1 00:00:01,620 --> 00:00:03,960 The following content is provided under a Creative 2 00:00:03,960 --> 00:00:05,380 Commons license. 3 00:00:05,380 --> 00:00:07,590 Your support will help MIT OpenCourseWare 4 00:00:07,590 --> 00:00:11,680 continue to offer high-quality educational resources for free. 5 00:00:11,680 --> 00:00:14,220 To make a donation or to view additional materials 6 00:00:14,220 --> 00:00:18,180 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,180 --> 00:00:19,050 at ocw.mit.edu. 8 00:00:24,120 --> 00:00:26,520 YEN-JIE LEE: So welcome back, everybody. 9 00:00:26,520 --> 00:00:29,340 This is the final exam checklist. 10 00:00:31,910 --> 00:00:34,560 For the single oscillator, we need 11 00:00:34,560 --> 00:00:37,830 to make sure that you know how to write down 12 00:00:37,830 --> 00:00:40,290 the Equation of Motion. 13 00:00:40,290 --> 00:00:44,760 We have discussed about damped, under-damped, critically 14 00:00:44,760 --> 00:00:45,870 damped, and over-damped. 15 00:00:45,870 --> 00:00:48,480 We did that. 16 00:00:48,480 --> 00:00:53,980 Oscillators, and we have tried to drive oscillators. 17 00:00:53,980 --> 00:00:58,090 We observed transient behavior in steady state solution. 18 00:00:58,090 --> 00:01:01,680 Resonance, right, so which we actually demonstrated that 19 00:01:01,680 --> 00:01:05,250 by breaking the glass. 20 00:01:05,250 --> 00:01:09,000 And then we moved on and tried to couple multiple objects 21 00:01:09,000 --> 00:01:10,000 together. 22 00:01:10,000 --> 00:01:13,020 And that brings us to the coupled system. 23 00:01:13,020 --> 00:01:14,850 What are the normal modes? 24 00:01:14,850 --> 00:01:17,470 And how to actually solve M minus 1 K 25 00:01:17,470 --> 00:01:21,600 matrix, the eigenvalue problem. 26 00:01:21,600 --> 00:01:25,970 What is actually the full solution for the description 27 00:01:25,970 --> 00:01:27,930 of coupled systems. 28 00:01:27,930 --> 00:01:32,840 And can we actually drive the coupled system, 29 00:01:32,840 --> 00:01:35,470 and we found out we can. 30 00:01:35,470 --> 00:01:38,490 So the system would respond as well 31 00:01:38,490 --> 00:01:42,140 similar to what we have seen in the single oscillator case. 32 00:01:42,140 --> 00:01:44,220 We see resonance as well. 33 00:01:44,220 --> 00:01:47,160 We can excite one of the normal modes 34 00:01:47,160 --> 00:01:50,640 by driving the coupled system. 35 00:01:50,640 --> 00:01:56,200 Then we put more and more objects until at some point, 36 00:01:56,200 --> 00:01:59,980 we have infinite number of coupled objects. 37 00:01:59,980 --> 00:02:03,120 What is actually the solution of refraction 38 00:02:03,120 --> 00:02:05,070 and the transmission-- 39 00:02:05,070 --> 00:02:08,789 refraction and the translation symmetric system. 40 00:02:08,789 --> 00:02:12,310 That is actually the discussion of symmetry. 41 00:02:12,310 --> 00:02:15,630 We go to the continuum then, and we actually 42 00:02:15,630 --> 00:02:18,360 found wave and wave equations. 43 00:02:18,360 --> 00:02:21,200 So we found that finally, we made the phase transition 44 00:02:21,200 --> 00:02:25,720 from single object vibration to waves, 45 00:02:25,720 --> 00:02:27,590 and that is actually an achievement 46 00:02:27,590 --> 00:02:30,250 we have done in 8.03. 47 00:02:30,250 --> 00:02:33,480 We have discussed about different systems, 48 00:02:33,480 --> 00:02:37,780 massive string, massive spring, sound wave, 49 00:02:37,780 --> 00:02:40,680 electromagnetic waves, and we have 50 00:02:40,680 --> 00:02:46,080 discussed a progressive wave and standing waves. 51 00:02:46,080 --> 00:02:50,940 For the bound system, we have also normal modes. 52 00:02:50,940 --> 00:02:53,470 We discussed about how to actually do 53 00:02:53,470 --> 00:02:56,120 Fourier decomposition, and what is actually 54 00:02:56,120 --> 00:03:03,460 the physical meaning of Fourier decomposition in 8.03. 55 00:03:03,460 --> 00:03:09,820 For the infinite system, we also learned about Fourier transform 56 00:03:09,820 --> 00:03:12,540 and uncertainty principles. 57 00:03:12,540 --> 00:03:17,280 And we learned to apply boundary conditions 58 00:03:17,280 --> 00:03:22,060 so that we constrain the possible wavelengths 59 00:03:22,060 --> 00:03:23,225 of the normal modes. 60 00:03:25,770 --> 00:03:28,020 Therefore, we also learned about how 61 00:03:28,020 --> 00:03:32,110 to put a system all together. 62 00:03:32,110 --> 00:03:35,760 Finally, how to determine the dispersion relation, 63 00:03:35,760 --> 00:03:39,030 which is omega as a function of K, the wave number. 64 00:03:42,260 --> 00:03:46,310 Until now, we discussed idealized systems, 65 00:03:46,310 --> 00:03:51,530 and we also moved on to discuss dispersive medium. 66 00:03:51,530 --> 00:03:55,100 We have learned some more, even more about dispersion 67 00:03:55,100 --> 00:03:59,780 relation for the dispersive medium and signal transmission, 68 00:03:59,780 --> 00:04:06,310 how to send signal through a highly dispersive medium. 69 00:04:06,310 --> 00:04:08,090 The solution we were proposing is 70 00:04:08,090 --> 00:04:15,440 to use an amplitude modulation radio and also 71 00:04:15,440 --> 00:04:18,690 the pattern of dispersion. 72 00:04:18,690 --> 00:04:24,050 The group velocity and phase velocity, we covered that. 73 00:04:24,050 --> 00:04:27,436 As I mentioned before, the uncertainty principle. 74 00:04:27,436 --> 00:04:30,410 A 2D/3D system. 75 00:04:30,410 --> 00:04:33,290 We have bound system, which we have 76 00:04:33,290 --> 00:04:36,770 normal modes for two-dimensional and three-dimensional systems, 77 00:04:36,770 --> 00:04:37,700 as well. 78 00:04:37,700 --> 00:04:39,770 Because we're all over the place, 79 00:04:39,770 --> 00:04:43,140 so just make sure that you know how to actually dewrap 80 00:04:43,140 --> 00:04:46,940 all those standing waves for different dimensions 81 00:04:46,940 --> 00:04:48,770 of systems. 82 00:04:48,770 --> 00:04:53,090 We showed and approved geometrical optics, 83 00:04:53,090 --> 00:04:57,170 which essentially is the direct consequence of waves. 84 00:04:57,170 --> 00:05:00,050 Wave function, a continuation of the wave function and boundary 85 00:05:00,050 --> 00:05:01,970 condition. 86 00:05:01,970 --> 00:05:06,380 We learned about the refraction rule and also Snell's law. 87 00:05:06,380 --> 00:05:11,260 We talked about polarized waves, linear, circularly polarized, 88 00:05:11,260 --> 00:05:14,220 elliptically polarized, and the polarizer 89 00:05:14,220 --> 00:05:17,420 and quarter-wave plate. 90 00:05:17,420 --> 00:05:20,750 At the end of the discussion of 2D/3D systems, 91 00:05:20,750 --> 00:05:26,120 we discussed about how to generate electromagnetic waves 92 00:05:26,120 --> 00:05:27,530 by accelerated charge. 93 00:05:30,260 --> 00:05:35,030 Finally, we went on and talked about how those EM 94 00:05:35,030 --> 00:05:40,250 waves propagate in dielectrics and again, boundary conditions, 95 00:05:40,250 --> 00:05:45,020 which leads to interesting phenomena, which belongs only 96 00:05:45,020 --> 00:05:47,370 to electromagnetic waves. 97 00:05:47,370 --> 00:05:51,100 For example, Brewster's angle. 98 00:05:51,100 --> 00:05:54,320 So the refraction amplitude-- 99 00:05:54,320 --> 00:05:56,510 refraction-- the wave amplitude is 100 00:05:56,510 --> 00:06:01,190 governed by the property of the electromagnetic waves, which 101 00:06:01,190 --> 00:06:05,900 is coming from the laws which governs 102 00:06:05,900 --> 00:06:10,170 electromagnetic waves, which is Maxwell's equations in matter. 103 00:06:10,170 --> 00:06:13,190 We were trying to also manipulate those waves 104 00:06:13,190 --> 00:06:16,130 by adding them together, and we see 105 00:06:16,130 --> 00:06:21,320 constructive and destructive interference and diffraction 106 00:06:21,320 --> 00:06:23,510 phenomena. 107 00:06:23,510 --> 00:06:27,290 Then we connect that to quantum mechanics 108 00:06:27,290 --> 00:06:32,990 by showing you a single electron interference experiment. 109 00:06:32,990 --> 00:06:35,900 That connects us to the beginning 110 00:06:35,900 --> 00:06:41,780 of the quantum mechanics, which is the probability waves, which 111 00:06:41,780 --> 00:06:45,980 behave very different from other waves we have been discussing. 112 00:06:45,980 --> 00:06:48,770 But you are going to learn a lot more in 8.04. 113 00:06:48,770 --> 00:06:52,381 OK, so don't worry. 114 00:06:52,381 --> 00:06:52,880 All right. 115 00:06:52,880 --> 00:06:55,920 So that is the checklist. 116 00:06:55,920 --> 00:06:58,640 You can see that I can write it in two pages, 117 00:06:58,640 --> 00:07:01,790 so it's not that bad, probably. 118 00:07:01,790 --> 00:07:05,930 I hope that there was nothing really sounds like new to you 119 00:07:05,930 --> 00:07:08,420 by now. 120 00:07:08,420 --> 00:07:12,890 If you find anything is new, you have to review that part. 121 00:07:12,890 --> 00:07:17,030 That means you missed a class. 122 00:07:17,030 --> 00:07:17,660 All right. 123 00:07:17,660 --> 00:07:21,920 So what I'm going to do now is to go through all the material 124 00:07:21,920 --> 00:07:24,140 faster than the speed of light. 125 00:07:24,140 --> 00:07:24,710 All right. 126 00:07:24,710 --> 00:07:26,220 So that you will get nauseated. 127 00:07:26,220 --> 00:07:30,560 No, you are going to get a list of the topics. 128 00:07:30,560 --> 00:07:33,020 You just have to feel it. 129 00:07:33,020 --> 00:07:37,780 If you feel good, like when you are having a cupcake, 130 00:07:37,780 --> 00:07:41,810 right, then you are good for the final. 131 00:07:41,810 --> 00:07:45,740 If you don't feel good, what is Professor Lee talking about? 132 00:07:45,740 --> 00:07:47,570 He's talking about nonsense now. 133 00:07:47,570 --> 00:07:51,290 Then you are in trouble, and you have to review that part. 134 00:07:51,290 --> 00:07:51,991 All right? 135 00:07:51,991 --> 00:07:52,490 OK. 136 00:07:52,490 --> 00:07:54,470 So that's what we'll do. 137 00:07:54,470 --> 00:07:55,810 So let's start. 138 00:07:55,810 --> 00:07:56,840 All right. 139 00:07:56,840 --> 00:07:57,980 Why 8.03? 140 00:07:57,980 --> 00:08:00,350 We started a discussion-- 141 00:08:00,350 --> 00:08:06,320 welcome, We started a discussion of 8.03 142 00:08:06,320 --> 00:08:10,200 and it's vibrations and wave systems, is name of this, 8.03. 143 00:08:10,200 --> 00:08:13,670 And the motivation is really simple, 144 00:08:13,670 --> 00:08:17,720 because we cannot even recognize the universe without using 145 00:08:17,720 --> 00:08:19,430 waves and vibration. 