WEBVTT 00:00:01.620 --> 00:00:03.960 The following content is provided under a Creative 00:00:03.960 --> 00:00:05.380 Commons license. 00:00:05.380 --> 00:00:07.590 Your support will help MIT OpenCourseWare 00:00:07.590 --> 00:00:11.680 continue to offer high-quality educational resources for free. 00:00:11.680 --> 00:00:14.220 To make a donation or to view additional materials 00:00:14.220 --> 00:00:18.180 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.180 --> 00:00:19.050 at ocw.mit.edu. 00:00:24.120 --> 00:00:26.520 YEN-JIE LEE: So welcome back, everybody. 00:00:26.520 --> 00:00:29.340 This is the final exam checklist. 00:00:31.910 --> 00:00:34.560 For the single oscillator, we need 00:00:34.560 --> 00:00:37.830 to make sure that you know how to write down 00:00:37.830 --> 00:00:40.290 the Equation of Motion. 00:00:40.290 --> 00:00:44.760 We have discussed about damped, under-damped, critically 00:00:44.760 --> 00:00:45.870 damped, and over-damped. 00:00:45.870 --> 00:00:48.480 We did that. 00:00:48.480 --> 00:00:53.980 Oscillators, and we have tried to drive oscillators. 00:00:53.980 --> 00:00:58.090 We observed transient behavior in steady state solution. 00:00:58.090 --> 00:01:01.680 Resonance, right, so which we actually demonstrated that 00:01:01.680 --> 00:01:05.250 by breaking the glass. 00:01:05.250 --> 00:01:09.000 And then we moved on and tried to couple multiple objects 00:01:09.000 --> 00:01:10.000 together. 00:01:10.000 --> 00:01:13.020 And that brings us to the coupled system. 00:01:13.020 --> 00:01:14.850 What are the normal modes? 00:01:14.850 --> 00:01:17.470 And how to actually solve M minus 1 K 00:01:17.470 --> 00:01:21.600 matrix, the eigenvalue problem. 00:01:21.600 --> 00:01:25.970 What is actually the full solution for the description 00:01:25.970 --> 00:01:27.930 of coupled systems. 00:01:27.930 --> 00:01:32.840 And can we actually drive the coupled system, 00:01:32.840 --> 00:01:35.470 and we found out we can. 00:01:35.470 --> 00:01:38.490 So the system would respond as well 00:01:38.490 --> 00:01:42.140 similar to what we have seen in the single oscillator case. 00:01:42.140 --> 00:01:44.220 We see resonance as well. 00:01:44.220 --> 00:01:47.160 We can excite one of the normal modes 00:01:47.160 --> 00:01:50.640 by driving the coupled system. 00:01:50.640 --> 00:01:56.200 Then we put more and more objects until at some point, 00:01:56.200 --> 00:01:59.980 we have infinite number of coupled objects. 00:01:59.980 --> 00:02:03.120 What is actually the solution of refraction 00:02:03.120 --> 00:02:05.070 and the transmission-- 00:02:05.070 --> 00:02:08.789 refraction and the translation symmetric system. 00:02:08.789 --> 00:02:12.310 That is actually the discussion of symmetry. 00:02:12.310 --> 00:02:15.630 We go to the continuum then, and we actually 00:02:15.630 --> 00:02:18.360 found wave and wave equations. 00:02:18.360 --> 00:02:21.200 So we found that finally, we made the phase transition 00:02:21.200 --> 00:02:25.720 from single object vibration to waves, 00:02:25.720 --> 00:02:27.590 and that is actually an achievement 00:02:27.590 --> 00:02:30.250 we have done in 8.03. 00:02:30.250 --> 00:02:33.480 We have discussed about different systems, 00:02:33.480 --> 00:02:37.780 massive string, massive spring, sound wave, 00:02:37.780 --> 00:02:40.680 electromagnetic waves, and we have 00:02:40.680 --> 00:02:46.080 discussed a progressive wave and standing waves. 00:02:46.080 --> 00:02:50.940 For the bound system, we have also normal modes. 00:02:50.940 --> 00:02:53.470 We discussed about how to actually do 00:02:53.470 --> 00:02:56.120 Fourier decomposition, and what is actually 00:02:56.120 --> 00:03:03.460 the physical meaning of Fourier decomposition in 8.03. 00:03:03.460 --> 00:03:09.820 For the infinite system, we also learned about Fourier transform 00:03:09.820 --> 00:03:12.540 and uncertainty principles. 00:03:12.540 --> 00:03:17.280 And we learned to apply boundary conditions 00:03:17.280 --> 00:03:22.060 so that we constrain the possible wavelengths 00:03:22.060 --> 00:03:23.225 of the normal modes. 00:03:25.770 --> 00:03:28.020 Therefore, we also learned about how 00:03:28.020 --> 00:03:32.110 to put a system all together. 00:03:32.110 --> 00:03:35.760 Finally, how to determine the dispersion relation, 00:03:35.760 --> 00:03:39.030 which is omega as a function of K, the wave number. 00:03:42.260 --> 00:03:46.310 Until now, we discussed idealized systems, 00:03:46.310 --> 00:03:51.530 and we also moved on to discuss dispersive medium. 00:03:51.530 --> 00:03:55.100 We have learned some more, even more about dispersion 00:03:55.100 --> 00:03:59.780 relation for the dispersive medium and signal transmission, 00:03:59.780 --> 00:04:06.310 how to send signal through a highly dispersive medium. 00:04:06.310 --> 00:04:08.090 The solution we were proposing is 00:04:08.090 --> 00:04:15.440 to use an amplitude modulation radio and also 00:04:15.440 --> 00:04:18.690 the pattern of dispersion. 00:04:18.690 --> 00:04:24.050 The group velocity and phase velocity, we covered that. 00:04:24.050 --> 00:04:27.436 As I mentioned before, the uncertainty principle. 00:04:27.436 --> 00:04:30.410 A 2D/3D system. 00:04:30.410 --> 00:04:33.290 We have bound system, which we have 00:04:33.290 --> 00:04:36.770 normal modes for two-dimensional and three-dimensional systems, 00:04:36.770 --> 00:04:37.700 as well. 00:04:37.700 --> 00:04:39.770 Because we're all over the place, 00:04:39.770 --> 00:04:43.140 so just make sure that you know how to actually dewrap 00:04:43.140 --> 00:04:46.940 all those standing waves for different dimensions 00:04:46.940 --> 00:04:48.770 of systems. 00:04:48.770 --> 00:04:53.090 We showed and approved geometrical optics, 00:04:53.090 --> 00:04:57.170 which essentially is the direct consequence of waves. 00:04:57.170 --> 00:05:00.050 Wave function, a continuation of the wave function and boundary 00:05:00.050 --> 00:05:01.970 condition. 00:05:01.970 --> 00:05:06.380 We learned about the refraction rule and also Snell's law. 00:05:06.380 --> 00:05:11.260 We talked about polarized waves, linear, circularly polarized, 00:05:11.260 --> 00:05:14.220 elliptically polarized, and the polarizer 00:05:14.220 --> 00:05:17.420 and quarter-wave plate. 00:05:17.420 --> 00:05:20.750 At the end of the discussion of 2D/3D systems, 00:05:20.750 --> 00:05:26.120 we discussed about how to generate electromagnetic waves 00:05:26.120 --> 00:05:27.530 by accelerated charge. 00:05:30.260 --> 00:05:35.030 Finally, we went on and talked about how those EM 00:05:35.030 --> 00:05:40.250 waves propagate in dielectrics and again, boundary conditions, 00:05:40.250 --> 00:05:45.020 which leads to interesting phenomena, which belongs only 00:05:45.020 --> 00:05:47.370 to electromagnetic waves. 00:05:47.370 --> 00:05:51.100 For example, Brewster's angle. 00:05:51.100 --> 00:05:54.320 So the refraction amplitude-- 00:05:54.320 --> 00:05:56.510 refraction-- the wave amplitude is 00:05:56.510 --> 00:06:01.190 governed by the property of the electromagnetic waves, which 00:06:01.190 --> 00:06:05.900 is coming from the laws which governs 00:06:05.900 --> 00:06:10.170 electromagnetic waves, which is Maxwell's equations in matter. 00:06:10.170 --> 00:06:13.190 We were trying to also manipulate those waves 00:06:13.190 --> 00:06:16.130 by adding them together, and we see 00:06:16.130 --> 00:06:21.320 constructive and destructive interference and diffraction 00:06:21.320 --> 00:06:23.510 phenomena. 00:06:23.510 --> 00:06:27.290 Then we connect that to quantum mechanics 00:06:27.290 --> 00:06:32.990 by showing you a single electron interference experiment. 00:06:32.990 --> 00:06:35.900 That connects us to the beginning 00:06:35.900 --> 00:06:41.780 of the quantum mechanics, which is the probability waves, which 00:06:41.780 --> 00:06:45.980 behave very different from other waves we have been discussing. 00:06:45.980 --> 00:06:48.770 But you are going to learn a lot more in 8.04. 00:06:48.770 --> 00:06:52.381 OK, so don't worry. 00:06:52.381 --> 00:06:52.880 All right. 00:06:52.880 --> 00:06:55.920 So that is the checklist. 00:06:55.920 --> 00:06:58.640 You can see that I can write it in two pages, 00:06:58.640 --> 00:07:01.790 so it's not that bad, probably. 00:07:01.790 --> 00:07:05.930 I hope that there was nothing really sounds like new to you 00:07:05.930 --> 00:07:08.420 by now. 00:07:08.420 --> 00:07:12.890 If you find anything is new, you have to review that part. 00:07:12.890 --> 00:07:17.030 That means you missed a class. 00:07:17.030 --> 00:07:17.660 All right. 00:07:17.660 --> 00:07:21.920 So what I'm going to do now is to go through all the material 00:07:21.920 --> 00:07:24.140 faster than the speed of light. 00:07:24.140 --> 00:07:24.710 All right. 00:07:24.710 --> 00:07:26.220 So that you will get nauseated. 00:07:26.220 --> 00:07:30.560 No, you are going to get a list of the topics. 00:07:30.560 --> 00:07:33.020 You just have to feel it. 00:07:33.020 --> 00:07:37.780 If you feel good, like when you are having a cupcake, 00:07:37.780 --> 00:07:41.810 right, then you are good for the final. 00:07:41.810 --> 00:07:45.740 If you don't feel good, what is Professor Lee talking about? 00:07:45.740 --> 00:07:47.570 He's talking about nonsense now. 00:07:47.570 --> 00:07:51.290 Then you are in trouble, and you have to review that part. 00:07:51.290 --> 00:07:51.991 All right? 00:07:51.991 --> 00:07:52.490 OK. 00:07:52.490 --> 00:07:54.470 So that's what we'll do. 00:07:54.470 --> 00:07:55.810 So let's start. 00:07:55.810 --> 00:07:56.840 All right. 00:07:56.840 --> 00:07:57.980 Why 8.03? 00:07:57.980 --> 00:08:00.350 We started a discussion-- 00:08:00.350 --> 00:08:06.320 welcome, We started a discussion of 8.03 00:08:06.320 --> 00:08:10.200 and it's vibrations and wave systems, is name of this, 8.03. 00:08:10.200 --> 00:08:13.670 And the motivation is really simple, 00:08:13.670 --> 00:08:17.720 because we cannot even recognize the universe without using 00:08:17.720 --> 00:08:19.430 waves and vibration. 