WEBVTT 00:00:00.060 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.820 under a Creative Commons license. 00:00:03.820 --> 00:00:06.540 Your support will help MIT OpenCourseWare continue 00:00:06.540 --> 00:00:10.120 to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.600 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.600 --> 00:00:17.275 at ocw.mit.edu. 00:00:20.592 --> 00:00:22.550 BARTON ZWIEBACH: [INAUDIBLE] of today's lecture 00:00:22.550 --> 00:00:28.220 is coherent states of the harmonic oscillator. 00:00:28.220 --> 00:00:33.860 So let me begin by telling you about some things 00:00:33.860 --> 00:00:40.730 we've learned in the last lecture, and here they are. 00:00:40.730 --> 00:00:46.080 We learned how to calculate the so-called Heisenberg operators. 00:00:46.080 --> 00:00:48.480 Remember, if you have a Schrodinger operator, 00:00:48.480 --> 00:00:52.710 you subject it to this transformation 00:00:52.710 --> 00:00:55.070 with a unitary operator. 00:00:55.070 --> 00:00:57.880 That creates time evolution and that gives you 00:00:57.880 --> 00:00:59.980 the Heisenberg operator. 00:00:59.980 --> 00:01:04.319 We learned things about Heisenberg expectation values. 00:01:04.319 --> 00:01:09.710 If the Hamiltonian is time independent, h, time 00:01:09.710 --> 00:01:14.550 independent, the formula is quite simple 00:01:14.550 --> 00:01:18.180 and gives you the Heisenberg operator at the later time. 00:01:18.180 --> 00:01:21.650 So we did this. 00:01:21.650 --> 00:01:23.970 We found, in fact, Heisenberg operators 00:01:23.970 --> 00:01:26.760 satisfy equations of motion. 00:01:26.760 --> 00:01:29.960 And we calculated the Heisenberg operators 00:01:29.960 --> 00:01:32.260 for the harmonic oscillator. 00:01:32.260 --> 00:01:35.430 That was our main achievement last time, 00:01:35.430 --> 00:01:40.410 a formula for the time development of the x 00:01:40.410 --> 00:01:44.830 and p operators in the Heisenberg picture. 00:01:44.830 --> 00:01:50.680 And that really contains all the information of the dynamics, 00:01:50.680 --> 00:01:56.000 as you will see today, when we will be using this stuff. 00:01:56.000 --> 00:02:02.242 Now, I suggested that you read-- and you may do it later. 00:02:02.242 --> 00:02:03.700 There's no need that you've done it 00:02:03.700 --> 00:02:07.560 for today-- the information on the time 00:02:07.560 --> 00:02:11.980 development of the creation and annihilation operators. 00:02:11.980 --> 00:02:18.340 You see, the a and a dagger are different inverses of x and p, 00:02:18.340 --> 00:02:19.880 are linear combinations. 00:02:19.880 --> 00:02:23.450 So the a and the a dagger operators 00:02:23.450 --> 00:02:26.776 also can be further Schrodinger operators 00:02:26.776 --> 00:02:29.530 that have no time dependence. 00:02:29.530 --> 00:02:33.150 And suddenly, if you go to the Heisenberg picture, 00:02:33.150 --> 00:02:35.290 the creation and annihilation operators 00:02:35.290 --> 00:02:39.240 become time dependent operators. 00:02:39.240 --> 00:02:41.700 So that's in the notes. 00:02:41.700 --> 00:02:44.070 You can read about it. 00:02:44.070 --> 00:02:49.070 So we define the time dependent operator, a hat, 00:02:49.070 --> 00:02:53.470 to be the Heisenberg version of a hat. 00:02:53.470 --> 00:02:56.100 And you're supposed to do a calculation 00:02:56.100 --> 00:03:03.770 and try it or read it, and the answer is very nice, 00:03:03.770 --> 00:03:07.030 simply a phase dependence. 00:03:07.030 --> 00:03:12.450 The a is a at time equals 0, the Schrodinger 1 times 00:03:12.450 --> 00:03:15.550 e to the minus i omega t. 00:03:15.550 --> 00:03:21.470 Then a dagger is just what you would expect, 00:03:21.470 --> 00:03:27.330 the dagger of this, which the face has an opposite sign and a 00:03:27.330 --> 00:03:29.660 becomes a dagger. 00:03:29.660 --> 00:03:35.160 Finally, if you substitute this a and a 00:03:35.160 --> 00:03:37.920 daggers in this formula. 00:03:37.920 --> 00:03:40.640 For example, you could say x Heisenberg 00:03:40.640 --> 00:03:44.450 is a Heisenberg plus a dagger Heisenberg. 00:03:44.450 --> 00:03:47.640 And you substitute those Heisenberg values there, 00:03:47.640 --> 00:03:49.780 you will obtain this. 00:03:49.780 --> 00:03:51.200 Same for the momentum. 00:03:51.200 --> 00:03:54.280 If you put Heisenberg, Heisenberg, Heisenberg, 00:03:54.280 --> 00:03:56.880 remember, if you have an equality of Schrodinger 00:03:56.880 --> 00:03:59.230 operators, it also holds when you 00:03:59.230 --> 00:04:03.270 put Heisenberg in every operator. 00:04:03.270 --> 00:04:07.180 And therefore, if you put the Heisenberg a, a, 00:04:07.180 --> 00:04:10.940 and use those values, you will recover this equation. 00:04:10.940 --> 00:04:17.640 So in a sense, these equations are equivalent to these ones. 00:04:21.040 --> 00:04:25.370 And that's basically our situation. 00:04:25.370 --> 00:04:28.550 This is what we've learned so far, 00:04:28.550 --> 00:04:35.480 and our goal today is to apply this to understand 00:04:35.480 --> 00:04:38.660 coherent states of the harmonic oscillator. 00:04:38.660 --> 00:04:40.570 Now, why do we want to understand 00:04:40.570 --> 00:04:44.400 coherent states of the harmonic oscillator? 00:04:44.400 --> 00:04:47.550 You want to understand coherent states because the energy 00:04:47.550 --> 00:04:53.150 eigenstates are extraordinarily quantum. 00:04:53.150 --> 00:04:56.720 The energy eigenstates of the harmonic oscillator 00:04:56.720 --> 00:05:00.600 don't look at all-- and you've seen the expectation 00:05:00.600 --> 00:05:02.120 value of the position. 00:05:02.120 --> 00:05:04.350 It's time independent. 00:05:04.350 --> 00:05:05.710 It just doesn't change. 00:05:05.710 --> 00:05:11.030 Expectation value of any operator in a stationary state 00:05:11.030 --> 00:05:12.400 is a constant. 00:05:12.400 --> 00:05:13.730 It just doesn't change. 00:05:13.730 --> 00:05:19.295 So you have any eigenstate, any energy eigenstate 00:05:19.295 --> 00:05:21.680 of the harmonic oscillator, you ask, 00:05:21.680 --> 00:05:24.660 what is the position of this particle doing? 00:05:24.660 --> 00:05:25.230 Nothing. 00:05:25.230 --> 00:05:27.680 What is the momentum of this particle doing? 00:05:27.680 --> 00:05:29.520 Nothing. 00:05:29.520 --> 00:05:33.230 So nevertheless, of course, it's an interesting state, 00:05:33.230 --> 00:05:38.770 but we want to construct quantum mechanical states that 00:05:38.770 --> 00:05:41.630 behave a little like the classical states we're 00:05:41.630 --> 00:05:43.830 accustomed to. 00:05:43.830 --> 00:05:48.100 And that's what coherent states do. 00:05:48.100 --> 00:05:51.260 We'll have an application of coherent states 00:05:51.260 --> 00:05:54.710 to light, photons, coherent photons. 00:05:54.710 --> 00:05:55.580 What are they? 00:05:55.580 --> 00:05:59.880 We'll see it later this week. 00:05:59.880 --> 00:06:03.200 So that's the reason we want to understand coherent states, 00:06:03.200 --> 00:06:07.640 because we want some states that in some ways 00:06:07.640 --> 00:06:12.980 behave classically, or close to classically. 00:06:12.980 --> 00:06:15.640 So they have many applications, these states, 00:06:15.640 --> 00:06:20.350 and you will see some of them in this lecture. 00:06:20.350 --> 00:06:24.430 I'm going to try to keep this blackboard there, untouched, 00:06:24.430 --> 00:06:26.336 so that we can refer to these equations. 00:06:40.860 --> 00:06:47.540 So our first step is considering translation operators. 00:06:47.540 --> 00:06:51.740 So let's consider the unitary translation operator. 00:06:51.740 --> 00:06:59.600 So translation operators. 00:06:59.600 --> 00:07:06.580 So this translation operator that I will write as T sub x0 00:07:06.580 --> 00:07:10.590 will be defined to be the exponential of e 00:07:10.590 --> 00:07:18.090 to the minus i p hat x0 over h bar. 00:07:18.090 --> 00:07:20.565 You have seen such operators before. 00:07:23.630 --> 00:07:27.380 We've seen a lot of them in the homework. 00:07:27.380 --> 00:07:30.770 So first of all, why is it unitary? 00:07:37.360 --> 00:07:43.655 well, it's unitary because x0 is supposed to be a real number. 00:07:48.040 --> 00:07:49.650 p is Hermitian. 00:07:49.650 --> 00:07:52.565 Therefore, this with the i is anti-Hermitian, 00:07:52.565 --> 00:07:55.580 and an exponential of anti-Hermitian operator 00:07:55.580 --> 00:07:57.940 is unitary. 00:07:57.940 --> 00:08:01.710 Now, it has, actually, a very simple property. 00:08:01.710 --> 00:08:09.585 The multiplication of two of those operators is what? 00:08:09.585 --> 00:08:14.130 Well, you have an exponential, e to the minus ipx0, 00:08:14.130 --> 00:08:17.790 and an exponential followed, e to the minus ipy0. 00:08:20.740 --> 00:08:24.120 Now, if you're well trained in 805, 00:08:24.120 --> 00:08:27.850 you should get a little nervous for a second 00:08:27.850 --> 00:08:30.610 because you don't know, can I treat it easily? 00:08:30.610 --> 00:08:33.690 And then you relax and say, yes, these two operators, 00:08:33.690 --> 00:08:38.929 whatsoever the numbers here, this with another one with a y0 00:08:38.929 --> 00:08:40.360 would commute. 00:08:40.360 --> 00:08:44.260 Therefore, they can be put together in the exponential, 00:08:44.260 --> 00:08:47.440 and this is T of x0 plus y0. 00:08:54.840 --> 00:09:00.000 No combo Baker-Hausdorff needed here. 00:09:00.000 --> 00:09:03.140 It's just straightforward. 00:09:03.140 --> 00:09:11.440 So what is Tx0 dagger? 00:09:11.440 --> 00:09:14.610 T x0 dagger, if you take the dagger, 00:09:14.610 --> 00:09:19.580 you change this i for a minus i, so it's 00:09:19.580 --> 00:09:23.920 exactly the same as changing the sign of x0. 00:09:23.920 --> 00:09:29.630 So this is T of minus x0. 00:09:29.630 --> 00:09:35.430 And by this identity, T of minus x0 with a T of x0 00:09:35.430 --> 00:09:39.830 would be T of 0, which is the unit operator. 00:09:39.830 --> 00:09:47.800 So T of minus x0 is the inverse of T of x0, 00:09:47.800 --> 00:09:53.082 confirming that the operator is unitary. 