WEBVTT 00:00:00.120 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.660 To make a donation or to view additional materials 00:00:12.660 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.850 at ocw.mit.edu. 00:00:21.310 --> 00:00:26.332 PROFESSOR: The outline for today is a little bit more review 00:00:26.332 --> 00:00:28.540 of feeling the power of the creation and annihilation 00:00:28.540 --> 00:00:31.420 operators. 00:00:31.420 --> 00:00:35.320 They enable you to do anything with harmonic oscillators, 00:00:35.320 --> 00:00:37.310 really fast. 00:00:37.310 --> 00:00:42.760 And so you don't spend time thinking 00:00:42.760 --> 00:00:44.860 about how to do the math. 00:00:44.860 --> 00:00:48.760 You think about the meaning of the problem and understanding 00:00:48.760 --> 00:00:51.940 how to manipulate, or how to understand 00:00:51.940 --> 00:00:54.780 harmonic oscillators. 00:00:54.780 --> 00:00:56.740 And then the exciting thing. 00:00:56.740 --> 00:00:59.790 The real Schrodinger equation is not the one 00:00:59.790 --> 00:01:01.160 we've been playing with. 00:01:01.160 --> 00:01:04.060 It's the time dependent Schrodinger equation. 00:01:04.060 --> 00:01:09.330 And I like to introduce the time independent one 00:01:09.330 --> 00:01:15.840 first, because it enables you to sort of develop some insight 00:01:15.840 --> 00:01:18.030 and assemble some of the tools before we 00:01:18.030 --> 00:01:21.450 hit the really serious stuff. 00:01:21.450 --> 00:01:25.970 And so I will talk about this and use the time dependent 00:01:25.970 --> 00:01:28.740 Schrodinger equation to show what 00:01:28.740 --> 00:01:37.600 it takes to get a motion of this product of the psi star psi. 00:01:37.600 --> 00:01:38.350 What does it take? 00:01:38.350 --> 00:01:42.750 It takes a superposition state consisting of at least 00:01:42.750 --> 00:01:45.660 two different energies. 00:01:45.660 --> 00:01:48.830 Then this normalization integral-- 00:01:48.830 --> 00:01:52.940 well, you'd expect normalization, or something 00:01:52.940 --> 00:01:56.150 like this to be preserved. 00:01:56.150 --> 00:02:00.910 And in fact, the normalization is time independent. 00:02:00.910 --> 00:02:04.910 Then, if we calculate-- 00:02:04.910 --> 00:02:08.570 now, this is a vector rather than just a simple coordinate. 00:02:08.570 --> 00:02:13.810 If we calculate the expectation value of the position, 00:02:13.810 --> 00:02:19.610 and the expectation value of the momentum, 00:02:19.610 --> 00:02:23.030 Ehrenfest's theorem tells you that this 00:02:23.030 --> 00:02:25.820 is going to describe the motion of the center of the wave 00:02:25.820 --> 00:02:27.170 packet. 00:02:27.170 --> 00:02:30.380 And the motion of the center of the wave packet 00:02:30.380 --> 00:02:32.720 is following Newton's laws. 00:02:32.720 --> 00:02:34.050 No surprise. 00:02:34.050 --> 00:02:37.490 And so we're seeing how classical mechanics 00:02:37.490 --> 00:02:40.680 is given back to you once we start 00:02:40.680 --> 00:02:43.470 making particle-like states. 00:02:43.470 --> 00:02:46.380 We can see how these particle-like states evolve. 00:02:46.380 --> 00:02:51.510 But there is a lot more than just particle-like behavior. 00:02:51.510 --> 00:02:56.070 You would, perhaps, challenge any outfielder for the Red Sox 00:02:56.070 --> 00:03:00.810 to go beyond calculating the center of the wave packet. 00:03:00.810 --> 00:03:02.149 That's what they know to do. 00:03:02.149 --> 00:03:03.690 But they don't know some other things 00:03:03.690 --> 00:03:06.280 like survival probability, how fast 00:03:06.280 --> 00:03:09.870 does the wave packet move away from its birthplace, 00:03:09.870 --> 00:03:15.360 or the wave packet does stuff and wanders around and comes 00:03:15.360 --> 00:03:18.450 back and rephases sometimes. 00:03:18.450 --> 00:03:22.080 And sometimes it just dephases. 00:03:22.080 --> 00:03:25.260 And these are words that you're going to want 00:03:25.260 --> 00:03:28.030 to put into your vocabulary. 00:03:28.030 --> 00:03:30.900 So there's a lot of really beautiful stuff 00:03:30.900 --> 00:03:33.480 when we start looking at the time dependent Schrodinger 00:03:33.480 --> 00:03:34.860 equation. 00:03:34.860 --> 00:03:40.770 And we are going to mostly consider the time dependent 00:03:40.770 --> 00:03:43.950 Schrodinger equation when the Hamiltonian is 00:03:43.950 --> 00:03:47.460 independent of time, because we can get our arms around that 00:03:47.460 --> 00:03:48.800 really easily. 00:03:48.800 --> 00:03:51.390 But when the Hamiltonian is dependent on time, 00:03:51.390 --> 00:03:54.450 then it opens up a world of complexity 00:03:54.450 --> 00:03:59.400 that is really best left to a more advanced course. 00:04:02.860 --> 00:04:09.210 But how do molecules get excited from one state to another? 00:04:09.210 --> 00:04:11.300 A time dependent Hamiltonian. 00:04:11.300 --> 00:04:15.440 So we are going to have to at least briefly talk 00:04:15.440 --> 00:04:18.800 about a time dependent Hamiltonian and what it does. 00:04:18.800 --> 00:04:20.959 And you'll see that eventually. 00:04:26.480 --> 00:04:29.800 I'm first going to do a little bit of review 00:04:29.800 --> 00:04:33.280 of the a's and a daggers. 00:04:33.280 --> 00:04:36.310 So the most important thing is, we're 00:04:36.310 --> 00:04:41.860 going to have some problem where we have this x operator 00:04:41.860 --> 00:04:44.810 to some integer power. 00:04:44.810 --> 00:04:48.220 And so we need to be able to relate that 00:04:48.220 --> 00:04:49.840 to the a's and a daggers. 00:05:00.750 --> 00:05:04.800 I'll drop the hats, because we know they're there. 00:05:04.800 --> 00:05:09.390 And one of the nice things is, if we have this operator 00:05:09.390 --> 00:05:16.010 to the nth power, we have this co-factor to the nth power. 00:05:16.010 --> 00:05:17.762 So we don't ever have to deal with it 00:05:17.762 --> 00:05:19.970 until the end of the problem when we say, oh, wow, we 00:05:19.970 --> 00:05:21.230 had x to the n. 00:05:21.230 --> 00:05:25.621 Well, we have then n h bar. 00:05:25.621 --> 00:05:26.120 I'm sorry. 00:05:26.120 --> 00:05:30.042 We have h bar over 2 mu n n over 2 power. 00:05:32.900 --> 00:05:34.940 So it really saves a lot of writing. 00:05:34.940 --> 00:05:38.150 And it means you solve one problem 00:05:38.150 --> 00:05:40.160 for a harmonic oscillator. 