WEBVTT 00:00:00.060 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.800 Commons license. 00:00:03.800 --> 00:00:06.060 Your support will help MIT OpenCourseWare 00:00:06.060 --> 00:00:10.110 continue to offer high quality educational resources for free. 00:00:10.110 --> 00:00:12.690 To make a donation or view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.245 at ocw.mit.edu. 00:00:22.510 --> 00:00:26.360 PROFESSOR: Today we're going to just continue 00:00:26.360 --> 00:00:28.040 what Allan Adams was doing. 00:00:28.040 --> 00:00:32.340 He's away on a trip in Europe, so he asked 00:00:32.340 --> 00:00:33.900 me to give their lecture today. 00:00:33.900 --> 00:00:39.650 And we'll just follow what he told me to do. 00:00:39.650 --> 00:00:44.280 He was sort of sad to give me this lecture, because it's one 00:00:44.280 --> 00:00:46.280 of the most interesting ones. 00:00:46.280 --> 00:00:49.980 This is the one where you get to see the Schrodinger equation. 00:00:49.980 --> 00:00:56.660 But anyway, had to be a way, so we'll do it here. 00:00:56.660 --> 00:00:59.980 He also told me to take off my shoes, but I won't do that. 00:01:02.780 --> 00:01:05.760 So let's go ahead. 00:01:05.760 --> 00:01:07.290 So what do we have so far. 00:01:07.290 --> 00:01:12.910 It's going to be a list of items that we have. 00:01:12.910 --> 00:01:15.310 And what have we learned? 00:01:15.310 --> 00:01:20.880 We know that particles, or systems, 00:01:20.880 --> 00:01:29.180 are governed by wave functions, described by wave functions. 00:01:29.180 --> 00:01:33.710 Wave functions. 00:01:33.710 --> 00:01:38.520 And those are psi of x at this moment. 00:01:38.520 --> 00:01:44.060 And these are complex numbers, belong to the complex numbers. 00:01:44.060 --> 00:01:48.500 And they're continuous and normalizable. 00:01:48.500 --> 00:01:51.030 Their derivatives need not be continuous, 00:01:51.030 --> 00:01:54.700 but the wave function has to be continuous. 00:01:54.700 --> 00:01:57.770 It should be also normalizable, and those 00:01:57.770 --> 00:01:59.900 are two important properties of it, 00:01:59.900 --> 00:02:02.223 continuous and normalizable. 00:02:07.910 --> 00:02:12.670 Second there's a probability associated with this thing. 00:02:12.670 --> 00:02:15.360 And this probability is described 00:02:15.360 --> 00:02:20.200 by this special p of x. 00:02:20.200 --> 00:02:24.700 And given that x is a continuous variable, you can say well, 00:02:24.700 --> 00:02:28.560 what is the probability that the particle is just at this point 00:02:28.560 --> 00:02:31.950 would be zero in general. 00:02:31.950 --> 00:02:34.670 You have to ask, typically, what's the probability that I 00:02:34.670 --> 00:02:37.040 find it in a range. 00:02:37.040 --> 00:02:38.800 It's any continuous variable that 00:02:38.800 --> 00:02:42.320 postulated to be given by the square of the wave 00:02:42.320 --> 00:02:45.870 function and the x. 00:02:45.870 --> 00:02:55.276 Third there's superposition of allowed states. 00:02:58.950 --> 00:03:01.720 So particles can be in superpositions. 00:03:01.720 --> 00:03:05.330 So a wave function that depends of x 00:03:05.330 --> 00:03:10.970 may quickly or generally be given as a superposition of two 00:03:10.970 --> 00:03:11.746 wave functions. 00:03:16.260 --> 00:03:18.590 And this is seen in many ways. 00:03:18.590 --> 00:03:19.760 You have these boxes. 00:03:19.760 --> 00:03:21.420 A particle was a superposition. 00:03:21.420 --> 00:03:24.370 A top, and side, and hard, and soft. 00:03:24.370 --> 00:03:26.800 And photon superposition of linearly 00:03:26.800 --> 00:03:29.960 polarized here or that way. 00:03:29.960 --> 00:03:33.300 That's always possible to explain. 00:03:33.300 --> 00:03:38.270 Now in addition to this, to motivate the next one 00:03:38.270 --> 00:03:41.606 we talk about relations between operators. 00:03:44.150 --> 00:03:50.820 There was an abstraction going on in this course 00:03:50.820 --> 00:03:53.580 in the previous lectures in which 00:03:53.580 --> 00:03:56.260 the idea of the momentum of a particle 00:03:56.260 --> 00:04:00.580 became that of an operator on the wave function. 00:04:03.800 --> 00:04:11.320 So as an aside, operators momentum, 00:04:11.320 --> 00:04:14.050 we have the momentum of a particle 00:04:14.050 --> 00:04:17.850 has been associated with an operator h 00:04:17.850 --> 00:04:21.730 bar over i [INAUDIBLE] x. 00:04:21.730 --> 00:04:25.250 Now two things here. 00:04:25.250 --> 00:04:28.180 My taste, I like to put h bar over i. 00:04:28.180 --> 00:04:31.000 Allan likes to put I h bar. 00:04:31.000 --> 00:04:35.700 That's exactly the same thing, but no minus sign 00:04:35.700 --> 00:04:37.640 is a good thing in my opinion. 00:04:37.640 --> 00:04:41.150 So I avoid them when possible. 00:04:41.150 --> 00:04:45.280 Now there's is d dx here is a partial derivative, 00:04:45.280 --> 00:04:50.130 and there seems to be no need for partial derivatives here. 00:04:50.130 --> 00:04:51.820 Why partial derivatives? 00:04:51.820 --> 00:04:55.560 I only see functions of x. 00:04:55.560 --> 00:04:56.240 Anybody? 00:04:56.240 --> 00:04:56.930 Why? 00:04:56.930 --> 00:04:57.590 Yes. 00:04:57.590 --> 00:04:59.310 AUDIENCE: The complete wave function also 00:04:59.310 --> 00:05:01.905 depends on time, doesn't it? 00:05:01.905 --> 00:05:03.280 PROFESSOR: Complete wave function 00:05:03.280 --> 00:05:04.680 depends on time as well. 00:05:04.680 --> 00:05:06.230 Yes, exactly. 00:05:06.230 --> 00:05:08.590 That's where we're going to get to today. 00:05:08.590 --> 00:05:12.740 This is the description of this particle at some instant. 00:05:12.740 --> 00:05:16.480 So within [INAUDIBLE] time, the time here is implicit. 00:05:16.480 --> 00:05:20.570 It could be at some time, now, later, some time, 00:05:20.570 --> 00:05:22.090 but that's all we know. 00:05:22.090 --> 00:05:26.360 So in physics you're allowed to ask the question, well, 00:05:26.360 --> 00:05:28.440 if I know the wave function of this time 00:05:28.440 --> 00:05:31.290 and that seems to be what I need to describe the physics, 00:05:31.290 --> 00:05:33.150 what will it be later? 00:05:33.150 --> 00:05:36.190 And later time will come in, so therefore we'll 00:05:36.190 --> 00:05:40.950 stick to this partial d dx in here. 00:05:40.950 --> 00:05:45.250 All right, so how do we use that? 00:05:45.250 --> 00:05:50.980 We think of this operator as acting on the wave functions 00:05:50.980 --> 00:05:54.430 to give you roughly the momentum of the particle. 00:05:54.430 --> 00:05:58.540 And we've made it in such a way that when 00:05:58.540 --> 00:06:02.670 we talk about the expectation value of the momentum, 00:06:02.670 --> 00:06:06.290 the expected value of the momentum of the particle, 00:06:06.290 --> 00:06:08.800 we compute the following quantity. 00:06:08.800 --> 00:06:14.190 We compute the integral from minus infinity to infinity dx. 00:06:14.190 --> 00:06:17.740 We put the conjugate of the wave function here. 00:06:17.740 --> 00:06:23.730 And we put the operator, h bar over i d dx acting 00:06:23.730 --> 00:06:25.640 on the wave function here. 00:06:28.350 --> 00:06:31.260 And that's supposed to be sort of 00:06:31.260 --> 00:06:37.380 like saying that this evaluates the momentum of the wave 00:06:37.380 --> 00:06:39.820 function. 00:06:39.820 --> 00:06:41.580 Why is that so? 00:06:41.580 --> 00:06:47.680 It is because if you're trying to say, oh any wave function, 00:06:47.680 --> 00:06:51.300 a general wave function need not be 00:06:51.300 --> 00:06:54.410 state in which the particle has definite momentum. 00:06:54.410 --> 00:06:57.770 So I kind of just say the momentum of a particle 00:06:57.770 --> 00:07:00.940 is the value of this operator on the function. 00:07:00.940 --> 00:07:04.930 Because if I act with this operator on the function, 00:07:04.930 --> 00:07:08.190 on the wave function, it might not return me, 00:07:08.190 --> 00:07:09.310 the wave function. 00:07:09.310 --> 00:07:14.100 In fact in general, we've seen that, for special wave 00:07:14.100 --> 00:07:21.220 functions, wave functions of the form psi, a number, 00:07:21.220 --> 00:07:22.440 e to the ikx. 00:07:26.490 --> 00:07:34.900 Then p, let's think p hat as the operator on psi. 00:07:34.900 --> 00:07:40.180 Would be h bar over i d dx on psi. 00:07:42.710 --> 00:07:43.660 Gives you what? 00:07:43.660 --> 00:07:48.760 Well, this h over I, the h remains there. 00:07:48.760 --> 00:07:52.550 When you differentiate with respect to x, the ik goes down, 00:07:52.550 --> 00:07:59.040 and the i cancels, so you get hk times the same wave function. 00:07:59.040 --> 00:08:03.000 And for this, we think that this wave function is a wave 00:08:03.000 --> 00:08:07.340 function with momentum hk. 00:08:07.340 --> 00:08:10.880 Because if you act with the operator p on that wave 00:08:10.880 --> 00:08:14.260 function, it returns for you hk. 00:08:14.260 --> 00:08:23.750 So we think of this as has p equal hk, h bar k. 00:08:23.750 --> 00:08:26.850 So the thing that we want to do now 00:08:26.850 --> 00:08:29.830 is to make this a little more general. 00:08:29.830 --> 00:08:31.731 This is just talking about momentum, 00:08:31.731 --> 00:08:33.230 but in quantum mechanics we're going 00:08:33.230 --> 00:08:35.150 to have all kinds of operators. 00:08:35.150 --> 00:08:37.860 So we need to be more general. 00:08:37.860 --> 00:08:42.650 So this is going to be, as Allan calls it, a math interlude. 00:08:50.800 --> 00:08:55.040 Based on the following question, what this is an operator? 00:08:55.040 --> 00:09:00.050 And then the physics question, what do measurable things 00:09:00.050 --> 00:09:01.950 have to do with our operators? 00:09:01.950 --> 00:09:12.750 So about operators aren't measurable things, quantities. 00:09:16.980 --> 00:09:21.950 Now your view of operators is going to evolve. 00:09:21.950 --> 00:09:23.450 It's going to evolve in this course, 00:09:23.450 --> 00:09:25.310 It's going to evolve in 805. 00:09:25.310 --> 00:09:27.300 It probably will continue to evolve. 00:09:27.300 --> 00:09:32.130 So we need to think of what operators are. 00:09:32.130 --> 00:09:37.630 And there is a simple way of thinking 00:09:37.630 --> 00:09:41.280 of operators that is going to be the basis of much 00:09:41.280 --> 00:09:42.480 of the intuition. 00:09:42.480 --> 00:09:46.280 It's a mathematical way of thinking of operators, 00:09:46.280 --> 00:09:50.180 and what we'll sometimes use it as a crutch. 00:09:50.180 --> 00:09:55.010 And the idea is that basically operators 00:09:55.010 --> 00:10:00.390 are things that act on objects and scramble them. 00:10:00.390 --> 00:10:02.860 So whenever you have an operator, 00:10:02.860 --> 00:10:05.120 you really have to figure out what 00:10:05.120 --> 00:10:07.550 are the objects you're talking about. 