WEBVTT 00:00:00.100 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.820 under a Creative Commons license. 00:00:03.820 --> 00:00:06.530 Your support will help MIT OpenCourseWare continue 00:00:06.530 --> 00:00:10.150 to offer high quality educational resources for free. 00:00:10.150 --> 00:00:12.680 To make a donation or to view additional materials 00:00:12.680 --> 00:00:16.580 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.580 --> 00:00:17.275 at ocw.mit.edu. 00:00:23.970 --> 00:00:26.690 PROFESSOR: So today we're going to do our last lecture 00:00:26.690 --> 00:00:30.089 on scattering in 1D quantum mechanics, 00:00:30.089 --> 00:00:32.130 and we're going to introduce some powerful ideas, 00:00:32.130 --> 00:00:34.129 in particular, the phase shift and the S matrix, 00:00:34.129 --> 00:00:35.270 and they're cool. 00:00:35.270 --> 00:00:36.570 We'll use them for good. 00:00:36.570 --> 00:00:40.660 Before we get started, questions from last time? 00:00:43.424 --> 00:00:45.340 AUDIENCE: What was the music that was playing? 00:00:45.340 --> 00:00:46.180 PROFESSOR: Just now? 00:00:46.180 --> 00:00:48.820 It's a band called Saint Germain. 00:00:48.820 --> 00:00:52.280 It's actually a guy, but he refers to himself 00:00:52.280 --> 00:00:53.720 as a band called Saint Germain. 00:00:53.720 --> 00:00:59.200 Anyway, it's from an album I think called "Traveler." 00:00:59.200 --> 00:01:02.480 Physics questions? 00:01:02.480 --> 00:01:04.530 Anyone? 00:01:04.530 --> 00:01:05.817 OK, good. 00:01:05.817 --> 00:01:06.650 Well, bad, actually. 00:01:06.650 --> 00:01:08.316 I'd much prefer it if you had questions, 00:01:08.316 --> 00:01:11.296 but I'll take that as a sign of knowledge, competence, 00:01:11.296 --> 00:01:11.795 and mastery. 00:01:16.680 --> 00:01:19.250 So last time, we talked about scattering 00:01:19.250 --> 00:01:21.870 past a barrier with some height L 00:01:21.870 --> 00:01:24.777 and some height V, which I think we called V0, 00:01:24.777 --> 00:01:26.860 and we observed a bunch of nice things about this. 00:01:26.860 --> 00:01:29.570 First off, we computed the probability of transmission 00:01:29.570 --> 00:01:34.400 across this barrier as a function of the energy, 00:01:34.400 --> 00:01:35.900 and it had a bunch of nice features. 00:01:35.900 --> 00:01:39.160 First off, it asymptoted to 1 at high energies, 00:01:39.160 --> 00:01:39.910 which makes sense. 00:01:39.910 --> 00:01:43.700 Things shouldn't really care if you have a little tiny barrier. 00:01:43.700 --> 00:01:45.215 It went to 0 at 0 energy. 00:01:47.894 --> 00:01:49.560 I think we should all find that obvious. 00:01:49.560 --> 00:01:50.430 If you have extremely low energy, 00:01:50.430 --> 00:01:52.720 you're just going to bounce off this very hard wall. 00:01:52.720 --> 00:01:55.200 In between, though, there's some structure. 00:01:55.200 --> 00:01:57.950 In particular, at certain values of the energy 00:01:57.950 --> 00:02:00.540 corresponding to certain values of the wave 00:02:00.540 --> 00:02:04.280 number in the barrier region, at certain values of the energy, 00:02:04.280 --> 00:02:06.841 we saw that the transmission was perfect. 00:02:06.841 --> 00:02:07.340 Oops. 00:02:07.340 --> 00:02:08.464 This should have been here. 00:02:08.464 --> 00:02:09.740 Sorry. 00:02:09.740 --> 00:02:18.220 The transition was perfect at special values of the energy, 00:02:18.220 --> 00:02:21.980 and that the reflection at those points was 0. 00:02:21.980 --> 00:02:25.260 You transmitted perfectly. 00:02:25.260 --> 00:02:27.100 But meanwhile, the reflection, which 00:02:27.100 --> 00:02:30.360 is 1 minus the transmission, hit maxima at special points, 00:02:30.360 --> 00:02:31.740 and those special points turn out 00:02:31.740 --> 00:02:37.440 to be half integer shifted away from the special points 00:02:37.440 --> 00:02:38.875 for perfect transmission. 00:02:38.875 --> 00:02:41.000 So at certain points, we have perfect transmission. 00:02:41.000 --> 00:02:45.049 At certain points, we have extremely efficient reflection. 00:02:45.049 --> 00:02:46.590 One of our goals today is going to be 00:02:46.590 --> 00:02:50.430 to understand this physics, the physics of resonance 00:02:50.430 --> 00:02:53.667 in scattering off of a potential. 00:02:53.667 --> 00:02:55.000 So the asymptotes we understand. 00:02:55.000 --> 00:02:56.010 Classically, that makes sense. 00:02:56.010 --> 00:02:57.259 Classically, that makes sense. 00:02:57.259 --> 00:03:00.732 But classically, this is a really weird structure to see. 00:03:00.732 --> 00:03:02.690 And notice that it's also not something that we 00:03:02.690 --> 00:03:05.970 saw when we looked at scattering off of a simple step. 00:03:05.970 --> 00:03:08.130 When we looked at a simple step, what we got 00:03:08.130 --> 00:03:13.582 was something that looked like 0 and then this. 00:03:13.582 --> 00:03:14.540 There was no structure. 00:03:14.540 --> 00:03:16.740 It was just a nice, simple curve. 00:03:16.740 --> 00:03:18.680 So something is happening when we 00:03:18.680 --> 00:03:22.352 have a barrier as opposed to a step. 00:03:22.352 --> 00:03:24.060 So our job is going to be, in some sense, 00:03:24.060 --> 00:03:26.590 to answer why are they so different. 00:03:26.590 --> 00:03:32.060 Before we get going, questions about the step barrier. 00:03:32.060 --> 00:03:32.680 Yeah? 00:03:32.680 --> 00:03:36.408 AUDIENCE: That little line that you drew over your other graph, 00:03:36.408 --> 00:03:37.810 is that to scale? 00:03:37.810 --> 00:03:39.310 PROFESSOR: Oh, sorry. 00:03:39.310 --> 00:03:40.212 This one? 00:03:40.212 --> 00:03:43.570 AUDIENCE: No, the little line that you drew right there. 00:03:43.570 --> 00:03:44.530 PROFESSOR: This? 00:03:44.530 --> 00:03:46.030 Sorry, I should drawn it separately. 00:03:46.030 --> 00:03:49.570 That was the transmission as a function of energy 00:03:49.570 --> 00:03:54.310 across a single step, and so that looked like this. 00:03:54.310 --> 00:03:55.937 AUDIENCE: So if we were to overlap that 00:03:55.937 --> 00:03:57.260 onto our resonance one-- 00:03:57.260 --> 00:03:58.051 PROFESSOR: Exactly. 00:03:58.051 --> 00:04:00.150 AUDIENCE: Would it [INAUDIBLE] to? 00:04:00.150 --> 00:04:01.600 PROFESSOR: Oh, sorry. 00:04:01.600 --> 00:04:03.590 So I just meant to compare the two. 00:04:03.590 --> 00:04:05.923 I just wanted to think of them as two different systems. 00:04:05.923 --> 00:04:08.820 In this system, the transmission curve is a nice, simple curve. 00:04:08.820 --> 00:04:09.920 It has no structure. 00:04:09.920 --> 00:04:13.120 In the case of a step barrier, we get non-zero transmission 00:04:13.120 --> 00:04:15.370 at low energies instead of going to 0 00:04:15.370 --> 00:04:17.431 and we get this resonant structure. 00:04:17.431 --> 00:04:18.514 So it's just for contrast. 00:04:22.540 --> 00:04:24.460 Before going into that in detail, 00:04:24.460 --> 00:04:27.820 I want to do one slight variation of this problem. 00:04:27.820 --> 00:04:29.680 I'm not going to do any of the computations. 00:04:29.680 --> 00:04:31.221 I'm just going to tell you how to get 00:04:31.221 --> 00:04:33.600 the answer from the answers we've already computed. 00:04:33.600 --> 00:04:36.160 So consider a square well. 00:04:36.160 --> 00:04:38.015 You guys have solved the finite square well 00:04:38.015 --> 00:04:39.104 on your problem sets. 00:04:39.104 --> 00:04:41.020 Consider a finite square well again of width L 00:04:41.020 --> 00:04:42.990 and of depth now minus V0. 00:04:42.990 --> 00:04:46.030 It's the same thing with V goes to minus V. 00:04:46.030 --> 00:04:48.030 And again, I want to consider scattering states. 00:04:48.030 --> 00:04:50.480 So I want to consider states with positive energy, energy 00:04:50.480 --> 00:04:51.950 above the asymptotic potential. 00:04:54.530 --> 00:04:57.270 Now, at this point, by now we all know how to solve this. 00:04:57.270 --> 00:04:59.755 We write plain waves here, plain waves here, 00:04:59.755 --> 00:05:00.880 and plain waves in between. 00:05:00.880 --> 00:05:03.750 We solve the potential in each region where it's constant, 00:05:03.750 --> 00:05:06.392 and then we impose continuity of the wave function 00:05:06.392 --> 00:05:08.600 and continuity of the derivative of the wave function 00:05:08.600 --> 00:05:10.080 at the matching points. 00:05:10.080 --> 00:05:11.580 So we know how to solve this problem 00:05:11.580 --> 00:05:13.890 and deduce the transmission reflection coefficients. 00:05:13.890 --> 00:05:15.890 We've done it for this problem, and it's exactly 00:05:15.890 --> 00:05:16.760 the same algebra. 00:05:16.760 --> 00:05:18.712 And in fact, it's so exactly the same algebra 00:05:18.712 --> 00:05:20.670 that we can just take the results from this one 00:05:20.670 --> 00:05:23.840 and take V to minus V and we'll get the right answer. 00:05:23.840 --> 00:05:24.970 You kind of have to. 00:05:24.970 --> 00:05:32.360 So if we do, what we get for the transition probability-- 00:05:32.360 --> 00:05:35.920 and now this is the transmission probability for a square well-- 00:05:35.920 --> 00:05:39.650 and again, I'm going to use the same dimensionless constants. 00:05:39.650 --> 00:05:45.530 g0 squared is equal to 2mL squared over h bar squared 00:05:45.530 --> 00:05:46.660 times V0. 00:05:46.660 --> 00:05:48.180 This is 1 over an energy. 00:05:48.180 --> 00:05:48.930 This is an energy. 00:05:48.930 --> 00:05:49.960 This is dimensionless. 00:05:49.960 --> 00:05:55.390 And the dimensionless energy epsilon is equal to E over V. 00:05:55.390 --> 00:05:57.580 To express my transmission probability, 00:05:57.580 --> 00:05:59.940 life is better when it's dimensionless. 00:05:59.940 --> 00:06:02.140 So T, the transmission probability, 00:06:02.140 --> 00:06:04.190 again, it's one of these horrible 1 overs. 00:06:04.190 --> 00:06:13.780 1 over 1 plus 1 over 4 epsilon, epsilon plus 1, 00:06:13.780 --> 00:06:18.240 sine squared of g0, square root of epsilon plus 1. 00:06:24.840 --> 00:06:27.580 This is again the same. 00:06:27.580 --> 00:06:30.081 You can see what we got last time by now taking V to minus V 00:06:30.081 --> 00:06:30.580 again. 00:06:30.580 --> 00:06:32.160 That takes epsilon to minus epsilon. 00:06:32.160 --> 00:06:36.360 So we get epsilon, 1 minus epsilon, or epsilon minus 1 00:06:36.360 --> 00:06:38.894 picking up the minus sign from this epsilon, 00:06:38.894 --> 00:06:40.810 and here we get a 1 minus epsilon instead of 1 00:06:40.810 --> 00:06:41.450 plus epsilon. 00:06:41.450 --> 00:06:43.390 That's precisely what we got last time. 00:06:43.390 --> 00:06:46.609 So if you're feeling punchy at home tonight, check this. 00:06:46.609 --> 00:06:48.400 In some sense, you're going to re-derive it 00:06:48.400 --> 00:06:49.486 on the problem set. 00:06:52.690 --> 00:06:55.290 So it looks basically the same as before. 00:06:55.290 --> 00:06:57.480 We have a sine function downstairs, 00:06:57.480 --> 00:07:00.011 which again will sometimes be 0. 00:07:00.011 --> 00:07:01.510 The sine will occasionally be 0 when 00:07:01.510 --> 00:07:03.250 its argument is a multiple of pi. 00:07:03.250 --> 00:07:05.290 And when that sine function is 0, 00:07:05.290 --> 00:07:07.940 then the transmission probability is 1 over 1 plus 0. 00:07:07.940 --> 00:07:09.880 It's also known as 1. 00:07:09.880 --> 00:07:10.900 Transmission is perfect. 00:07:10.900 --> 00:07:12.480 So we again get a resonant structure, 00:07:12.480 --> 00:07:14.900 and it's in fact exactly the same plot, or almost exactly 00:07:14.900 --> 00:07:17.100 the same plot. 00:07:17.100 --> 00:07:19.885 I'm going to plot transmission-- let's go ahead 00:07:19.885 --> 00:07:22.090 and do this-- transmission probability 00:07:22.090 --> 00:07:25.870 as a function, again, of the dimensionless energy. 00:07:25.870 --> 00:07:37.450 And what we get is, again, not very well drawn resonances 00:07:37.450 --> 00:07:39.720 where transmission goes to 1 thanks 00:07:39.720 --> 00:07:43.030 to my beautiful artistic skills. 00:07:43.030 --> 00:07:46.690 Points where the reflection hits a local maximum. 00:07:50.470 --> 00:07:52.540 But we know one more thing about this system, 00:07:52.540 --> 00:07:55.930 which is that, in addition to having scattering states whose 00:07:55.930 --> 00:07:58.160 transition probability are indicated by this plot, 00:07:58.160 --> 00:08:02.480 we know we also have, for negative energy, bound states. 00:08:02.480 --> 00:08:04.910 Unlike the step barrier, for the step well, 00:08:04.910 --> 00:08:06.430 we also have bound states. 00:08:06.430 --> 00:08:07.960 So here's the transmission curve, 00:08:07.960 --> 00:08:10.660 but I just want to remind you that we have energies. 00:08:10.660 --> 00:08:14.650 At special values of energies, we also have bound states. 00:08:14.650 --> 00:08:17.190 And precisely what energies depends 00:08:17.190 --> 00:08:19.260 on the structure of the well and the depth, 00:08:19.260 --> 00:08:24.240 but if the depth of the well is, say, V0 so this is minus 1, 00:08:24.240 --> 00:08:27.230 we know that the lowest bound state is always 00:08:27.230 --> 00:08:29.770 greater energy than the bottom of the well. 00:08:29.770 --> 00:08:31.320 Cool? 00:08:31.320 --> 00:08:33.669 So epsilon, remember, is E over V0, 00:08:33.669 --> 00:08:36.289 and in the square well of depth V0, 00:08:36.289 --> 00:08:38.830 the lowest bound state cannot possibly have energy lower than 00:08:38.830 --> 00:08:41.299 the bottom of the well, so its epsilon must have epsilon 00:08:41.299 --> 00:08:44.864 greater than minus 1. 00:08:44.864 --> 00:08:46.280 Anyway, this is just to remind you 00:08:46.280 --> 00:08:49.360 that there are bound states. 00:08:49.360 --> 00:08:51.030 Think of this like a gun put down 00:08:51.030 --> 00:08:52.920 on a table in a play in Act One. 00:08:56.600 --> 00:08:58.744 It's that dramatic. 00:08:58.744 --> 00:08:59.660 It will show up again. 00:08:59.660 --> 00:09:03.677 It will come back and play well with us. 00:09:03.677 --> 00:09:05.260 I want to talk about these resonances. 00:09:05.260 --> 00:09:07.280 Let's understand why they're there. 00:09:07.280 --> 00:09:09.030 There are a bunch of ways of understanding 00:09:09.030 --> 00:09:11.770 why these resonances are present, and let 00:09:11.770 --> 00:09:15.040 me just give you a couple. 