WEBVTT 00:00:00.060 --> 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 view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.305 at ocw.mit.edu. 00:00:22.180 --> 00:00:26.300 PROFESSOR: OK, last time we were talking about uncertainty. 00:00:26.300 --> 00:00:28.590 We gave a picture for uncertainty-- 00:00:28.590 --> 00:00:34.320 it was a neat picture, I think of the uncertainty, 00:00:34.320 --> 00:00:40.200 refer to the uncertainty measuring an operator 00:00:40.200 --> 00:00:43.540 A that was a Hermitian operator. 00:00:43.540 --> 00:00:47.360 And that uncertainty depended on the state 00:00:47.360 --> 00:00:49.550 that you were measuring. 00:00:49.550 --> 00:00:52.590 If the state was an eigenstate of A, 00:00:52.590 --> 00:00:54.320 there would be no uncertainty. 00:00:54.320 --> 00:00:56.300 If the state is not an eigenstate of A, 00:00:56.300 --> 00:00:58.680 there was an uncertainty. 00:00:58.680 --> 00:01:02.640 And this uncertainty was defined as the norm 00:01:02.640 --> 00:01:10.390 of A minus the expectation value of A acting on psi. 00:01:14.060 --> 00:01:18.330 So that was our definition of uncertainty. 00:01:18.330 --> 00:01:20.790 And it had nice properties. 00:01:20.790 --> 00:01:24.540 In fact, it was zero if and only if the state was 00:01:24.540 --> 00:01:27.400 an eigenstate of the operator. 00:01:27.400 --> 00:01:30.710 We proved a couple of things as well-- 00:01:30.710 --> 00:01:34.880 that, in particular, one that is kind of practical 00:01:34.880 --> 00:01:43.010 is that delta A of psi squared is the expectation 00:01:43.010 --> 00:01:48.290 value of A squared on the state psi minus the expectation 00:01:48.290 --> 00:01:52.055 value of A on the state psi squared. 00:01:55.590 --> 00:02:01.590 So that was also proven, which, since this number is 00:02:01.590 --> 00:02:07.130 greater than or equal to 0, this is greater than or equal to 0. 00:02:07.130 --> 00:02:10.460 And in particular, the expectation value of A squared 00:02:10.460 --> 00:02:15.686 is bigger than the expectation of A squared. 00:02:19.650 --> 00:02:23.740 So let's do a trivial example for a computation. 00:02:23.740 --> 00:02:30.750 Suppose somebody tells you in an example 00:02:30.750 --> 00:02:36.010 that the spin is in an eigenstate of Sz. 00:02:36.010 --> 00:02:45.470 So the state psi it's what we called the plus state, or the z 00:02:45.470 --> 00:02:48.890 plus state. 00:02:48.890 --> 00:02:56.125 And you want to know what is uncertainty delta of Sx. 00:03:02.890 --> 00:03:07.220 So you know if you're in an eigenstate of z, 00:03:07.220 --> 00:03:09.650 you are not in an eigenstate of x-- in fact, 00:03:09.650 --> 00:03:13.870 you're in a superposition of two eigenstates of Sx. 00:03:13.870 --> 00:03:17.250 Therefore, there should be some uncertainty here. 00:03:17.250 --> 00:03:19.760 And the question is, what is the quickest way in which you 00:03:19.760 --> 00:03:25.080 compute this uncertainty, and how much is it? 00:03:25.080 --> 00:03:30.280 So many times, the simplest way is to just use this formula. 00:03:33.500 --> 00:03:40.200 So let's do that. 00:03:40.200 --> 00:03:48.990 So what is the expectation value of Sx in that state? 00:03:48.990 --> 00:03:53.510 So it's Sx expectation value would 00:03:53.510 --> 00:03:57.530 be given by Sx on this thing. 00:03:57.530 --> 00:04:03.290 Now, actually, it's relatively clear 00:04:03.290 --> 00:04:08.070 to see that this expectation value is going to be 0, 00:04:08.070 --> 00:04:14.180 because Sx really in the state plus 00:04:14.180 --> 00:04:18.480 is equal amplitude to be Sx equal plus h bar over 2, 00:04:18.480 --> 00:04:20.740 or minus h bar over 2. 00:04:20.740 --> 00:04:24.340 But suppose you don't remember that. 00:04:24.340 --> 00:04:26.790 In order to compute this, it may come 00:04:26.790 --> 00:04:32.720 handy to recall the matrix presentation of Sx, which 00:04:32.720 --> 00:04:35.280 you don't need to know by heart. 00:04:35.280 --> 00:04:41.190 So this state plus is the first state, 00:04:41.190 --> 00:04:44.780 and the basis state is the state 1 0. 00:04:44.780 --> 00:04:51.960 And then we have Sx on plus is equal to h bar over 2 0 00:04:51.960 --> 00:04:55.440 1 1 0, acting on 1 0. 00:04:55.440 --> 00:05:00.880 Zero and that's equal to h bar over 2. 00:05:00.880 --> 00:05:05.810 The first thing gives you 0, and the second one gives you 1. 00:05:05.810 --> 00:05:10.090 So that's, in fact, equal to h bar over 2, the state of minus. 00:05:13.060 --> 00:05:18.990 So here you go to h bar over 2 plus minus, 00:05:18.990 --> 00:05:26.470 and you know plus and minus are orthogonal, so 0 is expected. 00:05:26.470 --> 00:05:28.550 Well, are we going to get zero uncertainty? 00:05:28.550 --> 00:05:33.660 No, because Sx squared, however, does 00:05:33.660 --> 00:05:35.540 have some expectation value. 00:05:35.540 --> 00:05:38.313 So what is the expectation value of Sx squared? 00:05:42.930 --> 00:05:45.650 Well, there's an advantage here. 00:05:45.650 --> 00:05:50.640 You may remember that this Sx squared is a funny matrix. 00:05:50.640 --> 00:05:54.110 It's a multiple of the identity, because if you square 00:05:54.110 --> 00:05:56.810 this matrix, you get the multiple of the identity. 00:05:56.810 --> 00:06:02.780 So Sx squared is h over 2 squared times the identity 00:06:02.780 --> 00:06:07.740 matrix-- the two by two identity matrix. 00:06:07.740 --> 00:06:10.620 So the expectation value of Sx squared 00:06:10.620 --> 00:06:15.340 is h bar over 2 squared times expectation value 00:06:15.340 --> 00:06:17.832 of the identity. 00:06:17.832 --> 00:06:20.450 And on any state, the expectation value 00:06:20.450 --> 00:06:23.350 on any normalized state, the expectation value 00:06:23.350 --> 00:06:26.770 of the identity will be equal to 1. 00:06:26.770 --> 00:06:31.210 So this is just h squared over 2 squared. 00:06:31.210 --> 00:06:37.310 So back to our uncertainty, delta Sx squared 00:06:37.310 --> 00:06:41.350 would be equal to the expectation value of Sx squared 00:06:41.350 --> 00:06:44.800 minus the expectation value of Sx squared. 00:06:44.800 --> 00:06:46.720 This was 0. 00:06:46.720 --> 00:06:52.990 This thing was equal to h bar over 2 squared, 00:06:52.990 --> 00:06:58.450 and therefore, delta Sx is equal to h bar over 2. 00:07:04.760 --> 00:07:11.500 So just I wanted to make you familiar with that. 00:07:11.500 --> 00:07:14.840 You can compute these things-- these norms and all 00:07:14.840 --> 00:07:20.770 these equations are pretty practical, and easy to use. 00:07:20.770 --> 00:07:24.600 So today what we have to do is the following-- 00:07:24.600 --> 00:07:28.740 we're going to establish the uncertainty principle. 00:07:28.740 --> 00:07:31.390 We're going to just prove it. 00:07:31.390 --> 00:07:35.930 And then, once we have the uncertainty principle, 00:07:35.930 --> 00:07:40.670 we'll try to find some applications for it. 00:07:40.670 --> 00:07:43.610 So before doing an application, we 00:07:43.610 --> 00:07:47.130 will discuss the case of the energy time uncertainty 00:07:47.130 --> 00:07:51.530 principle, which is slightly more subtle 00:07:51.530 --> 00:07:55.310 and has interestingly connotations that we 00:07:55.310 --> 00:07:56.480 will develop today. 00:07:56.480 --> 00:07:59.650 And finally, we'll use the uncertainty principle 00:07:59.650 --> 00:08:04.970 to learn how to find bounds for energies of ground states. 00:08:04.970 --> 00:08:08.400 So we might make a rigorous application of the uncertainty 00:08:08.400 --> 00:08:10.090 principle. 00:08:10.090 --> 00:08:16.410 So the uncertainty principle talks about two operators 00:08:16.410 --> 00:08:21.870 that are both Hermitian, and states 00:08:21.870 --> 00:08:32.070 the following-- so given the theorem, or uncertainty 00:08:32.070 --> 00:08:47.610 principle, given two Hermitian operators A and B, 00:08:47.610 --> 00:08:59.970 and a state psi normalized, then the following inequality holds. 00:08:59.970 --> 00:09:03.950 And we're going to write it in one way, then in another way. 00:09:03.950 --> 00:09:12.910 Delta A psi squared times delta B-- sometimes people 00:09:12.910 --> 00:09:15.870 in order to avoid cluttering don't put the psi. 00:09:15.870 --> 00:09:17.910 I don't know whether to put it or not. 00:09:17.910 --> 00:09:21.130 It does look a little more messy with the psi there, 00:09:21.130 --> 00:09:24.420 but it's something you have to keep in mind. 00:09:24.420 --> 00:09:26.940 Each time you have an uncertainty, 00:09:26.940 --> 00:09:29.550 you are talking about some specific state 00:09:29.550 --> 00:09:31.380 that should not be forgotten. 00:09:31.380 --> 00:09:35.940 So maybe I'll erase it to make it look a little nicer. 00:09:35.940 --> 00:09:41.140 Delta B squared-- now it's an inequality. 00:09:41.140 --> 00:09:44.260 So not just equality, but inequality. 00:09:44.260 --> 00:09:48.270 That product of uncertainties must exceed a number-- 00:09:48.270 --> 00:09:52.410 a computable number-- which is given by the following thing. 00:10:03.030 --> 00:10:04.695 OK, so here it is. 00:10:07.530 --> 00:10:11.220 This is a number, is the expectation value 00:10:11.220 --> 00:10:17.236 of this strange operator in the state psi squared. 00:10:22.700 --> 00:10:27.490 So even such a statement is somewhat quite confusing, 00:10:27.490 --> 00:10:34.690 because you wish to know what kind of number is this. 00:10:34.690 --> 00:10:36.430 Could this be a complex number? 00:10:36.430 --> 00:10:42.320 If it were a complex number, why am I squaring? 00:10:42.320 --> 00:10:44.870 That doesn't make any sense. 00:10:44.870 --> 00:10:47.720 Inequalities-- these are real numbers. 