00:10:07.550 --> 00:10:10.820 And then the operator is some instruction 00:10:10.820 --> 00:10:14.290 on how to scramble it, scramble this object. 00:10:14.290 --> 00:10:16.760 So for example, you have an operator. 00:10:16.760 --> 00:10:20.560 You must see what it does to any object here. 00:10:20.560 --> 00:10:24.530 The operator acts on the object and gives you another object. 00:10:24.530 --> 00:10:28.390 So in a picture you have all kinds 00:10:28.390 --> 00:10:37.020 of objects sets, a set of objects. 00:10:37.020 --> 00:10:40.300 And the operators are things. 00:10:40.300 --> 00:10:42.680 You can have a list of operators. 00:10:42.680 --> 00:10:45.780 And they come here and move those objects, 00:10:45.780 --> 00:10:48.078 scramble them, do something to them. 00:10:50.890 --> 00:10:53.960 And we should distinguish them, because the objects are not 00:10:53.960 --> 00:10:57.610 operators, and the operators are not the objects. 00:10:57.610 --> 00:11:03.070 So what is the simplest example in mathematics of this thing 00:11:03.070 --> 00:11:08.850 is vectors and matrices. 00:11:08.850 --> 00:11:09.865 Simplest example. 00:11:15.090 --> 00:11:20.060 Objects are vectors. 00:11:20.060 --> 00:11:24.480 Operators are matrices. 00:11:24.480 --> 00:11:26.920 And how does that work? 00:11:26.920 --> 00:11:31.780 Well, you have a two by two, the case where 00:11:31.780 --> 00:11:37.440 you have a set in which you have vectors with two components. 00:11:37.440 --> 00:11:47.250 So example, a vector that has two components v1 v2. 00:11:47.250 --> 00:11:53.170 And the matrices, this is the object. 00:11:53.170 --> 00:11:57.060 And the operator is a matrix. 00:11:57.060 --> 00:12:06.390 a 11, a 12, a 21, a 22 as an operator. 00:12:06.390 --> 00:12:11.090 An m on a vector is a vector. 00:12:14.480 --> 00:12:17.870 If you are with a matrix on a vector, this 2 00:12:17.870 --> 00:12:23.430 by 2 matrix on this common vector, you get another vector. 00:12:23.430 --> 00:12:29.455 So that's this simplest example of operators acting on objects. 00:12:31.980 --> 00:12:34.980 In our case, we're going to talk about a more-- we're 00:12:34.980 --> 00:12:37.170 going to have to begin, in quantum mechanics 00:12:37.170 --> 00:12:42.240 we're required to begin with a more sophisticated one in which 00:12:42.240 --> 00:12:48.340 the objects are going to be-- objects 00:12:48.340 --> 00:12:49.746 are going to be functions. 00:12:53.450 --> 00:13:02.165 In fact, I will write complex functions of x. 00:13:06.220 --> 00:13:10.090 So let's see it, the list of operators. 00:13:10.090 --> 00:13:12.680 And what do the operators do? 00:13:12.680 --> 00:13:18.010 The operators act on the functions. 00:13:25.450 --> 00:13:28.330 So what is an operator? 00:13:28.330 --> 00:13:32.110 It's a rule on how to take any function, 00:13:32.110 --> 00:13:34.710 and you must give a rule on how to obtain from that 00:13:34.710 --> 00:13:37.940 function another function. 00:13:37.940 --> 00:13:40.500 So let's start with an examples. 00:13:40.500 --> 00:13:43.200 It's probably the easiest thing to do. 00:13:43.200 --> 00:13:47.130 So an operator acts on the functions. 00:13:47.130 --> 00:13:52.340 So an operator for any function f of x 00:13:52.340 --> 00:13:54.033 will give you another function. 00:13:57.990 --> 00:14:01.760 Function of x. 00:14:04.600 --> 00:14:10.140 And here's operator o, we put a hat sometimes for operators. 00:14:10.140 --> 00:14:21.750 So the simplest operator, the operator one. 00:14:21.750 --> 00:14:27.220 We always, mathematicians, love to begin with trivial examples. 00:14:27.220 --> 00:14:29.910 Illustrate anything almost, and just kind of 00:14:29.910 --> 00:14:31.780 confuse you many times. 00:14:31.780 --> 00:14:35.170 But actually it's good to get them of the way. 00:14:35.170 --> 00:14:38.710 So what is the operator one? 00:14:38.710 --> 00:14:41.350 One possibility it takes any function 00:14:41.350 --> 00:14:44.660 and gives you the number one. 00:14:44.660 --> 00:14:45.850 Do you think that's it? 00:14:45.850 --> 00:14:48.950 Who thinks that's it? 00:14:48.950 --> 00:14:49.710 Nobody? 00:14:49.710 --> 00:14:50.470 Very good. 00:14:50.470 --> 00:14:53.000 that definitely is not a good thing 00:14:53.000 --> 00:14:54.960 to do to give you the number one. 00:14:54.960 --> 00:14:57.260 So this operator does the following. 00:14:57.260 --> 00:14:58.480 I will write it like that. 00:14:58.480 --> 00:15:05.517 The operator one takes f of x and gives you what? 00:15:05.517 --> 00:15:06.225 AUDIENCE: f of x. 00:15:06.225 --> 00:15:06.975 PROFESSOR: f of x. 00:15:06.975 --> 00:15:08.070 Correct. 00:15:08.070 --> 00:15:08.570 Good. 00:15:08.570 --> 00:15:12.440 So it's a very simple operator, but it's an operator. 00:15:12.440 --> 00:15:15.640 It's like what matrix? 00:15:15.640 --> 00:15:17.980 The identity matrix. 00:15:17.980 --> 00:15:18.770 Very good. 00:15:18.770 --> 00:15:22.370 There could be a zero operator that gives you nothing 00:15:22.370 --> 00:15:23.980 and would be the zero matrix. 00:15:23.980 --> 00:15:27.330 So let's write the more interesting operator. 00:15:27.330 --> 00:15:30.164 The operator would d dx. 00:15:30.164 --> 00:15:30.955 That's interesting. 00:15:30.955 --> 00:15:35.240 The derivative can be thought of as an operator 00:15:35.240 --> 00:15:38.670 because if you start with f of x, 00:15:38.670 --> 00:15:43.390 it gives you another function, d dx of f of x. 00:15:46.040 --> 00:15:48.830 And that's a rule to get from one function to another. 00:15:48.830 --> 00:15:53.890 Therefore that's an operator, qualifies as an operator. 00:15:53.890 --> 00:15:57.010 Another operator that typically can confuse you 00:15:57.010 --> 00:16:01.560 is the operator x. 00:16:01.560 --> 00:16:02.850 x an operator? 00:16:02.850 --> 00:16:04.820 What does that mean? 00:16:04.820 --> 00:16:07.040 Well, you just have to define it. 00:16:07.040 --> 00:16:10.500 At this moment, it could mean many things. 00:16:10.500 --> 00:16:13.220 But you will see that [INAUDIBLE] is the only thing 00:16:13.220 --> 00:16:15.820 that probably makes some sense. 00:16:15.820 --> 00:16:18.450 So what is the operator x? 00:16:18.450 --> 00:16:21.310 Well, it's the operator that, given f of x, 00:16:21.310 --> 00:16:27.090 gives you the function x times f of x. 00:16:27.090 --> 00:16:28.930 That's a reasonable thing to do. 00:16:28.930 --> 00:16:30.350 It's multiplying by x. 00:16:30.350 --> 00:16:32.710 It changes the function. 00:16:32.710 --> 00:16:35.590 You could define the operator x squared that multiplies 00:16:35.590 --> 00:16:36.610 by x squared. 00:16:36.610 --> 00:16:41.090 And the only reasonable thing is to multiply it by. 00:16:41.090 --> 00:16:43.330 You could divide by it, and you may 00:16:43.330 --> 00:16:45.630 need to divide by it as well. 00:16:45.630 --> 00:16:47.460 And you could define the operator 1 00:16:47.460 --> 00:16:51.650 over x gives you the function times 1 over x. 00:16:51.650 --> 00:16:54.940 We will need that sometime, but not now. 00:16:54.940 --> 00:16:59.410 Let's see another set of operators where we give a name. 00:16:59.410 --> 00:17:01.380 It doesn't have a name because it's not 00:17:01.380 --> 00:17:03.800 all that's useful in fact. 00:17:03.800 --> 00:17:05.640 But it's good to illustrate things. 00:17:05.640 --> 00:17:11.810 They operator s sub q for squared. 00:17:11.810 --> 00:17:15.710 S q for the first two letters of the word square. 00:17:15.710 --> 00:17:19.910 That takes f of x into f of x squared. 00:17:19.910 --> 00:17:21.069 That's another function. 00:17:21.069 --> 00:17:23.960 You could define more functions like that. 00:17:23.960 --> 00:17:30.020 The operator p 42. 00:17:30.020 --> 00:17:33.295 That's another silly operator. 00:17:33.295 --> 00:17:35.960 Well certainly a lot more silly than this one. 00:17:35.960 --> 00:17:37.330 That's not too bad. 00:17:37.330 --> 00:17:47.880 But the p 42 takes f of x And gives you the number 42 times 00:17:47.880 --> 00:17:50.760 the constant function. 00:17:50.760 --> 00:17:52.300 So that's a function of x. 00:17:52.300 --> 00:17:54.045 It's trivial function of x. 00:17:57.860 --> 00:18:00.360 Now enough examples. 00:18:00.360 --> 00:18:01.660 So you get the idea. 00:18:01.660 --> 00:18:05.400 Operators act on functions and give you functions. 00:18:05.400 --> 00:18:08.260 And we just need to define them, and then we 00:18:08.260 --> 00:18:09.760 know what we're talking about. 00:18:09.760 --> 00:18:11.645 Yes? 00:18:11.645 --> 00:18:14.400 AUDIENCE: Is the Dirac delta and operator? 00:18:14.400 --> 00:18:18.430 PROFESSOR: The Dirac delta, well, you 00:18:18.430 --> 00:18:20.226 can think of it as an operator. 00:18:23.600 --> 00:18:26.650 So it all depends how you define things. 00:18:26.650 --> 00:18:30.110 So how could I do find the Dirac delta function 00:18:30.110 --> 00:18:32.140 to be an operator? 00:18:32.140 --> 00:18:34.590 So delta of x minus a. 00:18:34.590 --> 00:18:39.410 Can I call it the operator delta hat of a? 00:18:39.410 --> 00:18:42.500 Well, I would have to tell you what it does in functions. 00:18:42.500 --> 00:18:45.110 And probably I would say delta had 00:18:45.110 --> 00:18:50.670 on a on a function of x is equal to delta of x minus a times 00:18:50.670 --> 00:18:53.920 the function of x. 00:18:53.920 --> 00:18:55.820 And I'd say it's an operator. 00:18:55.820 --> 00:18:59.910 Now the question is, really, is it a useful operator? 00:18:59.910 --> 00:19:04.350 And sometimes it will be useful in fact. 00:19:04.350 --> 00:19:07.080 This is a more general case of another operator 00:19:07.080 --> 00:19:08.330 that maybe I could define. 00:19:11.340 --> 00:19:17.960 o sub h of x is the operator that 00:19:17.960 --> 00:19:23.825 takes f of x into h of x times f of x. 00:19:27.330 --> 00:19:30.870 So that would be another operator. 00:19:30.870 --> 00:19:34.900 Now there are operators that are particularly nice, 00:19:34.900 --> 00:19:38.040 and there are the so-called linear operators. 00:19:40.700 --> 00:19:42.685 So what is a linear operator? 00:19:45.610 --> 00:19:47.970 It's one that respects superposition. 00:19:47.970 --> 00:19:54.590 So linear operator respects superposition. 00:19:59.420 --> 00:20:03.780 So o hat is linear. 00:20:03.780 --> 00:20:08.980 o hat is a linear operator. 00:20:08.980 --> 00:20:16.420 If o hat on a times f of x plus b times g of x 00:20:16.420 --> 00:20:19.310 does what you would imagine it should do, 00:20:19.310 --> 00:20:22.790 it that's on the first, and then it acts on the second. 00:20:22.790 --> 00:20:25.800 Acting on the first, the number goes out 00:20:25.800 --> 00:20:28.620 and doesn't do anything, say, on the number. 00:20:28.620 --> 00:20:29.510 It's linear. 