WEBVTT 00:00:00.060 --> 00:00:01.670 The following content is provided 00:00:01.670 --> 00:00:03.810 under a Creative Commons license. 00:00:03.810 --> 00:00:06.530 Your support will help MIT OpenCourseWare continue 00:00:06.530 --> 00:00:10.120 to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.590 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.590 --> 00:00:17.305 at ocw.mit.edu. 00:00:22.896 --> 00:00:23.770 PROFESSOR: All right. 00:00:27.000 --> 00:00:28.664 Hi, everyone. 00:00:28.664 --> 00:00:30.276 AUDIENCE: Hi. 00:00:30.276 --> 00:00:32.650 PROFESSOR: We're getting towards the end of the semester. 00:00:32.650 --> 00:00:37.000 Things are starting to cohere and come together. 00:00:37.000 --> 00:00:40.540 We have one more midterm exam. 00:00:40.540 --> 00:00:44.880 So there is an exam next Thursday, the 18th. 00:00:44.880 --> 00:00:45.380 OK? 00:00:45.380 --> 00:00:47.730 There will be a problem set due. 00:00:47.730 --> 00:00:50.310 It'll be posted later today, and it will be due next week 00:00:50.310 --> 00:00:51.920 on Tuesday as usual. 00:00:51.920 --> 00:00:55.840 Of course, next week on Tuesday is a holiday technically, 00:00:55.840 --> 00:00:58.530 so we'll actually make the due date be on Wednesday. 00:00:58.530 --> 00:01:01.280 So on Wednesday at 10 o'clock. 00:01:01.280 --> 00:01:02.870 You should think of this problem set 00:01:02.870 --> 00:01:04.930 as part of the review for the exam. 00:01:04.930 --> 00:01:09.820 Material that is covered today and on Thursday 00:01:09.820 --> 00:01:11.900 will be fair game for the exam. 00:01:14.560 --> 00:01:16.070 So the format of the exam is going 00:01:16.070 --> 00:01:17.722 to be much more canonical. 00:01:17.722 --> 00:01:19.430 It's going to be a series of short answer 00:01:19.430 --> 00:01:21.450 plus a series of computations. 00:01:21.450 --> 00:01:25.070 They'll be, roughly speaking, at the level of the problem 00:01:25.070 --> 00:01:26.990 sets, and a practice exam will be 00:01:26.990 --> 00:01:28.650 posted in the next couple of days. 00:01:32.130 --> 00:01:35.350 The practice exam is going to be of the same general 00:01:35.350 --> 00:01:36.920 intellectual difficulty, but it's 00:01:36.920 --> 00:01:40.140 going to be considerably longer than the actual exam. 00:01:40.140 --> 00:01:42.560 So as a check for yourself here's what I would recommend. 00:01:42.560 --> 00:01:44.820 I would recommend sitting down and giving yourself 00:01:44.820 --> 00:01:48.660 an hour and a half or two hours with the practice exam 00:01:48.660 --> 00:01:51.510 and see how that goes. 00:01:51.510 --> 00:01:53.380 OK? 00:01:53.380 --> 00:01:55.290 Try to give yourself the time constraint. 00:01:55.290 --> 00:01:59.240 And here's one of the things that-- there's 00:01:59.240 --> 00:02:01.810 a very strong correlation between having 00:02:01.810 --> 00:02:05.540 nailed the problem sets, and worked through all the practice 00:02:05.540 --> 00:02:07.780 exams, and just done a lot of problems, 00:02:07.780 --> 00:02:09.130 and doing well on exams. 00:02:09.130 --> 00:02:10.694 You'll be more confident. 00:02:10.694 --> 00:02:12.610 The only way to study for these things is just 00:02:12.610 --> 00:02:13.660 do a lot of problems. 00:02:13.660 --> 00:02:17.380 So on Stellar, for example, are a bunch of previous problem 00:02:17.380 --> 00:02:19.780 sets, and previous exams and practice 00:02:19.780 --> 00:02:22.080 exams from previous years. 00:02:22.080 --> 00:02:25.700 I would encourage you to look at those also on OCW 00:02:25.700 --> 00:02:27.240 and use those as practice. 00:02:27.240 --> 00:02:28.794 OK, any practical questions before we 00:02:28.794 --> 00:02:29.710 get on to the physics. 00:02:29.710 --> 00:02:30.417 Yeah? 00:02:30.417 --> 00:02:32.802 AUDIENCE: Will the exam cover all the material, 00:02:32.802 --> 00:02:34.427 or all the material [INAUDIBLE]? 00:02:34.427 --> 00:02:36.010 PROFESSOR: This is a cumulative topic. 00:02:36.010 --> 00:02:37.570 So the question is does it cover everything, or just 00:02:37.570 --> 00:02:38.560 last couple of weeks. 00:02:38.560 --> 00:02:40.800 And it's a cumulative topic, so the entire semester 00:02:40.800 --> 00:02:44.820 is necessary in order to answer the problems. 00:02:44.820 --> 00:02:47.620 Other questions? 00:02:47.620 --> 00:02:48.621 OK. 00:02:48.621 --> 00:02:50.980 Anything else? 00:02:50.980 --> 00:02:57.350 So quick couple of review questions 00:02:57.350 --> 00:03:00.234 before we launch into the main material of today. 00:03:00.234 --> 00:03:01.900 These are going to turn out to be useful 00:03:01.900 --> 00:03:04.970 reminders later on in today's lecture. 00:03:04.970 --> 00:03:06.890 So first, suppose I tell you I have 00:03:06.890 --> 00:03:11.700 a system with energy operator E, which has an operator A such 00:03:11.700 --> 00:03:18.380 that the commutator of E with A would say plus h bar A. OK? 00:03:18.380 --> 00:03:21.120 What does that tell you about the spectrum of the energy 00:03:21.120 --> 00:03:21.620 operator? 00:03:24.876 --> 00:03:26.064 AUDIENCE: It's a ladder. 00:03:26.064 --> 00:03:27.480 PROFESSOR: It's a ladder, exactly. 00:03:27.480 --> 00:03:30.400 So this tells you that the spectrum of E, or the energy 00:03:30.400 --> 00:03:34.060 eigenvalues En, are evenly spaced-- 00:03:34.060 --> 00:03:40.030 and we need a dimensional constant here, 00:03:40.030 --> 00:03:49.050 omega-- evenly spaced by h bar omega. 00:03:49.050 --> 00:03:54.530 And more precisely, that given a state phi E, 00:03:54.530 --> 00:03:58.030 we can act on it with the operator A 00:03:58.030 --> 00:04:02.310 to give us a new state, which is also an energy eigenstate, 00:04:02.310 --> 00:04:06.250 with energy E plus h bar omega. 00:04:06.250 --> 00:04:07.900 right? 00:04:07.900 --> 00:04:10.460 So any time you see that commutation relation, 00:04:10.460 --> 00:04:13.610 you know this fact to be true. 00:04:13.610 --> 00:04:14.540 Second statement. 00:04:14.540 --> 00:04:17.220 Suppose I have an operator B which 00:04:17.220 --> 00:04:19.860 commutes with the energy operator. 00:04:19.860 --> 00:04:20.360 OK? 00:04:20.360 --> 00:04:22.547 So that the commutator vanishes. 00:04:22.547 --> 00:04:24.255 What does that tell you about the system? 00:04:27.814 --> 00:04:30.130 AUDIENCE: Simultaneous eigenfunctions [INAUDIBLE]. 00:04:30.130 --> 00:04:31.005 PROFESSOR: Excellent. 00:04:31.005 --> 00:04:32.970 So one one consequence is that there 00:04:32.970 --> 00:04:37.100 exists simultaneous eigenfunctions phi sub E, B, 00:04:37.100 --> 00:04:46.576 which are simultaneous eigenfunctions of both E and B. 00:04:46.576 --> 00:04:47.700 What else does it tell you? 00:04:52.090 --> 00:04:56.557 Well, notice the following. 00:04:56.557 --> 00:04:58.890 Notice that if we took this computation relation and set 00:04:58.890 --> 00:05:01.670 omega to 0, we get this commutation relation. 00:05:01.670 --> 00:05:04.440 So this commutation relation is of the form of this commutation 00:05:04.440 --> 00:05:06.770 relation with omega equals 0. 00:05:06.770 --> 00:05:09.890 So what does that tell you about the states? 00:05:09.890 --> 00:05:11.210 About the energy eigenvalues? 00:05:14.490 --> 00:05:17.740 What happens if I take a state phi sub E 00:05:17.740 --> 00:05:20.060 and I act on it with B? 00:05:20.060 --> 00:05:20.780 What do I get? 00:05:25.819 --> 00:05:27.360 What can you say about this function? 00:05:31.300 --> 00:05:34.287 Well, what is its eigenvalue under E? 00:05:34.287 --> 00:05:36.370 It's E. It's the same thing, because they commute. 00:05:36.370 --> 00:05:38.300 You can pull the E through, multiply it, 00:05:38.300 --> 00:05:40.760 you get the constant E, you pull it back out. 00:05:40.760 --> 00:05:44.770 This is still an eigenfunction of phi 00:05:44.770 --> 00:05:47.345 of the energy operator with the same eigenvalue. 00:05:47.345 --> 00:05:49.220 But is it necessarily the same eigenfunction? 00:05:52.610 --> 00:05:54.590 No, because B may act on phi and just give you 00:05:54.590 --> 00:05:55.340 a different state. 00:05:55.340 --> 00:06:00.520 So this is some eigenfunction with the same energy, 00:06:00.520 --> 00:06:03.994 but it may be a different eigenfunction. 00:06:03.994 --> 00:06:04.669 OK? 00:06:04.669 --> 00:06:06.210 So let's think of an example of this. 00:06:06.210 --> 00:06:08.300 An example of this is consider a free particle. 00:06:11.150 --> 00:06:13.210 In the case of a free particle, we 00:06:13.210 --> 00:06:18.590 have E is equal to P squared upon 2m. 00:06:18.590 --> 00:06:21.500 And as a consequence E and P commute. 00:06:26.510 --> 00:06:29.160 Everyone agree with that? 00:06:29.160 --> 00:06:30.630 Everyone happy with that statement? 00:06:30.630 --> 00:06:31.420 They commute. 00:06:31.420 --> 00:06:32.950 So plus 0 if you will. 00:06:35.750 --> 00:06:38.600 And here what I wanted to say is when you have an operator 00:06:38.600 --> 00:06:41.990 that commutes with the energy, then 00:06:41.990 --> 00:06:45.860 there can be multiple states with the same energy which 00:06:45.860 --> 00:06:48.200 are different states, right? 00:06:48.200 --> 00:06:49.430 Different states entirely. 00:06:49.430 --> 00:06:51.230 So for example, in this case, what 00:06:51.230 --> 00:06:51.880 are the energy eigenfunctions? 00:06:51.880 --> 00:06:52.850 Well, e to the ikx. 00:06:55.718 --> 00:07:00.391 And this has energy h bar squared k squared upon 2m. 00:07:00.391 --> 00:07:02.890 But there's another state, which is a different state, which 00:07:02.890 --> 00:07:05.325 has the same energy. 00:07:05.325 --> 00:07:08.340 e to the minus ikx. 00:07:08.340 --> 00:07:09.610 OK? 00:07:09.610 --> 00:07:13.320 So when you have an operator that commutes with the energy 00:07:13.320 --> 00:07:19.770 operator, you can have simultaneous eigenfunctions. 00:07:19.770 --> 00:07:24.950 And you can also have multiple eigenfunctions 00:07:24.950 --> 00:07:26.900 that have the same energy eigenvalue, 00:07:26.900 --> 00:07:28.080 but are different functions. 00:07:28.080 --> 00:07:29.920 For example, this. 00:07:29.920 --> 00:07:31.390 Everyone cool with that? 00:07:31.390 --> 00:07:34.040 Now, just to make clear that it's actually 00:07:34.040 --> 00:07:36.320 the commuting that matters, imagine 00:07:36.320 --> 00:07:39.020 we took not the free particle, but the harmonic oscillator. 00:07:42.940 --> 00:07:51.700 E is p squared upon 2m plus m omega squared upon 2 x squared. 00:07:51.700 --> 00:07:57.170 Is it true that E-- I'll call this harmonic oscillator-- 00:07:57.170 --> 00:08:00.869 does E harmonic oscillator commute with P? 00:08:00.869 --> 00:08:02.410 No, you're going to get a term that's 00:08:02.410 --> 00:08:06.282 m omega squared x from the commutator. 00:08:06.282 --> 00:08:08.240 The potential is not translationally invariant. 00:08:08.240 --> 00:08:09.780 It does not commute with momentum. 00:08:09.780 --> 00:08:11.860 So this is not equal to 0. 00:08:11.860 --> 00:08:14.865 So that suggests that this degeneracy, two states having 00:08:14.865 --> 00:08:16.490 the same energy, should not be present. 00:08:16.490 --> 00:08:18.490 And indeed, are the states degenerate 00:08:18.490 --> 00:08:20.530 for the harmonic oscillator. 00:08:20.530 --> 00:08:22.100 No. 00:08:22.100 --> 00:08:22.800 No degeneracy. 00:08:28.310 --> 00:08:28.930 Yeah? 00:08:28.930 --> 00:08:31.798 AUDIENCE: We did something [INAUDIBLE] 00:08:31.798 --> 00:08:34.409 where I think it said that the eigenfunctions were complete. 00:08:34.409 --> 00:08:35.034 PROFESSOR: Yes. 00:08:35.034 --> 00:08:36.020 AUDIENCE: What does that mean? 00:08:36.020 --> 00:08:37.340 PROFESSOR: What does it mean for the eigenfunctions 00:08:37.340 --> 00:08:38.110 to be complete? 00:08:38.110 --> 00:08:39.860 What that means is that they form a basis. 00:08:42.369 --> 00:08:44.119 AUDIENCE: So the basis doesn't necessarily 00:08:44.119 --> 00:08:46.674 mean not [INAUDIBLE]. 00:08:46.674 --> 00:08:49.340 PROFESSOR: Yeah, no one told you the basis had to be degenerate, 00:08:49.340 --> 00:08:50.700 and in particular, that's a excellent-- 00:08:50.700 --> 00:08:52.324 so the question here is, wait a minute, 00:08:52.324 --> 00:08:54.726 I thought a basis had to be a complete set-- if you had 00:08:54.726 --> 00:08:57.100 an energy operator and you constructed the energy eigen-- 00:08:57.100 --> 00:08:57.970 this is a very good question. 00:08:57.970 --> 00:08:58.620 Thank you. 00:08:58.620 --> 00:09:01.220 If I have the energy operator and I construct it's energy 00:09:01.220 --> 00:09:03.220 eigenfunctions, then those energy eigenfunctions 00:09:03.220 --> 00:09:06.060 form a complete basis for any arbitrary function. 00:09:06.060 --> 00:09:07.927 Any function can be expanded in it, right? 00:09:07.927 --> 00:09:09.510 So for example, for the free particle, 00:09:09.510 --> 00:09:11.301 the energy eigenfunctions are e to the ikx, 00:09:11.301 --> 00:09:13.890 the momentum eigenfunctions for any value of k. 00:09:13.890 --> 00:09:15.800 But wait, how can there be multiple states 00:09:15.800 --> 00:09:17.804 with the same energy? 00:09:17.804 --> 00:09:19.470 Isn't that double counting or something? 00:09:19.470 --> 00:09:21.680 And the important thing is those guys have the same energy, 00:09:21.680 --> 00:09:23.150 but they have different momenta. 00:09:23.150 --> 00:09:24.630 They're different states. 00:09:24.630 --> 00:09:26.390 One has momentum h bar k. 00:09:26.390 --> 00:09:28.037 One has momentum minus h bar k. 00:09:28.037 --> 00:09:29.620 And you know that that has to be true, 00:09:29.620 --> 00:09:31.400 because the Fourier theorem tells you, 00:09:31.400 --> 00:09:33.830 in order to form a complete basis, you need all of them, 00:09:33.830 --> 00:09:36.080 all possible values of k. 00:09:36.080 --> 00:09:38.340 So there's no problem with being a complete basis 00:09:38.340 --> 00:09:43.120 and having states have the same energy eigenvalue, OK? 00:09:43.120 --> 00:09:44.720 It's a good question. 00:09:44.720 --> 00:09:45.920 Other questions? 00:09:45.920 --> 00:09:46.420 Yeah? 00:09:46.420 --> 00:09:49.300 AUDIENCE: So if we have a potential that only admits 00:09:49.300 --> 00:09:51.940 bound states, we'll never have this commutation 00:09:51.940 --> 00:09:52.670 happen basically? 00:09:52.670 --> 00:09:53.550 PROFESSOR: Yeah, exactly. 00:09:53.550 --> 00:09:54.050 Excellent. 00:09:54.050 --> 00:09:55.360 So the observation is this. 00:09:55.360 --> 00:09:58.810 Look, imagine I have a potential that's not trivial. 00:09:58.810 --> 00:10:02.200 It's not 0, OK? 00:10:02.200 --> 00:10:05.860 Will the momentum commute with the energy operator. 00:10:05.860 --> 00:10:08.029 No, because it's got a potential that's 00:10:08.029 --> 00:10:10.570 going to be acted upon by P, so you'll get a derivative term. 00:10:10.570 --> 00:10:14.940 But more precisely, if I have a system with bound states, 00:10:14.940 --> 00:10:17.980 I have to have a potential, right? 00:10:17.980 --> 00:10:21.360 And then I can't have P commuting with the energy 00:10:21.360 --> 00:10:24.160 operator, which means I can't have degeneracies. 