00:05:40.160 --> 00:05:43.110 And you've solved it for all harmonic oscillators. 00:05:43.110 --> 00:05:45.990 And this is just details about what is the mass 00:05:45.990 --> 00:05:50.830 and what is the force constant, which you need to know. 00:05:50.830 --> 00:05:52.430 And there is a similar thing for p. 00:05:54.990 --> 00:05:57.360 So that opens the door. 00:05:57.360 --> 00:06:00.720 It says, OK, we have some problem involving 00:06:00.720 --> 00:06:03.765 the coordinate and the momentum, some function of the coordinate 00:06:03.765 --> 00:06:08.130 and momentum, and we want to know things 00:06:08.130 --> 00:06:10.290 about the coordinate and momentum. 00:06:10.290 --> 00:06:15.070 And so we use this operator algebra. 00:06:15.070 --> 00:06:27.760 And so a operating on psi v gives v square root v minus 1. 00:06:27.760 --> 00:06:29.490 And a dagger, we know that. 00:06:29.490 --> 00:06:31.080 So that's hardwired. 00:06:31.080 --> 00:06:35.400 But suppose we wanted a to the 5th, 00:06:35.400 --> 00:06:40.810 operating on psi v. Well, what do we do? 00:06:40.810 --> 00:06:46.720 Well, we start, and we start operating on... 00:06:46.720 --> 00:06:48.900 You know, if this were a complicated operator, 00:06:48.900 --> 00:06:52.440 we would take the rightmost piece of that operator 00:06:52.440 --> 00:06:54.430 and operate on the wave function. 00:06:54.430 --> 00:07:02.260 And that will give us v and then v minus 1 and then v minus 2, 00:07:02.260 --> 00:07:04.154 v minus 3. 00:07:04.154 --> 00:07:05.070 I've got five of them. 00:07:05.070 --> 00:07:06.495 1, 2, 3, 4, 5. 00:07:06.495 --> 00:07:08.640 v minus 4. 00:07:08.640 --> 00:07:12.080 Square root psi v minus 5. 00:07:15.080 --> 00:07:15.860 OK. 00:07:15.860 --> 00:07:17.450 It's mechanical. 00:07:17.450 --> 00:07:19.370 You don't remember this. 00:07:19.370 --> 00:07:23.720 You generate this one step at a time. 00:07:23.720 --> 00:07:25.570 And it's automatic. 00:07:25.570 --> 00:07:27.110 And so it doesn't stress your brain. 00:07:27.110 --> 00:07:28.430 You can be thinking about the next thing 00:07:28.430 --> 00:07:29.846 while you're writing that garbage. 00:07:37.290 --> 00:07:40.320 We have this number operator, which 00:07:40.320 --> 00:07:43.920 is a friend, because it enables you to just 00:07:43.920 --> 00:07:45.840 get rid of a bunch of terms. 00:07:45.840 --> 00:07:50.340 The number operator is a dagger a, and the number operator 00:07:50.340 --> 00:07:58.660 operating on psi v gives v psi v. 00:07:58.660 --> 00:08:03.790 If we want any harmonic oscillator function, 00:08:03.790 --> 00:08:15.190 we can operate on psi 0 with a dagger to the v power. 00:08:15.190 --> 00:08:18.580 And then we have to correct, because you 00:08:18.580 --> 00:08:24.420 know that this operating v times on this will give psi v. 00:08:24.420 --> 00:08:27.730 But it will also give a bunch of garbage, right? 00:08:27.730 --> 00:08:29.800 And you want to cancel that garbage. 00:08:29.800 --> 00:08:32.919 And so you write v factorial minus 1/2. 00:08:36.980 --> 00:08:39.409 And so that gives you the normalized function. 00:08:39.409 --> 00:08:43.100 So these are really, really simple things. 00:08:43.100 --> 00:08:45.954 And most of them, once you've thought about it a little bit, 00:08:45.954 --> 00:08:47.120 you can figure out yourself. 00:08:51.790 --> 00:08:53.580 OK. 00:08:53.580 --> 00:09:01.590 Now, many problems involve x, x cubed, x to the 4th. 00:09:01.590 --> 00:09:05.340 And we know that x is this. 00:09:05.340 --> 00:09:08.046 And x squared is this squared. 00:09:08.046 --> 00:09:10.020 And x cubed is this cubed. 00:09:10.020 --> 00:09:12.580 And so we have to do a little algebra to simplify. 00:09:12.580 --> 00:09:24.060 And what we want to do is simplify to sum of terms 00:09:24.060 --> 00:09:31.000 according to delta v selection rule. 00:09:35.790 --> 00:09:38.460 So there is some algebra that we do, 00:09:38.460 --> 00:09:44.780 when we have, say, a dagger, a dagger, a, a, a dagger. 00:09:44.780 --> 00:09:49.010 But we know that three a daggers and two a's means 00:09:49.010 --> 00:09:50.330 delta v of plus 1. 00:09:54.780 --> 00:09:59.110 And that came from x to the 5th power. 00:09:59.110 --> 00:10:02.290 But you get a lot of terms from x to the 5th power. 00:10:02.290 --> 00:10:04.130 And you have to simplify them. 00:10:04.130 --> 00:10:08.260 And in order to do that, you use this commutation rule-- 00:10:08.260 --> 00:10:13.857 a, a dagger is equal to 1. 00:10:13.857 --> 00:10:14.940 And that rearranges terms. 00:10:14.940 --> 00:10:16.830 So all of the work you do when you're 00:10:16.830 --> 00:10:20.820 faced with a problem involving integrals 00:10:20.820 --> 00:10:24.990 of integer powers of x and p for a harmonic oscillator 00:10:24.990 --> 00:10:29.700 is playing around with moving the a's and a daggers 00:10:29.700 --> 00:10:32.220 around, so that you have all the terms that 00:10:32.220 --> 00:10:37.680 have the same selection rule compressed into one term. 00:10:37.680 --> 00:10:38.650 That's the work. 00:10:38.650 --> 00:10:39.990 It's not much. 00:10:39.990 --> 00:10:44.430 And once you've done it for x squared, x cubed and x 4th, 00:10:44.430 --> 00:10:47.040 you've done it as much as you'll ever need to do. 00:10:47.040 --> 00:10:48.220 And that's it. 00:10:48.220 --> 00:10:49.370 That's the end. 00:10:49.370 --> 00:10:49.870 OK. 00:10:54.390 --> 00:11:00.070 Now here's an example of a problem 00:11:00.070 --> 00:11:01.850 that's a little bit tricky. 00:11:01.850 --> 00:11:06.350 And it's sort of right at the borderline of what 00:11:06.350 --> 00:11:08.920 I might use on the exam. 00:11:08.920 --> 00:11:15.840 So we have this, we have a dagger to the m power. 00:11:15.840 --> 00:11:18.430 A to the n power. 00:11:18.430 --> 00:11:20.330 Psi. 00:11:20.330 --> 00:11:20.910 OK. 00:11:20.910 --> 00:11:22.870 Now, here. 00:11:22.870 --> 00:11:26.290 What v is going to give a non-zero integral? 00:11:26.290 --> 00:11:28.690 We have a dagger to the m. 00:11:28.690 --> 00:11:31.120 And we have a to the n. 00:11:31.120 --> 00:11:41.570 And so that's going to be v minus n plus m, right? 00:11:41.570 --> 00:11:47.430 Because we lose n quanta, because of this. 00:11:47.430 --> 00:11:49.440 And we gain m quanta because of that. 00:11:52.180 --> 00:11:54.220 And now, well, that's good. 00:11:54.220 --> 00:11:55.810 You've used the selection rule. 00:11:55.810 --> 00:11:58.890 Now, how do we write out this interval? 