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 --> 00:00:12.700 To make a donation or to view additional materials 00:00:12.700 --> 00:00:16.600 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.600 --> 00:00:17.275 at ocw.mit.edu. 00:00:21.400 --> 00:00:27.970 PROFESSOR: Now, a theorem that was quite powerful and applied 00:00:27.970 --> 00:00:33.060 to complex vector spaces for old V longing 00:00:33.060 --> 00:00:40.560 to V complex vector space. 00:00:40.560 --> 00:00:46.340 This implied that the operator was zero, 00:00:46.340 --> 00:00:51.010 and it's not true for real vector spaces. 00:00:51.010 --> 00:00:54.620 And we gave a proof that made it clear that indeed, 00:00:54.620 --> 00:00:59.970 a proof wouldn't work for a real vector space. 00:00:59.970 --> 00:01:04.489 I will ask you in the homework to do something that presumably 00:01:04.489 --> 00:01:07.020 would be the first thing you would do if you had 00:01:07.020 --> 00:01:11.800 to try to understand why this is true-- take two by two 00:01:11.800 --> 00:01:16.420 matrices, and just see why it has to be 0 in one case, 00:01:16.420 --> 00:01:19.410 and it doesn't have to be 0 in the other case. 00:01:19.410 --> 00:01:22.140 And I think that will give you a better 00:01:22.140 --> 00:01:25.210 perspective on why this happens. 00:01:25.210 --> 00:01:27.700 And then once you do it for a two by two, 00:01:27.700 --> 00:01:32.600 and you see how it works, you can do it for n by n matrices, 00:01:32.600 --> 00:01:34.250 and it will be clear as well. 00:01:34.250 --> 00:01:37.220 So we'll use this theorem. 00:01:37.220 --> 00:01:42.320 Our immediate application of this theorem 00:01:42.320 --> 00:01:50.150 was a well-known result that we could prove now 00:01:50.150 --> 00:01:55.680 rigorously-- that t is equal to t dagger, which 00:01:55.680 --> 00:01:59.200 is to say the operator is equal to the adjoined. 00:01:59.200 --> 00:02:03.740 And I think I will call this-- and in the notes 00:02:03.740 --> 00:02:07.570 you will see always the adjoint, as opposed to Hermitian 00:02:07.570 --> 00:02:08.780 congregate. 00:02:08.780 --> 00:02:10.789 And I will say whenever the operator 00:02:10.789 --> 00:02:14.860 is equal to the adjoint, that it is Hermitian. 00:02:14.860 --> 00:02:19.150 So a Hermitian operator is equivalent to saying 00:02:19.150 --> 00:02:29.480 that v Tv is a real number for all v. And we proved that. 00:02:34.760 --> 00:02:39.450 In other words, physically, Hermitian operators 00:02:39.450 --> 00:02:43.320 have real expectation values. 00:02:43.320 --> 00:02:47.420 This is an expectation value, because-- as you remember 00:02:47.420 --> 00:02:51.480 in the bracket notation-- v Tv, you 00:02:51.480 --> 00:02:57.230 can write it v T v-- the same thing. 00:02:57.230 --> 00:03:01.480 So it is an expectation value, so it's an important thing, 00:03:01.480 --> 00:03:04.890 because we usually deal with Hermitian operators, 00:03:04.890 --> 00:03:08.240 and we want expectation values of Hermitian operators 00:03:08.240 --> 00:03:10.770 to be real. 00:03:10.770 --> 00:03:13.770 Now that we're talking about Hermitian operators, 00:03:13.770 --> 00:03:20.360 I delay a complete discussion of diagonalization, 00:03:20.360 --> 00:03:23.170 and diagonalization of several operators 00:03:23.170 --> 00:03:26.440 simultaneously, for the next lecture. 00:03:26.440 --> 00:03:31.600 Today, I want to move forward a bit and do some other things. 00:03:31.600 --> 00:03:36.310 And this way we spread out a little more the math, 00:03:36.310 --> 00:03:41.070 and we can begin to look more at physical issues, 00:03:41.070 --> 00:03:42.720 and how they apply here. 00:03:42.720 --> 00:03:46.750 But at any rate, we're just here already, 00:03:46.750 --> 00:03:50.200 and we can prove two basic things that 00:03:50.200 --> 00:03:55.010 are the kinds of things that an 805 student should 00:03:55.010 --> 00:03:57.630 be able to prove at any time. 00:03:57.630 --> 00:03:59.960 They're really very simple. 00:03:59.960 --> 00:04:05.020 And it's a kind of proof that is very short-- couple of lines-- 00:04:05.020 --> 00:04:08.760 something you should be able to reproduce at any time. 00:04:08.760 --> 00:04:16.070 So the first theorem says the eigenvalues 00:04:16.070 --> 00:04:22.600 of Hermitian operators-- H is for Hermitian-- are real. 00:04:25.430 --> 00:04:34.080 And I will do a little bit of notation, in which I will start 00:04:34.080 --> 00:04:39.030 with an expression, and evaluate it sort of to the left 00:04:39.030 --> 00:04:40.130 and to the right. 00:04:40.130 --> 00:04:42.820 So when you have an equation, you start here, 00:04:42.820 --> 00:04:44.380 and you start evaluating there. 00:04:44.380 --> 00:04:51.270 So I will start with this-- consider v Tv. 00:04:51.270 --> 00:04:55.040 And I will box it as being the origin, 00:04:55.040 --> 00:04:57.830 and I will start evaluating it. 00:04:57.830 --> 00:05:12.230 Now if v is an eigenvector-- so let v be an eigenvector so 00:05:12.230 --> 00:05:17.530 that Tv is equal to lambda v. 00:05:17.530 --> 00:05:22.830 And now we say consider that expression in the box. 00:05:22.830 --> 00:05:24.590 And you try to evaluate it. 00:05:24.590 --> 00:05:27.600 So one way to evaluate it-- I evaluate it to the left, 00:05:27.600 --> 00:05:29.320 and then evaluate to the right. 00:05:29.320 --> 00:05:32.830 Is the naive evaluation-- t on v is lambda v, 00:05:32.830 --> 00:05:34.940 so substitute it there. 00:05:34.940 --> 00:05:38.690 v, lambda v-- we know it. 00:05:38.690 --> 00:05:42.740 And then by homogeneity, this lambda goes out, 00:05:42.740 --> 00:05:48.230 and therefore it's lambda v v. 00:05:48.230 --> 00:05:55.730 On the other hand, we have that we can go to the right. 00:05:55.730 --> 00:06:00.970 And what is the way you move an operator to the first position? 00:06:00.970 --> 00:06:02.580 By putting a dagger. 00:06:02.580 --> 00:06:04.480 That's a definition. 00:06:04.480 --> 00:06:05.610 So this is by definition. 00:06:10.140 --> 00:06:15.190 Now, we use that the operator is Hermitian. 00:06:15.190 --> 00:06:26.700 So this is equal to Tv v. And this is by T Hermitian. 00:06:26.700 --> 00:06:30.960 Then you can apply again the equation of the eigenvalues, 00:06:30.960 --> 00:06:39.220 so this is lambda v v. And by conjugate homogeneity 00:06:39.220 --> 00:06:46.760 of the first input, this is lambda star v v. 00:06:46.760 --> 00:06:48.550 So at the end of the day, you have 00:06:48.550 --> 00:06:52.300 something on the extreme left, and something 00:06:52.300 --> 00:06:53.405 on the extreme right. 00:06:57.240 --> 00:07:00.230 v-- if there is an eigenvector, v 00:07:00.230 --> 00:07:03.810 can be assumed to be non-zero. 00:07:03.810 --> 00:07:06.250 The way we are saying things in a sense, 00:07:06.250 --> 00:07:09.340 0-- we also think of it as an eigenvector, 00:07:09.340 --> 00:07:11.700 but it's a trivial one. 00:07:11.700 --> 00:07:14.690 But the fact that there's an eigenvalue 00:07:14.690 --> 00:07:17.570 means there's a non-zero v that solves this equation. 00:07:17.570 --> 00:07:21.000 So we're using that non-zero v. And therefore, this 00:07:21.000 --> 00:07:24.670 is a number that is non-zero. 00:07:24.670 --> 00:07:28.510 You bring one to the other side, and you 00:07:28.510 --> 00:07:35.640 have lambda minus lambda star times v v equals 0. 00:07:35.640 --> 00:07:37.490 This is different from 0. 00:07:40.140 --> 00:07:42.480 This is different from 0. 00:07:42.480 --> 00:07:45.340 And therefore, lambda is equal to lambda star. 00:07:49.030 --> 00:07:57.170 So it's a classic proof-- relatively straightforward. 00:07:57.170 --> 00:08:02.190 The second theorem is as simple to prove. 00:08:02.190 --> 00:08:05.300 And it's already interesting. 00:08:05.300 --> 00:08:09.660 And it states that different eigenvectors of a Hermitian 00:08:09.660 --> 00:08:14.490 operator-- well, different eigenvalues 00:08:14.490 --> 00:08:16.380 of Hermitian operators correspond 00:08:16.380 --> 00:08:19.630 to orthogonal eigenfunctions, or eigenvectors. 00:08:19.630 --> 00:08:35.140 So different eigenvalues of Hermitian ops correspond 00:08:35.140 --> 00:08:43.080 to orthogonal eigenfunctions-- eigenvectors, I'm sorry. 00:08:47.510 --> 00:08:50.080 So what are we saying here? 00:08:50.080 --> 00:08:55.340 We're saying that suppose you have a v1 00:08:55.340 --> 00:08:59.480 that [INAUDIBLE] T gives you a lambda1 v1. 00:08:59.480 --> 00:09:02.160 That's one eigenvalue. 00:09:02.160 --> 00:09:08.110 You have another one-- v2 is equal to lambda 2 v2, 00:09:08.110 --> 00:09:11.430 and lambda 1 is different from lambda 2. 00:09:14.960 --> 00:09:19.990 Now, just focusing on a fact that is going to show up later, 00:09:19.990 --> 00:09:24.630 is going to make life interesting, 00:09:24.630 --> 00:09:28.460 is that some eigenvalues may have 00:09:28.460 --> 00:09:32.180 a multiplicity of eigenvectors. 00:09:32.180 --> 00:09:37.190 In other words, If a vector v is an eigenvector, 00:09:37.190 --> 00:09:40.530 minus v is an eigenvector, square root of three v 00:09:40.530 --> 00:09:45.240 is an eigenvector, but that's a one-dimensional subspace. 00:09:45.240 --> 00:09:48.180 But sometimes for a given eigenvalue, 00:09:48.180 --> 00:09:50.510 there may be a higher dimensional 00:09:50.510 --> 00:09:53.850 subspace of eigenvectors. 