00:20:29.510 --> 00:20:31.720 It's part of that idea. 00:20:31.720 --> 00:20:42.690 And it gives you o on f of x plus b times o on g of x. 00:20:42.690 --> 00:20:48.005 So your operator may be linear, or it may not be linear. 00:20:54.670 --> 00:20:56.890 And we have to just guess them. 00:20:56.890 --> 00:21:00.680 And you would imagine that we can decide that, 00:21:00.680 --> 00:21:02.930 of the list of operators that we have, 00:21:02.930 --> 00:21:12.910 let's see, one d dx-- how much? 00:21:12.910 --> 00:21:13.470 Which one? 00:21:13.470 --> 00:21:21.750 Sq, p 42, and o sub h of x. 00:21:25.770 --> 00:21:27.070 Let's see. 00:21:27.070 --> 00:21:31.130 Let's vote on each one whether it's linear or not. 00:21:31.130 --> 00:21:35.190 A shouting match whether I hear a stronger yes or no. 00:21:35.190 --> 00:21:35.750 OK? 00:21:35.750 --> 00:21:38.700 One is a linear operator, yes? 00:21:38.700 --> 00:21:39.340 AUDIENCE: Yes. 00:21:39.340 --> 00:21:40.860 PROFESSOR: No? 00:21:40.860 --> 00:21:42.350 Yes. 00:21:42.350 --> 00:21:43.680 All right. 00:21:43.680 --> 00:21:45.050 d dx linear. 00:21:45.050 --> 00:21:45.782 Yes? 00:21:45.782 --> 00:21:46.554 AUDIENCE: Yes. 00:21:46.554 --> 00:21:47.220 PROFESSOR: Good. 00:21:47.220 --> 00:21:48.660 That's strong enough. 00:21:48.660 --> 00:21:50.800 Don't need to hear the other one. 00:21:50.800 --> 00:21:51.800 x hat. 00:21:51.800 --> 00:21:52.580 Linear operator? 00:21:52.580 --> 00:21:53.690 Yes or no? 00:21:53.690 --> 00:21:54.410 AUDIENCE: Yes. 00:21:54.410 --> 00:21:56.870 PROFESSOR: Yes, good. 00:21:56.870 --> 00:21:58.850 Squaring, linear operator? 00:21:58.850 --> 00:21:59.550 AUDIENCE: No. 00:21:59.550 --> 00:22:00.390 PROFESSOR: No. 00:22:00.390 --> 00:22:03.540 No way it could be a linear operator. 00:22:03.540 --> 00:22:06.520 It just doesn't happen. 00:22:06.520 --> 00:22:14.330 If you have sq on f plus g, it would be f plus g squared, 00:22:14.330 --> 00:22:18.470 which is f squared plus g squared plus, unfortunately 00:22:18.470 --> 00:22:20.200 too, fg. 00:22:20.200 --> 00:22:28.600 And this thing ruins it, because this is sq of f plus sq of g. 00:22:28.600 --> 00:22:30.210 It's even worse than that. 00:22:30.210 --> 00:22:37.680 You put sq on af, by linearity it 00:22:37.680 --> 00:22:39.880 should be a times the operator. 00:22:39.880 --> 00:22:44.060 But when you square a times f, you get a squared f squared. 00:22:44.060 --> 00:22:46.260 So you don't even need two functions 00:22:46.260 --> 00:22:48.110 to see that it's not real. 00:22:48.110 --> 00:22:52.600 So definitely no. 00:22:52.600 --> 00:22:55.070 How about p 42? 00:22:55.070 --> 00:22:55.685 AUDIENCE: No. 00:22:55.685 --> 00:22:57.100 PROFESSOR: No, of course not. 00:22:57.100 --> 00:22:59.930 Because if you add two functions, 00:22:59.930 --> 00:23:01.460 it still gives you 42. 00:23:01.460 --> 00:23:05.130 It doesn't get you 84, so no. 00:23:05.130 --> 00:23:07.840 How about oh of x? 00:23:07.840 --> 00:23:08.810 AUDIENCE: Yes. 00:23:08.810 --> 00:23:10.080 PROFESSOR: Yes, it does that. 00:23:10.080 --> 00:23:15.100 If you act with this operator on a sum of functions distributive 00:23:15.100 --> 00:23:16.070 law, it works. 00:23:16.070 --> 00:23:18.210 So this is linear. 00:23:18.210 --> 00:23:18.710 Good. 00:23:18.710 --> 00:23:21.080 Linear operators are important to us 00:23:21.080 --> 00:23:24.250 because we have some superposition 00:23:24.250 --> 00:23:25.690 of allowed states. 00:23:25.690 --> 00:23:28.390 So if this is a state and this is a state, 00:23:28.390 --> 00:23:29.910 this is also good state. 00:23:29.910 --> 00:23:35.120 So if we want superposition to work well with our theory, 00:23:35.120 --> 00:23:36.860 we want linear operator. 00:23:36.860 --> 00:23:38.470 So that's good. 00:23:38.470 --> 00:23:41.730 So we have those linear operators. 00:23:41.730 --> 00:23:45.420 And now operators have another thing that 00:23:45.420 --> 00:23:48.980 makes them something special. 00:23:48.980 --> 00:23:56.380 It is the idea that there's simpler object they can act on. 00:23:56.380 --> 00:23:59.920 We don't assume you've studied linear algebra in this course, 00:23:59.920 --> 00:24:02.770 so whatever I'm going to say, take it 00:24:02.770 --> 00:24:07.630 as motivation to learn some linear algebra at some stage. 00:24:07.630 --> 00:24:10.830 You will be a little more linear algebra in 805. 00:24:10.830 --> 00:24:13.970 But at this moment, just basic idea. 00:24:13.970 --> 00:24:19.660 So whenever you have matrices, one thing that people do 00:24:19.660 --> 00:24:23.450 is to see if there are special vectors. 00:24:23.450 --> 00:24:27.170 Any arbitrary vector, when you act within the matrix, 00:24:27.170 --> 00:24:30.060 is going to just jump and go somewhere else, point 00:24:30.060 --> 00:24:31.920 in another direction. 00:24:31.920 --> 00:24:33.970 But there are some special vectors 00:24:33.970 --> 00:24:38.040 that do act-- if you have a given matrix m, 00:24:38.040 --> 00:24:43.560 there are some funny vectors sometimes that acted by n 00:24:43.560 --> 00:24:45.390 remain the same direction. 00:24:45.390 --> 00:24:47.950 They may grow a little or become smaller, 00:24:47.950 --> 00:24:50.020 but they remain the same direction. 00:24:50.020 --> 00:24:52.340 These are called eigenvectors. 00:24:52.340 --> 00:24:55.000 And that constant of proportionality, 00:24:55.000 --> 00:24:58.530 proportional to the action of the operator on the vector, 00:24:58.530 --> 00:25:00.650 is called the eigenvalue. 00:25:00.650 --> 00:25:04.930 So these things have generalizations 00:25:04.930 --> 00:25:06.290 for our operators. 00:25:06.290 --> 00:25:16.410 So operators can have special functions, eigenfunctions. 00:25:23.630 --> 00:25:25.110 What are these eigenfunctions? 00:25:25.110 --> 00:25:28.640 So let's consider that operator a hat. 00:25:28.640 --> 00:25:29.700 It's some operator. 00:25:29.700 --> 00:25:31.950 I don't know which one of these, be we're 00:25:31.950 --> 00:25:33.710 going to talk about linear operator. 00:25:33.710 --> 00:25:37.020 So linear operators have eigenfunctions. 00:25:37.020 --> 00:25:38.560 A hat. 00:25:38.560 --> 00:25:41.040 So a hat. 00:25:41.040 --> 00:25:45.790 There may be functions that, when you act with the operator, 00:25:45.790 --> 00:25:48.330 you sort of get the same function up 00:25:48.330 --> 00:25:51.770 to possibly a constant a. 00:25:51.770 --> 00:25:55.250 So you may get a times the function. 00:25:55.250 --> 00:25:58.690 And that's a pretty unusual function, 00:25:58.690 --> 00:26:01.680 because, on most functions, any given operator 00:26:01.680 --> 00:26:03.880 is going to make a mess out of the function. 00:26:03.880 --> 00:26:05.560 But sometimes it does that. 00:26:05.560 --> 00:26:10.100 So to label them better with respect to operator, 00:26:10.100 --> 00:26:13.280 I would put a subscript a, which means that there's 00:26:13.280 --> 00:26:17.050 some special function that has a parameter a, 00:26:17.050 --> 00:26:19.450 for which this operator gives you 00:26:19.450 --> 00:26:21.870 a times that special function. 00:26:21.870 --> 00:26:24.670 And that special function is called-- 00:26:24.670 --> 00:26:29.526 this is the eigenfunction and this is the eigenvalue. 00:26:36.830 --> 00:26:40.040 And that the eigenvalue is a number. 00:26:40.040 --> 00:26:46.120 It's a complex number c there over there. 00:26:46.120 --> 00:26:48.270 So these are special things. 00:26:48.270 --> 00:26:51.180 They don't necessarily happen all the time to exist, 00:26:51.180 --> 00:26:54.500 but sometimes they do, and then they're pretty useful. 00:26:54.500 --> 00:26:59.130 And we have one example of them that is quite nice. 00:26:59.130 --> 00:27:12.650 For the operator a equal p, we have eigenfunctions 00:27:12.650 --> 00:27:23.710 e to the ikx with eigenvalue hk. 00:27:23.710 --> 00:27:26.750 So this is the connection to this whole thing. 00:27:26.750 --> 00:27:30.660 We wanted to make clear for you that what you saw here, 00:27:30.660 --> 00:27:33.170 that this operator acting on this function 00:27:33.170 --> 00:27:37.400 gives you something times this function is a general fact 00:27:37.400 --> 00:27:39.580 about the operators. 00:27:39.580 --> 00:27:41.920 Operators have eigenfunctions. 00:27:41.920 --> 00:27:48.000 So eigenfunction e of x with eigenvalue hk, because indeed 00:27:48.000 --> 00:27:54.000 p hat on this e to ikx, as you see this h bar 00:27:54.000 --> 00:27:58.060 k times e to the ikx. 00:27:58.060 --> 00:28:00.160 So here you have p hat is the a. 00:28:00.160 --> 00:28:05.220 This is the function labeled a would be like k. 00:28:05.220 --> 00:28:08.540 Here is something like k again. 00:28:08.540 --> 00:28:09.760 And here is this thing. 00:28:09.760 --> 00:28:14.140 But the main thing operator on the function number 00:28:14.140 --> 00:28:15.990 times the function is an eigenfunction. 00:28:15.990 --> 00:28:17.230 Yes? 00:28:17.230 --> 00:28:22.250 AUDIENCE: For a given operator, is the eigenvalue [INAUDIBLE]? 00:28:22.250 --> 00:28:25.940 PROFESSOR: Well, for a given operator good question. 00:28:25.940 --> 00:28:29.390 a is a list of values. 00:28:29.390 --> 00:28:32.580 So there may be many, many, many eigenfunctions. 00:28:32.580 --> 00:28:35.410 Many cases infinitely many eigenfunctions. 00:28:35.410 --> 00:28:40.730 In fact, here I can put for k any number I want, 00:28:40.730 --> 00:28:42.500 and I get a different function. 00:28:42.500 --> 00:28:55.390 So a belongs to c and may take many, or even infinite, values. 00:28:55.390 --> 00:29:01.650 If you remember for nice matrices, n by n matrix 00:29:01.650 --> 00:29:05.950 may be a nice n by n matrix because n eigenvectors 00:29:05.950 --> 00:29:09.730 and eigenvalues are sometimes hard to generate, 00:29:09.730 --> 00:29:12.690 sometimes eigenvalues have the same numbers and things 00:29:12.690 --> 00:29:13.330 like that. 00:29:17.510 --> 00:29:18.080 OK. 00:29:18.080 --> 00:29:24.680 Linearity is this some of two eigenvectors and eigenvector. 00:29:24.680 --> 00:29:26.960 Yes? 00:29:26.960 --> 00:29:28.169 No? 00:29:28.169 --> 00:29:28.710 AUDIENCE: No. 00:29:28.710 --> 00:29:29.980 PROFESSOR: No, no. 00:29:29.980 --> 00:29:30.650 Correct. 00:29:30.650 --> 00:29:33.380 That's not necessarily true. 00:29:33.380 --> 00:29:39.400 If you have two eigenvectors, they 00:29:39.400 --> 00:29:41.080 have different eigenvalues. 00:29:41.080 --> 00:29:44.580 So things don't work out well necessarily. 00:29:44.580 --> 00:29:47.680 So an eigenvector plus another eigenvector 00:29:47.680 --> 00:29:49.020 is not an eigenvector. 00:29:49.020 --> 00:29:56.920 So you have here, for example, A f1 equals a1f1. 00:29:56.920 --> 00:30:07.340 And A f2 equal a2f2, then a on f1 plus f2 00:30:07.340 --> 00:30:13.970 would be a1f1 plus a2f2, and that's 00:30:13.970 --> 00:30:20.940 not equal to something times f1 plus f2. 