WEBVTT 00:00:00.050 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.810 under a Creative Commons license. 00:00:03.810 --> 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 --> 00:00:17.305 at ocw.mit.edu. 00:00:21.670 --> 00:00:27.620 PROFESSOR: OK, so let me go back to what we were doing. 00:00:27.620 --> 00:00:30.090 The plan for today is as follows. 00:00:30.090 --> 00:00:34.240 We're going to look at this unitary time evolution 00:00:34.240 --> 00:00:39.800 and calculate this operator u, given the Hamiltonian. 00:00:39.800 --> 00:00:44.170 That will be the first order of business today. 00:00:44.170 --> 00:00:46.660 Then we will look at the Heisenberg 00:00:46.660 --> 00:00:50.090 picture of quantum mechanics. 00:00:50.090 --> 00:00:53.270 And the Heisenberg picture of quantum mechanics 00:00:53.270 --> 00:00:57.140 is one where the operators, the Schrodinger operators, 00:00:57.140 --> 00:00:59.510 acquire time dependence. 00:00:59.510 --> 00:01:07.480 And it's a pretty useful way of seeing things, a pretty useful 00:01:07.480 --> 00:01:11.620 way of calculating things as well, 00:01:11.620 --> 00:01:16.360 and makes the relation between classical mechanics and quantum 00:01:16.360 --> 00:01:19.270 mechanics more obvious. 00:01:19.270 --> 00:01:22.530 So it's a very important tool. 00:01:22.530 --> 00:01:23.890 So we'll discuss that. 00:01:23.890 --> 00:01:27.020 We'll find the Heisenberg equations of motion 00:01:27.020 --> 00:01:30.270 and solve them for a particular case today. 00:01:30.270 --> 00:01:36.290 All this material is not so to be covered in the test. 00:01:36.290 --> 00:01:38.740 The only part-- of course, the first few things 00:01:38.740 --> 00:01:42.750 I will say today about solving for the unitary operator 00:01:42.750 --> 00:01:46.770 you've done in other ways, and I will do it again this time. 00:01:49.790 --> 00:01:53.510 So going back to what we were saying last time, 00:01:53.510 --> 00:01:56.570 we postulated unitary time evolution. 00:01:56.570 --> 00:02:06.300 We said that psi at t was given by some operator U of t t0 psi 00:02:06.300 --> 00:02:08.800 t0. 00:02:08.800 --> 00:02:12.170 And then we found that this equation 00:02:12.170 --> 00:02:16.870 implied the Schrodinger equation with a Hamiltonian 00:02:16.870 --> 00:02:21.480 given by the following expression. 00:02:21.480 --> 00:02:32.830 ih dU dt of t t0 u dagger of t t0. 00:02:32.830 --> 00:02:36.740 So that was our derivation of the Schrodinger equation. 00:02:36.740 --> 00:02:38.820 We start with the time evolution. 00:02:38.820 --> 00:02:43.070 We found that, whenever we declare that states evolve 00:02:43.070 --> 00:02:46.870 in time in that way, they satisfy a first order time 00:02:46.870 --> 00:02:51.230 differential equation of the Schrodinger form 00:02:51.230 --> 00:02:56.491 in which the Hamiltonian is given in terms of U 00:02:56.491 --> 00:02:58.240 by this equation. 00:02:58.240 --> 00:03:01.160 And we talked about this operator. 00:03:01.160 --> 00:03:04.890 First we showed that it doesn't depend really on t0. 00:03:04.890 --> 00:03:07.250 Then we showed that it's Hermitian. 00:03:07.250 --> 00:03:09.430 It has units of energy. 00:03:09.430 --> 00:03:15.020 And as you may have seen already in the notes, 00:03:15.020 --> 00:03:18.610 there is a very clear correspondence 00:03:18.610 --> 00:03:22.330 between this operator and the way 00:03:22.330 --> 00:03:29.870 the dynamics follows with the ideas of Poisson brackets 00:03:29.870 --> 00:03:33.740 that are the precursors of commutators 00:03:33.740 --> 00:03:35.480 from classical mechanics. 00:03:35.480 --> 00:03:37.040 So that's in the notes. 00:03:37.040 --> 00:03:40.050 I will not go in detail in this. 00:03:40.050 --> 00:03:43.130 Many of you may have not heard of Poisson brackets. 00:03:43.130 --> 00:03:46.880 It's an interesting thing, and really that 00:03:46.880 --> 00:03:48.680 will be good enough. 00:03:48.680 --> 00:03:53.180 So our goal today is to find U given 00:03:53.180 --> 00:03:57.740 H, because as we mentioned last time, for physics 00:03:57.740 --> 00:04:01.380 it is typically more easy to invent a quantum 00:04:01.380 --> 00:04:04.340 system by postulating a Hamiltonian 00:04:04.340 --> 00:04:09.290 and then solving it than postulating a time evolution 00:04:09.290 --> 00:04:10.180 operator. 00:04:10.180 --> 00:04:24.270 So our goal in general is to find U of t t0 given H of t. 00:04:26.820 --> 00:04:29.060 That's what we're supposed to do. 00:04:29.060 --> 00:04:31.320 So the first thing I'm going to do 00:04:31.320 --> 00:04:34.510 is multiply this equation by u. 00:04:37.340 --> 00:04:41.500 By multiplying this equation by a u from the right, 00:04:41.500 --> 00:04:50.110 I will write first this term. ih dU dt of t t0 00:04:50.110 --> 00:04:55.012 is equal to H of t U of t t0. 00:04:57.610 --> 00:05:01.990 So I multiplied this equation by u from the right. 00:05:01.990 --> 00:05:06.090 This operator is unitary, so u dagger u is one. 00:05:06.090 --> 00:05:09.430 That's why this equation cleaned up to this. 00:05:09.430 --> 00:05:13.420 Now there's no confusion really here with derivatives, 00:05:13.420 --> 00:05:17.930 so I might this well write them with normal derivatives. 00:05:17.930 --> 00:05:27.886 So I'll write this equation as d dt of U t t0 00:05:27.886 --> 00:05:31.780 is equal to H of t U of t t0. 00:05:43.140 --> 00:05:45.960 You should be able to look at that equation 00:05:45.960 --> 00:05:50.230 and say I see the Schrodinger equation there. 00:05:50.230 --> 00:05:51.930 How? 00:05:51.930 --> 00:05:59.720 Imagine that you have a psi of t0 here, and you put it in. 00:05:59.720 --> 00:06:06.730 Then the right hand side becomes h and t acting on psi of t. 00:06:06.730 --> 00:06:14.110 And on the left hand side, this psi of t0 00:06:14.110 --> 00:06:18.200 can be put inside the derivative because it doesn't depend on t. 00:06:18.200 --> 00:06:23.640 Therefore this becomes ih bar d dt of psi of t. 00:06:23.640 --> 00:06:29.530 So the Schrodinger equation is there. 00:06:29.530 --> 00:06:31.850 OK so now let's solve this. 00:06:31.850 --> 00:06:33.680 We'll go through three cases. 00:06:33.680 --> 00:06:37.400 Case one, h is time independent. 00:06:43.560 --> 00:06:48.480 So we're doing this sort of quickly. 00:06:48.480 --> 00:06:54.240 So H of t is really H like that. 00:06:54.240 --> 00:06:58.700 No explicit time dependence there. 00:06:58.700 --> 00:07:00.510 So what do we have? 00:07:00.510 --> 00:07:02.730 ih bar. 00:07:02.730 --> 00:07:13.210 Let's write dU dt is equal H times U. 00:07:13.210 --> 00:07:18.020 And we tried to write a solution of the form U 00:07:18.020 --> 00:07:24.630 use equal to e to the minus iHt over h bar times U0. 00:07:29.500 --> 00:07:32.600 Does that work? 00:07:32.600 --> 00:07:40.030 Well, we can think du dt and ih. 00:07:40.030 --> 00:07:42.090 So we get ih. 00:07:42.090 --> 00:07:48.420 When I take dU dt, I have to differentiate this exponential. 00:07:48.420 --> 00:07:53.530 And now in this exponential, this full operator H is there. 00:07:53.530 --> 00:07:57.850 But we are differentiating with respect to time. 00:07:57.850 --> 00:08:00.970 And H doesn't depend on time, so this is not 00:08:00.970 --> 00:08:03.330 a very difficult situation. 00:08:03.330 --> 00:08:06.640 You could imagine the power series expansion. 00:08:06.640 --> 00:08:10.340 And H, as far as this derivative goes, 00:08:10.340 --> 00:08:12.340 is like if it would be even a number. 00:08:12.340 --> 00:08:15.200 It wouldn't make any difference if it's an operator. 00:08:15.200 --> 00:08:20.040 So the derivative with respect to time of this thing 00:08:20.040 --> 00:08:25.600 is minus iH over h times the same exponential. 00:08:32.000 --> 00:08:38.159 Moreover, the position of this h could be here, 00:08:38.159 --> 00:08:41.419 or it could be to the right. 00:08:41.419 --> 00:08:45.030 It cannot be to the right of U0 though, 00:08:45.030 --> 00:08:49.630 because this is a matrix, a constant matrix that we've put 00:08:49.630 --> 00:08:54.590 in here as a possible thing for boundary condition. 00:08:54.590 --> 00:08:58.320 So so far we've taken this derivative, 00:08:58.320 --> 00:09:03.706 and then i's cancel, the h bar cancels, and you get H. 00:09:03.706 --> 00:09:07.340 But this whole thing is, again, U. 00:09:07.340 --> 00:09:10.940 So the equation has been solved. 00:09:10.940 --> 00:09:14.440 So try this. 00:09:14.440 --> 00:09:15.630 And it works. 00:09:15.630 --> 00:09:25.120 So having this solution we can write, for example, 00:09:25.120 --> 00:09:31.410 that U of t t0 is going to be e to the minus 00:09:31.410 --> 00:09:38.110 iHt over h bar, some constant matrix. 00:09:38.110 --> 00:09:44.940 When t is equal to t0, this matrix becomes the unit matrix. 00:09:44.940 --> 00:09:51.180 So this is e to the minus iHt0 over h bar times U0. 00:09:54.500 --> 00:09:59.140 And therefore from here, U0 is the inverse 00:09:59.140 --> 00:10:06.520 of this matrix, which is nothing else but e to the iHt0 over h 00:10:06.520 --> 00:10:08.140 bar. 00:10:08.140 --> 00:10:18.910 So I can substitute back here what U0 is and finally 00:10:18.910 --> 00:10:31.170 obtain U of t t0 is e to the minus iH over h bar t minus t0. 