1 00:00:00,115 --> 00:00:02,410 The following content is provided under a Creative 2 00:00:02,410 --> 00:00:03,830 Commons license. 3 00:00:03,830 --> 00:00:06,040 Your support will help MIT Open Courseware 4 00:00:06,040 --> 00:00:10,130 continue to offer high quality educational resources for free. 5 00:00:10,130 --> 00:00:12,670 To make a donation or to view additional materials 6 00:00:12,670 --> 00:00:16,630 from hundreds of MIT courses, visit MIT Open Courseware 7 00:00:16,630 --> 00:00:17,503 at ocw.mit.edu. 8 00:00:22,157 --> 00:00:24,740 ROBERT FIELD: That's the outline of what we're going to cover. 9 00:00:24,740 --> 00:00:26,880 But before we get started on that, 10 00:00:26,880 --> 00:00:29,910 I want to talk about a couple of things. 11 00:00:29,910 --> 00:00:33,750 First of all, last time, we talked about the two slit 12 00:00:33,750 --> 00:00:36,080 experiment. 13 00:00:36,080 --> 00:00:37,870 And it's mostly classical. 14 00:00:37,870 --> 00:00:40,220 There is only a little bit of quantum in it 15 00:00:40,220 --> 00:00:45,840 where we talk about momentum as being 16 00:00:45,840 --> 00:00:48,161 determined by h over lambda. 17 00:00:48,161 --> 00:00:48,660 OK. 18 00:00:48,660 --> 00:00:53,400 But what were the two surprising things 19 00:00:53,400 --> 00:00:54,930 about the two slit experiment? 20 00:00:54,930 --> 00:00:57,860 There are two of them, two surprises. 21 00:00:57,860 --> 00:00:58,360 Yes? 22 00:00:58,360 --> 00:01:01,600 AUDIENCE: [INAUDIBLE] That a single particle 23 00:01:01,600 --> 00:01:02,920 can interfere with itself. 24 00:01:02,920 --> 00:01:03,670 ROBERT FIELD: Yes. 25 00:01:03,670 --> 00:01:07,360 That's the most surprising thing. 26 00:01:07,360 --> 00:01:13,310 And when you go to really low intensity-- 27 00:01:13,310 --> 00:01:17,200 so there's only one photon, which is quantum, 28 00:01:17,200 --> 00:01:20,470 in the apparatus, somehow, it knows 29 00:01:20,470 --> 00:01:23,220 enough to interfere with itself. 30 00:01:23,220 --> 00:01:29,110 And this is the most mysterious aspect. 31 00:01:29,110 --> 00:01:32,550 But then there's one other aspect, 32 00:01:32,550 --> 00:01:37,275 which is how it communicates that interference with itself. 33 00:01:39,799 --> 00:01:40,340 What is that? 34 00:01:46,510 --> 00:01:47,010 You're hot. 35 00:01:47,010 --> 00:01:48,375 You want to do another one? 36 00:01:48,375 --> 00:01:51,080 AUDIENCE: What do you mean by how it communicates? 37 00:01:51,080 --> 00:01:56,600 ROBERT FIELD: Well, here we have the screen on which 38 00:01:56,600 --> 00:02:00,070 the information is deposited. 39 00:02:00,070 --> 00:02:07,580 And you have possibly some sort of probability distribution, 40 00:02:07,580 --> 00:02:09,820 which is a kind of a continuous thing. 41 00:02:09,820 --> 00:02:10,960 But you don't observe that. 42 00:02:14,780 --> 00:02:16,740 What do you observe? 43 00:02:16,740 --> 00:02:21,320 So you could say the state of a particle 44 00:02:21,320 --> 00:02:26,430 has amplitude everywhere on this screen. 45 00:02:26,430 --> 00:02:32,010 But what do you see in the experiment? 46 00:02:32,010 --> 00:02:33,410 Yes? 47 00:02:33,410 --> 00:02:36,290 AUDIENCE: So you just see one [INAUDIBLE] point. 48 00:02:36,290 --> 00:02:38,310 [INAUDIBLE] thousands and thousands. 49 00:02:38,310 --> 00:02:40,645 And eventually, you see it mimic that probability 50 00:02:40,645 --> 00:02:41,270 [? sequence. ?] 51 00:02:41,270 --> 00:02:43,530 ROBERT FIELD: That's exactly right. 52 00:02:43,530 --> 00:02:47,660 So this state of the system which is distributed 53 00:02:47,660 --> 00:02:51,490 collapses to a single point. 54 00:02:51,490 --> 00:02:55,310 We say that what we are observing-- 55 00:02:55,310 --> 00:02:56,500 and this is mysterious. 56 00:02:56,500 --> 00:02:59,800 And it should bother you now. 57 00:02:59,800 --> 00:03:05,950 We're seeing an eigenvalue of the measurement operator. 58 00:03:05,950 --> 00:03:08,080 So we have this sort of thing. 59 00:03:08,080 --> 00:03:10,060 It goes into the measurement operator. 60 00:03:10,060 --> 00:03:12,010 And the measurement operator says, 61 00:03:12,010 --> 00:03:16,930 this is one of the answers I'm permitted to give you. 62 00:03:16,930 --> 00:03:21,970 And another aspect that's really disturbing or puzzling 63 00:03:21,970 --> 00:03:27,160 is that you do many identical experiments. 64 00:03:27,160 --> 00:03:28,992 And the answer is always different. 65 00:03:33,820 --> 00:03:36,090 There is no determinant. 66 00:03:36,090 --> 00:03:37,920 It's all probabilistic. 67 00:03:37,920 --> 00:03:41,610 So this really should bother you. 68 00:03:41,610 --> 00:03:44,710 And eventually, it won't. 69 00:03:44,710 --> 00:03:46,970 OK. 70 00:03:46,970 --> 00:03:49,430 Now, there's another question I have. 71 00:03:49,430 --> 00:03:54,940 When I described the two slit experiment, I intentionally 72 00:03:54,940 --> 00:03:58,880 put something up on the diagram that should bother you, 73 00:03:58,880 --> 00:04:02,420 that should have said, this is ridiculous. 74 00:04:02,420 --> 00:04:04,760 I used a candle in the lecture. 75 00:04:04,760 --> 00:04:08,990 And I used a light bulb in the notes. 76 00:04:08,990 --> 00:04:11,490 And why is that ridiculous? 77 00:04:11,490 --> 00:04:12,240 Yes? 78 00:04:12,240 --> 00:04:14,250 AUDIENCE: You said many frequencies. 79 00:04:14,250 --> 00:04:15,375 ROBERT FIELD: That's right. 80 00:04:15,375 --> 00:04:22,890 So the sources of light that I misled you with intentionally 81 00:04:22,890 --> 00:04:26,890 have a continuous frequency distribution. 82 00:04:26,890 --> 00:04:34,640 And the interference depends on the same frequency. 83 00:04:34,640 --> 00:04:41,260 So the only way you would see any kind of diffraction 84 00:04:41,260 --> 00:04:44,440 pattern, any kind of pattern on the two slit experiment, 85 00:04:44,440 --> 00:04:48,180 is if you had monochromatic light. 86 00:04:48,180 --> 00:04:50,220 It would all wash out. 87 00:04:50,220 --> 00:04:52,170 You'd still get dots. 88 00:04:52,170 --> 00:04:53,640 But the dots would never give you 89 00:04:53,640 --> 00:04:58,290 anything except perhaps a superposition of two or three 90 00:04:58,290 --> 00:05:02,521 or an infinite number of patterns. 91 00:05:02,521 --> 00:05:03,020 OK. 92 00:05:03,020 --> 00:05:08,300 This is, again, something that's really bothersome. 93 00:05:08,300 --> 00:05:13,610 Now, I also use the crude illustration of an uncertainty 94 00:05:13,610 --> 00:05:18,320 principle, that the uncertainty in position z-axis 95 00:05:18,320 --> 00:05:21,950 and the uncertainty in the momentum along the z-axis 96 00:05:21,950 --> 00:05:23,630 was greater than h. 97 00:05:26,720 --> 00:05:29,240 And I froze. 