1 00:00:00,840 --> 00:00:04,476 Welcome back everyone. I'm Charles Clark, and this is exploring 2 00:00:04,476 --> 00:00:07,923 Quantum Physics. Now we're going to continue on from the 3 00:00:07,923 --> 00:00:12,278 rather si, simple example of zero Angular Momentum, that we looked at, in the 4 00:00:12,278 --> 00:00:16,804 previous part. And to go on to see how to systematically 5 00:00:16,804 --> 00:00:21,060 develop things to look at more and more complex systems. 6 00:00:21,060 --> 00:00:25,260 Just to refresh your memory in the previous part we just looked at the 7 00:00:25,260 --> 00:00:32,000 problem of a purely radial function and found that it has zero angular momentum. 8 00:00:32,000 --> 00:00:37,304 So if you think about angular momentum as an operator that induces rotations. 9 00:00:37,304 --> 00:00:41,792 perhaps that's and intuitive way of understanding why a function that just 10 00:00:41,792 --> 00:00:48,670 is, is, has no variation as one goes out in in different directions of the origin. 11 00:00:48,670 --> 00:00:53,025 It's just, it's just something if you draw the, the countours of the function, 12 00:00:53,025 --> 00:00:57,440 they're circles. Pardon my draftmanship. 13 00:00:57,440 --> 00:01:00,790 maybe this is a good way of thinking about why. 14 00:01:00,790 --> 00:01:04,850 is the case the radial function has no angular momentum. 15 00:01:04,850 --> 00:01:08,630 Well, having settled that, let's look among other familiar things, to find 16 00:01:08,630 --> 00:01:12,540 functions with definite nontrivial angular momentum. 17 00:01:12,540 --> 00:01:17,460 It's always a good idea when starting with basic concepts to, look at the 18 00:01:17,460 --> 00:01:24,480 simplest variables that you have at hand. So, in our case certainly the, the 19 00:01:24,480 --> 00:01:29,680 coordinates of a particle, the three spatial coordinates are candidates for 20 00:01:29,680 --> 00:01:36,920 that sort of investigation. So let's, let's see how they what happens 21 00:01:36,920 --> 00:01:42,802 to them when they're acted upon by the a minimum operator. 22 00:01:42,802 --> 00:01:47,340 Now here, look at this thing here, I just want to make it very clear to you. 23 00:01:48,380 --> 00:01:52,906 This is a case where the index is not repeated, we have 1 index u and one index 24 00:01:52,906 --> 00:01:56,715 a. So as a stand alone expression, this is 25 00:01:56,715 --> 00:02:03,437 really just an indication of what happens when you pick a value of u like 1, right? 26 00:02:03,437 --> 00:02:08,117 And pick a value of a like 2. Then you're, you're going to get, you 27 00:02:08,117 --> 00:02:11,879 should, stick with an expression that depends specifically on the variables 1 28 00:02:11,879 --> 00:02:19,375 and 2, which are taking fixed values. Okay, so we just put in the, the fran 29 00:02:19,375 --> 00:02:26,698 levitivity answer thusly. And now the momentum, acting on a 30 00:02:26,698 --> 00:02:32,033 position operate, on a position coordinate, is h bar over i, if the 31 00:02:32,033 --> 00:02:39,890 momentum has the same coordinate as the position, delta w a. 32 00:02:40,910 --> 00:02:44,362 other words, well it's a delta b a. It's one if w is equal to a, it's zero 33 00:02:44,362 --> 00:02:49,855 otherwise. And so when we then now contract, we're 34 00:02:49,855 --> 00:02:56,434 now so, summing over w here. So you see what that means is that we 35 00:02:56,434 --> 00:03:04,660 replaced w by a. e u v w goes to e u v a. 36 00:03:04,660 --> 00:03:09,390 And then that is well, that, this is, this is, this is the answer. 