00:09:53.850 --> 00:09:57.970 That's a problem of degeneracy, and it's very interesting-- 00:09:57.970 --> 00:10:00.840 makes life really interesting in quantum mechanics. 00:10:00.840 --> 00:10:05.800 So if you have degeneracy, and that set of eigenvectors 00:10:05.800 --> 00:10:08.950 form a subspace, and you can choose a basis, 00:10:08.950 --> 00:10:12.850 and you could have several vectors here. 00:10:12.850 --> 00:10:14.630 Now what do you do in that case? 00:10:14.630 --> 00:10:18.880 The theorem doesn't say much, so it means choose any one. 00:10:18.880 --> 00:10:21.790 If you had the bases there, choose any one. 00:10:21.790 --> 00:10:26.490 The fact remains that if these two eigenvalues are different, 00:10:26.490 --> 00:10:30.900 then you will be able to show that the eigenvectors are 00:10:30.900 --> 00:10:32.220 orthogonal. 00:10:32.220 --> 00:10:39.530 So if you have some space of eigenvectors-- 00:10:39.530 --> 00:10:42.320 a [INAUDIBLE] higher-dimensional space of eigenvectors, 00:10:42.320 --> 00:10:45.700 one eigenvalue, and another space with another-- 00:10:45.700 --> 00:10:50.420 any vector here is orthogonal to any vector there. 00:10:50.420 --> 00:10:52.055 So how do you show this? 00:10:54.710 --> 00:10:56.150 How do you show this property? 00:10:56.150 --> 00:11:01.420 Well you have to involve v1 and v2, 00:11:01.420 --> 00:11:05.950 so you're never going to be using the property that 00:11:05.950 --> 00:11:08.950 gives Hermitian, unless you have an inner product. 00:11:08.950 --> 00:11:12.240 So if you don't have any idea how to prove that, 00:11:12.240 --> 00:11:15.810 you presumably at some stage realize 00:11:15.810 --> 00:11:18.720 that you probably have to use an inner product. 00:11:18.720 --> 00:11:21.760 And we should mix the vectors, so maybe 00:11:21.760 --> 00:11:25.100 a V2 inner product with this. 00:11:25.100 --> 00:11:30.295 So we'll take a v2 inner product with T v1. 00:11:33.050 --> 00:11:37.260 And this is interesting, because we can use it, 00:11:37.260 --> 00:11:44.960 that T v1 is lambda 1 v1 to show that this is just lambda 1 00:11:44.960 --> 00:11:45.495 v2 v1. 00:11:48.700 --> 00:11:51.900 And that already brings all kinds of good things. 00:11:51.900 --> 00:11:54.110 You're interested in this inner product. 00:11:54.110 --> 00:11:56.420 You want to show it's 0, so it shows up. 00:11:56.420 --> 00:11:57.640 So it's a good idea. 00:11:57.640 --> 00:12:00.440 So we have evaluated this, and now you 00:12:00.440 --> 00:12:04.700 have to think of evaluating it in a different way. 00:12:04.700 --> 00:12:07.040 Again, the operator is Hermitian, 00:12:07.040 --> 00:12:10.120 so it's asking you to move it to the other side 00:12:10.120 --> 00:12:12.200 and exploit to that. 00:12:12.200 --> 00:12:14.810 So we'll move it to the other side a little quicker 00:12:14.810 --> 00:12:15.430 this time. 00:12:15.430 --> 00:12:18.720 It goes as T dagger, but T dagger is equal to T, 00:12:18.720 --> 00:12:20.380 because it's Hermitian. 00:12:20.380 --> 00:12:24.230 So this is the center of the equation. 00:12:24.230 --> 00:12:25.140 We go one way. 00:12:25.140 --> 00:12:27.880 We go the other way-- this time down. 00:12:27.880 --> 00:12:37.100 So we'll put T v2 v1, and this is equal to lambda-- 00:12:37.100 --> 00:12:45.520 let me go a little slow here-- lambda 2 v2 v1. 00:12:45.520 --> 00:12:49.930 Your impulse should be it goes out as lambda 2 star, 00:12:49.930 --> 00:12:52.980 but the eigenvalues are already real, 00:12:52.980 --> 00:13:05.210 so it goes out as lambda 2 v2 v1, 00:13:05.210 --> 00:13:07.640 because the operator is Hermitian. 00:13:07.640 --> 00:13:10.440 So at this moment, you have these two equations. 00:13:10.440 --> 00:13:13.800 You bring, say, this to the right-hand side, 00:13:13.800 --> 00:13:23.670 and you get lambda 1 minus lambda 2 v1 v2 is equal to 0. 00:13:23.670 --> 00:13:26.780 And since the eigenvalues are supposed to be different, 00:13:26.780 --> 00:13:33.810 you conclude that v1 inner product with v2 is 0. 00:13:33.810 --> 00:13:35.405 So that's the end of the proof. 00:13:46.240 --> 00:13:48.160 And those are the two properties that 00:13:48.160 --> 00:13:56.660 are very quickly proven with rather little effort. 00:13:56.660 --> 00:14:02.240 So where do we go from now? 00:14:02.240 --> 00:14:05.750 Well there's one more class of operators 00:14:05.750 --> 00:14:08.320 that are crucial in the physics. 00:14:08.320 --> 00:14:13.280 They are perhaps as important as the Hermitian operators, 00:14:13.280 --> 00:14:15.070 if not more. 00:14:15.070 --> 00:14:20.350 They are some operators that are called unitary operators, 00:14:20.350 --> 00:14:22.580 and the way I will introduce them 00:14:22.580 --> 00:14:35.130 is as follows-- so I will say-- it's an economical way 00:14:35.130 --> 00:14:39.523 to introduce them-- so we'll talk about unitary operators. 00:14:45.700 --> 00:14:54.210 If S is unitary, and mathematicians 00:14:54.210 --> 00:15:07.180 call it anisometry-- if you find that S acting on any vector-- 00:15:07.180 --> 00:15:12.940 if you take the norm, it's equal to the norm 00:15:12.940 --> 00:15:17.820 of the vector for all u in the vector space. 00:15:24.140 --> 00:15:32.600 So let's follow this, and make a couple of comments. 00:15:32.600 --> 00:15:38.740 An example-- a trivial example-- this operator lambda 00:15:38.740 --> 00:15:39.560 times the identity. 00:15:42.960 --> 00:15:45.770 Lambda times the identity acts on vectors. 00:15:45.770 --> 00:15:47.810 What does it do, lambda times identity? 00:15:47.810 --> 00:15:50.040 The identity does nothing on the vector, 00:15:50.040 --> 00:15:52.800 and lambda stretches it. 00:15:52.800 --> 00:16:00.170 So lambda, in order not to change the length of any vector 00:16:00.170 --> 00:16:02.240 should be kind of 1. 00:16:02.240 --> 00:16:09.100 Well, in fact, it suffices-- it's unitary-- 00:16:09.100 --> 00:16:14.430 if the absolute value of lambda is equal to 1. 00:16:14.430 --> 00:16:19.400 Because then lambda is a phase, and it just rotates the vector. 00:16:19.400 --> 00:16:24.110 Or in other words, you know that the norm of av 00:16:24.110 --> 00:16:28.630 is equal to the absolute value of a times 00:16:28.630 --> 00:16:31.550 the norm of v, where this is a number. 00:16:34.920 --> 00:16:37.480 And remember these two norms are different. 00:16:37.480 --> 00:16:39.210 This is the norm of a vector. 00:16:39.210 --> 00:16:44.210 This is the normal of a complex number. 00:16:44.210 --> 00:16:51.090 And therefore, if you take lambda i u-- 00:16:51.090 --> 00:16:58.260 norm-- is lambda u is equal absolute value of lambda u, 00:16:58.260 --> 00:17:02.400 and absolute value of lambda is equal to 1 is the answer. 00:17:02.400 --> 00:17:09.599 So that's a simple unitary operator, but an important one. 00:17:09.599 --> 00:17:15.180 Another observation-- what are the vectors 00:17:15.180 --> 00:17:17.670 annihilated by this operator u? 00:17:24.150 --> 00:17:27.540 Zero-- it's the only vector, because any other vector that's 00:17:27.540 --> 00:17:30.570 nonzero has some length, so it's not killed. 00:17:30.570 --> 00:17:33.510 So it kills only zero. 00:17:33.510 --> 00:17:42.790 So the null space of S is equal to the 0 vector. 00:17:42.790 --> 00:17:48.480 So this operator has no kernel, nothing nontrivial 00:17:48.480 --> 00:17:49.820 is put to zero. 00:17:49.820 --> 00:17:52.740 It's an invertible operator. 00:17:52.740 --> 00:17:53.870 So s is invertible. 00:18:00.970 --> 00:18:05.680 So that's a few things that you get very cheaply. 00:18:05.680 --> 00:18:09.980 Now from this equation, S u equals 00:18:09.980 --> 00:18:12.270 u-- if you square that equation, you 00:18:12.270 --> 00:18:21.060 would have S u S u is equal to u u. 00:18:21.060 --> 00:18:22.700 Maybe I should probably call it v. 00:18:22.700 --> 00:18:26.890 I don't know why I called it u, but let's stick to u. 00:18:26.890 --> 00:18:31.980 Now, remember that we can move operators 00:18:31.980 --> 00:18:33.690 from one side to the other. 00:18:33.690 --> 00:18:36.970 So I'll move this one to that side. 00:18:36.970 --> 00:18:40.580 If you move an S here, you would put an S dagger. 00:18:40.580 --> 00:18:46.290 But since the dagger of an S dagger is S, 00:18:46.290 --> 00:18:50.860 you can move also the S to that side as S dagger. 00:18:50.860 --> 00:18:56.920 So u S dagger, S u-- you see that. 00:18:56.920 --> 00:18:58.940 If you want to move this one, you 00:18:58.940 --> 00:19:03.130 can move it by putting another dagger, and you get that one. 00:19:03.130 --> 00:19:07.910 And this is u u, and therefore you 00:19:07.910 --> 00:19:16.810 get u S dagger, S minus the identity acting on u 00:19:16.810 --> 00:19:19.210 is equal to 0 for all u. 00:19:22.210 --> 00:19:26.500 So for every vector, this is true, because this is true. 00:19:26.500 --> 00:19:29.200 We just squared it. 00:19:29.200 --> 00:19:32.100 And now you have our favorite theorem, 00:19:32.100 --> 00:19:35.020 that says if this is true in a complex vector space, 00:19:35.020 --> 00:19:37.830 this is 0, and therefore, you've shown 00:19:37.830 --> 00:19:41.035 that S dagger S is equal to 1. 00:19:44.680 --> 00:19:49.050 So that's another property of unitary operators. 00:19:49.050 --> 00:19:52.830 In fact that's the way it's many times defined. 