00:11:58.890 --> 00:12:01.960 And there is a little bit of art there too. 00:12:01.960 --> 00:12:07.240 Because we have a whole bunch of terms. 00:12:07.240 --> 00:12:11.360 We have n plus m terms in the square root. 00:12:11.360 --> 00:12:15.250 And how do you generate them without getting lost? 00:12:18.721 --> 00:12:19.220 Yes? 00:12:19.220 --> 00:12:20.886 AUDIENCE: I think you have it backwards. 00:12:20.886 --> 00:12:24.512 Shouldn't it be plus n minus n? 00:12:24.512 --> 00:12:26.920 It means you're going to lower it n times. 00:12:26.920 --> 00:12:27.950 PROFESSOR: Yes. 00:12:27.950 --> 00:12:30.410 Yes. 00:12:30.410 --> 00:12:32.190 I wonder what I had in my notes. 00:12:32.190 --> 00:12:33.290 This is wrong. 00:12:39.980 --> 00:12:41.690 OK. 00:12:41.690 --> 00:12:49.900 So this has to withstand this and this. 00:12:49.900 --> 00:12:51.730 And so, yes. 00:12:51.730 --> 00:12:52.230 OK. 00:12:52.230 --> 00:12:54.810 However you remember it, you've got to do it right. 00:12:54.810 --> 00:13:00.430 And now we have the actual matrix element. 00:13:00.430 --> 00:13:07.950 And so the first term is going to be what does a n do to that? 00:13:07.950 --> 00:13:10.080 So you start on the right. 00:13:10.080 --> 00:13:13.460 And you start building up this way. 00:13:13.460 --> 00:13:19.830 And so the first term is going to be, what does a do to this? 00:13:19.830 --> 00:13:24.950 Well, it's going to leave it alone. 00:13:24.950 --> 00:13:27.440 But it's going to lower v. 00:13:27.440 --> 00:13:32.060 So we have v plus n minus m. 00:13:32.060 --> 00:13:37.790 And then we have v plus n minus m minus 1, 00:13:37.790 --> 00:13:39.995 et cetera, until we have n terms. 00:13:45.740 --> 00:13:49.940 And then we start going back up, because we're now 00:13:49.940 --> 00:13:53.160 dealing with this. 00:13:53.160 --> 00:13:56.410 And so I'm not going to write the rest of this. 00:13:56.410 --> 00:14:00.760 Maybe this is going to be a problem I start the exam with. 00:14:04.180 --> 00:14:05.500 So you don't want to get lost. 00:14:05.500 --> 00:14:08.080 And the main thing is, you have these operators. 00:14:08.080 --> 00:14:12.910 And you start operating on the right, and one step at a time. 00:14:12.910 --> 00:14:16.060 And this is a little tricky because you're 00:14:16.060 --> 00:14:18.130 changing the quantum number, and you're 00:14:18.130 --> 00:14:20.100 changing the wave function. 00:14:20.100 --> 00:14:22.420 And you have to keep both in mind, 00:14:22.420 --> 00:14:25.610 but you're only writing down this. 00:14:25.610 --> 00:14:26.110 OK? 00:14:35.330 --> 00:14:38.330 I want to save enough time so that we can actually 00:14:38.330 --> 00:14:41.510 do the time dependent Schrodinger equation. 00:14:41.510 --> 00:14:43.470 The time dependent Schrodinger equation. 00:14:43.470 --> 00:14:45.340 H psi. 00:14:55.250 --> 00:14:57.530 It still looks pretty simple. 00:14:57.530 --> 00:15:02.130 Instead of e psi here, we have this thing. 00:15:02.130 --> 00:15:04.500 This is the time dependent Schrodinger equation. 00:15:04.500 --> 00:15:05.460 This is it. 00:15:05.460 --> 00:15:07.170 This is quantum mechanics. 00:15:07.170 --> 00:15:09.900 Everything that comes from quantum mechanics 00:15:09.900 --> 00:15:11.850 starts with this. 00:15:11.850 --> 00:15:15.270 When we don't care about time, we 00:15:15.270 --> 00:15:17.760 can use the time independent Schrodinger equation. 00:15:17.760 --> 00:15:19.950 But when we do care about time, we 00:15:19.950 --> 00:15:22.480 have to be a little bit careful. 00:15:22.480 --> 00:15:25.030 So this is the real Schrodinger equation. 00:15:25.030 --> 00:15:27.250 And notice that I'm using a capital 00:15:27.250 --> 00:15:33.900 psi, rather than a lower case, or less decorated psi. 00:15:33.900 --> 00:15:37.890 And so this is usually used to indicate the time dependent 00:15:37.890 --> 00:15:38.910 Schrodinger equation. 00:15:38.910 --> 00:15:40.680 It's time dependent wave function. 00:15:40.680 --> 00:15:46.200 This is used to indicate the time independent equation. 00:15:46.200 --> 00:15:49.520 Now, if the Hamiltonian-- 00:15:49.520 --> 00:15:51.180 and this is wonderful-- 00:15:51.180 --> 00:15:54.940 if the Hamiltonian is independent of time, 00:15:54.940 --> 00:16:03.010 then if we know the solutions psi n en. 00:16:03.010 --> 00:16:05.950 If we know all of these solutions, 00:16:05.950 --> 00:16:07.940 then there's nothing new. 00:16:07.940 --> 00:16:10.900 We just are repackaging the stuff 00:16:10.900 --> 00:16:13.900 that we know from the time independent Schrodinger 00:16:13.900 --> 00:16:15.430 equation. 00:16:15.430 --> 00:16:17.840 So the first thing I want to do is to show you 00:16:17.840 --> 00:16:24.180 that if the Hamiltonian is independent, 00:16:24.180 --> 00:16:30.090 we can always write a solution to the time 00:16:30.090 --> 00:16:36.180 dependent Schrodinger equation-- e to the minus i e n 00:16:36.180 --> 00:16:42.690 t over h bar psi n x. 00:16:45.330 --> 00:16:52.110 So this, for a time independent Hamiltonian, 00:16:52.110 --> 00:16:55.661 is always the solution of the time dependent Schrodinger 00:16:55.661 --> 00:16:56.160 equation. 00:17:02.010 --> 00:17:04.319 So we're just using this stuff that we know, 00:17:04.319 --> 00:17:06.210 or at least we barely know, because we just 00:17:06.210 --> 00:17:08.450 started playing the game. 00:17:08.450 --> 00:17:12.089 But we can manipulate to see all sorts of useful stuff. 00:17:12.089 --> 00:17:12.589 OK. 00:17:12.589 --> 00:17:16.319 So let's show if this form satisfies the Schrodinger 00:17:16.319 --> 00:17:17.359 equation. 00:17:17.359 --> 00:17:25.339 So we have ih bar partial PSI with respect to t. 00:17:25.339 --> 00:17:28.339 So we get an ih bar. 00:17:28.339 --> 00:17:32.660 And then we take the partial with respect to t. 00:17:32.660 --> 00:17:34.370 This is independent of time. 00:17:34.370 --> 00:17:36.290 This has time [INAUDIBLE] And so we 00:17:36.290 --> 00:17:42.830 get a minus i e n over h bar. 00:17:42.830 --> 00:17:50.060 And then we get e to the minus i e n t over h bar psi n of x. 00:17:56.090 --> 00:17:56.900 Well. 00:17:56.900 --> 00:17:59.040 So, let's put this together. 00:17:59.040 --> 00:18:01.520 We have an i times i times minus 1. 00:18:01.520 --> 00:18:02.840 So that's plus 1. 00:18:02.840 --> 00:18:04.540 We have an h bar in the numerator and h 00:18:04.540 --> 00:18:06.180 bar in the denominator. 