1 00:00:02,090 --> 00:00:05,000 Hello everyone, welcome back to exploring quantum physics. 2 00:00:05,000 --> 00:00:09,830 I'm Charles Clark, and we'll now start using some of the tools that we developed 3 00:00:09,830 --> 00:00:11,920 in the previous section to get the 4 00:00:11,920 --> 00:00:18,976 basic commutation relations for the angular momentum operators. 5 00:00:18,976 --> 00:00:21,760 Now, once again, I have to emphasize, and I've 6 00:00:21,760 --> 00:00:28,010 been there myself, the only way to develop facility with 7 00:00:28,010 --> 00:00:31,520 these techniques is to actually practice them. 8 00:00:31,520 --> 00:00:36,640 So in the interest of time, I'm going to outline a number of the 9 00:00:36,640 --> 00:00:42,639 key things that have to be done to develop a working set of angular momentum tools. 10 00:00:44,330 --> 00:00:47,000 but really, if you want to understand it, it's not 11 00:00:47,000 --> 00:00:49,290 going to do you good just to look at that. 12 00:00:49,290 --> 00:00:52,940 You will have to do some of the computations. 13 00:00:52,940 --> 00:00:56,830 And that's, for some people, those sorts of calculations 14 00:00:56,830 --> 00:01:00,990 turn out to be enjoyable. Others can't stand them. 15 00:01:00,990 --> 00:01:05,560 So you'll just, you'll, you'll, you'll get a sense of what it;s like and some of you, 16 00:01:05,560 --> 00:01:12,078 I think will not only benefit from this but come to enjoy it and value 17 00:01:12,078 --> 00:01:18,290 it, at least I hope so. Okay, so, once again here's the definition 18 00:01:18,290 --> 00:01:21,890 of the angular momentum operator. And, now I've, break it down. 19 00:01:21,890 --> 00:01:25,620 So, here's, here's what we started with at the beginning of the first part. 20 00:01:25,620 --> 00:01:27,790 Here's the Einstein summation condition. 21 00:01:27,790 --> 00:01:31,810 Here's the Einstein summation condition with the Levi-Civita symbol. 22 00:01:31,810 --> 00:01:35,990 Now by the way i here is the square root of minus 1. 23 00:01:35,990 --> 00:01:39,302 And I try to avoid try to avoid using i as 24 00:01:39,302 --> 00:01:43,442 a running index, when we have the square root of minus 25 00:01:43,442 --> 00:01:47,600 1 present. So just keep your i's out for that. 26 00:01:47,600 --> 00:01:53,162 But, there's no, no significance as to whether the symbol's 27 00:01:53,162 --> 00:01:56,048 a u, v, w, a, b, c, d, e, f. 28 00:01:56,048 --> 00:02:00,548 Anything that appears here is a running, is a running index, and the, 29 00:02:00,548 --> 00:02:03,398 the particular choice of symbol that's 30 00:02:03,398 --> 00:02:06,909 used to representative is just something arbitrary. 31 00:02:06,909 --> 00:02:08,009 Now in fact, 32 00:02:08,009 --> 00:02:13,409 everything that's going to be done in this part is based on the, 33 00:02:13,409 --> 00:02:19,218 the commutation relationship between position momentum. 34 00:02:19,218 --> 00:02:23,187 but as you can see when you have, when you have products, then there's a, 35 00:02:23,187 --> 00:02:26,148 there's a, a bit of bookkeeping that's required 36 00:02:26,148 --> 00:02:29,620 to make sure that everything turns out right. 37 00:02:29,620 --> 00:02:33,170 So just to just to start with something very 38 00:02:33,170 --> 00:02:37,370 basic, let's try a simple application of 39 00:02:37,370 --> 00:02:40,607 this commutation relationship in the, in-video quiz. 40 00:02:43,680 --> 00:02:45,090 Okay. So I hope you're able to 41 00:02:47,590 --> 00:02:52,810 to dev-, to, to derive this expression from the uncertainty 42 00:02:52,810 --> 00:02:58,460 principle, and we'll see, we'll see many examples like this later on. 43 00:03:02,010 --> 00:03:05,600 Okay, so here's, here's our basic definition again, just to, 44 00:03:05,600 --> 00:03:08,100 just to keep everything clear before you on the screen. 45 00:03:10,110 --> 00:03:12,750 here is the fundamental commutation relation 46 00:03:12,750 --> 00:03:15,030 that, on which everything else is based. 47 00:03:15,030 --> 00:03:19,600 Oh yes, now here is this this identity that you just derived. 48 00:03:21,420 --> 00:03:25,760 So now here's a very important expression. 49 00:03:27,430 --> 00:03:31,420 What does the angular moment, how, what's the commutation relation 50 00:03:31,420 --> 00:03:34,840 of the angular momentum with, let's say, the position operator? 