00:09:15.040 --> 00:09:16.790 First, a heuristic picture, and then I 00:09:16.790 --> 00:09:19.560 want to give you a very precise computational picture of why 00:09:19.560 --> 00:09:21.300 these resonances are happening. 00:09:21.300 --> 00:09:23.600 So the first is imagine we have a state where 00:09:23.600 --> 00:09:24.960 the transmission is perfect. 00:09:24.960 --> 00:09:26.980 What that tells you is KL is a multiple of pi. 00:09:29.520 --> 00:09:32.060 If this is the distance L of the well, 00:09:32.060 --> 00:09:35.240 in here the wave is exactly one period 00:09:35.240 --> 00:09:38.960 if its KL is equal to a multiple of pi. 00:09:47.150 --> 00:09:49.810 Let me simplify my life and consider the case KL is 2 pi. 00:10:01.550 --> 00:10:02.710 Sorry. 00:10:02.710 --> 00:10:04.300 4 pi. 00:10:04.300 --> 00:10:05.950 I switched notations for you. 00:10:05.950 --> 00:10:10.120 I was using in my head, rather in my notes, width of the well 00:10:10.120 --> 00:10:16.070 is 2L for reasons you'll see on the notes 00:10:16.070 --> 00:10:18.380 if you look at the notes, but let's ignore that. 00:10:18.380 --> 00:10:19.820 Let's focus on these guys. 00:10:19.820 --> 00:10:24.280 So consider the state, configuration in energy, 00:10:24.280 --> 00:10:28.020 such that in the well, we have exactly one period of the wave 00:10:28.020 --> 00:10:28.740 function. 00:10:28.740 --> 00:10:31.780 That means that the value of the wave function at the two ends 00:10:31.780 --> 00:10:35.400 is the same and the slope is the same. 00:10:35.400 --> 00:10:39.420 So whatever the energy is out here, 00:10:39.420 --> 00:10:41.820 if it matches smoothly and continuously 00:10:41.820 --> 00:10:44.485 and its derivative is continuous here, it will match smoothly 00:10:44.485 --> 00:10:49.480 and continuously out here as well with the same amplitude 00:10:49.480 --> 00:10:53.954 and the same period inside and outside. 00:10:53.954 --> 00:10:56.120 The amplitude is the same and the phase is the same. 00:10:56.120 --> 00:10:58.161 That means this wave must have the same amplitude 00:10:58.161 --> 00:10:59.700 and the same period. 00:10:59.700 --> 00:11:01.210 It has to have the same period because it's the same energy, 00:11:01.210 --> 00:11:03.460 but it must have the same amplitude and the same slope 00:11:03.460 --> 00:11:05.100 at that point. 00:11:05.100 --> 00:11:07.450 And that means that this wave has 00:11:07.450 --> 00:11:09.280 the same amplitude as this wave. 00:11:09.280 --> 00:11:11.449 The norm squared is the transmission probability, 00:11:11.449 --> 00:11:13.990 the norm squared of this divided by the norm squared of this. 00:11:13.990 --> 00:11:16.430 That means the transmission probability has to be 1. 00:11:16.430 --> 00:11:18.730 What would have happened had the system, 00:11:18.730 --> 00:11:21.899 instead of being perfectly periodic inside the well-- 00:11:21.899 --> 00:11:23.190 actually, let me leave that up. 00:11:28.207 --> 00:11:29.040 Let's do it as this. 00:11:36.501 --> 00:11:37.000 Shoot. 00:11:37.000 --> 00:11:39.208 I'm even getting my qualitative wave functions wrong. 00:11:39.208 --> 00:11:40.970 Let's try this again. 00:11:40.970 --> 00:11:45.710 Start at the top, go down, and then 00:11:45.710 --> 00:11:49.367 its deeper inside the well so the amplitude out here, 00:11:49.367 --> 00:11:51.950 the difference in energy between the energy and the potential, 00:11:51.950 --> 00:11:54.390 is less, which means the period is longer 00:11:54.390 --> 00:11:56.176 and the amplitude is greater. 00:11:56.176 --> 00:11:58.300 The way I've drawn it, it's got a particular value. 00:11:58.300 --> 00:12:07.510 It's got zero derivative at this point, 00:12:07.510 --> 00:12:15.547 so it's got to-- so there's our wave function. 00:12:15.547 --> 00:12:16.380 Same thing out here. 00:12:20.162 --> 00:12:21.870 The important thing is the amplitude here 00:12:21.870 --> 00:12:23.328 has to be the same as the amplitude 00:12:23.328 --> 00:12:26.080 here because the amplitude and the amplitude and the phase 00:12:26.080 --> 00:12:29.760 were exactly as if there had been no intervening region. 00:12:29.760 --> 00:12:31.670 Everyone agree with that? 00:12:31.670 --> 00:12:35.280 By contrast, if we had looked at a situation where 00:12:35.280 --> 00:12:42.080 inside the well, it was not the same amplitude, so for example, 00:12:42.080 --> 00:12:44.780 something like this, came up with some slope which 00:12:44.780 --> 00:12:46.750 is different and a value which is different, 00:12:46.750 --> 00:12:49.820 then this is going to match onto something with the same period 00:12:49.820 --> 00:12:53.165 but with a different amplitude than it would have over here. 00:12:56.015 --> 00:12:57.890 Same period because it's got the same energy, 00:12:57.890 --> 00:12:59.680 but a different amplitude because it 00:12:59.680 --> 00:13:03.810 has to match on with the value and the derivative. 00:13:03.810 --> 00:13:06.850 So you can think through this and pretty quickly 00:13:06.850 --> 00:13:10.790 convince yourself of the necessity of the transmission 00:13:10.790 --> 00:13:14.050 aptitude being 1 if this is exactly periodic. 00:13:14.050 --> 00:13:16.220 Again, if this is exactly one period inside, 00:13:16.220 --> 00:13:17.770 you can just imagine this is gone 00:13:17.770 --> 00:13:19.450 and you get a continuous wave function, 00:13:19.450 --> 00:13:20.908 so the amplitude and the derivative 00:13:20.908 --> 00:13:22.701 must be the same on both sides as 00:13:22.701 --> 00:13:23.950 if there was no barrier there. 00:13:27.110 --> 00:13:29.660 When the wave function doesn't have exactly one 00:13:29.660 --> 00:13:33.620 period inside the well, you can't do that, 00:13:33.620 --> 00:13:37.120 so the amplitudes can't be the same on both sides. 00:13:37.120 --> 00:13:38.996 But that's not a very satisfying explanation. 00:13:38.996 --> 00:13:40.828 That's really an explanation about solutions 00:13:40.828 --> 00:13:42.110 to the differential equation. 00:13:42.110 --> 00:13:44.609 I'm just telling you properties of second order differential 00:13:44.609 --> 00:13:45.120 equations. 00:13:45.120 --> 00:13:46.440 Let's think of a more physical, more 00:13:46.440 --> 00:13:47.731 quantum mechanical explanation. 00:13:47.731 --> 00:13:50.360 Why are we getting these resonances? 00:13:50.360 --> 00:13:52.500 Well, I want to think about this in the same way 00:13:52.500 --> 00:13:55.070 as we thought about the boxes in the very first lecture. 00:13:57.740 --> 00:14:03.620 Suppose we have this square well, 00:14:03.620 --> 00:14:06.889 and I know I have some amplitude here. 00:14:06.889 --> 00:14:07.930 I've got a wave function. 00:14:07.930 --> 00:14:09.157 It's got some amplitude here. 00:14:09.157 --> 00:14:10.740 It's got some momentum going this way, 00:14:10.740 --> 00:14:12.160 some positive momentum. 00:14:12.160 --> 00:14:15.770 And I want to ask, what's the probability that I will scatter 00:14:15.770 --> 00:14:18.020 past the potential, let's say to this point right here 00:14:18.020 --> 00:14:19.686 just on the other side of the potential? 00:14:19.686 --> 00:14:22.849 What's the probability that I will scatter across? 00:14:22.849 --> 00:14:24.640 Well, you say, we've done this calculation. 00:14:24.640 --> 00:14:27.000 We know the probability to transmit across this step 00:14:27.000 --> 00:14:27.870 potential. 00:14:27.870 --> 00:14:29.520 We did that last time. 00:14:29.520 --> 00:14:30.710 So that's T step. 00:14:35.410 --> 00:14:37.170 We know the probability of scattering 00:14:37.170 --> 00:14:43.134 across this potential step, and that's transmission up. 00:14:43.134 --> 00:14:45.550 And so the probability that you transmit from here to here 00:14:45.550 --> 00:14:47.883 is the probability that you transmit here first and then 00:14:47.883 --> 00:14:51.090 the probability that you transmit here, 00:14:51.090 --> 00:14:52.757 the product of the probabilities. 00:14:52.757 --> 00:14:53.465 Sound reasonable? 00:14:56.491 --> 00:14:56.990 Let's vote. 00:14:56.990 --> 00:14:59.700 How many people think that this is equal to the transmission 00:14:59.700 --> 00:15:04.170 probability across the potential well? 00:15:04.170 --> 00:15:05.047 Any votes? 00:15:05.047 --> 00:15:06.630 You have to vote one way or the other. 00:15:06.630 --> 00:15:07.370 No, it is not. 00:15:07.370 --> 00:15:09.000 How many people vote? 00:15:09.000 --> 00:15:09.500 OK. 00:15:09.500 --> 00:15:11.500 Yes, it is. 00:15:11.500 --> 00:15:12.000 OK. 00:15:12.000 --> 00:15:13.950 The no's have it, and that's not terribly 00:15:13.950 --> 00:15:16.660 surprising because, of course, these have no resonance 00:15:16.660 --> 00:15:18.035 structure, so where did that come 00:15:18.035 --> 00:15:20.080 from if it's just that thing squared? 00:15:20.080 --> 00:15:23.440 So that probably can't be it, but here's the bigger problem. 00:15:23.440 --> 00:15:24.910 Why is this the wrong argument? 00:15:27.379 --> 00:15:28.920 AUDIENCE: Because there's reflection. 00:15:28.920 --> 00:15:29.640 PROFESSOR: Yeah, exactly. 00:15:29.640 --> 00:15:31.810 There's reflection, but that's only one step in the answer 00:15:31.810 --> 00:15:33.018 or why it's the wrong answer. 00:15:33.018 --> 00:15:34.007 Why else? 00:15:34.007 --> 00:15:35.590 AUDIENCE: Your transmission operations 00:15:35.590 --> 00:15:37.197 have to do it far away? 00:15:37.197 --> 00:15:39.530 PROFESSOR: That's, true but I just want the probability, 00:15:39.530 --> 00:15:41.280 and if I got here with positive momentum, 00:15:41.280 --> 00:15:43.113 I'm eventually going to get out to infinity, 00:15:43.113 --> 00:15:45.080 so it's the same probability because it's 00:15:45.080 --> 00:15:46.330 just an E to the ikx out here. 00:15:46.330 --> 00:15:48.300 The wave function is just E to the ikx 00:15:48.300 --> 00:15:51.490 so the probability is going to be the same. 00:15:51.490 --> 00:15:53.439 Other reasons? 00:15:53.439 --> 00:15:55.772 AUDIENCE: Well, the width of the well is very important, 00:15:55.772 --> 00:15:57.272 but the first argument ignores that. 00:15:57.272 --> 00:15:58.063 PROFESSOR: Exactly. 00:15:58.063 --> 00:15:59.090 That's also true. 00:15:59.090 --> 00:16:01.673 So far, the reasons we have are we need the width of the well, 00:16:01.673 --> 00:16:03.120 doesn't appear. 00:16:03.120 --> 00:16:04.405 That seems probably wrong. 00:16:07.024 --> 00:16:09.440 The second is it's possible that you could have reflected. 00:16:09.440 --> 00:16:10.898 We haven't really incorporated that 00:16:10.898 --> 00:16:12.480 in any sort of elegant way. 00:16:12.480 --> 00:16:14.420 AUDIENCE: There are other ways to transmit 00:16:14.420 --> 00:16:14.905 by reflecting twice. 00:16:14.905 --> 00:16:16.130 PROFESSOR: That's absolutely true. 00:16:16.130 --> 00:16:17.160 There are other ways to transmit, 00:16:17.160 --> 00:16:19.320 so you could transmit, then you could reflect. 00:16:19.320 --> 00:16:22.581 So we could transmit then reflect, and reflect again, 00:16:22.581 --> 00:16:23.330 and then transmit. 00:16:23.330 --> 00:16:26.190 We could transmit, reflect, reflect, transmit. 00:16:26.190 --> 00:16:28.170 What else? 00:16:28.170 --> 00:16:30.590 Do probabilities add in quantum mechanics? 00:16:33.929 --> 00:16:35.470 And when you have products of events, 00:16:35.470 --> 00:16:37.470 do probabilities multiply? 00:16:37.470 --> 00:16:39.210 What adds in quantum mechanics? 00:16:39.210 --> 00:16:40.479 AUDIENCE: [INAUDIBLE]. 00:16:40.479 --> 00:16:41.520 PROFESSOR: The amplitude. 00:16:41.520 --> 00:16:42.740 The wave function. 00:16:42.740 --> 00:16:45.520 We do not take the product of probabilities. 00:16:45.520 --> 00:16:47.410 What we do is we ask, what's the amplitude 00:16:47.410 --> 00:16:49.237 to get here from there, and we take 00:16:49.237 --> 00:16:51.320 the amplitude norm squared to get the probability. 00:16:54.290 --> 00:16:56.830 So the correct question is what's 00:16:56.830 --> 00:16:59.099 the amplitude to get from here to here? 00:16:59.099 --> 00:17:00.890 How does the wave function of the amplitude 00:17:00.890 --> 00:17:03.610 change as you move from here to here? 00:17:03.610 --> 00:17:06.534 And for that, think back to the two slit experiment 00:17:06.534 --> 00:17:08.160 or think back to the boxes. 00:17:08.160 --> 00:17:11.134 We asked the following question. 00:17:11.134 --> 00:17:15.069 The amplitude that you should transmit across this well 00:17:15.069 --> 00:17:18.740 has a bunch of components, is a sum of a bunch of terms. 00:17:18.740 --> 00:17:22.564 You could transmit down this well. 00:17:22.564 --> 00:17:24.230 Inside here, you know your wave function 00:17:24.230 --> 00:17:30.910 is e to the i k prime L-- or I think I'm calling this k2 x. 00:17:30.910 --> 00:17:34.220 And in moving across the well, your wave function 00:17:34.220 --> 00:17:37.850 evolves by an e to the i k2 L, and then you 00:17:37.850 --> 00:17:41.010 could transmit again with some transmission amplitude. 00:17:41.010 --> 00:17:43.800 So this would be the transmission down times e 00:17:43.800 --> 00:17:49.114 to the i k2 L times the transmit up. 00:17:49.114 --> 00:17:50.780 As we saw last time, these are the same, 00:17:50.780 --> 00:17:51.900 but I just want to keep them separate 00:17:51.900 --> 00:17:53.520 so you know which one talking about. 00:17:53.520 --> 00:17:54.330 This is a contribution. 00:17:54.330 --> 00:17:56.360 This is something that could contribute to the amplitude. 00:17:56.360 --> 00:17:58.790 Is it the only thing that could contribute to the amplitude? 00:17:58.790 --> 00:17:59.289 No. 00:17:59.289 --> 00:18:01.590 What else could contribute? 00:18:01.590 --> 00:18:02.450 Bounce, right? 00:18:02.450 --> 00:18:06.120 So to get from here, to here I could transmit, evolve, 00:18:06.120 --> 00:18:07.030 transmit. 00:18:07.030 --> 00:18:11.280 I could also transmit, evolve, reflect, evolve, 00:18:11.280 --> 00:18:13.840 reflect, evolve, transmit. 00:18:13.840 --> 00:18:18.490 So there's also a term that's t, e to the ikL, r, 00:18:18.490 --> 00:18:24.030 e to the minus ikL, r, e to the iKL. 00:18:26.697 --> 00:18:28.280 Sorry, e to the plus ikL because we're 00:18:28.280 --> 00:18:30.509 increasing the evolution of the phase. 00:18:30.509 --> 00:18:32.050 And then transmit finally at the end. 00:18:36.060 --> 00:18:38.370 And these k's are all k2, but I could 00:18:38.370 --> 00:18:41.242 have done that many times. 