00:10:33.890 --> 00:10:41.180 And this is for h time independent. 00:10:41.180 --> 00:10:43.900 And that's our solution. 00:10:43.900 --> 00:10:46.435 There's very little to add to this. 00:10:50.790 --> 00:10:53.710 We discussed that in recitation on Thursday. 00:10:53.710 --> 00:10:56.560 This unitary operator you've been seeing that 00:10:56.560 --> 00:10:59.420 from the beginning of the course in some sense, 00:10:59.420 --> 00:11:02.800 that you evolve energy eigenstate. 00:11:02.800 --> 00:11:07.110 If this acts on any energy eigenstate, 00:11:07.110 --> 00:11:13.380 h is an energy-- if you act here on an energy eigenstate, 00:11:13.380 --> 00:11:17.030 the energy eigenstate is an eigenstate precisely for H, 00:11:17.030 --> 00:11:19.670 you can put just the number here. 00:11:19.670 --> 00:11:26.700 That is e to the, say, alpha h on a state psi 00:11:26.700 --> 00:11:38.290 n is equal to e to the alpha en psi n if h on psi n 00:11:38.290 --> 00:11:43.110 is equal to en on psi n. 00:11:43.110 --> 00:11:48.910 So the function of an operator acting on an eigenstate 00:11:48.910 --> 00:11:54.560 is just the function evaluated at the eigenvalue. 00:11:54.560 --> 00:12:01.600 So this is a rule that you've been using a really long time. 00:12:01.600 --> 00:12:05.140 OK, so when h is time independent, that's what it is. 00:12:05.140 --> 00:12:10.110 How about when h has a little time dependence? 00:12:10.110 --> 00:12:13.880 What do I call a little time dependence? 00:12:13.880 --> 00:12:19.590 A little time dependence is an idea, the sign 00:12:19.590 --> 00:12:24.050 to make it possible for you to solve the equation, 00:12:24.050 --> 00:12:26.800 even though it has some time dependence. 00:12:26.800 --> 00:12:31.180 So you could have Hamiltonians that are time dependent, 00:12:31.180 --> 00:12:33.870 but still have a simplifying virtue. 00:12:33.870 --> 00:12:40.290 So H of t is time dependent. 00:12:40.290 --> 00:12:54.655 But assume that H at t1 and h at t2 commute for all t1 and t2. 00:12:58.080 --> 00:13:00.140 So what could that be? 00:13:00.140 --> 00:13:03.890 For example, you know that the particle in a magnetic field, 00:13:03.890 --> 00:13:15.435 the spin in a magnetic field is minus gamma B dot the spin. 00:13:18.210 --> 00:13:24.970 And you could have a time dependent magnetic field, B 00:13:24.970 --> 00:13:28.380 of t times the spin. 00:13:28.380 --> 00:13:30.680 I'm not sure this is the constant gamma that they 00:13:30.680 --> 00:13:35.570 usually call gamma, but it may be. 00:13:35.570 --> 00:13:41.500 Now then if the magnetic field is time dependent, but imagine 00:13:41.500 --> 00:13:46.350 its direction is not time dependent. 00:13:46.350 --> 00:13:51.690 So if its direction is not time dependent, then, for example, 00:13:51.690 --> 00:13:59.450 you would have here minus gamma Bz of t times Sz. 00:13:59.450 --> 00:14:02.250 And the Hamiltonian at different times commute 00:14:02.250 --> 00:14:06.260 because Sz commutes with itself, and the fact that it's time 00:14:06.260 --> 00:14:12.770 independent doesn't make it fail to commute. 00:14:12.770 --> 00:14:17.900 So if you have a magnetic field that is fixed in one direction 00:14:17.900 --> 00:14:21.330 but change in time, you can have a situation where 00:14:21.330 --> 00:14:24.370 your Hamiltonian is time dependent, 00:14:24.370 --> 00:14:26.970 but still at different times it commutes. 00:14:26.970 --> 00:14:30.640 And you will discuss such case because it's interesting. 00:14:30.640 --> 00:14:34.290 But later on as we do nuclear magnetic resonance, 00:14:34.290 --> 00:14:36.550 we will have the more interesting case 00:14:36.550 --> 00:14:41.400 in which a magnetic field rotates and therefore it's 00:14:41.400 --> 00:14:43.900 not that simple. 00:14:43.900 --> 00:14:49.540 So what happens if you have a time dependent Hamiltonian that 00:14:49.540 --> 00:14:51.560 actually commutes? 00:14:51.560 --> 00:14:56.780 Well, the claim is that U of t t0 00:14:56.780 --> 00:15:02.660 is given by a natural extension of what we had before. 00:15:02.660 --> 00:15:08.000 You would want to put exponential of minus iHt, 00:15:08.000 --> 00:15:11.880 but the reason this worked was because the derivative 00:15:11.880 --> 00:15:15.880 with respect to time brought down an iH over h bar. 00:15:15.880 --> 00:15:24.140 So one way to fix this is to put t t0 H of t prime dt prime. 00:15:28.980 --> 00:15:34.635 So this is an answer to try this. 00:15:39.760 --> 00:15:40.630 Look at this. 00:15:40.630 --> 00:15:44.810 If the Hamiltonian were to be time independent, 00:15:44.810 --> 00:15:46.520 you could take it out. 00:15:46.520 --> 00:15:48.480 And then you would get t minus t0. 00:15:48.480 --> 00:15:53.390 That brings you back to this case, so this looks reasonable. 00:15:53.390 --> 00:15:58.280 So let me call this quantity R of t. 00:16:01.030 --> 00:16:05.660 And then you notice that R dot of t, 00:16:05.660 --> 00:16:10.050 the derivative of this quantity with respect to time. 00:16:10.050 --> 00:16:14.090 Well, when you differentiate an integral the upper argument, 00:16:14.090 --> 00:16:16.460 you get just the integrand evaluated 00:16:16.460 --> 00:16:21.350 at the time represented by the upper argument 00:16:21.350 --> 00:16:24.140 of the upper limit of integration. 00:16:24.140 --> 00:16:26.645 So this is H of t. 00:16:29.610 --> 00:16:36.370 And now here comes a crucial point. 00:16:36.370 --> 00:16:38.900 You're trying to differentiate. 00:16:38.900 --> 00:16:43.580 This U is really e to the R. And you're 00:16:43.580 --> 00:16:49.910 trying to differentiate to see if the equation holds dU dt. 00:16:49.910 --> 00:16:51.480 So what is the dU dt? 00:16:55.960 --> 00:17:05.897 Would be d dt of 1 plus R plus RR plus 1 3 factor RRR. 00:17:12.140 --> 00:17:15.520 And now what happens? 00:17:15.520 --> 00:17:23.280 You differentiate here, and the first term is R dot. 00:17:23.280 --> 00:17:29.520 Here You, would have one half R dot R plus R R dot. 00:17:34.220 --> 00:17:39.480 And then 1 over 3 factorial, but three factors. 00:17:39.480 --> 00:17:46.910 R dot RR plus R R dot R plus RR R dot. 00:17:53.900 --> 00:17:58.720 But here is the claim R dot commutes 00:17:58.720 --> 00:18:09.050 with R. Claim R dot and R commute. 00:18:09.050 --> 00:18:10.470 Why is that? 00:18:10.470 --> 00:18:16.320 Well, R dot depends on H. And R is an integral 00:18:16.320 --> 00:18:19.850 of H as well, but the H at different times commute 00:18:19.850 --> 00:18:22.610 anyway, so this must be true. 00:18:22.610 --> 00:18:25.250 There's no place where you can get a contribution, 00:18:25.250 --> 00:18:30.100 because R dot is like an H, and here's an integral of H. 00:18:30.100 --> 00:18:34.270 So since the Hamiltonians are assumed to commute, 00:18:34.270 --> 00:18:38.470 R dot commutes with R. And this becomes 00:18:38.470 --> 00:18:41.820 like a normal derivative of an exponential in which you 00:18:41.820 --> 00:18:45.840 can move the R dot to the left everywhere. 00:18:45.840 --> 00:18:48.270 And you're differentiating the usual thing. 00:18:48.270 --> 00:18:59.240 So this is R dot and times the exponential of R. 00:18:59.240 --> 00:19:01.800 So actually that means that we've 00:19:01.800 --> 00:19:07.010 got pretty much our answer, because R dot is minus 00:19:07.010 --> 00:19:11.300 i over h bar H of t. 00:19:11.300 --> 00:19:17.810 And e to the R is U, so we got dU dt equals 00:19:17.810 --> 00:19:21.330 this, which is the same as this equation. 00:19:34.430 --> 00:19:39.140 The only reason a derivative with respect to time will not 00:19:39.140 --> 00:19:45.880 give the usual thing is if R and R dot fail to commute, 00:19:45.880 --> 00:19:46.790 and they don't. 00:19:46.790 --> 00:19:48.810 So you could put the R dot here. 00:19:48.810 --> 00:19:50.840 You can put R dot on the other side, 00:19:50.840 --> 00:19:54.660 because it commutes with R, but it's better here. 00:19:54.660 --> 00:20:01.800 And therefore you've got this very nice solution. 00:20:01.800 --> 00:20:04.650 So the solution is not that bad. 00:20:04.650 --> 00:20:09.790 Now finally, I want to discuss for a second the general case. 00:20:12.710 --> 00:20:20.806 So that's case-- there was a 1, a 2, a 3 H of t general. 00:20:23.740 --> 00:20:25.270 What can you do? 00:20:25.270 --> 00:20:28.570 Well, if H of t is general, there's 00:20:28.570 --> 00:20:31.560 not too much you can do. 00:20:31.560 --> 00:20:34.350 You can write something that will 00:20:34.350 --> 00:20:38.570 get you started doing things, but it's not 00:20:38.570 --> 00:20:42.310 obviously terribly useful. 00:20:42.310 --> 00:20:45.060 But it's interesting anyway that there's 00:20:45.060 --> 00:20:47.530 a way to write something that makes sense. 00:20:47.530 --> 00:20:50.830 So here it is. 00:20:50.830 --> 00:20:52.450 U of t and t0. 00:20:52.450 --> 00:20:58.230 I'll write the answer and explain how it looks, 00:20:58.230 --> 00:21:04.090 and then you will see that it's OK. 00:21:04.090 --> 00:21:05.200 It's interesting. 