00:08:19.430 --> 00:08:22.490 You cannot see me, and you cannot hear anything, 00:08:22.490 --> 00:08:25.230 and you cannot feel the vibrations-- 00:08:25.230 --> 00:08:28.520 sorry, the rotation of a black hole by your body 00:08:28.520 --> 00:08:31.460 anymore, then it's not very cool. 00:08:31.460 --> 00:08:37.280 Therefore, we study 8.03 to understand the basic ideas 00:08:37.280 --> 00:08:39.110 about waves and vibrations. 00:08:39.110 --> 00:08:43.490 And we found that waves and vibrations 00:08:43.490 --> 00:08:45.590 are interesting phenomena. 00:08:45.590 --> 00:08:48.170 Waves are connected to vibrations. 00:08:48.170 --> 00:08:50.960 Because if you look at only, for example, 00:08:50.960 --> 00:08:55.130 a single object on these waves, you 00:08:55.130 --> 00:08:58.880 see that it is actually a single object which is oscillating 00:08:58.880 --> 00:09:01.500 up and down, oscillating up and down, 00:09:01.500 --> 00:09:02.870 and this is your vibration. 00:09:02.870 --> 00:09:06.710 So there's a close connection between single particle 00:09:06.710 --> 00:09:08.920 vibration and the waves. 00:09:08.920 --> 00:09:11.420 And that is the first thing that you learned. 00:09:11.420 --> 00:09:13.865 Therefore, we need to first understand the evolution 00:09:13.865 --> 00:09:17.460 of a single particle system. 00:09:17.460 --> 00:09:20.370 And we make use of this opportunity 00:09:20.370 --> 00:09:24.870 to start the discussion of scientific matter. 00:09:24.870 --> 00:09:27.830 So using this opportunity, basically, 00:09:27.830 --> 00:09:31.840 what we have been doing for the whole class is the following. 00:09:31.840 --> 00:09:35.490 So the first step is always to translate 00:09:35.490 --> 00:09:37.800 the physical situation which we are 00:09:37.800 --> 00:09:41.580 interested into mathematics, right? 00:09:41.580 --> 00:09:46.470 Because mathematics is the only language which we know which 00:09:46.470 --> 00:09:49.350 describes the nature. 00:09:49.350 --> 00:09:53.550 If you come out with a new language, 00:09:53.550 --> 00:09:59.470 and that is going to be a super duper breakthrough, 00:09:59.470 --> 00:10:02.360 it cannot be estimated by Nobel Prize. 00:10:02.360 --> 00:10:05.400 But the problem we are facing now 00:10:05.400 --> 00:10:09.990 is that this is the only language we know which works. 00:10:09.990 --> 00:10:15.550 Therefore, we really follow this recipe, 00:10:15.550 --> 00:10:18.900 which is similar to many, many other physics classes. 00:10:18.900 --> 00:10:21.090 And we have a physical situation, 00:10:21.090 --> 00:10:25.800 we use laws of nature or models, and we 00:10:25.800 --> 00:10:28.380 have a mathematical description, which 00:10:28.380 --> 00:10:30.307 is the Equation of Motion. 00:10:30.307 --> 00:10:31.890 And this is actually the hardest part, 00:10:31.890 --> 00:10:36.510 because you need to first define a coordinate system so that we 00:10:36.510 --> 00:10:39.990 can express everything in a system in that system, 00:10:39.990 --> 00:10:43.570 then you can make use of the physical laws you have learned 00:10:43.570 --> 00:10:47.100 from the previous 8.01 and 8.02 to write down 00:10:47.100 --> 00:10:48.330 the Equation of Motion. 00:10:48.330 --> 00:10:52.010 And most of the mistakes, and also most of the problems 00:10:52.010 --> 00:10:57.330 or difficulties you are facing is always in this step. 00:10:57.330 --> 00:11:01.230 Then we can solve the Equation of Motion, which is, strictly 00:11:01.230 --> 00:11:02.980 speaking, not my problem. 00:11:02.980 --> 00:11:06.660 It's the math department's problem. 00:11:06.660 --> 00:11:08.610 Yeah, that's their problem. 00:11:08.610 --> 00:11:10.980 Then we solve the Equation of Motion, 00:11:10.980 --> 00:11:13.350 and you will be given the formula. 00:11:13.350 --> 00:11:17.340 Then we use initial conditions and then make predictions. 00:11:17.340 --> 00:11:19.020 And then we would like to compare that 00:11:19.020 --> 00:11:20.680 to experimental results. 00:11:20.680 --> 00:11:23.310 And that is the general thing which 00:11:23.310 --> 00:11:25.680 we have been doing for physics. 00:11:25.680 --> 00:11:29.250 So let's take a look at those examples. 00:11:29.250 --> 00:11:33.840 Those are examples of simple harmonics motions. 00:11:33.840 --> 00:11:36.510 And you can see that these, all these systems 00:11:36.510 --> 00:11:40.860 have one object, which is oscillating. 00:11:40.860 --> 00:11:45.330 And you can see that their Equations of Motion 00:11:45.330 --> 00:11:47.480 are really similar to each other. 00:11:47.480 --> 00:11:51.570 It's theta double dot plus omega, zero squared, theta 00:11:51.570 --> 00:11:54.840 equal to zero for those idealized simple harmonic 00:11:54.840 --> 00:11:55.860 motions. 00:11:55.860 --> 00:12:00.710 And we learned that the solution of those equations 00:12:00.710 --> 00:12:07.250 are the same, which is a cosine function. 00:12:07.250 --> 00:12:13.430 Then we went ahead and added more craziness to the system. 00:12:13.430 --> 00:12:16.220 So basically, what we tried to do 00:12:16.220 --> 00:12:20.330 is to add a drag force into the game. 00:12:20.330 --> 00:12:27.560 And we were wondering if this more realistic description can 00:12:27.560 --> 00:12:31.500 match with experimental data. 00:12:31.500 --> 00:12:33.050 So this is the Equation of Motion, 00:12:33.050 --> 00:12:36.370 and the additional turn is the one in the middle, 00:12:36.370 --> 00:12:38.750 the gamma theta dot turn. 00:12:38.750 --> 00:12:42.950 And after entering these turns, not only is this 00:12:42.950 --> 00:12:46.340 an interesting model to describe the physical system we 00:12:46.340 --> 00:12:49.760 are talking about, but the mathematical solution 00:12:49.760 --> 00:12:53.870 is far more richer than what we talked about 00:12:53.870 --> 00:12:57.730 in the single harmonic oscillator case. 00:12:57.730 --> 00:13:01.790 Basically, you see that a general solution depends 00:13:01.790 --> 00:13:06.770 on the size of the gamma compared to omega dot zero, 00:13:06.770 --> 00:13:08.870 which is the oscillation-- 00:13:08.870 --> 00:13:11.410 the natural frequency of the system. 00:13:11.410 --> 00:13:11.930 OK. 00:13:11.930 --> 00:13:16.570 And then you can see there are three distinct different kinds 00:13:16.570 --> 00:13:17.870 of solutions. 00:13:17.870 --> 00:13:20.690 They have different mathematical forms. 00:13:20.690 --> 00:13:25.010 And we call them under damped solution, 00:13:25.010 --> 00:13:29.810 critically damped solution, and over damped solution. 00:13:29.810 --> 00:13:32.950 So those equations will be given to you. 00:13:32.950 --> 00:13:36.320 And the excitement is the following. 00:13:36.320 --> 00:13:39.210 So you can see that those solutions, 00:13:39.210 --> 00:13:43.100 if you plot the solution as a function of time, 00:13:43.100 --> 00:13:47.450 they look completely different as a function of time. 00:13:47.450 --> 00:13:50.760 So in the case of no damping, the amplitude is actually 00:13:50.760 --> 00:13:55.400 the constant, it's not actually reducing as a function of time. 00:13:55.400 --> 00:13:58.635 But when the damped system, the damping 00:13:58.635 --> 00:14:01.970 is turned on, then in the under damped situation, 00:14:01.970 --> 00:14:05.966 you can see that they end up reducing as a function of time. 00:14:05.966 --> 00:14:08.540 And if you have too much damping, 00:14:08.540 --> 00:14:14.310 you put the whole oscillator into some liquid, for example, 00:14:14.310 --> 00:14:19.580 and you see that oscillator disappear. 00:14:19.580 --> 00:14:21.550 The cool thing is the following. 00:14:21.550 --> 00:14:24.540 The excitement from-- as a physicist 00:14:24.540 --> 00:14:27.740 is that all of those crazy mathematical solutions 00:14:27.740 --> 00:14:32.330 actually match with experimental results. 00:14:32.330 --> 00:14:33.080 Wow. 00:14:33.080 --> 00:14:34.810 That is really cool. 00:14:34.810 --> 00:14:39.000 Because there is nobody saying that these should match 00:14:39.000 --> 00:14:42.980 and how, naturally, I should learn that OK, 00:14:42.980 --> 00:14:46.730 when should I change the behavior of the system. 00:14:46.730 --> 00:14:51.590 So this is really a miracle that this complicated mathematical 00:14:51.590 --> 00:14:56.390 description is useful and that it is super useful 00:14:56.390 --> 00:15:00.740 to describe the nature. 00:15:00.740 --> 00:15:06.950 Once we have learned that, we can now add a driving force 00:15:06.950 --> 00:15:09.260 into ligand. 00:15:09.260 --> 00:15:11.570 From the equation here, we can see 00:15:11.570 --> 00:15:15.680 that there is a natural frequency, omega dot zero, 00:15:15.680 --> 00:15:19.190 of this system, and there is a drag force turn, which 00:15:19.190 --> 00:15:23.150 is actually to quantify how much drag we have, 00:15:23.150 --> 00:15:24.320 we have a gamma there. 00:15:27.170 --> 00:15:32.150 We are driving it at a driving frequency omega t. 00:15:32.150 --> 00:15:33.620 So what we have learned from here 00:15:33.620 --> 00:15:38.690 is that if you are driving this system, you are-- for example, 00:15:38.690 --> 00:15:43.820 I am shaking that student, shaking you. 00:15:43.820 --> 00:15:47.510 OK, in the beginning, this student is going to resist. 00:15:47.510 --> 00:15:48.830 No, don't shake me. 00:15:48.830 --> 00:15:49.940 Come on. 00:15:49.940 --> 00:15:54.000 But at some point, he knows that Professor Lee 00:15:54.000 --> 00:15:57.090 is really determined. 00:15:57.090 --> 00:16:03.420 Therefore, he is going to be shaked at the frequency I like. 00:16:03.420 --> 00:16:03.920 OK. 00:16:03.920 --> 00:16:07.560 So that is actually what is happening here. 00:16:07.560 --> 00:16:11.480 This is so-called transient behavior. 00:16:11.480 --> 00:16:15.800 So in the beginning, the system doesn't like it. 00:16:15.800 --> 00:16:19.790 So this is making use of the superposition principle. 