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 Open Courseware 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 Open Courseware 00:00:17.250 --> 00:00:18.200 at ocw.mit.edu. 00:00:22.090 --> 00:00:23.250 ROBERT FIELD: OK. 00:00:23.250 --> 00:00:26.580 So this is going to be a fun lecture because it's mostly 00:00:26.580 --> 00:00:29.130 pictures and language. 00:00:29.130 --> 00:00:33.780 And it's also an area of physical chemistry, 00:00:33.780 --> 00:00:35.880 which is pretty much-- 00:00:35.880 --> 00:00:39.780 I mean, there's three areas of physical chemistry 00:00:39.780 --> 00:00:42.150 where people have trouble talking to each other. 00:00:42.150 --> 00:00:44.007 There's statistical mechanics. 00:00:44.007 --> 00:00:45.090 There's quantum mechanics. 00:00:45.090 --> 00:00:48.000 And there's the time dependent, time independent forms 00:00:48.000 --> 00:00:49.470 of quantum mechanics. 00:00:49.470 --> 00:00:51.990 And these three communities have to learn 00:00:51.990 --> 00:00:53.520 how to talk to each other. 00:00:53.520 --> 00:00:57.370 And I'm hoping that I will help to bridge that gap. 00:00:57.370 --> 00:00:57.870 OK. 00:00:57.870 --> 00:01:01.721 So last time, we talked about the time dependent Schrodinger 00:01:01.721 --> 00:01:02.220 equation. 00:01:02.220 --> 00:01:05.060 It's a simple looking little thing. 00:01:05.060 --> 00:01:07.610 Remember though, it's an extra level of complexity 00:01:07.610 --> 00:01:10.670 on what you already understand. 00:01:10.670 --> 00:01:15.020 Now, in the special case where the Hamiltonian is time 00:01:15.020 --> 00:01:21.650 independent, then, if you know all of the energy levels 00:01:21.650 --> 00:01:26.360 and wave functions for the time independent Hamiltonian, 00:01:26.360 --> 00:01:29.180 you can immediately write down the solution 00:01:29.180 --> 00:01:33.470 to the time dependent Hamiltonian. 00:01:33.470 --> 00:01:37.970 And so you have a complete set of wave functions. 00:01:37.970 --> 00:01:40.910 And you have a complete set of energy levels. 00:01:40.910 --> 00:01:44.900 And with that, you can basically describe something 00:01:44.900 --> 00:01:47.480 that satisfies this thing. 00:01:47.480 --> 00:01:49.730 Another set of coefficients is often something 00:01:49.730 --> 00:01:56.090 that you arrange for simplicity or for insight. 00:01:56.090 --> 00:01:57.260 And I want to understand-- 00:01:57.260 --> 00:01:59.240 I want to talk about that. 00:01:59.240 --> 00:02:00.020 OK. 00:02:00.020 --> 00:02:02.180 And so in the previous lecture, we 00:02:02.180 --> 00:02:10.250 talked about the probability density, which is psi star psi. 00:02:10.250 --> 00:02:12.300 And that can move. 00:02:12.300 --> 00:02:16.230 And that can move in two ways. 00:02:19.800 --> 00:02:24.270 In the wave function, if there are two or more states 00:02:24.270 --> 00:02:29.140 belonging to two or more different energies, 00:02:29.140 --> 00:02:30.780 then you can have motion. 00:02:30.780 --> 00:02:36.700 And you can have motion like breathing, where probability 00:02:36.700 --> 00:02:41.890 moves towards the extremes, or motion 00:02:41.890 --> 00:02:44.920 where there is actually a wave packet going from one side 00:02:44.920 --> 00:02:47.580 to another. 00:02:47.580 --> 00:02:50.600 So for anything interesting to happen, 00:02:50.600 --> 00:02:53.870 this probability density has to include 00:02:53.870 --> 00:02:56.945 at least two different eigenstates belonging 00:02:56.945 --> 00:02:59.300 to two different energy levels. 00:02:59.300 --> 00:03:02.540 Now, the total probability is just the integral 00:03:02.540 --> 00:03:08.540 over all coordinates of the probability density. 00:03:08.540 --> 00:03:12.180 And probability is conserved. 00:03:12.180 --> 00:03:14.000 So one of the things that happens-- 00:03:14.000 --> 00:03:16.550 when you do an integral, the wave functions go away. 00:03:16.550 --> 00:03:20.660 And in this particular case, integral psi star psi, 00:03:20.660 --> 00:03:22.550 all you get is one. 00:03:22.550 --> 00:03:25.800 Or you get a constant, depending on how you normalize things. 00:03:25.800 --> 00:03:28.930 But it's not time dependent. 00:03:28.930 --> 00:03:34.640 Now, in understanding motion from classical mechanics, 00:03:34.640 --> 00:03:36.310 we know Newton's equations. 00:03:36.310 --> 00:03:42.880 We know how the coordinate and the momentum depend on time. 00:03:42.880 --> 00:03:47.290 And so if we calculate the expectation 00:03:47.290 --> 00:03:49.900 value of the coordinate or the expectation 00:03:49.900 --> 00:03:53.650 value of the momentum, we get some results. 00:03:53.650 --> 00:03:58.180 And it turns out that these things describe the motion 00:03:58.180 --> 00:04:00.490 of the center of a wave packet. 00:04:00.490 --> 00:04:04.240 A wave packet is anything that's not just a single eigenstate, 00:04:04.240 --> 00:04:05.740 so it moves. 00:04:05.740 --> 00:04:10.560 And so the motion of a single eigenstate-- 00:04:10.560 --> 00:04:15.130 the motion of a wave packet is described by Newton's laws. 00:04:15.130 --> 00:04:17.050 And this is Ehrenfest theorem. 00:04:17.050 --> 00:04:18.940 And it can easily be proven. 00:04:18.940 --> 00:04:20.709 But we're not going to prove it. 00:04:20.709 --> 00:04:22.240 We're just going to use it. 00:04:22.240 --> 00:04:27.010 So we are very interested in expectation values 00:04:27.010 --> 00:04:29.410 of the coordinate and the momentum. 00:04:29.410 --> 00:04:33.700 And for a harmonic oscillator, this is duck soup. 00:04:33.700 --> 00:04:38.500 Then we have nothing but trivial integrals involving 00:04:38.500 --> 00:04:40.870 the a's and a-daggers. 00:04:40.870 --> 00:04:45.580 And so even though in some ways the harmonic oscillator 00:04:45.580 --> 00:04:49.880 is a more complicated problem than the particle in a box, 00:04:49.880 --> 00:04:51.610 the harmonic oscillator is the problem 00:04:51.610 --> 00:05:00.190 of choice for dealing with motion and developing insight. 00:05:00.190 --> 00:05:02.110 There's another useful-- 00:05:02.110 --> 00:05:02.990 OK. 00:05:02.990 --> 00:05:05.780 We have this wave function here. 00:05:05.780 --> 00:05:09.290 There is a huge amount of information embedded in it, 00:05:09.290 --> 00:05:12.050 too much to just look at. 00:05:12.050 --> 00:05:16.230 We need to have ways of reducing that information. 00:05:16.230 --> 00:05:19.880 And one of the nice ways is the survival probability. 00:05:19.880 --> 00:05:22.010 The survival probability tells you 00:05:22.010 --> 00:05:27.600 how fast does the initial state move away from itself. 00:05:27.600 --> 00:05:30.240 And this is very revealing. 00:05:30.240 --> 00:05:34.766 Another very revealing thing is if this survival probability, 00:05:34.766 --> 00:05:36.390 which is a function of time and nothing 00:05:36.390 --> 00:05:40.350 else, because we've integrated over the wave functions-- 00:05:40.350 --> 00:05:42.840 if this survival probability goes up and down 00:05:42.840 --> 00:05:47.160 and up and down, there are recurrences. 