00:21:05.200 --> 00:21:08.710 But it probably is not the most practical way 00:21:08.710 --> 00:21:09.895 you can solve this problem. 00:21:15.310 --> 00:21:16.790 So here it is. 00:21:16.790 --> 00:21:19.680 There's an acronym for this thing. 00:21:19.680 --> 00:21:23.900 T it's called the time ordered exponential. 00:21:23.900 --> 00:21:27.750 This operator does something to the exponential function. 00:21:27.750 --> 00:21:29.470 So it's a definition. 00:21:29.470 --> 00:21:31.350 So I have to say what this time ordered 00:21:31.350 --> 00:21:37.830 exponential is, and it's the following. 00:21:37.830 --> 00:21:40.970 You take the exponential and just begin to expand. 00:21:40.970 --> 00:21:45.217 So 1 minus i over h bar-- or I'll 00:21:45.217 --> 00:21:51.790 put like this, plus minus i over h bar integral 00:21:51.790 --> 00:22:01.360 from t0 to t of dt1 H of t1. 00:22:01.360 --> 00:22:03.130 So far, so good. 00:22:03.130 --> 00:22:07.270 I've just expanded this. 00:22:07.270 --> 00:22:10.960 Now if I would continue expanding, 00:22:10.960 --> 00:22:15.460 I would get something that doesn't provide the solution. 00:22:15.460 --> 00:22:20.110 You see, this thing is the solution 00:22:20.110 --> 00:22:23.580 when the Hamiltonian at different times commute. 00:22:23.580 --> 00:22:26.910 So it's unlikely to be the solution when 00:22:26.910 --> 00:22:28.000 they don't commute. 00:22:28.000 --> 00:22:29.610 In fact, it's not the solution. 00:22:29.610 --> 00:22:32.730 So what is the next term here? 00:22:32.730 --> 00:22:36.390 The next term is you think of the exponential 00:22:36.390 --> 00:22:38.180 as you would expand as usual. 00:22:38.180 --> 00:22:43.010 So you will have here plus one half of this thing squared. 00:22:43.010 --> 00:22:48.320 So I will put something and then erase it, so maybe don't copy. 00:22:48.320 --> 00:22:53.370 One half minus i over h bar squared. 00:22:53.370 --> 00:23:02.700 And you would say, well, t0 to t dt prime H of t prime. 00:23:02.700 --> 00:23:10.680 t0 to t dt double prime H of double prime. 00:23:10.680 --> 00:23:13.010 Well, that would be just an exponential. 00:23:13.010 --> 00:23:16.620 So what is a time ordered exponential? 00:23:16.620 --> 00:23:18.480 You erase the one half. 00:23:21.040 --> 00:23:26.885 And then for notation call this t1 and t1. 00:23:30.010 --> 00:23:37.210 And then the next integral do it only up to time t1, 00:23:37.210 --> 00:23:38.375 and call this t2. 00:23:44.370 --> 00:23:49.670 So t1 will always be greater than t2, 00:23:49.670 --> 00:23:53.760 because t2 is integrated from t0 to t1. 00:23:53.760 --> 00:24:00.410 And as you integrate here over the various t1's, you just 00:24:00.410 --> 00:24:03.740 integrate up to that value. 00:24:03.740 --> 00:24:06.580 So you're doing less of the full integral 00:24:06.580 --> 00:24:08.210 then you should be doing, and that's 00:24:08.210 --> 00:24:12.080 why the factor of one half has disappeared. 00:24:12.080 --> 00:24:14.350 This can be continued. 00:24:14.350 --> 00:24:19.330 I can write the next one would be minus i over h bar cubed 00:24:19.330 --> 00:24:33.080 integral t0 to t H of t1 integral t0 to t1 dt2 H of t2. 00:24:33.080 --> 00:24:36.370 And then they next integral goes up to t2. 00:24:36.370 --> 00:24:41.315 So t0 to t2 dt3 H of t3. 00:24:46.620 --> 00:24:50.190 Anyway, that's a time ordered exponential. 00:24:50.190 --> 00:24:58.970 And I leave it to you to take the time derivative, 00:24:58.970 --> 00:25:02.660 at least to see that the first few terms are working exactly 00:25:02.660 --> 00:25:04.550 the way they should. 00:25:04.550 --> 00:25:07.940 That is, if you take a time derivative of this, 00:25:07.940 --> 00:25:12.710 you will get H times that thing. 00:25:12.710 --> 00:25:14.670 So since it's a power series, you 00:25:14.670 --> 00:25:16.950 will differentiate the first term, 00:25:16.950 --> 00:25:18.680 and you will get the right thing. 00:25:18.680 --> 00:25:22.710 Then the second term and you will start getting everything 00:25:22.710 --> 00:25:25.310 that you need. 00:25:25.310 --> 00:25:27.770 So it's a funny object. 00:25:27.770 --> 00:25:30.950 It's reassuring that something like this success, 00:25:30.950 --> 00:25:33.150 but in general, you would want to be 00:25:33.150 --> 00:25:37.560 able to do all these integrals and to sum them up. 00:25:37.560 --> 00:25:41.420 And in general, it's not that easy. 00:25:41.420 --> 00:25:45.370 So it's of limited usefulness. 00:25:45.370 --> 00:25:47.670 It's a nice thing that you can write it, 00:25:47.670 --> 00:25:52.570 and you can prove things about it and manipulate it. 00:25:52.570 --> 00:25:54.750 But when you have a practical problem, 00:25:54.750 --> 00:25:57.820 generally that's not the way you solve it. 00:25:57.820 --> 00:26:01.920 In fact, when we will discuss the rotating magnetic fields 00:26:01.920 --> 00:26:06.680 for magnetic resonance, we will not solve it in this way. 00:26:06.680 --> 00:26:10.870 We will try to figure out the solution some other way. 00:26:10.870 --> 00:26:13.290 But in terms of completeness, it's 00:26:13.290 --> 00:26:16.150 kind of pretty in that you go from 00:26:16.150 --> 00:26:19.230 the exponential to the time ordered exponential. 00:26:19.230 --> 00:26:24.080 And I think you'll see more of this in 806. 00:26:24.080 --> 00:26:29.100 So that's basically our solution for H 00:26:29.100 --> 00:26:35.200 and for the unitary operator U in terms of H. 00:26:35.200 --> 00:26:36.810 And what we're going to do now is 00:26:36.810 --> 00:26:39.170 turn to the Heisenberg picture of quantum mechanics. 00:26:39.170 --> 00:26:40.300 Yes, questions? 00:26:40.300 --> 00:26:42.176 AUDIENCE: Why does R dot [INAUDIBLE]? 00:26:47.210 --> 00:26:53.090 PROFESSOR: Because that's really a property of integrals. 00:26:53.090 --> 00:26:59.180 d dx integral up to x from x0 g of x 00:26:59.180 --> 00:27:07.360 prime dx prime is just equal to g of x. 00:27:07.360 --> 00:27:09.570 This is a constant here, so you're not 00:27:09.570 --> 00:27:12.310 varying the integral over in this limit. 00:27:12.310 --> 00:27:16.170 So if this limit would also be x dependent, 00:27:16.170 --> 00:27:17.830 you would get another contribution, 00:27:17.830 --> 00:27:20.290 but we only get the contribution from here. 00:27:20.290 --> 00:27:22.640 What's really happening is you're integrating up 00:27:22.640 --> 00:27:25.740 to x, then up to x plus epsilon subtracting, 00:27:25.740 --> 00:27:30.810 so you pick up the value of the function of the upper limit. 00:27:30.810 --> 00:27:31.800 Yes? 00:27:31.800 --> 00:27:35.740 AUDIENCE: So what happens to the T that was pre factor? 00:27:35.740 --> 00:27:37.452 PROFESSOR: What happens to this T? 00:27:37.452 --> 00:27:38.660 AUDIENCE: Yeah, what happens? 00:27:38.660 --> 00:27:40.830 PROFESSOR: That's just a symbol. 00:27:40.830 --> 00:27:45.280 It says time order the following exponential. 00:27:45.280 --> 00:27:49.370 So at this stage, this is a definition 00:27:49.370 --> 00:27:52.670 of what t on an exponential means. 00:27:52.670 --> 00:27:54.140 AUDIENCE: OK. 00:27:54.140 --> 00:27:57.260 PROFESSOR: It's not-- let me say T is not 00:27:57.260 --> 00:28:00.290 an operator in the usual sense of quantum mechanics 00:28:00.290 --> 00:28:01.490 or anything like that. 00:28:01.490 --> 00:28:03.680 It's an instruction. 00:28:03.680 --> 00:28:07.560 Whenever you have an exponential of this form, 00:28:07.560 --> 00:28:10.380 the time ordered exponential is this series 00:28:10.380 --> 00:28:13.050 that we've written down. 00:28:13.050 --> 00:28:14.250 It's just a definition. 00:28:14.250 --> 00:28:14.926 Yes? 00:28:14.926 --> 00:28:17.050 AUDIENCE: So when we have operators in differential 00:28:17.050 --> 00:28:20.850 equations, do we still get [INAUDIBLE]? 00:28:20.850 --> 00:28:22.600 PROFESSOR: If we have what? 00:28:22.600 --> 00:28:24.933 AUDIENCE: If we have operators in differential equations 00:28:24.933 --> 00:28:27.500 do we still get unique [INAUDIBLE] solutions? 00:28:27.500 --> 00:28:29.680 PROFESSOR: Yes, pretty much. 00:28:29.680 --> 00:28:32.930 Because at the end of the day, this 00:28:32.930 --> 00:28:37.940 is a first order matrix differential equation. 00:28:37.940 --> 00:28:41.730 So it's a collection of first order differential equations 00:28:41.730 --> 00:28:43.445 for every element of a matrix. 00:28:47.600 --> 00:28:51.760 It's pretty much the same as you have before. 00:28:51.760 --> 00:28:57.440 If you know the operator at any time, initial time, 00:28:57.440 --> 00:28:59.980 with the differential equation you know the operator 00:28:59.980 --> 00:29:01.510 at a little bit time later. 00:29:01.510 --> 00:29:04.460 So the operator is completely determined 00:29:04.460 --> 00:29:07.280 if you know it initially and the differential equation. 00:29:07.280 --> 00:29:09.930 So I think it's completely analogous. 00:29:09.930 --> 00:29:13.890 It's just that it's harder to solve. 00:29:13.890 --> 00:29:15.150 Nothing else. 00:29:15.150 --> 00:29:16.556 One last question. 