WEBVTT 00:00:00.060 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.820 under a Creative Commons license. 00:00:03.820 --> 00:00:06.540 Your support will help MIT OpenCourseWare continue 00:00:06.540 --> 00:00:10.130 to offer high quality educational resources for free. 00:00:10.130 --> 00:00:12.700 To make a donation, or to view additional materials 00:00:12.700 --> 00:00:16.580 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.580 --> 00:00:17.210 at ocw.mit.edu. 00:00:22.560 --> 00:00:24.960 PROFESSOR: It's good to be back. 00:00:24.960 --> 00:00:28.930 I really want to thank both Aaron and Will who 00:00:28.930 --> 00:00:34.890 took my teaching duties over last week. 00:00:34.890 --> 00:00:39.340 You've been receiving updates of the lecture notes, 00:00:39.340 --> 00:00:42.700 and, in particular, as I don't want 00:00:42.700 --> 00:00:45.980 to go back over some things, I would 00:00:45.980 --> 00:00:50.980 like you to read some of the material you have there. 00:00:50.980 --> 00:00:54.500 In particular, the part on projectors 00:00:54.500 --> 00:00:57.460 has been developed further. 00:00:57.460 --> 00:01:02.670 We will meet projectors a lot in this space, 00:01:02.670 --> 00:01:07.420 because in quantum mechanics, whenever you do a measurement, 00:01:07.420 --> 00:01:11.392 the effect of a measurement is to act 00:01:11.392 --> 00:01:14.260 on a stage with a projector. 00:01:14.260 --> 00:01:17.530 So projectors are absolutely important. 00:01:17.530 --> 00:01:19.920 And orthogonal projectors are the ones 00:01:19.920 --> 00:01:21.990 that we're going to use-- are the ones that 00:01:21.990 --> 00:01:25.850 are relevant in quantum mechanics. 00:01:25.850 --> 00:01:28.470 There's a property, for example, of projectors 00:01:28.470 --> 00:01:33.250 that is quite neat that is used in maximization and fitting 00:01:33.250 --> 00:01:33.990 problems. 00:01:33.990 --> 00:01:37.800 And you will see that in the PSET. 00:01:37.800 --> 00:01:39.820 In the PSET, the last problem has 00:01:39.820 --> 00:01:44.330 to do with using a projector to find 00:01:44.330 --> 00:01:48.750 best approximations to some functions using polynomials. 00:01:48.750 --> 00:01:52.010 So there's lots of things to say about projectors, 00:01:52.010 --> 00:01:57.870 and we'll find them along when we go and do later stuff 00:01:57.870 --> 00:01:58.490 in the course. 00:01:58.490 --> 00:02:03.820 So please read that part on projectors. 00:02:03.820 --> 00:02:08.490 The other thing is that much of what we're going to do 00:02:08.490 --> 00:02:15.740 uses the notation that we have for describing inner products-- 00:02:15.740 --> 00:02:22.370 for example, u, v. And then, as we've mentioned, 00:02:22.370 --> 00:02:27.250 and this is in the notes-- this, in the bracket notation, 00:02:27.250 --> 00:02:29.920 becomes something like this. 00:02:29.920 --> 00:02:32.800 And the bracket notation of quantum mechanics 00:02:32.800 --> 00:02:36.610 is fairly nice for many things, and it's 00:02:36.610 --> 00:02:40.560 used sometimes for some applications. 00:02:40.560 --> 00:02:44.150 Everybody uses the bracket notation for some applications. 00:02:44.150 --> 00:02:47.910 I hope to get to one of those today. 00:02:47.910 --> 00:02:53.660 So much of the theory we've developed 00:02:53.660 --> 00:02:58.120 is done with this as the inner product. 00:02:58.120 --> 00:03:02.260 Nevertheless, the translation to the language of bras and kets 00:03:02.260 --> 00:03:04.190 is very quick. 00:03:04.190 --> 00:03:06.830 So the way the notes are going to be structured-- 00:03:06.830 --> 00:03:09.160 and we're still working on the notes, 00:03:09.160 --> 00:03:10.720 and they're going to chang a bit-- 00:03:10.720 --> 00:03:13.670 is that everything regarding the math 00:03:13.670 --> 00:03:17.070 is being developed more in this notation, 00:03:17.070 --> 00:03:19.480 but then we turn into bras and kets 00:03:19.480 --> 00:03:22.490 and just go quickly over all you've 00:03:22.490 --> 00:03:26.130 seen, just how it looks with bras and kets 00:03:26.130 --> 00:03:28.970 so that you're familiar. 00:03:28.970 --> 00:03:31.080 Then, in the later part of the course, 00:03:31.080 --> 00:03:34.820 we'll use sometimes bras and kets, and sometimes this. 00:03:34.820 --> 00:03:37.220 And sometimes some physicisists use 00:03:37.220 --> 00:03:39.223 this notation with parentheses. 00:03:42.110 --> 00:03:44.560 So for example, Weinberg's recent book 00:03:44.560 --> 00:03:46.735 on quantum mechanics uses this notation. 00:03:46.735 --> 00:03:49.905 It doesn't use bras and kets I think at all. 00:03:53.570 --> 00:03:56.890 So you have to be ready to work with any notation. 00:03:56.890 --> 00:04:01.290 The bra and ket notation has some nice properties 00:04:01.290 --> 00:04:05.140 that make it very fast to do things with it. 00:04:05.140 --> 00:04:06.740 It is very efficient. 00:04:06.740 --> 00:04:12.410 Nevertheless, in some ways this notation is a little clearer. 00:04:12.410 --> 00:04:17.160 So many of the things we'll develop is with this notation. 00:04:17.160 --> 00:04:19.350 So today I'm going to develop the idea 00:04:19.350 --> 00:04:22.190 of the Hermitian conjugator for an operator, 00:04:22.190 --> 00:04:24.440 or the adjoint of an operator. 00:04:24.440 --> 00:04:31.630 And this idea is generally a little subtle, a little hard 00:04:31.630 --> 00:04:33.140 to understand. 00:04:33.140 --> 00:04:38.200 But we'll just go at it slowly and try to make it very clear. 00:04:38.200 --> 00:04:44.100 So adjoints or Hermitian operators, or Hermitian 00:04:44.100 --> 00:04:54.260 conjugates-- adjoints or Hermitian conjugates. 00:04:59.680 --> 00:05:06.250 So the idea of Adjoints, or Hermition conjugates, 00:05:06.250 --> 00:05:14.220 really begins with some necessary background 00:05:14.220 --> 00:05:18.640 on what they're called-- linear functionals. 00:05:18.640 --> 00:05:20.800 It sounds complicated, but it's not. 00:05:20.800 --> 00:05:23.270 What is a linear functional? 00:05:23.270 --> 00:05:33.440 A linear functional on V-- on a vector 00:05:33.440 --> 00:05:44.250 space V-- is a linear map from V to the numbers 00:05:44.250 --> 00:05:48.870 F. We've always been calling F the numbers. 00:05:48.870 --> 00:05:53.060 So it's just that, something that, once you have a vector, 00:05:53.060 --> 00:05:55.120 you get a number and it's linear. 00:05:58.120 --> 00:06:02.900 So a linear function of Phi, if it's a linear functional, 00:06:02.900 --> 00:06:11.880 Phi on v belongs to F. Phi acts on a vector, v, 00:06:11.880 --> 00:06:15.720 that belongs to the vector space and gives you a number. 00:06:15.720 --> 00:06:25.710 So "linear" means Phi of v1 plus v2 is Phi of v1 plus Phi of v2. 00:06:25.710 --> 00:06:33.950 And Phi of av, for a is number, is a Phi of v. 00:06:33.950 --> 00:06:39.120 So seems simple, and indeed it is. 00:06:39.120 --> 00:06:43.430 And we can construct examples of linear functionals, 00:06:43.430 --> 00:06:45.515 some trivial ones, for example. 00:06:48.350 --> 00:06:58.030 Let Phi be a map that takes the vector space, reals in three 00:06:58.030 --> 00:07:00.840 dimensions, to the real numbers. 00:07:00.840 --> 00:07:02.950 So how does it act? 00:07:02.950 --> 00:07:09.000 Phi acts on a vector, which is x1, x2, and x3-- three 00:07:09.000 --> 00:07:10.190 components. 00:07:10.190 --> 00:07:13.680 And it must give a numbers, so it 00:07:13.680 --> 00:07:21.170 could be 3x1 minus x2 plus 7x3, as simple as that. 00:07:24.850 --> 00:07:26.480 It's linear. 00:07:26.480 --> 00:07:31.560 x1, x2, and x3 are the coordinates of a single vector. 00:07:31.560 --> 00:07:34.780 And whenever you have this vector, that is, 00:07:34.780 --> 00:07:37.530 this triplet-- now, I could have written it 00:07:37.530 --> 00:07:42.445 like this-- Phi of x1, x2, and x3, as a vector. 00:07:42.445 --> 00:07:43.770 It looks like that. 00:07:43.770 --> 00:07:47.600 But it's easier to use horizontal notation, 00:07:47.600 --> 00:07:51.785 so we'll write it like that. 00:07:51.785 --> 00:07:55.060 And, if you have an inner product 00:07:55.060 --> 00:08:01.170 on this space-- on this three dimensional vector space-- 00:08:01.170 --> 00:08:03.570 there's something you can say. 00:08:03.570 --> 00:08:07.810 Actually this Phi is equal-- and this we 00:08:07.810 --> 00:08:15.900 call the vector V-- is actually equal to u, 00:08:15.900 --> 00:08:22.020 inner product with v, where u is the vector that has components 00:08:22.020 --> 00:08:26.860 3, minus 1, and 7, because if you 00:08:26.860 --> 00:08:30.810 take the inner product of this vector with this vector, 00:08:30.810 --> 00:08:34.659 in three dimensions real vector spaces-- 00:08:34.659 --> 00:08:37.380 inner product is a dot product. 00:08:37.380 --> 00:08:42.970 And then we make the dot product of u with the vector V. 00:08:42.970 --> 00:08:46.270 Maybe I should have called it v1, v2, v3. 00:08:46.270 --> 00:08:53.360 I'll change that-- v1, v2, v3 here 00:08:53.360 --> 00:09:01.950 are components of the vector-- v1, v2, and v3, 00:09:01.950 --> 00:09:04.190 not to be confused with three vectors. 