00:09:53.082 --> 00:09:55.600 The inverse is the dagger. 00:09:59.580 --> 00:10:03.040 So I used here that this is the inverse 00:10:03.040 --> 00:10:08.860 because T minus x0 times T x0 is T of 0 is equal to 1. 00:10:08.860 --> 00:10:13.980 So I could mention here, T of 0 is equal to the unit operator. 00:10:18.030 --> 00:10:21.170 So these are our translation operators, 00:10:21.170 --> 00:10:23.570 but you don't get the intuition of what 00:10:23.570 --> 00:10:27.210 they do unless you compute a little more. 00:10:27.210 --> 00:10:30.700 And a little more than you should compute is this. 00:10:30.700 --> 00:10:36.960 What is T x0 dagger x T x0? 00:10:43.420 --> 00:10:49.610 And what is T x0 dagger p T x0? 00:10:54.210 --> 00:10:58.300 Now, why do we ask for these particular things? 00:10:58.300 --> 00:11:02.410 Why don't I ask, what is x hat multiplied by T x0? 00:11:02.410 --> 00:11:04.180 Why do I ask this? 00:11:04.180 --> 00:11:07.660 It is because an operator acting on an operator 00:11:07.660 --> 00:11:08.820 always does this. 00:11:08.820 --> 00:11:12.490 If you say an operator is acting on another operator, 00:11:12.490 --> 00:11:14.580 the first operator that is acting, 00:11:14.580 --> 00:11:16.720 you put it here with its inverse. 00:11:16.720 --> 00:11:20.420 It happens to be unitary, so you put the dagger, 00:11:20.420 --> 00:11:23.750 and you put the operator here. 00:11:23.750 --> 00:11:26.580 And this is the right thing to do. 00:11:26.580 --> 00:11:30.200 It has a simple answer and a simple interpretation, 00:11:30.200 --> 00:11:31.190 as we'll see now. 00:11:31.190 --> 00:11:35.910 So what is T, this commutator, supposed to be? 00:11:35.910 --> 00:11:40.220 Well, you can probably imagine what this is. 00:11:40.220 --> 00:11:42.220 You've calculated it in homework, 00:11:42.220 --> 00:11:44.960 so I will not do it again. 00:11:44.960 --> 00:11:48.470 This is x plus x0. 00:11:48.470 --> 00:11:54.080 So you get the operator, x, plus x0 times the unit operator. 00:11:54.080 --> 00:11:57.490 That was done before. 00:11:57.490 --> 00:12:03.730 And here, you get just p. 00:12:03.730 --> 00:12:04.360 Why? 00:12:04.360 --> 00:12:08.690 Because p hat is the only operator 00:12:08.690 --> 00:12:13.880 that exists in this translation thing, so p commutes with p. 00:12:13.880 --> 00:12:19.110 So these two operators commute and the T tagger hits the T, 00:12:19.110 --> 00:12:21.960 and it's equal to 1, so that's a simple thing. 00:12:24.780 --> 00:12:30.400 So why is this reasonable? 00:12:30.400 --> 00:12:34.160 It's because of the following situation. 00:12:34.160 --> 00:12:40.840 If you have a state, psi, you can ask, for example, 00:12:40.840 --> 00:12:46.120 what is the expectation value of x in the state psi? 00:12:50.930 --> 00:12:54.970 And if this state represents a particle that 00:12:54.970 --> 00:13:01.610 is sitting somewhere here, roughly, the expectation value 00:13:01.610 --> 00:13:05.860 of x is basically that vector that 00:13:05.860 --> 00:13:09.190 tells you where the particle is. 00:13:09.190 --> 00:13:16.260 So you could ask, then, what is the expectation 00:13:16.260 --> 00:13:21.560 value of x in the state T x0 psi? 00:13:28.500 --> 00:13:33.690 So you want to know, what does T x0 really do? 00:13:33.690 --> 00:13:38.330 Here, it seems to say something, takes the operator 00:13:38.330 --> 00:13:41.440 and displaces it, but that seems abstract. 00:13:41.440 --> 00:13:46.530 If you ask this question, this seems more physical. 00:13:46.530 --> 00:13:50.470 You had a state, you act with an operator, it's another state. 00:13:50.470 --> 00:13:52.350 How does it look? 00:13:52.350 --> 00:13:58.140 Well, this expectation value would be the expectation value 00:13:58.140 --> 00:14:07.470 of x on T x0 psi, and the [? brau ?] 00:14:07.470 --> 00:14:11.240 would be psi T x0 dagger. 00:14:14.330 --> 00:14:17.280 So actually, that expectation value 00:14:17.280 --> 00:14:22.220 builds precisely this combination, 00:14:22.220 --> 00:14:24.660 and that's why it's meaningful. 00:14:24.660 --> 00:14:38.790 And since you know what this is, this is psi x plus x0 psi. 00:14:38.790 --> 00:14:41.910 This is equal to the expectation value 00:14:41.910 --> 00:14:50.150 of x in the original state plus x0 times 1. 00:14:53.510 --> 00:15:00.790 So the expectation value of x in the new state, the x0 psi, 00:15:00.790 --> 00:15:05.920 is the expectation value of x in the old state plus x0. 00:15:05.920 --> 00:15:12.050 So indeed, if this is x, you could do this for vectors, 00:15:12.050 --> 00:15:15.100 and here is x0. 00:15:15.100 --> 00:15:22.680 Well, the expectation value of x in the new state, the T x0 00:15:22.680 --> 00:15:28.800 operator, took the state and moved it by a displacement x0 00:15:28.800 --> 00:15:33.840 so that the new expectation value is the old one plus x0. 00:15:37.450 --> 00:15:42.320 So that's physically why these things are relevant. 00:15:42.320 --> 00:15:49.420 A couple of other things you've shown in the homework, 00:15:49.420 --> 00:15:56.900 and you could retry doing them, is that T x0 on the x state, 00:15:56.900 --> 00:16:04.050 by this intuition, should be the x plus x0 state. 00:16:04.050 --> 00:16:07.810 It moves the state to the right. 00:16:07.810 --> 00:16:18.745 And if psi has a wave function, psi of x, T x0 of psi 00:16:18.745 --> 00:16:24.180 has a wave function, psi of x minus x0, 00:16:24.180 --> 00:16:27.860 since you know that psi of x minus x0 00:16:27.860 --> 00:16:33.300 is the wave function translated by x0 to the right. 00:16:33.300 --> 00:16:35.480 The sign is always the opposite one. 00:16:35.480 --> 00:16:37.930 When you write psi of x minus x0, 00:16:37.930 --> 00:16:42.600 the function has been moved to the right x0. 00:16:42.600 --> 00:16:48.410 So this is our whole discussion and reminder 00:16:48.410 --> 00:16:52.480 of what the translation operators are. 00:16:52.480 --> 00:16:55.170 So we've got our translation operator. 00:16:55.170 --> 00:16:57.670 Let's see how we can use it. 00:16:57.670 --> 00:17:01.880 And we'll use it to define the coherent states. 00:17:01.880 --> 00:17:05.715 So here comes the definition of what the coherent state is. 00:17:08.250 --> 00:17:12.069 It's a beginning definition, or a working definition, 00:17:12.069 --> 00:17:16.250 until we understand it enough that we can generalize it. 00:17:16.250 --> 00:17:18.400 By the time we finish the lecture, 00:17:18.400 --> 00:17:23.329 this definition will be generalized in a very nice way, 00:17:23.329 --> 00:17:25.829 in a very elegant way. 00:17:25.829 --> 00:17:28.975 So coherent states. 00:17:36.110 --> 00:17:37.480 So here it goes. 00:17:37.480 --> 00:17:43.010 I'm going to take the vacuum state of the harmonic 00:17:43.010 --> 00:17:46.840 oscillator, the ground state of the harmonic oscillator, 00:17:46.840 --> 00:17:54.430 and simply displace it with a translation operator by x0. 00:17:54.430 --> 00:18:04.260 So this is going to be e to the minus i p hat x0 over h bar 0. 00:18:04.260 --> 00:18:06.680 And I want a name for this state, 00:18:06.680 --> 00:18:09.190 and that's the worst part of it. 00:18:09.190 --> 00:18:11.590 There's no great name for it. 00:18:15.160 --> 00:18:19.000 I don't know if any notation is very good. 00:18:19.000 --> 00:18:23.760 If it's very good, it's cumbersome, 00:18:23.760 --> 00:18:25.890 so I'll write it like this. 00:18:25.890 --> 00:18:27.230 A little misleading. 00:18:27.230 --> 00:18:30.510 I'll put a tilde over the state. 00:18:30.510 --> 00:18:32.490 You could say it's a tilde over the x, 00:18:32.490 --> 00:18:34.430 but it really, morally speaking, is 00:18:34.430 --> 00:18:36.480 a tilde over the whole state. 00:18:36.480 --> 00:18:40.440 It means that this thing, you should 00:18:40.440 --> 00:18:44.820 read there's an x0 here used for the translation operator that 00:18:44.820 --> 00:18:46.480 appears here. 00:18:46.480 --> 00:18:50.770 So that's the state, x tilde 0. 00:18:50.770 --> 00:18:54.100 Intuitively, you know what it is. 00:18:54.100 --> 00:18:57.450 You have the harmonic oscillator potential. 00:18:57.450 --> 00:18:58.910 Here is x. 00:18:58.910 --> 00:19:02.640 The ground state is some wave function like that. 00:19:02.640 --> 00:19:07.670 This state has been moved to position x0, 00:19:07.670 --> 00:19:11.880 and presumably some sort of wave function like that, 00:19:11.880 --> 00:19:15.670 because this translates the wave function. 00:19:15.670 --> 00:19:19.900 So the ground state moves it up there to the right. 00:19:19.900 --> 00:19:21.260 That's what it is. 00:19:21.260 --> 00:19:23.240 That's a coherent state. 00:19:23.240 --> 00:19:27.800 And there's no time dependence here so far, 00:19:27.800 --> 00:19:31.020 so this is the state at some instant of time. 00:19:31.020 --> 00:19:34.580 The coherent state, maybe call it at time equals zero. 00:19:34.580 --> 00:19:38.850 Let's leave time frozen for a little while 00:19:38.850 --> 00:19:41.640 until we understand what this state does. 00:19:41.640 --> 00:19:45.190 Then we'll put the time back. 00:19:45.190 --> 00:19:48.475 So a few remarks on this. 00:19:51.220 --> 00:19:58.085 x0 x0 is how much? 00:20:04.350 --> 00:20:07.670 Now, don't think these are position eigenstates. 00:20:07.670 --> 00:20:09.380 That's a possible mistake. 00:20:09.380 --> 00:20:13.550 That's not a position eigenstate. 00:20:13.550 --> 00:20:16.780 This is a coherent state. 00:20:16.780 --> 00:20:18.680 If these would be position eigenstate, 00:20:18.680 --> 00:20:21.220 you say delta of this minus that, 00:20:21.220 --> 00:20:24.890 but it's nothing to do with that. 00:20:24.890 --> 00:20:27.840 Can you tell without doing any computation what 00:20:27.840 --> 00:20:29.330 is this number? 00:20:29.330 --> 00:20:30.890 How much should be? 00:20:30.890 --> 00:20:31.650 Yes? 00:20:31.650 --> 00:20:32.587 AUDIENCE: 1. 00:20:32.587 --> 00:20:33.920 BARTON ZWIEBACH: It should be 1. 