00:10:24.160 --> 00:10:25.770 So indeed, if you have bound states, 00:10:25.770 --> 00:10:27.626 you cannot have degeneracies. 00:10:27.626 --> 00:10:28.500 That's exactly right. 00:10:28.500 --> 00:10:29.102 Yeah? 00:10:29.102 --> 00:10:31.560 AUDIENCE: But doesn't this break down in higher dimensions? 00:10:31.560 --> 00:10:33.060 PROFESSOR: Excellent, so we're going 00:10:33.060 --> 00:10:35.150 to come back to higher dimensions later. 00:10:35.150 --> 00:10:37.069 So the question predicts what's going 00:10:37.069 --> 00:10:38.610 to happen in the rest of the lecture. 00:10:38.610 --> 00:10:39.810 What we're going to do in just a minute 00:10:39.810 --> 00:10:41.060 is we're going to start working in three 00:10:41.060 --> 00:10:42.330 dimensions for the first time. 00:10:42.330 --> 00:10:43.840 We're going to leave 1D behind. 00:10:43.840 --> 00:10:45.690 We're going to take our tripped out tricycle 00:10:45.690 --> 00:10:49.020 and replace it with a Yamaha. 00:10:49.020 --> 00:10:52.170 As you'll see, it has the same basic physics driving 00:10:52.170 --> 00:10:54.800 its, well, self. 00:10:54.800 --> 00:10:56.312 It's the same dynamics. 00:10:56.312 --> 00:10:58.145 But I want to emphasize a couple things that 00:10:58.145 --> 00:10:59.360 are going to show up. 00:10:59.360 --> 00:11:01.130 So the question is, isn't this story 00:11:01.130 --> 00:11:02.590 different in three dimensions? 00:11:02.590 --> 00:11:06.192 And we shall see exactly what happens in higher dimensions. 00:11:06.192 --> 00:11:07.400 We'll work in two dimensions. 00:11:07.400 --> 00:11:08.691 We'll work in three dimensions. 00:11:08.691 --> 00:11:10.882 We'll work in more. 00:11:10.882 --> 00:11:13.090 Doesn't really matter how many dimensions we work in. 00:11:13.090 --> 00:11:14.900 You'll see it. 00:11:14.900 --> 00:11:16.394 OK, third thing. 00:11:16.394 --> 00:11:18.560 You studied this in some detail in your problem set. 00:11:18.560 --> 00:11:20.040 Suppose I have an energy operator 00:11:20.040 --> 00:11:23.240 that commutes with a unitary operator, U, OK? 00:11:23.240 --> 00:11:25.800 So it commutes to 0. 00:11:25.800 --> 00:11:30.540 And U is unitary, so U dagger U is one, is the identity. 00:11:30.540 --> 00:11:34.100 So what does this tell you? 00:11:34.100 --> 00:11:38.420 Well, first off from these guys it 00:11:38.420 --> 00:11:48.510 tells us that we can have simultaneous eigenfunctions. 00:11:55.400 --> 00:12:00.190 It also tells us too that if we take our state phi 00:12:00.190 --> 00:12:02.860 and we act on it with U, this could give us 00:12:02.860 --> 00:12:09.310 a new state, phi tilde sub E, which will necessarily 00:12:09.310 --> 00:12:12.210 have the same energy eigenvalue, because U and E commute. 00:12:12.210 --> 00:12:13.930 But it may be a different state. 00:12:16.510 --> 00:12:18.472 We'll come to this in more detail later. 00:12:18.472 --> 00:12:20.555 But the third thing, and I want to emphasize this, 00:12:20.555 --> 00:12:23.116 is this tells us, look, we have a unitary operator. 00:12:23.116 --> 00:12:24.990 We can always write the unitary operator as e 00:12:24.990 --> 00:12:26.720 to the i of a Hermitian operator. 00:12:31.490 --> 00:12:34.410 So what is the meaning of the Hermitian operator? 00:12:34.410 --> 00:12:35.530 What is this guy? 00:12:35.530 --> 00:12:37.180 So in your problem set, you looked 00:12:37.180 --> 00:12:40.660 at what unitary operators are. 00:12:40.660 --> 00:12:43.365 And in the problem set, it's discussed in some detail 00:12:43.365 --> 00:12:45.230 that there's a relationship between 00:12:45.230 --> 00:12:47.630 a unitary transformation, or unitary operator, 00:12:47.630 --> 00:12:49.455 and the symmetry. 00:12:49.455 --> 00:12:51.080 A symmetry is when you take your system 00:12:51.080 --> 00:12:55.207 and you do something to it, like a rotation or translation, 00:12:55.207 --> 00:12:57.290 and it's a symmetry if it doesn't change anything, 00:12:57.290 --> 00:12:59.034 if the energy remains invariant. 00:12:59.034 --> 00:13:01.450 So if the energy doesn't change under this transformation, 00:13:01.450 --> 00:13:03.880 we call that a symmetry. 00:13:03.880 --> 00:13:07.080 And we also showed that symmetries, or translations, 00:13:07.080 --> 00:13:08.850 are generated by unitary operators. 00:13:08.850 --> 00:13:12.350 For example, my favorite examples are the translate by L 00:13:12.350 --> 00:13:14.350 operator, which is unitary. 00:13:14.350 --> 00:13:18.060 And also e to the minus dxl. 00:13:20.980 --> 00:13:25.890 And the boost by q operator, which 00:13:25.890 --> 00:13:28.460 similarly is e to the minus qdp. 00:13:37.760 --> 00:13:40.610 And the time translation operator, U sub t, 00:13:40.610 --> 00:13:46.690 which is equal to e to the minus i t over h bar energy operator. 00:13:46.690 --> 00:13:47.310 OK? 00:13:47.310 --> 00:13:50.470 So these are transformation operators. 00:13:50.470 --> 00:13:53.420 These are symmetry operators, which translate you by L, 00:13:53.420 --> 00:13:55.900 boost or speed you up by momentum q, 00:13:55.900 --> 00:13:57.510 evolve you forward in time t. 00:13:57.510 --> 00:14:00.920 And they can all be expressed as e to the unitary operator. 00:14:00.920 --> 00:14:11.250 So this in particular is l i over h bar p. 00:14:11.250 --> 00:14:14.530 And this is similarly x. 00:14:14.530 --> 00:14:19.402 And earlier we understood the role of momentum 00:14:19.402 --> 00:14:21.360 having to do translations in the following way. 00:14:21.360 --> 00:14:23.060 There's a beautiful theorem about this. 00:14:23.060 --> 00:14:25.900 If you take a system and its translation invariant, 00:14:25.900 --> 00:14:27.890 the classical statement of Noether's theorem 00:14:27.890 --> 00:14:30.670 is that there's a conserved quantity associated 00:14:30.670 --> 00:14:31.780 with that translation. 00:14:31.780 --> 00:14:34.260 That conserved quantity is the momentum. 00:14:34.260 --> 00:14:36.100 And quantum mechanically, the generator 00:14:36.100 --> 00:14:38.770 of that transformation the Hermitian operator 00:14:38.770 --> 00:14:40.770 that goes upstairs in the unitary 00:14:40.770 --> 00:14:43.260 is the operator associated to that conserved quantity, 00:14:43.260 --> 00:14:45.280 associated to that observable. 00:14:45.280 --> 00:14:47.010 You have translations. 00:14:47.010 --> 00:14:49.011 There's a conserved quantity, which is momentum. 00:14:49.011 --> 00:14:51.510 And the thing that generates translations, the operator that 00:14:51.510 --> 00:14:54.300 generates translations, is the operator representing momentum. 00:15:00.637 --> 00:15:02.845 So each of these are going to come up later in today, 00:15:02.845 --> 00:15:06.250 and I just wanted to flag them down before the moment. 00:15:06.250 --> 00:15:10.345 OK, questions before we move on? 00:15:10.345 --> 00:15:11.315 Yeah? 00:15:11.315 --> 00:15:14.952 AUDIENCE: So you made the claim that every unitary operator can 00:15:14.952 --> 00:15:17.195 be expressed as p to the eigenfunction. 00:15:17.195 --> 00:15:19.570 PROFESSOR: OK, I should be a little bit careful, but yes. 00:15:19.570 --> 00:15:21.910 That's right. 00:15:21.910 --> 00:15:25.134 AUDIENCE: But if I take the [INAUDIBLE] I 00:15:25.134 --> 00:15:26.800 should be able to figure out what it is, 00:15:26.800 --> 00:15:28.230 but you can't take the [INAUDIBLE] 00:15:28.230 --> 00:15:29.300 PROFESSOR: The more precise statement 00:15:29.300 --> 00:15:31.140 is that any unitary-- any one parameter 00:15:31.140 --> 00:15:34.142 family of unitary operators can be expressed in that form. 00:15:34.142 --> 00:15:35.600 And then you can take a derivative. 00:15:35.600 --> 00:15:38.060 And that's the theory of [INAUDIBLE], 00:15:38.060 --> 00:15:40.350 which is beyond the scope. 00:15:43.080 --> 00:15:45.350 Let me make a very specific statement, which 00:15:45.350 --> 00:15:47.190 is that one parameter of [INAUDIBLE] 00:15:47.190 --> 00:15:48.148 unitary transformation. 00:15:48.148 --> 00:15:50.240 So translations by l, where you can vary l, 00:15:50.240 --> 00:15:52.150 can be expressed in that form. 00:15:52.150 --> 00:15:53.900 And that's a very general statement. 00:15:53.900 --> 00:15:58.535 OK, so with all that as prelude, let's go back to 3D. 00:16:03.360 --> 00:16:07.540 So in 3D, the energy operator-- so what's going to change? 00:16:07.540 --> 00:16:13.180 Now instead of just having position and its momentum, 00:16:13.180 --> 00:16:16.090 we now also have-- I'll call this P sub x-- we can also 00:16:16.090 --> 00:16:19.090 have a y-coordinate and we have a z-coordinate. 00:16:19.090 --> 00:16:20.820 And each of them has its momentum. 00:16:20.820 --> 00:16:23.610 P sub z and P sub y. 00:16:26.290 --> 00:16:28.740 And here's just a quick practical question. 00:16:28.740 --> 00:16:36.020 We know that x with Px is equal to i h bar. 00:16:44.170 --> 00:16:46.750 So what do you expect to be true of x with y? 00:16:50.089 --> 00:16:51.050 AUDIENCE: 0. 00:16:51.050 --> 00:16:52.729 PROFESSOR: Why? 00:16:52.729 --> 00:16:53.645 AUDIENCE: [INAUDIBLE]. 00:16:56.967 --> 00:16:58.800 PROFESSOR: What does this equation tell you? 00:16:58.800 --> 00:17:00.008 What is its physical content? 00:17:02.170 --> 00:17:03.670 Well, that they don't commute, good. 00:17:03.670 --> 00:17:05.140 What does that tell you physically? 00:17:05.140 --> 00:17:05.479 Yes? 00:17:05.479 --> 00:17:07.437 AUDIENCE: That there's an uncertainty principle 00:17:07.437 --> 00:17:08.420 connecting the two. 00:17:08.420 --> 00:17:08.720 PROFESSOR: Excellent. 00:17:08.720 --> 00:17:09.550 So that's one statement. 00:17:09.550 --> 00:17:11.000 So the consequence of this is that there's 00:17:11.000 --> 00:17:12.041 an uncertainty principle. 00:17:12.041 --> 00:17:16.130 Delta x delta Px must be greater than or equal to h bar upon 2. 00:17:16.130 --> 00:17:17.550 What's another way of saying this? 00:17:22.349 --> 00:17:24.999 Do there exist simultaneous eigenfunctions of x and P? 00:17:24.999 --> 00:17:25.540 AUDIENCE: No. 00:17:25.540 --> 00:17:26.123 PROFESSOR: No. 00:17:26.123 --> 00:17:28.400 No simultaneous eigenfunctions. 00:17:28.400 --> 00:17:32.280 OK, so you can't have a definite value 00:17:32.280 --> 00:17:34.234 of x and a definite value of P simultaneously. 00:17:34.234 --> 00:17:35.150 There's no such state. 00:17:35.150 --> 00:17:36.358 It's not that you can't know. 00:17:36.358 --> 00:17:39.180 It's that there's no such state. 00:17:39.180 --> 00:17:42.416 Do you expect to be able to know the position in x 00:17:42.416 --> 00:17:45.710 and the position in y simultaneously? 00:17:45.710 --> 00:17:46.240 Sure. 00:17:46.240 --> 00:17:48.551 OK, so this turns out to be 0. 00:17:48.551 --> 00:17:50.050 And in some sense, you can take that 00:17:50.050 --> 00:17:51.850 as a definition of quantum mechanics. 00:17:51.850 --> 00:17:54.500 x and y need to be 0. 00:17:54.500 --> 00:18:00.780 And similarly, Px and Py commute. 00:18:00.780 --> 00:18:02.340 The momenta are independent. 00:18:02.340 --> 00:18:10.890 However, Py and y should be equal to minus i h bar. 00:18:10.890 --> 00:18:11.390 Good. 00:18:11.390 --> 00:18:11.740 Exactly. 00:18:11.740 --> 00:18:13.290 So the commutators work out exactly 00:18:13.290 --> 00:18:15.600 as you'd naively expect. 00:18:15.600 --> 00:18:20.450 Every pair of position and its momenta commute canonically 00:18:20.450 --> 00:18:21.920 to i h bar. 00:18:21.920 --> 00:18:25.480 And every pair of coordinates commute to 0. 00:18:25.480 --> 00:18:27.670 Every pair of momenta commute to 0. 00:18:27.670 --> 00:18:28.170 Cool? 00:18:33.240 --> 00:18:36.835 So what kind of systems are we going to interested in? 00:18:36.835 --> 00:18:38.710 Well, we're going to be interested in systems 00:18:38.710 --> 00:18:41.430 where the energy operator is equal to P 00:18:41.430 --> 00:18:49.790 vector hat squared upon 2m plus U of x and y and z, hat, hat. 00:18:49.790 --> 00:18:57.770 You can see why dropping the hats becomes almost / 00:18:57.770 --> 00:19:03.710 So in this language, we can write the Schrodinger equation. 00:19:03.710 --> 00:19:07.220 This is just a direct extension of the 1D Schrodinger equation. 00:19:07.220 --> 00:19:09.040 i h bar dt of psi. 00:19:09.040 --> 00:19:12.770 Now our wave function is a function of x and y and z. 00:19:12.770 --> 00:19:14.350 There's some finite probability then 00:19:14.350 --> 00:19:15.850 to find a particle at some position. 00:19:15.850 --> 00:19:18.850 That position is labeled by the three coordinates. 00:19:18.850 --> 00:19:21.630 Is equal to-- and of t. 00:19:24.890 --> 00:19:26.767 Is equal to-- well, I'm actually write 00:19:26.767 --> 00:19:28.100 this in slightly different form. 00:19:28.100 --> 00:19:30.510 This is going to be easier if I use vector notation. 00:19:30.510 --> 00:19:33.750 So I'm going to write this as psi of r and t, 00:19:33.750 --> 00:19:36.260 where r denotes the position vector, 00:19:36.260 --> 00:19:39.460 is equal to the energy operator acting on it. 00:19:39.460 --> 00:19:46.790 And P is just equal to minus i h bar the gradient. 00:19:49.830 --> 00:19:53.780 So this is minus i h bar squared, or minus h bar squared 00:19:53.780 --> 00:20:04.960 upon 2m gradient squared plus u of x or now u of r psi of r 00:20:04.960 --> 00:20:05.950 and t. 00:20:11.770 --> 00:20:13.520 Quick question, what are the units or what 00:20:13.520 --> 00:20:18.940 are the dimensions of psi of r in 3D? 00:20:22.692 --> 00:20:24.570 AUDIENCE: [INAUDIBLE]. 00:20:24.570 --> 00:20:29.170 PROFESSOR: Yeah, one over length to the root three halves. 00:20:29.170 --> 00:20:32.240 And the reason is this norm squared gives us a probability 00:20:32.240 --> 00:20:34.500 density, something that when we integrated 00:20:34.500 --> 00:20:38.660 over all positions in a region integral d 3x is going 00:20:38.660 --> 00:20:40.620 to give us a number, a probability. 00:20:40.620 --> 00:20:42.290 So its actual magnitude must be-- 00:20:42.290 --> 00:20:45.670 or its dimension must be 1 over L to the 3/2. 00:20:45.670 --> 00:20:47.580 Just the cube of what it was in 1D. 00:20:50.230 --> 00:20:54.415 And as you'll see on the problems set 00:20:54.415 --> 00:20:57.580 and as we'll do in a couple lectures down the road, 00:20:57.580 --> 00:21:02.430 it's convenient sometimes to work in Cartesian, 00:21:02.430 --> 00:21:04.395 but it's also sometimes convenient 00:21:04.395 --> 00:21:05.770 to work in spherical coordinates. 00:21:09.632 --> 00:21:10.590 And it does not matter. 00:21:10.590 --> 00:21:12.006 And here's a really deep statement 00:21:12.006 --> 00:21:13.750 that goes way beyond quantum mechanics. 00:21:13.750 --> 00:21:15.170 It does not matter which coordinates you work in. 00:21:15.170 --> 00:21:17.020 You cannot possibly get a different answer by using 00:21:17.020 --> 00:21:18.140 different coordinates. 00:21:18.140 --> 00:21:20.760 So we're going to be ruthless in exploiting coordinates that 00:21:20.760 --> 00:21:22.870 will simplify our problem throughout the rest 00:21:22.870 --> 00:21:24.600 of this course. 