1 00:00:00,970 --> 00:00:05,710 Welcome back everyone, I'm Charles Clark, this is Exploring Quantum Physics. 2 00:00:05,710 --> 00:00:09,742 today this lecture we're continuing the study of angular momentum and now getting 3 00:00:09,742 --> 00:00:14,097 to a really important part. Which is to see how the angular momentum 4 00:00:14,097 --> 00:00:18,168 appears in any prov of motion independent of whatever the, type of external 5 00:00:18,168 --> 00:00:22,180 potential is, as an effective potential, so as how of the angular momentum, and is 6 00:00:22,180 --> 00:00:25,720 a fundamental way into the equations of motion, so a particle moving at a 7 00:00:25,720 --> 00:00:33,426 potential. So in particular, here we're going to 8 00:00:33,426 --> 00:00:38,631 show how the angular momentum operator. Contributes to the kinetic energy 9 00:00:38,631 --> 00:00:42,584 operator, and will do this based on just the basic commutation relations and the 10 00:00:42,584 --> 00:00:48,204 identities we've done, preformed so far. I consider this to be a fundamental and 11 00:00:48,204 --> 00:00:51,984 important derivation because it shows something deep about the effects of 12 00:00:51,984 --> 00:00:56,004 angular momentum in quantum mechanical systems And there's a part which I think 13 00:00:56,004 --> 00:01:02,168 you really need to work through yourself in order to understand it. 14 00:01:02,168 --> 00:01:07,930 I'll, I'll, I'll outline the steps, but it's really up to you to implement them. 15 00:01:07,930 --> 00:01:13,880 And I can only say that if you complete this derivation and are happy with it. 16 00:01:13,880 --> 00:01:17,660 you'll have done something that's a pretty fundamental calculation in quantum 17 00:01:17,660 --> 00:01:21,170 physics, and I think that the results of it will give you an insight into a lot of 18 00:01:21,170 --> 00:01:28,360 other physical problems, as well. So, let's proceed. 19 00:01:28,360 --> 00:01:32,266 Now, just to review where we are from this definition [UNKNOWN] minimum 20 00:01:32,266 --> 00:01:36,970 operator. And this identity alone everything else 21 00:01:36,970 --> 00:01:41,845 follows but here are some important weight stations which simplify life a lot 22 00:01:41,845 --> 00:01:47,320 in thinking about how to proceed on more complicated calculations then that, these 23 00:01:47,320 --> 00:01:52,045 show that the the of the commutator of the angle incremental with either 24 00:01:52,045 --> 00:01:56,620 position or the momentum operator leads to a common, effectively sort of 25 00:01:56,620 --> 00:02:01,720 rotation, in the space indeed the same the same result as the change for the 26 00:02:01,720 --> 00:02:12,657 angle incremental operate itself. And then we have this important fact 27 00:02:12,657 --> 00:02:17,972 that. The commutative any compliment angle 28 00:02:17,972 --> 00:02:25,075 momentum with the square root angle momentum vanishes. 29 00:02:25,075 --> 00:02:30,430 I just emphasize that in some sense if you just have these first 2 pieces, you 30 00:02:30,430 --> 00:02:36,776 can do all the rest. But you'll find it useful to have A 31 00:02:36,776 --> 00:02:43,796 recollection of these, these staged identities,the staged in order of 32 00:02:43,796 --> 00:02:49,854 complexity. The key result of this part is to see how 33 00:02:49,854 --> 00:02:55,266 the square of the angular momentum Enters in to the kinetic energy operator in a 34 00:02:55,266 --> 00:03:04,002 very fundamental way. So there's some algebra associated with, 35 00:03:04,002 --> 00:03:11,610 with with deriving this result. So we start, as before, by. 36 00:03:11,610 --> 00:03:17,645 Expanding everything into the simplest possible expansion, that is, we'll just 37 00:03:17,645 --> 00:03:23,510 write out in all components, the the value of L squared just by expanding 38 00:03:23,510 --> 00:03:30,494 this. So in fact, the Einstein notation give us 39 00:03:30,494 --> 00:03:40,595 one rather a simple way of writing out of this, fairly dense in terms of indices. 40 00:03:40,595 --> 00:03:45,651 But so here I've actually I've taken the H bar squared and divided L squared by 41 00:03:45,651 --> 00:03:49,460 it. Just for further simplicity just so we 42 00:03:49,460 --> 00:03:53,550 don't keep getting factors of the H bar around. 