 01:02:50,765 as we'll see later in the course. 1207 01:02:50,765 --> 01:02:51,733 Yes. 1208 01:02:51,733 --> 01:02:55,677 AUDIENCE: So we lowered the energy [INAUDIBLE]. 1209 01:02:55,677 --> 01:02:59,621 And the next energy is 0, right? 1210 01:02:59,621 --> 01:03:00,607 PROFESSOR: Yes. 1211 01:03:00,607 --> 01:03:03,565 AUDIENCE: Isn't that supposed to mean that the [INAUDIBLE] is 1212 01:03:03,565 --> 01:03:05,680 h omega, not h omega over 2? 1213 01:03:05,680 --> 01:03:06,690 PROFESSOR: Sorry? 1214 01:03:06,690 --> 01:03:09,430 No, good. 1215 01:03:09,430 --> 01:03:11,920 So what this says is given some state with energy e, 1216 01:03:11,920 --> 01:03:13,697 when we lower it we get h bar omega less. 1217 01:03:13,697 --> 01:03:15,030 That doesn't tell you what e is. 1218 01:03:18,200 --> 01:03:20,096 Other questions? 1219 01:03:20,096 --> 01:03:21,470 AUDIENCE: Just to expand on that. 1220 01:03:21,470 --> 01:03:24,080 So if you get the 0 function, that doesn't actually 1221 01:03:24,080 --> 01:03:27,890 correspond to any energy eigenvalue, let alone 0, right? 1222 01:03:27,890 --> 01:03:28,990 PROFESSOR: Correct. 1223 01:03:28,990 --> 01:03:30,930 The function 0 is an eigenfunction 1224 01:03:30,930 --> 01:03:33,150 of every operator. 1225 01:03:33,150 --> 01:03:35,460 But it's a stupid eigenfunction of every operator. 1226 01:03:35,460 --> 01:03:37,030 In particuar, it has nothing to do 1227 01:03:37,030 --> 01:03:38,529 with the states we're interested in. 1228 01:03:38,529 --> 01:03:41,679 We are interested in normalizable states, 1229 01:03:41,679 --> 01:03:43,220 and that is not a normalizable state. 1230 01:03:43,220 --> 01:03:45,240 You can't multiply it by any finite coefficient 1231 01:03:45,240 --> 01:03:47,580 and get 1 when you square integrate it. 1232 01:03:53,350 --> 01:03:55,134 One last question. 1233 01:03:55,134 --> 01:03:56,550 AUDIENCE: Maybe a stupid question. 1234 01:03:56,550 --> 01:03:59,451 But does that mean that the only particle that can have 0 energy 1235 01:03:59,451 --> 01:04:03,530 is the particle that doesn't exist? 1236 01:04:03,530 --> 01:04:06,704 PROFESSOR: Well, we need to add more work to that. 1237 01:04:06,704 --> 01:04:08,120 So the question is, does that mean 1238 01:04:08,120 --> 01:04:10,910 that the only particle that could really have 0 energy 1239 01:04:10,910 --> 01:04:14,560 is some imaginary particle that doesn't exist. 1240 01:04:14,560 --> 01:04:17,210 First off, imaginary particles may exist. 1241 01:04:17,210 --> 01:04:19,080 We just haven't seen them. 1242 01:04:19,080 --> 01:04:22,042 AUDIENCE: [INAUDIBLE] real particle [INAUDIBLE] 0 energy. 1243 01:04:22,042 --> 01:04:22,750 PROFESSOR: Right. 1244 01:04:22,750 --> 01:04:24,640 OK, so I need to say two things about this. 1245 01:04:24,640 --> 01:04:26,080 So there's a technical complaint I 1246 01:04:26,080 --> 01:04:27,170 need to make about your question, which 1247 01:04:27,170 --> 01:04:27,990 is a very fair question. 1248 01:04:27,990 --> 01:04:29,310 But I need to make it all the same. 1249 01:04:29,310 --> 01:04:31,030 And the second is a more physical answer. 1250 01:04:31,030 --> 01:04:35,230 So the technical complaint is, what do you mean by 0 energy? 1251 01:04:35,230 --> 01:04:37,280 Do you ever measure energy directly? 1252 01:04:37,280 --> 01:04:38,740 You measure energy differences. 