00:20:33.920 --> 00:20:34.440 Why? 00:20:34.440 --> 00:20:38.630 Because it's a unitary operator acting on this thing, 00:20:38.630 --> 00:20:40.160 so it preserve length. 00:20:40.160 --> 00:20:44.100 So this should be equal to 0 0, should be 1. 00:20:44.100 --> 00:20:46.680 Very good. 00:20:46.680 --> 00:20:48.550 No need to do the computation. 00:20:48.550 --> 00:20:50.710 It's just 1. 00:20:50.710 --> 00:21:02.260 Psi associated to this state is the ground state wave function 00:21:02.260 --> 00:21:04.380 at x minus x0. 00:21:04.380 --> 00:21:14.435 Where this refers to the wave function, x0 is psi 0 of x. 00:21:17.440 --> 00:21:20.400 So this is what I was saying here. 00:21:20.400 --> 00:21:22.830 The wave function has been translated 00:21:22.830 --> 00:21:27.670 to x0, the remark over there. 00:21:30.180 --> 00:21:33.450 So these are our coherent states and we 00:21:33.450 --> 00:21:39.240 want to understand the first few basic things about them 00:21:39.240 --> 00:21:42.435 so we can do the following simple computations. 00:21:51.130 --> 00:21:53.540 So if I have to do the following, 00:21:53.540 --> 00:21:55.500 if I have to compute the expectation 00:21:55.500 --> 00:22:04.850 value of any operator, A, on a coherent state, 00:22:04.850 --> 00:22:09.180 I use the fact that I want to go back to the vacuum, 00:22:09.180 --> 00:22:15.820 so I put T x0 dagger A T x0 0. 00:22:19.160 --> 00:22:25.360 Because that way, I trace back to what the vacuum is doing. 00:22:25.360 --> 00:22:29.840 It's much easier to do that than to try to calculate something 00:22:29.840 --> 00:22:31.250 from scratch. 00:22:31.250 --> 00:22:46.830 So for example, we have here that x0 x x0, well, you 00:22:46.830 --> 00:22:53.090 would replace it by T x T, T dagger x T, which you know 00:22:53.090 --> 00:22:56.140 is x hat plus x0. 00:22:56.140 --> 00:23:03.960 We calculated it a few seconds ago, top blackboard. 00:23:03.960 --> 00:23:07.570 And therefore, you got what is the expectation 00:23:07.570 --> 00:23:10.800 value of x on the ground state? 00:23:10.800 --> 00:23:13.210 x0, very good. 00:23:13.210 --> 00:23:18.560 And therefore, we just got x0, which is what you would expect. 00:23:18.560 --> 00:23:23.100 The expectation value of x on the coherent state is x0. 00:23:23.100 --> 00:23:24.270 You're there. 00:23:24.270 --> 00:23:26.410 You've been displaced. 00:23:26.410 --> 00:23:37.080 How about the momentum, x0 p hat x0? 00:23:37.080 --> 00:23:43.430 Well, p acted by the translation operator is unchanged. 00:23:43.430 --> 00:23:49.570 Therefore, we got 0 p 0, and again that 0, so this state 00:23:49.570 --> 00:23:53.330 still has no momentum. 00:23:53.330 --> 00:23:57.860 It represents a T equals 0, a state that is over here. 00:23:57.860 --> 00:24:02.780 And just by looking at it, it's just sitting there, 00:24:02.780 --> 00:24:05.140 has no momentum whatsoever. 00:24:08.440 --> 00:24:11.310 Another question that is interesting, 00:24:11.310 --> 00:24:16.201 what is the expectation value of the Hamiltonian 00:24:16.201 --> 00:24:18.010 on the coherent state? 00:24:21.080 --> 00:24:28.640 Well, this should be, now you imagine in your head, 00:24:28.640 --> 00:24:37.350 T dagger H T. Now, H is p squared over 2m, 00:24:37.350 --> 00:24:41.800 and that p squared over 2m gets unchanged. 00:24:41.800 --> 00:24:49.520 p squared over 2m is not changed because T dagger and T does 00:24:49.520 --> 00:24:53.330 nothing to it, T dagger from the left, T. 00:24:53.330 --> 00:24:54.120 Nevertheless. 00:24:54.120 --> 00:25:01.240 the Hamiltonian has a 1/2 m omega squared x hat, 00:25:01.240 --> 00:25:06.960 and x hat is changed by becoming x hat plus x0. 00:25:16.120 --> 00:25:20.840 Well, we don't want to compute too hard, do 00:25:20.840 --> 00:25:22.280 too much effort here. 00:25:22.280 --> 00:25:26.920 So first, we realize that here's the p squared over 2m 00:25:26.920 --> 00:25:31.070 and here's the m omega squared x hat squared, so that's 00:25:31.070 --> 00:25:32.980 the whole Hamiltonian. 00:25:32.980 --> 00:25:44.500 So we got 0 H 0 plus the extra terms that come here. 00:25:44.500 --> 00:25:46.720 But what terms come here? 00:25:46.720 --> 00:25:54.250 There's a product of an x0 and an x between 0 and 0. 00:25:54.250 --> 00:26:01.220 x0 is a number, so you have an x between 0 and 0, and that's 0. 00:26:01.220 --> 00:26:05.810 So the cross product here won't contribute to the expectation 00:26:05.810 --> 00:26:10.240 value, so the last term that is there is 1/2 m 00:26:10.240 --> 00:26:16.310 is a number omega squared, x0 squared. 00:26:16.310 --> 00:26:18.200 And what is the expectation value 00:26:18.200 --> 00:26:19.890 of the Hamiltonian on the vacuum? 00:26:19.890 --> 00:26:28.450 It's h omega over 2 plus 1/2 m omega squared, x0 squared. 00:26:28.450 --> 00:26:35.000 And you start seeing classical behavior. 00:26:35.000 --> 00:26:38.400 The expectation value of the energy at this point 00:26:38.400 --> 00:26:44.840 is a little quantum thing plus the whole cost 00:26:44.840 --> 00:26:49.080 of stretching something all the way to x0. 00:26:49.080 --> 00:26:57.690 1/2 of k squared, k for the oscillator, x0 squared. 00:26:57.690 --> 00:27:02.500 So the energy of this thing is quite reasonably approximated, 00:27:02.500 --> 00:27:07.250 if x0 is large enough, by the second term, 00:27:07.250 --> 00:27:09.870 and this is the cost of energy of having 00:27:09.870 --> 00:27:13.450 a particle of the potential. 00:27:13.450 --> 00:27:16.280 So it's behaving in a reasonable way. 00:27:19.100 --> 00:27:26.910 You can do a couple more little exercises that I'll 00:27:26.910 --> 00:27:35.613 put here as things for you to check. 00:27:35.613 --> 00:27:36.113 Exercise. 00:27:38.660 --> 00:27:44.770 x0 tilde x squared x0 tilde. 00:27:47.480 --> 00:27:48.330 Just calculate. 00:27:48.330 --> 00:27:51.250 It's just useful to have. 00:27:51.250 --> 00:27:56.650 x0 squared plus h bar over 2m omega. 00:27:56.650 --> 00:28:09.290 And x0 tilde p squared x0 tilde is mh omega over 2. 00:28:09.290 --> 00:28:21.535 And finally, x0 tilde xp plus px x0 tilde is equal to 0. 00:28:29.740 --> 00:28:31.520 Any questions? 00:28:31.520 --> 00:28:34.580 These are exercises for you to practice 00:28:34.580 --> 00:28:36.072 a little these expectation values. 00:28:38.700 --> 00:28:40.700 Questions on what we've done so far? 00:28:40.700 --> 00:28:42.254 Yes? 00:28:42.254 --> 00:28:44.739 AUDIENCE: You said these coherent states is 00:28:44.739 --> 00:28:47.224 most significant only in the ground state, 00:28:47.224 --> 00:28:53.964 or is it also important to use them for [INAUDIBLE]? 00:28:56.770 --> 00:29:01.770 BARTON ZWIEBACH: Well, we've defined the coherent state 00:29:01.770 --> 00:29:05.110 by taking the ground state and moving it, 00:29:05.110 --> 00:29:07.420 and these are particularly interesting. 00:29:07.420 --> 00:29:10.800 You could try to figure out what would happen if you would take 00:29:10.800 --> 00:29:13.320 an excited state and you move it. 00:29:13.320 --> 00:29:16.410 Things are a little more complicated. 00:29:16.410 --> 00:29:18.360 PROFESSOR: And in a sense, they can all 00:29:18.360 --> 00:29:21.790 be understood in terms of what we do to the ground state. 00:29:21.790 --> 00:29:25.190 So we will not focus on them too much. 00:29:25.190 --> 00:29:27.860 In a sense, you will see when we generalize 00:29:27.860 --> 00:29:33.430 this how what we're doing is very special, in at least 00:29:33.430 --> 00:29:35.660 one simple way. 00:29:35.660 --> 00:29:41.040 So we'll always focus on translating the grounds. 00:29:41.040 --> 00:29:43.560 Other questions? 00:29:43.560 --> 00:29:44.690 Yes. 00:29:44.690 --> 00:29:47.180 AUDIENCE: Where does the term [INAUDIBLE] arise 00:29:47.180 --> 00:29:50.120 and why does it persevere [INAUDIBLE] 00:29:50.120 --> 00:29:54.940 PROFESSOR: OK, here is the thing of the coherent state. 00:29:54.940 --> 00:30:00.000 Is this an energy eigenstate at this moment? 00:30:00.000 --> 00:30:01.030 What do you think? 00:30:01.030 --> 00:30:06.090 Is this an energy eigenstate-- this state over here? 00:30:06.090 --> 00:30:09.250 No, it won't be an energy eigenstate. 00:30:09.250 --> 00:30:11.050 There's something funny about it. 00:30:11.050 --> 00:30:15.010 Energy eigenstates are always diffuse things. 00:30:15.010 --> 00:30:18.050 They never look like that. 00:30:18.050 --> 00:30:19.970 So this is not an energy eigenstate, 00:30:19.970 --> 00:30:22.820 and you've done things with non-energy eigenstates. 00:30:22.820 --> 00:30:25.430 They change shape. 00:30:25.430 --> 00:30:28.950 As they evolve, they change shape. 00:30:28.950 --> 00:30:32.160 What we will see very soon is that this state, 00:30:32.160 --> 00:30:37.170 if we let it go, it will start moving back and forth 00:30:37.170 --> 00:30:39.630 without changing shape. 00:30:39.630 --> 00:30:43.450 It's going to do an amazing thing. 00:30:43.450 --> 00:30:46.120 Energy eigenstates-- you're super-close to energy 00:30:46.120 --> 00:30:46.870 eigenstates. 00:30:46.870 --> 00:30:50.710 You get something that changes in time and the shape changes, 00:30:50.710 --> 00:30:53.210 and you've even done problems like that. 00:30:53.210 --> 00:30:58.620 But this state is so exceptional that even 00:30:58.620 --> 00:31:01.815 as we let it go in time, it's going to change, 00:31:01.815 --> 00:31:05.740 but the shape is not going to spread out. 00:31:05.740 --> 00:31:08.440 Do you remember when you considered a pulse 00:31:08.440 --> 00:31:13.270 in a free particle, how it disappears and stretches away? 00:31:13.270 --> 00:31:15.460 Well, in the harmonic oscillator, 00:31:15.460 --> 00:31:19.340 this has been so well prepared that this thing, as time goes 00:31:19.340 --> 00:31:23.660 by, will just move and oscillate like a particle. 00:31:23.660 --> 00:31:26.350 And it does so coherently. 00:31:26.350 --> 00:31:28.430 It doesn't change shape. 