1 00:00:02,420 --> 00:00:04,760 The following content is provided under a Creative 2 00:00:04,760 --> 00:00:06,180 Commons license. 3 00:00:06,180 --> 00:00:08,390 Your support will help MIT OpenCourseWare 4 00:00:08,390 --> 00:00:12,480 continue to offer high quality educational resources for free. 5 00:00:12,480 --> 00:00:15,020 To make a donation or to view additional materials 6 00:00:15,020 --> 00:00:17,510 from hundreds of MIT courses, visit 7 00:00:17,510 --> 00:00:19,625 MITOpenCourseWare@ocw.mit.edu. 8 00:00:23,510 --> 00:00:27,840 YEN-JIE LEE: OK, happy to see you again. 9 00:00:27,840 --> 00:00:31,260 Welcome back to 8.03. 10 00:00:31,260 --> 00:00:34,290 Today, as you see on the slide, we're 11 00:00:34,290 --> 00:00:37,710 going to continue the discussion of dispersive medium-- 12 00:00:37,710 --> 00:00:40,200 how the waves and vibration should 13 00:00:40,200 --> 00:00:44,090 be sent through this medium. 14 00:00:44,090 --> 00:00:50,370 And also, we will learn about uncertainty principle today. 15 00:00:50,370 --> 00:00:52,200 Kind of interesting. 16 00:00:52,200 --> 00:00:57,120 That is connected back here to what we discuss here. 17 00:00:57,120 --> 00:01:00,970 And finally, if we have time, we'll 18 00:01:00,970 --> 00:01:05,570 move to two-dimensional system and three-dimensional system 19 00:01:05,570 --> 00:01:09,500 to look at two-dimensional waves and three-dimensional waves. 20 00:01:09,500 --> 00:01:13,420 OK, that's the plan for today. 21 00:01:13,420 --> 00:01:17,210 Just a quick review about what we have learned so far. 22 00:01:17,210 --> 00:01:21,590 Last time, we discussed about shaking 23 00:01:21,590 --> 00:01:24,870 one end of this dispersive medium which is actually 24 00:01:24,870 --> 00:01:28,160 a string with stiffness. 25 00:01:28,160 --> 00:01:32,210 And basically you would see that the strategy that we have been 26 00:01:32,210 --> 00:01:38,120 following is to do a Fourier transform to actually decompose 27 00:01:38,120 --> 00:01:42,710 the motion of the hand, which is actually holding 28 00:01:42,710 --> 00:01:49,580 one end of the string, and then decompose that into wave 29 00:01:49,580 --> 00:01:53,030 population in frequency space. 30 00:01:53,030 --> 00:01:55,280 OK, so that's what we have been doing. 31 00:01:55,280 --> 00:02:00,510 And then, we know based on the property of this medium, 32 00:02:00,510 --> 00:02:04,790 the dispersion relation, which is omega as a function of k, 33 00:02:04,790 --> 00:02:11,750 we can propagate waves with different frequency 34 00:02:11,750 --> 00:02:13,340 at different speeds. 35 00:02:13,340 --> 00:02:15,650 Then we can see how this system will 36 00:02:15,650 --> 00:02:17,970 evolve as a function of time. 37 00:02:17,970 --> 00:02:22,360 That's the whole idea and the strategy 38 00:02:22,360 --> 00:02:28,420 we approach this interesting problem. 39 00:02:28,420 --> 00:02:32,095 Last time, we also introduced AM radio. 40 00:02:35,430 --> 00:02:39,550 As we discussed before, if we have a very simple-minded 41 00:02:39,550 --> 00:02:41,800 strategy to just send the pulse-- 42 00:02:41,800 --> 00:02:44,260 which is containing information-- directly 43 00:02:44,260 --> 00:02:47,170 through this medium, due to the dispersion 44 00:02:47,170 --> 00:02:49,600 relation which we have this medium, 45 00:02:49,600 --> 00:02:53,860 different component would be traveling at different speed. 46 00:02:53,860 --> 00:02:57,710 Therefore, the information is smeared out 47 00:02:57,710 --> 00:02:59,950 after it travels through a long distance. 48 00:02:59,950 --> 00:03:00,760 OK? 49 00:03:00,760 --> 00:03:01,860 That's the problem. 50 00:03:01,860 --> 00:03:10,350 And then the solution was to use this approach, which 51 00:03:10,350 --> 00:03:16,380 is amplitude modulation mixer. 52 00:03:16,380 --> 00:03:18,980 That's actually how AM radio works. 53 00:03:18,980 --> 00:03:26,380 So basically, we have a slowly oscillating message or signal 54 00:03:26,380 --> 00:03:29,910 like music or voice which we want to send, 55 00:03:29,910 --> 00:03:35,250 and then as we multiply that by a really fast oscillating 56 00:03:35,250 --> 00:03:37,570 cosine tan. 57 00:03:37,570 --> 00:03:40,380 If we do this, assuming that omega of 0 58 00:03:40,380 --> 00:03:45,750 is actually much, much higher or larger than the typical scale 59 00:03:45,750 --> 00:03:47,200 of your signal-- 60 00:03:47,200 --> 00:03:48,790 which is omega s-- 61 00:03:48,790 --> 00:03:51,330 then, what is going to happen is the following. 62 00:03:51,330 --> 00:03:54,350 Up to all the calculations we have done last time, 63 00:03:54,350 --> 00:03:59,440 we found that the resulting wave which 64 00:03:59,440 --> 00:04:04,830 is the amplitude as a function of time and space, 65 00:04:04,830 --> 00:04:08,770 you can see that this can factorize into two components. 66 00:04:08,770 --> 00:04:12,160 The first component is virtually the original signal 67 00:04:12,160 --> 00:04:13,750 you are trying to send. 68 00:04:13,750 --> 00:04:17,630 Since you're traveling at the speed of group velocity, 69 00:04:17,630 --> 00:04:21,130 and finally, the right hand side-- the second component-- 70 00:04:21,130 --> 00:04:28,300 is actually the contribution, the really small structure 71 00:04:28,300 --> 00:04:29,830 of these high frequency oscillation. 72 00:04:29,830 --> 00:04:32,200 We call it carrier, and the carrier 73 00:04:32,200 --> 00:04:36,740 is still traveling at the of face velocity. 74 00:04:36,740 --> 00:04:39,560 That's how we actually finally understand 75 00:04:39,560 --> 00:04:42,010 what is the meaning of group velocity 76 00:04:42,010 --> 00:04:47,050 and the face velocity through this example. 77 00:04:47,050 --> 00:04:50,950 What I am going to do today is to guide you 78 00:04:50,950 --> 00:04:53,720 through another example which will 79 00:04:53,720 --> 00:04:58,870 ensure we can learn some more insight from this calculation. 80 00:04:58,870 --> 00:05:03,250 Today, we are going to have another test of function, 81 00:05:03,250 --> 00:05:09,250 which actually I can do Fourier transforms really easily. 82 00:05:09,250 --> 00:05:12,700 And this function I'm trying to introduce here, 83 00:05:12,700 --> 00:05:16,300 I have this functional form exponential 84 00:05:16,300 --> 00:05:22,000 minus gamma times absolute value of t, OK? 85 00:05:22,000 --> 00:05:24,850 The reason why I choose absolute value of t 86 00:05:24,850 --> 00:05:28,440 is because I would like to make it symmetric around 0. 87 00:05:32,760 --> 00:05:36,340 I can now do the usual Fourier transform 88 00:05:36,340 --> 00:05:38,680 and then to extract the wave population. 89 00:05:38,680 --> 00:05:43,110 The function of angular frequency, c omega. 90 00:05:43,110 --> 00:05:45,010 c as a function of omega. 91 00:05:45,010 --> 00:05:47,500 And according to the formula here, 92 00:05:47,500 --> 00:05:50,560 which we introduced last time, we 93 00:05:50,560 --> 00:05:52,750 can quickly write it down like this. 94 00:05:52,750 --> 00:05:56,410 Basically you get 1 over 2 pi integration from minus 95 00:05:56,410 --> 00:06:02,080 to infinity to infinity integrating over time. 96 00:06:02,080 --> 00:06:05,950 This is the original function, f of t. 97 00:06:05,950 --> 00:06:09,140 And multiply that by exponential I omega t. 98 00:06:09,140 --> 00:06:13,060 And that's the way we extract c omega. 99 00:06:13,060 --> 00:06:14,180 OK? 100 00:06:14,180 --> 00:06:16,090 Since we have this absolute value 101 00:06:16,090 --> 00:06:23,960 here, basically the trick is to change the interval, 102 00:06:23,960 --> 00:06:26,440 split the interval into two pieces. 103 00:06:26,440 --> 00:06:30,850 So, y is actually the negative t part, 104 00:06:30,850 --> 00:06:35,440 therefore, you get the exponential plus t here, 105 00:06:35,440 --> 00:06:39,220 and the other part is from 0 to infinity. 106 00:06:39,220 --> 00:06:42,310 Then the absolute value doesn't change side. 107 00:06:42,310 --> 00:06:44,050 You have the original exponential 108 00:06:44,050 --> 00:06:47,110 minus gamma times t. 109 00:06:47,110 --> 00:06:50,230 Then you can go ahead and do with integration, 110 00:06:50,230 --> 00:06:53,680 and you get the two turns and you get the functional 111 00:06:53,680 --> 00:06:58,000 form, which is c omega equal to gamma 112 00:06:58,000 --> 00:07:02,300 over pi times gamma square plus omega square. 113 00:07:02,300 --> 00:07:03,160 OK? 114 00:07:03,160 --> 00:07:07,440 From this simple exercise, they are interesting things 115 00:07:07,440 --> 00:07:10,380 which we can learn from here. 116 00:07:10,380 --> 00:07:21,160 If I go ahead and draw f of t as a function of time, 117 00:07:21,160 --> 00:07:24,850 this is what you will get. 118 00:07:24,850 --> 00:07:32,060 Suppose I set gamma to be equal to 0.1. 119 00:07:32,060 --> 00:07:33,950 And I would like to visualize this function 120 00:07:33,950 --> 00:07:36,070 and that's what we did here. 121 00:07:36,070 --> 00:07:40,930 You can see that from the left hand side here, 122 00:07:40,930 --> 00:07:44,440 is f of t as a function of time. 123 00:07:44,440 --> 00:07:48,980 And you can see that this like exponential 124 00:07:48,980 --> 00:07:54,490 of t k but symmetric that mirror at the t equal to 0. 125 00:07:54,490 --> 00:07:58,990 And with a small gamma value I choose, 126 00:07:58,990 --> 00:08:01,300 that means this exponential decay will 127 00:08:01,300 --> 00:08:08,090 be really slow, therefore, you have a pretty wide distribution 128 00:08:08,090 --> 00:08:11,430 as a function of time. 129 00:08:11,430 --> 00:08:14,634 However, if you look at the right hand side, what did 130 00:08:14,634 --> 00:08:16,050 I show you in the right hand side? 131 00:08:16,050 --> 00:08:21,650 Right hand side is c omega, c as a function of omega, 132 00:08:21,650 --> 00:08:25,480 it's the population in frequency space. 