00:21:24.600 --> 00:21:27.800 In the notes is it a short discussion of the form 00:21:27.800 --> 00:21:30.540 of the Laplacian, or the gradient 00:21:30.540 --> 00:21:34.532 squared in Cartesian spherical and cylindrical coordinates. 00:21:34.532 --> 00:21:36.740 You should feel free to use any coordinate system you 00:21:36.740 --> 00:21:37.630 want at any point. 00:21:37.630 --> 00:21:39.690 You just have to be consistent about it. 00:21:42.630 --> 00:21:44.345 So let's work out a couple of examples. 00:21:49.160 --> 00:21:51.010 And here are all we're going to do 00:21:51.010 --> 00:21:53.120 is apply exactly the same logic that we see over 00:21:53.120 --> 00:21:56.410 and over in 1D to our 3D problems. 00:21:56.410 --> 00:22:07.070 So the first example is a free particle in 3D. 00:22:07.070 --> 00:22:09.192 So before I get started on this, any questions? 00:22:09.192 --> 00:22:10.400 Just in general 3D questions? 00:22:13.150 --> 00:22:14.150 OK. 00:22:14.150 --> 00:22:15.447 So this stuff starts off easy. 00:22:15.447 --> 00:22:17.405 And I'm going to work in Cartesian coordinates. 00:22:21.220 --> 00:22:24.990 And a fun problem is to repeat this analysis 00:22:24.990 --> 00:22:28.740 in spherical coordinates, and we'll do that later on. 00:22:28.740 --> 00:22:30.420 OK, so free particle in 3D, so what 00:22:30.420 --> 00:22:32.378 is the energy eigenfunction equation look like? 00:22:32.378 --> 00:22:34.480 We want to find-- the Schrodinger equation has 00:22:34.480 --> 00:22:36.790 exactly the same structure as before. 00:22:36.790 --> 00:22:38.840 It's a linear differential equation. 00:22:38.840 --> 00:22:41.800 So if we find the eigenfunctions of the energy operator, 00:22:41.800 --> 00:22:45.030 we can use superposition to construct the general solution, 00:22:45.030 --> 00:22:46.160 right? 00:22:46.160 --> 00:22:49.980 So exactly as in 1D, I'm going to construct first the energy 00:22:49.980 --> 00:22:52.700 eigenfunctions , and then use them in superposition to find 00:22:52.700 --> 00:22:54.960 a general solution to the Schrodinger equation. 00:22:54.960 --> 00:22:55.640 OK? 00:22:55.640 --> 00:22:57.910 So let's construct the energy eigenfunctions. 00:22:57.910 --> 00:23:00.540 So what is the energy eigenvalue equation look like? 00:23:00.540 --> 00:23:06.000 Well, E on psi is equal to minus h bar squared upon 2m. 00:23:06.000 --> 00:23:08.860 And in Cartesian, the Laplacian is derivative respect 00:23:08.860 --> 00:23:10.700 to x squared plus derivative with respect 00:23:10.700 --> 00:23:15.280 to y squared plus derivative with respect to z squared. 00:23:15.280 --> 00:23:17.904 And we have no potential, so this is just psi. 00:23:17.904 --> 00:23:19.820 So that would be energy operator acting on it. 00:23:19.820 --> 00:23:21.694 And the eigenvalue equation is at a constant, 00:23:21.694 --> 00:23:24.886 the energy E on psi satisfies this equation. 00:23:24.886 --> 00:23:26.260 I'm going to write this phi sub e 00:23:26.260 --> 00:23:28.560 to continue with our notation of phi 00:23:28.560 --> 00:23:30.070 being the energy eigenfunctions. 00:23:30.070 --> 00:23:31.903 It of course, doesn't matter what I call it, 00:23:31.903 --> 00:23:36.270 but just for consistency I'm going to use the letter phi. 00:23:36.270 --> 00:23:39.240 So this is a very easy equation to solve. 00:23:41.770 --> 00:23:45.240 In particular, it has a lovely property, 00:23:45.240 --> 00:23:46.560 which is that it's separable. 00:23:50.600 --> 00:23:51.150 OK? 00:23:51.150 --> 00:23:52.525 So separable means the following. 00:23:52.525 --> 00:23:58.350 It means, look, I note, I just observed that this differential 00:23:58.350 --> 00:24:00.749 equation can be written as a sum of terms 00:24:00.749 --> 00:24:03.290 where there's a derivative with respect to only one variable. 00:24:03.290 --> 00:24:04.390 There's a differential operator with respect 00:24:04.390 --> 00:24:06.640 to only one variable and another differential operator 00:24:06.640 --> 00:24:08.930 with respect to only one variable added together. 00:24:08.930 --> 00:24:12.230 And when you see that, you can separate. 00:24:12.230 --> 00:24:14.140 And here's what I mean by separate. 00:24:14.140 --> 00:24:15.550 I'm going to just construct. 00:24:15.550 --> 00:24:17.250 I'm not going to say that this is a general solution. 00:24:17.250 --> 00:24:18.666 I'm just going to try to construct 00:24:18.666 --> 00:24:20.470 a set of solutions of the following form. 00:24:20.470 --> 00:24:24.610 Psi E of x, y, and z is equal to psi 00:24:24.610 --> 00:24:33.800 E-- I will call this psi sub x of x times phi sub 00:24:33.800 --> 00:24:38.678 y of y times phi sub z of z. 00:24:41.460 --> 00:24:44.250 And if we take this and we plug it in, let's see what we get. 00:24:44.250 --> 00:24:48.940 This gives us that E on phi E is equal to minus h bar squared 00:24:48.940 --> 00:24:49.440 upon 2m. 00:24:52.210 --> 00:24:55.240 Well the dx squared acting on phi sub e 00:24:55.240 --> 00:24:56.610 is only going to hit this guy. 00:24:56.610 --> 00:25:03.580 So I'm going to get phi x prime prime phi y phi z. 00:25:03.580 --> 00:25:08.870 Plus from the next term phi y prime prime phi x phi z. 00:25:08.870 --> 00:25:13.301 And from the next term phi z prime prime phi x phi y. 00:25:15.817 --> 00:25:18.400 But now I can do a sneaky thing and divide the entire equation 00:25:18.400 --> 00:25:19.470 by phi sub e. 00:25:19.470 --> 00:25:21.800 Phi sub e is phi x phi y phi z. 00:25:21.800 --> 00:25:28.420 So if I do so, I lose the phi y phi z, and I divide by phi x. 00:25:28.420 --> 00:25:30.740 And in this term, when I divide by phi x phi y phi z, 00:25:30.740 --> 00:25:36.100 I lose the x and z, and I have a left over phi sub y, or phi y. 00:25:36.100 --> 00:25:38.240 And similarly here, phi sub z upon phi z. 00:25:42.994 --> 00:25:44.780 Everyone cool with that? 00:25:44.780 --> 00:25:46.036 Yeah? 00:25:46.036 --> 00:25:50.350 AUDIENCE: Can we also lose the [INAUDIBLE]? 00:25:50.350 --> 00:25:51.790 PROFESSOR: I don't think so. 00:25:51.790 --> 00:25:54.980 Minus h bar squared over 2m just hangs out for the ride. 00:25:58.870 --> 00:26:01.390 So when I take the derivatives, I get these guys. 00:26:01.390 --> 00:26:04.080 And I have E times the function. 00:26:04.080 --> 00:26:06.836 We could certainly write this as 2m over h bar squared 00:26:06.836 --> 00:26:07.710 and put it over here. 00:26:07.710 --> 00:26:10.630 That's fine. 00:26:10.630 --> 00:26:13.460 OK, so this is the form of the equation we have, 00:26:13.460 --> 00:26:15.680 and what does this give us? 00:26:15.680 --> 00:26:17.580 What content does this give us? 00:26:17.580 --> 00:26:19.120 Well, note the following. 00:26:19.120 --> 00:26:20.480 This is a funny system. 00:26:20.480 --> 00:26:24.020 This is a function of x. 00:26:24.020 --> 00:26:28.080 This is a function of y, g of y only, and not of x or z. 00:26:28.080 --> 00:26:33.280 And this is a function only of z, and not of x or y. 00:26:33.280 --> 00:26:35.750 Yeah? 00:26:35.750 --> 00:26:38.440 So we have that E, and let's put this 00:26:38.440 --> 00:26:42.160 as minus 2m over h bar squared is 00:26:42.160 --> 00:26:46.300 equal to a function of x plus a function of y 00:26:46.300 --> 00:26:47.310 plus a function of h. 00:26:51.100 --> 00:26:52.120 What does this tell you? 00:26:52.120 --> 00:26:53.570 AUDIENCE: They're all constant. 00:26:53.570 --> 00:26:54.930 PROFESSOR: They're all constant, right. 00:26:54.930 --> 00:26:56.320 So the important thing is this equation 00:26:56.320 --> 00:26:58.260 has to be true for every value of x, y, and z. 00:26:58.260 --> 00:26:59.130 It's a differential equation. 00:26:59.130 --> 00:27:00.005 It's true everywhere. 00:27:00.005 --> 00:27:00.650 It's true here. 00:27:00.650 --> 00:27:01.316 It's true there. 00:27:01.316 --> 00:27:03.288 It's true at every point. 00:27:03.288 --> 00:27:04.100 Yeah? 00:27:04.100 --> 00:27:07.450 So for any value of x, y, and z, this equation must be true. 00:27:07.450 --> 00:27:10.117 So now imagine I have a particular solution g at h. 00:27:10.117 --> 00:27:12.200 I'm going to fix y and z to some particular point. 00:27:12.200 --> 00:27:13.630 I'm going to look right here. 00:27:13.630 --> 00:27:15.570 And here that fixes y and z. 00:27:15.570 --> 00:27:17.510 So these are just some numbers. 00:27:17.510 --> 00:27:19.970 And suppose we satisfy this equation. 00:27:19.970 --> 00:27:23.280 Then there's a very x leaving y and z fixed. 00:27:23.280 --> 00:27:27.054 What must be true of f of x? 00:27:27.054 --> 00:27:27.930 AUDIENCE: Constant. 00:27:27.930 --> 00:27:29.763 PROFESSOR: It's got to be constant, exactly. 00:27:29.763 --> 00:27:31.870 So this tells us that f of x is a constant. 00:27:31.870 --> 00:27:32.620 I.e. 00:27:32.620 --> 00:27:36.920 phi x prime prime of over phi x is a constant. 00:27:36.920 --> 00:27:38.100 I'll call it epsilon x. 00:27:40.910 --> 00:27:46.560 And phi y prime prime of over phi y and this 00:27:46.560 --> 00:27:51.950 is a function of y of x is equal to epsilon y. 00:27:51.950 --> 00:27:56.840 And actually for-- yeah, that's fine. 00:27:56.840 --> 00:27:58.340 For fun I'm going to put in a minus. 00:27:58.340 --> 00:28:00.298 It doesn't matter what I call this coefficient. 00:28:00.298 --> 00:28:04.674 And similarly for phi z prime prime z over phi z 00:28:04.674 --> 00:28:05.840 is equal to minus epsilon z. 00:28:08.410 --> 00:28:13.910 So this tells us that minus 2m upon h bar squared 00:28:13.910 --> 00:28:19.400 e is equal to minus epsilon x minus epsilon y minus epsilon 00:28:19.400 --> 00:28:21.080 z. 00:28:21.080 --> 00:28:23.160 And any solutions of these equations 00:28:23.160 --> 00:28:25.316 with some constant value of epsilon x, epsilon 00:28:25.316 --> 00:28:26.880 y, and epsilon z is going to give me 00:28:26.880 --> 00:28:29.880 a solution of my original energy eigenvalue equation, where 00:28:29.880 --> 00:28:33.000 the value of capital E is equal to the sum. 00:28:33.000 --> 00:28:37.700 And I can take the minus signs make this plus plus. 00:28:37.700 --> 00:28:38.200 Yeah? 00:28:41.539 --> 00:28:46.360 AUDIENCE: Can epsilon x, epsilon y, 00:28:46.360 --> 00:28:50.495 and epsilon z-- can one of them be negative 00:28:50.495 --> 00:28:52.942 if the other's are sufficiently positive or vice versa? 00:28:52.942 --> 00:28:53.900 Or is that [INAUDIBLE]? 00:28:53.900 --> 00:28:54.890 PROFESSOR: Let's check. 00:28:54.890 --> 00:28:55.570 Let's check. 00:28:55.570 --> 00:28:57.640 So what are the solutions of this equation? 00:29:03.170 --> 00:29:03.670 Yeah. 00:29:03.670 --> 00:29:05.357 So solutions to this equation phi 00:29:05.357 --> 00:29:07.940 double-- so let's write this in a slightly more familiar form. 00:29:07.940 --> 00:29:12.140 This says that phi prime prime plus epsilon x 00:29:12.140 --> 00:29:15.800 phi is equal to 0, OK? 00:29:15.800 --> 00:29:18.730 But this just tells you that phi is exponential. 00:29:18.730 --> 00:29:25.670 Phi is equal to a e to the Ikx kxx plus B 00:29:25.670 --> 00:29:34.580 e to the minus ikxx, where k squared is 00:29:34.580 --> 00:29:37.609 equal to kx squared is equal to epsilon x. 00:29:37.609 --> 00:29:38.108 OK? 00:29:41.840 --> 00:29:47.340 So this becomes-- and similarly for epsilon y and epsilon z, 00:29:47.340 --> 00:29:50.270 each with their own value of ky and kz 00:29:50.270 --> 00:29:54.260 who squares the epsilon accordingly. 00:29:54.260 --> 00:29:57.900 So what can you say about these epsilons? 00:29:57.900 --> 00:30:01.945 Well, the epsilons are strictly positive numbers. 00:30:01.945 --> 00:30:03.070 So to answer your question. 00:30:03.070 --> 00:30:05.710 So the epsilons have to be positive. 00:30:05.710 --> 00:30:08.325 OK, so this equation becomes, though, E is equal to-- I'm 00:30:08.325 --> 00:30:10.908 going to put the h bar squared over 2m back on the other side. 00:30:10.908 --> 00:30:13.150 h bar squared upon 2m. 00:30:13.150 --> 00:30:15.150 Epsilon x, epsilon y, epsilon z. 00:30:15.150 --> 00:30:17.000 But those are just Kx squared plus Ky 00:30:17.000 --> 00:30:18.420 squared plus Kz squared. 00:30:18.420 --> 00:30:21.650 Kx squared plus Ky squared plus Kz 00:30:21.650 --> 00:30:32.680 squared, also known as h bar squared upon 2m k 00:30:32.680 --> 00:30:36.890 vector squared, where the wave function, phi sub 00:30:36.890 --> 00:30:41.550 E is equal to some overall normalization constant times, 00:30:41.550 --> 00:30:46.320 for the first function-- where did the definition go? 00:30:46.320 --> 00:30:46.970 Right here. 00:30:46.970 --> 00:30:49.430 So from here, phi x is the exponential with Kx. 00:30:49.430 --> 00:30:50.920 This is an exponential y with Ky. 00:30:50.920 --> 00:30:53.060 And then the exponential in z with Kz. 00:30:53.060 --> 00:31:00.700 e to the ikx times x plus Ky times y plus Kz times z. 00:31:04.510 --> 00:31:06.850 Let me write this as e to the i. 00:31:06.850 --> 00:31:09.070 e to the i. 00:31:09.070 --> 00:31:13.970 Also known as some normalization constant e to the i k vector 00:31:13.970 --> 00:31:17.772 dot r vector. 00:31:17.772 --> 00:31:18.272 OK? 00:31:21.580 --> 00:31:25.090 So the energy, if we have a free particle, the energy 00:31:25.090 --> 00:31:28.066 eigenfunctions can be put in the form, 00:31:28.066 --> 00:31:30.770 or at least we can build energy eigenfunctions of the form, 00:31:30.770 --> 00:31:36.520 plane waves with some 3D momentum with energy E 00:31:36.520 --> 00:31:41.370 is equal to h bar squared k squared upon 2m, 00:31:41.370 --> 00:31:44.286 just as we saw for the free particle in one dimension. 00:31:44.286 --> 00:31:45.910 And the actual wave function is nothing 00:31:45.910 --> 00:31:48.580 but a product of wave functions in 1D. 00:31:48.580 --> 00:31:49.530 Yeah? 00:31:49.530 --> 00:31:53.751 AUDIENCE: What happened to the minus ikx minus iky 00:31:53.751 --> 00:31:55.280 minus ikz terms? 00:31:55.280 --> 00:31:57.710 PROFESSOR: Good, so here I just dropped these guys. 00:31:57.710 --> 00:32:02.682 So I just picked examples where we just picked e to the ikz. 00:32:02.682 --> 00:32:03.890 That's an excellent question. 00:32:03.890 --> 00:32:05.750 So I've done something here. 00:32:05.750 --> 00:32:08.654 In particular, I looked at a special case. 00:32:08.654 --> 00:32:10.570 And here's an important lesson from the theory 00:32:10.570 --> 00:32:13.010 of the separable equations, which 00:32:13.010 --> 00:32:22.750 is that once I separate-- so if I have a separable equation 00:32:22.750 --> 00:32:25.010 and I find it separated solution, 00:32:25.010 --> 00:32:33.740 phi E is equal to phi x of x, phi y of y phi z of z, 00:32:33.740 --> 00:32:36.310 not all functions-- not all solutions of the equation 00:32:36.310 --> 00:32:37.520 are of this form. 00:32:37.520 --> 00:32:38.020 They're not. 00:32:38.020 --> 00:32:39.269 I had to make that assumption. 00:32:39.269 --> 00:32:42.070 I said suppose it's of this form right here. 00:32:42.070 --> 00:32:43.150 So this is an assumption. 