00:30:20.940 --> 00:30:24.000 It would have to be something times f1 plus f2 00:30:24.000 --> 00:30:25.650 to be an eigenvector. 00:30:25.650 --> 00:30:29.420 So this is not necessarily an eigenvector. 00:30:29.420 --> 00:30:33.290 And it doesn't help to put a constant in front of here. 00:30:33.290 --> 00:30:34.320 Nothing helps. 00:30:34.320 --> 00:30:37.980 There's no way to construct an eigenvector 00:30:37.980 --> 00:30:41.690 from two eigenvectors by adding or subtracting. 00:30:41.690 --> 00:30:45.720 The size of the eigenvector is not fixed either. 00:30:45.720 --> 00:30:56.516 If f is an eigenvector, then 3 times f is also an eigenvector. 00:30:59.700 --> 00:31:02.270 And we call it the same eigenvector. 00:31:02.270 --> 00:31:05.700 Nobody would call it a different eigenvector. 00:31:05.700 --> 00:31:07.860 It's really the same. 00:31:07.860 --> 00:31:10.710 OK, so how does that relate to physics? 00:31:10.710 --> 00:31:12.700 Well, we've seen it here already. 00:31:12.700 --> 00:31:16.640 that one operator that we've learned to work with 00:31:16.640 --> 00:31:17.870 is the momentum operator. 00:31:17.870 --> 00:31:20.460 It has those eigenfunctions. 00:31:20.460 --> 00:31:25.370 So back to physics. 00:31:25.370 --> 00:31:27.190 We have other operators. 00:31:27.190 --> 00:31:29.370 Therefore we have the P operator. 00:31:29.370 --> 00:31:30.660 That's good. 00:31:30.660 --> 00:31:32.970 We have the X operator. 00:31:32.970 --> 00:31:34.730 That's nice. 00:31:34.730 --> 00:31:36.850 It's multiplication by x. 00:31:36.850 --> 00:31:39.000 And why do we use it? 00:31:39.000 --> 00:31:41.560 Because sometimes you have the energy operator. 00:31:46.430 --> 00:31:48.560 And what is the energy operator? 00:31:48.560 --> 00:31:51.640 The energy operator is just the energy 00:31:51.640 --> 00:31:55.790 that you've always known, but think of it as an operator. 00:31:55.790 --> 00:31:58.040 So how do we do that? 00:31:58.040 --> 00:32:01.750 Well, what is the energy of a particle we've written 00:32:01.750 --> 00:32:07.300 p squared over 2m plus v of x. 00:32:07.300 --> 00:32:09.920 Well, that was the energy of a particle, 00:32:09.920 --> 00:32:13.200 the momentum squared over 2m plus v of x. 00:32:13.200 --> 00:32:17.470 So the energy operator is hat here, hat there. 00:32:24.250 --> 00:32:28.460 And now it becomes an interesting object. 00:32:28.460 --> 00:32:32.300 This energy operator will be called E hat. 00:32:32.300 --> 00:32:34.190 It acts and functions. 00:32:34.190 --> 00:32:35.810 It is not a number. 00:32:35.810 --> 00:32:39.020 The energy is a number, but the energy operator 00:32:39.020 --> 00:32:40.060 is not a number. 00:32:40.060 --> 00:32:43.220 Far from a number in fact. 00:32:43.220 --> 00:32:49.340 The energy operator is minus h squared over 2m. 00:32:49.340 --> 00:32:52.340 d second the x squared. 00:32:52.340 --> 00:32:53.440 Why that? 00:32:53.440 --> 00:32:59.270 Well, because p was h bar over i d dx as an operator. 00:32:59.270 --> 00:33:04.270 So this sort of arrow here, it sort of the introduction. 00:33:04.270 --> 00:33:14.090 But after a while you just say P hat is h bar over a i d dx. 00:33:14.090 --> 00:33:15.160 End of story. 00:33:15.160 --> 00:33:18.250 It's not like double arrow. 00:33:18.250 --> 00:33:19.670 It's just what it is. 00:33:19.670 --> 00:33:20.740 That operator. 00:33:20.740 --> 00:33:21.710 That's what we call it. 00:33:21.710 --> 00:33:26.390 So when we square it, the i squares, the minus h squares, 00:33:26.390 --> 00:33:31.400 and d dx and d dx applied twice is the second derivative. 00:33:31.400 --> 00:33:34.850 And here we get v of X hat, which 00:33:34.850 --> 00:33:38.820 is your good potential, whatever potential you're interested in, 00:33:38.820 --> 00:33:43.380 in which, whenever you see an x, you put an X hat. 00:33:43.380 --> 00:33:45.310 And now this is an operator. 00:33:45.310 --> 00:33:50.000 So you see this is not a number, not the function. 00:33:50.000 --> 00:33:52.020 It's just an operator. 00:33:52.020 --> 00:33:55.900 The operator has this sort of operator v of x. 00:33:55.900 --> 00:34:00.590 Now what is this v of x here as an operator? 00:34:00.590 --> 00:34:04.130 This is v of x as an operator is just multiplication 00:34:04.130 --> 00:34:07.120 by the function v of x. 00:34:07.120 --> 00:34:12.120 You see, you have here that the operator x is f of x like that. 00:34:12.120 --> 00:34:16.630 I could have written the operator X hat to the n. 00:34:16.630 --> 00:34:18.380 What would it be? 00:34:18.380 --> 00:34:23.239 Well, if I add to the function, this 00:34:23.239 --> 00:34:29.750 is a lot of X hats acting on the function. 00:34:29.750 --> 00:34:31.440 Well, let the first one out. 00:34:31.440 --> 00:34:33.510 You let x times f of x. 00:34:33.510 --> 00:34:36.090 The second, that's another x, another x. 00:34:36.090 --> 00:34:42.590 So this is just x to the n times f of x. 00:34:42.590 --> 00:34:44.550 So lots of X hats. 00:34:44.550 --> 00:34:47.469 X hats To the 100th on a function 00:34:47.469 --> 00:34:49.889 is just X to the 100th times a function. 00:34:49.889 --> 00:34:54.650 So v of x on a function is just v 00:34:54.650 --> 00:34:56.520 of the number x on a function. 00:34:56.520 --> 00:35:00.240 It's just like this operator, the O in which you 00:35:00.240 --> 00:35:03.550 multiply by a function. 00:35:03.550 --> 00:35:07.780 So please I hope this is completely clear what 00:35:07.780 --> 00:35:10.500 this means as an operator. 00:35:10.500 --> 00:35:13.220 You take the wave function, take two derivatives, 00:35:13.220 --> 00:35:18.790 and add the product of the wave function times v of plane x. 00:35:18.790 --> 00:35:20.970 So I'll write it here maybe. 00:35:20.970 --> 00:35:22.040 So important. 00:35:22.040 --> 00:35:28.160 E hat and psi of x is therefore minus h 00:35:28.160 --> 00:35:33.920 squared over 2m the [INAUDIBLE] the x squared of psi of x 00:35:33.920 --> 00:35:38.122 plus just plain v of x times psi of x. 00:35:42.157 --> 00:35:42.990 That's what it does. 00:35:42.990 --> 00:35:45.610 That's an operator. 00:35:45.610 --> 00:35:52.100 And for these operators in general. 00:35:52.100 --> 00:35:53.830 Math interlude, is it over? 00:35:53.830 --> 00:35:55.560 Not quite. 00:35:55.560 --> 00:35:56.470 Wow. 00:35:56.470 --> 00:35:57.860 No, yes. 00:35:57.860 --> 00:36:00.196 Allan said at this moment it's over, 00:36:00.196 --> 00:36:01.320 when you introduce it here. 00:36:04.760 --> 00:36:09.810 I'll say something more here, but it's going to be over now. 00:36:09.810 --> 00:36:12.970 Our three continues here then with four. 00:36:19.910 --> 00:36:37.563 Four, to each observable we have an associated operator. 00:36:43.810 --> 00:36:53.420 So for momentum, we have the operator P hat. 00:36:53.420 --> 00:37:00.460 And for position we have the operator X hat. 00:37:00.460 --> 00:37:04.780 And for energy we have the operator E hat. 00:37:08.150 --> 00:37:12.240 And these are examples of operators. 00:37:12.240 --> 00:37:17.840 Example operator A hat could be any of those. 00:37:17.840 --> 00:37:21.340 And there may be more observables depending 00:37:21.340 --> 00:37:23.460 on the system you're working. 00:37:23.460 --> 00:37:25.790 If you have particles in a line, there's 00:37:25.790 --> 00:37:28.719 not too many more observables at this moment. 00:37:28.719 --> 00:37:30.135 If you have a particle in general, 00:37:30.135 --> 00:37:32.840 you can have angular momentum. 00:37:32.840 --> 00:37:35.050 That's an interesting observable. 00:37:35.050 --> 00:37:36.110 It can be others. 00:37:36.110 --> 00:37:41.240 So for any of those, our definition 00:37:41.240 --> 00:37:44.360 is just like with it for momentum. 00:37:44.360 --> 00:37:48.490 The expectation value of the operator 00:37:48.490 --> 00:37:53.400 is computed by doing what you did for momentum. 00:37:53.400 --> 00:37:56.210 You act with the operator on the wave function 00:37:56.210 --> 00:38:02.950 here and multiply by the compass conjugate function. 00:38:02.950 --> 00:38:06.150 And integrate just like you did for momentum. 00:38:06.150 --> 00:38:11.080 This is going to be the value that you expect. 00:38:11.080 --> 00:38:14.680 After many trials on this wave function, 00:38:14.680 --> 00:38:19.040 you would expect the measured value of this exhibit 00:38:19.040 --> 00:38:23.270 a distribution which its expectation value, the mean, 00:38:23.270 --> 00:38:25.640 is given by this. 00:38:25.640 --> 00:38:29.430 Now there are other definitions. 00:38:29.430 --> 00:38:32.650 One definition that already has been mentioned 00:38:32.650 --> 00:38:41.010 is that the uncertainty of the operator on the state 00:38:41.010 --> 00:38:45.810 psi, the uncertainty, is computed 00:38:45.810 --> 00:38:49.880 by taking the square root of the expectation 00:38:49.880 --> 00:39:04.030 value of A squared minus the expectation value of A, 00:39:04.030 --> 00:39:07.230 as a number, squared. 00:39:07.230 --> 00:39:10.810 Now the expectation value of A squared, just simply 00:39:10.810 --> 00:39:12.830 here instead of A you put A squared, 00:39:12.830 --> 00:39:14.830 so you've got A squared here. 00:39:14.830 --> 00:39:19.430 That unless the function is very special, it's very different 00:39:19.430 --> 00:39:24.400 whole is bigger than the expectation value of A squared. 00:39:24.400 --> 00:39:28.160 So this is a number, and it's called the uncertainty. 00:39:28.160 --> 00:39:32.270 And that's the uncertainty of the uncertainty principle. 00:39:32.270 --> 00:39:37.530 So for operators, we need to have another observation that 00:39:37.530 --> 00:39:39.930 comes from matrices that is going to be crucial 00:39:39.930 --> 00:39:45.580 for us is the observation that operators don't necessarily 00:39:45.580 --> 00:39:47.510 commute. 00:39:47.510 --> 00:39:52.120 And we'll do the most important example of that. 00:39:52.120 --> 00:39:56.470 So we'll try to see in the operators associated 00:39:56.470 --> 00:40:00.530 with momentum and position commute. 00:40:00.530 --> 00:40:04.050 And what we mean by commute or don't communicate? 00:40:04.050 --> 00:40:07.420 Whether the order of multiplication matters. 00:40:07.420 --> 00:40:10.820 Now we talked about matrices at the beginning, 00:40:10.820 --> 00:40:14.800 and we said matrices act on vectors to give you vectors. 00:40:14.800 --> 00:40:16.560 So do they commute? 00:40:16.560 --> 00:40:18.145 Well, matrices don't commute. 00:40:18.145 --> 00:40:21.260 The order matters for matrices multiplication. 00:40:21.260 --> 00:40:25.500 So these operators we're inventing here for physics, 00:40:25.500 --> 00:40:27.900 the order does matter as well. 00:40:27.900 --> 00:40:29.526 So commutation. 00:40:36.480 --> 00:40:42.210 So let's try to see if we compute the operator p 00:40:42.210 --> 00:40:43.230 and x hat. 00:40:43.230 --> 00:40:49.360 Is it equal to the operator x hat times p? 00:40:49.360 --> 00:40:51.220 This is a very good question. 