WEBVTT 00:00:00.090 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:04.030 Commons license. 00:00:04.030 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.690 continue to offer high quality, educational resources for free. 00:00:10.690 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.250 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.250 --> 00:00:18.210 at ocw.mit.edu. 00:00:21.250 --> 00:00:25.120 ROBERT FIELD: Last time, I talked a lot 00:00:25.120 --> 00:00:30.970 about the semiclassical method, where we generalize 00:00:30.970 --> 00:00:35.830 on this wonderful relationship to say, well, 00:00:35.830 --> 00:00:41.180 if the potential is not constant, then we can say, 00:00:41.180 --> 00:00:45.550 well, the wavelength changes with position. 00:00:45.550 --> 00:00:50.050 And we can say that the momentum changes with position. 00:00:50.050 --> 00:00:51.970 But we're using this as the guide. 00:00:51.970 --> 00:00:54.550 And so the basis is really just saying, 00:00:54.550 --> 00:00:57.730 OK, we're going to take this kind of a relationship 00:00:57.730 --> 00:00:59.200 seriously. 00:00:59.200 --> 00:01:02.066 Because v is not constant. 00:01:02.066 --> 00:01:05.470 We have V of x. 00:01:05.470 --> 00:01:12.790 And we also know that the kinetic energy, 00:01:12.790 --> 00:01:18.180 which is an operator, is p squared over 2m. 00:01:18.180 --> 00:01:24.780 And the energy minus the potential 00:01:24.780 --> 00:01:27.940 is the kinetic energy. 00:01:27.940 --> 00:01:38.400 And so we can use this to get a classical, mechanical function 00:01:38.400 --> 00:01:45.520 that this p of x is going to be 2m E minus V of x. 00:01:48.515 --> 00:01:50.640 So why are we doing all this when we can just solve 00:01:50.640 --> 00:01:52.840 the differential equations? 00:01:52.840 --> 00:01:55.560 And the answer is we want insight. 00:01:55.560 --> 00:01:59.130 And we want to build our insight on what we know. 00:01:59.130 --> 00:02:02.830 And so we have this momentum function, 00:02:02.830 --> 00:02:05.910 which gives us a wavelength function, which tells us 00:02:05.910 --> 00:02:08.690 how far apart the waves are. 00:02:08.690 --> 00:02:10.699 But we also have another thing, which 00:02:10.699 --> 00:02:14.000 was demonstrated by my running across the room, 00:02:14.000 --> 00:02:18.020 and that the momentum is related to the velocity, which 00:02:18.020 --> 00:02:20.690 is related to the probability of finding the system 00:02:20.690 --> 00:02:22.400 at a particular place. 00:02:22.400 --> 00:02:29.010 And so we have, if we commit the travesty of saying, OK, 00:02:29.010 --> 00:02:32.870 we have a classical function-- it's not the momentum-- 00:02:32.870 --> 00:02:37.450 but it's going to somehow encode the classical behavior, 00:02:37.450 --> 00:02:42.130 we can determine without solving any differential equation what 00:02:42.130 --> 00:02:46.840 the spacing between nodes in the exact wave function is, 00:02:46.840 --> 00:02:50.320 and the amplitude in that region. 00:02:50.320 --> 00:02:54.220 And that's a lot because then we can 00:02:54.220 --> 00:02:59.840 use our knowledge of classical mechanics to say, 00:02:59.840 --> 00:03:03.200 oh, this is what we expect in quantum mechanics. 00:03:03.200 --> 00:03:05.000 And that's very powerful. 00:03:05.000 --> 00:03:10.090 And I keep stressing that you want to draw cartoons. 00:03:10.090 --> 00:03:15.480 And this is the way you get into those cartoons. 00:03:15.480 --> 00:03:23.190 OK, now, for the free particle, we 00:03:23.190 --> 00:03:28.860 have these kinds of wave functions, e to the ikx. 00:03:28.860 --> 00:03:33.210 And they're really great because often, we 00:03:33.210 --> 00:03:40.250 want to evaluate things like the integral of e to the ikx times 00:03:40.250 --> 00:03:43.490 e to the minus ikx dx. 00:03:43.490 --> 00:03:47.300 Well, that's one. 00:03:47.300 --> 00:03:50.330 And so instead of remembering that we 00:03:50.330 --> 00:03:55.460 have to evaluate trigonometric integrals, sine, sine theta, 00:03:55.460 --> 00:03:57.440 cosine, we can do this. 00:03:57.440 --> 00:04:00.440 And it really simplifies life. 00:04:00.440 --> 00:04:08.240 Now, using that insight, I'm asking you, OK, 00:04:08.240 --> 00:04:12.200 the expectation value for the momentum could be-- 00:04:22.540 --> 00:04:27.640 OK, if this is the expectation value for the momentum, what 00:04:27.640 --> 00:04:29.950 is psi of x? 00:04:33.400 --> 00:04:36.100 I promised I would ask you this question. 00:04:36.100 --> 00:04:38.630 I don't know if anybody really thought about it. 00:04:38.630 --> 00:04:44.880 But first of all, we're talking about the free particle. 00:04:44.880 --> 00:04:48.500 And this is some sort of eigenfunction 00:04:48.500 --> 00:04:51.540 of the Hamiltonian for the free particle. 00:04:51.540 --> 00:04:55.305 So what can we say about k? 00:05:00.600 --> 00:05:02.940 We have two parts to the wave function. 00:05:02.940 --> 00:05:04.580 We have an e to the ikx. 00:05:04.580 --> 00:05:06.990 And we have a e to the minus ikx. 00:05:06.990 --> 00:05:10.470 And the k is the same for both of them. 00:05:10.470 --> 00:05:14.840 So with all those hints, what is this? 00:05:14.840 --> 00:05:19.130 We have e to the ikx times something. 00:05:19.130 --> 00:05:25.354 And we have plus e to the minus ikx. 00:05:25.354 --> 00:05:27.020 So what are the somethings that go here? 00:05:34.291 --> 00:05:34.790 Yes? 00:05:34.790 --> 00:05:35.540 AUDIENCE: A and B? 00:05:35.540 --> 00:05:37.490 ROBERT FIELD: Right. 00:05:37.490 --> 00:05:42.080 And that didn't involve very much mental gymnastics 00:05:42.080 --> 00:05:44.960 if you really have done a little bit of practicing 00:05:44.960 --> 00:05:47.850 of integrals involving these sorts of things. 00:05:47.850 --> 00:05:49.110 Now there's two things. 