98 00:05:29,240 --> 00:05:34,760 And I didn't realize that I had delta s with the slit. 99 00:05:34,760 --> 00:05:37,580 But it really is the same thing as the z. 100 00:05:37,580 --> 00:05:40,670 Because the slit is how you define 101 00:05:40,670 --> 00:05:43,640 the position in the z-axis. 102 00:05:43,640 --> 00:05:48,710 And so this is the first taste of the uncertainty principle. 103 00:05:48,710 --> 00:05:49,910 And I said I didn't like it. 104 00:05:56,050 --> 00:05:57,770 OK. 105 00:05:57,770 --> 00:06:03,890 In quantum mechanics, you're not allowed 106 00:06:03,890 --> 00:06:05,600 to look inside small stuff. 107 00:06:05,600 --> 00:06:09,830 You're not allowed to see the microscopic structure. 108 00:06:09,830 --> 00:06:12,050 You're only able to do experiments, 109 00:06:12,050 --> 00:06:16,070 usually thought experiments, an infinite number 110 00:06:16,070 --> 00:06:17,540 of identical experiments. 111 00:06:17,540 --> 00:06:25,580 And they reveal the structure in some complicated, encoded way. 112 00:06:25,580 --> 00:06:29,950 And this is really not what the textbooks are about. 113 00:06:29,950 --> 00:06:34,330 Textbooks don't tell you how you actually 114 00:06:34,330 --> 00:06:36,640 think about a problem with quantum mechanics. 115 00:06:36,640 --> 00:06:39,940 They tell you, here are some exactly solved problems. 116 00:06:39,940 --> 00:06:42,214 Memorize them. 117 00:06:42,214 --> 00:06:43,130 And I don't want you-- 118 00:06:43,130 --> 00:06:44,130 I don't want to do that. 119 00:06:47,480 --> 00:06:49,870 OK. 120 00:06:49,870 --> 00:06:53,240 Last part of the introduction here-- 121 00:06:53,240 --> 00:06:58,810 suppose we have a circular drum, a square drum, 122 00:06:58,810 --> 00:07:01,390 and a rectangular drum. 123 00:07:05,320 --> 00:07:11,510 Have you ever seen a square drum or a rectangular drum? 124 00:07:11,510 --> 00:07:13,850 Do you have an idea how a square drum would sound? 125 00:07:16,620 --> 00:07:17,120 Yes. 126 00:07:17,120 --> 00:07:18,191 You do have an idea. 127 00:07:18,191 --> 00:07:19,190 It would sound terrible. 128 00:07:22,460 --> 00:07:27,450 Because the frequencies, you get are not integer multiples. 129 00:07:27,450 --> 00:07:29,970 It would just sound amazingly terrible. 130 00:07:29,970 --> 00:07:35,860 But if you had a square drum or a rectangular drum, 131 00:07:35,860 --> 00:07:38,080 you could do an experiment with some kind 132 00:07:38,080 --> 00:07:40,660 of acoustic instrument to find out 133 00:07:40,660 --> 00:07:45,820 what the frequency distribution is of the noise you make. 134 00:07:45,820 --> 00:07:47,470 And you would be able to tell. 135 00:07:47,470 --> 00:07:49,300 It's not round. 136 00:07:49,300 --> 00:07:51,310 It might be square. 137 00:07:51,310 --> 00:07:54,220 Or it might have a certain ratio of dimensions. 138 00:07:54,220 --> 00:07:57,880 This is what we're talking about as far as internal structure is 139 00:07:57,880 --> 00:07:58,960 concerned. 140 00:07:58,960 --> 00:08:01,420 And it's very much like what you would do as a musician. 141 00:08:04,100 --> 00:08:06,890 I mean, certainly, when a musical instrument 142 00:08:06,890 --> 00:08:11,780 is arranged correctly, it's not like a square drum. 143 00:08:11,780 --> 00:08:13,980 It sounds good. 144 00:08:13,980 --> 00:08:16,800 And that's because you get harmonics 145 00:08:16,800 --> 00:08:22,320 or you get integer multiples of some standard frequency. 146 00:08:24,850 --> 00:08:25,350 OK. 147 00:08:25,350 --> 00:08:31,250 So now, we're going to talk about the classical wave 148 00:08:31,250 --> 00:08:34,960 equation, which is not quantum. 149 00:08:34,960 --> 00:08:39,880 But it's a very similar sort of equation to the Schrodinger 150 00:08:39,880 --> 00:08:41,240 equation. 151 00:08:41,240 --> 00:08:44,740 And so the methods for solving this differential equation 152 00:08:44,740 --> 00:08:46,960 are on display. 153 00:08:46,960 --> 00:08:52,960 And so the trick is-- well, first of all, where 154 00:08:52,960 --> 00:08:55,800 does this equation come from? 155 00:08:55,800 --> 00:08:58,020 And it's always force is equal to mass times 156 00:08:58,020 --> 00:09:00,481 acceleration in disguise. 157 00:09:00,481 --> 00:09:00,980 OK. 158 00:09:00,980 --> 00:09:05,700 And then you have tricks for how you solve this. 159 00:09:05,700 --> 00:09:10,220 And one of the most frequently used and powerful tricks 160 00:09:10,220 --> 00:09:12,110 is separation of variables. 161 00:09:12,110 --> 00:09:15,160 You need to know how that works. 162 00:09:15,160 --> 00:09:20,760 Then once you solve the problem, you have the general solution. 163 00:09:20,760 --> 00:09:25,170 And you then say, well, OK, for the specific case 164 00:09:25,170 --> 00:09:29,740 we have, like a string tied down at both ends, 165 00:09:29,740 --> 00:09:32,460 we have boundary conditions. 166 00:09:32,460 --> 00:09:35,480 And we impose those boundary conditions. 167 00:09:35,480 --> 00:09:38,950 And then we have basically what we 168 00:09:38,950 --> 00:09:42,620 would call the normal modes of the problem. 169 00:09:42,620 --> 00:09:46,550 And then we would ask, OK, well, suppose 170 00:09:46,550 --> 00:09:51,710 we're doing a specific experiment 171 00:09:51,710 --> 00:09:56,660 or doing a specific preparation of the system. 172 00:09:56,660 --> 00:09:59,850 And we can call that the pluck of the system. 173 00:09:59,850 --> 00:10:04,400 And you might pluck several normal modes. 174 00:10:04,400 --> 00:10:06,430 You get a superposition state. 175 00:10:06,430 --> 00:10:12,170 And that superposition state behaves in a dynamic way. 176 00:10:12,170 --> 00:10:15,500 And you want to be able to understand that dynamics. 177 00:10:15,500 --> 00:10:19,010 And the most important thing that I want you to do 178 00:10:19,010 --> 00:10:24,410 is, instead of trying to draw the solutions to a differential 179 00:10:24,410 --> 00:10:27,200 equation, which is a mathematical equation, 180 00:10:27,200 --> 00:10:31,010 I want you to draw cartoons, cartoons 181 00:10:31,010 --> 00:10:34,077 that embody your understanding of the problem. 182 00:10:34,077 --> 00:10:35,910 And I'm going to be trying to do that today. 183 00:10:46,624 --> 00:10:47,130 OK. 184 00:10:47,130 --> 00:10:49,820 In this course, for the first half of the course, 185 00:10:49,820 --> 00:10:51,580 most of what we're going to be doing 186 00:10:51,580 --> 00:10:55,590 is solving for exactly soluble problems-- 187 00:10:55,590 --> 00:11:00,060 the particle in a box, the harmonic oscillator, 188 00:11:00,060 --> 00:11:07,777 the rigid rotor, and the hydrogen atom. 189 00:11:10,980 --> 00:11:14,700 With these four problems, most of the things 190 00:11:14,700 --> 00:11:19,170 that we will encounter in quantum mechanics 191 00:11:19,170 --> 00:11:20,740 are somehow related to these. 