WEBVTT 00:00:00.050 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.810 under a Creative Commons license. 00:00:03.810 --> 00:00:06.540 Your support will help MIT OpenCourseWare continue 00:00:06.540 --> 00:00:10.120 to offer high-quality educational resources for free. 00:00:10.120 --> 00:00:12.700 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 --> 00:00:17.282 at ocw.mit.edu. 00:00:21.282 --> 00:00:22.310 PROFESSOR: All right. 00:00:22.310 --> 00:00:30.750 So today, we'll continue our kind of review that included, 00:00:30.750 --> 00:00:34.130 of course, the last lecture, the variational principle that's 00:00:34.130 --> 00:00:38.560 supposed to be new stuff you didn't see in 804. 00:00:38.560 --> 00:00:42.690 And today, as we continue, we'll talk 00:00:42.690 --> 00:00:47.930 about position and momentum for about 30 minutes or 40 minutes, 00:00:47.930 --> 00:00:54.430 and then begin the study of spin. 00:00:54.430 --> 00:00:59.110 That will be spin-1/2 with a Stern-Gerlach experiment 00:00:59.110 --> 00:01:02.430 and the mathematics that comes out of it. 00:01:02.430 --> 00:01:05.310 Now, we will talk about the Stern-Gerlach experiment 00:01:05.310 --> 00:01:07.440 in quite some detail so that you can 00:01:07.440 --> 00:01:10.770 appreciate what was going on there. 00:01:10.770 --> 00:01:18.060 And then we will extract a few of the mathematical lessons 00:01:18.060 --> 00:01:23.410 that this experiment tells us about quantum mechanics. 00:01:23.410 --> 00:01:25.530 Immediately after that, which will 00:01:25.530 --> 00:01:30.030 be probably middle of next lecture, we will pivot. 00:01:30.030 --> 00:01:32.580 And as we learn this mathematics that 00:01:32.580 --> 00:01:37.790 the Stern-Gerlach experiment is telling us or asking us for, 00:01:37.790 --> 00:01:42.700 we will go in some detail on the necessary mathematics 00:01:42.700 --> 00:01:44.190 for quantum mechanics. 00:01:44.190 --> 00:01:48.790 We'll talk about vector spaces, linear operators, 00:01:48.790 --> 00:01:54.200 Hermitian operators, unitary operators, [INAUDIBLE], 00:01:54.200 --> 00:01:57.740 matrix representations, all kinds of things. 00:01:57.740 --> 00:02:02.120 That probably will be about two weeks, three lectures at least. 00:02:02.120 --> 00:02:05.940 So it will be a nice study. 00:02:05.940 --> 00:02:10.190 And in that way, people that don't 00:02:10.190 --> 00:02:12.370 have a background in linear algebra 00:02:12.370 --> 00:02:15.360 will feel more comfortable with what we're going to be doing. 00:02:15.360 --> 00:02:16.880 And I think even for the people that 00:02:16.880 --> 00:02:19.140 have a background in linear algebra, 00:02:19.140 --> 00:02:22.470 you will gain a new appreciation about the concepts 00:02:22.470 --> 00:02:23.950 that we meet here. 00:02:23.950 --> 00:02:28.790 So today, we begin, therefore, with position and momentum, 00:02:28.790 --> 00:02:31.640 and these are operators in quantum mechanics. 00:02:31.640 --> 00:02:36.260 And they have letters to denote them. x, 00:02:36.260 --> 00:02:39.900 we put a hat with it, that's a position operator. 00:02:39.900 --> 00:02:42.730 p, we put a hat on it. 00:02:42.730 --> 00:02:52.460 And the position and momentum operators don't commute. 00:02:52.460 --> 00:02:58.250 And the commutator is given by ih bar. 00:02:58.250 --> 00:03:02.990 Now, we have been dealing so far with wave functions. 00:03:02.990 --> 00:03:07.920 Our wave functions, where these functions of x and t, 00:03:07.920 --> 00:03:11.210 they represent the dynamics of your system, the dynamics 00:03:11.210 --> 00:03:13.410 of your particle as it moves in time. 00:03:13.410 --> 00:03:18.140 But time, as you are seeing in quantum mechanics, 00:03:18.140 --> 00:03:19.602 is a little bit of a spectator. 00:03:19.602 --> 00:03:21.980 It's an arena where things happen. 00:03:21.980 --> 00:03:25.670 But the operators, and most of the interesting things, 00:03:25.670 --> 00:03:30.570 are going on without reference to time. 00:03:30.570 --> 00:03:32.780 Time evolution, you have an expansion 00:03:32.780 --> 00:03:35.140 of a wave function in terms of energy, 00:03:35.140 --> 00:03:38.340 eigenstates, at a given time. 00:03:38.340 --> 00:03:40.650 And then you can evolve it easily 00:03:40.650 --> 00:03:43.700 with the way we've learned, adding e 00:03:43.700 --> 00:03:50.340 to the minus i et over h bar for each energy eigenstate. 00:03:50.340 --> 00:03:52.710 So time will play no role here. 00:03:52.710 --> 00:03:55.380 So when I talk about the wave function, at this moment 00:03:55.380 --> 00:03:57.630 you could put the time, but we will 00:03:57.630 --> 00:04:01.400 talk about the wave functions that have no time dependence. 00:04:01.400 --> 00:04:04.700 So, say, a psi of x wave function. 00:04:11.010 --> 00:04:16.970 So this psi of x may be the true wave function at time equals 0, 00:04:16.970 --> 00:04:23.510 or you could just simply think of it as the psi of x. 00:04:23.510 --> 00:04:26.770 Now, this wave function means that we're 00:04:26.770 --> 00:04:29.420 treating x in a particular way, and we 00:04:29.420 --> 00:04:33.310 say that we're working in the x representation, the position 00:04:33.310 --> 00:04:35.570 representation. 00:04:35.570 --> 00:04:39.870 Now, this means that we have an easy way 00:04:39.870 --> 00:04:43.800 to figure out what this operator does 00:04:43.800 --> 00:04:46.700 when it acts on this function. 00:04:46.700 --> 00:04:48.320 So what it acts on this function, 00:04:48.320 --> 00:04:50.840 it will give you another function, 00:04:50.840 --> 00:04:53.730 and the definition of this is that the position 00:04:53.730 --> 00:04:58.810 operator acting on the function psi of x 00:04:58.810 --> 00:05:03.070 is defined to be another function, 00:05:03.070 --> 00:05:06.845 which is the function x times psi of x. 00:05:16.390 --> 00:05:20.025 Well, we're talking about these wave functions and operators 00:05:20.025 --> 00:05:21.380 on wave functions. 00:05:21.380 --> 00:05:25.550 And a recurrent theme in quantum mechanics 00:05:25.550 --> 00:05:28.660 is that we will think of wave functions, 00:05:28.660 --> 00:05:31.160 sometimes we call them states. 00:05:31.160 --> 00:05:34.200 Sometimes we call them vectors. 00:05:34.200 --> 00:05:39.640 And we basically think of wave functions as vectors. 00:05:39.640 --> 00:05:42.550 And things that act on wave functions 00:05:42.550 --> 00:05:44.480 are the things that act on vectors. 00:05:44.480 --> 00:05:46.390 And the things that act on vectors, 00:05:46.390 --> 00:05:50.110 as you know in mathematics, is matrices. 00:05:50.110 --> 00:05:54.530 So we're compelled, even at this early stage, 00:05:54.530 --> 00:05:57.600 to get a picture of how that language would 00:05:57.600 --> 00:06:02.410 go if we're talking about these things. 00:06:02.410 --> 00:06:07.020 So how do we think of a wave function as a vector? 00:06:07.020 --> 00:06:12.000 And how do we think of x as a matrix? 00:06:12.000 --> 00:06:15.550 So there's a way to do that. 00:06:15.550 --> 00:06:21.580 It will not be totally precise, but it's clear enough. 00:06:21.580 --> 00:06:24.340 So suppose you have a wave function, 00:06:24.340 --> 00:06:28.545 and we're interested in its values from 0 up to a. 00:06:33.120 --> 00:06:38.230 This wave function is a function of x between 0 and a. 00:06:38.230 --> 00:06:45.250 So it's the psi of x for x between a and 0. 00:06:45.250 --> 00:06:47.420 That's all the information. 00:06:47.420 --> 00:06:50.410 What we're going to do is we're going 00:06:50.410 --> 00:06:55.540 to divide this thing, this line, this segment, 00:06:55.540 --> 00:06:56.940 into a lot of pieces. 00:06:56.940 --> 00:06:58.610 And we're going to say, look, instead 00:06:58.610 --> 00:07:03.040 of writing a function like sine of x or cosine of x, 00:07:03.040 --> 00:07:05.910 let's just give the values and organize them 00:07:05.910 --> 00:07:09.780 as if this will be a vector of many components. 00:07:09.780 --> 00:07:15.800 So let's divide this in sizes epsilon, 00:07:15.800 --> 00:07:20.510 such that N times epsilon is equal to a. 00:07:20.510 --> 00:07:23.700 So there are N of these intervals. 00:07:23.700 --> 00:07:29.696 So we think of psi as a vector whose first component 00:07:29.696 --> 00:07:31.390 is psi at 0. 00:07:31.390 --> 00:07:34.190 The second is psi at epsilon. 00:07:34.190 --> 00:07:38.500 The third is psi at 2 epsilon. 00:07:38.500 --> 00:07:43.560 And the last one is psi at N epsilon. 00:07:43.560 --> 00:07:47.370 And depending on how much accuracy you want to work with, 00:07:47.370 --> 00:07:54.390 you take epsilon smaller and larger, keeping a constant. 00:07:54.390 --> 00:07:59.110 And this would be like summarizing all the information 00:07:59.110 --> 00:08:00.705 of a function in a vector. 00:08:03.870 --> 00:08:08.800 Now, that's intuitively a nice way to think of it. 00:08:08.800 --> 00:08:13.960 May look, with your background in classical physics, 00:08:13.960 --> 00:08:18.910 a little strange that we sort of put the value at 0 00:08:18.910 --> 00:08:22.740 along the x-axis, first component, the value at epsilon 00:08:22.740 --> 00:08:26.260 along the y, the value of 2 epsilon along the z. 00:08:26.260 --> 00:08:27.630 But we need more axes. 00:08:27.630 --> 00:08:30.270 So you need many axes here. 00:08:30.270 --> 00:08:39.299 In this case, this is a N plus 1 column vector. 00:08:39.299 --> 00:08:43.480 It has N plus 1 entries, because 0 up 00:08:43.480 --> 00:08:45.740 to N, that's N plus 1 entries. 00:08:49.060 --> 00:08:53.900 But that's a fine way of thinking of it. 00:08:53.900 --> 00:08:56.700 Not exact because we have an epsilon. 00:08:56.700 --> 00:09:00.020 In this way of thinking about the wave function, 00:09:00.020 --> 00:09:08.110 we can then ask, what does the matrix x hat look like? 00:09:08.110 --> 00:09:13.110 So x hat is an operator, and it acts this way. 00:09:13.110 --> 00:09:15.650 So here is how it looks like. 00:09:15.650 --> 00:09:25.180 We would think of x hat as an N plus 1 times N plus 1 matrix. 00:09:30.110 --> 00:09:33.150 And its entries are 0 everywhere, 00:09:33.150 --> 00:09:35.190 except in the diagonal, where they 00:09:35.190 --> 00:09:41.980 are 0 epsilon, 2 epsilon, up to N epsilon. 00:09:44.685 --> 00:09:47.265 And here is a big 0 and a big 0. 