00:18:06.180 --> 00:18:07.160 That's 1. 00:18:07.160 --> 00:18:14.690 And so what we end up getting is e n e to the minus i e n 00:18:14.690 --> 00:18:17.090 t over h bar psi n. 00:18:20.054 --> 00:18:24.980 Well, this is psi. 00:18:28.320 --> 00:18:37.020 This function here is the solution to the time 00:18:37.020 --> 00:18:38.850 dependent Schrodinger equation. 00:18:38.850 --> 00:18:40.080 How do we know that? 00:18:40.080 --> 00:18:44.700 Well, we have this factor, h psi. 00:18:44.700 --> 00:18:50.200 Well, h doesn't operate on either the i e n t. 00:18:50.200 --> 00:18:57.280 So what we have is that we get E n e to the i omega t. 00:19:01.118 --> 00:19:01.618 Sorry. 00:19:01.618 --> 00:19:03.546 I'm jumping ahead. 00:19:03.546 --> 00:19:07.386 i e n t over h bar psi. 00:19:16.810 --> 00:19:18.190 All right. 00:19:18.190 --> 00:19:23.770 So, if we apply the Hamiltonian to this function, 00:19:23.770 --> 00:19:26.500 we get just e n times the function. 00:19:31.530 --> 00:19:35.240 And that's what we got when we did i h bar times a partial 00:19:35.240 --> 00:19:37.040 with respect to t. 00:19:37.040 --> 00:19:41.900 So what that shows is that this form always 00:19:41.900 --> 00:19:46.220 satisfies the time dependent Schrodinger equation, provided 00:19:46.220 --> 00:19:49.630 that the Hamiltonian is independent of time. 00:19:49.630 --> 00:19:53.400 Now, that's a large range of problems, things 00:19:53.400 --> 00:19:55.630 that we need to understand. 00:19:55.630 --> 00:19:58.920 But it's not the whole potato, because the Hamiltonian 00:19:58.920 --> 00:20:01.560 is often dependent on time. 00:20:01.560 --> 00:20:06.570 But it enables us to build up insight, and then 00:20:06.570 --> 00:20:10.890 treat the time dependence as a perturbation. 00:20:10.890 --> 00:20:13.860 And we're going to do perturbation theory in the time 00:20:13.860 --> 00:20:15.994 independent world. 00:20:15.994 --> 00:20:17.910 And then we're going to do perturbation theory 00:20:17.910 --> 00:20:21.470 a little bit in the time dependent world. 00:20:21.470 --> 00:20:21.970 OK. 00:20:21.970 --> 00:20:26.440 So what we always do here is we solve a familiar problem, 00:20:26.440 --> 00:20:29.050 and then we say, OK, well, there's something more 00:20:29.050 --> 00:20:31.400 to this familiar problem. 00:20:31.400 --> 00:20:34.410 And so we treat that as something extra. 00:20:34.410 --> 00:20:36.250 And we work out the formalism for dealing 00:20:36.250 --> 00:20:38.210 with that extra thing. 00:20:38.210 --> 00:20:39.700 But before we do the extra thing, 00:20:39.700 --> 00:20:42.460 we have to really kill the problem that 00:20:42.460 --> 00:20:45.950 is within our grasp. 00:20:45.950 --> 00:20:46.450 OK. 00:20:49.470 --> 00:20:54.630 So now, our job is to just explore 00:20:54.630 --> 00:20:55.920 what we've really got here. 00:20:58.840 --> 00:21:04.750 So the first problem is motion. 00:21:08.140 --> 00:21:16.975 So we have psi star x and t psi x and t. 00:21:20.640 --> 00:21:22.350 So we have this thing, which we're not 00:21:22.350 --> 00:21:23.640 going to integrate yet. 00:21:29.790 --> 00:21:32.220 Well, when is this thing going to move? 00:21:35.000 --> 00:21:39.010 Well, the only way this probability density 00:21:39.010 --> 00:21:42.190 is going to evolve in time is going 00:21:42.190 --> 00:21:46.650 to be if we have a wave function, psi, 00:21:46.650 --> 00:21:54.400 which is of x and t is equal to c1 psi 1, e to the minus i e 1, 00:21:54.400 --> 00:22:00.720 e over h bar, plus c2, psi 2, e to the minus i e 00:22:00.720 --> 00:22:10.290 2, t over h bar, where e1 is not equal to e2. 00:22:10.290 --> 00:22:12.780 So this is the first, most elementary step. 00:22:12.780 --> 00:22:18.040 And remember, we have this notation e to the i something. 00:22:18.040 --> 00:22:20.140 And e to the minus i something. 00:22:20.140 --> 00:22:24.200 And when we take a plus 1 and a minus and put them together, 00:22:24.200 --> 00:22:26.040 we get 1. 00:22:26.040 --> 00:22:28.820 So this notation, this exponential notation, 00:22:28.820 --> 00:22:30.450 is really valuable. 00:22:30.450 --> 00:22:30.950 OK. 00:22:30.950 --> 00:22:34.910 So now let's just look at this quantity, psi star psi. 00:22:39.050 --> 00:22:39.550 OK. 00:22:39.550 --> 00:22:40.870 Psi star psi. 00:22:44.980 --> 00:22:54.040 Well, it's going to be c1 squared, psi 1 squared-- 00:22:54.040 --> 00:23:01.430 square modulus-- and we get c2 square modulus, 00:23:01.430 --> 00:23:05.360 psi 2 square modulus. 00:23:05.360 --> 00:23:06.120 Are we done? 00:23:06.120 --> 00:23:07.280 No. 00:23:07.280 --> 00:23:14.360 So, what we did is, I put in c1, psi 1 plus c2, psi 2. 00:23:14.360 --> 00:23:17.150 And I looked at the easy terms, the terms 00:23:17.150 --> 00:23:20.980 where the exponential factor goes away. 00:23:20.980 --> 00:23:22.690 And then there's two cross terms. 00:23:22.690 --> 00:23:25.730 Those two cross terms are c1 star-- 00:23:25.730 --> 00:23:49.080 whoops-- c2 e to the minus i e 2 minus e1 t over h bar 00:23:49.080 --> 00:23:54.370 psi 1 star, psi 2. 00:23:54.370 --> 00:24:04.780 And we have c1, c2 star e to the plus i e2 minus e1 00:24:04.780 --> 00:24:10.930 t over h bar, psi 1 star. 00:24:10.930 --> 00:24:14.370 psi 1, psi 2 star. 00:24:14.370 --> 00:24:17.371 Now, this is just the automatic writing. 00:24:17.371 --> 00:24:17.870 OK. 00:24:17.870 --> 00:24:23.760 So we have two terms that are time independent. 00:24:23.760 --> 00:24:26.570 So, this is no big surprise. 00:24:26.570 --> 00:24:28.520 But then we have this stuff here. 00:24:28.520 --> 00:24:38.970 And if we say e2 minus e1 over h bar is omega 2, 1, 00:24:38.970 --> 00:24:42.830 everything becomes very transparent, 00:24:42.830 --> 00:24:44.630 because now we have something that 00:24:44.630 --> 00:24:52.070 looks like it looks like it's trying to be a cosine omega t. 00:24:52.070 --> 00:24:55.050 We have to be a little bit careful. 00:24:55.050 --> 00:24:57.440 These are the two time dependent terms. 00:24:57.440 --> 00:25:01.180 And they are the complex conjugate of each other. 00:25:01.180 --> 00:25:12.690 And so we know that if we have two complex numbers, 00:25:12.690 --> 00:25:21.840 c plus c star, we're going to get twice the real part of c. 00:25:27.140 --> 00:25:31.460 So this enables us to take these two terms and combine them. 00:25:31.460 --> 00:25:35.900 And we know it had better be real, 00:25:35.900 --> 00:25:38.360 because we're talking about a probability here. 00:25:38.360 --> 00:25:42.010 This probability has got to be real. 00:25:42.010 --> 00:25:44.410 And it's got to be positive. 