00:32:46.840 --> 00:32:47.340 OK? 00:32:47.340 --> 00:32:51.520 However, these form a good basis. 00:32:51.520 --> 00:32:54.390 By taking suitable linear combinations 00:32:54.390 --> 00:32:56.140 of them, suitable super positions, 00:32:56.140 --> 00:32:58.320 I can build a completely general solution. 00:32:58.320 --> 00:33:02.360 For example, as was noted, the true solution of this equation, 00:33:02.360 --> 00:33:04.690 even just focusing on the x, is a superposition 00:33:04.690 --> 00:33:08.030 of plus and minus waves, waves with 00:33:08.030 --> 00:33:09.650 plus positive negative momentum. 00:33:09.650 --> 00:33:10.920 So how do we get that? 00:33:10.920 --> 00:33:12.810 Well, we could write that as phi is 00:33:12.810 --> 00:33:18.940 equal to e to the ikx phi y phi z plus e 00:33:18.940 --> 00:33:29.550 to the minus ikx phi y phi z with the same phi y and phi z. 00:33:29.550 --> 00:33:30.930 So these are actually in there. 00:33:34.390 --> 00:33:34.950 Yeah? 00:33:34.950 --> 00:33:38.280 AUDIENCE: My other question is so I still 00:33:38.280 --> 00:33:39.828 don't see why any of the epsilons 00:33:39.828 --> 00:33:40.952 can't have a negative sign. 00:33:40.952 --> 00:33:44.482 You have an exponential, a real exponential as one 00:33:44.482 --> 00:33:45.580 of your products. 00:33:45.580 --> 00:33:51.230 PROFESSOR: OK, so if we had a negative epsilon, 00:33:51.230 --> 00:33:55.130 is that wave function going to be normalizable? 00:33:55.130 --> 00:33:57.100 AUDIENCE: Oh, as r goes to-- but can you 00:33:57.100 --> 00:33:59.440 just keep the minus term? 00:33:59.440 --> 00:34:01.417 PROFESSOR: In which direction? 00:34:01.417 --> 00:34:02.250 AUDIENCE: Oh, right. 00:34:02.250 --> 00:34:03.950 PROFESSOR: So if it's converging in this direction, 00:34:03.950 --> 00:34:05.790 it's got to be growing in this direction. 00:34:05.790 --> 00:34:07.590 And that's not going to be normalizable. 00:34:07.590 --> 00:34:14.000 And so as usual with the plane wave, 00:34:14.000 --> 00:34:17.830 we can pick the oscillating solutions that are also 00:34:17.830 --> 00:34:19.760 not normalizable to one, but they're 00:34:19.760 --> 00:34:20.929 delta function normalizable. 00:34:20.929 --> 00:34:21.902 And so that's what we've done here. 00:34:21.902 --> 00:34:23.280 It's exactly the same thing as in 1D. 00:34:23.280 --> 00:34:23.780 Yeah? 00:34:23.780 --> 00:34:27.560 AUDIENCE: So does this mean that any superposition of plane 00:34:27.560 --> 00:34:30.913 waves with wave vector equal in magnitude 00:34:30.913 --> 00:34:33.620 will also be an eigenfunction of the same energy? 00:34:33.620 --> 00:34:34.850 PROFESSOR: Absolutely. 00:34:34.850 --> 00:34:35.380 Awesome. 00:34:35.380 --> 00:34:36.130 Great observation. 00:34:36.130 --> 00:34:37.530 So the observation is this. 00:34:37.530 --> 00:34:41.199 Suppose I take K, I'll call it K1. 00:34:41.199 --> 00:34:48.530 So this is a vector such that K1 squared 00:34:48.530 --> 00:34:55.320 times h bar squared over 2m is equal to E. 00:34:55.320 --> 00:34:59.890 Now there are many vectors k which have the same magnitude, 00:34:59.890 --> 00:35:01.540 but not the same direction. 00:35:01.540 --> 00:35:05.010 So we could also make this equal to h bar squared K2 squared 00:35:05.010 --> 00:35:12.320 vector squared over 2m where K2 is not equal to K1 as a vector, 00:35:12.320 --> 00:35:14.125 although they share the same magnitude. 00:35:16.660 --> 00:35:17.940 So that's interesting. 00:35:17.940 --> 00:35:19.400 So that looks a lot like before. 00:35:19.400 --> 00:35:22.555 In 1D, we saw that if we have k or minus k, 00:35:22.555 --> 00:35:23.680 these have the same energy. 00:35:26.750 --> 00:35:28.000 All right? 00:35:28.000 --> 00:35:32.800 Now if we have any K, K1-- so this is 1D. 00:35:32.800 --> 00:35:44.920 In 3D, if we have K1 and K2 with the same magnitude 00:35:44.920 --> 00:35:51.730 and the same energy, they're degenerate. 00:35:57.010 --> 00:35:58.730 That's interesting. 00:35:58.730 --> 00:35:59.656 Why? 00:35:59.656 --> 00:36:01.962 Why do we have this gigantic degeneracy 00:36:01.962 --> 00:36:04.420 of the energy eigenfunctions for the free particle in three 00:36:04.420 --> 00:36:04.919 dimensions? 00:36:07.450 --> 00:36:07.950 Yeah? 00:36:07.950 --> 00:36:09.600 AUDIENCE: Well, there are an infinite number 00:36:09.600 --> 00:36:11.910 of directions it could be going in with the same momentum. 00:36:11.910 --> 00:36:12.370 PROFESSOR: Awesome. 00:36:12.370 --> 00:36:14.744 So this is clearly true that there are an infinite number 00:36:14.744 --> 00:36:17.330 of momenta with the same magnitude. 00:36:17.330 --> 00:36:19.935 So there are many, many, but why? 00:36:19.935 --> 00:36:22.480 Why do they have the same energy? 00:36:22.480 --> 00:36:24.390 Couldn't they have different energy? 00:36:24.390 --> 00:36:26.784 Couldn't this one have a different energy? 00:36:26.784 --> 00:36:27.700 AUDIENCE: [INAUDIBLE]. 00:36:27.700 --> 00:36:28.740 PROFESSOR: Excellent, it's symmetric. 00:36:28.740 --> 00:36:30.360 The system is invariant under rotation. 00:36:30.360 --> 00:36:32.151 Who are you to say this is the x direction? 00:36:32.151 --> 00:36:33.470 I call it y. 00:36:33.470 --> 00:36:34.910 Right? 00:36:34.910 --> 00:36:38.907 So the system has a symmetry. 00:36:38.907 --> 00:36:39.990 The symmetry is rotations. 00:36:42.560 --> 00:36:44.710 And when we have a symmetry, that 00:36:44.710 --> 00:36:47.070 means there's an operator, a unitary operator, which 00:36:47.070 --> 00:36:50.119 affects that rotation, the rotation operator. 00:36:50.119 --> 00:36:51.660 It's the operator that takes a vector 00:36:51.660 --> 00:36:54.250 and rotates in some particular way. 00:36:54.250 --> 00:36:56.040 We have a unitary operator that's 00:36:56.040 --> 00:36:59.070 a symmetry that means it commutes with the energy 00:36:59.070 --> 00:36:59.807 operator. 00:36:59.807 --> 00:37:01.640 But if it commutes with the energy operator, 00:37:01.640 --> 00:37:03.040 we get can degeneracies. 00:37:03.040 --> 00:37:06.004 We can get states that are different states mapped 00:37:06.004 --> 00:37:08.545 to each other under our unitary operator, under our rotation. 00:37:08.545 --> 00:37:11.749 We get states which are different states manifestly. 00:37:11.749 --> 00:37:13.290 But which have the same energy, which 00:37:13.290 --> 00:37:16.060 are shared energy eigenvalues. 00:37:16.060 --> 00:37:17.160 Cool? 00:37:17.160 --> 00:37:21.210 And this is a really lovely example, both in 1D and 3D, 00:37:21.210 --> 00:37:25.230 that when you have a symmetry, you get degeneracies. 00:37:29.570 --> 00:37:32.107 And when you have a degeneracy, you 00:37:32.107 --> 00:37:33.690 should be very suspicious that there's 00:37:33.690 --> 00:37:39.110 a symmetry hanging around, lurking around ensuring it, OK? 00:37:39.110 --> 00:37:40.770 And this is an important general lesson 00:37:40.770 --> 00:37:45.300 that goes way beyond the specifics of the free particle. 00:37:45.300 --> 00:37:46.210 Yeah? 00:37:46.210 --> 00:37:48.370 AUDIENCE: So that occurs in systems 00:37:48.370 --> 00:37:50.439 with bound states [INAUDIBLE]? 00:37:50.439 --> 00:37:52.730 PROFESSOR: Yeah, it occurs in systems with bound states 00:37:52.730 --> 00:37:54.146 and systems with non bound states. 00:37:54.146 --> 00:37:56.460 So here we're talking about a free particle. 00:37:56.460 --> 00:37:57.600 Certainly not bound. 00:37:57.600 --> 00:37:59.430 And its true. 00:37:59.430 --> 00:38:01.320 For bound states, we'll also see that there 00:38:01.320 --> 00:38:03.612 will be a degeneracy associated with symmetry. 00:38:03.612 --> 00:38:05.570 Now your question is a really, really good one, 00:38:05.570 --> 00:38:07.778 because what we found-- let me rephrase the question. 00:38:07.778 --> 00:38:10.790 The question is, look, in 1D when we had bound states, 00:38:10.790 --> 00:38:12.150 there was no degeneracy. 00:38:12.150 --> 00:38:13.858 Didn't matter what you did to the system. 00:38:13.858 --> 00:38:16.980 When you had bound states, bound states were non degenerate. 00:38:16.980 --> 00:38:19.339 In 3D, we see that when you have a free particle, 00:38:19.339 --> 00:38:20.380 you again get degeneracy. 00:38:20.380 --> 00:38:22.260 In fact, you get a heck of a lot more degeneracy. 00:38:22.260 --> 00:38:23.140 You get a sphere's worth. 00:38:23.140 --> 00:38:25.290 Although actually, that's a sphere in 0 dimension, right? 00:38:25.290 --> 00:38:26.962 It's a 0 dimensional sphere, two points. 00:38:26.962 --> 00:38:29.270 So you get a sphere's worth of degenerate states 00:38:29.270 --> 00:38:29.850 for the free particle. 00:38:29.850 --> 00:38:31.100 Well, what about bound states? 00:38:31.100 --> 00:38:34.110 Are bound states non degenerate still? 00:38:34.110 --> 00:38:35.840 Fantastic question. 00:38:35.840 --> 00:38:36.910 Let's find out. 00:38:36.910 --> 00:38:38.410 So let's do the harmonic oscillator. 00:38:38.410 --> 00:38:40.310 Let's do the 3D harmonic oscillator to check. 00:38:44.990 --> 00:38:48.240 So the 3D harmonic oscillator, the potential is h bar squared. 00:38:48.240 --> 00:38:52.020 And let's pick for fun the rotationally symmetric 3D 00:38:52.020 --> 00:38:53.575 harmonic oscillator. 00:38:53.575 --> 00:39:01.050 m omega squared upon 2 x squared plus y squared plus z squared. 00:39:01.050 --> 00:39:08.845 This could also be written m omega squared upon 2 r squared. 00:39:11.287 --> 00:39:13.120 So I could write it in spherical coordinates 00:39:13.120 --> 00:39:16.310 or in Cartesian coordinates. 00:39:16.310 --> 00:39:18.256 This is really in vector notation. 00:39:18.256 --> 00:39:18.880 Doesn't matter. 00:39:18.880 --> 00:39:19.713 It's the same thing. 00:39:19.713 --> 00:39:22.780 3D harmonic oscillator. 00:39:22.780 --> 00:39:25.250 So what you immediately deduce about the form of the energy 00:39:25.250 --> 00:39:25.875 eigenfunctions? 00:39:28.590 --> 00:39:33.400 Well, we have that E phi E is equal to P squared over 00:39:33.400 --> 00:39:35.830 minus h bar squared over 2m-- so here's 00:39:35.830 --> 00:39:42.430 the energy eigenvalue equation-- times 00:39:42.430 --> 00:39:50.680 dx squared plus dy squared plus dz squared plus m omega squared 00:39:50.680 --> 00:40:03.440 upon 2 times x squared plus y squared plus z squared phi E. 00:40:03.440 --> 00:40:05.610 But I can put this together in a nice way. 00:40:05.610 --> 00:40:11.330 This is minus h bar squared upon 2m dx squared 00:40:11.330 --> 00:40:18.620 plus m omega squared upon 2 x squared plus ditto for y 00:40:18.620 --> 00:40:26.940 plus ditto for z phi E. Yeah? 00:40:26.940 --> 00:40:28.241 Everyone agree? 00:40:28.241 --> 00:40:30.490 So this is differential operator that only involves x. 00:40:30.490 --> 00:40:31.930 Doesn't involve y or z. 00:40:31.930 --> 00:40:33.480 Ditto y, but no x or z. 00:40:33.480 --> 00:40:35.710 And ditto z, but no x or y. 00:40:35.710 --> 00:40:39.597 Aha, this is separable just as before. 00:40:39.597 --> 00:40:41.180 So now we have a nice separable system 00:40:41.180 --> 00:40:44.180 where I want to solve the equations 3 times, once 00:40:44.180 --> 00:40:44.960 for x, y, and z. 00:40:44.960 --> 00:40:46.626 And I'm just going to write it for x, y, 00:40:46.626 --> 00:40:51.530 and z epsilon sub x phi x is equal to minus h bar squared 00:40:51.530 --> 00:40:57.680 over 2m dx squared plus m omega squared upon 2 x 00:40:57.680 --> 00:41:02.000 squared phi sub x. 00:41:02.000 --> 00:41:04.180 And ditto for x, y, z. 00:41:04.180 --> 00:41:08.400 And then phi E is going to be equal to phi 00:41:08.400 --> 00:41:15.600 x of x, phi y of y, and phi z of z, OK? 00:41:15.600 --> 00:41:21.797 Where E is equal to epsilon x plus epsilon y plus epsilon z. 00:41:21.797 --> 00:41:24.380 Note that I've used a slightly different definition of epsilon 00:41:24.380 --> 00:41:25.230 here as before. 00:41:25.230 --> 00:41:28.715 Here it's explicitly the energy eigenvalue. 00:41:28.715 --> 00:41:30.090 So what this is telling is, look, 00:41:30.090 --> 00:41:31.130 we know what this equation is. 00:41:31.130 --> 00:41:32.530 This equation is the same equation 00:41:32.530 --> 00:41:34.280 we ran into in the 1D harmonic oscillator. 00:41:34.280 --> 00:41:38.590 It's exactly the 1D harmonic oscillator problem. 00:41:38.590 --> 00:41:41.140 So the solution to the 3D harmonic oscillator problem 00:41:41.140 --> 00:41:43.010 can be written for energy eigenfunctions, 00:41:43.010 --> 00:41:45.850 can be constructed by taking a harmonic 00:41:45.850 --> 00:41:48.750 oscillator in the x direction-- and we know what those are. 00:41:48.750 --> 00:41:49.750 There's a tower of them. 00:41:49.750 --> 00:41:52.335 There's a ladder of them created by the raising operator 00:41:52.335 --> 00:41:56.160 and lowered by the lowering operator. 00:41:56.160 --> 00:41:59.862 And similarly for y, and similarly for z. 00:41:59.862 --> 00:42:01.820 So I'm going to write this slightly differently 00:42:01.820 --> 00:42:10.540 in the same place as phi sub E is equal to phi sub nx of x. 00:42:10.540 --> 00:42:13.560 The state with energy E sub nx. 00:42:13.560 --> 00:42:16.530 Phi sub ny of y. 00:42:16.530 --> 00:42:18.820 Phi sub nz of z. 00:42:18.820 --> 00:42:21.030 Where these are all the single dimension, 00:42:21.030 --> 00:42:23.970 one dimensional harmonic oscillator eigenfunctions. 00:42:23.970 --> 00:42:34.658 And E is equal to Ex E sub nx plus E sub ny plus E sub nz. 00:42:34.658 --> 00:42:35.158 OK? 00:42:43.380 --> 00:42:46.845 So let's look at the consequences of this. 00:42:46.845 --> 00:42:48.720 So first off, does anyone have any questions? 00:42:48.720 --> 00:42:50.870 I went through the kind of quick. 00:42:50.870 --> 00:42:52.485 Any questions about that? 00:42:52.485 --> 00:42:54.610 If you're not comfortable with separable equations, 00:42:54.610 --> 00:42:56.970 you need to become super comfortable with separable 00:42:56.970 --> 00:42:58.084 equations. 00:42:58.084 --> 00:42:59.250 It's an important technique. 00:42:59.250 --> 00:43:01.450 We're going to use it a lot. 00:43:01.450 --> 00:43:04.099 So the upshot is that if I write-- in fact, 00:43:04.099 --> 00:43:06.390 I'm going to write that in slightly different notation. 00:43:06.390 --> 00:43:12.410 If I write phi E is equal to phi n 00:43:12.410 --> 00:43:22.474 of x, where it's the-- and phi l of y and phi m of z-- actually 00:43:22.474 --> 00:43:23.515 that's a stupid ordering. 00:43:23.515 --> 00:43:24.470 Let's try that again. 00:43:28.830 --> 00:43:30.670 l, m, n. 00:43:30.670 --> 00:43:32.350 That is the alphabetical ordering. 00:43:32.350 --> 00:43:34.500 With the energy is now equal to-- we 00:43:34.500 --> 00:43:37.610 know the energy is of a state with of the harmonic 00:43:37.610 --> 00:43:39.970 oscillator with excitation number l. 00:43:39.970 --> 00:43:47.480 It's h bar omega, the overall omega, times l plus 1/2. 00:43:47.480 --> 00:43:50.840 But from this guy it's got excitation number m, 00:43:50.840 --> 00:43:56.140 so energy of that is h bar omega m plus 1/2 plus m. 00:43:56.140 --> 00:43:59.010 And so now that's plus 1. 