00:09:49.980 --> 00:09:55.700 This, I claim, is the way you should think of the x operator 00:09:55.700 --> 00:10:01.640 if you thought of the wave function the way we wrote it. 00:10:01.640 --> 00:10:03.380 And how do we check that? 00:10:03.380 --> 00:10:09.500 Well, x operator acting on psi should be this acting on that. 00:10:09.500 --> 00:10:13.200 And then, indeed, we see that if x hat 00:10:13.200 --> 00:10:18.560 is acting on psi of x, what do we get? 00:10:18.560 --> 00:10:25.350 Well, it's easy to multiply a diagonal matrix times a vector. 00:10:25.350 --> 00:10:31.190 Here you get 0 times psi of 0. 00:10:31.190 --> 00:10:36.390 You get a vector, so let me make this thinner. 00:10:36.390 --> 00:10:45.180 Then I get epsilon times psi of epsilon, 2 epsilon times 00:10:45.180 --> 00:10:53.090 psi of 2 epsilon, up to N epsilon times psi of N epsilon. 00:10:53.090 --> 00:10:58.490 And indeed, that matrix looks like the matrix associated 00:10:58.490 --> 00:11:02.890 with this wave function because here 00:11:02.890 --> 00:11:05.950 is the value at 0 of this wave function. 00:11:05.950 --> 00:11:11.040 Here is the value at epsilon of this wave function, and so on. 00:11:11.040 --> 00:11:15.410 So this has worked out all right. 00:11:15.410 --> 00:11:20.560 We can think of the wave function as a column vector, 00:11:20.560 --> 00:11:24.540 and then the position operator as this vector as well. 00:11:27.270 --> 00:11:32.670 Now, given that we know how the x operator is defined, 00:11:32.670 --> 00:11:37.250 we can also think easily about what 00:11:37.250 --> 00:11:40.810 is the expectation value of x on a wave function. 00:11:40.810 --> 00:11:43.800 Something that you really know, but now 00:11:43.800 --> 00:11:46.370 maybe becomes a little clearer. 00:11:46.370 --> 00:11:49.850 Here you're supposed to do psi star 00:11:49.850 --> 00:11:56.590 of x times the x operator acting on psi of x. 00:11:56.590 --> 00:11:59.220 But we have the definition of this, 00:11:59.220 --> 00:12:05.040 so this is, as you imagine, dx-- and I should put primes maybe, 00:12:05.040 --> 00:12:09.430 well, I don't have to put primes-- dx psi 00:12:09.430 --> 00:12:14.910 star of x x psi of x, which is what 00:12:14.910 --> 00:12:16.575 you would have done anyway. 00:12:20.960 --> 00:12:26.200 Well, given that we've started with this, we can ask also, 00:12:26.200 --> 00:12:29.960 is there eigenstates of the x operator? 00:12:39.360 --> 00:12:40.970 Yes, there are. 00:12:40.970 --> 00:12:44.790 but then fortunately, are a bit singular. 00:12:44.790 --> 00:12:48.150 So what should be an eigenstate of x? 00:12:48.150 --> 00:12:51.860 It's some sort of state. 00:12:51.860 --> 00:12:55.480 Intuitively, it has a definite value of the position. 00:12:55.480 --> 00:12:57.970 So it just exists for some value of x. 00:12:57.970 --> 00:13:01.390 So it's naturally thought as a delta function. 00:13:01.390 --> 00:13:09.370 So let me define a function, psi sub x0 of x. 00:13:09.370 --> 00:13:13.530 So it's a function of x labeled by x0, 00:13:13.530 --> 00:13:17.320 and define it to be delta of x minus x0. 00:13:21.050 --> 00:13:24.790 So I claim that is an eigenstate of x hat. 00:13:24.790 --> 00:13:31.750 x hat on psi x0 of x is equal, by definition, 00:13:31.750 --> 00:13:42.990 to x times psi x0 of x, which is x times delta of x minus x0. 00:13:42.990 --> 00:13:45.220 And when you multiply a function of x 00:13:45.220 --> 00:13:48.250 times a delta function in x, it is 00:13:48.250 --> 00:13:50.660 possible to evaluate the function that 00:13:50.660 --> 00:13:53.550 is being multiplied by the delta function 00:13:53.550 --> 00:13:57.350 at the place where the delta function fires. 00:13:57.350 --> 00:14:00.880 It has the same effect on integrals or anything 00:14:00.880 --> 00:14:01.730 that you would do. 00:14:01.730 --> 00:14:08.230 So here, this is equal to x0 times delta x minus x0. 00:14:08.230 --> 00:14:11.950 You evaluate the x at x0. 00:14:11.950 --> 00:14:18.550 And finally, this is x0 times that function psi x0 of x. 00:14:18.550 --> 00:14:23.490 And therefore, you've shown that this operator acting 00:14:23.490 --> 00:14:27.510 on this function reproduces the function-- that's 00:14:27.510 --> 00:14:30.880 the definition of eigenstate as an operator-- 00:14:30.880 --> 00:14:36.170 and the eigenvalue is the number that appears here, and it's x0. 00:14:36.170 --> 00:14:44.150 So this function is an eigenstate 00:14:44.150 --> 00:14:51.950 of x hat with eigenvalue, e.v., x0. 00:14:59.010 --> 00:15:03.260 The only complication with this eigenfunction 00:15:03.260 --> 00:15:05.780 is that it's not normalizable. 00:15:05.780 --> 00:15:09.060 So it doesn't represent the particle. 00:15:09.060 --> 00:15:13.690 It can be used to represent the particle, 00:15:13.690 --> 00:15:16.490 but it's a useful function. 00:15:16.490 --> 00:15:19.000 You can think of it as something that 00:15:19.000 --> 00:15:22.360 can help you do physics, and don't insist 00:15:22.360 --> 00:15:26.000 that it represents a particle. 00:15:26.000 --> 00:15:28.640 So this is the story for position. 00:15:28.640 --> 00:15:32.680 And the position gets actually more interesting as soon 00:15:32.680 --> 00:15:37.380 as you introduce the dual quantity, momentum. 00:15:37.380 --> 00:15:40.660 So what is momentum here? 00:15:40.660 --> 00:15:55.020 So momentum is an operator, and this operator must be defined. 00:15:55.020 --> 00:15:59.890 Now, you had a shorthand for it in 804, 00:15:59.890 --> 00:16:04.790 which is p hat equal h bar over i d dx. 00:16:08.070 --> 00:16:10.910 And this shorthand means actually 00:16:10.910 --> 00:16:17.220 that, in what we call the position representation where 00:16:17.220 --> 00:16:22.420 we're using wave functions that depend on x, well, 00:16:22.420 --> 00:16:25.190 the momentum is given by this operator. 00:16:25.190 --> 00:16:27.880 And the story of why this was the case 00:16:27.880 --> 00:16:30.150 was sort of something that was elaborated 00:16:30.150 --> 00:16:33.870 on in 804, the work of de Broglie, that 00:16:33.870 --> 00:16:38.430 saw that the wavelength of the wave 00:16:38.430 --> 00:16:40.620 has to do with the momentum of a wave. 00:16:40.620 --> 00:16:42.850 And finally, people understood that this 00:16:42.850 --> 00:16:46.080 would measure the momentum of the wave. 00:16:46.080 --> 00:16:47.800 So this is the operator. 00:16:47.800 --> 00:16:51.570 And therefore, in the representation that 00:16:51.570 --> 00:16:56.750 we're working-- representation is a word that has a lot 00:16:56.750 --> 00:17:00.250 of precise meaning, but now I'm just using it in the sense 00:17:00.250 --> 00:17:03.430 that, well, we're working either with x's or with p's. 00:17:03.430 --> 00:17:05.140 And we're working with x's. 00:17:05.140 --> 00:17:09.440 That's why p looks like something to do with x. 00:17:09.440 --> 00:17:13.369 So what is p hat on a wave function? 00:17:13.369 --> 00:17:17.739 Well, that's what this means. 00:17:17.739 --> 00:17:20.119 It's another wave function obtained 00:17:20.119 --> 00:17:23.990 by taking the x derivative. 00:17:23.990 --> 00:17:28.620 So that's the definition of it acting on a wave function. 00:17:28.620 --> 00:17:33.310 The one thing that must be verified, of course, 00:17:33.310 --> 00:17:38.340 is that this definition is consistent or implies 00:17:38.340 --> 00:17:39.780 this commutation relation. 00:17:43.330 --> 00:17:46.840 So you've defined it as an operator. 00:17:46.840 --> 00:17:49.770 x, we've defined it as an operator. 00:17:49.770 --> 00:17:51.330 But most of us think that it doesn't 00:17:51.330 --> 00:17:53.850 look like an operator is multiplying. 00:17:53.850 --> 00:17:56.580 But it is an operator. 00:17:56.580 --> 00:17:59.230 So this one does look like an operator. 00:17:59.230 --> 00:18:02.000 It's a differential operator. 00:18:02.000 --> 00:18:08.250 And you can try to see if this equation is true. 00:18:08.250 --> 00:18:10.900 And the way to test these commutators 00:18:10.900 --> 00:18:13.110 is something that, again, I don't think 00:18:13.110 --> 00:18:17.630 is unfamiliar to you, but let's go through it, 00:18:17.630 --> 00:18:24.070 is that you try to evaluate this product of operators 00:18:24.070 --> 00:18:27.510 acting on a wave function. 00:18:27.510 --> 00:18:32.400 And if things work out well, we'll 00:18:32.400 --> 00:18:40.174 see you should get ih bar times that wave function. 00:18:40.174 --> 00:18:43.090 If that is the case, you say, OK, I've 00:18:43.090 --> 00:18:47.670 proven that equation, because it's an operator equation. 00:18:47.670 --> 00:18:50.440 The left-hand side of that equation 00:18:50.440 --> 00:18:53.510 is the product in different orders of two operators, 00:18:53.510 --> 00:18:55.290 therefore it's an operator. 00:18:55.290 --> 00:18:57.710 The right-hand side is another operator. 00:18:57.710 --> 00:19:02.270 It's the operator multiplied by ih, anything that you'll get. 00:19:02.270 --> 00:19:07.530 Well, if this is an operator identity, 00:19:07.530 --> 00:19:10.250 the operator on the left must be equal to the operator 00:19:10.250 --> 00:19:14.440 on the right, which just means that, acting on anything, 00:19:14.440 --> 00:19:16.080 they must give the same answer. 00:19:16.080 --> 00:19:20.000 So if I managed to prove that this is equal to this, 00:19:20.000 --> 00:19:24.550 I've proven that for anything that is the answer. 00:19:24.550 --> 00:19:28.330 And therefore, I can write the top one. 00:19:28.330 --> 00:19:31.890 And let me just do it, even though this 00:19:31.890 --> 00:19:37.240 may be kind of familiar to many of you. 00:19:37.240 --> 00:19:42.510 It's good to do this slowly once in your life. 00:19:42.510 --> 00:19:44.800 So let's go through this. 00:19:44.800 --> 00:19:50.360 So this says x operator p operator on psi 00:19:50.360 --> 00:19:54.840 minus p operator of x operator on psi. 00:19:54.840 --> 00:20:01.460 When you have several operators, like ABC acting on psi, 00:20:01.460 --> 00:20:04.990 this really means let C act on psi, 00:20:04.990 --> 00:20:11.960 and then let B act on C psi, and then let A act on that. 00:20:11.960 --> 00:20:13.970 The operators act one by one. 00:20:13.970 --> 00:20:16.360 The closest one acts first. 00:20:16.360 --> 00:20:21.540 So here I'm supposed to let B act on psi, 00:20:21.540 --> 00:20:23.160 but that means that thing. 