00:05:47.160 --> 00:05:49.860 So at a maximum, which is usually 00:05:49.860 --> 00:05:53.520 a maximum at t equals 0 because the wave function is it 00:05:53.520 --> 00:05:57.060 at its birthplace, the survival probability 00:05:57.060 --> 00:06:00.630 will start high and go down, and come up, and go down. 00:06:00.630 --> 00:06:03.780 And there's all sorts of information about recurrences. 00:06:03.780 --> 00:06:07.800 As the wave function returns to its birthplace, 00:06:07.800 --> 00:06:11.040 it might not return completely. 00:06:11.040 --> 00:06:13.420 And so we get partial recurrences. 00:06:13.420 --> 00:06:15.910 And these tell us something. 00:06:15.910 --> 00:06:20.830 And the times at which the maxima in the survival 00:06:20.830 --> 00:06:27.120 probability occur tell us a lot about the potential. 00:06:27.120 --> 00:06:29.280 And there are some grand recurrences 00:06:29.280 --> 00:06:33.390 where, because all of the energy level differences 00:06:33.390 --> 00:06:36.130 are an integer multiple of a common factor, 00:06:36.130 --> 00:06:39.120 then you get a perfect recurrence. 00:06:39.120 --> 00:06:43.150 And these are wonderful for drawing pictures. 00:06:43.150 --> 00:06:50.590 Now, you really want to understand the concepts of time 00:06:50.590 --> 00:06:53.900 dependent quantum mechanics pictorially 00:06:53.900 --> 00:06:56.720 and to develop a language that describes it. 00:06:56.720 --> 00:06:58.220 And there are a lot of little pieces 00:06:58.220 --> 00:07:02.904 here that you need to convince yourself you understand 00:07:02.904 --> 00:07:04.070 so you can use the language. 00:07:21.230 --> 00:07:22.080 OK. 00:07:22.080 --> 00:07:24.840 One of the things I said at the beginning of the course 00:07:24.840 --> 00:07:32.310 is the central object in quantum mechanics is the wave function. 00:07:32.310 --> 00:07:36.900 Or we could generalize to this. 00:07:36.900 --> 00:07:39.760 This is actually the truth. 00:07:39.760 --> 00:07:42.360 This is a partial truth. 00:07:42.360 --> 00:07:46.050 This is everything that we can possibly know. 00:07:46.050 --> 00:07:47.910 But we can't know this. 00:07:47.910 --> 00:07:50.560 We can never observe the wave function. 00:07:50.560 --> 00:07:53.490 However, we do observations. 00:07:53.490 --> 00:07:58.980 And we construct a picture, which 00:07:58.980 --> 00:08:03.040 is what we call the effective Hamiltonian. 00:08:03.040 --> 00:08:06.660 This effective Hamiltonian describes everything 00:08:06.660 --> 00:08:09.930 we know, mostly energy levels. 00:08:09.930 --> 00:08:12.640 We have formulas for the energy levels. 00:08:12.640 --> 00:08:14.490 And we determine the constants. 00:08:14.490 --> 00:08:18.130 There's all sorts of stuff that we collect by experiment. 00:08:18.130 --> 00:08:20.820 And so we create an object which we think 00:08:20.820 --> 00:08:23.730 is like the true Hamiltonian. 00:08:23.730 --> 00:08:27.660 And by having the effective Hamiltonian, 00:08:27.660 --> 00:08:32.309 we can get these things. 00:08:32.309 --> 00:08:36.330 So we observe we make some observations. 00:08:36.330 --> 00:08:40.320 And then this is a reduced version of the truth. 00:08:40.320 --> 00:08:43.770 And we can get a pretty good representation 00:08:43.770 --> 00:08:46.440 of the thing that is supposedly hidden from us. 00:08:50.201 --> 00:08:50.700 OK. 00:08:53.352 --> 00:08:58.085 So let's start now at t equals 0. 00:08:58.085 --> 00:09:01.430 At t equals 0, which I like to call the pluck-- 00:09:01.430 --> 00:09:04.430 and it really makes it contact with those of you who 00:09:04.430 --> 00:09:08.330 are musicians, that you know how to create 00:09:08.330 --> 00:09:12.700 a particular kind of sound, which will evolve in time. 00:09:12.700 --> 00:09:18.200 And it depends on the details of how you handle the instrument. 00:09:18.200 --> 00:09:24.080 So at t equals 0, we have this thing 00:09:24.080 --> 00:09:29.030 which, if we have a complete set of eigenfunctions 00:09:29.030 --> 00:09:34.160 of the Hamiltonian, of the time independent Hamiltonian, 00:09:34.160 --> 00:09:39.146 we know we can write this as cj psi j of x. 00:09:45.930 --> 00:09:48.960 Now, often we don't need an infinite number of terms. 00:09:48.960 --> 00:09:53.190 But we know that there are lots of different kinds of plucks. 00:09:53.190 --> 00:10:00.390 And you can find out what these coefficients are 00:10:00.390 --> 00:10:05.190 by just calculating overlap integral of these functions 00:10:05.190 --> 00:10:06.570 with this initial state. 00:10:10.961 --> 00:10:11.460 OK. 00:10:11.460 --> 00:10:14.730 So this is telling you what happens at t equals 0. 00:10:14.730 --> 00:10:18.630 And if you can write the initial state 00:10:18.630 --> 00:10:20.970 as a superposition of eigenstates, 00:10:20.970 --> 00:10:24.030 then you can immediately write the time 00:10:24.030 --> 00:10:46.620 evolving object this way. 00:10:46.620 --> 00:10:52.190 Now, we have a minus i. 00:10:52.190 --> 00:10:55.190 Well, why do we need a minus i? 00:10:55.190 --> 00:10:56.700 Well, because we're going to take 00:10:56.700 --> 00:10:59.760 the time derivative of the Hamiltonian 00:10:59.760 --> 00:11:02.430 and multiply it by iH bar. 00:11:02.430 --> 00:11:06.780 We're going to want to get the energy, not minus the energy. 00:11:09.380 --> 00:11:12.440 And so we need a minus i here. 00:11:12.440 --> 00:11:15.970 And we need a divided by H bar here 00:11:15.970 --> 00:11:20.240 in order to satisfy the time independence or time 00:11:20.240 --> 00:11:22.130 dependence. 00:11:22.130 --> 00:11:24.681 So this always bothers people. 00:11:24.681 --> 00:11:25.930 Why should there be an i here? 00:11:25.930 --> 00:11:28.210 And why should it be minus i? 00:11:28.210 --> 00:11:31.330 And I recommend not trying to memorize it, 00:11:31.330 --> 00:11:32.890 but to just convince yourself. 00:11:32.890 --> 00:11:37.060 I need the minus i because I'm going to bring down a minus i 00:11:37.060 --> 00:11:39.880 over H bar times e. 00:11:39.880 --> 00:11:42.310 And we have this iH bar. 00:11:42.310 --> 00:11:48.170 The minus i times i gives plus 1. 00:11:48.170 --> 00:11:49.600 OK. 00:11:49.600 --> 00:11:59.570 So now, in understanding how the time dependent Hamiltonian can 00:11:59.570 --> 00:12:05.660 be sampled, we build the superposition states 00:12:05.660 --> 00:12:10.550 with a minimum number of terms, like two or three. 00:12:10.550 --> 00:12:15.530 Even though, in order to create a very, very sharply localized 00:12:15.530 --> 00:12:20.420 state, it needs a huge number of terms here. 00:12:20.420 --> 00:12:23.300 But you always build your insight 00:12:23.300 --> 00:12:25.890 with something you can do in your head. 00:12:25.890 --> 00:12:32.010 And so for some problems, you need only two states. 