43 00:03:53,550 --> 00:04:03,454 And so this is now R cross grad is just xjdk, where dk refers to d dxk. 44 00:04:03,454 --> 00:04:11,538 just a simplified notation here. And so here are these again, now look at 45 00:04:11,538 --> 00:04:15,442 here, here's the repeated sum Because we've got the ith, we're taking a, a 46 00:04:15,442 --> 00:04:20,072 scalar product. So the first index is repeated, and then 47 00:04:20,072 --> 00:04:24,500 we have the others in a fairly general way. 48 00:04:24,500 --> 00:04:31,028 this gets moved over into simplified form, and applying this contraction 49 00:04:31,028 --> 00:04:37,450 identity. you see we get to this. 50 00:04:37,450 --> 00:04:43,177 And in fact, in just a few steps we have what looks like it might be pretty much 51 00:04:43,177 --> 00:04:49,950 as far as one can go. Okay, but, let's Keep moving because 52 00:04:49,950 --> 00:04:56,190 there, there's still is an issue with this expression in the sense that we have 53 00:04:56,190 --> 00:05:04,284 the let's say here djxk. And potentially when j and, when j and k 54 00:05:04,284 --> 00:05:10,160 are the same that doesn't commute. We want to, we want to basically move all 55 00:05:10,160 --> 00:05:16,358 the parts over so there's. As little ambiguity about the, presence 56 00:05:16,358 --> 00:05:20,820 of non-moving bodies. In other words, so that we have to 57 00:05:20,820 --> 00:05:24,750 evaluate the thing in the appropriate, in the simplest possible order. 58 00:05:24,750 --> 00:05:35,102 Which means, for example, getting, something like xk xk dj dj. 59 00:05:35,102 --> 00:05:42,398 If we could, that would be something like x squared times, or something like r 60 00:05:42,398 --> 00:05:50,350 squared, d squared, dr squared if you see what I mean. 61 00:05:50,350 --> 00:05:57,676 Okay, so this, I think, is something that You now have to start looking at and you 62 00:05:57,676 --> 00:06:04,780 can go and simplify using these expressions. 63 00:06:04,780 --> 00:06:13,030 This this general rule for exchanging the order of operators /g. 64 00:06:13,030 --> 00:06:18,190 And let me suggest that you just Just pause for a while and try your hand at 65 00:06:18,190 --> 00:06:22,898 this. so the next slide is going to show the 66 00:06:22,898 --> 00:06:29,396 central resolved and then we'll work out some implications of that. 67 00:06:29,396 --> 00:06:37,476 Well, here we have. The a the the the answer, this is the the 68 00:06:37,476 --> 00:06:46,493 the answer can be expressed in this form, so this is equal to minus l squared over 69 00:06:46,493 --> 00:06:56,755 h squared. Well now, you see what can be done 70 00:06:56,755 --> 00:07:09,800 because this here, the Dell squared is a Kinetic Energy operator. 71 00:07:09,800 --> 00:07:13,688 So from that we can write. The Hamiltonian for particle three 72 00:07:13,688 --> 00:07:18,245 dimensional potential has formed. This is, this is the familiar form that 73 00:07:18,245 --> 00:07:23,870 you've seen earlier many times, earlier. And now, we just substitute the result 74 00:07:23,870 --> 00:07:29,565 that we obtained, and that we see that we get we get equational motion That has 75 00:07:29,565 --> 00:07:36,080 derivatives in the root, in the radial coordinates. 76 00:07:36,080 --> 00:07:41,540 So r squared is equal to x squared plus y squared plus z squared. 77 00:07:41,540 --> 00:07:46,708 And that the angle minimum operator basically appears in that expression, if 78 00:07:46,708 --> 00:07:51,344 you, if you, if you extract this out, it's something that has to do with the 79 00:07:51,344 --> 00:07:57,892 derivatives here. Which is the kinetic energy operator and 80 00:07:57,892 --> 00:08:02,516 the angular momentum is also part of the kinetic energy operator but it looks like 81 00:08:02,516 --> 00:08:07,701 a potential of the type l squared. Keep in mind this is still an operate, we 82 00:08:07,701 --> 00:08:10,450 haven't said anything more about it there. 