WEBVTT 00:00:01.360 --> 00:00:04.240 PROFESSOR: We're finished with WKB. 00:00:06.870 --> 00:00:11.470 In recitation, you saw some transmission across the barrier 00:00:11.470 --> 00:00:13.740 and that's also included in the notes. 00:00:13.740 --> 00:00:22.550 That's an important application of WKB and should look at it. 00:00:22.550 --> 00:00:26.800 And today we're going to start with a new topic. 00:00:26.800 --> 00:00:29.590 It's time-dependent perturbation theory. 00:00:29.590 --> 00:00:31.590 And time-dependent perturbation theory 00:00:31.590 --> 00:00:36.790 is going to keep us busy for a number of lectures. 00:00:36.790 --> 00:00:40.270 There's a lot of applications of these ideas 00:00:40.270 --> 00:00:45.010 and the methods are rather general. 00:00:45.010 --> 00:00:48.970 Here is the place where we will learn about Fermi's Golden 00:00:48.970 --> 00:00:49.660 Rule. 00:00:49.660 --> 00:00:53.290 The kind of rule that is useful for radiation 00:00:53.290 --> 00:00:57.250 problems, ionization problems, transitions. 00:00:57.250 --> 00:01:00.790 It's all very applied. 00:01:00.790 --> 00:01:05.530 Nevertheless, we have to develop the theory carefully and see 00:01:05.530 --> 00:01:06.990 what are the main concepts. 00:01:06.990 --> 00:01:12.450 So time-dependent perturbation theory is our subject. 00:01:12.450 --> 00:01:23.670 Time-dependent perturbation theory. 00:01:31.070 --> 00:01:36.650 Again, we will begin with a Hamiltonian 00:01:36.650 --> 00:01:39.890 that is time independent. 00:01:39.890 --> 00:01:41.500 And we know about it. 00:01:41.500 --> 00:01:44.730 We call this Hamiltonian H zero. 00:01:44.730 --> 00:01:48.670 The same name we used for time-independent perturbation 00:01:48.670 --> 00:01:50.290 theory. 00:01:50.290 --> 00:01:52.030 We have H zero. 00:01:52.030 --> 00:01:56.740 This time, however, we will have a perturbation. 00:01:56.740 --> 00:02:01.660 The perturbation will also be called delta H. 00:02:01.660 --> 00:02:07.390 But the big difference is this time the perturbation will 00:02:07.390 --> 00:02:10.750 be time dependent. 00:02:10.750 --> 00:02:13.810 And this will be our whole Hamiltonian. 00:02:19.250 --> 00:02:25.460 So this is the subject we're trying to understand. 00:02:25.460 --> 00:02:31.520 Whenever you have a Hamiltonian, this time independent, 00:02:31.520 --> 00:02:34.670 we already know how we're supposed to deal with it. 00:02:34.670 --> 00:02:38.540 We're supposed to find the energy eigenstates. 00:02:38.540 --> 00:02:41.390 And then we'll have the whole collection 00:02:41.390 --> 00:02:42.620 of energy eigenstates. 00:02:42.620 --> 00:02:44.150 We can solve any problem. 00:02:44.150 --> 00:02:47.480 In the initial condition of the wave function, 00:02:47.480 --> 00:02:49.940 you expand it in energy eigenstates. 00:02:49.940 --> 00:02:52.280 You evolve the state in time. 00:02:52.280 --> 00:02:56.210 Everything is nice and simple. 00:02:56.210 --> 00:02:59.900 In a sense, for the first time in your studies 00:02:59.900 --> 00:03:01.760 of quantum mechanics at MIT, we're 00:03:01.760 --> 00:03:05.180 going to face very directly the difficulties of time 00:03:05.180 --> 00:03:06.800 dependence. 00:03:06.800 --> 00:03:12.110 And the first difficulty of a time dependent Hamiltonian is 00:03:12.110 --> 00:03:16.130 that you cannot define energy eigenstates anymore. 00:03:16.130 --> 00:03:20.270 The whole concept is gone. 00:03:20.270 --> 00:03:21.920 That's pretty radical. 00:03:21.920 --> 00:03:24.690 But it's unfortunately the truth. 