37 00:03:09,390 --> 00:03:15,986 So it depends upon the two, the two fixed index values, u and a, and shows how you 38 00:03:15,986 --> 00:03:24,972 basically, those, at most one value of v. For which this, relativity center doesn't 39 00:03:24,972 --> 00:03:28,964 vanish. Okay, so now, and this is probably 40 00:03:28,964 --> 00:03:33,690 something, It's a good idea for you to try to do 41 00:03:33,690 --> 00:03:38,350 these manipulations, satisfy yourselves that you understand how they work. 42 00:03:38,350 --> 00:03:41,720 Occasionally, there are things like factors of two that show up. 43 00:03:41,720 --> 00:03:45,980 That aren't necessarily so obvious, but again, this is strictly mechanical. 44 00:03:45,980 --> 00:03:51,245 So to, to, to calculate the L squared acting on x of a, we just do this Lu Lu 45 00:03:51,245 --> 00:03:55,517 xa. So now the, the repeated index u is 46 00:03:55,517 --> 00:04:03,016 summed over, implicitly summed over. And so just by expanding, we see this 47 00:04:03,016 --> 00:04:09,378 expression. We have a, a, a, a product of two 48 00:04:09,378 --> 00:04:16,280 levitivitus symbols. And a, a, again, a contraction on 49 00:04:16,280 --> 00:04:21,493 indices. and so this in particular here is an 50 00:04:21,493 --> 00:04:25,970 example that you might want to investigate. 51 00:04:25,970 --> 00:04:30,240 I think something like this came up in the derivation of the relationship 52 00:04:30,240 --> 00:04:35,110 between the angular momentum And the Laplacian operator. 53 00:04:35,110 --> 00:04:39,250 There's a factor of three there that one had to, that, that emerged in the 54 00:04:39,250 --> 00:04:45,590 calculation that required a little bit of thinking about what you're doing. 55 00:04:45,590 --> 00:04:48,060 Basically taking the trace of the identity matrix there. 56 00:04:48,060 --> 00:04:53,455 but in any event, what we see is that indeed if you apply the square of the 57 00:04:53,455 --> 00:05:00,410 angular momentum operator on the, position coordinate alone. 58 00:05:00,410 --> 00:05:04,502 You get back that coordinate. So, each val-, each xa, for a equal one, 59 00:05:04,502 --> 00:05:11,450 two, or three, is an eigenfunction of the square of the angular momentum operator. 60 00:05:11,450 --> 00:05:20,152 corresponding to an L value of 2. Now in fact these functions that we've 61 00:05:20,152 --> 00:05:24,194 just found, are widely used in quantum chemistry. 62 00:05:24,194 --> 00:05:31,180 they're the so called the, the, p, well it's often the px, py the pz orbitals. 63 00:05:31,180 --> 00:05:38,250 So, if you have a if you have a, if you have a molecule with, with several 64 00:05:38,250 --> 00:05:49,200 several atoms, you often will, you'll put a, a line a, a p orbital in between them. 65 00:05:49,200 --> 00:05:52,125 And so you, you want a, a directed orbital. 66 00:05:52,125 --> 00:05:57,290 that has well, let's, let's see, let's see what it looks like. 67 00:05:57,290 --> 00:06:04,106 So just what I've done here is multiplied these functions by the common gals in 68 00:06:04,106 --> 00:06:11,340 here, just to get a compact, get a compact orbital. 69 00:06:11,340 --> 00:06:17,960 That has this form, so a blue means a positive sign, red a negative. 70 00:06:17,960 --> 00:06:22,820 And so the P1 here is oriented along the x1 axis. 71 00:06:22,820 --> 00:06:28,969 P2 along the x2 axis, P3 along the x3, and so on. 72 00:06:31,600 --> 00:06:37,385 Now I'm going to suggest that you answer a question about the parity of these free 73 00:06:37,385 --> 00:06:40,940 orbitals. Do they have, do they have a definite 74 00:06:40,940 --> 00:06:44,000 parity, that is, do they have a symmetry under the inversion of all the coordinate 75 00:06:44,000 --> 00:06:47,854 systems. And are their parities the same or 76 00:06:47,854 --> 00:06:50,903 different? Okay, so I think that the answer to that 77 00:06:50,903 --> 00:06:54,905 question about the parity should be, should have been the way to think about 78 00:06:54,905 --> 00:06:58,911 it, is. These are the functions of the coordinate 79 00:06:58,911 --> 00:07:02,158 themselves. So of course, they just change sine when 80 00:07:02,158 --> 00:07:05,469 you change the sin of all the coordinates. 