00:43:59.010 --> 00:44:00.850 And for this guy similarly, h bar omega 00:44:00.850 --> 00:44:04.475 n plus n and now plus 1/2 again plus 3/2. 00:44:10.380 --> 00:44:12.781 This is a basis of solutions of the energy eigenfunction 00:44:12.781 --> 00:44:13.280 equations. 00:44:13.280 --> 00:44:20.520 These are the solutions of the energy eigenfunctions 00:44:20.520 --> 00:44:24.104 for the 3D harmonic oscillator. 00:44:24.104 --> 00:44:25.270 And now here's the question. 00:44:25.270 --> 00:44:27.460 The question that was asked is, look, 00:44:27.460 --> 00:44:31.180 there are no degeneracies in bound states in 1D. 00:44:31.180 --> 00:44:33.970 Here we have manifestly a 3D bound state system. 00:44:33.970 --> 00:44:35.299 Are there degeneracies? 00:44:35.299 --> 00:44:35.882 AUDIENCE: Yes. 00:44:35.882 --> 00:44:37.610 PROFESSOR: Yeah, obviously, right? 00:44:37.610 --> 00:44:40.205 So for example, if I call l1 this 0 and this 0. 00:44:40.205 --> 00:44:44.780 Or if I call this 010 or 001, those all have the same energy. 00:44:44.780 --> 00:44:47.610 They have the energy h bar omega 0 times 1 plus 3/2 or 5/2. 00:44:50.320 --> 00:44:53.970 So let's look at this in a little more detail. 00:44:53.970 --> 00:44:57.394 Let's write a list of the degeneracies 00:44:57.394 --> 00:44:58.560 as a function of the energy. 00:45:03.120 --> 00:45:06.660 So at energy what's the ground state energy 00:45:06.660 --> 00:45:09.520 for the 3D harmonic oscillator? 00:45:09.520 --> 00:45:10.466 3 halves h bar omega. 00:45:10.466 --> 00:45:11.840 It's three times the ground state 00:45:11.840 --> 00:45:13.830 energy for the single 1D harmonic oscillator. 00:45:13.830 --> 00:45:15.990 So 3/2 h bar omega. 00:45:19.420 --> 00:45:19.974 Yeah? 00:45:19.974 --> 00:45:20.890 AUDIENCE: [INAUDIBLE]. 00:45:23.830 --> 00:45:26.150 PROFESSOR: Good, the way we arrived that this was we 00:45:26.150 --> 00:45:31.300 found that the energy-- so the energy operator acting 00:45:31.300 --> 00:45:34.040 on the 3D wave function is what I 00:45:34.040 --> 00:45:36.870 get by taking the energy operator in 1D 00:45:36.870 --> 00:45:41.186 and acting on the wave function, and the energy operator in y 00:45:41.186 --> 00:45:43.685 acting on the wave function, the energy operator in z acting 00:45:43.685 --> 00:45:45.000 on the wave function, where the energy operator for each 00:45:45.000 --> 00:45:46.660 of those is the 1D harmonic oscillator 00:45:46.660 --> 00:45:48.560 with the same frequency. 00:45:48.560 --> 00:45:49.150 OK? 00:45:49.150 --> 00:45:50.120 And then I separated. 00:45:50.120 --> 00:45:51.600 I said look, let the wave function, 00:45:51.600 --> 00:45:55.400 the 3D wave function, since I know this is separable 00:45:55.400 --> 00:45:58.560 and each separated part of the wave function 00:45:58.560 --> 00:46:00.990 satisfies the 1D harmonic oscillator equation, 00:46:00.990 --> 00:46:04.380 I know what the eigenfunctions of the 1D harmonic oscillator 00:46:04.380 --> 00:46:05.790 energy eigenvalue problem are. 00:46:05.790 --> 00:46:08.780 They are the phi n's. 00:46:08.780 --> 00:46:10.840 And so I can just take my 3D wave function. 00:46:10.840 --> 00:46:13.510 I can say the 3D wave function is going to be the separated 00:46:13.510 --> 00:46:15.180 form, the product of 1D wave function 00:46:15.180 --> 00:46:19.610 in x, a 1D wave function in y, and 1D wave function in z. 00:46:19.610 --> 00:46:20.240 Cool? 00:46:20.240 --> 00:46:22.220 So now let's look back at what let's think 00:46:22.220 --> 00:46:23.740 about what happens when I take the energy operator 00:46:23.740 --> 00:46:25.270 and I act on that [INAUDIBLE]. 00:46:25.270 --> 00:46:27.160 When I take the 3D energy operator, which 00:46:27.160 --> 00:46:29.209 is the sum of the three 1D energy operators, 00:46:29.209 --> 00:46:30.750 harmonic oscillator energy operators. 00:46:30.750 --> 00:46:32.541 When thinking I'm going to act on this guy, 00:46:32.541 --> 00:46:35.150 the first one, which only knows about the x direction, 00:46:35.150 --> 00:46:38.060 sees the y and z parts as constants. 00:46:38.060 --> 00:46:40.210 And it's a phi x, and what does it give us back? 00:46:40.210 --> 00:46:44.440 E on phi x is just h bar omega n plus one half. 00:46:44.440 --> 00:46:45.654 Ditto for this guy. 00:46:45.654 --> 00:46:47.070 And then the energy operator in 3D 00:46:47.070 --> 00:46:49.640 is the sum of the three 1D energy operators. 00:46:49.640 --> 00:46:52.240 So that tells us the energy is the sum of the three energies. 00:46:52.240 --> 00:46:53.840 Is that cool? 00:46:53.840 --> 00:46:54.460 OK good. 00:46:54.460 --> 00:46:55.370 Other questions? 00:46:55.370 --> 00:46:56.193 Yeah? 00:46:56.193 --> 00:46:58.508 AUDIENCE: Is the number of degeneracies essentially 00:46:58.508 --> 00:47:01.760 [INAUDIBLE] of number theory. 00:47:01.760 --> 00:47:03.230 PROFESSOR: Ask me that after class. 00:47:07.120 --> 00:47:08.810 So let's look at the degeneracies 00:47:08.810 --> 00:47:11.530 as a function of the energy. 00:47:11.530 --> 00:47:15.160 So at the lowest possible energy, 3/2 omega, what states 00:47:15.160 --> 00:47:17.584 can I possibly have? 00:47:17.584 --> 00:47:20.250 I'm going to label the states by the three numbers, l, m, and n. 00:47:20.250 --> 00:47:22.670 So this is just the ground state 0 0 0. 00:47:22.670 --> 00:47:24.740 So is that degenerate? 00:47:24.740 --> 00:47:27.720 No, because there's just the one state. 00:47:27.720 --> 00:47:29.040 What about at the next level? 00:47:29.040 --> 00:47:30.727 What's the next allowed energy? 00:47:30.727 --> 00:47:31.310 AUDIENCE: 5/2. 00:47:31.310 --> 00:47:32.340 PROFESSOR: 5/2. 00:47:32.340 --> 00:47:33.660 OK, 5/2. 00:47:33.660 --> 00:47:35.660 So at 5/2 what states do we have? 00:47:35.660 --> 00:47:36.620 Well, we have 1 0 0. 00:47:36.620 --> 00:47:38.520 But we also have 0, 1, 0. 00:47:38.520 --> 00:47:41.410 And we also 0 0 1. 00:47:41.410 --> 00:47:44.044 Aha, this is looking good. 00:47:44.044 --> 00:47:45.710 What does this correspond to physically? 00:47:45.710 --> 00:47:48.860 This says you have excitation, so you've 00:47:48.860 --> 00:47:51.110 got a node in the x direction. 00:47:51.110 --> 00:47:55.220 But your Gaussian in the y and z directions. 00:47:55.220 --> 00:48:00.340 This one says your Gaussian in the x direction. 00:48:00.340 --> 00:48:02.590 You have a node in the y direction as a function of y, 00:48:02.590 --> 00:48:05.280 because it's phi 1 of y. 00:48:05.280 --> 00:48:07.440 And you're Gaussian in the z directions. 00:48:07.440 --> 00:48:11.310 And this one says you have 1 excitation in the z direction. 00:48:11.310 --> 00:48:15.290 So they sound sort of rotated from each other. 00:48:15.290 --> 00:48:18.500 That sounds promising. 00:48:18.500 --> 00:48:20.740 But in particular, what we just discovered sort of 00:48:20.740 --> 00:48:22.370 by construction is that there can 00:48:22.370 --> 00:48:35.580 be degeneracies among bound states in 3D. 00:48:35.580 --> 00:48:38.310 This was not possible in 1D, but it 00:48:38.310 --> 00:48:40.950 is possible in 3D, which is cool. 00:48:40.950 --> 00:48:42.360 But we've actually learned more. 00:48:42.360 --> 00:48:43.570 What's the form of the degeneracies? 00:48:43.570 --> 00:48:46.028 So here it looks like they're just rotations of each other. 00:48:46.028 --> 00:48:48.460 You call this x, I call it y, someone else calls it z. 00:48:48.460 --> 00:48:50.510 These can't possibly look different functions 00:48:50.510 --> 00:48:53.330 because they're just rotations of each other. 00:48:53.330 --> 00:48:55.017 However, things get a little more messy 00:48:55.017 --> 00:48:56.850 when you write, well, what's the next level? 00:48:56.850 --> 00:48:59.100 What's the next energy after 5/2 h bar omega? 00:48:59.100 --> 00:49:00.750 7/2 h bar omega, exactly. 00:49:00.750 --> 00:49:01.354 7/2. 00:49:01.354 --> 00:49:02.270 And that's not so bad. 00:49:02.270 --> 00:49:08.790 So for that one, we get 2 0 0, 0 2 0, 0 0 2. 00:49:08.790 --> 00:49:09.610 But is that it? 00:49:09.610 --> 00:49:10.370 AUDIENCE: No. 00:49:10.370 --> 00:49:12.410 PROFESSOR: What else do we get? 00:49:12.410 --> 00:49:17.560 1 1 0, 0 1 1, 1 0 1. 00:49:17.560 --> 00:49:19.200 So first off, let's look at the number. 00:49:19.200 --> 00:49:21.270 The degeneracy number here-- I'll call this d sub 00:49:21.270 --> 00:49:26.250 0-- the degeneracy of the ground state is 1, OK? 00:49:26.250 --> 00:49:27.790 The degeneracy-- and in fact, I'm 00:49:27.790 --> 00:49:31.530 going to write this as a table-- the degeneracy as level n. 00:49:31.530 --> 00:49:35.150 So for d0 is equal to 1. 00:49:35.150 --> 00:49:37.990 d1 is equal to 3. 00:49:37.990 --> 00:49:44.150 And d2 is equal to 6. 00:49:44.150 --> 00:49:46.590 Now it's less clear here what's going on, 00:49:46.590 --> 00:49:50.650 because is this just this guy relabelled? 00:49:50.650 --> 00:49:52.100 No. 00:49:52.100 --> 00:49:53.870 So this is weird, because we already 00:49:53.870 --> 00:50:00.280 said that the reason we expect that there might be degeneracy, 00:50:00.280 --> 00:50:02.369 is because of rotational symmetry. 00:50:02.369 --> 00:50:03.910 The system is rotationally invariant. 00:50:03.910 --> 00:50:06.326 The potential, which is the harmonic oscillator potential, 00:50:06.326 --> 00:50:15.600 doesn't care in what direction the radial displacement 00:50:15.600 --> 00:50:17.380 vector is pointing. 00:50:17.380 --> 00:50:18.630 It's rotationally symmetrical. 00:50:18.630 --> 00:50:21.180 When we have symmetry-- on general grounds, when 00:50:21.180 --> 00:50:23.350 we have a symmetry, we expect to have degeneracies. 00:50:26.422 --> 00:50:28.130 But this are kind of weird, because these 00:50:28.130 --> 00:50:30.129 don't seem to be simple rotations of each other, 00:50:30.129 --> 00:50:32.410 and yet they're degenerate. 00:50:32.410 --> 00:50:35.200 So what's up with that? 00:50:35.200 --> 00:50:35.711 Question? 00:50:35.711 --> 00:50:36.210 Yeah? 00:50:36.210 --> 00:50:40.230 AUDIENCE: [INAUDIBLE] Gaussian in certain directions? 00:50:40.230 --> 00:50:41.350 PROFESSOR: Yeah, sure OK. 00:50:41.350 --> 00:50:44.420 So let me just explain what this notation means again. 00:50:44.420 --> 00:50:50.260 So by 1 0 0, what I mean is that the number l is equal to 1, 00:50:50.260 --> 00:50:52.095 the number m is equal to 0, n is equal to 0. 00:50:52.095 --> 00:50:53.470 That means that the wave function 00:50:53.470 --> 00:51:03.280 phi 3D is equal to phi 1 of x, phi 0 of y, phi 0 of z. 00:51:03.280 --> 00:51:05.230 But what's phi 0 of z? 00:51:05.230 --> 00:51:07.350 What's a Gaussian in the z direction? 00:51:07.350 --> 00:51:08.790 Phi 0 of y. 00:51:08.790 --> 00:51:10.550 That's the ground state in the y direction 00:51:10.550 --> 00:51:11.260 of the harmonic oscillator. 00:51:11.260 --> 00:51:12.710 It's Gaussian in the y direction. 00:51:12.710 --> 00:51:15.255 If I wanted x, that's not the ground state. 00:51:15.255 --> 00:51:16.350 That's the excited state. 00:51:16.350 --> 00:51:19.450 And in particular, sort of being a Gaussian it goes through 0. 00:51:19.450 --> 00:51:20.900 It has a node. 00:51:20.900 --> 00:51:24.250 So this wave function is not rotationally invariant. 00:51:24.250 --> 00:51:26.890 It as a node in the x direction, but no nodes and y and z 00:51:26.890 --> 00:51:29.000 direction. 00:51:29.000 --> 00:51:30.750 And similarly for these guys. 00:51:30.750 --> 00:51:32.280 Did that answer your questions? 00:51:32.280 --> 00:51:33.510 Great. 00:51:33.510 --> 00:51:38.596 OK, so we have these degeneracies, 00:51:38.596 --> 00:51:40.160 and they beg an explanation. 00:51:40.160 --> 00:51:41.650 And if you look at the next level, 00:51:41.650 --> 00:51:43.930 it turns out that d3-- and you can do this quickly 00:51:43.930 --> 00:51:48.300 on a scrap of paper-- d3 is 10, OK? 00:51:48.300 --> 00:51:49.440 And they go on. 00:51:49.440 --> 00:51:52.802 And if you keep writing this list out, 00:51:52.802 --> 00:51:54.510 I guess it goes up-- what's the next one? 00:51:54.510 --> 00:51:57.260 15. 00:51:57.260 --> 00:51:58.870 21, yeah. 00:51:58.870 --> 00:52:01.260 So this has a simple mathematical structure, 00:52:01.260 --> 00:52:03.150 and you can very quickly convince yourself 00:52:03.150 --> 00:52:05.740 of the form of this degeneracy. 00:52:05.740 --> 00:52:09.680 dn is n n plus 1 over 2. 00:52:09.680 --> 00:52:12.090 So let's just make sure that works. 00:52:12.090 --> 00:52:15.940 1 1 plus 1 over 2. 00:52:15.940 --> 00:52:17.606 Sorry I should really call this 1. 00:52:20.868 --> 00:52:25.740 n plus 1, n plus 2 if I count from 0. 00:52:25.740 --> 00:52:30.620 So for 0, this is going to give us 1 times 2 over 2. 00:52:30.620 --> 00:52:31.190 That's 1. 00:52:31.190 --> 00:52:31.940 That works. 00:52:31.940 --> 00:52:35.620 So for 1 that gives 2 times 3 over 2, 00:52:35.620 --> 00:52:38.594 which is 3, and so on and so forth. 00:52:38.594 --> 00:52:39.760 So where did this come from? 00:52:39.760 --> 00:52:42.140 This is something we're going to have to answer. 00:52:42.140 --> 00:52:43.950 Why that degeneracy? 00:52:43.950 --> 00:52:45.420 That seems important. 00:52:45.420 --> 00:52:46.630 Why is it that number? 00:52:46.630 --> 00:52:48.210 Why do we have that much degeneracy? 00:52:48.210 --> 00:52:50.460 But the thing I really want to emphasize at this point 00:52:50.460 --> 00:52:55.050 is that there's an absolutely essential deep connection 00:52:55.050 --> 00:52:57.770 between symmetries and degeneracies. 00:52:57.770 --> 00:53:00.140 If we didn't have symmetry, we wouldn't have degeneracy, 00:53:00.140 --> 00:53:03.870 and we can see that very easily here. 00:53:03.870 --> 00:53:07.640 Imagine that this potential was not exactly symmetric. 00:53:07.640 --> 00:53:10.530 Imagine we made it slightly different by adding 00:53:10.530 --> 00:53:17.200 a little bit of extra frequency to z direction. 00:53:17.200 --> 00:53:19.300 Make the z frequency slightly different. 00:53:19.300 --> 00:53:24.210 Plus m omega tilde squared upon 2 z 00:53:24.210 --> 00:53:29.310 squared, where omega tilde is not equal to omega 0. 00:53:29.310 --> 00:53:30.540 OK? 00:53:30.540 --> 00:53:33.230 The system is still separable, but this guy 00:53:33.230 --> 00:53:34.982 has frequency omega 0. 00:53:34.982 --> 00:53:36.970 The x part has omega 0. 00:53:36.970 --> 00:53:38.040 This has omega 0. 00:53:38.040 --> 00:53:41.205 But this has omega tilde. 00:53:41.205 --> 00:53:42.690 OK? 00:53:42.690 --> 00:53:45.390 And so exactly the same argument is going to go through, 00:53:45.390 --> 00:53:48.530 but the energy now is going to have a different form. 00:53:48.530 --> 00:53:53.500 The energy is going to have h bar omega-- h bar omega 0 times 00:53:53.500 --> 00:53:57.060 l plus m plus 1. 00:53:57.060 --> 00:53:59.890 But from the z part it's going to have 00:53:59.890 --> 00:54:05.010 plus h bar omega tilde times n plus 1/2. 