00:10:47.720 --> 00:10:50.550 Deltas are defined to be real numbers. 00:10:50.550 --> 00:10:52.010 They're the norms. 00:10:52.010 --> 00:10:55.020 So this is real positive. 00:10:55.020 --> 00:11:00.260 This would make no sense if this would be a complex number. 00:11:00.260 --> 00:11:03.410 So this number better be real. 00:11:03.410 --> 00:11:06.200 And the way it's written, it seems 00:11:06.200 --> 00:11:08.490 to be particularly confusing, because there 00:11:08.490 --> 00:11:10.660 seems to be an i here. 00:11:10.660 --> 00:11:16.530 So at first sight, you might say, well, can it be real? 00:11:16.530 --> 00:11:19.060 But the thing that you should really focus here 00:11:19.060 --> 00:11:20.930 is this whole thing. 00:11:20.930 --> 00:11:22.115 This is some operator. 00:11:27.010 --> 00:11:32.890 And against all first impressions, this operator 00:11:32.890 --> 00:11:37.700 formed by taking the commutator of A and B-- 00:11:37.700 --> 00:11:40.850 this is the commutator A B minus B 00:11:40.850 --> 00:11:46.580 A-- is Hermitian, because, in fact, 00:11:46.580 --> 00:11:50.640 if you have two operators, and you take 00:11:50.640 --> 00:11:54.880 the commutator, if the two of them are Hermitian, 00:11:54.880 --> 00:11:58.080 the answer is not Hermitian. 00:11:58.080 --> 00:12:04.850 And that you know already-- x with p is equal to i h bar. 00:12:04.850 --> 00:12:09.500 These are Hermitian operators, and suddenly the commutator 00:12:09.500 --> 00:12:11.630 is not a Hermitian operator. 00:12:11.630 --> 00:12:13.050 You have the unit here. 00:12:13.050 --> 00:12:16.090 A Hermitian operator with a number 00:12:16.090 --> 00:12:18.210 here would have to be a real things. 00:12:18.210 --> 00:12:21.380 So there's an extra i, that's your first hint 00:12:21.380 --> 00:12:22.905 that this i is important. 00:12:25.480 --> 00:12:31.450 So the fact is that this operator as defind 00:12:31.450 --> 00:12:40.980 here is Hermitian, because if you take 1 over i A B-- 00:12:40.980 --> 00:12:46.980 and we're going to try to take its Hermitian conjugate-- 00:12:46.980 --> 00:12:54.790 we have 1 over i A B minus B A. And we're 00:12:54.790 --> 00:12:57.320 taking the Hermitian conjugate. 00:12:57.320 --> 00:13:01.940 Now, the i is going to get complex conjugated, 00:13:01.940 --> 00:13:05.930 so you're going to get 1 over minus i. 00:13:05.930 --> 00:13:09.810 The Hermitian conjugate of a product 00:13:09.810 --> 00:13:12.850 is the Hermitian conjugate in opposite order. 00:13:12.850 --> 00:13:19.065 So it would be B dagger A dagger minus A dagger B dagger. 00:13:23.340 --> 00:13:26.950 And of course, these operators are Hermitian, 00:13:26.950 --> 00:13:33.600 so 1 over minus i is minus 1 over i. 00:13:33.600 --> 00:13:40.450 And here I get B A minus A B. So with a minus sign, 00:13:40.450 --> 00:13:44.310 this is 1 over i A B again. 00:13:47.030 --> 00:13:52.720 So the operator is equal to its dagger-- its adjoint. 00:13:52.720 --> 00:13:55.170 And therefore, this operator is Hermitian. 00:14:02.380 --> 00:14:06.850 And as we proved, the expectation value 00:14:06.850 --> 00:14:12.920 of any Hermitian operator is real. 00:14:12.920 --> 00:14:15.500 And we're in good shape. 00:14:15.500 --> 00:14:17.040 We have a real number. 00:14:17.040 --> 00:14:19.950 This could be negative. 00:14:19.950 --> 00:14:22.150 And a number, when you square it, 00:14:22.150 --> 00:14:23.910 is going to be a positive number. 00:14:23.910 --> 00:14:25.650 So this makes sense. 00:14:25.650 --> 00:14:29.740 We're writing something that at least makes sense. 00:14:29.740 --> 00:14:32.370 Another way, of course, to write this equation, 00:14:32.370 --> 00:14:35.065 if you prefer-- this inequality, I 00:14:35.065 --> 00:14:38.750 mean-- is to take the square root. 00:14:38.750 --> 00:14:43.040 So you could write it delta A times delta 00:14:43.040 --> 00:14:50.460 B. Since this is a real number, I can take the square root 00:14:50.460 --> 00:14:57.300 and write just this as absolute value of psi, 1 over 2i i 00:14:57.300 --> 00:15:02.050 A B psi. 00:15:02.050 --> 00:15:05.685 And these bars here are absolute value. 00:15:09.060 --> 00:15:12.880 They're not norm of a vector. 00:15:12.880 --> 00:15:16.080 They are not norm of a complex number. 00:15:16.080 --> 00:15:20.460 They are just absolute value, because the thing inside 00:15:20.460 --> 00:15:22.400 is a real thing. 00:15:22.400 --> 00:15:27.160 So if you prefer, whatever you like better, 00:15:27.160 --> 00:15:32.070 you've got here the statement of the uncertainty principle. 00:15:32.070 --> 00:15:36.450 So the good thing about this uncertainty principle 00:15:36.450 --> 00:15:41.160 formulated this way is that it's completely precise, 00:15:41.160 --> 00:15:44.700 because you've defined uncertainties precisely. 00:15:44.700 --> 00:15:48.250 Many times, when you first study the uncertainty principle, 00:15:48.250 --> 00:15:51.020 you don't define uncertainties precisely, 00:15:51.020 --> 00:15:53.050 and the uncertainty principle is something 00:15:53.050 --> 00:15:59.600 that goes with [? sim ?] is approximately equal to this. 00:15:59.600 --> 00:16:02.830 And you make statements that are intuitively interesting, 00:16:02.830 --> 00:16:04.670 but are not thoroughly precise. 00:16:04.670 --> 00:16:07.060 Yes, question, yes. 00:16:07.060 --> 00:16:08.810 AUDIENCE: Should that be greater or equal? 00:16:08.810 --> 00:16:15.020 PROFESSOR: Greater than or equal to, yes-- no miracles here. 00:16:15.020 --> 00:16:18.000 Other question? 00:16:18.000 --> 00:16:18.630 Other question? 00:16:25.850 --> 00:16:29.960 So we have to prove this. 00:16:29.960 --> 00:16:31.920 And why do you have to prove this? 00:16:31.920 --> 00:16:34.640 This is a case, actually, in which 00:16:34.640 --> 00:16:39.020 many interesting questions are based on the proof. 00:16:39.020 --> 00:16:42.100 Why would that be the case? 00:16:42.100 --> 00:16:47.630 Well, a question that is always of great interest 00:16:47.630 --> 00:16:50.080 is reducing uncertainties. 00:16:50.080 --> 00:16:58.140 Now, if two operators commute, this right-hand side is 0 00:16:58.140 --> 00:17:00.420 and it just says that the uncertainty 00:17:00.420 --> 00:17:03.830 could be made perhaps equal to 0. 00:17:03.830 --> 00:17:06.839 It doesn't mean that the uncertainty is 0. 00:17:06.839 --> 00:17:10.010 It may depend on the state, even if the operators commute. 00:17:10.010 --> 00:17:13.190 This is just telling you it's bigger than 0, 00:17:13.190 --> 00:17:18.579 and perhaps by being clever, you can make it equal to 0. 00:17:18.579 --> 00:17:21.230 Similarly, when you have two operators that 00:17:21.230 --> 00:17:24.670 just don't commute, it is of great importance 00:17:24.670 --> 00:17:28.920 to try to figure out if there is some states for which 00:17:28.920 --> 00:17:32.690 the uncertainty relation is saturated. 00:17:32.690 --> 00:17:36.390 So this is the question that, in fact, you could not 00:17:36.390 --> 00:17:42.490 answer if you just know this theorem written like this, 00:17:42.490 --> 00:17:44.940 because there's no statement here 00:17:44.940 --> 00:17:50.430 of what are the conditions for which this inequality is 00:17:50.430 --> 00:17:51.960 saturated. 00:17:51.960 --> 00:17:55.410 So as we'll do the proof, we'll find those conditions. 00:17:55.410 --> 00:17:59.350 And in fact, they go a little beyond what 00:17:59.350 --> 00:18:02.690 the Schwarz inequality would say. 00:18:02.690 --> 00:18:07.180 I mentioned last time that this is a classic example 00:18:07.180 --> 00:18:09.880 of something that looks like the Schwarz inequality, 00:18:09.880 --> 00:18:12.020 and indeed, that will be the central part 00:18:12.020 --> 00:18:13.540 of the demonstration. 00:18:13.540 --> 00:18:18.570 But there's one extra step there that we will have to do. 00:18:18.570 --> 00:18:23.230 And therefore, if you want to understand 00:18:23.230 --> 00:18:28.020 when this is saturated, when do you have minimum uncertainty 00:18:28.020 --> 00:18:33.230 states, then you need to know the proof. 00:18:33.230 --> 00:18:37.830 So before we do, of course, even the proof, 00:18:37.830 --> 00:18:40.750 there's an example-- the classic illustration 00:18:40.750 --> 00:18:50.509 that should be mentioned-- A equal x and B equals p, 00:18:50.509 --> 00:18:53.880 xp equal i h bar. 00:18:53.880 --> 00:18:55.960 That's the identity. 00:18:55.960 --> 00:19:02.930 So delta x squared delta p squared 00:19:02.930 --> 00:19:09.965 is greater or equal than psi 1 over 2i-- the commutator-- i h 00:19:09.965 --> 00:19:16.940 bar 1 psi squared. 00:19:16.940 --> 00:19:18.740 And what do we get here? 00:19:18.740 --> 00:19:24.580 We get the i's cancel, the h bar over 2 goes out, gets squared, 00:19:24.580 --> 00:19:28.356 and everything else is equal to 1, because h is normalized. 00:19:31.130 --> 00:19:36.860 So the precise version of the uncertainty principle 00:19:36.860 --> 00:19:43.340 is this one for x and p. 00:19:46.060 --> 00:19:49.240 And we will, of course, try to figure out 00:19:49.240 --> 00:19:50.970 when we can saturate this. 00:19:50.970 --> 00:19:53.320 What kind of wave functions saturate them? 00:19:53.320 --> 00:19:59.015 You know the ones that are just sort of strange-- if x 00:19:59.015 --> 00:20:02.100 is totally localized, the uncertainty of momentum 00:20:02.100 --> 00:20:07.210 must be infinite, because if delta x is 0, well, 00:20:07.210 --> 00:20:09.850 to make this something that at least doesn't contradict 00:20:09.850 --> 00:20:12.970 the identity, delta p better be infinite. 