81 00:07:05,469 --> 00:07:11,610 Now, we made an easy choice, and found three eigenfunctions that have l equal to 82 00:07:11,610 --> 00:07:17,200 one. And if you remember the summary from a 83 00:07:17,200 --> 00:07:24,464 past lecture, and the bonus homework The num, the number of independent number 84 00:07:24,464 --> 00:07:30,340 of independent states with a given angular momentum Is 2L plus 1. 85 00:07:30,340 --> 00:07:33,808 So they're, there are just three 4L equal 1 and these are, these are perfectly good 86 00:07:33,808 --> 00:07:38,125 independent choices. But there's another choice of 87 00:07:38,125 --> 00:07:43,070 eigenfunctions which is more often used by physicists than by the chemists. 88 00:07:44,770 --> 00:07:48,805 they're complex value. So I've, I've just made up a notation 89 00:07:48,805 --> 00:07:54,943 here, chi 1, 1. Is minus the quantity x1 plus ix2 divided 90 00:07:54,943 --> 00:08:00,520 by the root 2. Chi 1,0 is x 3, as before. 91 00:08:00,520 --> 00:08:05,530 And the chi 1 minus 1 is x1 minus ix2, divided by root 2. 92 00:08:05,530 --> 00:08:13,342 And know there's a sign change, sign change between here and here, and that 93 00:08:13,342 --> 00:08:22,380 is, introduced, so that the well, let's just see. 94 00:08:22,380 --> 00:08:28,297 You can verify, I suggest that you do this, that when you apply L3 to any one 95 00:08:28,297 --> 00:08:35,035 of these, chi 1,m. You get an eigenfunction, which is mh bar 96 00:08:35,035 --> 00:08:40,405 times chi 1m. And then, these, the raising and lowering 97 00:08:40,405 --> 00:08:47,245 operators, which are useful for actually generating the entire set of angular 98 00:08:47,245 --> 00:08:56,500 momentum I, eigenfunctions for a given L. If you have one of them alone with this, 99 00:08:56,500 --> 00:09:02,140 this choice of phases, we get the correct, the conventional, factor 100 00:09:02,140 --> 00:09:09,055 associated with the raising and lowering operator. 101 00:09:09,055 --> 00:09:16,740 In principle, this, this factor can have random phase, random phase. 102 00:09:16,740 --> 00:09:23,814 It's magnitude is it's magnitude is determined. 103 00:09:23,814 --> 00:09:28,613 But by this choice of convention here, we make sure that the phase is uniform. 104 00:09:28,613 --> 00:09:33,232 Now there's a very important property of the chi 1. 105 00:09:35,110 --> 00:09:39,860 And this again is something that is fairly easy to work out for yourself. 106 00:09:39,860 --> 00:09:45,800 That is, if you take this to the Lth power, then, then that it is, turns out 107 00:09:45,800 --> 00:09:53,610 to be an eigenfunction of with of, of L3. With the projections L, which is the 108 00:09:53,610 --> 00:09:59,660 maximum projection allowed. And then it also is an eigenfunction of 109 00:09:59,660 --> 00:10:05,174 the total angular momentum. And this is, this is worth thinking about 110 00:10:05,174 --> 00:10:13,235 for a while. and you can use see how this, is affected 111 00:10:13,235 --> 00:10:23,780 by L minus and L plus. And construct L squared from L plus L 112 00:10:23,780 --> 00:10:34,790 minus, or, the other way around, and Lz squared. 