1253 01:04:38,740 --> 01:04:39,791 So relative to what? 1254 01:04:39,791 --> 01:04:41,290 We have to decide, relative to what? 1255 01:04:41,290 --> 01:04:42,880 So for example, in the harmonic oscillator 1256 01:04:42,880 --> 01:04:44,963 what we mean when we say the [INAUDIBLE] energy is 1257 01:04:44,963 --> 01:04:47,060 non-zero is if we draw the potential, 1258 01:04:47,060 --> 01:04:49,250 and we call the minimum of the potential e is 0, 1259 01:04:49,250 --> 01:04:51,740 classical e is 0, then what we discover 1260 01:04:51,740 --> 01:04:54,420 is that the energy of the quantum mechanical ground state 1261 01:04:54,420 --> 01:04:58,480 is 1/2 h bar omega above 0. 1262 01:05:01,110 --> 01:05:03,270 So is there any state of the harmonic oscillator 1263 01:05:03,270 --> 01:05:05,108 with energy 0? 1264 01:05:05,108 --> 01:05:06,510 No. 1265 01:05:06,510 --> 01:05:11,220 If we have a free particle, is there 1266 01:05:11,220 --> 01:05:14,210 a configuration with energy 0? 1267 01:05:14,210 --> 01:05:16,510 Well, that's an interesting question. 1268 01:05:16,510 --> 01:05:18,300 So what are the eigenstates, e to the ikx. 1269 01:05:22,860 --> 01:05:28,370 When the energy is 0-- so the energy is-- because it's just 1270 01:05:28,370 --> 01:05:31,070 p squared upon 2m, which is h bar squared k squared upon 2m, 1271 01:05:31,070 --> 01:05:33,510 the energy is h bar squared k squared up 1272 01:05:33,510 --> 01:05:36,750 on 2m for a free particle, no potential. 1273 01:05:36,750 --> 01:05:40,960 And by 0 we mean the asymptotic energy equals 0. 1274 01:05:40,960 --> 01:05:44,250 Can you have a state with energy 0? 1275 01:05:44,250 --> 01:05:48,290 This one's a little subtle, because if we take the energy 1276 01:05:48,290 --> 01:05:50,120 0, what must k be? 1277 01:05:50,120 --> 01:05:50,990 0, OK great. 1278 01:05:50,990 --> 01:05:54,180 So what's the wave function? 1279 01:05:54,180 --> 01:05:55,700 Constant. 1280 01:05:55,700 --> 01:05:57,370 Is that normalizable? 1281 01:05:57,370 --> 01:05:58,020 Not so much. 1282 01:05:58,020 --> 01:06:08,100 So can you put a single particle in a state with energy 0? 1283 01:06:08,100 --> 01:06:09,810 No, that would not seem to be possible, 1284 01:06:09,810 --> 01:06:11,726 because that would be a non-normalzable state. 1285 01:06:11,726 --> 01:06:14,250 Well you say, we deal with these exponentials all the time. 1286 01:06:14,250 --> 01:06:15,500 We know how to deal with that. 1287 01:06:15,500 --> 01:06:17,830 We don't use a single wave-- plane 1288 01:06:17,830 --> 01:06:20,470 wave-- we use a wave packet with some width. 1289 01:06:20,470 --> 01:06:24,170 So I can build a wave packet with some average width, 1290 01:06:24,170 --> 01:06:27,320 but will that average width ever have an expectation value of k 1291 01:06:27,320 --> 01:06:29,307 equals 0? 1292 01:06:29,307 --> 01:06:31,890 No, because it will always have contributions from k not equal 1293 01:06:31,890 --> 01:06:35,040 0 so that it's specially localized. 1294 01:06:35,040 --> 01:06:37,000 So can you build a normalizable state 1295 01:06:37,000 --> 01:06:39,450 with energy 0, which is an eigenfunction of the energy 0? 1296 01:06:39,450 --> 01:06:41,020 No, not in this case either. 1297 01:06:41,020 --> 01:06:42,500 So there's the technical complaint 1298 01:06:42,500 --> 01:06:44,550 that you need to talk about what you mean relative to what. 1299 01:06:44,550 --> 01:06:45,758 That was about your question. 