00:20:12.970 --> 00:20:15.450 Similarly, if you have an eigenstate 00:20:15.450 --> 00:20:19.420 of p, which is a wave, is totally delocalized, 00:20:19.420 --> 00:20:22.270 and you have infinite here and 0 here. 00:20:22.270 --> 00:20:25.370 Well, they're interesting states that have both, 00:20:25.370 --> 00:20:28.580 and we're going to try to find the ones 00:20:28.580 --> 00:20:30.960 of minimum uncertainty. 00:20:30.960 --> 00:20:34.810 So OK, we've stated the principle. 00:20:34.810 --> 00:20:36.860 We've given an example. 00:20:36.860 --> 00:20:38.620 We've calculated an uncertainty. 00:20:38.620 --> 00:20:43.320 Let us prove the theorem. 00:20:43.320 --> 00:20:49.580 So as we mentioned before, this idea that the uncertainty is 00:20:49.580 --> 00:20:51.810 a norm, is a good one. 00:20:51.810 --> 00:20:55.240 So let's define two auxilliary variables-- 00:20:55.240 --> 00:21:03.300 f, a state f, which is going to be A minus the expectation 00:21:03.300 --> 00:21:08.180 value of A on psi. 00:21:08.180 --> 00:21:10.810 And we can put the ket here. 00:21:10.810 --> 00:21:15.960 And g, which is going to be B minus the expectation 00:21:15.960 --> 00:21:20.505 value of B, psi. 00:21:23.330 --> 00:21:28.100 Now what do we know about this? 00:21:28.100 --> 00:21:32.700 Well the uncertainties are the norms of these states, 00:21:32.700 --> 00:21:35.250 so the norm squared of these states 00:21:35.250 --> 00:21:36.940 are the uncertainty squared. 00:21:36.940 --> 00:21:46.580 So delta A squared is f f, the norm squared. 00:21:46.580 --> 00:21:51.712 And delta B squared is g g. 00:21:55.100 --> 00:22:02.490 And Schwarz' inequality says that the norm 00:22:02.490 --> 00:22:06.000 of f times the normal of g is greater than 00:22:06.000 --> 00:22:10.460 or equal than the absolute value of the inner product of f 00:22:10.460 --> 00:22:12.520 with g. 00:22:12.520 --> 00:22:17.870 So squaring this thing, which is convenient perhaps 00:22:17.870 --> 00:22:24.950 at this moment, we have f f-- norm squared of f-- norm 00:22:24.950 --> 00:22:34.800 squared of g must be greater than or equal than f g squared, 00:22:34.800 --> 00:22:36.170 absolute value squared. 00:22:40.490 --> 00:22:41.920 So this is Schwarz. 00:22:49.820 --> 00:22:55.540 And this is going to just make a note-- here 00:22:55.540 --> 00:22:58.360 we know when this is saturated. 00:22:58.360 --> 00:23:02.400 It will be saturated if f is parallel to g. 00:23:02.400 --> 00:23:05.230 If these two vectors are parallel to each other, 00:23:05.230 --> 00:23:07.250 the Schwarz inequality is saturated. 00:23:07.250 --> 00:23:09.980 So that's something to keep in mind. 00:23:09.980 --> 00:23:12.510 We'll use it soon enough. 00:23:12.510 --> 00:23:18.480 But at this moment, we can simply rewrite this as delta 00:23:18.480 --> 00:23:24.280 A squared times delta B squared-- after all, those 00:23:24.280 --> 00:23:29.520 were definitions-- are greater than or equal-- 00:23:29.520 --> 00:23:35.420 and this is going to be a complex number in general, 00:23:35.420 --> 00:23:41.780 so f g in Schwarz' inequality is just a complex number. 00:23:41.780 --> 00:23:53.540 So this is real of f g squared, plus the imaginary part 00:23:53.540 --> 00:23:57.345 of f g squared. 00:24:02.300 --> 00:24:07.720 So that's what we have-- real and imaginary part. 00:24:07.720 --> 00:24:11.680 So let's try to get what f g is. 00:24:11.680 --> 00:24:16.610 So what is f g? 00:24:16.610 --> 00:24:18.800 Let's compute it. 00:24:18.800 --> 00:24:24.180 Well we must take the bra corresponding to this, 00:24:24.180 --> 00:24:25.730 so this is psi. 00:24:25.730 --> 00:24:28.180 Since the operator is Hermitian, you 00:24:28.180 --> 00:24:33.130 have A minus expectation value of A, 00:24:33.130 --> 00:24:39.740 and here you have B minus expectation value of B psi. 00:24:49.090 --> 00:24:53.650 Now we can expand this, and it will be useful to expand. 00:24:53.650 --> 00:24:58.690 But at the same time, I will invent a little notation here. 00:24:58.690 --> 00:25:05.390 I'll call this A check, and this B check. 00:25:05.390 --> 00:25:14.360 And for reference, I'll put that this is psi A check B check 00:25:14.360 --> 00:25:14.860 psi. 00:25:18.970 --> 00:25:23.000 On the other hand, let's just compute what we get. 00:25:23.000 --> 00:25:26.190 So what do we get? 00:25:26.190 --> 00:25:31.010 Well, let's expand this. 00:25:31.010 --> 00:25:35.670 Well, the first term is A times B on psi psi, 00:25:35.670 --> 00:25:37.610 and we're not going to be able to do 00:25:37.610 --> 00:25:42.960 much about that-- A B psi. 00:25:42.960 --> 00:25:49.050 And then we start getting funny terms-- A cross with B, 00:25:49.050 --> 00:25:51.910 and that's-- if you think about it a second, 00:25:51.910 --> 00:25:55.655 this is just going to be equal to the expectation value of A 00:25:55.655 --> 00:25:58.280 times the expectation of B, because the expectation value 00:25:58.280 --> 00:26:03.080 of B is a number, and then A is sandwich between two psi. 00:26:03.080 --> 00:26:07.770 So from this cross product, you get expectation value 00:26:07.770 --> 00:26:11.400 of A, expectation value of B, with a minus sign. 00:26:11.400 --> 00:26:14.450 From this cross product, you get the expectation value of A 00:26:14.450 --> 00:26:18.030 and expectation value of B-- another one with a minus sign. 00:26:18.030 --> 00:26:20.500 And then one with a plus sign. 00:26:20.500 --> 00:26:27.790 So the end result is a single one with a minus sign. 00:26:27.790 --> 00:26:42.740 So expectation value of A, expectation value of B. Now, 00:26:42.740 --> 00:26:47.250 if I change f and g, I would like 00:26:47.250 --> 00:26:52.220 to compute not only fg inner product, but gf inner product. 00:26:52.220 --> 00:26:54.090 And you may say why? 00:26:54.090 --> 00:26:58.370 Well, I want it because I need the real part 00:26:58.370 --> 00:27:04.390 and the imaginary parts, and gf is the complex conjugate 00:27:04.390 --> 00:27:08.070 of f g, so might as well compute it. 00:27:08.070 --> 00:27:11.230 So what is gf? 00:27:11.230 --> 00:27:14.120 Now you don't have to do the calculation again, 00:27:14.120 --> 00:27:17.510 because basically you change g to f or f 00:27:17.510 --> 00:27:20.990 to g by exchanging A and B. So I can just 00:27:20.990 --> 00:27:32.360 say that this is psi B A psi minus A B. 00:27:32.360 --> 00:27:35.060 And if I write it this way, I say 00:27:35.060 --> 00:27:41.300 it's just psi B check A check psi. 00:27:44.530 --> 00:27:49.670 OK so we've done some work, and the reason we've done this work 00:27:49.670 --> 00:27:51.825 is because we actually need to write 00:27:51.825 --> 00:27:57.880 the right-hand side of the inequality. 00:27:57.880 --> 00:28:01.400 And let's, therefore, explore what these ones are. 00:28:01.400 --> 00:28:09.200 So for example, the imaginary part of f g 00:28:09.200 --> 00:28:19.682 is 1 over 2i f g minus its complex conjugate-- gf. 00:28:27.590 --> 00:28:29.680 Imaginary part of a complex number 00:28:29.680 --> 00:28:34.040 is z minus z star divided by 2i. 00:28:34.040 --> 00:28:38.740 now, fg minus gf is actually simple, 00:28:38.740 --> 00:28:43.660 because this product of expectation values cancel, 00:28:43.660 --> 00:28:48.210 and this gives me the commutator of A with B. 00:28:48.210 --> 00:28:55.650 So this is 1 over 2i, and you have psi expectation 00:28:55.650 --> 00:29:00.040 value of A B commutator. 00:29:00.040 --> 00:29:04.550 So actually, that looks exactly like what we want. 00:29:04.550 --> 00:29:09.470 And we're not going to be able to simplify it more. 00:29:09.470 --> 00:29:12.550 We can put the 1 over 2i inside. 00:29:12.550 --> 00:29:13.450 That fine. 00:29:13.450 --> 00:29:15.800 It's sort of in the operator. 00:29:15.800 --> 00:29:21.490 It can go out, but we're not going to do better than that. 00:29:21.490 --> 00:29:24.820 You already recognize, in some sense, 00:29:24.820 --> 00:29:29.390 the inequality we want to prove, because if this 00:29:29.390 --> 00:29:33.340 is that, you could ignore this and say, well, 00:29:33.340 --> 00:29:36.380 it's anyway greater than this thing. 00:29:36.380 --> 00:29:39.770 And that's this term. 00:29:39.770 --> 00:29:43.040 But let's write the other one, at least for a little while. 00:29:43.040 --> 00:29:57.490 Real of fg would be 1/2 of fg plus gf. 00:29:57.490 --> 00:30:00.110 And now it is your choice how you write this. 00:30:00.110 --> 00:30:03.870 There's nothing great that you can do. 00:30:03.870 --> 00:30:09.890 The sum of these two things have AB plus BA and then twice 00:30:09.890 --> 00:30:13.000 of this expectation value, so it's not 00:30:13.000 --> 00:30:17.080 nothing particularly inspiring. 00:30:17.080 --> 00:30:22.420 So you put these two terms and just 00:30:22.420 --> 00:30:29.450 write it like this-- 1/2 of psi anti-commutator 00:30:29.450 --> 00:30:32.520 off A check with B check. 00:30:32.520 --> 00:30:37.710 Anti-commutator, remember, is this combination 00:30:37.710 --> 00:30:40.310 of operators in which you take the product in one way, 00:30:40.310 --> 00:30:42.450 and add the product in the other way. 00:30:42.450 --> 00:30:45.870 So I've used this formula to write this, 00:30:45.870 --> 00:30:49.280 and you could write it as an anti-commutator of A and B 00:30:49.280 --> 00:30:54.200 minus 2 times the expectation values, 00:30:54.200 --> 00:30:56.490 or whichever way you want it. 00:30:56.490 --> 00:31:00.830 But at the end of the day, that's what it is. 00:31:00.830 --> 00:31:03.140 And you cannot simplify it much. 00:31:03.140 --> 00:31:08.850 So your uncertainty principle has become delta 00:31:08.850 --> 00:31:12.960 A squared delta B squared greater 00:31:12.960 --> 00:31:25.340 than or equal to expectation value of psi 1 over 2i A B psi 00:31:25.340 --> 00:31:36.130 squared plus expectation value of psi 1 over 2 00:31:36.130 --> 00:31:42.840 A check B check psi squared. 