00:40:51.220 --> 00:40:54.890 These are two operators that we've defined. 00:40:54.890 --> 00:40:59.160 And we just want to know if the order matters 00:40:59.160 --> 00:41:01.560 or if it doesn't matter. 00:41:01.560 --> 00:41:02.900 So how can I check it? 00:41:02.900 --> 00:41:06.260 I cannot just check it like this, 00:41:06.260 --> 00:41:11.110 because operators are only clear what they do is when they act 00:41:11.110 --> 00:41:12.540 on functions. 00:41:12.540 --> 00:41:15.220 So the only thing that I can do is test 00:41:15.220 --> 00:41:19.430 if this thing acting on functions give the same. 00:41:19.430 --> 00:41:23.710 So I'm going act with this on the function f of x. 00:41:23.710 --> 00:41:26.960 And I'm going to have act with this on the function f of x. 00:41:29.620 --> 00:41:33.040 Now what do I mean by acting with p times 00:41:33.040 --> 00:41:35.010 x hat on the function f of x. 00:41:35.010 --> 00:41:39.060 This is by definition you act first 00:41:39.060 --> 00:41:41.925 with the operator that is next to the f and then 00:41:41.925 --> 00:41:42.550 with the other. 00:41:42.550 --> 00:41:48.285 So this is p hat on the function x hat times f of x. 00:41:51.440 --> 00:41:57.050 So here I would have, this is x hat on the function p hat 00:41:57.050 --> 00:41:58.700 f of x. 00:41:58.700 --> 00:42:04.160 See, if you have a series of matrices, m1, m2, m3 acting 00:42:04.160 --> 00:42:06.330 on a vector, what do you mean? 00:42:06.330 --> 00:42:09.070 Act with this on the vector, then with this on the vector, 00:42:09.070 --> 00:42:10.590 then with this. 00:42:10.590 --> 00:42:12.090 That's multiplication. 00:42:12.090 --> 00:42:14.370 So we're doing that. 00:42:14.370 --> 00:42:15.500 So let's evaluate. 00:42:15.500 --> 00:42:17.990 What is x operator on f of x? 00:42:17.990 --> 00:42:22.300 This is p hat on x times f of x. 00:42:25.510 --> 00:42:28.140 That's what the x operator in the function is. 00:42:28.140 --> 00:42:30.570 Here, what this x hat? 00:42:30.570 --> 00:42:39.100 And now I have this, so I have here h over i d dx of f. 00:42:39.100 --> 00:42:41.530 Let's go one more step here. 00:42:41.530 --> 00:42:50.460 This is h over i d ddx now of this function, x f of x. 00:42:50.460 --> 00:42:55.440 And here I have just the x function times this function. 00:42:55.440 --> 00:43:00.080 So h over i x df dx. 00:43:03.630 --> 00:43:05.840 Well, are these the same? 00:43:05.840 --> 00:43:11.750 No, because this d dx here is not only acting on f like here. 00:43:11.750 --> 00:43:12.940 It's acting on the x. 00:43:12.940 --> 00:43:14.860 So this gives you two terms. 00:43:14.860 --> 00:43:18.100 One extra term on the d dx acts on the x. 00:43:18.100 --> 00:43:20.390 And then one term that is equal to this. 00:43:20.390 --> 00:43:22.060 So you don't get the same. 00:43:22.060 --> 00:43:28.340 So you get from here h over i f of x, when you [INAUDIBLE] 00:43:28.340 --> 00:43:34.980 the x plus h over i x the df dx. 00:43:38.470 --> 00:43:39.790 So you don't get the same. 00:43:39.790 --> 00:43:43.050 So when I subtract them, so when I 00:43:43.050 --> 00:43:54.280 do xp minus px acting on the function f of x, what do I get? 00:43:54.280 --> 00:43:57.670 Well, I put them in this order, x before the p. 00:43:57.670 --> 00:43:59.540 Doesn't matter which one you take, 00:43:59.540 --> 00:44:01.280 but many people like this. 00:44:01.280 --> 00:44:05.150 Well, these terms cancel and I get minus this thing. 00:44:05.150 --> 00:44:16.600 So I get minus h over i f of x, or i h bar f of x. 00:44:16.600 --> 00:44:18.460 Wow. 00:44:18.460 --> 00:44:21.470 You got something very strange. 00:44:21.470 --> 00:44:26.740 The x times p minus p times x gives you a number-- 00:44:26.740 --> 00:44:30.900 an imaginary number, even worse-- times f of x. 00:44:30.900 --> 00:44:34.630 So from this we write the following. 00:44:34.630 --> 00:44:38.830 We say look, operators are defined by the action 00:44:38.830 --> 00:44:42.970 and function, but for any function, the only effect of xp 00:44:42.970 --> 00:44:51.600 minus px, which we call the commutator of x with p. 00:44:51.600 --> 00:44:56.010 This definition, the bracket of two things, of A B. 00:44:56.010 --> 00:45:00.830 Is defined to be A B minus B A. It's called the commutator. 00:45:00.830 --> 00:45:06.280 x p is an operator acting on f of x, gives you 00:45:06.280 --> 00:45:09.240 i h bar times f of x. 00:45:09.240 --> 00:45:13.530 So our kind of silly operator that does nothing 00:45:13.530 --> 00:45:14.670 has appeared here. 00:45:14.670 --> 00:45:20.140 Because I could now say that x hat with p 00:45:20.140 --> 00:45:24.070 is equal to i h bar times the unit operator. 00:45:28.700 --> 00:45:30.790 Apart from the Schrodinger equation, 00:45:30.790 --> 00:45:33.160 this is probably the most important equation 00:45:33.160 --> 00:45:34.190 in quantum mechanics. 00:45:37.260 --> 00:45:41.960 It's the fact that x and b are incompatible operators 00:45:41.960 --> 00:45:43.010 as you will see later. 00:45:43.010 --> 00:45:45.270 They don't commute. 00:45:45.270 --> 00:45:47.270 Their order matters. 00:45:47.270 --> 00:45:50.210 What's going to mean is that when you measure one, 00:45:50.210 --> 00:45:52.280 you have difficulties measuring the other. 00:45:52.280 --> 00:45:53.310 They interfere. 00:45:53.310 --> 00:45:55.560 They cannot be measured simultaneously. 00:45:55.560 --> 00:45:58.950 All those things are encapsulated 00:45:58.950 --> 00:46:02.530 in a very lovely mathematical formula, 00:46:02.530 --> 00:46:06.355 which says that this is the way these operators work. 00:46:09.560 --> 00:46:12.460 Any questions? 00:46:12.460 --> 00:46:14.410 Yes? 00:46:14.410 --> 00:46:17.140 AUDIENCE: When x-- the commutator of x and p 00:46:17.140 --> 00:46:18.592 is itself an operator, right? 00:46:18.592 --> 00:46:19.300 PROFESSOR: RIght. 00:46:19.300 --> 00:46:22.190 AUDIENCE: So is that what we're saying? 00:46:22.190 --> 00:46:27.370 When we had operators before, we can't simply 00:46:27.370 --> 00:46:29.190 just cancel the f of x. 00:46:29.190 --> 00:46:31.520 I mean we're not really canceling it, but it just 00:46:31.520 --> 00:46:35.820 because I h bar is the only eigenvalue of the operator? 00:46:35.820 --> 00:46:40.240 PROFESSOR: Well, basically what we've shown by this calculation 00:46:40.240 --> 00:46:43.910 is that this operator, this combination 00:46:43.910 --> 00:46:46.890 is really the same as the identity operator. 00:46:46.890 --> 00:46:50.480 That's all we've shown, that some particular combination is 00:46:50.480 --> 00:46:52.530 the identity operator. 00:46:52.530 --> 00:46:56.200 Now this is very deep, this equation. 00:46:56.200 --> 00:46:59.750 In fact, that's the way Heisenberg 00:46:59.750 --> 00:47:02.800 invented quantum mechanics. 00:47:02.800 --> 00:47:05.610 He called it the matrix mechanics, 00:47:05.610 --> 00:47:11.370 because he knew that operators were related to matrices. 00:47:11.370 --> 00:47:14.420 It's a beautiful story how he came up with this idea. 00:47:14.420 --> 00:47:16.660 It's very different from what we're doing today 00:47:16.660 --> 00:47:19.340 that we're going to follow Schrodinger today. 00:47:19.340 --> 00:47:25.850 But basically his analysis led very quickly to this idea. 00:47:25.850 --> 00:47:28.030 And this is deep. 00:47:34.530 --> 00:47:35.590 Why is it deep? 00:47:38.350 --> 00:47:40.440 Depends who you ask. 00:47:40.440 --> 00:47:44.370 If you ask a mathematician, they would probably tell you 00:47:44.370 --> 00:47:45.560 this equation is not deep. 00:47:45.560 --> 00:47:47.970 This is scary equation. 00:47:47.970 --> 00:47:50.730 And why is it scary? 00:47:50.730 --> 00:47:54.890 Because whenever a mathematician see operators, 00:47:54.890 --> 00:47:58.630 they want to write matrices. 00:47:58.630 --> 00:48:02.120 So the mathematician, you show him this equation, 00:48:02.120 --> 00:48:05.470 will say OK, Let me try to figure out which 00:48:05.470 --> 00:48:08.010 matrices you're talking about. 00:48:08.010 --> 00:48:10.580 And this mathematician will start doing calculations 00:48:10.580 --> 00:48:13.660 with two by two matrices, and will say, no, 00:48:13.660 --> 00:48:15.850 I can't find two by two matrices that 00:48:15.850 --> 00:48:18.550 behave like these operators. 00:48:18.550 --> 00:48:21.140 I can't find three by three matrices either. 00:48:21.140 --> 00:48:22.570 And four by four. 00:48:22.570 --> 00:48:23.990 And five by five. 00:48:23.990 --> 00:48:26.400 And finds no matrix really can do 00:48:26.400 --> 00:48:32.300 that, except if the matrix is infinite dimensional. 00:48:32.300 --> 00:48:34.130 Infinite by infinite matrices. 00:48:34.130 --> 00:48:38.070 So that's why it's very hard for a mathematician. 00:48:38.070 --> 00:48:40.070 This is the beginning of quantum mechanics. 00:48:40.070 --> 00:48:43.960 This looks like a trivial equation, 00:48:43.960 --> 00:48:47.090 and mathematicians get scared by it. 00:48:47.090 --> 00:48:51.930 You show them for physicists there will be angular momentum. 00:48:51.930 --> 00:48:56.010 The operators are like this, and there's complicated 00:48:56.010 --> 00:48:59.720 into the [INAUDIBLE]. 00:48:59.720 --> 00:49:02.170 The three components of angular momentum 00:49:02.170 --> 00:49:04.340 have this commutation relation. 00:49:04.340 --> 00:49:07.100 And h bar here as well. 00:49:07.100 --> 00:49:08.000 Complicated. 00:49:08.000 --> 00:49:09.540 Three operators. 00:49:09.540 --> 00:49:10.930 They mix with each other. 00:49:10.930 --> 00:49:14.000 Show it to a mathematician, he starts laughing at you. 00:49:14.000 --> 00:49:18.380 He says that best, the simplest case, this is easy. 00:49:18.380 --> 00:49:19.390 This is complicated. 00:49:22.010 --> 00:49:23.550 It's very strange. 00:49:23.550 --> 00:49:26.420 But the reason this is easier, the mathematician 00:49:26.420 --> 00:49:29.530 goes and, after five minutes, comes to you with three 00:49:29.530 --> 00:49:33.450 by three matrices that satisfies this relation. 00:49:33.450 --> 00:49:35.150 And here there weren't. 00:49:35.150 --> 00:49:38.250 And four by four that satisfy, and five by five, 00:49:38.250 --> 00:49:40.310 and two by two, and all of them. 00:49:40.310 --> 00:49:43.010 We can calculate all of them for you. 00:49:43.010 --> 00:49:45.770 But this one it's infinite dimensional matrices, 00:49:45.770 --> 00:49:50.570 and it's very surprising, very interesting, and very deep. 00:49:50.570 --> 00:49:55.300 All right, so we move on a little bit more 00:49:55.300 --> 00:49:56.820 to the other observable. 00:49:56.820 --> 00:50:08.930 So after this, we have more general observable. 