00:12:32.010 --> 00:12:34.440 And for others, you need three. 00:12:34.440 --> 00:12:37.440 And those are the ones you want to kill. 00:12:37.440 --> 00:12:39.330 You want to understand everything 00:12:39.330 --> 00:12:43.166 you can build with two and three state superpositions. 00:12:46.290 --> 00:12:47.130 OK. 00:12:47.130 --> 00:12:53.505 So there's two classes of problems. 00:13:07.500 --> 00:13:12.740 OK so for a half harmonic oscillator-- 00:13:24.710 --> 00:13:28.400 so for a half harmonic oscillator, 00:13:28.400 --> 00:13:30.440 we start with harmonic oscillator 00:13:30.440 --> 00:13:35.670 and divide it in half and say this goes to infinity. 00:13:35.670 --> 00:13:39.840 And so this side is not accessible. 00:13:39.840 --> 00:13:43.640 And so if the potential is half of a harmonic oscillator 00:13:43.640 --> 00:13:46.490 potential, then we know immediately what 00:13:46.490 --> 00:13:50.540 the eigenvalues and eigenfunctions are. 00:13:50.540 --> 00:13:52.519 They are the harmonic oscillator functions 00:13:52.519 --> 00:13:53.810 that have a node in the middle. 00:13:56.760 --> 00:14:07.230 So we know that we have v equals 1, v equals 3. 00:14:07.230 --> 00:14:10.600 No v equals 0 or 2. 00:14:10.600 --> 00:14:13.220 I mean, this corresponds to the lowest level, v 00:14:13.220 --> 00:14:14.740 equals 0 for the half oscillator. 00:14:14.740 --> 00:14:16.630 But this is basically-- now, what 00:14:16.630 --> 00:14:20.740 we're going to do is we create some state of a half 00:14:20.740 --> 00:14:23.250 oscillator. 00:14:23.250 --> 00:14:26.050 And then we take away the barrier. 00:14:26.050 --> 00:14:29.250 So that's a cheap way of creating localization. 00:14:29.250 --> 00:14:31.760 Of course, we can't do this experimentally. 00:14:31.760 --> 00:14:34.840 But you could, in principle, do it. 00:14:34.840 --> 00:14:46.650 And so basically, you want an initial state, 00:14:46.650 --> 00:14:50.885 which is a superposition of vibrational levels. 00:14:56.290 --> 00:14:56.970 OK. 00:14:56.970 --> 00:15:02.660 And this initial state needs to have the wave 00:15:02.660 --> 00:15:04.525 function be 0 at the barrier. 00:15:07.860 --> 00:15:13.740 So since we're going to be looking at the full potential, 00:15:13.740 --> 00:15:15.480 we're taking away the barrier. 00:15:15.480 --> 00:15:20.460 We're allowed to use the even and odd states. 00:15:20.460 --> 00:15:24.200 And so let's take the three lowest states. 00:15:24.200 --> 00:15:32.390 And so we have c0 psi 0 plus c1 psi 1 plus c2 psi 2. 00:15:34.811 --> 00:15:35.310 OK. 00:15:35.310 --> 00:15:37.140 This guy is OK. 00:15:37.140 --> 00:15:38.310 Yes? 00:15:38.310 --> 00:15:42.260 AUDIENCE: When you create the states of the half harmonic 00:15:42.260 --> 00:15:45.360 oscillator, do you have to re-scale them 00:15:45.360 --> 00:15:46.600 so that they normalize? 00:15:46.600 --> 00:15:47.350 ROBERT FIELD: Yes. 00:15:50.400 --> 00:15:52.980 We play fast and loose with normalization constants. 00:15:52.980 --> 00:15:55.530 We already always know that, whenever 00:15:55.530 --> 00:15:57.480 you want to calculate anything, you're 00:15:57.480 --> 00:16:02.130 going to divide by the normalization integral. 00:16:02.130 --> 00:16:04.890 However, it's OK. 00:16:04.890 --> 00:16:08.010 Because we're going to be using the full harmonic oscillator 00:16:08.010 --> 00:16:12.390 functions, which do exist over here. 00:16:12.390 --> 00:16:15.960 And it's fine once we remove the barrier. 00:16:15.960 --> 00:16:21.810 But we want to choose a problem which is as simple as possible. 00:16:21.810 --> 00:16:26.460 OK, now, this guy is not 0 at x equals 0. 00:16:26.460 --> 00:16:28.530 And this guy is not 0 at x equals 0. 00:16:28.530 --> 00:16:40.710 But you can say 0 is equal to c0 psi 0 of 0 plus c2 psi 2 of 0. 00:16:40.710 --> 00:16:46.590 We choose the coefficients so that these two guys together 00:16:46.590 --> 00:16:49.890 make 0 at the boundary. 00:16:49.890 --> 00:16:50.850 And why do we do this? 00:16:50.850 --> 00:16:54.720 We're going to be calculating expectation values of x and p. 00:16:54.720 --> 00:16:59.970 And if we only have half of the energy levels, 00:16:59.970 --> 00:17:03.940 all of the intricacies are going to be 0. 00:17:03.940 --> 00:17:05.950 So we have to have three consecutive levels 00:17:05.950 --> 00:17:07.511 to have anything interesting. 00:17:10.220 --> 00:17:14.640 OK, so the artifice of making these two-- 00:17:14.640 --> 00:17:20.190 arranging the coefficients so that you get a temporary node 00:17:20.190 --> 00:17:24.170 at x equals 0-- 00:17:24.170 --> 00:17:25.210 it won't stay a node. 00:17:25.210 --> 00:17:27.760 Because when we let things oscillate with time, 00:17:27.760 --> 00:17:30.670 these two will oscillate differently. 00:17:30.670 --> 00:17:33.409 And the node will go away. 00:17:33.409 --> 00:17:34.450 But that's how you do it. 00:17:34.450 --> 00:17:37.571 That's how you build a superposition. 00:17:37.571 --> 00:17:38.070 OK. 00:17:46.111 --> 00:17:46.610 OK. 00:17:46.610 --> 00:17:51.710 Now, what we want to do is to be able to draw pictures 00:17:51.710 --> 00:17:57.240 of what's happening and also to calculate what's going on. 00:17:57.240 --> 00:18:00.230 And one of the things we use for our pictures 00:18:00.230 --> 00:18:05.990 is the expectation value of x and the expectation value of p. 00:18:09.100 --> 00:18:12.460 So you know how to calculate the expectation value of x. 00:18:12.460 --> 00:18:18.650 We have this capital psi star x capital psi star capital 00:18:18.650 --> 00:18:21.610 psi integrate over x. 00:18:21.610 --> 00:18:35.535 And so we get to c0 c1 x01 cosine omega t plus 2 00:18:35.535 --> 00:18:43.710 c1 c2 x12 cosine omega t. 00:18:43.710 --> 00:18:45.000 Now, how did I do that? 00:18:48.784 --> 00:18:49.290 Oh, come on. 00:18:49.290 --> 00:18:51.330 You can do it, too. 00:18:51.330 --> 00:18:52.550 So what is this x01? 00:18:52.550 --> 00:19:03.020 Well, x01 is the integral psi 0 star x psi 1 dx. 00:19:03.020 --> 00:19:06.050 And we know we can replace x by a plus a-dagger 00:19:06.050 --> 00:19:08.370 times the constant. 00:19:08.370 --> 00:19:10.050 We always like to forget that concept. 00:19:10.050 --> 00:19:13.080 We only bring it in at the end anyway. 00:19:13.080 --> 00:19:16.700 And so, well, this has a value-- 00:19:16.700 --> 00:19:22.370 it's a constant, the constant that we're forgetting-- 00:19:22.370 --> 00:19:26.990 times 1 square root, right? 00:19:30.150 --> 00:19:34.080 And x12 is the same sort of thing. 00:19:34.080 --> 00:19:41.520 It's going to be the same constant times 2 square root. 00:19:41.520 --> 00:19:43.380 This is easy. 00:19:43.380 --> 00:19:48.450 So getting from this symbol to this, 00:19:48.450 --> 00:19:52.640 that just requires a little practice, 00:19:52.640 --> 00:19:54.730 and then simplifying further to know 00:19:54.730 --> 00:20:00.110 what these x's are, and then the constraints on c2 and c0. 