133 00:08:25,480 --> 00:08:29,610 And you can see that, if I plug in gamma equal to 0.1 134 00:08:29,610 --> 00:08:31,590 into that equation, then you would 135 00:08:31,590 --> 00:08:34,140 get a distribution which is actually 136 00:08:34,140 --> 00:08:39,409 pretty narrow, around 0. 137 00:08:39,409 --> 00:08:41,120 That's actually quite interesting. 138 00:08:41,120 --> 00:08:46,460 And now, if I change gamma, I increase the gamma slowly 139 00:08:46,460 --> 00:08:52,370 so it changes to 0.2, you see aha, that's what I expect-- 140 00:08:52,370 --> 00:08:56,780 the f function graphed in the coordinate space 141 00:08:56,780 --> 00:08:58,800 becomes narrower. 142 00:08:58,800 --> 00:09:00,800 But, on the other hand, you pay the price 143 00:09:00,800 --> 00:09:07,200 that the wave population in the frequency space becomes wider. 144 00:09:07,200 --> 00:09:10,270 OK, the distribution become wider. 145 00:09:10,270 --> 00:09:11,650 I can increase and increase. 146 00:09:11,650 --> 00:09:14,200 Now it's gamma equal to 0.5. 147 00:09:14,200 --> 00:09:16,370 Gamma equal to 1. 148 00:09:16,370 --> 00:09:19,450 And now I have a rather large gamma. 149 00:09:19,450 --> 00:09:25,750 Now it says 2.0, and you can see that as a function of gamma, 150 00:09:25,750 --> 00:09:32,650 if I set the gamma to be 5, and you can see that the wave, 151 00:09:32,650 --> 00:09:38,350 or say the waves of the wave in the coordinate space, 152 00:09:38,350 --> 00:09:40,450 becomes really small. 153 00:09:40,450 --> 00:09:45,820 But if you look at the corresponding c function, 154 00:09:45,820 --> 00:09:49,570 you can see that waves becomes really large. 155 00:09:49,570 --> 00:09:53,680 This seems to be telling us something interesting. 156 00:09:53,680 --> 00:09:58,840 It seems to me that I could not choose a gamma value which 157 00:09:58,840 --> 00:10:03,430 simultaneously make waves in a coordinate space 158 00:10:03,430 --> 00:10:07,000 narrow and those wave populations in the frequency 159 00:10:07,000 --> 00:10:10,360 space narrow at the same time. 160 00:10:10,360 --> 00:10:15,670 I cannot actually do that based on this simple-minded exercise. 161 00:10:15,670 --> 00:10:17,950 And what you are going to do in your p-set 162 00:10:17,950 --> 00:10:20,680 is to go through another parameterization, which 163 00:10:20,680 --> 00:10:22,570 is a Gaussian distribution. 164 00:10:22,570 --> 00:10:26,710 And you will see very similar, hope 165 00:10:26,710 --> 00:10:31,450 for these very similar conclusion from your exercise. 166 00:10:31,450 --> 00:10:33,560 So what is going on? 167 00:10:33,560 --> 00:10:37,660 And how do we interpret this result? 168 00:10:37,660 --> 00:10:40,120 And why is this result actually related 169 00:10:40,120 --> 00:10:42,850 to uncertainty principle? 170 00:10:42,850 --> 00:10:45,460 That's the first part of the lecture, which 171 00:10:45,460 --> 00:10:48,700 we are going to discuss today. 172 00:10:48,700 --> 00:10:57,810 We can demonstrate this in fact by one example of f 173 00:10:57,810 --> 00:11:00,670 of t, which is showing here. 174 00:11:00,670 --> 00:11:04,930 And we go through and change the waves of this distribution. 175 00:11:04,930 --> 00:11:07,450 Of course, we can also try to show 176 00:11:07,450 --> 00:11:13,180 this in a much more precise mathematical definition. 177 00:11:13,180 --> 00:11:15,480 That's what we are going to do now. 178 00:11:18,080 --> 00:11:20,600 The first thing which we would need to do 179 00:11:20,600 --> 00:11:24,800 is to define how to quantify the waves 180 00:11:24,800 --> 00:11:33,170 of the distribution in frequency space and in coordinate space. 181 00:11:33,170 --> 00:11:41,650 First, we define that the intensity of the signal 182 00:11:41,650 --> 00:11:49,040 is proportional to f of t squared. 183 00:11:49,040 --> 00:11:53,805 OK, that is the to estimate the size of the intensity. 184 00:11:53,805 --> 00:11:57,360 It kind of makes sense because, for example, 185 00:11:57,360 --> 00:12:00,500 the energy of the electromagnetic wave 186 00:12:00,500 --> 00:12:05,150 is actually proportional to the wave function squared. 187 00:12:05,150 --> 00:12:10,220 That's kind of reasonable to choose this definition. 188 00:12:10,220 --> 00:12:14,000 And then, once we have that the definition of intensity, 189 00:12:14,000 --> 00:12:19,240 then I can now calculate the average of some operator 190 00:12:19,240 --> 00:12:20,610 function. 191 00:12:20,610 --> 00:12:23,900 For example, I can calculate g of t 192 00:12:23,900 --> 00:12:27,210 is a average of the g function. 193 00:12:27,210 --> 00:12:30,170 And in this definition of intensity, 194 00:12:30,170 --> 00:12:34,770 how to calculate the average is to do integration 195 00:12:34,770 --> 00:12:40,700 over minus infinity to infinity over t. 196 00:12:40,700 --> 00:12:43,220 And this g function is put right there 197 00:12:43,220 --> 00:12:46,280 and all the components are weighted 198 00:12:46,280 --> 00:12:52,180 by this intensity estimator, which f of t squared. 199 00:12:55,070 --> 00:12:59,720 Of course, since we are actually calculating the average, 200 00:12:59,720 --> 00:13:02,930 we need to take out the sum of the intensity. 201 00:13:02,930 --> 00:13:05,000 So, the sum of all the intensity is 202 00:13:05,000 --> 00:13:08,440 an integration from minus infinity to infinity-- 203 00:13:08,440 --> 00:13:13,820 dt f of t squared. 204 00:13:13,820 --> 00:13:18,690 With this definition, we can calculate the average. 205 00:13:18,690 --> 00:13:28,020 OK and don't forget our goal is to have an estimator estimate 206 00:13:28,020 --> 00:13:31,510 the waves of sum distribution. 207 00:13:31,510 --> 00:13:35,320 Therefore, you are probably very familiar with that. 208 00:13:35,320 --> 00:13:37,960 We use standard deviation. 209 00:13:37,960 --> 00:13:41,560 So basically that's also usually associated with the exam, 210 00:13:41,560 --> 00:13:46,900 but this time it's associated with some physical quantity. 211 00:13:46,900 --> 00:13:53,740 what is the estimator of spread of time? 212 00:13:56,660 --> 00:13:58,800 Right. 213 00:13:58,800 --> 00:14:02,460 We can actually make use of this definition 214 00:14:02,460 --> 00:14:06,630 and I can write this notation that 215 00:14:06,630 --> 00:14:12,330 t-squared to be a quantity which is associated 216 00:14:12,330 --> 00:14:16,080 with the size of spread in time. 217 00:14:16,080 --> 00:14:24,866 And now as you define to be the average of t minus average of t 218 00:14:24,866 --> 00:14:25,366 squared. 219 00:14:27,960 --> 00:14:30,510 Basically you calculate that difference with respect 220 00:14:30,510 --> 00:14:36,920 to the mean value, square it, and then do the two averages 221 00:14:36,920 --> 00:14:39,620 again. 222 00:14:39,620 --> 00:14:42,110 All right, everybody is following? 223 00:14:42,110 --> 00:14:43,960 Any questions? 224 00:14:43,960 --> 00:14:45,020 OK. 225 00:14:45,020 --> 00:14:48,830 If I have this so-called standard deviation or spread 226 00:14:48,830 --> 00:14:54,210 of time definition here, then I can write it down explicitly, 227 00:14:54,210 --> 00:14:57,790 and this will become minus infinity to infinity, 228 00:14:57,790 --> 00:15:07,370 to disintegration over t, and I have t minus n value of t, 229 00:15:07,370 --> 00:15:10,670 half of t squared. 230 00:15:10,670 --> 00:15:13,700 And of course I would take out that normalization, which 231 00:15:13,700 --> 00:15:24,830 is a minus infinity to infinity dt f of t squared 232 00:15:24,830 --> 00:15:31,670 And I can also do a similar exercise for the frequency 233 00:15:31,670 --> 00:15:32,660 space. 234 00:15:32,660 --> 00:15:38,990 Basically, I can define the spread 235 00:15:38,990 --> 00:15:47,143 of the frequency spectrum. 236 00:15:50,600 --> 00:15:53,770 And that I define it to be delta omega 237 00:15:53,770 --> 00:15:56,700 squared, and this will be defined 238 00:15:56,700 --> 00:16:02,100 as the average of omega minus the mean value of omega 239 00:16:02,100 --> 00:16:02,600 squared. 240 00:16:07,800 --> 00:16:13,230 And with this definition, we have an estimator 241 00:16:13,230 --> 00:16:18,150 of this spread of time, and we have an estimator 242 00:16:18,150 --> 00:16:20,610 of spread the frequency. 243 00:16:20,610 --> 00:16:24,360 The phenomenon which we see from here, from this exercise, 244 00:16:24,360 --> 00:16:30,660 going from low gamma value to a large gamma value is that it 245 00:16:30,660 --> 00:16:37,290 seems to us that the spread of the time or in coordinate space 246 00:16:37,290 --> 00:16:40,920 and the spread of the distribution in the frequency 247 00:16:40,920 --> 00:16:45,190 space cannot simultaneously be small. 248 00:16:45,190 --> 00:16:46,170 OK. 249 00:16:46,170 --> 00:16:50,880 Therefore, based on this mathematical definition, 250 00:16:50,880 --> 00:16:54,950 our goal is now to show that we can 251 00:16:54,950 --> 00:17:02,826 prove that delta omega times delta t 252 00:17:02,826 --> 00:17:06,599 will be larger or equal to 1/2. 253 00:17:06,599 --> 00:17:10,650 That is an interesting consequence based 254 00:17:10,650 --> 00:17:13,710 on this definition of spread. 255 00:17:13,710 --> 00:17:16,290 We can actually achieve the lecture today. 256 00:17:19,230 --> 00:17:20,550 That's our goal. 257 00:17:20,550 --> 00:17:25,800 And we are going to try to achieve this goal. 258 00:17:25,800 --> 00:17:30,450 Before we go ahead and prove this relation 259 00:17:30,450 --> 00:17:35,670 delta omega times delta t greater or equal to 1/2, 260 00:17:35,670 --> 00:17:41,010 we also realize that when we discuss 261 00:17:41,010 --> 00:17:45,780 this spread of the frequencies spectrum, 262 00:17:45,780 --> 00:17:49,140 if I write it down here, if I try 263 00:17:49,140 --> 00:17:53,310 to calculate the average of omega, 264 00:17:53,310 --> 00:17:56,880 then what I'm going to do is to do the integration from 265 00:17:56,880 --> 00:18:00,870 minus infinity to infinity, d omega. 