146 00:08:19,430 --> 00:08:22,490 You cannot see me, and you cannot hear anything, 147 00:08:22,490 --> 00:08:25,230 and you cannot feel the vibrations-- 148 00:08:25,230 --> 00:08:28,520 sorry, the rotation of a black hole by your body 149 00:08:28,520 --> 00:08:31,460 anymore, then it's not very cool. 150 00:08:31,460 --> 00:08:37,280 Therefore, we study 8.03 to understand the basic ideas 151 00:08:37,280 --> 00:08:39,110 about waves and vibrations. 152 00:08:39,110 --> 00:08:43,490 And we found that waves and vibrations 153 00:08:43,490 --> 00:08:45,590 are interesting phenomena. 154 00:08:45,590 --> 00:08:48,170 Waves are connected to vibrations. 155 00:08:48,170 --> 00:08:50,960 Because if you look at only, for example, 156 00:08:50,960 --> 00:08:55,130 a single object on these waves, you 157 00:08:55,130 --> 00:08:58,880 see that it is actually a single object which is oscillating 158 00:08:58,880 --> 00:09:01,500 up and down, oscillating up and down, 159 00:09:01,500 --> 00:09:02,870 and this is your vibration. 160 00:09:02,870 --> 00:09:06,710 So there's a close connection between single particle 161 00:09:06,710 --> 00:09:08,920 vibration and the waves. 162 00:09:08,920 --> 00:09:11,420 And that is the first thing that you learned. 163 00:09:11,420 --> 00:09:13,865 Therefore, we need to first understand the evolution 164 00:09:13,865 --> 00:09:17,460 of a single particle system. 165 00:09:17,460 --> 00:09:20,370 And we make use of this opportunity 166 00:09:20,370 --> 00:09:24,870 to start the discussion of scientific matter. 167 00:09:24,870 --> 00:09:27,830 So using this opportunity, basically, 168 00:09:27,830 --> 00:09:31,840 what we have been doing for the whole class is the following. 169 00:09:31,840 --> 00:09:35,490 So the first step is always to translate 170 00:09:35,490 --> 00:09:37,800 the physical situation which we are 171 00:09:37,800 --> 00:09:41,580 interested into mathematics, right? 172 00:09:41,580 --> 00:09:46,470 Because mathematics is the only language which we know which 173 00:09:46,470 --> 00:09:49,350 describes the nature. 174 00:09:49,350 --> 00:09:53,550 If you come out with a new language, 175 00:09:53,550 --> 00:09:59,470 and that is going to be a super duper breakthrough, 176 00:09:59,470 --> 00:10:02,360 it cannot be estimated by Nobel Prize. 177 00:10:02,360 --> 00:10:05,400 But the problem we are facing now 178 00:10:05,400 --> 00:10:09,990 is that this is the only language we know which works. 179 00:10:09,990 --> 00:10:15,550 Therefore, we really follow this recipe, 180 00:10:15,550 --> 00:10:18,900 which is similar to many, many other physics classes. 181 00:10:18,900 --> 00:10:21,090 And we have a physical situation, 182 00:10:21,090 --> 00:10:25,800 we use laws of nature or models, and we 183 00:10:25,800 --> 00:10:28,380 have a mathematical description, which 184 00:10:28,380 --> 00:10:30,307 is the Equation of Motion. 185 00:10:30,307 --> 00:10:31,890 And this is actually the hardest part, 186 00:10:31,890 --> 00:10:36,510 because you need to first define a coordinate system so that we 187 00:10:36,510 --> 00:10:39,990 can express everything in a system in that system, 188 00:10:39,990 --> 00:10:43,570 then you can make use of the physical laws you have learned 189 00:10:43,570 --> 00:10:47,100 from the previous 8.01 and 8.02 to write down 190 00:10:47,100 --> 00:10:48,330 the Equation of Motion. 191 00:10:48,330 --> 00:10:52,010 And most of the mistakes, and also most of the problems 192 00:10:52,010 --> 00:10:57,330 or difficulties you are facing is always in this step. 193 00:10:57,330 --> 00:11:01,230 Then we can solve the Equation of Motion, which is, strictly 194 00:11:01,230 --> 00:11:02,980 speaking, not my problem. 195 00:11:02,980 --> 00:11:06,660 It's the math department's problem. 196 00:11:06,660 --> 00:11:08,610 Yeah, that's their problem. 197 00:11:08,610 --> 00:11:10,980 Then we solve the Equation of Motion, 198 00:11:10,980 --> 00:11:13,350 and you will be given the formula. 199 00:11:13,350 --> 00:11:17,340 Then we use initial conditions and then make predictions. 200 00:11:17,340 --> 00:11:19,020 And then we would like to compare that 201 00:11:19,020 --> 00:11:20,680 to experimental results. 202 00:11:20,680 --> 00:11:23,310 And that is the general thing which 203 00:11:23,310 --> 00:11:25,680 we have been doing for physics. 204 00:11:25,680 --> 00:11:29,250 So let's take a look at those examples. 205 00:11:29,250 --> 00:11:33,840 Those are examples of simple harmonics motions. 206 00:11:33,840 --> 00:11:36,510 And you can see that these, all these systems 207 00:11:36,510 --> 00:11:40,860 have one object, which is oscillating. 208 00:11:40,860 --> 00:11:45,330 And you can see that their Equations of Motion 209 00:11:45,330 --> 00:11:47,480 are really similar to each other. 210 00:11:47,480 --> 00:11:51,570 It's theta double dot plus omega, zero squared, theta 211 00:11:51,570 --> 00:11:54,840 equal to zero for those idealized simple harmonic 212 00:11:54,840 --> 00:11:55,860 motions. 213 00:11:55,860 --> 00:12:00,710 And we learned that the solution of those equations 214 00:12:00,710 --> 00:12:07,250 are the same, which is a cosine function. 215 00:12:07,250 --> 00:12:13,430 Then we went ahead and added more craziness to the system. 216 00:12:13,430 --> 00:12:16,220 So basically, what we tried to do 217 00:12:16,220 --> 00:12:20,330 is to add a drag force into the game. 218 00:12:20,330 --> 00:12:27,560 And we were wondering if this more realistic description can 219 00:12:27,560 --> 00:12:31,500 match with experimental data. 220 00:12:31,500 --> 00:12:33,050 So this is the Equation of Motion, 221 00:12:33,050 --> 00:12:36,370 and the additional turn is the one in the middle, 222 00:12:36,370 --> 00:12:38,750 the gamma theta dot turn. 223 00:12:38,750 --> 00:12:42,950 And after entering these turns, not only is this 224 00:12:42,950 --> 00:12:46,340 an interesting model to describe the physical system we 225 00:12:46,340 --> 00:12:49,760 are talking about, but the mathematical solution 226 00:12:49,760 --> 00:12:53,870 is far more richer than what we talked about 227 00:12:53,870 --> 00:12:57,730 in the single harmonic oscillator case. 228 00:12:57,730 --> 00:13:01,790 Basically, you see that a general solution depends 229 00:13:01,790 --> 00:13:06,770 on the size of the gamma compared to omega dot zero, 230 00:13:06,770 --> 00:13:08,870 which is the oscillation-- 231 00:13:08,870 --> 00:13:11,410 the natural frequency of the system. 232 00:13:11,410 --> 00:13:11,930 OK. 233 00:13:11,930 --> 00:13:16,570 And then you can see there are three distinct different kinds 234 00:13:16,570 --> 00:13:17,870 of solutions. 235 00:13:17,870 --> 00:13:20,690 They have different mathematical forms. 236 00:13:20,690 --> 00:13:25,010 And we call them under damped solution, 237 00:13:25,010 --> 00:13:29,810 critically damped solution, and over damped solution. 238 00:13:29,810 --> 00:13:32,950 So those equations will be given to you. 239 00:13:32,950 --> 00:13:36,320 And the excitement is the following. 240 00:13:36,320 --> 00:13:39,210 So you can see that those solutions, 241 00:13:39,210 --> 00:13:43,100 if you plot the solution as a function of time, 242 00:13:43,100 --> 00:13:47,450 they look completely different as a function of time. 243 00:13:47,450 --> 00:13:50,760 So in the case of no damping, the amplitude is actually 244 00:13:50,760 --> 00:13:55,400 the constant, it's not actually reducing as a function of time. 245 00:13:55,400 --> 00:13:58,635 But when the damped system, the damping 246 00:13:58,635 --> 00:14:01,970 is turned on, then in the under damped situation, 247 00:14:01,970 --> 00:14:05,966 you can see that they end up reducing as a function of time. 248 00:14:05,966 --> 00:14:08,540 And if you have too much damping, 249 00:14:08,540 --> 00:14:14,310 you put the whole oscillator into some liquid, for example, 250 00:14:14,310 --> 00:14:19,580 and you see that oscillator disappear. 251 00:14:19,580 --> 00:14:21,550 The cool thing is the following. 252 00:14:21,550 --> 00:14:24,540 The excitement from-- as a physicist 253 00:14:24,540 --> 00:14:27,740 is that all of those crazy mathematical solutions 254 00:14:27,740 --> 00:14:32,330 actually match with experimental results. 255 00:14:32,330 --> 00:14:33,080 Wow. 256 00:14:33,080 --> 00:14:34,810 That is really cool. 257 00:14:34,810 --> 00:14:39,000 Because there is nobody saying that these should match 258 00:14:39,000 --> 00:14:42,980 and how, naturally, I should learn that OK, 259 00:14:42,980 --> 00:14:46,730 when should I change the behavior of the system. 260 00:14:46,730 --> 00:14:51,590 So this is really a miracle that this complicated mathematical 261 00:14:51,590 --> 00:14:56,390 description is useful and that it is super useful 262 00:14:56,390 --> 00:15:00,740 to describe the nature. 263 00:15:00,740 --> 00:15:06,950 Once we have learned that, we can now add a driving force 264 00:15:06,950 --> 00:15:09,260 into ligand. 265 00:15:09,260 --> 00:15:11,570 From the equation here, we can see 266 00:15:11,570 --> 00:15:15,680 that there is a natural frequency, omega dot zero, 267 00:15:15,680 --> 00:15:19,190 of this system, and there is a drag force turn, which 268 00:15:19,190 --> 00:15:23,150 is actually to quantify how much drag we have, 269 00:15:23,150 --> 00:15:24,320 we have a gamma there. 270 00:15:27,170 --> 00:15:32,150 We are driving it at a driving frequency omega t. 271 00:15:32,150 --> 00:15:33,620 So what we have learned from here 272 00:15:33,620 --> 00:15:38,690 is that if you are driving this system, you are-- for example, 273 00:15:38,690 --> 00:15:43,820 I am shaking that student, shaking you. 274 00:15:43,820 --> 00:15:47,510 OK, in the beginning, this student is going to resist. 275 00:15:47,510 --> 00:15:48,830 No, don't shake me. 276 00:15:48,830 --> 00:15:49,940 Come on. 277 00:15:49,940 --> 00:15:54,000 But at some point, he knows that Professor Lee 278 00:15:54,000 --> 00:15:57,090 is really determined. 279 00:15:57,090 --> 00:16:03,420 Therefore, he is going to be shaked at the frequency I like. 280 00:16:03,420 --> 00:16:03,920 OK. 281 00:16:03,920 --> 00:16:07,560 So that is actually what is happening here. 282 00:16:07,560 --> 00:16:11,480 This is so-called transient behavior. 283 00:16:11,480 --> 00:16:15,800 So in the beginning, the system doesn't like it. 284 00:16:15,800 --> 00:16:19,790 So this is making use of the superposition principle. 285 00:16:19,790 --> 00:16:24,210 So you can solve that homogeneous solution, which 286 00:16:24,210 --> 00:16:26,750 is on the right-hand side. 287 00:16:26,750 --> 00:16:29,094 It depends on the physical situation 288 00:16:29,094 --> 00:16:30,010 you are talking about. 