00:03:24.690 --> 00:03:29.300 When you first learn in 804, how to work 00:03:29.300 --> 00:03:32.000 with energy eigenstates, it was all 00:03:32.000 --> 00:03:35.270 dependent on a factorization. 00:03:35.270 --> 00:03:38.330 A possible factorization of the solution 00:03:38.330 --> 00:03:41.930 in the factor that depends on position and a function that 00:03:41.930 --> 00:03:43.490 depends on time. 00:03:43.490 --> 00:03:47.150 And that time-dependence was always very simple. 00:03:47.150 --> 00:03:52.490 E to the minus iEt over h-bar, where 00:03:52.490 --> 00:03:55.580 E was the energy which was a solution 00:03:55.580 --> 00:04:01.390 of the time-independent, spatial part of the problem. 00:04:01.390 --> 00:04:04.210 So separating the differential equation 00:04:04.210 --> 00:04:08.680 was possible because, for example, the potentials never 00:04:08.680 --> 00:04:11.270 depended on time. 00:04:11.270 --> 00:04:13.420 If the potential depends on time, 00:04:13.420 --> 00:04:18.130 imagine the equation H psi equal e psi. 00:04:18.130 --> 00:04:20.529 This supposed to be time independent. 00:04:20.529 --> 00:04:23.570 But H has a time dependent potential. 00:04:23.570 --> 00:04:25.250 It's just impossible. 00:04:25.250 --> 00:04:30.280 So we don't have energy eigenstates, any more, 00:04:30.280 --> 00:04:34.120 for this potential, for the new potential, 00:04:34.120 --> 00:04:36.036 for the new Hamiltonian. 00:04:36.036 --> 00:04:37.660 And we'll have to think how we're going 00:04:37.660 --> 00:04:39.870 to face these difficulties. 00:04:39.870 --> 00:04:54.330 So no energy eigenstates for H of t. 00:04:54.330 --> 00:05:00.190 Of course, there are energy eigenstates for H0. 00:05:00.190 --> 00:05:05.730 H0 is supposed to be your time-independent Hamiltonian. 00:05:05.730 --> 00:05:07.485 So how do we think of this? 00:05:11.280 --> 00:05:17.010 We will think of time, evolving here, 00:05:17.010 --> 00:05:23.460 and typically we will have an initial time t0. 00:05:23.460 --> 00:05:31.550 And throughout this region, the Hamiltonian, H is equal to H0, 00:05:31.550 --> 00:05:34.530 for time less than 10 0. 00:05:34.530 --> 00:05:38.490 Then we imagine that the perturbation turns on, 00:05:38.490 --> 00:05:44.250 and suddenly things start happening, up to time tf. 00:05:44.250 --> 00:05:51.917 Here the Hamiltonian is H0 plus delta H t. 00:05:54.660 --> 00:06:01.890 And after, the time tf, the Hamiltonian, is back to H0. 00:06:06.220 --> 00:06:11.770 So that's a nice way of thinking about the problem, in which we 00:06:11.770 --> 00:06:16.420 imagine, OK, perturbation is localized 00:06:16.420 --> 00:06:20.890 that some time t begins there before you 00:06:20.890 --> 00:06:26.140 have the known Hamiltonian, after you have the known 00:06:26.140 --> 00:06:27.970 Hamiltonian. 00:06:27.970 --> 00:06:34.720 That allows you to rephrase questions in a clearer way, 00:06:34.720 --> 00:06:39.490 because there are energy eigenstates here. 00:06:39.490 --> 00:06:42.430 If you sit at that energy eigenstate 00:06:42.430 --> 00:06:48.160 before time equal t0, you will remain in an energy eigenstate. 00:06:48.160 --> 00:06:49.900 There's no reason why it changes. 00:06:49.900 --> 00:06:53.710 That's what time-independent Hamiltonians do. 00:06:53.710 --> 00:06:57.550 The energy eigenstate changes by a phase. 00:06:57.550 --> 00:06:59.200 And that's all it does. 00:06:59.200 --> 00:07:03.790 So here, we can speak about energy eigenstates, 00:07:03.790 --> 00:07:08.440 and after time tf, we can speak about energy eigenstates. 