266 00:18:00,870 --> 00:18:03,390 because now I'm trying to calculate 267 00:18:03,390 --> 00:18:06,090 the mean value of omega. 268 00:18:06,090 --> 00:18:09,810 I have the omega times c omega. 269 00:18:12,600 --> 00:18:15,390 And the exponential is i omega t. 270 00:18:24,700 --> 00:18:31,820 If I go ahead and evaluate this integral, 271 00:18:31,820 --> 00:18:37,025 I integrate over omega, and I have omega times c omega 272 00:18:37,025 --> 00:18:42,820 times exponential i omega t. 273 00:18:42,820 --> 00:18:47,500 And you can see that this omega can actually be extracted 274 00:18:47,500 --> 00:18:50,110 from this exponential function. 275 00:18:50,110 --> 00:18:55,760 If I do differentiation, which is spread through time, 276 00:18:55,760 --> 00:18:57,970 then I can actually extract one omega out 277 00:18:57,970 --> 00:19:00,470 of the initial function. 278 00:19:00,470 --> 00:19:03,580 Therefore, what I'm going to get is this will be equal 279 00:19:03,580 --> 00:19:14,390 to i partial t minus infinity to infinity d omega, c omega, 280 00:19:14,390 --> 00:19:18,640 exponential minus i omega t. 281 00:19:18,640 --> 00:19:20,800 So you can see that this is the design 282 00:19:20,800 --> 00:19:24,760 if I do a partiality relative to with respect to t, 283 00:19:24,760 --> 00:19:29,260 then I take minus i omega out of this exponential function 284 00:19:29,260 --> 00:19:33,690 and this i will make minus i become 1. 285 00:19:33,690 --> 00:19:39,490 Therefore, you can see that this integral, which I construct, 286 00:19:39,490 --> 00:19:43,790 is equal to i partial, partial t. 287 00:19:43,790 --> 00:19:45,790 This function. 288 00:19:45,790 --> 00:19:52,210 OK, and you can quickly realize that we know what 289 00:19:52,210 --> 00:19:54,330 this integral is doing right. 290 00:19:54,330 --> 00:19:59,470 According to the form which I just did here, f of t 291 00:19:59,470 --> 00:20:02,900 is equal to this integral which I actually just highlight 292 00:20:02,900 --> 00:20:03,910 there. 293 00:20:03,910 --> 00:20:08,366 Therefore, this is just f of t. 294 00:20:10,870 --> 00:20:16,500 that's kind of interesting because that would give me i 295 00:20:16,500 --> 00:20:20,715 partial, partial t, f of t. 296 00:20:25,770 --> 00:20:30,570 Basically you can see that I don't need to deal with omega, 297 00:20:30,570 --> 00:20:35,100 I can actually do a partial relative with respect to time, 298 00:20:35,100 --> 00:20:38,490 then I can take one omega out of the function which 299 00:20:38,490 --> 00:20:41,736 I have constructed. 300 00:20:41,736 --> 00:20:45,440 Any questions? 301 00:20:45,440 --> 00:20:52,490 All right, now I can calculate what will be the mean omega. 302 00:20:52,490 --> 00:20:53,750 What would be the mean omega? 303 00:20:53,750 --> 00:20:58,570 The mean omega, according to this definition here, 304 00:20:58,570 --> 00:21:03,590 this is how we calculate the mean of some quantity, 305 00:21:03,590 --> 00:21:09,360 mean omega will be equal to minus infinity to infinity, 306 00:21:09,360 --> 00:21:18,990 tt f star, t i partial, partial t, f of t. 307 00:21:18,990 --> 00:21:22,430 OK sorry that this is kind of close to here. 308 00:21:27,180 --> 00:21:32,100 The original definition I should put omega got here, right? 309 00:21:32,100 --> 00:21:34,770 But instead of putting omega there, 310 00:21:34,770 --> 00:21:39,810 I used the trick that this i times partial partial t 311 00:21:39,810 --> 00:21:43,440 can generate an omega for me. 312 00:21:43,440 --> 00:21:46,860 Therefore, instead of putting omega explicitly 313 00:21:46,860 --> 00:21:50,940 into the integral, I put i partial partial t 314 00:21:50,940 --> 00:21:53,880 into the integral, then I get 1 omega out of it, 315 00:21:53,880 --> 00:21:57,970 and that's equivalent to the calculation 316 00:21:57,970 --> 00:22:00,930 with g of t equal to omega. 317 00:22:00,930 --> 00:22:02,606 OK, everybody's following? 318 00:22:07,470 --> 00:22:10,430 Therefore, I of course still need 319 00:22:10,430 --> 00:22:14,420 to normalize the calculation. 320 00:22:14,420 --> 00:22:20,100 This is the denominator, which is minus infinity to infinity, 321 00:22:20,100 --> 00:22:26,450 integral over tt f of t squared. 322 00:22:26,450 --> 00:22:30,280 OK, you can see that instead of using omega directly, 323 00:22:30,280 --> 00:22:36,500 the I used this trick to use i partial partial t to extract 324 00:22:36,500 --> 00:22:41,040 1 omega and I can calculate the mean value of omega. 325 00:22:46,930 --> 00:22:49,630 Therefore I can also calculate explicitly 326 00:22:49,630 --> 00:22:54,590 what would be the delta omega square based on the definition 327 00:22:54,590 --> 00:22:56,080 which I outlined before. 328 00:22:56,080 --> 00:23:01,600 This would be the average value of omega minus mean omega. 329 00:23:01,600 --> 00:23:06,880 Mean omega is a number, and if I'd write it down 330 00:23:06,880 --> 00:23:15,050 explicitly I get minus infinity to infinity, tt, i 331 00:23:15,050 --> 00:23:22,206 partial partial t, minus average value omega, f 332 00:23:22,206 --> 00:23:30,520 of t squared divided by minus infinity 333 00:23:30,520 --> 00:23:39,530 to infinity disintegration over dt f of t squared. 334 00:23:39,530 --> 00:23:41,930 The take home message is that I'm 335 00:23:41,930 --> 00:23:46,430 using this trick to replace all the omega 336 00:23:46,430 --> 00:23:48,860 by i partial partial t. 337 00:23:48,860 --> 00:23:52,060 Therefore, in my formula, you will 338 00:23:52,060 --> 00:23:56,570 see that originally, this is supposed to be omega 339 00:23:56,570 --> 00:23:59,480 and now we were using that trick. 340 00:23:59,480 --> 00:24:02,720 Therefore, it can be written as pi partial partial t. 341 00:24:02,720 --> 00:24:08,450 And you'll realize what this is used for afterwards. 342 00:24:08,450 --> 00:24:11,960 All right, so those are just preparation. 343 00:24:11,960 --> 00:24:14,960 What we have done is that my goal 344 00:24:14,960 --> 00:24:19,460 is to show that delta omega times delta t 345 00:24:19,460 --> 00:24:22,670 is greater than or equal to 1/2. 346 00:24:22,670 --> 00:24:25,980 OK, that's my goal and I'm preparing for that. 347 00:24:25,980 --> 00:24:29,590 And I have that definition of delta t and delta omega. 348 00:24:29,590 --> 00:24:30,090 Yes? 349 00:24:30,090 --> 00:24:35,660 AUDIENCE: What do you think [INAUDIBLE] 350 00:24:35,660 --> 00:24:39,940 YEN-JIE LEE: Oh sorry, there should be-- 351 00:24:39,940 --> 00:24:41,570 it should be like this. 352 00:24:41,570 --> 00:24:44,630 So I am taking partial partial t out of f. 353 00:24:44,630 --> 00:24:46,040 OK, sorry. 354 00:24:46,040 --> 00:24:48,230 Good question. 355 00:24:48,230 --> 00:24:51,020 Any other mistakes? 356 00:24:51,020 --> 00:24:52,250 Very good. 357 00:24:52,250 --> 00:24:53,170 Not yet? 358 00:24:53,170 --> 00:24:54,380 All right. 359 00:24:54,380 --> 00:24:58,470 So now you can see that I have the definition in my hand, 360 00:24:58,470 --> 00:25:00,620 and I am almost there to show you 361 00:25:00,620 --> 00:25:03,500 that delta omega times delta t is going 362 00:25:03,500 --> 00:25:08,030 to be greater or equal to 1/2. 363 00:25:08,030 --> 00:25:13,160 And what I'm going to do after this-- 364 00:25:13,160 --> 00:25:17,460 maybe you will be even more mad at me-- 365 00:25:17,460 --> 00:25:21,170 is to use exactly the same trick which 366 00:25:21,170 --> 00:25:25,130 would be used to show Heisenberg's Uncertainty 367 00:25:25,130 --> 00:25:29,300 Principle in quantum mechanics. 368 00:25:29,300 --> 00:25:33,590 basically what I'm going to do is to consider a function which 369 00:25:33,590 --> 00:25:34,910 is r of t. 370 00:25:39,805 --> 00:25:44,840 r as a function kappa and t. 371 00:25:44,840 --> 00:25:48,830 and the definition of this r function is like this. 372 00:25:48,830 --> 00:26:04,060 I define this r function to be t minus average t minus i kappa i 373 00:26:04,060 --> 00:26:07,310 partial partial t minus omega. 374 00:26:10,202 --> 00:26:11,648 f of t. 375 00:26:14,550 --> 00:26:21,150 If you don't know where is this relationship coming from, 376 00:26:21,150 --> 00:26:24,780 don't be worried because you don't really need to. 377 00:26:24,780 --> 00:26:31,230 This is just to guide us through this mathematical calculation. 378 00:26:31,230 --> 00:26:34,685 But if you can see directly how this will help, 379 00:26:34,685 --> 00:26:36,433 the maybe you are Heisenberg. 380 00:26:39,243 --> 00:26:39,743 Maybe. 381 00:26:39,743 --> 00:26:40,980 So that's very nice. 382 00:26:40,980 --> 00:26:41,580 It's a test. 383 00:26:45,030 --> 00:26:48,390 What I am going to do is to employ this r 384 00:26:48,390 --> 00:26:51,780 as a function of kappa of t. 385 00:26:51,780 --> 00:26:55,960 And the 2 for our purpose to show that the delta omega 386 00:26:55,960 --> 00:26:58,370 and delta t greater than 1/2. 387 00:26:58,370 --> 00:27:01,330 And first, to make my life easier, 388 00:27:01,330 --> 00:27:05,490 I would define this to be capital T, 389 00:27:05,490 --> 00:27:12,090 and I would define this thing to be capital omega. 390 00:27:12,090 --> 00:27:15,120 So that my mathematical expression doesn't explode. 391 00:27:19,680 --> 00:27:27,210 Now I can consider this ratio function r of kappa. 392 00:27:27,210 --> 00:27:30,870 This is defined as minus infinity 393 00:27:30,870 --> 00:27:40,070 to infinity integrating over t r kappa t divided 394 00:27:40,070 --> 00:27:48,700 by minus infinity to infinity dt f of t squared. 395 00:27:48,700 --> 00:27:56,650 This is r function which is the ratio of the area of r function 396 00:27:56,650 --> 00:28:00,067 and the area of the f function. 397 00:28:00,067 --> 00:28:02,150 You may say that, professor, this is really crazy. 