289 00:16:30,010 --> 00:16:33,230 You choose the corresponding homogeneous solution. 290 00:16:33,230 --> 00:16:38,270 And lamba and psi is the driving force from E and G, right, 291 00:16:38,270 --> 00:16:44,060 and that is going to win at the end of the experiment, 292 00:16:44,060 --> 00:16:46,280 because I'm going to shake it forever, 293 00:16:46,280 --> 00:16:48,560 until the end of the universe. 294 00:16:48,560 --> 00:16:53,140 So you can see that at the end, you-- what is left over 295 00:16:53,140 --> 00:16:57,600 is really the steady state solution. 296 00:16:57,600 --> 00:17:02,990 And it has this structure, A omega d, depends on omega. 297 00:17:02,990 --> 00:17:05,280 And you get resonance behavior. 298 00:17:05,280 --> 00:17:07,690 Don't forget to review that. 299 00:17:07,690 --> 00:17:13,160 So you have a delay in phase because when 300 00:17:13,160 --> 00:17:17,099 I shake the student, the student needs some time to respond. 301 00:17:17,099 --> 00:17:24,420 Therefore, the delta is non-zero if the student is damped. 302 00:17:24,420 --> 00:17:25,349 All right. 303 00:17:25,349 --> 00:17:29,070 So now we have learned all the secrets about a single object 304 00:17:29,070 --> 00:17:30,060 system. 305 00:17:30,060 --> 00:17:34,470 Then we can now go ahead and study coupled oscillators. 306 00:17:34,470 --> 00:17:37,140 There are a few examples here, which 307 00:17:37,140 --> 00:17:41,520 is coupled pendulums or coupled spring-mass systems. 308 00:17:41,520 --> 00:17:45,720 And we found that a very useful description 309 00:17:45,720 --> 00:17:48,030 of this kind of system is to make 310 00:17:48,030 --> 00:17:52,290 use of the matrix language. 311 00:17:52,290 --> 00:17:56,940 So originally, if you have n objects in a system, 312 00:17:56,940 --> 00:18:02,840 you have n Equations of Motion, and that looks horrible. 313 00:18:02,840 --> 00:18:09,740 But what is done in 8.03 is that we introduce a notation 314 00:18:09,740 --> 00:18:11,280 with a matrix. 315 00:18:11,280 --> 00:18:13,830 Basically, if you write everything in terms of matrix, 316 00:18:13,830 --> 00:18:18,210 then it looks really friendly, and it looks really 317 00:18:18,210 --> 00:18:21,330 like a single oscillator. 318 00:18:21,330 --> 00:18:22,050 OK? 319 00:18:22,050 --> 00:18:25,860 Although solving this equation is still 320 00:18:25,860 --> 00:18:29,190 a little bit more work. 321 00:18:29,190 --> 00:18:32,140 And basically, you can see that from this example, 322 00:18:32,140 --> 00:18:35,880 we can actually derive M minus 1 K matrix, 323 00:18:35,880 --> 00:18:39,300 and the whole equation won't be-- 324 00:18:39,300 --> 00:18:43,110 the Equation of Motion problem solving problem 325 00:18:43,110 --> 00:18:47,480 becomes an M minus 1 K matrix eigenvalue problem. 326 00:18:47,480 --> 00:18:50,740 What is an M minus 1 K matrix? 327 00:18:50,740 --> 00:18:54,390 This is describing how each component in the system 328 00:18:54,390 --> 00:18:57,610 interacts with each other. 329 00:18:57,610 --> 00:19:02,610 Once we have this, we can solve the eigenvalue problem, 330 00:19:02,610 --> 00:19:05,250 and we are going to be able to figure out 331 00:19:05,250 --> 00:19:08,620 the normal modes of those systems. 332 00:19:08,620 --> 00:19:11,140 So what is a normal mode? 333 00:19:11,140 --> 00:19:14,950 Normal modes is a situation where 334 00:19:14,950 --> 00:19:19,330 all the components in the system are oscillating 335 00:19:19,330 --> 00:19:23,830 at the same frequency and they are also at the same phase. 336 00:19:23,830 --> 00:19:26,320 So that is the definition of normal mode. 337 00:19:26,320 --> 00:19:31,710 And those are what is used in a deviation, 338 00:19:31,710 --> 00:19:34,960 also, which leads us to the eigenvalue problem. 339 00:19:34,960 --> 00:19:39,310 We define Z equal to X 1 H or I omega t plus 5. 340 00:19:39,310 --> 00:19:43,440 Everybody is oscillating at omega and also at phase 5, 341 00:19:43,440 --> 00:19:43,960 right? 342 00:19:43,960 --> 00:19:46,450 So that is what we actually learned. 343 00:19:46,450 --> 00:19:48,760 And what is actually the physical meaning 344 00:19:48,760 --> 00:19:50,980 of those normal modes? 345 00:19:50,980 --> 00:19:55,030 So if we plot the locus of the two coupled 346 00:19:55,030 --> 00:19:58,490 pendulum problem, what we see is the following. 347 00:19:58,490 --> 00:20:02,590 So basically, you will see that the locus looks 348 00:20:02,590 --> 00:20:06,100 like really complicated as a function of time 349 00:20:06,100 --> 00:20:10,630 if you plot X1 X2 versus time. 350 00:20:10,630 --> 00:20:14,000 But if we rotate this system a bit, 351 00:20:14,000 --> 00:20:18,410 then we find that there's a really interesting projection, 352 00:20:18,410 --> 00:20:21,510 which is the principal coordinate. 353 00:20:21,510 --> 00:20:28,240 You see that all those crazy strange phenomena 354 00:20:28,240 --> 00:20:32,390 we see with coupled systems are just illusions. 355 00:20:32,390 --> 00:20:35,170 Actually, you can understand then 356 00:20:35,170 --> 00:20:39,320 by really using the right projection. 357 00:20:39,320 --> 00:20:43,180 To one-- to the right coordinate system. 358 00:20:43,180 --> 00:20:47,230 Then you will see that actually the system is doing still 359 00:20:47,230 --> 00:20:49,570 simple harmonic motion. 360 00:20:49,570 --> 00:20:51,100 So that is actually the core thing 361 00:20:51,100 --> 00:20:54,560 which we learned from coupled systems. 362 00:20:54,560 --> 00:20:58,120 So we learned about how to solve the coupled system, 363 00:20:58,120 --> 00:21:04,030 and we also learned about going to an infinite number 364 00:21:04,030 --> 00:21:06,370 of coupled systems. 365 00:21:06,370 --> 00:21:09,470 So then this is an example here. 366 00:21:09,470 --> 00:21:15,160 So for example, I can have pendulum and springs, 367 00:21:15,160 --> 00:21:17,520 and we connect them all together, 368 00:21:17,520 --> 00:21:23,890 and I need to hire many, many students so that they plays it, 369 00:21:23,890 --> 00:21:27,190 plays until it fills up the whole universe. 370 00:21:27,190 --> 00:21:32,680 So this is the idea of an infinite system. 371 00:21:32,680 --> 00:21:35,320 You can see that that means my M minus 372 00:21:35,320 --> 00:21:37,350 1 K matrix is going to be an infinite 373 00:21:37,350 --> 00:21:40,930 times infinite long matrix. 374 00:21:40,930 --> 00:21:42,370 It's two dimensional. 375 00:21:42,370 --> 00:21:45,460 And the A is infinitely long. 376 00:21:45,460 --> 00:21:47,710 And that sounds really scary. 377 00:21:47,710 --> 00:21:53,020 And in general, we don't know how to deal with this, really. 378 00:21:53,020 --> 00:22:00,700 And it can be as arbitrarily crazy as you can imagine. 379 00:22:00,700 --> 00:22:04,510 What we discuss 8.03 is a special case. 380 00:22:04,510 --> 00:22:07,540 Basically, we are discussing about systems 381 00:22:07,540 --> 00:22:11,420 which are having a spatial kind of symmetry, which 382 00:22:11,420 --> 00:22:17,110 is translation symmetry, as you can see from all those figures. 383 00:22:17,110 --> 00:22:20,110 And you can see that all those figures will 384 00:22:20,110 --> 00:22:28,530 have to all have the same normal modes because of this base 385 00:22:28,530 --> 00:22:30,890 translation symmetry. 386 00:22:30,890 --> 00:22:37,700 What we discussed about is that we introduce an S matrix, which 387 00:22:37,700 --> 00:22:43,130 is used to describe the kind of symmetry 388 00:22:43,130 --> 00:22:46,220 that this system satisfies. 389 00:22:46,220 --> 00:22:52,400 And if we calculate the commutator S and M minus 1 K 390 00:22:52,400 --> 00:22:58,100 matrix, if this commutator shows that the evaluate-- 391 00:22:58,100 --> 00:23:01,010 if you evaluate this commutator and you get zero, 392 00:23:01,010 --> 00:23:03,470 now it means they commute. 393 00:23:03,470 --> 00:23:09,230 And the consequence is that the S matrix and the S M minus 1 K 394 00:23:09,230 --> 00:23:13,970 matrix will share the same eigenvectors. 395 00:23:13,970 --> 00:23:19,700 So you don't really need to know how to derive this-- 396 00:23:19,700 --> 00:23:21,560 to arrive at this conclusion, but it 397 00:23:21,560 --> 00:23:24,380 is a very useful conclusion. 398 00:23:24,380 --> 00:23:29,500 So that means instead of solving M minus 1 K matrix eigenvalue 399 00:23:29,500 --> 00:23:34,370 problem, I can now go ahead to solve the S matrix eigenvalue 400 00:23:34,370 --> 00:23:34,940 problem. 401 00:23:34,940 --> 00:23:37,520 And usually, that's much easier. 402 00:23:37,520 --> 00:23:39,920 So for the exam, you need to know 403 00:23:39,920 --> 00:23:42,830 how to write down S matrix. 404 00:23:42,830 --> 00:23:46,860 You need to know how to solve eigenvalue problems, including 405 00:23:46,860 --> 00:23:50,090 M minus 1 K matrix and the S matrix. 406 00:23:50,090 --> 00:23:54,950 And then we can get to normal mode frequency, omega squared, 407 00:23:54,950 --> 00:23:58,460 and we can also solve the corresponding normal modes. 408 00:23:58,460 --> 00:24:01,670 And here is telling you what would 409 00:24:01,670 --> 00:24:05,840 be the solution for space translation of the matrix 410 00:24:05,840 --> 00:24:06,656 system. 411 00:24:06,656 --> 00:24:08,030 And basically what we will see is 412 00:24:08,030 --> 00:24:12,350 that making use of the S matrix should be-- 413 00:24:12,350 --> 00:24:14,880 brings you to the conclusion that A, 414 00:24:14,880 --> 00:24:19,880 j must be proportional to exponential i, j, k, a, 415 00:24:19,880 --> 00:24:25,160 where this A is the length scale of this system, the distance 416 00:24:25,160 --> 00:24:27,140 between all those little mass. 417 00:24:27,140 --> 00:24:31,530 And the j is a label which tells you which little mass 418 00:24:31,530 --> 00:24:33,020 I am talking about. 419 00:24:33,020 --> 00:24:36,500 And k is the-- 420 00:24:36,500 --> 00:24:39,520 some arbitrary constant. 421 00:24:39,520 --> 00:24:43,520 But by now, you should have the idea basically that's-- 422 00:24:43,520 --> 00:24:44,590 that's what? 423 00:24:44,590 --> 00:24:48,260 That, essentially, is the wave number, right? 424 00:24:48,260 --> 00:24:49,790 So that is really cool. 425 00:24:49,790 --> 00:24:52,970 So that's all planned in advance. 426 00:24:52,970 --> 00:24:58,850 And basically, you can see that we can also write down the A, k 427 00:24:58,850 --> 00:25:01,940 because we know that A, j will be proportional to exponential 428 00:25:01,940 --> 00:25:06,340 i, j, k, A, after solving the eigenvalue problem for S 429 00:25:06,340 --> 00:25:08,930 matrix. 