00:19:52.830 --> 00:19:55.360 Unitary operators sometimes are said 00:19:55.360 --> 00:20:00.590 to be operators whose inverse is S dagger. 00:20:00.590 --> 00:20:02.710 I will not go into the subtleties 00:20:02.710 --> 00:20:07.270 of what steps in all these things I'm saying 00:20:07.270 --> 00:20:11.260 are true or not true for infinite dimensional 00:20:11.260 --> 00:20:15.340 operators-- infinite dimensional vector spaces. 00:20:15.340 --> 00:20:19.070 So I will assume, and it will be true in our examples, 00:20:19.070 --> 00:20:22.170 that if S dagger is an inverse from the left, 00:20:22.170 --> 00:20:23.880 it's also an inverse from the right. 00:20:23.880 --> 00:20:30.420 And perhaps everything is true for infinite dimensional vector 00:20:30.420 --> 00:20:35.240 spaces, but I'm not 100% positive. 00:20:35.240 --> 00:20:50.580 So S dagger is the inverse of S. And that's 00:20:50.580 --> 00:20:54.920 a pretty important thing. 00:20:54.920 --> 00:21:04.160 So one last comment on unitary operators has to do with basis. 00:21:04.160 --> 00:21:13.980 So suppose you have an orthonormal basis, e1 up to en. 00:21:17.880 --> 00:21:20.015 Now you can define another basis. 00:21:28.280 --> 00:21:33.550 fi equal-- I'll change to a letter 00:21:33.550 --> 00:21:39.910 U-- U e i where U is unitary, so it's 00:21:39.910 --> 00:21:45.720 like the S. In fact, most books in physics 00:21:45.720 --> 00:21:47.450 call it U for unitary. 00:21:47.450 --> 00:21:50.210 So maybe I should have changed that letter in there, too, 00:21:50.210 --> 00:21:51.340 as well. 00:21:51.340 --> 00:21:55.190 So suppose you change basis. 00:21:55.190 --> 00:22:02.580 You put-- oh, there was something else 00:22:02.580 --> 00:22:05.930 I wanted to say before. 00:22:05.930 --> 00:22:09.170 Thanks to this equation, consider now 00:22:09.170 --> 00:22:14.830 the following thing-- S U Sv. 00:22:19.530 --> 00:22:24.040 SUSv-- you can move this S, for example, 00:22:24.040 --> 00:22:30.370 to the other side-- S dagger S U v, 00:22:30.370 --> 00:22:34.160 and S dagger S is equal to 1, and it's Uv. 00:22:34.160 --> 00:22:36.770 So this is a pretty nice property. 00:22:36.770 --> 00:22:41.030 We started from the fact that it preserved 00:22:41.030 --> 00:22:45.200 the norm of a single vector, of all vectors, 00:22:45.200 --> 00:22:49.670 and now you see that in fact, it preserved the inner product. 00:22:49.670 --> 00:22:53.160 So if you have two vectors, to compare their inner product, 00:22:53.160 --> 00:22:57.700 compute them after action with U or before action with U, 00:22:57.700 --> 00:23:00.620 and it doesn't make a difference. 00:23:00.620 --> 00:23:05.670 So suppose you define a second basis here. 00:23:05.670 --> 00:23:07.630 You have one orthonormal basis. 00:23:07.630 --> 00:23:10.980 You define basis vectors like this. 00:23:10.980 --> 00:23:21.660 Then the claim is that the f1 up to fn is orthonormal. 00:23:21.660 --> 00:23:26.480 And for that you simply do the following-- you just 00:23:26.480 --> 00:23:36.840 check f i f j is equal to U e i, U e j. 00:23:36.840 --> 00:23:40.540 By this property, you can delete both U's, rules, 00:23:40.540 --> 00:23:45.550 and therefore this is e i, e j. 00:23:45.550 --> 00:23:48.800 And that's delta i j. 00:23:48.800 --> 00:23:50.920 So the new basis is orthonormal. 00:23:56.400 --> 00:23:59.480 If you play with these things, it's 00:23:59.480 --> 00:24:06.460 easy to get some extra curious fact here. 00:24:06.460 --> 00:24:15.150 Let's think of the matrix representation of the operator 00:24:15.150 --> 00:24:24.310 U. Well, we know how these things 00:24:24.310 --> 00:24:28.770 are, and let's think of this in the basis e basis. 00:24:28.770 --> 00:24:37.480 So U k i is equal to ek U e i. 00:24:40.800 --> 00:24:46.300 That's the definition of U in the basis 00:24:46.300 --> 00:24:58.160 e-- the matrix elements of U. You can try to figure out 00:24:58.160 --> 00:25:02.320 what is Uki in the f basis. 00:25:02.320 --> 00:25:06.106 How does operator U look in the f basis? 00:25:08.700 --> 00:25:12.320 Well, let's just do it without thinking. 00:25:12.320 --> 00:25:18.720 So in the f basis, I would put fk U fi. 00:25:21.850 --> 00:25:30.750 Well, but fk is U ek, so I'll put Uek Ufi. 00:25:34.080 --> 00:25:44.000 Now we can delete both U's, and it's ek fi. 00:25:44.000 --> 00:25:53.930 And I can remember what fi was, which is ek U ei. 00:25:53.930 --> 00:25:57.700 And it's just the same as the one we had there. 00:25:57.700 --> 00:26:00.790 So the operator, unitary operator, 00:26:00.790 --> 00:26:02.456 looks the same in both bases. 00:26:06.030 --> 00:26:09.690 That might seem strange or a coincidence, but it's not. 00:26:09.690 --> 00:26:12.200 So I leave it to you to think about, 00:26:12.200 --> 00:26:17.570 and visualize why did that happen. 00:26:17.570 --> 00:26:20.740 What's the reason? 00:26:20.740 --> 00:26:30.000 So the bracket notation-- we've been using it here and there-- 00:26:30.000 --> 00:26:34.650 and I will ask you to please read the notes. 00:26:34.650 --> 00:26:37.410 The notes will be posted this afternoon, 00:26:37.410 --> 00:26:41.180 and they will have-- not maybe all we've done today, 00:26:41.180 --> 00:26:44.360 but they will have some of what we'll do today, 00:26:44.360 --> 00:26:47.190 and all of what we've been doing. 00:26:47.190 --> 00:26:48.900 And the way it's done-- it's first 00:26:48.900 --> 00:26:51.800 done in this sort of inner product language, 00:26:51.800 --> 00:26:55.790 and then things are done in the bracket language. 00:26:55.790 --> 00:26:58.210 And it's a little repetitious, and I'm 00:26:58.210 --> 00:27:00.630 trying to take out some things here and there, 00:27:00.630 --> 00:27:02.250 so it's less repetitious. 00:27:02.250 --> 00:27:06.260 But at this moment it's probably worth reading it, 00:27:06.260 --> 00:27:07.600 and reading it again. 00:27:07.600 --> 00:27:08.748 Yes. 00:27:08.748 --> 00:27:13.230 AUDIENCE: [INAUDIBLE] if you have two orthonormal bases, 00:27:13.230 --> 00:27:17.370 is the transformation between them necessarily unitary? 00:27:17.370 --> 00:27:18.597 PROFESSOR: Yes, yes. 00:27:23.170 --> 00:27:23.850 All right. 00:27:23.850 --> 00:27:28.590 So as I was saying we're going to go into the Dirac notation 00:27:28.590 --> 00:27:29.090 again. 00:27:29.090 --> 00:27:33.160 And here's an example of a place where everybody, I think, 00:27:33.160 --> 00:27:35.940 tends to use Dirac notation. 00:27:35.940 --> 00:27:40.060 And the reason is a little curious, 00:27:40.060 --> 00:27:44.553 and you will appreciate it quite fast. 00:27:47.510 --> 00:27:50.050 So this will be the case of where 00:27:50.050 --> 00:27:57.273 we return to x and p operators, on a non-denumerable basis. 00:28:08.470 --> 00:28:12.830 So we're going to try to do x and p. 00:28:12.830 --> 00:28:15.730 now this is the classic of Dirac notation. 00:28:15.730 --> 00:28:17.890 It's probably-- as I said-- the place 00:28:17.890 --> 00:28:21.370 where everybody likes to use Dirac notation. 00:28:21.370 --> 00:28:24.750 And the reason it's efficient is because it prevents you 00:28:24.750 --> 00:28:27.850 from confusing two things. 00:28:27.850 --> 00:28:32.340 So I've written in the notes, and we 00:28:32.340 --> 00:28:36.150 have all these v's that belong to the vector space. 00:28:36.150 --> 00:28:38.580 And then we put this, and we still 00:28:38.580 --> 00:28:41.150 say it belongs to the vector space. 00:28:41.150 --> 00:28:46.310 And this is just a decoration that doesn't do much. 00:28:46.310 --> 00:28:48.050 And we can play with this. 00:28:48.050 --> 00:28:50.660 Now, in the non-denumerable basis, 00:28:50.660 --> 00:28:53.490 the catch-- and the possible confusion-- 00:28:53.490 --> 00:29:02.030 is that the label is not quite a vector in the vector space. 00:29:02.030 --> 00:29:06.490 So that is the reason why the notation is helpful, 00:29:06.490 --> 00:29:11.300 because it helps you distinguish two things that you 00:29:11.300 --> 00:29:13.150 could confuse. 00:29:13.150 --> 00:29:14.580 So here we go. 00:29:14.580 --> 00:29:19.080 We're going to talk about coordinate x, 00:29:19.080 --> 00:29:24.260 and the x operator, and the states. 00:29:24.260 --> 00:29:26.500 Well, this is a state space. 00:29:26.500 --> 00:29:30.360 So what kind of states do we have here? 00:29:30.360 --> 00:29:32.625 Well, we've talked about wave functions, 00:29:32.625 --> 00:29:34.250 and we could give the value of the wave 00:29:34.250 --> 00:29:36.430 function of different places. 00:29:36.430 --> 00:29:40.560 We're going to go for a more intrinsic definition. 00:29:40.560 --> 00:29:44.525 We're going to try to introduce position states. 00:29:50.970 --> 00:29:56.060 And position states will be called this-- x. 00:30:03.650 --> 00:30:07.060 Now, what is the meaning of this position state? 00:30:07.060 --> 00:30:15.490 We should think of this intuitively as a particle at x. 00:30:18.230 --> 00:30:21.760 Now here's how you can go wrong with this thing, 00:30:21.760 --> 00:30:25.860 if you stop thinking for a second. 00:30:32.660 --> 00:30:37.330 What is, then, ax? 00:30:37.330 --> 00:30:48.250 Is it ax, a being a number. 00:30:48.250 --> 00:30:51.140 Is it the same thing? 