1300 01:06:45,758 --> 01:06:48,182 And the second of, can you ever have an energy eigenstate 1301 01:06:48,182 --> 01:06:49,890 which is at the minimum of the potential? 1302 01:06:49,890 --> 01:06:50,750 And that's also 0. 1303 01:06:50,750 --> 01:06:52,710 That's also no. 1304 01:06:52,710 --> 01:06:55,890 You'll actually prove that on a later problem set. 1305 01:06:55,890 --> 01:06:58,990 So with all that said, we have a mere-- 1306 01:06:58,990 --> 01:07:02,890 that's awesome-- 15 minutes for a 14-page lecture. 1307 01:07:02,890 --> 01:07:05,090 So can I do this fast? 1308 01:07:05,090 --> 01:07:05,972 No, I'm kidding. 1309 01:07:05,972 --> 01:07:06,930 I won't do that to you. 1310 01:07:06,930 --> 01:07:09,140 So I want to just-- instead of going through it. 1311 01:07:09,140 --> 01:07:12,660 So this was not directly relevant for the exam. 1312 01:07:12,660 --> 01:07:16,820 And this can come a little bit later. 1313 01:07:16,820 --> 01:07:21,510 So let me show you a couple of quick things. 1314 01:07:21,510 --> 01:07:23,139 So the notes are posted online. 1315 01:07:23,139 --> 01:07:23,680 Look at them. 1316 01:07:23,680 --> 01:07:27,560 I'll go over them again in the future. 1317 01:07:27,560 --> 01:07:29,480 What do I want to do? 1318 01:07:29,480 --> 01:07:31,570 So I want to introduce to you two ideas. 1319 01:07:31,570 --> 01:07:34,230 One is Dirac notation. 1320 01:07:34,230 --> 01:07:37,330 And this is hearkening back to basics of vector spaces. 1321 01:07:37,330 --> 01:07:39,400 And the second is, I want to tell you 1322 01:07:39,400 --> 01:07:42,395 something about what the commutator means. 1323 01:07:42,395 --> 01:07:44,270 So first off, let me ask you guys a question. 1324 01:07:48,900 --> 01:07:50,608 I have one more clicker question for you. 1325 01:08:11,524 --> 01:08:13,755 All right, are you all ready? 1326 01:08:17,300 --> 01:08:20,211 Everyone up with the clickers? 1327 01:08:20,211 --> 01:08:21,169 Here's is the question. 1328 01:08:28,703 --> 01:08:31,830 [MUSIC PLAYING - JEOPARDY THEME] 1329 01:08:41,994 --> 01:08:43,530 OK, go ahead and start clicking now. 1330 01:08:50,810 --> 01:08:51,810 Sigma's the uncertainty. 1331 01:08:51,810 --> 01:08:53,260 Sorry, sigma is the uncertainty. 1332 01:08:58,184 --> 01:08:58,850 This is awesome. 1333 01:08:58,850 --> 01:09:03,470 I swear the clicks are going in beat with the music. 1334 01:09:03,470 --> 01:09:05,200 So let me give you three more seconds. 1335 01:09:05,200 --> 01:09:07,760 1-2-3. 1336 01:09:07,760 --> 01:09:09,100 This is awesome. 1337 01:09:09,100 --> 01:09:11,960 You guys are totally at chance. 1338 01:09:11,960 --> 01:09:14,370 You have even probability distribution across the three. 1339 01:09:14,370 --> 01:09:16,203 So this is good, because we haven't actually 1340 01:09:16,203 --> 01:09:18,649 introduced this idea yet in 804. 1341 01:09:18,649 --> 01:09:22,100 So let me quickly talk you through this. 1342 01:09:22,100 --> 01:09:25,479 So I'll leave it off. 1343 01:09:25,479 --> 01:09:27,170 So let me quickly talk you through this. 1344 01:09:27,170 --> 01:09:28,086 And this is important. 1345 01:09:28,086 --> 01:09:31,380 This is going to give you some intuition for what uncertainty 1346 01:09:31,380 --> 01:09:31,880 means. 1347 01:09:31,880 --> 01:09:34,450 And it's also going to give you some intuition for what 1348 01:09:34,450 --> 01:09:35,627 the commutator means. 