00:20:00.110 --> 00:20:03.030 We have all that stuff. 00:20:03.030 --> 00:20:03.530 OK. 00:20:03.530 --> 00:20:10.230 And now, we can draw a picture of what's going to happen. 00:20:10.230 --> 00:20:12.380 So here we have the full oscillator. 00:20:12.380 --> 00:20:15.950 And let's just draw some energy which is-- 00:20:15.950 --> 00:20:17.070 now that's complicated. 00:20:17.070 --> 00:20:20.790 We have a time dependent wave function, 00:20:20.790 --> 00:20:25.910 which is composed of several different energy eigenstates. 00:20:25.910 --> 00:20:27.470 So what is its energy? 00:20:27.470 --> 00:20:29.210 Well, you can evaluate what the energy 00:20:29.210 --> 00:20:34.380 is by taking the expectation value of the Hamiltonian. 00:20:34.380 --> 00:20:38.300 And so you can do that. 00:20:38.300 --> 00:20:40.149 So what we've made at t equals 0 is 00:20:40.149 --> 00:20:41.440 something that looks like this. 00:20:47.060 --> 00:20:48.904 It's localized on the left side. 00:20:48.904 --> 00:20:50.570 Or it's more localized on the left side. 00:20:50.570 --> 00:20:54.655 Now, sometimes, you're going to worry about phase. 00:20:57.360 --> 00:21:02.280 And so many times when you're working symbolically 00:21:02.280 --> 00:21:05.160 rather than actually evaluating integrals, 00:21:05.160 --> 00:21:12.860 there are symbolic phase choices that what you're using 00:21:12.860 --> 00:21:13.990 has made. 00:21:13.990 --> 00:21:15.980 And for example, for the harmonic oscillator, 00:21:15.980 --> 00:21:18.740 if you look in the book, you'll see 00:21:18.740 --> 00:21:24.410 that, for all of the harmonic oscillator functions, 00:21:24.410 --> 00:21:26.780 the outer lobe is always positive. 00:21:26.780 --> 00:21:30.080 And the inner lobe is alternating 00:21:30.080 --> 00:21:33.330 with the quantum number. 00:21:33.330 --> 00:21:38.090 So that's a phase convention that's implicit in everything 00:21:38.090 --> 00:21:41.150 that people have derived. 00:21:41.150 --> 00:21:44.180 And as long as the different things 00:21:44.180 --> 00:21:48.560 you combine in doing a calculation involve 00:21:48.560 --> 00:21:52.190 the same phase convention, which is implicit-- we don't 00:21:52.190 --> 00:21:54.330 want to look at what functions. 00:21:54.330 --> 00:21:57.731 We want to look at these xij's. 00:21:57.731 --> 00:21:58.230 OK. 00:21:58.230 --> 00:22:03.570 And so those things that we're manipulating 00:22:03.570 --> 00:22:07.020 have a phase implicit in how you define them. 00:22:07.020 --> 00:22:09.350 But that's gone in your manipulation. 00:22:09.350 --> 00:22:11.340 So you want to be a little careful. 00:22:11.340 --> 00:22:13.350 OK. 00:22:13.350 --> 00:22:21.090 So we start out and the expectation value 00:22:21.090 --> 00:22:23.460 is on this side. 00:22:23.460 --> 00:22:30.370 And the psi star psi is localized on this side mostly. 00:22:30.370 --> 00:22:33.330 And at a later time-- 00:22:33.330 --> 00:22:35.950 so this is a later time. 00:22:35.950 --> 00:22:39.320 I shouldn't make it bigger. 00:22:39.320 --> 00:22:42.380 At a later time, this thing has moved to the other turning 00:22:42.380 --> 00:22:44.260 point. 00:22:44.260 --> 00:22:46.360 Back and forth, back and forth. 00:22:49.690 --> 00:22:55.610 Now, since we have three energy levels-- 00:22:55.610 --> 00:22:59.780 and well, actually, we have a coherence term 00:22:59.780 --> 00:23:04.970 which involves the product of psi 0 and psi 1 00:23:04.970 --> 00:23:07.610 and psi 1 and psi 2-- 00:23:07.610 --> 00:23:11.420 they differ in energy both by omega. 00:23:11.420 --> 00:23:15.680 So these two things have the same oscillation frequency. 00:23:15.680 --> 00:23:18.950 And so what's going to happen is the wave function, the wave 00:23:18.950 --> 00:23:21.620 packet, is going to move from this side to that side, 00:23:21.620 --> 00:23:26.690 back and forth always forever at the same frequency, 00:23:26.690 --> 00:23:28.660 no dephasing. 00:23:28.660 --> 00:23:32.470 In the middle, I can't tell you what it's going to look like. 00:23:32.470 --> 00:23:33.670 I don't want to tell you. 00:23:33.670 --> 00:23:34.650 Because you don't care. 00:23:37.320 --> 00:23:41.580 You care mostly about what's it going to look like at a turning 00:23:41.580 --> 00:23:42.870 point. 00:23:42.870 --> 00:23:47.920 Or what is it going to look like when it returns to home base? 00:23:47.920 --> 00:23:50.050 And all sorts of insights come from that. 00:23:52.600 --> 00:23:54.340 OK. 00:23:54.340 --> 00:23:59.940 Now, I said, well, if we want to know the energy of this wave 00:23:59.940 --> 00:24:03.300 packet, well, we take the expectation value 00:24:03.300 --> 00:24:04.890 of the Hamiltonian. 00:24:04.890 --> 00:24:07.020 And the expectation of the Hamiltonian 00:24:07.020 --> 00:24:29.450 is c0 squared e0 plus c1 squared e1 plus c2 squared e2. 00:24:35.240 --> 00:24:37.010 These things are easy once you've gone 00:24:37.010 --> 00:24:38.930 through it a couple of times. 00:24:38.930 --> 00:24:42.760 Because what's happening is you have these factors, 00:24:42.760 --> 00:24:50.310 e to the minus i ej t over H bar. 00:24:50.310 --> 00:24:54.960 And they cancel when you do a psi star psi. 00:24:54.960 --> 00:24:58.830 Or they generate a difference, omega H bar omega, 00:24:58.830 --> 00:25:01.750 when you have different values of the energy. 00:25:01.750 --> 00:25:05.700 So this is something that you can derive really quickly. 00:25:05.700 --> 00:25:09.090 And the fact that your eigenfunction-- 00:25:09.090 --> 00:25:13.780 the psi's in your linear combination 00:25:13.780 --> 00:25:16.530 up there are eigenfunctions of the Hamiltonian. 00:25:16.530 --> 00:25:21.460 So every time the Hamiltonian operates in a wave function, 00:25:21.460 --> 00:25:24.510 it gives the energy times that wave function. 00:25:24.510 --> 00:25:27.930 And then you're taking the expectation value. 00:25:27.930 --> 00:25:32.010 And so we have the wave function time itself integrated. 00:25:32.010 --> 00:25:33.490 And that goes away. 00:25:33.490 --> 00:25:36.870 So after doing this a few times, you 00:25:36.870 --> 00:25:40.000 don't need to write the intermediate steps. 00:25:40.000 --> 00:25:41.760 And you shouldn't. 00:25:41.760 --> 00:25:45.397 Because you'll get lost in the forest of notation. 00:25:45.397 --> 00:25:46.980 Because one of the things you probably 00:25:46.980 --> 00:25:50.790 noticed in the last lecture is the equations got really big. 00:25:50.790 --> 00:25:52.500 And then we calculate something else. 00:25:52.500 --> 00:25:54.660 And it gets twice as big. 00:25:54.660 --> 00:25:56.750 And then all of a sudden, it all goes away. 00:25:56.750 --> 00:25:59.588 And that's what you want to be able to anticipate. 00:26:09.160 --> 00:26:09.660 OK. 00:26:09.660 --> 00:26:14.260 So you can really kill this half oscillator problem. 00:26:14.260 --> 00:26:16.210 There's nothing much happening except you 00:26:16.210 --> 00:26:18.430 create a localization. 