00:16:19.790 --> 00:16:24.210 So you can solve that homogeneous solution, which 00:16:24.210 --> 00:16:26.750 is on the right-hand side. 00:16:26.750 --> 00:16:29.094 It depends on the physical situation 00:16:29.094 --> 00:16:30.010 you are talking about. 00:16:30.010 --> 00:16:33.230 You choose the corresponding homogeneous solution. 00:16:33.230 --> 00:16:38.270 And lamba and psi is the driving force from E and G, right, 00:16:38.270 --> 00:16:44.060 and that is going to win at the end of the experiment, 00:16:44.060 --> 00:16:46.280 because I'm going to shake it forever, 00:16:46.280 --> 00:16:48.560 until the end of the universe. 00:16:48.560 --> 00:16:53.140 So you can see that at the end, you-- what is left over 00:16:53.140 --> 00:16:57.600 is really the steady state solution. 00:16:57.600 --> 00:17:02.990 And it has this structure, A omega d, depends on omega. 00:17:02.990 --> 00:17:05.280 And you get resonance behavior. 00:17:05.280 --> 00:17:07.690 Don't forget to review that. 00:17:07.690 --> 00:17:13.160 So you have a delay in phase because when 00:17:13.160 --> 00:17:17.099 I shake the student, the student needs some time to respond. 00:17:17.099 --> 00:17:24.420 Therefore, the delta is non-zero if the student is damped. 00:17:24.420 --> 00:17:25.349 All right. 00:17:25.349 --> 00:17:29.070 So now we have learned all the secrets about a single object 00:17:29.070 --> 00:17:30.060 system. 00:17:30.060 --> 00:17:34.470 Then we can now go ahead and study coupled oscillators. 00:17:34.470 --> 00:17:37.140 There are a few examples here, which 00:17:37.140 --> 00:17:41.520 is coupled pendulums or coupled spring-mass systems. 00:17:41.520 --> 00:17:45.720 And we found that a very useful description 00:17:45.720 --> 00:17:48.030 of this kind of system is to make 00:17:48.030 --> 00:17:52.290 use of the matrix language. 00:17:52.290 --> 00:17:56.940 So originally, if you have n objects in a system, 00:17:56.940 --> 00:18:02.840 you have n Equations of Motion, and that looks horrible. 00:18:02.840 --> 00:18:09.740 But what is done in 8.03 is that we introduce a notation 00:18:09.740 --> 00:18:11.280 with a matrix. 00:18:11.280 --> 00:18:13.830 Basically, if you write everything in terms of matrix, 00:18:13.830 --> 00:18:18.210 then it looks really friendly, and it looks really 00:18:18.210 --> 00:18:21.330 like a single oscillator. 00:18:21.330 --> 00:18:22.050 OK? 00:18:22.050 --> 00:18:25.860 Although solving this equation is still 00:18:25.860 --> 00:18:29.190 a little bit more work. 00:18:29.190 --> 00:18:32.140 And basically, you can see that from this example, 00:18:32.140 --> 00:18:35.880 we can actually derive M minus 1 K matrix, 00:18:35.880 --> 00:18:39.300 and the whole equation won't be-- 00:18:39.300 --> 00:18:43.110 the Equation of Motion problem solving problem 00:18:43.110 --> 00:18:47.480 becomes an M minus 1 K matrix eigenvalue problem. 00:18:47.480 --> 00:18:50.740 What is an M minus 1 K matrix? 00:18:50.740 --> 00:18:54.390 This is describing how each component in the system 00:18:54.390 --> 00:18:57.610 interacts with each other. 00:18:57.610 --> 00:19:02.610 Once we have this, we can solve the eigenvalue problem, 00:19:02.610 --> 00:19:05.250 and we are going to be able to figure out 00:19:05.250 --> 00:19:08.620 the normal modes of those systems. 00:19:08.620 --> 00:19:11.140 So what is a normal mode? 00:19:11.140 --> 00:19:14.950 Normal modes is a situation where 00:19:14.950 --> 00:19:19.330 all the components in the system are oscillating 00:19:19.330 --> 00:19:23.830 at the same frequency and they are also at the same phase. 00:19:23.830 --> 00:19:26.320 So that is the definition of normal mode. 00:19:26.320 --> 00:19:31.710 And those are what is used in a deviation, 00:19:31.710 --> 00:19:34.960 also, which leads us to the eigenvalue problem. 00:19:34.960 --> 00:19:39.310 We define Z equal to X 1 H or I omega t plus 5. 00:19:39.310 --> 00:19:43.440 Everybody is oscillating at omega and also at phase 5, 00:19:43.440 --> 00:19:43.960 right? 00:19:43.960 --> 00:19:46.450 So that is what we actually learned. 00:19:46.450 --> 00:19:48.760 And what is actually the physical meaning 00:19:48.760 --> 00:19:50.980 of those normal modes? 00:19:50.980 --> 00:19:55.030 So if we plot the locus of the two coupled 00:19:55.030 --> 00:19:58.490 pendulum problem, what we see is the following. 00:19:58.490 --> 00:20:02.590 So basically, you will see that the locus looks 00:20:02.590 --> 00:20:06.100 like really complicated as a function of time 00:20:06.100 --> 00:20:10.630 if you plot X1 X2 versus time. 00:20:10.630 --> 00:20:14.000 But if we rotate this system a bit, 00:20:14.000 --> 00:20:18.410 then we find that there's a really interesting projection, 00:20:18.410 --> 00:20:21.510 which is the principal coordinate. 00:20:21.510 --> 00:20:28.240 You see that all those crazy strange phenomena 00:20:28.240 --> 00:20:32.390 we see with coupled systems are just illusions. 00:20:32.390 --> 00:20:35.170 Actually, you can understand then 00:20:35.170 --> 00:20:39.320 by really using the right projection. 00:20:39.320 --> 00:20:43.180 To one-- to the right coordinate system. 00:20:43.180 --> 00:20:47.230 Then you will see that actually the system is doing still 00:20:47.230 --> 00:20:49.570 simple harmonic motion. 00:20:49.570 --> 00:20:51.100 So that is actually the core thing 00:20:51.100 --> 00:20:54.560 which we learned from coupled systems. 00:20:54.560 --> 00:20:58.120 So we learned about how to solve the coupled system, 00:20:58.120 --> 00:21:04.030 and we also learned about going to an infinite number 00:21:04.030 --> 00:21:06.370 of coupled systems. 00:21:06.370 --> 00:21:09.470 So then this is an example here. 00:21:09.470 --> 00:21:15.160 So for example, I can have pendulum and springs, 00:21:15.160 --> 00:21:17.520 and we connect them all together, 00:21:17.520 --> 00:21:23.890 and I need to hire many, many students so that they plays it, 00:21:23.890 --> 00:21:27.190 plays until it fills up the whole universe. 00:21:27.190 --> 00:21:32.680 So this is the idea of an infinite system. 00:21:32.680 --> 00:21:35.320 You can see that that means my M minus 00:21:35.320 --> 00:21:37.350 1 K matrix is going to be an infinite 00:21:37.350 --> 00:21:40.930 times infinite long matrix. 00:21:40.930 --> 00:21:42.370 It's two dimensional. 00:21:42.370 --> 00:21:45.460 And the A is infinitely long. 00:21:45.460 --> 00:21:47.710 And that sounds really scary. 00:21:47.710 --> 00:21:53.020 And in general, we don't know how to deal with this, really. 00:21:53.020 --> 00:22:00.700 And it can be as arbitrarily crazy as you can imagine. 00:22:00.700 --> 00:22:04.510 What we discuss 8.03 is a special case. 00:22:04.510 --> 00:22:07.540 Basically, we are discussing about systems 00:22:07.540 --> 00:22:11.420 which are having a spatial kind of symmetry, which 00:22:11.420 --> 00:22:17.110 is translation symmetry, as you can see from all those figures. 00:22:17.110 --> 00:22:20.110 And you can see that all those figures will 00:22:20.110 --> 00:22:28.530 have to all have the same normal modes because of this base 00:22:28.530 --> 00:22:30.890 translation symmetry. 00:22:30.890 --> 00:22:37.700 What we discussed about is that we introduce an S matrix, which 00:22:37.700 --> 00:22:43.130 is used to describe the kind of symmetry 00:22:43.130 --> 00:22:46.220 that this system satisfies. 00:22:46.220 --> 00:22:52.400 And if we calculate the commutator S and M minus 1 K 00:22:52.400 --> 00:22:58.100 matrix, if this commutator shows that the evaluate-- 00:22:58.100 --> 00:23:01.010 if you evaluate this commutator and you get zero, 00:23:01.010 --> 00:23:03.470 now it means they commute. 00:23:03.470 --> 00:23:09.230 And the consequence is that the S matrix and the S M minus 1 K 00:23:09.230 --> 00:23:13.970 matrix will share the same eigenvectors. 00:23:13.970 --> 00:23:19.700 So you don't really need to know how to derive this-- 00:23:19.700 --> 00:23:21.560 to arrive at this conclusion, but it 00:23:21.560 --> 00:23:24.380 is a very useful conclusion. 00:23:24.380 --> 00:23:29.500 So that means instead of solving M minus 1 K matrix eigenvalue 00:23:29.500 --> 00:23:34.370 problem, I can now go ahead to solve the S matrix eigenvalue 00:23:34.370 --> 00:23:34.940 problem. 00:23:34.940 --> 00:23:37.520 And usually, that's much easier. 00:23:37.520 --> 00:23:39.920 So for the exam, you need to know 00:23:39.920 --> 00:23:42.830 how to write down S matrix. 00:23:42.830 --> 00:23:46.860 You need to know how to solve eigenvalue problems, including 00:23:46.860 --> 00:23:50.090 M minus 1 K matrix and the S matrix. 00:23:50.090 --> 00:23:54.950 And then we can get to normal mode frequency, omega squared, 00:23:54.950 --> 00:23:58.460 and we can also solve the corresponding normal modes. 00:23:58.460 --> 00:24:01.670 And here is telling you what would 00:24:01.670 --> 00:24:05.840 be the solution for space translation of the matrix 00:24:05.840 --> 00:24:06.656 system. 00:24:06.656 --> 00:24:08.030 And basically what we will see is 00:24:08.030 --> 00:24:12.350 that making use of the S matrix should be-- 00:24:12.350 --> 00:24:14.880 brings you to the conclusion that A, 00:24:14.880 --> 00:24:19.880 j must be proportional to exponential i, j, k, a, 00:24:19.880 --> 00:24:25.160 where this A is the length scale of this system, the distance 00:24:25.160 --> 00:24:27.140 between all those little mass. 00:24:27.140 --> 00:24:31.530 And the j is a label which tells you which little mass 00:24:31.530 --> 00:24:33.020 I am talking about. 00:24:33.020 --> 00:24:36.500 And k is the-- 00:24:36.500 --> 00:24:39.520 some arbitrary constant. 00:24:39.520 --> 00:24:43.520 But by now, you should have the idea basically that's-- 00:24:43.520 --> 00:24:44.590 that's what? 00:24:44.590 --> 00:24:48.260 That, essentially, is the wave number, right? 00:24:48.260 --> 00:24:49.790 So that is really cool. 00:24:49.790 --> 00:24:52.970 So that's all planned in advance. 