1349 01:09:35,627 --> 01:09:37,710 So this is a little more experience, a little more 1350 01:09:37,710 --> 01:09:39,580 practice with operators. 1351 01:09:39,580 --> 01:09:42,010 And we'll pick up on the Dirac notation and everything 1352 01:09:42,010 --> 01:09:42,510 next time. 1353 01:09:42,510 --> 01:09:45,450 Because I just want to get through this physics. 1354 01:09:45,450 --> 01:09:49,200 So consider two operators, A and B. So A and B 1355 01:09:49,200 --> 01:09:51,810 are my two operators. 1356 01:09:51,810 --> 01:09:52,469 They have hats. 1357 01:09:54,975 --> 01:09:58,120 He has a top hot. 1358 01:09:58,120 --> 01:10:01,407 So we have-- you've got to make this stuff a little more 1359 01:10:01,407 --> 01:10:01,990 light hearted. 1360 01:10:01,990 --> 01:10:07,240 So we have two operators, A and B. And I want to ask, 1361 01:10:07,240 --> 01:10:12,170 is it possible for there to be a function phi little a little 1362 01:10:12,170 --> 01:10:16,240 b which is simultaneously an eigenfunction of a 1363 01:10:16,240 --> 01:10:18,480 and an eigenfunction of b? 1364 01:10:18,480 --> 01:10:20,310 So now this is a pure math question. 1365 01:10:20,310 --> 01:10:23,500 Given two operators, a and b, can you 1366 01:10:23,500 --> 01:10:26,250 build a function which is simultaneously 1367 01:10:26,250 --> 01:10:28,520 an eigenfunction of a and an eigenfunction of b? 1368 01:10:28,520 --> 01:10:31,370 I will call this phi sub ab. 1369 01:10:31,370 --> 01:10:31,870 Why not? 1370 01:10:31,870 --> 01:10:33,800 OK, phi sub ab has this property. 1371 01:10:33,800 --> 01:10:36,460 And phi ab is equal to little a. 1372 01:10:36,460 --> 01:10:37,460 Why not? 1373 01:10:37,460 --> 01:10:42,020 And b on phi ab is equal to little b on phi ab. 1374 01:10:42,020 --> 01:10:46,140 You can't stop me, I have now created this object. 1375 01:10:46,140 --> 01:10:48,150 From the existence of this state, 1376 01:10:48,150 --> 01:10:51,007 what can you deduce about the operators a and b? 1377 01:10:51,007 --> 01:10:53,340 And if you've already seen this before, that's cheating. 1378 01:10:53,340 --> 01:10:54,260 So don't raise your hand. 1379 01:10:54,260 --> 01:10:56,634 But if you haven't seen it before, just think through it. 1380 01:10:56,634 --> 01:10:59,214 What does this tell you about the operators a and b? 1381 01:10:59,214 --> 01:11:00,130 AUDIENCE: [INAUDIBLE]. 1382 01:11:04,462 --> 01:11:05,170 PROFESSOR: Great. 1383 01:11:05,170 --> 01:11:05,730 That's a nice guess. 1384 01:11:05,730 --> 01:11:06,710 I like that guess. 1385 01:11:06,710 --> 01:11:08,600 So let's check. 1386 01:11:08,600 --> 01:11:13,970 So what would it mean for the commutator of a and b to be 0? 1387 01:11:13,970 --> 01:11:17,625 So this is equal to ab minus ba. 1388 01:11:17,625 --> 01:11:19,900 And what the commutator does is it takes two operators 1389 01:11:19,900 --> 01:11:23,280 and it gives you a new operator called bracket ab. 1390 01:11:23,280 --> 01:11:30,050 And sometimes it's useful to call it c, for commutator. 1391 01:11:30,050 --> 01:11:31,650 So it gives you a new operator. 1392 01:11:31,650 --> 01:11:33,630 Given two operators you build a new operator. 1393 01:11:36,280 --> 01:11:41,600 And suppose now that we have a common eigenfunction 1394 01:11:41,600 --> 01:11:42,270 of a and b. 1395 01:11:42,270 --> 01:11:43,270 What does that tell you? 1396 01:11:43,270 --> 01:11:47,920 Well, let's take the commutator-- a, b-- 1397 01:11:47,920 --> 01:11:50,380 and let's take it and act on phi of sub ab 1398 01:11:50,380 --> 01:11:51,839 on this common eigenstate. 