00:18:41.242 --> 00:18:43.450 But notice that each time what I'm going to do is I'm 00:18:43.450 --> 00:18:47.110 going to transmit, reflect, reflect, transmit or transmit, 00:18:47.110 --> 00:18:50.950 reflect, reflect, reflect, reflect, transmit. 00:18:50.950 --> 00:18:58.890 So I'm always going to do this some number of times. 00:18:58.890 --> 00:19:01.420 I do this once, I do this twice, I do it thrice. 00:19:01.420 --> 00:19:03.080 This gives me a geometric series. 00:19:03.080 --> 00:19:09.560 This is t e to the i k2 L t times 00:19:09.560 --> 00:19:13.370 1 plus this quantity plus this quantity squared. 00:19:13.370 --> 00:19:16.900 That's a geometric series, 1 over 1 plus this quantity 00:19:16.900 --> 00:19:19.610 squared. 00:19:19.610 --> 00:19:22.017 Sorry, 1 minus because it's a geometric sum. 00:19:22.017 --> 00:19:23.100 And what is this quantity? 00:19:23.100 --> 00:19:24.900 Well, it's r squared, and remember the r's 00:19:24.900 --> 00:19:26.733 in both directions are the same, so I'm just 00:19:26.733 --> 00:19:30.452 going to write it as r squared e to the 2i. 00:19:33.826 --> 00:19:34.686 r squared. 00:19:34.686 --> 00:19:36.060 It's a real number, but I'm going 00:19:36.060 --> 00:19:38.259 to put the absolute value on anyway. 00:19:38.259 --> 00:19:39.550 It's going to simplify my life. 00:19:39.550 --> 00:19:47.750 e to the 2i k2 L. So this is our prediction 00:19:47.750 --> 00:19:50.002 from multiple bounces. 00:19:50.002 --> 00:19:51.960 What we're doing here is we're taking seriously 00:19:51.960 --> 00:19:55.870 the superposition principle that says given any process, any way 00:19:55.870 --> 00:19:58.440 that that process could happen, you sum up the amplitudes 00:19:58.440 --> 00:20:00.106 and the probability is the norm squared. 00:20:00.106 --> 00:20:03.100 If we have a source and we have two slits 00:20:03.100 --> 00:20:06.226 and I ask you, what's the probability that you land here, 00:20:06.226 --> 00:20:08.350 the probability is not the sum of the probabilities 00:20:08.350 --> 00:20:09.690 for each individual transit. 00:20:09.690 --> 00:20:13.230 The probability is the square of the amplitude where 00:20:13.230 --> 00:20:15.710 the amplitude is the sum, amplitude top 00:20:15.710 --> 00:20:17.730 plus amplitude bottom. 00:20:17.730 --> 00:20:19.395 Here, there are many, many slits. 00:20:19.395 --> 00:20:21.420 There are many different ways this could happen. 00:20:21.420 --> 00:20:23.345 You could reflect multiple times. 00:20:23.345 --> 00:20:25.809 Everyone cool with that? 00:20:25.809 --> 00:20:27.350 By the same token, we could have done 00:20:27.350 --> 00:20:28.780 the same thing for reflection, but let's stick 00:20:28.780 --> 00:20:30.154 with transmission for the moment. 00:20:30.154 --> 00:20:31.930 This is what we get for the transmission, 00:20:31.930 --> 00:20:34.800 and again, the transmission amplitudes across the step, 00:20:34.800 --> 00:20:36.470 as we saw last time, are the same. 00:20:36.470 --> 00:20:37.855 So this is, in fact, t squared. 00:20:42.720 --> 00:20:47.590 And this gives us a result for transmission down the potential 00:20:47.590 --> 00:20:58.930 well, and if we use what the reflection and transmission 00:20:58.930 --> 00:21:03.290 amplitudes were for our step wells, 00:21:03.290 --> 00:21:10.040 the answer that this gives is 1 over e to the i k2 L minus 2i 00:21:10.040 --> 00:21:19.390 upon the transmission for a step times sine of k2 L. 00:21:19.390 --> 00:21:24.799 Now, this isn't the same as the probability 00:21:24.799 --> 00:21:26.340 that we derived over here, but that's 00:21:26.340 --> 00:21:27.440 because this isn't the probability. 00:21:27.440 --> 00:21:28.398 This was the amplitude. 00:21:28.398 --> 00:21:30.030 We just computed the total amplitude. 00:21:30.030 --> 00:21:31.920 To get the probability, we have to take 00:21:31.920 --> 00:21:34.300 the norm squared of the amplitude. 00:21:34.300 --> 00:21:36.175 And when we take the norm squared, 00:21:36.175 --> 00:21:43.940 what we get is 1 upon 1 plus 1 over 4 epsilon, epsilon plus 1, 00:21:43.940 --> 00:21:48.170 sine squared of g0 root epsilon plus 1. 00:21:52.892 --> 00:21:53.850 We get the same answer. 00:21:59.317 --> 00:22:02.310 AUDIENCE: Is that an equality? 00:22:02.310 --> 00:22:04.480 PROFESSOR: This is an equality. 00:22:04.480 --> 00:22:05.010 Oh, sorry. 00:22:08.530 --> 00:22:10.297 Thank you, Barton. 00:22:10.297 --> 00:22:11.880 We get to the transmission probability 00:22:11.880 --> 00:22:13.230 when we take the norm squared of the amplitude. 00:22:13.230 --> 00:22:13.860 Thank you. 00:22:13.860 --> 00:22:17.050 Is equal to this, which is the same as we got before. 00:22:17.050 --> 00:22:17.550 Thanks. 00:22:21.280 --> 00:22:22.445 Yeah, please? 00:22:22.445 --> 00:22:25.770 AUDIENCE: Tell me why kL equals pi doesn't work. 00:22:28.269 --> 00:22:29.810 PROFESSOR: kL equals pi doesn't work. 00:22:29.810 --> 00:22:30.350 It does. 00:22:30.350 --> 00:22:31.891 You just have to be careful what kL-- 00:22:35.554 --> 00:22:37.012 AUDIENCE: On the first drawing, you 00:22:37.012 --> 00:22:38.678 changed your kL [INAUDIBLE]. 00:22:43.247 --> 00:22:44.580 PROFESSOR: Because I'm an idiot. 00:22:44.580 --> 00:22:45.810 Because I got a factor of 2 wrong. 00:22:45.810 --> 00:22:46.309 Thank you. 00:22:52.280 --> 00:22:55.200 Thank you, Matt. 00:22:55.200 --> 00:22:57.750 Thank you. 00:22:57.750 --> 00:22:58.422 Answer analysis. 00:22:58.422 --> 00:22:59.380 It's a wonderful thing. 00:23:06.800 --> 00:23:10.140 This does something really nice for us. 00:23:10.140 --> 00:23:15.230 Why are we getting a resonance at special values 00:23:15.230 --> 00:23:16.630 at the energy? 00:23:16.630 --> 00:23:17.890 What's happening? 00:23:17.890 --> 00:23:22.090 Well, in this quantum mechanical process 00:23:22.090 --> 00:23:25.250 of multiple interactions, multiple scatterings, 00:23:25.250 --> 00:23:29.659 there are many terms in the amplitude for transmitting. 00:23:29.659 --> 00:23:31.450 There are terms that involve no reflection, 00:23:31.450 --> 00:23:33.425 there are terms that involve two reflections, 00:23:33.425 --> 00:23:36.570 there are terms that involve four reflections, 00:23:36.570 --> 00:23:41.170 and they all come with an actual magnitude and a phase. 00:23:41.170 --> 00:23:47.270 And when the phase is the same, they add constructively, 00:23:47.270 --> 00:23:51.200 and when the phases are not the same, they interfere. 00:23:51.200 --> 00:23:53.850 And when the phases are exactly off, 00:23:53.850 --> 00:23:57.500 they interfere destructively, and that 00:23:57.500 --> 00:23:59.980 is why you're getting a resonance. 00:23:59.980 --> 00:24:02.760 Multiple terms in your superposition 00:24:02.760 --> 00:24:04.620 interfere with each other, something 00:24:04.620 --> 00:24:07.060 that does not happen classically. 00:24:07.060 --> 00:24:09.120 Classically, the probabilities are products. 00:24:09.120 --> 00:24:11.440 Quantum mechanically, we have superposition 00:24:11.440 --> 00:24:14.110 and probabilities are the squares of the amplitude, 00:24:14.110 --> 00:24:16.630 and we get interference effects in the probabilities. 00:24:16.630 --> 00:24:18.590 Cool? 00:24:18.590 --> 00:24:22.530 To me, this is a nice, glorious version of the two slit 00:24:22.530 --> 00:24:25.260 experiment, and we're going to bump up again into it later 00:24:25.260 --> 00:24:31.610 when we talk about the physics of solids in the real world. 00:24:31.610 --> 00:24:33.870 Questions at this point? 00:24:33.870 --> 00:24:34.689 Yeah? 00:24:34.689 --> 00:24:36.730 AUDIENCE: Question on something you said earlier. 00:24:36.730 --> 00:24:38.399 What is the k2 L equals [INAUDIBLE]? 00:24:38.399 --> 00:24:40.190 PROFESSOR: Yeah, what's special about that? 00:24:40.190 --> 00:24:46.660 What's special about that is at this point, where kL is n pi, 00:24:46.660 --> 00:24:48.670 we get perfect transmission. 00:24:48.670 --> 00:24:52.200 When kL is equal to n plus 1/2 pi, 00:24:52.200 --> 00:24:55.054 the reflection is as good as it gets. 00:24:55.054 --> 00:24:56.720 What that's really doing is it's saying, 00:24:56.720 --> 00:25:00.550 when is this locally largest? 00:25:00.550 --> 00:25:03.324 So that's the special points when the transmission 00:25:03.324 --> 00:25:05.740 is as small as possible, which means the reflection, which 00:25:05.740 --> 00:25:07.406 is 1 minus the transmission probability, 00:25:07.406 --> 00:25:09.620 is as large as possible. 00:25:09.620 --> 00:25:13.659 And you can get that, again, from these expressions. 00:25:13.659 --> 00:25:14.325 Other questions? 00:25:28.790 --> 00:25:31.780 So I want to think a little more about these square barriers. 00:25:31.780 --> 00:25:34.080 And in particular, in thinking about the square well 00:25:34.080 --> 00:25:36.360 barrier, what we've been talking about 00:25:36.360 --> 00:25:39.650 all along are monochromatic wave packets. 00:25:39.650 --> 00:25:42.180 We've been talking about plain waves, just simply 00:25:42.180 --> 00:25:48.990 e to the ikx, but you can't put a single particle 00:25:48.990 --> 00:25:52.630 in a state which is a plain wave. 00:25:52.630 --> 00:25:53.710 It's not normalizable. 00:25:53.710 --> 00:25:55.490 What we really mean at the end of the day 00:25:55.490 --> 00:25:57.198 when we talk about single particles is we 00:25:57.198 --> 00:26:00.940 put them in some well localized wave packet, which at time 0, 00:26:00.940 --> 00:26:05.200 let's say, is at position x0, which in this case is negative, 00:26:05.200 --> 00:26:09.840 and which has some well defined average momentum, k0. 00:26:09.840 --> 00:26:12.340 I'll say it's the expectation value of momentum in this wave 00:26:12.340 --> 00:26:14.077 packet. 00:26:14.077 --> 00:26:15.660 And the question we really want to ask 00:26:15.660 --> 00:26:18.970 when we talk about scattering is, what happens to this beast 00:26:18.970 --> 00:26:23.310 as it hits the barrier, which I'm 00:26:23.310 --> 00:26:24.810 going to put the left hand side at 0 00:26:24.810 --> 00:26:27.170 and the right hand side at L and let 00:26:27.170 --> 00:26:29.810 the depth be minus v0 again. 00:26:29.810 --> 00:26:32.740 What happens as this incident wave packet hits the barrier 00:26:32.740 --> 00:26:33.859 and then scatters off? 00:26:33.859 --> 00:26:36.150 Well, we know what would happen if it was a plain wave, 00:26:36.150 --> 00:26:37.520 but a plain wave wouldn't be localized. 00:26:37.520 --> 00:26:39.160 So this is the question I want to ask, 00:26:39.160 --> 00:26:42.530 and I want to use the results that we already have. 00:26:42.530 --> 00:26:43.750 Now, here's the key thing. 00:26:46.490 --> 00:26:49.100 Consider to begin with just a wave 00:26:49.100 --> 00:26:51.890 packet for a free particle centered 00:26:51.890 --> 00:26:58.520 at x0 and with momentum k0, just look for a free particle. 00:26:58.520 --> 00:27:01.350 We know how to write this. 00:27:01.350 --> 00:27:02.980 We can put the system, for example, 00:27:02.980 --> 00:27:04.650 we can take our wave function at time 0 00:27:04.650 --> 00:27:07.560 to be a Gaussian, some normalization times 00:27:07.560 --> 00:27:13.270 e to the minus x minus x0 squared over 2a squared. 00:27:13.270 --> 00:27:15.309 And we want to give it some momentum k0, 00:27:15.309 --> 00:27:17.600 and you know how to do that after the last problem set, 00:27:17.600 --> 00:27:20.066 e to the i k0 x. 00:27:20.066 --> 00:27:22.830 Everyone cool with that? 00:27:22.830 --> 00:27:26.995 So there's our initial wave function, 00:27:26.995 --> 00:27:28.870 and we want to know how it evolves with time, 00:27:28.870 --> 00:27:30.130 and we know how to do that. 00:27:30.130 --> 00:27:31.930 To evolve it in time, we first expand it 00:27:31.930 --> 00:27:33.820 in energy eigenstates. 00:27:33.820 --> 00:27:39.260 So psi of x0 is equal to-- well, the energy eigenstates 00:27:39.260 --> 00:27:46.150 in this case are plain waves, dk e to the ikx over root 2 pi 00:27:46.150 --> 00:27:49.990 times some coefficients, f of k, the expansion coefficients. 00:27:49.990 --> 00:27:52.400 But these are just the Fourier transform 00:27:52.400 --> 00:27:55.550 of our initial Gaussian wave packet, 00:27:55.550 --> 00:28:02.740 and we know what the form of f of k is equal to. 00:28:02.740 --> 00:28:05.560 Well, it's a Gaussian of width 1 upon alpha 00:28:05.560 --> 00:28:10.130 e to the minus k minus k0 because it has momentum k0, 00:28:10.130 --> 00:28:13.460 so it's centered around k0, squared over 2 times a squared, 00:28:13.460 --> 00:28:15.310 and the a goes upstairs. 00:28:15.310 --> 00:28:19.700 And the position of the initial wave packet 00:28:19.700 --> 00:28:21.300 is encoded in the Fourier transform, 00:28:21.300 --> 00:28:23.050 and I'm going to put a normalization here, 00:28:23.050 --> 00:28:26.560 which I'm not going to worry about, with an overall phase 00:28:26.560 --> 00:28:31.900 e to the minus ik x0. 00:28:31.900 --> 00:28:34.360 So in just the same say that adding on a phase, 00:28:34.360 --> 00:28:36.667 e to the i k0 x, in the position space 00:28:36.667 --> 00:28:39.250 wave function tells you what the expectation value of momentum 00:28:39.250 --> 00:28:41.285 is, tacking on the phase-- and we can get this 00:28:41.285 --> 00:28:43.368 from just Fourier transform-- tacking on the phase 00:28:43.368 --> 00:28:45.470 e to the ik x0 in the Fourier transform 00:28:45.470 --> 00:28:48.210 tells you the center point, the central position, 00:28:48.210 --> 00:28:49.292 of the wave packet. 00:28:49.292 --> 00:28:50.750 So this you did on the problem set, 00:28:50.750 --> 00:28:52.650 and this is a trivial momentum space 00:28:52.650 --> 00:28:55.170 version of the same thing. 00:28:55.170 --> 00:28:56.890 So here's our wave packet expanded 00:28:56.890 --> 00:28:58.890 in plain waves, which are energy eigenstates. 00:28:58.890 --> 00:29:01.015 And the statement that these are energy eigenstates 00:29:01.015 --> 00:29:03.650 is equivalent to the statement that under time evolution, 00:29:03.650 --> 00:29:06.207 they do nothing but rotate by a phase. 00:29:06.207 --> 00:29:08.165 So if we want to know what the wave function is 00:29:08.165 --> 00:29:12.540 as a function of time, psi of x and t is equal to the integral, 00:29:12.540 --> 00:29:18.470 dk, the Fourier mode f of k. 00:29:18.470 --> 00:29:20.052 I'll write it out. 00:29:20.052 --> 00:29:22.610 Actually, I'm going to write this out explicitly. 