00:30:51.140 --> 00:30:53.630 No, not at all. 00:30:53.630 --> 00:30:58.930 This is a particle at the coordinate ax, 00:30:58.930 --> 00:31:04.530 and this is a particle at x with some different amplitude-- very 00:31:04.530 --> 00:31:05.260 different. 00:31:05.260 --> 00:31:08.500 So this is not true-- typical mistake. 00:31:08.500 --> 00:31:14.770 This is not minus x. 00:31:14.770 --> 00:31:16.450 That's totally different. 00:31:16.450 --> 00:31:19.595 So there's no such thing as this, either. 00:31:25.760 --> 00:31:29.310 It doesn't mean anything. 00:31:29.310 --> 00:31:34.990 And the reason is that these things are not our vectors. 00:31:34.990 --> 00:31:40.670 Our vector is this whole thing that says a particle at x. 00:31:40.670 --> 00:31:45.250 Maybe to make a clearer impression, 00:31:45.250 --> 00:31:47.310 imagine you're in three dimensions, 00:31:47.310 --> 00:31:50.210 and you have an x vector. 00:31:50.210 --> 00:31:53.965 So then you have to ket this. 00:31:56.690 --> 00:32:01.370 This is the ket particle at x. 00:32:01.370 --> 00:32:02.930 x is now a vector. 00:32:02.930 --> 00:32:05.320 It's a three-dimensional vector. 00:32:05.320 --> 00:32:08.270 This is a vector, but it's a vector 00:32:08.270 --> 00:32:11.910 in an infinite dimensional space, 00:32:11.910 --> 00:32:15.480 because the particle can be anywhere. 00:32:15.480 --> 00:32:18.990 So this is a vector in quantum mechanics. 00:32:18.990 --> 00:32:21.090 This is a complex vector space. 00:32:21.090 --> 00:32:25.880 This is a real vector space, and it's the label here. 00:32:25.880 --> 00:32:33.730 So again, minus x is not minus x vector. 00:32:33.730 --> 00:32:35.575 It's not the vector. 00:32:35.575 --> 00:32:41.060 The addition of the bra has moved you from vectors 00:32:41.060 --> 00:32:43.720 that you're familiar with, to states 00:32:43.720 --> 00:32:45.830 that are a little more abstract. 00:32:49.270 --> 00:32:52.990 So the reason this notation is quite good is because this is 00:32:52.990 --> 00:32:56.490 the number, but this i--- or this is a coordinate, 00:32:56.490 --> 00:33:00.090 and this is a vector already. 00:33:00.090 --> 00:33:02.572 So these are going to be our basis states, 00:33:02.572 --> 00:33:03.780 and they are non-denumerable. 00:33:09.790 --> 00:33:13.260 And here you can have that all x must 00:33:13.260 --> 00:33:16.390 belong to the real numbers, because we have particles 00:33:16.390 --> 00:33:22.270 in a line, while this thing can be changed by real numbers. 00:33:22.270 --> 00:33:24.982 The states can be multiplied by complex numbers, 00:33:24.982 --> 00:33:26.565 because we're doing quantum mechanics. 00:33:29.210 --> 00:33:33.260 So if you want to define a vector space-- now, 00:33:33.260 --> 00:33:34.990 this is all infinite dimension. 00:33:34.990 --> 00:33:36.730 It's a little worse in this sense 00:33:36.730 --> 00:33:39.150 the basis is non-denumerable. 00:33:39.150 --> 00:33:43.800 If I use this basis, I cannot make a list of all the basis 00:33:43.800 --> 00:33:46.540 vectors. 00:33:46.540 --> 00:33:49.930 So for an inner product, we will take the following-- 00:33:49.930 --> 00:33:58.220 we will take x with y to be delta of x minus y. 00:33:58.220 --> 00:34:00.090 That will be our inner product. 00:34:00.090 --> 00:34:06.550 And it has all the properties of the inner product 00:34:06.550 --> 00:34:09.250 that we may want. 00:34:09.250 --> 00:34:13.449 And what else? 00:34:13.449 --> 00:34:18.370 Well at this moment, we can try to-- this 00:34:18.370 --> 00:34:20.800 is physically sensible, let me say, 00:34:20.800 --> 00:34:23.830 because if you have a particle at one point 00:34:23.830 --> 00:34:26.159 and a particle at another point, the amplitude 00:34:26.159 --> 00:34:31.440 that this particle at one point is at this other point is 0. 00:34:31.440 --> 00:34:33.710 And these states are not normalizable. 00:34:33.710 --> 00:34:35.880 They correspond to a particle at the point, 00:34:35.880 --> 00:34:38.710 so once you try to normalize them, you get infinity, 00:34:38.710 --> 00:34:41.000 and you can't do much. 00:34:41.000 --> 00:34:45.179 But what you can do here is state more of the properties, 00:34:45.179 --> 00:34:46.860 and learn how to manipulate this. 00:34:46.860 --> 00:34:57.590 So remember we had one was the sum of all e i e i. 00:34:57.590 --> 00:35:00.890 The unit operator was that. 00:35:00.890 --> 00:35:05.490 Well, let's try to write a similar one. 00:35:05.490 --> 00:35:09.505 The unit operator will be the sum over all x's. 00:35:12.940 --> 00:35:16.640 And you could say, well, looks reasonable, 00:35:16.640 --> 00:35:20.210 but maybe there's a 1/2 in here, or some factor. 00:35:20.210 --> 00:35:22.260 Well, no factor is needed. 00:35:22.260 --> 00:35:30.240 You can check that-- that you've defined this thing properly. 00:35:30.240 --> 00:35:34.650 So let me do it. 00:35:34.650 --> 00:35:39.010 So act to on this so-called resolution 00:35:39.010 --> 00:35:48.060 of the identity with the vector y, so 1 on y is equal to y. 00:35:48.060 --> 00:35:51.777 And now let's add on the right xxy. 00:35:55.320 --> 00:35:58.670 This is delta of x minus y. 00:35:58.670 --> 00:36:02.300 And then when you integrate, you get y. 00:36:02.300 --> 00:36:04.750 So we're fine. 00:36:09.610 --> 00:36:13.120 So this looks a little too abstract, 00:36:13.120 --> 00:36:17.520 but it's not the abstract if you now introduce wave functions. 00:36:17.520 --> 00:36:19.830 So let's do wave functions. 00:36:19.830 --> 00:36:26.930 So you have a particle, a state of the particle psi. 00:36:26.930 --> 00:36:30.790 Time would be random, so I will put just this psi like that 00:36:30.790 --> 00:36:34.690 without the bottom line. 00:36:34.690 --> 00:36:37.350 And let's look at it. 00:36:37.350 --> 00:36:39.343 Oh, I want to say one more thing. 00:36:44.070 --> 00:36:52.500 The x operator acts on the x states to give x x. 00:36:52.500 --> 00:36:56.400 So these are eigenstates of the x operator. 00:36:56.400 --> 00:36:58.400 We declare them to be eigenstates 00:36:58.400 --> 00:37:00.680 of the x operator with eigenvalue x. 00:37:00.680 --> 00:37:04.330 That's their physical interpretation. 00:37:04.330 --> 00:37:08.430 I probably should have said before. 00:37:08.430 --> 00:37:15.030 Now, if we have a psi as a state or a vector, how do 00:37:15.030 --> 00:37:16.530 we get the wave function? 00:37:16.530 --> 00:37:19.860 Well, in this language the wave function, 00:37:19.860 --> 00:37:23.260 which we call psi of x, is defined 00:37:23.260 --> 00:37:27.365 to be the overlap of x with psi. 00:37:31.660 --> 00:37:37.810 And that makes sense, because this overlap 00:37:37.810 --> 00:37:42.420 is a function of this label here, where the particle is. 00:37:42.420 --> 00:37:46.310 And therefore, the result is a complex number 00:37:46.310 --> 00:37:49.200 that is dependent on x. 00:37:49.200 --> 00:37:51.590 So this belongs to the complex numbers, 00:37:51.590 --> 00:37:55.480 because inner products can have complex numbers. 00:37:55.480 --> 00:37:58.040 Now, I didn't put any complex number here, 00:37:58.040 --> 00:38:01.200 but when you form states, you can superpose states 00:38:01.200 --> 00:38:02.610 with complex numbers. 00:38:02.610 --> 00:38:07.060 So this psi of x will come out this way. 00:38:07.060 --> 00:38:11.110 And now that you are armed with that, 00:38:11.110 --> 00:38:14.440 you can even think of this in a nicer way. 00:38:14.440 --> 00:38:19.290 The state psi is equal to 1 times psi. 00:38:19.290 --> 00:38:22.740 And then use the rest of this formula, 00:38:22.740 --> 00:38:30.940 so this is integral-- dx x x psi. 00:38:30.940 --> 00:38:34.460 And again, the bracket notation is quite nice, 00:38:34.460 --> 00:38:37.360 because the bra already meets the ket. 00:38:37.360 --> 00:38:43.875 This is a number, and this is dx x psi of x. 00:38:46.780 --> 00:38:49.760 This equation has a nice interpretation. 00:38:49.760 --> 00:38:55.980 It says that the state is a superposition of the basis 00:38:55.980 --> 00:39:00.130 states, the position states, and the component 00:39:00.130 --> 00:39:03.690 of your original state along the basis state 00:39:03.690 --> 00:39:08.170 x is precisely the value of the wave function at x. 00:39:08.170 --> 00:39:10.270 So the wave function at x is giving you 00:39:10.270 --> 00:39:14.370 the weight of the state x as it enters into the sum. 00:39:17.310 --> 00:39:23.620 So one can compute more things. 00:39:23.620 --> 00:39:27.750 You will get practice in this type of computations. 00:39:27.750 --> 00:39:31.730 There are just a limited type of variations that you can do, 00:39:31.730 --> 00:39:37.120 so it's not that complicated. 00:39:37.120 --> 00:39:41.180 Basically, you can introduce resolutions of the identity 00:39:41.180 --> 00:39:42.710 wherever you need them. 00:39:42.710 --> 00:39:47.050 And if you introduce too many, you waste time, 00:39:47.050 --> 00:39:51.220 but you typically get the answer anyway. 00:39:51.220 --> 00:39:54.970 So it's not too serious. 00:39:54.970 --> 00:39:56.810 So suppose you want to understand 00:39:56.810 --> 00:39:59.213 what is the inner product of two states. 00:40:02.130 --> 00:40:05.000 Put the resolution of the identity in between. 