00:24:52.970 --> 00:24:58.850 And basically, you can see that we can also write down the A, k 00:24:58.850 --> 00:25:01.940 because we know that A, j will be proportional to exponential 00:25:01.940 --> 00:25:06.340 i, j, k, A, after solving the eigenvalue problem for S 00:25:06.340 --> 00:25:08.930 matrix. 00:25:08.930 --> 00:25:14.240 Then we actually went one step forward to make it continuous. 00:25:14.240 --> 00:25:20.670 So basically, we made the space between particles very, very 00:25:20.670 --> 00:25:22.170 small. 00:25:22.170 --> 00:25:24.350 And also, at the same time, we make 00:25:24.350 --> 00:25:30.240 sure that the string doesn't become supermassive. 00:25:30.240 --> 00:25:36.380 And we concluded that we get some kind of equation 00:25:36.380 --> 00:25:39.170 popping out from this exercise. 00:25:39.170 --> 00:25:43.730 M minus 1 K matrix becomes minus T over rho L partial square, 00:25:43.730 --> 00:25:45.650 partial x squared. 00:25:45.650 --> 00:25:48.830 You don't have to really derive this for the exam, 00:25:48.830 --> 00:25:50.900 but you would need to know the conclusion 00:25:50.900 --> 00:25:57.560 and that psi j becomes psi as a function of x and t. 00:25:57.560 --> 00:26:01.110 And the magical function appeared, 00:26:01.110 --> 00:26:02.710 which is the wave equation. 00:26:02.710 --> 00:26:05.090 Oh my god, this is the whole craziness 00:26:05.090 --> 00:26:09.650 we have been dealing with the whole 8.03. 00:26:09.650 --> 00:26:11.750 This is actually really remarkable 00:26:11.750 --> 00:26:16.340 that we can come from single object oscillation, 00:26:16.340 --> 00:26:18.920 putting it all together, making it continuous, 00:26:18.920 --> 00:26:21.050 then this equation really popped out. 00:26:21.050 --> 00:26:27.200 And this equation really describes multiple systems. 00:26:27.200 --> 00:26:30.830 Then we went ahead to actually discuss the property 00:26:30.830 --> 00:26:32.510 of the wave equation. 00:26:32.510 --> 00:26:33.620 It looks like this. 00:26:33.620 --> 00:26:39.290 Basically, I replaced the t over rho L by v, p squared. 00:26:39.290 --> 00:26:41.450 By now, you know the meaning of v, 00:26:41.450 --> 00:26:43.580 p is actually the phase velocity. 00:26:43.580 --> 00:26:48.790 And we discussed two kinds of solutions, 00:26:48.790 --> 00:26:50.390 special kinds of solutions. 00:26:50.390 --> 00:26:54.320 The first kind is normal modes. 00:26:54.320 --> 00:26:57.830 The second one is progressive wave solution, 00:26:57.830 --> 00:27:00.050 or traveling wave solution, whatever name 00:27:00.050 --> 00:27:02.080 you want to call it. 00:27:02.080 --> 00:27:07.090 Let's take a look at the normal modes, what have we learned. 00:27:07.090 --> 00:27:13.400 So if you have a bound system, a bound continuous system, 00:27:13.400 --> 00:27:17.790 the normal mode is your distending waves for the wave 00:27:17.790 --> 00:27:19.680 equation we discussed. 00:27:19.680 --> 00:27:22.350 And basically, the functional form 00:27:22.350 --> 00:27:28.560 is A, m, sine, k, m, x plus alpha, m and sine, omega, m, t 00:27:28.560 --> 00:27:30.780 plus beta, m. 00:27:30.780 --> 00:27:35.940 So what we actually learned from the previous lecture 00:27:35.940 --> 00:27:37.030 is the following. 00:27:37.030 --> 00:27:42.930 So basically, you can decide the k, m and alpha, 00:27:42.930 --> 00:27:45.180 m by just boundary conditions. 00:27:45.180 --> 00:27:48.810 So before you introduce boundary conditions, which 00:27:48.810 --> 00:27:52.890 are the conditions allow you to describe 00:27:52.890 --> 00:27:56.520 multiple nearby systems consistently. 00:27:56.520 --> 00:27:59.520 So that is the meaning of boundary condition. 00:27:59.520 --> 00:28:02.100 Before you introduce that, k, m and alpha, 00:28:02.100 --> 00:28:04.420 m are arbitrary numbers. 00:28:04.420 --> 00:28:08.340 Whatever number you choose is the-- 00:28:08.340 --> 00:28:11.160 can satisfy the wave equation. 00:28:11.160 --> 00:28:15.180 But after you introduce the boundary condition, 00:28:15.180 --> 00:28:21.660 you figure that out from the problem you are given, then k, 00:28:21.660 --> 00:28:25.020 m and alpha, m cannot be arbitrary anymore. 00:28:25.020 --> 00:28:28.330 And they usually become discrete numbers. 00:28:28.330 --> 00:28:28.830 OK. 00:28:28.830 --> 00:28:33.000 So that is what we learned from the previous lectures. 00:28:33.000 --> 00:28:37.030 And finally, we also see that omega, 00:28:37.030 --> 00:28:41.790 m is determined by the property of the system, 00:28:41.790 --> 00:28:45.030 by a so-called dispersion relation. 00:28:45.030 --> 00:28:49.170 In this case, it's linear, it's proportional to k, m, 00:28:49.170 --> 00:28:51.840 because we are talking about non-dispersive medium 00:28:51.840 --> 00:28:52.830 for the moment. 00:28:52.830 --> 00:28:57.510 And we have this beta, m, which is related 00:28:57.510 --> 00:28:59.370 to the initial condition. 00:28:59.370 --> 00:29:03.150 And the a, m, which can be determined 00:29:03.150 --> 00:29:06.120 by a Fourier decomposition. 00:29:06.120 --> 00:29:08.620 So if you are not familiar with this, 00:29:08.620 --> 00:29:12.900 you have to really review how to do Fourier decomposition. 00:29:12.900 --> 00:29:16.630 I know most of did very well on the midterm, 00:29:16.630 --> 00:29:21.150 but maybe some of you forgot how to determine a, m 00:29:21.150 --> 00:29:22.950 and it will be very, very important 00:29:22.950 --> 00:29:27.330 to review that for the preparation for the final. 00:29:27.330 --> 00:29:30.610 Now the second set of solutions is the following. 00:29:30.610 --> 00:29:31.950 So you have progress-- 00:29:31.950 --> 00:29:33.510 progressing waves. 00:29:33.510 --> 00:29:37.390 And the functional form is really interesting. 00:29:37.390 --> 00:29:42.390 So you can see this can be written as F, 00:29:42.390 --> 00:29:47.990 F is some arbitrary function, x plus-minus v, p, t. 00:29:47.990 --> 00:29:50.790 Basically that is that you're describing a wave which 00:29:50.790 --> 00:29:53.250 is traveling to the positive-- 00:29:53.250 --> 00:29:59.580 to a negative or positive direction in the x direction. 00:29:59.580 --> 00:30:01.080 Or you can actually write it down 00:30:01.080 --> 00:30:04.380 as G function k, x plus-minus omega, t. 00:30:04.380 --> 00:30:09.730 Actually, they all work for wave equations. 00:30:09.730 --> 00:30:13.780 Now we went ahead and applied approach 00:30:13.780 --> 00:30:17.520 which we learned from the general solution of wave 00:30:17.520 --> 00:30:23.750 equation to massive strings, and we discussed about sound waves. 00:30:23.750 --> 00:30:29.100 For the sound waves, it will be important to review 00:30:29.100 --> 00:30:33.640 what are the boundary conditions for the displacement 00:30:33.640 --> 00:30:38.430 of the molecules in the sound wave, 00:30:38.430 --> 00:30:43.050 compare that to the pressure deviation 00:30:43.050 --> 00:30:44.850 from the room pressure. 00:30:44.850 --> 00:30:47.100 So I think it's important to make sure 00:30:47.100 --> 00:30:50.090 that you understand the difference between these two, 00:30:50.090 --> 00:30:55.950 what are the boundary conditions and basically it should be 00:30:55.950 --> 00:30:58.460 very similar to the solution-- 00:30:58.460 --> 00:31:01.620 the boundary condition for the massive strings. 00:31:01.620 --> 00:31:06.450 And we also talked about electromagnetic waves. 00:31:06.450 --> 00:31:13.180 And that is another topic which you will really have to review. 00:31:13.180 --> 00:31:15.820 Several things which are especially interesting is that 00:31:15.820 --> 00:31:22.110 an electric field cannot be without a magnetic field. 00:31:22.110 --> 00:31:25.240 They are always together, no matter what. 00:31:25.240 --> 00:31:28.050 So if you have trouble with the electric field, 00:31:28.050 --> 00:31:31.410 then there must be trouble in the magnetic field. 00:31:31.410 --> 00:31:36.930 And that is governed by the Maxwell's equation. 00:31:36.930 --> 00:31:40.280 Before we go into the detail of those, 00:31:40.280 --> 00:31:44.400 we also discussed about dispersive medium. 00:31:44.400 --> 00:31:46.950 So in the case of dispersive medium, 00:31:46.950 --> 00:31:51.300 we used a special kind of example, which 00:31:51.300 --> 00:31:55.060 is strings with stiffness. 00:31:55.060 --> 00:31:57.810 So basically, what we found is that if you 00:31:57.810 --> 00:32:16.460 have a certain kind of wave equation, like this one, 00:32:16.460 --> 00:32:18.340 I am writing this one here. 00:32:18.340 --> 00:32:21.190 Basically, if I add the additional term 00:32:21.190 --> 00:32:27.860 to describe the stiffness, then what is going to happen 00:32:27.860 --> 00:32:31.310 is that the dispersion relation, when 00:32:31.310 --> 00:32:34.850 I ask you to plot the dispersion relation, you will be-- 00:32:34.850 --> 00:32:40.520 I am requesting you to find the relation between omega and K. 00:32:40.520 --> 00:32:42.620 And I'm going over this in more detail 00:32:42.620 --> 00:32:46.310 because I see so many similar mistakes on the midterm. 00:32:46.310 --> 00:32:51.020 So basically what I'm asking is omega versus K. 00:32:51.020 --> 00:32:55.640 And in the-- if we don't have this turn, then basically, 00:32:55.640 --> 00:32:57.050 you have a straight line. 00:32:57.050 --> 00:33:01.560 Straight line means you have a non-dispersive medium. 00:33:01.560 --> 00:33:04.190 And if you add this turn, you need 00:33:04.190 --> 00:33:08.090 to know how to evaluate the dispersion relation. 00:33:08.090 --> 00:33:13.350 The quickest way to evaluate the dispersion relation is 00:33:13.350 --> 00:33:16.850 to just simply plug in the progresssing wave 00:33:16.850 --> 00:33:21.260 solution for the G function or harmonic progressing 00:33:21.260 --> 00:33:23.680 wave solution, find omega-- 00:33:23.680 --> 00:33:28.670 K, x plus-minus omega t, into this equation, 00:33:28.670 --> 00:33:32.500 then you will be able to figure out the dispersion relation. 00:33:32.500 --> 00:33:34.770 And what we figure out is the following. 