1399 01:11:51,839 --> 01:11:52,880 So what is this equal to? 1400 01:11:52,880 --> 01:12:00,060 This is equal to ab minus ba phi b, which 1401 01:12:00,060 --> 01:12:06,550 I can write as ab phi ab minus ba phi ab 1402 01:12:06,550 --> 01:12:11,020 by linearity-- by addition, really. 1403 01:12:11,020 --> 01:12:15,390 But b acting on phi ab gives me, by hypothesis, little b. 1404 01:12:18,630 --> 01:12:21,650 And then a acting on a constant times phi ab 1405 01:12:21,650 --> 01:12:26,060 gives me a little a times that constant minus-- now 1406 01:12:26,060 --> 01:12:29,370 here a acting on phi ab gives me little a phi ab. 1407 01:12:29,370 --> 01:12:34,190 And then b acting on constant times phi ab gives me little b. 1408 01:12:34,190 --> 01:12:35,750 And now I can join these together. 1409 01:12:35,750 --> 01:12:43,030 This is equal to little ab minus ba phi ab. 1410 01:12:48,300 --> 01:12:50,460 But little a and little b are numbers. 1411 01:12:50,460 --> 01:12:53,380 If you take 7 times 5 and 5 times 7 and you subtract them, 1412 01:12:53,380 --> 01:12:55,230 what do you get? 1413 01:12:55,230 --> 01:12:56,210 0. 1414 01:12:56,210 --> 01:13:03,887 So in order for a and b to share a single eigenfunction phi ab, 1415 01:13:03,887 --> 01:13:05,470 what must be true of their commutator? 1416 01:13:08,390 --> 01:13:10,990 It must have an a non-zero kernel, exactly. 1417 01:13:10,990 --> 01:13:15,400 The commutator must annihilate that particular common 1418 01:13:15,400 --> 01:13:16,940 eigenfunction. 1419 01:13:16,940 --> 01:13:19,380 Does it tell you that it kills every function? 1420 01:13:19,380 --> 01:13:21,000 No. 1421 01:13:21,000 --> 01:13:24,570 But it must at the very least annihilate the shared 1422 01:13:24,570 --> 01:13:25,375 eigenfunction. 1423 01:13:25,375 --> 01:13:25,875 True? 1424 01:13:28,920 --> 01:13:35,490 Consider two operators of the following form. 1425 01:13:35,490 --> 01:13:40,370 Consider two operators who commutator ab 1426 01:13:40,370 --> 01:13:43,245 is equal to the identity times a constant. 1427 01:13:47,800 --> 01:13:52,650 Do these operators share any common eigenfunctions? 1428 01:13:52,650 --> 01:13:56,090 They can't, because the commutator-- the identity-- 1429 01:13:56,090 --> 01:13:58,980 doesn't annihilate any wave functions. 1430 01:13:58,980 --> 01:14:00,130 Any functions. 1431 01:14:00,130 --> 01:14:02,510 Nothing is 0 when acted upon by the identity. 1432 01:14:02,510 --> 01:14:05,490 That's the definition of the identity. 1433 01:14:05,490 --> 01:14:11,290 The identity takes any function, gives it that function back. 1434 01:14:11,290 --> 01:14:12,920 Cool? 1435 01:14:12,920 --> 01:14:16,050 Do a and b share any common eigenfunctions? 1436 01:14:16,050 --> 01:14:22,010 Now let's think about what that means in terms of observables. 1437 01:14:22,010 --> 01:14:25,700 Observables are represented by operators. 1438 01:14:25,700 --> 01:14:28,614 What is the meaning of the eigenfunctions 1439 01:14:28,614 --> 01:14:29,405 of those operators? 1440 01:14:34,560 --> 01:14:36,510 AUDIENCE: It corresponds to the [INAUDIBLE]. 1441 01:14:36,510 --> 01:14:39,070 PROFESSOR: It corresponds to the state 1442 01:14:39,070 --> 01:14:41,580 with a definite value of that observable. 