00:31:42.840 --> 00:31:46.840 And some people call this the generalized uncertainty 00:31:46.840 --> 00:31:48.120 principle. 00:31:48.120 --> 00:31:50.940 You may find some textbooks that tell you 00:31:50.940 --> 00:31:54.710 "Prove the generalized uncertainty principle," 00:31:54.710 --> 00:31:56.880 because that's really what you get 00:31:56.880 --> 00:32:01.860 if you follow the rules and Schwarz' inequality. 00:32:01.860 --> 00:32:03.570 So it is of some interest. 00:32:03.570 --> 00:32:09.200 It is conceivable that sometimes you may want to use this. 00:32:09.200 --> 00:32:14.530 But the fact is that this is a real number. 00:32:14.530 --> 00:32:17.800 This is a Hermitian operator as well. 00:32:17.800 --> 00:32:19.420 This is a real number. 00:32:19.420 --> 00:32:21.250 This is a positive number. 00:32:21.250 --> 00:32:27.350 So if you ignore it, you still have the inequality holding. 00:32:27.350 --> 00:32:31.020 And many times-- and that's the interesting thing-- 00:32:31.020 --> 00:32:33.650 you really are justified to ignore it. 00:32:33.650 --> 00:32:36.900 In fact, I don't know of a single example-- 00:32:36.900 --> 00:32:40.640 perhaps somebody can tell me-- in which that second term is 00:32:40.640 --> 00:32:41.140 useful. 00:32:44.300 --> 00:32:53.300 So what you say at this moment is go ahead, drop that term, 00:32:53.300 --> 00:32:56.900 and get an inequality. 00:32:56.900 --> 00:33:07.380 So it follows directly from that, from this inequality, 00:33:07.380 --> 00:33:12.710 that delta A squared delta B squared 00:33:12.710 --> 00:33:15.720 is greater than or equal-- you might say, well, 00:33:15.720 --> 00:33:16.930 how do you know it's equal? 00:33:16.930 --> 00:33:19.460 Maybe that thing cannot be 0. 00:33:19.460 --> 00:33:22.330 Well, it can be 0 in some examples. 00:33:22.330 --> 00:33:29.460 So it's still greater than or equal to psi 1 over 2i 00:33:29.460 --> 00:33:36.080 A B psi squared. 00:33:39.430 --> 00:33:43.090 And that's by ignoring the positive quantity. 00:33:43.090 --> 00:33:49.130 So that is really the proof of the uncertainty principle. 00:33:49.130 --> 00:33:54.960 But now we can ask what are the things that 00:33:54.960 --> 00:33:58.030 have to happen for the uncertainty principle 00:33:58.030 --> 00:34:00.290 to be saturated? 00:34:00.290 --> 00:34:08.199 That you really have delta A delta B equal to this quantity, 00:34:08.199 --> 00:34:09.649 so when can we saturate? 00:34:19.530 --> 00:34:23.280 OK, what do we need? 00:34:23.280 --> 00:34:28.350 First we need Schwarz inequality saturation. 00:34:28.350 --> 00:34:33.730 So f and g must be states that are proportional to each other. 00:34:33.730 --> 00:34:42.623 So we need one, that Schwarz is saturated. 00:34:49.080 --> 00:34:55.679 Which means that g is some number times f, 00:34:55.679 --> 00:35:00.660 where beta is a complex number. 00:35:00.660 --> 00:35:03.780 This is complex vector space, so parallel 00:35:03.780 --> 00:35:06.100 means multiply by a complex number. 00:35:06.100 --> 00:35:09.430 That's still a parallel vector. 00:35:09.430 --> 00:35:13.130 So this is the saturation of Schwarz. 00:35:13.130 --> 00:35:15.920 Now, what else do we need? 00:35:15.920 --> 00:35:18.780 Well, we need that this quantity be 00:35:18.780 --> 00:35:25.570 0 as well, that the real part of this thing is equal to 0. 00:35:25.570 --> 00:35:28.130 Otherwise, you really cannot reach it. 00:35:28.130 --> 00:35:32.010 The true inequality is this, so if you have Schwarz, 00:35:32.010 --> 00:35:33.070 you've saturated. 00:35:33.070 --> 00:35:36.390 This thing is equal to this thing. 00:35:36.390 --> 00:35:39.440 The left-hand side is equal to this whole right-hand side. 00:35:39.440 --> 00:35:41.680 Schwarz buys you that. 00:35:41.680 --> 00:35:45.440 But now we want this to be just equal to that. 00:35:45.440 --> 00:35:52.010 So this thing must be 0, so the real part of f 00:35:52.010 --> 00:36:00.020 overlap g-- of fg must be 0. 00:36:00.020 --> 00:36:01.160 What does that mean? 00:36:01.160 --> 00:36:11.180 It means that fg plus gf has to be 0. 00:36:11.180 --> 00:36:16.600 But now we know what g is, so we can plug it here. 00:36:16.600 --> 00:36:19.310 So g is beta times f. 00:36:19.310 --> 00:36:26.850 Beta goes out, and you get beta f f. 00:36:26.850 --> 00:36:31.120 Now when you form the bra g, beta becomes beta star. 00:36:31.120 --> 00:36:38.730 So you get beta star f f equals 0. 00:36:38.730 --> 00:36:45.640 And since f need not have zero norm, because there 00:36:45.640 --> 00:36:48.530 is some uncertainty presumably, you 00:36:48.530 --> 00:36:59.460 have that beta plus beta star is equal to 0, or real of beta 00:36:59.460 --> 00:37:02.470 is equal to 0. 00:37:02.470 --> 00:37:07.190 So that said, it's not that bad. 00:37:07.190 --> 00:37:12.410 You need two things-- that the f and g vectors 00:37:12.410 --> 00:37:16.920 be parallel with a complex constant, 00:37:16.920 --> 00:37:21.500 but actually, that constant must be purely imaginary. 00:37:21.500 --> 00:37:31.260 So beta is purely imaginary-- that this beta 00:37:31.260 --> 00:37:35.395 is equal to i lambda, with lambda real. 00:37:40.600 --> 00:37:44.560 And we then are in shape. 00:37:44.560 --> 00:37:53.410 So for saturation, we need just g 00:37:53.410 --> 00:37:57.120 to be that, and g to be beta f. 00:37:57.120 --> 00:38:08.220 So let me write that equation over here. 00:38:08.220 --> 00:38:11.730 So g-- what was g? 00:38:11.730 --> 00:38:31.960 It's B, B minus absolute value of B on psi, which is g, 00:38:31.960 --> 00:38:39.320 must be equal to beta, which is i lambda 00:38:39.320 --> 00:38:43.963 A minus absolute value of A on psi. 00:38:50.320 --> 00:38:54.186 Condition-- so this is the final condition for saturation. 00:39:05.190 --> 00:39:09.380 now, that's a strange-looking equation. 00:39:09.380 --> 00:39:11.810 It's not all that obvious how you're even 00:39:11.810 --> 00:39:15.450 supposed to begin solving it. 00:39:15.450 --> 00:39:16.390 Why is that? 00:39:16.390 --> 00:39:20.080 Well, you're trying to look for a psi, 00:39:20.080 --> 00:39:22.360 and you have a constraint on the psi. 00:39:22.360 --> 00:39:24.485 The psi must satisfy this. 00:39:27.990 --> 00:39:32.640 I actually will tell both Arum and Will 00:39:32.640 --> 00:39:37.990 to discuss some of these things in recitation-- 00:39:37.990 --> 00:39:42.030 how to calculate minimum uncertainty wave packets based 00:39:42.030 --> 00:39:44.460 on this equation, and what it means. 00:39:44.460 --> 00:39:46.810 But in principle, what do you have to do? 00:39:46.810 --> 00:39:50.000 You have some kind of differential equation, 00:39:50.000 --> 00:39:53.280 because you have, say, x and p, and you want to saturate. 00:39:53.280 --> 00:39:56.330 So this is x, and this is p. 00:39:56.330 --> 00:39:59.760 Since p, you want to use a coordinate representation, 00:39:59.760 --> 00:40:03.110 this will be a derivative, and this will be a multiplication, 00:40:03.110 --> 00:40:06.460 so you'll get a differential equation on the wave function. 00:40:06.460 --> 00:40:10.260 So you write an answer for the wave function. 00:40:10.260 --> 00:40:13.260 You must calculate the expectation value of B. 00:40:13.260 --> 00:40:15.480 You must calculate the expectation value of A, 00:40:15.480 --> 00:40:17.620 and then plug into this equation, 00:40:17.620 --> 00:40:20.570 and try to see if your answer allows 00:40:20.570 --> 00:40:26.410 a solution-- and a solution with some number here, lambda. 00:40:26.410 --> 00:40:28.590 At least one thing I can tell you 00:40:28.590 --> 00:40:32.600 before you try this too hard-- this lambda is essentially 00:40:32.600 --> 00:40:38.470 fixed, because we can take the norm of this equation. 00:40:38.470 --> 00:40:41.275 And that's an interesting fact-- take the norm. 00:40:45.330 --> 00:40:48.190 And what is the norm of this? 00:40:48.190 --> 00:40:55.210 This is delta B, the norm of this state. 00:40:55.210 --> 00:40:59.880 And the norm of i lambda--, well norm of i is 1. 00:40:59.880 --> 00:41:02.740 Norm of lambda is absolute value of lambda, 00:41:02.740 --> 00:41:04.910 because lambda was real. 00:41:04.910 --> 00:41:12.720 And you have delta A here of psi, of course. 00:41:12.720 --> 00:41:19.410 So lambda can be either plus or minus delta B 00:41:19.410 --> 00:41:23.910 of psi over delta A of psi. 00:41:23.910 --> 00:41:26.200 So that's not an arbitrary constant. 00:41:26.200 --> 00:41:28.590 It's fixed by the equation already, 00:41:28.590 --> 00:41:30.345 in terms of things that you know. 00:41:35.760 --> 00:41:38.980 And therefore, this will be a subject 00:41:38.980 --> 00:41:43.850 of problems in a little bit of your recitation, in which you, 00:41:43.850 --> 00:41:50.980 hopefully, discuss how to find minimum uncertainty packets. 00:41:50.980 --> 00:41:56.310 All right, so that's it for the proof 00:41:56.310 --> 00:41:58.920 of the uncertainty principle. 00:41:58.920 --> 00:42:02.830 And as I told you, the proof is useful in particular 00:42:02.830 --> 00:42:06.520 to find those special states of saturated uncertainty. 00:42:06.520 --> 00:42:09.550 We'll have a lot to say about them for the harmonic 00:42:09.550 --> 00:42:15.100 oscillator later on, and in fact throughout the course. 00:42:15.100 --> 00:42:18.710 So are there any questions? 00:42:18.710 --> 00:42:19.950 Yes. 00:42:19.950 --> 00:42:22.450 AUDIENCE: So if we have one of the states and an eigenstate, 00:42:22.450 --> 00:42:26.960 we know that [INAUDIBLE] is 0 and we then 00:42:26.960 --> 00:42:29.214 mandate that the uncertainty of the other variable 00:42:29.214 --> 00:42:29.922 must be infinite. 