00:29:16.556 --> 00:29:18.306 AUDIENCE: So let's say that we can somehow 00:29:18.306 --> 00:29:21.620 fly in this unitary operator, and then we have a differential 00:29:21.620 --> 00:29:23.859 equation, and we somehow, let's say, get 00:29:23.859 --> 00:29:27.400 a wave function out of it. 00:29:27.400 --> 00:29:30.140 What is the interpretation of that wave function? 00:29:30.140 --> 00:29:31.515 PROFESSOR: Well, it's not that we 00:29:31.515 --> 00:29:34.040 get the wave function out of this. 00:29:34.040 --> 00:29:38.300 What really is happening is that you 00:29:38.300 --> 00:29:45.020 have learned how to calculate this operator given H. 00:29:45.020 --> 00:29:49.330 And therefore now you're able to evolve any wave function. 00:29:49.330 --> 00:29:51.940 So you have solved the dynamical system. 00:29:51.940 --> 00:29:55.690 If somebody tells you a time equals 0, your system is here, 00:29:55.690 --> 00:29:59.270 you can now calculate where it's going to be at the later time. 00:29:59.270 --> 00:30:01.200 So that's really all you have achieved. 00:30:01.200 --> 00:30:05.537 You now know the solution. 00:30:05.537 --> 00:30:07.370 When you're doing mechanics and they ask you 00:30:07.370 --> 00:30:10.060 for an orbit problem, they say at this time 00:30:10.060 --> 00:30:11.130 the planet is here. 00:30:11.130 --> 00:30:12.980 What are you supposed to find? 00:30:12.980 --> 00:30:14.830 x is a function of time. 00:30:14.830 --> 00:30:16.910 You now know how it's going to develop. 00:30:16.910 --> 00:30:19.040 You've solved equations of motion. 00:30:19.040 --> 00:30:19.860 Here it's the same. 00:30:19.860 --> 00:30:22.020 You know the wave function of time equals. 00:30:22.020 --> 00:30:26.490 If you know it at any time, you've solved problem. 00:30:26.490 --> 00:30:29.650 OK, so Heisenberg picture of quantum mechanics. 00:30:34.920 --> 00:30:38.230 Heisenberg picture. 00:30:43.050 --> 00:30:48.540 So basically the Heisenberg picture 00:30:48.540 --> 00:30:54.210 exists thanks to the existence of the Schrodinger picture. 00:30:54.210 --> 00:30:56.240 Heisenberg picture of quantum mechanics 00:30:56.240 --> 00:30:58.690 is not something that you necessarily 00:30:58.690 --> 00:31:00.380 invent from the beginning. 00:31:00.380 --> 00:31:05.190 The way we think of it is we assume there is a Schrodinger 00:31:05.190 --> 00:31:10.020 picture that we've developed in which we have operators like x, 00:31:10.020 --> 00:31:14.790 p, spin, Hamiltonians, and wave functions. 00:31:14.790 --> 00:31:19.020 And then we are going to define a new way of thinking 00:31:19.020 --> 00:31:21.840 about this, which is called the Heisenberg 00:31:21.840 --> 00:31:24.040 picture of the quantum mechanics. 00:31:24.040 --> 00:31:29.690 So it all begins by considering a Schrodinger operator 00:31:29.690 --> 00:31:35.070 As hat, which is s is for Schrodinger. 00:31:42.940 --> 00:31:48.700 And the motivation comes from expectation values. 00:31:48.700 --> 00:31:51.190 Suppose you have time dependent states, 00:31:51.190 --> 00:31:53.440 in fact, matrix elements. 00:31:53.440 --> 00:31:58.780 One time dependent state alpha of t, one time dependent state 00:31:58.780 --> 00:32:00.640 beta of t. 00:32:00.640 --> 00:32:06.000 Two independent time dependent states. 00:32:06.000 --> 00:32:17.090 So you could ask what is the matrix element of A 00:32:17.090 --> 00:32:20.520 between these two time dependent states, a matrix element. 00:32:26.180 --> 00:32:31.800 But then, armed with our unitary operator, 00:32:31.800 --> 00:32:37.980 we know that As is here, and this state beta at time t 00:32:37.980 --> 00:32:47.190 is equal to U of t comma 0 beta at time 0. 00:32:47.190 --> 00:32:53.390 And alpha t is equal to alpha at 0 U dagger of t0. 00:32:56.160 --> 00:33:00.590 So the states have time dependence. 00:33:00.590 --> 00:33:02.280 But the time dependence has already 00:33:02.280 --> 00:33:06.850 been found, say, in principle, if you know U dagger. 00:33:06.850 --> 00:33:13.200 And then you can speak about the time dependent matrix elements 00:33:13.200 --> 00:33:19.510 of the operator As or the matrix element 00:33:19.510 --> 00:33:24.960 of this time dependent operator between the time 00:33:24.960 --> 00:33:28.480 equals 0 states. 00:33:28.480 --> 00:33:31.820 And this operator is sufficiently important 00:33:31.820 --> 00:33:35.520 that this operator is called the Heisenberg 00:33:35.520 --> 00:33:38.760 version of the operator s. 00:33:38.760 --> 00:33:44.685 Has time dependence, and it's defined by this equation. 00:33:52.820 --> 00:33:57.560 So whenever you have Schrodinger operator, 00:33:57.560 --> 00:34:02.190 whether it be time dependent or time independent, 00:34:02.190 --> 00:34:04.790 whatever the Schrodinger operator is, 00:34:04.790 --> 00:34:07.150 I have now a definition of what I 00:34:07.150 --> 00:34:11.409 will call the Heisenberg operator. 00:34:11.409 --> 00:34:17.719 And it is obtained by acting with a unitary operator, U. 00:34:17.719 --> 00:34:23.900 And operators always act on operators from the left 00:34:23.900 --> 00:34:25.360 and from the right. 00:34:25.360 --> 00:34:30.000 That's something that operators act on states from the left. 00:34:30.000 --> 00:34:31.850 They act on the state. 00:34:31.850 --> 00:34:35.730 But operators act on operator from the left 00:34:35.730 --> 00:34:38.360 and from the right, as you see them here, 00:34:38.360 --> 00:34:41.739 is the natural, ideal thing to happen. 00:34:41.739 --> 00:34:45.510 If you have an operator that's on another from the right only 00:34:45.510 --> 00:34:47.969 or from the left only, I think you 00:34:47.969 --> 00:34:50.270 have grounds to be suspicious that maybe you're 00:34:50.270 --> 00:34:53.000 not doing things right. 00:34:53.000 --> 00:34:55.440 So this is the Heisenberg operator. 00:34:55.440 --> 00:34:57.360 And as you can imagine, there's a lot 00:34:57.360 --> 00:35:01.090 of things to be said about this operator. 00:35:01.090 --> 00:35:03.060 So let's begin with a remark. 00:35:03.060 --> 00:35:07.970 Are there questions about this Heisenberg operator. 00:35:07.970 --> 00:35:08.700 Yes? 00:35:08.700 --> 00:35:12.180 AUDIENCE: Do we know anything about the Schrodinger operator? 00:35:12.180 --> 00:35:13.680 PROFESSOR: You have to speak louder. 00:35:13.680 --> 00:35:15.221 AUDIENCE: Is the Schrodinger operator 00:35:15.221 --> 00:35:17.850 related to the Hamiltonian [INAUDIBLE]? 00:35:17.850 --> 00:35:19.830 PROFESSOR: Any Schrodinger operator, 00:35:19.830 --> 00:35:22.260 this could be the Hamiltonian, this 00:35:22.260 --> 00:35:27.550 could be x hat, it could be Sz, could 00:35:27.550 --> 00:35:29.090 be any of the operators you know. 00:35:29.090 --> 00:35:34.420 All the operators you know are Schrodinger operators. 00:35:34.420 --> 00:35:38.330 So remarks, comments. 00:35:38.330 --> 00:35:39.306 OK, comments. 00:35:43.130 --> 00:35:57.730 One, at t equals 0 A Heisenberg becomes 00:35:57.730 --> 00:36:05.630 identical to A Schrodinger at t equals 0. 00:36:14.190 --> 00:36:16.300 So look why. 00:36:16.300 --> 00:36:22.260 Because when t is equal to 0, U of t-- of 0 0 is the operator 00:36:22.260 --> 00:36:26.020 propagates no state, so it's equal to the identity. 00:36:26.020 --> 00:36:28.980 So this is a wonderful relation that 00:36:28.980 --> 00:36:34.220 tell us you that time equals 0 the two operators are really 00:36:34.220 --> 00:36:36.190 the same. 00:36:36.190 --> 00:36:38.880 And another simple remark. 00:36:38.880 --> 00:36:44.440 If you have the unit operator in the Schrodinger picture, 00:36:44.440 --> 00:36:49.210 what is the unit operator in the Heisenberg picture? 00:36:49.210 --> 00:37:00.790 Well, it would be U t 0 dagger 1 U t 0. 00:37:00.790 --> 00:37:03.300 But 1 doesn't matter. 00:37:03.300 --> 00:37:05.890 U dagger with U is 1. 00:37:05.890 --> 00:37:09.460 This is a 1 Schrodinger, and therefore it's 00:37:09.460 --> 00:37:11.300 the same operator. 00:37:11.300 --> 00:37:16.170 So the unit operator is the same. 00:37:16.170 --> 00:37:20.300 It just doesn't change whatsoever. 00:37:20.300 --> 00:37:23.080 OK, so that's good. 00:37:23.080 --> 00:37:26.470 But now this is something interesting also happens. 00:37:26.470 --> 00:37:31.380 Suppose you have Schrodinger operator 00:37:31.380 --> 00:37:37.526 C that is equal to the product of A with B, two Schrodingers. 00:37:41.200 --> 00:37:47.460 If I try to figure out what is CH, 00:37:47.460 --> 00:37:53.640 I would put U dagger-- avoid all the letters, the t 0. 00:37:53.640 --> 00:37:55.750 It's supposed to be t 0. 00:37:55.750 --> 00:38:07.520 Cs U. But that's equal U dagger As Bs U. 00:38:07.520 --> 00:38:10.270 But now, in between the two operators, 00:38:10.270 --> 00:38:14.980 you can put a U U dagger, which is equal to 1. 00:38:14.980 --> 00:38:23.670 So As U U dagger Bs U. And then you 00:38:23.670 --> 00:38:26.780 see why this is really nice. 00:38:26.780 --> 00:38:31.890 Because what do you get is that C Heisenberg is just 00:38:31.890 --> 00:38:34.650 A Heisenberg times B Heisenberg. 00:38:34.650 --> 00:38:38.882 So if you have C Schrodinger equals A Schrodinger, B 00:38:38.882 --> 00:38:45.330 Schrodinger, C Heisenberg is A Heisenberg B Heisenberg. 