1 00:00:00,830 --> 00:00:03,110 Welcome back everybody. I'm Charles Clark. 2 00:00:03,110 --> 00:00:06,248 This is Exploring Quantum Physics. And today we're going to turn to the 3 00:00:06,248 --> 00:00:09,512 problem of angular momentum, one of most fundamental symmetries in quantum 4 00:00:09,512 --> 00:00:13,812 physics. We say everything revolves around angular 5 00:00:13,812 --> 00:00:18,860 momentum and we're going to start looking at some of the explicit solutions. 6 00:00:18,860 --> 00:00:22,143 We'll try to show you different ways of calculating them from first principles, 7 00:00:22,143 --> 00:00:25,377 and then with pointers to the literature so that you can, you can develop better 8 00:00:25,377 --> 00:00:30,180 understanding if you so choose. I hope you enjoy it. 9 00:00:30,180 --> 00:00:34,338 Now, home works of week six had some problems of tensor analysis, a lot of 10 00:00:34,338 --> 00:00:38,842 vector algebra. some of you found that quite frustrating, 11 00:00:38,842 --> 00:00:42,720 I think. others, I believe, found it enjoyable. 12 00:00:42,720 --> 00:00:45,840 in any event, we're going to do a little bit more of that. 13 00:00:45,840 --> 00:00:49,340 And please don't be too alarmed by everything that's up here. 14 00:00:49,340 --> 00:00:53,580 These are really the core, core identities that I'm putting for 15 00:00:53,580 --> 00:01:00,140 completeness. the uh, [INAUDIBLE] tensor. 16 00:01:00,140 --> 00:01:05,340 Just the basic notation about, our change in notation in order to simplify 17 00:01:05,340 --> 00:01:09,622 calculations. the definition and the position vector, 18 00:01:09,622 --> 00:01:14,190 in terms of the old and the new notation, the radius. 19 00:01:14,190 --> 00:01:17,050 And then the definition of the angular momentum operator and the square of the 20 00:01:17,050 --> 00:01:21,476 angular momentum operator. So we're going to now use some of these 21 00:01:21,476 --> 00:01:26,340 to generate Eigenfunctions of the angular momentum operator. 22 00:01:27,640 --> 00:01:31,000 And I think you'll find this approach very useful. 23 00:01:31,000 --> 00:01:35,380 Especially if you work in other fields so electrical engineering, or anything that 24 00:01:35,380 --> 00:01:39,640 studies acoustics or wave motions or o, optics, you'll find a lot of counterparts 25 00:01:39,640 --> 00:01:44,110 in those areas to what we're studying here. 26 00:01:45,710 --> 00:01:49,700 We'll start by showing what I hope you think is quite an amazing result. 27 00:01:49,700 --> 00:01:54,150 If we take any function of the radial coordinate only. 28 00:01:54,150 --> 00:01:59,912 it, it turns out to be an Eigenfunction of any given complement of the angular 29 00:01:59,912 --> 00:02:05,470 momentum operator and of the square itself. 30 00:02:05,470 --> 00:02:08,090 Okay. So let's see how this goes. 31 00:02:08,090 --> 00:02:14,860 we start with, Just applying the l, l sub u upon psi. 32 00:02:14,860 --> 00:02:20,518 And then that is, very straightforward, expansion of the angular momentum 33 00:02:20,518 --> 00:02:26,165 operator. And now what you see is that the, effect 34 00:02:26,165 --> 00:02:31,861 of the derivative on a, a radial function, function of the radius only can 35 00:02:31,861 --> 00:02:40,555 be evaluated by the chain rule. And so you see we get an expression that 36 00:02:40,555 --> 00:02:46,096 boils down to this. So we'll go with a little in-line quiz to 37 00:02:46,096 --> 00:02:49,540 see It get, get, give you a gauge of your 38 00:02:49,540 --> 00:02:54,520 understanding of how to deal with an expression encountered like this. 39 00:02:57,140 --> 00:03:00,800 So I hope that you saw that this vanished, this term vanishes by symmetry 40 00:03:00,800 --> 00:03:04,704 relations because it's just a sum over dummy indices but with a, a sign change 41 00:03:04,704 --> 00:03:10,862 associated with the permutation of. The two indices, so that term must 42 00:03:10,862 --> 00:03:16,210 necessarily be zero. So we've recorded that fact. 43 00:03:17,220 --> 00:03:23,172 And then the next one is is, is easy because if l u of psi of r zero, then 44 00:03:23,172 --> 00:03:31,133 anything else applied to that l u. Of psi of r is also 0, including lu 45 00:03:31,133 --> 00:03:35,794 squared, and then the sum of the squares of the l sub u, which is the total 46 00:03:35,794 --> 00:03:41,380 angular momentum vector is also equal to 0. 