192 00:11:23,924 --> 00:11:28,140 And in the textbooks, they treat these things as sacred. 193 00:11:28,140 --> 00:11:30,480 And they say, OK, well, now that you've solved them, 194 00:11:30,480 --> 00:11:32,580 you understand quantum mechanics. 195 00:11:32,580 --> 00:11:35,160 But these are really tools for understanding 196 00:11:35,160 --> 00:11:37,509 more complicated situations. 197 00:11:37,509 --> 00:11:39,300 I mean, you might have a particle in a box. 198 00:11:39,300 --> 00:11:44,280 Instead of with a square bottom, it might have a tilted bottom. 199 00:11:44,280 --> 00:11:47,080 Or it might have a double minimum. 200 00:11:47,080 --> 00:11:50,520 But if you understand that, you then 201 00:11:50,520 --> 00:11:52,800 can begin to build an understanding of, 202 00:11:52,800 --> 00:11:56,190 what are the things in the experiment that tell you 203 00:11:56,190 --> 00:12:01,200 about these distortions of the standard problem? 204 00:12:01,200 --> 00:12:03,960 And the same thing for a harmonic oscillator. 205 00:12:03,960 --> 00:12:05,760 Almost everything that's vibrating 206 00:12:05,760 --> 00:12:07,890 is harmonic approximately. 207 00:12:07,890 --> 00:12:09,600 But there's a little bit of distortion 208 00:12:09,600 --> 00:12:11,220 as you stretch it more. 209 00:12:11,220 --> 00:12:13,560 And again, you can understand how 210 00:12:13,560 --> 00:12:17,730 to measure the distortions from harmonicity by understanding 211 00:12:17,730 --> 00:12:18,890 the harmonic oscillator. 212 00:12:18,890 --> 00:12:20,400 We did rotor, H atom. 213 00:12:20,400 --> 00:12:21,610 It's all the same. 214 00:12:21,610 --> 00:12:25,650 So I would like to tell you that these standard problems are 215 00:12:25,650 --> 00:12:27,540 really important. 216 00:12:27,540 --> 00:12:30,500 But nothing is like that. 217 00:12:30,500 --> 00:12:34,650 And what's important is how it's different from that. 218 00:12:34,650 --> 00:12:37,160 And this is my unique perspective. 219 00:12:37,160 --> 00:12:43,030 And you won't get that from McQuarrie or any textbook. 220 00:12:43,030 --> 00:12:45,190 But this is MIT. 221 00:12:45,190 --> 00:12:47,740 So there are templates for understanding real quantum 222 00:12:47,740 --> 00:12:49,400 mechanical system. 223 00:12:49,400 --> 00:12:59,800 And the big thing, the most important technique 224 00:12:59,800 --> 00:13:01,930 for doing that is perturbation theory. 225 00:13:06,270 --> 00:13:09,690 And so perturbation theory is just 226 00:13:09,690 --> 00:13:15,450 a way of building beyond the oversimplification. 227 00:13:15,450 --> 00:13:17,820 And it's mathematically really ugly. 228 00:13:17,820 --> 00:13:20,710 But it's tremendously powerful. 229 00:13:20,710 --> 00:13:23,710 And it's where you get insight. 230 00:13:23,710 --> 00:13:24,620 OK. 231 00:13:24,620 --> 00:13:31,250 Now, many people have complained that they found 5.61 hard. 232 00:13:31,250 --> 00:13:32,780 Because it's so mathematical. 233 00:13:36,500 --> 00:13:39,230 And maybe this is going to be the most mathematical lecture 234 00:13:39,230 --> 00:13:40,790 in the course. 235 00:13:40,790 --> 00:13:43,310 But I don't want it to be hard. 236 00:13:43,310 --> 00:13:49,550 Now, chemists usually derive insights from pictorial 237 00:13:49,550 --> 00:13:51,950 rather than mathematical views of a problem. 238 00:13:55,190 --> 00:14:00,060 So what are the pictures that describe these differential 239 00:14:00,060 --> 00:14:00,930 equations? 240 00:14:00,930 --> 00:14:04,350 How do we convert what seems to be 241 00:14:04,350 --> 00:14:07,800 just straight mathematics to pictures that mean something 242 00:14:07,800 --> 00:14:08,790 to us? 243 00:14:08,790 --> 00:14:12,150 And that's my goal, to get you to be drawing 244 00:14:12,150 --> 00:14:15,900 freehand pictures that embody the important features 245 00:14:15,900 --> 00:14:19,300 of the solutions to the problems. 246 00:14:19,300 --> 00:14:21,060 OK. 247 00:14:21,060 --> 00:14:25,560 So we're going to be looking at a differential equation. 248 00:14:25,560 --> 00:14:28,380 And one of the first questions you ask, well, where 249 00:14:28,380 --> 00:14:29,670 did that equation come from? 250 00:14:32,510 --> 00:14:35,330 And you're not going to derive a differential equation ever 251 00:14:35,330 --> 00:14:36,140 in this course. 252 00:14:36,140 --> 00:14:37,990 But you're going to want to think, well, 253 00:14:37,990 --> 00:14:43,850 I pretty much understand what's in this differential equation. 254 00:14:43,850 --> 00:14:46,820 And then we'll use standard methods 255 00:14:46,820 --> 00:14:47,930 for solving that equation. 256 00:14:52,630 --> 00:14:56,830 And one such differential equation is this-- 257 00:14:56,830 --> 00:14:58,720 second derivative of some function 258 00:14:58,720 --> 00:15:03,580 with respect to a variable is equal to a constant times 259 00:15:03,580 --> 00:15:05,200 that function. 260 00:15:05,200 --> 00:15:07,250 Now, that you know. 261 00:15:07,250 --> 00:15:09,980 You know sines and cosines are solutions to that. 262 00:15:09,980 --> 00:15:12,190 And you know that exponentials are solutions to that. 263 00:15:14,890 --> 00:15:17,730 Now, that pretty much takes you through a lot of problems 264 00:15:17,730 --> 00:15:18,740 in quantum mechanics. 265 00:15:21,340 --> 00:15:22,990 But now, one of the important things 266 00:15:22,990 --> 00:15:27,982 is this is a second-order differential equation. 267 00:15:27,982 --> 00:15:29,440 And that means that there are going 268 00:15:29,440 --> 00:15:35,010 to be two linearly independent solutions. 269 00:15:35,010 --> 00:15:38,490 And you need to know both of them. 270 00:15:38,490 --> 00:15:41,460 I'll talk about this some more later. 271 00:15:41,460 --> 00:15:44,480 Now, sometimes, the differential equations 272 00:15:44,480 --> 00:15:47,210 look much more complicated than this. 273 00:15:47,210 --> 00:15:52,160 And so the goal is usually to rewrite it 274 00:15:52,160 --> 00:15:56,390 in a form which corresponds to a differential equation that 275 00:15:56,390 --> 00:16:00,200 is well known and solved by mathematicians 276 00:16:00,200 --> 00:16:02,650 whose business is doing that. 277 00:16:02,650 --> 00:16:06,100 But we won't be doing that. 278 00:16:06,100 --> 00:16:08,140 OK. 279 00:16:08,140 --> 00:16:12,590 But usually, when you have a differential equation, 280 00:16:12,590 --> 00:16:16,400 the function is of more than one variable. 