00:07:08.440 --> 00:07:12.130 So we can ask the question, suppose 00:07:12.130 --> 00:07:15.760 you were sitting on this particular energy eigenstate, 00:07:15.760 --> 00:07:16.510 here. 00:07:16.510 --> 00:07:20.960 Then the world shakes for a few moments. 00:07:20.960 --> 00:07:24.640 Which state are you going to find yourself, 00:07:24.640 --> 00:07:27.800 after this process? 00:07:27.800 --> 00:07:31.480 And this is a question of going from one energy aide and state 00:07:31.480 --> 00:07:34.510 to some collection of energy eigenstates. 00:07:34.510 --> 00:07:36.590 By the time the lecture will be finished, 00:07:36.590 --> 00:07:42.040 we will have solved this problem in principle, 00:07:42.040 --> 00:07:47.870 and set up how you would do it, in practice, for any case. 00:07:47.870 --> 00:07:52.900 This is a very physical way of thinking, as well. 00:07:52.900 --> 00:07:56.530 You can imagine, you have a hydrogen atom 00:07:56.530 --> 00:07:59.600 in its ground state. 00:07:59.600 --> 00:08:03.550 So the original system is a Hamiltonian 00:08:03.550 --> 00:08:08.240 for a particle, an electron, and a proton. 00:08:08.240 --> 00:08:10.690 And it happens to be in a ground state. 00:08:10.690 --> 00:08:14.530 Then you send in an electromagnetic wave, something 00:08:14.530 --> 00:08:17.130 we will do next lecture. 00:08:17.130 --> 00:08:19.240 No, in a couple of lectures. 00:08:19.240 --> 00:08:23.770 And then, it's possible that the atom gets ionized, 00:08:23.770 --> 00:08:27.640 or that the electron is kicked up to a higher level. 00:08:27.640 --> 00:08:31.840 You will be able to calculate those transition functions. 00:08:31.840 --> 00:08:36.610 You will be able to calculate the probability of ionization. 00:08:36.610 --> 00:08:43.330 In which, after the perturbation is all said and done, you ask, 00:08:43.330 --> 00:08:45.190 what is the probability that you're 00:08:45.190 --> 00:08:48.730 in an eigenstate, that this a higher excited state 00:08:48.730 --> 00:08:50.740 of the hydrogen atom? 00:08:50.740 --> 00:08:55.240 And with a little bit of flexibility in your mind, 00:08:55.240 --> 00:08:57.220 you can think of the hydrogen atom 00:08:57.220 --> 00:09:00.310 as a system that has bound states, 00:09:00.310 --> 00:09:03.220 and continuum states, in which you 00:09:03.220 --> 00:09:07.600 have a proton and an electron traveling a wave. 00:09:07.600 --> 00:09:11.350 If you your electron is far enough from the proton, 00:09:11.350 --> 00:09:13.480 it's like a plane wave. 00:09:13.480 --> 00:09:15.525 If it's not that far, it can travel. 00:09:15.525 --> 00:09:17.800 And its wave is deformed a little. 00:09:17.800 --> 00:09:21.470 Those are the continuum states of the hydrogen atom. 00:09:21.470 --> 00:09:24.940 So you could ask, what is the probability that it's ionized, 00:09:24.940 --> 00:09:28.890 and it's a transition to a continuum eigenstate 00:09:28.890 --> 00:09:31.640 in the hydrogen atom? 00:09:31.640 --> 00:09:34.690 This sounds more complicated. 00:09:34.690 --> 00:09:36.850 It's a little more complicated. 00:09:36.850 --> 00:09:37.350 Why? 00:09:37.350 --> 00:09:40.370 Because when you go from one state to another, 00:09:40.370 --> 00:09:41.480 you can count it. 00:09:41.480 --> 00:09:44.590 But when you ask what is the probability that the electron 00:09:44.590 --> 00:09:47.680 goes into the continuum of plane wave, 00:09:47.680 --> 00:09:51.130 you'll have infinitely many continuum states. 