00:31:28.430 --> 00:31:32.780 When we talk about light, coherent light 00:31:32.780 --> 00:31:34.610 is what you get from lasers. 00:31:34.610 --> 00:31:36.500 And so if you want understand lasers, 00:31:36.500 --> 00:31:38.230 you have to understand coherent states. 00:31:41.020 --> 00:31:46.140 OK, so this brings us there to time evolution. 00:31:46.140 --> 00:31:48.376 So let's do time evolution. 00:32:05.480 --> 00:32:06.760 So what will happen? 00:32:06.760 --> 00:32:16.440 We'll have a state x0 goes to x0 comma t. 00:32:16.440 --> 00:32:17.880 So that's the notation. 00:32:17.880 --> 00:32:23.530 That's what we'll mean by the state at a later time. 00:32:23.530 --> 00:32:26.510 And how are we going to explore this? 00:32:26.510 --> 00:32:30.290 Well, we're all set with our Heisenberg operator, there. 00:32:30.290 --> 00:32:33.410 We'll take expectation values of things 00:32:33.410 --> 00:32:36.420 to figure out how things look. 00:32:36.420 --> 00:32:39.300 So what do we have here? 00:32:39.300 --> 00:32:46.550 We'll ask for X0 t, and we'll put the Schrodinger operator 00:32:46.550 --> 00:32:50.910 in between here-- X0 t, and this is 00:32:50.910 --> 00:32:54.760 what we'll call the expectation value of A 00:32:54.760 --> 00:32:58.230 as time goes by in the X0 0 state. 00:33:01.510 --> 00:33:03.120 This is what we call this. 00:33:03.120 --> 00:33:06.720 But then, we have the time evolution. 00:33:06.720 --> 00:33:14.320 So this is equal to the original state, Heisenberg operator 00:33:14.320 --> 00:33:17.785 of A-- original state. 00:33:21.650 --> 00:33:26.460 And if you wish, you could then put the t operator-- 00:33:26.460 --> 00:33:30.280 as we have in the top blackboard to the right-- 00:33:30.280 --> 00:33:32.130 and reduce it even more. 00:33:32.130 --> 00:33:37.930 But we've computed a lot of this coherent state expectation 00:33:37.930 --> 00:33:41.510 value, so let's leave it like that. 00:33:41.510 --> 00:33:44.810 So you could, if you wish, say this 00:33:44.810 --> 00:33:56.970 is equal to 0-- T X0 dagger A T X0 0. 00:33:56.970 --> 00:34:01.080 So you can ultimately reduce the expectation values 00:34:01.080 --> 00:34:02.810 of things on the vacuum. 00:34:05.830 --> 00:34:07.680 So OK, we're all set. 00:34:07.680 --> 00:34:09.125 Let's try to do one. 00:34:17.210 --> 00:34:19.960 And the reason this is a nice calculation 00:34:19.960 --> 00:34:21.820 is that the time evolution of this state 00:34:21.820 --> 00:34:23.370 is a little complicated. 00:34:23.370 --> 00:34:26.540 We'll figure it out later, but it's easier 00:34:26.540 --> 00:34:28.449 to work with the time evolved state. 00:34:28.449 --> 00:34:31.820 So here it goes-- what is the expectation 00:34:31.820 --> 00:34:40.120 value of X as a function of time on the X0 state? 00:34:40.120 --> 00:34:45.620 Well, it says here take the X0 state, 00:34:45.620 --> 00:34:49.739 and take the Heisenberg value of X. 00:34:49.739 --> 00:34:56.960 So we have it up there-- X hat cosine omega t plus b hat 00:34:56.960 --> 00:35:03.260 over M omega sine omega t X0. 00:35:03.260 --> 00:35:06.630 Forget about time evolution of the coherent states. 00:35:06.630 --> 00:35:09.240 We evolved the operator. 00:35:09.240 --> 00:35:13.510 On the other hand, we have that the expectation value of p 00:35:13.510 --> 00:35:19.090 is 0 in the coherent state, and the expectation value of X 00:35:19.090 --> 00:35:20.980 is X0. 00:35:20.980 --> 00:35:27.760 So end of story-- calculation over-- X0 cosine of omega t. 00:35:27.760 --> 00:35:30.630 That's expectation value in time. 00:35:30.630 --> 00:35:32.806 This thing is oscillating classically. 00:35:36.110 --> 00:35:39.300 That's nice as can be. 00:35:39.300 --> 00:35:46.720 So classical behavior again, of a quantum state. 00:35:46.720 --> 00:35:55.690 How about expectation value of p X0 of t? 00:35:55.690 --> 00:35:58.535 If it's oscillating, it better be moving, 00:35:58.535 --> 00:36:00.720 and it better have some momentum. 00:36:00.720 --> 00:36:04.430 So let's put the momentum operator here, 00:36:04.430 --> 00:36:07.110 the Heisenberg one. 00:36:07.110 --> 00:36:14.500 So we'll have p hat cosine omega t minus m omega x 00:36:14.500 --> 00:36:22.160 hat sine omega t X0 tilde. 00:36:22.160 --> 00:36:27.080 And this is 0, but X has X0 there, 00:36:27.080 --> 00:36:35.510 so minus m omega X0 sine of omega t, which 00:36:35.510 --> 00:36:46.110 is equal to m d dt of the expectation value of X. 00:36:46.110 --> 00:36:50.430 Here it is-- expectation value of X. m d dt of that 00:36:50.430 --> 00:36:53.900 is minus m omega X0 sine omega t. 00:36:53.900 --> 00:36:57.220 That's what it should be. 00:36:57.220 --> 00:37:00.760 And this thing is really oscillating classically-- 00:37:00.760 --> 00:37:04.330 not only the exposition, but the momentum is doing that. 00:37:07.130 --> 00:37:09.750 Now, the other thing that we can compute-- 00:37:09.750 --> 00:37:15.650 and we want to compute-- is the key thing. 00:37:15.650 --> 00:37:17.110 You have this state. 00:37:17.110 --> 00:37:20.420 We said it's coherent evolution. 00:37:20.420 --> 00:37:23.800 So the ground state is this state 00:37:23.800 --> 00:37:27.190 that is a minimum uncertainty packet. 00:37:27.190 --> 00:37:31.800 It has a delta X uncertainty mix and a delta p. 00:37:31.800 --> 00:37:37.280 Their product saturates the uncertainty in equality. 00:37:37.280 --> 00:37:41.710 And when we move the state X0, well, the delta X 00:37:41.710 --> 00:37:42.650 will be the same. 00:37:42.650 --> 00:37:45.970 The delta p will be the same, and it's that. 00:37:45.970 --> 00:37:49.730 But now as it starts to move, we want 00:37:49.730 --> 00:37:52.630 to see if the shape is kept the same. 00:37:52.630 --> 00:37:56.100 Maybe it fattens up, and shrinks down, 00:37:56.100 --> 00:37:58.770 and does things in the middle. 00:37:58.770 --> 00:38:02.200 So the issue of coherency of this state 00:38:02.200 --> 00:38:07.270 is the issue whether the uncertainties remain the same. 00:38:07.270 --> 00:38:09.670 If the uncertainties remain the same, 00:38:09.670 --> 00:38:13.070 and they are saturated-- the product is saturating 00:38:13.070 --> 00:38:17.700 the inequality, you know that the shape has to be Gaussian, 00:38:17.700 --> 00:38:22.210 and it must be the same shape that is running around. 00:38:22.210 --> 00:38:28.570 So what we need to compute is the uncertainty in X, 00:38:28.570 --> 00:38:29.800 for example. 00:38:29.800 --> 00:38:42.130 So how do delta X of t and delta p of p behave? 00:38:45.610 --> 00:38:48.220 That's our question. 00:38:48.220 --> 00:38:50.305 And let's see how they do. 00:38:54.210 --> 00:38:57.060 Well, we have this computation-- actually, 00:38:57.060 --> 00:38:59.640 if you don't have the Heisenberg picture, 00:38:59.640 --> 00:39:02.930 it's kind of a nightmare. 00:39:02.930 --> 00:39:07.450 With the Heisenberg picture, it's a lot easier. 00:39:07.450 --> 00:39:12.250 Delta x squared of t would be the expectation 00:39:12.250 --> 00:39:21.120 value of X0 t of X squared, X0 t, 00:39:21.120 --> 00:39:30.790 minus the expectation value of X0 t X, X0 t squared. 00:39:35.200 --> 00:39:39.396 I wrote what the definition of the uncertainty squared is. 00:39:39.396 --> 00:39:43.450 It's the expectation value of the operator squared, 00:39:43.450 --> 00:39:46.330 minus the square of the expectation 00:39:46.330 --> 00:39:49.250 value of the operator. 00:39:49.250 --> 00:39:52.390 And of course, everything is going to turn Heisenberg 00:39:52.390 --> 00:39:57.490 immediately, so this thing-- maybe I can go one more 00:39:57.490 --> 00:40:08.070 line here-- would be X0 X Heisenberg squared of t, 00:40:08.070 --> 00:40:16.170 X0 minus-- this is simple-- this we've calculated. 00:40:16.170 --> 00:40:20.010 it's that expectation value at the top 00:40:20.010 --> 00:40:23.530 is the expectation value of X in time. 00:40:23.530 --> 00:40:24.250 It's that. 00:40:24.250 --> 00:40:29.650 So this is minus X0 squared cosine squared of omega t. 00:40:37.950 --> 00:40:39.680 So what do we have to do? 00:40:39.680 --> 00:40:42.670 We have to focus on this term. 00:40:45.860 --> 00:40:50.960 So this term is equal to X0. 00:40:50.960 --> 00:40:53.430 And you have X Heisenberg squared, 00:40:53.430 --> 00:40:57.590 so let's do it-- X squared cosine squared 00:40:57.590 --> 00:41:04.550 omega t plus p hat squared over m 00:41:04.550 --> 00:41:12.810 squared w squared sine squared omega t plus 1 00:41:12.810 --> 00:41:28.240 over mw cosine omega t sine omega t X p plus pX X0 tilde. 00:41:32.290 --> 00:41:36.500 That shows that term, and I just squared that thing, 00:41:36.500 --> 00:41:41.260 but that I suggested a few exercises here. 00:41:41.260 --> 00:41:44.340 This is 0. 00:41:44.340 --> 00:41:48.770 In fact, it's 0 in the ground state as well, so this is 0. 00:41:51.850 --> 00:41:57.410 X squared gives you the top equation-- 00:41:57.410 --> 00:42:05.420 X0 squared plus h bar over 2 m omega cosine 00:42:05.420 --> 00:42:21.040 squared omega t-- plus p squared over m squared w squared, 00:42:21.040 --> 00:42:25.730 so p squared is m h omega over 2. 00:42:25.730 --> 00:42:28.590 And then you have m squared omega 00:42:28.590 --> 00:42:31.860 squared sine squared omega t. 00:42:35.590 --> 00:42:38.470 And that's this whole term. 00:42:38.470 --> 00:42:41.020 And the thing that we're supposed to do 00:42:41.020 --> 00:42:45.370 is subtract this here. 00:42:45.370 --> 00:42:51.210 You see that the X0 squared cosine squared of omega t 00:42:51.210 --> 00:42:52.130 cancels here. 00:42:56.790 --> 00:42:58.810 So what do we get? 00:42:58.810 --> 00:43:03.680 h bar over 2 mw cosine squared omega t. 00:43:03.680 --> 00:43:13.980 But this thing is also h bar over 2 mw sine squared omega t. 00:43:13.980 --> 00:43:16.890 So this whole thing, all the times 00:43:16.890 --> 00:43:20.840 have disappeared-- delta X squared-- 00:43:20.840 --> 00:43:24.410 the time dependence here has disappeared with that, 00:43:24.410 --> 00:43:28.090 and the cosine squared with sine squared have combined, 00:43:28.090 --> 00:43:33.050 and you get h bar over 2 m omega, which 00:43:33.050 --> 00:43:38.500 was-- this is, I'm sorry, of t. 00:43:38.500 --> 00:43:41.000 We work very hard to put the t there. 