00:20:23.160 --> 00:20:30.180 So now x is acting on h over i d psi dx. 00:20:32.750 --> 00:20:40.090 On this one, I have p acting on x psi, 00:20:40.090 --> 00:20:43.900 because that's what x hat psi is. 00:20:43.900 --> 00:20:47.770 Here, this is multiplication by x of a function of x. 00:20:47.770 --> 00:20:52.120 So this is just h over i x d psi dx. 00:20:55.310 --> 00:21:01.920 And here, I have h over i d dx of this whole thing x psi. 00:21:04.950 --> 00:21:07.750 And you can see that when you act here, 00:21:07.750 --> 00:21:12.080 you act first on the x, and you get something. 00:21:12.080 --> 00:21:16.580 And then you act on the psi, and you get this same term. 00:21:16.580 --> 00:21:24.130 So the only contribution here is equal to minus h over 00:21:24.130 --> 00:21:28.330 i, the d dx on x times psi, which 00:21:28.330 --> 00:21:32.530 is ih bar psi, which is what I wanted to show. 00:21:36.050 --> 00:21:41.170 So this is true. 00:21:41.170 --> 00:21:44.880 And therefore, you could say that this definition is 00:21:44.880 --> 00:21:47.800 consistent with your definition of x, 00:21:47.800 --> 00:21:51.710 and they represent this operator. 00:21:51.710 --> 00:21:57.010 One more thing you could try to do, and it's fun to do it, 00:21:57.010 --> 00:22:02.930 is we had a matrix representation for x. 00:22:02.930 --> 00:22:05.353 Can I think of p as a matrix? 00:22:07.980 --> 00:22:09.180 How would you do it? 00:22:09.180 --> 00:22:11.880 What kind of matrix would p look like? 00:22:15.680 --> 00:22:17.386 Well, yes? 00:22:17.386 --> 00:22:20.410 AUDIENCE: You just generate a finite difference equation. 00:22:20.410 --> 00:22:22.620 PROFESSOR: You could do it, exactly, 00:22:22.620 --> 00:22:25.920 with taking finite differences. 00:22:25.920 --> 00:22:28.590 So for example, if you think that you 00:22:28.590 --> 00:22:35.555 want to produce the wave function psi prime at 0, 00:22:35.555 --> 00:22:39.200 psi prime at epsilon, psi prime, that's 00:22:39.200 --> 00:22:41.240 what the derivative gives you, you'll 00:22:41.240 --> 00:22:47.970 write this as 1 over epsilon, say, psi at epsilon 00:22:47.970 --> 00:22:50.040 minus psi at 0. 00:22:50.040 --> 00:22:53.950 That's the derivative at 0 roughly. 00:22:53.950 --> 00:23:00.720 It would be psi at 2 epsilon minus psi at 0 over 2. 00:23:00.720 --> 00:23:03.759 And you could build it. 00:23:03.759 --> 00:23:04.550 You could build it. 00:23:04.550 --> 00:23:05.600 I'm not going to do it. 00:23:05.600 --> 00:23:08.920 You may want to do it and try and see 00:23:08.920 --> 00:23:13.550 how the derivative operator looks as a matrix. 00:23:13.550 --> 00:23:18.100 And then if you really want to spend some time thinking 00:23:18.100 --> 00:23:21.770 about it, you could try to see if this matrix 00:23:21.770 --> 00:23:27.000 and this matrix commute to give the right answer. 00:23:27.000 --> 00:23:28.800 And as you try it, you will figure out 00:23:28.800 --> 00:23:32.430 all kinds of funny things that we 00:23:32.430 --> 00:23:34.580 will talk about later in the course. 00:23:34.580 --> 00:23:39.010 So you can represent the momentum operator as a matrix 00:23:39.010 --> 00:23:41.580 indeed, and there are interesting things 00:23:41.580 --> 00:23:47.960 to say about it, and it's a good subject. 00:23:47.960 --> 00:23:52.090 So let's continue with the momentum 00:23:52.090 --> 00:23:55.380 and ask for eigenstates of the momentum. 00:23:55.380 --> 00:24:06.796 So eigenstates of p, you know them. 00:24:06.796 --> 00:24:09.250 They're e to the ipx things. 00:24:09.250 --> 00:24:13.650 So let's write them with some convenient normalization. 00:24:16.180 --> 00:24:18.780 This is an [INAUDIBLE] wave function 00:24:18.780 --> 00:24:21.880 that depends on x with momentum p. 00:24:21.880 --> 00:24:24.250 And we'll write it, as a definition, 00:24:24.250 --> 00:24:32.000 as e to the ipx over h bar, and I'll put it a 2 pi h bar here. 00:24:34.540 --> 00:24:38.440 It's kind of a useful normalization. 00:24:38.440 --> 00:24:46.530 Then p hat on psi p of x, well, p hat is supposed 00:24:46.530 --> 00:24:56.615 to take h over i d dx, and take h over i d dx of psi p. 00:24:59.290 --> 00:25:03.330 And h over i cancels the i over h. 00:25:03.330 --> 00:25:05.680 When you take the d dx, you get p out, 00:25:05.680 --> 00:25:07.520 and you get the same wave function. 00:25:07.520 --> 00:25:12.590 So indeed, you get p times psi p of x. 00:25:12.590 --> 00:25:19.010 So indeed, this is the eigenstate of the momentum 00:25:19.010 --> 00:25:21.655 operator, and it has momentum p. 00:25:24.980 --> 00:25:28.150 Well, what is the use of this? 00:25:28.150 --> 00:25:33.370 Well, say you have a representation, what 00:25:33.370 --> 00:25:36.180 we call the position representation, 00:25:36.180 --> 00:25:38.880 of the wave function and operators. 00:25:38.880 --> 00:25:42.450 Let us think now of the momentum representation. 00:25:42.450 --> 00:25:45.200 So what does all that mean? 00:25:45.200 --> 00:25:49.760 Well, there is the Fourier transform operation 00:25:49.760 --> 00:25:54.780 in which we have psi of p. 00:25:54.780 --> 00:26:00.780 Well, let me write it this way, actually. 00:26:00.780 --> 00:26:05.930 I'll write any psi of x physically 00:26:05.930 --> 00:26:11.340 can be represented as a sum of momentum eigenstates. 00:26:11.340 --> 00:26:15.040 Therefore, that's Fourier's theorem, 00:26:15.040 --> 00:26:23.600 minus infinity to infinity dp e to the ipx over h bar 00:26:23.600 --> 00:26:29.060 square root of 2 pi h psi tilde of p. 00:26:31.880 --> 00:26:38.870 That's Fourier transformation, defines psi tilde of p. 00:26:38.870 --> 00:26:44.450 And Fourier's theorem is the fact that not only you 00:26:44.450 --> 00:26:49.020 can do that, but you can invert it so that psi tilde of p 00:26:49.020 --> 00:26:52.935 can also be written as an integral, this time over x 00:26:52.935 --> 00:27:00.620 from minus infinity to infinity e to the minus ipx over h 00:27:00.620 --> 00:27:09.190 bar, also 2 pi h bar psi of x. 00:27:09.190 --> 00:27:12.690 So let's ponder this equation for a couple of minutes. 00:27:15.640 --> 00:27:21.820 Well, as a physicist, you think of this, well, 00:27:21.820 --> 00:27:25.910 this is telling me that any wave function could 00:27:25.910 --> 00:27:30.380 be written as a superposition of momentum eigenstates. 00:27:30.380 --> 00:27:33.050 Here are the momentum eigenstates. 00:27:33.050 --> 00:27:35.140 And for each value of momentum, you 00:27:35.140 --> 00:27:39.040 have some coefficient here that tells me how much of that 00:27:39.040 --> 00:27:40.640 momentum eigenstate I have. 00:27:44.340 --> 00:27:46.600 Now, here is the opposite one. 00:27:46.600 --> 00:27:52.050 Psi tilde of p and psi of x are related in this way. 00:27:52.050 --> 00:27:55.340 So these coefficients, if you want to calculate them, 00:27:55.340 --> 00:27:57.900 you calculate them this way. 00:27:57.900 --> 00:28:03.100 But now let's think of it as a change of representation. 00:28:06.810 --> 00:28:10.210 The physics is contained in psi of x. 00:28:10.210 --> 00:28:15.060 All what you wish to know about this physical system in quantum 00:28:15.060 --> 00:28:18.040 mechanics is there in psi of x. 00:28:18.040 --> 00:28:21.600 But it's also there in psi of p, because they 00:28:21.600 --> 00:28:24.270 contain the same information. 00:28:24.270 --> 00:28:28.370 So there are different ways of encoding the same information. 00:28:31.540 --> 00:28:36.150 What is the relation between them? 00:28:36.150 --> 00:28:39.240 This, we thought of it as a vector, 00:28:39.240 --> 00:28:44.210 vector in position space, an infinite dimensional space 00:28:44.210 --> 00:28:46.880 that is talking about positions. 00:28:46.880 --> 00:28:51.580 This is another vector in momentum space. 00:28:51.580 --> 00:28:53.720 Think of it now the infinite line. 00:28:53.720 --> 00:28:58.390 So this is an infinite vector with all those points little 00:28:58.390 --> 00:29:02.030 by little, from minus infinity to plus infinity, all of them 00:29:02.030 --> 00:29:05.300 there, gigantic vector. 00:29:05.300 --> 00:29:08.410 And here is another gigantic vector with p 00:29:08.410 --> 00:29:11.250 from minus infinity to infinity. 00:29:11.250 --> 00:29:13.420 And in between, there's an integral. 00:29:13.420 --> 00:29:17.530 But now, with your picture of quantum mechanics, 00:29:17.530 --> 00:29:21.740 you see an integral, but you also see a matrix. 00:29:21.740 --> 00:29:23.580 And what is this matrix? 00:29:23.580 --> 00:29:28.650 Think of this as some sort of psi sub p. 00:29:31.180 --> 00:29:38.590 And this as some sort of matrix, px 00:29:38.590 --> 00:29:47.020 psi x, in which if you have a product-- you'll remember when 00:29:47.020 --> 00:29:51.230 you multiply matrices, a matrix on a vector, 00:29:51.230 --> 00:29:54.520 you sum over the second index. 00:29:54.520 --> 00:29:56.360 That's the product for matrix. 00:29:56.360 --> 00:29:59.400 And then the first index is the index here. 00:29:59.400 --> 00:30:04.330 So here is what it, more or less, is like. 00:30:04.330 --> 00:30:07.040 Psi tilde of p [? subtend ?] by this, 00:30:07.040 --> 00:30:11.750 and this matrix depends on two labels, p and x, and it's that. 00:30:11.750 --> 00:30:16.340 So it's a matrix full of phases. 00:30:16.340 --> 00:30:20.020 So how do you pass from the coordinate representation 00:30:20.020 --> 00:30:23.490 of the information, a vector of all values of the wave 00:30:23.490 --> 00:30:25.340 function in all positions? 00:30:25.340 --> 00:30:31.140 By multiplying with this matrix of phases that is here, 00:30:31.140 --> 00:30:33.070 and it gives you this representation. 00:30:33.070 --> 00:30:38.670 So different representations means using different vectors 00:30:38.670 --> 00:30:42.140 to represent the physics. 00:30:42.140 --> 00:30:45.610 And this vector is a very nice one. 00:30:45.610 --> 00:30:49.070 And because of these properties of the momentum operator 00:30:49.070 --> 00:30:52.940 and all these things, this vector is also a very nice one. 00:30:52.940 --> 00:30:55.690 And there's an integral transform 00:30:55.690 --> 00:30:59.480 or some sort of infinite matrix product that relates them. 00:31:02.450 --> 00:31:05.700 And we shouldn't be uncomfortable about it. 00:31:05.700 --> 00:31:07.260 That's all fine. 00:31:07.260 --> 00:31:12.190 So we say that we have, for example, 00:31:12.190 --> 00:31:22.260 psi of x as one representation of the state and psi tilde of p 00:31:22.260 --> 00:31:25.635 as another representation of the same physics. 00:31:30.950 --> 00:31:38.780 We can do one more thing here, If I continue. 