281 00:16:16,400 --> 00:16:18,990 And frequently, it's position and time. 282 00:16:18,990 --> 00:16:23,289 And so the first thing you do is you try to separate variables. 283 00:16:23,289 --> 00:16:24,830 And so that's what we're going to do. 284 00:16:28,247 --> 00:16:29,705 So we have a differential equation. 285 00:16:32,250 --> 00:16:35,520 And the first thing is a general solution. 286 00:16:41,260 --> 00:16:47,270 And one of the things that this solution will have is nodes. 287 00:16:47,270 --> 00:16:50,120 And the distance between nodes-- 288 00:16:53,047 --> 00:16:53,630 here's a node. 289 00:16:53,630 --> 00:16:54,680 Here's a node. 290 00:16:54,680 --> 00:16:59,160 That's half the wavelength. 291 00:16:59,160 --> 00:17:01,650 And we know that in quantum mechanics, 292 00:17:01,650 --> 00:17:05,400 if you know the wavelength, you know the momentum. 293 00:17:05,400 --> 00:17:06,994 So nodes are really important. 294 00:17:06,994 --> 00:17:09,160 Because it's telling you how fast things are moving. 295 00:17:12,220 --> 00:17:17,109 We can also look at the envelope. 296 00:17:17,109 --> 00:17:19,270 And this would be some kind of classical, 297 00:17:19,270 --> 00:17:22,210 as opposed to a quantum mechanical, probability 298 00:17:22,210 --> 00:17:23,780 distribution. 299 00:17:23,780 --> 00:17:29,470 And so it might look like this. 300 00:17:29,470 --> 00:17:31,720 But the important thing about the envelope 301 00:17:31,720 --> 00:17:33,250 is that it's always positive. 302 00:17:33,250 --> 00:17:35,770 Because it's probability, as opposed 303 00:17:35,770 --> 00:17:38,200 to a probability amplitude, which 304 00:17:38,200 --> 00:17:41,610 can be positive and negative. 305 00:17:41,610 --> 00:17:44,280 Interference is really important in quantum mechanics. 306 00:17:44,280 --> 00:17:49,350 But sometimes, the envelope tells you all you need to know. 307 00:17:49,350 --> 00:17:56,475 And the other thing is the velocity of a stationary phase. 308 00:18:00,550 --> 00:18:01,470 So you have a wave. 309 00:18:01,470 --> 00:18:02,340 And it's moving. 310 00:18:02,340 --> 00:18:04,380 And you sit at a point on that wave. 311 00:18:04,380 --> 00:18:07,390 And you ask, how fast does that point move? 312 00:18:07,390 --> 00:18:09,700 And I did that last time. 313 00:18:09,700 --> 00:18:10,510 And OK. 314 00:18:13,390 --> 00:18:19,390 So I've already talked a little bit about what we do next. 315 00:18:19,390 --> 00:18:23,140 But the important thing is always, at the end, 316 00:18:23,140 --> 00:18:24,880 you draw a cartoon. 317 00:18:24,880 --> 00:18:31,060 And you endow that cartoon with your insights. 318 00:18:31,060 --> 00:18:35,320 And that enables you to remember and to understand 319 00:18:35,320 --> 00:18:37,290 and to organize questions about the problem. 320 00:18:40,640 --> 00:18:41,480 OK. 321 00:18:41,480 --> 00:18:43,160 So let's get to work on a real problem. 322 00:18:51,820 --> 00:18:58,020 So we have a string that's tied down to two points. 323 00:18:58,020 --> 00:19:06,600 And so let's look at the distortion of that string. 324 00:19:06,600 --> 00:19:12,490 And so we chop this region of space up into regions. 325 00:19:15,670 --> 00:19:20,460 So this might be the region at x minus 1. 326 00:19:20,460 --> 00:19:22,230 And this might be the region at x0. 327 00:19:22,230 --> 00:19:26,100 And this might be the region of x1. 328 00:19:26,100 --> 00:19:28,270 And we're interested in-- 329 00:19:28,270 --> 00:19:36,150 OK, suppose we have the value of the displacement of the wave 330 00:19:36,150 --> 00:19:42,410 here at x minus 1 and here and here. 331 00:19:42,410 --> 00:19:45,780 OK, so these would be the amount that the wave 332 00:19:45,780 --> 00:19:48,560 is displaced from equilibrium. 333 00:19:48,560 --> 00:19:53,205 And we call those u of x. 334 00:19:55,820 --> 00:20:03,230 And so the first segment here, the minus 1 segment, 335 00:20:03,230 --> 00:20:08,770 this segment is pulling down on this segment of the string 336 00:20:08,770 --> 00:20:11,180 by this amount. 337 00:20:11,180 --> 00:20:17,260 And this one is pulling up on the segment by that amount. 338 00:20:17,260 --> 00:20:22,650 So we want to know, what is the force acting on each segment? 339 00:20:22,650 --> 00:20:30,240 And so we have the force constant times 340 00:20:30,240 --> 00:20:36,930 the displacement at x0 minus the displacement at x minus 1. 341 00:20:36,930 --> 00:20:40,360 So this is the difference between the displacements. 342 00:20:40,360 --> 00:20:41,730 And this is the force constant. 343 00:20:41,730 --> 00:20:44,530 We're talking about Hooke's law. 344 00:20:44,530 --> 00:20:48,830 Hooke's law is the force is equal to minus 345 00:20:48,830 --> 00:20:51,198 k times the displacement. 346 00:20:54,070 --> 00:21:00,270 And so we collect the forces felt by each particle. 347 00:21:00,270 --> 00:21:06,690 And the forces felt by each particle are, again, the force 348 00:21:06,690 --> 00:21:15,360 constant times the difference in u at 0 and minus 1 349 00:21:15,360 --> 00:21:17,730 minus the difference-- 350 00:21:17,730 --> 00:21:23,746 plus 1 minus the difference in u at 0 and plus 1. 351 00:21:23,746 --> 00:21:27,400 And this is a second derivative. 352 00:21:27,400 --> 00:21:32,520 This is the second derivative of u with respect to x. 353 00:21:32,520 --> 00:21:36,310 So we've derived a wave equation. 354 00:21:36,310 --> 00:21:39,450 And we know it's going to involve a second derivative. 355 00:21:39,450 --> 00:21:43,990 So force is equal to mass times acceleration. 356 00:21:43,990 --> 00:21:46,950 Well, we already know the force is 357 00:21:46,950 --> 00:21:53,610 going to be related to the second derivative of u 358 00:21:53,610 --> 00:21:57,030 with respect to x. 359 00:21:57,030 --> 00:21:58,860 And now, this is something. 360 00:21:58,860 --> 00:22:01,320 And we know what this is. 361 00:22:01,320 --> 00:22:03,544 This is going to be the time derivative. 362 00:22:08,390 --> 00:22:08,900 OK. 363 00:22:08,900 --> 00:22:11,320 And this is just something that gets the units right. 364 00:22:15,160 --> 00:22:18,610 And it has physical significance. 365 00:22:18,610 --> 00:22:22,490 But in the case of this particle on a string-- 366 00:22:22,490 --> 00:22:25,690 this wave on a string, it's related 367 00:22:25,690 --> 00:22:29,010 to the mass of the string and the tension of the string. 368 00:22:31,780 --> 00:22:35,820 And it's also related to the velocity that things move. 369 00:22:35,820 --> 00:22:36,320 OK. 370 00:22:36,320 --> 00:22:38,600 So we have a differential equation 371 00:22:38,600 --> 00:22:42,200 that is related to forces equal to mass times acceleration. 