00:33:34.770 --> 00:33:38.150 If we include stiffness, then you 00:33:38.150 --> 00:33:43.610 can see that the dispersion relation is not a line anymore 00:33:43.610 --> 00:33:47.360 and is actually some kind of curve, 00:33:47.360 --> 00:33:51.340 and the slope is actually changing. 00:33:51.340 --> 00:33:56.380 And there are dramatic consequence from this thing. 00:33:56.380 --> 00:34:02.150 That means if I have a traveling wave 00:34:02.150 --> 00:34:09.120 with different wavelengths, that means the phase velocity v, 00:34:09.120 --> 00:34:13.930 p equal to omega over K is going to be 00:34:13.930 --> 00:34:17.940 different for waves with different frequencies, 00:34:17.940 --> 00:34:20.100 or different wavelengths. 00:34:20.100 --> 00:34:22.540 So that is how you clear the problem. 00:34:22.540 --> 00:34:26.449 Because if I have initially produced a signal which 00:34:26.449 --> 00:34:30.530 is a triangle and I let it propagate, 00:34:30.530 --> 00:34:35.370 what is going to happen is that the slow component 00:34:35.370 --> 00:34:37.670 will be lagging behind. 00:34:37.670 --> 00:34:40.469 Those are the slow components. 00:34:40.469 --> 00:34:45.960 And the fast components will go ahead of the nominal speed. 00:34:45.960 --> 00:34:49.690 So there will be a spread of the signal. 00:34:49.690 --> 00:34:54.239 Originally, maybe you have some kind of a square wave, 00:34:54.239 --> 00:35:01.050 and this thing will become something which is actually 00:35:01.050 --> 00:35:08.100 smeared out in space, and then you lose the information. 00:35:08.100 --> 00:35:10.420 And we are going to talk about that later. 00:35:10.420 --> 00:35:14.190 And we also learned about group velocity. 00:35:14.190 --> 00:35:17.000 So what is your group velocity? 00:35:17.000 --> 00:35:19.920 Group velocity v, p-- oh, sorry, v, 00:35:19.920 --> 00:35:25.050 g is actually partial omega, partial K, 00:35:25.050 --> 00:35:31.000 which is the slope of a tangential line here. 00:35:31.000 --> 00:35:35.400 And where the phase velocity is connecting 00:35:35.400 --> 00:35:39.360 this point to that point, and the slope of this line 00:35:39.360 --> 00:35:44.280 is the phase velocity, and the slope of the line 00:35:44.280 --> 00:35:47.730 cutting through this point, which is giving you 00:35:47.730 --> 00:35:49.770 the group velocity. 00:35:49.770 --> 00:35:54.480 And we actually learned the definition of-- 00:35:54.480 --> 00:35:57.330 the consequence of group velocity and phase 00:35:57.330 --> 00:36:01.140 velocity by introducing you a bit phenomena. 00:36:01.140 --> 00:36:09.480 Basically, we add two waves with similar wavelengths, or wave 00:36:09.480 --> 00:36:10.120 numbers. 00:36:10.120 --> 00:36:11.910 Basically, what we see is the following. 00:36:11.910 --> 00:36:15.050 So basically, you see some behavior like this. 00:36:17.730 --> 00:36:22.530 We see this-- the superposition of these two waves 00:36:22.530 --> 00:36:27.440 which produce a bit phenomena can be understood by something 00:36:27.440 --> 00:36:32.230 which is oscillating really fast modulated by a much 00:36:32.230 --> 00:36:39.840 slower more variating envelope. 00:36:39.840 --> 00:36:44.780 Basically, you can actually understand the bit phenomena 00:36:44.780 --> 00:36:50.670 by actually identifying these two interesting structures. 00:36:50.670 --> 00:36:54.990 And the speed of all those little peaks 00:36:54.990 --> 00:36:59.760 is traveling at phase velocity. 00:36:59.760 --> 00:37:02.940 And the speed of the envelope is found 00:37:02.940 --> 00:37:07.140 to be traveling at group velocity. 00:37:07.140 --> 00:37:09.600 So that is what we have learned. 00:37:09.600 --> 00:37:14.450 And we can have group velocity and the phase velocity 00:37:14.450 --> 00:37:17.280 traveling in the same direction. 00:37:17.280 --> 00:37:21.130 And we can also have a negative group velocity. 00:37:21.130 --> 00:37:25.010 So that is a technique which is really, really very difficult. 00:37:25.010 --> 00:37:27.680 And I'm still trying to practice and make sure 00:37:27.680 --> 00:37:30.180 they I can demo that in 8.03. 00:37:30.180 --> 00:37:33.710 Basically, it's like the whole system, 00:37:33.710 --> 00:37:38.970 the whole detailed structure moving in a positive direction. 00:37:38.970 --> 00:37:42.360 But the body, or say the envelope, 00:37:42.360 --> 00:37:46.570 is actually moving in the negative x direction. 00:37:46.570 --> 00:37:48.880 So that is also possible. 00:37:48.880 --> 00:37:51.210 And you can actually construct a system which 00:37:51.210 --> 00:37:54.420 has a negative group velocity. 00:37:54.420 --> 00:38:00.150 So once we have done that, we also 00:38:00.150 --> 00:38:05.400 tried to understand further the description of the solution 00:38:05.400 --> 00:38:08.980 for the dispersive medium. 00:38:08.980 --> 00:38:12.660 So basically, what we actually went over during the class 00:38:12.660 --> 00:38:17.190 is that OK, now, if the f function f of t 00:38:17.190 --> 00:38:22.000 is describing Yen-Jie's hand, and I'm holding an infinitely 00:38:22.000 --> 00:38:27.690 long string and I shake it as a function of time, 00:38:27.690 --> 00:38:29.900 and that essentially, this motion, 00:38:29.900 --> 00:38:33.620 is actually described by this f function. 00:38:33.620 --> 00:38:39.720 What we know is that this oscillation, OK, I can do one, 00:38:39.720 --> 00:38:43.980 but I won't, but all kinds of f functions 00:38:43.980 --> 00:38:47.340 can be described as superpositions 00:38:47.340 --> 00:38:53.490 of many, many, many waves with different angular frequencies. 00:38:53.490 --> 00:38:55.740 So that's a miracle which we borrowed 00:38:55.740 --> 00:38:57.720 from the math department again. 00:38:57.720 --> 00:38:59.730 And you can see that f function can 00:38:59.730 --> 00:39:04.950 be written as the sum of all kinds of different waves 00:39:04.950 --> 00:39:07.350 with different angular frequencies 00:39:07.350 --> 00:39:12.290 with population c omega. 00:39:12.290 --> 00:39:19.080 This is the weight which makes that become the f function. 00:39:19.080 --> 00:39:23.790 And we can figure out the c omega 00:39:23.790 --> 00:39:27.480 by doing a Fourier transform. 00:39:27.480 --> 00:39:34.550 And finally, what will be the resulting wave function, psi, 00:39:34.550 --> 00:39:38.610 x, t, which is the wave function generated 00:39:38.610 --> 00:39:41.100 by the oscillation of my hand. 00:39:41.100 --> 00:39:45.070 And those are governed by the wave equation, which gives you 00:39:45.070 --> 00:39:47.880 the relation between omega and the k 00:39:47.880 --> 00:39:51.810 can be returned in that functional form. 00:39:51.810 --> 00:39:57.570 So the good news is that with the help of Fourier transform, 00:39:57.570 --> 00:40:01.170 we can also describe and predict what 00:40:01.170 --> 00:40:06.570 is going to happen no matter if this system is dispersive 00:40:06.570 --> 00:40:09.820 or not dispersive using this approach. 00:40:09.820 --> 00:40:10.320 OK. 00:40:10.320 --> 00:40:12.000 So that is really cool. 00:40:12.000 --> 00:40:17.750 And you can of course can do a cross-check just to-- 00:40:17.750 --> 00:40:22.260 assuring that this is a non-dispersive medium. 00:40:22.260 --> 00:40:24.420 And you are also going to get back 00:40:24.420 --> 00:40:27.540 to what you should expect the solution 00:40:27.540 --> 00:40:31.070 to non-dispersive medium for the psi, x, t. 00:40:31.070 --> 00:40:36.720 So that is one thing which is really remarkable. 00:40:36.720 --> 00:40:40.530 And I think what is needed to know 00:40:40.530 --> 00:40:44.520 is not a deviation of all those formulas, 00:40:44.520 --> 00:40:51.540 but how the plotting and the derived c omega by using 00:40:51.540 --> 00:40:56.870 the formula you are given and how to then put together 00:40:56.870 --> 00:41:01.710 all the solutions and it becomes the resulting solution 00:41:01.710 --> 00:41:04.410 for the psi, x, t, which is really 00:41:04.410 --> 00:41:06.930 the solution we really care. 00:41:06.930 --> 00:41:11.000 So for that, you need to know how to do the integration. 00:41:11.000 --> 00:41:16.000 You need to know how to derive the dispersion relation. 00:41:16.000 --> 00:41:19.760 Then one thing left over is to put the problem 00:41:19.760 --> 00:41:24.500 into that equation, which is also given to you in a formula. 00:41:24.500 --> 00:41:27.530 And we will not ask you to do a very, very 00:41:27.530 --> 00:41:32.600 complicated integration for sure on the final. 00:41:32.600 --> 00:41:35.650 So what is the consequence? 00:41:35.650 --> 00:41:38.810 Basically, one thing which is interesting to know 00:41:38.810 --> 00:41:43.100 is that if you have a wave in a coordinate space, which 00:41:43.100 --> 00:41:49.880 is really widely spread out, and you can do a Fourier transform 00:41:49.880 --> 00:41:54.170 to get the wave population in the frequency space, what 00:41:54.170 --> 00:41:58.610 we find is that when this wave is really, really 00:41:58.610 --> 00:42:01.550 wide in the space, then what we find 00:42:01.550 --> 00:42:05.510 is that the wave population in the frequency space 00:42:05.510 --> 00:42:09.490 is very narrow by using a Fourier transform. 00:42:09.490 --> 00:42:12.900 And that just gives you the result. And on the other hand, 00:42:12.900 --> 00:42:14.380 if you have a really-- 00:42:14.380 --> 00:42:18.410 a very narrow pulse in the coordinate space, 00:42:18.410 --> 00:42:21.950 for example, I do this-- shwhew --very, very-- really quickly. 00:42:21.950 --> 00:42:24.460 I create a very narrow pulse. 00:42:24.460 --> 00:42:26.630 And then what is actually happening 00:42:26.630 --> 00:42:33.800 is that I will have to use a very wide range of frequency 00:42:33.800 --> 00:42:38.120 space to describe this very narrow pulse. 00:42:38.120 --> 00:42:39.410 So that leads to-- 00:42:39.410 --> 00:42:43.310 direct consequences of that is uncertainty principle. 00:42:43.310 --> 00:42:47.660 And this is closely connected to the uncertainty principle 00:42:47.660 --> 00:42:50.390 we talk about in quantum mechanics. 00:42:50.390 --> 00:42:54.620 Delta, p times delta, x greater or equal to h bar over 2. 