00:42:32.700 --> 00:42:35.770 But is it even possible to talk about the uncertainty? 00:42:35.770 --> 00:42:38.125 And if so, are we still guaranteed-- 00:42:38.125 --> 00:42:40.096 we know that it's infinite, but it's 00:42:40.096 --> 00:42:44.710 possible for 0 and an infinite number to multiply [INAUDIBLE] 00:42:44.710 --> 00:42:48.860 PROFESSOR: Right, so you're in a somewhat uncomfortable position 00:42:48.860 --> 00:42:51.470 if you have zero uncertainty. 00:42:51.470 --> 00:42:53.630 Then you need the other one to be infinite. 00:42:53.630 --> 00:42:57.760 So the way, presumably, you should think of that, 00:42:57.760 --> 00:43:01.590 is that you should take limits of sequences of wave functions 00:43:01.590 --> 00:43:04.640 in which the uncertainty in x is going to 0, 00:43:04.640 --> 00:43:07.710 and you will find that as you take the limit, 00:43:07.710 --> 00:43:11.510 and delta x is going to 0, and delta p is going to infinity, 00:43:11.510 --> 00:43:12.773 you can still have that. 00:43:16.160 --> 00:43:16.833 Other questions? 00:43:23.800 --> 00:43:29.720 Well, having done this, let's try the more subtle case 00:43:29.720 --> 00:43:35.700 of the uncertainty principle for energy and time. 00:43:35.700 --> 00:43:40.880 So that is a pretty interesting subject, actually. 00:43:40.880 --> 00:43:44.520 And should I erase here? 00:43:44.520 --> 00:43:45.675 Yes, I think so. 00:43:52.550 --> 00:43:56.100 Actually, [? Griffith ?] says that it's usually 00:43:56.100 --> 00:44:00.430 badly misunderstood, this energy-time uncertainty 00:44:00.430 --> 00:44:04.960 principle, but seldom your misunderstanding 00:44:04.960 --> 00:44:07.860 leads to a serious mistake. 00:44:07.860 --> 00:44:10.390 So you're saved. 00:44:10.390 --> 00:44:17.150 It's used in a hand-wavy way, and it's roughly correct, 00:44:17.150 --> 00:44:19.835 although people say all kinds of funny things 00:44:19.835 --> 00:44:21.670 that are not exactly right. 00:44:21.670 --> 00:44:37.670 So energy time uncertainty-- so let 00:44:37.670 --> 00:44:42.910 me give a small motivation-- a hand-wavy motivation, 00:44:42.910 --> 00:44:46.290 so it doesn't get us very far, but at least it 00:44:46.290 --> 00:44:49.060 gives you a picture of what's going on. 00:44:49.060 --> 00:44:55.910 And these uncertainty relations, in some sense, 00:44:55.910 --> 00:45:02.260 have a basis on some simple statements that 00:45:02.260 --> 00:45:06.790 are totally classical, and maybe a little imprecise, 00:45:06.790 --> 00:45:11.690 but incontrovertible, about looking at waveforms, 00:45:11.690 --> 00:45:14.060 and trying to figure out what's going on. 00:45:14.060 --> 00:45:20.850 So for example, suppose in time you detect a fluctuation 00:45:20.850 --> 00:45:27.250 that as time progresses, just suddenly turns on. 00:45:27.250 --> 00:45:31.660 Some wave that just dies off after a little while. 00:45:31.660 --> 00:45:34.120 And you have a good understanding 00:45:34.120 --> 00:45:36.640 of when it started, and when it ended. 00:45:36.640 --> 00:45:45.610 And there's a time T. 00:45:45.610 --> 00:45:49.480 So whenever you have a situation like that, 00:45:49.480 --> 00:45:53.560 you can try to count the number of waves-- full waves 00:45:53.560 --> 00:45:56.260 that you see here. 00:45:56.260 --> 00:46:04.990 So the number of waves would be equal to-- 00:46:04.990 --> 00:46:11.790 or periods, number of full waves-- 00:46:11.790 --> 00:46:22.670 would be the total time divided by the period of this wave. 00:46:22.670 --> 00:46:25.830 So sometimes T is called the period. 00:46:25.830 --> 00:46:28.240 But here, T is the total time here, 00:46:28.240 --> 00:46:30.690 and the period is 2 pi over omega. 00:46:30.690 --> 00:46:39.540 So we say this is omega t over 2 pi. 00:46:39.540 --> 00:46:45.510 Now, the problem with these waves that begin and end, 00:46:45.510 --> 00:46:48.330 is that you can't quite see or make 00:46:48.330 --> 00:46:50.950 sure that you've got the full wave here. 00:46:50.950 --> 00:46:56.210 So in the hand-wavy way, we say that even 00:46:56.210 --> 00:46:59.090 as we looked at the perfectly well-defined, 00:46:59.090 --> 00:47:01.170 and you know the shape exactly-- it's 00:47:01.170 --> 00:47:04.070 been measured-- you can't quite tell whether you've 00:47:04.070 --> 00:47:07.840 got the full wave here or a quarter of a wave more, 00:47:07.840 --> 00:47:14.890 so there's an uncertainty in delta n which is of order 1. 00:47:14.890 --> 00:47:18.700 You miss half on one side, and half on the other side. 00:47:18.700 --> 00:47:22.350 So if you have an uncertainty here of order 1, 00:47:22.350 --> 00:47:25.760 and you have no uncertainty in T, 00:47:25.760 --> 00:47:29.090 you would claim that you have, actually, 00:47:29.090 --> 00:47:33.990 in some sense, an uncertainty in what omega is. 00:47:33.990 --> 00:47:36.670 Omega might be well measured here, but somehow 00:47:36.670 --> 00:47:39.480 towards the end you can't quite see. 00:47:39.480 --> 00:47:45.260 T we said was precise, so over 2 pi is equal to 1. 00:47:45.260 --> 00:47:49.700 I just took a delta of here, and I said P is precise, 00:47:49.700 --> 00:47:52.050 so it's delta omega. 00:47:52.050 --> 00:47:55.610 So this is a classical statement. 00:47:55.610 --> 00:47:59.100 An electrical engineer would not need 00:47:59.100 --> 00:48:03.820 to know any quantum mechanics to say that's about right, 00:48:03.820 --> 00:48:07.240 and you can make it more or less precise. 00:48:07.240 --> 00:48:10.160 But that's a classical statement. 00:48:10.160 --> 00:48:12.650 In quantum mechanics, all that happens 00:48:12.650 --> 00:48:17.110 is that something has become quantum, and the idea 00:48:17.110 --> 00:48:19.520 that you have something like this, 00:48:19.520 --> 00:48:24.310 we can associate it with a particle, a photon, 00:48:24.310 --> 00:48:28.630 and in which case, the uncertainty in omega 00:48:28.630 --> 00:48:30.350 is uncertainty in energy. 00:48:30.350 --> 00:48:41.020 So for a photon, the uncertainty is equal to h bar omega, 00:48:41.020 --> 00:48:48.970 so delta omega times h bar is equal to the uncertainty 00:48:48.970 --> 00:48:49.470 in energy. 00:48:52.010 --> 00:49:00.150 So if you plug it in here, you multiply it by h bar here, 00:49:00.150 --> 00:49:10.280 and you would get delta E times T is equal to 2 pi h bar. 00:49:14.800 --> 00:49:17.350 And then you have to add words. 00:49:17.350 --> 00:49:19.140 What is T? 00:49:19.140 --> 00:49:22.430 Well, this T is the time it takes the photon 00:49:22.430 --> 00:49:24.710 to go through your detector. 00:49:24.710 --> 00:49:26.520 You've been seeing it. 00:49:26.520 --> 00:49:27.560 You saw a wave. 00:49:27.560 --> 00:49:30.500 You recorded it, and took a time T-- began, ended. 00:49:30.500 --> 00:49:34.260 And it so it's the time it took you 00:49:34.260 --> 00:49:38.170 to have the pulse go through. 00:49:38.170 --> 00:49:42.600 And that time is related to an uncertainty 00:49:42.600 --> 00:49:44.790 in the energy of the photon. 00:49:44.790 --> 00:49:48.500 And that's sort of the beginning of a time energy uncertainty 00:49:48.500 --> 00:49:49.830 relationship. 00:49:49.830 --> 00:49:52.640 This is quantum, because the idea 00:49:52.640 --> 00:49:56.110 that photons carry energies and they're quantized-- 00:49:56.110 --> 00:50:00.230 this is a single photon-- and this connection with energy 00:50:00.230 --> 00:50:01.105 is quantum mechanics. 00:50:04.170 --> 00:50:08.110 So this is good and reasonable intuition, perhaps. 00:50:08.110 --> 00:50:11.480 And it can be the basis of all kinds of things. 00:50:11.480 --> 00:50:16.470 But it points out the fact that the more delicate part here 00:50:16.470 --> 00:50:23.100 is T. How could I speak of a time uncertainty? 00:50:23.100 --> 00:50:27.420 And the fact is that you can't speak of a time uncertainty 00:50:27.420 --> 00:50:29.750 really precisely. 00:50:29.750 --> 00:50:31.830 And the reason is, because there's 00:50:31.830 --> 00:50:34.950 no Hermitian operator for which we could say, 00:50:34.950 --> 00:50:39.630 OK the eigenstates of this Hermitian operator are times, 00:50:39.630 --> 00:50:44.100 and then you have a norm, and it's an uncertainty. 00:50:44.100 --> 00:50:45.210 So you can't do it. 00:50:45.210 --> 00:50:49.530 So you have to do something different this time. 00:50:49.530 --> 00:50:52.160 And happily, there's something you 00:50:52.160 --> 00:50:55.360 can do that is precise and makes sense. 00:50:55.360 --> 00:51:00.460 So we'll do it. 00:51:00.460 --> 00:51:04.330 So what we have to do is just try to use the uncertainty 00:51:04.330 --> 00:51:08.730 principle that we have, and at least one operator. 00:51:08.730 --> 00:51:11.370 We can use something that is good for us. 00:51:11.370 --> 00:51:14.740 We want uncertainty in energy, and we have the Hamiltonian. 00:51:14.740 --> 00:51:15.850 It's an operator. 00:51:15.850 --> 00:51:19.660 So for that one, we can use it, and that's the clue. 00:51:19.660 --> 00:51:31.900 So you'll take A to be the Hamiltonian, and B 00:51:31.900 --> 00:51:35.870 to be some operator Q that may depend 00:51:35.870 --> 00:51:41.070 on some things-- for example, x and p, or whatever you want. 00:51:41.070 --> 00:51:43.820 But the one thing I want to ask from this operator 00:51:43.820 --> 00:51:58.210 is that Q has no explicit time dependence-- no explicit time 00:51:58.210 --> 00:51:59.255 dependence whatsoever. 00:52:02.800 --> 00:52:08.620 So let's see what this gives us as an uncertainty relationship. 