00:09:04.190 --> 00:09:10.890 This whole thing is a vector V. So this linear functional, 00:09:10.890 --> 00:09:14.650 that, given a vector gives me a number. 00:09:14.650 --> 00:09:18.050 The clever thing is that the inner product 00:09:18.050 --> 00:09:20.730 is this thing that gives you numbers out of vectors. 00:09:20.730 --> 00:09:25.100 So you've reconstructed this linear functional 00:09:25.100 --> 00:09:30.120 as the inner product of some vector with the vector you're 00:09:30.120 --> 00:09:33.490 acting on, so, where u is given by that. 00:09:39.440 --> 00:09:42.770 The most important result about linear functionals 00:09:42.770 --> 00:09:45.610 is that this is not an accident. 00:09:45.610 --> 00:09:48.190 This kind be that very generally. 00:09:48.190 --> 00:09:51.960 So any time you give me a linear functional, 00:09:51.960 --> 00:09:57.930 I can find a vector that, using the inner product, 00:09:57.930 --> 00:10:01.620 acts on the vector you're acting on the same way 00:10:01.620 --> 00:10:05.480 as the linear function of thus. 00:10:05.480 --> 00:10:08.540 The most general linear functional 00:10:08.540 --> 00:10:13.320 is just some most general vector acting this way. 00:10:13.320 --> 00:10:16.800 So let's state that and prove it. 00:10:16.800 --> 00:10:20.190 So this is a theorem, it's not a definition or anything 00:10:20.190 --> 00:10:22.370 like that. 00:10:22.370 --> 00:10:34.820 Let Phi be a linear functional on v. Then 00:10:34.820 --> 00:10:48.800 there is a unique vector u belonging to the vector space 00:10:48.800 --> 00:11:10.370 such that Phi acting on v is equal to u, v. 00:11:10.370 --> 00:11:13.360 Since this is such a canonical thing, 00:11:13.360 --> 00:11:15.490 you could even invent a notation. 00:11:15.490 --> 00:11:20.760 Call this the linear functional created by u, 00:11:20.760 --> 00:11:25.250 acting on v. Everybody doesn't use this, 00:11:25.250 --> 00:11:27.860 but you could call it like that. 00:11:27.860 --> 00:11:30.870 This is a linear functional acting on v, 00:11:30.870 --> 00:11:34.180 but it's labeled by u, which is the vector that you've 00:11:34.180 --> 00:11:34.900 use there. 00:11:38.945 --> 00:11:43.210 This is important enough that we better understand why it works. 00:11:43.210 --> 00:11:44.930 So I'll prove it. 00:11:52.370 --> 00:11:56.070 We're going to use an orthonormal basis, 00:11:56.070 --> 00:12:07.582 say e1 up to en is an orthonormal, O-N, basis. 00:12:07.582 --> 00:12:09.040 AUDIENCE: That means we're assuming 00:12:09.040 --> 00:12:10.532 v is finite dimensional here? 00:12:10.532 --> 00:12:11.240 PROFESSOR: Sorry? 00:12:11.240 --> 00:12:13.656 AUDIENCE: We're assuming V is finite dimensional, correct? 00:12:16.459 --> 00:12:18.125 PROFESSOR: Yeah, it's finite dimensional 00:12:18.125 --> 00:12:22.776 I'm going to prove it using a finite basis like that. 00:12:22.776 --> 00:12:24.790 Is true finite dimensional? 00:12:24.790 --> 00:12:27.050 I presume yes. 00:12:27.050 --> 00:12:28.529 AUDIENCE: If it's not [INAUDIBLE]. 00:12:33.813 --> 00:12:34.938 PROFESSOR: What hypothesis? 00:12:34.938 --> 00:12:37.403 AUDIENCE: You say continuous when you're talking 00:12:37.403 --> 00:12:38.890 [INAUDIBLE]. 00:12:38.890 --> 00:12:41.745 PROFESSOR: OK, I'll check. 00:12:41.745 --> 00:12:46.280 But let's just prove this one finite dimensional like this. 00:12:50.990 --> 00:12:51.980 Let's take that. 00:12:51.980 --> 00:12:57.800 And now write the vector as a superposition of these vectors. 00:12:57.800 --> 00:12:59.900 Now we know how to do that. 00:12:59.900 --> 00:13:07.560 We just have the components of v along each basis vector. 00:13:07.560 --> 00:13:11.700 For example, the component of v along e1 00:13:11.700 --> 00:13:15.560 is precisely e1, v. So then you go on 00:13:15.560 --> 00:13:22.350 like that until you go en, v, en. 00:13:28.420 --> 00:13:31.150 I think you've derived this a couple of times already, 00:13:31.150 --> 00:13:33.920 but this is a statement you can review, 00:13:33.920 --> 00:13:38.410 and let's take it to be correct. 00:13:38.410 --> 00:13:44.540 Now let's consider what is Phi acting on a v like that. 00:13:44.540 --> 00:13:48.960 Well, it's a linear map, so it takes on a sum of vectors 00:13:48.960 --> 00:13:51.630 by acting on the vectors, each one. 00:13:51.630 --> 00:13:53.970 So it should act on from this plus that, plus that, 00:13:53.970 --> 00:13:54.800 plus that. 00:13:54.800 --> 00:13:57.610 Now, it acts on this vector. 00:13:57.610 --> 00:13:59.200 Well, this is a number. 00:13:59.200 --> 00:14:00.840 The number goes out. 00:14:00.840 --> 00:14:02.100 It's a linear function. 00:14:02.100 --> 00:14:19.190 So this is e1, v, Phi of e1, all the way up to en, v Phi of en. 00:14:24.260 --> 00:14:30.320 Now this is a number, so let's bring it 00:14:30.320 --> 00:14:32.060 into the inner product. 00:14:32.060 --> 00:14:35.820 Now, if you brought it in on the side of V as a number 00:14:35.820 --> 00:14:37.850 it would go in just like the number. 00:14:37.850 --> 00:14:40.230 If you bring it into the left side, 00:14:40.230 --> 00:14:43.200 remember it's conjugate homogeneous, 00:14:43.200 --> 00:14:45.750 so this enters as a complex number. 00:14:45.750 --> 00:15:05.730 So this would be e1, Phi of e1 star times V plus en, Phi 00:15:05.730 --> 00:15:23.240 of en star, v. And then we have our result that this Phi of v 00:15:23.240 --> 00:15:25.050 has been written now. 00:15:25.050 --> 00:15:29.690 The left input is different on each of these terms, 00:15:29.690 --> 00:15:31.630 but the right input is the same. 00:15:31.630 --> 00:15:36.120 So at this moment linearity on the first input 00:15:36.120 --> 00:15:46.240 says that you can put here e1, Phi of e1 star plus up to en, 00:15:46.240 --> 00:15:57.950 Phi of en star, v. And this is the vector you were looking 00:15:57.950 --> 00:16:05.030 for, the vector U. Kind of simple, at the end of the day 00:16:05.030 --> 00:16:09.300 you just used the basis and made it clearer. 00:16:09.300 --> 00:16:11.180 It can always be constructed. 00:16:11.180 --> 00:16:21.590 Basically, the vector you want is e1 times Phi of u1 star 00:16:21.590 --> 00:16:24.270 plus en up to Phi of en star. 00:16:24.270 --> 00:16:27.900 So if you know what the linear map does to the basis vectors, 00:16:27.900 --> 00:16:31.640 you construct the vector this way. 00:16:31.640 --> 00:16:33.780 Vector is done. 00:16:33.780 --> 00:16:37.760 The only thing to be proven is that it's unique. 00:16:37.760 --> 00:16:43.210 Uniqueness is rather easy to prove at this stage. 00:16:43.210 --> 00:16:48.260 Suppose you know that u with v works and gives you 00:16:48.260 --> 00:16:49.240 the right answer. 00:16:49.240 --> 00:16:51.660 Well, you ask, is there a u prime 00:16:51.660 --> 00:16:56.570 that also gives the right answer for all v? 00:16:56.570 --> 00:16:58.970 Well, pass it to the other side, and you 00:16:58.970 --> 00:17:04.550 would have u minus u prime, would have zero inner product 00:17:04.550 --> 00:17:11.740 with v for all v. Pass to the other side, 00:17:11.740 --> 00:17:14.010 take the difference, and it's that. 00:17:14.010 --> 00:17:16.410 So u minus u prime is a vector that 00:17:16.410 --> 00:17:19.720 has zero inner product with any vector. 00:17:19.720 --> 00:17:23.319 And any such thing as always zero. 00:17:23.319 --> 00:17:27.109 And perhaps the easiest way to show that, in case you 00:17:27.109 --> 00:17:30.730 haven't seen that before, if x with v 00:17:30.730 --> 00:17:39.053 equals 0 for all for all v. What can you say about x? 00:17:41.590 --> 00:17:46.880 Well, take v is the value for any v. So take v equal x. 00:17:46.880 --> 00:17:50.050 So you take x, x is equal to 0. 00:17:50.050 --> 00:17:52.840 And by the axioms of the inner product, 00:17:52.840 --> 00:17:59.130 if a vector has 0 inner product with itself, it's 0. 00:17:59.130 --> 00:18:04.290 So at this stage, you go u minus u prime equals 0, 00:18:04.290 --> 00:18:06.470 and u is equal to u prime. 00:18:06.470 --> 00:18:10.070 So it's definitely unique, you can't 00:18:10.070 --> 00:18:11.755 find another one that works. 00:18:16.720 --> 00:18:20.250 So we have this thing. 00:18:20.250 --> 00:18:21.875 This theorem is proven. 00:18:25.900 --> 00:18:29.220 And now let's use to define this the adjoint, 00:18:29.220 --> 00:18:32.310 which is a very interesting thing. 00:18:32.310 --> 00:18:36.700 So the adjoing, or Hermitian conjugate, 00:18:36.700 --> 00:18:42.130 sometimes called adjoint-- physicists 00:18:42.130 --> 00:18:48.320 use the name Hermitian conjugate, which 00:18:48.320 --> 00:18:49.487 is more appropriate. 00:18:49.487 --> 00:18:51.320 Well, I don't know if it's more appropriate. 00:18:51.320 --> 00:18:57.260 It's more pictorial if you have a complex vector space. 00:18:57.260 --> 00:19:00.460 And if you're accustomed with linear algebra 00:19:00.460 --> 00:19:03.490 about Hermition matrices, and what they are, 00:19:03.490 --> 00:19:06.085 and that will show up a little later, 00:19:06.085 --> 00:19:10.610 although with a very curious twist. 