00:38:45.330 --> 00:38:49.260 So there's a nice correspondence between those operators. 00:38:52.360 --> 00:38:58.670 Also you can do is for commutators. 00:38:58.670 --> 00:39:02.350 So you don't have to worry about this thing. 00:39:02.350 --> 00:39:13.500 So for example, if A Schrodinger with B Schrodinger 00:39:13.500 --> 00:39:19.380 is equal to C Schrodinger, then by doing exactly 00:39:19.380 --> 00:39:27.760 the same things, you see that A Heisenberg with B Heisenberg 00:39:27.760 --> 00:39:31.400 would be the commutator equal to C Heisenberg. 00:39:34.140 --> 00:39:35.670 Yes? 00:39:35.670 --> 00:39:38.828 AUDIENCE: That argument for the identity operators 00:39:38.828 --> 00:39:41.120 being the same in both pictures. 00:39:41.120 --> 00:39:43.970 If the Hamiltonian is time independent, 00:39:43.970 --> 00:39:46.470 does that work for any operator that commutes 00:39:46.470 --> 00:39:48.727 with the Hamiltonian? 00:39:48.727 --> 00:39:50.310 PROFESSOR: Hamiltonian is [INAUDIBLE]. 00:39:56.070 --> 00:39:58.280 AUDIENCE: Because then you can push the operator just 00:39:58.280 --> 00:40:00.850 through the exponential of the Hamiltonian. 00:40:00.850 --> 00:40:05.010 PROFESSOR: Yeah, we'll see things like that. 00:40:05.010 --> 00:40:08.430 We could discuss that maybe a little later. 00:40:08.430 --> 00:40:11.950 But there are some cases, as we will see immediately, 00:40:11.950 --> 00:40:16.650 in which some operators are the same in the two pictures. 00:40:16.650 --> 00:40:20.700 So basically operators that commute with the Hamiltonian 00:40:20.700 --> 00:40:26.320 as you say, since U involves the Hamiltonian, 00:40:26.320 --> 00:40:28.790 and this is the Hamiltonian, if the operator 00:40:28.790 --> 00:40:31.310 commutes with the Hamiltonian and you can move them across, 00:40:31.310 --> 00:40:32.268 then they are the same. 00:40:32.268 --> 00:40:35.880 So I think it's definitely true. 00:40:35.880 --> 00:40:40.640 So we will have an interesting question, 00:40:40.640 --> 00:40:43.450 in fact, whether the Heisenberg Hamiltonian is 00:40:43.450 --> 00:40:45.070 equal to the Schrodinger Hamiltonian, 00:40:45.070 --> 00:40:48.290 and we'll answer that very soon. 00:40:48.290 --> 00:40:52.720 So the one example that here I think you should keep in mind 00:40:52.720 --> 00:40:54.860 is this one. 00:40:54.860 --> 00:40:56.900 You know this is true. 00:40:56.900 --> 00:41:00.730 So what do you knowing the Heisenberg picture? 00:41:00.730 --> 00:41:08.150 That X Heisenberg of t times P Heisenberg of t commutator 00:41:08.150 --> 00:41:12.210 is equal to the Heisenberg version of this. 00:41:12.210 --> 00:41:15.000 But here was the unit operator. 00:41:15.000 --> 00:41:20.510 And therefore this is just ih bar times the unit operator 00:41:20.510 --> 00:41:22.790 again, because the units operator 00:41:22.790 --> 00:41:25.900 is the same in all pictures. 00:41:25.900 --> 00:41:31.840 So these commutation relation is true for any Heisenberg 00:41:31.840 --> 00:41:33.950 operator. 00:41:33.950 --> 00:41:37.100 Whatever commutation relation you have of Schrodinger, 00:41:37.100 --> 00:41:38.990 it's true for Heisenberg as well. 00:41:42.000 --> 00:41:46.670 OK, so then let's talk about Hamiltonians. 00:41:46.670 --> 00:41:53.180 Three, Hamiltonians. 00:41:53.180 --> 00:42:00.160 So Heisenberg Hamiltonian by definition 00:42:00.160 --> 00:42:07.890 would be equal to U dagger t 0 Schrodinger Hamiltonian 00:42:07.890 --> 00:42:09.980 times U of t 0. 00:42:14.500 --> 00:42:18.100 So if the Schrodinger Hamiltonian-- actually, 00:42:18.100 --> 00:42:25.490 if Hs at t1 commutes units with Hs at t2, 00:42:25.490 --> 00:42:31.270 the Schrodinger Hamiltonian is such that for all t1 and t2 00:42:31.270 --> 00:42:34.890 they commute with each other. 00:42:34.890 --> 00:42:38.560 Remember, if that is the case, the unitary operator 00:42:38.560 --> 00:42:42.790 is any way built by an exponential. 00:42:42.790 --> 00:42:43.880 It's this one. 00:42:47.590 --> 00:42:51.010 And the Schrodinger Hamiltonians commute. 00:42:51.010 --> 00:42:53.770 So as was asked in the question before, 00:42:53.770 --> 00:42:57.700 this thing commutes with that, and you 00:42:57.700 --> 00:43:00.180 get that they are the same. 00:43:00.180 --> 00:43:05.730 So if this is happening, the two Hamiltonians are identical. 00:43:12.030 --> 00:43:17.110 And we'll have the chance to check 00:43:17.110 --> 00:43:21.220 this today in a nice example. 00:43:21.220 --> 00:43:29.230 So I will write in this as saying the Heisenberg 00:43:29.230 --> 00:43:31.550 Hamiltonian as a function of time 00:43:31.550 --> 00:43:34.580 then is equal to the Schrodinger Hamiltonian 00:43:34.580 --> 00:43:37.160 as a function of time. 00:43:37.160 --> 00:43:45.160 And this goes Hs of t1 and Hs of t2 commute. 00:43:48.130 --> 00:43:55.990 OK, now I want you to notice this thing. 00:43:59.700 --> 00:44:15.260 Suppose the Hs of t is some Hs of x,p, and t, for example. 00:44:15.260 --> 00:44:21.150 OK, now you come and turn it into Heisenberg 00:44:21.150 --> 00:44:26.930 by putting a U dagger from the left and a U from the right. 00:44:26.930 --> 00:44:29.560 What will that do? 00:44:29.560 --> 00:44:32.360 It will put U dagger from the left, U dagger on the right. 00:44:32.360 --> 00:44:36.750 And then it will start working it's way inside, 00:44:36.750 --> 00:44:41.990 and any x that it will find will turn into a Heisenberg x. 00:44:41.990 --> 00:44:44.690 Any p will turn into Heisenberg p. 00:44:44.690 --> 00:44:47.160 Imagine, for example, any Hamiltonian 00:44:47.160 --> 00:44:49.630 is some function of x. 00:44:49.630 --> 00:44:51.090 It has an x squared. 00:44:51.090 --> 00:44:53.970 Well the U dagger and U come and turn 00:44:53.970 --> 00:44:57.390 this into x Heisenberg squared. 00:44:57.390 --> 00:45:05.720 So what I claim here happens is that H Heisenberg of t 00:45:05.720 --> 00:45:15.540 is equal to U dagger H Schrodinger of x, p, t, U. 00:45:15.540 --> 00:45:22.770 And therefore this becomes H Schrodinger of x 00:45:22.770 --> 00:45:29.260 Heisenberg of t, P Heisenberg of t, and t. 00:45:34.040 --> 00:45:41.550 So here is what the Heisenberg Hamiltonian is. 00:45:41.550 --> 00:45:46.040 It's the Schrodinger Hamiltonian where X's, and P's, or spins 00:45:46.040 --> 00:45:49.700 and everything has become Heisenberg. 00:45:49.700 --> 00:45:53.300 So the equality of the two Hamiltonians 00:45:53.300 --> 00:45:58.110 is a very funny condition on the Schrodinger Hamiltonian, 00:45:58.110 --> 00:46:02.010 because this is supposed to be equal to the Schrodinger 00:46:02.010 --> 00:46:05.900 Hamiltonian, which is of x, p, and t. 00:46:11.710 --> 00:46:16.420 So you have a function of x, p, and t. 00:46:16.420 --> 00:46:19.430 And you put X Heisenberg P Heisenberg, 00:46:19.430 --> 00:46:22.040 and somehow the whole thing is the same. 00:46:27.030 --> 00:46:34.470 So this is something very useful and we'll need it. 00:46:34.470 --> 00:46:38.170 One more comment, expectation values. 00:46:38.170 --> 00:46:41.300 So this is three. 00:46:41.300 --> 00:46:47.400 Comment number four on expectation values, which 00:46:47.400 --> 00:46:51.220 is something you've already-- it's sort of the way 00:46:51.220 --> 00:46:57.770 we began the discussion and wanted to make sure it's clear. 00:46:57.770 --> 00:47:02.032 So four, expectation values. 00:47:05.430 --> 00:47:09.670 So we started with this with alpha and beta, 00:47:09.670 --> 00:47:12.610 two arbitrary states, matrix elements. 00:47:12.610 --> 00:47:17.170 Take them equal and to be equal to psi of t. 00:47:17.170 --> 00:47:26.120 So you would have psi t As psi t is, in fact, 00:47:26.120 --> 00:47:39.080 equal to psi 0 A Heisenberg psi 0. 00:47:39.080 --> 00:47:43.790 Now that is a key equation. 00:47:43.790 --> 00:47:47.920 You know you're doing expectation value at any given 00:47:47.920 --> 00:47:52.640 time of a Schrodinger operator, turn it into Heisenberg 00:47:52.640 --> 00:47:55.320 and work at time equals 0. 00:47:55.320 --> 00:47:58.170 It simplifies life tremendously. 00:47:58.170 --> 00:48:00.370 Now this is the key identity. 00:48:00.370 --> 00:48:03.810 It's the way we motivated everything in a way. 00:48:03.810 --> 00:48:08.180 And it's written in a way that maybe it's 00:48:08.180 --> 00:48:14.460 a little too schematic, but we write it this way. 00:48:14.460 --> 00:48:16.620 We just say the expectation value 00:48:16.620 --> 00:48:23.640 of As is equal to the expectation value of AH. 00:48:28.140 --> 00:48:31.660 And this, well, we save time like that, 00:48:31.660 --> 00:48:33.770 but you have to know what you mean. 00:48:33.770 --> 00:48:36.290 When you're computing the expectation 00:48:36.290 --> 00:48:37.980 value for a Schrodinger operator, 00:48:37.980 --> 00:48:40.464 you're using time dependent states. 