00:09:51.130 --> 00:09:53.130 And we will have to deal with that. 00:09:53.130 --> 00:09:56.520 Where we will know-- 00:09:56.520 --> 00:09:59.380 we will not be able to avoid this complication. 00:09:59.380 --> 00:10:02.680 And that complication makes for a very interesting result, 00:10:02.680 --> 00:10:06.460 transitions to the continuum. 00:10:06.460 --> 00:10:08.740 So this is what we want to do. 00:10:08.740 --> 00:10:13.750 And if given, that in general, this 00:10:13.750 --> 00:10:18.470 is not a problem of finding energy eigenstates, 00:10:18.470 --> 00:10:21.510 the energy eigenstates are known, 00:10:21.510 --> 00:10:32.300 we want to find the wave function, psi of t. 00:10:32.300 --> 00:10:34.055 That's our real unknown. 00:10:37.940 --> 00:10:41.240 And to do that, we will use something 00:10:41.240 --> 00:10:43.130 called the interaction picture. 00:10:49.370 --> 00:10:53.780 Just like we have Heisenberg picture, Schrodinger picture, 00:10:53.780 --> 00:10:58.100 we'll have an interaction picture. 00:10:58.100 --> 00:10:58.720 Don't worry. 00:10:58.720 --> 00:11:03.220 It's not more complicated than anything you've seen. 00:11:03.220 --> 00:11:05.620 It's, in fact, a very sensible way 00:11:05.620 --> 00:11:08.770 of doing things, in which you combine 00:11:08.770 --> 00:11:11.350 good things from the Heisenberg picture, 00:11:11.350 --> 00:11:15.910 and nice things from the Schrodinger picture, together. 00:11:15.910 --> 00:11:24.900 So remember, a little of what was going on 00:11:24.900 --> 00:11:29.790 with the Heisenberg picture. 00:11:29.790 --> 00:11:35.490 It all began by saying that, if you have the expectation 00:11:35.490 --> 00:11:46.300 value of a Schrodinger operator, that was the same thing. 00:11:46.300 --> 00:11:52.990 A s for A Schrodinger, that was possible to compute 00:11:52.990 --> 00:12:00.310 as the expectation value of the Heisenberg operator on the time 00:12:00.310 --> 00:12:03.450 equals 0 states, in the states that don't vary. 00:12:03.450 --> 00:12:10.880 You don't need to compute expectation values of operators 00:12:10.880 --> 00:12:13.060 used in the time involved states. 00:12:13.060 --> 00:12:16.300 You can think of time involved operators 00:12:16.300 --> 00:12:19.355 and compute in this expectation value 00:12:19.355 --> 00:12:21.280 in the time equals 0 states. 00:12:21.280 --> 00:12:24.640 For that, you use the unitary operator, 00:12:24.640 --> 00:12:29.965 that does time evolution, u of t psi 00:12:29.965 --> 00:12:42.640 at 0 u dagger of t psi at 0. 00:12:42.640 --> 00:12:46.630 And this unitary operator, it's in general 00:12:46.630 --> 00:12:49.310 difficult to calculate. 00:12:49.310 --> 00:12:54.220 It's a unitary operator that does time evolution 00:12:54.220 --> 00:12:58.030 and, in our case, it's hard because the Hamiltonian 00:12:58.030 --> 00:12:58.930 is complicated. 00:12:58.930 --> 00:13:04.840 But in general, we call this the Heisenberg operator. 00:13:04.840 --> 00:13:08.890 A Heisenberg of t. 00:13:08.890 --> 00:13:11.116 That's the definition of the Heisenberg operator. 00:13:14.430 --> 00:13:18.180 Another concept that is kind of useful, 00:13:18.180 --> 00:13:23.610 is the idea of operators that brings states to rest. 00:13:23.610 --> 00:13:35.720 So suppose you have the state psi at sum time t. 00:13:35.720 --> 00:13:42.530 I want to act with an operator that will bring it to rest. 00:13:42.530 --> 00:13:45.210 That means that this is time-dependent. 