00:50:08.930 --> 00:50:11.460 So let's talk a little about them. 00:50:11.460 --> 00:50:15.290 That's another postulate of quantum mechanics 00:50:15.290 --> 00:50:18.640 that continues with this one, postulate number five. 00:50:18.640 --> 00:50:27.650 So once you measure, upon measuring 00:50:27.650 --> 00:50:42.640 an observable A associated with the operator A hat, 00:50:42.640 --> 00:50:43.605 two things happen. 00:50:46.130 --> 00:50:49.340 You measure this quantity that could be momentum, 00:50:49.340 --> 00:50:55.360 could be energy, could be position, you name it. 00:50:55.360 --> 00:51:02.560 The measured value must be a number. 00:51:02.560 --> 00:51:13.460 It's one of the eigenvalues of A hat. 00:51:13.460 --> 00:51:17.450 So actually those eigenvalues, remember the definition 00:51:17.450 --> 00:51:18.920 of the eigenvalues. 00:51:18.920 --> 00:51:19.990 It's there. 00:51:19.990 --> 00:51:23.940 I said many, but whenever you measure, 00:51:23.940 --> 00:51:27.750 the only possibilities that you get this number. 00:51:27.750 --> 00:51:32.600 So you measure the momentum, you must get this hk, for example. 00:51:32.600 --> 00:51:37.870 So observables, we have an associated operator, 00:51:37.870 --> 00:51:40.480 and the measured values are the eigenvalues. 00:51:43.630 --> 00:51:47.120 Now these eigenvalues, in order to be observable, 00:51:47.120 --> 00:51:48.940 they should be a real numbers. 00:51:48.940 --> 00:51:51.160 And we said oh, they can be complex. 00:51:51.160 --> 00:51:56.190 Well, we will limit the kind of observables 00:51:56.190 --> 00:51:59.380 to things that have real eigenvalues, 00:51:59.380 --> 00:52:03.830 and these are going to be called later on permission operators. 00:52:03.830 --> 00:52:07.770 At this moment, the notes, they don't mention them. 00:52:07.770 --> 00:52:11.240 You're going to get them confused. 00:52:11.240 --> 00:52:17.650 So anyway, special operators that have real eigenvalues. 00:52:17.650 --> 00:52:21.350 So we mentioned here they will have to be a real. 00:52:21.350 --> 00:52:26.050 Have to be real. 00:52:26.050 --> 00:52:30.330 And then the second one, which is an even stranger thing that 00:52:30.330 --> 00:52:34.000 happens is something you've already seen in examples. 00:52:34.000 --> 00:52:37.360 After you measure, the whole wave function 00:52:37.360 --> 00:52:42.970 goes into the state which is the eigenfunction of the operator. 00:52:42.970 --> 00:53:00.950 So after measurement system collapses into psi a. 00:53:04.060 --> 00:53:10.080 The measure value is one over the eigenvalues a of A. 00:53:10.080 --> 00:53:14.020 And the system collapses into psi a. 00:53:14.020 --> 00:53:21.780 So psi a is such that A hat psi a is a psi a. 00:53:21.780 --> 00:53:26.960 So this is the eigenvector with eigenvalue a that you measured. 00:53:26.960 --> 00:53:29.570 So after you measure the momentum 00:53:29.570 --> 00:53:34.160 and you found that its h bar k, the wave function 00:53:34.160 --> 00:53:37.140 is the wave function of momentum h bar k. 00:53:37.140 --> 00:53:40.450 If at the beginning, it was a superposition of many, 00:53:40.450 --> 00:53:44.140 as Fourier told you, then after measuring, 00:53:44.140 --> 00:53:46.420 if you get one component of momentum, 00:53:46.420 --> 00:53:48.670 that's all that is left of the wave function. 00:53:48.670 --> 00:53:50.560 It collapses. 00:53:50.560 --> 00:53:53.950 This collapse is a very strange thing, 00:53:53.950 --> 00:53:57.370 and is something about quantum mechanics 00:53:57.370 --> 00:54:02.740 that people are a little uncomfortable with, 00:54:02.740 --> 00:54:07.230 and try to understand better, but surprisingly nobody 00:54:07.230 --> 00:54:14.170 has understood it better after 60 years of thinking about it. 00:54:14.170 --> 00:54:15.670 And it works very well. 00:54:15.670 --> 00:54:18.250 It's a very strange thing. 00:54:18.250 --> 00:54:22.320 Because for example, if you have a wave function that 00:54:22.320 --> 00:54:28.240 says your particle can be anywhere, after you measure it 00:54:28.240 --> 00:54:30.710 where it is, the whole wave function 00:54:30.710 --> 00:54:35.240 becomes a delta function at the position that you measure. 00:54:35.240 --> 00:54:38.640 So everything on the wave function, 00:54:38.640 --> 00:54:41.540 when you do a measurement, basically 00:54:41.540 --> 00:54:45.110 collapses as we'll see. 00:54:45.110 --> 00:54:49.650 Now for example, let's do an example. 00:54:49.650 --> 00:54:50.150 Position. 00:54:54.420 --> 00:54:59.130 So you have a wave function psi of x. 00:54:59.130 --> 00:55:09.810 You find measure and find the particle at x0. 00:55:13.350 --> 00:55:16.830 Measure and you find the particle at x0. 00:55:16.830 --> 00:55:21.930 So measure what? 00:55:21.930 --> 00:55:22.900 I should be clear. 00:55:22.900 --> 00:55:23.630 Measure position. 00:55:27.000 --> 00:55:28.580 So we said two things. 00:55:28.580 --> 00:55:32.860 The measured value is one of the eigenvalues of a, 00:55:32.860 --> 00:55:34.690 and after measurement, the system 00:55:34.690 --> 00:55:36.790 collapses to eigenfunctions. 00:55:36.790 --> 00:55:42.310 Now here we really need a little of your intuition. 00:55:42.310 --> 00:55:47.010 Our position eigenstate is a particle a localized at one 00:55:47.010 --> 00:55:47.510 place. 00:55:47.510 --> 00:55:49.850 What is the best function associated 00:55:49.850 --> 00:55:51.430 to a position eigenstate? 00:55:51.430 --> 00:55:52.990 It's a delta function. 00:55:52.990 --> 00:55:57.440 The function that says it's at some point and nowhere else. 00:55:57.440 --> 00:56:04.040 So eigenfunctions delta of x minus x0, it's 00:56:04.040 --> 00:56:08.510 a function as a function of x. 00:56:08.510 --> 00:56:14.140 It peaks at x0, and it's 0 everywhere else. 00:56:14.140 --> 00:56:18.330 And this is, when you find a particle at x0, 00:56:18.330 --> 00:56:19.810 this is the wave function. 00:56:19.810 --> 00:56:24.830 The wave function must be proportional to this quantity. 00:56:24.830 --> 00:56:27.595 Now you can't normalize this wave function. 00:56:27.595 --> 00:56:30.665 It's a small complication, but we shouldn't worry about it 00:56:30.665 --> 00:56:31.165 too much. 00:56:33.680 --> 00:56:37.760 Basically you really can't localize a particle perfectly, 00:56:37.760 --> 00:56:40.890 so that's the little problem with this wave function. 00:56:40.890 --> 00:56:43.540 You've studied how you can represent delta functions 00:56:43.540 --> 00:56:47.130 as limits and probably intuitively those limits 00:56:47.130 --> 00:56:48.740 are the best things. 00:56:48.740 --> 00:56:52.430 But this is the wave function, so after you 00:56:52.430 --> 00:56:56.400 measure the system, you go into an eigenstate of the operator. 00:56:56.400 --> 00:56:59.420 Is this an eigenstate of the x operator? 00:56:59.420 --> 00:57:01.740 What a strange question. 00:57:01.740 --> 00:57:02.850 But it is. 00:57:02.850 --> 00:57:10.310 Look, if you put the x operator on delta of x minus x zero, 00:57:10.310 --> 00:57:11.870 what is it supposed to do? 00:57:11.870 --> 00:57:14.620 It's supposed to multiply by x. 00:57:14.620 --> 00:57:21.160 So it's x times delta of x minus x zero. 00:57:21.160 --> 00:57:23.950 If you had a little experience with delta functions, 00:57:23.950 --> 00:57:27.270 you'd know that this function is 0 everywhere, 00:57:27.270 --> 00:57:33.230 except when x is equal to x0, so this x can be turned into x0. 00:57:33.230 --> 00:57:37.220 It just never at any other place it contributes. 00:57:37.220 --> 00:57:44.490 This x really can be turned into x0 times delta of x minus x0. 00:57:44.490 --> 00:57:47.680 Because delta functions are really used to do integrals. 00:57:47.680 --> 00:57:51.160 And if you do the integral of this function, 00:57:51.160 --> 00:57:53.860 you will see that it gives you the same value as the integral 00:57:53.860 --> 00:57:55.770 of this function. 00:57:55.770 --> 00:57:57.230 So there you have it. 00:57:57.230 --> 00:58:00.290 The operator acting on the eigenfunction 00:58:00.290 --> 00:58:01.790 is a number times this. 00:58:01.790 --> 00:58:04.970 So these are indeed eigenfunctions 00:58:04.970 --> 00:58:07.390 of the x operator. 00:58:07.390 --> 00:58:11.610 And what you measured was an eigenvalue of the x operator. 00:58:11.610 --> 00:58:16.780 Eigenvalue of x. 00:58:16.780 --> 00:58:20.970 And this is an eigenfunction of x. 00:58:26.980 --> 00:58:34.570 So we can do the same with the momentum. 00:58:34.570 --> 00:58:39.255 Eigenvalues and eigenfunctions, we've seen them more properly. 00:58:42.000 --> 00:58:46.270 Now we'll go to the sixth postulate, the last postulate 00:58:46.270 --> 00:58:55.220 that we'll want to talk about is the one of general operators, 00:58:55.220 --> 00:58:59.970 and general eigenfunctions, and what happens with them. 00:58:59.970 --> 00:59:10.590 So let's take now our operator A and its functions that 00:59:10.590 --> 00:59:12.310 can be found. 00:59:12.310 --> 00:59:32.300 So six, given an observable A hat and its eigenfunctions phi 00:59:32.300 --> 00:59:34.620 a of x. 00:59:34.620 --> 00:59:40.285 So an a runs over many values, many values. 00:59:43.910 --> 00:59:47.730 OK so let's consider this case. 00:59:47.730 --> 00:59:51.310 Now eigenfunctions of an operator 00:59:51.310 --> 00:59:54.610 are very interesting objects. 00:59:54.610 --> 01:00:00.290 You see the eigenfunctions of momentum were of this form. 01:00:00.290 --> 01:00:04.610 And they allow you to expand via the Fourier 01:00:04.610 --> 01:00:08.750 any wave function as super positions of these things. 01:00:08.750 --> 01:00:12.140 Fourier told you you can't expand any function 01:00:12.140 --> 01:00:14.940 in eigenfunctions of the momentum, 01:00:14.940 --> 01:00:17.900 or the result is more general. 01:00:17.900 --> 01:00:21.790 For observables in general you can 01:00:21.790 --> 01:00:24.660 expand functions, arbitrary functions, 01:00:24.660 --> 01:00:27.630 in terms of the eigenfunctions. 01:00:27.630 --> 01:00:31.940 Now for that, remember, an eigenfunction 01:00:31.940 --> 01:00:35.070 is not determined up to scale. 01:00:35.070 --> 01:00:39.030 You change multiplied by three, it's an eigenfunction still. 01:00:39.030 --> 01:00:43.470 So people like to normalize them nicely, the eigenfunctions. 01:00:43.470 --> 01:00:47.580 You construct them like this, and you normalize them nicely. 01:00:47.580 --> 01:00:50.300 So how do you normalize them? 01:00:50.300 --> 01:00:59.150 Normalize them by saying that the integral over x 01:00:59.150 --> 01:01:06.930 of psi a star of x psi b of x is going to be what? 01:01:06.930 --> 01:01:12.560 OK, basically what you want is that these eigenfunctions 01:01:12.560 --> 01:01:14.190 be basically orthogonal. 01:01:14.190 --> 01:01:17.020 Each one orthogonal to the next. 01:01:17.020 --> 01:01:20.360 So you want this to be 0 unless these two 01:01:20.360 --> 01:01:22.260 different eigenfunctions are different. 