00:25:44.410 --> 00:25:48.825 It's got to be real and positive everywhere and forever. 00:25:51.580 --> 00:25:55.410 Because there's no such thing as a negative probability. 00:25:55.410 --> 00:25:57.650 There's no such thing as a negative probability 00:25:57.650 --> 00:26:00.170 in a little region of space, and we say, well, 00:26:00.170 --> 00:26:04.170 we integrate over all space, and so that goes away. 00:26:04.170 --> 00:26:12.330 This psi star psi is going to be real everywhere for all time. 00:26:12.330 --> 00:26:14.060 And so that's a good thing, because we 00:26:14.060 --> 00:26:21.450 have a sum of a plus its complex conjugate. 00:26:21.450 --> 00:26:23.270 And so this is real. 00:26:23.270 --> 00:26:32.900 And we can simplify everything, and we can write it simply. 00:26:32.900 --> 00:26:35.400 But suppose we choose a particular case-- 00:26:35.400 --> 00:26:42.980 c1 star is equal to c1, which is equal to 1 00:26:42.980 --> 00:26:45.560 over square root of 2. 00:26:45.560 --> 00:26:52.910 And we can say psi 1 and psi 2 are real. 00:26:52.910 --> 00:26:57.330 When we do that, then this complicated-looking thing, psi 00:26:57.330 --> 00:27:03.760 star psi, becomes 1/2 psi 1 squared, 00:27:03.760 --> 00:27:15.430 plus 1/2 psi 2 squared, plus cosine omega 1 2 t psi 1 psi 2. 00:27:20.950 --> 00:27:22.760 OK. 00:27:22.760 --> 00:27:25.490 Well, these two guys aren't moving. 00:27:25.490 --> 00:27:27.540 And they're real. 00:27:27.540 --> 00:27:28.590 And positive. 00:27:28.590 --> 00:27:29.240 Yeah. 00:27:29.240 --> 00:27:32.160 AUDIENCE: Wait, is c2 also 1 over square root of 2? 00:27:35.932 --> 00:27:37.640 PROFESSOR: I had 1 over square root of 2, 00:27:37.640 --> 00:27:38.490 one over square root of 2. 00:27:38.490 --> 00:27:39.060 That's 1/2. 00:27:41.880 --> 00:27:46.560 But then when you add the two terms, 00:27:46.560 --> 00:27:48.600 you get twice the real part. 00:27:48.600 --> 00:27:50.640 So we get a 1/2 times 2. 00:27:50.640 --> 00:27:54.000 And so this is not a mistake. 00:27:54.000 --> 00:27:56.730 It comes out this way, OK? 00:27:56.730 --> 00:27:58.800 AUDIENCE: I think you meant c2. 00:27:58.800 --> 00:28:05.340 You're only finding c1 star and c1 to be 2 over square root-- 00:28:05.340 --> 00:28:07.580 1 over square root of 2. 00:28:07.580 --> 00:28:09.467 You didn't say anything about c2. 00:28:09.467 --> 00:28:10.050 PROFESSOR: Oh. 00:28:10.050 --> 00:28:10.860 OK. 00:28:10.860 --> 00:28:12.010 I want c1 and c2. 00:28:16.710 --> 00:28:18.640 That's what I wanted. 00:28:18.640 --> 00:28:19.140 OK. 00:28:24.450 --> 00:28:25.560 Thank you. 00:28:25.560 --> 00:28:27.360 That shows you're listening. 00:28:27.360 --> 00:28:30.570 And it shows that I'm sufficiently here 00:28:30.570 --> 00:28:32.460 to understand your questions, which 00:28:32.460 --> 00:28:34.350 is another wonderful thing. 00:28:34.350 --> 00:28:36.960 So this is now, we have something 00:28:36.960 --> 00:28:40.460 that's positive everywhere. 00:28:40.460 --> 00:28:41.860 It's time independent. 00:28:41.860 --> 00:28:45.800 And we have something that's oscillating. 00:28:45.800 --> 00:28:49.080 And so this term can be negative. 00:28:49.080 --> 00:28:51.150 But you can show-- 00:28:51.150 --> 00:28:52.750 I don't choose to do that. 00:28:52.750 --> 00:28:55.500 You can show that this term is never larger 00:28:55.500 --> 00:28:57.480 than psi 1 squared plus psi 2 squared. 00:29:00.770 --> 00:29:05.720 And so even though this term can be negative at some points, 00:29:05.720 --> 00:29:10.970 it never is a negative enough to make the evolving probability 00:29:10.970 --> 00:29:12.211 go negative. 00:29:15.880 --> 00:29:18.780 Now, you may want to play with that just to convince yourself. 00:29:18.780 --> 00:29:19.780 And it's an easy proof. 00:29:19.780 --> 00:29:22.600 And I'm just not going to do it. 00:29:22.600 --> 00:29:25.080 OK. 00:29:25.080 --> 00:29:29.400 So we have something that says, just 00:29:29.400 --> 00:29:35.270 like for the wave equation, what does it take to get motion? 00:29:35.270 --> 00:29:37.040 And to get motion you had to have 00:29:37.040 --> 00:29:41.810 a superposition of two waves of different energy 00:29:41.810 --> 00:29:43.070 or different wave vector. 00:29:45.630 --> 00:29:51.180 And so here we have, if e1 is not equal to e2, 00:29:51.180 --> 00:29:53.695 we have a non-zero omega 1, 2. 00:29:53.695 --> 00:29:56.340 It doesn't matter whether it's omega 1, 2, or omega 2, 1, 00:29:56.340 --> 00:29:59.100 because it's cosine. 00:29:59.100 --> 00:30:02.700 And so that's motion. 00:30:02.700 --> 00:30:07.890 So we get a standing wave sort of situation. 00:30:07.890 --> 00:30:10.950 And then we get this motion. 00:30:10.950 --> 00:30:14.430 OK, now, the fun begins. 00:30:14.430 --> 00:30:23.614 Suppose we want to calculate the expectation value of x and p. 00:30:30.290 --> 00:30:30.790 OK. 00:30:30.790 --> 00:30:35.160 And let's just take it as a 1D problem. 00:30:35.160 --> 00:30:37.300 And so x of t. 00:30:40.090 --> 00:30:41.640 No, there was no question. 00:30:41.640 --> 00:30:42.870 All right. 00:30:42.870 --> 00:30:43.410 We have-- 00:30:59.122 --> 00:30:59.940 OK. 00:30:59.940 --> 00:31:02.100 And again, we take this thing apart. 00:31:02.100 --> 00:31:05.460 And we say, all right, suppose we have the same superposition 00:31:05.460 --> 00:31:09.160 of psi is equal to c1. 00:31:09.160 --> 00:31:10.330 psi 1. 00:31:21.640 --> 00:31:23.050 OK. 00:31:23.050 --> 00:31:30.650 Actually, these should be the time independent. 00:31:30.650 --> 00:31:35.120 So what we get is, when we do this integral, we get c1 00:31:35.120 --> 00:31:47.940 squared integral psi star x, psi 1 dx. 00:31:47.940 --> 00:31:59.890 And we get c2 squared integral psi 2 star x, psi 2 dx. 00:31:59.890 --> 00:32:03.260 And then we get cross terms. c1 star, c2. 00:32:05.980 --> 00:32:18.970 e to the minus i omega 2, 1 t times the integral psi 1 star 00:32:18.970 --> 00:32:23.170 x psi 2 dx. 00:32:23.170 --> 00:32:31.330 And we have c1, c2 star, e to the plus i omega 2 1 00:32:31.330 --> 00:32:38.090 t integral psi 2 star x psi 1 dx. 00:32:40.741 --> 00:32:43.170 A lot of stuff here. 00:32:43.170 --> 00:32:47.810 Now, we're talking about the harmonic oscillator. 