00:05:49.110 --> 00:05:53.900 One is when you have the same k in e to the ikx 00:05:53.900 --> 00:05:55.340 and e to the minus ikx. 00:05:55.340 --> 00:05:58.150 The product is one. 00:05:58.150 --> 00:06:03.000 If you have different k's and you're 00:06:03.000 --> 00:06:08.970 integrating over all space, or over some cleverly chosen 00:06:08.970 --> 00:06:11.165 region, that integral is zero. 00:06:14.080 --> 00:06:17.830 Because these are eigenfunctions of-- 00:06:17.830 --> 00:06:20.050 these are different eigenfunctions 00:06:20.050 --> 00:06:24.200 of an operator belonging to different eigenvalues. 00:06:24.200 --> 00:06:27.110 And you always can count on those things being zero. 00:06:27.110 --> 00:06:31.270 Now, quantum mechanics is full of integrals. 00:06:31.270 --> 00:06:34.410 Basically, there's an infinite number of them. 00:06:34.410 --> 00:06:37.140 And most of them are zero. 00:06:37.140 --> 00:06:39.540 And you want to be able to look at an integral 00:06:39.540 --> 00:06:42.035 and say, oh, I don't need to evaluate that. 00:06:42.035 --> 00:06:44.160 And often, you want to look at an integral and say, 00:06:44.160 --> 00:06:46.140 I do know how to evaluate that. 00:06:46.140 --> 00:06:49.900 And I know an infinite number of those like that. 00:06:49.900 --> 00:06:53.460 And all of a sudden, it starts to be transparent again. 00:06:53.460 --> 00:06:59.250 Because the barrier between insight and quantum 00:06:59.250 --> 00:07:03.110 mechanics is usually a whole bunch of integrals. 00:07:03.110 --> 00:07:05.280 And they're all yours. 00:07:05.280 --> 00:07:12.000 And so we like problems where the wave functions 00:07:12.000 --> 00:07:14.260 have simple forms. 00:07:14.260 --> 00:07:16.710 And this is true for free particle. 00:07:16.710 --> 00:07:18.455 It's true for the particle in a box. 00:07:21.114 --> 00:07:22.530 We're going to start talking about 00:07:22.530 --> 00:07:24.880 their harmonic oscillator. 00:07:24.880 --> 00:07:30.200 And it seems like those integrals are not simple, 00:07:30.200 --> 00:07:31.580 but they are. 00:07:31.580 --> 00:07:33.690 I have to teach you why they're not simple. 00:07:33.690 --> 00:07:39.740 So today, we're starting on the classical mechanical treatment 00:07:39.740 --> 00:07:41.570 of the harmonic oscillator. 00:07:41.570 --> 00:07:46.020 Then we'll do the traditional quantum mechanical treatment. 00:07:46.020 --> 00:07:49.430 And then, we'll come back and use these creation 00:07:49.430 --> 00:07:51.600 and annililation operators, which 00:07:51.600 --> 00:07:54.270 are the magic decoders for essentially evaluating 00:07:54.270 --> 00:07:57.940 all the integrals trivially. 00:07:57.940 --> 00:08:00.640 And then, with all that in hand, we're 00:08:00.640 --> 00:08:03.370 going to make our first step into time-dependent quantum 00:08:03.370 --> 00:08:05.420 mechanics. 00:08:05.420 --> 00:08:07.400 And we're going to use time-dependent quantum 00:08:07.400 --> 00:08:08.280 mechanics. 00:08:08.280 --> 00:08:11.860 Well, we're going to use our facility with integrals 00:08:11.860 --> 00:08:15.700 to describe the properties of some particle-like state we 00:08:15.700 --> 00:08:17.350 can construct. 00:08:17.350 --> 00:08:21.160 And these constructions are really simple 00:08:21.160 --> 00:08:24.970 for the particle in a box or the harmonic oscillator, 00:08:24.970 --> 00:08:28.020 depending on which properties you want. 00:08:28.020 --> 00:08:33.450 So your ability to draw cartoons and to use classical insights 00:08:33.450 --> 00:08:37.620 for the particle in a box and the harmonic oscillator 00:08:37.620 --> 00:08:41.100 will be incredibly valuable once we take our first step 00:08:41.100 --> 00:08:43.799 into the reality of time-dependent quantum 00:08:43.799 --> 00:08:46.270 mechanics. 00:08:46.270 --> 00:08:50.380 Now, one of the things that's going to happen 00:08:50.380 --> 00:08:56.260 is that we're going to describe real situations, 00:08:56.260 --> 00:08:59.580 real situations that are not one of the standard solved 00:08:59.580 --> 00:09:00.880 problems. 00:09:00.880 --> 00:09:09.440 The standard solved problems are the particle in a box, 00:09:09.440 --> 00:09:13.970 or a particle in the infinite box, the harmonic oscillator, 00:09:13.970 --> 00:09:19.970 the hydrogen atom, and the rigid rotor. 00:09:19.970 --> 00:09:22.520 And the particle in a box-- 00:09:22.520 --> 00:09:25.770 so the potential for each of these is different. 00:09:25.770 --> 00:09:27.220 It looks like this. 00:09:27.220 --> 00:09:28.860 It looks like this. 00:09:28.860 --> 00:09:31.110 It looks like this. 00:09:31.110 --> 00:09:34.520 And it looks like that. 00:09:34.520 --> 00:09:38.230 So there is no stretching in a rigid rotor. 00:09:38.230 --> 00:09:42.250 And so all of the complexity is in the kinetic energy, not 00:09:42.250 --> 00:09:43.240 the potential. 00:09:43.240 --> 00:09:47.350 But each of these has a different potential energy. 00:09:47.350 --> 00:09:51.486 And the energy levels for the particle in a box 00:09:51.486 --> 00:09:52.735 are proportional to n squared. 00:09:57.390 --> 00:09:59.170 For the harmonic oscillator, they're 00:09:59.170 --> 00:10:03.280 proportional to n plus 1/2. 00:10:03.280 --> 00:10:06.310 For the hydrogen atom, they are proportional to 1 00:10:06.310 --> 00:10:09.470 over n squared. 00:10:09.470 --> 00:10:12.650 And for the rigid rotor, they're proportional to n times n 00:10:12.650 --> 00:10:14.390 plus 1. 00:10:14.390 --> 00:10:16.640 So one of the valuable things you 00:10:16.640 --> 00:10:21.020 get from looking at these exactly soluble problems 00:10:21.020 --> 00:10:24.350 is that the energy level patterns for each of them 00:10:24.350 --> 00:10:26.710 are slightly different. 00:10:26.710 --> 00:10:30.160 And you can tell what you've got from the energy level pattern. 00:10:30.160 --> 00:10:33.210 And so when you take a spectrum, so often you want to know 00:10:33.210 --> 00:10:34.460 what kind of spectrum is this? 00:10:34.460 --> 00:10:38.170 Sometimes you can tell just by what frequency region it is. 00:10:38.170 --> 00:10:40.210 But usually, in a spectrum, there's 00:10:40.210 --> 00:10:42.040 a pattern of energy levels. 