00:40:05.000 --> 00:40:14.300 So put phi, and then put the integral dx x x psi. 00:40:14.300 --> 00:40:23.380 Well, the integral goes out, and you get phi x x psi. 00:40:23.380 --> 00:40:29.740 And remember, if x psi is psi of x, 00:40:29.740 --> 00:40:36.450 phi x is the complex conjugate, so it's phi star of x. 00:40:36.450 --> 00:40:39.240 And you knew that. 00:40:39.240 --> 00:40:41.100 If you have two wave functions, and you 00:40:41.100 --> 00:40:45.700 want to compute the overlap, you integrate the complex conjugate 00:40:45.700 --> 00:40:49.700 of one against the other. 00:40:49.700 --> 00:40:54.235 So this notation is doing all what you want from this. 00:40:58.040 --> 00:41:01.560 You want to compute a matrix element of x. 00:41:06.920 --> 00:41:12.920 Well, put another resolution of the identity here. 00:41:12.920 --> 00:41:19.740 So this would be integral dx phi-- the x hat is here. 00:41:19.740 --> 00:41:23.960 And then you put x x psi. 00:41:26.820 --> 00:41:30.670 The x hat on x is x. 00:41:30.670 --> 00:41:33.610 That's what this operator does, so you 00:41:33.610 --> 00:41:46.830 get integral dx of-- I'll put x phi x x psi, which 00:41:46.830 --> 00:41:50.160 is what you expect it to be-- integral 00:41:50.160 --> 00:41:55.260 of x phi star of x, psi of x. 00:42:00.990 --> 00:42:06.930 Now we can do exactly the same thing with momentum states. 00:42:06.930 --> 00:42:09.790 So I don't want to bore you, so I just 00:42:09.790 --> 00:42:15.520 list the properties-- basis states are 00:42:15.520 --> 00:42:19.830 momenta where the momenta is real. 00:42:19.830 --> 00:42:26.520 p prime p is equal delta of p minus p prime. 00:42:26.520 --> 00:42:33.900 One is the integral dp of p p. 00:42:33.900 --> 00:42:38.725 And p hat p is equal to p p. 00:42:44.170 --> 00:42:47.350 So these are the momentum bases. 00:42:47.350 --> 00:42:49.530 They're exactly analogous. 00:42:49.530 --> 00:42:53.600 So all what we've done for x is true. 00:42:53.600 --> 00:42:57.440 The completeness and normalization 00:42:57.440 --> 00:42:59.930 work well together, like we checked there, 00:42:59.930 --> 00:43:02.610 and everything is true. 00:43:02.610 --> 00:43:07.830 The only thing that you need to make this more interesting 00:43:07.830 --> 00:43:12.860 is a relation between the x basis and the p basis. 00:43:12.860 --> 00:43:15.440 And that's where physics comes in. 00:43:15.440 --> 00:43:20.250 Anybody can define these two, but then a physical assumption 00:43:20.250 --> 00:43:27.560 as to what you really mean by momentum is necessary. 00:43:27.560 --> 00:43:31.070 And what we've said is that the wave function 00:43:31.070 --> 00:43:36.091 of a particle with momentum p is e to the i 00:43:36.091 --> 00:43:42.370 px over h bar over square root of 2 pi 00:43:42.370 --> 00:43:49.280 h-- convenient normalization, but that was it. 00:43:49.280 --> 00:43:52.090 That was our physical interpretation 00:43:52.090 --> 00:43:55.010 of the wave function of a particle with some momentum. 00:43:55.010 --> 00:43:58.665 And therefore, if this is a wave function, that's xp. 00:44:01.340 --> 00:44:06.650 A state of momentum p has this wave function. 00:44:06.650 --> 00:44:08.400 So we write this. 00:44:16.180 --> 00:44:21.130 OK, there are tricks you can do, and please read the notes. 00:44:21.130 --> 00:44:24.270 But let's do a little computation. 00:44:24.270 --> 00:44:29.930 Suppose you want to compute what is p on psi. 00:44:29.930 --> 00:44:32.350 You could say, well, I don't know 00:44:32.350 --> 00:44:34.770 why would I want to do something with that? 00:44:34.770 --> 00:44:36.370 Looks simple enough. 00:44:36.370 --> 00:44:38.000 Well, it's simple enough, but you 00:44:38.000 --> 00:44:42.840 could say I want to see that in terms of wave functions, 00:44:42.840 --> 00:44:44.660 coordinate space wave functions. 00:44:44.660 --> 00:44:47.530 Well, if you want to see them in terms of coordinate space wave 00:44:47.530 --> 00:44:51.330 functions, you have to introduce a complete set of states. 00:44:51.330 --> 00:44:58.430 So introduce p x x psi. 00:44:58.430 --> 00:45:00.240 Then you have this wave function, 00:45:00.240 --> 00:45:03.380 and oh, this is sort of known, because it's 00:45:03.380 --> 00:45:10.566 the complex conjugate of this, so it's integral dx 00:45:10.566 --> 00:45:19.300 px over h bar, square root of 2 pi h bar times psi of x. 00:45:19.300 --> 00:45:22.140 And this was the Fourier transform-- 00:45:22.140 --> 00:45:26.260 what we call the Fourier transform of the wave function. 00:45:26.260 --> 00:45:33.840 So we can call it psi tilde of p, just to distinguish it, 00:45:33.840 --> 00:45:37.320 because we called psi with x, psi of x. 00:45:37.320 --> 00:45:39.190 So if I didn't put a tilde, you might 00:45:39.190 --> 00:45:41.450 think it's the same functional form, 00:45:41.450 --> 00:45:45.400 but it's the momentum space wave function. 00:45:45.400 --> 00:45:51.180 So here is the wave function in the p basis. 00:45:51.180 --> 00:45:53.690 It's the Fourier transform of the wave 00:45:53.690 --> 00:45:57.220 function in the x basis. 00:45:57.220 --> 00:46:01.020 One last computation, and then we change subjects again. 00:46:05.710 --> 00:46:10.360 It's the classic computation that you 00:46:10.360 --> 00:46:15.530 have now a mixed situation, in which you have the momentum 00:46:15.530 --> 00:46:20.430 operator states and the coordinate bra. 00:46:20.430 --> 00:46:34.753 So what is the following expression-- X p hat psi? 00:46:44.696 --> 00:46:45.195 OK. 00:46:49.900 --> 00:46:51.750 What is your temptation? 00:46:51.750 --> 00:46:57.170 Your temptation is to say, look, this is like the momentum 00:46:57.170 --> 00:47:02.490 operator acting on the wave function in the x basis. 00:47:02.490 --> 00:47:09.790 It can only be h bar over i d dx of psi of x. 00:47:09.790 --> 00:47:13.020 That's probably what it means. 00:47:13.020 --> 00:47:15.730 But the notation is clear enough, 00:47:15.730 --> 00:47:19.120 so we can check if that is exactly what it is. 00:47:19.120 --> 00:47:22.360 We can manipulate things already. 00:47:22.360 --> 00:47:24.250 So let's do it. 00:47:24.250 --> 00:47:28.900 So for that, I first have to try to get rid of this operator. 00:47:28.900 --> 00:47:32.840 Now the only way I know how to get rid of this operator p 00:47:32.840 --> 00:47:36.090 is because it has eigenstates. 00:47:36.090 --> 00:47:40.890 So it suggests very strongly that we should introduce 00:47:40.890 --> 00:47:43.430 momentum states, complete them. 00:47:43.430 --> 00:47:53.080 So I'll put v p x p hat p p psi. 00:48:00.660 --> 00:48:03.770 And now I can evaluate the little-- 00:48:03.770 --> 00:48:09.050 because p hat and p is little p, or p without the hat. 00:48:09.050 --> 00:48:17.570 So this is p xp p psi. 00:48:22.470 --> 00:48:28.990 Now you can look at that, and think 00:48:28.990 --> 00:48:32.800 carefully what should you do. 00:48:32.800 --> 00:48:35.250 And there's one thing that you can 00:48:35.250 --> 00:48:40.200 do is look at the equation on top. 00:48:40.200 --> 00:48:44.100 And this is a way to avoid working very hard. 00:48:44.100 --> 00:48:48.250 So look at the equation on top-- x p is equal to that. 00:48:48.250 --> 00:48:53.240 How do I get a p to multiply this? 00:48:53.240 --> 00:48:57.390 I can get a p to multiply this xp 00:48:57.390 --> 00:49:06.530 by doing h bar over i d dx of x p. 00:49:11.930 --> 00:49:17.790 Because if I see it there, I see that differentiating by d dx 00:49:17.790 --> 00:49:21.160 brings down an ip over h bar. 00:49:21.160 --> 00:49:24.760 So if I multiply by h bar over i, I get that. 00:49:24.760 --> 00:49:26.055 So let's do this. 00:49:36.670 --> 00:49:43.730 Now I claim we can take the h over i d 00:49:43.730 --> 00:49:47.690 dx out of this integral. 00:49:47.690 --> 00:49:51.380 And the reason is that first, it's not an x integral. 00:49:51.380 --> 00:49:55.440 It's a p integral, and nothing else except this factor 00:49:55.440 --> 00:49:56.630 depends on x. 00:49:56.630 --> 00:49:59.170 So I take it out and I want to bring it back, 00:49:59.170 --> 00:50:01.970 it will only act on this, because this is not 00:50:01.970 --> 00:50:03.920 x dependent. 00:50:03.920 --> 00:50:10.930 So you should think of psi, psi doesn't have an x dependence. 00:50:10.930 --> 00:50:15.050 Psi is a state, and here is p-- doesn't have an x dependence? 00:50:15.050 --> 00:50:18.280 You say no, it does, it looks here. 00:50:18.280 --> 00:50:22.060 No, but it doesn't have it, because it's been integrated. 00:50:22.060 --> 00:50:24.030 It really doesn't have x dependence. 00:50:24.030 --> 00:50:27.140 So we can take this out. 00:50:27.140 --> 00:50:31.490 We'll have h over i d dx. 00:50:31.490 --> 00:50:37.140 And now we have vp x p p psi. 00:50:40.750 --> 00:50:44.850 And now by completeness, this is just 1. 00:50:44.850 --> 00:50:47.070 So this becomes x psi. 00:50:47.070 --> 00:50:55.750 So h bar over i d dx of x psi, which is what we claimed it 00:50:55.750 --> 00:50:56.340 would be. 00:50:59.480 --> 00:51:06.083 So this is rigorous-- a rigorous derivation. 00:51:08.880 --> 00:51:11.260 There's no guessing. 00:51:11.260 --> 00:51:13.670 We've introduced complete states until you 00:51:13.670 --> 00:51:17.490 can see how things act. 00:51:17.490 --> 00:51:19.860 But the moral is here that you shouldn't 00:51:19.860 --> 00:51:22.730 have to go through this more than once in your life, 00:51:22.730 --> 00:51:23.750 or practice it. 00:51:23.750 --> 00:51:27.270 But once you see something like that, you think. 