398 00:28:02,150 --> 00:28:04,990 Today is telling about all the crazy things, 399 00:28:04,990 --> 00:28:07,232 but that is because I would like to let 400 00:28:07,232 --> 00:28:11,200 you know that we are going to see a very interesting result. 401 00:28:11,200 --> 00:28:14,050 So that's why I'm doing this. 402 00:28:14,050 --> 00:28:18,580 And if I construct this r function, 403 00:28:18,580 --> 00:28:22,150 this r function will have an interesting property. 404 00:28:22,150 --> 00:28:24,080 What is the interesting property? 405 00:28:24,080 --> 00:28:26,710 I entered an integral over something 406 00:28:26,710 --> 00:28:32,080 squared in the numerator and the denominator. 407 00:28:32,080 --> 00:28:37,590 Now it means, what would be the value of this r function? 408 00:28:37,590 --> 00:28:42,290 The r function would be always positive. 409 00:28:42,290 --> 00:28:43,430 Right? 410 00:28:43,430 --> 00:28:45,920 Because this is a square, this is a square, therefore, 411 00:28:45,920 --> 00:28:48,890 r is going to be positive. 412 00:28:48,890 --> 00:28:55,780 That means r is going to be greater than or equal to 0. 413 00:28:55,780 --> 00:28:58,320 That's why we have this r function. 414 00:28:58,320 --> 00:29:02,470 And the miracle will happen because if I go ahead 415 00:29:02,470 --> 00:29:05,020 and calculate this r-- 416 00:29:05,020 --> 00:29:08,350 before I calculate this capital R function-- what's 417 00:29:08,350 --> 00:29:14,380 the function of kappa, I need to actually deal with this small r 418 00:29:14,380 --> 00:29:18,250 as a function of kappa and t squared. 419 00:29:18,250 --> 00:29:24,885 If I extract this component and then calculate that, 420 00:29:24,885 --> 00:29:30,190 r kappa t squared. 421 00:29:30,190 --> 00:29:34,150 What I am going to do is to use this expression r 422 00:29:34,150 --> 00:29:46,150 is equal to t, capital T minus i kappa omega times f of t. 423 00:29:46,150 --> 00:29:48,940 So that my life would be easier. 424 00:29:48,940 --> 00:29:57,690 Then basically you get t minus i kappa omega f. 425 00:29:57,690 --> 00:30:00,750 And then you need tje complex conjugate. 426 00:30:00,750 --> 00:30:13,510 Basically, you get T cross i kappa omega star f star. 427 00:30:13,510 --> 00:30:18,000 You can have T star, but T is a real number. 428 00:30:18,000 --> 00:30:20,095 Therefore, it doesn't do anything. 429 00:30:23,200 --> 00:30:26,884 Then, I can now go ahead and collect all the terms. 430 00:30:26,884 --> 00:30:28,550 Then the first terms which I can collect 431 00:30:28,550 --> 00:30:32,450 is everything related to T times f. 432 00:30:32,450 --> 00:30:37,710 Then basically you get the T f squared. 433 00:30:37,710 --> 00:30:43,920 That is coming from this T times f times T times f. 434 00:30:46,680 --> 00:30:50,870 This term times this term times this term. 435 00:30:50,870 --> 00:30:53,640 to give you the first term. 436 00:30:53,640 --> 00:30:55,600 And you also you can connect another term 437 00:30:55,600 --> 00:31:00,740 which omega f squared. 438 00:31:00,740 --> 00:31:01,580 Right. 439 00:31:01,580 --> 00:31:09,601 Basically, you can find that contribution. 440 00:31:17,590 --> 00:31:22,360 Use should have a kappa square in front of it. 441 00:31:22,360 --> 00:31:24,160 Any questions so far? 442 00:31:24,160 --> 00:31:29,740 Basically, I collect the terms related to omega times f 443 00:31:29,740 --> 00:31:31,780 and put it here. 444 00:31:31,780 --> 00:31:33,610 Finally, you have the third term, 445 00:31:33,610 --> 00:31:52,020 which is i kappa T f omega star f star minus omega f T f star. 446 00:31:52,020 --> 00:31:56,700 Basically, this small r function squared 447 00:31:56,700 --> 00:32:01,270 can be written in this functional form. 448 00:32:01,270 --> 00:32:04,150 We are almost there. 449 00:32:04,150 --> 00:32:06,840 What I'm going to discuss first is that now I 450 00:32:06,840 --> 00:32:10,180 have these three terms. 451 00:32:10,180 --> 00:32:14,860 Number one, number two, and number three. 452 00:32:17,380 --> 00:32:20,080 I can now attack number three first. 453 00:32:23,300 --> 00:32:33,750 Number three, I'm going to get i kappa Tf minus i partial 454 00:32:33,750 --> 00:32:42,100 partial t minus omega f star. 455 00:32:42,100 --> 00:32:47,710 Basically what I'm doing is to take this omega here. 456 00:32:47,710 --> 00:32:50,170 This is omega star. 457 00:32:50,170 --> 00:32:53,110 And then use that definition, write down 458 00:32:53,110 --> 00:32:55,420 the expression for omega-- 459 00:32:55,420 --> 00:32:57,550 typical omega-- explicitly. 460 00:32:57,550 --> 00:33:02,660 Since I am writing omega star, therefore, you get a minus i 461 00:33:02,660 --> 00:33:07,400 partial partial t minus average omega out of it. 462 00:33:07,400 --> 00:33:10,360 That's why here you have this expression 463 00:33:10,360 --> 00:33:13,615 and then multiple it by f, which is the original expression. 464 00:33:17,420 --> 00:33:21,680 I also write this omega capital Omega explicitly. 465 00:33:21,680 --> 00:33:31,290 I partial partial t minus average Omega. 466 00:33:31,290 --> 00:33:34,080 f t f star. 467 00:33:37,760 --> 00:33:41,340 And you can immediately realize that-- 468 00:33:41,340 --> 00:33:47,160 OK, this whole thing is multiplied by i times kappa. 469 00:33:47,160 --> 00:33:51,040 You can immediately recognize that this term actually 470 00:33:51,040 --> 00:33:53,240 canceled because they are-- 471 00:33:53,240 --> 00:33:56,300 actually they are literally the same. 472 00:33:56,300 --> 00:33:59,950 And then what is actually left over 473 00:33:59,950 --> 00:34:03,220 is the two terms, which is in the middle. 474 00:34:03,220 --> 00:34:06,040 So basically, you are going to get now I 475 00:34:06,040 --> 00:34:11,330 can multiply i and cancel this minus i. 476 00:34:11,330 --> 00:34:17,065 Basically what you get is kappa time 477 00:34:17,065 --> 00:34:21,461 T equals-- both terms have a T, so I can extract this T out 478 00:34:21,461 --> 00:34:21,960 of it. 479 00:34:25,150 --> 00:34:35,449 f partial f star partial T cross partial f partial T f star. 480 00:34:40,000 --> 00:34:43,199 After all those works, you can see that this one 481 00:34:43,199 --> 00:34:45,600 looks pretty nice. 482 00:34:45,600 --> 00:34:46,429 This says what? 483 00:34:49,080 --> 00:34:54,980 This is not bad at all after all those calculations basically 484 00:34:54,980 --> 00:35:02,880 these will be equal to kappa T partial partial t f f star. 485 00:35:07,860 --> 00:35:10,930 Everybody's following or everybody already lost? 486 00:35:14,280 --> 00:35:17,700 We are almost there. 487 00:35:17,700 --> 00:35:18,690 All right. 488 00:35:18,690 --> 00:35:20,170 Now, we have these three. 489 00:35:20,170 --> 00:35:21,990 Three originally is a beast. 490 00:35:21,990 --> 00:35:26,520 Looks really horrible and after I write it down explicitly, 491 00:35:26,520 --> 00:35:28,700 it looks OK, not perfect. 492 00:35:28,700 --> 00:35:29,458 Yes? 493 00:35:29,458 --> 00:35:32,117 AUDIENCE: [INAUDIBLE] 494 00:35:32,117 --> 00:35:34,325 YEN-JIE LEE: The complex conjugate of the f function. 495 00:35:36,960 --> 00:35:38,130 All right. 496 00:35:38,130 --> 00:35:44,070 Now I can put one, two, and three into this integral. 497 00:35:44,070 --> 00:35:46,410 Then we are done. 498 00:35:46,410 --> 00:35:51,510 Now let's put numbers 3 into the integral first. 499 00:35:51,510 --> 00:35:56,652 I do a minus infinity to infinity, number three, dt. 500 00:35:59,460 --> 00:36:02,130 What is going to happen? 501 00:36:02,130 --> 00:36:07,470 This will give you minus infinity to infinity kappa T 502 00:36:07,470 --> 00:36:12,030 partial partial t f f star. 503 00:36:15,050 --> 00:36:18,810 And I can use integration by parts so what I'm going to get 504 00:36:18,810 --> 00:36:28,620 is kappa T f f star evaluating minus infinity 505 00:36:28,620 --> 00:36:35,870 and then plus infinity minus kappa minus 506 00:36:35,870 --> 00:36:41,936 infinity to infinity f square partial t 507 00:36:41,936 --> 00:36:46,030 partial capital T partial t d t. 508 00:36:48,690 --> 00:36:50,070 Let's look at this. 509 00:36:50,070 --> 00:36:55,440 Basically, what I'm doing is to put in the numbers written back 510 00:36:55,440 --> 00:37:00,990 into this integral and then use integration by parts. 511 00:37:00,990 --> 00:37:05,190 Basically you can see that this is what you would expect. 512 00:37:08,170 --> 00:37:12,820 The interesting thing is that this function 513 00:37:12,820 --> 00:37:18,070 is evaluated at crossing at infinity and minus infinity. 514 00:37:18,070 --> 00:37:24,740 If you assume that your f function is localized-- 515 00:37:24,740 --> 00:37:30,110 it's confined in some specific range of time, 516 00:37:30,110 --> 00:37:35,460 instead of spreading out over the whole universe. 517 00:37:35,460 --> 00:37:41,230 That means this term will be equal to 0 518 00:37:41,230 --> 00:37:45,710 because it's evaluated at plus infinity time and minus 519 00:37:45,710 --> 00:37:48,890 infinity time. 520 00:37:48,890 --> 00:37:55,500 If the f function is localized, then at the boundary of time, 521 00:37:55,500 --> 00:37:57,520 you are going to get 0. 522 00:37:57,520 --> 00:37:58,660 This term disappears. 523 00:37:58,660 --> 00:37:59,200 Very good. 524 00:37:59,200 --> 00:38:01,510 We've solved one problem. 525 00:38:01,510 --> 00:38:07,140 And this looks horrible, but partial capital T, partial t, 526 00:38:07,140 --> 00:38:08,860 what is capital T? 527 00:38:08,860 --> 00:38:11,770 T is small t minus average of t. 528 00:38:11,770 --> 00:38:17,290 Average of t is a number and t is just t. 529 00:38:17,290 --> 00:38:23,740 Therefore, partial t partial small d is just 1. 530 00:38:23,740 --> 00:38:26,050 You can see that there are hopes, 531 00:38:26,050 --> 00:38:29,290 things are becoming simpler and simpler. 532 00:38:29,290 --> 00:38:33,430 Therefore, what I'm going to get is this-- 533 00:38:33,430 --> 00:38:41,940 minus kappa minus infinity to infinity t t f squared. 