00:43:41.000 --> 00:43:42.970 We should leave it. 00:43:42.970 --> 00:43:48.080 The uncertainty as a function of time has not changed. 00:43:48.080 --> 00:43:52.160 It is the original uncertainty of the ground state. 00:43:52.160 --> 00:43:55.030 So this is moving in a nice way. 00:43:55.030 --> 00:44:00.790 You're supposed to compute now as well the uncertainty in p. 00:44:00.790 --> 00:44:04.050 I leave that as an exercise-- delta 00:44:04.050 --> 00:44:14.180 p squared of t equal m h bar omega over 2. 00:44:14.180 --> 00:44:15.290 So this is an exercise. 00:44:19.340 --> 00:44:22.560 Practice with coherent states. 00:44:22.560 --> 00:44:24.520 It's worth doing it, I think. 00:44:24.520 --> 00:44:27.170 Actually there's going to be a problem in the homework 00:44:27.170 --> 00:44:31.570 set, in which you're going to ask to do most of these things, 00:44:31.570 --> 00:44:33.970 including things I'm doing here on the blackboard. 00:44:33.970 --> 00:44:37.350 So you will practice this. 00:44:37.350 --> 00:44:46.230 So between these two, delta p, delta X-- delta X of t, 00:44:46.230 --> 00:44:52.760 delta p of t is, in fact, equal to h bar over 2. 00:44:52.760 --> 00:44:57.190 And this is a minimum uncertainty thing. 00:44:57.190 --> 00:44:58.750 And it behaves quite nicely. 00:45:02.320 --> 00:45:05.950 All right, so the name coherent now should make sense. 00:45:05.950 --> 00:45:09.480 You've produced a quantum state that 00:45:09.480 --> 00:45:13.790 has about the energy of a state that you're familiar with, 00:45:13.790 --> 00:45:18.310 and it moves classically, and it doesn't change shape 00:45:18.310 --> 00:45:20.575 as it moves, so it moves coherent. 00:45:23.520 --> 00:45:26.870 So our next task, therefore, will 00:45:26.870 --> 00:45:30.390 be to understand this in the energy basis. 00:45:30.390 --> 00:45:34.990 Because in the energy basis, it looks like a miracle. 00:45:34.990 --> 00:45:36.980 You've suddenly managed to produce 00:45:36.980 --> 00:45:41.020 a set of states of different energies, 00:45:41.020 --> 00:45:44.150 created the superposition, and suddenly, it 00:45:44.150 --> 00:45:45.620 moves in a nice way. 00:45:45.620 --> 00:45:48.450 Why does that happen? 00:45:48.450 --> 00:45:51.550 So we need to understand the energy basis. 00:45:51.550 --> 00:45:53.560 And as we do that, we'll understand 00:45:53.560 --> 00:45:58.090 how to generalize the coherent states completely. 00:45:58.090 --> 00:46:02.690 So let's go on with that, and let's 00:46:02.690 --> 00:46:04.860 explore this in the energy basis. 00:46:22.690 --> 00:46:23.985 So what do we have? 00:46:28.140 --> 00:46:33.940 We have the coherent state-- no need to put the time yet-- 00:46:33.940 --> 00:46:42.000 is e to the exponential of minus i p hat X0 over h bar. 00:46:46.370 --> 00:46:48.940 There is, as you've seen already, 00:46:48.940 --> 00:46:53.040 at length scale in the harmonic oscillator-- famous length 00:46:53.040 --> 00:46:58.580 scale, and we'll have an abbreviation for it. 00:46:58.580 --> 00:47:06.060 It's the length scale d0 squared h bar over m omega. 00:47:06.060 --> 00:47:11.850 You can use the parameters of the harmonic oscillator-- h bar 00:47:11.850 --> 00:47:14.820 and m and omega-- to produce a length scale. 00:47:14.820 --> 00:47:18.110 And that length scale is d0. 00:47:18.110 --> 00:47:24.930 It's essentially the uncertainty in the position in the ground 00:47:24.930 --> 00:47:28.770 state, up to the square root of 2. 00:47:28.770 --> 00:47:32.930 It's the way-- you want to construct a length-- there 00:47:32.930 --> 00:47:36.060 it is-- the only way you can construct a length. 00:47:36.060 --> 00:47:38.480 So I'm going to use that notation. 00:47:38.480 --> 00:47:49.330 So let me put what the p is into that formula, and simplify it. 00:47:49.330 --> 00:47:53.830 So this is on the vacuum-- I'm sorry, I stopped half the way. 00:47:53.830 --> 00:48:02.590 So this is the exponential of X0 over square root of 2d a dagger 00:48:02.590 --> 00:48:07.325 minus a on the vacuum. 00:48:13.590 --> 00:48:18.800 Plug in the p, get the h bars, and you 00:48:18.800 --> 00:48:22.710 will see that d enters in that way. 00:48:22.710 --> 00:48:27.740 It's the way it has to enter, because this exponential should 00:48:27.740 --> 00:48:32.850 have no units, and therefore X0 over d0 has no units, 00:48:32.850 --> 00:48:36.820 and the a's and the a daggers have no units. 00:48:36.820 --> 00:48:40.640 So it couldn't be any way different like that. 00:48:40.640 --> 00:48:45.960 The i also shouldn't be there, because this operator-- 00:48:45.960 --> 00:48:49.550 the i was there to make this anti-Hermitian. 00:48:49.550 --> 00:48:55.942 But this, with this real, is already anti-Hermitian. 00:48:55.942 --> 00:48:57.150 You see, you take the dagger. 00:48:57.150 --> 00:48:59.450 It becomes minus itself. 00:48:59.450 --> 00:49:01.250 So this is anti-Hermitian. 00:49:01.250 --> 00:49:05.280 No need for an i-- in fact, an i would be wrong, 00:49:05.280 --> 00:49:07.660 so there's no i. 00:49:07.660 --> 00:49:08.630 And that's this. 00:49:11.940 --> 00:49:17.780 Now, we want to figure out how this looks in the energy basis, 00:49:17.780 --> 00:49:23.630 so what are we going to do? 00:49:23.630 --> 00:49:25.370 We're going to have to do something 00:49:25.370 --> 00:49:27.610 with that exponential. 00:49:27.610 --> 00:49:29.710 We're going to have to reorder it. 00:49:29.710 --> 00:49:31.520 This is a job for Baker-Campbell-Hausdorff. 00:49:37.570 --> 00:49:38.660 Which one? 00:49:38.660 --> 00:49:45.600 Well, this one-- e to the X plus Y is equal to e to X, 00:49:45.600 --> 00:49:50.670 e to the Y, e to the minus 1/2-- I 00:49:50.670 --> 00:49:59.510 don't know this by heart-- XY, and it stops there. 00:49:59.510 --> 00:50:06.460 If and only if X commutator with Y commutes with X, 00:50:06.460 --> 00:50:09.100 and commutes with Y. 00:50:09.100 --> 00:50:13.700 There was a problem in the test that there 00:50:13.700 --> 00:50:17.690 was an operator with Xp plus pX acting on X, 00:50:17.690 --> 00:50:22.630 and after you commuted, you get X again, 00:50:22.630 --> 00:50:25.830 and you have to keep including terms. 00:50:25.830 --> 00:50:40.410 So this stops here if XY commutes with X and Y. 00:50:40.410 --> 00:50:42.330 And why do I want this? 00:50:42.330 --> 00:50:46.990 Because I actually want to split the creation 00:50:46.990 --> 00:50:49.150 and the manipulation operators. 00:50:49.150 --> 00:50:53.630 I want them in separate exponentials. 00:50:53.630 --> 00:50:59.650 We have that energy eigenstates are creation operators 00:50:59.650 --> 00:51:05.080 on the vacuum, but here I have creation minus destruction. 00:51:05.080 --> 00:51:07.500 So if I expand the exponential, I'm 00:51:07.500 --> 00:51:10.030 going to get lots of creation and destruction, 00:51:10.030 --> 00:51:14.510 and I'm going to spend hours trying to sort it out. 00:51:14.510 --> 00:51:19.460 If you did expand it, I bet you won't see it through 00:51:19.460 --> 00:51:22.920 so easily-- probably will take you forever, 00:51:22.920 --> 00:51:26.440 and it might not work out. 00:51:26.440 --> 00:51:29.520 So expanding an exponential is something 00:51:29.520 --> 00:51:33.480 that we should be reluctant to do. 00:51:33.480 --> 00:51:36.500 On the other hand, this is a nice option, 00:51:36.500 --> 00:51:38.810 because then you think of this as e 00:51:38.810 --> 00:51:47.310 to the X0 over square root of 2d0 a dagger minus X0 00:51:47.310 --> 00:51:49.525 over square root of 2d0 a. 00:51:55.520 --> 00:52:03.820 And I chose this analogy X with this, and Y with this. 00:52:03.820 --> 00:52:08.300 It's like Y is this thing minus that, and X is that. 00:52:08.300 --> 00:52:10.360 You could have done it the other way around, 00:52:10.360 --> 00:52:13.510 but then you would run into trouble again. 00:52:13.510 --> 00:52:14.440 Why? 00:52:14.440 --> 00:52:21.040 Because I want a Y factor that has the manipulators 00:52:21.040 --> 00:52:26.480 to be to the right of the X factor that has the creation. 00:52:26.480 --> 00:52:27.240 Why? 00:52:27.240 --> 00:52:33.740 Because if I have annihilators closer to the vacuum, 00:52:33.740 --> 00:52:36.100 that's good. 00:52:36.100 --> 00:52:40.110 Annihilators closer to the vacuum is what you really want, 00:52:40.110 --> 00:52:42.840 because if you have creators close to the vacuum, 00:52:42.840 --> 00:52:45.290 they create state, but then you have the manipulators, 00:52:45.290 --> 00:52:47.640 and you have to start working them out. 00:52:47.640 --> 00:52:49.590 On the other hand, if the annihilators 00:52:49.590 --> 00:52:52.880 are close to the vacuum, they just kill the vacuum 00:52:52.880 --> 00:52:55.020 and you can forget them. 00:52:55.020 --> 00:53:00.130 So it's very important that you identify X with this, 00:53:00.130 --> 00:53:03.840 and Y with this whole thing. 00:53:07.170 --> 00:53:18.830 So that this is e to the X0 over square root of 2D0 a dagger, e 00:53:18.830 --> 00:53:26.470 to the minus X0 over square root of 2d0 a. 00:53:26.470 --> 00:53:29.860 And now you're supposed to do the commutator of these two 00:53:29.860 --> 00:53:30.600 things. 00:53:30.600 --> 00:53:32.560 And the commutator is the commutator 00:53:32.560 --> 00:53:39.140 of an a dagger with an a, and that is 1. 00:53:39.140 --> 00:53:44.860 So this commutator is a number-- crucial, 00:53:44.860 --> 00:53:46.850 because if it wasn't a number, if it 00:53:46.850 --> 00:53:49.680 would be an a or an a dagger, it would not 00:53:49.680 --> 00:53:54.320 commute with a, X, and Y, and you have to include more terms. 00:53:54.320 --> 00:53:57.190 So the fact that this commutator is a number 00:53:57.190 --> 00:53:59.680 allows you to use this formula. 