00:54:09.861 --> 00:54:11.360 And now these degeneracies are going 00:54:11.360 --> 00:54:13.420 to be broken, because this state will not 00:54:13.420 --> 00:54:16.430 have the same energy as these two. 00:54:16.430 --> 00:54:18.540 Everyone see that? 00:54:18.540 --> 00:54:20.810 When you have symmetry, you get to degeneracy. 00:54:20.810 --> 00:54:23.740 When you don't have a symmetry, you do not get degeneracy. 00:54:23.740 --> 00:54:25.770 This connection is extremely important, 00:54:25.770 --> 00:54:27.550 because it allows you to do two things. 00:54:27.550 --> 00:54:29.504 It allows you to first not solve things 00:54:29.504 --> 00:54:30.670 you don't need to solve for. 00:54:30.670 --> 00:54:33.253 If you know there's a symmetry, solve it once and then compute 00:54:33.253 --> 00:54:34.820 the degeneracy and you're done. 00:54:34.820 --> 00:54:36.620 On the other hand, if you have a system 00:54:36.620 --> 00:54:40.040 and you see just manifestly you measure the energies, 00:54:40.040 --> 00:54:42.990 and you measure that the energies are degenerate, 00:54:42.990 --> 00:54:46.279 you know there's a symmetry protecting those degeneracies. 00:54:46.279 --> 00:54:47.820 You actually can't be 100% confident, 00:54:47.820 --> 00:54:49.695 because I didn't prove that these are related 00:54:49.695 --> 00:54:52.760 to each other, but you should be highly suspicious. 00:54:52.760 --> 00:54:55.130 And in fact, this is an incredibly powerful tool 00:54:55.130 --> 00:54:57.140 in building models of physical systems. 00:54:57.140 --> 00:54:59.696 If you see a degeneracy or an approximate degeneracy, 00:54:59.696 --> 00:55:02.070 you can exploit that to learn things about the underlying 00:55:02.070 --> 00:55:02.680 system. 00:55:02.680 --> 00:55:03.263 Yeah? 00:55:03.263 --> 00:55:05.096 AUDIENCE: So we just add the different omega 00:55:05.096 --> 00:55:08.180 to each omega [INAUDIBLE] number there 00:55:08.180 --> 00:55:10.430 is still a possibility to get a degeneracy. 00:55:10.430 --> 00:55:12.320 PROFESSOR: Exactly. 00:55:12.320 --> 00:55:20.120 So it's possible for these omegas to be specially tuned 00:55:20.120 --> 00:55:24.010 so that rational combinations of them give you a degeneracy. 00:55:24.010 --> 00:55:26.330 But it's extraordinarily unlikely for that 00:55:26.330 --> 00:55:28.470 to happen accidentally, because they 00:55:28.470 --> 00:55:30.700 have to be rationally related to each other, 00:55:30.700 --> 00:55:34.220 and the rationals are a set of measures 0 in the reels. 00:55:34.220 --> 00:55:36.560 So if you just randomly pick some frequencies, 00:55:36.560 --> 00:55:37.380 they'll be totally incommensurate, 00:55:37.380 --> 00:55:38.900 and you'll never get a degeneracy. 00:55:38.900 --> 00:55:41.750 So it is possible to have an accidental degeneracy. 00:55:41.750 --> 00:55:44.240 Whoops, just pure coincidence. 00:55:44.240 --> 00:55:45.960 But it's extraordinarily unlikely. 00:55:45.960 --> 00:55:48.210 And as you'll see when you get to perturbation theory, 00:55:48.210 --> 00:55:50.110 it's more than unlikely. 00:55:50.110 --> 00:55:51.682 It's almost impossible. 00:55:51.682 --> 00:55:53.890 So it's very rare that you get accidental degeneracy. 00:55:53.890 --> 00:55:57.000 It happens, but it's rare. 00:55:57.000 --> 00:55:58.960 Other questions? 00:55:58.960 --> 00:56:00.950 OK, nothing? 00:56:00.950 --> 00:56:03.170 OK so here we're now going to launch into-- 00:56:03.170 --> 00:56:05.740 so this leads us into a very simple question. 00:56:05.740 --> 00:56:07.620 At the end of the day, the degeneracies 00:56:07.620 --> 00:56:11.420 that we see for the 3D free particle, which 00:56:11.420 --> 00:56:13.610 is a whole sphere's worth of degeneracy, 00:56:13.610 --> 00:56:16.130 and the degeneracy we see for the 3D harmonic oscillator, 00:56:16.130 --> 00:56:18.496 the bound states, which is discrete, 00:56:18.496 --> 00:56:19.870 but with more and more degeneracy 00:56:19.870 --> 00:56:22.610 the higher and higher energy you go. 00:56:22.610 --> 00:56:24.640 Those we're blaming, at the moment, 00:56:24.640 --> 00:56:26.540 on a symmetry, on rotational symmetry, 00:56:26.540 --> 00:56:29.050 rotational invariance. 00:56:29.050 --> 00:56:32.240 So it seems wise to study rotations, 00:56:32.240 --> 00:56:35.220 to study rotational invariance and rotational transformations 00:56:35.220 --> 00:56:36.110 in the first place. 00:56:36.110 --> 00:56:38.890 In the first part of the course, in 1D quantum mechanics, 00:56:38.890 --> 00:56:42.000 we got an awful lot of juice out of studying translations. 00:56:42.000 --> 00:56:44.354 And the generator of translations was momentum. 00:56:44.354 --> 00:56:46.020 So we're going to do the same thing now. 00:56:46.020 --> 00:56:48.370 We're going to study rotations and the generators 00:56:48.370 --> 00:56:50.700 of rotations, which are the angular momentum operators, 00:56:50.700 --> 00:56:52.824 and that's going to occupy us for the rest of today 00:56:52.824 --> 00:56:53.430 and Thursday. 00:56:53.430 --> 00:56:53.930 Yeah? 00:56:53.930 --> 00:56:56.720 AUDIENCE: So your rotational symmetry 00:56:56.720 --> 00:57:03.690 will explain a factor of three in your degeneracy, right? 00:57:03.690 --> 00:57:07.500 But what's the symmetry that explains the way this grows. 00:57:07.500 --> 00:57:10.150 Because this very clearly appears 00:57:10.150 --> 00:57:12.835 there's 1 up to a factor of three. 00:57:12.835 --> 00:57:14.626 And then there's 2 up to a factor of three. 00:57:14.626 --> 00:57:18.300 And there's even more that's not even a multiple of three. 00:57:18.300 --> 00:57:19.895 PROFESSOR: Right, actually so here's 00:57:19.895 --> 00:57:21.270 a very tempting bit of intuition. 00:57:21.270 --> 00:57:23.811 Very tempting bit of intuition is going to say the following. 00:57:23.811 --> 00:57:26.010 Look, rotational invariance, there's x, there's y, 00:57:26.010 --> 00:57:26.220 and there's z. 00:57:26.220 --> 00:57:28.386 It's going to explain rotations amongst those three. 00:57:28.386 --> 00:57:31.130 So that could only possibly give you a factor of three. 00:57:31.130 --> 00:57:33.415 But it's important to keep in mind 00:57:33.415 --> 00:57:35.100 that that's not correct intuition. 00:57:35.100 --> 00:57:36.580 It's tempting intuition, but it's not correct. 00:57:36.580 --> 00:57:38.371 And an easy way to see that its not correct 00:57:38.371 --> 00:57:41.540 is that for the free particle, there 00:57:41.540 --> 00:57:44.340 is a continuum, a whole sphere's worth 00:57:44.340 --> 00:57:46.600 of degenerate states in any energy. 00:57:46.600 --> 00:57:49.550 And all of those are related to each other by simple rotation 00:57:49.550 --> 00:57:52.430 of the k vector, of the wave vector, right? 00:57:52.430 --> 00:57:54.850 So the rotational symmetry is giving us 00:57:54.850 --> 00:57:56.400 a lot more than a factor of three. 00:57:56.400 --> 00:57:58.150 And in fact, as we'll see, it's going 00:57:58.150 --> 00:58:00.360 to explain exactly the n plus 1 n plus 2 over 2. 00:58:02.890 --> 00:58:05.450 OK, so with that motivation let's 00:58:05.450 --> 00:58:07.335 start talking about angular momentum. 00:58:11.900 --> 00:58:19.790 So I found this topic to be not obviously 00:58:19.790 --> 00:58:23.500 the most powerful or interesting thing in the world when I first 00:58:23.500 --> 00:58:24.150 studied it. 00:58:24.150 --> 00:58:25.697 And my professor was like no, no, no. 00:58:25.697 --> 00:58:26.780 This is the deepest thing. 00:58:26.780 --> 00:58:28.412 And recently I had a fun conversation with one 00:58:28.412 --> 00:58:29.786 of my colleagues, Frank Wilczeck, 00:58:29.786 --> 00:58:32.530 who said yeah, in intro to quantum mechanics 00:58:32.530 --> 00:58:35.310 the single most interesting thing is the angular momentum 00:58:35.310 --> 00:58:36.930 and the addition of angular momentum. 00:58:36.930 --> 00:58:39.890 And something has happened to me in the intervening 20 years 00:58:39.890 --> 00:58:41.470 that I totally agree with him. 00:58:41.470 --> 00:58:46.020 So I will attempt to convey to you the awesomeness of this. 00:58:46.020 --> 00:58:48.050 But you have to buy in a little. 00:58:48.050 --> 00:58:50.740 So work with me in the math at the beginning of this, 00:58:50.740 --> 00:58:53.380 and it has a great payoff. 00:58:53.380 --> 00:58:55.927 OK so the question is, what is the operator. 00:58:55.927 --> 00:58:58.010 So we're going to talk about angular momentum now. 00:59:01.770 --> 00:59:05.510 And I want to start with the following question. 00:59:05.510 --> 00:59:07.170 In the same sense as we started out 00:59:07.170 --> 00:59:08.640 by asking what represents position 00:59:08.640 --> 00:59:11.970 and momentum, linear momentum, in quantum mechanics, what 00:59:11.970 --> 00:59:17.020 represents what operator by our first, second, or third 00:59:17.020 --> 00:59:19.020 postulate-- I don't even remember the order now. 00:59:19.020 --> 00:59:26.060 What operator represents angular momentum in quantum mechanics? 00:59:30.980 --> 00:59:33.370 And let's start by remembering what angular momentum is 00:59:33.370 --> 00:59:35.550 in classical mechanics. 00:59:35.550 --> 00:59:41.070 So L in classical mechanics is r cross p. 00:59:41.070 --> 00:59:43.702 In classical mechanics. 00:59:43.702 --> 00:59:44.660 So let's just try this. 00:59:44.660 --> 00:59:45.950 Let's construct that operator. 00:59:45.950 --> 00:59:48.450 This is not the world's most beautiful way of deriving this, 00:59:48.450 --> 00:59:50.880 but let's just write down natural guess. 00:59:50.880 --> 00:59:53.320 For in quantum mechanics what's the operator we want? 00:59:53.320 --> 00:59:56.430 Well, we want a vectors worth of operators, 00:59:56.430 --> 00:59:58.267 because angular momentum is a vector. 00:59:58.267 --> 01:00:00.100 It's a vector of operators, three operators. 01:00:00.100 --> 01:00:06.280 And I'm going to write these as r vector the operators x, y, 01:00:06.280 --> 01:00:12.920 z cross p the vector of momentum operators. 01:00:12.920 --> 01:00:14.670 So at this point, you should really worry, 01:00:14.670 --> 01:00:18.050 because do r and p commute? 01:00:18.050 --> 01:00:19.770 Not so much, right? 01:00:19.770 --> 01:00:22.150 However, the situation is better than it first appears. 01:00:22.150 --> 01:00:24.840 Let's write this out in terms of components. 01:00:24.840 --> 01:00:27.387 So this is in components. 01:00:27.387 --> 01:00:28.970 And I'm going to work, for the moment, 01:00:28.970 --> 01:00:30.560 in Cartesian coordinates. 01:00:30.560 --> 01:00:34.120 So Lx is equal to? 01:00:34.120 --> 01:00:37.400 Lx is equal to? 01:00:37.400 --> 01:00:38.840 You all took mechanics. 01:00:38.840 --> 01:00:41.332 Lx is equal to? 01:00:41.332 --> 01:00:43.575 AUDIENCE: [INAUDIBLE]. 01:00:43.575 --> 01:00:44.450 PROFESSOR: Thank you. 01:00:44.450 --> 01:00:47.920 YPz minus ZPy. 01:00:47.920 --> 01:00:50.960 And that's the curl, the x component of the curl. 01:00:50.960 --> 01:00:55.650 And similarly, the x component-- so the way to remember this 01:00:55.650 --> 01:00:56.740 is that its cyclic. 01:00:56.740 --> 01:00:58.644 x, y, z. 01:00:58.644 --> 01:01:00.470 y, z, x. 01:01:00.470 --> 01:01:02.400 So z, p, y. 01:01:05.450 --> 01:01:10.310 PX minus XPz. 01:01:10.310 --> 01:01:14.030 And then we have z XPy minus YPx. 01:01:18.210 --> 01:01:21.200 So we were worried here about maybe an ordering problem. 01:01:21.200 --> 01:01:24.930 Is there an ordering problem here? 01:01:24.930 --> 01:01:27.923 Does it matter if I write YPz or PZy? 01:01:27.923 --> 01:01:28.465 AUDIENCE: No. 01:01:28.465 --> 01:01:30.631 PROFESSOR: No, because they commute with each other. 01:01:30.631 --> 01:01:32.570 PZ is momentum for the z-coordinate, not 01:01:32.570 --> 01:01:34.700 the y-coordinate, and they commute with each other. 01:01:34.700 --> 01:01:36.200 So there's no ambiguity. 01:01:36.200 --> 01:01:37.480 It's perfectly well defined. 01:01:37.480 --> 01:01:39.063 So we're just going to take this to be 01:01:39.063 --> 01:01:41.430 the definition of the components of the angular momentum 01:01:41.430 --> 01:01:44.710 operator Lx, Ly, and Lz. 01:01:50.460 --> 01:01:53.180 And just for fun, I want to write this out. 01:01:53.180 --> 01:01:56.870 So because we know that Px, Py, and Pz can be expressed 01:01:56.870 --> 01:01:59.600 in terms of derivatives or differential operators, 01:01:59.600 --> 01:02:01.970 we can write the same operator in Cartesian coordinates 01:02:01.970 --> 01:02:02.960 in the following way. 01:02:02.960 --> 01:02:06.880 So clearly we could write this as Y d dx i 01:02:06.880 --> 01:02:09.120 upon h bar-- or sorry, h bar upon i. 01:02:09.120 --> 01:02:12.232 And z h bar upon i d dy. 01:02:12.232 --> 01:02:14.190 So we could write that in Cartesian coordinates 01:02:14.190 --> 01:02:16.320 as a differential operator. 01:02:16.320 --> 01:02:19.350 But we can also write this in spherical coordinates. 01:02:19.350 --> 01:02:21.300 I'm just going to take a quick side note just 01:02:21.300 --> 01:02:22.350 to write down what it is. 01:02:22.350 --> 01:02:24.391 If we did this spherical coordinates, 01:02:24.391 --> 01:02:26.016 it's particularly convenient to write-- 01:02:26.016 --> 01:02:27.980 let me just write Lz for the moment. 01:02:27.980 --> 01:02:31.890 This is equal to minus i h bar derivative with respect to phi, 01:02:31.890 --> 01:02:33.430 where the coordinates, [INAUDIBLE] 01:02:33.430 --> 01:02:36.010 coordinates and spherical coordinates in this class 01:02:36.010 --> 01:02:38.670 is theta is going to be the angle down 01:02:38.670 --> 01:02:40.550 from the vertical axis, from the z-axis. 01:02:40.550 --> 01:02:43.370 And phi is going to be the angle of a period 2pi 01:02:43.370 --> 01:02:44.860 that goes around the equator, OK? 01:02:44.860 --> 01:02:45.860 Just to give you a name. 01:02:45.860 --> 01:02:47.510 This is typically what physicists call them. 01:02:47.510 --> 01:02:49.676 This is typically not what mathematicians call them. 01:02:49.676 --> 01:02:51.180 This leads to enormous confusion. 01:02:51.180 --> 01:02:52.520 I apologize for my field. 01:02:52.520 --> 01:02:55.420 So here it is. 01:02:55.420 --> 01:02:57.450 I can also construct the operator associated 01:02:57.450 --> 01:02:59.150 with the square of the momentum. 01:02:59.150 --> 01:03:01.640 And why would we care about the square of the momentum? 01:03:01.640 --> 01:03:03.848 Well, that's what shows up in the Hamiltonian, that's 01:03:03.848 --> 01:03:05.020 what shows up in the energy. 01:03:05.020 --> 01:03:07.603 So I can construct the operator for the square of the momentum 01:03:07.603 --> 01:03:10.280 and write it, and it takes a surprisingly simple form. 01:03:10.280 --> 01:03:12.780 When I see surprisingly simple, you might disagree with me, 01:03:12.780 --> 01:03:14.750 but if you actually do the derivation of this, 01:03:14.750 --> 01:03:16.190 it's much worse in between. 01:03:16.190 --> 01:03:21.340 1 over sine theta d theta sine theta 01:03:21.340 --> 01:03:28.020 d theta plus 1 over sine squared theta d phi. 01:03:30.620 --> 01:03:31.770 Couple quick things. 