1443 01:14:41,580 --> 01:14:46,040 And the value is the corresponding eigenvalue. 1444 01:14:46,040 --> 01:14:49,570 So the eigenfunction of an observable operator 1445 01:14:49,570 --> 01:14:51,882 is a possible state with a definite value 1446 01:14:51,882 --> 01:14:52,590 of that operator. 1447 01:14:52,590 --> 01:14:54,756 If I tell you that I have a state which is an energy 1448 01:14:54,756 --> 01:14:56,280 eigenfunction, then that means it 1449 01:14:56,280 --> 01:14:58,420 is a state with a definite value of the energy. 1450 01:14:58,420 --> 01:15:01,510 If I measure that energy, I know exactly what I will get. 1451 01:15:01,510 --> 01:15:03,240 Cool? 1452 01:15:03,240 --> 01:15:07,560 If a and b are two observables and their commutator 1453 01:15:07,560 --> 01:15:11,510 is proportional to the identity, is it possible 1454 01:15:11,510 --> 01:15:13,690 that there is a state with a definite value 1455 01:15:13,690 --> 01:15:18,140 of a and a definite value of b simultaneously? 1456 01:15:18,140 --> 01:15:20,060 No, because a state with a definite value of a 1457 01:15:20,060 --> 01:15:21,351 would be an eigenfunction of a. 1458 01:15:21,351 --> 01:15:22,970 And a state with a definite value of b 1459 01:15:22,970 --> 01:15:24,470 would be an eigenfunction of b. 1460 01:15:24,470 --> 01:15:28,410 And there are no common states, no common eigenfunctions 1461 01:15:28,410 --> 01:15:33,590 of a and b because the commutator never kills a state. 1462 01:15:33,590 --> 01:15:34,420 It has no 0's. 1463 01:15:38,280 --> 01:15:40,571 What is the commutator of x and p? 1464 01:15:44,500 --> 01:15:46,510 Commit this to memory, i h bar. 1465 01:15:46,510 --> 01:15:49,384 Now when we write i h bar, this is not really an operator. 1466 01:15:49,384 --> 01:15:50,050 That's a number. 1467 01:15:50,050 --> 01:15:51,300 What's the operator? 1468 01:15:51,300 --> 01:15:53,410 The identity. 1469 01:15:53,410 --> 01:15:56,910 Are there any states which are simultaneously eigenfunctions 1470 01:15:56,910 --> 01:16:00,386 of x-- the operator x-- and eigenfunctions of p? 1471 01:16:00,386 --> 01:16:02,990 Are there any states that have a definite position 1472 01:16:02,990 --> 01:16:05,370 and a definite momentum simultaneously? 1473 01:16:05,370 --> 01:16:07,390 Is that because we're ignorant? 1474 01:16:07,390 --> 01:16:08,700 Are we ignorant? 1475 01:16:08,700 --> 01:16:09,660 Yes, OK good. 1476 01:16:09,660 --> 01:16:13,140 So it is because the kinds of things 1477 01:16:13,140 --> 01:16:19,160 that position [INAUDIBLE] r forbid the existence of a state 1478 01:16:19,160 --> 01:16:21,690 with a definite value of x and with a definite value of p 1479 01:16:21,690 --> 01:16:23,000 simultaneously. 1480 01:16:23,000 --> 01:16:25,430 It is neither here, nor there, nor both, nor neither. 1481 01:16:31,060 --> 01:16:33,750 So this tells us something very lovely. 1482 01:16:33,750 --> 01:16:36,400 This tells us that if we're in a state with a definite value 1483 01:16:36,400 --> 01:16:41,970 of x, what must be true of our uncertainty in the value of p? 1484 01:16:41,970 --> 01:16:45,887 It can't be 0, because if our uncertainty in the knowledge p 1485 01:16:45,887 --> 01:16:47,970 were 0, that would mean we were in a p eigenstate. 1486 01:16:47,970 --> 01:16:49,980 We would have definite value of p. 1487 01:16:49,980 --> 01:16:53,639 So if a delta x is 0, delta p cannot be 0 for sure. 1488 01:16:53,639 --> 01:16:54,930 Now how big does it have to be? 1489 01:16:54,930 --> 01:16:57,416 Could it be arbitrarily small? 