00:10:42.040 --> 00:10:47.140 And that gets you focused on well, what kind of problem-- 00:10:47.140 --> 00:10:51.339 what is the nature of the building 00:10:51.339 --> 00:10:52.630 blocks that we're going to use? 00:10:56.660 --> 00:10:58.420 So this is wonderful. 00:10:58.420 --> 00:11:03.040 Now, we're also going to find that when 00:11:03.040 --> 00:11:05.950 we solve these problems in quantum mechanics, 00:11:05.950 --> 00:11:11.680 we get an infinite number of eigenfunctions and eigenvalues. 00:11:11.680 --> 00:11:16.450 And often, we get presented to us a lot of integrals 00:11:16.450 --> 00:11:23.230 involving the operators between different wave functions. 00:11:23.230 --> 00:11:24.790 And one of the beautiful things is 00:11:24.790 --> 00:11:27.670 when the theory gives you an infinite number 00:11:27.670 --> 00:11:29.230 of those integrals. 00:11:29.230 --> 00:11:31.750 So now we would have a collection 00:11:31.750 --> 00:11:33.340 of all sorts of integrals that are 00:11:33.340 --> 00:11:39.360 evaluated for us in a simple dependence on quantum numbers. 00:11:39.360 --> 00:11:41.350 So what do we do with them? 00:11:41.350 --> 00:11:44.780 Well, there's two main things we do with these infinite numbers 00:11:44.780 --> 00:11:46.310 of integrals. 00:11:46.310 --> 00:11:50.360 One is we say, well, this problem is not 00:11:50.360 --> 00:11:51.800 the standard problem. 00:11:51.800 --> 00:11:53.150 There are defects. 00:11:53.150 --> 00:11:56.230 Instead of having the potential for the harmonic oscillator 00:11:56.230 --> 00:11:59.450 that looked like this, it might look like that. 00:11:59.450 --> 00:12:02.950 There might be some anharmonicity. 00:12:02.950 --> 00:12:07.560 Or there can be all sorts of defects. 00:12:07.560 --> 00:12:10.160 And these defects can be expressed by integrals 00:12:10.160 --> 00:12:12.310 of some other operator. 00:12:12.310 --> 00:12:15.530 And that goes into perturbation theory. 00:12:15.530 --> 00:12:18.470 Or if we're going to want to look at wavepackets, 00:12:18.470 --> 00:12:20.810 creating a particle-like state and asking 00:12:20.810 --> 00:12:26.310 how it propagates in time, those integrals are useful. 00:12:26.310 --> 00:12:29.180 So there's all sorts of fantastic stuff, 00:12:29.180 --> 00:12:32.690 once we kill the standard problems. 00:12:32.690 --> 00:12:36.350 It's not just some thing you have to study, 00:12:36.350 --> 00:12:38.250 and it's over with. 00:12:38.250 --> 00:12:39.620 You're going to use this. 00:12:39.620 --> 00:12:42.440 As long as you're involved in physical chemistry, 00:12:42.440 --> 00:12:45.170 you're going to be using perturbation theory 00:12:45.170 --> 00:12:50.130 to understand what's going on beyond the simple stuff. 00:12:50.130 --> 00:12:56.595 OK, so let's get started. 00:12:59.480 --> 00:13:02.680 So why is the harmonic oscillator so special? 00:13:02.680 --> 00:13:06.910 If we look at any potential energy curve, any one 00:13:06.910 --> 00:13:11.200 dimensional problem, it's typically 00:13:11.200 --> 00:13:13.960 harmonic at the bottom, no matter what it does. 00:13:13.960 --> 00:13:18.330 I mean, this is a typical molecular diatomic molecule 00:13:18.330 --> 00:13:19.420 potential. 00:13:19.420 --> 00:13:22.000 We have a hard inner wall. 00:13:22.000 --> 00:13:25.570 And we have bond breaking at the outer wall. 00:13:25.570 --> 00:13:28.040 And so this is an anharmonic potential. 00:13:28.040 --> 00:13:33.120 But the bottom part is harmonic. 00:13:33.120 --> 00:13:37.800 And so we can use everything we learn 00:13:37.800 --> 00:13:39.660 about the harmonic oscillator to begin 00:13:39.660 --> 00:13:44.780 to draw a picture of arbitrary potentials. 00:13:44.780 --> 00:13:54.695 So this is hard wall bond breaking. 00:13:57.530 --> 00:13:59.300 Now, we're mostly chemists. 00:13:59.300 --> 00:14:02.540 And breaking a bond is-- 00:14:02.540 --> 00:14:04.440 how much energy does it take to break a bond? 00:14:04.440 --> 00:14:06.023 Well, is that encoded in the spectrum? 00:14:06.023 --> 00:14:07.650 Yeah. 00:14:07.650 --> 00:14:10.860 So this is the sort of thing we would care about. 00:14:10.860 --> 00:14:18.580 And now let me put some notation here. 00:14:18.580 --> 00:14:21.360 So this is the potential. 00:14:21.360 --> 00:14:26.440 And this is the potential of-- the internuclear resistance, 00:14:26.440 --> 00:14:30.000 which is traditionally called R. We're going to switch notations 00:14:30.000 --> 00:14:32.160 really quickly from R to x. 00:14:32.160 --> 00:14:36.460 But this, the minimum of the potential, is the equilibrium 00:14:36.460 --> 00:14:38.834 internuclear distance R sub e. 00:14:42.810 --> 00:14:52.680 So for R near R sub e the potential V of R 00:14:52.680 --> 00:14:59.730 looks simply like k over 2, a forced constant, R minus R sub 00:14:59.730 --> 00:15:00.345 e squared. 00:15:05.980 --> 00:15:09.580 So this is harmonic oscillator. 00:15:09.580 --> 00:15:12.920 And now we're going to change notation a little bit. 00:15:12.920 --> 00:15:18.010 We're going to use lowercase x to be R minus Re. 00:15:18.010 --> 00:15:21.040 In other words, it's the distortion away 00:15:21.040 --> 00:15:23.550 from equilibrium. 00:15:23.550 --> 00:15:28.830 And we can take any potential and write it as a power series. 00:15:51.262 --> 00:15:51.762 Sorry. 00:16:00.670 --> 00:16:01.290 And so on. 00:16:04.320 --> 00:16:10.035 So if we know these derivatives, we know what to do with them. 00:16:10.035 --> 00:16:10.535 OK. 00:16:20.900 --> 00:16:27.870 So one standard potential is called the Morse potential. 00:16:27.870 --> 00:16:29.710 Because it looks like this. 00:16:29.710 --> 00:16:33.530 It looks like what you need for a diatomic molecule. 