372 00:22:42,200 --> 00:22:47,126 And the differential equation has the form second derivative 373 00:22:47,126 --> 00:22:56,030 of u with respect to x is equal to 1 374 00:22:56,030 --> 00:23:01,980 over v squared times the second derivative of u 375 00:23:01,980 --> 00:23:05,540 with respect to t. 376 00:23:05,540 --> 00:23:08,880 That's the wave equation. 377 00:23:08,880 --> 00:23:12,260 So it is really f is equal to ma. 378 00:23:12,260 --> 00:23:13,790 But OK. 379 00:23:13,790 --> 00:23:15,820 And now, the units of this-- 380 00:23:15,820 --> 00:23:17,230 this is x. 381 00:23:17,230 --> 00:23:18,110 And this is t. 382 00:23:18,110 --> 00:23:20,480 In order to be dimensionally consistent, 383 00:23:20,480 --> 00:23:27,080 this has to be something that is x over t, OK? 384 00:23:27,080 --> 00:23:29,700 And so this may be a velocity. 385 00:23:29,700 --> 00:23:31,150 But it has units of velocity. 386 00:23:31,150 --> 00:23:33,690 That's the differential equation we want to solve. 387 00:23:37,400 --> 00:23:38,170 OK. 388 00:23:38,170 --> 00:23:41,710 Well, the original differential equation that I wrote-- 389 00:23:48,020 --> 00:23:49,640 but I'm getting ahead of myself. 390 00:23:49,640 --> 00:23:50,480 OK. 391 00:23:50,480 --> 00:23:53,975 So this U of x and t-- 392 00:23:56,590 --> 00:24:03,280 we'd like it to be X of x times T of t. 393 00:24:03,280 --> 00:24:07,780 We think we could separate the variables in this way. 394 00:24:07,780 --> 00:24:08,950 So we try it. 395 00:24:08,950 --> 00:24:13,050 If we fail, it says you can't do that. 396 00:24:13,050 --> 00:24:16,980 Failure is usually going to be a result 397 00:24:16,980 --> 00:24:20,670 that the solution to the differential equation 398 00:24:20,670 --> 00:24:22,200 in this form is nothing. 399 00:24:22,200 --> 00:24:23,440 It's a straight line. 400 00:24:23,440 --> 00:24:25,770 Nothing's happening. 401 00:24:25,770 --> 00:24:28,290 So failure is acceptable. 402 00:24:28,290 --> 00:24:31,730 But if we're successful, we're going 403 00:24:31,730 --> 00:24:36,100 to get two separate differential equations. 404 00:24:36,100 --> 00:24:36,610 OK. 405 00:24:36,610 --> 00:24:42,690 So what we do then is take this differential equation, 406 00:24:42,690 --> 00:24:45,480 substitute this in, and divide on the left. 407 00:24:45,480 --> 00:24:54,650 So we have 1 over X of x times T of t times 408 00:24:54,650 --> 00:25:02,150 the second derivative with respect to x of xt 409 00:25:02,150 --> 00:25:10,740 is equal to 1 over xt times the second derivative with respect 410 00:25:10,740 --> 00:25:13,200 to t of xt. 411 00:25:16,360 --> 00:25:18,610 OK. 412 00:25:18,610 --> 00:25:24,190 Well, on this side of the equation, 413 00:25:24,190 --> 00:25:27,710 the only thing that involves time is here. 414 00:25:27,710 --> 00:25:29,440 This doesn't operate on time. 415 00:25:29,440 --> 00:25:35,560 And so we can cancel the time dependence from this side. 416 00:25:35,560 --> 00:25:40,490 And over on this side, this derivative 417 00:25:40,490 --> 00:25:42,620 operates on t but not x. 418 00:25:42,620 --> 00:25:45,890 And so we can cancel the x part. 419 00:25:45,890 --> 00:25:50,150 And so what we have now is an equation 420 00:25:50,150 --> 00:25:56,310 X of x second derivative with respect to x squared 421 00:25:56,310 --> 00:26:07,640 of x is equal to this constant, 1 over v squared, times 1 422 00:26:07,640 --> 00:26:15,630 over t times the derivative of T with respect to little t, OK? 423 00:26:15,630 --> 00:26:17,550 So this is interesting. 424 00:26:17,550 --> 00:26:20,600 We have a function of x on this side 425 00:26:20,600 --> 00:26:23,330 and a function of t on this side. 426 00:26:23,330 --> 00:26:25,510 They are independent variables. 427 00:26:25,510 --> 00:26:28,510 This can only be true if both sides 428 00:26:28,510 --> 00:26:29,965 are equal to the same constant. 429 00:26:34,970 --> 00:26:38,710 So now, we have to differential equations. 430 00:26:38,710 --> 00:26:42,940 We have 1 over x second derivative with respect 431 00:26:42,940 --> 00:26:47,570 to x of x is equal to a concept. 432 00:26:47,570 --> 00:26:54,400 And we have 1 over v squared 1 over t second derivative 433 00:26:54,400 --> 00:27:00,160 with respect to t of T is equal to K. 434 00:27:00,160 --> 00:27:03,290 So now, we're on firmer ground. 435 00:27:03,290 --> 00:27:05,835 We know about solutions to this kind of equation. 436 00:27:11,200 --> 00:27:13,320 And so there's two cases. 437 00:27:13,320 --> 00:27:15,800 One is this K is greater than 0. 438 00:27:15,800 --> 00:27:19,440 And the other is K less than 0. 439 00:27:19,440 --> 00:27:21,120 So let's look at this equation. 440 00:27:21,120 --> 00:27:28,750 If K is greater than 0, then if we plug in a sine or a cosine, 441 00:27:28,750 --> 00:27:31,330 we get something that's less than 0. 442 00:27:31,330 --> 00:27:35,260 Because the derivative of sine with respect to its variable 443 00:27:35,260 --> 00:27:37,240 is negative cosine. 444 00:27:37,240 --> 00:27:40,430 And then we do it again, we get back to sine. 445 00:27:40,430 --> 00:27:44,830 And so sines and cosines are no good for this equation 446 00:27:44,830 --> 00:27:48,090 if K is greater than 0. 447 00:27:48,090 --> 00:27:57,100 But exponentials-- so we can have e to some constant x 448 00:27:57,100 --> 00:28:00,730 or e to the minus some constant x. 449 00:28:00,730 --> 00:28:05,980 Or here, for the negative value of K, 450 00:28:05,980 --> 00:28:09,690 we could have sine some constant x 451 00:28:09,690 --> 00:28:12,600 and cosine of some constant x. 452 00:28:12,600 --> 00:28:15,176 We know that. 453 00:28:15,176 --> 00:28:16,050 So we have two cases. 454 00:28:19,480 --> 00:28:24,600 So to make life simple, we say K is 455 00:28:24,600 --> 00:28:27,450 going to be equal to lowercase k squared. 456 00:28:30,400 --> 00:28:35,230 Because we want to use this lowercase k in our solutions. 457 00:28:35,230 --> 00:28:35,980 All right. 458 00:28:35,980 --> 00:28:39,610 So I may have confused matters. 459 00:28:39,610 --> 00:28:47,830 But the solution for the time equation and the position 460 00:28:47,830 --> 00:28:50,680 equation are clear. 461 00:28:50,680 --> 00:28:53,980 And so depending on whether K is positive or negative, 462 00:28:53,980 --> 00:29:00,511 we're dealing with sines and cosines or exponentials. 463 00:29:00,511 --> 00:29:01,010 OK. 464 00:29:01,010 --> 00:29:04,760 So I don't want to belabor this, but the next stage 465 00:29:04,760 --> 00:29:06,444 is boundary conditions. 466 00:29:13,220 --> 00:29:17,740 We don't know whether K positive or K negative is possible. 467 00:29:20,350 --> 00:29:22,360 But we do know boundary conditions. 