00:42:57.170 --> 00:42:57.740 All right. 00:42:57.740 --> 00:43:01.960 So we have done with the one-dimensional case. 00:43:01.960 --> 00:43:05.040 And we also talked about a two-dimensional 00:43:05.040 --> 00:43:06.990 and a three-dimensional case. 00:43:06.990 --> 00:43:11.450 And this is the example of two-dimensional membranes, 00:43:11.450 --> 00:43:14.730 and they actually are constrained so that 00:43:14.730 --> 00:43:17.580 their boundary condition at-- 00:43:17.580 --> 00:43:18.720 the boundary is equal-- 00:43:18.720 --> 00:43:21.300 no, the wave function is equal to zero. 00:43:21.300 --> 00:43:25.620 And you can identify all those normal modes. 00:43:25.620 --> 00:43:32.410 And we went ahead also to talk about geometrical optics laws. 00:43:32.410 --> 00:43:38.350 Basically, how we derive that is to have a plane wave. 00:43:38.350 --> 00:43:41.040 First, you have a plane wave propagating 00:43:41.040 --> 00:43:44.970 toward the boundary of two different mediums, 00:43:44.970 --> 00:43:49.160 and we were wondering well, what is the refracted wave 00:43:49.160 --> 00:43:50.730 and the transmitting wave. 00:43:50.730 --> 00:43:56.160 By using the-- by making sure just one point, which 00:43:56.160 --> 00:44:00.900 is that the membranes don't break, 00:44:00.900 --> 00:44:04.890 the wave function is continuous at this boundary. 00:44:04.890 --> 00:44:07.590 That's the only assumption which you use. 00:44:07.590 --> 00:44:10.290 We went through the mathematics, which you don't really 00:44:10.290 --> 00:44:12.630 need to remember all of them. 00:44:12.630 --> 00:44:16.260 But you really need to remember the consequence. 00:44:16.260 --> 00:44:18.060 The consequence is the following. 00:44:18.060 --> 00:44:23.190 Basically, what we see is that if you have incident plane 00:44:23.190 --> 00:44:27.570 wave with incident angle theta 1, 00:44:27.570 --> 00:44:31.350 the refractive wave will be having an angle of theta 1 00:44:31.350 --> 00:44:32.260 as well. 00:44:32.260 --> 00:44:36.270 So that's the first law of refraction, refraction law. 00:44:36.270 --> 00:44:38.090 And then the second one which we learned 00:44:38.090 --> 00:44:41.260 is that the transmitting wave will 00:44:41.260 --> 00:44:47.000 satisfy Snell's law, n, 1, sine, theta, 1 equal to n, 2, 00:44:47.000 --> 00:44:48.530 sine, theta, 2. 00:44:48.530 --> 00:44:55.930 And that is very interesting because this, Snell's law 00:44:55.930 --> 00:45:01.230 has also nothing to do with Maxwell's equation. 00:45:01.230 --> 00:45:02.400 You see? 00:45:02.400 --> 00:45:02.900 Right? 00:45:02.900 --> 00:45:05.840 That's actually what you can learn from here. 00:45:05.840 --> 00:45:09.600 We usually use electromagnetic waves 00:45:09.600 --> 00:45:12.450 to demonstrate Snell's law. 00:45:12.450 --> 00:45:15.390 But from 8.03, we learned that it has nothing 00:45:15.390 --> 00:45:20.090 to do with Maxwell's equation. 00:45:20.090 --> 00:45:23.910 It applies to all kinds of different systems, which 00:45:23.910 --> 00:45:24.890 you can-- 00:45:24.890 --> 00:45:27.850 which can be described by wave functions. 00:45:27.850 --> 00:45:32.490 So that is actually the very important consequence. 00:45:32.490 --> 00:45:36.690 But on the other hand, as we all discussed later, 00:45:36.690 --> 00:45:41.200 the relative amplitude of the incident wave, refracted wave, 00:45:41.200 --> 00:45:45.120 and transmitting wave, the relative amplitude 00:45:45.120 --> 00:45:48.450 is governed by Maxwell's equation. 00:45:48.450 --> 00:45:52.380 So I would like to make that really crystal clear. 00:45:52.380 --> 00:45:54.870 So the relative amplitude is governed 00:45:54.870 --> 00:45:58.500 by really the physical laws, which 00:45:58.500 --> 00:46:06.380 actually governs the propagation of those plane waves. 00:46:06.380 --> 00:46:07.070 OK. 00:46:07.070 --> 00:46:14.750 So I think we can take a five minute break to have some air. 00:46:14.750 --> 00:46:18.530 And of course, you can-- you are welcome to continue to use 00:46:18.530 --> 00:46:20.840 all this juice and coffee. 00:46:20.840 --> 00:46:22.260 And coming back at 38. 00:46:31.850 --> 00:46:32.350 OK. 00:46:32.350 --> 00:46:36.567 So welcome back, everybody, from the break. 00:46:36.567 --> 00:46:38.657 AUDIENCE: [INAUDIBLE] 00:46:38.657 --> 00:46:40.990 YEN-JIE LEE: So we are going to continue the discussion. 00:46:40.990 --> 00:46:45.880 We have learned about the two important laws 00:46:45.880 --> 00:46:48.040 for the geometrical optics. 00:46:48.040 --> 00:46:53.040 And we also went ahead to discuss the polarization that's 00:46:53.040 --> 00:46:55.940 solved in greater detail. 00:46:55.940 --> 00:47:00.260 So for example, we can have linear depolarized wave. 00:47:00.260 --> 00:47:03.640 So basically, the wave is essentially moving up and down, 00:47:03.640 --> 00:47:04.810 up and down. 00:47:04.810 --> 00:47:10.810 But the direction of the background field 00:47:10.810 --> 00:47:11.980 doesn't change. 00:47:11.980 --> 00:47:13.780 It's always, for example, initially, 00:47:13.780 --> 00:47:19.120 if it's in x direction, then it is x direction forever. 00:47:19.120 --> 00:47:24.700 And in that case, I call it linearly polarized. 00:47:24.700 --> 00:47:26.590 Of course, I can also have the case 00:47:26.590 --> 00:47:32.170 that I can have a superposition of two waves. 00:47:32.170 --> 00:47:36.920 One is having the electric field in the x direction. 00:47:36.920 --> 00:47:39.620 And the other one is in the y direction. 00:47:39.620 --> 00:47:45.820 And they are off by a phase of pi over 2. 00:47:45.820 --> 00:47:47.570 If that happens, then basically, you 00:47:47.570 --> 00:47:51.920 will see that it produces something really interesting. 00:47:51.920 --> 00:47:54.850 That direction of the electric field 00:47:54.850 --> 00:47:58.030 is going to be rotating as a function of time-- 00:47:58.030 --> 00:48:02.350 as a function of the space these waves travel. 00:48:02.350 --> 00:48:07.780 And we call it circularly polarized waves. 00:48:07.780 --> 00:48:12.260 And we can also have elliptically polarized wave. 00:48:12.260 --> 00:48:18.440 Then we learned about how to do a filtering, which 00:48:18.440 --> 00:48:20.330 is the polarizer. 00:48:20.330 --> 00:48:22.870 So suppose I have a perfect conductor here, 00:48:22.870 --> 00:48:28.660 where I have the easy axis, which is described 00:48:28.660 --> 00:48:31.450 by the green arrow there. 00:48:31.450 --> 00:48:34.540 And you can see that easy axis means 00:48:34.540 --> 00:48:41.410 that if you have electric field parallel to the easy axis, 00:48:41.410 --> 00:48:44.530 and then since that's the easy axis, so it is supposed 00:48:44.530 --> 00:48:48.490 to be easy, therefore, this electric field is going to be 00:48:48.490 --> 00:48:51.880 passing through the polarizer. 00:48:51.880 --> 00:48:56.020 On the other hand, if the electric field 00:48:56.020 --> 00:49:00.620 is perpendicular to the direction of the easy axis, 00:49:00.620 --> 00:49:03.730 that means it's taking the perfect conductor 00:49:03.730 --> 00:49:05.110 in the hard way. 00:49:05.110 --> 00:49:08.830 Therefore, when it pass through-- 00:49:08.830 --> 00:49:10.670 when it is trying to pass through 00:49:10.670 --> 00:49:16.300 with the perfect conductor, the electrons in those conductors 00:49:16.300 --> 00:49:22.330 are going to be working like crazy to deflect this wave when 00:49:22.330 --> 00:49:25.420 the direction of the electric field 00:49:25.420 --> 00:49:30.220 is perpendicular to the direction of the easy axis. 00:49:30.220 --> 00:49:33.070 So that is how this works. 00:49:33.070 --> 00:49:36.160 For example, in the first example, 00:49:36.160 --> 00:49:39.370 you can see that in this case, you 00:49:39.370 --> 00:49:44.610 have an easy axis which is perpendicular to the direction 00:49:44.610 --> 00:49:48.310 of the electric field, which is the red field, 00:49:48.310 --> 00:49:51.010 then this wave actually got refracted. 00:49:51.010 --> 00:49:53.130 There will be no transmission-- 00:49:53.130 --> 00:49:58.300 sorry, no electromagnetic field passing the perfect conductor. 00:49:58.300 --> 00:50:01.305 And on the other hand, if you have another perfect conductor, 00:50:01.305 --> 00:50:03.540 in which you have easy axis which 00:50:03.540 --> 00:50:09.475 is parallel to the electric field, then you can-- 00:50:09.475 --> 00:50:15.160 you will see that it will pass through the perfect conductor. 00:50:15.160 --> 00:50:19.270 So that is the polarizer. 00:50:19.270 --> 00:50:23.380 And also, we discussed about quarter-weight plate, 00:50:23.380 --> 00:50:28.240 which I would suggest you to have a review about the concept 00:50:28.240 --> 00:50:31.075 which we have learned about polarizer 00:50:31.075 --> 00:50:34.780 and quarter-wave plate so that you make sure that you 00:50:34.780 --> 00:50:41.050 understand how to calculate the electric field after passing 00:50:41.050 --> 00:50:44.770 through a polarizer and quarter-wave plate 00:50:44.770 --> 00:50:48.720 and how the secondary, or the elliptically depolarized waves 00:50:48.720 --> 00:50:55.260 are created using all those wave plates, et cetera. 00:50:55.260 --> 00:50:56.620 All right. 00:50:56.620 --> 00:51:00.160 So the next thing which we discussed during the class 00:51:00.160 --> 00:51:04.390 is how do we produce electromagnetic waves. 00:51:04.390 --> 00:51:09.880 I think by now, you should know that a stationary charge 00:51:09.880 --> 00:51:13.570 doesn't produce electromagnetic waves. 00:51:13.570 --> 00:51:16.540 Even a moving charge at constant speed 00:51:16.540 --> 00:51:19.590 doesn't create electromagnetic waves. 00:51:19.590 --> 00:51:22.090 So how do we create an electromagnetic wave 00:51:22.090 --> 00:51:25.740 which propagates to the edge of the universe? 00:51:25.740 --> 00:51:33.100 That is-- the trick is to create a kink in the fuel line. 00:51:33.100 --> 00:51:37.400 So you have to accelerate and stop it. 00:51:37.400 --> 00:51:42.130 Accelerate and then try to actually stop the acceleration. 