00:19:10.610 --> 00:19:18.170 So given an operator T belonging to the set of linear operators 00:19:18.170 --> 00:19:26.950 on a vector space, you can define T dagger, 00:19:26.950 --> 00:19:33.730 also belonging to l of v. So this 00:19:33.730 --> 00:19:38.420 is the aim-- constructing an operator called the Hermitian 00:19:38.420 --> 00:19:40.816 conjugate. 00:19:40.816 --> 00:19:43.370 Now the way we're going to do it is 00:19:43.370 --> 00:19:47.940 going to be defining something that is a T star. 00:19:47.940 --> 00:19:51.130 Well, I said "T star" because mathematicians in fact 00:19:51.130 --> 00:19:52.810 call it star. 00:19:52.810 --> 00:19:55.320 And most mathematicians, they complex conjugate 00:19:55.320 --> 00:19:59.290 if a number is not z star but z bar. 00:19:59.290 --> 00:20:03.200 So that's why we call it T star and I 00:20:03.200 --> 00:20:06.060 may make this mistake a few times today. 00:20:06.060 --> 00:20:09.120 We're going to use dagger. 00:20:09.120 --> 00:20:12.150 And so I will make a definition that 00:20:12.150 --> 00:20:16.400 will tell you what T dagger is supposed to be, 00:20:16.400 --> 00:20:17.540 acting on things. 00:20:17.540 --> 00:20:20.860 But it might not be obvious, at least at first sight, 00:20:20.860 --> 00:20:23.070 that it's a linear operator. 00:20:23.070 --> 00:20:29.030 So let's see how does this go. 00:20:29.030 --> 00:20:30.390 Here is the claim. 00:20:30.390 --> 00:20:37.590 Consider the following thing-- u, 00:20:37.590 --> 00:20:51.380 T, v-- this inner product of u with T, v. And think of it 00:20:51.380 --> 00:20:53.510 as a linear functional. 00:20:53.510 --> 00:20:56.820 Well, it's certainly a linear functional 00:20:56.820 --> 00:21:05.380 of v. It's a linear functional because if you 00:21:05.380 --> 00:21:08.550 put a times v the a goes out. 00:21:08.550 --> 00:21:11.970 And if you put v1 plus v2 you get it's linear. 00:21:11.970 --> 00:21:15.150 So it's linear, but it's not the usual one's 00:21:15.150 --> 00:21:20.590 that we've been building, in which the linear functional 00:21:20.590 --> 00:21:26.700 looks like u with v. I just put an operator there. 00:21:26.700 --> 00:21:36.700 So by this theorem, there must be some vector 00:21:36.700 --> 00:21:40.210 that this can be represented as this acting 00:21:40.210 --> 00:21:46.180 with that vector inside here, because any linear operator is 00:21:46.180 --> 00:21:49.400 some vector acting on the vector-- on the vector 00:21:49.400 --> 00:21:54.720 v. Any linear functional, I'm sorry-- not linear operator. 00:21:54.720 --> 00:21:57.940 Any linear functional-- this is a linear functional. 00:21:57.940 --> 00:22:01.730 And every linear function can be written as some vector acting 00:22:01.730 --> 00:22:05.430 on v. So there must be a vector here. 00:22:05.430 --> 00:22:12.160 Now this vector surely will depend on what u is. 00:22:12.160 --> 00:22:13.760 So we'll give it a name. 00:22:19.080 --> 00:22:26.260 It's a vector that depends on U. I'll write it as T dagger u. 00:22:26.260 --> 00:22:37.360 At this moment, T dagger is just a map from v 00:22:37.360 --> 00:22:42.320 to v. We said that this thing that we must put here 00:22:42.320 --> 00:22:45.860 depends on u, and it must be a vector. 00:22:45.860 --> 00:22:49.310 So it's some thing that takes u and produces 00:22:49.310 --> 00:22:54.300 another vector called T dagger on u. 00:22:54.300 --> 00:22:56.440 But we don't know what T dagger is, 00:22:56.440 --> 00:22:58.225 and we don't even know that it's linear. 00:23:00.820 --> 00:23:05.530 So at this moment it's just a map, and it's a definition. 00:23:05.530 --> 00:23:13.020 This defines what T dagger u is, because some vector-- 00:23:13.020 --> 00:23:17.590 it could be calculated exactly the same way we calculated 00:23:17.590 --> 00:23:20.160 the other ones. 00:23:20.160 --> 00:23:27.270 So let's try to see why it is linear. 00:23:27.270 --> 00:23:38.280 Claim T dagger belongs to the linear operators in v. 00:23:38.280 --> 00:23:39.640 So how do we do that? 00:23:39.640 --> 00:23:41.120 Well, we can say the following. 00:23:41.120 --> 00:23:47.032 Consider u1 plus u1 acting on Tv. 00:23:49.960 --> 00:24:01.700 Well, by definition, this would be the T dagger of u1 plus u2, 00:24:01.700 --> 00:24:07.160 some function on u1 plus u2, because whatever is here 00:24:07.160 --> 00:24:12.570 gets acted by T dagger times v. On the other hand, 00:24:12.570 --> 00:24:20.970 this thing is equal to u1, Tv plus u2, 00:24:20.970 --> 00:24:39.200 Tv, which is equal to T dagger u1, v plus T dagger u2, v. 00:24:39.200 --> 00:24:48.700 And, by linearity, here you get equal to T dagger u1 plus T 00:24:48.700 --> 00:24:50.590 dagger on u2. 00:24:55.740 --> 00:24:59.680 And then comparing this too-- and this is true for arbitrary 00:24:59.680 --> 00:25:05.080 v-- you find that T dagger, acting on this sum of vectors, 00:25:05.080 --> 00:25:06.960 is the same as this thing. 00:25:10.930 --> 00:25:20.078 And similarly, how about au, Tv? 00:25:22.800 --> 00:25:41.060 Well, this is equal to T dagger on au, v. Now, T dagger on au, 00:25:41.060 --> 00:25:44.500 do you think the a goes out as a or as a bar? 00:25:47.830 --> 00:25:49.386 Sorry? 00:25:49.386 --> 00:25:51.110 a or a-bar? 00:25:51.110 --> 00:25:55.120 What do you think T dagger and au is supposed to be? 00:25:55.120 --> 00:26:01.730 a, because it's supposed to be a linear operator, so no dagger 00:26:01.730 --> 00:26:02.230 here. 00:26:02.230 --> 00:26:05.920 You see-- well, I didn't show it here. 00:26:05.920 --> 00:26:11.770 Any linear operator, T on av, is supposed to be a T of v. 00:26:11.770 --> 00:26:13.800 And we're saying T dagger is also 00:26:13.800 --> 00:26:17.360 a linear operator in the vector space. 00:26:17.360 --> 00:26:20.850 So this should be with an a. 00:26:20.850 --> 00:26:22.010 We'll see what we get. 00:26:22.010 --> 00:26:34.010 Well, the a can go out here, and it becomes a star u1, Tv, 00:26:34.010 --> 00:26:35.220 which is equal. 00:26:35.220 --> 00:26:37.250 I'm going through the left side. 00:26:37.250 --> 00:26:43.600 By definition, a bar T dagger of u, 00:26:43.600 --> 00:26:47.470 v. And now the constant can go in, 00:26:47.470 --> 00:26:54.330 and it goes back as a, T dagger u, v. So 00:26:54.330 --> 00:26:59.100 this must be equal to that, and you 00:26:59.100 --> 00:27:07.340 get what we're claiming here, which is T dagger on au, 00:27:07.340 --> 00:27:10.420 is equal to a T dagger of u. 00:27:13.010 --> 00:27:17.200 So the operator is linear. 00:27:17.200 --> 00:27:25.720 So we've defined something this way, and it's linear, 00:27:25.720 --> 00:27:30.030 and it's doing all the right things. 00:27:30.030 --> 00:27:33.470 Now, you really feel proud at this stage. 00:27:33.470 --> 00:27:36.470 This is still not all that intuitive. 00:27:36.470 --> 00:27:38.600 What does this all do? 00:27:38.600 --> 00:27:41.230 So we're going to do an example, and we're 00:27:41.230 --> 00:27:46.710 going to do one more property. 00:27:46.710 --> 00:27:49.660 Let me do one more property and then stop for a second. 00:27:49.660 --> 00:27:56.220 So here is one property-- ST dagger is supposed to be 00:27:56.220 --> 00:27:58.290 T dagger S dagger. 00:27:58.290 --> 00:28:00.210 So how do you get that? 00:28:00.210 --> 00:28:03.970 Not hard-- u, STv. 00:28:06.700 --> 00:28:12.300 Well, STv is really the same as S acting on Tv. 00:28:12.300 --> 00:28:15.330 Now the first S can be brought to the other side 00:28:15.330 --> 00:28:19.610 by the definition that you can bring something 00:28:19.610 --> 00:28:20.830 to the other side. 00:28:20.830 --> 00:28:22.220 Put in a dagger. 00:28:22.220 --> 00:28:24.550 So the S is brought there, and you 00:28:24.550 --> 00:28:33.060 get S dagger on u, T on v. And then the T can be brought here 00:28:33.060 --> 00:28:40.450 and act on this one, and you get T dagger S dagger u, 00:28:40.450 --> 00:28:48.670 v. So this thing is the dagger of this thing, 00:28:48.670 --> 00:28:50.230 and that's the statement here. 00:28:53.920 --> 00:28:57.540 There's yet one more simple property, 00:28:57.540 --> 00:29:09.320 that the dagger of S dagger is S. You take dagger twice 00:29:09.320 --> 00:29:11.550 and you're back to the same operator. 00:29:11.550 --> 00:29:13.570 Nothing has changed. 00:29:13.570 --> 00:29:17.062 So how do you do that? 00:29:17.062 --> 00:29:24.120 Take, for example, this-- take u, put S dagger here, and put 00:29:24.120 --> 00:29:33.740 v. Now, by definition, this is equal to-- you 00:29:33.740 --> 00:29:37.000 put the operator on the other side, adding a dagger. 00:29:37.000 --> 00:29:44.655 So that's why we put that one like this. 00:29:44.655 --> 00:29:47.540 The operator gets daggers, so now you've 00:29:47.540 --> 00:29:49.410 got the double dagger. 00:29:49.410 --> 00:29:52.540 So at this moment, however, you have 00:29:52.540 --> 00:29:57.170 to do something to simplify this. 00:29:57.170 --> 00:30:00.470 The easiest thing to do is probably the following-- 00:30:00.470 --> 00:30:02.760 to just flip these two, which you 00:30:02.760 --> 00:30:08.170 can do the order by putting a star. 00:30:08.170 --> 00:30:10.230 So this is equal. 