00:16:33.530 --> 00:16:36.200 And the Morse potential has an analytic form, 00:16:36.200 --> 00:16:41.400 where the potential v of x is equal to D sub 00:16:41.400 --> 00:16:46.185 e, the dissociation energy, 1 minus e to the minus ax. 00:16:49.050 --> 00:16:52.640 Well, that doesn't look like polynomial of x. 00:16:52.640 --> 00:16:56.930 But we power series expand this, and we get a polynomial x. 00:16:56.930 --> 00:16:59.610 OK, now, what is this D sub e? 00:16:59.610 --> 00:17:03.890 So if x is equal to 0-- 00:17:03.890 --> 00:17:05.810 this is 1 0. 00:17:05.810 --> 00:17:08.490 And so we get the energy. 00:17:08.490 --> 00:17:12.859 The potential at x equals 0 is 0. 00:17:12.859 --> 00:17:17.359 And if x is equal to infinity, v of x 00:17:17.359 --> 00:17:21.630 equals D sub e, so basically, this 00:17:21.630 --> 00:17:26.099 is the energy between the bottom of the well 00:17:26.099 --> 00:17:27.490 and where it breaks. 00:17:27.490 --> 00:17:30.050 This is D sub e. 00:17:30.050 --> 00:17:30.550 OK? 00:17:30.550 --> 00:17:34.890 So this is well interpreted. 00:17:34.890 --> 00:17:36.630 All the rest of the action is in here. 00:17:36.630 --> 00:17:41.300 AUDIENCE: [INAUDIBLE] 00:17:41.300 --> 00:17:42.300 ROBERT FIELD: I'm sorry? 00:17:42.300 --> 00:17:44.170 AUDIENCE: Do you just square the x? 00:17:44.170 --> 00:17:46.400 ROBERT FIELD: Yep! 00:17:46.400 --> 00:17:47.000 Thank you. 00:17:50.150 --> 00:17:51.580 It's an innocent factor. 00:17:51.580 --> 00:17:54.010 But it turns out to be very important. 00:17:54.010 --> 00:17:57.300 OK, so let's square this. 00:17:57.300 --> 00:18:02.740 We have v of x is equal to De times 1 00:18:02.740 --> 00:18:12.245 minus 2e to the minus ax plus e to the minus 2ax. 00:18:16.190 --> 00:18:19.640 OK, so we're going to do power series expansions of these. 00:18:19.640 --> 00:18:21.910 And you can do that. 00:18:21.910 --> 00:18:32.950 And so we have plus some function 00:18:32.950 --> 00:18:43.660 of x plus some function of x squared, et cetera. 00:18:43.660 --> 00:18:47.270 Now, what is the function of x? 00:18:47.270 --> 00:18:53.320 We're taking our derivatives at the equilibrium, at x equals 0. 00:18:53.320 --> 00:18:58.110 And we have a potential, which has a minimum. 00:18:58.110 --> 00:19:02.050 So what's the derivative of the potential at equilibrium? 00:19:02.050 --> 00:19:02.550 Yes? 00:19:02.550 --> 00:19:03.150 AUDIENCE: Zero. 00:19:03.150 --> 00:19:03.983 ROBERT FIELD: Right. 00:19:03.983 --> 00:19:06.650 And so the x term goes away. 00:19:06.650 --> 00:19:08.730 And so we have a constant term, which we usually 00:19:08.730 --> 00:19:11.630 choose to be zero. 00:19:11.630 --> 00:19:16.420 And we have this quadratic term. 00:19:16.420 --> 00:19:18.440 Looks like a harmonic oscillator. 00:19:18.440 --> 00:19:19.960 And then there are other terms which 00:19:19.960 --> 00:19:24.310 express the personality of the potential. 00:19:24.310 --> 00:19:25.240 This is universal. 00:19:25.240 --> 00:19:30.290 And the rest becomes a special case. 00:19:30.290 --> 00:19:39.850 So we have v of x is De a squared 00:19:39.850 --> 00:19:44.770 x squared plus other stuff. 00:19:44.770 --> 00:19:46.530 OK, we're going to call this 1/2k. 00:19:49.350 --> 00:19:49.850 why? 00:19:49.850 --> 00:19:53.210 Because we'd like it to look like a harmonic oscillator. 00:19:53.210 --> 00:19:55.430 I mean, we know that the potential 00:19:55.430 --> 00:19:59.930 for a harmonic oscillator is described in this way. 00:20:02.810 --> 00:20:07.510 So let's just draw a picture. 00:20:15.840 --> 00:20:19.740 So we have a spring and a mass. 00:20:19.740 --> 00:20:24.960 And I should have drawn this up a little higher. 00:20:24.960 --> 00:20:29.340 OK, so at equilibrium there is no force. 00:20:29.340 --> 00:20:33.850 If the mass is down here, there is a force pulling it up. 00:20:33.850 --> 00:20:36.740 And if it's up here, there's a force pushing it down. 00:20:36.740 --> 00:20:48.390 Hooke's Law says that the force is 00:20:48.390 --> 00:20:56.380 equal to minus k x minus x 0. 00:20:56.380 --> 00:21:01.420 OK, and now we're going to switch to just the lowercase x 00:21:01.420 --> 00:21:02.710 in a second. 00:21:02.710 --> 00:21:09.860 But now, the force according to Newton 00:21:09.860 --> 00:21:17.530 is minus gradient of the potential. 00:21:17.530 --> 00:21:20.970 So the potential for this problem 00:21:20.970 --> 00:21:28.382 is v of little x is equal to 1/2k x squared. 00:21:32.380 --> 00:21:39.310 OK, so we have what we expect for a harmonic oscillator. 00:21:39.310 --> 00:21:44.700 And we're going to say the small displacement part of the Morse 00:21:44.700 --> 00:21:46.950 oscillator looks like 1/2k. 00:21:46.950 --> 00:21:48.910 x squared looks harmonic. 00:21:48.910 --> 00:21:50.550 And so what do we do now? 00:21:54.970 --> 00:22:02.250 Well, one of Newton's laws, force 00:22:02.250 --> 00:22:04.770 is equal to the mass times the acceleration. 00:22:04.770 --> 00:22:14.490 And so we can say, oh, well, the acceleration 00:22:14.490 --> 00:22:22.190 is the second derivative of x with respect to t. 00:22:22.190 --> 00:22:29.850 And the force is a minus gradient of the potential. 00:22:29.850 --> 00:22:32.300 So it's minus 1/2kx squared. 00:22:37.620 --> 00:22:38.370 Minus kx. 00:22:41.680 --> 00:22:45.900 OK, so this is the gradient of the potential. 00:22:45.900 --> 00:22:48.790 This is the mass-- 00:22:48.790 --> 00:22:50.570 well, I need the mass-- 00:22:50.570 --> 00:22:52.116 times the acceleration. 00:22:52.116 --> 00:22:53.990 And so this is the equation we have to solve. 00:23:00.590 --> 00:23:05.680 And so we want to find the solution x of t. 00:23:05.680 --> 00:23:08.080 And this is a second order differential equation 00:23:08.080 --> 00:23:10.760 because we have a second derivative. 00:23:10.760 --> 00:23:12.830 So there's going to be two terms. 00:23:12.830 --> 00:23:21.530 And they'll be-- we'll have some sine and some cosine function. 00:23:21.530 --> 00:23:24.940 And we're going to want a derivative, 00:23:24.940 --> 00:23:30.460 a second derivative, that brings down a constant k. 00:23:30.460 --> 00:23:32.780 And so we know that these are going 00:23:32.780 --> 00:23:45.500 to be things that have the form a sine kx k or m square root 00:23:45.500 --> 00:23:56.840 x plus b cosine k m x. 00:23:56.840 --> 00:23:58.620 So this is the solution. 