00:31:38.780 --> 00:31:46.160 We can take that boxed equation on the blackboard up there 00:31:46.160 --> 00:31:54.430 and act with h bar over i d dx on psi of x. 00:31:57.120 --> 00:32:02.160 So that is equal to h i d dx, and I'll 00:32:02.160 --> 00:32:08.070 write what psi of x is, is minus infinity to infinity dp e 00:32:08.070 --> 00:32:12.680 to the ipx over h bar square root 00:32:12.680 --> 00:32:17.950 of 2 pi h bar psi tilde of p. 00:32:22.580 --> 00:32:28.190 Now, when we act on this, as you know, h bar over i d dx 00:32:28.190 --> 00:32:32.020 just acts on this and produces the factor of p. 00:32:32.020 --> 00:32:38.410 So this is equal to minus infinity to infinity dp e 00:32:38.410 --> 00:32:45.890 to the ipx over h bar over square root of 2 pi h bar p 00:32:45.890 --> 00:32:48.140 times psi tilde of p. 00:32:54.720 --> 00:32:59.240 So look at this equation again. 00:32:59.240 --> 00:33:02.940 This double arrow is to mean that there 00:33:02.940 --> 00:33:05.080 are equivalent physics in them. 00:33:05.080 --> 00:33:06.790 They have the same information. 00:33:06.790 --> 00:33:10.300 It's the same data encoded in a different way. 00:33:10.300 --> 00:33:13.329 And that different way, this arrow 00:33:13.329 --> 00:33:14.412 is Fourier transformation. 00:33:17.420 --> 00:33:20.750 And this Fourier transformation is explained here. 00:33:20.750 --> 00:33:26.920 So now you have Fourier transformation the same way. 00:33:26.920 --> 00:33:30.270 So here we have-- what we've learned 00:33:30.270 --> 00:33:41.120 is that h over i d dx of psi is represented in momentum space 00:33:41.120 --> 00:33:47.060 by p psi tilde of p. 00:33:47.060 --> 00:33:52.620 And this was p hat acting on psi of x. 00:33:55.520 --> 00:34:00.900 So the corresponding thing in momentum space of p hat 00:34:00.900 --> 00:34:08.530 acting on psi of x is p multiplying psi tilde of p, 00:34:08.530 --> 00:34:15.590 which is to say that we can think of the abstract operator 00:34:15.590 --> 00:34:25.620 p hat acting on psi tilde of p as just p psi tilde of p. 00:34:33.040 --> 00:34:39.639 So in momentum space, the operator p hat 00:34:39.639 --> 00:34:41.260 acts in a very easy way. 00:34:44.280 --> 00:34:48.300 In coordinate space, it takes derivatives. 00:34:48.300 --> 00:34:51.940 In momentum space, it's multiplicative. 00:34:51.940 --> 00:34:58.620 So in position space, x is multiplicative. 00:34:58.620 --> 00:35:05.380 But in momentum space, x would not be multiplicative. 00:35:05.380 --> 00:35:07.780 x would also be a derivative. 00:35:07.780 --> 00:35:11.870 So I leave it for you as an exercise 00:35:11.870 --> 00:35:19.540 to show that or convince yourself in several ways, 00:35:19.540 --> 00:35:34.270 that x hat is really i h bar d dp in p space, in i h bar d dp. 00:35:36.960 --> 00:35:37.760 All right. 00:35:37.760 --> 00:35:43.590 So that's really all I wanted to say about position and momentum 00:35:43.590 --> 00:35:45.850 operators at this moment. 00:35:45.850 --> 00:35:49.960 They will come back when we'll introduce bra-ket notation 00:35:49.960 --> 00:35:50.460 in detail. 00:35:50.460 --> 00:35:52.550 We'll revisit this a little. 00:35:52.550 --> 00:35:56.690 But the main concepts have been illustrated. 00:35:56.690 --> 00:35:57.980 Are there questions? 00:35:57.980 --> 00:36:01.330 We're about to leave this, so if you 00:36:01.330 --> 00:36:02.750 have any questions at this moment. 00:36:02.750 --> 00:36:03.310 Yes? 00:36:03.310 --> 00:36:04.795 AUDIENCE: Could you explain again 00:36:04.795 --> 00:36:10.240 how you used this [INAUDIBLE] h bar over i d dx 00:36:10.240 --> 00:36:12.730 assign to [INAUDIBLE]? 00:36:12.730 --> 00:36:13.920 PROFESSOR: Right. 00:36:13.920 --> 00:36:18.760 So the question was, why did I associate these things? 00:36:18.760 --> 00:36:24.340 So it really goes back here to what the meaning of this arrow 00:36:24.340 --> 00:36:25.420 is. 00:36:25.420 --> 00:36:28.510 The meaning of this arrow is Fourier transformation. 00:36:28.510 --> 00:36:34.420 So this psi tilde and psi of x are related in this way. 00:36:34.420 --> 00:36:36.540 That's Fourier transformation, and that's 00:36:36.540 --> 00:36:38.850 what we mean by this arrow. 00:36:38.850 --> 00:36:43.950 It also means that whatever physics you have here, 00:36:43.950 --> 00:36:45.760 you have it there. 00:36:45.760 --> 00:36:52.570 So really, when you have something acting on a state, 00:36:52.570 --> 00:36:56.050 for example, if you have some operator acting in here, well, 00:36:56.050 --> 00:36:58.270 you get a new wave function. 00:36:58.270 --> 00:37:00.410 And there should be one on the right that 00:37:00.410 --> 00:37:03.810 corresponds to it, that has the same information as the one 00:37:03.810 --> 00:37:06.030 in which you've acted with something. 00:37:06.030 --> 00:37:10.330 So what we claim here is that, also in the sense of Fourier 00:37:10.330 --> 00:37:15.420 transformation or having the same information, h bar over i, 00:37:15.420 --> 00:37:20.130 the derivative of psi, is encoded by this. 00:37:20.130 --> 00:37:24.660 So we say, thinking abstractly, what is this? 00:37:24.660 --> 00:37:28.970 This is the momentum operator. 00:37:28.970 --> 00:37:35.350 Therefore, I'm going to say that the momentum operator really 00:37:35.350 --> 00:37:39.000 is the same momentum operator, whether it acts 00:37:39.000 --> 00:37:42.810 on wave functions that you show them to mean this way or wave 00:37:42.810 --> 00:37:45.680 functions that, because you're in another mood, 00:37:45.680 --> 00:37:48.660 you decide to give them to me in momentum space. 00:37:48.660 --> 00:37:53.690 So as you change your mood, the operator takes different forms 00:37:53.690 --> 00:37:56.380 but is doing the same thing. 00:37:56.380 --> 00:37:58.030 It's totally reversible. 00:37:58.030 --> 00:38:02.460 It's acting on that-- you see, the operator is always 00:38:02.460 --> 00:38:06.370 the same, but you give me the data in two different ways, 00:38:06.370 --> 00:38:10.170 then the operator has to do the thing in a different way. 00:38:10.170 --> 00:38:12.770 So that's what it means that the operator has 00:38:12.770 --> 00:38:15.030 different representations. 00:38:15.030 --> 00:38:16.770 In the [INAUDIBLE] representation, 00:38:16.770 --> 00:38:18.790 it looks like a derivative. 00:38:18.790 --> 00:38:21.915 In the momentum representation, it looks like multiplying. 00:38:25.130 --> 00:38:27.828 Other questions? 00:38:27.828 --> 00:38:28.806 Yes? 00:38:28.806 --> 00:38:31.904 AUDIENCE: So by saying that they sort of represent 00:38:31.904 --> 00:38:33.320 [INAUDIBLE] to the same positions, 00:38:33.320 --> 00:38:36.605 does that mean that h bar over i p e to the xi and p psi p 00:38:36.605 --> 00:38:38.620 are like the same [INAUDIBLE]? 00:38:38.620 --> 00:38:43.690 PROFESSOR: That h bar over d dx psi and p-- yeah. 00:38:43.690 --> 00:38:47.400 They are the same data, the same state 00:38:47.400 --> 00:38:49.345 represented in different ways. 00:38:49.345 --> 00:38:49.845 Yeah. 00:38:54.800 --> 00:38:55.470 All right. 00:38:55.470 --> 00:38:58.470 So time for a change. 00:38:58.470 --> 00:39:03.460 We're going to talk about Stern-Gerlach and spin. 00:39:03.460 --> 00:39:09.580 Now, spin will keep us busy the biggest chunk of this semester. 00:39:09.580 --> 00:39:14.310 So it will be spin-1/2, and we're really going to go 00:39:14.310 --> 00:39:17.200 into enormous detail on it. 00:39:17.200 --> 00:39:20.830 So this is just the beginning of the story that 00:39:20.830 --> 00:39:25.030 will be elaborated at various stages. 00:39:25.030 --> 00:39:30.420 So at this moment, I will talk about this experiment 00:39:30.420 --> 00:39:33.090 that led to the discovery of spin, 00:39:33.090 --> 00:39:37.200 and if you try to invent the theory that describes 00:39:37.200 --> 00:39:42.050 this experiment, what you would possibly begin doing. 00:39:42.050 --> 00:39:44.340 And then we go through the mathematics, 00:39:44.340 --> 00:39:48.720 as I mentioned to you, for maybe a week and a half or two weeks, 00:39:48.720 --> 00:39:52.260 and then return to the spin with more tools 00:39:52.260 --> 00:39:55.960 to understand it well. 00:39:55.960 --> 00:40:00.420 So the subject is the Stern-Gerlach experiment, 00:40:00.420 --> 00:40:06.050 Stern-Gerlach experiment. 00:40:12.650 --> 00:40:21.080 So the Stern-Gerlach experiment was done in Frankfurt, 1922. 00:40:21.080 --> 00:40:23.930 It was an experiment that, in fact, people 00:40:23.930 --> 00:40:25.672 were extraordinarily confused. 00:40:25.672 --> 00:40:29.080 It was not clear why they were doing it. 00:40:29.080 --> 00:40:33.590 And for quite a while, people didn't 00:40:33.590 --> 00:40:37.170 understand what they were getting, 00:40:37.170 --> 00:40:39.400 what was happening with it. 00:40:39.400 --> 00:40:46.760 In fact, Pauli had thought that the electron has 00:40:46.760 --> 00:40:48.890 like two degrees of freedom and didn't 00:40:48.890 --> 00:40:53.410 know what it was, those two degrees of freedom. 00:40:53.410 --> 00:40:57.605 Kronig suggested that it had to do somehow 00:40:57.605 --> 00:41:01.350 with the rotation of the electron. 00:41:01.350 --> 00:41:05.620 Now, Pauli said that's nonsense. 00:41:05.620 --> 00:41:09.950 How can an electron rotate and have angular momentum 00:41:09.950 --> 00:41:11.770 because it has a rotation? 00:41:11.770 --> 00:41:15.450 It would have to rotate so fast, even faster 00:41:15.450 --> 00:41:18.200 than the speed of light to have the angular momentum, 00:41:18.200 --> 00:41:21.510 and then this little ball that would be the electron 00:41:21.510 --> 00:41:22.790 would disintegrate. 00:41:22.790 --> 00:41:27.060 And it made no sense to him that there would be such a thing. 00:41:27.060 --> 00:41:30.890 So Kronig didn't publish this. 00:41:30.890 --> 00:41:35.986 Then there were another two people, Uhlenbeck and Goudsmit, 00:41:35.986 --> 00:41:40.190 at the same time, around 1925, had 00:41:40.190 --> 00:41:44.870 the same idea, angular momentum of this particle. 00:41:44.870 --> 00:41:48.860 And their advisor was Ehrenfest, and said 00:41:48.860 --> 00:41:52.285 it doesn't make too much sense, but you should publish it. 00:41:52.285 --> 00:41:54.390 [LAUGHTER] 00:41:54.390 --> 00:41:57.080 And thanks to their publishing, they 00:41:57.080 --> 00:42:00.430 are given credit for discovering the spin of the electron. 00:42:00.430 --> 00:42:03.390 And Pauli, a couple of years later, 00:42:03.390 --> 00:42:05.260 decided, after all, I was wrong. 