1490 01:16:57,416 --> 01:16:58,790 So here's a commutation relation. 1491 01:16:58,790 --> 01:17:00,600 You will prove the following relation 1492 01:17:00,600 --> 01:17:02,529 later on in the course. 1493 01:17:02,529 --> 01:17:04,070 But I want to tell you now, because I 1494 01:17:04,070 --> 01:17:06,528 want to give you some intuition for what the commutator is. 1495 01:17:06,528 --> 01:17:07,320 What it means. 1496 01:17:07,320 --> 01:17:08,870 What the physics of it is. 1497 01:17:08,870 --> 01:17:13,016 If I take two operators, a and b, 1498 01:17:13,016 --> 01:17:15,140 which have a commutator-- which is a constant times 1499 01:17:15,140 --> 01:17:20,320 the identity-- then the uncertainty in any state 1500 01:17:20,320 --> 01:17:29,700 psi of a times the uncertainty in the same state psi of b 1501 01:17:29,700 --> 01:17:36,530 must be greater than or equal to 1/2 the absolute value 1502 01:17:36,530 --> 01:17:40,210 of the expectation value of the commutator-- that's 1503 01:17:40,210 --> 01:17:42,600 an amazing set of symbols-- of a with b. 1504 01:17:51,990 --> 01:17:52,856 Square root. 1505 01:17:52,856 --> 01:17:54,730 AUDIENCE: You've got yo take the square root. 1506 01:17:54,730 --> 01:17:56,106 PROFESSOR: Thank you. 1507 01:17:56,106 --> 01:17:57,860 AUDIENCE: Oh, I was kidding. 1508 01:17:57,860 --> 01:18:00,367 PROFESSOR: So the easier way to-- let's see a,b. 1509 01:18:00,367 --> 01:18:02,200 Let's just make sure I'm getting this right. 1510 01:18:02,200 --> 01:18:03,449 I'm the commutator of a and b. 1511 01:18:06,190 --> 01:18:06,690 Oh no, no. 1512 01:18:06,690 --> 01:18:07,648 There's no square root. 1513 01:18:07,648 --> 01:18:08,200 Sorry. 1514 01:18:08,200 --> 01:18:09,366 Why are you messing with me? 1515 01:18:13,720 --> 01:18:16,260 Expectation value of a and b. 1516 01:18:16,260 --> 01:18:17,770 Delta a. 1517 01:18:17,770 --> 01:18:18,540 Delta p. 1518 01:18:18,540 --> 01:18:19,890 H bar. 1519 01:18:19,890 --> 01:18:21,640 Yes, good. 1520 01:18:21,640 --> 01:18:23,650 So in particular, these expectation values 1521 01:18:23,650 --> 01:18:25,720 should be taken with the state psi. 1522 01:18:25,720 --> 01:18:29,084 Sorry, it's late and I'm tired. 1523 01:18:29,084 --> 01:18:30,750 And this should be taken into state psi. 1524 01:18:35,470 --> 01:18:37,080 So we compute the commutator. 1525 01:18:37,080 --> 01:18:39,290 We take the expectation value in our state. 1526 01:18:39,290 --> 01:18:42,510 We take the norm, multiply it by 1/2, 1527 01:18:42,510 --> 01:18:45,960 and this is the bound This is the claim. 1528 01:18:45,960 --> 01:18:48,310 So let's check this in the case of x and p. 1529 01:18:48,310 --> 01:18:49,940 And do I have this using just math? 1530 01:18:49,940 --> 01:18:51,810 This will come from just linear algebra. 1531 01:18:57,699 --> 01:18:59,490 So let's check this in the case of x and p. 1532 01:18:59,490 --> 01:19:02,790 What's the commutator? 1533 01:19:02,790 --> 01:19:03,560 1. 1534 01:19:03,560 --> 01:19:06,160 So what's the expectation value in a state, psi, 1535 01:19:06,160 --> 01:19:13,230 of i h bar times operator 1 psi? 1536 01:19:13,230 --> 01:19:15,011 It's i h bar. 1537 01:19:15,011 --> 01:19:15,510 Great. 