00:13:45.210 --> 00:13:47.450 I want to act with something that 00:13:47.450 --> 00:13:49.190 will make it time-independent. 00:13:49.190 --> 00:13:53.480 So the answer is U dagger. 00:13:53.480 --> 00:13:57.650 U dagger is a unitary operator, so it's the inverse of U. 00:13:57.650 --> 00:14:03.200 So look at this expression, U dagger on this state 00:14:03.200 --> 00:14:05.850 gives you U dagger. 00:14:05.850 --> 00:14:11.540 The state is U on the state at time equals 0. 00:14:11.540 --> 00:14:14.990 U dagger U, being unitary, it's just 00:14:14.990 --> 00:14:20.060 the unit matrix, and your 2 0. 00:14:20.060 --> 00:14:27.380 So this operator, U dagger, removes the time-dependence. 00:14:27.380 --> 00:14:29.120 The [INAUDIBLE] uses the expression, 00:14:29.120 --> 00:14:33.110 brings the operator to rest. 00:14:33.110 --> 00:14:39.710 So we can think of doing something of this kind. 00:14:39.710 --> 00:14:45.490 You see, the whole Hamiltonian is H0 plus delta H. 00:14:45.490 --> 00:14:47.900 H0 you understand well. 00:14:47.900 --> 00:14:50.120 Delta H is complicated. 00:14:50.120 --> 00:14:58.430 So how about doing the time evolution through Heisenberg, 00:14:58.430 --> 00:15:01.160 with H0, with what you know. 00:15:01.160 --> 00:15:04.580 You don't know the full U, but you 00:15:04.580 --> 00:15:07.430 know the U that would do the time 00:15:07.430 --> 00:15:11.120 evolution, for the time-independent Hamiltonian, 00:15:11.120 --> 00:15:12.200 H0. 00:15:12.200 --> 00:15:18.830 So let's attempt to do the part that is easy. 00:15:18.830 --> 00:15:20.960 You see, there's going to be time evolution 00:15:20.960 --> 00:15:23.150 as you go from t0 to tf. 00:15:23.150 --> 00:15:26.780 Some of that is going to be generated by H0, 00:15:26.780 --> 00:15:32.390 some of that by delta H. Let's let Heisenberg do the work 00:15:32.390 --> 00:15:40.040 for H0, and Schrodinger do the work for delta H. 00:15:40.040 --> 00:15:43.130 That's basically the idea. . 00:15:43.130 --> 00:15:47.120 You see, you know that what is difficult about this problem 00:15:47.120 --> 00:15:50.940 is the delta H. So you solve the Schrodinger equation. 00:15:50.940 --> 00:15:55.400 Let's solve the Schrodinger equation that just has delta H, 00:15:55.400 --> 00:15:59.480 doesn't have H anymore. 00:15:59.480 --> 00:16:03.680 So motivated by this, we'll do exactly that. 00:16:03.680 --> 00:16:18.310 Think of for H0, the operator U is e to the minus i H0 00:16:18.310 --> 00:16:19.900 t over h-bar. 00:16:19.900 --> 00:16:22.180 This is the operator that generates 00:16:22.180 --> 00:16:26.890 time evolution for H0. 00:16:33.460 --> 00:16:43.400 So we will take the state psi of t 00:16:43.400 --> 00:16:53.810 and remove the time-dependence associated to H0. 00:16:53.810 --> 00:16:57.140 So try to bring the state to rest. 00:16:57.140 --> 00:17:00.650 So we'll put here, e to the minus-- 00:17:03.261 --> 00:17:03.760 I'm sorry. 00:17:03.760 --> 00:17:05.540 I'm supposed to put U dagger. 00:17:05.540 --> 00:17:14.069 So I'll put e to the I H0 t over H-bar. 00:17:14.069 --> 00:17:17.450 And look at that. 00:17:17.450 --> 00:17:19.010 This is U dagger. 00:17:19.010 --> 00:17:22.849 That's the kind of thing that brings the state to rest. 00:17:22.849 --> 00:17:29.510 If the Hamiltonian had only been H0, only H0, 00:17:29.510 --> 00:17:32.480 this would be time-independent. 00:17:32.480 --> 00:17:35.420 H0 brings it to rest. 00:17:35.420 --> 00:17:39.140 Because the Hamilton is not just H0, 00:17:39.140 --> 00:17:42.800 this will not be, in general, time-independent. 