430 00:25:08,930 --> 00:25:14,240 Then we actually went one step forward to make it continuous. 431 00:25:14,240 --> 00:25:20,670 So basically, we made the space between particles very, very 432 00:25:20,670 --> 00:25:22,170 small. 433 00:25:22,170 --> 00:25:24,350 And also, at the same time, we make 434 00:25:24,350 --> 00:25:30,240 sure that the string doesn't become supermassive. 435 00:25:30,240 --> 00:25:36,380 And we concluded that we get some kind of equation 436 00:25:36,380 --> 00:25:39,170 popping out from this exercise. 437 00:25:39,170 --> 00:25:43,730 M minus 1 K matrix becomes minus T over rho L partial square, 438 00:25:43,730 --> 00:25:45,650 partial x squared. 439 00:25:45,650 --> 00:25:48,830 You don't have to really derive this for the exam, 440 00:25:48,830 --> 00:25:50,900 but you would need to know the conclusion 441 00:25:50,900 --> 00:25:57,560 and that psi j becomes psi as a function of x and t. 442 00:25:57,560 --> 00:26:01,110 And the magical function appeared, 443 00:26:01,110 --> 00:26:02,710 which is the wave equation. 444 00:26:02,710 --> 00:26:05,090 Oh my god, this is the whole craziness 445 00:26:05,090 --> 00:26:09,650 we have been dealing with the whole 8.03. 446 00:26:09,650 --> 00:26:11,750 This is actually really remarkable 447 00:26:11,750 --> 00:26:16,340 that we can come from single object oscillation, 448 00:26:16,340 --> 00:26:18,920 putting it all together, making it continuous, 449 00:26:18,920 --> 00:26:21,050 then this equation really popped out. 450 00:26:21,050 --> 00:26:27,200 And this equation really describes multiple systems. 451 00:26:27,200 --> 00:26:30,830 Then we went ahead to actually discuss the property 452 00:26:30,830 --> 00:26:32,510 of the wave equation. 453 00:26:32,510 --> 00:26:33,620 It looks like this. 454 00:26:33,620 --> 00:26:39,290 Basically, I replaced the t over rho L by v, p squared. 455 00:26:39,290 --> 00:26:41,450 By now, you know the meaning of v, 456 00:26:41,450 --> 00:26:43,580 p is actually the phase velocity. 457 00:26:43,580 --> 00:26:48,790 And we discussed two kinds of solutions, 458 00:26:48,790 --> 00:26:50,390 special kinds of solutions. 459 00:26:50,390 --> 00:26:54,320 The first kind is normal modes. 460 00:26:54,320 --> 00:26:57,830 The second one is progressive wave solution, 461 00:26:57,830 --> 00:27:00,050 or traveling wave solution, whatever name 462 00:27:00,050 --> 00:27:02,080 you want to call it. 463 00:27:02,080 --> 00:27:07,090 Let's take a look at the normal modes, what have we learned. 464 00:27:07,090 --> 00:27:13,400 So if you have a bound system, a bound continuous system, 465 00:27:13,400 --> 00:27:17,790 the normal mode is your distending waves for the wave 466 00:27:17,790 --> 00:27:19,680 equation we discussed. 467 00:27:19,680 --> 00:27:22,350 And basically, the functional form 468 00:27:22,350 --> 00:27:28,560 is A, m, sine, k, m, x plus alpha, m and sine, omega, m, t 469 00:27:28,560 --> 00:27:30,780 plus beta, m. 470 00:27:30,780 --> 00:27:35,940 So what we actually learned from the previous lecture 471 00:27:35,940 --> 00:27:37,030 is the following. 472 00:27:37,030 --> 00:27:42,930 So basically, you can decide the k, m and alpha, 473 00:27:42,930 --> 00:27:45,180 m by just boundary conditions. 474 00:27:45,180 --> 00:27:48,810 So before you introduce boundary conditions, which 475 00:27:48,810 --> 00:27:52,890 are the conditions allow you to describe 476 00:27:52,890 --> 00:27:56,520 multiple nearby systems consistently. 477 00:27:56,520 --> 00:27:59,520 So that is the meaning of boundary condition. 478 00:27:59,520 --> 00:28:02,100 Before you introduce that, k, m and alpha, 479 00:28:02,100 --> 00:28:04,420 m are arbitrary numbers. 480 00:28:04,420 --> 00:28:08,340 Whatever number you choose is the-- 481 00:28:08,340 --> 00:28:11,160 can satisfy the wave equation. 482 00:28:11,160 --> 00:28:15,180 But after you introduce the boundary condition, 483 00:28:15,180 --> 00:28:21,660 you figure that out from the problem you are given, then k, 484 00:28:21,660 --> 00:28:25,020 m and alpha, m cannot be arbitrary anymore. 485 00:28:25,020 --> 00:28:28,330 And they usually become discrete numbers. 486 00:28:28,330 --> 00:28:28,830 OK. 487 00:28:28,830 --> 00:28:33,000 So that is what we learned from the previous lectures. 488 00:28:33,000 --> 00:28:37,030 And finally, we also see that omega, 489 00:28:37,030 --> 00:28:41,790 m is determined by the property of the system, 490 00:28:41,790 --> 00:28:45,030 by a so-called dispersion relation. 491 00:28:45,030 --> 00:28:49,170 In this case, it's linear, it's proportional to k, m, 492 00:28:49,170 --> 00:28:51,840 because we are talking about non-dispersive medium 493 00:28:51,840 --> 00:28:52,830 for the moment. 494 00:28:52,830 --> 00:28:57,510 And we have this beta, m, which is related 495 00:28:57,510 --> 00:28:59,370 to the initial condition. 496 00:28:59,370 --> 00:29:03,150 And the a, m, which can be determined 497 00:29:03,150 --> 00:29:06,120 by a Fourier decomposition. 498 00:29:06,120 --> 00:29:08,620 So if you are not familiar with this, 499 00:29:08,620 --> 00:29:12,900 you have to really review how to do Fourier decomposition. 500 00:29:12,900 --> 00:29:16,630 I know most of did very well on the midterm, 501 00:29:16,630 --> 00:29:21,150 but maybe some of you forgot how to determine a, m 502 00:29:21,150 --> 00:29:22,950 and it will be very, very important 503 00:29:22,950 --> 00:29:27,330 to review that for the preparation for the final. 504 00:29:27,330 --> 00:29:30,610 Now the second set of solutions is the following. 505 00:29:30,610 --> 00:29:31,950 So you have progress-- 506 00:29:31,950 --> 00:29:33,510 progressing waves. 507 00:29:33,510 --> 00:29:37,390 And the functional form is really interesting. 508 00:29:37,390 --> 00:29:42,390 So you can see this can be written as F, 509 00:29:42,390 --> 00:29:47,990 F is some arbitrary function, x plus-minus v, p, t. 510 00:29:47,990 --> 00:29:50,790 Basically that is that you're describing a wave which 511 00:29:50,790 --> 00:29:53,250 is traveling to the positive-- 512 00:29:53,250 --> 00:29:59,580 to a negative or positive direction in the x direction. 513 00:29:59,580 --> 00:30:01,080 Or you can actually write it down 514 00:30:01,080 --> 00:30:04,380 as G function k, x plus-minus omega, t. 515 00:30:04,380 --> 00:30:09,730 Actually, they all work for wave equations. 516 00:30:09,730 --> 00:30:13,780 Now we went ahead and applied approach 517 00:30:13,780 --> 00:30:17,520 which we learned from the general solution of wave 518 00:30:17,520 --> 00:30:23,750 equation to massive strings, and we discussed about sound waves. 519 00:30:23,750 --> 00:30:29,100 For the sound waves, it will be important to review 520 00:30:29,100 --> 00:30:33,640 what are the boundary conditions for the displacement 521 00:30:33,640 --> 00:30:38,430 of the molecules in the sound wave, 522 00:30:38,430 --> 00:30:43,050 compare that to the pressure deviation 523 00:30:43,050 --> 00:30:44,850 from the room pressure. 524 00:30:44,850 --> 00:30:47,100 So I think it's important to make sure 525 00:30:47,100 --> 00:30:50,090 that you understand the difference between these two, 526 00:30:50,090 --> 00:30:55,950 what are the boundary conditions and basically it should be 527 00:30:55,950 --> 00:30:58,460 very similar to the solution-- 528 00:30:58,460 --> 00:31:01,620 the boundary condition for the massive strings. 529 00:31:01,620 --> 00:31:06,450 And we also talked about electromagnetic waves. 530 00:31:06,450 --> 00:31:13,180 And that is another topic which you will really have to review. 531 00:31:13,180 --> 00:31:15,820 Several things which are especially interesting is that 532 00:31:15,820 --> 00:31:22,110 an electric field cannot be without a magnetic field. 533 00:31:22,110 --> 00:31:25,240 They are always together, no matter what. 534 00:31:25,240 --> 00:31:28,050 So if you have trouble with the electric field, 535 00:31:28,050 --> 00:31:31,410 then there must be trouble in the magnetic field. 536 00:31:31,410 --> 00:31:36,930 And that is governed by the Maxwell's equation. 537 00:31:36,930 --> 00:31:40,280 Before we go into the detail of those, 538 00:31:40,280 --> 00:31:44,400 we also discussed about dispersive medium. 539 00:31:44,400 --> 00:31:46,950 So in the case of dispersive medium, 540 00:31:46,950 --> 00:31:51,300 we used a special kind of example, which 541 00:31:51,300 --> 00:31:55,060 is strings with stiffness. 542 00:31:55,060 --> 00:31:57,810 So basically, what we found is that if you 543 00:31:57,810 --> 00:32:16,460 have a certain kind of wave equation, like this one, 544 00:32:16,460 --> 00:32:18,340 I am writing this one here. 545 00:32:18,340 --> 00:32:21,190 Basically, if I add the additional term 546 00:32:21,190 --> 00:32:27,860 to describe the stiffness, then what is going to happen 547 00:32:27,860 --> 00:32:31,310 is that the dispersion relation, when 548 00:32:31,310 --> 00:32:34,850 I ask you to plot the dispersion relation, you will be-- 549 00:32:34,850 --> 00:32:40,520 I am requesting you to find the relation between omega and K. 550 00:32:40,520 --> 00:32:42,620 And I'm going over this in more detail 551 00:32:42,620 --> 00:32:46,310 because I see so many similar mistakes on the midterm. 552 00:32:46,310 --> 00:32:51,020 So basically what I'm asking is omega versus K. 553 00:32:51,020 --> 00:32:55,640 And in the-- if we don't have this turn, then basically, 554 00:32:55,640 --> 00:32:57,050 you have a straight line. 555 00:32:57,050 --> 00:33:01,560 Straight line means you have a non-dispersive medium. 556 00:33:01,560 --> 00:33:04,190 And if you add this turn, you need 557 00:33:04,190 --> 00:33:08,090 to know how to evaluate the dispersion relation. 558 00:33:08,090 --> 00:33:13,350 The quickest way to evaluate the dispersion relation is 559 00:33:13,350 --> 00:33:16,850 to just simply plug in the progresssing wave 560 00:33:16,850 --> 00:33:21,260 solution for the G function or harmonic progressing 561 00:33:21,260 --> 00:33:23,680 wave solution, find omega-- 562 00:33:23,680 --> 00:33:28,670 K, x plus-minus omega t, into this equation, 563 00:33:28,670 --> 00:33:32,500 then you will be able to figure out the dispersion relation. 564 00:33:32,500 --> 00:33:34,770 And what we figure out is the following. 565 00:33:34,770 --> 00:33:38,150 If we include stiffness, then you 566 00:33:38,150 --> 00:33:43,610 can see that the dispersion relation is not a line anymore 567 00:33:43,610 --> 00:33:47,360 and is actually some kind of curve, 568 00:33:47,360 --> 00:33:51,340 and the slope is actually changing. 