00:30:10.230 --> 00:30:12.155 The left hand side is equal to this. 00:30:14.980 --> 00:30:22.406 And now this S dagger can be moved here and becomes an S. 00:30:22.406 --> 00:30:32.290 So this is u, Sv, and you still have the star. 00:30:32.290 --> 00:30:37.680 And now reverse this by eliminating the star, 00:30:37.680 --> 00:30:45.050 so you have S-- I'm sorry, I have 00:30:45.050 --> 00:30:46.640 this notation completely wrong. 00:30:46.640 --> 00:30:50.600 Sv-- this is u. 00:30:50.600 --> 00:30:52.960 The u's v's are easily confused. 00:30:52.960 --> 00:30:57.630 So this is v, and this is u. 00:31:00.340 --> 00:31:08.260 I move the S, and then finally I have Su, v without a star. 00:31:08.260 --> 00:31:12.240 I flipped it again. 00:31:12.240 --> 00:31:19.175 So then you compare these two, and you get the desired result. 00:31:23.560 --> 00:31:26.760 OK, so we've gone through this thing, which 00:31:26.760 --> 00:31:30.252 is the main result of daggers, and I 00:31:30.252 --> 00:31:31.960 would like to see if there are questions. 00:31:31.960 --> 00:31:36.730 Anything that has been unclear as we've gone along here? 00:31:36.730 --> 00:31:37.355 And question? 00:31:46.970 --> 00:31:48.316 OK. 00:31:48.316 --> 00:31:50.230 No questions. 00:31:50.230 --> 00:31:59.070 So let's do a simple example, and it's good 00:31:59.070 --> 00:32:04.500 because it's useful to practice with explicit things. 00:32:04.500 --> 00:32:05.630 So here's an example. 00:32:08.190 --> 00:32:16.290 There's a vector space V, which is three complex numbers, 00:32:16.290 --> 00:32:19.090 three component vectors-- complex vectors. 00:32:19.090 --> 00:32:29.706 So a v is equal to v1, v2, v3-- three numbers are all the vi. 00:32:29.706 --> 00:32:32.380 Each one belongs to the complex number. 00:32:32.380 --> 00:32:37.130 So three complex numbers makes a vector space like this. 00:32:37.130 --> 00:32:39.210 So somebody comes along and gives you 00:32:39.210 --> 00:32:47.400 the following linear map-- T on a vector, v1, v1, v3, 00:32:47.400 --> 00:32:48.870 gives you another vector. 00:32:48.870 --> 00:32:49.940 It's a linear map. 00:32:49.940 --> 00:32:51.740 So what is it? 00:32:51.740 --> 00:33:00.720 It's 0 times v1 plus 2v2 plus iv3 for the first component. 00:33:00.720 --> 00:33:02.780 The first component of the new vector-- I 00:33:02.780 --> 00:33:09.450 put the 0v1 just so you see that it just depends on v2 and v3. 00:33:09.450 --> 00:33:15.800 The second component is v1 minus iv2 plus 0v3. 00:33:19.500 --> 00:33:20.990 Those are not vectors. 00:33:20.990 --> 00:33:22.100 These are components. 00:33:22.100 --> 00:33:22.950 These are numbers. 00:33:22.950 --> 00:33:24.740 So this is just a complex number. 00:33:24.740 --> 00:33:27.980 This is another complex number, as it should be. 00:33:27.980 --> 00:33:30.090 Acting on three complex numbers gives 00:33:30.090 --> 00:33:33.400 you, linearly, three other ones. 00:33:33.400 --> 00:33:36.350 And then the third component-- they don't have space there, 00:33:36.350 --> 00:33:43.340 so I'll put it here-- 3iv1 plus v2 plus 7v3. 00:33:49.640 --> 00:33:54.090 And the question is two questions. 00:33:54.090 --> 00:34:06.990 Find T dagger, and write the matrix representations 00:34:06.990 --> 00:34:09.969 of T and T dagger. 00:34:09.969 --> 00:34:19.830 Write the matrices T and T dagger 00:34:19.830 --> 00:34:23.730 using the standard basis in which the three basis 00:34:23.730 --> 00:34:33.440 vectors are 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, and 0, 0, 1. 00:34:33.440 --> 00:34:38.920 These are the three basis vectors-- e1, e2, and e3. 00:34:38.920 --> 00:34:42.590 You know, to write the matrix you need the basis vectors. 00:34:42.590 --> 00:34:45.300 So that's a problem. 00:34:45.300 --> 00:34:49.080 It's a good problem in order to practice, 00:34:49.080 --> 00:34:52.739 to see that you understand how to turn 00:34:52.739 --> 00:34:54.690 an operator into a matrix. 00:34:54.690 --> 00:34:56.639 And you don't get confused. 00:34:56.639 --> 00:34:57.540 Is it a row? 00:34:57.540 --> 00:34:58.540 Is it a column? 00:34:58.540 --> 00:35:01.340 How does it go? 00:35:01.340 --> 00:35:05.700 So let's do this. 00:35:05.700 --> 00:35:10.820 So first we're going to try to find the rules for T dagger. 00:35:10.820 --> 00:35:14.230 So we have the following. 00:35:14.230 --> 00:35:16.130 You see, you use the basic property. 00:35:16.130 --> 00:35:29.312 u on Tv is equal to T dagger u on v. 00:35:29.312 --> 00:35:36.870 So let's try to compute the left hand side, and then look at it 00:35:36.870 --> 00:35:41.640 and try to see if we could derive the right hand side. 00:35:41.640 --> 00:35:45.350 So what is u supposed to be a three component vector? 00:35:45.350 --> 00:35:53.360 So for that use, u equals u1, u2, u3. 00:35:53.360 --> 00:36:00.520 OK, now implicit in all that is that when somebody tells you-- 00:36:00.520 --> 00:36:03.650 OK, you've got a three dimensional complex vector 00:36:03.650 --> 00:36:06.610 space what is the inner product? 00:36:06.610 --> 00:36:08.800 The inner product is complex conjugate 00:36:08.800 --> 00:36:09.880 of the first component. 00:36:09.880 --> 00:36:12.850 That's first component of the second, plus complex conjugate 00:36:12.850 --> 00:36:16.390 of the second times star, times star. 00:36:16.390 --> 00:36:19.770 So it's just a generalization of the dot product, 00:36:19.770 --> 00:36:21.865 but you complex conjugate the first entries. 00:36:25.280 --> 00:36:27.080 So what is this? 00:36:27.080 --> 00:36:30.430 I should take the complex conjugate of the first term 00:36:30.430 --> 00:36:34.350 here-- u1-- times the first one. 00:36:34.350 --> 00:36:38.940 So I have 2v2 plus iv3. 00:36:41.570 --> 00:36:45.660 This is the left hand side, plus the complex conjugate 00:36:45.660 --> 00:36:47.260 of the second component-- there's 00:36:47.260 --> 00:36:56.490 the second component-- so u2 times v1 minus iv2 00:36:56.490 --> 00:36:59.380 plus-- well, 0v3-- his time I won't write 00:36:59.380 --> 00:37:07.010 it-- plus u3 bar times the last vector, 00:37:07.010 --> 00:37:11.710 which is 3iv1 plus v2 plus 7v3. 00:37:15.460 --> 00:37:17.450 OK, that's the left hand side. 00:37:20.810 --> 00:37:24.320 I think I'm going to use this blackboard here, 00:37:24.320 --> 00:37:26.490 because otherwise the numbers are going 00:37:26.490 --> 00:37:29.160 to be hard to see from one side to the other. 00:37:29.160 --> 00:37:36.235 So this information, those two little proofs, 00:37:36.235 --> 00:37:36.985 are to be deleted. 00:37:40.150 --> 00:37:45.190 And now we have this left hand side. 00:37:45.190 --> 00:37:49.090 Now, somehow when you say, OK, now I'm 00:37:49.090 --> 00:37:54.940 going to try to figure out this right hand side your head goes 00:37:54.940 --> 00:37:56.330 and looks in there and says well, 00:37:56.330 --> 00:38:00.250 in the left hand side the u's are sort of the ones that 00:38:00.250 --> 00:38:05.000 are alone, and the v's are acted upon. 00:38:05.000 --> 00:38:07.230 Here the v's must be alone. 00:38:07.230 --> 00:38:12.760 So what I should do is collect along v. 00:38:12.760 --> 00:38:20.160 So let's collect along v. So let's put "something" times v1 00:38:20.160 --> 00:38:26.150 plus "something" like v2 plus "something" like v3. 00:38:26.150 --> 00:38:31.400 And then I will know what is the vector T star this. 00:38:31.400 --> 00:38:32.710 So let's do that. 00:38:32.710 --> 00:38:34.530 So v1, let's collect. 00:38:34.530 --> 00:38:41.750 So you get u2 bar for this v1, and 3iu3 bar. 00:38:46.530 --> 00:39:05.220 v2 will have 2u1 bar minus iu2 bar plus u3 bar. 00:39:11.180 --> 00:39:12.860 I think I got them right. 00:39:12.860 --> 00:39:14.430 OK. 00:39:14.430 --> 00:39:21.840 And then v3, let's collect-- iu1 bar, nothing here, 00:39:21.840 --> 00:39:28.300 and v3 7u3 bar. 00:39:28.300 --> 00:39:34.240 OK, and now I must say, OK, this is the inner product 00:39:34.240 --> 00:39:38.320 of T dagger u times v3. 00:39:38.320 --> 00:39:46.900 So actually, T dagger on u, which is u1, u2, u3, 00:39:46.900 --> 00:39:52.980 must be this vector with three components for which this thing 00:39:52.980 --> 00:40:02.120 is the inner product of this vector with the vector V. 00:40:02.120 --> 00:40:04.290 So I look at this I say, well, what 00:40:04.290 --> 00:40:06.600 was the formula for the inner product? 00:40:06.600 --> 00:40:10.960 Well, you complex conjugate the first entry of this 00:40:10.960 --> 00:40:12.840 and multiply by the first entry of that. 00:40:12.840 --> 00:40:14.900 Complex conjugate the second entry. 00:40:14.900 --> 00:40:23.970 So here I should put u2 minus 3iu3, 00:40:23.970 --> 00:40:25.940 because the complex conjugate of that 00:40:25.940 --> 00:40:28.340 is that as multiplied by v1. 00:40:28.340 --> 00:40:35.675 So here I continue-- 2u1 plus iu2 plus u3. 00:40:38.540 --> 00:40:43.680 And, finally, minus iu1 plus 7u3. 00:40:46.690 --> 00:40:48.960 And that's the answer for this operator. 00:41:00.260 --> 00:41:01.990 So the operator is there for you. 00:41:01.990 --> 00:41:04.300 The only thing we haven't done is the matrices. 00:41:07.870 --> 00:41:11.180 Let me do a little piece of one, and you 00:41:11.180 --> 00:41:12.710 try to compute the rest. 00:41:12.710 --> 00:41:14.890 Make sure you understand it. 00:41:14.890 --> 00:41:22.190 So suppose you get T on the basis vector e1. 