00:29:22.610 --> 00:29:28.540 f of k 1 over root 2 pi e to the ikx 00:29:28.540 --> 00:29:33.410 minus omega t where omega, of course, is a function of t. 00:29:33.410 --> 00:29:39.750 It's a free particle, so for our free particle, 00:29:39.750 --> 00:29:45.462 h bar squared k squared upon 2m is equal to h bar omega. 00:29:48.410 --> 00:29:50.300 We've done this before. 00:29:50.300 --> 00:29:52.680 But now what I want to do is I want to take exactly 00:29:52.680 --> 00:29:54.910 the same system and I want to add, at position 00:29:54.910 --> 00:30:00.120 zero, a well of depth v0 and width L. 00:30:00.120 --> 00:30:03.553 How is our story going to change? 00:30:03.553 --> 00:30:05.886 Well, we want our initial wave packet, which is up to us 00:30:05.886 --> 00:30:08.330 to choose, we want our initial wave packet to be the same. 00:30:08.330 --> 00:30:09.871 We want to start with a Gaussian far, 00:30:09.871 --> 00:30:12.359 far, far away from the barrier. 00:30:12.359 --> 00:30:14.150 We want it to be well localized in position 00:30:14.150 --> 00:30:17.090 and well localized in momentum space, not perfectly localized, 00:30:17.090 --> 00:30:17.659 of course. 00:30:17.659 --> 00:30:19.700 It's a finite Gaussian to satisfy the uncertainty 00:30:19.700 --> 00:30:22.520 principle, but it's well localized. 00:30:22.520 --> 00:30:24.096 So how does this story change? 00:30:24.096 --> 00:30:25.470 Well, this doesn't change at all. 00:30:25.470 --> 00:30:27.080 It's the same wave function. 00:30:27.080 --> 00:30:29.980 However, when we expand in energy eigenstates, 00:30:29.980 --> 00:30:37.350 the energy eigenstates are no longer simple plain waves. 00:30:37.350 --> 00:30:40.600 The energy eigenstates, as we know for the system, 00:30:40.600 --> 00:30:43.570 take a different form. 00:30:43.570 --> 00:30:48.990 For the square well and positive energy scattering states, 00:30:48.990 --> 00:30:51.170 the plain wave, or the energy eigenstates, 00:30:51.170 --> 00:30:55.470 which I will label by k, just because I'm going to call e 00:30:55.470 --> 00:31:01.121 is equal to h bar squared k squared upon 2m-- 00:31:01.121 --> 00:31:03.620 the energy is a constant-- which is the k asymptotically far 00:31:03.620 --> 00:31:04.620 away from the potential. 00:31:06.850 --> 00:31:12.050 The wave function I can write as 1 over root 2 pi times 00:31:12.050 --> 00:31:16.300 e to the ikx when we're on the left hand side, 00:31:16.300 --> 00:31:18.920 but being on the left hand side is equivalent to multiplying 00:31:18.920 --> 00:31:21.870 by a theta function of minus x. 00:31:21.870 --> 00:31:24.750 This is the function which is 0 when its argument is negative 00:31:24.750 --> 00:31:30.350 and 1 when its argument is positive. 00:31:33.530 --> 00:31:36.930 Plus we have the reflected term, which 00:31:36.930 --> 00:31:39.680 has an amplitude, r, which is again a function of k. 00:31:39.680 --> 00:31:41.250 This is the reflection amplitude, 00:31:41.250 --> 00:31:43.850 also known as c upon a. 00:31:43.850 --> 00:31:48.020 e to the minus ikx-- again, on the left hand side-- 00:31:48.020 --> 00:31:53.980 theta of minus x plus a transition amplitude whose 00:31:53.980 --> 00:31:57.070 norm squared is transmission probability e to the ikx when 00:31:57.070 --> 00:32:01.470 we're on the right, theta of x. 00:32:06.280 --> 00:32:08.930 So this is just a slightly different notation 00:32:08.930 --> 00:32:11.810 than what we usually write with left and right separated. 00:32:17.950 --> 00:32:20.565 So what we want to do now is we want to decompose our wave 00:32:20.565 --> 00:32:22.690 function in terms of the actual energy eigenstates. 00:32:29.440 --> 00:32:33.860 The way we're going to do that, and having done that, having 00:32:33.860 --> 00:32:38.275 expanded our wave function in this basis, 00:32:38.275 --> 00:32:40.650 we can determine the time evolution in the following way. 00:32:40.650 --> 00:32:43.420 First, we expand the wave function 00:32:43.420 --> 00:32:48.230 at time 0 as an integral, dk, and I'm 00:32:48.230 --> 00:32:50.176 going to pull the root 2 pi out. 00:32:53.880 --> 00:32:55.340 And we have some Fourier transform, 00:32:55.340 --> 00:32:56.890 which I'm now going to call f tilde 00:32:56.890 --> 00:32:59.223 because is slightly different than the f we used before, 00:32:59.223 --> 00:33:01.500 but it's what the expansion coefficients have to be, 00:33:01.500 --> 00:33:11.260 f tilde of k times this beast, phi of k. 00:33:24.745 --> 00:33:26.120 Let me actually take this product 00:33:26.120 --> 00:33:27.995 and write it out in terms of the three terms. 00:33:27.995 --> 00:33:33.680 So those three terms are going to be f of k, again tilde, 00:33:33.680 --> 00:33:43.340 e to the ikx, theta of minus x, plus f tilde r e to the minus 00:33:43.340 --> 00:33:53.272 ikx, theta of minus x plus f tilde t 00:33:53.272 --> 00:33:57.100 e to the ikx, theta of x. 00:34:02.490 --> 00:34:03.590 Let's look at these terms. 00:34:06.510 --> 00:34:08.639 Finally, we want to look at the time evolution, 00:34:08.639 --> 00:34:11.300 but we started out as a superposition of states 00:34:11.300 --> 00:34:13.159 with definite energy labeled by k, 00:34:13.159 --> 00:34:17.346 so we know the time evolution is e to the ikx minus omega t, e 00:34:17.346 --> 00:34:22.000 to the i minus ikx plus omega t, so minus omega t, 00:34:22.000 --> 00:34:24.130 and kx minus omega t. 00:34:24.130 --> 00:34:27.310 So we can immediately, from this time evolving wave function, 00:34:27.310 --> 00:34:29.699 identify these two terms as terms 00:34:29.699 --> 00:34:33.350 with a central peak moving to the right, 00:34:33.350 --> 00:34:36.588 and this, central peak moving to the left, kx plus omega t. 00:34:40.040 --> 00:34:41.040 Everyone cool with that? 00:34:44.530 --> 00:34:45.030 Questions? 00:34:49.326 --> 00:34:50.826 AUDIENCE: Can you explain real quick 00:34:50.826 --> 00:34:53.666 one more time how that's the [INAUDIBLE]? 00:34:53.666 --> 00:34:55.290 PROFESSOR: How that-- sorry, say again. 00:34:55.290 --> 00:34:57.166 AUDIENCE: How the top equation [INAUDIBLE]. 00:34:57.166 --> 00:35:00.450 PROFESSOR: How the top equation led to this, 00:35:00.450 --> 00:35:01.697 or just where this came from? 00:35:01.697 --> 00:35:02.950 AUDIENCE: [INAUDIBLE]. 00:35:02.950 --> 00:35:04.800 PROFESSOR: Good. 00:35:04.800 --> 00:35:06.710 This is just a notational thing. 00:35:06.710 --> 00:35:11.900 Usually when I say phi sub k is equal to, on the left, 00:35:11.900 --> 00:35:16.560 e to the ikx with some overall amplitude. 00:35:16.560 --> 00:35:21.870 Let's say 1 over 2 pi e to the ikx 00:35:21.870 --> 00:35:28.040 plus c over a e to the minus ikx on the left, 00:35:28.040 --> 00:35:30.770 and because this is the reflected wave, 00:35:30.770 --> 00:35:35.460 I'm just going to call this R. And then on the right, 00:35:35.460 --> 00:35:38.099 we have e to the ikx. 00:35:38.099 --> 00:35:39.890 There's only a wave travelling to the right 00:35:39.890 --> 00:35:42.162 and the coefficient is the transmission amplitude. 00:35:42.162 --> 00:35:44.120 So this is what we normally write, but then I'm 00:35:44.120 --> 00:35:46.760 using the so-called theta function. 00:35:46.760 --> 00:35:51.500 Theta of x is defined as 0 when x is less than 0 and 1 00:35:51.500 --> 00:35:53.680 when x is greater than 0. 00:35:53.680 --> 00:35:55.520 This is a function 1. 00:35:55.520 --> 00:35:57.310 Using theta function allows me to write 00:35:57.310 --> 00:35:59.510 this thing as a single function without having 00:35:59.510 --> 00:36:02.490 to goose around with lots of terms. 00:36:02.490 --> 00:36:03.440 Is that cool? 00:36:03.440 --> 00:36:04.035 Great. 00:36:04.035 --> 00:36:06.460 AUDIENCE: Are we just thinking about one 00:36:06.460 --> 00:36:09.612 step here or the whole well, because always, we 00:36:09.612 --> 00:36:11.320 should have something [INAUDIBLE]. 00:36:11.320 --> 00:36:11.900 PROFESSOR: Fantastic. 00:36:11.900 --> 00:36:12.640 Excellent, excellent. 00:36:12.640 --> 00:36:13.390 Thank you so much. 00:36:16.180 --> 00:36:19.130 What I want to do is I want to think of the wave function. 00:36:19.130 --> 00:36:22.539 This is a good description when the particle is far, far away, 00:36:22.539 --> 00:36:24.830 and this is a good description when the particle is not 00:36:24.830 --> 00:36:27.240 in the potential. 00:36:27.240 --> 00:36:27.740 Sorry. 00:36:27.740 --> 00:36:28.050 Thank you. 00:36:28.050 --> 00:36:29.520 I totally glossed over this step. 00:36:29.520 --> 00:36:31.860 I want to imagine this as a potential 00:36:31.860 --> 00:36:40.422 where all of the matching is implemented at x equals 0, 00:36:40.422 --> 00:36:41.880 so when I write it in this fashion. 00:36:44.816 --> 00:36:46.565 Another equivalent way to think about this 00:36:46.565 --> 00:36:48.648 is this is a good description of the wave function 00:36:48.648 --> 00:36:49.940 when we're not inside the well. 00:36:49.940 --> 00:36:52.273 So for the purposes of the rest of the analysis that I'm 00:36:52.273 --> 00:36:53.970 going to do, this is exactly what 00:36:53.970 --> 00:36:59.010 the form of the wave function is when we're not inside the well, 00:36:59.010 --> 00:37:01.060 and then let's just not use this to ask 00:37:01.060 --> 00:37:03.380 about questions inside the well. 00:37:03.380 --> 00:37:04.849 When I write it in this theta form, 00:37:04.849 --> 00:37:06.890 or for that matter, when I write it in this form, 00:37:06.890 --> 00:37:08.130 this is not the form. 00:37:08.130 --> 00:37:13.780 What I mean is left of the well and right of the well, 00:37:13.780 --> 00:37:20.180 and inside-- t e to the ikx-- inside, 00:37:20.180 --> 00:37:23.789 it's doing something else, but we 00:37:23.789 --> 00:37:25.330 don't want to ask questions about it. 00:37:28.080 --> 00:37:31.140 That's just going to simplify my life. 00:37:31.140 --> 00:37:31.810 Yeah? 00:37:31.810 --> 00:37:34.250 AUDIENCE: And is the lowercase t then 00:37:34.250 --> 00:37:35.977 just the square root of capital T? 00:37:35.977 --> 00:37:37.810 PROFESSOR: It's the square root of capital T 00:37:37.810 --> 00:37:40.060 but you've got to be careful because there can be a phase. 00:37:40.060 --> 00:37:42.185 Remember, this is the amplitude, and what it really 00:37:42.185 --> 00:37:45.430 is is this reflection is B over A, 00:37:45.430 --> 00:37:47.240 and these guys are complex numbers. 00:37:47.240 --> 00:37:48.990 And it's true that B over A norm squared 00:37:48.990 --> 00:37:52.660 is the transmission probability, but B over A 00:37:52.660 --> 00:37:55.090 has a phase, and that quantity I'm going to call t, 00:37:55.090 --> 00:37:57.500 and we'll interpret that in more detail in a few minutes. 00:38:01.610 --> 00:38:03.520 So it's of this form. 00:38:03.520 --> 00:38:06.420 I just want to look at each of these three terms. 00:38:06.420 --> 00:38:09.040 In particular, I want to focus on them at time 0. 00:38:09.040 --> 00:38:11.380 So at t equals 0, what does this look like? 00:38:11.380 --> 00:38:17.740 Well, that first term is integral dk of f of k 00:38:17.740 --> 00:38:25.630 e to the ikx minus omega t, theta of minus x. 00:38:25.630 --> 00:38:28.065 Notice that theta of minus x is independent of k. 00:38:28.065 --> 00:38:29.440 It's independent of the integral, 00:38:29.440 --> 00:38:33.930 so this is just a function times theta of minus x. 00:38:33.930 --> 00:38:35.917 And this function was constructed 00:38:35.917 --> 00:38:37.250 to give us the initial Gaussian. 00:38:42.210 --> 00:38:43.650 And this was at t equals 0. 00:38:43.650 --> 00:38:49.090 So this is just our Gaussian at position x0 centered 00:38:49.090 --> 00:38:55.292 around value k0 at time equals 0, theta of minus x. 00:39:01.555 --> 00:39:07.830 Sorry, at position x0. 00:39:07.830 --> 00:39:10.880 All this function is, this is the Fourier transform 00:39:10.880 --> 00:39:13.270 of our Gaussian and we're undoing the Fourier transform. 00:39:13.270 --> 00:39:16.090 This is just giving us our Gaussian back. 00:39:16.090 --> 00:39:18.830 And as long as the particle is far away from the well-- 00:39:18.830 --> 00:39:21.992 so here's the well and here's our wave packet-- 00:39:21.992 --> 00:39:23.700 that theta function is totally irrelevant 00:39:23.700 --> 00:39:26.272 because the Gaussian makes it 0 away from the center 00:39:26.272 --> 00:39:27.230 of the Gaussian anyway. 00:39:34.800 --> 00:39:36.520 And so just as a quick question, if we 00:39:36.520 --> 00:39:39.260 look at this is a function of t so we put back in the minus 00:39:39.260 --> 00:39:49.760 omega t, so if we put back in the time dependence, 00:39:49.760 --> 00:39:52.870 this is a wave that's moving to the right, 00:39:52.870 --> 00:39:54.705 and to be more precise about that, 00:39:54.705 --> 00:39:57.080 what does it mean to say it's a wave moving to the right? 00:39:57.080 --> 00:40:02.700 Well, this is an envelope on this set of plain waves, 00:40:02.700 --> 00:40:04.890 and the envelope, by construction, 00:40:04.890 --> 00:40:06.770 was well localized around position x 00:40:06.770 --> 00:40:09.994 but it was also well localized in momentum, 00:40:09.994 --> 00:40:11.660 and in particular, the Fourier transform 00:40:11.660 --> 00:40:14.130 is well localized around the momentum k0. 00:40:17.930 --> 00:40:20.370 And so using the method of stationary phase, 00:40:20.370 --> 00:40:27.660 or just asking, where is the phase constant stationary, 00:40:27.660 --> 00:40:33.440 we get that the central peak of the wave function 00:40:33.440 --> 00:40:36.890 satisfies the equation d d k0. 00:40:36.890 --> 00:40:38.940 If you're not familiar with stationary phase, 00:40:38.940 --> 00:40:42.240 let the recitation instructors know 00:40:42.240 --> 00:40:45.110 and they will discuss it for you. 00:40:45.110 --> 00:40:49.890 The points of stationary phase of this superposition 00:40:49.890 --> 00:40:58.690 of this wave packet lie at the position d dk of kx 00:40:58.690 --> 00:41:04.930 minus omega of kt evaluated at the peak, k0, 00:41:04.930 --> 00:41:06.530 of the distribution. 00:41:06.530 --> 00:41:10.860 But d dk of kx minus omega t is, for the first term x, 00:41:10.860 --> 00:41:13.350 and for the second term, d dk of omega. 