01:03:31.770 --> 01:03:33.615 What are the dimensions of angular momentum? 01:03:36.890 --> 01:03:38.170 Length and momentum. 01:03:38.170 --> 01:03:40.880 What else has units of angular momentum? 01:03:40.880 --> 01:03:41.960 AUDIENCE: h bar. 01:03:41.960 --> 01:03:44.226 PROFESSOR: Solid. h bar, dimensionless. 01:03:44.226 --> 01:03:46.350 Angular momentum squared, angular momentum squared. 01:03:46.350 --> 01:03:46.880 OK, great. 01:03:46.880 --> 01:03:48.463 So that's going to be very convenient. 01:03:48.463 --> 01:03:52.540 h bars are just going to float around willy nilly. 01:03:52.540 --> 01:03:57.800 OK, so suppose I ask you the following-- bless you. 01:03:57.800 --> 01:03:59.680 Suppose I ask you the following questions. 01:03:59.680 --> 01:04:02.950 I say look, here are the operators of angular momentum. 01:04:02.950 --> 01:04:03.550 This is Lz. 01:04:03.550 --> 01:04:05.040 We could have written down the same expression for Lx, 01:04:05.040 --> 01:04:06.490 and a Ly, and L squared. 01:04:06.490 --> 01:04:08.650 What are the eigenfunctions of these operators? 01:04:08.650 --> 01:04:10.070 Suppose I ask you this question. 01:04:10.070 --> 01:04:12.270 You all know how to answer this question. 01:04:12.270 --> 01:04:14.020 You take these operators-- so for example, 01:04:14.020 --> 01:04:17.360 if I ask you what are the eigenfunctions of Lz? 01:04:17.360 --> 01:04:18.910 Well, that's not so bad, right? 01:04:18.910 --> 01:04:20.409 The eigenfunction of Lz is something 01:04:20.409 --> 01:04:25.010 where Lz on phi-- I'll call little m-- 01:04:25.010 --> 01:04:29.680 is equal to minus I h bar d d theta phi sub m. 01:04:29.680 --> 01:04:31.550 But I want the eigenvalue, so I'll 01:04:31.550 --> 01:04:35.130 call this h bar times some number. 01:04:35.130 --> 01:04:37.920 Let's call it m, because Lz is an angular momentum. 01:04:37.920 --> 01:04:40.230 It carries units of h bar, and its h bar 01:04:40.230 --> 01:04:41.800 times some number which is dimensional, so we'll call it 01:04:41.800 --> 01:04:42.350 m. 01:04:42.350 --> 01:04:44.510 And we all know the solution of this equation. 01:04:44.510 --> 01:04:47.290 The derivative is equal to a constant times-- 01:04:47.290 --> 01:04:49.340 we can lose the h bar. 01:04:49.340 --> 01:04:51.335 We get a minus i, so we pick up an i. 01:04:51.335 --> 01:04:55.640 So therefore phi sub m is equal to some constant times e 01:04:55.640 --> 01:04:58.200 to the im phi. 01:04:58.200 --> 01:05:00.590 What can we say about m? 01:05:00.590 --> 01:05:02.570 Well, heres an important thing-- oh shoot. 01:05:02.570 --> 01:05:05.139 I'm using phi in so many different ways here. 01:05:05.139 --> 01:05:06.680 Let's call this not phi, because it's 01:05:06.680 --> 01:05:08.180 going to confuse the heck out of us. 01:05:08.180 --> 01:05:12.600 Let's call it Y. So we'll call it Y sub m. 01:05:12.600 --> 01:05:13.200 Why not? 01:05:16.650 --> 01:05:17.970 It's not my joke. 01:05:17.970 --> 01:05:22.070 This goes back to a bunch of-- yeah well, it goes way back. 01:05:22.070 --> 01:05:23.380 So phi is the variable. 01:05:23.380 --> 01:05:26.670 Y is the deciding eigenfunction of Lz, and that's great. 01:05:26.670 --> 01:05:30.170 But what can you say about m? 01:05:30.170 --> 01:05:32.970 Well phi is the variable around the equator 01:05:32.970 --> 01:05:34.957 is periodic with period 2pi. 01:05:34.957 --> 01:05:37.040 And our wave function had better be single valued. 01:05:37.040 --> 01:05:40.040 So what does that tell you about m? 01:05:40.040 --> 01:05:41.850 Well, under phi goes to phi plus 2pi. 01:05:41.850 --> 01:05:44.859 This shifts by im 2pi. 01:05:44.859 --> 01:05:46.650 And that's only one to make a single valued 01:05:46.650 --> 01:05:48.707 if m is an integer. 01:05:48.707 --> 01:05:49.790 So m has to be an integer. 01:05:52.654 --> 01:05:53.320 m is an integer. 01:05:53.320 --> 01:05:55.050 Now we did that for Lz. 01:05:55.050 --> 01:05:56.500 We found the eigenfunctions of Lz. 01:05:56.500 --> 01:05:59.310 What about finding the eigenfunctions L squared? 01:05:59.310 --> 01:06:00.360 Exactly the same thing. 01:06:00.360 --> 01:06:02.234 We're going to solve the eigenvalue equation, 01:06:02.234 --> 01:06:04.970 but it's going to be horrible, horrible to find 01:06:04.970 --> 01:06:05.965 these functions, right? 01:06:05.965 --> 01:06:06.840 Because look at this. 01:06:06.840 --> 01:06:08.370 1 over sine squared d d phi. 01:06:08.370 --> 01:06:11.230 And then 1 over sine d theta sine d theta. 01:06:11.230 --> 01:06:13.590 This is not going to be a fun thing to do. 01:06:13.590 --> 01:06:17.020 So we could just brute force this, but let's not. 01:06:17.020 --> 01:06:20.730 Let's all agree that that's probably a bad idea. 01:06:20.730 --> 01:06:23.340 Let's find a better way to construct the eigenfunctions 01:06:23.340 --> 01:06:25.720 of the angular momentum operators. 01:06:25.720 --> 01:06:26.840 So let's do it. 01:06:26.840 --> 01:06:28.590 So we ran into a situation like this 01:06:28.590 --> 01:06:31.625 before when we dealt with the harmonic oscillator. 01:06:31.625 --> 01:06:33.000 There was a differential equation 01:06:33.000 --> 01:06:34.290 that we wanted to solve. 01:06:34.290 --> 01:06:37.760 And OK, this one isn't nearly as bad, not nearly as bad 01:06:37.760 --> 01:06:40.950 as that one would have been. 01:06:40.950 --> 01:06:44.740 But still it was more useful to work with operator methods. 01:06:44.740 --> 01:06:48.270 So let's take a hint from that and work with operator methods. 01:06:48.270 --> 01:06:53.090 So now we need to study the operators of angular momentum. 01:06:53.090 --> 01:06:54.924 So let's study them in a little more detail. 01:06:54.924 --> 01:06:57.131 So something you're going to show in your problem set 01:06:57.131 --> 01:06:58.010 is the following. 01:06:58.010 --> 01:07:03.280 The commutator of Lx with Ly takes a really simple form. 01:07:03.280 --> 01:07:07.409 This is equal to i h bar-- let's just do this out. 01:07:07.409 --> 01:07:08.450 Let's do this commutator. 01:07:08.450 --> 01:07:09.510 We're OK. 01:07:09.510 --> 01:07:12.210 So Lx with Ly, this is equal to the commutator 01:07:12.210 --> 01:07:21.400 of YPz minus ZPy with Ly ZPx minus XPz. 01:07:24.830 --> 01:07:26.630 So let's look at these term by term. 01:07:26.630 --> 01:07:29.790 So the first one is YPz ZPx. 01:07:29.790 --> 01:07:33.130 YPz ZPx. 01:07:33.130 --> 01:07:34.310 That's a Px. 01:07:34.310 --> 01:07:34.810 Sorry, ZPy. 01:07:37.540 --> 01:07:40.230 ZPx, this term. 01:07:40.230 --> 01:07:41.900 X, good. 01:07:41.900 --> 01:07:43.360 That's the first commutator. 01:07:43.360 --> 01:07:51.310 The second commutator, it can be YPz an XPz minus YPz XPz. 01:07:54.470 --> 01:07:56.665 And then these commutators, the next two, 01:07:56.665 --> 01:08:03.400 are going to be ZPy with ZPx. 01:08:06.060 --> 01:08:08.390 And finally ZPy. 01:08:08.390 --> 01:08:10.260 And that's minus and this is a plus. 01:08:10.260 --> 01:08:12.030 ZPy and XPz. 01:08:15.810 --> 01:08:19.240 OK, so let's look at these. 01:08:19.240 --> 01:08:21.180 These look kind of scary at first. 01:08:21.180 --> 01:08:22.760 But in this one notice the following. 01:08:22.760 --> 01:08:28.630 This is XPz ZPx minus ZPx YPz. 01:08:28.630 --> 01:08:33.779 But what you say about Y with all these other operators? 01:08:33.779 --> 01:08:35.210 Y commutes with all of them. 01:08:35.210 --> 01:08:36.740 So in each term I could just pull Y 01:08:36.740 --> 01:08:38.689 all the way out to one side. 01:08:38.689 --> 01:08:39.505 Yeah? 01:08:39.505 --> 01:08:43.460 So I could just pull out this Y. So for the first term, 01:08:43.460 --> 01:08:48.582 I'm going to write this as Y commutator PZ with ZPx. 01:08:48.582 --> 01:08:50.040 And let me just do that explicitly. 01:08:50.040 --> 01:08:51.200 There's no reason to. 01:08:51.200 --> 01:09:02.580 So this is YPz ZPx minus ZPx YPz. 01:09:02.580 --> 01:09:05.560 And I can pull the Y out front, because this commutes with Px 01:09:05.560 --> 01:09:12.979 and with Z. So I can make this Y times PZ ZPx minus ZPx Pz. 01:09:12.979 --> 01:09:15.260 But now note that I can do exactly the same thing 01:09:15.260 --> 01:09:16.179 with the Px. 01:09:16.179 --> 01:09:16.970 Px commutes with z. 01:09:16.970 --> 01:09:18.240 Px commutes with Pz. 01:09:18.240 --> 01:09:19.529 And it commutes with y. 01:09:19.529 --> 01:09:22.310 So I can pull the px from each term out. 01:09:22.310 --> 01:09:27.415 Px Y. And now I lose the Px. 01:09:27.415 --> 01:09:28.620 I lose the Px. 01:09:28.620 --> 01:09:30.569 But now this is looking good. 01:09:30.569 --> 01:09:38.210 This is Px times Y. And Pz minus ZPz PZz minus ZPz. 01:09:38.210 --> 01:09:47.047 This is also known as PXy times commutator of PZ with Z. 01:09:47.047 --> 01:09:48.130 And what is this equal to? 01:09:51.497 --> 01:09:52.580 Let's get our signs right. 01:09:52.580 --> 01:09:55.930 This is Px with Y time-- OK, we all 01:09:55.930 --> 01:09:57.930 agree that this is going to have an h bar in it. 01:09:57.930 --> 01:09:59.290 Let's write units. 01:09:59.290 --> 01:10:00.950 There's going to be an i. 01:10:00.950 --> 01:10:04.030 And is it a plus or minus? 01:10:04.030 --> 01:10:05.020 Minus. 01:10:05.020 --> 01:10:05.770 OK, good. 01:10:05.770 --> 01:10:08.730 So we get minus h bar XPy. 01:10:08.730 --> 01:10:17.060 So this term is going to give us minus i h bar Py XPx times Y. 01:10:17.060 --> 01:10:18.480 And let's look at this term. 01:10:18.480 --> 01:10:24.440 OK, this is Y, and Y commutes with PZx and PZ, right? 01:10:24.440 --> 01:10:26.080 So I could just pull out the y. 01:10:26.080 --> 01:10:28.700 And x commutes with everything, so I can pull out the x. 01:10:28.700 --> 01:10:31.410 And I'm left with the commutator of Pz with Pz. 01:10:31.410 --> 01:10:32.910 What's the commutator of Pz with Pz? 01:10:32.910 --> 01:10:33.410 AUDIENCE: 0. 01:10:33.410 --> 01:10:33.951 PROFESSOR: 0. 01:10:33.951 --> 01:10:35.170 This term gives me a 0. 01:10:35.170 --> 01:10:37.150 Similarly here, Py commutes with everything. 01:10:37.150 --> 01:10:38.696 Z ZPx. 01:10:38.696 --> 01:10:40.100 Px commutes with everything. 01:10:40.100 --> 01:10:41.590 Z ZPy. 01:10:41.590 --> 01:10:44.870 So I can pull out the Px Py, and I get Z commutator Z, 01:10:44.870 --> 01:10:45.610 and what's that? 01:10:45.610 --> 01:10:46.159 AUDIENCE: 0. 01:10:46.159 --> 01:10:46.700 PROFESSOR: 0. 01:10:46.700 --> 01:10:47.710 So this gives me 0. 01:10:47.710 --> 01:10:51.480 And now this term, ZPy XPz, the only two things 01:10:51.480 --> 01:10:54.600 that don't commute with each other are the Z and the Pz. 01:10:54.600 --> 01:10:56.580 The Py and the X I pull out, so I 01:10:56.580 --> 01:11:00.640 get it a term that's XPy and the commutator of Z with Pz. 01:11:00.640 --> 01:11:03.190 And what is that going to give me? 01:11:03.190 --> 01:11:05.250 PyX i h bar. 01:11:11.685 --> 01:11:14.870 Aha, look at what we got. 01:11:14.870 --> 01:11:22.050 This is equal to i h bar times-- did I screw up the signs? 01:11:22.050 --> 01:11:24.765 XPy minus YPz. 01:11:30.060 --> 01:11:33.130 Oh sorry, Px. 01:11:33.130 --> 01:11:35.036 And what is this equal to? 01:11:35.036 --> 01:11:36.380 AUDIENCE: [INAUDIBLE]. 01:11:36.380 --> 01:11:37.801 PROFESSOR: Yeah, i h bar Lz. 01:11:43.580 --> 01:11:46.964 And more generally, as you'll show on the problem set, 01:11:46.964 --> 01:11:48.380 you get the following commutators. 01:11:48.380 --> 01:11:51.510 Once you've done this once, you can do the rest very easily. 01:11:51.510 --> 01:11:57.980 Lx Ly is-- so Lx with Ly is i h bar Lz. 01:12:03.010 --> 01:12:06.000 And then the rest can be got from cyclic rotations 01:12:06.000 --> 01:12:16.970 Ly with Lz is i h bar Lx and Lz with Lx is i h bar Ly. 01:12:21.460 --> 01:12:22.930 And now here's a fancier one. 01:12:22.930 --> 01:12:25.536 This is less obvious, but exactly the same machinations 01:12:25.536 --> 01:12:27.660 will give you this result, and you'll do this again 01:12:27.660 --> 01:12:28.580 on the problems set. 01:12:28.580 --> 01:12:31.840 If I take L squared, k and L squared 01:12:31.840 --> 01:12:36.150 here is going to be L squared, I just 01:12:36.150 --> 01:12:42.520 mean Lx squared plus Ly squared plus Lz squared. 01:12:42.520 --> 01:12:46.130 This is the norm squared of the vector, the operator form. 01:12:46.130 --> 01:12:49.470 L squared with Lz-- or sorry, with Lx. 01:12:49.470 --> 01:12:50.350 Yeah, fine. 01:12:50.350 --> 01:12:52.570 Lx is equal to 0. 01:12:52.570 --> 01:12:55.910 So Lx commutes with the magnitude L squared. 01:12:55.910 --> 01:12:58.910 Similarly L squared, now just by rotational invariance, 01:12:58.910 --> 01:13:00.814 if Lx commutes with it, Ly and Lz 01:13:00.814 --> 01:13:02.480 had better also commute with it, because 01:13:02.480 --> 01:13:06.180 of who you to see what's Lx. 01:13:06.180 --> 01:13:09.290 And L squared with Lz must be equal to 0. 01:13:15.340 --> 01:13:15.840 Yeah? 01:13:15.840 --> 01:13:16.860 Everyone cool with that? 01:13:21.160 --> 01:13:23.320 And the thing is we've just-- and I'm 01:13:23.320 --> 01:13:24.950 going to put big flags around this. 01:13:24.950 --> 01:13:26.620 We've just learned a tremendous amount 01:13:26.620 --> 01:13:29.520 about the eigenfunctions of the angular momentum operators. 01:13:29.520 --> 01:13:31.411 Why? 01:13:31.411 --> 01:13:33.900 Let's leave that up. 01:13:33.900 --> 01:13:36.590 So what have we just learned about the eigenfunctions 01:13:36.590 --> 01:13:39.221 of the angular momentum operators Lx, Ly, Lz, and L 01:13:39.221 --> 01:13:39.720 squared? 01:13:44.890 --> 01:13:45.562 Anyone? 01:13:45.562 --> 01:13:47.770 We just learned something totally awesome about them. 01:13:57.620 --> 01:13:58.750 Anyone? 01:13:58.750 --> 01:14:03.174 So can you find simultaneous eigenfunctions of Lx and Ly? 01:14:03.174 --> 01:14:03.912 AUDIENCE: No. 01:14:03.912 --> 01:14:04.870 PROFESSOR: Not so much. 01:14:04.870 --> 01:14:06.000 They don't commute to 0. 01:14:06.000 --> 01:14:07.810 What about Ly and Lz? 01:14:07.810 --> 01:14:08.800 Nope. 01:14:08.800 --> 01:14:11.080 Lz Lx, nope. 01:14:11.080 --> 01:14:14.060 So you cannot find simultaneous eigenfunctions of Lx and Ly. 01:14:17.010 --> 01:14:20.680 What about Lx and L squared? 01:14:20.680 --> 01:14:22.120 Yes. 01:14:22.120 --> 01:14:24.480 So we can find simultaneously eigenfunctions 01:14:24.480 --> 01:14:29.740 of L squared and Lx. 01:14:29.740 --> 01:14:32.850 OK, what about can we find simultaneous eigenfunctions 01:14:32.850 --> 01:14:35.370 of L squared and Ly? 01:14:35.370 --> 01:14:35.990 Yep. 01:14:35.990 --> 01:14:36.489 Exactly. 01:14:36.489 --> 01:14:39.840 What about L squared, Ly, and Lz? 01:14:39.840 --> 01:14:40.580 Nope. 01:14:40.580 --> 01:14:42.530 No such luck. 01:14:42.530 --> 01:14:44.720 And this leads us to the following idea. 01:14:44.720 --> 01:14:51.130 The idea is a complete set of commuting observables. 01:14:54.790 --> 01:14:58.180 And here's what this idea is meant to contain. 