00:41:22.190 --> 00:41:23.980 It's easier than what it looks. 00:41:23.980 --> 00:41:26.620 I'm going to have to write some things in order 00:41:26.620 --> 00:41:28.950 to give you a few components, but then 00:41:28.950 --> 00:41:32.450 once you get a little practice, or you look what it means, 00:41:32.450 --> 00:41:33.670 it will become clear. 00:41:33.670 --> 00:41:35.520 So what is T on e1? 00:41:35.520 --> 00:41:40.990 Well, it's T on the vector 1, 0, 0. 00:41:40.990 --> 00:41:45.750 T on the vector 1, 0, 0-- look at the top formula ther3-- 00:41:45.750 --> 00:41:50.870 is equal to 0, 1, and 3i. 00:41:56.710 --> 00:42:02.660 Top formula-- the v1 is 1, and all others are 0. 00:42:02.660 --> 00:42:06.995 And this is e2 plus 3ie3. 00:42:10.330 --> 00:42:14.150 So how do you read, now, matrix elements? 00:42:14.150 --> 00:42:20.230 You remember the formula that T on ei 00:42:20.230 --> 00:42:27.560 is supposed to be Tkiek-- sum over k. 00:42:27.560 --> 00:42:36.770 So this thing is supposed to be equal to T11e1 00:42:36.770 --> 00:42:41.840 plus T21e2 plus T31e3. 00:42:45.340 --> 00:42:51.350 Your sum over the first index, T of e1, is there for that. 00:42:51.350 --> 00:42:58.280 So then I read this, and I see that T21 is equal to 1. 00:42:58.280 --> 00:43:01.050 This is equal to 3i. 00:43:01.050 --> 00:43:03.580 And this is equal to 0. 00:43:03.580 --> 00:43:07.140 So you've got a piece of the matrix, 00:43:07.140 --> 00:43:13.140 and the rest I will just tell you how you see it. 00:43:13.140 --> 00:43:17.240 But you should check it. 00:43:17.240 --> 00:43:19.790 You don't have to write that much after you 00:43:19.790 --> 00:43:23.130 have a little practice with this. 00:43:23.130 --> 00:43:27.370 But, the matrix T-- what you've learned 00:43:27.370 --> 00:43:29.640 is that you have 0, 1, and 3i. 00:43:33.070 --> 00:43:38.890 So 0, 1, and 3i are these numbers, 00:43:38.890 --> 00:43:42.280 in fact-- 0, 1, and 3i. 00:43:42.280 --> 00:43:43.460 And they go vertical. 00:43:43.460 --> 00:43:48.580 So 2, minus i, and 1 is the next column. 00:43:52.140 --> 00:43:56.070 2, minus i, and 1 is the next column, 00:43:56.070 --> 00:44:00.580 and the third one would be i-- look at the v3 there. 00:44:00.580 --> 00:44:06.500 It has an i for the first entry, a 0 for the second, and a 7. 00:44:06.500 --> 00:44:07.620 So this is the matrix. 00:44:13.790 --> 00:44:16.360 How about the matrix T dagger? 00:44:19.760 --> 00:44:23.470 Same thing-- once you've done one, don't worry. 00:44:23.470 --> 00:44:24.370 Don't do the one. 00:44:24.370 --> 00:44:28.040 So this you look for the first column. 00:44:28.040 --> 00:44:32.505 It's going to be a 0-- no u1 here-- a 2, and a minus i. 00:44:38.070 --> 00:44:48.390 0, 2, and a minus i, then 1, i, and 0, minus 3i, 1, and 7. 00:44:48.390 --> 00:44:51.190 And those are it. 00:44:51.190 --> 00:44:54.150 And look how nice. 00:44:54.150 --> 00:44:58.430 The second one is in fact the Hermitian conjugate 00:44:58.430 --> 00:44:59.120 of the other. 00:44:59.120 --> 00:45:01.890 Transpose and complex conjugate gives it to you. 00:45:04.750 --> 00:45:09.090 So that example suggests that that, of course, 00:45:09.090 --> 00:45:11.830 is not an accident. 00:45:11.830 --> 00:45:14.225 So what do you need for that to happen? 00:45:16.810 --> 00:45:21.020 Nobody said that what you're supposed to do to find T dagger 00:45:21.020 --> 00:45:23.260 is transpose some complex conjugate, 00:45:23.260 --> 00:45:27.840 but somehow that's what you do once you have the matrix, 00:45:27.840 --> 00:45:29.550 or at least what it seems that you 00:45:29.550 --> 00:45:30.940 do when you have the matrix. 00:45:30.940 --> 00:45:35.040 So let's see if we can get that more generally. 00:45:35.040 --> 00:45:37.100 So end of example. 00:45:37.100 --> 00:45:48.170 Look at T dagger u, v is equal to u, Tv. 00:45:48.170 --> 00:45:50.230 We know this is the key equation. 00:45:50.230 --> 00:45:53.130 Everything comes from this. 00:45:53.130 --> 00:45:56.650 Now take u and v to be orthonormal vectors, 00:45:56.650 --> 00:46:03.930 so u equal ei, and v equal ej. 00:46:03.930 --> 00:46:05.990 And these are orthonormal. 00:46:05.990 --> 00:46:08.730 The e's are going to be orthonormal each time 00:46:08.730 --> 00:46:12.300 we say basis vectors-- e, orthonormal. 00:46:12.300 --> 00:46:21.942 So put them here, so you get T dagger on ei times ej 00:46:21.942 --> 00:46:25.810 is equal to ei, Tej. 00:46:29.480 --> 00:46:36.590 Now use the matrix action on these operators. 00:46:36.590 --> 00:46:43.848 So T dagger on ei is supposed to be T dagger kiek. 00:46:48.700 --> 00:46:52.870 The equation is something worth knowing by heart. 00:46:52.870 --> 00:46:55.330 What is the matrix representation? 00:46:55.330 --> 00:46:58.530 If the index of the vector goes here, 00:46:58.530 --> 00:47:00.610 the sum index goes like that. 00:47:00.610 --> 00:47:06.910 So then you have ej here, and here you have ei, 00:47:06.910 --> 00:47:08.345 and you have Tkjek. 00:47:33.530 --> 00:47:36.470 So now this basis orthonormal. 00:47:36.470 --> 00:47:41.570 This is a number, and this is the basis. 00:47:41.570 --> 00:47:43.860 The number goes out. 00:47:43.860 --> 00:47:50.520 T dagger ki-- remember, it's on the left side, 00:47:50.520 --> 00:47:53.540 so it should go out with a star. 00:47:53.540 --> 00:47:56.970 And then you have ekej. 00:47:56.970 --> 00:48:00.880 That's orthonormal, so it's delta kej. 00:48:00.880 --> 00:48:03.910 The number here goes out as well, 00:48:03.910 --> 00:48:06.220 and the inner product gives delta ik. 00:48:09.360 --> 00:48:10.585 So what do we get? 00:48:14.470 --> 00:48:24.140 T dagger ji star is equal to Tij. 00:48:29.090 --> 00:48:33.495 First, change i for j, so it looks more familiar. 00:48:33.495 --> 00:48:39.500 So then you have T dagger ij star is equal to Tji. 00:48:42.710 --> 00:48:46.730 And then take complex conjugate, so that finally you 00:48:46.730 --> 00:48:53.843 have T dagger ij is equal to Tji star. 00:48:56.860 --> 00:49:04.050 And that shows that, as long as you have an orthonormal basis 00:49:04.050 --> 00:49:10.750 you can see the Hermitian conjugate of the operator 00:49:10.750 --> 00:49:13.600 by taking the matrix, and then what you usually 00:49:13.600 --> 00:49:16.730 call the Hermitian conjugate of the matrix. 00:49:16.730 --> 00:49:20.360 But I want to emphasize that, if you 00:49:20.360 --> 00:49:23.880 didn't have an orthonormal basis-- if you 00:49:23.880 --> 00:49:30.640 have your operator, and you want to calculate the dagger of it, 00:49:30.640 --> 00:49:32.600 and you find its matrix representation. 00:49:32.600 --> 00:49:35.790 You take the Hermitian conjugate of the matrix. 00:49:35.790 --> 00:49:41.910 It would be wrong if your basis vectors are not orthonormal. 00:49:41.910 --> 00:49:42.880 It just fails. 00:49:42.880 --> 00:49:46.050 So what would happen if the basis vectors are not 00:49:46.050 --> 00:49:47.390 orthonormal? 00:49:47.390 --> 00:49:57.460 Instead of having ei with ej giving you delta iej, 00:49:57.460 --> 00:50:02.930 you have that ei with ej is some number. 00:50:02.930 --> 00:50:07.830 And you can call it aij, or alpha iej, or gij, I think, 00:50:07.830 --> 00:50:09.510 is maybe a better name. 00:50:09.510 --> 00:50:21.550 So if the basis is not orthonormal, then ei with ej 00:50:21.550 --> 00:50:25.540 is some sort of gij. 00:50:28.190 --> 00:50:30.020 And then you go back here. 00:50:30.020 --> 00:50:36.970 And, instead of having deltas here, you would have g's. 00:50:36.970 --> 00:50:48.240 So you would have the T dagger star ki with gkj 00:50:48.240 --> 00:50:53.808 is equal to Tkj, gik. 00:50:58.380 --> 00:51:01.480 And there's no such simple thing as saying, 00:51:01.480 --> 00:51:04.310 oh, well you just take the matrix and complex 00:51:04.310 --> 00:51:06.260 conjugate and transpose. 00:51:06.260 --> 00:51:08.250 That's not the dagger. 00:51:08.250 --> 00:51:11.920 It's more complicated than that. 00:51:11.920 --> 00:51:15.180 If this matrix should be invertible, 00:51:15.180 --> 00:51:17.320 you could pass this to the other side 00:51:17.320 --> 00:51:19.590 using the inverse of this matrix. 00:51:19.590 --> 00:51:23.450 And you can find a formula for the dagger 00:51:23.450 --> 00:51:27.970 in terms of the g matrix, its inverses and multiplications. 00:51:27.970 --> 00:51:29.935 So what do you learn from here? 00:51:29.935 --> 00:51:35.730 You learn a fundamental fact, that the statement 00:51:35.730 --> 00:51:39.010 that an operator-- for example, you have T. 00:51:39.010 --> 00:51:43.510 And you can find T dagger as the adjoint. 00:51:43.510 --> 00:51:48.360 The adjoint operator, or the Hermitian conjugate operator, 00:51:48.360 --> 00:51:52.420 has a basis independent definition. 00:51:52.420 --> 00:51:56.350 It just needs that statement that we've 00:51:56.350 --> 00:52:02.050 written many times now, that T dagger u, 00:52:02.050 --> 00:52:06.106 v is defined via this relation. 00:52:06.106 --> 00:52:08.680 And it has nothing to do with a basis. 00:52:08.680 --> 00:52:11.030 It's true for arbitrary vectors. 