00:23:58.620 --> 00:24:00.560 And we have to find A and B. 00:24:00.560 --> 00:24:02.870 Now, why do we have k over m? 00:24:02.870 --> 00:24:05.550 Well, you can look at this differential equation. 00:24:05.550 --> 00:24:07.385 And you can see that we have an m here. 00:24:07.385 --> 00:24:08.510 We have a k here. 00:24:08.510 --> 00:24:11.180 And that's what you need in order to solve it. 00:24:11.180 --> 00:24:14.340 So now our job is simply to find A and B. 00:24:14.340 --> 00:24:19.740 AUDIENCE: Should these be functions of t and not x? 00:24:19.740 --> 00:24:21.560 ROBERT FIELD: You know, sometimes 00:24:21.560 --> 00:24:27.480 I go onto automatic pilot because it's so familiar to me. 00:24:27.480 --> 00:24:31.030 I'm just writing what comes up in my subconscious. 00:24:31.030 --> 00:24:34.860 But yes, that's a good point. 00:24:34.860 --> 00:24:37.350 All right, so we want x of t. 00:24:37.350 --> 00:24:40.470 And it is this combination. 00:24:40.470 --> 00:24:43.635 OK, so now we put in some insights. 00:24:52.920 --> 00:24:56.340 We want to know what the period of oscillation is. 00:24:56.340 --> 00:25:04.860 And so x t plus tau has to be equal to x of t. 00:25:07.520 --> 00:25:12.210 And so when we do that, we discover-- 00:25:12.210 --> 00:25:14.700 and I'm just going to skip a lot of steps-- 00:25:14.700 --> 00:25:26.210 that k over m square root times tau has to be equal to 2 pi 00:25:26.210 --> 00:25:27.350 in order to satisfy that. 00:25:35.490 --> 00:25:42.230 We call k over m square root omega, just 00:25:42.230 --> 00:25:47.090 to simplify the equations. 00:25:47.090 --> 00:25:49.610 But we also discover that this actually 00:25:49.610 --> 00:25:51.650 is an angular frequency. 00:25:55.980 --> 00:26:04.390 So if we say omega tau is equal to 2 pi, 00:26:04.390 --> 00:26:09.690 then tau equals 2 pi over omega, which is 00:26:09.690 --> 00:26:13.140 equal to 1 over the frequency. 00:26:13.140 --> 00:26:14.680 Just exactly what we expect. 00:26:19.150 --> 00:26:26.430 So we have the beginning of a solution and omega, tau, 00:26:26.430 --> 00:26:32.190 and frequency make sense. 00:26:35.640 --> 00:26:38.790 Everything is what we sort of expect. 00:26:38.790 --> 00:26:41.460 OK, so now the next step is to determine 00:26:41.460 --> 00:26:43.560 the values of the constants. 00:26:48.950 --> 00:26:52.810 So normally, when we have a differential equation, 00:26:52.810 --> 00:27:01.530 after we find the general form, we apply boundary conditions. 00:27:01.530 --> 00:27:06.550 And so we're going to apply some boundary conditions. 00:27:06.550 --> 00:27:09.130 So here we have the potential. 00:27:09.130 --> 00:27:26.450 And-- so this is the potential as a function of coordinate. 00:27:26.450 --> 00:27:30.740 And this is the turning point, this 00:27:30.740 --> 00:27:33.160 is the inner turning point at energy E. 00:27:33.160 --> 00:27:36.860 This is the outer turning point. 00:27:36.860 --> 00:27:39.380 Well, what's true at the turning point? 00:27:39.380 --> 00:27:42.020 The turning point, the potential is 00:27:42.020 --> 00:27:49.100 equal to the energy at the two turning points. 00:27:49.100 --> 00:27:58.790 So 1/2k x plus or minus of E squared. 00:27:58.790 --> 00:28:02.900 So if we know E, we know where the turning points are. 00:28:02.900 --> 00:28:12.030 And so x of plus minus of E, we can solve this. 00:28:12.030 --> 00:28:19.740 And so we have 2E over k. 00:28:19.740 --> 00:28:20.240 Yeah. 00:28:27.200 --> 00:28:28.602 So this is the turning points. 00:28:28.602 --> 00:28:29.810 These are the turning points. 00:28:32.750 --> 00:28:36.810 Now, this is a fairly frequent exercise in quantum mechanics. 00:28:36.810 --> 00:28:41.180 You're going to want to know where are the turning points. 00:28:41.180 --> 00:28:44.420 Because this is how you impose boundary conditions easily. 00:28:44.420 --> 00:28:47.780 And so knowing that a turning point corresponds 00:28:47.780 --> 00:28:50.570 to where the potential is equal to the total energy 00:28:50.570 --> 00:28:54.820 is enough to be able to solve for this. 00:28:54.820 --> 00:29:07.600 OK, so suppose we start at x equals x plus. 00:29:07.600 --> 00:29:14.620 And so x of 0 is equal to x plus. 00:29:14.620 --> 00:29:19.730 And that determines one of the coefficients. 00:29:19.730 --> 00:29:22.690 So x of 0-- 00:29:22.690 --> 00:29:29.500 so we have at t equals 0 the sine term is 0 00:29:29.500 --> 00:29:32.580 and the cosine term is 1. 00:29:32.580 --> 00:29:39.440 And so the first thing we get at t 00:29:39.440 --> 00:29:48.307 equals 0 is that B is equal to x plus. 00:29:48.307 --> 00:29:48.806 Right? 00:29:52.070 --> 00:29:57.224 The next step is to find a. 00:29:57.224 --> 00:29:58.640 There are several ways to do this. 00:29:58.640 --> 00:30:03.220 But it's useful to draw a little picture. 00:30:03.220 --> 00:30:06.550 So x plus is here. 00:30:06.550 --> 00:30:12.190 And that occurs at t equals 0. 00:30:12.190 --> 00:30:17.880 And x minus occurs at tau over 2. 00:30:17.880 --> 00:30:20.020 And in the middle we have tau over 4. 00:30:22.710 --> 00:30:29.190 So let's ask for, what is the value of the wave function 00:30:29.190 --> 00:30:33.150 when x is equal to 0? 00:30:33.150 --> 00:30:38.010 So how do we make the x be 0 at tau over 4? 00:30:38.010 --> 00:30:45.750 And that determines the value of B. I mean, the value of A. 00:30:45.750 --> 00:30:49.906 So we have everything we need. 00:30:49.906 --> 00:30:54.480 And now, before just rushing on, let me just say, 00:30:54.480 --> 00:31:02.160 well, this just gives A is equal to zero. 00:31:02.160 --> 00:31:05.650 OK, there's a different approach. 00:31:05.650 --> 00:31:10.230 When we have a sine plus a cosine term, 00:31:10.230 --> 00:31:14.100 we can always re-express it as x of t 00:31:14.100 --> 00:31:22.550 is equal to some other constant times sine omega t plus phi. 00:31:28.620 --> 00:31:31.480 And so the same sort of analysis gives 00:31:31.480 --> 00:31:39.330 c is equal to E over k square root. 00:31:39.330 --> 00:31:44.900 And phi is equal to minus pi over 2. 