00:52:08.620 --> 00:52:15.140 Well, it would give us that delta H squared-- 00:52:15.140 --> 00:52:20.770 that's delta Q squared-- would be 00:52:20.770 --> 00:52:27.780 greater than or equal to the square of psi 1 00:52:27.780 --> 00:52:36.780 over 2i H with Q psi. 00:52:46.070 --> 00:52:49.160 OK, that's it. 00:52:49.160 --> 00:52:54.520 Well, but in order to get some intuition from here, 00:52:54.520 --> 00:52:59.170 we better be able to interpret this. 00:52:59.170 --> 00:53:00.820 This doesn't seem to have anything 00:53:00.820 --> 00:53:03.050 to do with energy and time. 00:53:03.050 --> 00:53:08.260 So is there something to do with time here? 00:53:08.260 --> 00:53:15.500 That is, in fact, a very well-known result 00:53:15.500 --> 00:53:20.220 in quantum mechanics-- that somehow commutators 00:53:20.220 --> 00:53:27.460 with the Hamiltonian test the time derivative of operators. 00:53:27.460 --> 00:53:31.970 So whenever you see an H with Q commutator, 00:53:31.970 --> 00:53:35.800 you think ah, that's roughly dQ dt. 00:53:41.229 --> 00:53:42.770 And we'll see what happens with that. 00:53:42.770 --> 00:53:45.157 And say, oh, dQ dt, but it doesn't 00:53:45.157 --> 00:53:46.850 depend on T-- you said 0. 00:53:46.850 --> 00:53:50.080 No it's not 0. 00:53:50.080 --> 00:53:54.350 There's no explicit dependence, but we'll see what happens. 00:53:54.350 --> 00:53:58.080 So at this moment, you really have to stop for one second 00:53:58.080 --> 00:54:05.030 and derive a familiar result-- that may or may not 00:54:05.030 --> 00:54:08.480 be that familiar to you from 804. 00:54:08.480 --> 00:54:11.370 I don't think it was all that emphasized. 00:54:11.370 --> 00:54:15.380 Consider expectation value of Q. 00:54:15.380 --> 00:54:17.780 And then the expectation of Q-- let 00:54:17.780 --> 00:54:24.920 me write it as psi Q psi, like this. 00:54:24.920 --> 00:54:29.890 Now let's try to take the time derivative of this thing. 00:54:29.890 --> 00:54:34.960 So what is the time derivative of the expectation value of q? 00:54:34.960 --> 00:54:38.440 And the idea being that look, the operator 00:54:38.440 --> 00:54:42.900 depends on some things, and it can 00:54:42.900 --> 00:54:45.850 have time-dependent expectation value, 00:54:45.850 --> 00:54:48.980 because the state is changing in time. 00:54:48.980 --> 00:54:52.360 So operators can have time-dependent expectation 00:54:52.360 --> 00:54:56.260 values even though the operators don't depend on time. 00:54:56.260 --> 00:54:58.900 So for example, this depends on x and p, 00:54:58.900 --> 00:55:03.770 and the x and p in a harmonic oscillator are time dependent. 00:55:03.770 --> 00:55:09.970 They're moving around, and this could have time dependence. 00:55:09.970 --> 00:55:11.730 So what do we get from here? 00:55:11.730 --> 00:55:15.630 Well, if I have to take the time derivative of this, 00:55:15.630 --> 00:55:27.280 I have d psi dt here, Q psi, plus psi Q d psi dt. 00:55:27.280 --> 00:55:31.990 And in doing this, and not differentiating Q itself, 00:55:31.990 --> 00:55:35.100 I've used the fact that this is an operator 00:55:35.100 --> 00:55:37.710 and there's no time anywhere there. 00:55:37.710 --> 00:55:54.530 I didn't have to differentiate Q. 00:55:54.530 --> 00:55:59.370 So how do we evaluate this? 00:55:59.370 --> 00:56:01.890 Well, you remember the Schrodinger equation. 00:56:01.890 --> 00:56:04.180 Here the Schrodinger equation comes in, 00:56:04.180 --> 00:56:07.280 because you have time derivatives of your state. 00:56:07.280 --> 00:56:15.590 So i d psi dt, i H bar d psi dt is equal to H psi. 00:56:15.590 --> 00:56:20.450 That's a full time-dependent Schrodinger equation. 00:56:20.450 --> 00:56:23.345 So here, maybe, I should write this 00:56:23.345 --> 00:56:27.380 like that-- this is all time-dependent stuff. 00:56:27.380 --> 00:56:31.780 At this moment, I don't ignore the time dependence. 00:56:31.780 --> 00:56:34.670 The states are not stationary states. 00:56:34.670 --> 00:56:36.770 If they would be stationary states, 00:56:36.770 --> 00:56:40.030 there would be no energy uncertainty. 00:56:40.030 --> 00:56:45.350 So I have this, and therefore, I plug this in here, 00:56:45.350 --> 00:56:55.100 and what do we get? i H bar h psi Q psi plus psi 00:56:55.100 --> 00:56:57.975 Q i H bar H psi. 00:57:01.690 --> 00:57:06.020 Now, I got the i H in the wrong place-- sorry-- 1 00:57:06.020 --> 00:57:11.180 over i H bar, and 1 over i H bar. 00:57:14.710 --> 00:57:18.690 Now the first term-- this thing comes out 00:57:18.690 --> 00:57:22.280 as its complex conjugate-- 1 minus i H 00:57:22.280 --> 00:57:25.470 bar, because it's on the first input. 00:57:25.470 --> 00:57:29.040 H is Hermitian, so I can send it to the other side, 00:57:29.040 --> 00:57:32.165 so psi, HQ psi. 00:57:35.480 --> 00:57:39.790 Second term-- the 1 over i H just goes out, 00:57:39.790 --> 00:57:42.140 and I don't have to move anybody. 00:57:42.140 --> 00:57:44.635 QH is there, psi. 00:57:48.210 --> 00:57:52.120 So actually, this is i over H bar, 00:57:52.120 --> 00:57:55.780 because minus i down goes up with i. 00:57:55.780 --> 00:58:01.902 And I have here psi HQ, and this is minus i 00:58:01.902 --> 00:58:09.570 over H bar, so I get HQ minus QH psi. 00:58:09.570 --> 00:58:15.975 So this is your final result-- the expectation value d 00:58:15.975 --> 00:58:25.400 dt of the expectation value of Q is equal to i over H bar, 00:58:25.400 --> 00:58:43.060 expectation value of the commutator of H with Q. 00:58:43.060 --> 00:58:49.490 So this is neat, and it should always stick in your mind. 00:58:49.490 --> 00:58:50.890 This is true. 00:58:50.890 --> 00:58:54.520 We will see the Heisenberg way of writing 00:58:54.520 --> 00:58:57.500 this equation in a little while-- not today, 00:58:57.500 --> 00:58:59.680 but in a couple of weeks. 00:58:59.680 --> 00:59:05.935 But maybe even write it even more briefly 00:59:05.935 --> 00:59:10.386 as i over H bar expectation value of HQ. 00:59:15.920 --> 00:59:23.790 So what do we get from here? 00:59:23.790 --> 00:59:28.250 Well, we can go back to our uncertainty principle, 00:59:28.250 --> 00:59:33.710 and rewrite it, having learned that we have time derivative. 00:59:33.710 --> 00:59:38.300 So time finally showed up, and that's good news. 00:59:38.300 --> 00:59:42.110 So we're maybe not too far from a clear interpretation 00:59:42.110 --> 00:59:44.050 of the uncertainty principle. 00:59:44.050 --> 00:59:48.920 So we're going back to that top equation, 00:59:48.920 --> 00:59:52.900 so that what we have now is delta 00:59:52.900 --> 01:00:00.090 H squared delta Q squared is that thing over there, 01:00:00.090 --> 01:00:03.500 the expectation value of 1 over 2i. 01:00:03.500 --> 01:00:05.550 There's some signs there, so what 01:00:05.550 --> 01:00:24.690 do we have-- equals 1 over 2i H bar over i d dt of Q. 01:00:24.690 --> 01:00:30.960 So what I did here was to say that this expectation value was 01:00:30.960 --> 01:00:35.250 H bar over i d dt of Q, and I plugged it in there. 01:00:38.000 --> 01:00:44.290 So you square this thing, so there's not too much really 01:00:44.290 --> 01:00:44.910 to be done. 01:00:44.910 --> 01:00:47.460 The i don't matter at the end of the day. 01:00:47.460 --> 01:00:49.910 It's a minus 1 that gets squared. 01:00:49.910 --> 01:00:56.020 So the H bar over 2-- I'm sorry-- the H bar over 2 01:00:56.020 --> 01:00:59.440 does remain here, squared. 01:00:59.440 --> 01:01:06.485 And you have dQ dt squared. 01:01:09.930 --> 01:01:12.790 Q is a Hermitian operator. 01:01:12.790 --> 01:01:14.310 B was supposed to be Hermitian. 01:01:14.310 --> 01:01:16.630 The expectation value is real. 01:01:16.630 --> 01:01:18.345 The time derivative is real. 01:01:18.345 --> 01:01:20.920 It could be going up or down. 01:01:20.920 --> 01:01:23.720 So at the end of the day, you have 01:01:23.720 --> 01:01:30.620 delta H delta Q is greater than or equal to H bar 01:01:30.620 --> 01:01:36.050 over 2, the absolute value of dQ over dt. 01:01:43.910 --> 01:01:46.830 There we go. 01:01:46.830 --> 01:01:50.110 This is, in a sense, the best you can do. 01:01:50.110 --> 01:01:55.550 Let's try to interpret what we've got. 01:01:55.550 --> 01:01:59.580 Well, we've got something that still doesn't quite 01:01:59.580 --> 01:02:02.720 look like a time uncertainty relationship, 01:02:02.720 --> 01:02:04.980 but there's time in there. 01:02:04.980 --> 01:02:07.700 But it's a matter of a definition now. 01:02:11.600 --> 01:02:22.410 You see, if you have delta Q, and you divide it by dQ dt, 01:02:22.410 --> 01:02:26.030 first it is some sort of time. 01:02:26.030 --> 01:02:28.570 It has the units of time. 01:02:28.570 --> 01:02:32.900 And we can define it, if you wish, to be sub delta t. 01:02:35.660 --> 01:02:41.620 And what physically, does this delta t represent? 01:02:41.620 --> 01:02:46.190 Well, it's roughly-- you see, things change in time. 01:02:46.190 --> 01:02:49.750 The rate of change of the expectation value of Q 01:02:49.750 --> 01:02:51.090 may not be uniform. 01:02:51.090 --> 01:02:55.160 It make change fast, or it may change slowly. 01:02:55.160 --> 01:02:58.990 But suppose it's changing. 01:02:58.990 --> 01:03:01.850 Roughly, this ratio, of this would be constant, 01:03:01.850 --> 01:03:06.060 is the time it takes the expectation value of Q 01:03:06.060 --> 01:03:12.730 to change by delta Q. It is like a distance divided 01:03:12.730 --> 01:03:13.313 by a velocity. 01:03:16.710 --> 01:03:25.480 So this is roughly the time needed for the expectation 01:03:25.480 --> 01:03:38.280 value of Q to change by delta Q, by the uncertainty. 