534 00:38:44,650 --> 00:38:56,400 And then if you divide this by this term, you can see that 3-- 535 00:38:56,400 --> 00:39:02,110 number 3 term-- will give you a contribution of minus kappa. 536 00:39:02,110 --> 00:39:03,870 That's all. 537 00:39:03,870 --> 00:39:07,260 Because once you plug this integral back 538 00:39:07,260 --> 00:39:10,460 into this function, the third term contribution 539 00:39:10,460 --> 00:39:13,140 gives you minus kappa. 540 00:39:13,140 --> 00:39:19,140 That's a very good news because it's actually pretty simple. 541 00:39:19,140 --> 00:39:20,280 Any questions? 542 00:39:26,418 --> 00:39:33,840 AUDIENCE: [INAUDIBLE] 543 00:39:33,840 --> 00:39:35,972 YEN-JIE LEE: Oh, you mean this one? 544 00:39:35,972 --> 00:39:37,418 AUDIENCE: No. 545 00:39:37,418 --> 00:39:38,382 YEN-JIE LEE: This one? 546 00:39:38,382 --> 00:39:40,310 AUDIENCE: To the left. 547 00:39:40,310 --> 00:39:41,274 YEN-JIE LEE: Oh, yeah. 548 00:39:41,274 --> 00:39:41,857 You are right. 549 00:39:41,857 --> 00:39:43,087 I missed a dt. 550 00:39:43,087 --> 00:39:43,920 Thank you very much. 551 00:39:43,920 --> 00:39:44,750 Very good. 552 00:39:44,750 --> 00:39:45,660 Yeah. 553 00:39:45,660 --> 00:39:49,140 Basically what I'm trying to do is plug in the expression 554 00:39:49,140 --> 00:39:50,960 here into the integral. 555 00:39:54,886 --> 00:39:56,260 You can see that the contribution 556 00:39:56,260 --> 00:39:59,040 from the third term that number 2 is rather simple. 557 00:39:59,040 --> 00:40:04,030 It's just minus kappa. 558 00:40:04,030 --> 00:40:06,134 Let's also take a look at the computation 559 00:40:06,134 --> 00:40:08,520 from the first and the second. 560 00:40:08,520 --> 00:40:17,130 Wife Number one, will give you minus infinity to infinity t 561 00:40:17,130 --> 00:40:27,016 minus average of t squared f of t squared dt. 562 00:40:27,016 --> 00:40:30,886 And this is divided by minus infinity 563 00:40:30,886 --> 00:40:36,460 to infinity dt, f of t. 564 00:40:36,460 --> 00:40:41,852 This is not crazy at all because this just the definition 565 00:40:41,852 --> 00:40:44,636 of delta t squared. 566 00:40:44,636 --> 00:40:48,785 Just a reminder that the definition of delta t squared 567 00:40:48,785 --> 00:40:51,290 is written here. 568 00:40:51,290 --> 00:40:56,880 Therefore, this is just delta t squared-- 569 00:40:56,880 --> 00:40:59,740 the first term, which looks really strange there, 570 00:40:59,740 --> 00:41:05,604 but in reality, it's actually very simple. 571 00:41:05,604 --> 00:41:07,532 Let's look at the second term. 572 00:41:07,532 --> 00:41:14,430 This is kappa squared minus infinity to infinity i 573 00:41:14,430 --> 00:41:22,828 partial partial t minus average of omega f of t. 574 00:41:22,828 --> 00:41:24,820 And then square that. 575 00:41:24,820 --> 00:41:34,300 Divide it by minus infinity to infinity dt, f of t squared. 576 00:41:34,300 --> 00:41:39,708 And that will give you kappa squared delta omega squared. 577 00:41:44,000 --> 00:41:48,916 Basically, our conclusion that this r function 578 00:41:48,916 --> 00:41:51,380 is a function of kappa. 579 00:41:51,380 --> 00:41:57,911 Essentially equal to the first terms here delta t squared, 580 00:41:57,911 --> 00:42:03,315 the second term is plus kappa squared of delta omega squared 581 00:42:03,315 --> 00:42:03,900 . 582 00:42:03,900 --> 00:42:07,440 And finally, the third term is there. 583 00:42:07,440 --> 00:42:08,140 Minus kappa. 584 00:42:14,160 --> 00:42:19,020 And this would be greater or equal to 0. 585 00:42:19,020 --> 00:42:22,000 Because what I am doing is just summing 586 00:42:22,000 --> 00:42:26,820 all those positive functions. 587 00:42:26,820 --> 00:42:28,580 Then, take the rest. 588 00:42:28,580 --> 00:42:29,730 . 589 00:42:29,730 --> 00:42:33,208 Any questions? 590 00:42:33,208 --> 00:42:35,698 AUDIENCE: Why does the integral from negative infinity 591 00:42:35,698 --> 00:42:38,805 to infinity dt f squared equal? 592 00:42:41,720 --> 00:42:43,740 YEN-JIE LEE: This one? 593 00:42:43,740 --> 00:42:44,240 This one? 594 00:42:47,040 --> 00:42:49,430 This is equal to zero, right? 595 00:42:49,430 --> 00:42:50,070 Oh, here? 596 00:42:50,070 --> 00:42:52,320 AUDIENCE: Yeah. 597 00:42:52,320 --> 00:42:53,392 Why does that-- 598 00:42:53,392 --> 00:42:54,350 YEN-JIE LEE: Oh, I see. 599 00:42:54,350 --> 00:42:56,030 I see your point. 600 00:42:56,030 --> 00:42:59,330 This is an integrated minus infinity to infinity number 601 00:42:59,330 --> 00:43:00,690 3 dt. 602 00:43:00,690 --> 00:43:02,210 It's the contribution here. 603 00:43:02,210 --> 00:43:06,710 Then, if I take a ratio between this term and that term, 604 00:43:06,710 --> 00:43:10,050 then this is canceled by the denominator. 605 00:43:10,050 --> 00:43:12,667 Therefore, what is actually left over is minus kappa. 606 00:43:12,667 --> 00:43:14,750 AUDIENCE: OK. 607 00:43:14,750 --> 00:43:18,110 YEN-JIE LEE: This 3, the contribution of 3 in green 608 00:43:18,110 --> 00:43:22,820 is already taking the ratio when I evaluate the capital R 609 00:43:22,820 --> 00:43:24,870 function. 610 00:43:24,870 --> 00:43:27,530 Good question. 611 00:43:27,530 --> 00:43:30,800 Now you can see that you can safely 612 00:43:30,800 --> 00:43:33,680 ignore what I have said so far. 613 00:43:33,680 --> 00:43:34,940 Everything you can ignore. 614 00:43:34,940 --> 00:43:38,120 Those are just mathematics tricks. 615 00:43:38,120 --> 00:43:42,580 But what is very important is that now I have this relation-- 616 00:43:42,580 --> 00:43:45,940 delta t squared plus kappa square plus delta omega 617 00:43:45,940 --> 00:43:48,800 squared minus k. 618 00:43:48,800 --> 00:43:51,380 This is a function of k. 619 00:43:51,380 --> 00:43:53,040 And I can actually minimize it. 620 00:43:57,760 --> 00:44:04,030 I can minimize R if I carefully choose a kappa value. 621 00:44:04,030 --> 00:44:09,490 This kappa equal to kappa mean value 622 00:44:09,490 --> 00:44:14,110 which makes the minimize the R function is 623 00:44:14,110 --> 00:44:18,640 equal to 1/2 delta omega squared, which I would not 624 00:44:18,640 --> 00:44:22,420 go over this calculation because this is just a minimization 625 00:44:22,420 --> 00:44:22,920 problem. 626 00:44:25,450 --> 00:44:29,860 That means if plug that in, what I'm getting 627 00:44:29,860 --> 00:44:39,865 is R kappa min will be equal to delta T squared minus 1 628 00:44:39,865 --> 00:44:43,290 over 4 delta omega squared. 629 00:44:43,290 --> 00:44:45,594 That is greater or equal to. 630 00:44:45,594 --> 00:44:46,094 0. 631 00:44:52,110 --> 00:44:55,380 We arrive there. 632 00:44:55,380 --> 00:45:00,240 If I multiple both sides by 4 delta omega 633 00:45:00,240 --> 00:45:09,140 squared you get delta t squared delta omega squared 634 00:45:09,140 --> 00:45:13,780 greater or equal to 1 over 4. 635 00:45:13,780 --> 00:45:16,090 If you take the square root of that, 636 00:45:16,090 --> 00:45:22,220 the you get delta t delta omega greater or equal to 1/2. 637 00:45:22,220 --> 00:45:28,390 That's actually what we started to try to prove right? 638 00:45:28,390 --> 00:45:30,850 You can see that after all those works 639 00:45:30,850 --> 00:45:35,290 a lot of complicated mathematic calculations, 640 00:45:35,290 --> 00:45:40,180 you can see that we make no assumption, 641 00:45:40,180 --> 00:45:45,460 we are just using the definition of the spread of time 642 00:45:45,460 --> 00:45:48,520 and the spread of frequency. 643 00:45:48,520 --> 00:45:52,600 We follow that definition and the use of mathematical trick 644 00:45:52,600 --> 00:45:57,370 which we used to prove Heisenberg's Uncertainty 645 00:45:57,370 --> 00:46:01,210 Principle and we arrive there. 646 00:46:01,210 --> 00:46:07,000 This means that this is an intrinsic property 647 00:46:07,000 --> 00:46:09,160 of wave function. 648 00:46:09,160 --> 00:46:12,630 Intrinsic property means that it's a mathematic property 649 00:46:12,630 --> 00:46:13,540 of wave function. 650 00:46:16,860 --> 00:46:22,310 What do I mean by this equation, which we finally did right? 651 00:46:26,470 --> 00:46:28,360 After all those hard work, we have 652 00:46:28,360 --> 00:46:31,160 to enjoy what we have learned right 653 00:46:31,160 --> 00:46:34,330 from all of those crazy things. 654 00:46:34,330 --> 00:46:36,550 What do we learn? 655 00:46:36,550 --> 00:46:38,000 Look at this function. 656 00:46:38,000 --> 00:46:42,770 Delta t times delta omega, greater or equal to 1/2. 657 00:46:42,770 --> 00:46:49,450 That means if I construct a function, which 658 00:46:49,450 --> 00:46:53,410 is how I oscillate the stream as a function of time, 659 00:46:53,410 --> 00:47:00,020 if I construct a really narrow one to this very fast 660 00:47:00,020 --> 00:47:02,200 and then I stop-- 661 00:47:02,200 --> 00:47:05,470 very narrow-- then you will have a very small delta t. 662 00:47:08,660 --> 00:47:10,010 Now it sounds really nice. 663 00:47:10,010 --> 00:47:12,380 I produce a delta function, delta t, 664 00:47:12,380 --> 00:47:20,540 but the delta omega space is going to be a mess. 665 00:47:20,540 --> 00:47:24,640 It's going to be a super wide distribution because delta t is 666 00:47:24,640 --> 00:47:26,510 really very, very small. 667 00:47:26,510 --> 00:47:28,880 That means you have to compensate that 668 00:47:28,880 --> 00:47:34,410 by a rather large delta omega because if you multiple delta t 669 00:47:34,410 --> 00:47:38,190 times delta omega, that is going to be great or equal to 1/2. 670 00:47:41,540 --> 00:47:43,910 And is the consequence of this, for example, 671 00:47:43,910 --> 00:47:49,340 for the discussion of AM radio. 672 00:47:49,340 --> 00:47:58,140 If I have an AM radio with bandwidth delta omega. 673 00:47:58,140 --> 00:48:09,530 This is 2 pi delta nu and that is something like 3 times 10 674 00:48:09,530 --> 00:48:14,790 to the 4 Hz. 675 00:48:14,790 --> 00:48:17,460 If I have some kind of bandwidth which 676 00:48:17,460 --> 00:48:20,820 is actually roughly this value. 