00:42:05.260 --> 00:42:08.380 Yes, it is spin, and it's all working out. 00:42:08.380 --> 00:42:14.160 And 1927, five years after the experiment basically, 00:42:14.160 --> 00:42:17.110 people understood what was going on. 00:42:17.110 --> 00:42:20.040 So what were these people trying to do? 00:42:20.040 --> 00:42:23.400 First, Stern and Gerlach were atomic physicists, 00:42:23.400 --> 00:42:26.550 and they were just interested in measuring 00:42:26.550 --> 00:42:34.220 speeds of thermal motion of ions. 00:42:34.220 --> 00:42:37.460 So they would send beams of these ions 00:42:37.460 --> 00:42:41.530 and put magnetic fields and deflect them and measure 00:42:41.530 --> 00:42:43.550 their velocities. 00:42:43.550 --> 00:42:46.170 And eventually, they were experts 00:42:46.170 --> 00:42:47.580 doing this kind of thing. 00:42:47.580 --> 00:42:51.630 And they heard of Bohr, that said that the electron has 00:42:51.630 --> 00:42:56.440 angular momentum and is going around the proton in circles, 00:42:56.440 --> 00:42:58.280 so it might have angular momentum. 00:42:58.280 --> 00:43:01.120 They said, oh, if it has angular momentum because it's 00:43:01.120 --> 00:43:05.330 going around the proton, maybe we can detect it. 00:43:05.330 --> 00:43:09.080 And when they did the experiment, they got something. 00:43:09.080 --> 00:43:11.510 And they said, well, we're seeing it. 00:43:11.510 --> 00:43:14.520 But it was not that. 00:43:14.520 --> 00:43:17.700 They were not seeing the orbital angular momentum 00:43:17.700 --> 00:43:24.650 of the electron because that electron in these silver atoms 00:43:24.650 --> 00:43:27.310 actually has no angular momentum, 00:43:27.310 --> 00:43:30.340 as we will see, no orbital angular momentum. 00:43:30.340 --> 00:43:31.370 It only has spin. 00:43:31.370 --> 00:43:34.220 So they were actually seeing the spin. 00:43:34.220 --> 00:43:36.090 So it was a big confusion. 00:43:36.090 --> 00:43:37.650 It took some time. 00:43:37.650 --> 00:43:41.660 Basically, they took the beam, and they split it 00:43:41.660 --> 00:43:45.770 with a magnetic field, and the clean split 00:43:45.770 --> 00:43:47.510 was something nobody understood. 00:43:47.510 --> 00:43:51.440 So they called it space quantization, 00:43:51.440 --> 00:43:54.500 as of it's separated in space. 00:43:54.500 --> 00:43:56.710 Space is quantized. 00:43:56.710 --> 00:43:59.570 A pretty awful name, of course. 00:43:59.570 --> 00:44:03.200 There's nothing quantized about space here. 00:44:03.200 --> 00:44:07.600 But it reflects that when you don't know what's really 00:44:07.600 --> 00:44:12.500 happening, your names don't come out too well. 00:44:12.500 --> 00:44:17.460 So what we have to understand here, 00:44:17.460 --> 00:44:20.540 our goal today is to just see what's 00:44:20.540 --> 00:44:27.310 happening in that experiment, quantify a bit the results, 00:44:27.310 --> 00:44:33.050 and then extract the quantum mechanical lessons from it. 00:44:33.050 --> 00:44:35.900 So let us begin with the important thing. 00:44:35.900 --> 00:44:39.380 You don't see the spin directly. 00:44:39.380 --> 00:44:43.930 What you see is magnetic moments. 00:44:43.930 --> 00:44:45.600 So what's that? 00:44:45.600 --> 00:44:47.680 So what are magnetic moments? 00:44:47.680 --> 00:45:00.340 Magnetic moments, mu, is the analog, the magnetic analog 00:45:00.340 --> 00:45:01.780 of an electric dipole. 00:45:01.780 --> 00:45:03.740 A mu is called a magnetic dipole. 00:45:03.740 --> 00:45:05.740 You say it has a magnetic moment. 00:45:08.680 --> 00:45:14.160 And the magnetic moment is given by I times the area. 00:45:14.160 --> 00:45:15.490 What does that mean? 00:45:15.490 --> 00:45:20.360 Well, a precise discussion would take some time. 00:45:20.360 --> 00:45:24.110 But roughly, you can simplify when 00:45:24.110 --> 00:45:28.150 you think of a loop that is in a plane, in which case 00:45:28.150 --> 00:45:31.270 there's an area associated to it. 00:45:31.270 --> 00:45:33.570 And if the loop is this one, the area 00:45:33.570 --> 00:45:38.280 is defined as the normal vector to the oriented loop. 00:45:38.280 --> 00:45:42.140 So an oriented loop has an area vector. 00:45:42.140 --> 00:45:43.960 And the orientation could be focused 00:45:43.960 --> 00:45:45.500 the direction of the current. 00:45:45.500 --> 00:45:46.450 There is some area. 00:45:46.450 --> 00:45:49.970 And the magnetic moment is given by this thing. 00:45:49.970 --> 00:45:52.520 It points up in the circumstances 00:45:52.520 --> 00:45:55.580 when this current goes like that. 00:45:55.580 --> 00:45:58.630 So that's a magnetic moment. 00:45:58.630 --> 00:46:01.930 A little bit of units. 00:46:01.930 --> 00:46:06.020 The way units work out is that mu B-- magnetic moments 00:46:06.020 --> 00:46:11.190 and magnetic fields have units of energy. 00:46:14.680 --> 00:46:19.140 So magnetic moments you could define 00:46:19.140 --> 00:46:26.450 as energy, which is joules, divided by tesla, 00:46:26.450 --> 00:46:32.520 or ergs divided by gauss, because mu B has 00:46:32.520 --> 00:46:35.490 units of energy. 00:46:35.490 --> 00:46:39.300 So how do magnetic moments originate 00:46:39.300 --> 00:46:43.340 in a charge configuration? 00:46:43.340 --> 00:46:46.230 Well, you can simply have a little current like that. 00:46:46.230 --> 00:46:49.630 But let's consider a different situation 00:46:49.630 --> 00:46:58.040 in which you have a ring of charge, 00:46:58.040 --> 00:47:08.945 a ring of charge of some radius R. It has a total charge Q, 00:47:08.945 --> 00:47:12.350 and it has a linear charge density lambda. 00:47:12.350 --> 00:47:17.980 It's uniform, and it's rotating with some velocity 00:47:17.980 --> 00:47:24.220 v. If you wish, it also has a mass M. There are all 00:47:24.220 --> 00:47:25.720 kinds of [? parameters. ?] How many? 00:47:25.720 --> 00:47:29.490 Mass, charge, radius, and velocity. 00:47:29.490 --> 00:47:30.100 Here we go. 00:47:30.100 --> 00:47:35.250 We have our solid ring of charge rotating, 00:47:35.250 --> 00:47:39.030 and we want to figure out something quite fundamental, 00:47:39.030 --> 00:47:41.470 which is the origin of this principle. 00:47:41.470 --> 00:47:46.300 We said, you really never see spins directly. 00:47:46.300 --> 00:47:49.960 You never see this intrinsic angular momentum directly. 00:47:49.960 --> 00:47:53.280 You see magnetic moments. 00:47:53.280 --> 00:47:55.800 But then actually what happens is 00:47:55.800 --> 00:47:58.310 that there's a universal relation 00:47:58.310 --> 00:48:00.615 between magnetic moments and angular momentum. 00:48:00.615 --> 00:48:04.370 This is a key concept in physics. 00:48:04.370 --> 00:48:05.890 Maybe you've seen it before. 00:48:05.890 --> 00:48:07.080 Maybe you haven't. 00:48:07.080 --> 00:48:11.540 Probably you might have seen that in 802. 00:48:11.540 --> 00:48:13.040 So how does that go? 00:48:13.040 --> 00:48:16.020 Let's calculate the magnetic moment. 00:48:16.020 --> 00:48:22.830 So the current is the linear charge density 00:48:22.830 --> 00:48:23.650 times the velocity. 00:48:28.440 --> 00:48:32.760 The linear charge density is the total charge 00:48:32.760 --> 00:48:35.705 divided by 2 pi R times the velocity. 00:48:38.360 --> 00:48:41.060 Now the area, to give the magnetic moment, 00:48:41.060 --> 00:48:44.120 we'll have mu is equal to I times the area. 00:48:44.120 --> 00:48:49.300 So it would be this Q times 2 pi R v 00:48:49.300 --> 00:48:54.700 times the area, which would be pi R squared. 00:48:54.700 --> 00:49:00.605 So the pi's cancel, and we get 1/2 QvR. 00:49:08.520 --> 00:49:10.550 OK. 00:49:10.550 --> 00:49:14.510 1/2 QvR, and that's fine and interesting. 00:49:14.510 --> 00:49:20.230 But OK, depends on the radius, depends on the velocity. 00:49:20.230 --> 00:49:26.200 So here is the magnetic moment is supposed to be going up. 00:49:26.200 --> 00:49:27.730 But what else is going up? 00:49:27.730 --> 00:49:31.020 The angular momentum of this thing is also going up. 00:49:31.020 --> 00:49:35.026 So what is the magnitude of the angular momentum L? 00:49:35.026 --> 00:49:36.476 L is angular momentum. 00:49:39.700 --> 00:49:43.690 Well, it's the mass times the momentum-- 00:49:43.690 --> 00:49:47.970 it's the mass momentum cross R, so MvR. 00:49:53.790 --> 00:49:58.270 The momentum of R cross p for each piece, 00:49:58.270 --> 00:50:02.910 contributes the same, so you just take the total momentum. 00:50:02.910 --> 00:50:06.260 This really is 0, but add them up little by little, 00:50:06.260 --> 00:50:08.270 and you've got your MvR. 00:50:11.050 --> 00:50:22.780 So here you have vR, so here you put 1/2 Q over M MvR. 00:50:25.330 --> 00:50:38.600 And you discover that mu is equal to 1/2 Q over M L. 00:50:38.600 --> 00:50:49.430 So maybe write it better-- Q over 2M L. I'm sorry, 00:50:49.430 --> 00:50:50.835 this is the normal. 00:50:50.835 --> 00:50:54.710 The M shouldn't change, M. 00:50:54.710 --> 00:50:59.680 And I box this relation because an interesting thing 00:50:59.680 --> 00:51:00.440 has happened. 00:51:00.440 --> 00:51:04.510 All kinds of incidentals have dropped out. 00:51:04.510 --> 00:51:08.090 Like the velocity has dropped out. 00:51:08.090 --> 00:51:11.140 The radius has dropped out as well. 00:51:11.140 --> 00:51:14.140 So if I have one ring with this radius 00:51:14.140 --> 00:51:16.390 and another ring with a bigger radius, 00:51:16.390 --> 00:51:19.740 the relation between mu and L is the same, 00:51:19.740 --> 00:51:23.090 as long as it's rotating with the same speed. 00:51:23.090 --> 00:51:26.900 So this is actually a universal relation. 00:51:26.900 --> 00:51:29.330 It is not just true for a little ring. 00:51:29.330 --> 00:51:33.790 It's true for a solid sphere or any solid object axially 00:51:33.790 --> 00:51:35.880 symmetric. 00:51:35.880 --> 00:51:36.830 It would be true. 00:51:36.830 --> 00:51:43.030 You could consider any object that is axially symmetric, 00:51:43.030 --> 00:51:45.870 and then you start considering all the little rings that 00:51:45.870 --> 00:51:46.710 can be built. 00:51:46.710 --> 00:51:49.340 And for every ring, mu over L is the same, 00:51:49.340 --> 00:51:51.880 and they all point in the same direction. 00:51:51.880 --> 00:51:56.220 Therefore, it's true under very general grounds. 00:51:56.220 --> 00:51:59.790 And that is a very famous relation. 00:51:59.790 --> 00:52:03.770 So now you could speculate that, indeed, 00:52:03.770 --> 00:52:12.390 the reason that a particle may have a magnetic moment if it's 00:52:12.390 --> 00:52:15.610 made by a little ball of charge that is rotating. 