00:48:40.464 --> 00:48:42.005 When you're computing the expectation 00:48:42.005 --> 00:48:44.590 value of the Heisenberg operator, 00:48:44.590 --> 00:48:48.360 you're using the time equals 0 version of the states, 00:48:48.360 --> 00:48:49.680 but they are the same. 00:48:49.680 --> 00:48:53.530 So we say that the Schrodinger expectation value 00:48:53.530 --> 00:48:55.680 is equal to the Heisenberg expectation value. 00:48:55.680 --> 00:49:01.760 We right it in the bottom, but we mean the top equation. 00:49:01.760 --> 00:49:06.140 And we use it that way. 00:49:06.140 --> 00:49:12.320 So the Heisenberg operators, at this moment, 00:49:12.320 --> 00:49:14.640 are a little mysterious. 00:49:14.640 --> 00:49:18.920 They're supposed to be given by this formula, 00:49:18.920 --> 00:49:22.940 but we've seen that calculating U can be difficult. 00:49:22.940 --> 00:49:25.860 So calculating the Heisenberg operator 00:49:25.860 --> 00:49:28.010 can be difficult sometimes. 00:49:28.010 --> 00:49:31.740 So what we try to do in order to simplify 00:49:31.740 --> 00:49:35.950 that is find an equation that is satisfied by the Heisenberg 00:49:35.950 --> 00:49:40.270 operator, a time derivative equation. 00:49:40.270 --> 00:49:43.170 So let's try to find an equation that 00:49:43.170 --> 00:49:47.830 is satisfied by the Heisenberg operator rather than a formula. 00:49:47.830 --> 00:49:50.150 You'll say, well, this is better. 00:49:50.150 --> 00:49:54.996 But the fact is that seldom you know U. And even if you know U, 00:49:54.996 --> 00:49:59.010 you have to do this simplification, which is hard. 00:49:59.010 --> 00:50:03.080 So finding a differential equation for the operator 00:50:03.080 --> 00:50:04.090 is useful. 00:50:04.090 --> 00:50:11.300 So differential equation for Heisenberg operators. 00:50:14.730 --> 00:50:16.300 So what do we want to do? 00:50:16.300 --> 00:50:28.775 We want to calculate ih bar d dt of the Heisenberg operator. 00:50:32.670 --> 00:50:35.710 And so what do we get? 00:50:35.710 --> 00:50:40.160 Well, we have several things. 00:50:40.160 --> 00:50:42.050 Remember, the Schrodinger operator 00:50:42.050 --> 00:50:45.160 can have a bit of time dependence. 00:50:45.160 --> 00:50:48.220 The time dependence would be an explicit time dependence. 00:50:48.220 --> 00:50:51.360 So let's take the time derivative of all this. 00:50:51.360 --> 00:50:53.680 So you would have three terms. 00:50:53.680 --> 00:51:06.670 ih bar dU dagger dt As U plus U dagger 00:51:06.670 --> 00:51:21.455 As dU dt plus-- with an ih bar-- U dagger ih bar dAs minus dt. 00:51:26.050 --> 00:51:40.670 dAs dt and U. 00:51:40.670 --> 00:51:44.060 Well, you have these equations. 00:51:44.060 --> 00:51:45.910 Those were the Schrodinger equations 00:51:45.910 --> 00:51:49.100 we started with today. 00:51:49.100 --> 00:51:53.600 The derivatives of U, or the derivatives of U dagger. 00:51:53.600 --> 00:51:56.790 so what did we have? 00:51:56.790 --> 00:52:11.800 Well, we have that ih bar dU dt was HU-- H Schrodinger times U. 00:52:11.800 --> 00:52:20.160 And therefore ih bar dU dagger dt. 00:52:20.160 --> 00:52:21.770 I take the dagger of this. 00:52:21.770 --> 00:52:23.510 I would get a minus sign. 00:52:23.510 --> 00:52:25.290 I would put it on the other side. 00:52:25.290 --> 00:52:32.310 Is equal to U dagger Hs with a minus here. 00:52:32.310 --> 00:52:37.020 And all the U's are U's of t and t0. 00:52:37.020 --> 00:52:41.900 I ran out of this thick chalk. 00:52:41.900 --> 00:52:43.890 So we'll continue with thin chalk. 00:52:46.500 --> 00:52:48.050 All right, so we are here. 00:52:51.890 --> 00:52:54.290 We wrote the time derivative, and we 00:52:54.290 --> 00:52:58.870 have three terms to work out. 00:52:58.870 --> 00:53:00.450 So what are they? 00:53:00.450 --> 00:53:04.000 Well we have this thing, ih bar this. 00:53:04.000 --> 00:53:16.060 So let's write it. ih bar d d dt of As-- of A Heisenberg, 00:53:16.060 --> 00:53:27.310 I'm sorry-- Is equal to that term is minus U dagger Hs A 00:53:27.310 --> 00:53:31.830 Schrodinger U. 00:53:31.830 --> 00:53:37.940 The next term plus ih bar dU dt on the right. 00:53:37.940 --> 00:53:55.840 So we have plus U dagger As Hs dU dt, so U. Well, 00:53:55.840 --> 00:53:57.020 that's not bad. 00:53:57.020 --> 00:53:59.250 It's actually quite nice. 00:53:59.250 --> 00:54:03.830 And then the last term, which I have very little to say, 00:54:03.830 --> 00:54:08.600 because in general, this is a derivative of a time 00:54:08.600 --> 00:54:11.370 dependent operator. 00:54:11.370 --> 00:54:16.470 Partial with respect to time, it would be 0 if As depends, 00:54:16.470 --> 00:54:21.160 just say, on X, on P, on Sx, or any of those things, 00:54:21.160 --> 00:54:23.320 has to have a particular t. 00:54:23.320 --> 00:54:36.800 So I will just leave this as plus ih bar dAs dt Heisenberg. 00:54:36.800 --> 00:54:40.760 The Heisenberg version of this operator using 00:54:40.760 --> 00:54:44.790 the definition that anything, any operator that we have 00:54:44.790 --> 00:54:47.880 a U dagger in front, a U to the right, 00:54:47.880 --> 00:54:52.620 is the Heisenberg version of the operator. 00:54:52.620 --> 00:54:58.460 So I think I'm doing all right with this equation. 00:54:58.460 --> 00:55:00.850 So what did we have? 00:55:00.850 --> 00:55:08.230 Here it is. ih bar d dt of A Heisenberg of t. 00:55:08.230 --> 00:55:14.850 And now comes the nice thing, of course. 00:55:14.850 --> 00:55:16.670 This thing, look at it. 00:55:16.670 --> 00:55:21.150 U dagger U. This turns everything here 00:55:21.150 --> 00:55:22.460 into Heisenberg. 00:55:22.460 --> 00:55:25.170 H Heisenberg, A Heisenberg. 00:55:25.170 --> 00:55:30.160 Here you have A Heisenberg H Heisenberg, and what you got 00:55:30.160 --> 00:55:32.590 is the commutator between them. 00:55:32.590 --> 00:55:39.000 So this thing is A Heisenberg commutator with H Heisenberg. 00:55:45.070 --> 00:55:46.330 That whole thing. 00:55:46.330 --> 00:55:54.215 And then you have plus ih bar dAs dt Heisenberg. 00:56:00.320 --> 00:56:05.280 So that is the Heisenberg equation of motion. 00:56:05.280 --> 00:56:11.640 That is how you can calculate a Heisenberg operator 00:56:11.640 --> 00:56:12.290 if you want. 00:56:14.950 --> 00:56:18.020 You tried to solve this differential equation, 00:56:18.020 --> 00:56:20.060 and many times that's the simplest 00:56:20.060 --> 00:56:23.320 way to calculate the Heisenberg operator. 00:56:23.320 --> 00:56:25.840 So there you go. 00:56:25.840 --> 00:56:28.400 It's a pretty important equation. 00:56:28.400 --> 00:56:34.390 So let's consider particular cases immediately 00:56:34.390 --> 00:56:36.560 to just get some intuition. 00:56:36.560 --> 00:56:40.175 So remarks. 00:56:43.450 --> 00:56:55.700 Suppose As has no explicit time dependence. 00:56:55.700 --> 00:56:58.765 So basically, there's no explicit t, 00:56:58.765 --> 00:57:02.310 and therefore this derivative goes away. 00:57:02.310 --> 00:57:11.090 So the equation becomes ih bar dAh, of course, 00:57:11.090 --> 00:57:21.406 dt is equal to Ah Heisenberg sub h of t. 00:57:21.406 --> 00:57:24.150 And you know the Heisenberg operator 00:57:24.150 --> 00:57:27.110 is supposed to be simpler. 00:57:27.110 --> 00:57:27.850 Simple. 00:57:27.850 --> 00:57:30.730 If the Schrodinger operator is time independent, 00:57:30.730 --> 00:57:34.810 it's identical to the Schrodinger Hamiltonian. 00:57:34.810 --> 00:57:39.020 Even if the Schrodinger operator has time dependence, 00:57:39.020 --> 00:57:43.620 but they commute, this will become the Schrodinger 00:57:43.620 --> 00:57:44.470 Hamiltonian. 00:57:44.470 --> 00:57:46.040 But we can leave it like that. 00:57:46.040 --> 00:57:50.870 It's a nice thing anyway. 00:57:50.870 --> 00:57:53.470 Time dependence of expectation value. 00:57:53.470 --> 00:57:59.841 So let me do a little remark on time dependence of expectation 00:57:59.841 --> 00:58:00.340 values. 00:58:00.340 --> 00:58:05.310 So suppose you have the usual thing that you want to compute. 00:58:05.310 --> 00:58:11.620 How does the expectation value of a Schrodinger operator 00:58:11.620 --> 00:58:12.880 depend on time? 00:58:15.590 --> 00:58:20.050 You're faced with that expectation value of As, 00:58:20.050 --> 00:58:22.670 and it changes in time, and you want 00:58:22.670 --> 00:58:24.530 to know how you can compute that. 00:58:24.530 --> 00:58:30.100 Well, you first say, OK, ih bar d dt. 00:58:30.100 --> 00:58:41.960 But this thing is nothing but psi 0 A Heisenberg of t psi 0. 00:58:41.960 --> 00:58:46.230 Now I can let the derivative go in. 00:58:46.230 --> 00:58:59.690 So this becomes psi 0 ih bar dAh dt psi 0. 00:58:59.690 --> 00:59:09.335 And using this, assuming that A is still no time dependence, 00:59:09.335 --> 00:59:18.700 A has no explicit time dependence, 00:59:18.700 --> 00:59:21.910 then you can use just this equation, 00:59:21.910 --> 00:59:31.935 which give you psi 0 Ah Hh psi 0. 00:59:34.940 --> 00:59:38.760 So all in all, what have you gotten? 00:59:44.840 --> 00:59:53.100 You've gotten a rather simple thing, the time derivative 00:59:53.100 --> 00:59:54.930 of the expectation values. 