51 00:03:36,060 --> 00:03:40,987 Well, alright that's here's the computated La,xb. 52 00:03:40,987 --> 00:03:44,420 There's just a very simple expression, what that means, 53 00:03:46,640 --> 00:03:51,624 and now we, we we now 54 00:03:51,624 --> 00:03:57,023 decompose the angular momentum operator into its 55 00:03:57,023 --> 00:04:02,510 basic parts, its, you know, r cross p, and then, 56 00:04:05,260 --> 00:04:09,045 then we, we 57 00:04:09,045 --> 00:04:14,990 apply this 58 00:04:16,330 --> 00:04:22,320 from this expression up here. We apply contraction, and we get this 59 00:04:22,320 --> 00:04:27,410 this very significant identity, which you will see is generic. 60 00:04:27,410 --> 00:04:31,170 In other words, if we have any vector-valued operator 61 00:04:31,170 --> 00:04:34,450 in here, like we've got the same identity 62 00:04:34,450 --> 00:04:38,050 for, for the momentum, we get this common here. 63 00:04:38,050 --> 00:04:41,010 La with b is ih bar epsilon abc xc. 64 00:04:42,970 --> 00:04:45,740 I, I rather like express, I rather liking ex, 65 00:04:45,740 --> 00:04:48,750 writing the, expressing that in this way because you. 66 00:04:48,750 --> 00:04:53,010 It emphasizes the, the difference between classical mechanics and quantum mechanics. 67 00:04:53,010 --> 00:04:53,820 Do you see that 68 00:04:55,940 --> 00:04:58,510 in class of the mechanics, this expression 69 00:04:58,510 --> 00:05:03,110 would vanish identically because, because a cross b 70 00:05:05,690 --> 00:05:11,350 is equal to minus b cross a, for classical operators? 71 00:05:11,350 --> 00:05:15,780 So, here we have like, these two real observable operators. 72 00:05:15,780 --> 00:05:16,280 The 73 00:05:18,120 --> 00:05:19,760 the sum of these two cross products is an 74 00:05:19,760 --> 00:05:23,850 imaginary number, times the vector, so this really is, this 75 00:05:23,850 --> 00:05:31,140 is about as quantum as it gets, and indeed note that in the limit, h bar goes to 0. 76 00:05:31,140 --> 00:05:33,180 This produces the right answer, in that 77 00:05:33,180 --> 00:05:36,730 you get back the same, the classical cross-product. 78 00:05:36,730 --> 00:05:39,970 So, there are, there are number of identities that 79 00:05:39,970 --> 00:05:43,320 you'll see that look like this in quantum mechanics. 80 00:05:47,530 --> 00:05:48,840 So, once again, where are we? 81 00:05:48,840 --> 00:05:52,710 Okay, we had, we had we have here, we 82 00:05:52,710 --> 00:05:56,520 have the basic relations, the one that you derived. 83 00:05:56,520 --> 00:06:00,910 So we're about to conclude here strongly 84 00:06:00,910 --> 00:06:04,390 recommend, that you, you verify the following things. 85 00:06:04,390 --> 00:06:13,610 So for example, here's the the analog, the identity we proved on the previous slide. 86 00:06:13,610 --> 00:06:16,580 With x, we get the same type of thing for p. 87 00:06:18,240 --> 00:06:21,090 Now here's, and here's the sort of the most important one. 88 00:06:21,090 --> 00:06:24,478 The commutator of the different components of angle 89 00:06:24,478 --> 00:06:25,017 [UNKNOWN] 90 00:06:25,017 --> 00:06:29,098 with themselves, again, it has, it is really of the same 91 00:06:29,098 --> 00:06:32,998 form, because L is a vector. So this, this 92 00:06:32,998 --> 00:06:36,802 allows us to write another, amusingly quantum expression. 93 00:06:36,802 --> 00:06:43,020 The cross-product of something with itself is an imaginary number times that thing. 94 00:06:43,020 --> 00:06:51,140 So, this is, is truly a quantum identity. And then finally, so, I'd say 95 00:06:53,190 --> 00:06:55,170 these two are more or less the same thing. 96 00:06:55,170 --> 00:06:58,850 But this is, this is of central importance. 97 00:06:58,850 --> 00:07:03,070 That any com, so any two components of the 98 00:07:03,070 --> 00:07:05,900 angular momentum operator in general do not com, they, 99 00:07:05,900 --> 00:07:11,260 in general they don't commute but any component of 100 00:07:11,260 --> 00:07:15,490 the square of the angular momentum operator, so L squared. 101 00:07:15,490 --> 00:07:20,360 I should have, should have written this as like Lk, Lk, 102 00:07:20,360 --> 00:07:25,608 just so as not to confuse you. Or, La, 103 00:07:25,608 --> 00:07:30,700 La. so let me, let me just write this. 104 00:07:30,700 --> 00:07:35,780 La, Lb, Lb equals 0 105 00:07:35,780 --> 00:07:40,380 is the identity that you are being asked to 106 00:07:40,380 --> 00:07:45,960 verify in other words, that any component of the angular momentum 107 00:07:45,960 --> 00:07:49,780 operator commutes with the, the square 108 00:07:49,780 --> 00:07:53,560 magnitude of the angular momentum operator. 109 00:07:53,560 --> 00:07:54,330 Okay. 110 00:07:54,330 --> 00:07:59,710 Do take some time to work on those. I think you'll find it rewarding.