00:51:42.130 --> 00:51:44.560 So then you can create a kink. 00:51:44.560 --> 00:51:50.050 And this kink is going to be propagating out of the-- 00:51:50.050 --> 00:51:51.780 as a function of time. 00:51:51.780 --> 00:51:58.630 And this kink is creating the so-called radiation 00:51:58.630 --> 00:52:01.540 from this accelerated charge. 00:52:01.540 --> 00:52:04.790 So you don't really need to remember all the deviations, 00:52:04.790 --> 00:52:08.510 but you really need to know the conclusion. 00:52:08.510 --> 00:52:10.760 So what is the conclusion is the following. 00:52:10.760 --> 00:52:15.820 The radiated electric field is equal to minus-- 00:52:15.820 --> 00:52:18.250 very important that there's a minus sign in front 00:52:18.250 --> 00:52:21.990 of it, which is a common mistake to drop it, and the q 00:52:21.990 --> 00:52:24.910 is the charge of the oscillating-- 00:52:24.910 --> 00:52:29.380 the accelerated charge, proportional to the charge. 00:52:29.380 --> 00:52:33.250 If the particle is more charged, then you have more radiation. 00:52:33.250 --> 00:52:41.270 Aperp is the acceleration projected to, which is-- 00:52:43.960 --> 00:52:48.970 the perpendicular projection of the acceleration 00:52:48.970 --> 00:52:54.860 of the particle with respect to the direction of propagation 00:52:54.860 --> 00:52:57.040 is so-called the Aperp. 00:52:57.040 --> 00:53:02.840 And only the perpendicular direction acceleration counts. 00:53:02.840 --> 00:53:06.880 The one which is parallel to the direction of propagation 00:53:06.880 --> 00:53:10.920 doesn't really count, as you can see from this equation. 00:53:10.920 --> 00:53:14.200 And the t prime what is t prime? t prime 00:53:14.200 --> 00:53:18.820 is t minus r divided by c. 00:53:18.820 --> 00:53:24.100 So t prime is the retarded time, so that is telling you 00:53:24.100 --> 00:53:27.220 that it takes some time for the information 00:53:27.220 --> 00:53:30.100 to propagate from the origin, which 00:53:30.100 --> 00:53:37.010 is the position of the moving charge to the observer, which 00:53:37.010 --> 00:53:42.190 is r, this distance, away from the moving charge. 00:53:42.190 --> 00:53:44.920 So the information takes some time to propagate, 00:53:44.920 --> 00:53:49.360 and you cannot know what is really happening, for example, 00:53:49.360 --> 00:53:52.060 100 light years away from Earth. 00:53:52.060 --> 00:53:53.970 You have no idea about what is happening. 00:53:53.970 --> 00:53:56.030 Maybe a black hole is created there 00:53:56.030 --> 00:54:00.390 and is going to suck everybody up in a few years. 00:54:00.390 --> 00:54:04.490 But nobody knows, and we don't care because we cannot control 00:54:04.490 --> 00:54:06.760 it. 00:54:06.760 --> 00:54:09.640 All right so that is very important. 00:54:09.640 --> 00:54:12.730 And also very important to know the magnetic 00:54:12.730 --> 00:54:14.740 field must be there. 00:54:14.740 --> 00:54:16.960 You can see the relation between magnetic field 00:54:16.960 --> 00:54:18.340 and the electric field. 00:54:18.340 --> 00:54:25.310 And the Poynting vector is also its joint field. 00:54:25.310 --> 00:54:28.210 And when we went ahead, given all the knowledge 00:54:28.210 --> 00:54:30.520 we have learned, we discussed about how 00:54:30.520 --> 00:54:35.020 to take very beautiful photos using a polarizer filter. 00:54:35.020 --> 00:54:41.500 And we discussed about how to filter out the scattered light 00:54:41.500 --> 00:54:42.910 from the sun. 00:54:42.910 --> 00:54:46.750 And it would be nice to figure out 00:54:46.750 --> 00:54:50.940 why this is the case, how these polarizer lines, scatter lines 00:54:50.940 --> 00:54:52.960 are created. 00:54:52.960 --> 00:54:55.650 It's purely geometrical. 00:54:55.650 --> 00:54:59.830 And also, we discussed about Brewster's angle 00:54:59.830 --> 00:55:06.670 and also how it leads to the explanation of the filtering 00:55:06.670 --> 00:55:11.170 of the light, the refracted light from the, for example, 00:55:11.170 --> 00:55:15.010 window of a car or from the water. 00:55:15.010 --> 00:55:17.710 And this is the demonstration of-- 00:55:17.710 --> 00:55:20.440 the summary of Brewster's angle. 00:55:20.440 --> 00:55:28.450 So somebody reminded me that the amplitude should be given. 00:55:28.450 --> 00:55:32.560 So I think, this is the amplitude formula 00:55:32.560 --> 00:55:36.190 for Brewster's angle will be given to you. 00:55:36.190 --> 00:55:40.099 If not, it's asked in the final exam. 00:55:40.099 --> 00:55:41.890 So don't be worried about it, and you don't 00:55:41.890 --> 00:55:44.440 have to remember this formula. 00:55:44.440 --> 00:55:48.280 And I'm not going to ask you to derive that just 00:55:48.280 --> 00:55:52.490 in such a short time, the three hours in the final exam. 00:55:52.490 --> 00:55:57.160 But what is very important is to know how this Brewster's angle, 00:55:57.160 --> 00:56:03.250 why there's no refracted light polarizing in a way 00:56:03.250 --> 00:56:07.590 that the polarization should be-- 00:56:07.590 --> 00:56:11.020 why the refracted light is polarized, for example. 00:56:11.020 --> 00:56:19.850 And also why the transmitting wave is slightly polarized. 00:56:19.850 --> 00:56:24.140 And I think the conclusions you need to remember, 00:56:24.140 --> 00:56:29.020 and you need to know how to calculate the angle, at least. 00:56:29.020 --> 00:56:34.620 Because for this purely polarized light 00:56:34.620 --> 00:56:39.490 to be produced in a refracted light, 00:56:39.490 --> 00:56:43.960 you need to have normal angle between the direction 00:56:43.960 --> 00:56:45.840 of the refracted light and the direction 00:56:45.840 --> 00:56:47.380 of the transmitted light. 00:56:47.380 --> 00:56:51.340 And that, you should be able to remember. 00:56:51.340 --> 00:56:54.370 And you should be able to derive that also from your mind 00:56:54.370 --> 00:56:58.570 as well, because that means the direction of the oscillation 00:56:58.570 --> 00:57:01.780 of the molecule at the boundary will 00:57:01.780 --> 00:57:05.740 be in the direction of propagation 00:57:05.740 --> 00:57:07.720 of the refracted wave. 00:57:07.720 --> 00:57:12.640 Therefore, that cannot be the solution to the progressing 00:57:12.640 --> 00:57:13.790 electromagnetic wave. 00:57:13.790 --> 00:57:16.600 Therefore, the refracted waves are polarized. 00:57:16.600 --> 00:57:18.610 So if you follow this logic, then you 00:57:18.610 --> 00:57:24.360 don't really need to memorize all those formulas. 00:57:24.360 --> 00:57:25.640 All right. 00:57:25.640 --> 00:57:30.990 So finally, in the last part of the course, 00:57:30.990 --> 00:57:35.230 we focused on the superposition of many, many electromagnetic 00:57:35.230 --> 00:57:40.650 waves so you can produce constructive interference. 00:57:40.650 --> 00:57:44.840 Or that means all those waves are in phase. 00:57:44.840 --> 00:57:47.900 And you can have destructive interference 00:57:47.900 --> 00:57:50.010 when they are out of phase. 00:57:50.010 --> 00:57:52.455 And that is a very important topic, 00:57:52.455 --> 00:57:57.630 so you should review that for the preparation of the final. 00:57:57.630 --> 00:58:01.190 And you can see that there are three concrete examples which 00:58:01.190 --> 00:58:03.340 we used during the class. 00:58:03.340 --> 00:58:04.610 A laser beam. 00:58:04.610 --> 00:58:08.220 We talked about a water ripple in a demo. 00:58:08.220 --> 00:58:13.160 And we also studied how it make use of this phenomena 00:58:13.160 --> 00:58:16.110 to design a phased radar. 00:58:16.110 --> 00:58:21.860 So to detect this unknown object in the sky, what we really 00:58:21.860 --> 00:58:25.610 need to have is electromagnetic waves pointing 00:58:25.610 --> 00:58:27.560 to a specific direction. 00:58:27.560 --> 00:58:35.520 And that can be achieved by using multi-slit interference. 00:58:35.520 --> 00:58:41.650 And this is the property of the two-slit interference pattern. 00:58:41.650 --> 00:58:45.620 And you are going to have many, many peaks. 00:58:45.620 --> 00:58:50.360 They have equal height for two-slit interference 00:58:50.360 --> 00:58:56.300 If you ignore any effect coming from diffraction. 00:58:56.300 --> 00:59:04.250 So we've assume that the slit is infinitely small. 00:59:04.250 --> 00:59:07.280 The slit is super narrow. 00:59:07.280 --> 00:59:11.630 And then we can ignore the diffraction-- 00:59:11.630 --> 00:59:12.810 single-slit diffraction. 00:59:12.810 --> 00:59:17.240 In fact, then all the peaks due to this two-slit interference 00:59:17.240 --> 00:59:19.940 will have the same height. 00:59:19.940 --> 00:59:23.090 On the other hand, when we start to increase 00:59:23.090 --> 00:59:26.330 the number of slits, for example, unequal to 3, 00:59:26.330 --> 00:59:31.190 unequal to 4, unequal to 5, unequal to 6, 00:59:31.190 --> 00:59:37.880 as you can see that, the structure of the intensity 00:59:37.880 --> 00:59:42.830 as a function of delta, which is the phase difference, 00:59:42.830 --> 00:59:44.360 is actually changing. 00:59:44.360 --> 00:59:48.660 And you can see that the general structure is the following. 00:59:48.660 --> 00:59:52.280 So if you have unequal to 3, then basically, 00:59:52.280 --> 00:59:57.260 you have 2 of adult, and between them, you have 1 child. 00:59:57.260 --> 00:59:59.450 And if you have unequal to 6, then 00:59:59.450 --> 01:00:02.360 basically, you have 2 adults and somehow there 01:00:02.360 --> 01:00:06.260 are 4 children in this collection. 01:00:06.260 --> 01:00:10.910 So basically, that is what we learned from the solution 01:00:10.910 --> 01:00:13.400 of the multi-slit interference. 01:00:13.400 --> 01:00:15.640 And in this way, we can actually make 01:00:15.640 --> 01:00:20.960 the width of the principal maxima as narrow as you want. 01:00:20.960 --> 01:00:25.800 So that is why phased radar works. 01:00:25.800 --> 01:00:29.030 And then we discussed about diffraction. 01:00:29.030 --> 01:00:33.470 So that is related, again, to the explanation of laser beams. 