569 00:33:51,340 --> 00:33:56,380 And there are dramatic consequence from this thing. 570 00:33:56,380 --> 00:34:02,150 That means if I have a traveling wave 571 00:34:02,150 --> 00:34:09,120 with different wavelengths, that means the phase velocity v, 572 00:34:09,120 --> 00:34:13,930 p equal to omega over K is going to be 573 00:34:13,930 --> 00:34:17,940 different for waves with different frequencies, 574 00:34:17,940 --> 00:34:20,100 or different wavelengths. 575 00:34:20,100 --> 00:34:22,540 So that is how you clear the problem. 576 00:34:22,540 --> 00:34:26,449 Because if I have initially produced a signal which 577 00:34:26,449 --> 00:34:30,530 is a triangle and I let it propagate, 578 00:34:30,530 --> 00:34:35,370 what is going to happen is that the slow component 579 00:34:35,370 --> 00:34:37,670 will be lagging behind. 580 00:34:37,670 --> 00:34:40,469 Those are the slow components. 581 00:34:40,469 --> 00:34:45,960 And the fast components will go ahead of the nominal speed. 582 00:34:45,960 --> 00:34:49,690 So there will be a spread of the signal. 583 00:34:49,690 --> 00:34:54,239 Originally, maybe you have some kind of a square wave, 584 00:34:54,239 --> 00:35:01,050 and this thing will become something which is actually 585 00:35:01,050 --> 00:35:08,100 smeared out in space, and then you lose the information. 586 00:35:08,100 --> 00:35:10,420 And we are going to talk about that later. 587 00:35:10,420 --> 00:35:14,190 And we also learned about group velocity. 588 00:35:14,190 --> 00:35:17,000 So what is your group velocity? 589 00:35:17,000 --> 00:35:19,920 Group velocity v, p-- oh, sorry, v, 590 00:35:19,920 --> 00:35:25,050 g is actually partial omega, partial K, 591 00:35:25,050 --> 00:35:31,000 which is the slope of a tangential line here. 592 00:35:31,000 --> 00:35:35,400 And where the phase velocity is connecting 593 00:35:35,400 --> 00:35:39,360 this point to that point, and the slope of this line 594 00:35:39,360 --> 00:35:44,280 is the phase velocity, and the slope of the line 595 00:35:44,280 --> 00:35:47,730 cutting through this point, which is giving you 596 00:35:47,730 --> 00:35:49,770 the group velocity. 597 00:35:49,770 --> 00:35:54,480 And we actually learned the definition of-- 598 00:35:54,480 --> 00:35:57,330 the consequence of group velocity and phase 599 00:35:57,330 --> 00:36:01,140 velocity by introducing you a bit phenomena. 600 00:36:01,140 --> 00:36:09,480 Basically, we add two waves with similar wavelengths, or wave 601 00:36:09,480 --> 00:36:10,120 numbers. 602 00:36:10,120 --> 00:36:11,910 Basically, what we see is the following. 603 00:36:11,910 --> 00:36:15,050 So basically, you see some behavior like this. 604 00:36:17,730 --> 00:36:22,530 We see this-- the superposition of these two waves 605 00:36:22,530 --> 00:36:27,440 which produce a bit phenomena can be understood by something 606 00:36:27,440 --> 00:36:32,230 which is oscillating really fast modulated by a much 607 00:36:32,230 --> 00:36:39,840 slower more variating envelope. 608 00:36:39,840 --> 00:36:44,780 Basically, you can actually understand the bit phenomena 609 00:36:44,780 --> 00:36:50,670 by actually identifying these two interesting structures. 610 00:36:50,670 --> 00:36:54,990 And the speed of all those little peaks 611 00:36:54,990 --> 00:36:59,760 is traveling at phase velocity. 612 00:36:59,760 --> 00:37:02,940 And the speed of the envelope is found 613 00:37:02,940 --> 00:37:07,140 to be traveling at group velocity. 614 00:37:07,140 --> 00:37:09,600 So that is what we have learned. 615 00:37:09,600 --> 00:37:14,450 And we can have group velocity and the phase velocity 616 00:37:14,450 --> 00:37:17,280 traveling in the same direction. 617 00:37:17,280 --> 00:37:21,130 And we can also have a negative group velocity. 618 00:37:21,130 --> 00:37:25,010 So that is a technique which is really, really very difficult. 619 00:37:25,010 --> 00:37:27,680 And I'm still trying to practice and make sure 620 00:37:27,680 --> 00:37:30,180 they I can demo that in 8.03. 621 00:37:30,180 --> 00:37:33,710 Basically, it's like the whole system, 622 00:37:33,710 --> 00:37:38,970 the whole detailed structure moving in a positive direction. 623 00:37:38,970 --> 00:37:42,360 But the body, or say the envelope, 624 00:37:42,360 --> 00:37:46,570 is actually moving in the negative x direction. 625 00:37:46,570 --> 00:37:48,880 So that is also possible. 626 00:37:48,880 --> 00:37:51,210 And you can actually construct a system which 627 00:37:51,210 --> 00:37:54,420 has a negative group velocity. 628 00:37:54,420 --> 00:38:00,150 So once we have done that, we also 629 00:38:00,150 --> 00:38:05,400 tried to understand further the description of the solution 630 00:38:05,400 --> 00:38:08,980 for the dispersive medium. 631 00:38:08,980 --> 00:38:12,660 So basically, what we actually went over during the class 632 00:38:12,660 --> 00:38:17,190 is that OK, now, if the f function f of t 633 00:38:17,190 --> 00:38:22,000 is describing Yen-Jie's hand, and I'm holding an infinitely 634 00:38:22,000 --> 00:38:27,690 long string and I shake it as a function of time, 635 00:38:27,690 --> 00:38:29,900 and that essentially, this motion, 636 00:38:29,900 --> 00:38:33,620 is actually described by this f function. 637 00:38:33,620 --> 00:38:39,720 What we know is that this oscillation, OK, I can do one, 638 00:38:39,720 --> 00:38:43,980 but I won't, but all kinds of f functions 639 00:38:43,980 --> 00:38:47,340 can be described as superpositions 640 00:38:47,340 --> 00:38:53,490 of many, many, many waves with different angular frequencies. 641 00:38:53,490 --> 00:38:55,740 So that's a miracle which we borrowed 642 00:38:55,740 --> 00:38:57,720 from the math department again. 643 00:38:57,720 --> 00:38:59,730 And you can see that f function can 644 00:38:59,730 --> 00:39:04,950 be written as the sum of all kinds of different waves 645 00:39:04,950 --> 00:39:07,350 with different angular frequencies 646 00:39:07,350 --> 00:39:12,290 with population c omega. 647 00:39:12,290 --> 00:39:19,080 This is the weight which makes that become the f function. 648 00:39:19,080 --> 00:39:23,790 And we can figure out the c omega 649 00:39:23,790 --> 00:39:27,480 by doing a Fourier transform. 650 00:39:27,480 --> 00:39:34,550 And finally, what will be the resulting wave function, psi, 651 00:39:34,550 --> 00:39:38,610 x, t, which is the wave function generated 652 00:39:38,610 --> 00:39:41,100 by the oscillation of my hand. 653 00:39:41,100 --> 00:39:45,070 And those are governed by the wave equation, which gives you 654 00:39:45,070 --> 00:39:47,880 the relation between omega and the k 655 00:39:47,880 --> 00:39:51,810 can be returned in that functional form. 656 00:39:51,810 --> 00:39:57,570 So the good news is that with the help of Fourier transform, 657 00:39:57,570 --> 00:40:01,170 we can also describe and predict what 658 00:40:01,170 --> 00:40:06,570 is going to happen no matter if this system is dispersive 659 00:40:06,570 --> 00:40:09,820 or not dispersive using this approach. 660 00:40:09,820 --> 00:40:10,320 OK. 661 00:40:10,320 --> 00:40:12,000 So that is really cool. 662 00:40:12,000 --> 00:40:17,750 And you can of course can do a cross-check just to-- 663 00:40:17,750 --> 00:40:22,260 assuring that this is a non-dispersive medium. 664 00:40:22,260 --> 00:40:24,420 And you are also going to get back 665 00:40:24,420 --> 00:40:27,540 to what you should expect the solution 666 00:40:27,540 --> 00:40:31,070 to non-dispersive medium for the psi, x, t. 667 00:40:31,070 --> 00:40:36,720 So that is one thing which is really remarkable. 668 00:40:36,720 --> 00:40:40,530 And I think what is needed to know 669 00:40:40,530 --> 00:40:44,520 is not a deviation of all those formulas, 670 00:40:44,520 --> 00:40:51,540 but how the plotting and the derived c omega by using 671 00:40:51,540 --> 00:40:56,870 the formula you are given and how to then put together 672 00:40:56,870 --> 00:41:01,710 all the solutions and it becomes the resulting solution 673 00:41:01,710 --> 00:41:04,410 for the psi, x, t, which is really 674 00:41:04,410 --> 00:41:06,930 the solution we really care. 675 00:41:06,930 --> 00:41:11,000 So for that, you need to know how to do the integration. 676 00:41:11,000 --> 00:41:16,000 You need to know how to derive the dispersion relation. 677 00:41:16,000 --> 00:41:19,760 Then one thing left over is to put the problem 678 00:41:19,760 --> 00:41:24,500 into that equation, which is also given to you in a formula. 679 00:41:24,500 --> 00:41:27,530 And we will not ask you to do a very, very 680 00:41:27,530 --> 00:41:32,600 complicated integration for sure on the final. 681 00:41:32,600 --> 00:41:35,650 So what is the consequence? 682 00:41:35,650 --> 00:41:38,810 Basically, one thing which is interesting to know 683 00:41:38,810 --> 00:41:43,100 is that if you have a wave in a coordinate space, which 684 00:41:43,100 --> 00:41:49,880 is really widely spread out, and you can do a Fourier transform 685 00:41:49,880 --> 00:41:54,170 to get the wave population in the frequency space, what 686 00:41:54,170 --> 00:41:58,610 we find is that when this wave is really, really 687 00:41:58,610 --> 00:42:01,550 wide in the space, then what we find 688 00:42:01,550 --> 00:42:05,510 is that the wave population in the frequency space 689 00:42:05,510 --> 00:42:09,490 is very narrow by using a Fourier transform. 690 00:42:09,490 --> 00:42:12,900 And that just gives you the result. And on the other hand, 691 00:42:12,900 --> 00:42:14,380 if you have a really-- 692 00:42:14,380 --> 00:42:18,410 a very narrow pulse in the coordinate space, 693 00:42:18,410 --> 00:42:21,950 for example, I do this-- shwhew --very, very-- really quickly. 694 00:42:21,950 --> 00:42:24,460 I create a very narrow pulse. 695 00:42:24,460 --> 00:42:26,630 And then what is actually happening 696 00:42:26,630 --> 00:42:33,800 is that I will have to use a very wide range of frequency 697 00:42:33,800 --> 00:42:38,120 space to describe this very narrow pulse. 698 00:42:38,120 --> 00:42:39,410 So that leads to-- 699 00:42:39,410 --> 00:42:43,310 direct consequences of that is uncertainty principle. 700 00:42:43,310 --> 00:42:47,660 And this is closely connected to the uncertainty principle 701 00:42:47,660 --> 00:42:50,390 we talk about in quantum mechanics. 702 00:42:50,390 --> 00:42:54,620 Delta, p times delta, x greater or equal to h bar over 2. 703 00:42:57,170 --> 00:42:57,740 All right. 