00:31:44.900 --> 00:31:46.377 OK, I'm just writing this. 00:31:46.377 --> 00:31:47.210 You want to do that. 00:31:54.740 --> 00:32:05.000 OK, so now we're going to be preparing 00:32:05.000 --> 00:32:09.830 to do quantum mechanics, the quantum mechanical solution 00:32:09.830 --> 00:32:12.300 of the harmonic oscillator. 00:32:12.300 --> 00:32:15.020 And so there are going to be other things that we care 00:32:15.020 --> 00:32:18.570 about, and one is the kinetic energy 00:32:18.570 --> 00:32:20.400 and one is the potential energy. 00:32:23.420 --> 00:32:27.230 And in particular, we'd like to know the kinetic energy, which 00:32:27.230 --> 00:32:28.670 we call t. 00:32:28.670 --> 00:32:33.700 And we'd like to know the expectation value of t 00:32:33.700 --> 00:32:37.280 as a function of time, or just T bar of t. 00:32:40.120 --> 00:32:46.940 And similarly, we'd like to know the expectation 00:32:46.940 --> 00:32:51.450 value of the potential energy as a function of time. 00:32:51.450 --> 00:32:58.190 And that's going to be V bar of t. 00:32:58.190 --> 00:33:00.680 And so from classical mechanics, we 00:33:00.680 --> 00:33:03.560 should be able to determine what these average values 00:33:03.560 --> 00:33:06.420 of the kinetic and potential energy. 00:33:06.420 --> 00:33:10.870 So what do we know? 00:33:10.870 --> 00:33:17.580 We know that the frequency is omega over 2 pi. 00:33:17.580 --> 00:33:22.360 We know the period is 1 over omega. 00:33:22.360 --> 00:33:35.280 OK, and so T of t is 1/2 mv squared of t, 00:33:35.280 --> 00:33:37.800 or p squared over 2m. 00:33:43.550 --> 00:33:50.120 But all right, v is the derivative 00:33:50.120 --> 00:33:52.670 of x with respect to t. 00:33:52.670 --> 00:33:55.500 And we have the solutions over here. 00:33:55.500 --> 00:34:07.480 And so we know that we can write x of t and v of t 00:34:07.480 --> 00:34:08.800 just by taking derivatives. 00:34:08.800 --> 00:34:17.710 And so we have x of t is 2e over k 00:34:17.710 --> 00:34:23.934 square root sine omega t plus phi. 00:34:26.780 --> 00:34:41.370 And v of t is omega times 2e over k square root cosine omega 00:34:41.370 --> 00:34:42.604 t plus phi. 00:34:45.510 --> 00:34:48.790 So we want v squared to be able to calculate 00:34:48.790 --> 00:34:50.550 the kinetic energy. 00:34:50.550 --> 00:34:51.659 And so we do that. 00:34:54.250 --> 00:35:01.860 And so the kinetic energy T of t is 00:35:01.860 --> 00:35:16.170 1/2m m omega squared 2e over k cosine squared 00:35:16.170 --> 00:35:18.430 omega t plus phi. 00:35:22.290 --> 00:35:27.780 This here, omega is k over m, the square root of k over m. 00:35:27.780 --> 00:35:30.740 S this is m times k over m. 00:35:30.740 --> 00:35:32.100 So we just get k out there. 00:35:40.070 --> 00:35:43.840 And so now we would like to know the average value 00:35:43.840 --> 00:35:45.460 of the kinetic energy. 00:35:55.210 --> 00:36:00.380 OK, so we have m omega squared. 00:36:00.380 --> 00:36:05.190 And so that's k over k. 00:36:05.190 --> 00:36:12.980 And so we just get E integral from 0 00:36:12.980 --> 00:36:27.470 to tau bt of cosine square omega t plus phi over tau. 00:36:27.470 --> 00:36:29.160 We want the time average. 00:36:29.160 --> 00:36:32.670 And so we calculate this integral and we divide by tau. 00:36:32.670 --> 00:36:36.000 That's how we take an average. 00:36:36.000 --> 00:36:47.870 And what we discover is that this integral is 00:36:47.870 --> 00:36:55.010 the numerator is 1/2 tau times E. No, I'm sorry. 00:36:55.010 --> 00:36:58.340 1/2 tau, we have E times 1/2 tau divided by tau. 00:36:58.340 --> 00:37:06.560 And this becomes E over 2, an important result. 00:37:06.560 --> 00:37:11.000 And we're going to discover that the average value 00:37:11.000 --> 00:37:13.820 of the momentum for a harmonic oscillator 00:37:13.820 --> 00:37:17.360 is E over 2 in quantum mechanics. 00:37:17.360 --> 00:37:21.400 We do the same thing for the potential. 00:37:21.400 --> 00:37:24.290 And we discover that it is also E over 2. 00:37:29.890 --> 00:37:40.240 So we know that E is equal to T of t plus V of t. 00:37:40.240 --> 00:37:44.470 But it's also true that the average is equal-- 00:37:44.470 --> 00:38:00.870 so T bar is equal to V bar which is equal to E over 2. 00:38:00.870 --> 00:38:04.080 So now we have an important interpretation 00:38:04.080 --> 00:38:06.130 of the harmonic oscillator. 00:38:06.130 --> 00:38:09.090 The harmonic oscillator is moving 00:38:09.090 --> 00:38:12.630 from turning point to the middle to the other turning point. 00:38:12.630 --> 00:38:15.690 And what's happening is energy is being exchanged 00:38:15.690 --> 00:38:21.570 between all potential energy at a turning point 00:38:21.570 --> 00:38:24.750 to all kinetic energy in the middle. 00:38:24.750 --> 00:38:27.420 So energy is going back and forth 00:38:27.420 --> 00:38:33.750 between kinetic energy and potential energy. 00:38:33.750 --> 00:38:40.490 We can solve for the relationship between T of t 00:38:40.490 --> 00:38:41.240 and V of t. 00:38:41.240 --> 00:38:53.770 And we can find that V of t is equal to T minus pi over 4. 00:38:58.730 --> 00:38:59.610 tau over 4. 00:39:04.170 --> 00:39:08.040 So this is telling us just what I said before, 00:39:08.040 --> 00:39:11.790 that energy is being exchanged between potential and kinetic 00:39:11.790 --> 00:39:12.630 energy. 00:39:12.630 --> 00:39:16.470 And that the potential energy is lagging by tau over 4 00:39:16.470 --> 00:39:20.220 behind the momentum. 00:39:24.734 --> 00:39:25.650 This is all very fast. 00:39:25.650 --> 00:39:29.210 But it's all classical mechanics, which you know. 00:39:29.210 --> 00:39:31.100 And we're going to be rediscovering all 00:39:31.100 --> 00:39:33.840 of this in quantum mechanics. 00:39:33.840 --> 00:39:36.080 And so we have to know what are we 00:39:36.080 --> 00:39:39.270 aiming for in quantum mechanics? 