01:03:38.280 --> 01:03:40.740 So it's a measure of the time needed 01:03:40.740 --> 01:03:48.930 for a significant change, if the expectation 01:03:48.930 --> 01:03:53.180 value, if the uncertainty of Q is significant, and is 01:03:53.180 --> 01:03:56.550 comparable to Q. Well, this is the time needed 01:03:56.550 --> 01:03:59.220 for significant change. 01:03:59.220 --> 01:04:02.960 Now this is pretty much all you can do, except that of course, 01:04:02.960 --> 01:04:05.980 once you write it like that, you pull this down, 01:04:05.980 --> 01:04:10.670 and you go up now, delta H delta t 01:04:10.670 --> 01:04:15.630 is greater or equal than H bar over 2. 01:04:15.630 --> 01:04:23.530 And this is the best you can do with this kind of approach. 01:04:23.530 --> 01:04:25.518 Yes? 01:04:25.518 --> 01:04:27.890 AUDIENCE: [INAUDIBLE] 01:04:27.890 --> 01:04:31.290 PROFESSOR: Yeah, I simply define this, 01:04:31.290 --> 01:04:35.180 which is a time that has some meaning if you know what 01:04:35.180 --> 01:04:38.330 the uncertainty of the operator is and how fast it's changing-- 01:04:38.330 --> 01:04:41.310 is the time needed for a change. 01:04:41.310 --> 01:04:47.160 Once I defined this, I simply brought this factor down here, 01:04:47.160 --> 01:04:51.210 so that delta Q over this derivative is delta t, 01:04:51.210 --> 01:04:55.110 and the equation just became this equation. 01:05:02.410 --> 01:05:07.120 So we'll try to figure out a little more of what this means 01:05:07.120 --> 01:05:12.390 right away, but you can make a few criticisms 01:05:12.390 --> 01:05:14.510 about this thing. 01:05:14.510 --> 01:05:17.380 You can say, look, this delta time 01:05:17.380 --> 01:05:19.290 uncertainty is not universal. 01:05:19.290 --> 01:05:23.090 It depends which operator Q you took. 01:05:23.090 --> 01:05:23.750 True enough. 01:05:26.440 --> 01:05:30.480 I cannot prove that it's independent of the operator Q, 01:05:30.480 --> 01:05:34.440 and many times I cannot even tell you which operator Q is 01:05:34.440 --> 01:05:37.880 the best operator to think about. 01:05:37.880 --> 01:05:39.570 But you can try. 01:05:39.570 --> 01:05:42.600 And it does give you-- first, it's 01:05:42.600 --> 01:05:47.660 a mathematical statement about how fast things can change. 01:05:47.660 --> 01:05:54.080 And that contains physics, and it contains a very precise fact 01:05:54.080 --> 01:05:54.580 as well. 01:05:57.490 --> 01:06:01.570 Actually, there's a version of the uncertainty principle 01:06:01.570 --> 01:06:06.580 that you will explore in the homework that is, maybe, 01:06:06.580 --> 01:06:12.130 an alternative picture of this, and asks the following thing-- 01:06:12.130 --> 01:06:15.630 if you have a state and a stationary state, 01:06:15.630 --> 01:06:18.160 nothing changes in the state. 01:06:18.160 --> 01:06:21.880 But if it's a stationary state, the energy uncertainty 01:06:21.880 --> 01:06:26.070 is 0, because the energy is an eigenstate of the energy. 01:06:26.070 --> 01:06:27.750 So nothing changes. 01:06:27.750 --> 01:06:30.780 So you have to wait infinite time for there to be a change, 01:06:30.780 --> 01:06:34.090 and this makes sense. 01:06:34.090 --> 01:06:36.860 Now you can ask the following question-- suppose 01:06:36.860 --> 01:06:43.220 I have a state that is not an eigenstate of energy. 01:06:43.220 --> 01:06:45.330 So therefore, for example, the simplest thing 01:06:45.330 --> 01:06:47.624 would be a superposition of two eigenstates 01:06:47.624 --> 01:06:48.540 of different energies. 01:06:51.100 --> 01:06:53.980 You can ask, well, there will be time evolution 01:06:53.980 --> 01:06:57.710 and this state will change in time. 01:06:57.710 --> 01:07:05.180 So how can I get a constraint on changes? 01:07:05.180 --> 01:07:06.870 How can I approach changes? 01:07:06.870 --> 01:07:11.050 And people discovered the following interesting fact-- 01:07:11.050 --> 01:07:18.120 that if you have a state, it has unit norm, and if it evolves, 01:07:18.120 --> 01:07:20.530 it may happen that at some stage, 01:07:20.530 --> 01:07:24.320 it becomes orthogonal to itself-- to the original one. 01:07:24.320 --> 01:07:26.150 And that is a big change. 01:07:26.150 --> 01:07:29.000 You become orthogonal to what you used to be. 01:07:29.000 --> 01:07:32.400 That's as big a change as can happen. 01:07:32.400 --> 01:07:37.350 And then you can ask, is there a minimum time 01:07:37.350 --> 01:07:40.720 for which this can happen? 01:07:40.720 --> 01:07:42.830 What is the minimum time in which 01:07:42.830 --> 01:07:46.410 a state can change so much that it 01:07:46.410 --> 01:07:49.090 becomes orthogonal to itself? 01:07:49.090 --> 01:07:51.860 And there is such an uncertainty principle. 01:07:51.860 --> 01:07:54.690 It's derived a little differently from that. 01:07:54.690 --> 01:08:03.120 And it says that if you take delta t to be the time 01:08:03.120 --> 01:08:23.760 it takes psi of x and t to become orthogonal to psi of x0, 01:08:23.760 --> 01:08:28.638 then this delta t times delta E-- 01:08:28.638 --> 01:08:32.630 the uncertainty of the energies is the uncertainty in h-- 01:08:32.630 --> 01:08:37.494 is greater than or equal to h bar over 4. 01:08:46.660 --> 01:08:50.040 Now a state may never become orthogonal to itself, 01:08:50.040 --> 01:08:51.120 but that's OK. 01:08:51.120 --> 01:08:54.970 Then it's a big number on the left-hand side. 01:08:54.970 --> 01:08:58.029 But the quickest it can do it is that. 01:08:58.029 --> 01:09:00.540 And that's an interesting thing. 01:09:00.540 --> 01:09:02.650 And it's a version of the uncertainty principle. 01:09:05.960 --> 01:09:08.270 I want to make a couple more remarks, 01:09:08.270 --> 01:09:10.640 because this thing is mysterious enough 01:09:10.640 --> 01:09:15.520 that it requires thinking. 01:09:15.520 --> 01:09:21.970 So let's make some precise claims about energy 01:09:21.970 --> 01:09:26.550 uncertainties and then give an example of what's 01:09:26.550 --> 01:09:29.439 happening in the physical situation. 01:09:29.439 --> 01:09:30.720 Was there a question? 01:09:30.720 --> 01:09:33.095 Yes. 01:09:33.095 --> 01:09:33.970 AUDIENCE: [INAUDIBLE] 01:09:33.970 --> 01:09:36.303 PROFESSOR: You're going to explore that in the homework. 01:09:36.303 --> 01:09:39.752 Actually, I don't think you're going to show it, but-- 01:09:39.752 --> 01:09:41.642 AUDIENCE: [INAUDIBLE] H bar [INAUDIBLE] 01:09:41.642 --> 01:09:43.600 it's even less than the uncertainty [INAUDIBLE] 01:09:49.776 --> 01:09:51.359 PROFESSOR: It's a different statement. 01:09:51.359 --> 01:09:56.110 It's a very precise way of measuring, creating a time. 01:09:56.110 --> 01:09:58.480 It's a precise definition of time, 01:09:58.480 --> 01:10:02.054 and therefore, there's no reason why 01:10:02.054 --> 01:10:03.220 it would have been the same. 01:10:07.280 --> 01:10:12.120 So here is a statement that is interesting-- is 01:10:12.120 --> 01:10:27.740 that the uncertainty delta E in an isolated system 01:10:27.740 --> 01:10:33.680 is constant-- doesn't change. 01:10:36.370 --> 01:10:39.770 And by an isolated system, a system 01:10:39.770 --> 01:10:42.040 in which there's no influences on it, 01:10:42.040 --> 01:10:47.900 a system in which you have actually time independent 01:10:47.900 --> 01:10:48.920 Hamiltonians. 01:10:48.920 --> 01:10:56.030 So H is a time independent Hamiltonian. 01:11:01.000 --> 01:11:04.370 Now that, of course, doesn't mean the physics is boring. 01:11:04.370 --> 01:11:07.090 Time- independent Hamiltonians are quite interesting, 01:11:07.090 --> 01:11:09.110 but you have a whole system. 01:11:09.110 --> 01:11:11.250 Let's take it to be isolated. 01:11:11.250 --> 01:11:14.050 There's no time dependent things acting on it, 01:11:14.050 --> 01:11:18.880 and H should be a time independent Hamiltonian. 01:11:18.880 --> 01:11:28.180 So I want to use this statement to say 01:11:28.180 --> 01:11:32.170 the following-- if I take Q equals 01:11:32.170 --> 01:11:36.020 H in that theorem over there, I get 01:11:36.020 --> 01:11:42.970 that d dt of the expectation value of H would be what? 01:11:42.970 --> 01:11:45.480 It would be i over H bar. 01:11:45.480 --> 01:11:48.420 Since H is time independent-- the condition 01:11:48.420 --> 01:11:50.700 here was that Q had no time dependence. 01:11:50.700 --> 01:11:54.640 But then I get H commutator with H. 01:11:54.640 --> 01:12:01.980 So I get here H commutator with H. And that commutator is 0. 01:12:01.980 --> 01:12:07.230 However complicated an operator is, it commutes with itself. 01:12:07.230 --> 01:12:13.150 So the expectation value of the energy doesn't change. 01:12:13.150 --> 01:12:15.920 We call that energy conservation. 01:12:15.920 --> 01:12:21.610 But still, if you take Q now equal to H squared, 01:12:21.610 --> 01:12:25.790 the time derivative of the expectation value of H 01:12:25.790 --> 01:12:29.760 squared, you get i over H bar. 01:12:29.760 --> 01:12:33.030 You're supposed to be H commutator with Q, 01:12:33.030 --> 01:12:35.370 which is H squared, now. 01:12:35.370 --> 01:12:36.555 And that's also 0. 01:12:40.290 --> 01:12:46.550 So no power of the expectation value of H vanishes. 01:12:46.550 --> 01:12:52.310 And therefore, we have that the time derivative 01:12:52.310 --> 01:12:56.580 of the uncertainty of H squared-- 01:12:56.580 --> 01:13:01.300 which is the time derivative of the expectation value of H 01:13:01.300 --> 01:13:06.270 squared minus the expectation value of H squared-- well, 01:13:06.270 --> 01:13:10.490 we've shown each one of the things on the right-hand side 01:13:10.490 --> 01:13:14.670 are 0, so this is 0. 01:13:14.670 --> 01:13:18.070 So delta H is constant. 