00:41:13.350 --> 00:41:16.650 Well, we know that omega, we're in the free particle regime. 00:41:16.650 --> 00:41:19.490 Omega, we all know what it is. 00:41:19.490 --> 00:41:21.952 It's h bar k squared upon 2m. 00:41:21.952 --> 00:41:23.660 We take the derivative with respect to k. 00:41:23.660 --> 00:41:25.300 We get h bar k over m. 00:41:25.300 --> 00:41:32.860 The 2's cancel, so x minus h bar k over m t. 00:41:32.860 --> 00:41:37.140 And a place where this phase is 0, 00:41:37.140 --> 00:41:40.920 where the phase is stationary, moves over time 00:41:40.920 --> 00:41:46.190 as x is equal to h bar k over m t. 00:41:46.190 --> 00:41:49.880 But h bar k over m, that's the momentum. 00:41:49.880 --> 00:41:52.930 p over m, that's the classical velocity. 00:41:52.930 --> 00:41:58.140 So this is v0 t, the velocity associated with that momentum. 00:41:58.140 --> 00:41:58.919 So this is good. 00:41:58.919 --> 00:42:01.210 We've done the right job of setting up our wave packet. 00:42:01.210 --> 00:42:03.120 We built a Gaussian that was far away that 00:42:03.120 --> 00:42:05.794 was moving in with a fixed velocity towards the barrier, 00:42:05.794 --> 00:42:08.210 and now we want to know what happens after it collides off 00:42:08.210 --> 00:42:09.300 the barrier. 00:42:09.300 --> 00:42:10.012 Cool? 00:42:10.012 --> 00:42:11.970 So what we really want to ask is at late times, 00:42:11.970 --> 00:42:13.553 what does the wave function look like? 00:42:15.794 --> 00:42:17.460 Again, we're focusing on the first term. 00:42:17.460 --> 00:42:20.450 The position of this wave packet at late times, a positive t, 00:42:20.450 --> 00:42:22.932 is positive, and when it's positive, 00:42:22.932 --> 00:42:24.890 then this theta function kills its contribution 00:42:24.890 --> 00:42:27.860 to the overall wave function. 00:42:27.860 --> 00:42:30.410 This theta is now theta of minus a positive number 00:42:30.410 --> 00:42:34.060 and this Gaussian is gone. 00:42:34.060 --> 00:42:35.510 What is it replaced by? 00:42:35.510 --> 00:42:38.060 Well, these two terms aren't necessarily 0. 00:42:38.060 --> 00:42:40.350 In particular, this one is moving to the left. 00:42:40.350 --> 00:42:43.120 So as time goes forward, x is moving further and further 00:42:43.120 --> 00:42:47.190 to the left, and so this theta function starts turning on. 00:42:47.190 --> 00:42:49.490 And similarly, as x goes positive, 00:42:49.490 --> 00:42:53.290 this theta function starts turning on as well. 00:42:53.290 --> 00:42:59.959 Let's focus on this third term, the transmitted term. 00:42:59.959 --> 00:43:01.250 Let's focus on this third term. 00:43:08.120 --> 00:43:12.400 In particular, that term looks like integral dk over root 2 00:43:12.400 --> 00:43:18.130 pi, f times the transmission amplitude times e 00:43:18.130 --> 00:43:21.590 to the ikx minus omega t. 00:43:24.540 --> 00:43:26.680 And there's an overall theta of x outside, 00:43:26.680 --> 00:43:28.810 but for late times where the center of the wave 00:43:28.810 --> 00:43:31.510 packet where the transmitted wave packet should be positive, 00:43:31.510 --> 00:43:34.255 this theta is just going to be 1 so we can safely ignore it. 00:43:34.255 --> 00:43:37.740 It's just saying we're far off to the right. 00:43:37.740 --> 00:43:41.700 And now I want to do one last thing. 00:43:41.700 --> 00:43:44.310 This was an overall envelope. 00:43:44.310 --> 00:43:46.025 This t was our scattering amplitude, 00:43:46.025 --> 00:43:47.900 and I want to write it in the following form. 00:43:47.900 --> 00:43:53.410 I want to write it as root T e to the minus i phi 00:43:53.410 --> 00:43:54.287 where phi is k. 00:43:54.287 --> 00:43:55.870 So what this is saying is that indeed, 00:43:55.870 --> 00:43:57.703 as was pointed out earlier, the norm squared 00:43:57.703 --> 00:44:01.010 of this coefficient T is the transmission probability, 00:44:01.010 --> 00:44:02.540 but it has a phase. 00:44:02.540 --> 00:44:05.520 And what I want to know is what does this phase mean? 00:44:05.520 --> 00:44:08.360 What information is contained in this phase? 00:44:08.360 --> 00:44:10.110 And that's what we're about to find. 00:44:10.110 --> 00:44:11.470 So let's put that in. 00:44:11.470 --> 00:44:18.030 Here we have root T and minus phi 00:44:18.030 --> 00:44:20.952 where omega and phi are both functions of k 00:44:20.952 --> 00:44:22.410 because the transmission amplitudes 00:44:22.410 --> 00:44:27.260 depend on the momentum, or the energy. 00:44:27.260 --> 00:44:30.840 And now I again want to know, how does this wave packet move? 00:44:30.840 --> 00:44:33.930 If I look at this wave packet, how does it move en masse? 00:44:33.930 --> 00:44:37.510 As a group of waves, how does this wave packet move? 00:44:37.510 --> 00:44:40.130 In particular, with what velocity? 00:44:40.130 --> 00:44:42.560 I'll again make an argument by stationary phase. 00:44:42.560 --> 00:44:44.960 I'll look at a point of phase equals 0 00:44:44.960 --> 00:44:46.565 and ask how it moves over time. 00:44:46.565 --> 00:44:47.940 And the point of stationary phase 00:44:47.940 --> 00:44:54.860 is again given by d dk of the phase kx minus omega 00:44:54.860 --> 00:45:01.220 of kt minus phi of k is equal to 0 evaluated at k0, which 00:45:01.220 --> 00:45:03.200 is where our envelope is sharply peaked. 00:45:07.470 --> 00:45:09.420 This expression is equal to-- this 00:45:09.420 --> 00:45:12.730 is, again, x minus d omega dk. 00:45:12.730 --> 00:45:22.250 That's the classical velocity, v0 t-- minus d phi dk. 00:45:22.250 --> 00:45:28.495 But just as a note, d phi dk is equal to d omega dk, 00:45:28.495 --> 00:45:34.290 d phi d omega, but this is equal to-- this 00:45:34.290 --> 00:45:36.110 is just the chain rule. d omega dk, that's 00:45:36.110 --> 00:45:38.310 the classical velocity. 00:45:38.310 --> 00:45:45.230 d phi d omega, that's d phi dE times h bar. 00:45:45.230 --> 00:45:48.510 I just multiplied by h bar on the top and bottom. 00:45:48.510 --> 00:45:57.280 So this is equal to x minus v0 t plus h bar d phi dE. 00:46:00.480 --> 00:46:03.100 So first off, let's just make sure that the units make sense. 00:46:03.100 --> 00:46:04.997 That's a length, that's a velocity, 00:46:04.997 --> 00:46:06.580 so this had better have units of time. 00:46:06.580 --> 00:46:07.950 Time, that's good. 00:46:07.950 --> 00:46:10.890 So h bar times d phase over dE. 00:46:10.890 --> 00:46:13.970 Well, phase is dimensionless, energy is units of energy, 00:46:13.970 --> 00:46:17.390 h is energy times time, so this dimensionally works out. 00:46:17.390 --> 00:46:18.500 So this is 0. 00:46:18.500 --> 00:46:20.630 So the claim is the point of stationary phase 00:46:20.630 --> 00:46:24.260 has this derivative equal to 0, so setting this equal to 0 00:46:24.260 --> 00:46:26.580 tells us that the peak of the wave function 00:46:26.580 --> 00:46:29.650 moves according to this equation. 00:46:33.420 --> 00:46:34.915 So this is really satisfying. 00:46:34.915 --> 00:46:37.290 This should be really satisfying for a couple of reasons. 00:46:37.290 --> 00:46:41.440 First off, it tells you that the peak of the transmitted wave 00:46:41.440 --> 00:46:44.460 packet, not just a plain wave, the peak of the actual wave 00:46:44.460 --> 00:46:48.100 packet, well localized, moves with overall velocity 00:46:48.100 --> 00:46:50.860 v0, the constant velocity that we started with, 00:46:50.860 --> 00:46:51.530 and that's good. 00:46:51.530 --> 00:46:53.280 If it was moving with some other velocity, 00:46:53.280 --> 00:46:54.696 we would have lost energy somehow. 00:46:54.696 --> 00:46:57.600 That would be not so sensible. 00:46:57.600 --> 00:47:00.240 So this wave packet is moving with an overall velocity v0. 00:47:00.240 --> 00:47:06.650 However, it doesn't move along just 00:47:06.650 --> 00:47:09.300 as the original wave packet's peak had. 00:47:09.300 --> 00:47:11.550 It moves as if shifted in time. 00:47:14.560 --> 00:47:16.920 So the phase, and more to the point, the gradient 00:47:16.920 --> 00:47:18.420 of the phase with respect to energy, 00:47:18.420 --> 00:47:22.150 the rate of change with energy of the phase times h bar 00:47:22.150 --> 00:47:28.690 is giving us a shift in the time of where the wave packet is. 00:47:28.690 --> 00:47:29.530 What does that mean? 00:47:32.730 --> 00:47:33.910 Let's be precise about this. 00:47:43.120 --> 00:47:45.545 Let's interpret this. 00:47:45.545 --> 00:47:47.420 Here's the interpretation I want to give you. 00:47:51.160 --> 00:47:53.070 Consider classically the system. 00:47:53.070 --> 00:47:57.180 Classically, we have an object with some energy, 00:47:57.180 --> 00:48:01.390 and it rolls along and it finds a potential well, and what 00:48:01.390 --> 00:48:04.200 happens when it gets into the potential well? 00:48:04.200 --> 00:48:05.494 It speeds up. 00:48:05.494 --> 00:48:06.910 because it's got a lot more energy 00:48:06.910 --> 00:48:07.790 relative to the potential. 00:48:07.790 --> 00:48:10.290 So it goes much faster in here and it gets to the other side 00:48:10.290 --> 00:48:11.460 and it slows down again. 00:48:11.460 --> 00:48:13.571 So if I had taken a particle with velocity-- 00:48:13.571 --> 00:48:15.070 let's call this position 0 and let's 00:48:15.070 --> 00:48:17.680 say it gets to this wall at time 0, 00:48:17.680 --> 00:48:21.390 so it's moving with x is equal to v0 t, 00:48:21.390 --> 00:48:23.060 if there had been no barrier there, 00:48:23.060 --> 00:48:25.018 then at subsequent times, it would get out here 00:48:25.018 --> 00:48:29.160 in a time that distance over v0, right? 00:48:29.160 --> 00:48:32.570 However, imagine this well was extraordinarily deep. 00:48:32.570 --> 00:48:34.330 If this well were extraordinarily deep, 00:48:34.330 --> 00:48:35.530 what would happen? 00:48:35.530 --> 00:48:38.840 Well basically, in here, its velocity is arbitrarily large, 00:48:38.840 --> 00:48:42.145 and it would just immediately jump across this well. 00:48:45.180 --> 00:48:48.460 This would be a perfectly good description of the motion 00:48:48.460 --> 00:48:51.020 before it gets to the well, but after it leaves the well, 00:48:51.020 --> 00:48:57.290 the position is going to be v0 t plus-- well, 00:48:57.290 --> 00:48:58.520 what's the time shift? 00:48:58.520 --> 00:49:00.430 The time shift is the time that we 00:49:00.430 --> 00:49:04.080 didn't need to cross this gap. 00:49:04.080 --> 00:49:06.540 And how much would that time have been? 00:49:06.540 --> 00:49:10.220 Well, it's the distance divided by the velocity. 00:49:10.220 --> 00:49:13.300 That's the time we didn't need. 00:49:13.300 --> 00:49:16.740 So t plus-- it moves as if it's at a later time-- 00:49:16.740 --> 00:49:20.430 t plus L over v0. 00:49:20.430 --> 00:49:22.944 Cool? 00:49:22.944 --> 00:49:25.360 So if we had a really deep well and we watched a particle, 00:49:25.360 --> 00:49:29.120 we would watch it move x0, x v0 t, v0 t, v0 t, v0 t 00:49:29.120 --> 00:49:33.550 plus L over v, v0 t plus L over v. 00:49:33.550 --> 00:49:36.560 So it's the time that we made up by being deep in the well. 00:49:36.560 --> 00:49:40.450 So there's a classical picture because it goes faster inside. 00:49:40.450 --> 00:49:44.520 So comparing these, here we've done a calculation 00:49:44.520 --> 00:49:48.760 of the time shift due to the quantum particle 00:49:48.760 --> 00:49:54.390 quantum mechanically transiting the potential well. 00:49:54.390 --> 00:49:55.430 So let's compare these. 00:49:58.420 --> 00:50:02.530 This says the classical prediction is the time it took, 00:50:02.530 --> 00:50:08.130 delta t, classical is equal to L over v0, 00:50:08.130 --> 00:50:10.940 and the question is, is this the same? 00:50:10.940 --> 00:50:13.880 One way to phrase this question is, is it the same as h bar d 00:50:13.880 --> 00:50:21.350 phi dE evaluated at k0? 00:50:28.100 --> 00:50:35.290 From our results last time, for the amplitude c over a, 00:50:35.290 --> 00:50:39.860 we get that phi is equal to-- which is just minus 00:50:39.860 --> 00:50:46.430 the argument, or the phase, of c over a, or of the transmission 00:50:46.430 --> 00:50:48.254 amplitude little t-- phi turns out 00:50:48.254 --> 00:50:58.670 to be equal to k2 L minus arctan of k1 squared plus k2 squared 00:50:58.670 --> 00:51:06.290 over 2 k1 k2, tan of k2 L. I look at this 00:51:06.290 --> 00:51:07.870 and it doesn't tell me all that much. 00:51:07.870 --> 00:51:10.444 It's a little bit bewildering, so let's unpack this. 00:51:10.444 --> 00:51:11.860 What we really want to know is, is 00:51:11.860 --> 00:51:13.318 this close to the classical result? 00:51:20.050 --> 00:51:22.120 Here's a quick way to check. 00:51:22.120 --> 00:51:24.630 We know this expression is going to simplify near resonance 00:51:24.630 --> 00:51:27.100 where the sine vanishes, so let's look, just 00:51:27.100 --> 00:51:31.130 for simplicity, near the resonance. 00:51:31.130 --> 00:51:34.380 And in particular, let's look near the resonance k2 L 00:51:34.380 --> 00:51:36.974 is equal to n pi. 00:51:36.974 --> 00:51:39.140 Then it turns out that a quick calculation gives you 00:51:39.140 --> 00:51:43.840 that h bar d phi dE at this value of k, 00:51:43.840 --> 00:51:50.540 at that value of the energy, goes as L over 2 v0 times 00:51:50.540 --> 00:51:56.840 1 plus the energy over the depth of the well, v0. 00:51:56.840 --> 00:52:00.070 Now remember, the classical approximation was L over v0. 00:52:00.070 --> 00:52:01.330 We just did this very quickly. 00:52:01.330 --> 00:52:03.310 We did it assuming an arbitrarily deep well. 00:52:03.310 --> 00:52:05.820 So v0 is arbitrarily larger in magnitude than E, 00:52:05.820 --> 00:52:07.222 so this term is negligible. 00:52:07.222 --> 00:52:09.180 Where we should compare that very simple, naive 00:52:09.180 --> 00:52:12.510 classical result is here, L over 2 v0. 00:52:12.510 --> 00:52:15.180 And what we see is that the quantum mechanical result 00:52:15.180 --> 00:52:18.030 gives a time shift which is down by a factor of 2. 00:52:21.920 --> 00:52:22.880 So what's going on? 00:52:22.880 --> 00:52:26.950 Well, apparently, the things slow down inside. 00:52:26.950 --> 00:52:29.630 The time that it took us to cross 00:52:29.630 --> 00:52:31.810 was greater than you would have naively guessed 00:52:31.810 --> 00:52:34.530 by making it arbitrarily deep. 00:52:34.530 --> 00:52:37.590 And we can make that a little more sharp 00:52:37.590 --> 00:52:46.380 by plotting, as a function of E over v0, the actual phase 00:52:46.380 --> 00:52:47.984 shift. 