677 00:48:20,820 --> 00:48:24,090 I can now immediately calculate what 678 00:48:24,090 --> 00:48:26,130 will be the resulting delta t. 679 00:48:26,130 --> 00:48:33,230 The resulting delta t will be a few times 10 680 00:48:33,230 --> 00:48:40,190 to the minus 5 seconds based on this equation. 681 00:48:42,700 --> 00:48:58,470 This means that if I'm trying to send two signals in sequence 682 00:48:58,470 --> 00:49:02,030 through this AM radio. 683 00:49:02,030 --> 00:49:05,940 that mean if the delta t-- 684 00:49:05,940 --> 00:49:08,880 the time difference between the first 685 00:49:08,880 --> 00:49:11,040 and the second information-- 686 00:49:11,040 --> 00:49:16,710 if the time difference is large, if that delta-t 687 00:49:16,710 --> 00:49:23,000 between these two much, much larger than 10 to the minus 688 00:49:23,000 --> 00:49:24,090 to the minus 5 seconds. 689 00:49:26,670 --> 00:49:31,270 Then I can actually easily separate these two signals. 690 00:49:34,310 --> 00:49:38,060 On the other hand, if I send then really, 691 00:49:38,060 --> 00:49:41,442 the two signal really close to each other, 692 00:49:41,442 --> 00:49:45,780 if it looks like this, then the receiver, 693 00:49:45,780 --> 00:49:48,350 the ones who will receive the signal, 694 00:49:48,350 --> 00:49:51,410 will not be able to separate, if this is just 695 00:49:51,410 --> 00:49:56,690 one signal or two signals, or one pulse or two pulse 696 00:49:56,690 --> 00:49:58,160 which you are trying to send. 697 00:50:01,220 --> 00:50:03,840 Any questions so far? 698 00:50:03,840 --> 00:50:05,640 So you can see that we can actually 699 00:50:05,640 --> 00:50:10,020 quantify what will be the limitation in the resolution, 700 00:50:10,020 --> 00:50:15,340 tiny resolution, due to the limitation of bandwidth delta 701 00:50:15,340 --> 00:50:15,840 omega. 702 00:50:18,570 --> 00:50:21,390 Before we take a break, I would like 703 00:50:21,390 --> 00:50:24,730 to make a connection to quantum physics. 704 00:50:24,730 --> 00:50:28,500 So if I look at this delta t times delta omega greater than 705 00:50:28,500 --> 00:50:34,380 or equal to 1 over 2, this expression, I can rewrite it. 706 00:50:34,380 --> 00:50:43,140 I can multiply t by velocity v. And I get v times velocity 707 00:50:43,140 --> 00:50:47,820 and I can have omega divided by v. 708 00:50:47,820 --> 00:50:51,180 And this would be better or equal to 1 over 2. 709 00:50:51,180 --> 00:50:54,300 So I just multiply v and divide by v, 710 00:50:54,300 --> 00:50:57,400 then actually you can solve. 711 00:50:57,400 --> 00:51:03,530 And that means this will become delta x. 712 00:51:03,530 --> 00:51:08,570 And that, the second term, will become delta k. 713 00:51:08,570 --> 00:51:12,140 And that would be greater or equal to 1 over 2. 714 00:51:14,650 --> 00:51:24,240 In the quantum physics, momentum is equal to h bar times k. 715 00:51:28,780 --> 00:51:34,372 Momentum will be equal to h bar times k. 716 00:51:34,372 --> 00:51:39,160 And h bar is actually the Planck constant. 717 00:51:39,160 --> 00:51:46,700 So that, actually you will see that a few times in L4. 718 00:51:46,700 --> 00:51:47,650 OK. 719 00:51:47,650 --> 00:51:53,090 So if I have p equal to h bar times k, 720 00:51:53,090 --> 00:52:00,520 that means I have delta x times delta p greater or equal to h 721 00:52:00,520 --> 00:52:02,020 bar over 2. 722 00:52:04,690 --> 00:52:13,140 That is exactly the uncertainty principle, which was actually 723 00:52:13,140 --> 00:52:16,170 introduced by Heisenberg. 724 00:52:16,170 --> 00:52:17,990 And what is actually the meaning of this? 725 00:52:17,990 --> 00:52:23,070 So if we describe all those particles 726 00:52:23,070 --> 00:52:30,070 we see by quantum mechanical waves, 727 00:52:30,070 --> 00:52:36,240 if I have momentum p, now it corresponds to a wave function, 728 00:52:36,240 --> 00:52:38,530 with wave number k. 729 00:52:38,530 --> 00:52:44,260 And the constant, which is associated with p and the k 730 00:52:44,260 --> 00:52:48,040 is the Planck constant. 731 00:52:48,040 --> 00:52:54,930 So this means that if I measure one particle really, 732 00:52:54,930 --> 00:53:00,260 really precisely in a position, due 733 00:53:00,260 --> 00:53:04,380 to the nature of wave function that 734 00:53:04,380 --> 00:53:09,130 means I will not have a lot of information 735 00:53:09,130 --> 00:53:12,730 about the momentum of that particle. 736 00:53:12,730 --> 00:53:17,580 And where this uncertainty principle is coming from, 737 00:53:17,580 --> 00:53:24,850 it's coming from purely the mathematics related to waves. 738 00:53:24,850 --> 00:53:27,520 As you can see there, there's really nothing 739 00:53:27,520 --> 00:53:31,300 to do with quantum so far. 740 00:53:31,300 --> 00:53:34,690 Quantum I'm saying actually only goes in 741 00:53:34,690 --> 00:53:39,110 after we prove the uncertainty principle, delta omega 742 00:53:39,110 --> 00:53:40,510 times delta t. 743 00:53:40,510 --> 00:53:45,820 You can cannot have a very precise frequency and a very 744 00:53:45,820 --> 00:53:50,020 precise position in a coordinated space over time 745 00:53:50,020 --> 00:53:51,990 at the same time. 746 00:53:51,990 --> 00:53:55,230 And that actually has direct consequence. 747 00:53:55,230 --> 00:53:59,660 That means if you are considered in quantum mechanics, that 748 00:53:59,660 --> 00:54:02,760 is essentially the limitation which will be posted, 749 00:54:02,760 --> 00:54:05,730 the uncertainty principle. 750 00:54:05,730 --> 00:54:07,960 So we will take a five minute break. 751 00:54:07,960 --> 00:54:14,210 And we come back and we take a look at 2-3 dimensional waves. 752 00:54:14,210 --> 00:54:16,932 And let me know if you have any questions. 753 00:54:26,260 --> 00:54:27,670 So welcome, back everybody. 754 00:54:27,670 --> 00:54:31,930 So before we actually moved to 2-3 dimensional waves, 755 00:54:31,930 --> 00:54:34,960 we will discuss a very interesting topic, 756 00:54:34,960 --> 00:54:36,910 which is related to the dispersion 757 00:54:36,910 --> 00:54:42,520 relation of the light actually. 758 00:54:42,520 --> 00:54:47,270 So if you use spatial relativity, 759 00:54:47,270 --> 00:54:54,400 basically you can relate energy to momentum and the mass. 760 00:54:54,400 --> 00:54:59,360 So E square will be equal to a p square c square plus m square c 761 00:54:59,360 --> 00:55:01,180 to the 4. 762 00:55:01,180 --> 00:55:08,080 And you actually interpret light as a photon, 763 00:55:08,080 --> 00:55:11,410 then basically E is actually equal-- to the energy 764 00:55:11,410 --> 00:55:15,400 of the photon will be equal to h bar times omega. 765 00:55:15,400 --> 00:55:18,640 So we are actually really going really forward a bit. 766 00:55:18,640 --> 00:55:20,140 Because maybe some of you actually 767 00:55:20,140 --> 00:55:21,610 haven't seen this before. 768 00:55:21,610 --> 00:55:25,030 But if you just believe what I have said, 769 00:55:25,030 --> 00:55:28,090 basically you can actually divide everything 770 00:55:28,090 --> 00:55:31,030 from the first formula, which is the spatial relativity 771 00:55:31,030 --> 00:55:33,420 formula, by h bar square. 772 00:55:33,420 --> 00:55:35,830 Then you will be able to derive and arrive 773 00:55:35,830 --> 00:55:39,490 the second formula, which is omega square equal to c 774 00:55:39,490 --> 00:55:43,540 square k square plus omega 0 square. 775 00:55:43,540 --> 00:55:45,250 And the omega 0 is actually defined 776 00:55:45,250 --> 00:55:50,020 as mc square over h bar, just for simplicity. 777 00:55:50,020 --> 00:55:54,200 So if we look at this equation, this is essentially 778 00:55:54,200 --> 00:55:56,920 a dispersion relation. 779 00:55:56,920 --> 00:55:59,570 Now you have seen this so many times. 780 00:55:59,570 --> 00:56:04,300 And this omega square equal to c square k square plus omega 0 781 00:56:04,300 --> 00:56:06,940 square, this formula is actually reminding you 782 00:56:06,940 --> 00:56:10,330 that this is actually a dispersion relation. 783 00:56:10,330 --> 00:56:17,820 So what I mean by a photon having mass here? 784 00:56:17,820 --> 00:56:22,920 That means the m term in this special relativity formula 785 00:56:22,920 --> 00:56:25,540 is not 0. 786 00:56:25,540 --> 00:56:29,820 Therefore omega 0 will be non-zero. 787 00:56:29,820 --> 00:56:31,380 What is going to happen? 788 00:56:31,380 --> 00:56:35,850 That means the space of velocity of light 789 00:56:35,850 --> 00:56:37,770 is going to be different. 790 00:56:37,770 --> 00:56:42,990 It depends on what value of k you choose. 791 00:56:42,990 --> 00:56:44,820 That's kind of interesting. 792 00:56:44,820 --> 00:56:50,400 Because that means light with different frequency 793 00:56:50,400 --> 00:56:51,990 or different wavelengths is going 794 00:56:51,990 --> 00:56:56,460 to be traveling through the vacuum at different speeds, 795 00:56:56,460 --> 00:56:57,210 if that's true. 796 00:56:59,860 --> 00:57:02,220 Everybody get it? 797 00:57:02,220 --> 00:57:03,240 Very good. 798 00:57:03,240 --> 00:57:05,880 So how do we actually test this? 799 00:57:05,880 --> 00:57:11,040 So that means I need a light source, which are very, 800 00:57:11,040 --> 00:57:13,880 very far away from earth. 801 00:57:13,880 --> 00:57:17,540 Then I would like to measure the delta t 802 00:57:17,540 --> 00:57:22,430 as a function of frequency, for example, and analyzing. 803 00:57:22,430 --> 00:57:25,820 So how do we do that? 804 00:57:25,820 --> 00:57:30,590 So this is actually possible if you actually 805 00:57:30,590 --> 00:57:35,562 use a natural light source, which is the pulsar. 806 00:57:35,562 --> 00:57:36,770 So what is actually a pulsar? 807 00:57:36,770 --> 00:57:38,840 So what we actually use, essentially a 808 00:57:38,840 --> 00:57:40,970 millisecond pulsar. 