00:52:15.610 --> 00:52:18.900 But that was exactly what Pauli didn't like, of course. 00:52:21.928 --> 00:52:24.600 And you would like to see what's really 00:52:24.600 --> 00:52:25.980 happening with particles. 00:52:25.980 --> 00:52:31.770 So when you think of a true quantum mechanical particle, 00:52:31.770 --> 00:52:37.550 let's think of a particle in general, a solid particle 00:52:37.550 --> 00:52:38.110 rotating. 00:52:38.110 --> 00:52:43.160 We'll change the name to S for spin angular momentum. 00:52:43.160 --> 00:52:46.480 Because that little part, this is just one particle. 00:52:46.480 --> 00:52:48.920 We're not thinking of that little particle 00:52:48.920 --> 00:52:51.700 going around a nucleus. 00:52:51.700 --> 00:52:55.310 We're thinking of that little particle rotating. 00:52:55.310 --> 00:52:57.620 So this is a little piece of that little particle 00:52:57.620 --> 00:52:59.270 that is rotating. 00:52:59.270 --> 00:53:08.640 So you could ask, if, for the electron, 00:53:08.640 --> 00:53:12.150 for example, is it true that mu is 00:53:12.150 --> 00:53:17.595 equal to e over 2 mass of the electron times its spin? 00:53:22.030 --> 00:53:28.040 So this would be a vindication of this classical analysis. 00:53:28.040 --> 00:53:34.770 It might be that it's related in this way. 00:53:34.770 --> 00:53:48.910 So actually, it's not quite true, 00:53:48.910 --> 00:53:53.080 but let's still improve this a little bit. 00:53:53.080 --> 00:54:00.590 In terms of units, we like to put an h bar here and a 2Me. 00:54:00.590 --> 00:54:05.700 And put spin here, angular momentum, divided by h. 00:54:05.700 --> 00:54:11.960 Because this has no units, h bar has the units 00:54:11.960 --> 00:54:17.000 of angular momentum, x times p. 00:54:17.000 --> 00:54:19.390 It's the same units, so units of angular momentum. 00:54:19.390 --> 00:54:24.430 So h bar would be convenient. 00:54:24.430 --> 00:54:33.270 So that over here, you would have units of a dipole moment, 00:54:33.270 --> 00:54:38.300 or magnetic moment, magnetic moment units. 00:54:43.300 --> 00:54:47.810 So what does happen for the electron? 00:54:47.810 --> 00:54:52.810 Well, it's almost true, but not quite. 00:54:52.810 --> 00:54:57.680 In fact, what you get is that you need a fudge factor. 00:54:57.680 --> 00:55:00.700 The fudge factor is that, actually, 00:55:00.700 --> 00:55:02.800 for elementary particles, you have 00:55:02.800 --> 00:55:06.980 a g, which is a constant, which is the fudge factor, 00:55:06.980 --> 00:55:15.850 e h bar 2 over M of the particle S over h bar. 00:55:15.850 --> 00:55:19.410 Sometimes called the Lande factor. 00:55:19.410 --> 00:55:22.410 You must put a number there. 00:55:22.410 --> 00:55:28.140 Now, the good thing is that the number sometimes 00:55:28.140 --> 00:55:31.030 can be calculated and predicted. 00:55:31.030 --> 00:55:34.100 So when people did this, they figured out 00:55:34.100 --> 00:55:38.950 that for the electron the number is actually a 2. 00:55:38.950 --> 00:55:46.210 So for the electron, g of the electron is equal to 2. 00:55:46.210 --> 00:55:49.770 Now that, you would say, cannot be an accident. 00:55:49.770 --> 00:55:54.720 It's twice what you would predict sort of classically. 00:55:54.720 --> 00:55:59.250 And the Dirac equation, the relativistic equation 00:55:59.250 --> 00:56:02.010 of the electron that you have not studied yet 00:56:02.010 --> 00:56:07.200 but you will study soon, predicts this g equal to 2. 00:56:07.200 --> 00:56:08.880 It was considered a great success 00:56:08.880 --> 00:56:12.670 that that equation gave the right answer, that people 00:56:12.670 --> 00:56:16.400 understood that this number was going to be 2. 00:56:16.400 --> 00:56:19.670 So for the electron, this is 2. 00:56:19.670 --> 00:56:25.210 So this quantity is called-- it's a magnetic dipole 00:56:25.210 --> 00:56:32.210 moment-- is called mu B for Bohr magneton. 00:56:39.900 --> 00:56:44.330 So how big is a mu B? 00:56:44.330 --> 00:56:51.880 It's about 9.3 times 10 to the minus 24 joules per tesla. 00:56:55.454 --> 00:56:56.380 AUDIENCE: Professor. 00:56:56.380 --> 00:56:57.520 PROFESSOR: Yes? 00:56:57.520 --> 00:56:58.966 AUDIENCE: [INAUDIBLE]. 00:56:58.966 --> 00:57:01.376 So where exactly does the fudge factor come in? 00:57:01.376 --> 00:57:05.112 Is it just merely because [INAUDIBLE]? 00:57:05.112 --> 00:57:05.820 PROFESSOR: Right. 00:57:05.820 --> 00:57:11.590 So the classical analysis is not valid. 00:57:11.590 --> 00:57:14.800 So it's pretty invalid, in fact. 00:57:14.800 --> 00:57:19.530 You see, the picture of an electron, 00:57:19.530 --> 00:57:22.480 as of today, is that it's a point particle. 00:57:22.480 --> 00:57:27.200 And a point particle literally means no size. 00:57:27.200 --> 00:57:30.880 The electron is not a little ball of charge. 00:57:30.880 --> 00:57:32.740 Otherwise, it would have parts. 00:57:32.740 --> 00:57:34.970 So an electron is a point particle. 00:57:34.970 --> 00:57:38.250 Therefore, a point particle cannot be rotating and have 00:57:38.250 --> 00:57:39.020 a spin. 00:57:39.020 --> 00:57:43.670 So how does the electron manage to have spin? 00:57:43.670 --> 00:57:45.720 That you can't answer in physics. 00:57:45.720 --> 00:57:47.590 It just has it. 00:57:47.590 --> 00:57:51.640 Just like a point particle that has no size can have mass. 00:57:51.640 --> 00:57:54.670 How do you have mass if you have no size? 00:57:54.670 --> 00:57:56.130 You get accustomed to the idea. 00:57:56.130 --> 00:57:58.550 The mathematics says it's possible. 00:57:58.550 --> 00:58:00.420 You don't run into trouble. 00:58:00.420 --> 00:58:06.510 So this particle has no size, but it has an angular spin, 00:58:06.510 --> 00:58:09.070 angular momentum, as if it would be rotating. 00:58:09.070 --> 00:58:12.520 But it's definitely not the case that it's rotating. 00:58:12.520 --> 00:58:17.420 And therefore, this 2 confirms that it was a pointless idea 00:58:17.420 --> 00:58:19.260 to believe that it would be true. 00:58:19.260 --> 00:58:22.720 Nevertheless, kind of unit analyses 00:58:22.720 --> 00:58:26.770 or maybe some truth to the fact that quantum mechanics 00:58:26.770 --> 00:58:29.290 changes classical mechanics. 00:58:29.290 --> 00:58:32.580 Turns out that it's closely related. 00:58:32.580 --> 00:58:36.910 For the proton, for example, the magnetic moment of the proton 00:58:36.910 --> 00:58:40.510 is quite complicated as well because the proton is made out 00:58:40.510 --> 00:58:43.110 of quarks that are rotating inside. 00:58:43.110 --> 00:58:47.070 And how do you get the spin of the proton 00:58:47.070 --> 00:58:48.790 and the magnetic moment of the proton? 00:58:48.790 --> 00:58:50.000 It's complicated. 00:58:50.000 --> 00:58:54.560 The neutron, that has no charge, has a magnetic moment, 00:58:54.560 --> 00:58:58.450 because somehow the quarks inside arrange in a way 00:58:58.450 --> 00:59:01.900 that their angular momentum doesn't quite cancel. 00:59:05.580 --> 00:59:08.690 So for example, the value for a neutron, I believe, 00:59:08.690 --> 00:59:12.390 is minus 2.78 or something like that. 00:59:12.390 --> 00:59:14.770 It's a strange number. 00:59:14.770 --> 00:59:19.070 Another thing that is sort of interesting that is also true 00:59:19.070 --> 00:59:22.390 is that this mass is the mass of a particle. 00:59:22.390 --> 00:59:25.250 So if you're talking about the magnetic moment 00:59:25.250 --> 00:59:28.530 of the proton or the neutron, it's 00:59:28.530 --> 00:59:31.740 suppressed with respect to the one of the electron. 00:59:31.740 --> 00:59:34.580 The electron one is much bigger because, actually, the mass 00:59:34.580 --> 00:59:36.440 shows up here. 00:59:36.440 --> 00:59:40.860 So for a neutron or a proton, the magnetic moment 00:59:40.860 --> 00:59:44.210 is much, much smaller. 00:59:44.210 --> 00:59:48.560 So, in fact, for an electron then, you 00:59:48.560 --> 00:59:49.870 would have the following. 00:59:49.870 --> 01:00:01.720 Mu is equal to minus g, which is 2, mu B S bar over h. 01:00:01.720 --> 01:00:04.280 And actually, we put the minus sign 01:00:04.280 --> 01:00:07.820 because the electron has negative charge. 01:00:07.820 --> 01:00:12.270 So the magnetic moment actually points opposite. 01:00:12.270 --> 01:00:16.120 If you rotate this way, the angular momentum is always up. 01:00:16.120 --> 01:00:18.770 But if you rotate this way and you're negative, 01:00:18.770 --> 01:00:21.740 it's as if the current goes in the other direction. 01:00:21.740 --> 01:00:25.871 So this is due to the fact that the electron is negatively 01:00:25.871 --> 01:00:26.370 charged. 01:00:26.370 --> 01:00:28.870 And that's the final expression. 01:00:31.740 --> 01:00:38.765 So OK, so that's the general story with magnetic moments. 01:00:42.390 --> 01:00:45.360 So the next thing is, how do magnetic moments 01:00:45.360 --> 01:00:49.530 react when you have magnetic fields? 01:00:49.530 --> 01:00:54.710 So that is something that you can calculate, 01:00:54.710 --> 01:00:57.790 or you can decide if you have a picture. 01:00:57.790 --> 01:01:03.650 For example, if you have a loop of charge like this, 01:01:03.650 --> 01:01:10.050 and you have magnetic field lines that go like this, 01:01:10.050 --> 01:01:11.795 they diverge a bit. 01:01:15.830 --> 01:01:18.220 Let me see you use your right-hand rule 01:01:18.220 --> 01:01:20.950 and tell me whether that loop of current 01:01:20.950 --> 01:01:22.925 will feel a force up or down. 01:01:26.936 --> 01:01:31.090 I'll give you 30 seconds, and I take a vote. 01:01:31.090 --> 01:01:32.880 Let's see how we're doing with that. 01:01:39.547 --> 01:01:43.280 And I'll prepare these blackboards in the meantime. 01:01:59.420 --> 01:02:00.310 All right. 01:02:00.310 --> 01:02:03.060 Who votes up? 01:02:03.060 --> 01:02:03.560 Nobody. 01:02:03.560 --> 01:02:05.503 Who votes down? 01:02:05.503 --> 01:02:06.405 Yeah, [INAUDIBLE]. 01:02:06.405 --> 01:02:08.640 Down, exactly. 01:02:08.640 --> 01:02:10.150 How do you see down? 01:02:10.150 --> 01:02:14.250 Well, one way to see this, look at the cross-section. 01:02:14.250 --> 01:02:18.980 You would have this wire here like that. 01:02:18.980 --> 01:02:22.870 The current is coming in on this side and going out this way. 01:02:22.870 --> 01:02:26.470 Here you have the field lines that go through those two 01:02:26.470 --> 01:02:33.570 edges, and the magnetic field is like that. 01:02:33.570 --> 01:02:38.550 And the force goes like I cross B. So I goes in, B goes out. 01:02:38.550 --> 01:02:42.720 The force must be like that, a little bit of force. 