00:53:59.680 --> 00:54:05.960 So now we'll put this factor here, minus 1/2. 00:54:05.960 --> 00:54:08.370 Then you have X with Y, and that's 00:54:08.370 --> 00:54:17.740 X0 minus X0 over square root of 2d squared, 00:54:17.740 --> 00:54:20.610 minus-- and this factor squared-- an a dagger 00:54:20.610 --> 00:54:22.620 with a, which is minus 1. 00:54:25.340 --> 00:54:26.860 So that's that whole operator. 00:54:31.800 --> 00:54:36.860 So let's write it. 00:54:36.860 --> 00:54:40.550 The coherent state, therefore, X0 tilde, 00:54:40.550 --> 00:54:47.610 is equal to e to the X0 over square root of 2d0 a hat 00:54:47.610 --> 00:54:55.390 dagger, e to the minus X0 over square root of 2d0 a, 00:54:55.390 --> 00:55:04.130 and this factor that seems to be e to the minus 1/4 X0 00:55:04.130 --> 00:55:07.420 squared over d0 squared. 00:55:07.420 --> 00:55:12.340 And here is this nice vacuum. 00:55:12.340 --> 00:55:14.770 Yes, factor is right. 00:55:14.770 --> 00:55:17.680 So what is this? 00:55:17.680 --> 00:55:22.540 Well, this is a number, so I can pull it to the left. 00:55:22.540 --> 00:55:27.280 And here is the exponential of the annihilator operator. 00:55:27.280 --> 00:55:29.480 Now expand the exponential. 00:55:29.480 --> 00:55:30.680 It's 1. 00:55:30.680 --> 00:55:34.900 That survives, but the first term has an a-- kills it. 00:55:34.900 --> 00:55:38.120 The second term has an a squared-- kills it. 00:55:38.120 --> 00:55:40.260 Everything kills it. 00:55:40.260 --> 00:55:44.640 This thing acting on the vacuum is just 1. 00:55:44.640 --> 00:55:47.380 That's why this is simple. 00:55:47.380 --> 00:55:48.920 So what have we got? 00:55:48.920 --> 00:55:56.290 In the state X0 tilde is e to the minus 1/4 X0 squared 00:55:56.290 --> 00:56:04.660 over d0 squared times e to the X0 over square root of 2d0 00:56:04.660 --> 00:56:07.115 a dagger on the vacuum. 00:56:17.640 --> 00:56:22.700 Well this is nice-- not quite energy eigenstates, 00:56:22.700 --> 00:56:25.480 but we're almost there. 00:56:25.480 --> 00:56:27.050 What is this? 00:56:27.050 --> 00:56:33.780 e to the minus 1/4, X0 squared over d0 squared. 00:56:33.780 --> 00:56:35.210 And now expand. 00:56:35.210 --> 00:56:39.850 This is the sum from n equals 1 to infinity, 00:56:39.850 --> 00:56:43.600 1 over n factorial. 00:56:43.600 --> 00:56:47.948 X0 over square root of 2d0 to the n, 00:56:47.948 --> 00:56:52.530 a hat dagger to the n on the vacuum. 00:56:58.220 --> 00:57:02.680 And what was the nth energy eigenstate? 00:57:02.680 --> 00:57:04.000 You probably remember. 00:57:04.000 --> 00:57:07.130 The nth energy eigenstate is a dagger 00:57:07.130 --> 00:57:12.370 to the n on the vacuum over square root of n factorial. 00:57:12.370 --> 00:57:16.320 So we've got a little more than the square root 00:57:16.320 --> 00:57:17.560 of 2 n factorial. 00:57:17.560 --> 00:57:22.480 So maybe I'll do it here. 00:57:22.480 --> 00:57:31.450 We get e to the minus 1/4 X0 squared over d0 squared, 00:57:31.450 --> 00:57:42.300 sum from n equals 1 to infinity, 1 00:57:42.300 --> 00:57:51.410 over square root of n factorial, X0 over the square root of 2 d0 00:57:51.410 --> 00:57:55.610 to the n times the nth energy eigenstate. 00:57:58.730 --> 00:58:07.180 It's a little messy, but not so bad. 00:58:07.180 --> 00:58:10.080 I think actually I won't need that anymore. 00:58:10.080 --> 00:58:11.445 Well no, I may. 00:58:11.445 --> 00:58:11.945 I will. 00:58:14.670 --> 00:58:22.650 So let's write it maybe again. 00:58:22.650 --> 00:58:25.630 Well, it's OK. 00:58:25.630 --> 00:58:30.540 Let's write it as follows-- Cn n. 00:58:37.144 --> 00:58:42.950 OK, so I got some cn's and n's. 00:58:42.950 --> 00:58:46.600 So this is a very precise superposition 00:58:46.600 --> 00:58:51.040 of energy eigenstates, very delicate superposition 00:58:51.040 --> 00:58:54.290 of energy eigenstates. 00:58:54.290 --> 00:59:00.610 Let me write it in the following way-- cn squared. 00:59:03.440 --> 00:59:07.200 Why would I care about cn squared? 00:59:07.200 --> 00:59:13.940 cn squared is the probability to find the coherent state 00:59:13.940 --> 00:59:17.020 in the nth energy eigenstate. 00:59:17.020 --> 00:59:21.540 The amplitude to have it in the nth energy eigenstate is cn. 00:59:21.540 --> 00:59:26.020 So that probability to find it in the nth energy eigenstate 00:59:26.020 --> 00:59:34.080 is cn squared, is the probability 4x tilde 0 00:59:34.080 --> 00:59:39.680 2b in the nth energy eigenstate. 00:59:39.680 --> 00:59:40.800 That is what? 00:59:40.800 --> 00:59:50.220 Exponential of minus 1/2 x0 squared over d0 squared. 00:59:50.220 --> 00:59:51.770 And I have to square that. 00:59:51.770 --> 00:59:56.322 So I have 1 over n factorial. 00:59:56.322 --> 00:59:58.980 I have to square that coefficient there. 00:59:58.980 --> 01:00:03.970 So it's x0 squared over 2d0 squared-- that's 01:00:03.970 --> 01:00:06.700 nice, it's the same one here-- to the n. 01:00:12.670 --> 01:00:17.210 So it's easier to think of this if you 01:00:17.210 --> 01:00:28.410 invent a new letter, lambda, to be x0 squared over 2d0 squared. 01:00:30.940 --> 01:00:39.850 Then, cn squared is equal to e to the minus lambda lambda 01:00:39.850 --> 01:00:42.170 to the n over n factorial. 01:01:06.400 --> 01:01:07.680 Yes. 01:01:07.680 --> 01:01:10.039 AUDIENCE: Is that something that we 01:01:10.039 --> 01:01:13.532 should expect to be true for any time of [INAUDIBLE], 01:01:13.532 --> 01:01:15.528 or is that something that we [INAUDIBLE]? 01:01:18.720 --> 01:01:24.120 PROFESSOR: Well, let me say it this way. 01:01:24.120 --> 01:01:26.400 In a second, it will become clear 01:01:26.400 --> 01:01:29.280 that this was almost necessary. 01:01:29.280 --> 01:01:35.050 I actually don't know very deeply why this is true. 01:01:35.050 --> 01:01:38.570 And I'm always a little puzzled and uncomfortable at this point 01:01:38.570 --> 01:01:40.840 in 805. 01:01:40.840 --> 01:01:43.925 So what is really strange about this 01:01:43.925 --> 01:01:47.245 is that this is the so-called Poisson distribution. 01:01:49.990 --> 01:01:53.290 So there's something about this energy eigenstate 01:01:53.290 --> 01:01:58.270 that their Poisson distributed in a coherent state. 01:01:58.270 --> 01:02:04.150 So these are probabilities, as I claimed, to find an n. 01:02:04.150 --> 01:02:09.750 And indeed, let's check the sum of the cn squareds from n 01:02:09.750 --> 01:02:12.410 equals 1 to infinity. 01:02:12.410 --> 01:02:13.500 Let's see what it is. 01:02:13.500 --> 01:02:17.734 And you will see, you cannot tinker with this. 01:02:17.734 --> 01:02:21.530 This is e to the minus lambda the sum from n 01:02:21.530 --> 01:02:26.850 equals 1 to infinity lambda to the n over n factorial. 01:02:26.850 --> 01:02:29.810 And that sum-- it's not from n equals 1. 01:02:29.810 --> 01:02:32.370 It's 0 to infinity, I'm sorry. 01:02:32.370 --> 01:02:34.750 Did I write once anywhere? 01:02:34.750 --> 01:02:37.940 Yeah, it should be 0. 01:02:37.940 --> 01:02:39.425 OK, this is 0. 01:02:42.250 --> 01:02:49.290 There is the ground state, so from n equals 0 to infinity. 01:02:49.290 --> 01:02:52.090 And this is e to the minus lambda e 01:02:52.090 --> 01:02:54.030 to the lambda, which is 1. 01:02:58.140 --> 01:03:01.390 So yes, this is Poisson distributed. 01:03:04.170 --> 01:03:07.930 It's some sort of distribution like that. 01:03:07.930 --> 01:03:18.470 So if you have the n's, the cn's, Poisson distributions 01:03:18.470 --> 01:03:22.485 have to do with if you have a radioactive material. 01:03:22.485 --> 01:03:24.030 It has a lifetime. 01:03:24.030 --> 01:03:26.870 And you say, how many events should I-- 01:03:26.870 --> 01:03:28.690 the lifetime is five years. 01:03:28.690 --> 01:03:32.850 How many events should you expect to happen in a week? 01:03:32.850 --> 01:03:34.130 These are Poisson distributed. 01:03:37.100 --> 01:03:40.740 So it's a Poisson distribution. 01:03:40.740 --> 01:03:43.930 It's a very nice thing. 01:03:43.930 --> 01:03:49.160 So let me just make one more remark about it. 01:03:49.160 --> 01:03:56.530 And it's quite something. 01:04:03.040 --> 01:04:08.380 So one question that you could ask 01:04:08.380 --> 01:04:13.130 is, what is the most probable n? 01:04:13.130 --> 01:04:15.030 That's a good question. 01:04:15.030 --> 01:04:17.026 You have a coherent state. 01:04:17.026 --> 01:04:18.650 So it's going to have the superposition 01:04:18.650 --> 01:04:20.570 of the vacuum, the first. 01:04:20.570 --> 01:04:27.300 What is the most probable n, so the expectation value of n? 01:04:27.300 --> 01:04:30.960 Now, I'm thinking of it probabilistically. 01:04:30.960 --> 01:04:33.916 So I'm thinking this is a probability distribution. 01:04:36.860 --> 01:04:40.020 Then, I will show it for you that this is really 01:04:40.020 --> 01:04:41.390 computing what you want. 01:04:41.390 --> 01:04:46.270 But probabilistically, what is the expectation value of n? 01:04:46.270 --> 01:04:49.995 You should sum n times the probability that you get n. 01:04:53.590 --> 01:05:02.160 So this is sum over ne to the minus lambda lambda 01:05:02.160 --> 01:05:04.570 to the n over n factorial. 01:05:04.570 --> 01:05:07.152 So you got an n there. 01:05:07.152 --> 01:05:11.410 And the way you get an n there-- well, the e to the minus lambda 01:05:11.410 --> 01:05:14.100 goes out. 01:05:14.100 --> 01:05:18.230 And the n can be reproduced by doing lambda d d 01:05:18.230 --> 01:05:20.670 lambda on the sum. 01:05:23.590 --> 01:05:27.630 Because lambda d d lambda on this sum brings down this n, 01:05:27.630 --> 01:05:31.380 puts back the lambda so it gives you the thing you had, 01:05:31.380 --> 01:05:32.700 and that's what it is. 01:05:32.700 --> 01:05:38.485 So here, you get e to the minus lambda lambda 01:05:38.485 --> 01:05:44.280 d d lambda of e to the lambda. 01:05:44.280 --> 01:05:47.730 And that is lambda, OK? 01:05:51.220 --> 01:05:59.765 So the expectation value of the most sort of not the peak, 01:05:59.765 --> 01:06:05.590 but the expected value of n in this distribution, the level 01:06:05.590 --> 01:06:08.980 that you're going to be excited is basically lambda. 