809 00:57:40,970 --> 00:57:46,320 So those are actually coming from rapidly rotating neutron 810 00:57:46,320 --> 00:57:50,590 stars, and that those rotating neutron stars will 811 00:57:50,590 --> 00:57:55,010 emit pulses of radiation like x-ray and radio waves, 812 00:57:55,010 --> 00:57:56,354 at regular intervals. 813 00:57:56,354 --> 00:57:57,770 Because it's essentially rotating, 814 00:57:57,770 --> 00:58:01,710 rotating, rotating again and again. 815 00:58:01,710 --> 00:58:06,560 Based on this movie, basically what it's showing here 816 00:58:06,560 --> 00:58:11,000 is a very old neutron star. 817 00:58:11,000 --> 00:58:13,430 It's actually in a binary system. 818 00:58:13,430 --> 00:58:16,520 And this neutron star can absorb the material 819 00:58:16,520 --> 00:58:18,500 from the other partner. 820 00:58:18,500 --> 00:58:20,230 So that actually is-- 821 00:58:20,230 --> 00:58:23,400 the rotation speed actually increased. 822 00:58:23,400 --> 00:58:29,300 And finally at the speed of a millisecond per turn. 823 00:58:29,300 --> 00:58:30,980 So this actually really happened. 824 00:58:30,980 --> 00:58:33,230 And we can actually observe this. 825 00:58:33,230 --> 00:58:38,010 And if we are lucky, the earth is essentially 826 00:58:38,010 --> 00:58:42,410 somehow in a spatial direction such that the emitting radio 827 00:58:42,410 --> 00:58:46,520 wave actually pointing from the pulsar to earth, 828 00:58:46,520 --> 00:58:51,800 then I can see the pulsar, the amplitude of the light 829 00:58:51,800 --> 00:58:56,120 from pulsar essentially changing rapidly as a function of time. 830 00:58:56,120 --> 00:58:59,740 And another very good news is that typically those pulsars 831 00:58:59,740 --> 00:59:01,250 are really far away. 832 00:59:01,250 --> 00:59:07,850 For example, in this example, pulsar B1937+21, 833 00:59:07,850 --> 00:59:12,840 this is essentially a pulsar with rotation period of just 834 00:59:12,840 --> 00:59:15,830 1.6 milliseconds. 835 00:59:15,830 --> 00:59:19,250 And this is actually something which is really 836 00:59:19,250 --> 00:59:22,310 happening really far away from the Earth, which essentially 837 00:59:22,310 --> 00:59:25,250 is 16,000 light years away. 838 00:59:25,250 --> 00:59:27,520 And that we can actually observe this. 839 00:59:27,520 --> 00:59:30,380 This is actually pretty close to Sagitta, 840 00:59:30,380 --> 00:59:35,200 and you can actually see this pulsar. 841 00:59:35,200 --> 00:59:38,510 And how does that actually associate 842 00:59:38,510 --> 00:59:41,300 with the original question we were posting? 843 00:59:41,300 --> 00:59:44,610 The original question is, does the light 844 00:59:44,610 --> 00:59:49,052 with different frequency travel at different speed. 845 00:59:49,052 --> 00:59:50,760 And this is essentially a very nice tool. 846 00:59:50,760 --> 00:59:51,260 Right? 847 00:59:51,260 --> 00:59:53,610 Because it is emitting the radio wave. 848 00:59:53,610 --> 00:59:57,520 And now I can just measure the spectra as a function of time. 849 00:59:57,520 --> 01:00:01,280 And I will be able to see if we actually 850 01:00:01,280 --> 01:00:04,190 can observe different speed. 851 01:00:04,190 --> 01:00:08,930 Because we know the rotation in the world, and et cetera. 852 01:00:08,930 --> 01:00:15,127 And it also emits a wide spectra of the frequency, 853 01:00:15,127 --> 01:00:15,960 the light frequency. 854 01:00:15,960 --> 01:00:18,380 Therefore, I can use this as a light source 855 01:00:18,380 --> 01:00:23,760 far, far away from the Earth, to see what will happen. 856 01:00:23,760 --> 01:00:27,300 So somebody actually did this measurement, 857 01:00:27,300 --> 01:00:28,680 and this is that what they found. 858 01:00:31,640 --> 01:00:34,740 They found a non-zero omega 0. 859 01:00:34,740 --> 01:00:37,580 A non-zero omega 0 was found. 860 01:00:37,580 --> 01:00:42,500 So that means the mass will be 1.3 times 10 861 01:00:42,500 --> 01:00:47,360 to the minus 49 gram. 862 01:00:47,360 --> 01:00:48,830 That sounds really small. 863 01:00:48,830 --> 01:00:51,590 But it's not small at all. 864 01:00:51,590 --> 01:00:54,630 That's actually destroying the whole understanding of light. 865 01:00:57,160 --> 01:00:58,866 What is going on? 866 01:00:58,866 --> 01:00:59,740 So we are in trouble. 867 01:01:02,950 --> 01:01:06,370 So after all this discussion, et cetera, 868 01:01:06,370 --> 01:01:08,260 and also other measurements which 869 01:01:08,260 --> 01:01:11,730 are sensitive to photon mass, they actually 870 01:01:11,730 --> 01:01:14,392 threw out this possible contribution. 871 01:01:14,392 --> 01:01:17,290 This is essentially is just simply too large 872 01:01:17,290 --> 01:01:21,210 based on, for example, measurement of magnetic field 873 01:01:21,210 --> 01:01:22,960 in the galaxy, et cetera. 874 01:01:22,960 --> 01:01:24,350 It doesn't really work. 875 01:01:24,350 --> 01:01:26,860 So what essentially is really happening? 876 01:01:26,860 --> 01:01:34,150 The explanation is that the path from the pulsar to the earth 877 01:01:34,150 --> 01:01:36,860 it's really not vacuum. 878 01:01:36,860 --> 01:01:39,250 There are a lot of-- 879 01:01:39,250 --> 01:01:44,920 not a lot, but we have very few or very dilute electrons, 880 01:01:44,920 --> 01:01:48,670 very diluted free electrons all over the place. 881 01:01:48,670 --> 01:01:52,750 And that will change the frequency and the speed 882 01:01:52,750 --> 01:01:55,210 of light slightly. 883 01:01:55,210 --> 01:01:57,830 Therefore you observe the interesting-- 884 01:01:57,830 --> 01:01:59,180 observe the effect. 885 01:01:59,180 --> 01:02:01,300 And we are going to actually also talk 886 01:02:01,300 --> 01:02:04,330 about how the material actually changes 887 01:02:04,330 --> 01:02:07,030 the behavior of the electromagnetic wave 888 01:02:07,030 --> 01:02:09,010 in the coming lectures. 889 01:02:09,010 --> 01:02:11,820 I hope you find this interesting. 890 01:02:11,820 --> 01:02:15,000 Any questions? 891 01:02:15,000 --> 01:02:16,540 All right. 892 01:02:16,540 --> 01:02:19,560 So we are going to move on. 893 01:02:19,560 --> 01:02:22,310 So far what we have been discussing 894 01:02:22,310 --> 01:02:26,280 is always 1-dimensional waves. 895 01:02:26,280 --> 01:02:33,150 So for example, a string, and also the sound 896 01:02:33,150 --> 01:02:34,480 save in a tube, et cetera. 897 01:02:34,480 --> 01:02:38,590 We always discuss things which are in one dimension. 898 01:02:38,590 --> 01:02:42,190 But we are actually not one dimensional animal. 899 01:02:42,190 --> 01:02:46,080 We are 3-dimensional And of course, for example, 900 01:02:46,080 --> 01:02:49,490 these objects the surface is 2-dimensional 901 01:02:49,490 --> 01:02:50,990 So there are many, many things which 902 01:02:50,990 --> 01:02:53,740 are more than one dimension. 903 01:02:53,740 --> 01:02:56,080 So can-- the question that I'm trying 904 01:02:56,080 --> 01:03:01,240 to ask is, can we actually understand this kind of object, 905 01:03:01,240 --> 01:03:07,000 and how actually to understand those objects 906 01:03:07,000 --> 01:03:10,330 and how do we actually derive the normal amounts, 907 01:03:10,330 --> 01:03:12,730 and how do we actually write down 908 01:03:12,730 --> 01:03:16,090 the general solution, which describes a 2-dimensional 909 01:03:16,090 --> 01:03:17,610 or a 3-dimensional wave. 910 01:03:17,610 --> 01:03:22,510 That's actually the next topic which I would like to discuss. 911 01:03:22,510 --> 01:03:28,480 So that's actually gets started with a plate like this. 912 01:03:28,480 --> 01:03:35,770 So basically that plate is actually a 2-dimensional. 913 01:03:35,770 --> 01:03:42,940 And assuming that this plate is infinitely long, for a moment, 914 01:03:42,940 --> 01:03:45,820 very, very long. 915 01:03:45,820 --> 01:03:47,300 So what does that mean? 916 01:03:47,300 --> 01:03:54,460 This means that if I define my x and y-coordinate, which 917 01:03:54,460 --> 01:04:01,000 is actually used to describe the position of a specific point 918 01:04:01,000 --> 01:04:03,925 on this plate, then basically you 919 01:04:03,925 --> 01:04:08,540 will see that they are beautiful symmetries, which 920 01:04:08,540 --> 01:04:13,600 you can actually identify from this simple example. 921 01:04:13,600 --> 01:04:17,290 What is actually the symmetry which we can identify? 922 01:04:17,290 --> 01:04:20,650 Can anybody help me with that? 923 01:04:20,650 --> 01:04:21,760 AUDIENCE: x and y. 924 01:04:21,760 --> 01:04:22,510 YEN-JIE LEE: Yeah. 925 01:04:22,510 --> 01:04:25,300 So yeah, x and y are symmetric, yes. 926 01:04:25,300 --> 01:04:28,600 And the other function of x, what kind of symmetry to you 927 01:04:28,600 --> 01:04:29,716 have? 928 01:04:29,716 --> 01:04:30,720 AUDIENCE: Reflection. 929 01:04:30,720 --> 01:04:31,470 YEN-JIE LEE: Yeah. 930 01:04:31,470 --> 01:04:34,570 Also reflection, and what I'm looking 931 01:04:34,570 --> 01:04:39,330 for is if I change x and change y, what kind of symmetry 932 01:04:39,330 --> 01:04:40,095 do you have? 933 01:04:40,095 --> 01:04:41,214 AUDIENCE: Translation. 934 01:04:41,214 --> 01:04:42,630 YEN-JIE LEE: Translation symmetry. 935 01:04:42,630 --> 01:04:44,720 Well, all of you are correct. 936 01:04:44,720 --> 01:04:46,940 But what I am trying to focus on now 937 01:04:46,940 --> 01:04:48,615 is the translation symmetry. 938 01:04:51,150 --> 01:04:54,800 So if I use translation symmetry, what I'm going to get 939 01:04:54,800 --> 01:04:58,280 is that I can already know the functional 940 01:04:58,280 --> 01:05:01,410 form of the normal mode. 941 01:05:01,410 --> 01:05:05,000 Because essentially if it's translation symmetric, 942 01:05:05,000 --> 01:05:07,940 as a function of x, it's translation symmetric 943 01:05:07,940 --> 01:05:09,770 as a function of y. 944 01:05:09,770 --> 01:05:17,700 Then I can say is in the x direction will be proportional 945 01:05:17,700 --> 01:05:20,360 to exponential iKxX. 946 01:05:24,030 --> 01:05:29,640 K underscore x is essentially the wave number associated 947 01:05:29,640 --> 01:05:32,539 with the wave in the x direction. 