01:01:22.260 --> 01:01:24.840 And when they are the same, you want 01:01:24.840 --> 01:01:26.820 them to be just like wave functions, 01:01:26.820 --> 01:01:30.640 that their total integral of psi squared is equal to 1. 01:01:30.640 --> 01:01:32.695 So what you put here is delta ab. 01:01:35.380 --> 01:01:38.050 Now this is something that you can always 01:01:38.050 --> 01:01:40.200 do with eigenfunctions. 01:01:40.200 --> 01:01:42.680 It's proven in mathematics books. 01:01:42.680 --> 01:01:44.760 It's not all that simple to prove, 01:01:44.760 --> 01:01:47.980 but this can always be done. 01:01:47.980 --> 01:01:53.740 And when we need examples, we'll do it ourselves. 01:01:53.740 --> 01:01:58.370 So given an operator that you have its eigenfunctions 01:01:58.370 --> 01:02:00.230 like that, two things happen. 01:02:00.230 --> 01:02:12.800 One can expand psi as psi of x. 01:02:12.800 --> 01:02:17.160 Any arbitrary wave function as the sum. 01:02:17.160 --> 01:02:18.770 Or sometimes an integral. 01:02:18.770 --> 01:02:22.340 So some people like to write this and put them an integral 01:02:22.340 --> 01:02:24.910 on top of that. 01:02:24.910 --> 01:02:26.990 You can write it whichever way you want. 01:02:26.990 --> 01:02:28.150 It doesn't matter. 01:02:28.150 --> 01:02:33.850 Of coefficients times the eigenfunctions. 01:02:33.850 --> 01:02:38.330 So just like any wave could be a written, a Fourier coefficient 01:02:38.330 --> 01:02:40.330 [INAUDIBLE] Fourier function. 01:02:40.330 --> 01:02:47.180 Any state can be written a superposition of these things. 01:02:47.180 --> 01:02:49.580 So that's one. 01:02:49.580 --> 01:03:10.940 And two, the probability of measuring A hat and getting a. 01:03:10.940 --> 01:03:16.150 So a one of the particular values that you can get. 01:03:16.150 --> 01:03:22.390 That probability is given by the square of this coefficient. 01:03:24.930 --> 01:03:29.010 Ca squared. 01:03:29.010 --> 01:03:36.870 So this is P, the probability, to measure in psi and to get a. 01:03:36.870 --> 01:03:40.150 I think, actually, let's put an a0 there. 01:03:44.430 --> 01:03:49.360 So here we go. 01:03:49.360 --> 01:03:52.600 So it's a very interesting thing. 01:03:52.600 --> 01:03:55.190 Basically you expand the wave function 01:03:55.190 --> 01:03:57.950 in terms of these eigenfunctions, 01:03:57.950 --> 01:04:02.070 and these coefficients give you the probabilities 01:04:02.070 --> 01:04:05.130 of measuring these numbers. 01:04:05.130 --> 01:04:10.380 So we can illustrate that again with the delta functions, 01:04:10.380 --> 01:04:12.450 and we'll do it quick, because we 01:04:12.450 --> 01:04:17.000 get to the punchline of this lecture with the Schrodinger 01:04:17.000 --> 01:04:18.720 equation. 01:04:18.720 --> 01:04:24.540 So what do we have? 01:04:24.540 --> 01:04:29.920 Well, let me think of the operator X example. 01:04:29.920 --> 01:04:33.220 Operator X, the eigenfunctions are 01:04:33.220 --> 01:04:38.812 delta of x minus x0 for all x0. 01:04:38.812 --> 01:04:40.020 These are the eigenfunctions. 01:04:44.290 --> 01:04:47.270 And we'll write sort of a trivial equation, 01:04:47.270 --> 01:04:50.730 but it sort of illustrates what's going on. 01:04:50.730 --> 01:04:59.410 Psi of x as a superposition over an integral over x0 01:04:59.410 --> 01:05:02.930 of delta of x minus x0. 01:05:06.260 --> 01:05:07.520 Psi of x0. 01:05:10.520 --> 01:05:13.550 Delta of x minus x zero. 01:05:20.280 --> 01:05:23.725 OK, first let's check that this make sense. 01:05:23.725 --> 01:05:26.500 Here we're integrating over x0. 01:05:26.500 --> 01:05:28.290 x0 is the variable. 01:05:28.290 --> 01:05:32.430 This thing shoots and fires whenever 01:05:32.430 --> 01:05:37.030 x is equal to x-- whenever x0 is equal to x. 01:05:37.030 --> 01:05:38.890 Therefore the whole result of that integral 01:05:38.890 --> 01:05:42.590 is psi of x is a little funny how it's written, 01:05:42.590 --> 01:05:46.250 because you have x minus 0, which is the same 01:05:46.250 --> 01:05:50.660 as delta of x0 minus x is just the same thing. 01:05:50.660 --> 01:05:54.690 And you integrate over x0, and you get just psi of x. 01:05:54.690 --> 01:05:57.490 But what have you achieved here? 01:05:57.490 --> 01:06:03.410 You've achieved the analogue of this equation in which these 01:06:03.410 --> 01:06:08.660 are the psi a. 01:06:08.660 --> 01:06:10.945 These are the coefficients Ca. 01:06:14.700 --> 01:06:19.980 And this is the sum, this integral. 01:06:19.980 --> 01:06:21.420 So there you go. 01:06:21.420 --> 01:06:26.000 Any wave function can be written as the sum of coefficients 01:06:26.000 --> 01:06:30.010 times the eigenfunctions of the operator. 01:06:30.010 --> 01:06:33.920 And what is the probability to find the particle at x0? 01:06:33.920 --> 01:06:35.340 Well, it's from here. 01:06:35.340 --> 01:06:37.040 The coefficients, the a squared. 01:06:37.040 --> 01:06:41.910 That's exactly what we had before. 01:06:41.910 --> 01:06:46.080 So this is getting basically what we want. 01:06:46.080 --> 01:06:48.440 So this brings us to the final stage 01:06:48.440 --> 01:06:53.430 of this lecture in which we have to get the time 01:06:53.430 --> 01:06:55.580 evolution finally. 01:06:55.580 --> 01:07:01.180 So how does it happen? 01:07:01.180 --> 01:07:04.270 Well it happens in a very interesting way. 01:07:04.270 --> 01:07:07.175 So maybe I'll call it seven, Schrodinger equation. 01:07:13.950 --> 01:07:21.910 So as with any fundamental equation in physics, 01:07:21.910 --> 01:07:27.530 there's experimental evidence and suddenly, however, you 01:07:27.530 --> 01:07:31.100 have to do a conceptual leap. 01:07:31.100 --> 01:07:34.440 Experimental evidence doesn't tell you the equation. 01:07:34.440 --> 01:07:36.790 It suggests the equation. 01:07:36.790 --> 01:07:40.500 And it tells you probably what you're doing is right. 01:07:40.500 --> 01:07:44.110 So what we're going to do now is collect some of the evidence 01:07:44.110 --> 01:07:49.220 we had and look at an equation, and then just have 01:07:49.220 --> 01:07:53.670 a flash of inspiration, change something very little, 01:07:53.670 --> 01:07:55.545 and suddenly that's the Schrodinger equation. 01:07:59.260 --> 01:08:02.240 Allan told me, in fact, still sometimes 01:08:02.240 --> 01:08:05.540 are disappointed that we don't derive the Schrodinger 01:08:05.540 --> 01:08:06.160 equation. 01:08:06.160 --> 01:08:08.690 Now let's derive it mathematically. 01:08:08.690 --> 01:08:11.700 But you also don't derive Newton's equations. 01:08:11.700 --> 01:08:14.520 F equal ma. 01:08:14.520 --> 01:08:16.800 You have an inspiration, you get it. 01:08:16.800 --> 01:08:17.979 Newton got it. 01:08:17.979 --> 01:08:21.319 And then you use it and you see it makes sense. 01:08:21.319 --> 01:08:23.580 It has to be a sensible equation, 01:08:23.580 --> 01:08:26.069 and you can test very quickly whether your equation is 01:08:26.069 --> 01:08:27.020 sensible. 01:08:27.020 --> 01:08:29.300 But you can't quite derive it. 01:08:29.300 --> 01:08:33.020 In 805 we come a little closer to deriving the Schrodinger 01:08:33.020 --> 01:08:37.390 equation, which we say unitary time evolution, something 01:08:37.390 --> 01:08:39.819 that I haven't explained what it is, 01:08:39.819 --> 01:08:41.910 implies the Schrodinger equation. 01:08:41.910 --> 01:08:43.740 And that's a mathematical fact. 01:08:43.740 --> 01:08:47.720 And you can begin unitary time evolution, define it, 01:08:47.720 --> 01:08:50.279 and you derive the Schrodinger equation. 01:08:50.279 --> 01:08:53.180 But that's just saying that you've 01:08:53.180 --> 01:08:55.840 substituted the Schrodinger equation by saying there 01:08:55.840 --> 01:08:57.950 is unitary time evolution. 01:08:57.950 --> 01:09:00.689 The Schrodinger question really comes from something 01:09:00.689 --> 01:09:03.660 a little deeper than that. 01:09:03.660 --> 01:09:06.279 Experimentally it comes from something else. 01:09:06.279 --> 01:09:08.790 So how does it come? 01:09:08.790 --> 01:09:12.220 Well, you've studied some of the history of this subject, 01:09:12.220 --> 01:09:18.979 and you've seen that Planck postulated quantized energies 01:09:18.979 --> 01:09:24.660 in multiples of h bar omega. 01:09:24.660 --> 01:09:26.770 And then came Einstein and said look, 01:09:26.770 --> 01:09:32.000 in fact, the energy of a photon is h bar omega. 01:09:32.000 --> 01:09:36.960 And the momentum of the photon was h bar k. 01:09:36.960 --> 01:09:44.510 So all these people, starting with Planck and then Einstein, 01:09:44.510 --> 01:09:47.290 understood what the photon is. 01:09:47.290 --> 01:09:53.399 The quantum of photons for photons, 01:09:53.399 --> 01:09:58.890 you have E is equal h bar omega, and the momentum 01:09:58.890 --> 01:10:02.090 is equal to h bar k. 01:10:02.090 --> 01:10:04.610 I write them as a vector, because the momentum 01:10:04.610 --> 01:10:09.160 is a vector, but we also write them in this because p equal h 01:10:09.160 --> 01:10:13.530 bar k, assuming you move just in one direction. 01:10:13.530 --> 01:10:15.280 And that's the way it's been written. 01:10:15.280 --> 01:10:22.710 So this is the result of much work beginning by Planck, 01:10:22.710 --> 01:10:26.110 Einstein, and Compton. 01:10:29.230 --> 01:10:32.480 So you may recall Einstein said in 1905 01:10:32.480 --> 01:10:36.050 for that there seemed to be this quantum of light that 01:10:36.050 --> 01:10:38.350 carry energy h omega. 01:10:38.350 --> 01:10:41.130 Planck didn't quite like that. 01:10:41.130 --> 01:10:43.080 And people were not all that convinced. 01:10:43.080 --> 01:10:47.620 Experiments were done by Millikan in 1915, 01:10:47.620 --> 01:10:51.350 and people were still not quite convinced. 01:10:51.350 --> 01:10:54.640 And then came Compton and did Compton scattering. 01:10:54.640 --> 01:10:57.250 And then people said, yeah, they seem to be particles. 01:10:57.250 --> 01:10:58.790 No way out of that. 01:10:58.790 --> 01:11:03.960 And they satisfy such a relation. 01:11:03.960 --> 01:11:10.200 Now there was something about this 01:11:10.200 --> 01:11:15.460 that was quite nice, that these photons are associated 01:11:15.460 --> 01:11:21.140 with waves, and that was not too surprising, 01:11:21.140 --> 01:11:28.960 because people understood that electromagnetic waves are waves 01:11:28.960 --> 01:11:30.700 that correspond to photons. 01:11:30.700 --> 01:11:33.550 So you can also see that this says 01:11:33.550 --> 01:11:40.220 that E p is equal to h bar omega k 01:11:40.220 --> 01:11:42.950 as an equation between vectors. 01:11:42.950 --> 01:11:48.000 You see the E is the first, and the p is the second equation. 01:11:48.000 --> 01:11:52.540 And this is actually a relativistic equation. 