01:06:08.980 --> 01:06:16.140 So if x0 is 1,000 times bigger than d0, 01:06:16.140 --> 01:06:22.290 you've moved this thing 1,000 times the quantum uncertainty. 01:06:22.290 --> 01:06:26.010 Then, you're occupying most strongly the levels 01:06:26.010 --> 01:06:29.160 at 1 million. 01:06:29.160 --> 01:06:37.590 You get x0 over d0 controls which n is the most likely. 01:06:37.590 --> 01:06:41.140 Indeed, look, this n-- suppose you 01:06:41.140 --> 01:06:48.375 would compute x0 tilde n hat x0 tilde. 01:06:53.120 --> 01:06:56.290 This is what you would think is an occupation number. 01:06:56.290 --> 01:06:59.110 This sounds a little hand wavy. 01:06:59.110 --> 01:07:02.890 But this is the number operator, the expected value 01:07:02.890 --> 01:07:06.030 of the number operator, in the coherent state. 01:07:06.030 --> 01:07:12.410 But this is-- you have that the coherent state is this. 01:07:12.410 --> 01:07:15.640 So let's substitute that in there. 01:07:15.640 --> 01:07:19.860 So you get two sums over n and over m. 01:07:19.860 --> 01:07:32.500 And you would have cm star m N n cn. 01:07:32.500 --> 01:07:37.030 I've substituted x0 and x0 dagger. 01:07:37.030 --> 01:07:40.140 The c's in fact are real. 01:07:40.140 --> 01:07:44.240 And then, the number operator on here is little n. 01:07:44.240 --> 01:07:46.380 And then, you get the Kronecker delta. 01:07:46.380 --> 01:07:53.690 So this is sum over n and m cmcn-- it's real. 01:07:53.690 --> 01:07:57.880 And then, you get n delta m, n. 01:07:57.880 --> 01:08:03.800 So this is in fact the sum over ncn squared. 01:08:03.800 --> 01:08:06.780 So what we wrote here, this is really 01:08:06.780 --> 01:08:09.195 the expectation value of the number operator. 01:08:12.260 --> 01:08:17.040 And one can do more calculations here. 01:08:17.040 --> 01:08:20.229 A calculation that is particularly interesting 01:08:20.229 --> 01:08:24.109 to discover what these states look like 01:08:24.109 --> 01:08:30.069 is the uncertainty in the energy. 01:08:30.069 --> 01:08:36.060 So that's another sort of relevant measure. 01:08:36.060 --> 01:08:39.180 How big is the uncertainty in the energy? 01:08:41.800 --> 01:08:48.200 What are, basically, the delta E associated 01:08:48.200 --> 01:08:49.510 to the coherent state? 01:08:49.510 --> 01:08:52.720 How does it look like? 01:08:52.720 --> 01:08:53.795 Is it very sharp? 01:08:56.460 --> 01:08:59.490 So it's a good question. 01:08:59.490 --> 01:09:02.220 And it's in the notes. 01:09:02.220 --> 01:09:05.399 I leave it for you to try to calculate it. 01:09:05.399 --> 01:09:14.080 Delta E in the coherent state x0, how much is it? 01:09:14.080 --> 01:09:18.029 And it turns out to be the following-- h omega 01:09:18.029 --> 01:09:20.384 x0 over square root of 2d. 01:09:24.689 --> 01:09:28.160 So actually, maybe this is a little surprising. 01:09:28.160 --> 01:09:37.529 But delta E over h omega is equal to x0 over d. 01:09:40.040 --> 01:09:46.689 So actually, the energy uncertainty 01:09:46.689 --> 01:09:50.180 for a classical look in coherent state-- I'm sorry, 01:09:50.180 --> 01:09:52.189 I'm missing a square root of 2 there. 01:09:56.480 --> 01:10:00.020 So what is a classical looking coherent state? 01:10:00.020 --> 01:10:05.580 It's a state in which x0 is much bigger than the quantum d. 01:10:05.580 --> 01:10:09.450 So x0 is much bigger than d. 01:10:09.450 --> 01:10:17.530 So in that case, this is a large number for a classical state-- 01:10:17.530 --> 01:10:21.480 "classical" state. 01:10:21.480 --> 01:10:26.480 But in that case, the uncertainty in delta E 01:10:26.480 --> 01:10:31.280 is really big compared to the spacing 01:10:31.280 --> 01:10:33.540 of the harmonic oscillator. 01:10:33.540 --> 01:10:37.150 So you have-- here is the ground state. 01:10:37.150 --> 01:10:39.270 Here is h omega. 01:10:39.270 --> 01:10:43.040 Here is the coherent state, maybe. 01:10:43.040 --> 01:10:48.740 And you have a lot of energy levels that are excited. 01:10:48.740 --> 01:10:52.730 So if x0 over this is 1,000, well, 01:10:52.730 --> 01:10:57.740 at least 1,000 energy levels are excited. 01:10:57.740 --> 01:11:01.810 But you shouldn't fear that too much. 01:11:01.810 --> 01:11:07.960 Because at the same time, the expectation value of E 01:11:07.960 --> 01:11:11.980 over delta E-- the expectation of E 01:11:11.980 --> 01:11:15.120 is something we calculated at the beginning of the lecture. 01:11:15.120 --> 01:11:17.520 You have the oscillator displaced. 01:11:17.520 --> 01:11:23.320 So this is roughly 1/2 m omega squared x0 squared. 01:11:23.320 --> 01:11:25.310 Throw away the ground state energy. 01:11:25.310 --> 01:11:27.570 That's supposed to be very little. 01:11:27.570 --> 01:11:33.920 Delta E is h bar omega x0 over square root of 2. 01:11:33.920 --> 01:11:36.265 And this is, again, the same ratio. 01:11:39.980 --> 01:11:46.620 So yes, this state is very funny. 01:11:46.620 --> 01:11:52.990 It contains an uncertainty that measured in harmonic oscillator 01:11:52.990 --> 01:11:56.080 levels contains many levels. 01:11:56.080 --> 01:12:01.280 But still, the uncertainty is much smaller 01:12:01.280 --> 01:12:05.490 by the same amount than the average energy. 01:12:05.490 --> 01:12:10.510 So this state is a state of some almost definite energy, 01:12:10.510 --> 01:12:12.830 the uncertainty being much smaller. 01:12:12.830 --> 01:12:15.020 But even though it's much smaller, 01:12:15.020 --> 01:12:19.500 it still contains a lot of levels of the oscillator. 01:12:22.490 --> 01:12:28.510 So that I think gives you a reasonable picture of this. 01:12:28.510 --> 01:12:32.010 So you're ready for a generalization. 01:12:32.010 --> 01:12:37.890 This is a time to generalize the coherent states 01:12:37.890 --> 01:12:40.220 and produce the set of coherent states 01:12:40.220 --> 01:12:45.073 that are most useful eventually, and most flexible. 01:12:48.050 --> 01:12:50.560 And we do them as follows. 01:12:50.560 --> 01:12:59.290 We basically are inspired by this formula 01:12:59.290 --> 01:13:01.020 to write the following operator. 01:13:05.500 --> 01:13:07.760 And here, we change notation. 01:13:07.760 --> 01:13:09.500 This x0 was here. 01:13:09.500 --> 01:13:11.420 But now we'll introduce what is called 01:13:11.420 --> 01:13:16.190 the alpha coherent state. 01:13:16.190 --> 01:13:19.860 Most general coherent state is going 01:13:19.860 --> 01:13:26.070 to be obtained by acted with a unitary operator on the vacuum. 01:13:26.070 --> 01:13:30.126 So far so good-- D of alpha unitary. 01:13:34.060 --> 01:13:39.070 But now generalize what you have there. 01:13:39.070 --> 01:13:42.630 Here, you put a minus a dagger minus a, 01:13:42.630 --> 01:13:46.030 because that was anti-Hermitian, and you put the real constant. 01:13:46.030 --> 01:13:52.580 Now this alpha will belong to the complex numbers. 01:13:52.580 --> 01:13:56.430 Quantum mechanics is all about complex numbers. 01:13:56.430 --> 01:14:00.570 You've got complex vector spaces, complex numbers. 01:14:00.570 --> 01:14:02.040 It's all over the place. 01:14:02.040 --> 01:14:04.660 So how do we do this? 01:14:04.660 --> 01:14:10.630 We do this exponential of alpha a dagger. 01:14:10.630 --> 01:14:13.360 And I want it to be anti-Hermitian. 01:14:13.360 --> 01:14:24.580 So I should put minus alpha star a on the vacuum. 01:14:24.580 --> 01:14:29.180 This thing for alpha equals-- this real number 01:14:29.180 --> 01:14:31.060 reduces to that. 01:14:31.060 --> 01:14:34.720 But now, with alpha complex, it's 01:14:34.720 --> 01:14:38.210 a little more complicated operator. 01:14:38.210 --> 01:14:39.460 And it's more general. 01:14:39.460 --> 01:14:42.770 But it's still unitary. 01:14:42.770 --> 01:14:44.840 And it preserves a norm. 01:14:44.840 --> 01:14:49.280 And its most of what you want from these states. 01:14:49.280 --> 01:14:52.660 So the first thing you do to figure out 01:14:52.660 --> 01:14:58.740 what this operator does is to calculate something 01:14:58.740 --> 01:15:03.410 that maybe you would not expect it to be simple. 01:15:03.410 --> 01:15:06.740 But it's worth doing. 01:15:06.740 --> 01:15:11.330 What is a acting on the alpha state? 01:15:14.560 --> 01:15:19.590 Well, I would have to do a acting 01:15:19.590 --> 01:15:26.060 on this exponential of alpha a dagger minus alpha 01:15:26.060 --> 01:15:29.480 star a on the vacuum. 01:15:34.470 --> 01:15:39.350 Now, a kills the vacuum. 01:15:39.350 --> 01:15:43.460 So maybe you're accustomed already to the next step. 01:15:43.460 --> 01:15:47.470 I can replace that product by a commutator. 01:15:47.470 --> 01:15:50.640 Because the other ordering is 0. 01:15:50.640 --> 01:15:54.050 So this is equal to the commutator. 01:15:54.050 --> 01:16:00.060 Because the other term when a is on the other side is 0 anyway. 01:16:00.060 --> 01:16:02.900 And now I have to compute the commutator 01:16:02.900 --> 01:16:04.535 of a with an exponential. 01:16:08.000 --> 01:16:11.160 Again, it's a little scary. 01:16:11.160 --> 01:16:16.586 But A with an exponential e to the B-- 01:16:16.586 --> 01:16:19.990 it's in the formula sheet, this Campbell-Baker-Hausdorff 01:16:19.990 --> 01:16:37.080 again-- is A, B e to the B if A, B commutes with B. 01:16:37.080 --> 01:16:47.310 Well, this is A. This is B. A with B 01:16:47.310 --> 01:16:53.920 is-- A with B, the exponent-- just alpha times 1 01:16:53.920 --> 01:16:54.670 because of this. 01:16:54.670 --> 01:16:55.590 So it's a number. 01:16:55.590 --> 01:16:57.770 So this is safe. 01:16:57.770 --> 01:17:01.306 So you get alpha times the same exponential. 01:17:04.270 --> 01:17:07.180 But the same exponential means the state 01:17:07.180 --> 01:17:14.120 alpha-- a little quick, isn't it? 01:17:14.120 --> 01:17:20.190 OK, A with B, this factor, was alpha. 01:17:20.190 --> 01:17:26.160 And e to the B anyway on the vacuum is the state alpha. 