704 00:42:57,740 --> 00:43:01,960 So we have done with the one-dimensional case. 705 00:43:01,960 --> 00:43:05,040 And we also talked about a two-dimensional 706 00:43:05,040 --> 00:43:06,990 and a three-dimensional case. 707 00:43:06,990 --> 00:43:11,450 And this is the example of two-dimensional membranes, 708 00:43:11,450 --> 00:43:14,730 and they actually are constrained so that 709 00:43:14,730 --> 00:43:17,580 their boundary condition at-- 710 00:43:17,580 --> 00:43:18,720 the boundary is equal-- 711 00:43:18,720 --> 00:43:21,300 no, the wave function is equal to zero. 712 00:43:21,300 --> 00:43:25,620 And you can identify all those normal modes. 713 00:43:25,620 --> 00:43:32,410 And we went ahead also to talk about geometrical optics laws. 714 00:43:32,410 --> 00:43:38,350 Basically, how we derive that is to have a plane wave. 715 00:43:38,350 --> 00:43:41,040 First, you have a plane wave propagating 716 00:43:41,040 --> 00:43:44,970 toward the boundary of two different mediums, 717 00:43:44,970 --> 00:43:49,160 and we were wondering well, what is the refracted wave 718 00:43:49,160 --> 00:43:50,730 and the transmitting wave. 719 00:43:50,730 --> 00:43:56,160 By using the-- by making sure just one point, which 720 00:43:56,160 --> 00:44:00,900 is that the membranes don't break, 721 00:44:00,900 --> 00:44:04,890 the wave function is continuous at this boundary. 722 00:44:04,890 --> 00:44:07,590 That's the only assumption which you use. 723 00:44:07,590 --> 00:44:10,290 We went through the mathematics, which you don't really 724 00:44:10,290 --> 00:44:12,630 need to remember all of them. 725 00:44:12,630 --> 00:44:16,260 But you really need to remember the consequence. 726 00:44:16,260 --> 00:44:18,060 The consequence is the following. 727 00:44:18,060 --> 00:44:23,190 Basically, what we see is that if you have incident plane 728 00:44:23,190 --> 00:44:27,570 wave with incident angle theta 1, 729 00:44:27,570 --> 00:44:31,350 the refractive wave will be having an angle of theta 1 730 00:44:31,350 --> 00:44:32,260 as well. 731 00:44:32,260 --> 00:44:36,270 So that's the first law of refraction, refraction law. 732 00:44:36,270 --> 00:44:38,090 And then the second one which we learned 733 00:44:38,090 --> 00:44:41,260 is that the transmitting wave will 734 00:44:41,260 --> 00:44:47,000 satisfy Snell's law, n, 1, sine, theta, 1 equal to n, 2, 735 00:44:47,000 --> 00:44:48,530 sine, theta, 2. 736 00:44:48,530 --> 00:44:55,930 And that is very interesting because this, Snell's law 737 00:44:55,930 --> 00:45:01,230 has also nothing to do with Maxwell's equation. 738 00:45:01,230 --> 00:45:02,400 You see? 739 00:45:02,400 --> 00:45:02,900 Right? 740 00:45:02,900 --> 00:45:05,840 That's actually what you can learn from here. 741 00:45:05,840 --> 00:45:09,600 We usually use electromagnetic waves 742 00:45:09,600 --> 00:45:12,450 to demonstrate Snell's law. 743 00:45:12,450 --> 00:45:15,390 But from 8.03, we learned that it has nothing 744 00:45:15,390 --> 00:45:20,090 to do with Maxwell's equation. 745 00:45:20,090 --> 00:45:23,910 It applies to all kinds of different systems, which 746 00:45:23,910 --> 00:45:24,890 you can-- 747 00:45:24,890 --> 00:45:27,850 which can be described by wave functions. 748 00:45:27,850 --> 00:45:32,490 So that is actually the very important consequence. 749 00:45:32,490 --> 00:45:36,690 But on the other hand, as we all discussed later, 750 00:45:36,690 --> 00:45:41,200 the relative amplitude of the incident wave, refracted wave, 751 00:45:41,200 --> 00:45:45,120 and transmitting wave, the relative amplitude 752 00:45:45,120 --> 00:45:48,450 is governed by Maxwell's equation. 753 00:45:48,450 --> 00:45:52,380 So I would like to make that really crystal clear. 754 00:45:52,380 --> 00:45:54,870 So the relative amplitude is governed 755 00:45:54,870 --> 00:45:58,500 by really the physical laws, which 756 00:45:58,500 --> 00:46:06,380 actually governs the propagation of those plane waves. 757 00:46:06,380 --> 00:46:07,070 OK. 758 00:46:07,070 --> 00:46:14,750 So I think we can take a five minute break to have some air. 759 00:46:14,750 --> 00:46:18,530 And of course, you can-- you are welcome to continue to use 760 00:46:18,530 --> 00:46:20,840 all this juice and coffee. 761 00:46:20,840 --> 00:46:22,260 And coming back at 38. 762 00:46:31,850 --> 00:46:32,350 OK. 763 00:46:32,350 --> 00:46:36,567 So welcome back, everybody, from the break. 764 00:46:36,567 --> 00:46:38,657 AUDIENCE: [INAUDIBLE] 765 00:46:38,657 --> 00:46:40,990 YEN-JIE LEE: So we are going to continue the discussion. 766 00:46:40,990 --> 00:46:45,880 We have learned about the two important laws 767 00:46:45,880 --> 00:46:48,040 for the geometrical optics. 768 00:46:48,040 --> 00:46:53,040 And we also went ahead to discuss the polarization that's 769 00:46:53,040 --> 00:46:55,940 solved in greater detail. 770 00:46:55,940 --> 00:47:00,260 So for example, we can have linear depolarized wave. 771 00:47:00,260 --> 00:47:03,640 So basically, the wave is essentially moving up and down, 772 00:47:03,640 --> 00:47:04,810 up and down. 773 00:47:04,810 --> 00:47:10,810 But the direction of the background field 774 00:47:10,810 --> 00:47:11,980 doesn't change. 775 00:47:11,980 --> 00:47:13,780 It's always, for example, initially, 776 00:47:13,780 --> 00:47:19,120 if it's in x direction, then it is x direction forever. 777 00:47:19,120 --> 00:47:24,700 And in that case, I call it linearly polarized. 778 00:47:24,700 --> 00:47:26,590 Of course, I can also have the case 779 00:47:26,590 --> 00:47:32,170 that I can have a superposition of two waves. 780 00:47:32,170 --> 00:47:36,920 One is having the electric field in the x direction. 781 00:47:36,920 --> 00:47:39,620 And the other one is in the y direction. 782 00:47:39,620 --> 00:47:45,820 And they are off by a phase of pi over 2. 783 00:47:45,820 --> 00:47:47,570 If that happens, then basically, you 784 00:47:47,570 --> 00:47:51,920 will see that it produces something really interesting. 785 00:47:51,920 --> 00:47:54,850 That direction of the electric field 786 00:47:54,850 --> 00:47:58,030 is going to be rotating as a function of time-- 787 00:47:58,030 --> 00:48:02,350 as a function of the space these waves travel. 788 00:48:02,350 --> 00:48:07,780 And we call it circularly polarized waves. 789 00:48:07,780 --> 00:48:12,260 And we can also have elliptically polarized wave. 790 00:48:12,260 --> 00:48:18,440 Then we learned about how to do a filtering, which 791 00:48:18,440 --> 00:48:20,330 is the polarizer. 792 00:48:20,330 --> 00:48:22,870 So suppose I have a perfect conductor here, 793 00:48:22,870 --> 00:48:28,660 where I have the easy axis, which is described 794 00:48:28,660 --> 00:48:31,450 by the green arrow there. 795 00:48:31,450 --> 00:48:34,540 And you can see that easy axis means 796 00:48:34,540 --> 00:48:41,410 that if you have electric field parallel to the easy axis, 797 00:48:41,410 --> 00:48:44,530 and then since that's the easy axis, so it is supposed 798 00:48:44,530 --> 00:48:48,490 to be easy, therefore, this electric field is going to be 799 00:48:48,490 --> 00:48:51,880 passing through the polarizer. 800 00:48:51,880 --> 00:48:56,020 On the other hand, if the electric field 801 00:48:56,020 --> 00:49:00,620 is perpendicular to the direction of the easy axis, 802 00:49:00,620 --> 00:49:03,730 that means it's taking the perfect conductor 803 00:49:03,730 --> 00:49:05,110 in the hard way. 804 00:49:05,110 --> 00:49:08,830 Therefore, when it pass through-- 805 00:49:08,830 --> 00:49:10,670 when it is trying to pass through 806 00:49:10,670 --> 00:49:16,300 with the perfect conductor, the electrons in those conductors 807 00:49:16,300 --> 00:49:22,330 are going to be working like crazy to deflect this wave when 808 00:49:22,330 --> 00:49:25,420 the direction of the electric field 809 00:49:25,420 --> 00:49:30,220 is perpendicular to the direction of the easy axis. 810 00:49:30,220 --> 00:49:33,070 So that is how this works. 811 00:49:33,070 --> 00:49:36,160 For example, in the first example, 812 00:49:36,160 --> 00:49:39,370 you can see that in this case, you 813 00:49:39,370 --> 00:49:44,610 have an easy axis which is perpendicular to the direction 814 00:49:44,610 --> 00:49:48,310 of the electric field, which is the red field, 815 00:49:48,310 --> 00:49:51,010 then this wave actually got refracted. 816 00:49:51,010 --> 00:49:53,130 There will be no transmission-- 817 00:49:53,130 --> 00:49:58,300 sorry, no electromagnetic field passing the perfect conductor. 818 00:49:58,300 --> 00:50:01,305 And on the other hand, if you have another perfect conductor, 819 00:50:01,305 --> 00:50:03,540 in which you have easy axis which 820 00:50:03,540 --> 00:50:09,475 is parallel to the electric field, then you can-- 821 00:50:09,475 --> 00:50:15,160 you will see that it will pass through the perfect conductor. 822 00:50:15,160 --> 00:50:19,270 So that is the polarizer. 823 00:50:19,270 --> 00:50:23,380 And also, we discussed about quarter-weight plate, 824 00:50:23,380 --> 00:50:28,240 which I would suggest you to have a review about the concept 825 00:50:28,240 --> 00:50:31,075 which we have learned about polarizer 826 00:50:31,075 --> 00:50:34,780 and quarter-wave plate so that you make sure that you 827 00:50:34,780 --> 00:50:41,050 understand how to calculate the electric field after passing 828 00:50:41,050 --> 00:50:44,770 through a polarizer and quarter-wave plate 829 00:50:44,770 --> 00:50:48,720 and how the secondary, or the elliptically depolarized waves 830 00:50:48,720 --> 00:50:55,260 are created using all those wave plates, et cetera. 831 00:50:55,260 --> 00:50:56,620 All right. 832 00:50:56,620 --> 00:51:00,160 So the next thing which we discussed during the class 833 00:51:00,160 --> 00:51:04,390 is how do we produce electromagnetic waves. 834 00:51:04,390 --> 00:51:09,880 I think by now, you should know that a stationary charge 835 00:51:09,880 --> 00:51:13,570 doesn't produce electromagnetic waves. 836 00:51:13,570 --> 00:51:16,540 Even a moving charge at constant speed 837 00:51:16,540 --> 00:51:19,590 doesn't create electromagnetic waves. 838 00:51:19,590 --> 00:51:22,090 So how do we create an electromagnetic wave 839 00:51:22,090 --> 00:51:25,740 which propagates to the edge of the universe? 840 00:51:25,740 --> 00:51:33,100 That is-- the trick is to create a kink in the fuel line. 841 00:51:33,100 --> 00:51:37,400 So you have to accelerate and stop it. 842 00:51:37,400 --> 00:51:42,130 Accelerate and then try to actually stop the acceleration. 