83 00:08:10,450 --> 00:08:16,660 L squared over 2 mr squared so it's so we can actually incorporate, we can think 84 00:08:16,660 --> 00:08:23,140 about a combined potential that consists both of the externally applied potential 85 00:08:23,140 --> 00:08:32,210 and the angular momentum. So a couple of notes about this. 86 00:08:32,210 --> 00:08:37,535 first of all we didn't obtain this by transforming to spherical coordinates in 87 00:08:37,535 --> 00:08:42,710 any explicit way, but by just taking considering the properties of the angular 88 00:08:42,710 --> 00:08:47,435 momentum operator, we see that it motivates a description of the problem in 89 00:08:47,435 --> 00:08:52,160 terms of the spherical coordinate r and then the, the angles of the source of 90 00:08:52,160 --> 00:09:00,624 theta a and b. So basically the way that the angular 91 00:09:00,624 --> 00:09:04,912 momentum enters directly in the Hamiltonian, does, indicate the, an 92 00:09:04,912 --> 00:09:10,899 appropriate choice of perspective from the spherical coordinate system. 93 00:09:13,570 --> 00:09:17,665 So the second point I'd like to make is that there are many other equations in 94 00:09:17,665 --> 00:09:22,435 physics that a involve a Laplacian operator dell squared. 95 00:09:22,435 --> 00:09:28,620 Electromagnetism, acoustics, motion in fluids optics and so on. 96 00:09:28,620 --> 00:09:34,015 So any time that you have An equation with a Laplacian, you have something like 97 00:09:34,015 --> 00:09:40,861 quantum angular momentum. in fact, there's, the, expansion of the 98 00:09:40,861 --> 00:09:47,002 electromagnetic field in, in states of definite or angular momentum actually 99 00:09:47,002 --> 00:09:53,232 resembles the way that we describe the short, the the angular momentum of the 100 00:09:53,232 --> 00:10:00,819 Schrodinger equation. So, your better understanding of angular 101 00:10:00,819 --> 00:10:05,375 momentum and quantum mechanics, I think, it will give you, as it did me, a renewed 102 00:10:05,375 --> 00:10:10,422 appreciation of many classical problems of physics. 103 00:10:10,422 --> 00:10:15,120 third point I'd like to say it this way. Angular momentum is more powerful than 104 00:10:15,120 --> 00:10:17,450 gravity. It's more powerful than electricity. 105 00:10:17,450 --> 00:10:24,380 By this, I mean the the angular, the effect of potential angular momentum, V 106 00:10:24,380 --> 00:10:33,310 effective, goes as one over r squared, whereas for gravity. 107 00:10:33,310 --> 00:10:37,040 The effective potential goes as 1 over r, and the same for electricity. 108 00:10:37,040 --> 00:10:43,773 So this is a highly repulsive potential. It to, it has a greater, approach to 109 00:10:43,773 --> 00:10:49,302 infinity as r goes to 0 than does any of the, do any of the conventional 110 00:10:49,302 --> 00:10:56,675 potentials. In in physics so that, that it shows how 111 00:10:56,675 --> 00:11:02,638 the, that small r, the angular momentum will always dominate over any other type 112 00:11:02,638 --> 00:11:10,430 of the pairwise potential in the problem, as it turns out. 113 00:11:11,790 --> 00:11:15,780 finally we want to generalize this approach to N dimensional systems. 114 00:11:15,780 --> 00:11:19,066 So instead of thinking about, of a particle moving in three dimensions, we 115 00:11:19,066 --> 00:11:23,590 have to think of a particle moving in, in, well hundreds of dimensions. 116 00:11:23,590 --> 00:11:31,610 Now you may ask, well how relevant to the world in which we live. 117 00:11:31,610 --> 00:11:35,831 And the answer is that in some cases to discuss collective particle motions, for 118 00:11:35,831 --> 00:11:40,030 example lets say you have a motion of two particles. 119 00:11:40,030 --> 00:11:45,055 Well there are six dimensions in which two particles move and in some cases if 120 00:11:45,055 --> 00:11:49,330 you want to identify collective coordinates it makes sense to start 121 00:11:49,330 --> 00:11:54,280 thinking of the, the two particle has something that's being described by a 122 00:11:54,280 --> 00:11:59,230 six, six dimension and to see about the most general possible description that 123 00:11:59,230 --> 00:12:08,326 one can get in a six dimension. And this leads to a generalized angular 124 00:12:08,326 --> 00:12:11,316 momentum. Which has to do with the behavior, the 125 00:12:11,316 --> 00:12:14,846 wave function. Under rotations in the space that 126 00:12:14,846 --> 00:12:18,748 includes all particles. So this is something that's been applied 127 00:12:18,748 --> 00:12:22,336 in nuclear and atomic physics, and it introduces many of the same concepts that 128 00:12:22,336 --> 00:12:26,330 we find in the quantum theory of the angular momentum.