00:52:11.030 --> 00:52:14.170 Nevertheless, how you construct T dagger, 00:52:14.170 --> 00:52:19.360 if you have a basis-- well, sometimes 00:52:19.360 --> 00:52:22.360 it's a Hermitian conjugate matrix, 00:52:22.360 --> 00:52:25.110 if your basis is orthonormal. 00:52:25.110 --> 00:52:28.700 But that statement, that the dagger is the Hermitian 00:52:28.700 --> 00:52:33.090 conjugate basis, is a little basis dependent, 00:52:33.090 --> 00:52:38.030 is not a universal fact about the adjoint. 00:52:38.030 --> 00:52:40.330 It's not always constructed that way. 00:52:40.330 --> 00:52:44.596 And there will be examples where you will see that. 00:52:44.596 --> 00:52:45.096 Questions? 00:52:51.806 --> 00:52:52.802 No questions? 00:53:00.290 --> 00:53:04.500 Well, let's do brackets for a few minutes 00:53:04.500 --> 00:53:08.490 so that you see a few properties of them. 00:53:08.490 --> 00:53:13.520 With the same language, I'll write formulas that we've-- OK, 00:53:13.520 --> 00:53:16.190 I wrote a formula here, in fact. 00:53:16.190 --> 00:53:21.160 So for example, this formula-- if I 00:53:21.160 --> 00:53:28.110 want to write it with bras and kets, I would write u Tv. 00:53:28.110 --> 00:53:34.360 And I could also write it as u T v, 00:53:34.360 --> 00:53:38.620 because remember this means-- the bra and the ket-- 00:53:38.620 --> 00:53:43.270 just says a way to make clear that this object is a vector. 00:53:43.270 --> 00:53:47.060 But this vector is obtained by acting T on the vector 00:53:47.060 --> 00:53:52.770 v. So it's T on the vector v, because a vector v is just 00:53:52.770 --> 00:53:55.770 something, and when you put it like that that's still 00:53:55.770 --> 00:54:01.850 the vector v. The kit doesn't do much to it. 00:54:01.850 --> 00:54:03.670 It's almost like putting an arrow, 00:54:03.670 --> 00:54:08.560 so that's why this thing is really this thing as well. 00:54:08.560 --> 00:54:10.500 Now, on the other hand, this thing-- 00:54:10.500 --> 00:54:19.350 let's say that this is equal to v, T dagger u star. 00:54:19.350 --> 00:54:26.295 So then you would put here that this is v T dagger u star. 00:54:31.470 --> 00:54:35.230 So this formula is something that most people remember 00:54:35.230 --> 00:54:39.750 in physics, written perhaps a little differently. 00:54:39.750 --> 00:54:44.460 Change v and u so that this left hand side now 00:54:44.460 --> 00:54:52.520 reads u T dagger v. And it has a star, 00:54:52.520 --> 00:54:58.780 and the right hand side would become v T u. 00:54:58.780 --> 00:55:01.170 And just complex conjugated it. 00:55:01.170 --> 00:55:16.550 So u T dagger v is equal to v T u star-- a nice formula that 00:55:16.550 --> 00:55:22.530 says how do you get to understand what T dagger is. 00:55:22.530 --> 00:55:25.870 Well, if you know T dagger's value 00:55:25.870 --> 00:55:29.014 in between any set of states, then you 00:55:29.014 --> 00:55:34.300 know-- well, if you know T between any set of states 00:55:34.300 --> 00:55:36.400 u and v, then you can figure out what 00:55:36.400 --> 00:55:43.100 T dagger is between any same two states by using this formula. 00:55:43.100 --> 00:55:45.210 What you have to do is that this thing 00:55:45.210 --> 00:55:47.230 is equal to the reverse thing. 00:55:47.230 --> 00:55:50.600 So you go from right to left and reverse it here. 00:55:50.600 --> 00:55:54.990 So you go v, then T, then u, and you put a star, 00:55:54.990 --> 00:55:58.595 and that gives you that object. 00:56:01.400 --> 00:56:04.380 Another thing that we've been doing all the time when 00:56:04.380 --> 00:56:11.605 we calculate, for example, ei, T on ej. 00:56:14.400 --> 00:56:17.900 What is this? 00:56:17.900 --> 00:56:20.050 Well, you know what this is. 00:56:20.050 --> 00:56:23.160 Let's write it like that-- ei. 00:56:23.160 --> 00:56:32.368 Now T on ej is the matrix T kjek. 00:56:32.368 --> 00:56:39.460 If this is an orthonormal basis, here is a delta iek. 00:56:39.460 --> 00:56:43.930 So this is nothing else but Tij. 00:56:43.930 --> 00:56:49.200 So another way of writing that matrix element, ij, of a matrix 00:56:49.200 --> 00:56:54.840 is to put an ei, an ej here, and a T here. 00:56:54.840 --> 00:57:07.240 So people write it like that-- Tij is ei comma Tej. 00:57:07.240 --> 00:57:13.960 Or, in bracket language, they put ei T ej. 00:57:21.330 --> 00:57:24.860 So I need it to be flexible and just 00:57:24.860 --> 00:57:27.720 be able to pass from one notation to the other, 00:57:27.720 --> 00:57:31.890 because it helps you. 00:57:31.890 --> 00:57:39.900 One of the most helpful things in this object 00:57:39.900 --> 00:57:44.960 is to understand, for example, in bra and ket notation, what 00:57:44.960 --> 00:57:48.070 is the following object? 00:57:48.070 --> 00:57:52.910 What is ei ei? 00:57:57.370 --> 00:58:00.950 This seems like the wrong kind of thing, 00:58:00.950 --> 00:58:05.880 because you were supposed to have bras acting on vectors. 00:58:05.880 --> 00:58:07.980 So this would be on the left of that, 00:58:07.980 --> 00:58:10.000 but otherwise it would be too trivial. 00:58:10.000 --> 00:58:13.140 If it would be on the left of it, it would give you a number. 00:58:13.140 --> 00:58:18.920 But think of this thing as a object that stands there. 00:58:18.920 --> 00:58:23.470 And it's repeated endlessly, so it's summed. 00:58:23.470 --> 00:58:25.680 So what is this object? 00:58:25.680 --> 00:58:29.210 Well, this object is a sum of things like that, 00:58:29.210 --> 00:58:41.200 so this is really e1 e1 plus e2 e2, and it goes on like that. 00:58:43.790 --> 00:58:47.310 Well, let it act on a vector. 00:58:47.310 --> 00:58:50.160 This kind of object is an operator. 00:58:50.160 --> 00:58:52.210 Whenever you have the bra and the ket 00:58:52.210 --> 00:58:54.930 sort of in this wrong position-- the ket first, 00:58:54.930 --> 00:58:59.210 and the bra afterwards-- this is, in Dirac's notation, 00:58:59.210 --> 00:59:03.160 an operator, a particular operator. 00:59:03.160 --> 00:59:06.610 And you will see in general how it is the general operator 00:59:06.610 --> 00:59:07.660 very soon. 00:59:07.660 --> 00:59:09.140 So look at this. 00:59:09.140 --> 00:59:12.120 You have something like that, and why do we 00:59:12.120 --> 00:59:14.210 call it an operator? 00:59:14.210 --> 00:59:16.130 We call it an operator business if it 00:59:16.130 --> 00:59:19.355 acts on a vector-- you put a vector here, 00:59:19.355 --> 00:59:23.760 a bra-- this becomes a number, and there's still 00:59:23.760 --> 00:59:25.310 a vector left. 00:59:25.310 --> 00:59:34.480 So this kind of structure, acting on something like that, 00:59:34.480 --> 00:59:38.660 gives a vector, because this thing goes in here, 00:59:38.660 --> 00:59:41.840 produces a number, and the vector is left there. 00:59:41.840 --> 00:59:53.041 So for example, if you act with this thing on the vector a-- 00:59:53.041 --> 01:00:00.720 an arbitrary vector a-- what do you get? 01:00:03.810 --> 01:00:07.226 Whatever this operator is is acted on a. 01:00:07.226 --> 01:00:12.230 Well, you remember that these thing are the components of a, 01:00:12.230 --> 01:00:13.870 and these are the basis vectors. 01:00:13.870 --> 01:00:17.795 So this is nothing else but the vector a again. 01:00:22.710 --> 01:00:29.530 You see, you can start with a equals some alpha i's with 01:00:29.530 --> 01:00:33.540 ei's, and then you calculate what are the alpha i's. 01:00:33.540 --> 01:00:41.610 You put an ej a, and this ej on that gives you alpha j. 01:00:41.610 --> 01:00:44.430 So alpha j-- these numbers are nothing else 01:00:44.430 --> 01:00:47.190 but these things, these numbers. 01:00:47.190 --> 01:00:51.050 So here you have the number times the vector. 01:00:51.050 --> 01:00:55.470 The only difference is that this is like ei alpha i. 01:00:55.470 --> 01:00:58.650 The number has been to the right. 01:00:58.650 --> 01:01:02.680 So this thing acting on any vector is the vector itself. 01:01:02.680 --> 01:01:07.280 So this is perhaps the most fundamental relation 01:01:07.280 --> 01:01:15.270 in bracket notation, is that the identity operator is this. 01:01:19.664 --> 01:01:20.163 Yes. 01:01:20.163 --> 01:01:23.040 AUDIENCE: Is that just 1 e of i, or sum over all e of i? 01:01:23.040 --> 01:01:24.475 PROFESSOR: It's sum of over all. 01:01:24.475 --> 01:01:33.568 So here implicit sum is the sum of all up to en en. 01:01:33.568 --> 01:01:36.430 You will see, if you take just one of them, 01:01:36.430 --> 01:01:41.050 you will get what is an orthogonal projector. 01:01:41.050 --> 01:01:43.590 Now this allows you to do another piece 01:01:43.590 --> 01:01:49.860 of very nice Dirac notation. 01:01:49.860 --> 01:01:52.005 So let's do that. 01:01:59.240 --> 01:02:09.610 Suppose you have an operator T. You put a 1 01:02:09.610 --> 01:02:15.140 in front of it-- a T and a 1 in front of it. 01:02:15.140 --> 01:02:22.070 And then you say, OK, this 1, I'll put ei ei. 01:02:22.070 --> 01:02:31.650 Then comes the T, and then comes the ej ej-- another 1. 01:02:35.460 --> 01:02:37.850 And then you look at that and you suddenly 01:02:37.850 --> 01:02:40.851 see a number lying there. 