113 00:10:34,790 --> 00:10:39,822 And I think you'll, you'll be able to show this relationship with a little, 114 00:10:39,822 --> 00:10:48,950 with a little bit of practice. So here's a summary sheet if you want to 115 00:10:48,950 --> 00:10:53,875 work on that. And this enables you I think you, you may 116 00:10:53,875 --> 00:11:02,080 be able to show with some thought that these two identities are true. 117 00:11:02,080 --> 00:11:05,850 And so, you see, we haven't actually, we've only specified a rather primitive 118 00:11:05,850 --> 00:11:09,300 object here. It's, it's actually just a, a Linear 119 00:11:09,300 --> 00:11:12,716 function of X1 and X2. When we find it, when we ta, when we 120 00:11:12,716 --> 00:11:16,345 raise it to a certain power. We get an eigenfunction of the angular 121 00:11:16,345 --> 00:11:20,266 momentum of the operator of the direction three, and of the square. 122 00:11:20,266 --> 00:11:29,566 And then, from that we can construct the general, an arbitrary state of angular 123 00:11:29,566 --> 00:11:35,571 momentum. Like arbitrary, eigenfunction angular 124 00:11:35,571 --> 00:11:40,460 momentum as follows. we just, we just we have the LL state 125 00:11:40,460 --> 00:11:45,160 which is given by this, this primitive here. 126 00:11:45,160 --> 00:11:50,848 The L kind one, one. And then we just apply L minus to it, and 127 00:11:50,848 --> 00:11:57,190 applying L minus to any object is quite straight forward. 128 00:12:00,290 --> 00:12:03,910 And we know that the action must be of this form. 129 00:12:03,910 --> 00:12:08,870 So from L, L we can generate. The next date down by application of 130 00:12:08,870 --> 00:12:14,624 lowering operator, and keep on going. I'd like to point out that, by the way, 131 00:12:14,624 --> 00:12:23,840 that we have built these up starting with something which is just, you know, a 132 00:12:23,840 --> 00:12:35,967 power of the sum of two coordinates. If you think about it, this means that 133 00:12:35,967 --> 00:12:42,736 the the LL item function. Is a polynomial of homogeneous degree in 134 00:12:42,736 --> 00:12:46,240 the spatial coordinates. But let me say it another way. 135 00:12:46,240 --> 00:12:52,243 It's a multinomial of homogeneous degree, meaning, it's, consists of a, of a sum of 136 00:12:52,243 --> 00:12:58,480 our terms, Actually there is I, there is no X3 here. 137 00:12:58,480 --> 00:13:03,990 I mean I've just, I've just well what I guess I, let me generalize this. 138 00:13:03,990 --> 00:13:15,990 But when, when we when we the expression is valued is, is, the expression here is 139 00:13:15,990 --> 00:13:23,490 valid LM. So but basically, the way that you can 140 00:13:23,490 --> 00:13:30,980 think of an eigenfunction of angular momentum is in terms of polynomial form. 141 00:13:30,980 --> 00:13:36,770 That sometimes leads to great advantages in solving certain types of problems. 142 00:13:36,770 --> 00:13:42,290 now the actual, the actual set of, range of coefficients is fairly complicated. 143 00:13:42,290 --> 00:13:45,860 And these things have been extensively studied mathematically. 144 00:13:45,860 --> 00:13:50,480 And so the, the best way to get a better impression of what's going on is to 145 00:13:50,480 --> 00:13:58,353 consult reference works. Now, here's another remark that you can 146 00:13:58,353 --> 00:14:03,155 infer from our discussion. So, we were, we were, constructing 147 00:14:03,155 --> 00:14:09,615 eigenfunctions of the angular momentum operator from various sums and products 148 00:14:09,615 --> 00:14:16,019 of these coordinates. Well every one of these contains an r. 149 00:14:17,140 --> 00:14:22,100 So, we know that the, the radial coordinate itself has no angular momentum 150 00:14:22,100 --> 00:14:28,480 connotations. It can just basically be removed from the 151 00:14:28,480 --> 00:14:35,198 from the definition of an eigenfunction. And so, the, the most useful set of 152 00:14:35,198 --> 00:14:39,245 eigenfunctions, for many applications of angular momentum operation or functions 153 00:14:39,245 --> 00:14:42,854 of the angles. Only in the conventional, folder 154 00:14:42,854 --> 00:14:47,318 coordinate system. So, I think you can see that if you, you 155 00:14:47,318 --> 00:14:56,002 just take the previous argument about having products of these various things. 