01:13:25.650 --> 01:13:32.910 So the uncertainty-- delta E or delta H of the system 01:13:32.910 --> 01:13:35.340 is constant. 01:13:35.340 --> 01:13:38.310 So what do we do with that? 01:13:38.310 --> 01:13:44.480 Well it helps us think a little about time dependent processes. 01:13:44.480 --> 01:13:48.420 And the example we must have in mind 01:13:48.420 --> 01:13:52.220 is perhaps the one of a decay that 01:13:52.220 --> 01:13:57.010 leads to a radiation of a photon, so a transition 01:13:57.010 --> 01:14:00.140 that leads to a photon radiation. 01:14:00.140 --> 01:14:03.096 So let's consider that example. 01:14:08.900 --> 01:14:12.460 So we have an atom in some excited state, 01:14:12.460 --> 01:14:16.020 decays to the ground state and shoots out the photon. 01:14:28.340 --> 01:14:41.320 Then it's an unstable state, because if it would be stable, 01:14:41.320 --> 01:14:44.190 it wouldn't change in time. 01:14:44.190 --> 01:14:47.300 And the excited state of an atom is an unstable state, 01:14:47.300 --> 01:14:49.505 decays into-- goes into the ground state. 01:14:55.590 --> 01:14:56.680 And it makes a photon. 01:15:02.090 --> 01:15:07.190 Now this idea of the conservation 01:15:07.190 --> 01:15:12.980 of energy uncertainty at least helps you in this situation 01:15:12.980 --> 01:15:16.610 that you would typically do it with a lot of hand-waving, 01:15:16.610 --> 01:15:18.740 organize your thoughts. 01:15:18.740 --> 01:15:23.230 So what happens in such decay? 01:15:23.230 --> 01:15:34.090 There's a lifetime, which is a typical time you 01:15:34.090 --> 01:15:38.480 have to wait for that excited state to decay. 01:15:38.480 --> 01:15:41.590 And these lifetime is called tau. 01:15:41.590 --> 01:15:45.830 And certainly as the lifetime goes through, 01:15:45.830 --> 01:15:51.250 and the decay happens, some observable changes a lot. 01:15:51.250 --> 01:15:53.950 Some observable Q must change a lot. 01:15:53.950 --> 01:15:57.940 Maybe a position of the electron in an orbit, 01:15:57.940 --> 01:16:02.480 or the angular momentum of it, or some squared 01:16:02.480 --> 01:16:05.330 of the momentum-- some observable 01:16:05.330 --> 01:16:10.030 that we could do an atomic calculation in more detail 01:16:10.030 --> 01:16:11.460 must change a lot. 01:16:11.460 --> 01:16:15.280 So there will be associated with some observable that 01:16:15.280 --> 01:16:18.680 changes a lot during the lifetime, 01:16:18.680 --> 01:16:23.140 because it takes that long for this thing to change. 01:16:23.140 --> 01:16:26.150 There will be an energy uncertainty 01:16:26.150 --> 01:16:28.130 associated to a lifetime. 01:16:28.130 --> 01:16:32.340 So how does the energy uncertainty reflect itself? 01:16:32.340 --> 01:16:33.975 Well, you have a ground state. 01:16:38.830 --> 01:16:40.830 And you have this excited state. 01:16:40.830 --> 01:16:44.650 But generally, when you have an excited state 01:16:44.650 --> 01:16:48.320 due to some interactions that produce instability, 01:16:48.320 --> 01:16:52.380 you actually have a lot of states here 01:16:52.380 --> 01:16:55.040 that are part of the excited state. 01:16:55.040 --> 01:17:00.370 So you have an excited state, but you do have, 01:17:00.370 --> 01:17:03.290 typically, a lot of uncertainty-- but not 01:17:03.290 --> 01:17:06.380 a lot-- some uncertainty of the energy here. 01:17:06.380 --> 01:17:09.140 The state is not a particular one. 01:17:09.140 --> 01:17:10.865 If it would be a particular one, it 01:17:10.865 --> 01:17:14.790 would be a stationary state-- would stay there forever. 01:17:14.790 --> 01:17:18.280 Nevertheless, it's a combination of some things, 01:17:18.280 --> 01:17:20.630 so it's not quite a stationary state. 01:17:20.630 --> 01:17:22.440 It couldn't be a stationary state, 01:17:22.440 --> 01:17:24.160 because it would be eternal. 01:17:24.160 --> 01:17:27.580 So somehow, the dynamics of this atom 01:17:27.580 --> 01:17:31.625 must be such that there's interactions between, say, 01:17:31.625 --> 01:17:36.850 the electron and the nucleus, or possibly a radiation field that 01:17:36.850 --> 01:17:43.660 makes the state of this electron unstable, 01:17:43.660 --> 01:17:47.680 and associated to it an uncertainty in the energy. 01:17:47.680 --> 01:17:52.510 So there's an uncertainty here, and this particle-- 01:17:52.510 --> 01:17:56.040 this electron goes eventually to the ground state, 01:17:56.040 --> 01:17:57.270 and it meets a photon. 01:18:03.130 --> 01:18:09.670 So there is, associated to this lifetime, an uncertainty delta 01:18:09.670 --> 01:18:18.390 E times tau, and I will put similar to H bar over 2. 01:18:18.390 --> 01:18:24.060 And this would be the delta E here, 01:18:24.060 --> 01:18:26.430 because your state must be a superposition 01:18:26.430 --> 01:18:28.880 of some states over there. 01:18:28.880 --> 01:18:31.740 And then what happens later? 01:18:31.740 --> 01:18:34.360 Well, this particle goes to the ground state-- 01:18:34.360 --> 01:18:40.060 no uncertainty any more about what its energy is. 01:18:40.060 --> 01:18:42.670 So the only possibility at this moment 01:18:42.670 --> 01:18:47.320 consistent with the conservation of uncertainty in the system 01:18:47.320 --> 01:18:49.780 is that the photon carries the uncertainty. 01:18:49.780 --> 01:18:54.120 So that photon must have an uncertainty as well. 01:18:54.120 --> 01:19:00.020 So delta energy of the photon will 01:19:00.020 --> 01:19:09.470 be equal to h bar delta omega, or h delta nu. 01:19:17.220 --> 01:19:23.230 So the end result is that in a physical decay process, 01:19:23.230 --> 01:19:24.230 there are uncertainties. 01:19:24.230 --> 01:19:27.700 And the uncertainty gets carried out, 01:19:27.700 --> 01:19:30.970 and it's always there-- the delta E here 01:19:30.970 --> 01:19:34.170 and the photon having some uncertainty. 01:19:34.170 --> 01:19:38.820 Now one of the most famous applications of this thing 01:19:38.820 --> 01:19:42.570 is related to the hyperfine transition of hydrogen. 01:19:42.570 --> 01:19:45.750 And we're very lucky in physics. 01:19:45.750 --> 01:19:47.210 Physicists are very lucky. 01:19:47.210 --> 01:19:52.220 This is a great break for astronomy and cosmology, 01:19:52.220 --> 01:19:56.370 and it's all based on this uncertainty principle. 01:19:56.370 --> 01:20:06.440 You have the hyperfine transition of hydrogen. 01:20:09.510 --> 01:20:12.960 So we will study later in this course 01:20:12.960 --> 01:20:18.430 that because of the proton and electron 01:20:18.430 --> 01:20:21.300 spins in the hydrogen atom, there's 01:20:21.300 --> 01:20:24.330 a splitting of energies having to do 01:20:24.330 --> 01:20:26.280 with the hyperfine interaction. 01:20:26.280 --> 01:20:29.120 It's a magnetic dipole interaction 01:20:29.120 --> 01:20:31.560 between the proton and the electron. 01:20:31.560 --> 01:20:34.380 And there's going to be a splitting. 01:20:34.380 --> 01:20:38.510 And there's a transition associated with this splitting. 01:20:38.510 --> 01:20:42.160 So there's a hyperfine splitting-- the ground state 01:20:42.160 --> 01:20:45.270 of the hyperfine splitting of some states. 01:20:45.270 --> 01:20:49.360 And it's the top state and the bottom state. 01:20:49.360 --> 01:20:54.940 And as the system decays, it emits a photon. 01:20:54.940 --> 01:21:03.220 This photon is approximately a 21 centimeter wavelength-- 01:21:03.220 --> 01:21:07.140 is the famous 21 centimeter line of hydrogen. 01:21:07.140 --> 01:21:15.740 And it corresponds to about 1420 megahertz. 01:21:15.740 --> 01:21:20.150 So how about so far so good. 01:21:20.150 --> 01:21:23.960 There's an energy splitting here, 01:21:23.960 --> 01:21:30.670 21 centimeters wavelength, 5.9 times 10 to the minus 6 01:21:30.670 --> 01:21:34.110 eV in here. 01:21:34.110 --> 01:21:37.150 But that's not the energy difference 01:21:37.150 --> 01:21:39.130 that matters for the uncertainty, 01:21:39.130 --> 01:21:41.580 just like this is not the energy difference that 01:21:41.580 --> 01:21:43.250 matters for the uncertainty. 01:21:43.250 --> 01:21:47.370 What matters for the uncertainty is how broad this state 01:21:47.370 --> 01:21:51.630 is, due to interactions that will produce the decay. 01:21:51.630 --> 01:21:55.120 It's a very funny, magnetic transition. 01:21:55.120 --> 01:21:58.630 And how long is the lifetime of this state? 01:21:58.630 --> 01:22:02.018 Anybody know? 01:22:02.018 --> 01:22:09.410 A second, a millisecond, a day? 01:22:09.410 --> 01:22:09.910 Nobody? 01:22:14.580 --> 01:22:26.230 Ten million years-- a long time-- 10 million years-- 01:22:26.230 --> 01:22:27.930 lifetime tau. 01:22:27.930 --> 01:22:33.510 A year is about pi times 10 to the 7 seconds 01:22:33.510 --> 01:22:34.370 is pretty accurate. 01:22:38.410 --> 01:22:42.150 Anyway, 10 million years is a lot of time. 01:22:42.150 --> 01:22:45.410 It's such a large time that it corresponds 01:22:45.410 --> 01:22:50.540 to an energy uncertainty that is so extraordinarily 01:22:50.540 --> 01:22:54.930 small, that the wavelength uncertainty, or the frequency 01:22:54.930 --> 01:23:00.880 uncertainty, is so small that corresponding to this 1420, 01:23:00.880 --> 01:23:04.770 it's I think, the uncertainty in lambda-- and lambda 01:23:04.770 --> 01:23:09.250 is of the order of 10 to the minus 8. 01:23:09.250 --> 01:23:14.190 The line is extremely sharp, so it's not a fussy line 01:23:14.190 --> 01:23:16.050 that it's hard to measure. 01:23:16.050 --> 01:23:19.280 It's the sharpest possible line. 01:23:19.280 --> 01:23:24.260 And it's so sharp because of this 10 million years lifetime, 01:23:24.260 --> 01:23:27.360 and the energy time uncertainty relationship. 01:23:27.360 --> 01:23:29.590 That's it for today.