00:51:27.270 --> 00:51:29.780 You're using x representation, and you're 00:51:29.780 --> 00:51:32.040 talking about the operator p. 00:51:32.040 --> 00:51:35.140 It cannot be anything like that. 00:51:35.140 --> 00:51:39.650 If you want to practice something different, 00:51:39.650 --> 00:51:49.860 show that the analogue p x hat psi is equal i h bar d dp 00:51:49.860 --> 00:51:52.920 of psi tilde. 00:51:52.920 --> 00:51:55.670 So it's the opposite relation. 00:51:59.140 --> 00:52:00.550 All right. 00:52:00.550 --> 00:52:01.050 Questions? 00:52:04.240 --> 00:52:05.212 Yes. 00:52:05.212 --> 00:52:09.256 AUDIENCE: So how's one supposed to-- so what it appears 00:52:09.256 --> 00:52:11.755 is happening is you're basically taking some state like psi, 00:52:11.755 --> 00:52:15.486 and you're basically writing in terms of some basis. 00:52:15.486 --> 00:52:18.430 And then you're basically using the [INAUDIBLE] coordinates 00:52:18.430 --> 00:52:19.470 of this thing. 00:52:19.470 --> 00:52:23.780 But the question is, what does this basis actually look like? 00:52:23.780 --> 00:52:27.100 Like, what do these vectors-- because if you put them 00:52:27.100 --> 00:52:29.400 in their own coordinates, they're just infinite. 00:52:29.400 --> 00:52:30.430 PROFESSOR: Yup. 00:52:30.430 --> 00:52:32.290 AUDIENCE: They're not even delta-- I mean-- 00:52:32.290 --> 00:52:33.790 PROFESSOR: They are delta functions. 00:52:33.790 --> 00:52:34.980 AUDIENCE: [INAUDIBLE] 00:52:34.980 --> 00:52:36.980 PROFESSOR: These vectors are delta functions 00:52:36.980 --> 00:52:43.720 because if you have a state that has this as the position 00:52:43.720 --> 00:52:45.550 state of a particle, you find the wave 00:52:45.550 --> 00:52:47.580 function by doing x on it. 00:52:47.580 --> 00:52:50.120 That's our definition of a wave function. 00:52:50.120 --> 00:52:52.670 And its infinite. 00:52:52.670 --> 00:52:59.150 So there's is not too much one can say about this. 00:52:59.150 --> 00:53:02.830 If people want to work more mathematically, 00:53:02.830 --> 00:53:05.155 the more comfortable way, what you do 00:53:05.155 --> 00:53:07.950 is, instead of taking infinite things, 00:53:07.950 --> 00:53:10.540 you put everything on a big circle. 00:53:10.540 --> 00:53:13.220 And then you have a Fourier series 00:53:13.220 --> 00:53:18.400 and they transform as sums, and everything goes into sums. 00:53:18.400 --> 00:53:20.210 But there's no real need. 00:53:20.210 --> 00:53:22.510 These operations are safe. 00:53:22.510 --> 00:53:27.920 And we managed to do them, and we're OK with them. 00:53:27.920 --> 00:53:30.150 Other questions? 00:53:30.150 --> 00:53:31.334 Yes. 00:53:31.334 --> 00:53:32.209 AUDIENCE: [INAUDIBLE] 00:53:38.926 --> 00:53:39.925 PROFESSOR: Probably not. 00:53:39.925 --> 00:53:45.000 You know, infinite bases are delicate. 00:53:45.000 --> 00:53:48.510 Hilbert spaces are infinite dimensional vector spaces, 00:53:48.510 --> 00:53:53.670 and they-- not every infinite dimensional space 00:53:53.670 --> 00:53:55.550 is a Hilbert space. 00:53:55.550 --> 00:53:57.930 The most important thing of a Hilbert space 00:53:57.930 --> 00:54:00.840 is this norm, this inner product. 00:54:00.840 --> 00:54:05.980 But the other important thing is some convergence facts 00:54:05.980 --> 00:54:09.170 about sequences of vectors that converge 00:54:09.170 --> 00:54:11.220 to points that are on the space. 00:54:11.220 --> 00:54:12.900 So it's delicate. 00:54:12.900 --> 00:54:15.880 Infinite dimensional spaces can be pretty bad. 00:54:15.880 --> 00:54:19.550 A Banach space is not a Hilbert space. 00:54:19.550 --> 00:54:20.190 It's more com-- 00:54:20.190 --> 00:54:21.800 AUDIENCE: [INAUDIBLE] 00:54:21.800 --> 00:54:24.640 PROFESSOR: Only for Hilbert spaces, 00:54:24.640 --> 00:54:31.505 and basically, this problem of a particle in a line, 00:54:31.505 --> 00:54:35.630 or a particle in three space is sufficiently well known 00:54:35.630 --> 00:54:40.410 that we're totally comfortable with this somewhat 00:54:40.410 --> 00:54:42.100 singular operation. 00:54:42.100 --> 00:54:46.110 So the operator x or the operator p 00:54:46.110 --> 00:54:48.960 may not be what mathematicians like them 00:54:48.960 --> 00:54:51.950 to be-- bounded operators in Hilbert spaces. 00:54:51.950 --> 00:54:56.260 But we know how not to make mistakes with them. 00:54:56.260 --> 00:55:00.230 And if you have a very subtle problem, one day 00:55:00.230 --> 00:55:01.950 you probably have to be more careful. 00:55:01.950 --> 00:55:04.740 But for the problems we're interested in now, we don't. 00:55:07.300 --> 00:55:15.270 So our last topic today is uncertainties and uncertainty 00:55:15.270 --> 00:55:17.040 relations. 00:55:17.040 --> 00:55:21.095 I probably won't get through all of it, but we'll get started. 00:55:27.340 --> 00:55:29.750 And so we'll have uncertainties. 00:55:38.800 --> 00:55:43.850 And we will talk about operators, 00:55:43.850 --> 00:55:46.760 and Hermitian operators. 00:55:46.760 --> 00:55:49.560 So here is the question, basically-- 00:55:49.560 --> 00:55:53.550 if you have a state, we know the result 00:55:53.550 --> 00:55:57.380 of a measurement of an observable 00:55:57.380 --> 00:56:02.090 is the eigenvalue of a Hermitian operator. 00:56:02.090 --> 00:56:06.730 Now, if the state is an eigenstate of the Hermitian 00:56:06.730 --> 00:56:09.100 operator, you measure the observable, 00:56:09.100 --> 00:56:12.860 and out comes eigenvalue. 00:56:12.860 --> 00:56:17.290 And there's no uncertainty in the measured observable, 00:56:17.290 --> 00:56:20.000 because the measured observable is an eigenvalue 00:56:20.000 --> 00:56:23.420 and its state is an eigenstate. 00:56:23.420 --> 00:56:26.480 The problem arises when the state that you're 00:56:26.480 --> 00:56:30.600 trying to measure this property is not 00:56:30.600 --> 00:56:34.240 an eigenstate of the observable. 00:56:34.240 --> 00:56:36.510 So you know that the interpretation of quantum 00:56:36.510 --> 00:56:40.040 mechanics is a probabilistic distribution. 00:56:40.040 --> 00:56:43.610 You sometimes get one thing, sometimes get another thing, 00:56:43.610 --> 00:56:46.280 depending on the amplitudes of the states 00:56:46.280 --> 00:56:49.500 to be in those particular eigenstates. 00:56:49.500 --> 00:56:51.450 But there's an uncertainty. 00:56:51.450 --> 00:56:54.420 At this time, you don't know what 00:56:54.420 --> 00:56:57.690 the measured value will be. 00:56:57.690 --> 00:57:01.620 So we'll define the uncertainty associated to a Hermitian 00:57:01.620 --> 00:57:05.855 operator, and we want to define this uncertainty. 00:57:05.855 --> 00:57:08.000 So A will be a Hermitian operator. 00:57:16.890 --> 00:57:20.400 And you were talking about the uncertainty. 00:57:20.400 --> 00:57:22.812 Now the uncertainty of that operator-- 00:57:22.812 --> 00:57:24.770 the first thing that you should remember is you 00:57:24.770 --> 00:57:26.870 can't talk about the uncertainty of the operator 00:57:26.870 --> 00:57:29.500 unless you give me a state. 00:57:29.500 --> 00:57:32.370 So all the formulas we're going to write for uncertainties 00:57:32.370 --> 00:57:36.790 are uncertainties of operators in some state. 00:57:36.790 --> 00:57:39.635 So let's call the state psi. 00:57:47.250 --> 00:57:50.510 And time will not be relevant, so maybe I 00:57:50.510 --> 00:57:56.890 should delete the-- well, I'll leave that bar there, 00:57:56.890 --> 00:57:58.470 just in case. 00:57:58.470 --> 00:58:04.300 So we're going to try to define uncertainty. 00:58:04.300 --> 00:58:08.970 But before we do that, let's try to define 00:58:08.970 --> 00:58:11.350 another thing-- the expectation value. 00:58:11.350 --> 00:58:14.240 Well, the expectation value-- you know it. 00:58:14.240 --> 00:58:17.740 The expectation value of A, and you could put a psi here 00:58:17.740 --> 00:58:24.600 if you wish, to remind you that it depends on the state-- is, 00:58:24.600 --> 00:58:29.080 well, psi A psi. 00:58:29.080 --> 00:58:31.520 That's what we call expectation value. 00:58:31.520 --> 00:58:36.710 In the inner product notation would be psi A psi. 00:58:41.520 --> 00:58:48.570 And one thing you know-- that this thing is real, 00:58:48.570 --> 00:58:55.490 because the expectation values of Hermitian operators is real. 00:58:55.490 --> 00:58:58.280 That's something we reviewed at the beginning of the lecture 00:58:58.280 --> 00:59:01.080 today. 00:59:01.080 --> 00:59:06.070 So now comes the question, what can I 00:59:06.070 --> 00:59:10.240 do to define an uncertainty of an operator? 00:59:10.240 --> 00:59:15.120 And an uncertainty-- now we've said already something. 00:59:15.120 --> 00:59:18.970 I wish to define an uncertainty that is such 00:59:18.970 --> 00:59:25.310 that the uncertainty is 0 if the state is an eigenstate, 00:59:25.310 --> 00:59:27.720 and the uncertainty is different from 0 00:59:27.720 --> 00:59:29.910 if it's not an eigenstate. 00:59:29.910 --> 00:59:33.850 In fact, I wish that the uncertainty is 0 if 00:59:33.850 --> 00:59:38.130 and only if the state is an eigenstate. 