00:26:18.430 --> 00:26:20.080 And when you take away the other-- 00:26:20.080 --> 00:26:23.410 when you restore the full oscillator, 00:26:23.410 --> 00:26:28.320 everything oscillates at omega. 00:26:28.320 --> 00:26:30.350 And so you have a whole bunch of terms 00:26:30.350 --> 00:26:35.240 contributing to the motion of the wave packet. 00:26:35.240 --> 00:26:36.410 And they're all very simple. 00:26:36.410 --> 00:26:38.659 Because they're all oscillating at the same frequency. 00:26:42.780 --> 00:26:47.420 Now, as an aside, I want to say there's a huge number of stuff 00:26:47.420 --> 00:26:49.590 that's up in this lecture that's going to be 00:26:49.590 --> 00:26:53.850 on the exam, a huge amount. 00:26:53.850 --> 00:26:58.260 So if for example you wanted to calculate something 00:26:58.260 --> 00:27:02.770 like x squared, well, fine. 00:27:02.770 --> 00:27:06.750 You know what the selection rule for x squared is. 00:27:06.750 --> 00:27:10.560 It's delta v of 0 plus or minus 2. 00:27:10.560 --> 00:27:14.820 And so this thing is going to generate some constant terms 00:27:14.820 --> 00:27:16.320 and some terms at 2 omega. 00:27:21.940 --> 00:27:22.620 OK. 00:27:22.620 --> 00:27:26.750 So there are a lot of things about the harmonic oscillator 00:27:26.750 --> 00:27:31.370 that make it really wonderful to consider a problem, even 00:27:31.370 --> 00:27:34.100 a complicated problem, which is not explicitly 00:27:34.100 --> 00:27:35.690 a harmonic oscillator problem. 00:27:35.690 --> 00:27:39.740 Because you can get everything so quickly without any 00:27:39.740 --> 00:27:42.215 thought after a little bit of investment. 00:27:44.830 --> 00:27:46.640 OK. 00:27:46.640 --> 00:27:50.120 Now, we haven't talked about electronic transitions 00:27:50.120 --> 00:27:51.590 and potential energy curves. 00:27:51.590 --> 00:27:52.910 But we will. 00:27:52.910 --> 00:27:55.130 And I think you know about them. 00:27:55.130 --> 00:27:57.850 And so we have some electronic ground state. 00:27:57.850 --> 00:28:01.020 And we have some electronically excited state. 00:28:01.020 --> 00:28:07.790 And so each of these states has a vibrational coordinate. 00:28:07.790 --> 00:28:13.420 And we can pretend that it's harmonic even if it's not. 00:28:13.420 --> 00:28:15.520 Because we build a framework treating 00:28:15.520 --> 00:28:17.260 them as harmonic, and then discover 00:28:17.260 --> 00:28:23.320 that there's discrepancies which we can fit to a model. 00:28:23.320 --> 00:28:25.780 And we can determine from the time 00:28:25.780 --> 00:28:27.340 dependence what that model is. 00:28:27.340 --> 00:28:28.090 OK. 00:28:28.090 --> 00:28:34.220 So we start out with a molecule in v equals 0. 00:28:37.890 --> 00:28:45.420 And there is a much repeated truism. 00:28:45.420 --> 00:28:48.870 Electrons move fast, nuclei slow. 00:28:48.870 --> 00:28:52.450 Transition is instantaneous, or nearly instantaneous, 00:28:52.450 --> 00:28:54.400 because it involves the electrons. 00:28:54.400 --> 00:28:59.280 So what ends up happening is you draw a vertical line. 00:28:59.280 --> 00:29:05.750 And now, you transfer this wave function to the upper state. 00:29:05.750 --> 00:29:08.930 You just move-- this is the probability amplitude 00:29:08.930 --> 00:29:15.170 distribution of the vibration. 00:29:15.170 --> 00:29:17.360 We transfer that to the excited state. 00:29:17.360 --> 00:29:19.880 And that's not an eigenstate, the excited state. 00:29:19.880 --> 00:29:23.770 It's a localized state, localized at a turning point. 00:29:23.770 --> 00:29:26.560 Now, you want to know-- 00:29:26.560 --> 00:29:34.330 so there's the Franck-Condon principle, 00:29:34.330 --> 00:29:36.930 that is just another way of saying electrons 00:29:36.930 --> 00:29:38.950 move fast, nuclei slow. 00:29:38.950 --> 00:29:44.460 And there is delta x equals 0 delta p equals 0. 00:29:47.980 --> 00:29:51.160 If the nuclear state can't change while the electron is 00:29:51.160 --> 00:29:58.810 jumping, then the coordinate and the momentum are both constant. 00:29:58.810 --> 00:30:01.480 So you're creating a wave packet. 00:30:01.480 --> 00:30:04.600 And the best place to create it is near a turning point. 00:30:04.600 --> 00:30:08.080 Because then, you can match the momentum of this guy 00:30:08.080 --> 00:30:11.890 to that zero point momentum, which you can calculate. 00:30:11.890 --> 00:30:13.660 You know how to calculate this. 00:30:13.660 --> 00:30:25.540 Because the potential energy curve is 1/2 k x squared. 00:30:25.540 --> 00:30:34.100 And the momentum is given by this energy difference here. 00:30:34.100 --> 00:30:35.530 And so you can work it out. 00:30:35.530 --> 00:30:36.280 It's in the notes. 00:30:36.280 --> 00:30:38.290 I don't want to write it down. 00:30:38.290 --> 00:30:40.330 So you're going to create something 00:30:40.330 --> 00:30:43.960 which is vertical and is not quite exactly 00:30:43.960 --> 00:30:45.550 at the turning point. 00:30:45.550 --> 00:30:48.130 Because you have to have a little bit of momentum 00:30:48.130 --> 00:30:51.270 to match the zero point momentum here. 00:30:51.270 --> 00:30:55.220 So you know everything about the initial state. 00:30:55.220 --> 00:30:59.035 And so you can calculate what it is by taking the overlap of v 00:30:59.035 --> 00:31:03.550 equals 0 of the ground state with all 00:31:03.550 --> 00:31:06.620 of the vibrational levels of the excited state. 00:31:06.620 --> 00:31:09.730 And so we have the coefficients of each 00:31:09.730 --> 00:31:12.760 of those vibrational levels in the excited state that 00:31:12.760 --> 00:31:15.530 makes this wave packet. 00:31:15.530 --> 00:31:17.690 Now, of course, this wave packet is 00:31:17.690 --> 00:31:20.660 going to move back and forth, back and forth. 00:31:23.450 --> 00:31:25.570 And if this were a harmonic oscillator, 00:31:25.570 --> 00:31:28.070 it would move harmonically. 00:31:28.070 --> 00:31:30.400 And so the only thing that would appear 00:31:30.400 --> 00:31:33.700 in the expectation value of x is going 00:31:33.700 --> 00:31:38.450 to be this motion at omega. 00:31:38.450 --> 00:31:41.000 Now, this is usually a relatively high 00:31:41.000 --> 00:31:42.110 vibrational level. 00:31:42.110 --> 00:31:45.860 And the molecule is not being harmonic here. 00:31:45.860 --> 00:31:49.460 So there are correction terms called anharmonicity terms. 00:31:52.550 --> 00:32:05.320 So we have the energy level expression plus H bar omega e 00:32:05.320 --> 00:32:06.620 xe-- 00:32:06.620 --> 00:32:13.600 that's one number-- plus 1/2 squared. 00:32:13.600 --> 00:32:15.960 So we have a linear term and a quadratic term. 00:32:15.960 --> 00:32:24.180 And this omega e xe is on the order of 0.02 times omega 00:32:24.180 --> 00:32:26.550 e, 2%. 00:32:26.550 --> 00:32:28.870 So it's a small thing. 