00:59:54.930 --> 00:59:57.900 So ih bar d dt. 00:59:57.900 --> 01:00:00.420 And now I write the left hand side 01:00:00.420 --> 01:00:06.780 as just expectation value of H Heisenberg of t. 01:00:10.588 --> 01:00:15.330 And on the left hand side has to the A Schrodinger expectation 01:00:15.330 --> 01:00:18.360 value, but we call those expectation values 01:00:18.360 --> 01:00:22.080 the same thing as a Heisenberg expectation value. 01:00:22.080 --> 01:00:30.350 So this thing becomes the right hand side is the expectation 01:00:30.350 --> 01:00:37.050 value of A Heisenberg H Heisenberg like that. 01:00:40.354 --> 01:00:47.690 And just the way we say that Heisenberg expectation 01:00:47.690 --> 01:00:50.140 values are the same as Schrodinger expectation values, 01:00:50.140 --> 01:00:52.760 you could as well write, if you prefer, 01:00:52.760 --> 01:00:59.600 as d dt of A Schrodinger is equal to the expectation 01:00:59.600 --> 01:01:02.756 value of A Schrodinger with H Schrodinger. 01:01:06.990 --> 01:01:09.720 It's really the same equation. 01:01:09.720 --> 01:01:14.240 This equation we derived a couple of lectures ago. 01:01:16.830 --> 01:01:19.570 And now we know that the expectation values 01:01:19.570 --> 01:01:22.450 of Schrodinger operators are the same as the expectation 01:01:22.450 --> 01:01:25.050 value of their Heisenberg counterparts, 01:01:25.050 --> 01:01:28.890 except that the states are taking up time equals 0. 01:01:28.890 --> 01:01:34.590 So you can use either form of this equation. 01:01:34.590 --> 01:01:38.590 The bottom one is one that you've already seen. 01:01:38.590 --> 01:01:43.340 The top one now looks almost obvious from the bottom one, 01:01:43.340 --> 01:01:48.140 but it really took quite a bit to get it. 01:01:48.140 --> 01:01:50.453 One last comment on these operators. 01:01:54.850 --> 01:01:56.900 How about conserved operators? 01:01:56.900 --> 01:01:58.950 What are those things? 01:01:58.950 --> 01:02:10.440 A time independent As is set to be conserved 01:02:10.440 --> 01:02:12.400 if it commutes with a Schrodinger Hamiltonian. 01:02:16.390 --> 01:02:27.190 If As commutes with As equals 0. 01:02:27.190 --> 01:02:41.830 Now you know that if As with Hs is 0, Ah with Hh is 0, 01:02:41.830 --> 01:02:45.950 because the map between Heisenberg and Schrodinger 01:02:45.950 --> 01:02:49.650 pictures is a commutator that is valued at the Schrodinger 01:02:49.650 --> 01:02:52.540 picture is valued in the Heisenberg picture by putting 01:02:52.540 --> 01:02:53.650 H's. 01:02:53.650 --> 01:02:58.020 So what you realize from this is that this thing, 01:02:58.020 --> 01:03:05.690 this implies Ah commutes with Hh. 01:03:05.690 --> 01:03:12.260 And therefore by point 1, by 1, you 01:03:12.260 --> 01:03:20.810 have to dAh dt is equal to 0. 01:03:20.810 --> 01:03:25.150 And this is nice, actually. 01:03:25.150 --> 01:03:32.530 The Heisenberg operator is actually time independent. 01:03:32.530 --> 01:03:34.590 It just doesn't depend on time. 01:03:34.590 --> 01:03:39.230 So a Schrodinger operator, it's a funny operator. 01:03:39.230 --> 01:03:41.020 It doesn't have time in there. 01:03:41.020 --> 01:03:47.790 It has X's, P's, spins, and you don't know in general, 01:03:47.790 --> 01:03:52.530 if it's time independent in the sense of conserve 01:03:52.530 --> 01:03:55.430 of expectation values. 01:03:55.430 --> 01:04:00.420 But whenever As commutes with Hs, 01:04:00.420 --> 01:04:04.740 well, the expectation values don't change in time. 01:04:04.740 --> 01:04:08.100 But as you know, this d dt can be brought in, 01:04:08.100 --> 01:04:10.740 because the states are not time dependent. 01:04:10.740 --> 01:04:14.420 So the fact that this is 0 means the operator, 01:04:14.420 --> 01:04:18.810 Heisenberg operator, is really time independent. 01:04:18.810 --> 01:04:22.710 Whenever you have a Schrodinger operator, has no t, 01:04:22.710 --> 01:04:25.830 the Heisenberg one can have a lot of t. 01:04:25.830 --> 01:04:29.460 But if the operator is conserved, 01:04:29.460 --> 01:04:33.640 then the Heisenberg operator will have no t's after all. 01:04:33.640 --> 01:04:37.260 It will really be conserved. 01:04:37.260 --> 01:04:42.640 So let's use our last 10 minutes to do an example 01:04:42.640 --> 01:04:46.140 and illustrate much of this. 01:04:46.140 --> 01:04:48.560 In the notes, there will be three examples. 01:04:48.560 --> 01:04:51.860 I will do just one in lecture. 01:04:51.860 --> 01:04:56.580 You can do the other ones in recitation next week. 01:04:56.580 --> 01:05:00.495 There's no need really that you study these things 01:05:00.495 --> 01:05:01.990 at this moment. 01:05:01.990 --> 01:05:05.410 Just try to get whatever you can now from the lecture, 01:05:05.410 --> 01:05:09.840 and next week you'll go back to this. 01:05:09.840 --> 01:05:13.050 So the example is the harmonic oscillator. 01:05:21.080 --> 01:05:25.130 And it will illustrate the ideas very nicely, I think. 01:05:25.130 --> 01:05:28.420 The Schrodinger Hamiltonian is p squared 01:05:28.420 --> 01:05:35.400 over 2m plus 1/2 m omega squared x hat squared. 01:05:35.400 --> 01:05:41.350 OK, I could put x Schrodinger and p Schrodinger, but that 01:05:41.350 --> 01:05:43.880 would be just far too much. x and p 01:05:43.880 --> 01:05:46.380 are the operators you've always known. 01:05:46.380 --> 01:05:48.470 They are Schrodinger operators. 01:05:48.470 --> 01:05:51.980 So we leave them like that. 01:05:51.980 --> 01:05:57.150 Now I have to write the Heisenberg Hamiltonian. 01:05:57.150 --> 01:06:00.490 Well, what is the Heisenberg Hamiltonian? 01:06:07.540 --> 01:06:08.520 yes? 01:06:08.520 --> 01:06:09.852 AUDIENCE: It's identical. 01:06:09.852 --> 01:06:10.560 PROFESSOR: Sorry? 01:06:10.560 --> 01:06:12.140 AUDIENCE: It's identical. 01:06:12.140 --> 01:06:14.800 PROFESSOR: Identical, yes. 01:06:14.800 --> 01:06:18.280 But I will leave that for a little later. 01:06:18.280 --> 01:06:21.670 I will just assume, well, I'm supposed 01:06:21.670 --> 01:06:26.340 to do U dagger U. As you said, this 01:06:26.340 --> 01:06:28.000 is a time independent Hamiltonian. 01:06:28.000 --> 01:06:29.960 It better be the same, but it will 01:06:29.960 --> 01:06:35.000 be clearer if we now write what it should be in general. 01:06:35.000 --> 01:06:37.650 Have a U dagger and a U from the right. 01:06:37.650 --> 01:06:43.550 They come here, and they turn this into P Heisenberg over 2m 01:06:43.550 --> 01:06:47.380 plus 1/2 m omega squared x Heisenberg. 01:06:50.300 --> 01:06:55.100 OK, that's your Heisenberg Hamiltonian. 01:06:55.100 --> 01:06:58.660 And we will check, in fact, that it's time independent. 01:06:58.660 --> 01:07:09.130 So how about the operators X Heisenberg and P Heisenberg. 01:07:09.130 --> 01:07:10.810 What are they? 01:07:10.810 --> 01:07:13.040 Well, I don't know how to get them, 01:07:13.040 --> 01:07:16.240 unless I do this sort of U thing. 01:07:16.240 --> 01:07:22.170 That doesn't look too bad but certainly would be messy. 01:07:22.170 --> 01:07:25.430 You would have to do an exponential of e 01:07:25.430 --> 01:07:30.225 to the minus iHt over t with the x operator 01:07:30.225 --> 01:07:31.990 and another exponential. 01:07:31.990 --> 01:07:34.290 Sounds a little complicated. 01:07:34.290 --> 01:07:41.150 So let's do it the way the equations of the Heisenberg 01:07:41.150 --> 01:07:42.740 operators tell you. 01:07:42.740 --> 01:07:46.570 Well, X and P are time independent Schrodinger 01:07:46.570 --> 01:07:49.950 operators, so that equation that I boxed holds. 01:07:49.950 --> 01:07:57.290 So ih dx Heisenberg dt is nothing else 01:07:57.290 --> 01:08:02.910 than X Heisenberg commuted with H Heisenberg. 01:08:02.910 --> 01:08:05.860 OK, can we do that commutator? 01:08:05.860 --> 01:08:07.030 Yes. 01:08:07.030 --> 01:08:09.580 Because X Heisenberg, as you remember, 01:08:09.580 --> 01:08:11.670 just commutes with P Heisenberg. 01:08:11.670 --> 01:08:15.422 So instead of the Hamiltonian, you can put this. 01:08:15.422 --> 01:08:20.720 This is X Heisenberg P Heisenberg squared over 2m. 01:08:27.352 --> 01:08:30.330 OK well, X Heisenberg P Heisenberg 01:08:30.330 --> 01:08:34.240 is like you had X and P. So what is this commutator? 01:08:34.240 --> 01:08:36.200 You probably know it by now. 01:08:36.200 --> 01:08:39.600 You act with this and these two p. 01:08:39.600 --> 01:08:43.350 So it acts on one, acts on the other, gives the same on each. 01:08:43.350 --> 01:08:49.100 So you get P Heisenberg times the commutator of X and P, 01:08:49.100 --> 01:08:53.830 which is ih bar times a factor of 2. 01:08:57.250 --> 01:09:00.760 So we could put hats. 01:09:00.760 --> 01:09:03.689 Better maybe. 01:09:03.689 --> 01:09:05.960 And then what do we get? 01:09:05.960 --> 01:09:09.069 The ih there and ih cancels. 01:09:09.069 --> 01:09:11.029 And we get some nice equation that 01:09:11.029 --> 01:09:17.960 says dX Heisenberg dt is 1 over m P Heisenberg. 01:09:21.430 --> 01:09:23.729 Well, it actually looks like an equation 01:09:23.729 --> 01:09:25.310 in classical mechanics. 01:09:25.310 --> 01:09:28.109 dx dt is P over m. 01:09:28.109 --> 01:09:31.529 So that's a good thing about the Heisenberg equations of motion. 01:09:31.529 --> 01:09:34.960 They look like ordinary equations 01:09:34.960 --> 01:09:37.890 for dynamical variables. 