00:59:38.130 --> 00:59:41.700 So actually, we can achieve that. 00:59:41.700 --> 00:59:50.140 And in some sense, I think, the most intuitive definition 00:59:50.140 --> 00:59:55.670 is the one that I will show here. 00:59:55.670 --> 01:00:00.080 It's that we define the uncertainty, delta A, 01:00:00.080 --> 01:00:01.950 and I'll put the psi here. 01:00:01.950 --> 01:00:10.060 So this is called the uncertainty of A 01:00:10.060 --> 01:00:11.285 in the state psi. 01:00:17.330 --> 01:00:19.390 So we'll define it a simple way. 01:00:19.390 --> 01:00:20.720 What else do we want? 01:00:20.720 --> 01:00:25.010 We said this should be 0 if and only 01:00:25.010 --> 01:00:28.490 if the state is an eigenstate. 01:00:28.490 --> 01:00:32.900 Second, I want this thing to be a real number-- 01:00:32.900 --> 01:00:35.790 in fact, a positive number. 01:00:35.790 --> 01:00:40.990 What function do we know in quantum mechanics 01:00:40.990 --> 01:00:43.000 that can do that magic? 01:00:43.000 --> 01:00:44.940 Well, it's the norm. 01:00:44.940 --> 01:00:48.610 The norm function is always real and positive. 01:00:48.610 --> 01:00:52.860 So this-- we'll try to set it equal to a norm. 01:00:52.860 --> 01:01:00.870 So it's the norm of the state A minus the expectation 01:01:00.870 --> 01:01:08.240 value of A times 1 acting on psi. 01:01:08.240 --> 01:01:12.170 This will be our definition of the uncertainty. 01:01:16.010 --> 01:01:18.233 So it's the norm of this vector. 01:01:23.280 --> 01:01:25.700 Now let's look at this. 01:01:25.700 --> 01:01:32.730 Suppose the norm uncertainty is 0. 01:01:36.230 --> 01:01:40.180 And if the uncertainty is 0, this vector must be 0. 01:01:40.180 --> 01:01:49.960 So A minus expectation value of A on psi is 0. 01:01:49.960 --> 01:02:00.330 Or A psi is equal to expectation value of A on psi. 01:02:00.330 --> 01:02:02.090 The 1 doesn't do much. 01:02:02.090 --> 01:02:04.460 Many people don't write the 1. 01:02:04.460 --> 01:02:08.030 I could get tired and stop writing it. 01:02:08.030 --> 01:02:13.630 You should-- probably it's good manners to write the i, 01:02:13.630 --> 01:02:16.910 but it's not all that necessary. 01:02:16.910 --> 01:02:18.500 You don't get that confused. 01:02:18.500 --> 01:02:21.970 If there's an operator and a number here, 01:02:21.970 --> 01:02:24.310 it must be an identity matrix. 01:02:24.310 --> 01:02:28.880 So the uncertainty is 0, the vector is 0, then this is true. 01:02:28.880 --> 01:02:32.200 Now, you say, well, this equation looks kind of funny, 01:02:32.200 --> 01:02:37.270 but it says that psi is an eigenstate of A, 01:02:37.270 --> 01:02:40.230 because this is a number. 01:02:40.230 --> 01:02:42.300 It looks a little funny, because we're 01:02:42.300 --> 01:02:46.580 accustomed to A psi lambda psi, but this is a number. 01:02:46.580 --> 01:02:49.410 And in fact, let me show you one thing. 01:02:49.410 --> 01:02:57.320 If you have A psi equal lambda psi-- oh, I 01:02:57.320 --> 01:03:00.426 should say here that psi is normalized. 01:03:04.220 --> 01:03:08.010 If psi would not be normalized, you change the normalization. 01:03:08.010 --> 01:03:10.765 You change the uncertainty. 01:03:10.765 --> 01:03:11.890 So it should be normalized. 01:03:14.530 --> 01:03:19.980 And look at this-- if you have a psi equal lambda psi, 01:03:19.980 --> 01:03:27.130 do the inner product with psi. 01:03:27.130 --> 01:03:33.510 Psi comma A psi would be equal to lambda, 01:03:33.510 --> 01:03:36.810 because psi inner product with psi is 1. 01:03:36.810 --> 01:03:38.310 But what is this? 01:03:38.310 --> 01:03:41.730 This is the expectation value of A. 01:03:41.730 --> 01:03:46.010 So actually, given our definition, 01:03:46.010 --> 01:03:50.750 the eigenvalue of some operator on this state 01:03:50.750 --> 01:03:53.720 is the expectation value of the operator in the state. 01:03:53.720 --> 01:03:58.910 So back to the argument-- if the uncertainty is 0, 01:03:58.910 --> 01:04:01.000 the state is an eigenstate. 01:04:01.000 --> 01:04:06.130 And the eigenvalue happens to be the expectation value-- that 01:04:06.130 --> 01:04:07.480 is, if the uncertainty is 0. 01:04:07.480 --> 01:04:12.220 On the other hand, if you are in an eigenstate, you're here. 01:04:12.220 --> 01:04:15.180 Then lambda is A, and this equation 01:04:15.180 --> 01:04:18.470 shows that this vector is 0, and therefore you get 0. 01:04:18.470 --> 01:04:25.860 So you've shown that this norm or this uncertainty is 0, if 01:04:25.860 --> 01:04:29.820 and only if the state is an eigenstate. 01:04:29.820 --> 01:04:32.550 And that's a very powerful statement. 01:04:32.550 --> 01:04:36.130 The statement that's always known by everybody 01:04:36.130 --> 01:04:40.480 is that if you have an eigenstate-- yes-- 01:04:40.480 --> 01:04:41.450 no uncertainty. 01:04:41.450 --> 01:04:45.700 But if there's no uncertainty, you must have an eigenstate. 01:04:45.700 --> 01:04:48.360 That's the second part, and uses the fact 01:04:48.360 --> 01:04:53.420 that the only vector with 0 norm is the zero vector-- a thing 01:04:53.420 --> 01:04:56.490 that we use over and over again. 01:04:56.490 --> 01:05:00.930 So let me make a couple more comments 01:05:00.930 --> 01:05:02.430 on how you compute this. 01:05:05.020 --> 01:05:09.525 So that's the uncertainty so far. 01:05:15.620 --> 01:05:17.970 So the uncertainty vanishes in that case. 01:05:17.970 --> 01:05:24.420 Now, we can square this equation to find a formula that 01:05:24.420 --> 01:05:30.130 is perhaps more familiar-- not necessarily more useful, 01:05:30.130 --> 01:05:31.440 but also good. 01:05:31.440 --> 01:05:34.500 For computations, it's pretty good-- delta A 01:05:34.500 --> 01:05:38.180 of psi, which is real-- we square it. 01:05:38.180 --> 01:05:42.100 Well, the norm square is the inner product 01:05:42.100 --> 01:05:58.560 of this A minus A psi A minus A psi. 01:05:58.560 --> 01:06:02.090 Norm squared is the inner product of these two vectors. 01:06:02.090 --> 01:06:04.800 Now, the thing that we like to do 01:06:04.800 --> 01:06:09.420 is to move this factor to that side. 01:06:09.420 --> 01:06:13.650 How do you move a factor on the first input to the other input? 01:06:13.650 --> 01:06:16.600 You take the adjoint. 01:06:16.600 --> 01:06:18.660 So I should move it with an adjoint. 01:06:18.660 --> 01:06:20.370 So what do I get? 01:06:20.370 --> 01:06:31.210 Psi, and then I get the adjoint and this factor again. 01:06:39.010 --> 01:06:44.320 Now, I should put a dagger here, but let me not put it, 01:06:44.320 --> 01:06:47.990 because A is Hermitian. 01:06:47.990 --> 01:06:53.800 And moreover, expectation value of A is real. 01:06:53.800 --> 01:06:57.140 Remember-- so no need for the dagger, 01:06:57.140 --> 01:06:58.920 so you can put the dagger, and then 01:06:58.920 --> 01:07:02.290 explain that this is Hermitian and this is real-- 01:07:02.290 --> 01:07:05.910 or just not put it. 01:07:05.910 --> 01:07:08.700 And now look at this. 01:07:08.700 --> 01:07:10.580 This is a typical calculation. 01:07:10.580 --> 01:07:12.255 You'll do it many, many times. 01:07:14.780 --> 01:07:18.440 You just spread out the things. 01:07:18.440 --> 01:07:20.800 So let me just do it once. 01:07:20.800 --> 01:07:27.360 Here you get A squared minus A expectation value of A minus 01:07:27.360 --> 01:07:32.960 expectation value of A A plus expectation 01:07:32.960 --> 01:07:36.135 value of A squared psi. 01:07:41.030 --> 01:07:44.850 So I multiplied everything, but you shouldn't be all 01:07:44.850 --> 01:07:49.820 that-- I should put a 1 here, probably-- 01:07:49.820 --> 01:07:51.450 shouldn't worry about this much. 01:07:51.450 --> 01:07:54.850 This is just a number and an A, a number and an A. 01:07:54.850 --> 01:07:56.010 The order doesn't matter. 01:07:56.010 --> 01:07:59.120 These two terms are really the same. 01:07:59.120 --> 01:08:04.520 Well, let me go slowly on this once. 01:08:04.520 --> 01:08:06.020 What is the first term? 01:08:06.020 --> 01:08:09.770 It's psi A squared psi, so it's the expectation value 01:08:09.770 --> 01:08:12.870 of A squared. 01:08:12.870 --> 01:08:15.260 Now, what is this term? 01:08:15.260 --> 01:08:18.960 Well, you have a number here, which is real. 01:08:18.960 --> 01:08:23.220 It goes out of whatever you're doing, and you have psi A psi. 01:08:23.220 --> 01:08:26.090 So this is expectation value of A. 01:08:26.090 --> 01:08:28.590 And from the leftover psi A psi, you 01:08:28.590 --> 01:08:36.640 get another expectation value of A. So this is A A. 01:08:36.640 --> 01:08:39.430 Here the same thing-- the number goes out, 01:08:39.430 --> 01:08:42.220 and you're left with a psi A psi, which 01:08:42.220 --> 01:08:49.010 is another expectation value of A, so you get minus A A. 01:08:49.010 --> 01:08:53.430 And you have a plus expectation value of A squared. 01:08:53.430 --> 01:08:56.890 And I don't need the i anymore, because the expectation values 01:08:56.890 --> 01:08:58.490 have been taken. 01:08:58.490 --> 01:09:00.689 And this always happens. 01:09:00.689 --> 01:09:03.920 It's a minus here, a minus here, and a plus 01:09:03.920 --> 01:09:06.964 here, so there's just one minus at the end of the day. 01:09:16.100 --> 01:09:21.529 One minus at the end of the day, and a familiar, or famous 01:09:21.529 --> 01:09:31.779 formula comes out that delta of A on psi squared 01:09:31.779 --> 01:09:36.250 is equal to the expectation value of A squared minus 01:09:36.250 --> 01:09:41.420 expectation value of A squared. 