00:32:47.810 --> 00:32:49.385 This integral is 0. 00:32:52.360 --> 00:32:55.340 Because x is a plus a dagger. 00:32:55.340 --> 00:32:58.750 The selection rule is delta v of plus and minus 1. 00:32:58.750 --> 00:33:01.190 This one is 0. 00:33:01.190 --> 00:33:05.790 For the particle in a box, one can also 00:33:05.790 --> 00:33:08.610 ask, what about this integral? 00:33:08.610 --> 00:33:11.880 And for the particle in a box, it's a little bit more 00:33:11.880 --> 00:33:12.510 complicated. 00:33:12.510 --> 00:33:15.510 Because we've chosen a mathematically simple way 00:33:15.510 --> 00:33:19.110 to solve the particle in a box, with the box having a zero 00:33:19.110 --> 00:33:21.500 left edge. 00:33:21.500 --> 00:33:25.310 If we make the box symmetric, then we can make judgments 00:33:25.310 --> 00:33:26.930 and say, oh, yeah. 00:33:26.930 --> 00:33:30.930 This integral is also 0 for the particle in a box. 00:33:36.010 --> 00:33:38.490 And that's a little bit more complicated, the argument, 00:33:38.490 --> 00:33:40.200 than what I just did. 00:33:40.200 --> 00:33:45.070 So these two terms are 0. 00:33:45.070 --> 00:33:51.100 And now we have motion of the expectation value, which is 00:33:51.100 --> 00:33:52.720 described by these two terms. 00:33:52.720 --> 00:33:56.180 And again, they're the complex conjugate of each other. 00:33:56.180 --> 00:34:13.949 And so what we have is x of t is equal to twice 00:34:13.949 --> 00:34:22.900 the real part of c1 star, c2 e to the minus i omega 2, 1 t. 00:34:28.500 --> 00:34:32.131 Times x 1, 2. 00:34:32.131 --> 00:34:32.630 OK. 00:34:32.630 --> 00:34:34.310 This x 1, 2 is an integral. 00:34:37.210 --> 00:34:42.969 So x 1, 2 if these are the vibrational quantum numbers, 00:34:42.969 --> 00:34:46.310 well, then this is non-zero. 00:34:46.310 --> 00:34:52.370 This is just the square root of 2, or square root of 1. 00:35:00.051 --> 00:35:00.550 OK. 00:35:00.550 --> 00:35:05.005 So we have motion described by this. 00:35:07.990 --> 00:35:14.440 And so the only time we get motion 00:35:14.440 --> 00:35:20.650 is if v1 is equal to v2 plus or minus 1, 00:35:20.650 --> 00:35:22.065 for the harmonic oscillator. 00:35:25.320 --> 00:35:29.720 For the particle in a box, there's different rules. 00:35:29.720 --> 00:35:32.010 And often, for these simple problems, 00:35:32.010 --> 00:35:35.090 you want to go through in your head all of the simple cases. 00:35:48.810 --> 00:35:52.630 Now, we already can see that-- 00:35:59.950 --> 00:36:02.270 I don't want to talk about the particle in a box. 00:36:02.270 --> 00:36:05.320 So now let's just take another step. 00:36:05.320 --> 00:36:17.780 And we're going to have Ehrenfest's theorem, which 00:36:17.780 --> 00:36:26.800 you can prove, says that m times the derivative of vector p-- 00:36:31.790 --> 00:36:35.210 so this is a time dependent expectation value-- 00:36:35.210 --> 00:36:37.681 is equal to-- 00:36:37.681 --> 00:36:38.180 I'm sorry. 00:36:38.180 --> 00:36:39.200 This is not vector p. 00:36:39.200 --> 00:36:41.420 This is vector r-- 00:36:41.420 --> 00:36:42.080 is equal to-- 00:36:48.550 --> 00:36:49.720 So this is the vector p. 00:36:49.720 --> 00:36:53.500 So this is the derivative of the coordinate, 00:36:53.500 --> 00:36:54.380 with respect to time. 00:36:54.380 --> 00:36:55.600 That's velocity. 00:36:55.600 --> 00:36:59.080 Velocity times mass is momentum. 00:36:59.080 --> 00:37:02.380 And so we have a relationship between the expectation 00:37:02.380 --> 00:37:06.070 value of the position and the expectation 00:37:06.070 --> 00:37:08.480 value of the momentum. 00:37:08.480 --> 00:37:10.520 And that's for Newton's first law. 00:37:10.520 --> 00:37:14.170 And then there is another Newton's equation translated 00:37:14.170 --> 00:37:16.580 into quantum mechanics. 00:37:16.580 --> 00:37:25.270 So we have the expectation value of the momentum, dt, 00:37:25.270 --> 00:37:30.520 is equal to minus the expectation 00:37:30.520 --> 00:37:33.250 value of the potential. 00:37:37.310 --> 00:37:43.540 While this is acceleration times mass, and this is force-- 00:37:43.540 --> 00:37:46.020 minus the gradient of a potential is the force-- 00:37:46.020 --> 00:37:47.860 these are Newton's two equations. 00:37:47.860 --> 00:37:58.280 And what they're saying is, if we know these things, 00:37:58.280 --> 00:38:01.520 we know something about the center of the wave packet, 00:38:01.520 --> 00:38:03.570 and how the center of the wave packet moves. 00:38:03.570 --> 00:38:06.540 Now, the wave packet might be localized at one time. 00:38:06.540 --> 00:38:09.210 It might be mushed out at another time. 00:38:09.210 --> 00:38:11.710 But you can always calculate the center of the wave packet. 00:38:11.710 --> 00:38:13.501 It's just that there are only certain times 00:38:13.501 --> 00:38:16.650 that it looks like a particle. 00:38:16.650 --> 00:38:19.830 But this thing, these quantities, 00:38:19.830 --> 00:38:24.380 which you define by an integral, they evolve classically. 00:38:24.380 --> 00:38:26.000 So I told you at the beginning, you 00:38:26.000 --> 00:38:28.790 had to give up classical mechanics. 00:38:28.790 --> 00:38:31.130 It's all coming back. 00:38:31.130 --> 00:38:34.320 But it's coming back in a quantum mechanical framework, 00:38:34.320 --> 00:38:38.360 because we're talking now about wave functions, which 00:38:38.360 --> 00:38:43.350 have amplitudes and phases, and can do terrible things. 00:38:43.350 --> 00:38:49.110 But at certain limits, they're going to act like particles. 00:38:49.110 --> 00:38:57.650 But if you were to ask a question, well-- 00:38:57.650 --> 00:39:00.230 suppose we do this experiment. 00:39:00.230 --> 00:39:06.770 And so here we have an electronic ground state 00:39:06.770 --> 00:39:07.970 potential surface. 00:39:07.970 --> 00:39:10.610 Now, I'm jumping way ahead. 00:39:10.610 --> 00:39:16.730 But this is the wave function for that vibrational level. 00:39:16.730 --> 00:39:21.780 And you excite the molecule with a time dependent Hamiltonian, 00:39:21.780 --> 00:39:25.200 a time dependent radiation field light. 00:39:25.200 --> 00:39:29.880 And you vertically transport this wave function 00:39:29.880 --> 00:39:32.520 to the upper state, until you get something 00:39:32.520 --> 00:39:35.690 which is not an eigenstate. 