468 00:29:22,360 --> 00:29:26,300 And so if we have a string which is tied down at the end-- 469 00:29:26,300 --> 00:29:28,560 so this is x equals 0. 470 00:29:28,560 --> 00:29:33,030 And this is x equals L. Then we input 471 00:29:33,030 --> 00:29:36,016 impose the boundary conditions. 472 00:29:36,016 --> 00:29:43,750 Well, the boundary conditions are u of 0t is equal to 0. 473 00:29:43,750 --> 00:29:49,960 And u of Lt is equal to 0. 474 00:29:49,960 --> 00:30:00,930 So if we take the K greater than 0 case, u of 0t is equal to-- 475 00:30:11,110 --> 00:30:13,390 well, let's just do this again. 476 00:30:13,390 --> 00:30:16,120 0t. 477 00:30:16,120 --> 00:30:22,570 We have x of 0 times T of t. 478 00:30:22,570 --> 00:30:25,180 OK, we don't really care about this. 479 00:30:25,180 --> 00:30:31,830 But x of 0 has to be 0-- 480 00:30:31,830 --> 00:30:32,620 I'm sorry. 481 00:30:32,620 --> 00:30:33,120 OK. 482 00:30:33,120 --> 00:30:36,310 So we have two solutions. 483 00:30:36,310 --> 00:30:44,030 If K is greater than 0, we have the exponential terms. 484 00:30:44,030 --> 00:30:50,830 So we have Ae to the 0 plus B to the minus 0. 485 00:30:54,800 --> 00:30:56,180 And this has to be equal to 0. 486 00:30:59,630 --> 00:31:01,610 That's x of 0. 487 00:31:04,295 --> 00:31:06,200 Well, e to the 0 is 1. 488 00:31:06,200 --> 00:31:07,850 E to the minus 0 is 1. 489 00:31:07,850 --> 00:31:09,590 And so this is good. 490 00:31:09,590 --> 00:31:15,850 A has to be equal to minus B. And then we 491 00:31:15,850 --> 00:31:22,060 have the other boundary condition, X of L. 492 00:31:22,060 --> 00:31:35,190 We have A e to the L plus B e to the minus L. Well, I'm sorry. 493 00:31:35,190 --> 00:31:38,290 Let's just put what we already know. 494 00:31:38,290 --> 00:31:43,320 Minus A. So we can write this as A e to the L minus 495 00:31:43,320 --> 00:31:47,400 e to the minus L. And that has to be equal to 0. 496 00:31:47,400 --> 00:31:49,530 Can't do it. 497 00:31:49,530 --> 00:31:50,610 This can never be 0. 498 00:31:56,580 --> 00:32:00,170 So that means the K greater than 0 solutions are illegal. 499 00:32:03,950 --> 00:32:06,170 Well, that's kind of bad news. 500 00:32:06,170 --> 00:32:08,240 Because it sounds like separation of variables 501 00:32:08,240 --> 00:32:09,660 is failing. 502 00:32:09,660 --> 00:32:10,340 But it doesn't. 503 00:32:10,340 --> 00:32:13,115 Because the K less than 0 solution works. 504 00:32:17,060 --> 00:32:28,980 So for K less than 0, X of 0 is equal to C sine 0 plus D cosine 505 00:32:28,980 --> 00:32:31,070 0. 506 00:32:31,070 --> 00:32:34,790 And so that means D is equal to 0. 507 00:32:34,790 --> 00:32:36,710 Things are dying. 508 00:32:36,710 --> 00:32:56,200 And X of L, boundary condition, is C sine KL is equal to 0. 509 00:32:56,200 --> 00:32:59,000 And this we can solve. 510 00:32:59,000 --> 00:33:12,660 So sine is equal to 0 when KL is equal to 0, 0 pi, 2 pi, et 511 00:33:12,660 --> 00:33:13,160 cetera. 512 00:33:13,160 --> 00:33:17,850 And so we have KL is equal to n pi. 513 00:33:22,250 --> 00:33:26,180 So we can write this as Kn is equal to n 514 00:33:26,180 --> 00:33:30,320 pi over L. We get quantization. 515 00:33:30,320 --> 00:33:32,990 This isn't quantum mechanics. 516 00:33:32,990 --> 00:33:38,670 But there are certain allowed values of this K constant. 517 00:33:38,670 --> 00:33:41,130 And we have a bunch of solutions. 518 00:33:43,940 --> 00:33:45,450 And so what do they look like? 519 00:33:45,450 --> 00:33:55,920 So n equals 0, n equals 1, n equals 2. 520 00:33:55,920 --> 00:33:57,990 So what does the 0 solution look like? 521 00:34:04,930 --> 00:34:05,848 Yes? 522 00:34:05,848 --> 00:34:07,184 AUDIENCE: No node. 523 00:34:07,184 --> 00:34:08,100 ROBERT FIELD: Nothing. 524 00:34:11,380 --> 00:34:13,070 So we don't even think about this. 525 00:34:13,070 --> 00:34:15,650 We say, n equals 0 is not a solution. 526 00:34:15,650 --> 00:34:19,909 Because the wave isn't there. 527 00:34:19,909 --> 00:34:24,760 No nodes, one node. 528 00:34:24,760 --> 00:34:27,320 And if we look at this carefully, 529 00:34:27,320 --> 00:34:30,139 the node is always at a cemetery point. 530 00:34:30,139 --> 00:34:31,699 It's in the middle. 531 00:34:31,699 --> 00:34:35,270 If we have the next one, we'll have two nodes. 532 00:34:35,270 --> 00:34:38,360 And they'll be at the 1/3 2/3 point. 533 00:34:38,360 --> 00:34:41,960 And so we know where the nodes are. 534 00:34:41,960 --> 00:34:45,469 We also know that the amplitude of each loop of the wave 535 00:34:45,469 --> 00:34:46,460 function is the same. 536 00:34:46,460 --> 00:34:50,199 But it alternates in sine. 537 00:34:50,199 --> 00:34:53,500 So you can draw cartoons now at will. 538 00:34:53,500 --> 00:35:00,610 Because this spatial part of the solution to this wave is clear. 539 00:35:00,610 --> 00:35:04,060 Any value of the quantum number, or the n, 540 00:35:04,060 --> 00:35:08,690 gives you a picture that you can draw in seconds. 541 00:35:08,690 --> 00:35:13,540 And there are a lot of quantum mechanical problems like that. 542 00:35:13,540 --> 00:35:16,590 But sometimes, you have to keep in mind 543 00:35:16,590 --> 00:35:19,860 that the node separation, in other words-- 544 00:35:23,390 --> 00:35:31,580 well, let's just say node separation is lambda over 2. 545 00:35:31,580 --> 00:35:41,820 And lambda over 2 is equal to 1/2 h over p. 546 00:35:41,820 --> 00:35:47,310 So if we know what the momentum is, or we 547 00:35:47,310 --> 00:35:50,430 know what the kinetic energy is, we know what the momentum is. 548 00:35:50,430 --> 00:35:54,750 We know how momentum is encoded in node separations. 549 00:35:57,470 --> 00:36:00,590 So everything we want to know about a one-dimensional problem 550 00:36:00,590 --> 00:36:05,510 is expressed in the spacing of nodes 551 00:36:05,510 --> 00:36:08,060 and the amplitude between nodes. 552 00:36:08,060 --> 00:36:10,070 And the amplitude between nodes have something 553 00:36:10,070 --> 00:36:13,076 to do with the momentum, too. 554 00:36:13,076 --> 00:36:14,450 Because if you're going from here 555 00:36:14,450 --> 00:36:22,150 to here at some high velocity, there's not much amplitude. 556 00:36:22,150 --> 00:36:26,690 And at a lower velocity, you get more amplitude. 557 00:36:26,690 --> 00:36:31,250 And so the amplitude in each of these node 558 00:36:31,250 --> 00:36:33,800 to node separate sections is related 559 00:36:33,800 --> 00:36:37,280 to the average momentum of the classical particle 560 00:36:37,280 --> 00:36:40,070 in that section. 561 00:36:40,070 --> 00:36:41,860 So the classical mechanics is going 562 00:36:41,860 --> 00:36:46,510 to be extremely important in drawing cartoons 563 00:36:46,510 --> 00:36:49,180 for quantum mechanical systems. 