00:39:39.270 --> 00:39:41.270 So that we can completely say, yes, it's 00:39:41.270 --> 00:39:45.450 consistent with classical mechanics. 00:39:45.450 --> 00:39:46.870 And there's some really-- 00:39:46.870 --> 00:39:48.510 now, there's another really neat thing. 00:39:52.030 --> 00:40:07.480 So if we look at X of t, X of t is oscillating. 00:40:07.480 --> 00:40:13.929 And suppose we start out at a turning 00:40:13.929 --> 00:40:15.220 point, the outer turning point. 00:40:19.450 --> 00:40:23.970 So we can tell from this, the derivative of x with respect 00:40:23.970 --> 00:40:29.786 to t at x equals x plus is going to be 0. 00:40:33.440 --> 00:40:35.510 So at a turning point, the particle 00:40:35.510 --> 00:40:38.090 is hardly moving, not moving. 00:40:41.080 --> 00:40:44.470 And so what about the momentum? 00:40:44.470 --> 00:40:49.245 Well, the momentum is going to look like this. 00:40:53.270 --> 00:40:54.810 Yeah, it's going to look like that. 00:40:57.820 --> 00:41:07.020 And so at the time that we are starting at a turning point, 00:41:07.020 --> 00:41:08.910 the time derivative of the momentum 00:41:08.910 --> 00:41:10.260 is at its maximum value. 00:41:13.570 --> 00:41:17.310 So this is going to be really important when 00:41:17.310 --> 00:41:21.030 we start looking at properties of time-evolving wave 00:41:21.030 --> 00:41:21.540 functions. 00:41:21.540 --> 00:41:24.090 Because what we're going to discover is, 00:41:24.090 --> 00:41:30.420 suppose we start our system here, at a turning point. 00:41:30.420 --> 00:41:33.750 And that's actually something that we can do very easily 00:41:33.750 --> 00:41:35.670 in an experiment. 00:41:35.670 --> 00:41:43.830 Because when we excite from one electronic state to another, 00:41:43.830 --> 00:41:47.760 you automatically create a wave packet, 00:41:47.760 --> 00:41:50.220 which is localized at a particular internuclear 00:41:50.220 --> 00:41:50.720 distance. 00:41:53.460 --> 00:41:57.400 And so you go typically, to a turning point. 00:41:57.400 --> 00:42:00.630 So then, well, what's going to happen? 00:42:00.630 --> 00:42:02.910 Well, there is a thing in quantum mechanics, which 00:42:02.910 --> 00:42:05.880 you will become familiar with, called the autocorrelation 00:42:05.880 --> 00:42:07.320 function. 00:42:07.320 --> 00:42:07.910 No, it's not. 00:42:07.910 --> 00:42:09.570 It's called the survival probability. 00:42:18.550 --> 00:42:23.400 And that's going to be the product 00:42:23.400 --> 00:42:32.760 of the time-dependent wave function at x and T, 00:42:32.760 --> 00:42:37.380 the time-dependent wave function at x and 0. 00:42:37.380 --> 00:42:40.620 So this is expressing somehow how 00:42:40.620 --> 00:42:43.620 the wave function that is created at t 00:42:43.620 --> 00:42:47.740 equals 0 gets away from itself. 00:42:47.740 --> 00:42:49.440 It's a very important idea. 00:42:49.440 --> 00:42:51.450 Because you make something. 00:42:51.450 --> 00:42:53.580 And it evolves. 00:42:53.580 --> 00:42:57.300 And for a harmonic oscillator, if you make it at a turning 00:42:57.300 --> 00:43:00.360 point, this thing changes. 00:43:00.360 --> 00:43:03.320 Because the momentum changes. 00:43:03.320 --> 00:43:08.670 And the contribution of the coordinate 00:43:08.670 --> 00:43:15.120 to the decay of the survival probability is-- 00:43:15.120 --> 00:43:19.950 it's all due to the momentum, and not the coordinate change. 00:43:19.950 --> 00:43:22.380 That's actually, a very important insight. 00:43:22.380 --> 00:43:27.210 Because the momentum, the time rate of change of the momentum, 00:43:27.210 --> 00:43:30.610 is minus the gradient of the potential. 00:43:30.610 --> 00:43:34.380 So this is one way we learn about the potential 00:43:34.380 --> 00:43:37.350 simply by starting at a turning point 00:43:37.350 --> 00:43:40.380 and knowing that this thing, which we can measure, 00:43:40.380 --> 00:43:43.980 is measuring the thing we want to know. 00:43:43.980 --> 00:43:45.981 Now, we will get to this very soon. 00:43:45.981 --> 00:43:48.480 Because I haven't even told you about time-dependent quantum 00:43:48.480 --> 00:43:49.410 mechanics. 00:43:49.410 --> 00:43:52.664 But those are the things that we expect to encounter. 00:43:58.100 --> 00:44:02.440 OK, now I want to give you-- 00:44:02.440 --> 00:44:04.280 I'm going to throw at you-- 00:44:04.280 --> 00:44:09.770 some useful stuff, which turns out to be really easy. 00:44:09.770 --> 00:44:14.450 Suppose we want to know the expectation 00:44:14.450 --> 00:44:20.030 values, or the average values, of x, x squared, 00:44:20.030 --> 00:44:21.620 p, and p squared. 00:44:25.310 --> 00:44:29.490 OK, so we have a harmonic oscillator. 00:44:29.490 --> 00:44:34.400 And do we know what this is? 00:44:34.400 --> 00:44:37.868 Do we know the expectation value of the coordinate? 00:44:37.868 --> 00:44:38.740 AUDIENCE: 0. 00:44:38.740 --> 00:44:39.740 ROBERT FIELD: We have 0. 00:44:39.740 --> 00:44:41.330 Why is it 0? 00:44:41.330 --> 00:44:44.130 There's two ways of answering that question. 00:44:44.130 --> 00:44:47.660 But these are easy questions, which on an exam, 00:44:47.660 --> 00:44:49.310 you don't want to evaluate an integral. 00:44:49.310 --> 00:44:51.530 You want to know why is it 0. 00:44:51.530 --> 00:44:53.030 And there's two answers. 00:44:53.030 --> 00:44:54.950 You said 0, right? 00:44:54.950 --> 00:44:56.328 Why did you say 0? 00:44:56.328 --> 00:44:58.240 AUDIENCE: Because it's symmetric. 00:44:58.240 --> 00:44:59.430 ROBERT FIELD: Yes. 00:44:59.430 --> 00:45:03.100 OK, so there is a symmetry argument. 00:45:03.100 --> 00:45:06.930 Another is well, is the potential moving? 00:45:06.930 --> 00:45:08.440 The particle is in a potential. 00:45:08.440 --> 00:45:09.523 It's going back and forth. 00:45:09.523 --> 00:45:11.390 The potential is stationary. 00:45:11.390 --> 00:45:13.530 So there's no way that x could move. 00:45:13.530 --> 00:45:14.910 X could be time dependent. 