156 00:14:56,002 --> 00:15:01,800 We would get an overall r to the k comma to everything. 157 00:15:01,800 --> 00:15:09,150 And then, then there would be products of, you know, sin to the a theta cosin to 158 00:15:09,150 --> 00:15:17,100 the b of theta. sin to the c of v, cosin to the d of v. 159 00:15:17,100 --> 00:15:25,640 so this means that in fact, there's a universal description that's valid for 160 00:15:25,640 --> 00:15:31,904 all central potentials. All potentials that depend on the radial 161 00:15:31,904 --> 00:15:34,432 coordinate only. That is, where angular momentum is a good 162 00:15:34,432 --> 00:15:37,839 quantum number. We can define it just in terms of the 163 00:15:37,839 --> 00:15:42,490 angles only is independent of the, of the distance from the origin. 164 00:15:45,750 --> 00:15:49,719 Now to check your understanding, I'm going to pose to you a question that has 165 00:15:49,719 --> 00:15:53,482 to do with identifying the angular momentum. 166 00:15:53,482 --> 00:15:57,512 By inspection of the equation of mo, of the equation of, of a system, in terms of 167 00:15:57,512 --> 00:16:04,751 these Cartesian coordinates. Okay, I hope that was comprehensible, 168 00:16:04,751 --> 00:16:11,279 And you could see that the, the one of those functions was radially symmetric, 169 00:16:11,279 --> 00:16:18,800 that what has an increment of zero, the other was 170 00:16:18,800 --> 00:16:29,140 The the, the product of the the two the X1 plus iX2 function. 171 00:16:29,140 --> 00:16:32,930 Then the third was actually just a weighted difference of those two. 172 00:16:32,930 --> 00:16:38,352 So the third one couldn't possibly have be an eigenfunction of angular momentum. 173 00:16:38,352 --> 00:16:42,256 Because it consisted of linear combination of eigenfunctions is two 174 00:16:42,256 --> 00:16:48,220 different values of angular momentum. Now, I mentioned that I've repeatedly 175 00:16:48,220 --> 00:16:54,060 said, you know, how, how, important this subject is, and it does occur in the most 176 00:16:54,060 --> 00:17:03,441 diverse type of systems. For example, here's a recent publication 177 00:17:03,441 --> 00:17:07,701 from April 2013, about the use of spherical harmonics in programming of 178 00:17:07,701 --> 00:17:13,405 video games. now, you might ask why, and here's what 179 00:17:13,405 --> 00:17:19,296 the, is stated, The use of Spherical Harmonics as a basis 180 00:17:19,296 --> 00:17:25,661 allows for very fast rendering and it sort of provides some optimal accuracy 181 00:17:25,661 --> 00:17:32,484 for a given speed of processing. So you might think of a spherical 182 00:17:32,484 --> 00:17:37,430 harmonic as being the equivalent over the surface of a sphere. 183 00:17:37,430 --> 00:17:40,470 To what the Fourier Transform is in space. 184 00:17:40,470 --> 00:17:43,942 And as you saw in Victor's lectures Fourier Transform is an incredibly 185 00:17:43,942 --> 00:17:48,328 powerful tool in quantum physics. Also widely used in engineering signal 186 00:17:48,328 --> 00:17:52,674 processing and to be like. And so again, there are many applications 187 00:17:52,674 --> 00:17:57,026 of spherical harmonics in, in fields such as imaging, mapping the Earth's 188 00:17:57,026 --> 00:18:02,411 gravitational potential. And wide variety of others, and so even 189 00:18:02,411 --> 00:18:06,386 if you're not going to be a professional physicist. 190 00:18:06,386 --> 00:18:10,351 You might benefit quite a bit from applying the knowledge you've got from 191 00:18:10,351 --> 00:18:16,015 the subject here to problems of characterization of motions on a sphere. 192 00:18:16,015 --> 00:18:19,630 Okay, hope to see you again. Be back soon.