01:00:33.470 --> 01:00:38.090 And we discussed about the design of a Star Trek 01:00:38.090 --> 01:00:41.990 ship, the gun for the ship. 01:00:41.990 --> 01:00:44.900 And we also talked about resolution. 01:00:44.900 --> 01:00:47.990 And what is actually happening here is the following. 01:00:47.990 --> 01:00:52.790 A single-slit diffraction essentially 01:00:52.790 --> 01:00:57.950 can be viewed as an infinite number of source interference. 01:00:57.950 --> 01:01:04.250 And you just need to integrate over all the point-like sources 01:01:04.250 --> 01:01:07.440 between the two walls. 01:01:07.440 --> 01:01:14.540 And all of them are acting like a spherical wave source. 01:01:14.540 --> 01:01:17.720 So basically, for every point-- 01:01:17.720 --> 01:01:22.952 continuously, every point between these two walls 01:01:22.952 --> 01:01:27.200 are a point source of spherical waves. 01:01:27.200 --> 01:01:29.720 And that is Huygens' principle. 01:01:29.720 --> 01:01:32.600 And we can see that the structures 01:01:32.600 --> 01:01:35.500 is-- of the intensity as a function of position 01:01:35.500 --> 01:01:36.420 is the following. 01:01:36.420 --> 01:01:38.720 So basically, you have a principal maxima, 01:01:38.720 --> 01:01:41.840 which is a peak in the middle. 01:01:41.840 --> 01:01:44.240 And at some angle, basically, you 01:01:44.240 --> 01:01:46.760 have destructive interference such 01:01:46.760 --> 01:01:49.560 that if you integrate over all the contributions 01:01:49.560 --> 01:01:53.000 from an infinite number of sources in this window, 01:01:53.000 --> 01:01:55.490 basically, you would see that they completely 01:01:55.490 --> 01:01:56.840 cancel each other. 01:01:56.840 --> 01:02:03.170 So that is the origin of all those deep structure minima. 01:02:03.170 --> 01:02:07.390 And then, after the minima, actually, you 01:02:07.390 --> 01:02:10.270 will see another peak, but the height of the peak 01:02:10.270 --> 01:02:13.970 is suppressed by 1 over beta squared. 01:02:13.970 --> 01:02:17.360 And it would be good to review that. 01:02:17.360 --> 01:02:19.370 And what is the consequence? 01:02:19.370 --> 01:02:23.980 So if you shoot a laser beam to the moon, 01:02:23.980 --> 01:02:28.950 the size of the laser beam will be very large. 01:02:28.950 --> 01:02:33.820 After you learn 8.03, you know that the size of the laser beam 01:02:33.820 --> 01:02:38.830 is going to be very, very large due to interference 01:02:38.830 --> 01:02:45.250 between all the point-like sources from the laser beam. 01:02:45.250 --> 01:02:48.320 And finally, we can put them all together. 01:02:48.320 --> 01:02:52.000 So the single-slit diffraction and 01:02:52.000 --> 01:02:55.630 the multi-slit interference, you can put them 01:02:55.630 --> 01:02:59.420 all together, and basically, what you get is the following. 01:02:59.420 --> 01:03:04.540 So basically, you have a multi-slit interference 01:03:04.540 --> 01:03:07.780 pattern, which is showing there. 01:03:07.780 --> 01:03:13.450 But now the intensity of the multi-slit pattern 01:03:13.450 --> 01:03:21.070 is modulated by the single-slit diffraction pattern. 01:03:21.070 --> 01:03:25.370 And of course, the full formula will be given to you. 01:03:25.370 --> 01:03:27.160 But on the other hand, you are also 01:03:27.160 --> 01:03:31.780 requested to know how to calculate, just 01:03:31.780 --> 01:03:36.250 to add the contribution from multi-slit 01:03:36.250 --> 01:03:41.370 together in case if we change the amplitude of the incident 01:03:41.370 --> 01:03:43.410 light or we change the phase, like what 01:03:43.410 --> 01:03:45.970 we did in the homework. 01:03:45.970 --> 01:03:49.690 And I think that is one important point, 01:03:49.690 --> 01:03:51.670 and you should review that. 01:03:51.670 --> 01:03:55.450 And if you are not sure about how to proceed with that, 01:03:55.450 --> 01:04:00.770 it would be good to review Lecture 22, Lecture 23. 01:04:00.770 --> 01:04:04.520 So finally, we talk about the connection 01:04:04.520 --> 01:04:07.340 to quantum mechanics. 01:04:07.340 --> 01:04:11.960 Einstein already told us that "I have said so many times, 01:04:11.960 --> 01:04:15.520 God doesn't play dice with the world." 01:04:15.520 --> 01:04:18.230 But what we actually find is that there 01:04:18.230 --> 01:04:21.860 are two very interesting things which we found. 01:04:21.860 --> 01:04:26.590 The first thing is that if we have a single photon source, 01:04:26.590 --> 01:04:31.550 and basically, if we don't play dice, 01:04:31.550 --> 01:04:38.120 we cannot explain the intensity of the-- 01:04:38.120 --> 01:04:42.260 after this single photon source passes through two polarizers. 01:04:42.260 --> 01:04:44.000 And what happens is the following. 01:04:44.000 --> 01:04:49.220 Basically, the result of a single photon source 01:04:49.220 --> 01:04:52.670 tells you that you really need to play dice 01:04:52.670 --> 01:04:59.660 so that you can get the resulting polarized light 01:04:59.660 --> 01:05:01.970 intensity. 01:05:01.970 --> 01:05:05.480 And also, the second pseudo-experiment 01:05:05.480 --> 01:05:10.450 we discussed is that if you have billiard balls, basically, 01:05:10.450 --> 01:05:15.920 you have them pass through the two-slit experiment, what 01:05:15.920 --> 01:05:18.260 you are going to get is two piles, 01:05:18.260 --> 01:05:20.870 Gaussian-like distribution. 01:05:20.870 --> 01:05:26.410 And if you have a single electron source, what it does 01:05:26.410 --> 01:05:32.050 is that it interferes with itself. 01:05:32.050 --> 01:05:36.080 An electron, a single electron, can interfere with itself 01:05:36.080 --> 01:05:38.900 and produce a pattern which is very 01:05:38.900 --> 01:05:43.550 similar to what we see in the double-slit interference 01:05:43.550 --> 01:05:44.460 pattern. 01:05:44.460 --> 01:05:46.950 So that is really remarkable. 01:05:46.950 --> 01:05:50.960 And also, we talked about a single-slit-- 01:05:50.960 --> 01:05:54.140 single electron experiment. 01:05:54.140 --> 01:05:58.730 That gives you also a diffraction pattern. 01:05:58.730 --> 01:06:02.550 We have to use the wave function to describe 01:06:02.550 --> 01:06:06.890 the position-- the probability density of the position 01:06:06.890 --> 01:06:09.220 of the electron on the screen. 01:06:09.220 --> 01:06:14.930 And know this issue closely connected to the uncertainty 01:06:14.930 --> 01:06:18.530 principle, which we discussed earlier, delta, p, delta, 01:06:18.530 --> 01:06:22.190 x is greater than or equal to h bar over 2. 01:06:22.190 --> 01:06:24.420 So if you have a very narrow window, 01:06:24.420 --> 01:06:27.830 that means you have very similar delta x, 01:06:27.830 --> 01:06:30.410 so you have very, very good confidence 01:06:30.410 --> 01:06:32.840 about the location of the electron. 01:06:32.840 --> 01:06:39.300 And then the momentum is in the x-- 01:06:39.300 --> 01:06:41.670 in the momentum in the x direction, 01:06:41.670 --> 01:06:45.830 you have large uncertainty, according to this equation. 01:06:45.830 --> 01:06:52.350 And that can be seen from this single-slit diffraction pattern 01:06:52.350 --> 01:06:56.640 and it is closely connected to what we have learned before. 01:06:56.640 --> 01:07:01.520 So where is this-- how to actually describe 01:07:01.520 --> 01:07:04.310 what this is really the dispersion 01:07:04.310 --> 01:07:08.660 relation of the probability density wave 01:07:08.660 --> 01:07:13.140 is actually coming from Schrodinger's Equation. 01:07:13.140 --> 01:07:15.560 And this is given here. 01:07:15.560 --> 01:07:18.560 We briefly talked about that. 01:07:18.560 --> 01:07:22.970 And the consequence is the following. 01:07:22.970 --> 01:07:27.560 You can describe the evolution of the wave 01:07:27.560 --> 01:07:35.750 function as a function of time by using this wave equation. 01:07:35.750 --> 01:07:38.690 And this wave equation is slightly different from what 01:07:38.690 --> 01:07:42.040 we have learned before. 01:07:42.040 --> 01:07:46.300 And we also can use what we have learned from 8.03 01:07:46.300 --> 01:07:51.460 to solve a particle in a box problem, which is covered 01:07:51.460 --> 01:07:54.970 in lecture number 23. 01:07:54.970 --> 01:07:56.890 And I just wanted to say that you 01:07:56.890 --> 01:07:59.530 need to know the general principle, 01:07:59.530 --> 01:08:02.710 but I'm not teaching 8.04, so I'm not 01:08:02.710 --> 01:08:07.510 expecting you to solve a quantum mechanics problem. 01:08:07.510 --> 01:08:11.490 But I would like to say that OK, from this point, 01:08:11.490 --> 01:08:15.190 it's motivating you to take 8.04, right? 01:08:15.190 --> 01:08:18.350 Because there can be a lot of fun there as well. 01:08:18.350 --> 01:08:25.010 And it is closely related to what we have learned from 8.03. 01:08:25.010 --> 01:08:28.060 So I just want to say, the last point 01:08:28.060 --> 01:08:31.410 is that this is really not the end of the vibrations 01:08:31.410 --> 01:08:32.109 and waves. 01:08:32.109 --> 01:08:33.609 It's just the beginning. 01:08:33.609 --> 01:08:36.069 And that there is a path toward the peak. 01:08:36.069 --> 01:08:41.784 And it may take a long time to reach the peak. 01:08:41.784 --> 01:08:42.729 All right. 01:08:42.729 --> 01:08:47.200 And I would like to let you know that I'm really, really 01:08:47.200 --> 01:08:52.240 very happy to be your lecturer this semester. 01:08:52.240 --> 01:08:55.630 And I really enjoyed teaching this class 01:08:55.630 --> 01:08:59.680 and getting your responses when I asked questions. 01:08:59.680 --> 01:09:01.939 Thank you for the support. 01:09:01.939 --> 01:09:05.600 And I would like to say good luck with the final exam. 01:09:05.600 --> 01:09:11.319 And we have 800 contributions on Piazza, many thanks to Yinan, 01:09:11.319 --> 01:09:15.340 who is actually doing all the hard work, day and night. 01:09:15.340 --> 01:09:21.500 And thank you very much, and see you around MIT in the future. 01:09:28.400 --> 01:09:30.250 Thank you.