00:17:42.800 --> 00:17:44.790 But it will depend on time. 00:17:44.790 --> 00:17:49.370 So this is a kind of a nice wave function in which you 00:17:49.370 --> 00:17:55.850 sort of have removed the time evolution having to do with H0. 00:17:55.850 --> 00:18:02.360 So we will define this as our auxiliary variable, psi of t. 00:18:05.180 --> 00:18:06.590 That's the definition. 00:18:06.590 --> 00:18:21.310 It's motivated by the idea that, if delta H was equal to 0, then 00:18:21.310 --> 00:18:27.665 psi tilde is constant in time. 00:18:31.750 --> 00:18:34.930 Because if delta H was equal to 0, 00:18:34.930 --> 00:18:37.810 all the evolution is created by H0. 00:18:37.810 --> 00:18:42.585 You would put here, oh, this is e to the minus i H0 times psi 00:18:42.585 --> 00:18:45.790 of t equal 0, the exponentials would cancel, 00:18:45.790 --> 00:18:47.920 and everything would be simple. 00:18:47.920 --> 00:18:51.550 So this is a wave function that is 00:18:51.550 --> 00:18:53.080 going to be our new variable. 00:18:53.080 --> 00:18:56.390 We wanted to find psi of t. 00:18:56.390 --> 00:19:02.330 Now you can say your task is find psi tilde of t. 00:19:05.030 --> 00:19:06.920 That's your new task. 00:19:06.920 --> 00:19:10.310 And it's an equally good task, because if you 00:19:10.310 --> 00:19:16.340 find psi tilde of t, then you can write psi of t 00:19:16.340 --> 00:19:19.820 as, from this equation, e to the minus 00:19:19.820 --> 00:19:26.740 i H0 t over H-bar psi tilde of t. 00:19:31.150 --> 00:19:37.260 So our task now will be to find psi tilde. 00:19:37.260 --> 00:19:40.350 If we have psi tilde, we have psi, as well. 00:19:40.350 --> 00:19:44.170 So we haven't lost any information. 00:19:44.170 --> 00:19:47.170 And this is all good. 00:19:47.170 --> 00:19:51.510 So let's try to see what equation is satisfied 00:19:51.510 --> 00:19:55.050 by psi tilde, what kind of Schrodinger equation 00:19:55.050 --> 00:19:57.050 is satisfied by it. 00:19:57.050 --> 00:20:09.330 So what is the Schrodinger equation for psi tilde. 00:20:09.330 --> 00:20:20.270 I'll just take i H-bar d dt of psi tilde and see what I get. 00:20:20.270 --> 00:20:20.780 OK. 00:20:20.780 --> 00:20:28.360 If I have to differentiate this term, i H-bar d dt. 00:20:28.360 --> 00:20:31.820 I have to differentiate this exponential. 00:20:31.820 --> 00:20:35.540 And the i is going to give you a minus sign. 00:20:35.540 --> 00:20:38.130 The H's are going to cancel. 00:20:38.130 --> 00:20:42.320 And this is just going to bring an H0 down. 00:20:42.320 --> 00:20:50.720 So I'm going to get minus H0 times that exponential. 00:20:50.720 --> 00:20:55.740 And that exponential times psi of t is psi tilde of t. 00:21:03.600 --> 00:21:06.870 So the derivative of the first term 00:21:06.870 --> 00:21:08.670 gives me something with H0. 00:21:08.670 --> 00:21:12.090 And the face times that is still psi tilde. 00:21:12.090 --> 00:21:16.050 Now I have to differentiate the second one so I have plus 00:21:16.050 --> 00:21:25.540 e to the i H0 t over H-bar. 00:21:25.540 --> 00:21:31.680 And i H-bar d dt of this cat. 00:21:31.680 --> 00:21:35.430 But that's the Schrodinger equation for the original cat. 00:21:35.430 --> 00:21:42.150 So I should put here the full Hamiltonian, H0 plus delta H 00:21:42.150 --> 00:21:43.830 times psi of t. 00:21:48.760 --> 00:21:53.890 So when the i dd H hits the state, 00:21:53.890 --> 00:21:58.630 you get the full Hamiltonian time psi of t. 00:21:58.630 --> 00:22:06.190 But I actually were right for psi of t e to the minus i H0 t 00:22:06.190 --> 00:22:11.080 over H-bar psi tilde of t. 00:22:11.080 --> 00:22:16.480 Because I'm looking for an equation for psi tilde. 