564 00:36:49,180 --> 00:36:50,140 Not in the textbooks. 565 00:36:53,550 --> 00:36:56,400 We supposedly know classical mechanics pretty well, 566 00:36:56,400 --> 00:36:59,160 and especially here at MIT. 567 00:36:59,160 --> 00:37:01,140 So you might as well use that in order 568 00:37:01,140 --> 00:37:06,240 to get an idea of how all of the quantum mechanical problems 569 00:37:06,240 --> 00:37:09,790 you're facing will be behaving. 570 00:37:09,790 --> 00:37:10,870 OK. 571 00:37:10,870 --> 00:37:18,220 So the next thing we want to do is finish the job. 572 00:37:18,220 --> 00:37:23,110 And so I can simply write down the time-dependent solutions. 573 00:37:23,110 --> 00:37:40,280 They are E sine vkn t plus F cosine vkn t. 574 00:37:44,840 --> 00:37:47,120 And we can say that-- 575 00:37:47,120 --> 00:37:49,760 rather than carrying around all this stuff, 576 00:37:49,760 --> 00:37:53,750 we can say omega n is vkn. 577 00:38:00,540 --> 00:38:03,050 Isn't that nice? 578 00:38:03,050 --> 00:38:07,570 So we have a frequency for the time-dependent part, which 579 00:38:07,570 --> 00:38:11,950 is an integer multiple of this constant V times this 580 00:38:11,950 --> 00:38:17,570 [? vector. ?] Or this you can think of as just k sub n. 581 00:38:17,570 --> 00:38:22,250 So we can rewrite this in a frequency and phase form. 582 00:38:22,250 --> 00:38:26,000 We have now the full solution. 583 00:38:26,000 --> 00:38:30,760 We have A n sine n pi L over x. 584 00:38:35,210 --> 00:38:46,830 And then this e N sine n pi-- 585 00:38:56,322 --> 00:38:58,350 I'm so used to the pictures I don't even 586 00:38:58,350 --> 00:39:01,393 want to look at the equations anymore-- 587 00:39:01,393 --> 00:39:11,270 n omega t plus F n cosine n omega t. 588 00:39:11,270 --> 00:39:11,770 OK. 589 00:39:20,590 --> 00:39:23,980 So we can also take this and rewrite it 590 00:39:23,980 --> 00:39:36,160 in a simpler form, E n prime cosine n omega t plus phi n. 591 00:39:36,160 --> 00:39:39,700 So we can combine these two terms as a single cosine 592 00:39:39,700 --> 00:39:40,630 with a phase vector. 593 00:39:46,500 --> 00:39:47,360 OK. 594 00:39:47,360 --> 00:39:53,860 So now, we're ready to actually go to the specific thing 595 00:39:53,860 --> 00:39:59,930 that you do in a real experiment or a real musical instrument. 596 00:40:02,870 --> 00:40:07,400 We say, OK, here is the actual initial condition, 597 00:40:07,400 --> 00:40:10,880 the pluck of the system. 598 00:40:10,880 --> 00:40:15,950 And the pluck usually occurs at t equals 0. 599 00:40:15,950 --> 00:40:19,440 But I'll just specify it here. 600 00:40:19,440 --> 00:40:26,630 And what we have is now a sum over as many normal modes 601 00:40:26,630 --> 00:40:27,800 as you want. 602 00:40:27,800 --> 00:40:46,030 We have A n E n prime times sine n pi over L x times cosine N 603 00:40:46,030 --> 00:40:51,600 omega t plus phi N. 604 00:40:51,600 --> 00:40:56,160 So we have a bunch of terms like this, a spacial factor, 605 00:40:56,160 --> 00:40:58,424 and a temporal factor. 606 00:40:58,424 --> 00:40:59,840 And you can draw pictures of both. 607 00:41:02,590 --> 00:41:05,000 Now, but there is another simplification. 608 00:41:05,000 --> 00:41:15,020 From trigonometry, sine A cosine B-- we have sine and cosine-- 609 00:41:15,020 --> 00:41:28,730 can be written as 1/2 times sine n pi L x plus 610 00:41:28,730 --> 00:41:46,810 n omega t plus phi n plus sine n pi L x 611 00:41:46,810 --> 00:41:51,250 minus n omega t minus phi n. 612 00:41:51,250 --> 00:41:55,450 So these are the two possible solutions. 613 00:41:55,450 --> 00:41:58,690 And we can write them now in terms of position factor 614 00:41:58,690 --> 00:42:04,760 at a time factor in the same sine or cosine function. 615 00:42:07,290 --> 00:42:08,550 So these are the things. 616 00:42:08,550 --> 00:42:10,070 Now, we're ready to make a picture. 617 00:42:12,610 --> 00:42:13,530 OK. 618 00:42:13,530 --> 00:42:16,050 So these are the actual things that you 619 00:42:16,050 --> 00:42:22,010 make by exciting the system not in an eigenfunction. 620 00:42:22,010 --> 00:42:24,352 But it's a superposition of eigenfunctions. 621 00:42:27,250 --> 00:42:29,760 And again, there are certain things you learn. 622 00:42:33,030 --> 00:42:37,570 If you have a pure eigenfunction, 623 00:42:37,570 --> 00:42:39,560 you have standing waves. 624 00:42:39,560 --> 00:42:41,360 There's no left-right motion. 625 00:42:41,360 --> 00:42:42,650 There's no breathing motion. 626 00:42:42,650 --> 00:42:48,760 There's only up-down motion of each loop of the wave function. 627 00:42:48,760 --> 00:42:54,360 If you have a superposition of two or more functions, which 628 00:42:54,360 --> 00:43:02,950 are all of even n, then what happens is you have no motions, 629 00:43:02,950 --> 00:43:04,450 you just have breathing. 630 00:43:04,450 --> 00:43:06,220 In other words, you have a function 631 00:43:06,220 --> 00:43:08,910 that might look sort of like this at one time 632 00:43:08,910 --> 00:43:12,570 and like that at another time. 633 00:43:12,570 --> 00:43:14,750 So amplitude is moving. 634 00:43:14,750 --> 00:43:18,740 So it can be moving. 635 00:43:18,740 --> 00:43:22,530 And in between, it's sort of like this. 636 00:43:22,530 --> 00:43:24,500 Now, if you have a function which 637 00:43:24,500 --> 00:43:29,410 involves both even and odd n, you have left-right motion. 638 00:43:32,030 --> 00:43:34,400 This is true in quantum mechanics, too. 639 00:43:34,400 --> 00:43:40,040 So you only get motion if you're making superposition of eigen-- 640 00:43:40,040 --> 00:43:40,610 Yes? 641 00:43:40,610 --> 00:43:42,693 AUDIENCE: What is the difference between breathing 642 00:43:42,693 --> 00:43:46,580 and the standing wave with no nodes? 643 00:43:46,580 --> 00:43:50,930 ROBERT FIELD: Well, for this picture, this is over-simple. 644 00:43:50,930 --> 00:43:53,120 So I mean, you could have-- 645 00:43:56,360 --> 00:43:59,390 basically, what's happening is amplitude is moving from middle 646 00:43:59,390 --> 00:44:01,340 to the edges and back. 647 00:44:01,340 --> 00:44:05,430 And so yes. 648 00:44:05,430 --> 00:44:08,530 But you want to develop your own language, 649 00:44:08,530 --> 00:44:12,970 your own set of drawings so that you understand these things. 650 00:44:12,970 --> 00:44:16,740 And the important thing is the understanding, the ability 651 00:44:16,740 --> 00:44:19,800 to draw these pictures which contain 652 00:44:19,800 --> 00:44:24,330 the critical information about node spacings, amplitudes, 653 00:44:24,330 --> 00:44:28,290 shapes, and to anticipate when you're 654 00:44:28,290 --> 00:44:31,330 going to have left-right motion or when 655 00:44:31,330 --> 00:44:35,200 you're just going to have complicated up-down motions. 