00:32:28.870 --> 00:32:31.800 But if you're going to be allowing a wave 00:32:31.800 --> 00:32:36.520 packet to be built out of many vibrational levels, 00:32:36.520 --> 00:32:38.500 this guy is going to de-phase a little bit. 00:32:38.500 --> 00:32:40.300 So you go around and come back. 00:32:40.300 --> 00:32:42.460 And you can't quite have everybody 00:32:42.460 --> 00:32:45.230 back where they started. 00:32:45.230 --> 00:32:53.900 And so you'll see a decreasing amplitude here. 00:32:53.900 --> 00:33:00.480 And that's best looked at by the survival probability. 00:33:00.480 --> 00:33:03.151 And you'll see characteristic behavior in the survival 00:33:03.151 --> 00:33:03.650 probably. 00:33:10.570 --> 00:33:12.010 OK. 00:33:12.010 --> 00:33:14.620 But let's go a little bit deeper before I 00:33:14.620 --> 00:33:17.240 move on to-- what time is it? 00:33:17.240 --> 00:33:20.860 Oh, I'm doing OK. 00:33:20.860 --> 00:33:23.380 So let's just say our superposition 00:33:23.380 --> 00:33:30.790 state, the electronically excited state, 00:33:30.790 --> 00:33:35.410 is a combination of v equals 10 and v equals 11. 00:33:51.710 --> 00:34:02.660 So we know immediately that psi star t psi t-- 00:34:02.660 --> 00:34:07.370 this probability amplitude-- probability, yes, 00:34:07.370 --> 00:34:16.389 this probability is c10 squared psi 10 00:34:16.389 --> 00:34:31.380 squared plus c11 squared psi 11 squared plus 2 c10 c11 00:34:31.380 --> 00:34:38.510 psi 10 psi 11 cosine omega t. 00:34:38.510 --> 00:34:40.100 That's not very legible. 00:34:45.540 --> 00:34:48.670 Now, I'm playing fast and loose here. 00:34:48.670 --> 00:34:50.050 Because I've done this before. 00:34:50.050 --> 00:34:52.000 I know what disappears. 00:34:52.000 --> 00:34:55.030 And I know, if it's harmonic, you just get this. 00:34:55.030 --> 00:34:57.250 Well, if there's two states, this thing 00:34:57.250 --> 00:35:02.380 is really just the frequency associated 00:35:02.380 --> 00:35:05.220 with these two levels. 00:35:05.220 --> 00:35:07.440 OK. 00:35:07.440 --> 00:35:13.560 Now, once we have this, we can also generate the survival 00:35:13.560 --> 00:35:15.240 probability. 00:35:15.240 --> 00:35:17.570 Well, the survival probability, capital P 00:35:17.570 --> 00:35:30.390 of t, that's defined as the square modulus of psi star xt 00:35:30.390 --> 00:35:34.768 psi x0 dx. 00:35:38.920 --> 00:35:39.420 OK. 00:35:39.420 --> 00:35:42.100 And you can immediately write what that is going to be. 00:35:42.100 --> 00:35:47.680 It's going to be c10 squared. 00:35:47.680 --> 00:35:53.010 And we've integrated so the wave functions go away. 00:35:53.010 --> 00:36:08.220 And so we have c10 e to the i H bar 10.5 omega 00:36:08.220 --> 00:36:29.970 t divided by H bar plus c11 squared e to the minus i plus H 00:36:29.970 --> 00:36:37.001 bar 11.5 omega e over H bar. 00:36:37.001 --> 00:36:37.500 OK. 00:36:37.500 --> 00:36:38.980 Why did I made the mistake here? 00:36:42.010 --> 00:36:43.350 Well, we have a psi star. 00:36:45.960 --> 00:36:53.490 And so we're going to get the complex conjugate of e 00:36:53.490 --> 00:36:56.660 to the minus i omega. 00:36:56.660 --> 00:36:59.730 So you end up getting this. 00:36:59.730 --> 00:37:00.755 Practice that. 00:37:00.755 --> 00:37:02.380 AUDIENCE: That whole thing is [? the ?] 00:37:02.380 --> 00:37:02.700 modulus squared, right? 00:37:02.700 --> 00:37:03.450 ROBERT FIELD: Yes. 00:37:03.450 --> 00:37:04.721 I have that right. 00:37:04.721 --> 00:37:05.220 Exactly. 00:37:08.100 --> 00:37:16.320 So and now, lo and behold, we know what to do with this too. 00:37:16.320 --> 00:37:23.790 And so we're going to get c1 0 to the 4 plus c-- 00:37:23.790 --> 00:37:33.700 not 1-0, 10, c11 to the 4 plus 2 c10 00:37:33.700 --> 00:37:39.430 squared c11 squared cosine omega t. 00:37:43.690 --> 00:37:45.200 Isn't that neat? 00:37:45.200 --> 00:37:48.872 So we've got a whole bunch of constant terms. 00:37:48.872 --> 00:37:49.580 This is constant. 00:37:49.580 --> 00:37:53.660 Because it's square modulus and it's to the fourth power. 00:37:53.660 --> 00:37:59.480 And so all of these coefficients are positive. 00:37:59.480 --> 00:38:02.640 At t equals 0, this is 1. 00:38:02.640 --> 00:38:09.110 And so at t equals 0, the survival probability 00:38:09.110 --> 00:38:09.800 is at a maximum. 00:38:13.680 --> 00:38:16.775 And at some later time, that survival probability 00:38:16.775 --> 00:38:17.650 will be at a minimum. 00:38:24.530 --> 00:38:25.030 OK. 00:38:25.030 --> 00:38:31.540 And so you can say the maximum will occur at integer when 00:38:31.540 --> 00:38:43.500 omega t is equal to 2 n pi. 00:38:48.290 --> 00:38:52.100 So then the exponential factor is always 1. 00:38:55.690 --> 00:39:04.210 And we have a minimum when omega t is an odd multiple of pi. 00:39:04.210 --> 00:39:06.900 And all the exponential factors are minus 1. 00:39:14.660 --> 00:39:19.120 So we can also now look at the expectation 00:39:19.120 --> 00:39:24.960 value for x of t and p of t. 00:39:24.960 --> 00:39:26.860 And I'm going to just draw sketches. 00:39:33.950 --> 00:39:41.690 So for x of t, we have something that looks like this. 00:39:41.690 --> 00:39:45.387 This is at the left turning point. 00:39:45.387 --> 00:39:47.470 This is the right turning point, or near the right 00:39:47.470 --> 00:39:48.970 turning point. 00:39:48.970 --> 00:39:56.250 And this is at pi over 2 omega. 00:39:56.250 --> 00:39:59.920 This is at pi over omega. 00:39:59.920 --> 00:40:01.950 So that's the half oscillator point. 00:40:01.950 --> 00:40:07.110 Now, it starts out and the expectation value 00:40:07.110 --> 00:40:08.730 is at the left turning point. 00:40:08.730 --> 00:40:11.210 And it's not changing. 00:40:11.210 --> 00:40:13.970 The derivative of the expectation value 00:40:13.970 --> 00:40:17.138 with respect to t is 0. 00:40:17.138 --> 00:40:24.970 The momentum, which has a different phase-- 00:40:24.970 --> 00:40:28.940 the momentum starts out at 0 also. 00:40:28.940 --> 00:40:35.270 And at pi over 2 omega, it reaches a maximum. 00:40:35.270 --> 00:40:40.460 And at pi over omega, it reaches 9 again. 00:40:40.460 --> 00:40:44.270 But the important thing here is at t 00:40:44.270 --> 00:40:47.420 equals 0 the derivative of the momentum 00:40:47.420 --> 00:40:48.770 is as big as it can get. 00:40:52.510 --> 00:40:56.600 So what's that telling you? 00:40:56.600 --> 00:40:58.660 It's telling you this wave packet, 00:40:58.660 --> 00:41:02.320 as far as coordinate space is concerned, it's not moving at t 00:41:02.320 --> 00:41:04.120 equals 0. 00:41:04.120 --> 00:41:09.470 As far as momentum is concerned, it's moving like crazy. 