47 00:03:41,380 --> 00:03:44,964 So we've shown that any, any function that is a function of the radius alone is 48 00:03:44,964 --> 00:03:48,580 an Eigenfunction. Well, it's an Eigenfunction of any 49 00:03:48,580 --> 00:03:52,080 complement of the angular momentum operator. 50 00:03:52,080 --> 00:03:58,930 It is conventional to take the complement along the third axis. 51 00:03:58,930 --> 00:04:02,230 This traditionally the z axis, as the quantization axis. 52 00:04:02,230 --> 00:04:04,710 But we'll see that that's not always the case. 53 00:04:05,830 --> 00:04:12,920 Anyway, let's proceed and see what the implications of this discovery are. 54 00:04:12,920 --> 00:04:16,375 Several weeks back, we tried a sort of simple constructive technique. 55 00:04:16,375 --> 00:04:20,170 It's not a general purpose technique, but just a way of getting some understanding 56 00:04:20,170 --> 00:04:23,910 of the Schrodinger equation by applying it to functions that we already know, the 57 00:04:23,910 --> 00:04:29,882 ones that we're familiar with. And seeing what, what type of potential 58 00:04:29,882 --> 00:04:35,364 would support those as an Eigenfunction. So what we want is to use a function of 59 00:04:35,364 --> 00:04:39,853 the radius that is, is, is normalized, or normalizable, that is it can be divided 60 00:04:39,853 --> 00:04:44,805 by another constant to make this identity. 61 00:04:44,805 --> 00:04:50,195 Correct. And in the previous case, we chose a a 62 00:04:50,195 --> 00:04:56,560 radial Gaussian wave function, of this, of this form, which is in fact the the 63 00:04:56,560 --> 00:05:02,735 solution for the ground state of a three-dimensional isotropic Harmonic 64 00:05:02,735 --> 00:05:09,608 oscillator. You see that this term here is just equal 65 00:05:09,608 --> 00:05:13,160 to r squared, so this is a radial solution. 66 00:05:13,160 --> 00:05:19,300 And it therefore has the anglar momentum, 0, that we just described. 67 00:05:19,300 --> 00:05:24,110 Well, how many functions do you know that are radially symmetric and die off in 68 00:05:24,110 --> 00:05:29,780 infinity, in a, in a way that's sufficient to, 69 00:05:29,780 --> 00:05:32,200 To allow the normalization integral to be computed. 70 00:05:32,200 --> 00:05:38,150 Well, of course, one can invent a whole number of, contrive functions that do 71 00:05:38,150 --> 00:05:41,210 that. But, to me, 72 00:05:41,210 --> 00:05:45,650 This one here sort of stands out. It's a simple, exponential decay in all 73 00:05:45,650 --> 00:05:50,420 directions. it's easy to, use, and so it's a good 74 00:05:50,420 --> 00:06:00,850 choice for this constructive approach. So here I show a cut through that 75 00:06:00,850 --> 00:06:04,984 function. This is radially symmetric, but I cut 76 00:06:04,984 --> 00:06:09,592 through the origin, r equals 0, along, it doesn't matter which axis, any axis looks 77 00:06:09,592 --> 00:06:14,226 the same. And this shows something that you might 78 00:06:14,226 --> 00:06:19,950 not have appreciated before, which is the wave function has a cusp at the origin. 79 00:06:19,950 --> 00:06:24,642 In fact the second derivative of the wave function, you know, with respect to x, is 80 00:06:24,642 --> 00:06:30,061 actually a singular there. Well that corresponds, in fact, to 81 00:06:30,061 --> 00:06:36,683 something else that we find if you just if you take this function and put it into 82 00:06:36,683 --> 00:06:43,219 this equation you will find that the the, the potential that is implied by the use 83 00:06:43,219 --> 00:06:52,802 of this function is just the Coulomb funct-, Coulomb potential. 