01:14:58.180 --> 01:15:00.350 You can always write down a lot of operators. 01:15:00.350 --> 01:15:02.710 So let me step back and ask a classical question. 01:15:02.710 --> 01:15:04.730 Classically, suppose I have a particle in three dimensions, 01:15:04.730 --> 01:15:07.063 a particle moving around in this room, non relativistic, 01:15:07.063 --> 01:15:08.380 familiar 801. 01:15:08.380 --> 01:15:10.720 I have a particle moving around in this room. 01:15:10.720 --> 01:15:12.835 How much data must I specify to specify 01:15:12.835 --> 01:15:14.210 the configuration of this system? 01:15:14.210 --> 01:15:16.290 Well, I have to tell you where the particle is, 01:15:16.290 --> 01:15:18.530 and I have to tell you what its momentum is, right? 01:15:18.530 --> 01:15:21.130 So I have to tell you the three coordinates and the three 01:15:21.130 --> 01:15:22.230 momenta. 01:15:22.230 --> 01:15:26.480 If I give you five numbers, that's not enough, right? 01:15:26.480 --> 01:15:29.902 I need to give you six bits of data. 01:15:29.902 --> 01:15:31.860 On the other hand, if I give you seven numbers, 01:15:31.860 --> 01:15:37.750 like I give x, y, z, Px, Py, Pz, and e, that's over complete. 01:15:37.750 --> 01:15:38.250 Right? 01:15:38.250 --> 01:15:41.214 That was unnecessary. 01:15:41.214 --> 01:15:42.630 So in classical mechanics, you can 01:15:42.630 --> 01:15:45.570 ask what data must you specify to completely specify 01:15:45.570 --> 01:15:46.570 the state of the system. 01:15:46.570 --> 01:15:47.600 And that's usually pretty easy. 01:15:47.600 --> 01:15:48.620 You specify the number coordinates 01:15:48.620 --> 01:15:49.802 and the number of momenta. 01:15:49.802 --> 01:15:52.260 In quantum mechanics, we ask a slightly different question. 01:15:52.260 --> 01:15:54.970 We ask, in order to specify the state of a system, 01:15:54.970 --> 01:15:57.860 we say we want to specify which state it is. 01:15:57.860 --> 01:15:59.930 You can specify that by saying which 01:15:59.930 --> 01:16:01.830 superposition in a particular basis. 01:16:01.830 --> 01:16:04.440 So you pick a basis, and you specify 01:16:04.440 --> 01:16:05.980 which particular superposition, what 01:16:05.980 --> 01:16:09.970 are the eigenvalues of the operators you've 01:16:09.970 --> 01:16:12.410 diagonalized in that basis? 01:16:12.410 --> 01:16:16.619 So another way to phrase this is for a 1D problem, 01:16:16.619 --> 01:16:18.160 say we have just a simple 1D problem. 01:16:18.160 --> 01:16:23.140 We have X and we have P. What is a complete set of commuting 01:16:23.140 --> 01:16:23.780 operators? 01:16:23.780 --> 01:16:25.405 Well, you have to have enough operators 01:16:25.405 --> 01:16:27.540 so that the eigenvalue specifies a state. 01:16:27.540 --> 01:16:30.130 So X would be-- is that enough? 01:16:32.650 --> 01:16:35.520 So if I take the operator X and I say look, 01:16:35.520 --> 01:16:37.881 my system is in the state with an eigenvalue X not of X, 01:16:37.881 --> 01:16:39.630 does that specify the state of the system? 01:16:42.817 --> 01:16:45.150 It tells you the particles are in a delta function state 01:16:45.150 --> 01:16:46.590 right here. 01:16:46.590 --> 01:16:47.820 Does that specify the state? 01:16:47.820 --> 01:16:48.130 AUDIENCE: Yes. 01:16:48.130 --> 01:16:49.380 PROFESSOR: Fantastic, it does. 01:16:49.380 --> 01:16:51.030 Now what if I tell you instead, oh it's 01:16:51.030 --> 01:16:55.460 in a state of definite P. Does that specify the state? 01:16:55.460 --> 01:16:56.570 Absolutely. 01:16:56.570 --> 01:16:59.910 Can I say it's in a state with definite X and definite P? 01:16:59.910 --> 01:17:00.730 AUDIENCE: No. 01:17:00.730 --> 01:17:01.480 PROFESSOR: No. 01:17:01.480 --> 01:17:03.839 These don't commute. 01:17:03.839 --> 01:17:06.130 So a complete set of commuting observables in this case 01:17:06.130 --> 01:17:09.300 would be either X or P, but not both. 01:17:09.300 --> 01:17:10.410 Yeah? 01:17:10.410 --> 01:17:15.090 Now if we're in three dimensions, is x a complete set 01:17:15.090 --> 01:17:16.466 of commuting observables? 01:17:16.466 --> 01:17:17.205 AUDIENCE: No. 01:17:17.205 --> 01:17:18.830 PROFESSOR: No, because it's not enough. 01:17:18.830 --> 01:17:20.880 You tell me that it's X, that doesn't tell me 01:17:20.880 --> 01:17:22.255 what state it is because it could 01:17:22.255 --> 01:17:24.420 have y dependence or z dependence. 01:17:24.420 --> 01:17:30.760 So in 3D, we take, for example, x, and y, and z. 01:17:30.760 --> 01:17:37.010 Or Px, and Py, and Pz. 01:17:37.010 --> 01:17:42.940 We could also pick z, and Px, and y. 01:17:42.940 --> 01:17:45.710 Are these complete? 01:17:45.710 --> 01:17:47.690 If I tell you I have definite position in z, 01:17:47.690 --> 01:17:50.050 definite position in y, and definite momentum in x? 01:17:50.050 --> 01:17:51.380 Does that completely specify my state? 01:17:51.380 --> 01:17:51.855 AUDIENCE: Yes. 01:17:51.855 --> 01:17:53.479 PROFESSOR: Yeah, totally unambiguously. 01:17:53.479 --> 01:18:00.030 e to the ikx delta of y delta of z totally fixes my function, 01:18:00.030 --> 01:18:01.030 completely specifies it. 01:18:01.030 --> 01:18:03.190 If I'd only picked two of these, it 01:18:03.190 --> 01:18:04.712 would not have been complete. 01:18:04.712 --> 01:18:06.170 It would have told me, for example, 01:18:06.170 --> 01:18:09.004 e to the ikx delta of y, but it doesn't tell me 01:18:09.004 --> 01:18:11.850 how it depends on z, how the wave functions on z. 01:18:11.850 --> 01:18:16.154 And if I added another operator, for example, Py, 01:18:16.154 --> 01:18:17.320 this is no longer commuting. 01:18:17.320 --> 01:18:18.751 These two operators don't commute. 01:18:18.751 --> 01:18:20.500 So a complete set of commuting observables 01:18:20.500 --> 01:18:22.270 can be thought of as the most operators 01:18:22.270 --> 01:18:25.020 you can write down that all commute with each other 01:18:25.020 --> 01:18:28.060 and the minimum number whose eigenvalues completely 01:18:28.060 --> 01:18:30.008 specify the state of the system. 01:18:30.008 --> 01:18:30.508 Cool? 01:18:30.508 --> 01:18:31.340 OK. 01:18:31.340 --> 01:18:33.750 So with all that said, what is a complete set 01:18:33.750 --> 01:18:36.400 of commuting observables for the angular momentum system? 01:18:40.730 --> 01:18:44.552 Well, it can't be any two of Lx, Ly, and Lz. 01:18:44.552 --> 01:18:45.510 So let's just pick one. 01:18:45.510 --> 01:18:47.210 I'll call it Lz. 01:18:47.210 --> 01:18:48.220 I could've called it Lx. 01:18:48.220 --> 01:18:48.650 It doesn't matter. 01:18:48.650 --> 01:18:50.210 It's up to you what axis is what. 01:18:50.210 --> 01:18:53.950 I'll just call it Lz conventionally. 01:18:53.950 --> 01:18:56.280 And then L squared also commutes, 01:18:56.280 --> 01:18:57.780 because L squared commutes with Lx. 01:18:57.780 --> 01:18:59.420 It commutes Ly and with Lz. 01:18:59.420 --> 01:19:01.100 So this actually forms a complete set 01:19:01.100 --> 01:19:06.010 of commuting observables for the angular momentum system. 01:19:06.010 --> 01:19:08.290 Complete set of commuting observables 01:19:08.290 --> 01:19:09.840 for angular momentum. 01:19:15.670 --> 01:19:21.000 So this idea will come up more in the future. 01:19:21.000 --> 01:19:28.860 And here is going to be the key [INAUDIBLE]. 01:19:28.860 --> 01:19:31.320 So we'd like to use the following fact. 01:19:31.320 --> 01:19:33.010 We want to construct the eigenfunctions 01:19:33.010 --> 01:19:34.731 of our complete set of-- yeah, question? 01:19:34.731 --> 01:19:36.314 AUDIENCE: Really quick can you explain 01:19:36.314 --> 01:19:38.470 how you got that L squared and Lx commute? 01:19:38.470 --> 01:19:39.970 PROFESSOR: Yeah, I got it by knowing 01:19:39.970 --> 01:19:42.010 what you're going to write on your solution set. 01:19:42.010 --> 01:19:43.530 So this is on your problem set. 01:19:43.530 --> 01:19:46.230 So the way it goes-- so there are fancy ways of doing it, 01:19:46.230 --> 01:19:48.170 but the just direct way of doing, 01:19:48.170 --> 01:19:49.930 how do you construct these commutators? 01:19:49.930 --> 01:19:51.388 Is you know what the operators are. 01:19:51.388 --> 01:19:52.802 You know what L squared is. 01:19:52.802 --> 01:19:54.510 And you know that L squared is Lx squared 01:19:54.510 --> 01:19:55.950 plus Ly squared plus Lz squared. 01:19:55.950 --> 01:19:59.310 It's built in that fashion out of x and Py. 01:19:59.310 --> 01:20:02.460 And then I literally just put in the definitions of Lx, Ly, 01:20:02.460 --> 01:20:04.910 and Lz into that expression for L squared 01:20:04.910 --> 01:20:06.910 and compute the commutator with Lx, again, 01:20:06.910 --> 01:20:10.880 using the definition in terms of Py and z. 01:20:10.880 --> 01:20:13.390 And then you just chug through the commutators. 01:20:13.390 --> 01:20:14.390 Yeah, it just works out. 01:20:14.390 --> 01:20:17.700 So it's not obvious from the way I just phrased it 01:20:17.700 --> 01:20:20.022 that it works out like that. 01:20:20.022 --> 01:20:21.980 Later on you'll probably develop some intuition 01:20:21.980 --> 01:20:23.370 that it should be obvious. 01:20:23.370 --> 01:20:24.786 But for the moment, I'm just going 01:20:24.786 --> 01:20:26.636 to call it a brute force computation. 01:20:26.636 --> 01:20:29.010 And that's how you're going to do it on your problem set. 01:20:29.010 --> 01:20:32.480 So let me tell you where we're going next. 01:20:32.480 --> 01:20:35.310 So the question I really want to deal with 01:20:35.310 --> 01:20:37.780 is what are the eigenfunctions? 01:20:37.780 --> 01:20:39.730 So this is where we're leaving off. 01:20:42.270 --> 01:20:47.530 What are the eigenfunctions of our complete set 01:20:47.530 --> 01:20:50.950 of commuting variable L squared and Lz. 01:20:55.270 --> 01:20:56.120 What are these guys? 01:20:56.120 --> 01:20:57.190 And we know that we could solve them 01:20:57.190 --> 01:20:58.960 by solving the differential equations using 01:20:58.960 --> 01:21:00.520 those operators, but that would be horrible. 01:21:00.520 --> 01:21:01.980 We'd like to do something a little smarter. 01:21:01.980 --> 01:21:04.438 We'd like to use the commutation relations and the algebra. 01:21:06.650 --> 01:21:11.690 And here a really beautiful thing is going to happen. 01:21:11.690 --> 01:21:13.601 When you look at these commutation relations, 01:21:13.601 --> 01:21:16.100 one thing they're telling us is that we can't simultaneously 01:21:16.100 --> 01:21:20.030 have eigenfunctions of Lx and Ly. 01:21:20.030 --> 01:21:24.919 However, the way that Lx and Ly commute together is to form Lz. 01:21:24.919 --> 01:21:26.210 That gives us some information. 01:21:26.210 --> 01:21:29.910 That gives us some magic, some power. 01:21:29.910 --> 01:21:32.339 And in particular, much like that moment in the harmonic 01:21:32.339 --> 01:21:34.380 oscillator I said, well look, we could write down 01:21:34.380 --> 01:21:37.040 these operators as a. 01:21:37.040 --> 01:21:40.170 Well look, we can write down these operators, 01:21:40.170 --> 01:21:43.320 which I'm going to call L plus and L minus. 01:21:43.320 --> 01:21:48.200 L plus is going to be equal to Lx plus i Ly. 01:21:48.200 --> 01:21:53.390 And L minus is going to be equal to Lx minus I Ly. 01:21:53.390 --> 01:21:55.880 Now Lx, Ly, those are observables? 01:21:55.880 --> 01:21:59.464 What can you say about them as operators? 01:21:59.464 --> 01:22:01.880 What kind of operators are they since they're observables? 01:22:01.880 --> 01:22:03.180 Hermitian, exactly. 01:22:03.180 --> 01:22:07.320 So what's the Hermitian adjoint of Hermitian plus i Hermitian? 01:22:07.320 --> 01:22:08.571 Hermitian minus i Hermitian. 01:22:08.571 --> 01:22:09.070 OK, good. 01:22:09.070 --> 01:22:11.120 So L minus is the adjoint of L plus. 01:22:15.392 --> 01:22:17.100 So this is just going to be a definition. 01:22:17.100 --> 01:22:19.308 Let's take these to be the definitions of these guys. 01:22:22.902 --> 01:22:25.110 If we take their commutator, something totally lovely 01:22:25.110 --> 01:22:26.080 happens. 01:22:26.080 --> 01:22:26.930 I'm not going to write all the commutators. 01:22:26.930 --> 01:22:28.305 I'm just going to write a couple. 01:22:28.305 --> 01:22:33.560 The first is if I take L squared and I commute with L plus, 01:22:33.560 --> 01:22:36.010 well, L plus is Lx plus Ly, and we already 01:22:36.010 --> 01:22:37.510 know that L squared commutes with Lx 01:22:37.510 --> 01:22:38.510 and it commutes with Ly. 01:22:38.510 --> 01:22:42.600 So L squared commutes with L plus. 01:22:42.600 --> 01:22:46.140 And similarly for L minus, it commutes with each term. 01:22:46.140 --> 01:22:48.737 Question? 01:22:48.737 --> 01:22:49.320 Oh, I'm sorry. 01:22:49.320 --> 01:22:50.360 Lx. 01:22:50.360 --> 01:22:51.320 Shoot, that was an x. 01:22:51.320 --> 01:22:52.630 It just didn't look like it. 01:22:52.630 --> 01:22:54.000 Lx minus i Ly. 01:22:56.760 --> 01:22:59.796 So it commutes both L plus and L minus. 01:22:59.796 --> 01:23:02.350 But here's the real beauty of it. 01:23:02.350 --> 01:23:07.760 Lz with L plus is equal to-- well, 01:23:07.760 --> 01:23:10.110 let's just do dimensional analysis. 01:23:10.110 --> 01:23:12.070 L plus is Lx plus i Ly. 01:23:12.070 --> 01:23:14.930 We know that Lz with Lx is something like Ly. 01:23:14.930 --> 01:23:17.930 And Lz with Ly is something like Lx. 01:23:17.930 --> 01:23:19.560 With factors of i's an h bars. 01:23:22.340 --> 01:23:26.930 And when you work out the commutator, which should only 01:23:26.930 --> 01:23:32.040 take you second, you get i h bar L plus. 01:23:32.040 --> 01:23:37.020 And similarly, when we construct Lz and L minus, 01:23:37.020 --> 01:23:42.630 we get minus h bar L minus. 01:23:42.630 --> 01:23:46.160 Are L plus an L minus Hermitian? 01:23:46.160 --> 01:23:49.310 No, they're each other's adjoints. 01:23:49.310 --> 01:23:50.620 Lz is Hermitian. 01:23:53.360 --> 01:23:56.111 And look at this commutation relation. 01:23:56.111 --> 01:23:57.110 What does that tell you? 01:24:00.520 --> 01:24:02.840 From the first observation, if we have an energy 01:24:02.840 --> 01:24:05.070 or if we have an operator e and an operator a 01:24:05.070 --> 01:24:06.850 that commute in this fashion, then this 01:24:06.850 --> 01:24:09.010 tells you that the eigenfunctions of this operator 01:24:09.010 --> 01:24:11.660 are staggered in a the ladder spaced by h bar. 01:24:11.660 --> 01:24:16.680 The eigenvalues of Lz come in a ladder spaced by h bar. 01:24:16.680 --> 01:24:20.030 We can raise with L plus, and we can 01:24:20.030 --> 01:24:22.670 lower with L minus just like in the harmonic oscillator 01:24:22.670 --> 01:24:25.440 problem. 01:24:25.440 --> 01:24:28.187 And we'll exploit the rest of the-- we'll 01:24:28.187 --> 01:24:30.520 deduce the rest of the structure of the angular momentum 01:24:30.520 --> 01:24:33.920 operator eigenfunctions next time using this computation 01:24:33.920 --> 01:24:34.826 relation. 01:24:34.826 --> 01:24:37.580 See you next time.