00:22:16.480 --> 00:22:18.370 I'm going a little-- 00:22:18.370 --> 00:22:20.935 I'm speaking slowly, but going a little fast. 00:22:25.930 --> 00:22:27.100 Now what happens? 00:22:29.800 --> 00:22:33.970 What you wanted to happen, happened. 00:22:33.970 --> 00:22:36.060 H0 is here. 00:22:36.060 --> 00:22:37.900 And look, H0 is here. 00:22:37.900 --> 00:22:40.480 Well it's accompanied by these two exponentials, 00:22:40.480 --> 00:22:42.130 but they have H0. 00:22:42.130 --> 00:22:45.040 So they commute through and these two exponentials 00:22:45.040 --> 00:22:49.960 cancel as far as this first term is concerned. 00:22:49.960 --> 00:22:56.440 So nice consolation over here. 00:22:56.440 --> 00:23:05.090 And then we get the following equation, 00:23:05.090 --> 00:23:15.050 i H-bar d dt of psi tilde is equal to, well, this delta H 00:23:15.050 --> 00:23:21.290 sandwiched in these two operators. 00:23:21.290 --> 00:23:25.670 So I'll just copy it. 00:23:25.670 --> 00:23:36.970 e to the H0 t over H-bar delta H e to the minus i H0 t 00:23:36.970 --> 00:23:42.310 over H-S bar times pi tilde of t. 00:23:47.510 --> 00:23:51.440 So this equation makes what we wanted manifest. 00:23:51.440 --> 00:23:58.310 If delta H vanishes, psi tilde is time-independent. 00:23:58.310 --> 00:24:01.100 But it's more than that. 00:24:01.100 --> 00:24:07.285 This which we will call delta H tilde. 00:24:10.170 --> 00:24:12.280 So tilde objects are objects that 00:24:12.280 --> 00:24:17.290 have been acted by H0, like the tilde state, 00:24:17.290 --> 00:24:20.200 it has an H0 with respect to the other. 00:24:20.200 --> 00:24:24.970 This delta H tilde is because it has been acted by similarity 00:24:24.970 --> 00:24:27.580 with those things. 00:24:27.580 --> 00:24:30.010 But even more, I think you should 00:24:30.010 --> 00:24:47.340 realize that this thing is really the Heisenberg 0 00:24:47.340 --> 00:24:55.830 version of delta H. A Heisenberg operator is obtained 00:24:55.830 --> 00:24:59.490 by taking the Schrodinger operator, putting U dagger 00:24:59.490 --> 00:25:02.850 and U. And that's exactly what you've done here. 00:25:02.850 --> 00:25:06.790 You've taken the Schrodinger operator and put U 00:25:06.790 --> 00:25:12.360 dagger with respect to H0, and U for H0. 00:25:12.360 --> 00:25:18.210 So this is the Heisenberg version of delta H 00:25:18.210 --> 00:25:21.890 relative to H0. 00:25:21.890 --> 00:25:29.180 So delta H has been "Heisenberg-ed" using H0. 00:25:29.180 --> 00:25:37.300 And then this whole thing looks like i H-bar d dt of psi 00:25:37.300 --> 00:25:43.220 tilde equal delta H tilde psi tilde. 00:25:47.750 --> 00:25:50.285 And this is a Schrodinger equation. 00:25:55.780 --> 00:26:01.950 So there it is, for you, the so-called interaction picture. 00:26:01.950 --> 00:26:05.050 The interaction picture says whatever 00:26:05.050 --> 00:26:08.830 is not an interaction will make it Heisenberg. 00:26:08.830 --> 00:26:14.320 Whatever is purely interactive will make it Schrodinger. 00:26:14.320 --> 00:26:18.140 And therefore, this state varies in time. 00:26:18.140 --> 00:26:20.320 And there are some operators that 00:26:20.320 --> 00:26:25.510 have acquired extra time-dependence, as well, due 00:26:25.510 --> 00:26:28.270 to the Heisenberg process. 00:26:28.270 --> 00:26:32.736 So this is the situation we are going to try to solve.