843 00:51:42,130 --> 00:51:44,560 So then you can create a kink. 844 00:51:44,560 --> 00:51:50,050 And this kink is going to be propagating out of the-- 845 00:51:50,050 --> 00:51:51,780 as a function of time. 846 00:51:51,780 --> 00:51:58,630 And this kink is creating the so-called radiation 847 00:51:58,630 --> 00:52:01,540 from this accelerated charge. 848 00:52:01,540 --> 00:52:04,790 So you don't really need to remember all the deviations, 849 00:52:04,790 --> 00:52:08,510 but you really need to know the conclusion. 850 00:52:08,510 --> 00:52:10,760 So what is the conclusion is the following. 851 00:52:10,760 --> 00:52:15,820 The radiated electric field is equal to minus-- 852 00:52:15,820 --> 00:52:18,250 very important that there's a minus sign in front 853 00:52:18,250 --> 00:52:21,990 of it, which is a common mistake to drop it, and the q 854 00:52:21,990 --> 00:52:24,910 is the charge of the oscillating-- 855 00:52:24,910 --> 00:52:29,380 the accelerated charge, proportional to the charge. 856 00:52:29,380 --> 00:52:33,250 If the particle is more charged, then you have more radiation. 857 00:52:33,250 --> 00:52:41,270 Aperp is the acceleration projected to, which is-- 858 00:52:43,960 --> 00:52:48,970 the perpendicular projection of the acceleration 859 00:52:48,970 --> 00:52:54,860 of the particle with respect to the direction of propagation 860 00:52:54,860 --> 00:52:57,040 is so-called the Aperp. 861 00:52:57,040 --> 00:53:02,840 And only the perpendicular direction acceleration counts. 862 00:53:02,840 --> 00:53:06,880 The one which is parallel to the direction of propagation 863 00:53:06,880 --> 00:53:10,920 doesn't really count, as you can see from this equation. 864 00:53:10,920 --> 00:53:14,200 And the t prime what is t prime? t prime 865 00:53:14,200 --> 00:53:18,820 is t minus r divided by c. 866 00:53:18,820 --> 00:53:24,100 So t prime is the retarded time, so that is telling you 867 00:53:24,100 --> 00:53:27,220 that it takes some time for the information 868 00:53:27,220 --> 00:53:30,100 to propagate from the origin, which 869 00:53:30,100 --> 00:53:37,010 is the position of the moving charge to the observer, which 870 00:53:37,010 --> 00:53:42,190 is r, this distance, away from the moving charge. 871 00:53:42,190 --> 00:53:44,920 So the information takes some time to propagate, 872 00:53:44,920 --> 00:53:49,360 and you cannot know what is really happening, for example, 873 00:53:49,360 --> 00:53:52,060 100 light years away from Earth. 874 00:53:52,060 --> 00:53:53,970 You have no idea about what is happening. 875 00:53:53,970 --> 00:53:56,030 Maybe a black hole is created there 876 00:53:56,030 --> 00:54:00,390 and is going to suck everybody up in a few years. 877 00:54:00,390 --> 00:54:04,490 But nobody knows, and we don't care because we cannot control 878 00:54:04,490 --> 00:54:06,760 it. 879 00:54:06,760 --> 00:54:09,640 All right so that is very important. 880 00:54:09,640 --> 00:54:12,730 And also very important to know the magnetic 881 00:54:12,730 --> 00:54:14,740 field must be there. 882 00:54:14,740 --> 00:54:16,960 You can see the relation between magnetic field 883 00:54:16,960 --> 00:54:18,340 and the electric field. 884 00:54:18,340 --> 00:54:25,310 And the Poynting vector is also its joint field. 885 00:54:25,310 --> 00:54:28,210 And when we went ahead, given all the knowledge 886 00:54:28,210 --> 00:54:30,520 we have learned, we discussed about how 887 00:54:30,520 --> 00:54:35,020 to take very beautiful photos using a polarizer filter. 888 00:54:35,020 --> 00:54:41,500 And we discussed about how to filter out the scattered light 889 00:54:41,500 --> 00:54:42,910 from the sun. 890 00:54:42,910 --> 00:54:46,750 And it would be nice to figure out 891 00:54:46,750 --> 00:54:50,940 why this is the case, how these polarizer lines, scatter lines 892 00:54:50,940 --> 00:54:52,960 are created. 893 00:54:52,960 --> 00:54:55,650 It's purely geometrical. 894 00:54:55,650 --> 00:54:59,830 And also, we discussed about Brewster's angle 895 00:54:59,830 --> 00:55:06,670 and also how it leads to the explanation of the filtering 896 00:55:06,670 --> 00:55:11,170 of the light, the refracted light from the, for example, 897 00:55:11,170 --> 00:55:15,010 window of a car or from the water. 898 00:55:15,010 --> 00:55:17,710 And this is the demonstration of-- 899 00:55:17,710 --> 00:55:20,440 the summary of Brewster's angle. 900 00:55:20,440 --> 00:55:28,450 So somebody reminded me that the amplitude should be given. 901 00:55:28,450 --> 00:55:32,560 So I think, this is the amplitude formula 902 00:55:32,560 --> 00:55:36,190 for Brewster's angle will be given to you. 903 00:55:36,190 --> 00:55:40,099 If not, it's asked in the final exam. 904 00:55:40,099 --> 00:55:41,890 So don't be worried about it, and you don't 905 00:55:41,890 --> 00:55:44,440 have to remember this formula. 906 00:55:44,440 --> 00:55:48,280 And I'm not going to ask you to derive that just 907 00:55:48,280 --> 00:55:52,490 in such a short time, the three hours in the final exam. 908 00:55:52,490 --> 00:55:57,160 But what is very important is to know how this Brewster's angle, 909 00:55:57,160 --> 00:56:03,250 why there's no refracted light polarizing in a way 910 00:56:03,250 --> 00:56:07,590 that the polarization should be-- 911 00:56:07,590 --> 00:56:11,020 why the refracted light is polarized, for example. 912 00:56:11,020 --> 00:56:19,850 And also why the transmitting wave is slightly polarized. 913 00:56:19,850 --> 00:56:24,140 And I think the conclusions you need to remember, 914 00:56:24,140 --> 00:56:29,020 and you need to know how to calculate the angle, at least. 915 00:56:29,020 --> 00:56:34,620 Because for this purely polarized light 916 00:56:34,620 --> 00:56:39,490 to be produced in a refracted light, 917 00:56:39,490 --> 00:56:43,960 you need to have normal angle between the direction 918 00:56:43,960 --> 00:56:45,840 of the refracted light and the direction 919 00:56:45,840 --> 00:56:47,380 of the transmitted light. 920 00:56:47,380 --> 00:56:51,340 And that, you should be able to remember. 921 00:56:51,340 --> 00:56:54,370 And you should be able to derive that also from your mind 922 00:56:54,370 --> 00:56:58,570 as well, because that means the direction of the oscillation 923 00:56:58,570 --> 00:57:01,780 of the molecule at the boundary will 924 00:57:01,780 --> 00:57:05,740 be in the direction of propagation 925 00:57:05,740 --> 00:57:07,720 of the refracted wave. 926 00:57:07,720 --> 00:57:12,640 Therefore, that cannot be the solution to the progressing 927 00:57:12,640 --> 00:57:13,790 electromagnetic wave. 928 00:57:13,790 --> 00:57:16,600 Therefore, the refracted waves are polarized. 929 00:57:16,600 --> 00:57:18,610 So if you follow this logic, then you 930 00:57:18,610 --> 00:57:24,360 don't really need to memorize all those formulas. 931 00:57:24,360 --> 00:57:25,640 All right. 932 00:57:25,640 --> 00:57:30,990 So finally, in the last part of the course, 933 00:57:30,990 --> 00:57:35,230 we focused on the superposition of many, many electromagnetic 934 00:57:35,230 --> 00:57:40,650 waves so you can produce constructive interference. 935 00:57:40,650 --> 00:57:44,840 Or that means all those waves are in phase. 936 00:57:44,840 --> 00:57:47,900 And you can have destructive interference 937 00:57:47,900 --> 00:57:50,010 when they are out of phase. 938 00:57:50,010 --> 00:57:52,455 And that is a very important topic, 939 00:57:52,455 --> 00:57:57,630 so you should review that for the preparation of the final. 940 00:57:57,630 --> 00:58:01,190 And you can see that there are three concrete examples which 941 00:58:01,190 --> 00:58:03,340 we used during the class. 942 00:58:03,340 --> 00:58:04,610 A laser beam. 943 00:58:04,610 --> 00:58:08,220 We talked about a water ripple in a demo. 944 00:58:08,220 --> 00:58:13,160 And we also studied how it make use of this phenomena 945 00:58:13,160 --> 00:58:16,110 to design a phased radar. 946 00:58:16,110 --> 00:58:21,860 So to detect this unknown object in the sky, what we really 947 00:58:21,860 --> 00:58:25,610 need to have is electromagnetic waves pointing 948 00:58:25,610 --> 00:58:27,560 to a specific direction. 949 00:58:27,560 --> 00:58:35,520 And that can be achieved by using multi-slit interference. 950 00:58:35,520 --> 00:58:41,650 And this is the property of the two-slit interference pattern. 951 00:58:41,650 --> 00:58:45,620 And you are going to have many, many peaks. 952 00:58:45,620 --> 00:58:50,360 They have equal height for two-slit interference 953 00:58:50,360 --> 00:58:56,300 If you ignore any effect coming from diffraction. 954 00:58:56,300 --> 00:59:04,250 So we've assume that the slit is infinitely small. 955 00:59:04,250 --> 00:59:07,280 The slit is super narrow. 956 00:59:07,280 --> 00:59:11,630 And then we can ignore the diffraction-- 957 00:59:11,630 --> 00:59:12,810 single-slit diffraction. 958 00:59:12,810 --> 00:59:17,240 In fact, then all the peaks due to this two-slit interference 959 00:59:17,240 --> 00:59:19,940 will have the same height. 960 00:59:19,940 --> 00:59:23,090 On the other hand, when we start to increase 961 00:59:23,090 --> 00:59:26,330 the number of slits, for example, unequal to 3, 962 00:59:26,330 --> 00:59:31,190 unequal to 4, unequal to 5, unequal to 6, 963 00:59:31,190 --> 00:59:37,880 as you can see that, the structure of the intensity 964 00:59:37,880 --> 00:59:42,830 as a function of delta, which is the phase difference, 965 00:59:42,830 --> 00:59:44,360 is actually changing. 966 00:59:44,360 --> 00:59:48,660 And you can see that the general structure is the following. 967 00:59:48,660 --> 00:59:52,280 So if you have unequal to 3, then basically, 968 00:59:52,280 --> 00:59:57,260 you have 2 of adult, and between them, you have 1 child. 969 00:59:57,260 --> 00:59:59,450 And if you have unequal to 6, then 970 00:59:59,450 --> 01:00:02,360 basically, you have 2 adults and somehow there 971 01:00:02,360 --> 01:00:06,260 are 4 children in this collection. 972 01:00:06,260 --> 01:00:10,910 So basically, that is what we learned from the solution 973 01:00:10,910 --> 01:00:13,400 of the multi-slit interference. 974 01:00:13,400 --> 01:00:15,640 And in this way, we can actually make 975 01:00:15,640 --> 01:00:20,960 the width of the principal maxima as narrow as you want. 976 01:00:20,960 --> 01:00:25,800 So that is why phased radar works. 977 01:00:25,800 --> 01:00:29,030 And then we discussed about diffraction. 978 01:00:29,030 --> 01:00:33,470 So that is related, again, to the explanation of laser beams. 