656 00:44:35,200 --> 00:44:36,810 Because there could be nodes. 657 00:44:36,810 --> 00:44:43,620 But there is no motion of the center of this wavepacket. 658 00:44:43,620 --> 00:44:46,500 Now, this is fantastic. 659 00:44:46,500 --> 00:44:51,170 Because I just said wavepacket. 660 00:44:51,170 --> 00:45:01,000 Quantum mechanics-- eigenfunctions don't move. 661 00:45:01,000 --> 00:45:04,420 Superpositions of eigenfunctions do move. 662 00:45:04,420 --> 00:45:07,040 If we make a superposition of many eigenfunctions, 663 00:45:07,040 --> 00:45:10,370 it is a particle-like state. 664 00:45:10,370 --> 00:45:12,060 What that particle-like state will do 665 00:45:12,060 --> 00:45:17,600 is exactly what you expect from 8.01. 666 00:45:17,600 --> 00:45:20,870 The particle-like states-- the center 667 00:45:20,870 --> 00:45:25,190 of the wavepacket for a particle-like state moves 668 00:45:25,190 --> 00:45:28,080 according to Newton's equations. 669 00:45:28,080 --> 00:45:30,710 So I'm saying I'm taking away your ability 670 00:45:30,710 --> 00:45:33,169 to look at microscopic stuff. 671 00:45:33,169 --> 00:45:34,710 And I'm going to give it back to you. 672 00:45:34,710 --> 00:45:36,590 By the end of the course, we're going 673 00:45:36,590 --> 00:45:39,300 to have the time-dependent Schrodinger equation. 674 00:45:39,300 --> 00:45:41,640 We're going to be able to see things move. 675 00:45:41,640 --> 00:45:43,560 And we're going to see why they move 676 00:45:43,560 --> 00:45:46,840 and how they encode that motion. 677 00:45:46,840 --> 00:45:49,041 Not in the textbooks, but I think 678 00:45:49,041 --> 00:45:51,040 it's something you really want to be able to do. 679 00:45:51,040 --> 00:45:53,560 If you're going to understand physical systems 680 00:45:53,560 --> 00:45:56,780 and use it to guide your understanding, 681 00:45:56,780 --> 00:45:59,140 you have to be able to draw these pictures 682 00:45:59,140 --> 00:46:01,130 and build a step at a time. 683 00:46:01,130 --> 00:46:05,110 And so this way the equation, the classical wave equation, 684 00:46:05,110 --> 00:46:09,040 gives you almost all of the tools for artistry 685 00:46:09,040 --> 00:46:10,040 as well as insight. 686 00:46:13,230 --> 00:46:15,200 OK. 687 00:46:15,200 --> 00:46:19,610 Now, in the notes, there is a time-lapse movie that shows 688 00:46:19,610 --> 00:46:23,972 what a two-state wave function-- 689 00:46:23,972 --> 00:46:27,290 what a two-state solution to the wave equation looks 690 00:46:27,290 --> 00:46:33,360 like if you have even and odd or only even and even terms. 691 00:46:33,360 --> 00:46:33,860 OK. 692 00:46:33,860 --> 00:46:38,740 Now, I'm going to make some assertions at the end. 693 00:46:38,740 --> 00:46:40,470 We're coming back to the drum problem. 694 00:46:43,000 --> 00:46:45,690 And suppose we have a rectangular drum. 695 00:46:48,970 --> 00:46:51,760 Well, solving the differential equation 696 00:46:51,760 --> 00:46:54,190 for this rectangular drum gives you 697 00:46:54,190 --> 00:46:56,440 a bunch of normal mode frequencies 698 00:46:56,440 --> 00:46:57,940 that depend on two indices. 699 00:47:15,871 --> 00:47:21,900 And so this is the geometric structure of the drum. 700 00:47:21,900 --> 00:47:24,410 And these are the quantum numbers. 701 00:47:24,410 --> 00:47:27,557 And these are the frequencies. 702 00:47:27,557 --> 00:47:29,890 And it's going to make a whole bunch of frequencies that 703 00:47:29,890 --> 00:47:31,810 are not integer multiples of each other 704 00:47:31,810 --> 00:47:34,734 or of any simpler thing. 705 00:47:34,734 --> 00:47:36,150 And that's why it sounds horrible. 706 00:47:39,980 --> 00:47:44,510 It's perhaps a little bit like playing a violin with a saw. 707 00:47:44,510 --> 00:47:47,200 It will sound terrible. 708 00:47:47,200 --> 00:47:48,700 You would never do it. 709 00:47:48,700 --> 00:47:52,490 But you would also never build a square or rectangular drum. 710 00:47:52,490 --> 00:47:57,910 But the noise that you make tells immediately not 711 00:47:57,910 --> 00:48:02,440 just what the shape of the instrument is 712 00:48:02,440 --> 00:48:03,940 but how it was played. 713 00:48:03,940 --> 00:48:09,320 For example, suppose you had an elliptical drum. 714 00:48:09,320 --> 00:48:10,940 That'll sound terrible, too. 715 00:48:10,940 --> 00:48:13,070 But here are the two foci. 716 00:48:13,070 --> 00:48:14,780 I'm not so sure it'll sound terrible 717 00:48:14,780 --> 00:48:16,130 if you hit it here or here. 718 00:48:18,650 --> 00:48:21,720 And certainly, if you are a circular drum, 719 00:48:21,720 --> 00:48:24,470 if you hit it in the middle as opposed to on the edges, 720 00:48:24,470 --> 00:48:26,670 it'll sound different. 721 00:48:26,670 --> 00:48:29,110 The spectral content will be the same. 722 00:48:29,110 --> 00:48:32,830 But the amplitudes of each component will be different. 723 00:48:32,830 --> 00:48:38,370 And so in quantum mechanics, you use the same sort of instinct 724 00:48:38,370 --> 00:48:41,070 as you develop as a musician in order 725 00:48:41,070 --> 00:48:45,135 to figure out how this system is going to respond 726 00:48:45,135 --> 00:48:46,010 to what you do to it. 727 00:48:48,411 --> 00:48:49,535 And that's pretty powerful. 728 00:48:52,320 --> 00:48:54,540 So many of you are musicians. 729 00:48:54,540 --> 00:48:57,640 And you know instinctively what's 730 00:48:57,640 --> 00:49:01,450 wrong when you do something that's not quite right, 731 00:49:01,450 --> 00:49:04,600 or your instrument is out of tune. 732 00:49:04,600 --> 00:49:07,690 But in quantum mechanics, all of those insights 733 00:49:07,690 --> 00:49:09,880 will come to bear. 734 00:49:09,880 --> 00:49:13,240 Not in the textbooks. 735 00:49:13,240 --> 00:49:17,140 Because the textbooks tell you about exactly solved problems. 736 00:49:17,140 --> 00:49:20,050 And then they tell you how to do spectra 737 00:49:20,050 --> 00:49:23,940 that are too perfect for anybody else to observe. 738 00:49:23,940 --> 00:49:26,490 And you won't see those spectra. 739 00:49:26,490 --> 00:49:29,970 And they don't tell you what the spectra you will observe 740 00:49:29,970 --> 00:49:32,610 tell you about the system in question. 741 00:49:32,610 --> 00:49:33,300 OK. 742 00:49:33,300 --> 00:49:35,070 So I should stop now. 743 00:49:35,070 --> 00:49:41,550 And I'm going to be generating Problem Set 2, which 744 00:49:41,550 --> 00:49:43,230 will be posted on Friday. 745 00:49:43,230 --> 00:49:46,130 Problem Set 1 is due this Friday. 746 00:49:46,130 --> 00:49:49,080 And-- Good. 747 00:49:49,080 --> 00:49:50,870 Thank you.