1538 01:19:15,510 --> 01:19:16,593 So what happens over here? 1539 01:19:16,593 --> 01:19:20,010 What do we get for delta x in the state psi times delta 1540 01:19:20,010 --> 01:19:21,350 p in the state psi? 1541 01:19:24,080 --> 01:19:27,470 This must be greater than or equal to 1/2 times 1542 01:19:27,470 --> 01:19:29,810 the absolute value of the expectation 1543 01:19:29,810 --> 01:19:30,810 value of the commutator. 1544 01:19:30,810 --> 01:19:33,580 But the expectation value of the commutator is i h bar. 1545 01:19:33,580 --> 01:19:35,960 And the norm of that is just h bar. 1546 01:19:39,150 --> 01:19:42,670 This is the uncertainty relation. 1547 01:19:42,670 --> 01:19:44,464 So given some uncertainty in delta x, 1548 01:19:44,464 --> 01:19:45,880 this tells us how much delta p is. 1549 01:19:45,880 --> 01:19:47,296 Now, how does this right hand side 1550 01:19:47,296 --> 01:19:49,970 depend on the wave function psi? 1551 01:19:49,970 --> 01:19:51,810 It turns out it doesn't. 1552 01:19:51,810 --> 01:19:54,200 It's explicitly independent of the particular wave 1553 01:19:54,200 --> 01:19:55,408 function we're interested in. 1554 01:19:55,408 --> 01:19:58,642 For any wave function whatsoever, 1555 01:19:58,642 --> 01:20:00,600 the uncertainty in x times the uncertainty in p 1556 01:20:00,600 --> 01:20:03,776 must be greater than or equal to 1/2 h bar. 1557 01:20:03,776 --> 01:20:05,650 And this is a consequence of the commutation, 1558 01:20:05,650 --> 01:20:07,710 and more importantly, the failure 1559 01:20:07,710 --> 01:20:12,070 of x and p to commute, because the ability to commute and get 1560 01:20:12,070 --> 01:20:16,520 0 is necessary for there to be common eigenstates. 1561 01:20:16,520 --> 01:20:20,350 Now imagine in this case a and b are that a and b that in fact 1562 01:20:20,350 --> 01:20:22,180 do share a common eigenstate. 1563 01:20:22,180 --> 01:20:27,050 So let's compute the uncertainty of a in the state phi sub 1564 01:20:27,050 --> 01:20:37,230 ab, phi sub ab, phi sub ab, phi sub ab. 1565 01:20:37,230 --> 01:20:38,580 And what is this equal to? 1566 01:20:38,580 --> 01:20:42,490 Well, what's the commutator acting on the state phi sub ab? 1567 01:20:42,490 --> 01:20:43,170 0. 1568 01:20:43,170 --> 01:20:45,790 So this is greater than or equal to 0. 1569 01:20:45,790 --> 01:20:47,834 And in fact, you can just check by going 1570 01:20:47,834 --> 01:20:50,000 through the proof of this that it's just equal to 0. 1571 01:20:50,000 --> 01:20:51,020 So what's the uncertainty of a? 1572 01:20:51,020 --> 01:20:53,020 Well, we know it's in the state with definite value of a. 1573 01:20:53,020 --> 01:20:53,600 0. 1574 01:20:53,600 --> 01:20:56,651 Uncertainty in b in this state, what's the uncertainty? 1575 01:20:56,651 --> 01:20:57,150 0. 1576 01:20:57,150 --> 01:20:59,300 We know that it's in an eigenstate of b. 1577 01:20:59,300 --> 01:21:01,850 So 0 is indeed greater than or equal to 0, 1578 01:21:01,850 --> 01:21:04,810 so it satisfies the uncertainty relation. 1579 01:21:04,810 --> 01:21:08,060 Commuting is telling you about the possibility of the state 1580 01:21:08,060 --> 01:21:11,400 having definite values of both operators simultaneously. 1581 01:21:11,400 --> 01:21:14,010 And this is going to turn out to be enormously valuable when 1582 01:21:14,010 --> 01:21:15,551 we talk about angular momentum, which 1583 01:21:15,551 --> 01:21:16,850 is coming in a couple of weeks. 1584 01:21:16,850 --> 01:21:20,340 OK, see you guys on Thursday here for the midterm.