00:52:47.984 --> 00:52:50.150 If do a better job than saying it's infinitely deep, 00:52:50.150 --> 00:52:55.720 the classical prediction looks something like this, 00:52:55.720 --> 00:53:01.109 and this is for delta t, the time shift, classical. 00:53:01.109 --> 00:53:03.400 When you look at the correct quantum mechanical result, 00:53:03.400 --> 00:53:19.920 here's what you find, where the difference is 00:53:19.920 --> 00:53:24.570 a factor of 2, 1/2 the height down, and again, 00:53:24.570 --> 00:53:26.450 1/2 the height down. 00:53:26.450 --> 00:53:28.810 So this is that factor of 2 downstairs. 00:53:28.810 --> 00:53:32.442 So the wave packet goes actually a little bit faster 00:53:32.442 --> 00:53:33.900 than the classical prediction would 00:53:33.900 --> 00:53:36.440 guess except near resonance, and these 00:53:36.440 --> 00:53:38.815 are at the resonant values of the momentum. 00:53:38.815 --> 00:53:40.440 At the resonant values of the momentum, 00:53:40.440 --> 00:53:41.971 it takes much longer to get across. 00:53:41.971 --> 00:53:43.470 Instead of going a little bit faster 00:53:43.470 --> 00:53:45.430 than the classical result, it goes a factor 00:53:45.430 --> 00:53:48.570 of 2 slower than the classical result. 00:53:48.570 --> 00:53:50.050 And so now I ask you the question, 00:53:50.050 --> 00:53:51.450 why is it going more slowly? 00:53:51.450 --> 00:53:55.190 Why does it spend so much more time inside the well quantum 00:53:55.190 --> 00:53:57.810 mechanically than it would have classically? 00:53:57.810 --> 00:54:00.830 Why is the particle effectively taking so much longer 00:54:00.830 --> 00:54:03.990 to transit the well near resonance? 00:54:03.990 --> 00:54:06.449 AUDIENCE: Because it can reflect and it can keep going 00:54:06.449 --> 00:54:08.490 and a classical particle is not going to do that. 00:54:08.490 --> 00:54:09.280 PROFESSOR: Yeah, exactly. 00:54:09.280 --> 00:54:10.950 So the classical particle just goes across. 00:54:10.950 --> 00:54:13.160 The quantum mechanical particle has a superposition 00:54:13.160 --> 00:54:14.534 of contributions to its amplitude 00:54:14.534 --> 00:54:17.792 where it transits-- transit, bounce, bounce, transit, 00:54:17.792 --> 00:54:20.560 transit, bounce, bounce, bounce, bounce, transit. 00:54:20.560 --> 00:54:22.250 And now you can ask, how much time 00:54:22.250 --> 00:54:25.540 was spent by each of those imaginary particles imaginarily 00:54:25.540 --> 00:54:27.005 moving across? 00:54:27.005 --> 00:54:29.380 And if you're careful about how you set up that question, 00:54:29.380 --> 00:54:33.740 you can recover this factor of 2, which is kind of beautiful. 00:54:33.740 --> 00:54:36.640 But the important thing here is when you're hitting resonance, 00:54:36.640 --> 00:54:39.780 the multiple scattering processes are important. 00:54:39.780 --> 00:54:41.040 They're not canceling out. 00:54:41.040 --> 00:54:42.270 They're not at random phase. 00:54:42.270 --> 00:54:44.030 They're not interfering destructively with each other. 00:54:44.030 --> 00:54:45.488 They're interfering constructively, 00:54:45.488 --> 00:54:47.700 and you get perfect transmission precisely 00:54:47.700 --> 00:54:51.160 because of the constructive interference 00:54:51.160 --> 00:54:53.050 of an infinite number of contributions 00:54:53.050 --> 00:54:54.882 to the quantum mechanical amplitude. 00:54:54.882 --> 00:54:56.840 And this is, again, we're seeing the same thing 00:54:56.840 --> 00:54:59.550 in this annoying slow down. 00:54:59.550 --> 00:55:01.150 This tells us another thing, though, 00:55:01.150 --> 00:55:04.110 which is that the scattering phase, the phase 00:55:04.110 --> 00:55:06.650 in the transmission amplitude, contains 00:55:06.650 --> 00:55:08.400 an awful lot of the physics of the system. 00:55:08.400 --> 00:55:12.350 It's telling us about how long it takes for the wave packet 00:55:12.350 --> 00:55:14.680 to transit across the potential, effectively. 00:55:14.680 --> 00:55:15.347 Yeah? 00:55:15.347 --> 00:55:16.600 AUDIENCE: What's the vertical axis being used for? 00:55:16.600 --> 00:55:17.308 PROFESSOR: Sorry. 00:55:17.308 --> 00:55:21.620 The vertical axis here is the shift 00:55:21.620 --> 00:55:24.300 in the time due to the fact that it went across this well 00:55:24.300 --> 00:55:26.700 and went a little bit faster inside. 00:55:26.700 --> 00:55:30.580 So empirically, what it means is when you get out very far away 00:55:30.580 --> 00:55:33.070 and you watch the motion of the wave packet, 00:55:33.070 --> 00:55:35.794 and you ask, how long has it been since it got 00:55:35.794 --> 00:55:37.210 to the barrier in the first place, 00:55:37.210 --> 00:55:38.640 it took less time than you would have guessed 00:55:38.640 --> 00:55:41.400 by knowing that its velocity is v0, and the amount of time less 00:55:41.400 --> 00:55:42.757 is this much time. 00:55:42.757 --> 00:55:43.840 That answer your question? 00:55:43.840 --> 00:55:44.720 Good. 00:55:44.720 --> 00:55:45.972 Yeah? 00:55:45.972 --> 00:55:48.710 AUDIENCE: Why is each of the amplitudes 1/2? 00:55:48.710 --> 00:55:50.880 Why is the first one, and it bounces around twice. 00:55:54.847 --> 00:55:56.930 PROFESSOR: When we compare the classical amplitude 00:55:56.930 --> 00:56:00.140 and the limit that v0 goes to infinity? 00:56:00.140 --> 00:56:03.660 The comparison is just a factor of 2. 00:56:03.660 --> 00:56:07.150 It's more complicated out here. 00:56:07.150 --> 00:56:10.428 In the limit that v0 is large, this is exactly 1/2. 00:56:10.428 --> 00:56:11.344 AUDIENCE: [INAUDIBLE]. 00:56:13.779 --> 00:56:15.320 PROFESSOR: The resonances are leading 00:56:15.320 --> 00:56:16.486 to this extra factor of 1/2. 00:56:18.907 --> 00:56:20.490 I have to say I don't remember exactly 00:56:20.490 --> 00:56:24.020 whether, as you include the sub-leading terms of 1 00:56:24.020 --> 00:56:26.590 over v0, whether it stays 1/2 or whether it doesn't, 00:56:26.590 --> 00:56:28.480 but in the limit that v0 is large, 00:56:28.480 --> 00:56:30.500 it remains either close to 1/2 or exactly 1/2, 00:56:30.500 --> 00:56:31.750 I just don't remember exactly. 00:56:31.750 --> 00:56:34.000 The important thing is that there's a sharp dip. 00:56:34.000 --> 00:56:38.680 It takes much longer to transit, and so you get less bonus time, 00:56:38.680 --> 00:56:40.240 as it were. 00:56:40.240 --> 00:56:43.370 You've gained less time in the quantum mechanical model 00:56:43.370 --> 00:56:44.822 than the classical model. 00:56:44.822 --> 00:56:47.280 And when you do the experiment, you get the quantum result. 00:56:50.940 --> 00:56:52.580 That's the crucial point. 00:56:52.580 --> 00:56:53.260 Other questions? 00:56:56.680 --> 00:56:59.290 So the phase contains an awful lot of the physics. 00:57:04.240 --> 00:57:08.440 So I want to generalize this whole story 00:57:08.440 --> 00:57:11.920 in a very particular way, this way of reorganizing 00:57:11.920 --> 00:57:13.140 the scattering in 1D. 00:57:13.140 --> 00:57:14.060 What we're doing right now is we're 00:57:14.060 --> 00:57:16.060 studying scattering problems in one dimension, 00:57:16.060 --> 00:57:17.900 but we live in three dimensions. 00:57:17.900 --> 00:57:19.983 The story is going to be more complicated in three 00:57:19.983 --> 00:57:20.483 dimensions. 00:57:20.483 --> 00:57:22.649 It's going to be more complicated in two dimensions, 00:57:22.649 --> 00:57:24.220 but the basic ideas are all the same. 00:57:24.220 --> 00:57:26.219 It's just the details are going to be different. 00:57:26.219 --> 00:57:29.420 And one thing that turns out to be very useful in organizing 00:57:29.420 --> 00:57:31.449 scattering, both in one dimension 00:57:31.449 --> 00:57:32.990 and in three dimensions, is something 00:57:32.990 --> 00:57:34.198 called the scattering matrix. 00:57:34.198 --> 00:57:35.870 I'm going to talk about that now. 00:57:35.870 --> 00:57:39.380 In three dimensions, it's essential, 00:57:39.380 --> 00:57:41.720 but even in one dimension, where it's usually not used, 00:57:41.720 --> 00:57:43.360 it's a very powerful way to organize 00:57:43.360 --> 00:57:49.350 our knowledge of the system as encoded by the scattering data. 00:57:49.350 --> 00:57:51.000 So here's the basic idea. 00:57:51.000 --> 00:57:52.800 As we discussed before, what we really 00:57:52.800 --> 00:57:55.420 want to do in the ideal world is take some unknown potential 00:57:55.420 --> 00:57:58.510 in some bounded region, some region, 00:57:58.510 --> 00:58:01.660 and outside, we have the potential is constant. 00:58:01.660 --> 00:58:03.300 [INAUDIBLE] my bad artistic skills. 00:58:03.300 --> 00:58:06.740 So potential is constant out here. 00:58:06.740 --> 00:58:08.990 And the potential could be some horrible thing in here 00:58:08.990 --> 00:58:10.490 that we don't happen to know, and we 00:58:10.490 --> 00:58:12.530 want to read off of the scattering process, 00:58:12.530 --> 00:58:15.500 we want to be able to deduce something about the potential. 00:58:15.500 --> 00:58:26.190 So for example, we can deduce the energy 00:58:26.190 --> 00:58:30.920 by looking at the position of the barriers, 00:58:30.920 --> 00:58:33.380 and we can disentangle the position of the energy 00:58:33.380 --> 00:58:37.350 and the depth and the width by looking at the phase shift, 00:58:37.350 --> 00:58:38.680 by looking at the time delay. 00:58:38.680 --> 00:58:40.940 So we can deduce all the parameters of our potential 00:58:40.940 --> 00:58:43.830 by looking at the resonance points and the phase shift. 00:58:43.830 --> 00:58:46.665 I want to do this more generally for general potential. 00:58:46.665 --> 00:58:48.040 And to set that up, we need to be 00:58:48.040 --> 00:58:50.150 a bit more general than we've been. 00:58:54.060 --> 00:58:56.730 So in general, if we solve this potential, 00:58:56.730 --> 00:59:02.270 as we talked about before, we have A and B, e to the ikx, 00:59:02.270 --> 00:59:05.850 e to the minus ikx out here, and out here, we 00:59:05.850 --> 00:59:11.278 have C e to the ikx and D e to the minus ikx. 00:59:14.235 --> 00:59:16.110 And I'm not going to ask what happens inside. 00:59:19.377 --> 00:59:20.960 Now we can do, as discussed, two kinds 00:59:20.960 --> 00:59:22.043 of scattering experiments. 00:59:22.043 --> 00:59:24.220 We can send things in from the left, in which case 00:59:24.220 --> 00:59:27.100 A is nonzero, and then things can either transmit or reflect, 00:59:27.100 --> 00:59:29.755 but nothing's going to come in from infinity, so D is 0. 00:59:29.755 --> 00:59:31.130 Or we can do the same in reverse. 00:59:31.130 --> 00:59:33.254 Send things in from here, and that corresponds to A 00:59:33.254 --> 00:59:36.630 is 0, nothing coming in this way but d0 and 0. 00:59:36.630 --> 00:59:39.900 And more generally, we can ask the following question. 00:59:39.900 --> 00:59:42.950 Suppose I send some amount of stuff in from the left 00:59:42.950 --> 00:59:46.010 and I send some amount of stuff in from the right, D. Then 00:59:46.010 --> 00:59:48.650 that will tell me how much stuff will be going out to the right 00:59:48.650 --> 00:59:52.150 and how much stuff will be going out to the left. 00:59:52.150 --> 00:59:54.040 If you tell me how much is coming in, 00:59:54.040 --> 00:59:59.440 I will tell you how much is coming out, B, C. 00:59:59.440 --> 01:00:02.630 So if you could solve this problem, 01:00:02.630 --> 01:00:05.480 the answer is just some paralinear relations 01:00:05.480 --> 01:00:09.080 between these, and we can write this as a matrix, which I will 01:00:09.080 --> 01:00:15.030 S11, S12, S21, S22. 01:00:15.030 --> 01:00:17.650 What this matrix is doing is it takes the amplitude you're 01:00:17.650 --> 01:00:19.120 sending in from the left to right 01:00:19.120 --> 01:00:21.344 and tells you the amplitude coming out to the left 01:00:21.344 --> 01:00:22.010 or to the right. 01:00:22.010 --> 01:00:22.833 Yes? 01:00:22.833 --> 01:00:25.870 AUDIENCE: How do we know this relation is linear? 01:00:25.870 --> 01:00:28.850 PROFESSOR: If you double the amount of stuff coming in, 01:00:28.850 --> 01:00:31.030 then you must double the amount of stuff going out 01:00:31.030 --> 01:00:32.420 or probability is not conserved. 01:00:35.070 --> 01:00:37.120 Also, we've derived these relations. 01:00:37.120 --> 01:00:38.730 You know how the relations work. 01:00:38.730 --> 01:00:42.769 The relations work by satisfying a series of linear equations 01:00:42.769 --> 01:00:44.310 between the various coefficients such 01:00:44.310 --> 01:00:46.855 that you have continuity and differentiability at all 01:00:46.855 --> 01:00:48.634 the matching points. 01:00:48.634 --> 01:00:50.050 But the crucial thing of linearity 01:00:50.050 --> 01:00:54.034 is probability is conserved and time evolution is linear. 01:00:54.034 --> 01:00:54.700 Other questions? 01:00:58.540 --> 01:01:01.100 So this is just some stupid matrix, and we call it, 01:01:01.100 --> 01:01:05.505 not surprisingly, the S matrix in all of its majesty. 01:01:09.510 --> 01:01:10.810 The basic idea is this. 01:01:10.810 --> 01:01:13.685 For scattering problems, if someone tells you the S matrix, 01:01:13.685 --> 01:01:15.310 and in particular, if someone tells you 01:01:15.310 --> 01:01:18.892 how all the coefficients of the S matrix vary with energy, 01:01:18.892 --> 01:01:21.100 then you've completely solved any scattering problem. 01:01:21.100 --> 01:01:22.950 You tell me what A and D are. 01:01:22.950 --> 01:01:23.450 Great. 01:01:23.450 --> 01:01:26.071 I'll tell you exactly what B and C are mode by mode, 01:01:26.071 --> 01:01:27.570 and I can do this for superposition. 01:01:27.570 --> 01:01:30.377 So this allows you to completely solve any scattering problem 01:01:30.377 --> 01:01:31.960 in quantum mechanics once you know it. 01:01:31.960 --> 01:01:35.176 So it suffices to know S to solve all scattering problems. 01:01:38.270 --> 01:01:40.520 I want to now spend just a little bit of time thinking 01:01:40.520 --> 01:01:43.310 about what properties the S matrix and its components 01:01:43.310 --> 01:01:45.580 must satisfy. 01:01:45.580 --> 01:01:47.210 What properties must the S matrix 01:01:47.210 --> 01:01:51.400 satisfy in order to jibe well with the rest 01:01:51.400 --> 01:01:54.610 of the rules of quantum mechanics? 