00:39:35.690 --> 00:39:37.470 It's a pluck. 00:39:37.470 --> 00:39:41.560 This pluck is a superposition of eigenstates. 00:39:41.560 --> 00:39:43.360 And we can ask, how does this evolve? 00:39:43.360 --> 00:39:45.130 And what it's going to do is, it's 00:39:45.130 --> 00:39:47.910 going to start out localized. 00:39:47.910 --> 00:39:51.280 And at some point, it'll do terrible things. 00:39:51.280 --> 00:39:53.140 And then at some other time it'll 00:39:53.140 --> 00:39:55.380 be localized again at the other turning point. 00:39:55.380 --> 00:39:57.780 And it will come back and forth. 00:39:57.780 --> 00:40:00.810 And now, if it's not a harmonic oscillator, 00:40:00.810 --> 00:40:03.780 it won't quite relocalize. 00:40:03.780 --> 00:40:05.810 It'll mush out a little bit, and it'll come back 00:40:05.810 --> 00:40:07.660 and it'll mush out more. 00:40:07.660 --> 00:40:11.900 And so again, we can use the evolution of the wave packet 00:40:11.900 --> 00:40:15.530 to sample the shape of the potential. 00:40:15.530 --> 00:40:17.940 We can measure the anharmonicity. 00:40:17.940 --> 00:40:20.780 And so let's now talk about that. 00:40:24.970 --> 00:40:27.400 What are other quantities that we 00:40:27.400 --> 00:40:34.039 can calculate from the wave function of the time 00:40:34.039 --> 00:40:35.330 dependent Schrodinger equation? 00:40:35.330 --> 00:40:38.960 So let's talk about the survival probability. 00:40:38.960 --> 00:40:41.500 So this is a capital P. Survival probability. 00:40:41.500 --> 00:40:45.790 It's going to be the wave function. 00:40:45.790 --> 00:40:50.440 The time dependent wave function is created at t equals 0. 00:40:50.440 --> 00:40:51.970 It has some shape. 00:40:51.970 --> 00:40:55.990 Now, we always like to have a simple shape at t equals 0. 00:40:55.990 --> 00:41:00.670 And we'd like to know how fast that thing moves away 00:41:00.670 --> 00:41:04.210 from its birthplace. 00:41:04.210 --> 00:41:10.900 So we have this survival probability. 00:41:10.900 --> 00:41:12.280 It's a probability. 00:41:12.280 --> 00:41:25.640 So we're going to have integral psi star xt psi xt dx. 00:41:25.640 --> 00:41:28.730 And this integral is going to look like that. 00:41:32.490 --> 00:41:40.610 Now, whoops. 00:41:40.610 --> 00:41:42.800 I knew I was going to screw up. 00:41:42.800 --> 00:41:46.370 If I put a t here, then it would just 00:41:46.370 --> 00:41:48.320 be the normalization interval. 00:41:48.320 --> 00:41:51.250 And we already know what that is. 00:41:51.250 --> 00:41:52.690 But this is the birthplace. 00:41:52.690 --> 00:41:55.460 This is what was created at t equals 0. 00:41:55.460 --> 00:41:58.300 And this is time evolving thing. 00:41:58.300 --> 00:42:01.150 And so we can calculate how that behaves. 00:42:01.150 --> 00:42:04.790 And I'm just going to write the solution. 00:42:09.770 --> 00:42:19.150 So the result is, if we have the same kind of two state c1 psi 00:42:19.150 --> 00:42:27.530 1, e to the minus i e1 t for h bar 00:42:27.530 --> 00:42:35.140 plus c2 psi 2 e to the minus i e2 e over h bar, 00:42:35.140 --> 00:42:41.330 well then, what we get is c1 to the 4th power 00:42:41.330 --> 00:42:55.380 plus c2 to the 4th power plus c1 squared, c2 squared, times 00:42:55.380 --> 00:43:06.210 e to the i omega 2, 1 t plus e to the minus i omega 2, 1 t. 00:43:10.390 --> 00:43:14.090 Now we're integrating. 00:43:14.090 --> 00:43:18.520 When we integrate, the wave functions go away. 00:43:18.520 --> 00:43:23.010 The wave functions become either 1 or 0. 00:43:23.010 --> 00:43:24.750 And so we're integrating. 00:43:24.750 --> 00:43:27.030 We're making the wave functions go away. 00:43:27.030 --> 00:43:32.490 And we have some amplitudes of the wave functions. 00:43:32.490 --> 00:43:36.210 And we have a time independent part and a time dependent part. 00:43:36.210 --> 00:43:41.472 And this is 2 cosine omega 2, 1 t. 00:43:44.070 --> 00:43:49.530 So what we see is this survival probability, the wave function, 00:43:49.530 --> 00:43:51.870 starts out at some place. 00:43:51.870 --> 00:43:53.560 And it goes away. 00:43:53.560 --> 00:43:54.700 And it comes back. 00:43:54.700 --> 00:43:55.800 And it goes away. 00:43:55.800 --> 00:43:59.060 And it comes back. 00:43:59.060 --> 00:43:59.930 And it does that. 00:43:59.930 --> 00:44:04.670 It completely rephases, because there's only two terms. 00:44:04.670 --> 00:44:07.070 But if there were three terms that are not 00:44:07.070 --> 00:44:09.400 satisfying a certain requirement, 00:44:09.400 --> 00:44:11.870 then when it comes back, it can't completely 00:44:11.870 --> 00:44:12.960 reconstruct itself. 00:44:17.080 --> 00:44:20.560 And this is the basis for doing experiments. 00:44:20.560 --> 00:44:26.770 One can observe the periodic rephasings 00:44:26.770 --> 00:44:29.500 of some initial pluck. 00:44:29.500 --> 00:44:31.840 And you can look at the decay of them, 00:44:31.840 --> 00:44:33.850 and that gives you something about the shape 00:44:33.850 --> 00:44:36.650 of the potential, the anharmonicity. 00:44:36.650 --> 00:44:40.490 And you can do all sorts of fantastic things. 00:44:40.490 --> 00:44:43.760 You can create a wave packet, and you 00:44:43.760 --> 00:44:47.250 can wait until it reaches the other turning point. 00:44:47.250 --> 00:44:48.920 And at the other turning point, it's 00:44:48.920 --> 00:44:50.495 possible that you could excite it 00:44:50.495 --> 00:44:53.900 to a different higher excited state. 00:44:53.900 --> 00:44:55.770 Ahmed Zewail got the Nobel Prize for that. 00:44:59.550 --> 00:45:06.690 So we're right at the frontier of what you can do, 00:45:06.690 --> 00:45:08.430 and what you can understand using 00:45:08.430 --> 00:45:11.810 this very simple problem of a time independent Hamiltonian 00:45:11.810 --> 00:45:17.400 and a 2 or a 3 term superposition. 00:45:17.400 --> 00:45:17.900 OK. 00:45:27.850 --> 00:45:28.510 Recurrence. 00:45:35.910 --> 00:45:38.750 This is a special property. 00:45:38.750 --> 00:45:50.960 When all of the wave functions, or all of the difference-- 00:45:50.960 --> 00:45:54.410 all of the energy levels, or differences in energy levels 00:45:54.410 --> 00:46:06.140 are integer multiples of a common factor, well then 00:46:06.140 --> 00:46:09.830 any coherent superposition state you make 00:46:09.830 --> 00:46:13.940 will rephase at a special time. 00:46:13.940 --> 00:46:19.440 And so what we can do is say, OK, for a particle in a box, 00:46:19.440 --> 00:46:26.360 the energy levels are e1 times n squared. 