84 00:06:52,802 --> 00:06:59,848 Again, Gaussian units, Gaussian units. and this is, in fact, the ground state 85 00:06:59,848 --> 00:07:09,188 wave function for the Hydrogen atom. it's [INAUDIBLE] with energy, given by 86 00:07:09,188 --> 00:07:13,434 this expression. in, whatever the Bohr radius is you've 87 00:07:13,434 --> 00:07:17,370 seen it before, but here it is made more explicit. 88 00:07:17,370 --> 00:07:21,834 Now there was a, one of the problems in, one of the questions in week five 89 00:07:21,834 --> 00:07:26,514 homework had to do with approximating using a Gaussian approximation to this 90 00:07:26,514 --> 00:07:33,354 function in a variational calculation. Well, a Gaussian, you see, is going to 91 00:07:33,354 --> 00:07:36,400 have a rather smooth profile of the origin. 92 00:07:36,400 --> 00:07:39,996 So this type of cusp behavior is not something that's well represented by a 93 00:07:39,996 --> 00:07:43,970 Gaussian wave function. As a matter of fact, there were two 94 00:07:43,970 --> 00:07:47,718 variational calculations in that week's homework. 95 00:07:47,718 --> 00:07:53,244 And in some sense, the. Functions that you were being asked to 96 00:07:53,244 --> 00:07:58,114 approximate were the same function. This is actually also, if you think about 97 00:07:58,114 --> 00:08:02,998 it, the one dimensional cut here that gives functions the same as a function of 98 00:08:02,998 --> 00:08:09,600 a ground state of an attractive delta function potential of one dimension. 99 00:08:09,600 --> 00:08:12,561 So this is a rather interesting correspondence between the delta function 100 00:08:12,561 --> 00:08:15,979 in one dimension and the Hydrogen atom in three dimensions. 101 00:08:18,820 --> 00:08:22,980 So just for convenience of further notation we'll just, we'll write that, 102 00:08:22,980 --> 00:08:27,782 this wave function psi sub 1s. The 1s is the conventional designation 103 00:08:27,782 --> 00:08:31,500 for the quantum numbers that bounced eight of Hydrogen. 104 00:08:31,500 --> 00:08:35,252 Maybe this is also a good time for you to refresh your recollection of the meaning 105 00:08:35,252 --> 00:08:39,130 of the Bohr radius. And so that will be the subject of an 106 00:08:39,130 --> 00:08:43,870 in-video quiz. Okay. 107 00:08:43,870 --> 00:08:49,850 So we're going to break at this point, and you might review this material. 108 00:08:49,850 --> 00:08:55,040 See if you can follow the derivations. I also, recommend that you use your time 109 00:08:55,040 --> 00:08:59,990 just to do a few simple calculations, like verifying that this wave function is 110 00:08:59,990 --> 00:09:06,278 appropriately normalized. You know, I could make a mistake and I'll 111 00:09:06,278 --> 00:09:11,120 give a nice reward to the first person who identifies that in the student form. 112 00:09:11,120 --> 00:09:15,655 to do this, it requires integration over polar coordinate angles. 113 00:09:15,655 --> 00:09:18,950 A theta in phi. Now we're going to discuss those later. 114 00:09:18,950 --> 00:09:23,302 We haven't really dealt with that yet, but you can find pointers on how to do 115 00:09:23,302 --> 00:09:29,310 three dimensional integrals many places in the open literature. 116 00:09:29,310 --> 00:09:33,500 And so this is a sort of exercise for those of you who want some practice. 117 00:09:33,500 --> 00:09:38,278 those of you who had trouble. With the homework problems of week five, 118 00:09:38,278 --> 00:09:44,440 maybe this is a good way a good approach to sharpening your skills. 119 00:09:44,440 --> 00:09:47,450 Okay, thanks for listening and hope to see you again.