01:09:41.420 --> 01:09:44.430 Which shows something quite powerful. 01:09:44.430 --> 01:09:48.120 This has connections, of course, with statistical mechanics 01:09:48.120 --> 01:09:49.193 and standard deviations. 01:09:52.920 --> 01:09:56.390 It's a probabilistic interpretation of this formula, 01:09:56.390 --> 01:10:00.940 but one fact that this has allowed us to prove 01:10:00.940 --> 01:10:04.050 is that the expectation value of A squared 01:10:04.050 --> 01:10:06.550 is always greater or equal than that, 01:10:06.550 --> 01:10:10.380 because this number is positive, because it 01:10:10.380 --> 01:10:13.670 is the square of a real positive number. 01:10:13.670 --> 01:10:17.610 So that's a slightly non-trivial thing, 01:10:17.610 --> 01:10:19.256 and it's good to know it. 01:10:23.290 --> 01:10:26.380 And this formula, of course, is very well known. 01:10:29.020 --> 01:10:36.030 Now, I'm going to leave a funny geometrical interpretation 01:10:36.030 --> 01:10:37.020 of the uncertainty. 01:10:37.020 --> 01:10:40.110 Maybe you will find it illuminating, 01:10:40.110 --> 01:10:42.580 in some ways turning into pictures 01:10:42.580 --> 01:10:45.460 all these calculations we've done. 01:10:45.460 --> 01:10:48.460 I think it actually adds value to it, 01:10:48.460 --> 01:10:52.680 and I don't think it's very well known, 01:10:52.680 --> 01:10:57.380 or it's kind of funny, because it must not be very well known. 01:10:57.380 --> 01:11:01.630 But maybe people don't find it that suggestive. 01:11:01.630 --> 01:11:04.020 I kind of find it suggestive. 01:11:04.020 --> 01:11:09.550 So here's what I want to say geometrically. 01:11:09.550 --> 01:11:12.110 You have this vector space, and you have a vector psi. 01:11:17.770 --> 01:11:22.180 Then you come along, and you add with the operator 01:11:22.180 --> 01:11:27.890 A. Now the fact that this thing is not 01:11:27.890 --> 01:11:32.510 and eigenstate means that after you add with A, 01:11:32.510 --> 01:11:35.500 you don't keep in the same direction. 01:11:35.500 --> 01:11:37.180 You go in different directions. 01:11:37.180 --> 01:11:41.670 So here is A psi. 01:11:48.350 --> 01:11:50.630 So what can we say here? 01:11:50.630 --> 01:11:55.700 Well, actually here is this thing. 01:11:55.700 --> 01:12:00.030 Think of this vector space spanned by psi. 01:12:03.740 --> 01:12:06.850 Let's call it U psi. 01:12:06.850 --> 01:12:10.860 So it's that line there. 01:12:10.860 --> 01:12:16.680 You can project this in here, orthogonally. 01:12:23.930 --> 01:12:27.030 Here is the first claim-- the vector 01:12:27.030 --> 01:12:30.130 that you get up to here-- this vector-- is 01:12:30.130 --> 01:12:37.210 nothing else but expectation value of A times psi. 01:12:37.210 --> 01:12:40.140 And that makes sense, because it's a number times psi. 01:12:40.140 --> 01:12:44.120 But precisely the orthogonal projection is this. 01:12:44.120 --> 01:12:46.850 And here, you get an orthogonal vector. 01:12:46.850 --> 01:12:48.740 We'll call it psi perp. 01:12:52.830 --> 01:12:56.000 And the funny thing about this psi perp 01:12:56.000 --> 01:13:00.863 is that its length is precisely the uncertainty. 01:13:05.890 --> 01:13:12.220 So all this, but you could prove-- I'm going to do it. 01:13:12.220 --> 01:13:14.220 I'm going to show you all these things are true, 01:13:14.220 --> 01:13:16.550 but it gives you a bit of an insight. 01:13:16.550 --> 01:13:17.870 you have a vector. 01:13:17.870 --> 01:13:19.980 A moves you out. 01:13:19.980 --> 01:13:24.524 What is the uncertainty is this vertical projection-- 01:13:24.524 --> 01:13:25.940 vertical thing is the uncertainty. 01:13:25.940 --> 01:13:30.560 If you're down there, you get nothing. 01:13:30.560 --> 01:13:35.130 So how do we prove that? 01:13:35.130 --> 01:13:41.440 Well, let's construct a projector 01:13:41.440 --> 01:13:49.440 down to the space U psi, which is psi psi. 01:13:49.440 --> 01:13:56.880 This is a projector, just like any e1. 01:13:56.880 --> 01:14:00.660 e1 is a projection into the direction of 1. 01:14:00.660 --> 01:14:04.450 Well, take your first basis vector to be psi, 01:14:04.450 --> 01:14:07.310 and that's a projection to psi. 01:14:07.310 --> 01:14:11.950 So let's see what it-- so the projection to psi. 01:14:11.950 --> 01:14:15.600 So now let's see what it gives you 01:14:15.600 --> 01:14:24.390 when it acts on A psi-- this project acting on A psi 01:14:24.390 --> 01:14:31.120 is equal to psi psi A psi. 01:14:34.280 --> 01:14:36.590 And again, the usefulness of bracket notation 01:14:36.590 --> 01:14:38.400 is kind of nice here. 01:14:38.400 --> 01:14:39.310 So what is this? 01:14:39.310 --> 01:14:44.780 The expectation value of A. So indeed psi expectation value 01:14:44.780 --> 01:14:49.870 of A is what you get when you project this down. 01:15:00.882 --> 01:15:05.550 So then, the rest is sort of simple. 01:15:05.550 --> 01:15:13.885 If you take psi, and subtract from psi-- well, 01:15:13.885 --> 01:15:31.360 I'll subtract from psi, psi times expectation value of A. 01:15:31.360 --> 01:15:34.270 I'm sorry, I was saying it wrong. 01:15:34.270 --> 01:15:41.040 If you think the original vector-- A psi, 01:15:41.040 --> 01:15:45.520 and subtract from it what we took out, 01:15:45.520 --> 01:15:50.610 which is psi times expectation value of A, the projected 01:15:50.610 --> 01:15:53.840 thing-- this is some vector. 01:15:53.840 --> 01:16:01.880 But the main thing is that this vector is orthogonal to psi. 01:16:01.880 --> 01:16:02.630 Why? 01:16:02.630 --> 01:16:09.450 If you take a psi on the left, this is orthogonal to psi. 01:16:09.450 --> 01:16:10.830 And how do you see it? 01:16:10.830 --> 01:16:13.980 Put the psi from the left. 01:16:13.980 --> 01:16:15.340 And what do you get here? 01:16:15.340 --> 01:16:19.180 Psi A psi, which is expectation value of A, 01:16:19.180 --> 01:16:23.840 psi psi, which is 1, and expectation value A is 0. 01:16:23.840 --> 01:16:30.110 So this is a vector psi perp. 01:16:30.110 --> 01:16:35.280 And this is, of course, A minus expectation value 01:16:35.280 --> 01:16:40.270 of A acting on the state psi. 01:16:40.270 --> 01:16:46.300 Well, precisely the norm of psi perp is the norm of this, 01:16:46.300 --> 01:16:48.510 but that's what we defined to be the uncertainty. 01:16:51.830 --> 01:17:04.840 So indeed, the norm of psi perp is delta A of psi. 01:17:04.840 --> 01:17:09.230 So our ideas of projectors and orthogonal projectors 01:17:09.230 --> 01:17:11.340 allow you to understand better what 01:17:11.340 --> 01:17:14.010 is the uncertainty-- more pictorially. 01:17:14.010 --> 01:17:18.080 You have pictures of vectors, and orthogonal projections, 01:17:18.080 --> 01:17:20.550 and you want to make the uncertainty 0, 01:17:20.550 --> 01:17:24.010 you have to push the A psi into psi. 01:17:24.010 --> 01:17:28.160 You have to be an eigenstate, and you're there. 01:17:28.160 --> 01:17:31.430 Now, the last thing of-- I'll use the last five minutes 01:17:31.430 --> 01:17:37.965 to motivate the uncertainty, the famous uncertainty theorem. 01:17:59.260 --> 01:18:01.470 And typically, the uncertainly theorem 01:18:01.470 --> 01:18:09.460 is useful for A and B-- two Hermitian operators. 01:18:13.220 --> 01:18:22.250 And it relates the uncertainty in A on the state psi 01:18:22.250 --> 01:18:27.570 to the uncertainty in B of psi, saying 01:18:27.570 --> 01:18:31.535 it must be greater than or equal than some number. 01:18:38.300 --> 01:18:42.600 Now, if you look at that, and you 01:18:42.600 --> 01:18:46.580 think of all the math we've been talking about, 01:18:46.580 --> 01:18:49.870 you maybe know exactly how you're 01:18:49.870 --> 01:18:51.860 supposed to prove the uncertainty theorem. 01:18:57.300 --> 01:19:00.070 Well, what does this remind you of? 01:19:02.830 --> 01:19:05.750 Cauchy-Schwarz-- Schwarz inequality, 01:19:05.750 --> 01:19:08.350 I'm sorry-- not Cauchy-Schwarz. 01:19:08.350 --> 01:19:08.850 Why? 01:19:08.850 --> 01:19:16.870 Because for Schwarz inequality, you have norm of u, norm of v 01:19:16.870 --> 01:19:21.600 is greater than or equal than the norm of the inner product 01:19:21.600 --> 01:19:29.290 of u and v-- absolute value of the inner product of u and v. 01:19:29.290 --> 01:19:31.660 Remember, in this thing, this is norm of a vector, 01:19:31.660 --> 01:19:34.910 this is norm of a vector, and this is value of a scalar. 01:19:34.910 --> 01:19:42.160 And our uncertainties are norms. 01:19:42.160 --> 01:19:44.720 So it better be that. 01:19:44.720 --> 01:19:46.640 That inequality is the only inequality 01:19:46.640 --> 01:19:49.800 that can possibly give you the answer. 01:19:49.800 --> 01:19:54.500 So how would you set this up? 01:19:54.500 --> 01:20:04.990 You would say define-- as we'll say f equal A minus A acting 01:20:04.990 --> 01:20:15.060 on psi, and g is equal to B minus B acting on psi. 01:20:15.060 --> 01:20:25.730 And then f f, or f f is delta A squared. 01:20:25.730 --> 01:20:32.806 f g g is delta B squared. 01:20:37.730 --> 01:20:42.090 And you just need to compute the inner product of f g, 01:20:42.090 --> 01:20:44.330 because you need the mixed one. 01:20:44.330 --> 01:20:48.420 So if you want to have fun, try it. 01:20:48.420 --> 01:20:50.380 We'll do it next time anyway. 01:20:50.380 --> 01:20:53.600 All right that's it for today.