00:46:26.360 --> 00:46:29.450 For our harmonic oscillator, the energy level difference 00:46:29.450 --> 00:46:39.360 is ev plus n minus ev our n h bar omega. 00:46:39.360 --> 00:46:41.970 For a rigid rotor, which we haven't seen yet, 00:46:41.970 --> 00:46:49.520 the energy levels are a function of this quantum number j hcb. 00:46:49.520 --> 00:46:54.410 This is the rotational constant-- times j plus 1. 00:46:56.940 --> 00:47:01.970 So all of the energy levels are related. 00:47:01.970 --> 00:47:03.590 I'm jumping ahead. 00:47:03.590 --> 00:47:04.550 Sorry. 00:47:04.550 --> 00:47:12.380 So the energy levels are j times j plus 1. 00:47:12.380 --> 00:47:18.980 And the differences, ej plus 1 minus ej, 00:47:18.980 --> 00:47:27.620 are given by 2 hcb times j plus 1. 00:47:27.620 --> 00:47:31.880 So for these three problems, we have this perfect situation 00:47:31.880 --> 00:47:34.820 where one can have these oscillating 00:47:34.820 --> 00:47:37.330 terms all have a common factor. 00:47:37.330 --> 00:47:42.130 And at certain times, the oscillating terms are all 1. 00:47:42.130 --> 00:47:44.510 Or some are 1 and some are minus 1. 00:47:44.510 --> 00:47:47.660 And we get a really wonderful simplification. 00:47:47.660 --> 00:47:52.190 So at a time which we call, or I call, 00:47:52.190 --> 00:47:59.860 the grand recurrence time, where it's 00:47:59.860 --> 00:48:09.010 equal to h over e1 for this case of the particle in a box, 00:48:09.010 --> 00:48:24.060 or h over h bar omega, or h over 2 hcb, 00:48:24.060 --> 00:48:27.540 we get all of the phase factors becoming 1. 00:48:33.040 --> 00:48:38.740 And sometimes, something special happens at the grand recurrence 00:48:38.740 --> 00:48:41.170 time divided by 2. 00:48:41.170 --> 00:48:44.410 And that's a little bit like this. 00:48:44.410 --> 00:48:47.900 We have a wave function here. 00:48:47.900 --> 00:48:50.010 And at half the recurrence time, it's over here. 00:48:53.720 --> 00:48:56.050 And so you want to work your way through the algebra 00:48:56.050 --> 00:49:00.590 to convince yourself that that is in fact true. 00:49:00.590 --> 00:49:04.280 And in between, if you have enough terms 00:49:04.280 --> 00:49:08.380 in your superposition, which is usually the case, 00:49:08.380 --> 00:49:09.610 in between you get this. 00:49:09.610 --> 00:49:13.080 You get garbage looking. 00:49:13.080 --> 00:49:15.330 It's still moving. 00:49:15.330 --> 00:49:18.390 It's still satisfying the Ehrenfest theorem, 00:49:18.390 --> 00:49:20.010 but it doesn't look like a particle. 00:49:20.010 --> 00:49:23.470 It just looks like garbage. 00:49:23.470 --> 00:49:26.140 But you get these wonderful things happening. 00:49:26.140 --> 00:49:28.480 And now, if they're not perfectly 00:49:28.480 --> 00:49:32.350 satisfying this integer rule, then each time you 00:49:32.350 --> 00:49:36.130 get to a turning point, the amplitude 00:49:36.130 --> 00:49:40.440 has decreased and decreased and decreased. 00:49:40.440 --> 00:49:46.830 And at infinite time, it might recur. 00:49:46.830 --> 00:49:48.580 But nobody waits around for infinite time, 00:49:48.580 --> 00:49:51.970 because other things happen and destroy the coherence that you 00:49:51.970 --> 00:49:55.005 built. So this is a glimpse. 00:49:59.310 --> 00:50:01.020 Time independent quantum mechanics 00:50:01.020 --> 00:50:03.980 is complicated enough. 00:50:03.980 --> 00:50:07.610 Well, we can embed what we understand 00:50:07.610 --> 00:50:10.730 in the time dependent mechanics. 00:50:10.730 --> 00:50:14.560 And there are a lot of beautiful things that we can anticipate. 00:50:14.560 --> 00:50:19.210 Now, we can use these beautiful things 00:50:19.210 --> 00:50:22.420 to do experiments which measure stuff 00:50:22.420 --> 00:50:27.020 that is related to, how does energy move in the molecule? 00:50:27.020 --> 00:50:28.550 What is going on in the molecule? 00:50:28.550 --> 00:50:31.964 What are the mechanisms for stuff happening? 00:50:31.964 --> 00:50:33.880 And in magnetic resonance, there are all sorts 00:50:33.880 --> 00:50:37.150 of pulse sequences that interrogate distances 00:50:37.150 --> 00:50:39.830 and correlated motions. 00:50:39.830 --> 00:50:43.570 And it is really a laboratory for time dependent quantum 00:50:43.570 --> 00:50:48.130 mechanics with a time dependent Hamiltonian. 00:50:48.130 --> 00:50:54.700 But a lot of stuff that we do with ordinary laser experiments 00:50:54.700 --> 00:51:01.880 are usually understandable in a time independent way. 00:51:01.880 --> 00:51:07.590 Now, we want to get rid of things that are complicated. 00:51:07.590 --> 00:51:11.220 And so one of the things that we do 00:51:11.220 --> 00:51:14.970 is, we map a problem onto something 00:51:14.970 --> 00:51:17.460 which is time independent. 00:51:17.460 --> 00:51:20.970 And so one of the tricks that you will see 00:51:20.970 --> 00:51:25.890 is that if we go into what's called a rotating coordinate 00:51:25.890 --> 00:51:28.890 system, this rotating coordinate system 00:51:28.890 --> 00:51:36.330 is rotating at an energy level different divided by h bar. 00:51:36.330 --> 00:51:39.984 And that converts the problem into a time independent problem 00:51:39.984 --> 00:51:41.400 in the rotating coordinate system. 00:51:41.400 --> 00:51:43.660 So that you're in a rotating coordinate system 00:51:43.660 --> 00:51:47.610 and, basically, you use the ordinary perturbation theory. 00:51:47.610 --> 00:51:50.670 And then you go back to the non-rotating coordinate system. 00:51:50.670 --> 00:51:53.880 So there's all sorts of tricks where we build on the stuff 00:51:53.880 --> 00:51:56.480 that we understood. 00:51:56.480 --> 00:52:00.560 And we can have a picture which is intuitive, 00:52:00.560 --> 00:52:02.870 because we want to strip away a lot of the mathematics 00:52:02.870 --> 00:52:05.540 and see the universal stuff. 00:52:05.540 --> 00:52:08.660 And so I'm going to try to present as much of that 00:52:08.660 --> 00:52:10.950 as I can during this course. 00:52:10.950 --> 00:52:13.882 But a lot of the time independent stuff 00:52:13.882 --> 00:52:16.340 is heavy lifting, and I'm going to have to do a lot of that 00:52:16.340 --> 00:52:17.250 too. 00:52:17.250 --> 00:52:18.080 OK. 00:52:18.080 --> 00:52:20.600 Have a nice long weekend. 00:52:20.600 --> 00:52:22.250 I'll see you on Monday.