01:09:37.890 --> 01:09:39.560 Well, we've got this one. 01:09:39.560 --> 01:09:46.479 Let's get P. Well, we didn't get the operator still, 01:09:46.479 --> 01:09:49.569 but we got an equation. 01:09:49.569 --> 01:09:53.100 So how about P dP dt. 01:09:53.100 --> 01:10:03.060 So ih dP Heisenberg dt would be P Heisenberg with H Heisenberg. 01:10:03.060 --> 01:10:06.550 And this time only the potential term in here matters. 01:10:06.550 --> 01:10:11.090 So it's P Heisenberg with 1/2 m omega 01:10:11.090 --> 01:10:14.730 squared X Heisenberg squared. 01:10:14.730 --> 01:10:15.695 So what do we? 01:10:15.695 --> 01:10:20.340 We get 1/2 m omega squared. 01:10:20.340 --> 01:10:23.870 Then we get again a factor of 2. 01:10:23.870 --> 01:10:27.420 Then we get one left over Xh. 01:10:27.420 --> 01:10:33.310 And then a P with Xh, which is a minus ih bar. 01:10:33.310 --> 01:10:43.302 So the ih bars cancel, and we get dPh dt is equal to-- the h 01:10:43.302 --> 01:10:48.380 bar cancelled-- m omega squared Xh. 01:10:48.380 --> 01:10:49.250 Minus m. 01:10:56.960 --> 01:11:01.616 All right, so these are our Heisenberg equations of motion. 01:11:04.570 --> 01:11:08.350 So how do we solve for them now? 01:11:08.350 --> 01:11:12.570 Well, you sort of have to try the kind of things 01:11:12.570 --> 01:11:14.690 that you would do classically. 01:11:14.690 --> 01:11:18.140 Take a second derivative of this equation. 01:11:18.140 --> 01:11:29.430 d second Xh dt squared would be 1 over m dPh dt. 01:11:29.430 --> 01:11:32.550 And the dPh dt would be [INAUDIBLE] 01:11:32.550 --> 01:11:39.530 1 over m times minus m omega squared Xh. 01:11:39.530 --> 01:11:50.710 So d second Xh dt squared is equal to minus omega squared 01:11:50.710 --> 01:11:55.420 Xh, exactly the equation of motion 01:11:55.420 --> 01:11:57.050 of a harmonic oscillator. 01:12:01.010 --> 01:12:05.060 It's really absolutely nice that you 01:12:05.060 --> 01:12:09.350 recover those equations that you had before, 01:12:09.350 --> 01:12:14.310 except that now you're talking operators. 01:12:14.310 --> 01:12:17.970 And it's going to simplify your life quite dramatically when 01:12:17.970 --> 01:12:23.400 you try to use these operators, because, in a sense, 01:12:23.400 --> 01:12:28.840 solving for the time dependent Heisenberg operators 01:12:28.840 --> 01:12:33.340 is the same as finding the time evolution of all states. 01:12:33.340 --> 01:12:35.730 This time the operators change, and you 01:12:35.730 --> 01:12:38.020 will know what they change like. 01:12:38.020 --> 01:12:39.500 So you have this. 01:12:39.500 --> 01:12:48.120 And then you write Xh is equal to A cosine omega t plus B 01:12:48.120 --> 01:12:54.910 sine omega t where A and B are some time 01:12:54.910 --> 01:12:56.560 independent operators. 01:12:56.560 --> 01:13:04.360 So Xh of t, well, that's a solution. 01:13:04.360 --> 01:13:07.880 How about what is P? 01:13:07.880 --> 01:13:18.590 Ph of t would be m dX m dx dt. 01:13:18.590 --> 01:13:31.190 So you get minus m omega sine omega tA plus m omega 01:13:31.190 --> 01:13:33.100 cosine omega tB. 01:13:38.053 --> 01:13:40.480 OK, so that's it. 01:13:40.480 --> 01:13:42.210 That is the most general solution. 01:13:44.780 --> 01:13:50.280 But it still doesn't look like what you would want, does it? 01:13:50.280 --> 01:13:54.580 No, because you haven't used the time equals 0 conditions. 01:13:54.580 --> 01:13:57.430 At time equals 0, the Heisenberg operators 01:13:57.430 --> 01:14:00.310 are identical they to the Schrodinger operators. 01:14:00.310 --> 01:14:09.070 So at t equals 0, Xh of t becomes A, 01:14:09.070 --> 01:14:15.440 but that must be X hat, the Schrodinger operator. 01:14:15.440 --> 01:14:21.630 And at t equals 0, Ph of t becomes 01:14:21.630 --> 01:14:30.880 equal to this is 0 m omega B. And that 01:14:30.880 --> 01:14:35.580 must be equal to the P hat operator. 01:14:35.580 --> 01:14:43.220 So actually we have already now A and B. So B from here 01:14:43.220 --> 01:14:47.890 is P hat over m omega. 01:14:47.890 --> 01:14:55.040 And therefore Xh of t is equal to A, 01:14:55.040 --> 01:15:00.370 which is X hat cosine omega t plus B, which 01:15:00.370 --> 01:15:06.560 is P hat over m omega sine omega t. 01:15:06.560 --> 01:15:12.660 An Ph of t is here. 01:15:12.660 --> 01:15:23.780 A is-- Ph of t is m omega B, which is [INAUDIBLE] P. 01:15:23.780 --> 01:15:33.563 So it's P hat cosine omega t minus m omega X 01:15:33.563 --> 01:15:36.050 hat sine omega t. 01:15:42.640 --> 01:15:43.800 So let's see. 01:15:43.800 --> 01:15:46.380 I hope I didn't make mistakes. 01:15:46.380 --> 01:15:50.360 P hat minus m omega X hat sine omega t. 01:15:50.360 --> 01:15:53.000 Yep, this is correct. 01:15:53.000 --> 01:15:59.400 This is your whole solution for the Heisenberg operators. 01:15:59.400 --> 01:16:04.990 So any expectation value of any power of X and P 01:16:04.990 --> 01:16:08.630 that you will want to find its time dependence, 01:16:08.630 --> 01:16:11.490 just put those Heisenberg operators, 01:16:11.490 --> 01:16:15.630 and you will calculate things with states at time equals 0. 01:16:15.630 --> 01:16:18.360 It will become very easy. 01:16:18.360 --> 01:16:20.760 So the last thing I want to do is 01:16:20.760 --> 01:16:26.730 complete the promise that we had about what 01:16:26.730 --> 01:16:30.520 is the Heisenberg Hamiltonian. 01:16:30.520 --> 01:16:34.440 Well, we had the Heisenberg Hamiltonian there. 01:16:34.440 --> 01:16:38.300 And now we know the Heisenberg operators 01:16:38.300 --> 01:16:43.800 in terms of the Schrodinger one. 01:16:43.800 --> 01:16:53.630 So Hh of t is equal to Ph-- 1/2m Ph squared. 01:16:53.630 --> 01:17:01.550 So I have P hat cosine omega t minus m omega X hat sine omega 01:17:01.550 --> 01:17:15.120 t squared plus 1/2 m omega squared Xh squared. 01:17:15.120 --> 01:17:19.940 So X hat cosine omega t plus P hat 01:17:19.940 --> 01:17:27.150 over m omega sine omega t squared. 01:17:27.150 --> 01:17:30.810 So that's what the Heisenberg Hamiltonian is. 01:17:30.810 --> 01:17:36.010 So let's simplify this. 01:17:36.010 --> 01:17:37.790 Well, let's square these things. 01:17:37.790 --> 01:17:47.310 You have 1/2m cosine squared omega t P hat squared. 01:17:47.310 --> 01:17:50.750 Let's do the square of this one. 01:17:50.750 --> 01:17:57.340 You would have plus 1/2m m squared 01:17:57.340 --> 01:18:07.440 omega squared sine squared omega t X squared. 01:18:10.000 --> 01:18:12.340 And then we have the cross product, 01:18:12.340 --> 01:18:17.465 which would be plus-- or actually minus 1/2m. 01:18:20.120 --> 01:18:22.910 The product of these two things. 01:18:22.910 --> 01:18:29.900 m omega sine omega t cosine omega t. 01:18:29.900 --> 01:18:35.670 And you have Px plus XP. 01:18:35.670 --> 01:18:45.000 OK, I squared the first terms. 01:18:45.000 --> 01:18:46.410 Now the second one. 01:18:46.410 --> 01:18:49.120 Well, let's square the P squared here. 01:18:49.120 --> 01:18:51.540 What do we have? 01:18:51.540 --> 01:18:57.950 1/2 m omega squared over m squared 01:18:57.950 --> 01:19:06.120 omega squared sine squared of omega t P squared. 01:19:06.120 --> 01:19:11.740 The x plus 1/2 m omega squared cosine 01:19:11.740 --> 01:19:14.965 squared omega t X squared. 01:19:17.740 --> 01:19:20.880 And the cross term. 01:19:20.880 --> 01:19:37.990 Plus 1/2 m omega squared over m omega times cosine omega t 01:19:37.990 --> 01:19:42.863 sine omega t XP plus PX. 01:19:45.540 --> 01:19:49.740 A little bit of work, but what do we get? 01:19:49.740 --> 01:19:52.330 Well, 1/2 m. 01:19:52.330 --> 01:19:56.170 And here we must have 1/2 m, correct. 01:19:56.170 --> 01:19:58.190 1/2 m. 01:19:58.190 --> 01:20:01.040 Sine squared omega t P squared. 01:20:01.040 --> 01:20:05.680 So this is equal 1/2 m P squared. 01:20:08.370 --> 01:20:14.180 These one's, hall here you have 1/2 m omega squared. 01:20:14.180 --> 01:20:18.440 So it's 1/2 m omega squared cosine and sine 01:20:18.440 --> 01:20:21.930 squared is X hat squared. 01:20:21.930 --> 01:20:26.430 And then here we have all being over 2. 01:20:26.430 --> 01:20:30.970 And here omega over 2, same factors, same factors, 01:20:30.970 --> 01:20:33.350 opposite signs. 01:20:33.350 --> 01:20:35.460 Very good. 01:20:35.460 --> 01:20:37.000 Schrodinger Hamiltonian. 01:20:37.000 --> 01:20:41.340 So you confirm that this theoretical expectation 01:20:41.340 --> 01:20:43.350 is absolutely correct. 01:20:43.350 --> 01:20:45.100 And what's the meaning? 01:20:45.100 --> 01:20:47.900 You have the Heisenberg Hamiltonian 01:20:47.900 --> 01:20:51.000 written in terms of the Heisenberg variables. 01:20:51.000 --> 01:20:57.040 But by the time you substitute these Heisenberg variables in, 01:20:57.040 --> 01:21:01.130 it just becomes identical to the Schrodinger Hamiltonian. 01:21:01.130 --> 01:21:03.530 All right, so that's all for today. 01:21:03.530 --> 01:21:07.420 I hope to see in office hours in the coming days. 01:21:07.420 --> 01:21:13.160 Be here Wednesday 12:30, maybe 12:25 would be better, 01:21:13.160 --> 01:21:16.800 and we'll see you then. 01:21:16.800 --> 01:21:18.350 [APPLAUSE]