01:11:52.540 --> 01:11:55.310 It's a wonderful relativistic equation, 01:11:55.310 --> 01:11:59.910 because energy and momentum form what is called 01:11:59.910 --> 01:12:03.340 a relativity of four vector. 01:12:03.340 --> 01:12:08.550 It's the four vector-- this is a little aside on relativity-- 01:12:08.550 --> 01:12:11.310 four vector. 01:12:11.310 --> 01:12:16.080 The index mew runs from 0, 1, 2, 3. 01:12:16.080 --> 01:12:20.760 Just like the x mews, which are t and x. 01:12:20.760 --> 01:12:25.790 Run from x0, which is t, x1, which is x, x2 which is y, 01:12:25.790 --> 01:12:28.290 three-- these are four vectors. 01:12:28.290 --> 01:12:30.500 And this is a four vector. 01:12:30.500 --> 01:12:31.760 This is a four vector. 01:12:31.760 --> 01:12:34.720 This all seemed quite pretty, and this 01:12:34.720 --> 01:12:38.200 was associated to photons. 01:12:38.200 --> 01:12:42.080 But then came De Broglie. 01:12:42.080 --> 01:12:46.020 And De Broglie had a very daring idea 01:12:46.020 --> 01:12:49.370 that even though this was written for photons, 01:12:49.370 --> 01:12:53.150 it was true for particles as well, for any particle. 01:12:53.150 --> 01:13:01.805 De Broglie says good for particles, all particles. 01:13:04.770 --> 01:13:08.810 And these particles are really waves. 01:13:08.810 --> 01:13:13.810 So what if he write-- he wrote psi of x and t 01:13:13.810 --> 01:13:17.530 is equal a wave associated to a matter particle. 01:13:17.530 --> 01:13:25.790 And it would be an e to the i kx minus omega t. 01:13:25.790 --> 01:13:26.505 That's a wave. 01:13:29.220 --> 01:13:37.970 And you know that this wave has momentum p equal h bar k. 01:13:37.970 --> 01:13:42.690 If k is positive, look at this sign. 01:13:42.690 --> 01:13:46.080 If this sign is like that, then k is positive. 01:13:46.080 --> 01:13:49.640 This is a wave that is moving to the right. 01:13:49.640 --> 01:13:51.885 So p being hk. 01:13:51.885 --> 01:13:55.080 If k is positive, p is positive, is moving to the right, 01:13:55.080 --> 01:14:03.890 this is a wave moving to the right, and has this momentum. 01:14:03.890 --> 01:14:07.160 So it should also have an energy. 01:14:07.160 --> 01:14:11.520 Compton said that this is relativistic because this all 01:14:11.520 --> 01:14:12.820 comes from photons. 01:14:12.820 --> 01:14:19.920 So if the momentum is given by that, and the energy 01:14:19.920 --> 01:14:23.160 must also be given by a similar relation. 01:14:23.160 --> 01:14:26.520 In fact, he mostly said, look, you 01:14:26.520 --> 01:14:32.120 must have the energy being equal to h bar omega. 01:14:32.120 --> 01:14:37.960 The momentum, therefore, would be equal to h bar k. 01:14:37.960 --> 01:14:42.560 And I will sometimes erase these things. 01:14:42.560 --> 01:14:46.310 So what happens with this thing? 01:14:46.310 --> 01:14:49.570 Well, momentum equal to hk. 01:14:49.570 --> 01:14:56.120 We've already understood this as momentum operator 01:14:56.120 --> 01:14:59.655 being h bar over i d dx. 01:15:04.510 --> 01:15:09.290 So this fact that these two must go together and be 01:15:09.290 --> 01:15:13.030 true for particles was De Broglie's insight, 01:15:13.030 --> 01:15:15.850 and the connection to relativity. 01:15:15.850 --> 01:15:17.790 Now here we have this. 01:15:17.790 --> 01:15:22.820 So now we just have to try to figure out what could we 01:15:22.820 --> 01:15:25.060 do for the energy. 01:15:25.060 --> 01:15:27.500 Could we have an energy operator? 01:15:27.500 --> 01:15:31.820 What would the energy operator have to do? 01:15:31.820 --> 01:15:34.730 Well, if the energy operator is supposed 01:15:34.730 --> 01:15:38.995 to give us h bar omega, the only thing it could be 01:15:38.995 --> 01:15:45.400 is that the energy is i h bar d dt. 01:15:45.400 --> 01:15:45.940 Why? 01:15:45.940 --> 01:15:50.400 Because you go again at the wave function. 01:15:50.400 --> 01:15:55.740 And you think i h bar d dt, and what do you get? 01:15:55.740 --> 01:16:08.675 i h bar d dt on the wave function is equal to i h bar. 01:16:08.675 --> 01:16:14.420 You take the d dt, you get minus i omega 01:16:14.420 --> 01:16:16.450 times the whole wave function. 01:16:16.450 --> 01:16:22.984 So this is equal h bar omega times the wave function, 01:16:22.984 --> 01:16:25.240 times the wave function like that. 01:16:25.240 --> 01:16:26.490 So here it is. 01:16:26.490 --> 01:16:30.380 This is the operator that realizes the energy, 01:16:30.380 --> 01:16:34.870 just like this is the operator that realizes the momentum. 01:16:34.870 --> 01:16:40.760 You could say these are the main relations that we have. 01:16:40.760 --> 01:16:46.110 So if you have this wave function, 01:16:46.110 --> 01:16:55.480 it corresponds to a particle with momentum hk and energy h 01:16:55.480 --> 01:16:56.680 omega. 01:16:56.680 --> 01:17:01.680 So now we write this. 01:17:01.680 --> 01:17:15.670 So for this psi that we have here, 01:17:15.670 --> 01:17:22.030 h bar over i d dx of psi of x and t 01:17:22.030 --> 01:17:27.890 is equal the value of the momentum times psi of x and t. 01:17:27.890 --> 01:17:30.880 That is something we've seen. 01:17:30.880 --> 01:17:32.810 But then there's a second one. 01:17:32.810 --> 01:17:42.940 For this psi, we also that i h bar d dt of psi of x and t 01:17:42.940 --> 01:17:48.270 is equal to the energy of that particle times x and t, 01:17:48.270 --> 01:17:52.010 because the energy of that particle is h bar omega. 01:17:56.430 --> 01:18:02.770 And look, this is familiar. 01:18:02.770 --> 01:18:08.820 And here the t plays no role, but here the t plays a role. 01:18:08.820 --> 01:18:14.670 And this is prescribing you how a wave function of energy E 01:18:14.670 --> 01:18:17.830 evolves in time. 01:18:17.830 --> 01:18:19.260 So you're almost there. 01:18:19.260 --> 01:18:22.930 You have something very deep in here. 01:18:22.930 --> 01:18:25.290 It's telling you if you know the wave function 01:18:25.290 --> 01:18:30.240 and it has energy E, this is how it looks later. 01:18:30.240 --> 01:18:33.620 You can take this derivative and solve this differential 01:18:33.620 --> 01:18:35.570 equation. 01:18:35.570 --> 01:18:38.200 Now this differential equation is kind of trivial 01:18:38.200 --> 01:18:40.260 because E is a number here. 01:18:45.330 --> 01:18:51.180 But if you know that you have a particle with energy E, 01:18:51.180 --> 01:18:54.800 that's how it evolves in time. 01:18:54.800 --> 01:18:59.190 So came Schrodinger and looked at this equation. 01:19:02.300 --> 01:19:09.970 Psi of x and t equal E psi of x and t. 01:19:09.970 --> 01:19:14.240 This is true for any particle that has energy E. 01:19:14.240 --> 01:19:18.580 How can I make out of this a full equation? 01:19:18.580 --> 01:19:22.390 Because maybe I don't know what is the energy E. 01:19:22.390 --> 01:19:26.410 The energy E might be anything in general. 01:19:26.410 --> 01:19:27.490 What can I do? 01:19:30.260 --> 01:19:32.090 Very simple. 01:19:32.090 --> 01:19:34.710 One single replacement in that equation. 01:19:38.670 --> 01:19:40.790 Done. 01:19:40.790 --> 01:19:41.490 It's over. 01:19:41.490 --> 01:19:43.000 That's the Schrodinger equation. 01:19:43.000 --> 01:19:48.308 It's the energy operator that we introduced before. 01:19:48.308 --> 01:19:48.807 Inspiration. 01:19:53.150 --> 01:19:57.110 Change E to E hat. 01:19:57.110 --> 01:19:58.920 This is the Schrodinger equation. 01:19:58.920 --> 01:20:02.570 Now what has really happened here, 01:20:02.570 --> 01:20:07.520 this equation that was trivial for a wave function that 01:20:07.520 --> 01:20:10.700 represented a particle with energy E, 01:20:10.700 --> 01:20:15.360 if this is the energy operator, this is not so easy anymore. 01:20:15.360 --> 01:20:18.970 Because remember, the energy operator, for example, 01:20:18.970 --> 01:20:26.590 was p squared over 2m plus v of x. 01:20:26.590 --> 01:20:33.590 And this was minus h squared over 2m d second dx squared 01:20:33.590 --> 01:20:39.640 plus v of x acting on wave functions. 01:20:39.640 --> 01:20:44.040 So now you've got a really interesting equation, 01:20:44.040 --> 01:20:49.070 because you don't assume that the energy is a number, 01:20:49.070 --> 01:20:50.740 because you don't know it. 01:20:50.740 --> 01:20:52.740 In general, if the particle is moving 01:20:52.740 --> 01:20:55.830 in a complicated potential, you don't know 01:20:55.830 --> 01:20:58.180 what are the possible energies. 01:20:58.180 --> 01:21:02.830 But this is symbolically what must be happening, 01:21:02.830 --> 01:21:06.630 because if this particle has a definite energy, 01:21:06.630 --> 01:21:11.040 then this energy operator gives you the energy acting 01:21:11.040 --> 01:21:13.560 on the function, and then you recover 01:21:13.560 --> 01:21:18.160 what you know is true for a particle of a given energy. 01:21:18.160 --> 01:21:20.820 So in general, the Schrodinger equation 01:21:20.820 --> 01:21:22.810 is a complicated equation. 01:21:22.810 --> 01:21:24.810 Let's write it now completely. 01:21:24.810 --> 01:21:26.540 So this is the Schrodinger equation. 01:21:30.390 --> 01:21:32.590 And if we write it completely, it 01:21:32.590 --> 01:21:41.030 will read i h bar d psi dt is equal to minus h bar squared 01:21:41.030 --> 01:21:49.880 over 2m d second dx squared of psi plus v of x times psi-- 01:21:49.880 --> 01:21:53.900 psi of x and t, psi of x and t. 01:21:59.650 --> 01:22:02.090 So it's an equation, a differential equation. 01:22:02.090 --> 01:22:07.920 It's first order in time, and second order in space. 01:22:07.920 --> 01:22:13.210 So let me say three things about this equation and finish. 01:22:13.210 --> 01:22:17.790 First, it requires complex numbers. 01:22:17.790 --> 01:22:21.330 If psi would be real, everything on the right hand side 01:22:21.330 --> 01:22:22.150 would be real. 01:22:22.150 --> 01:22:25.980 But with an i it would spoil it, so complex numbers 01:22:25.980 --> 01:22:28.300 have to be there. 01:22:28.300 --> 01:22:31.290 Second, it's a linear equation. 01:22:31.290 --> 01:22:33.310 It satisfies the proposition. 01:22:33.310 --> 01:22:36.270 So if one wave function satisfies the Schrodinger 01:22:36.270 --> 01:22:38.860 equation, the sum of wave functions, 01:22:38.860 --> 01:22:42.380 and another wave function does, the sum does. 01:22:42.380 --> 01:22:44.490 Third, it's deterministic. 01:22:44.490 --> 01:22:49.430 If you know psi at x and time equals 0, 01:22:49.430 --> 01:22:53.320 you can calculate psi at any later time, 01:22:53.320 --> 01:22:57.890 because this is a first order differential equation in time. 01:22:57.890 --> 01:23:00.030 This equation will be the subject of all 01:23:00.030 --> 01:23:02.300 what we'll do in this course. 01:23:02.300 --> 01:23:03.940 So that's it for today. 01:23:03.940 --> 01:23:05.620 Thank you. 01:23:05.620 --> 01:23:08.370 [APPLAUSE]