01:02:40.851 --> 01:02:41.350 Why? 01:02:41.350 --> 01:02:43.165 Because this thing is some number. 01:02:45.790 --> 01:02:47.901 So this is the magic of the Dirac notation. 01:02:47.901 --> 01:02:49.650 You write all this thing, and suddenly you 01:02:49.650 --> 01:02:53.750 see numbers have been created in between. 01:02:53.750 --> 01:02:59.770 This number is nothing else but this matrix representation 01:02:59.770 --> 01:03:01.140 of the operator. 01:03:01.140 --> 01:03:04.550 T, between this, is Tij. 01:03:04.550 --> 01:03:12.445 So this is ei Tij ej. 01:03:16.990 --> 01:03:22.610 So this formula is very fundamental. 01:03:22.610 --> 01:03:27.210 It shows that the most general operator that you can ever 01:03:27.210 --> 01:03:33.770 invent is some sort of ket before a bra, 01:03:33.770 --> 01:03:37.470 and then you superimpose them with these numbers which 01:03:37.470 --> 01:03:43.280 actually happen to be the matrix representation of the operator. 01:03:43.280 --> 01:03:45.860 So the operator can be written as a sum 01:03:45.860 --> 01:03:51.180 of, if this is an n by n matrix n squared thinks of this form-- 01:03:51.180 --> 01:03:55.300 1 with 1, 1 with 2, 1 with 3, and all of them. 01:03:55.300 --> 01:03:58.340 Bu then, you know this formula is so important 01:03:58.340 --> 01:04:00.770 that people make sure that you realize 01:04:00.770 --> 01:04:03.420 that you're summing over i and j. 01:04:03.420 --> 01:04:05.190 So just put it there. 01:04:08.010 --> 01:04:12.730 Given an operator, these are its matrix elements. 01:04:12.730 --> 01:04:19.340 And this is the operator written back in abstract notation. 01:04:19.340 --> 01:04:21.370 The whole operator is back there for you. 01:04:24.450 --> 01:04:27.440 I want to use the last part of the lecture 01:04:27.440 --> 01:04:35.790 to discuss a theorem that is pretty interesting, 01:04:35.790 --> 01:04:39.230 that allows you to understand things 01:04:39.230 --> 01:04:44.580 about all these Hermitian operators and unitary operators 01:04:44.580 --> 01:04:46.620 much more clearly. 01:04:46.620 --> 01:04:51.400 And it's a little mysterious, this theorem, 01:04:51.400 --> 01:04:53.720 and let's see how it goes. 01:04:58.140 --> 01:05:01.650 So any questions about this Dirac notation at this moment, 01:05:01.650 --> 01:05:03.730 anything that I wrote there? 01:05:03.730 --> 01:05:05.720 It takes a while to get accustomed 01:05:05.720 --> 01:05:06.940 to the Dirac notation. 01:05:06.940 --> 01:05:10.540 But once you get the hang of it, it's 01:05:10.540 --> 01:05:13.425 sort of fun and easy to manipulate. 01:05:16.070 --> 01:05:17.480 No questions? 01:05:17.480 --> 01:05:18.458 Can't be. 01:05:22.860 --> 01:05:24.670 You can prove all kinds of things 01:05:24.670 --> 01:05:29.230 with this matrix representation of the identity. 01:05:29.230 --> 01:05:32.140 For example, you can prove easily 01:05:32.140 --> 01:05:35.420 something you proved already, that when you multiply 01:05:35.420 --> 01:05:39.341 two operators the matrices multiply. 01:05:39.341 --> 01:05:41.120 You can prove all kinds of things. 01:05:41.120 --> 01:05:43.180 Pretty much everything we've done 01:05:43.180 --> 01:05:45.530 can also be proven this way. 01:05:45.530 --> 01:05:49.500 OK, so here comes the theorem I want to ask you about. 01:05:49.500 --> 01:05:54.870 Suppose somebody comes along, and they tell you, 01:05:54.870 --> 01:05:59.160 well, you know, here's a vector v, 01:05:59.160 --> 01:06:04.520 and I'm going to have a linear operator acting on this space. 01:06:04.520 --> 01:06:07.300 So the operator's going to be T, and I'm 01:06:07.300 --> 01:06:12.770 going act with the vector v. 01:06:12.770 --> 01:06:15.560 And moreover, I find that this is 01:06:15.560 --> 01:06:26.150 0 for all vectors v belonging to the vector space. 01:06:26.150 --> 01:06:30.280 And the question is-- what can we say about this operator? 01:06:35.950 --> 01:06:39.050 From all vectors it's just 0. 01:06:39.050 --> 01:06:43.290 So is this operator 0, maybe? 01:06:43.290 --> 01:06:44.680 Does it have to be 0? 01:06:44.680 --> 01:06:46.245 Can it be something else? 01:06:48.800 --> 01:06:53.120 OK, we've been talking about real and complex vector spaces. 01:06:53.120 --> 01:06:55.720 And we've seen that it's different. 01:06:55.720 --> 01:06:59.000 The inner product is a little different. 01:06:59.000 --> 01:07:01.410 But let's think about this. 01:07:01.410 --> 01:07:05.530 Take two dimensions, real vector space. 01:07:05.530 --> 01:07:07.760 The operator that takes any vector 01:07:07.760 --> 01:07:11.440 and rotates it by 90 degrees, that's a linear operator. 01:07:14.300 --> 01:07:20.570 And that is a non-trivial linear operator, and it gives you 0. 01:07:20.570 --> 01:07:26.710 So case settled-- there's no theorem here, nothing 01:07:26.710 --> 01:07:28.500 you can say about this operator. 01:07:28.500 --> 01:07:31.140 It may be non-zero. 01:07:31.140 --> 01:07:33.440 But here comes the catch. 01:07:33.440 --> 01:07:40.010 If you're talking complex vector spaces, T is 0. 01:07:40.010 --> 01:07:42.350 It just is 0, can't be anything else. 01:07:42.350 --> 01:07:44.460 Complex vector spaces are different. 01:07:44.460 --> 01:07:48.540 You can't quite do that thing-- rotate all vectors by something 01:07:48.540 --> 01:07:50.420 and do things. 01:07:50.420 --> 01:07:53.020 So that's a theorem we want to understand. 01:07:53.020 --> 01:08:05.830 Theorem-- let v be a complex inner product space. 01:08:10.270 --> 01:08:13.810 By that is a complex vector space with an inner product. 01:08:13.810 --> 01:08:25.890 Then v, Tv equals 0 for all v implies 01:08:25.890 --> 01:08:28.170 that the operator is just 0. 01:08:35.580 --> 01:08:39.930 I traced a lot of my confusions in quantum mechanics 01:08:39.930 --> 01:08:42.390 to not knowing about this theorem, 01:08:42.390 --> 01:08:46.109 that somehow it must be true. 01:08:46.109 --> 01:08:48.520 I don't know why it should be true, 01:08:48.520 --> 01:08:56.250 but somehow it's not, because it really has exceptions. 01:08:56.250 --> 01:08:57.300 So here it is. 01:08:57.300 --> 01:09:01.340 We tried to prove that. 01:09:01.340 --> 01:09:05.060 It's so important, I think, that it should be proven. 01:09:05.060 --> 01:09:08.590 And how could you prove that? 01:09:08.590 --> 01:09:12.330 And at first sight it seems it's going to be difficult, 01:09:12.330 --> 01:09:16.490 because, if I do just a formal proof, 01:09:16.490 --> 01:09:18.760 how is it going to know that I'm not 01:09:18.760 --> 01:09:21.069 talking real or complex vector spaces. 01:09:21.069 --> 01:09:24.430 So it must make a crucial difference in the proof 01:09:24.430 --> 01:09:27.359 whether it's real or complex. 01:09:27.359 --> 01:09:31.939 So this property really sets the complex vector spaces 01:09:31.939 --> 01:09:33.970 quite apart from the real ones. 01:09:33.970 --> 01:09:38.899 So let's see what you would need to do. 01:09:38.899 --> 01:10:01.630 Well, here's a strategy-- if I could prove that u, 01:10:01.630 --> 01:10:10.800 Tv is equal to 0 for all u and all v. You see, 01:10:10.800 --> 01:10:14.010 the problem here is that these two are the same vector. 01:10:14.010 --> 01:10:17.320 They're all vectors, but they're the same vector. 01:10:17.320 --> 01:10:23.230 If I could prove that this is 0 for all u and v, then 01:10:23.230 --> 01:10:24.380 what would I say? 01:10:24.380 --> 01:10:27.900 I would say, oh, if this is 0 for all u and v, 01:10:27.900 --> 01:10:34.040 then pick u equal to Tv. 01:10:34.040 --> 01:10:39.410 And then you find that Tv, Tv is 0, 01:10:39.410 --> 01:10:43.600 therefore Tv is the 0 vector. 01:10:43.600 --> 01:10:45.660 By the axiom of the inner product, 01:10:45.660 --> 01:10:50.580 for all v is a 0 vector, so T kills all vectors, 01:10:50.580 --> 01:10:53.630 therefore T is 0. 01:10:53.630 --> 01:11:00.060 So if I could prove this is true, I would be done. 01:11:03.150 --> 01:11:05.510 Now, of course, that's the difficulty. 01:11:05.510 --> 01:11:09.470 Well, I wouldn't say of course. 01:11:09.470 --> 01:11:13.120 This takes a leap of faith to believe 01:11:13.120 --> 01:11:16.060 that this is the way you're going to prove that. 01:11:16.060 --> 01:11:20.050 You could try to prove this, and then it would follow. 01:11:20.050 --> 01:11:21.890 But maybe that's difficult to prove. 01:11:21.890 --> 01:11:24.490 But actually that's possible to prove. 01:11:24.490 --> 01:11:28.780 But how could you ever prove that this is true? 01:11:28.780 --> 01:11:31.690 You could prove it if you could somehow 01:11:31.690 --> 01:11:40.626 rewrite u and Tv as some sort of something with a T 01:11:40.626 --> 01:11:47.530 and something plus some other thing with a T, 01:11:47.530 --> 01:11:52.450 and that other thing plus some-- all kinds of things like that. 01:11:52.450 --> 01:11:57.100 Because the things in which this is the same as that are 0. 01:11:59.710 --> 01:12:03.660 So if you can do that-- if you could re-express this left hand 01:12:03.660 --> 01:12:07.840 side as a sum of things of that kind-- that would be 0. 