01:01:54.610 --> 01:01:56.760 I'm not going to study any particular system. 01:01:56.760 --> 01:01:59.480 I just want to ask general questions. 01:01:59.480 --> 01:02:01.510 So the first thing that must be true 01:02:01.510 --> 01:02:04.285 is that stuff doesn't disappear. 01:02:11.950 --> 01:02:16.120 Disapparate is probably the appropriate. 01:02:16.120 --> 01:02:18.870 Stuff doesn't leak out of the world, 01:02:18.870 --> 01:02:20.490 so whatever goes in must come out. 01:02:20.490 --> 01:02:23.760 What that means is norm of A squared plus norm of D 01:02:23.760 --> 01:02:26.970 squared, which is the probability density in 01:02:26.970 --> 01:02:29.030 and probability density out, must 01:02:29.030 --> 01:02:32.000 be equal to B squared plus C squared. 01:02:37.160 --> 01:02:39.422 Everyone agree with that? 01:02:39.422 --> 01:02:42.340 AUDIENCE: Why can't stuff stay in the potential? 01:02:42.340 --> 01:02:44.960 PROFESSOR: Why can't stuff stay in the potential? 01:02:44.960 --> 01:02:46.350 That's a good. 01:03:00.750 --> 01:03:03.260 So if we're looking at fixed energy eigenstates, 01:03:03.260 --> 01:03:05.070 we know that we're in a stationary state, 01:03:05.070 --> 01:03:09.200 so whatever the amplitude going into the middle is, 01:03:09.200 --> 01:03:12.187 it must also be coming out of the middle. 01:03:12.187 --> 01:03:13.770 There's another way to say this, which 01:03:13.770 --> 01:03:16.520 is let's think about it not in terms of individual energy 01:03:16.520 --> 01:03:17.020 eigenstates. 01:03:17.020 --> 01:03:18.936 Let's think about it in terms of wave packets. 01:03:18.936 --> 01:03:20.590 So if we take a wave packet of stuff 01:03:20.590 --> 01:03:23.400 and we send in that wave packet, it has some momentum, right? 01:03:26.280 --> 01:03:28.190 This is going to get delicate and technical. 01:03:28.190 --> 01:03:31.040 Let me just stick with the first statement, which 01:03:31.040 --> 01:03:33.350 is that if stuff is going in, then 01:03:33.350 --> 01:03:36.140 it has to also be coming out by the fact 01:03:36.140 --> 01:03:38.150 that this is an energy eigenstate. 01:03:38.150 --> 01:03:39.650 The overall probability distribution 01:03:39.650 --> 01:03:40.810 is not changing in time. 01:03:43.324 --> 01:03:44.990 If stuff went in and it didn't come out, 01:03:44.990 --> 01:03:45.770 that would mean it's staying there. 01:03:45.770 --> 01:03:46.700 That would mean that the probability 01:03:46.700 --> 01:03:48.060 density is changing in time. 01:03:48.060 --> 01:03:50.060 That's not what happens in an energy eigenstate. 01:03:50.060 --> 01:03:53.631 The probability distribution is time independent. 01:03:53.631 --> 01:03:54.630 Everyone cool with that? 01:04:02.380 --> 01:04:04.570 So let's think, though, about what this is. 01:04:04.570 --> 01:04:07.080 I can write this in the following nice way. 01:04:07.080 --> 01:04:12.440 I can write this as A complex conjugate D complex 01:04:12.440 --> 01:04:16.522 conjugate, AD. 01:04:16.522 --> 01:04:17.980 And on the right hand side, this is 01:04:17.980 --> 01:04:23.525 equal to B complex conjugate, C complex conjugate, BC. 01:04:23.525 --> 01:04:25.400 I have done nothing other than write this out 01:04:25.400 --> 01:04:27.440 in some suggested form. 01:04:27.440 --> 01:04:31.760 But B and C are equal to the S matrix, 01:04:31.760 --> 01:04:36.380 so BC is the S matrix times A, and B complex conjugate, 01:04:36.380 --> 01:04:39.550 C complex conjugate is the transposed complex conjugate. 01:04:39.550 --> 01:04:44.970 So this is equal to A complex conjugate, D complex conjugate, 01:04:44.970 --> 01:04:48.620 S transpose complex conjugate, also known as adjoint, 01:04:48.620 --> 01:04:51.370 and this is S on AD. 01:04:55.010 --> 01:04:57.100 Yeah? 01:04:57.100 --> 01:05:02.360 But this has to be equal to this for any A and D. 01:05:02.360 --> 01:05:06.750 So what must be true of S dagger S as a matrix? 01:05:06.750 --> 01:05:11.230 It's got to be the identity as a matrix in order for this 01:05:11.230 --> 01:05:14.450 to be true for all A and D. Ah. 01:05:14.450 --> 01:05:15.170 That's cool. 01:05:15.170 --> 01:05:16.500 Stuff doesn't disappear. 01:05:16.500 --> 01:05:17.970 S is a unitary matrix. 01:05:23.640 --> 01:05:24.980 So S is a unitary matrix. 01:05:31.090 --> 01:05:33.290 Its inverse is its adjoint. 01:05:43.110 --> 01:05:47.530 You'll study this in a little more detail on the problem set. 01:05:47.530 --> 01:05:49.190 You studied the definition of unitary 01:05:49.190 --> 01:05:52.316 on the last problem set. 01:05:52.316 --> 01:05:53.690 So that's the first thing about S 01:05:53.690 --> 01:05:55.500 and it turns out to be completely general. 01:05:55.500 --> 01:05:57.990 Anytime, whether you're in one dimension or two dimensions 01:05:57.990 --> 01:06:01.550 or three, if you send stuff in, it should not get stuck. 01:06:01.550 --> 01:06:03.360 It should come out. 01:06:03.360 --> 01:06:06.064 And when it does come out, the statement 01:06:06.064 --> 01:06:07.730 that it comes out for energy eigenstates 01:06:07.730 --> 01:06:10.850 is the statement that S is a unitary matrix. 01:06:10.850 --> 01:06:11.876 Questions on that. 01:06:17.840 --> 01:06:29.760 So a consequence of that is that the eigenvalues of S 01:06:29.760 --> 01:06:32.510 are phases, pure phases. 01:06:35.020 --> 01:06:38.800 So I can write S-- I'll write them 01:06:38.800 --> 01:06:43.480 as S1 is equal to e to the i of phi 1 01:06:43.480 --> 01:06:47.824 and S2 is equal to the i of phi 2. 01:06:53.090 --> 01:06:56.970 So the statement that S is a unitary matrix 01:06:56.970 --> 01:07:00.362 leads to constraints on the coefficients, 01:07:00.362 --> 01:07:02.570 and you're going to derive these on your problem set. 01:07:02.570 --> 01:07:03.910 I'm just going to list them now. 01:07:03.910 --> 01:07:07.960 The first is that the magnitude of S11 01:07:07.960 --> 01:07:11.840 is equal to the magnitude of S22. 01:07:11.840 --> 01:07:15.970 The magnitude of S12 is equal to the magnitude of S21. 01:07:24.200 --> 01:07:31.880 More importantly, S12 norm squared plus S11 squared 01:07:31.880 --> 01:07:33.990 is equal to 1. 01:07:33.990 --> 01:07:43.670 And finally, S11, S12 complex conjugate, plus S21, S22 01:07:43.670 --> 01:07:47.010 complex conjugate is equal to 0. 01:07:47.010 --> 01:07:51.820 So what do these mean? 01:07:51.820 --> 01:07:53.453 What are these conditions telling us? 01:07:56.877 --> 01:07:58.460 They're telling us, of course, they're 01:07:58.460 --> 01:08:00.490 a consequence of conservational probability, 01:08:00.490 --> 01:08:01.950 but they have another meaning. 01:08:01.950 --> 01:08:03.491 To get that other meaning, let's look 01:08:03.491 --> 01:08:07.230 at the definition of the transmission amplitudes. 01:08:07.230 --> 01:08:09.890 In particular, consider the case that we send stuff 01:08:09.890 --> 01:08:13.390 in from the left and nothing in from the right. 01:08:13.390 --> 01:08:16.830 That corresponds to D equals 0. 01:08:16.830 --> 01:08:18.710 When D equals 0, what does this tell us? 01:08:18.710 --> 01:08:20.899 It tells us that B is equal to-- and A 01:08:20.899 --> 01:08:22.119 equals 1 for normalization. 01:08:24.740 --> 01:08:30.810 B is equal to-- well, D is equal to 0, so it's just S11 A, 01:08:30.810 --> 01:08:35.029 so B over A is S11. 01:08:35.029 --> 01:08:40.601 And similarly, C over A is S21. 01:08:40.601 --> 01:08:42.100 But C over A is the thing that we've 01:08:42.100 --> 01:08:45.720 been calling the transmission amplitude, little t, 01:08:45.720 --> 01:08:48.109 and this is the reflection amplitude, little r. 01:08:48.109 --> 01:08:51.140 So this is the reflection amplitude, if we send in stuff 01:08:51.140 --> 01:08:53.479 and it bounces off, reflects to the left, 01:08:53.479 --> 01:08:55.869 and this is the transmission amplitude for transmitting 01:08:55.869 --> 01:08:56.410 to the right. 01:08:58.529 --> 01:08:59.529 Everyone cool with that? 01:09:02.279 --> 01:09:04.380 This is reflection to the left, this 01:09:04.380 --> 01:09:06.085 is transmission to the right. 01:09:06.085 --> 01:09:08.210 By the same token, this is going to be transmission 01:09:08.210 --> 01:09:10.060 to the left and reflection to the right. 01:09:12.680 --> 01:09:14.840 So now let's look at these conditions. 01:09:14.840 --> 01:09:17.350 S11 is reflection and this is going to be reflection. 01:09:17.350 --> 01:09:19.350 This says that the reflection to the left 01:09:19.350 --> 01:09:22.875 is equal to the reflection to the right in magnitude. 01:09:25.910 --> 01:09:30.149 Before, what we saw was for the simple step, 01:09:30.149 --> 01:09:32.660 the reflection amplitude was equal from left to right, not 01:09:32.660 --> 01:09:34.430 just the magnitude, but the actual value 01:09:34.430 --> 01:09:36.100 was equal from the left and the right. 01:09:36.100 --> 01:09:38.350 It was a little bit of a cheat because they were real. 01:09:40.217 --> 01:09:41.800 And we saw that that was a consequence 01:09:41.800 --> 01:09:42.870 of just being the step. 01:09:42.870 --> 01:09:43.970 We didn't know anything more about it. 01:09:43.970 --> 01:09:45.595 But now we see that on general grounds, 01:09:45.595 --> 01:09:47.309 on conservation of probability grounds, 01:09:47.309 --> 01:09:49.100 the magnitude of the reflection to the left 01:09:49.100 --> 01:09:52.510 and to the right for any potential had better be equal. 01:09:52.510 --> 01:09:57.490 And similarly, the magnitude of the transmission for the left 01:09:57.490 --> 01:10:03.120 and the transmission to the right had better be equal, 01:10:03.120 --> 01:10:06.540 all other things being-- if you're sending in from the left 01:10:06.540 --> 01:10:08.140 and then transmitting to the left, 01:10:08.140 --> 01:10:10.598 or sending in from the right and transmitting to the right. 01:10:10.598 --> 01:10:12.170 And what does this one tell us? 01:10:12.170 --> 01:10:18.130 Well, S12 and S11, that tells us that little t squared, which 01:10:18.130 --> 01:10:23.890 is the total probability to transmit, plus little r squared 01:10:23.890 --> 01:10:27.040 is the total probability to reflect, is equal to 1, 01:10:27.040 --> 01:10:28.350 and we saw this last time, too. 01:10:32.360 --> 01:10:35.630 This was the earlier definition of nothing gets stuck. 01:10:35.630 --> 01:10:39.771 And this one you'll study on your problem set. 01:10:39.771 --> 01:10:41.254 It's a little more subtle. 01:10:45.440 --> 01:10:46.935 Questions? 01:10:46.935 --> 01:10:47.435 Yeah? 01:10:47.435 --> 01:10:50.345 AUDIENCE: Can you explain one more time why is S unitary? 01:10:50.345 --> 01:10:51.777 How did you get that? 01:10:51.777 --> 01:10:54.110 PROFESSOR: So the way we got S was unitary is first off, 01:10:54.110 --> 01:10:55.740 this is just the definition of S. S 01:10:55.740 --> 01:10:57.800 is the matrix that, for any energy, 01:10:57.800 --> 01:11:00.500 relates A and D, the ingoing amplitudes, 01:11:00.500 --> 01:11:03.830 to the outgoing amplitudes B and C. Just the definition. 01:11:03.830 --> 01:11:06.640 Meanwhile, I claim that stuff doesn't go away, 01:11:06.640 --> 01:11:09.010 nothing gets stuck, nothing disappears, 01:11:09.010 --> 01:11:11.350 so the total probability density of stuff going in 01:11:11.350 --> 01:11:12.550 must be equal to the total probability 01:11:12.550 --> 01:11:13.675 density of stuff going out. 01:11:15.826 --> 01:11:17.200 The probability of stuff going in 01:11:17.200 --> 01:11:19.970 can be expressed as this row vector times this column 01:11:19.970 --> 01:11:23.020 vector, and on the out, this row vector times this column 01:11:23.020 --> 01:11:25.560 vector, and then we use the definition of the S matrix. 01:11:25.560 --> 01:11:28.590 This column vector is equal to S times this row vector. 01:11:31.800 --> 01:11:34.090 BC is equal to S times AD. 01:11:34.090 --> 01:11:36.470 And when we take the transposed complex conjugate, 01:11:36.470 --> 01:11:39.050 I get AD transposed complex conjugate, 01:11:39.050 --> 01:11:42.040 S transposed complex conjugate, but that's the S adjoint. 01:11:42.040 --> 01:11:46.730 But in order for this to be true for any vectors A and D, 01:11:46.730 --> 01:11:50.340 it must be that S dagger S is unitary is the identity, 01:11:50.340 --> 01:11:53.340 but that's the definition of a unitary matrix. 01:11:53.340 --> 01:11:55.270 Cool? 01:11:55.270 --> 01:11:55.770 Others. 01:11:59.240 --> 01:12:01.100 You're going to prove a variety of things 01:12:01.100 --> 01:12:07.370 on the problem set about the scattering 01:12:07.370 --> 01:12:08.930 matrix and its coefficients, but I 01:12:08.930 --> 01:12:12.600 want to show you two properties of it. 01:12:12.600 --> 01:12:17.630 The first is reasonably tame, and it'll 01:12:17.630 --> 01:12:21.135 make a little sharper the step result we got earlier 01:12:21.135 --> 01:12:23.260 that the reflection in both directions off the step 01:12:23.260 --> 01:12:25.170 potential was in fact exactly the same. 01:12:34.974 --> 01:12:36.890 Suppose our system is time reversal invariant. 01:12:40.390 --> 01:12:43.990 So if t goes to minus t, nothing changes. 01:12:43.990 --> 01:12:45.490 This would not be true, for example, 01:12:45.490 --> 01:12:47.365 if we had electric currents in our system 01:12:47.365 --> 01:12:50.797 because as we take t to minus t, then the current reverses. 01:12:50.797 --> 01:12:52.630 So if the current shows up in the potential, 01:12:52.630 --> 01:12:54.140 or if a magnetic field due to a current 01:12:54.140 --> 01:12:55.620 shows up in the potential energy, 01:12:55.620 --> 01:12:57.090 then as we change t to minus t, we 01:12:57.090 --> 01:12:58.590 change the direction of the current, 01:12:58.590 --> 01:13:01.250 we change the direction of the magnetic field. 01:13:01.250 --> 01:13:04.879 In simple systems where we have time reversal invariance, 01:13:04.879 --> 01:13:06.420 for example, electrostatics, but not, 01:13:06.420 --> 01:13:09.590 for example, magnetostatics, suppose we have time reversal 01:13:09.590 --> 01:13:13.560 invariance, then what you've shown on a previous problem 01:13:13.560 --> 01:13:19.060 set is that psi is a solution, then psi star, 01:13:19.060 --> 01:13:20.820 psi complex conjugate is also a solution. 