00:41:09.470 --> 00:41:16.990 And so this survival probability is changing entirely dominated 00:41:16.990 --> 00:41:19.420 by the change in momentum, which is encoded in the wave 00:41:19.420 --> 00:41:23.780 function, which is neat. 00:41:23.780 --> 00:41:31.340 Because Newton's equation says that the time derivative 00:41:31.340 --> 00:41:39.200 of the momentum is equal to-- 00:41:39.200 --> 00:41:44.330 that's the mass times acceleration-- 00:41:44.330 --> 00:41:47.660 is equal to the force, which is minus 00:41:47.660 --> 00:41:55.912 the gradient of the potential-- 00:41:55.912 --> 00:41:57.860 the function of time. 00:41:57.860 --> 00:42:04.530 And for a harmonic oscillator, the gradient potential is x kx. 00:42:04.530 --> 00:42:15.760 And so it's telling you that we have this relationship 00:42:15.760 --> 00:42:19.060 between x and t and p. 00:42:19.060 --> 00:42:23.350 And we know what the momentum is. 00:42:23.350 --> 00:42:24.760 And I'm sorry. 00:42:24.760 --> 00:42:27.320 And so the change in the momentum, 00:42:27.320 --> 00:42:30.760 which is responsible for the change in the survival 00:42:30.760 --> 00:42:35.890 probability at t equals 0, is due to the gradient 00:42:35.890 --> 00:42:37.550 of the potential. 00:42:37.550 --> 00:42:40.390 So we're actually sampling what is 00:42:40.390 --> 00:42:44.320 the gradient of the potential at the left turning point. 00:42:44.320 --> 00:42:46.150 And often, for any kind of a problem, 00:42:46.150 --> 00:42:49.060 we want to know what kind of force 00:42:49.060 --> 00:42:53.470 is acting on the wave packet. 00:42:53.470 --> 00:42:57.540 There it is, classic mechanics embedded in quantum mechanics. 00:42:57.540 --> 00:43:00.640 It's really amazing. 00:43:00.640 --> 00:43:04.110 So if you have some way of measuring 00:43:04.110 --> 00:43:07.560 the survival probability near t equals 0, 00:43:07.560 --> 00:43:11.460 it's telling you what the slope of the potential 00:43:11.460 --> 00:43:12.660 is at the turning point. 00:43:22.575 --> 00:43:23.075 OK. 00:43:39.990 --> 00:43:44.490 Well, if we have an initial state which 00:43:44.490 --> 00:43:49.830 involves many eigenstates, the Hamiltonian, 00:43:49.830 --> 00:43:54.840 we still know that there's going to be some complicated behavior 00:43:54.840 --> 00:43:58.540 which is modulated by omega. 00:43:58.540 --> 00:44:03.620 So we have a cosine omega t always. 00:44:03.620 --> 00:44:05.610 And so no matter what kind of a wave 00:44:05.610 --> 00:44:09.750 function we make, if we start out at one turning point, 00:44:09.750 --> 00:44:13.230 it's going to go to the other turning point. 00:44:13.230 --> 00:44:16.230 And it'll keep coming back, and back, and back. 00:44:16.230 --> 00:44:20.610 And so one could imagine doing an experiment where-- 00:44:20.610 --> 00:44:24.881 let's just draw the excited state and some other repulsive 00:44:24.881 --> 00:44:25.380 state. 00:44:29.980 --> 00:44:33.520 So at this turning point, vertical transition 00:44:33.520 --> 00:44:37.690 to that repulsive state is within the range of your laser. 00:44:37.690 --> 00:44:41.230 And at this turning point, it's way high. 00:44:41.230 --> 00:44:43.810 And so what you can imagine doing 00:44:43.810 --> 00:44:52.830 is probing where this wave packet is as a function of time 00:44:52.830 --> 00:44:59.570 by having a probe pulse which creates dissociating fragments. 00:44:59.570 --> 00:45:05.040 And now, that is what Ahmed Zewail did. 00:45:05.040 --> 00:45:09.500 He talked about real dynamics in real time. 00:45:09.500 --> 00:45:12.620 So when the wave packet is here, it can't be dissociated. 00:45:12.620 --> 00:45:14.380 When it's here, it can. 00:45:14.380 --> 00:45:16.650 And you look at the fragments. 00:45:16.650 --> 00:45:17.890 Very simple. 00:45:17.890 --> 00:45:21.810 So there are lots of ways of taking these simple pictures 00:45:21.810 --> 00:45:26.080 of wave packets moving and saying, 00:45:26.080 --> 00:45:29.120 OK, I can use them to set up an experiment where 00:45:29.120 --> 00:45:32.490 I ask a very specific question. 00:45:32.490 --> 00:45:33.270 And I get answers. 00:45:33.270 --> 00:45:34.894 And I know what to do with the answers. 00:45:37.530 --> 00:45:47.560 Now, if it's not harmonic, you can still 00:45:47.560 --> 00:45:49.930 do Zewail's experiment or some other experiment. 00:45:49.930 --> 00:45:54.970 And you can ask, OK, the wave packet moved over here. 00:45:54.970 --> 00:45:56.960 And there's some dynamics that occurs. 00:45:56.960 --> 00:45:59.230 It dissociates or you do something. 00:45:59.230 --> 00:46:04.240 And when it comes back, it's not at the same amplitude. 00:46:04.240 --> 00:46:13.600 And so you can use the time history of these recurrences 00:46:13.600 --> 00:46:14.500 if it's harmonic. 00:46:14.500 --> 00:46:18.700 If it's not harmonic, then the recurrences-- even if nothing 00:46:18.700 --> 00:46:21.250 happens over here, the recurrences 00:46:21.250 --> 00:46:24.930 will be increasingly less perfect. 00:46:24.930 --> 00:46:27.202 And so you measure the anharmonicity. 00:46:35.240 --> 00:46:36.290 OK. 00:46:36.290 --> 00:46:36.990 Five minutes. 00:46:50.980 --> 00:46:54.460 Tunneling is a quantum mechanical phenomenon. 00:46:54.460 --> 00:46:58.450 And again, you want to use the simplest possible picture 00:46:58.450 --> 00:47:01.450 to understand the signature of tunneling. 00:47:01.450 --> 00:47:02.450 And so what do you do? 00:47:02.450 --> 00:47:04.300 Well, the simplest thing you can do 00:47:04.300 --> 00:47:07.460 is start with a harmonic oscillator. 00:47:07.460 --> 00:47:11.740 The next thing you do is you put a barrier in the middle. 00:47:11.740 --> 00:47:14.400 And you make it really thin. 00:47:14.400 --> 00:47:17.100 Thin is because you don't want to calculate the integral. 00:47:17.100 --> 00:47:22.410 You want to just say, oh yeah, I know that the wave function-- 00:47:22.410 --> 00:47:23.640 forget about the barrier. 00:47:23.640 --> 00:47:31.230 We know the magnitude of the wave function at the barrier, 00:47:31.230 --> 00:47:31.770 OK? 00:47:31.770 --> 00:47:41.040 And so if you have a state which has a maximum here, 00:47:41.040 --> 00:47:44.720 well, that means that it's feeling the barrier. 00:47:44.720 --> 00:47:47.480 And so you know what to do. 00:47:47.480 --> 00:47:51.760 If there is a node here, it doesn't know about the barrier. 00:47:51.760 --> 00:47:53.961 So you can do all sorts of things really fast. 00:47:53.961 --> 00:47:54.460 OK. 00:47:54.460 --> 00:47:58.520 So and it's harmonic because you're going to want to use 00:47:58.520 --> 00:47:59.835 a's and a-dagger's. 00:48:04.950 --> 00:48:09.930 So v equals 1, 3, 5. 00:48:09.930 --> 00:48:12.790 They have nodes here. 00:48:12.790 --> 00:48:14.620 They don't know about the barrier. 00:48:14.620 --> 00:48:19.460 Or if they do, it's a very modest effect. 00:48:19.460 --> 00:48:21.920 If you make this thin enough, you won't know about it all. 00:48:25.110 --> 00:48:36.650 And then v equals 0, 2, 2 4, et cetera-- 00:48:36.650 --> 00:48:37.850 well, they're maxima. 00:48:37.850 --> 00:48:40.857 Well, they have a local maximum here. 00:48:40.857 --> 00:48:42.690 And remember, this is a harmonic oscillator. 00:48:42.690 --> 00:48:45.770 So the big lobes are on the extremes. 