01:12:07.840 --> 01:12:11.700 So let's try. 01:12:11.700 --> 01:12:13.840 So what can you try? 01:12:13.840 --> 01:12:20.490 You can put u plus v here, and T of u plus v. That 01:12:20.490 --> 01:12:25.290 would be 0, because that's a vector, same vector here. 01:12:25.290 --> 01:12:29.450 But that's not equal to this, because it has the u, Tu, 01:12:29.450 --> 01:12:31.650 and it has the v Tv. 01:12:31.650 --> 01:12:34.130 And it has this in a different order. 01:12:34.130 --> 01:12:49.500 So maybe we can subtract u minus v, T of u minus v. Well, 01:12:49.500 --> 01:12:52.400 we're getting there, but all this 01:12:52.400 --> 01:12:59.570 is question marks-- u, Tu, v, Tv-- these cancel-- u, Tu, v, 01:12:59.570 --> 01:13:01.460 Tv. 01:13:01.460 --> 01:13:04.410 But, the cross-products, what are they? 01:13:04.410 --> 01:13:06.114 Well here you have a u, Tv. 01:13:08.904 --> 01:13:13.210 And here you have a v, Tu. 01:13:13.210 --> 01:13:16.660 And do they cancel? 01:13:16.660 --> 01:13:17.630 No. 01:13:17.630 --> 01:13:18.540 Let's see. 01:13:18.540 --> 01:13:22.160 u, Tv, and up here is u minus Tv about. 01:13:22.160 --> 01:13:25.000 But there's another minus, so there's another one there. 01:13:25.000 --> 01:13:28.050 And v, Tu has a minus, minus is a plus. 01:13:28.050 --> 01:13:32.010 So actually this gives me two of this plus two of that. 01:13:32.010 --> 01:13:34.290 OK, it shouldn't have been so easy anyway. 01:13:40.090 --> 01:13:43.673 So here is where you have to have the small inspiration. 01:13:48.070 --> 01:13:50.170 Somehow it shouldn't have worked, you know. 01:13:50.170 --> 01:13:53.860 If this had worked, the theorem would read different. 01:13:53.860 --> 01:13:57.240 You could use a real vector space. 01:13:57.240 --> 01:13:59.900 Nothing is imaginary there. 01:13:59.900 --> 01:14:02.600 So the fact that you have a complex vector space 01:14:02.600 --> 01:14:03.290 might help. 01:14:03.290 --> 01:14:06.150 So somehow you have to put i's there. 01:14:06.150 --> 01:14:08.260 So let's try i's here. 01:14:08.260 --> 01:14:16.000 So you put u plus iv and T of u plus iv. 01:14:19.090 --> 01:14:23.380 Well, then you probably have to subtract things as well, so 01:14:23.380 --> 01:14:29.740 u minus iv, T of u minus iv. 01:14:29.740 --> 01:14:34.030 These things will be 0 because of the general structure-- 01:14:34.030 --> 01:14:38.020 the same operator here as here. 01:14:38.020 --> 01:14:39.620 And let's see what they are. 01:14:39.620 --> 01:14:43.700 Well, there's u, Tu, and here's minus u, Tu, 01:14:43.700 --> 01:14:49.480 so the diagonal things go away-- the minus iv, minus iv, iv, 01:14:49.480 --> 01:14:53.940 and a T. You have minus iv, minus iv subtracted, 01:14:53.940 --> 01:14:55.800 so that also cancels. 01:14:55.800 --> 01:14:58.790 So there's the cross-products. 01:14:58.790 --> 01:15:05.790 Now you will say, well, just like the minus signs, 01:15:05.790 --> 01:15:09.050 you're not going to get anything because you're 01:15:09.050 --> 01:15:10.300 going to get 2 and 2. 01:15:10.300 --> 01:15:13.492 Let's see. 01:15:13.492 --> 01:15:15.640 Let's see what we get with this one. 01:15:15.640 --> 01:15:23.090 You get u with Tiv, so you get i u, Tv. 01:15:26.930 --> 01:15:32.870 But look, this i on the left, however, when you take it out, 01:15:32.870 --> 01:15:41.440 becomes a minus i, so you get minus i v, Tu. 01:15:46.780 --> 01:15:49.490 And the other products [INAUDIBLE]. 01:15:49.490 --> 01:15:56.050 So let's look what you get here-- a u with a minus iv 01:15:56.050 --> 01:16:00.366 and a minus here gives you a 2 here. 01:16:00.366 --> 01:16:03.450 And the other term, v, Tu-- well, 01:16:03.450 --> 01:16:06.570 this goes out as a plus i. 01:16:06.570 --> 01:16:11.020 But with a minus, it becomes a minus i, so v, Tu is this. 01:16:11.020 --> 01:16:12.185 So there's a 2 here. 01:16:17.850 --> 01:16:21.500 So that's what these terms give you. 01:16:21.500 --> 01:16:23.700 And now you've succeeded. 01:16:23.700 --> 01:16:24.220 Why? 01:16:24.220 --> 01:16:27.110 Because the relative sign is negative. 01:16:27.110 --> 01:16:28.830 So who cares? 01:16:28.830 --> 01:16:32.370 You can divide by i, and divide this by i. 01:16:32.370 --> 01:16:34.270 You are constructing something. 01:16:34.270 --> 01:16:37.070 So let me put here what you get. 01:16:42.230 --> 01:16:43.695 I can erase this blackboard. 01:16:50.020 --> 01:16:51.285 So what do we get? 01:16:55.860 --> 01:17:03.280 I claim that if you put one quarter of u plus v, 01:17:03.280 --> 01:17:15.230 T u plus v minus u minus v, T of u minus v, then, 01:17:15.230 --> 01:17:17.680 let's see, what do we need to keep? 01:17:17.680 --> 01:17:21.450 We need to keep u and Tv. 01:17:21.450 --> 01:17:27.250 So divide this by i plus 1 over i 01:17:27.250 --> 01:17:36.400 u plus iv, T of u plus iv minus 1 01:17:36.400 --> 01:17:42.730 over i, u minus iv, T of u minus iv. 01:17:46.027 --> 01:17:47.490 And close it. 01:17:50.700 --> 01:17:52.550 You've divided by i. 01:17:52.550 --> 01:17:57.180 You get here four of these ones, zero of these ones, 01:17:57.180 --> 01:18:00.820 and you got the answer you wanted. 01:18:00.820 --> 01:18:04.180 So this whole thing is written like that, 01:18:04.180 --> 01:18:11.270 and now, since this is equal to u with Tv, 01:18:11.270 --> 01:18:14.170 by the conditions of the theorem, 01:18:14.170 --> 01:18:20.500 any vector-- any vector here-- these are all 0. 01:18:20.500 --> 01:18:26.000 You've shown that this is 0, and therefore the operator is 0. 01:18:26.000 --> 01:18:28.030 And you should be very satisfied, 01:18:28.030 --> 01:18:30.940 because the proof made use of the fact 01:18:30.940 --> 01:18:32.840 that it was a complex vector space. 01:18:32.840 --> 01:18:36.790 Otherwise you could not add vectors 01:18:36.790 --> 01:18:38.250 with an imaginary number. 01:18:38.250 --> 01:18:41.880 And the imaginary number made it all work. 01:18:41.880 --> 01:18:45.010 So the theorem is there. 01:18:45.010 --> 01:18:48.370 It's a pretty useful theorem, so let's use it 01:18:48.370 --> 01:18:50.940 for the most obvious application. 01:18:54.890 --> 01:19:01.170 People say that, whenever you find that v, 01:19:01.170 --> 01:19:13.210 Tv is real for all v, then this operator 01:19:13.210 --> 01:19:16.050 is Hermitian, or self-adjoint. 01:19:16.050 --> 01:19:23.800 That is, then, it implies T dagger equals T. 01:19:23.800 --> 01:19:25.950 So let's show that. 01:19:28.590 --> 01:19:32.620 So let's take v, Tv. 01:19:36.730 --> 01:19:38.962 Proof. 01:19:38.962 --> 01:19:48.180 You take v, Tv, and now this thing is real. 01:19:48.180 --> 01:19:55.795 So since this is real, you can say it's equal to v, Tv star. 01:19:59.270 --> 01:20:02.750 Now, because it's real-- that's the assumption. 01:20:02.750 --> 01:20:03.990 The number is real. 01:20:03.990 --> 01:20:16.520 Now, the star off an inner product is Tv, v. 01:20:16.520 --> 01:20:19.530 But on the other hand, this operator, 01:20:19.530 --> 01:20:24.820 by the definition of adjoint, can be moved here. 01:20:24.820 --> 01:20:30.120 And this is equal to T dagger v, v. So 01:20:30.120 --> 01:20:34.530 now you have done this is equal to this. 01:20:34.530 --> 01:20:37.090 So if you put it to one side, you 01:20:37.090 --> 01:20:46.510 get that T dagger minus T on v times v is equal to 0. 01:20:46.510 --> 01:20:53.560 Or, since any inner product that is 00-- 01:20:53.560 --> 01:20:59.140 it's complex conjugate is 0-- you can write it as v, 01:20:59.140 --> 01:21:08.180 T dagger minus v is 0 for all v. 01:21:08.180 --> 01:21:11.430 And so this is an actually well known statement, 01:21:11.430 --> 01:21:17.460 that any operator that gives you real things must be Hermitian. 01:21:17.460 --> 01:21:23.110 But it's not obvious, because that theorem is not obvious. 01:21:23.110 --> 01:21:26.660 And now you can use a theorem and say, well, 01:21:26.660 --> 01:21:31.910 since this is true for all v, T dagger minus T is 0, 01:21:31.910 --> 01:21:37.010 and T dagger is equal to T. Then you can also show, of course, 01:21:37.010 --> 01:21:41.730 if T dagger is equal to T, this thing is real. 01:21:41.730 --> 01:21:47.100 So in fact, this arrow is both ways. 01:21:47.100 --> 01:21:53.320 And this way is very easy, but this way uses this theorem. 01:21:53.320 --> 01:21:56.445 There's another kind of operators 01:21:56.445 --> 01:21:58.630 that are called unitary operators. 01:21:58.630 --> 01:22:01.060 We'll talk a little more about them next time. 01:22:01.060 --> 01:22:04.940 And they preserve the norm of vectors. 01:22:04.940 --> 01:22:06.480 People define them from you, and you 01:22:06.480 --> 01:22:09.080 see that they preserve the norm of vectors. 01:22:09.080 --> 01:22:11.375 On the other hand, you sometimes find an operator 01:22:11.375 --> 01:22:14.460 that preserves every norm. 01:22:14.460 --> 01:22:16.200 Is it unitary? 01:22:16.200 --> 01:22:17.667 You will say, yes, must be. 01:22:17.667 --> 01:22:18.500 How do you prove it? 01:22:18.500 --> 01:22:20.550 You need again that theorem. 01:22:20.550 --> 01:22:23.120 So this theorem is really quite fundamental 01:22:23.120 --> 01:22:25.950 to understand the properties of operators. 01:22:25.950 --> 01:22:27.750 And we'll continue that next time. 01:22:27.750 --> 01:22:29.600 All right.