979 01:00:33,470 --> 01:00:38,090 And we discussed about the design of a Star Trek 980 01:00:38,090 --> 01:00:41,990 ship, the gun for the ship. 981 01:00:41,990 --> 01:00:44,900 And we also talked about resolution. 982 01:00:44,900 --> 01:00:47,990 And what is actually happening here is the following. 983 01:00:47,990 --> 01:00:52,790 A single-slit diffraction essentially 984 01:00:52,790 --> 01:00:57,950 can be viewed as an infinite number of source interference. 985 01:00:57,950 --> 01:01:04,250 And you just need to integrate over all the point-like sources 986 01:01:04,250 --> 01:01:07,440 between the two walls. 987 01:01:07,440 --> 01:01:14,540 And all of them are acting like a spherical wave source. 988 01:01:14,540 --> 01:01:17,720 So basically, for every point-- 989 01:01:17,720 --> 01:01:22,952 continuously, every point between these two walls 990 01:01:22,952 --> 01:01:27,200 are a point source of spherical waves. 991 01:01:27,200 --> 01:01:29,720 And that is Huygens' principle. 992 01:01:29,720 --> 01:01:32,600 And we can see that the structures 993 01:01:32,600 --> 01:01:35,500 is-- of the intensity as a function of position 994 01:01:35,500 --> 01:01:36,420 is the following. 995 01:01:36,420 --> 01:01:38,720 So basically, you have a principal maxima, 996 01:01:38,720 --> 01:01:41,840 which is a peak in the middle. 997 01:01:41,840 --> 01:01:44,240 And at some angle, basically, you 998 01:01:44,240 --> 01:01:46,760 have destructive interference such 999 01:01:46,760 --> 01:01:49,560 that if you integrate over all the contributions 1000 01:01:49,560 --> 01:01:53,000 from an infinite number of sources in this window, 1001 01:01:53,000 --> 01:01:55,490 basically, you would see that they completely 1002 01:01:55,490 --> 01:01:56,840 cancel each other. 1003 01:01:56,840 --> 01:02:03,170 So that is the origin of all those deep structure minima. 1004 01:02:03,170 --> 01:02:07,390 And then, after the minima, actually, you 1005 01:02:07,390 --> 01:02:10,270 will see another peak, but the height of the peak 1006 01:02:10,270 --> 01:02:13,970 is suppressed by 1 over beta squared. 1007 01:02:13,970 --> 01:02:17,360 And it would be good to review that. 1008 01:02:17,360 --> 01:02:19,370 And what is the consequence? 1009 01:02:19,370 --> 01:02:23,980 So if you shoot a laser beam to the moon, 1010 01:02:23,980 --> 01:02:28,950 the size of the laser beam will be very large. 1011 01:02:28,950 --> 01:02:33,820 After you learn 8.03, you know that the size of the laser beam 1012 01:02:33,820 --> 01:02:38,830 is going to be very, very large due to interference 1013 01:02:38,830 --> 01:02:45,250 between all the point-like sources from the laser beam. 1014 01:02:45,250 --> 01:02:48,320 And finally, we can put them all together. 1015 01:02:48,320 --> 01:02:52,000 So the single-slit diffraction and 1016 01:02:52,000 --> 01:02:55,630 the multi-slit interference, you can put them 1017 01:02:55,630 --> 01:02:59,420 all together, and basically, what you get is the following. 1018 01:02:59,420 --> 01:03:04,540 So basically, you have a multi-slit interference 1019 01:03:04,540 --> 01:03:07,780 pattern, which is showing there. 1020 01:03:07,780 --> 01:03:13,450 But now the intensity of the multi-slit pattern 1021 01:03:13,450 --> 01:03:21,070 is modulated by the single-slit diffraction pattern. 1022 01:03:21,070 --> 01:03:25,370 And of course, the full formula will be given to you. 1023 01:03:25,370 --> 01:03:27,160 But on the other hand, you are also 1024 01:03:27,160 --> 01:03:31,780 requested to know how to calculate, just 1025 01:03:31,780 --> 01:03:36,250 to add the contribution from multi-slit 1026 01:03:36,250 --> 01:03:41,370 together in case if we change the amplitude of the incident 1027 01:03:41,370 --> 01:03:43,410 light or we change the phase, like what 1028 01:03:43,410 --> 01:03:45,970 we did in the homework. 1029 01:03:45,970 --> 01:03:49,690 And I think that is one important point, 1030 01:03:49,690 --> 01:03:51,670 and you should review that. 1031 01:03:51,670 --> 01:03:55,450 And if you are not sure about how to proceed with that, 1032 01:03:55,450 --> 01:04:00,770 it would be good to review Lecture 22, Lecture 23. 1033 01:04:00,770 --> 01:04:04,520 So finally, we talk about the connection 1034 01:04:04,520 --> 01:04:07,340 to quantum mechanics. 1035 01:04:07,340 --> 01:04:11,960 Einstein already told us that "I have said so many times, 1036 01:04:11,960 --> 01:04:15,520 God doesn't play dice with the world." 1037 01:04:15,520 --> 01:04:18,230 But what we actually find is that there 1038 01:04:18,230 --> 01:04:21,860 are two very interesting things which we found. 1039 01:04:21,860 --> 01:04:26,590 The first thing is that if we have a single photon source, 1040 01:04:26,590 --> 01:04:31,550 and basically, if we don't play dice, 1041 01:04:31,550 --> 01:04:38,120 we cannot explain the intensity of the-- 1042 01:04:38,120 --> 01:04:42,260 after this single photon source passes through two polarizers. 1043 01:04:42,260 --> 01:04:44,000 And what happens is the following. 1044 01:04:44,000 --> 01:04:49,220 Basically, the result of a single photon source 1045 01:04:49,220 --> 01:04:52,670 tells you that you really need to play dice 1046 01:04:52,670 --> 01:04:59,660 so that you can get the resulting polarized light 1047 01:04:59,660 --> 01:05:01,970 intensity. 1048 01:05:01,970 --> 01:05:05,480 And also, the second pseudo-experiment 1049 01:05:05,480 --> 01:05:10,450 we discussed is that if you have billiard balls, basically, 1050 01:05:10,450 --> 01:05:15,920 you have them pass through the two-slit experiment, what 1051 01:05:15,920 --> 01:05:18,260 you are going to get is two piles, 1052 01:05:18,260 --> 01:05:20,870 Gaussian-like distribution. 1053 01:05:20,870 --> 01:05:26,410 And if you have a single electron source, what it does 1054 01:05:26,410 --> 01:05:32,050 is that it interferes with itself. 1055 01:05:32,050 --> 01:05:36,080 An electron, a single electron, can interfere with itself 1056 01:05:36,080 --> 01:05:38,900 and produce a pattern which is very 1057 01:05:38,900 --> 01:05:43,550 similar to what we see in the double-slit interference 1058 01:05:43,550 --> 01:05:44,460 pattern. 1059 01:05:44,460 --> 01:05:46,950 So that is really remarkable. 1060 01:05:46,950 --> 01:05:50,960 And also, we talked about a single-slit-- 1061 01:05:50,960 --> 01:05:54,140 single electron experiment. 1062 01:05:54,140 --> 01:05:58,730 That gives you also a diffraction pattern. 1063 01:05:58,730 --> 01:06:02,550 We have to use the wave function to describe 1064 01:06:02,550 --> 01:06:06,890 the position-- the probability density of the position 1065 01:06:06,890 --> 01:06:09,220 of the electron on the screen. 1066 01:06:09,220 --> 01:06:14,930 And know this issue closely connected to the uncertainty 1067 01:06:14,930 --> 01:06:18,530 principle, which we discussed earlier, delta, p, delta, 1068 01:06:18,530 --> 01:06:22,190 x is greater than or equal to h bar over 2. 1069 01:06:22,190 --> 01:06:24,420 So if you have a very narrow window, 1070 01:06:24,420 --> 01:06:27,830 that means you have very similar delta x, 1071 01:06:27,830 --> 01:06:30,410 so you have very, very good confidence 1072 01:06:30,410 --> 01:06:32,840 about the location of the electron. 1073 01:06:32,840 --> 01:06:39,300 And then the momentum is in the x-- 1074 01:06:39,300 --> 01:06:41,670 in the momentum in the x direction, 1075 01:06:41,670 --> 01:06:45,830 you have large uncertainty, according to this equation. 1076 01:06:45,830 --> 01:06:52,350 And that can be seen from this single-slit diffraction pattern 1077 01:06:52,350 --> 01:06:56,640 and it is closely connected to what we have learned before. 1078 01:06:56,640 --> 01:07:01,520 So where is this-- how to actually describe 1079 01:07:01,520 --> 01:07:04,310 what this is really the dispersion 1080 01:07:04,310 --> 01:07:08,660 relation of the probability density wave 1081 01:07:08,660 --> 01:07:13,140 is actually coming from Schrodinger's Equation. 1082 01:07:13,140 --> 01:07:15,560 And this is given here. 1083 01:07:15,560 --> 01:07:18,560 We briefly talked about that. 1084 01:07:18,560 --> 01:07:22,970 And the consequence is the following. 1085 01:07:22,970 --> 01:07:27,560 You can describe the evolution of the wave 1086 01:07:27,560 --> 01:07:35,750 function as a function of time by using this wave equation. 1087 01:07:35,750 --> 01:07:38,690 And this wave equation is slightly different from what 1088 01:07:38,690 --> 01:07:42,040 we have learned before. 1089 01:07:42,040 --> 01:07:46,300 And we also can use what we have learned from 8.03 1090 01:07:46,300 --> 01:07:51,460 to solve a particle in a box problem, which is covered 1091 01:07:51,460 --> 01:07:54,970 in lecture number 23. 1092 01:07:54,970 --> 01:07:56,890 And I just wanted to say that you 1093 01:07:56,890 --> 01:07:59,530 need to know the general principle, 1094 01:07:59,530 --> 01:08:02,710 but I'm not teaching 8.04, so I'm not 1095 01:08:02,710 --> 01:08:07,510 expecting you to solve a quantum mechanics problem. 1096 01:08:07,510 --> 01:08:11,490 But I would like to say that OK, from this point, 1097 01:08:11,490 --> 01:08:15,190 it's motivating you to take 8.04, right? 1098 01:08:15,190 --> 01:08:18,350 Because there can be a lot of fun there as well. 1099 01:08:18,350 --> 01:08:25,010 And it is closely related to what we have learned from 8.03. 1100 01:08:25,010 --> 01:08:28,060 So I just want to say, the last point 1101 01:08:28,060 --> 01:08:31,410 is that this is really not the end of the vibrations 1102 01:08:31,410 --> 01:08:32,109 and waves. 1103 01:08:32,109 --> 01:08:33,609 It's just the beginning. 1104 01:08:33,609 --> 01:08:36,069 And that there is a path toward the peak. 1105 01:08:36,069 --> 01:08:41,784 And it may take a long time to reach the peak. 1106 01:08:41,784 --> 01:08:42,729 All right. 1107 01:08:42,729 --> 01:08:47,200 And I would like to let you know that I'm really, really 1108 01:08:47,200 --> 01:08:52,240 very happy to be your lecturer this semester. 1109 01:08:52,240 --> 01:08:55,630 And I really enjoyed teaching this class 1110 01:08:55,630 --> 01:08:59,680 and getting your responses when I asked questions. 1111 01:08:59,680 --> 01:09:01,939 Thank you for the support. 1112 01:09:01,939 --> 01:09:05,600 And I would like to say good luck with the final exam. 1113 01:09:05,600 --> 01:09:11,319 And we have 800 contributions on Piazza, many thanks to Yinan, 1114 01:09:11,319 --> 01:09:15,340 who is actually doing all the hard work, day and night. 1115 01:09:15,340 --> 01:09:21,500 And thank you very much, and see you around MIT in the future. 1116 01:09:28,400 --> 01:09:30,250 Thank you.