00:48:45.770 --> 00:48:49.900 The smallest lobe is in the middle, but it's not 0. 00:48:49.900 --> 00:48:50.400 OK. 00:48:50.400 --> 00:48:52.600 There are also symmetry restrictions. 00:48:52.600 --> 00:48:56.230 This is a problem that has symmetry. 00:48:56.230 --> 00:48:59.280 We have even functions and odd functions. 00:48:59.280 --> 00:49:11.970 And for the even functions, d psi dx is equal to 0. 00:49:11.970 --> 00:49:18.970 And for the odd functions, psi of 0 is equal to 0. 00:49:24.080 --> 00:49:24.730 OK. 00:49:24.730 --> 00:49:28.121 So this is something that-- you choose the simplest problem. 00:49:28.121 --> 00:49:28.995 And you use symmetry. 00:49:31.670 --> 00:49:34.040 And bang, all of a sudden, you get fantastic insights. 00:49:45.500 --> 00:49:48.390 So let's look at what happens to the even functions. 00:49:48.390 --> 00:49:53.100 So let's just draw a picture of the harmonic oscillator energy 00:49:53.100 --> 00:49:53.600 levels. 00:49:59.940 --> 00:50:04.120 And we have a barrier, which we say maybe goes that high. 00:50:07.170 --> 00:50:10.180 And so we have 1. 00:50:10.180 --> 00:50:12.380 And we have 3. 00:50:12.380 --> 00:50:14.180 And we have 5. 00:50:14.180 --> 00:50:16.804 They're basically not affected by the barrier. 00:50:16.804 --> 00:50:18.470 Because they have a node at the barrier. 00:50:21.300 --> 00:50:25.060 And then we have v equals 0. 00:50:25.060 --> 00:50:26.250 What happens to v equals 0? 00:50:26.250 --> 00:50:29.000 Well, it hits this barrier. 00:50:29.000 --> 00:50:33.340 And it's got a large amplitude. 00:50:33.340 --> 00:50:38.890 And it can't accrue more phase as you go through the barrier. 00:50:38.890 --> 00:50:41.050 So when it hits the other turning point, 00:50:41.050 --> 00:50:43.130 it doesn't satisfy the boundary conditions. 00:50:43.130 --> 00:50:47.420 It'll go to infinity at x equals positive infinity. 00:50:47.420 --> 00:50:54.010 So it has to be shifted so that the boundary conditions 00:50:54.010 --> 00:50:56.230 at the turning points are met. 00:50:56.230 --> 00:50:58.450 And the only way that can happen is 00:50:58.450 --> 00:51:01.420 it gets shifted up in energy a lot. 00:51:01.420 --> 00:51:05.520 So v equals 0 is here. 00:51:05.520 --> 00:51:08.025 Now, it can't be shifted above v equals 1. 00:51:08.025 --> 00:51:10.710 Because that would violate the node rule. 00:51:10.710 --> 00:51:13.650 And if you draw a picture of the wave function for v 00:51:13.650 --> 00:51:17.400 equals 0 in the boundary region, the wave function 00:51:17.400 --> 00:51:19.710 has to look something-- well, it has 00:51:19.710 --> 00:51:23.110 to look something like that. 00:51:23.110 --> 00:51:25.830 So we have two lobes. 00:51:25.830 --> 00:51:29.550 And it's trying to make a node, but it can't. 00:51:29.550 --> 00:51:32.040 It's a decreasing exponential. 00:51:32.040 --> 00:51:36.180 I mean, if you have a wave going into this region 00:51:36.180 --> 00:51:39.300 of the barrier, you can have an increasing exponential 00:51:39.300 --> 00:51:41.770 or a decreasing exponential. 00:51:41.770 --> 00:51:44.400 You can never satisfy continuity of the wave function 00:51:44.400 --> 00:51:46.560 with the increasing exponential. 00:51:46.560 --> 00:51:49.440 But you can with the decreasing exponential. 00:51:49.440 --> 00:51:53.550 And so the height of the barrier determines how close this guy 00:51:53.550 --> 00:51:56.520 comes to having a node. 00:51:56.520 --> 00:51:57.360 It never will. 00:52:00.080 --> 00:52:03.890 And how does this compare to the wave function for v equals 1? 00:52:03.890 --> 00:52:07.670 Well, the wave function for v equals 1 looks like this. 00:52:07.670 --> 00:52:11.300 The amplitude in this lobe and the amplitude in that lobe 00:52:11.300 --> 00:52:12.170 are the same. 00:52:12.170 --> 00:52:14.360 But the sign is reversed. 00:52:14.360 --> 00:52:17.750 And so basically, this picture tells you 00:52:17.750 --> 00:52:20.900 that these two levels have almost exactly the same energy. 00:52:23.470 --> 00:52:27.010 So v equals 0 is shifted up. 00:52:27.010 --> 00:52:30.940 v equals 2 is shifted up, but not quite so much. 00:52:30.940 --> 00:52:36.410 And v 4 is shifted up hardly at all. 00:52:36.410 --> 00:52:40.070 And so you get what's called level staggering. 00:52:40.070 --> 00:52:45.360 And this level staggering is the signature of tunneling. 00:52:45.360 --> 00:52:50.070 This is how we know about tunneling, the only way we 00:52:50.070 --> 00:52:52.510 know about tunneling. 00:52:52.510 --> 00:52:54.210 So we measure the energy levels. 00:52:54.210 --> 00:52:59.040 And it tells us how are the wave functions sampling 00:52:59.040 --> 00:53:00.750 this barrier. 00:53:00.750 --> 00:53:03.710 Now, this is a childish barrier. 00:53:03.710 --> 00:53:05.820 And there is a real barrier-- 00:53:05.820 --> 00:53:08.610 molecules isomerize. 00:53:08.610 --> 00:53:10.770 And they isomerize over a barrier. 00:53:10.770 --> 00:53:13.410 And the barrier isn't necessarily at x equals 0. 00:53:13.410 --> 00:53:15.030 But if you understand this problem, 00:53:15.030 --> 00:53:17.650 you can deal with isomerization. 00:53:17.650 --> 00:53:20.400 Now, I'm an author of a paper that's 00:53:20.400 --> 00:53:23.570 just appearing in Science in the next few weeks 00:53:23.570 --> 00:53:28.510 on the isomerization from vinylidene to acetylene. 00:53:28.510 --> 00:53:29.635 There's a barrier involved. 00:53:32.470 --> 00:53:37.510 So these sorts of pictures are important for understanding 00:53:37.510 --> 00:53:39.650 those sorts of phenomena. 00:53:39.650 --> 00:53:43.210 Now, so I'm done really. 00:53:43.210 --> 00:53:48.340 So what we are now encountering with the time 00:53:48.340 --> 00:53:52.300 dependent Schrodinger equation is discovering 00:53:52.300 --> 00:53:58.680 how dynamics, like tunneling, is encoded 00:53:58.680 --> 00:54:01.060 in an eigenstate spectrum. 00:54:01.060 --> 00:54:06.850 Because the encoding is level staggering eigenstates. 00:54:06.850 --> 00:54:11.080 So people normally have this naive idea 00:54:11.080 --> 00:54:19.420 that eigenstate time independent Hamiltonian spectroscopy 00:54:19.420 --> 00:54:21.820 does not sample dynamics because it's only 00:54:21.820 --> 00:54:23.620 measuring energy levels. 00:54:23.620 --> 00:54:26.890 Real dynamical processes are time dependent. 00:54:26.890 --> 00:54:31.000 But the dynamics is encoded in energy level patterns. 00:54:31.000 --> 00:54:36.650 And that is actually my signature 00:54:36.650 --> 00:54:39.440 in experimental spectroscopy. 00:54:39.440 --> 00:54:42.050 I've looked for ways in which dynamics is 00:54:42.050 --> 00:54:45.140 encoded in eigenstate spectra. 00:54:45.140 --> 00:54:49.340 And chemists are interested in dynamics much more than 00:54:49.340 --> 00:54:51.520 in structure. 00:54:51.520 --> 00:54:53.540 So that's it for today. 00:54:53.540 --> 00:54:57.660 I will see you on Wednesday.