01:02:42.720 --> 01:02:46.700 In this one, I cross B would be like that, 01:02:46.700 --> 01:02:48.160 a little bit of force. 01:02:48.160 --> 01:02:49.070 Yep. 01:02:49.070 --> 01:02:52.105 Has a component down because the field lines are diverging. 01:02:56.080 --> 01:02:59.920 So what is the force really given by? 01:02:59.920 --> 01:03:05.540 The force is given by the gradient 01:03:05.540 --> 01:03:14.490 of mu dot B. This is derived in E&M. I will not derive it here. 01:03:14.490 --> 01:03:17.490 This is not really the point of this course. 01:03:17.490 --> 01:03:21.310 But you can see that it's consistent. 01:03:21.310 --> 01:03:26.080 This is saying that the force goes in the direction that 01:03:26.080 --> 01:03:29.810 makes mu dot B grow the fastest. 01:03:29.810 --> 01:03:34.830 Now mu, in this case, is up. 01:03:34.830 --> 01:03:40.170 So mu dot B is positive, because mu and the magnetic field 01:03:40.170 --> 01:03:41.550 go in the same direction. 01:03:41.550 --> 01:03:43.610 So mu dot b is positive. 01:03:43.610 --> 01:03:47.720 So the force will be towards the direction-- that's 01:03:47.720 --> 01:03:51.220 what the gradient is-- that this becomes bigger. 01:03:51.220 --> 01:03:55.210 So it becomes bigger here, because as the field lines come 01:03:55.210 --> 01:03:58.570 together, that means stronger magnetic field. 01:03:58.570 --> 01:04:03.550 And therefore, mu dot B would be larger, so it's pointing down. 01:04:03.550 --> 01:04:07.790 If you have a magnetic field that is roughly 01:04:07.790 --> 01:04:11.010 in the z direction, there will be a simplification, 01:04:11.010 --> 01:04:13.020 as we will see very soon. 01:04:13.020 --> 01:04:16.480 So what did Stern and Gerlach do? 01:04:16.480 --> 01:04:21.110 Well, they were working with silver atoms. 01:04:21.110 --> 01:04:25.410 And silver atoms have 47 electrons, 01:04:25.410 --> 01:04:30.070 out of which 46 fill up the levels and equal 1, 01:04:30.070 --> 01:04:33.620 2, 3, and 4. 01:04:33.620 --> 01:04:41.010 Just one lone electron, a 5s electron, the 47th electron, 01:04:41.010 --> 01:04:45.810 it's a lonely electron that is out in a spherical shell, 01:04:45.810 --> 01:04:49.760 we know now with zero orbital angular momentum. 01:04:49.760 --> 01:04:52.130 It's an S state. 01:04:52.130 --> 01:04:58.110 And therefore, throwing silver atoms through your apparatus 01:04:58.110 --> 01:05:03.320 was pretty much the same thing as throwing electrons, 01:05:03.320 --> 01:05:05.810 because all these other electrons 01:05:05.810 --> 01:05:08.440 are tied up with each other. 01:05:08.440 --> 01:05:11.970 We know now one has spin up, one spin down. 01:05:11.970 --> 01:05:15.710 Nothing contributes, no angular momentum as a whole. 01:05:15.710 --> 01:05:19.090 And then you have this last electron unpaired. 01:05:19.090 --> 01:05:21.060 It has a spin. 01:05:21.060 --> 01:05:23.066 So it's like throwing spins. 01:05:26.130 --> 01:05:32.820 So moreover, throwing spins, as far as we're concerned, 01:05:32.820 --> 01:05:34.510 Stern and Gerlach wouldn't care. 01:05:34.510 --> 01:05:39.370 Because of these relations, it's throwing in dipole moments. 01:05:39.370 --> 01:05:42.910 And they would care about that because magnetic fields 01:05:42.910 --> 01:05:46.740 push dipole moments up or down. 01:05:46.740 --> 01:05:52.690 Therefore, what is the apparatus these people had? 01:05:52.690 --> 01:06:01.760 It was sort of like this, with an oven, 01:06:01.760 --> 01:06:04.580 and you produce some silver atoms 01:06:04.580 --> 01:06:10.450 that come out as a gas, a collimating slit. 01:06:14.490 --> 01:06:19.490 Then you put axes here-- we put axes just 01:06:19.490 --> 01:06:22.060 to know the components we're talking about. 01:06:22.060 --> 01:06:39.850 And then there's magnets, some sort of magnet like this, 01:06:39.850 --> 01:06:41.780 and the screen over there. 01:06:45.320 --> 01:06:49.285 So basically, this form of this magnet 01:06:49.285 --> 01:06:53.330 that I've tried to draw there, although it's not so easy, if I 01:06:53.330 --> 01:06:56.945 would take a cross-section it would look like this. 01:07:01.740 --> 01:07:03.866 So the magnetic field has a gradient. 01:07:06.430 --> 01:07:09.330 The lines bend a bit, so there's a gradient 01:07:09.330 --> 01:07:10.445 of the magnetic field. 01:07:10.445 --> 01:07:16.870 And it's mostly in the z direction, so z direction 01:07:16.870 --> 01:07:18.280 being pointed out here. 01:07:18.280 --> 01:07:19.590 So there's the magnetic field. 01:07:19.590 --> 01:07:21.800 The beam then comes here. 01:07:21.800 --> 01:07:24.045 And the question is, what do you get on this screen? 01:07:27.570 --> 01:07:33.278 Now, I have it a little too low. 01:07:33.278 --> 01:07:37.110 The beam comes there and goes through there. 01:07:37.110 --> 01:07:41.840 So the analysis that we would have to do 01:07:41.840 --> 01:07:47.100 is basically an analysis of the forces. 01:07:47.100 --> 01:07:52.020 And relatively, we don't care too much. 01:07:52.020 --> 01:07:55.870 The fact is that there's basically, 01:07:55.870 --> 01:07:59.190 because the magnetic field is mostly in the z direction 01:07:59.190 --> 01:08:01.570 and varies in z direction, there will 01:08:01.570 --> 01:08:04.940 be a force basically in the z direction. 01:08:04.940 --> 01:08:05.790 Why is that? 01:08:05.790 --> 01:08:07.720 Because you take this, and you say, 01:08:07.720 --> 01:08:15.870 well, that's roughly mu z Bz, because it's mostly 01:08:15.870 --> 01:08:18.120 a magnetic field in the z direction. 01:08:18.120 --> 01:08:24.670 And mu is a constant, so it's basically gradient of Bz. 01:08:24.670 --> 01:08:26.029 Now, that's a vector. 01:08:26.029 --> 01:08:30.160 But we're saying also most of the gradient of Bz 01:08:30.160 --> 01:08:37.109 is in the z direction, so it's basically dBz dz. 01:08:45.569 --> 01:08:47.830 Now, there is some bending of the lines, 01:08:47.830 --> 01:08:50.750 so there's a little bit of gradient in other directions. 01:08:50.750 --> 01:08:54.220 But people have gone through the analysis, 01:08:54.220 --> 01:08:57.790 and they don't matter for any calculation that you do. 01:08:57.790 --> 01:09:00.830 They actually average out. 01:09:00.830 --> 01:09:04.850 So roughly, this gradient is in the z direction. 01:09:04.850 --> 01:09:09.040 I'm sorry, the gradient is supposed to be a vector. 01:09:09.040 --> 01:09:11.600 So you get a force in the z direction. 01:09:11.600 --> 01:09:16.350 And therefore, the thing that people expected 01:09:16.350 --> 01:09:17.870 was the following. 01:09:17.870 --> 01:09:24.200 You know, here comes one atom, and it has its magnetic moment. 01:09:24.200 --> 01:09:28.420 Well, they've all been boiling in this oven for a while. 01:09:28.420 --> 01:09:29.740 They're very disordered. 01:09:29.740 --> 01:09:33.055 Some have a z component of magnetic-- the magnetic moment 01:09:33.055 --> 01:09:37.029 is pointing like that, so they have some component, some down. 01:09:37.029 --> 01:09:37.880 Some are here. 01:09:37.880 --> 01:09:39.569 They have no component. 01:09:39.569 --> 01:09:44.260 It's all Boltzmann distributed all over the directions. 01:09:44.260 --> 01:09:49.870 Therefore, you're going to get a smudge like this. 01:09:49.870 --> 01:09:53.240 Some ones are going to be deflected a lot because they 01:09:53.240 --> 01:09:57.030 have lots of z component of angular 01:09:57.030 --> 01:10:00.240 momentum or z magnetic moment. 01:10:00.240 --> 01:10:04.720 Others are going to be deflected little. 01:10:04.720 --> 01:10:07.590 So this was the classical expectation. 01:10:07.590 --> 01:10:11.080 And the shock was that you got, actually, 01:10:11.080 --> 01:10:16.010 one peak here, an empty space, and another peak there. 01:10:16.010 --> 01:10:17.950 That was called space quantization. 01:10:22.720 --> 01:10:26.790 Stern and Gerlach worked with a magnetic field 01:10:26.790 --> 01:10:35.190 that was of about 0.1 tesla, a tenth of a tesla. 01:10:35.190 --> 01:10:38.600 And in their experiment, the space quantization, 01:10:38.600 --> 01:10:42.680 this difference, was 1/5 of a millimeter. 01:10:45.740 --> 01:10:51.110 So not that big, but it was a clear thing. 01:10:51.110 --> 01:10:53.700 It was there. 01:10:53.700 --> 01:11:00.350 So everybody was confused. 01:11:00.350 --> 01:11:04.340 They thought it was the orbital spin, angular momentum that 01:11:04.340 --> 01:11:06.600 somehow had been measured. 01:11:06.600 --> 01:11:08.880 At the end of the day, that was wrong. 01:11:08.880 --> 01:11:10.370 It couldn't have been that. 01:11:10.370 --> 01:11:13.400 People understood the Bohr atom, realized, no, there's 01:11:13.400 --> 01:11:15.220 no angular momentum there. 01:11:15.220 --> 01:11:17.810 The idea of the spin came back, and you 01:11:17.810 --> 01:11:20.820 would have to do a calculation to determine 01:11:20.820 --> 01:11:23.030 what is the value of the spin. 01:11:23.030 --> 01:11:28.960 So the exact factor took a while to get it right. 01:11:28.960 --> 01:11:36.660 But with the idea that mu z is equal to minus 2 Bohr magenton 01:11:36.660 --> 01:11:39.250 Sz over h bar, which we wrote before. 01:11:42.330 --> 01:11:46.900 Well, mu z, if you know the strength 01:11:46.900 --> 01:11:51.350 of your magnetic field, you can calculate the deflections. 01:11:51.350 --> 01:11:53.030 You know what mu B is. 01:11:53.030 --> 01:11:57.040 So therefore, you get the value for Sz over h. 01:11:57.040 --> 01:12:00.540 And experiments suggested that Sz over h 01:12:00.540 --> 01:12:04.165 was either plus or minus 1/2. 01:12:08.000 --> 01:12:15.280 And this kind of particle, it has Sz over h bar equal plus 01:12:15.280 --> 01:12:20.970 or minus 1/2, is called the spin-1/2 particle. 01:12:20.970 --> 01:12:26.400 So again, from this equation, this can be measured. 01:12:29.930 --> 01:12:34.725 And you then use this, and you get this value. 01:12:39.800 --> 01:12:43.160 So the experiment is a little confusing. 01:12:43.160 --> 01:12:44.760 Why did this happen? 01:12:44.760 --> 01:12:48.020 And how do we think of it quantum mechanically? 01:12:48.020 --> 01:12:53.340 Now 804 sort of began with these kind of things. 