948 01:05:32,539 --> 01:05:34,080 So that's essentially one consequence 949 01:05:34,080 --> 01:05:37,950 which we actually learned from the discussion of symmetry. 950 01:05:37,950 --> 01:05:42,300 And in the y direction, I can conclude also 951 01:05:42,300 --> 01:05:51,690 that the normal mode will be proportional to exponential iKy 952 01:05:51,690 --> 01:05:57,970 times Y. Therefore, I already know 953 01:05:57,970 --> 01:06:01,180 what will be the function form of the normal mode 954 01:06:01,180 --> 01:06:05,410 of this highly symmetric system. 955 01:06:05,410 --> 01:06:06,150 What is that? 956 01:06:06,150 --> 01:06:14,680 The psi xy will be equal to A times exponential iKx 957 01:06:14,680 --> 01:06:18,360 times X, exponential iKyY. 958 01:06:22,000 --> 01:06:24,710 So you can see that. 959 01:06:24,710 --> 01:06:28,320 And also I need to take the real part. 960 01:06:28,320 --> 01:06:32,650 Something like this will be possible in normal mode. 961 01:06:32,650 --> 01:06:36,650 Therefore without going into detail basically, 962 01:06:36,650 --> 01:06:41,060 we will see that the expected behavior of psi 963 01:06:41,060 --> 01:06:46,930 as a function of x and y will be something 964 01:06:46,930 --> 01:06:56,420 like a sine Kx times x, sin Ky times y. 965 01:06:56,420 --> 01:07:00,995 So that's actually the kind of normal mode, which 966 01:07:00,995 --> 01:07:05,780 we will expect based on the argument of translation 967 01:07:05,780 --> 01:07:07,880 symmetry. 968 01:07:07,880 --> 01:07:13,340 And of course if I now go back from infinitely long system 969 01:07:13,340 --> 01:07:17,870 to a finite system, then you can use the boundary condition 970 01:07:17,870 --> 01:07:21,730 to determine what would be the K value, 971 01:07:21,730 --> 01:07:28,780 Kx value, and allow the Kx value and allow the Ky value using 972 01:07:28,780 --> 01:07:30,710 boundary conditions. 973 01:07:30,710 --> 01:07:34,460 So actually without doing any calculation, 974 01:07:34,460 --> 01:07:39,470 we can already find that, so now if I 975 01:07:39,470 --> 01:07:43,820 have a plate with finite size, basically you 976 01:07:43,820 --> 01:07:49,310 expect that I can have some kind of normal mode, which this 977 01:07:49,310 --> 01:07:52,910 is the amplitude, a projection in the x direction, 978 01:07:52,910 --> 01:07:55,000 it can be a sine function. 979 01:07:55,000 --> 01:08:00,830 And that it can become 0 at the left-hand side edge 980 01:08:00,830 --> 01:08:02,540 and the right-hand side edge. 981 01:08:02,540 --> 01:08:04,610 And in the y direction it has to be 982 01:08:04,610 --> 01:08:11,780 also some kind of sine wave as a function of y. 983 01:08:11,780 --> 01:08:14,870 And of course it goes to 0 at the edge. 984 01:08:14,870 --> 01:08:18,620 Because if those are actually the fixed boundary, 985 01:08:18,620 --> 01:08:20,420 for example. 986 01:08:20,420 --> 01:08:22,930 And if those are actually not fixed boundary, 987 01:08:22,930 --> 01:08:24,859 then you expect that-- 988 01:08:24,859 --> 01:08:27,649 like open-end solution. 989 01:08:27,649 --> 01:08:32,970 So you expect that the distribution 990 01:08:32,970 --> 01:08:38,460 will be more like a cosine function 991 01:08:38,460 --> 01:08:42,100 for the first normal mode. 992 01:08:42,100 --> 01:08:44,109 And if you look at this, the structure 993 01:08:44,109 --> 01:08:47,890 of this kind of solution, it looks really complicated. 994 01:08:47,890 --> 01:08:51,500 Because you have x direction and you also have y direction. 995 01:08:51,500 --> 01:08:55,100 Both of them are actually sine functions. 996 01:08:55,100 --> 01:08:59,300 And how do we actually visualize this kind of sine function? 997 01:08:59,300 --> 01:09:03,790 And here is a demonstration, which I have prepared. 998 01:09:03,790 --> 01:09:09,399 It's really a 2-dimensional plate. 999 01:09:09,399 --> 01:09:13,630 And as you can see that under this plate, 1000 01:09:13,630 --> 01:09:18,040 I have a loudspeaker which actually produces a sound wave 1001 01:09:18,040 --> 01:09:23,850 to try to excite one of the normal mode. 1002 01:09:23,850 --> 01:09:29,910 And the one I am going to do is to turn on this loud speaker. 1003 01:09:29,910 --> 01:09:31,600 You can hear the sound. 1004 01:09:31,600 --> 01:09:35,705 And I would like to see the normal mode. 1005 01:09:35,705 --> 01:09:39,100 But it's very hard to see that, without doing anything. 1006 01:09:39,100 --> 01:09:42,520 Because it's vibrating, but its so fast that it is really 1007 01:09:42,520 --> 01:09:44,410 very difficult to see it. 1008 01:09:44,410 --> 01:09:49,359 So what I am going to do is to pour some sand on the surface, 1009 01:09:49,359 --> 01:09:51,220 and see what is going to happen. 1010 01:09:51,220 --> 01:09:57,720 And if we look at this, I am putting sand on it. 1011 01:09:57,720 --> 01:10:00,250 And you can see that, there is something happening. 1012 01:10:02,950 --> 01:10:09,052 If I change the frequency to one of the normal mode frequencies, 1013 01:10:09,052 --> 01:10:14,480 you can see that now we are reaching some kind of resonance 1014 01:10:14,480 --> 01:10:18,770 and exciting one of the normal mode. 1015 01:10:18,770 --> 01:10:21,890 And you can see that the sand actually it 1016 01:10:21,890 --> 01:10:25,440 doesn't like to stay on some of the plate. 1017 01:10:25,440 --> 01:10:28,050 Because it's vibrating like crazy 1018 01:10:28,050 --> 01:10:31,380 and it's not very comfortable to sit there. 1019 01:10:31,380 --> 01:10:35,670 So the sand, where will the sand actually sit? 1020 01:10:35,670 --> 01:10:42,226 They will set at the place where you don't have any vibration. 1021 01:10:42,226 --> 01:10:43,600 Because what we are talking here, 1022 01:10:43,600 --> 01:10:45,970 is essentially some kind of sine wave times 1023 01:10:45,970 --> 01:10:49,250 sine wave or cosine wave times cosine wave. 1024 01:10:49,250 --> 01:10:53,320 That means there will be nodes on the plate. 1025 01:10:53,320 --> 01:10:56,940 And those are 2-dimensional nodes. 1026 01:10:56,940 --> 01:11:00,680 In the 1-dimensional case, we are talking about nodes, 1027 01:11:00,680 --> 01:11:04,800 it's actually the place where you have zero amplitude. 1028 01:11:04,800 --> 01:11:07,980 And now I have cosine times cosine. 1029 01:11:07,980 --> 01:11:11,990 Therefore, there will be a complicated pattern appearing 1030 01:11:11,990 --> 01:11:16,290 which is essentially the place the plate is not actually 1031 01:11:16,290 --> 01:11:18,560 moving at all as a function of time. 1032 01:11:18,560 --> 01:11:21,270 And you can see that now I can actually excite one 1033 01:11:21,270 --> 01:11:22,380 with the normal node. 1034 01:11:22,380 --> 01:11:24,890 And you can see a really beautiful pattern. 1035 01:11:24,890 --> 01:11:30,280 And allow me to do this and increase the frequency. 1036 01:11:30,280 --> 01:11:35,790 So that if we see if I can excite another normal mode. 1037 01:11:35,790 --> 01:11:37,000 Look at what is happening. 1038 01:11:37,000 --> 01:11:44,820 So now you see that the number of lines actually increased. 1039 01:11:44,820 --> 01:11:52,800 So this is actually so-called Chladni figures. 1040 01:11:52,800 --> 01:11:55,950 Basically those figures are actually 1041 01:11:55,950 --> 01:12:03,260 produced by this trying to excite one of the normal mode. 1042 01:12:03,260 --> 01:12:08,690 And basically the sand will be collected in the nodal lines. 1043 01:12:08,690 --> 01:12:15,310 And you can see that this higher frequency input sound wave. 1044 01:12:15,310 --> 01:12:18,890 You can excite the higher frequency in normal mode. 1045 01:12:18,890 --> 01:12:24,370 And of course I can continue to increase and see what happens. 1046 01:12:24,370 --> 01:12:30,955 Now I'm increasing the frequency even higher and higher. 1047 01:12:30,955 --> 01:12:36,894 You can see that now the sound is actually rather loud. 1048 01:12:36,894 --> 01:12:39,890 And I am actually putting more sand. 1049 01:12:39,890 --> 01:12:42,720 You can see that there are more and more patterns. 1050 01:12:42,720 --> 01:12:45,450 Because now I am increasing the frequency, 1051 01:12:45,450 --> 01:12:48,810 so that actually the higher frequency normal modes 1052 01:12:48,810 --> 01:12:50,040 are excited. 1053 01:12:50,040 --> 01:12:55,190 And you will expect more nodes for higher frequency ones. 1054 01:12:55,190 --> 01:13:00,220 And now I can even go even higher 1055 01:13:00,220 --> 01:13:03,785 to see if I find success. 1056 01:13:03,785 --> 01:13:08,075 It's not easy now. 1057 01:13:08,075 --> 01:13:08,575 Look. 1058 01:13:12,900 --> 01:13:15,450 Probably this is a very good way to design 1059 01:13:15,450 --> 01:13:16,680 the pattern of your t-shirt. 1060 01:13:19,510 --> 01:13:20,010 OK. 1061 01:13:20,010 --> 01:13:23,100 So how do we actually understand all those patterns? 1062 01:13:23,100 --> 01:13:26,250 And we have already started. 1063 01:13:26,250 --> 01:13:29,260 This is actually something related to cosine and sine 1064 01:13:29,260 --> 01:13:30,900 multiplied to each other. 1065 01:13:30,900 --> 01:13:32,880 And the next time we are going to do 1066 01:13:32,880 --> 01:13:36,900 a more detailed calculation and show you a few more demos 1067 01:13:36,900 --> 01:13:38,730 and see what we can actually learn 1068 01:13:38,730 --> 01:13:41,310 from the 2-dimensional case. 1069 01:13:41,310 --> 01:13:42,430 Thank you very much. 1070 01:13:42,430 --> 01:13:44,109 I hope you enjoyed the lecture today. 1071 01:13:44,109 --> 01:13:45,900 And if you have any questions, let me know. 1072 01:13:48,940 --> 01:13:50,800 And you can actually come forward and play 1073 01:13:50,800 --> 01:13:53,590 with those demos if you want.