01:17:26.160 --> 01:17:28.910 So there you go. 01:17:28.910 --> 01:17:30.665 You have achieved the impossible. 01:17:33.510 --> 01:17:37.375 You've diagonalized a non-Hermitian operator. 01:17:40.000 --> 01:17:45.080 This is not Hermitian, and you found its eigenvalues. 01:17:45.080 --> 01:17:48.850 How could that happen? 01:17:48.850 --> 01:17:52.340 Well, it can happen. 01:17:52.340 --> 01:17:54.350 But then, all of the theorems that you 01:17:54.350 --> 01:17:58.290 like about Hermitian operators don't hold. 01:17:58.290 --> 01:17:59.810 So it's a fluke. 01:17:59.810 --> 01:18:01.350 This can be done. 01:18:01.350 --> 01:18:05.860 But then, states that correspond to different eigenvalues 01:18:05.860 --> 01:18:08.770 will not be orthogonal, and they will not 01:18:08.770 --> 01:18:11.680 form a complete set of states, and nothing 01:18:11.680 --> 01:18:16.220 will be quite as you may think it would be. 01:18:16.220 --> 01:18:19.450 But still, it's quite remarkable that this can be done. 01:18:22.020 --> 01:18:26.610 So this characterizes the coherent state in a nice way. 01:18:26.610 --> 01:18:29.600 They're eigenstates of the destruction operator. 01:18:29.600 --> 01:18:34.310 And they're the most general exponentials 01:18:34.310 --> 01:18:40.500 of creation and annihilation operators acting on the vacuum. 01:18:40.500 --> 01:18:49.000 Now, we knew that when alpha is real, it has to do with x0. 01:18:49.000 --> 01:18:53.520 So we've put a complex alpha. 01:18:53.520 --> 01:18:55.440 What will it do? 01:18:55.440 --> 01:18:59.650 A complex alpha, what it does is gives 01:18:59.650 --> 01:19:03.570 the original coherent state some momentum. 01:19:03.570 --> 01:19:06.760 Remember, the original state that we had was an x0. 01:19:06.760 --> 01:19:08.490 And how did it move? 01:19:08.490 --> 01:19:11.100 x0 cosine of omega t. 01:19:11.100 --> 01:19:14.550 So at time equals 0, it had 0 momentum. 01:19:14.550 --> 01:19:19.150 This creates a coherent state at x0, 01:19:19.150 --> 01:19:21.980 and it gives it a momentum controlled 01:19:21.980 --> 01:19:25.170 by the imaginary part of this thing. 01:19:25.170 --> 01:19:28.250 In fact, we can do this as follows. 01:19:28.250 --> 01:19:30.460 You can ask, what is the expectation 01:19:30.460 --> 01:19:33.960 value of x in this state? 01:19:33.960 --> 01:19:37.660 Well, x is written here. 01:19:37.660 --> 01:19:49.041 It's d over square root of 2 alpha a plus a dagger alpha. 01:19:49.041 --> 01:19:53.160 And look, these are easy to compute. 01:19:53.160 --> 01:19:57.110 a gives an alpha, gives you alpha. 01:19:57.110 --> 01:20:00.380 a dagger and alpha, you don't know what it is. 01:20:00.380 --> 01:20:05.620 But a dagger on [? brau ?] alpha is alpha star. 01:20:05.620 --> 01:20:07.940 So this one you know on the right. 01:20:07.940 --> 01:20:10.980 This one you know on the left. 01:20:10.980 --> 01:20:16.070 It gives you the over square root of 2 alpha plus 01:20:16.070 --> 01:20:17.880 alpha star. 01:20:17.880 --> 01:20:23.670 So it's square root of 2d real of alpha. 01:20:23.670 --> 01:20:28.535 So the real part of alpha is the expectation value of x. 01:20:31.400 --> 01:20:33.475 So I'll go here. 01:20:39.420 --> 01:20:43.840 I'm almost done-- waiting for the punch line. 01:20:47.560 --> 01:20:52.740 Similarly, you can calculate the expectation value 01:20:52.740 --> 01:20:55.111 of the momentum. 01:20:55.111 --> 01:21:00.010 It will be alpha p alpha. 01:21:00.010 --> 01:21:02.620 And p is a minus a dagger. 01:21:02.620 --> 01:21:07.230 So you're going to get alpha star minus alpha, so 01:21:07.230 --> 01:21:08.770 the imaginary part. 01:21:08.770 --> 01:21:15.310 So p is actually square root of 2 h 01:21:15.310 --> 01:21:20.570 bar over d imaginary part of alpha. 01:21:20.570 --> 01:21:22.250 So the physics is clear. 01:21:22.250 --> 01:21:24.920 Maybe the formulas are a little messy. 01:21:24.920 --> 01:21:31.005 But when you have an alpha, the real part of alpha 01:21:31.005 --> 01:21:34.590 is telling you where you're positioning the coherent state. 01:21:34.590 --> 01:21:37.020 The imaginary part of alpha is telling 01:21:37.020 --> 01:21:39.450 what kick are you giving to it. 01:21:39.450 --> 01:21:43.690 And you now have produced the most general coherent state. 01:21:43.690 --> 01:21:49.490 So how do we describe that geometrically? 01:21:49.490 --> 01:21:51.655 We imagine it, the alpha plane. 01:21:55.050 --> 01:21:57.970 And here it is. 01:21:57.970 --> 01:22:02.810 The alpha plane, here is the vector, the complex number 01:22:02.810 --> 01:22:06.530 alpha that you've chosen maybe for your state, 01:22:06.530 --> 01:22:09.970 some particular complex value alpha. 01:22:09.970 --> 01:22:13.100 On the x-axis, the real part of alpha 01:22:13.100 --> 01:22:20.020 is the expectation value of x over square root of 2d. 01:22:20.020 --> 01:22:22.970 The real part of alpha is the expectation value 01:22:22.970 --> 01:22:25.620 of x over square root of 2d. 01:22:25.620 --> 01:22:29.670 And the imaginary part of alpha is the expectation value 01:22:29.670 --> 01:22:35.950 of p over square root of 2 h bar d. 01:22:39.100 --> 01:22:43.120 And there it is, your state at time equals 0. 01:22:45.730 --> 01:22:49.880 What is it going to do a little later? 01:22:49.880 --> 01:22:52.080 Well, that will be the last thing 01:22:52.080 --> 01:22:54.610 I want to calculate for you. 01:22:54.610 --> 01:22:56.675 It's a nice answer. 01:22:56.675 --> 01:22:58.820 And you should see it. 01:22:58.820 --> 01:23:01.555 It's going to take me two minutes. 01:23:04.290 --> 01:23:05.990 And what is it? 01:23:05.990 --> 01:23:12.180 Well, alpha at time t, you have to evolve the state-- e 01:23:12.180 --> 01:23:18.580 to the minus iHt over h bar on the state, which 01:23:18.580 --> 01:23:27.970 is e to the alpha a dagger minus alpha star a e to the output 01:23:27.970 --> 01:23:37.090 and into the iHt over h bar, and an e to the minus iHt over h 01:23:37.090 --> 01:23:38.640 bar on the vacuum. 01:23:38.640 --> 01:23:40.420 So I put the one here. 01:23:40.420 --> 01:23:42.830 I evolve with this. 01:23:42.830 --> 01:23:47.210 But I take the state and put this and that. 01:23:47.210 --> 01:23:49.265 This is simple. 01:23:49.265 --> 01:23:54.980 It's e to the minus iH omega over 2. 01:23:54.980 --> 01:23:56.980 That's the energy of the ground state. 01:23:59.500 --> 01:24:01.850 But what is this part? 01:24:05.980 --> 01:24:11.080 It's pretty much the Heisenberg operator. 01:24:11.080 --> 01:24:13.020 But the sign came out wrong. 01:24:16.000 --> 01:24:18.530 Well, it didn't come out wrong. 01:24:18.530 --> 01:24:21.250 It's what it is. 01:24:21.250 --> 01:24:24.290 It just means that what I have to put here 01:24:24.290 --> 01:24:29.000 is the Heisenberg operator at minus t. 01:24:29.000 --> 01:24:31.250 Because I have t for minus t. 01:24:31.250 --> 01:24:36.010 So this is e to the alpha a Heisenberg at 01:24:36.010 --> 01:24:40.536 minus t dagger minus alpha star. 01:24:43.040 --> 01:24:45.910 I'm sorry, I have too many parentheses here. 01:24:48.690 --> 01:24:52.880 That's it, much better-- minus alpha 01:24:52.880 --> 01:25:00.050 star a Heisenberg of minus t acting on this thing. 01:25:00.050 --> 01:25:01.400 And what is this? 01:25:01.400 --> 01:25:04.800 Well, we have the formula for the Heisenberg states here. 01:25:04.800 --> 01:25:11.840 So you've got e to the alpha H a dagger of minus t 01:25:11.840 --> 01:25:20.390 would be e to the minus I omega t a dagger. 01:25:20.390 --> 01:25:23.950 And here, you have minus alpha star e 01:25:23.950 --> 01:25:30.210 to the i omega t a on e to the minus 01:25:30.210 --> 01:25:35.110 i omega h bar over 2 times the vacuum. 01:25:35.110 --> 01:25:36.430 And look what has happened. 01:25:39.130 --> 01:25:44.860 Alpha has become alpha times e to the minus i omega t. 01:25:44.860 --> 01:25:47.200 Because the star is here. 01:25:47.200 --> 01:25:50.200 It's minus the star one. 01:25:50.200 --> 01:25:52.460 So the only thing that has changed 01:25:52.460 --> 01:25:57.590 is that this state, alpha at time t, 01:25:57.590 --> 01:26:03.590 is e to the minus ih bar omega over 2. 01:26:03.590 --> 01:26:05.290 I'm sorry, I'm missing a t here. 01:26:07.810 --> 01:26:09.160 AUDIENCE: [INAUDIBLE] 01:26:09.160 --> 01:26:18.800 PROFESSOR: Yeah, I dropped-- yeah, minus i omega t over 2, 01:26:18.800 --> 01:26:29.370 minus i omega t over 2, minus i omega t over 2, 01:26:29.370 --> 01:26:35.880 times the coherent state, the time independent coherent state 01:26:35.880 --> 01:26:40.090 of value e to the minus i omega t alpha. 01:26:40.090 --> 01:26:41.745 That's a new complex number. 01:26:45.390 --> 01:26:48.290 That's what has happened. 01:26:48.290 --> 01:26:51.940 The number alpha has become e to the minus i omega t. 01:26:51.940 --> 01:26:55.410 Now, this is a face for the whole state multiplicative. 01:26:55.410 --> 01:26:56.900 It's irrelevant. 01:26:56.900 --> 01:26:59.040 So what has this alpha done? 01:26:59.040 --> 01:27:03.390 It has been rotated by e to the minus i omega t. 01:27:03.390 --> 01:27:09.880 So this at the time t is the state alpha t. 01:27:09.880 --> 01:27:11.220 Here is the state alpha. 01:27:11.220 --> 01:27:15.500 And it has rotated by omega t. 01:27:15.500 --> 01:27:18.650 So the coherent state can be visualized 01:27:18.650 --> 01:27:22.870 as a complex number in this complex plane. 01:27:22.870 --> 01:27:26.110 This real part is the expectation value of x at time 01:27:26.110 --> 01:27:28.760 equals 0 whose imaginary part is the expectation 01:27:28.760 --> 01:27:31.720 value of the momentum at time equals 0. 01:27:31.720 --> 01:27:33.700 And how does it evolve? 01:27:33.700 --> 01:27:37.480 This state just rotates with frequency omega 01:27:37.480 --> 01:27:40.510 all along and forever. 01:27:40.510 --> 01:27:43.120 All right, that's it for today. 01:27:43.120 --> 01:27:44.670 See you on Wednesday. 01:27:44.670 --> 01:27:46.470 [APPLAUSE] 01:27:46.470 --> 01:27:48.431 PROFESSOR: Thank you, thank you.