01:13:23.920 --> 01:13:26.882 And using these, what you'll see what you can show-- 01:13:26.882 --> 01:13:28.590 and I'm not going to go through the steps 01:13:28.590 --> 01:13:31.750 for this-- well, that's easy. 01:13:31.750 --> 01:13:35.010 If we do the time reversal, the wave function, 01:13:35.010 --> 01:13:39.160 looking on the left or on the right, so comparing these guys, 01:13:39.160 --> 01:13:40.650 what changes? 01:13:40.650 --> 01:13:43.244 Under time reversal, we get a solution. 01:13:43.244 --> 01:13:45.160 Given this solution, we have another solution, 01:13:45.160 --> 01:13:49.990 A star e to the minus ikx, B star e to the plus ikx. 01:13:49.990 --> 01:13:55.082 Minus star, star, plus. 01:13:55.082 --> 01:13:56.540 So we can run exactly the same game 01:13:56.540 --> 01:13:58.168 but now with this amplitude. 01:14:02.760 --> 01:14:05.680 And when you put the conditions together, 01:14:05.680 --> 01:14:08.850 which you'll do on the problem set, another way to say this 01:14:08.850 --> 01:14:12.220 is the same solution with k to minus k and with A and B 01:14:12.220 --> 01:14:13.780 replaced by their complex conjugates 01:14:13.780 --> 01:14:17.480 and C and D replaced by each other's complex conjugates. 01:14:17.480 --> 01:14:19.810 Then this implies that it must be true 01:14:19.810 --> 01:14:25.840 that A and D, which are now the outgoing guys because we've 01:14:25.840 --> 01:14:29.780 time reversed, is equal to S-- and I'll write this 01:14:29.780 --> 01:14:39.440 out explicitly-- S11, S12, S21, S22, B star, and C star. 01:14:42.530 --> 01:14:44.440 So these together give you that. 01:14:48.950 --> 01:14:53.500 Therefore, S complex conjugate S is equal to 1. 01:14:57.460 --> 01:14:59.780 If S complex conjugate S is equal to 1, 01:14:59.780 --> 01:15:05.160 then S inverse is equal to S transpose, 01:15:05.160 --> 01:15:06.840 just putting this on the right. 01:15:06.840 --> 01:15:12.027 So S transpose is equal to S inverse is equal to S adjoint 01:15:12.027 --> 01:15:13.110 because S is also unitary. 01:15:15.910 --> 01:15:26.960 So time reversal invariance implies, for example, 01:15:26.960 --> 01:15:28.590 that S dagger is equal to S star, 01:15:28.590 --> 01:15:29.673 or S equal to S transpose. 01:15:36.380 --> 01:15:37.880 This is what I wanted to write here. 01:15:37.880 --> 01:15:40.390 Gives us that S is equal to S transpose. 01:15:40.390 --> 01:15:46.110 And in particular, this tells us that S21 is equal to S12. 01:15:46.110 --> 01:15:48.950 The off diagonal terms are equal, 01:15:48.950 --> 01:15:52.650 not just in magnitude, which was insured by unitary, 01:15:52.650 --> 01:15:55.457 but if, in addition to being a unitary system, 01:15:55.457 --> 01:15:57.540 which of course, it should be, if in addition it's 01:15:57.540 --> 01:16:00.580 time reversal invariant, then we see that the off diagonal terms 01:16:00.580 --> 01:16:03.104 are equal not just in magnitude but also in phase. 01:16:03.104 --> 01:16:04.770 And as you know, the phase is important. 01:16:04.770 --> 01:16:06.480 The phase contains physics. 01:16:06.480 --> 01:16:09.930 It tells you about time delays and shifts in the scattering 01:16:09.930 --> 01:16:10.576 process. 01:16:10.576 --> 01:16:11.700 So the phases are the same. 01:16:11.700 --> 01:16:13.200 That statement is not a trivial one. 01:16:13.200 --> 01:16:16.080 It contains physics. 01:16:16.080 --> 01:16:18.960 So when the system is time reversal invariant, the phases 01:16:18.960 --> 01:16:20.890 as well as the amplitudes are the same. 01:16:20.890 --> 01:16:23.120 And you'll derive a series of related conditions 01:16:23.120 --> 01:16:27.482 or consequences for the S matrix from various properties 01:16:27.482 --> 01:16:28.940 of the system, for example, parity, 01:16:28.940 --> 01:16:31.670 if you could have a symmetric potential. 01:16:31.670 --> 01:16:33.375 But now in the last few minutes, I just 01:16:33.375 --> 01:16:35.920 want to tell you a really lovely thing. 01:16:35.920 --> 01:16:37.730 So it should be pretty clear at this point 01:16:37.730 --> 01:16:40.170 that all the information about scattering 01:16:40.170 --> 01:16:42.806 is contained in the S matrix and its dependence on the energy. 01:16:42.806 --> 01:16:44.680 If you know what the incident amplitudes are, 01:16:44.680 --> 01:16:47.236 you know what the outgoing amplitudes are, 01:16:47.236 --> 01:16:49.110 and that's cool because you can measure this. 01:16:49.110 --> 01:16:51.191 You can take a potential, you can literally just 01:16:51.191 --> 01:16:52.690 send in a beam of particles, and you 01:16:52.690 --> 01:16:54.970 can ask, how likely are they to get out. 01:16:54.970 --> 01:16:59.180 And more importantly, if I build a wave packet, on average, 01:16:59.180 --> 01:17:02.049 what's the time delay or acceleration 01:17:02.049 --> 01:17:03.340 of the transmitted wave packet? 01:17:03.340 --> 01:17:04.890 And that way, I can measure the phase as well. 01:17:04.890 --> 01:17:06.630 I can measure both the transmission probabilities 01:17:06.630 --> 01:17:09.110 and the phases, or at least the gradient of the phase 01:17:09.110 --> 01:17:10.800 with energy. 01:17:10.800 --> 01:17:11.300 Go ahead. 01:17:11.300 --> 01:17:12.883 AUDIENCE: Is there a special condition 01:17:12.883 --> 01:17:16.330 that we can [? pose ?] to see the resonance? 01:17:16.330 --> 01:17:17.580 PROFESSOR: Excellent question. 01:17:17.580 --> 01:17:20.569 Hold onto your question for a second. 01:17:20.569 --> 01:17:22.860 There's an enormous amount of the physics of scattering 01:17:22.860 --> 01:17:25.529 contained in the S matrix, and you can measure the S matrix, 01:17:25.529 --> 01:17:27.570 and you can measure its dependence on the energy. 01:17:27.570 --> 01:17:31.070 You can measure the coefficients S12 and S22, their phases 01:17:31.070 --> 01:17:33.780 and their amplitudes, as a function of energy, 01:17:33.780 --> 01:17:35.970 and you can plot them. 01:17:35.970 --> 01:17:37.660 Here's what I want to convince you of. 01:17:37.660 --> 01:17:41.480 If you plot those and look at how the functions behave 01:17:41.480 --> 01:17:45.560 as functions of energy and ask, how do those functions extend 01:17:45.560 --> 01:17:48.680 to negative energy by just drawing the line, 01:17:48.680 --> 01:17:51.260 continuing the lines, you can derive the energy 01:17:51.260 --> 01:17:55.320 of any bound states in the system, too. 01:17:55.320 --> 01:17:59.950 Knowledge of the scattering is enough to determine the bound 01:17:59.950 --> 01:18:02.952 state energies of a system, and let me show you that. 01:18:02.952 --> 01:18:05.410 And this is one of the coolest things in quantum mechanics. 01:18:05.410 --> 01:18:07.760 Here's how this works. 01:18:07.760 --> 01:18:09.740 We have, from the definition of the S matrix, 01:18:09.740 --> 01:18:13.930 that BC is equal to the S matrix on AD, 01:18:13.930 --> 01:18:15.790 where the wave function-- let me just 01:18:15.790 --> 01:18:20.930 put this back in the original form-- is CD, AB, e to the ikx, 01:18:20.930 --> 01:18:26.812 e to the minus ikx, and e to the plus ikx, e to the minus ikx. 01:18:26.812 --> 01:18:28.520 So that's the definition of the S matrix. 01:18:28.520 --> 01:18:31.730 The S matrix, at a given energy e, 01:18:31.730 --> 01:18:34.060 is a coefficient relation matrix between the ingoing 01:18:34.060 --> 01:18:36.820 and outgoing, or, more to the point, A and D. 01:18:36.820 --> 01:18:39.850 And in all of this, I've assumed that the energy was positive, 01:18:39.850 --> 01:18:42.840 that the k1 and k2 are positive and real. 01:18:42.840 --> 01:18:44.340 But now let's ask the question, what 01:18:44.340 --> 01:18:46.215 would have happened if, in the whole process, 01:18:46.215 --> 01:18:53.490 I had taken the energy less than 0? 01:18:53.490 --> 01:18:57.280 If the energy were less than 0, instead of k, 01:18:57.280 --> 01:18:59.490 k would be replaced by i alpha. 01:18:59.490 --> 01:19:01.240 Let's think about what that does. 01:19:01.240 --> 01:19:07.540 If k is i alpha, this is e to the ik is minus alpha 01:19:07.540 --> 01:19:11.350 and minus ik is plus alpha. 01:19:11.350 --> 01:19:15.890 Similarly, ik times i, that gives me a minus alpha 01:19:15.890 --> 01:19:18.730 and this gives me a plus alpha. 01:19:18.730 --> 01:19:19.770 Yeah? 01:19:19.770 --> 01:19:21.800 So as equations, they're the same equations 01:19:21.800 --> 01:19:26.800 with k replaced by i alpha. 01:19:26.800 --> 01:19:30.030 And now what must be true for these states 01:19:30.030 --> 01:19:32.870 to be normalizable? 01:19:32.870 --> 01:19:34.890 What must be true, for example, of A? 01:19:38.070 --> 01:19:40.770 A must be 0 because at minus infinity, there's divergence. 01:19:40.770 --> 01:19:42.416 Not normalizable. 01:19:42.416 --> 01:19:43.790 So in order to be at bound state, 01:19:43.790 --> 01:19:47.160 in order to have a physical state, A must be 0. 01:19:47.160 --> 01:19:49.416 What about D? 01:19:49.416 --> 01:19:50.220 Same reason. 01:19:50.220 --> 01:19:51.930 It's got to be 0 at positive infinity. 01:19:51.930 --> 01:19:54.470 These guys are convergent, so C and B can be non-zero. 01:19:54.470 --> 01:19:57.800 So now here's my question. 01:19:57.800 --> 01:19:59.760 We know that these relations must be true 01:19:59.760 --> 01:20:04.690 because all these relations are encoding is how a solution here 01:20:04.690 --> 01:20:07.150 matches to a solution here through a potential 01:20:07.150 --> 01:20:10.140 in between with continuity of the derivative and anything 01:20:10.140 --> 01:20:11.900 else that's true of that potential inside. 01:20:11.900 --> 01:20:13.810 All that S is doing from that point of view 01:20:13.810 --> 01:20:15.610 is telling me how these coefficients 01:20:15.610 --> 01:20:17.150 match onto these coefficients. 01:20:17.150 --> 01:20:18.480 Yes? 01:20:18.480 --> 01:20:21.960 Now what, for a bound state-- if we have e less than 0, 01:20:21.960 --> 01:20:23.510 what must be true? 01:20:23.510 --> 01:20:30.200 It must be true that AD is equal to 0, and in particular, 00. 01:20:30.200 --> 01:20:31.310 So what are B and C? 01:20:34.090 --> 01:20:37.000 Well, a matrix times 0 is equal to-- 01:20:37.000 --> 01:20:37.790 AUDIENCE: 0. 01:20:37.790 --> 01:20:40.508 PROFESSOR: Unless? 01:20:40.508 --> 01:20:41.867 AUDIENCE: [INAUDIBLE]. 01:20:41.867 --> 01:20:45.170 PROFESSOR: Unless the matrix itself is diverging, 01:20:45.170 --> 01:20:47.300 and then you have to be more careful, but let's 01:20:47.300 --> 01:20:49.340 be naive for the moment. 01:20:49.340 --> 01:20:56.720 If A is 00, then in order for B and C to be non-zero, 01:20:56.720 --> 01:20:57.650 S must have a pole. 01:21:01.580 --> 01:21:05.120 S must go like 1 over 0. 01:21:05.120 --> 01:21:08.560 S must diverge at some special value of the energy. 01:21:08.560 --> 01:21:09.389 Well, that's easy. 01:21:09.389 --> 01:21:11.930 That tells you that if you look at any particular coefficient 01:21:11.930 --> 01:21:15.440 in S, any of the matrix elements of S, 01:21:15.440 --> 01:21:17.940 the numerator can you whatever you want, some finite number, 01:21:17.940 --> 01:21:19.740 but the denominator had better be? 01:21:19.740 --> 01:21:20.600 AUDIENCE: 0. 01:21:20.600 --> 01:21:21.590 PROFESSOR: 0. 01:21:21.590 --> 01:21:23.830 So let's look at the denominator. 01:21:23.830 --> 01:21:30.466 If I compute S21-- actually, let me do this over here. 01:21:30.466 --> 01:21:32.890 No, it's all filled. 01:21:32.890 --> 01:21:35.280 Let's do it over here. 01:21:35.280 --> 01:21:39.060 If I look at S21 for the potential well, 01:21:39.060 --> 01:21:40.530 scattering off the potential well 01:21:40.530 --> 01:21:44.610 that we looked at at the beginning of today's lecture, 01:21:44.610 --> 01:21:46.840 this guy, and now I'm going to look 01:21:46.840 --> 01:21:51.420 at S21, one of the coefficients of this guy, 01:21:51.420 --> 01:21:55.990 also known as the scattering amplitude t for the well. 01:21:58.870 --> 01:22:05.020 This is equal to-- and it's a godawful expression-- 2 k1 k2 01:22:05.020 --> 01:22:12.210 e to the i k2 L over 2 k1 k2 cos of k 01:22:12.210 --> 01:22:18.920 prime L minus i k squared plus k1 squared plus k2 squared 01:22:18.920 --> 01:22:24.910 times sine of k2 L. This is some horrible thing. 01:22:24.910 --> 01:22:30.220 But now I ask the condition, when does this have a pole? 01:22:30.220 --> 01:22:32.810 When the energy has continued to be negative, 01:22:32.810 --> 01:22:34.650 for what values does this have a pole 01:22:34.650 --> 01:22:36.410 or does the denominator have a 0? 01:22:36.410 --> 01:22:38.170 And the answer is if you take this 01:22:38.170 --> 01:22:39.780 and you massage the equation, this 01:22:39.780 --> 01:22:43.753 is equal to 0 a little bit, you get the following expression, 01:22:43.753 --> 01:22:53.870 k2 L upon 2 tangent of k2 L upon 2 is equal to k1 L upon 2. 01:22:53.870 --> 01:22:56.310 This is the condition for the bound state 01:22:56.310 --> 01:22:58.980 energies of the square well, and we 01:22:58.980 --> 01:23:03.080 computed it using knowledge only of the scattering states. 01:23:03.080 --> 01:23:05.320 If you took particles and a square well 01:23:05.320 --> 01:23:08.180 and roll the particles across the square well potential 01:23:08.180 --> 01:23:10.857 and measure it as a function of energy, the scattering 01:23:10.857 --> 01:23:12.440 amplitude, the transmission amplitude, 01:23:12.440 --> 01:23:15.720 and in particular S21, an element of the S matrix, 01:23:15.720 --> 01:23:17.910 and you plotted it as a function of energy, 01:23:17.910 --> 01:23:23.860 and then you approximated that by a function of energy that 01:23:23.860 --> 01:23:27.196 satisfies the basic properties of unitarity, what you would 01:23:27.196 --> 01:23:28.570 find is that when you then extend 01:23:28.570 --> 01:23:31.276 that function in mathematica to minus 01:23:31.276 --> 01:23:32.650 a particular value of the energy, 01:23:32.650 --> 01:23:36.000 the denominator diverges at that energy. 01:23:36.000 --> 01:23:38.912 You know that there will be a bound state. 01:23:38.912 --> 01:23:40.620 And so from scattering, you've determined 01:23:40.620 --> 01:23:42.236 the existence of a bound state. 01:23:42.236 --> 01:23:44.610 This is how we find an awful lot of the particles that we 01:23:44.610 --> 01:23:49.050 actually deduce must exist in the real world. 01:23:49.050 --> 01:23:52.050 We'll pick up next time. 01:23:52.050 --> 01:23:53.600 [APPLAUSE]