00:45:14.910 --> 00:45:19.110 So we know this is 0. 00:45:19.110 --> 00:45:21.510 What about p? 00:45:21.510 --> 00:45:27.540 Same thing, whether you use symmetry or just 00:45:27.540 --> 00:45:29.784 physical insight. 00:45:29.784 --> 00:45:30.825 But what about x squared? 00:45:36.150 --> 00:45:44.050 Well, x squared is equal to V of x over k over 2. 00:45:48.650 --> 00:45:52.370 So expectation value of x squared 00:45:52.370 --> 00:46:00.800 is equal to the expectation value of V over k over 2. 00:46:00.800 --> 00:46:04.850 But we know the expectation value of V. It's E over 2. 00:46:13.920 --> 00:46:17.880 So we know without doing any integrals what 00:46:17.880 --> 00:46:21.300 the expectation value of x squared is, and similarly, 00:46:21.300 --> 00:46:22.050 for p squared. 00:46:26.980 --> 00:46:32.300 And that's just going to be m times E. OK, so why 00:46:32.300 --> 00:46:33.990 do I care about these things? 00:46:33.990 --> 00:46:37.360 Well, we have a little thing called the uncertainty 00:46:37.360 --> 00:46:39.190 principle. 00:46:39.190 --> 00:46:44.920 And we'd like to know the uncertainty in the coordinate 00:46:44.920 --> 00:46:46.090 and the momentum. 00:46:46.090 --> 00:46:48.230 And this is our classical view of it. 00:46:48.230 --> 00:46:52.510 But it's going to remind us of what 00:46:52.510 --> 00:46:54.400 we find quantum mechanically. 00:46:54.400 --> 00:46:59.530 So the uncertainty in x can be defined 00:46:59.530 --> 00:47:08.650 as the average value of x squared minus the average value 00:47:08.650 --> 00:47:11.910 of x squared square root. 00:47:11.910 --> 00:47:14.460 That's just the variance. 00:47:14.460 --> 00:47:17.085 And well, this is 0. 00:47:17.085 --> 00:47:19.090 And we know what this is. 00:47:19.090 --> 00:47:27.670 So we know that the uncertainty in x is E over k square root. 00:47:27.670 --> 00:47:30.400 And the uncertainty in p is going 00:47:30.400 --> 00:47:36.970 to be p squared average value minus p squared. 00:47:36.970 --> 00:47:38.330 This is still 0. 00:47:38.330 --> 00:47:44.060 And this one is then m times E square root. 00:47:47.010 --> 00:47:55.310 So delta x delta p, it looks like the uncertainty principle. 00:47:55.310 --> 00:48:01.235 This is classic mechanics, is just E over omega. 00:48:01.235 --> 00:48:02.710 But what's E over omega? 00:48:06.034 --> 00:48:06.534 AUDIENCE: h? 00:48:09.890 --> 00:48:10.950 h bar. 00:48:10.950 --> 00:48:12.450 ROBERT FIELD: Right. 00:48:12.450 --> 00:48:16.470 So it's all going to come around. 00:48:16.470 --> 00:48:21.360 This is related to the uncertainty principle 00:48:21.360 --> 00:48:22.500 in quantum mechanics. 00:48:22.500 --> 00:48:26.340 There is a minimum joint uncertainty between x and p. 00:48:26.340 --> 00:48:29.620 And it's just related to this constant. 00:48:29.620 --> 00:48:32.080 Now, this doesn't say the uncertainty grows 00:48:32.080 --> 00:48:34.510 as you go to higher and higher energy, 00:48:34.510 --> 00:48:36.280 as it will in quantum mechanics. 00:48:36.280 --> 00:48:39.080 But this is really a neat thing to see. 00:48:39.080 --> 00:48:41.020 OK, I've got two minutes left. 00:48:44.010 --> 00:48:49.760 So if we wanted to know the probability of finding 00:48:49.760 --> 00:49:03.279 x as a function of the coordinate, the turning points. 00:49:03.279 --> 00:49:04.570 So here are the turning points. 00:49:04.570 --> 00:49:07.330 And we can calculate what is going 00:49:07.330 --> 00:49:13.131 to be the probability of finding the particle at one turning 00:49:13.131 --> 00:49:13.630 point. 00:49:13.630 --> 00:49:15.640 Well, it comes down from infinity, 00:49:15.640 --> 00:49:18.620 goes back up to infinity. 00:49:18.620 --> 00:49:28.660 So here at the turning points, the particle is stopped. 00:49:28.660 --> 00:49:32.410 And so the probability of finding it at a turning point 00:49:32.410 --> 00:49:35.530 is infinite, times dx. 00:49:35.530 --> 00:49:39.636 So we don't have an infinite number. 00:49:39.636 --> 00:49:41.010 In quantum mechanics, we're going 00:49:41.010 --> 00:49:44.204 to discover that quantum mechanics is smarter than that. 00:49:44.204 --> 00:49:45.870 And what quantum mechanics does, suppose 00:49:45.870 --> 00:49:47.325 we have this is the energy. 00:49:51.040 --> 00:49:54.100 What quantum mechanics does is that the probability 00:49:54.100 --> 00:49:57.940 at a turning point is not 0. 00:49:57.940 --> 00:50:02.600 And there's something-- what happens 00:50:02.600 --> 00:50:08.490 is there's tunneling tails that instead of going to infinity, 00:50:08.490 --> 00:50:09.950 it has a finite value. 00:50:09.950 --> 00:50:14.270 And it reaches out into the forbidden region 00:50:14.270 --> 00:50:17.300 in an exponentially decreasing way. 00:50:17.300 --> 00:50:19.610 And that's basically the difference 00:50:19.610 --> 00:50:21.980 between classical mechanics and quantum mechanics. 00:50:21.980 --> 00:50:24.470 And there's an awful lot of important stuff 00:50:24.470 --> 00:50:25.780 that happens there. 00:50:25.780 --> 00:50:27.920 Now, I've been very fast through all of this 00:50:27.920 --> 00:50:32.084 because it's built on stuff that you're supposed to know. 00:50:32.084 --> 00:50:33.500 And it's built on stuff that we're 00:50:33.500 --> 00:50:36.840 going to work hard to understand from a quantum mechanical point 00:50:36.840 --> 00:50:37.340 of view. 00:50:37.340 --> 00:50:39.740 But this sets the stage for, what 00:50:39.740 --> 00:50:43.700 are the things we have to look for in quantum mechanics? 00:50:43.700 --> 00:50:47.880 So I do recommend that you look at the notes 00:50:47.880 --> 00:50:51.330 and make sure you can follow all the steps, which I went 00:50:51.330 --> 00:50:52.855 through incredibly rapidly. 00:50:56.260 --> 00:50:57.730 And it'll be really helpful. 00:50:57.730 --> 00:51:02.680 Because your job will be to draw cartoons. 00:51:02.680 --> 00:51:04.770 And these will guide you through it. 00:51:04.770 --> 00:51:09.989 OK, so I'll see you on Friday.