01:12:53.340 --> 01:12:56.680 And you know by now that what's happening 01:12:56.680 --> 01:13:01.090 is the following, that somehow, mathematically, every state is 01:13:01.090 --> 01:13:04.000 a superposition of a spin up and a spin down. 01:13:04.000 --> 01:13:06.410 So every particle that goes there 01:13:06.410 --> 01:13:10.230 has half of its brain in the spin up and half of its brain 01:13:10.230 --> 01:13:11.630 in the spin down. 01:13:11.630 --> 01:13:14.170 And then as it goes through the magnetic field, 01:13:14.170 --> 01:13:17.990 this thing splits, but each particle is in both beams 01:13:17.990 --> 01:13:19.880 still. 01:13:19.880 --> 01:13:22.140 And they just have this dual existence 01:13:22.140 --> 01:13:25.130 until there's a screen and there's detectors. 01:13:25.130 --> 01:13:27.460 So they have to decide what happens, and then 01:13:27.460 --> 01:13:33.260 either collapses in the top beam or lower beam. 01:13:33.260 --> 01:13:35.630 Nothing happens until you put the screen. 01:13:35.630 --> 01:13:38.790 That's what we think now is the interpretation 01:13:38.790 --> 01:13:39.980 of this experiment. 01:13:39.980 --> 01:13:43.720 But let's use the last few minutes 01:13:43.720 --> 01:13:50.440 to just write this in terms of boxes and get the right ideas. 01:13:50.440 --> 01:13:57.590 So instead of drawing all that stuff, 01:13:57.590 --> 01:14:01.620 we'll draw a little box called a z hat 01:14:01.620 --> 01:14:04.690 box, a Stern-Gerlach apparatus. 01:14:04.690 --> 01:14:10.350 In comes a beam, out would come two beams, 01:14:10.350 --> 01:14:17.150 Sz equal h bar over 2 and Sz equal minus h bar over 2. 01:14:17.150 --> 01:14:19.980 And the convention is that the plus goes up 01:14:19.980 --> 01:14:21.840 and the minus goes down, which I think 01:14:21.840 --> 01:14:24.555 is probably consistent with that drawing. 01:14:27.470 --> 01:14:29.980 And that's the Stern-Gerlach apparatus. 01:14:29.980 --> 01:14:34.150 It measures Sz, and it splits the beam. 01:14:34.150 --> 01:14:36.510 Each particle goes into both beams 01:14:36.510 --> 01:14:38.520 until there's a device that measures 01:14:38.520 --> 01:14:41.270 and decides where you go. 01:14:41.270 --> 01:14:44.440 So you can do the following arrangements. 01:14:44.440 --> 01:14:50.590 So here's arrangement number 1, a Stern-Gerlach device with z. 01:14:50.590 --> 01:14:54.000 Then you block the lower one and let 01:14:54.000 --> 01:15:03.650 the top one go as Sz equal h bar over 2. 01:15:03.650 --> 01:15:08.710 And then you put another Stern-Gerlach machine, z hat, 01:15:08.710 --> 01:15:12.220 that has two outputs. 01:15:12.220 --> 01:15:15.270 And then you ask, what's going to happen? 01:15:15.270 --> 01:15:19.680 And the experiment can be done and, actually, there's 01:15:19.680 --> 01:15:23.720 nothing here coming out, and all the particles come out here 01:15:23.720 --> 01:15:26.250 with Sz equal h bar over 2. 01:15:30.230 --> 01:15:34.280 What are we going to learn from this? 01:15:34.280 --> 01:15:36.710 In our picture of quantum mechanics, 01:15:36.710 --> 01:15:39.450 we're going to think of this as there 01:15:39.450 --> 01:15:43.700 are states of the electron that have-- 01:15:43.700 --> 01:15:46.610 and I will write them with respect 01:15:46.610 --> 01:15:51.950 to z-- they have plus h bar over 2 01:15:51.950 --> 01:15:58.060 and states that have minus h bar over 2. 01:15:58.060 --> 01:16:03.590 And what we will think is that these are really old basis 01:16:03.590 --> 01:16:09.080 states, that any other state, even one that points along x, 01:16:09.080 --> 01:16:11.950 is a superposition of those two. 01:16:11.950 --> 01:16:15.030 This is a very incredible physical assumption. 01:16:15.030 --> 01:16:20.310 It's saying this system is a 2-dimensional complex vector 01:16:20.310 --> 01:16:24.850 space, two vectors, two unit, two basis vectors. 01:16:24.850 --> 01:16:28.860 And from those two, all linear combinations that are infinite 01:16:28.860 --> 01:16:33.860 represent all possible spin configurations. 01:16:33.860 --> 01:16:36.090 And what is this saying? 01:16:36.090 --> 01:16:41.060 Well, as we will translate it into algebra, 01:16:41.060 --> 01:16:45.480 we will say that, look, here is a state plus. 01:16:45.480 --> 01:16:49.380 And when you try to measure, if it had any minus component, 01:16:49.380 --> 01:16:50.740 it had nothing. 01:16:50.740 --> 01:16:55.410 So we will state that as saying that these states are 01:16:55.410 --> 01:16:56.330 orthogonal. 01:16:56.330 --> 01:17:00.590 The minus state and the plus state have zero overlap. 01:17:03.330 --> 01:17:05.950 They are orthogonal basis states. 01:17:05.950 --> 01:17:10.910 And, for example, well, you could also do it this way. 01:17:10.910 --> 01:17:12.800 That would also be 0. 01:17:12.800 --> 01:17:17.540 And you could also say that z plus 01:17:17.540 --> 01:17:23.370 and z plus is 1, because every state that came in 01:17:23.370 --> 01:17:26.040 as a plus came out as a plus. 01:17:26.040 --> 01:17:27.910 They had perfect overlap. 01:17:27.910 --> 01:17:32.130 So these are two orthonormal basis vectors. 01:17:32.130 --> 01:17:34.395 That's what this seems to suggest. 01:17:34.395 --> 01:17:36.940 And it's a little strange, if you think, 01:17:36.940 --> 01:17:41.950 because there's a clash between arrows 01:17:41.950 --> 01:17:45.300 and the notion of orthonormality. 01:17:45.300 --> 01:17:48.110 In 3-dimensional vectors, you think 01:17:48.110 --> 01:17:50.489 of this vector being orthogonal to this. 01:17:50.489 --> 01:17:52.030 But you wouldn't think of this vector 01:17:52.030 --> 01:17:54.280 as being orthogonal to that one. 01:17:54.280 --> 01:18:00.130 And here is the spin is up, and this is the spin down. 01:18:00.130 --> 01:18:01.600 And those two are orthogonal. 01:18:01.600 --> 01:18:03.450 You say, no, they're anti-parallel. 01:18:03.450 --> 01:18:04.780 They're not orthogonal. 01:18:04.780 --> 01:18:06.820 No, they are orthogonal. 01:18:06.820 --> 01:18:12.070 And that's the endlessly confusing thing about spin-1/2. 01:18:12.070 --> 01:18:16.830 So these states, their pictures of the spins are arrows. 01:18:16.830 --> 01:18:20.020 But don't think that those arrows and the dot product 01:18:20.020 --> 01:18:23.370 give you the orthogonality, because this is up and down. 01:18:23.370 --> 01:18:26.280 If you would be doing the dot product of an up and down 01:18:26.280 --> 01:18:28.000 vector, you would not get 0. 01:18:28.000 --> 01:18:29.870 But this is 0. 01:18:29.870 --> 01:18:31.860 Then you do the following experiment. 01:18:36.550 --> 01:18:38.390 So let's do the next one. 01:18:42.820 --> 01:18:47.680 And the next one is, again, the z filter. 01:18:47.680 --> 01:18:50.920 Take this one, block it. 01:18:50.920 --> 01:18:53.025 Then you put an x filter. 01:18:55.610 --> 01:18:58.550 And what actually happens is that you would get states 01:18:58.550 --> 01:19:05.570 with Sx, now, h bar over 2 and Sx equal minus h bar over 2, 01:19:05.570 --> 01:19:07.070 because it's an x filter. 01:19:07.070 --> 01:19:10.490 The magnetic field is a line in the x direction. 01:19:10.490 --> 01:19:16.630 Now, all these things have Sz equal h bar over 2. 01:19:16.630 --> 01:19:18.860 And what happens in the experiment 01:19:18.860 --> 01:19:21.860 is that 50% of the particles come out here 01:19:21.860 --> 01:19:23.790 and 50% come out there. 01:19:23.790 --> 01:19:29.280 So a spin state along the x direction 01:19:29.280 --> 01:19:33.340 has some overlap with a spin state along the z direction. 01:19:33.340 --> 01:19:38.190 Normal vectors, a z vector and an x vector, are orthogonal. 01:19:38.190 --> 01:19:40.470 Not here for spins. 01:19:40.470 --> 01:19:44.350 The spin pointing in the z and the spin pointing in the x 01:19:44.350 --> 01:19:46.100 are not orthogonal states. 01:19:46.100 --> 01:19:47.890 They have overlaps. 01:19:47.890 --> 01:19:53.690 So this means that, for example, the x plus state and the z 01:19:53.690 --> 01:19:58.990 plus state have an overlap. 01:19:58.990 --> 01:20:02.840 This is notations that-- we're going to be precise later. 01:20:02.840 --> 01:20:11.170 But the same thing with the x minus state, it has an overlap, 01:20:11.170 --> 01:20:14.140 and somehow they're about the same. 01:20:14.140 --> 01:20:21.920 Finally, the last experiment is this, z hat, block again, 01:20:21.920 --> 01:20:26.640 x hat, but this time block one. 01:20:26.640 --> 01:20:32.040 So here is a state with Sx equals minus h bar over 2. 01:20:32.040 --> 01:20:35.830 Here is a state with Sz equal h bar over 2. 01:20:35.830 --> 01:20:38.400 And now you put the z machine again. 01:20:40.980 --> 01:20:44.160 And what happens? 01:20:44.160 --> 01:20:46.330 Well, there's two options. 01:20:46.330 --> 01:20:48.520 People who were inventing quantum mechanics 01:20:48.520 --> 01:20:50.190 no wonder thought about them. 01:20:50.190 --> 01:20:55.290 Here they could say, look, I filtered this thing, 01:20:55.290 --> 01:21:01.100 and now all these electrons have Sz equal h bar over 2. 01:21:01.100 --> 01:21:06.340 And now all these electrons have Sx equal minus h bar over 2. 01:21:06.340 --> 01:21:08.830 Maybe, actually, all these electrons 01:21:08.830 --> 01:21:13.420 have both Sz equal h over 2 and that 01:21:13.420 --> 01:21:15.090 because I filtered it twice. 01:21:15.090 --> 01:21:16.850 So it maybe satisfies both. 01:21:16.850 --> 01:21:20.530 So if all these electrons would have Sz equals 01:21:20.530 --> 01:21:23.690 h over 2 and this, then you would only 01:21:23.690 --> 01:21:26.080 get something from the top one. 01:21:26.080 --> 01:21:28.350 But no, that's not what happens. 01:21:28.350 --> 01:21:30.330 You get in both. 01:21:30.330 --> 01:21:36.870 So somehow, the memory of these states coming from Sz 01:21:36.870 --> 01:21:39.260 equals h over 2 has been destroyed 01:21:39.260 --> 01:21:41.020 by the time it turned into a state 01:21:41.020 --> 01:21:44.130 with Sx equal minus h over 2. 01:21:44.130 --> 01:21:48.970 And a state cannot have simultaneously this and that. 01:21:48.970 --> 01:21:54.840 That's two properties, because you get 50% here and 50% there. 01:21:54.840 --> 01:21:57.760 So we'll discuss next time a little more 01:21:57.760 --> 01:22:00.500 about these relations and how can the states 01:22:00.500 --> 01:22:04.480 be related, the ones that we use as the basis 01:22:04.480 --> 01:22:07.590 vectors and all the others along x and others 01:22:07.590 --> 01:22:09.790 that we could build some other way. 01:22:09.790 --> 01:22:10.430 All right. 01:22:10.430 --> 01:22:11.340 See you next week. 01:22:11.340 --> 01:22:18.110 There's office hours today, 5:00 to 6:00, Monday, 4:30 to 5:30.