1 00:00:00,050 --> 00:00:01,670 The following content is provided 2 00:00:01,670 --> 00:00:03,810 under a Creative Commons license. 3 00:00:03,810 --> 00:00:06,540 Your support will help MIT OpenCourseWare continue 4 00:00:06,540 --> 00:00:10,120 to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,700 To make a donation or to view additional materials 6 00:00:12,700 --> 00:00:18,105 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,105 --> 00:00:18,730 at ocw.mit.edu. 8 00:00:22,740 --> 00:00:25,940 PROFESSOR: All right, it is time to get started. 9 00:00:25,940 --> 00:00:30,750 Thanks for coming for this cold and rainy 10 00:00:30,750 --> 00:00:32,540 Wednesday before Thanksgiving. 11 00:00:35,490 --> 00:00:40,150 Today we're supposed to talk about the radial equation. 12 00:00:40,150 --> 00:00:43,710 That's our main subject today. 13 00:00:43,710 --> 00:00:47,640 We discussed last time the states 14 00:00:47,640 --> 00:00:51,430 of angular momentum from the abstract viewpoint, 15 00:00:51,430 --> 00:00:55,460 and now we make contact with some important problems, 16 00:00:55,460 --> 00:01:00,390 and differential equations, and things like that. 17 00:01:00,390 --> 00:01:05,730 And there's a few concepts I want to emphasize today. 18 00:01:05,730 --> 00:01:09,570 And basically, the main concept is 19 00:01:09,570 --> 00:01:13,480 that I want you to just become familiar with what we would 20 00:01:13,480 --> 00:01:18,330 call the diagram, the key diagram for the states 21 00:01:18,330 --> 00:01:23,250 of a theory, of a particle in a three dimensional potential. 22 00:01:23,250 --> 00:01:27,860 I think you have to have a good understanding of what it looks, 23 00:01:27,860 --> 00:01:30,700 and what is special about it, and when 24 00:01:30,700 --> 00:01:32,790 it shows particular properties. 25 00:01:32,790 --> 00:01:37,730 So to begin with, I'll have to do a little aside 26 00:01:37,730 --> 00:01:43,296 on a object that is covered in many courses. 27 00:01:43,296 --> 00:01:46,050 I don't know to what that it's covered, 28 00:01:46,050 --> 00:01:48,920 but it's the subject of spherical harmonics. 29 00:01:48,920 --> 00:01:53,730 So we'll talk about spherical harmonics for about 15 minutes. 30 00:01:53,730 --> 00:01:56,295 And then we'll do the radial equation. 31 00:01:56,295 --> 00:01:59,360 And for the radial equation, after we discuss it, 32 00:01:59,360 --> 00:02:01,550 we'll do three examples. 33 00:02:01,550 --> 00:02:05,610 And that will be the end of today's lecture. 34 00:02:05,610 --> 00:02:10,830 Next time, as you come back from the holiday next week, 35 00:02:10,830 --> 00:02:14,470 we are doing the addition of angular momentum basically. 36 00:02:14,470 --> 00:02:19,070 And then the last week, more examples and a few more 37 00:02:19,070 --> 00:02:23,420 things for emphasis to understand it all well. 38 00:02:23,420 --> 00:02:26,610 All right, so in terms of spherical harmonics, 39 00:02:26,610 --> 00:02:37,530 I wanted to emphasize that our algebraic analysis led 40 00:02:37,530 --> 00:02:42,380 to states that we called jm, but today I 41 00:02:42,380 --> 00:02:47,750 will call lm, because they will refer to orbital angular 42 00:02:47,750 --> 00:02:49,180 momentum. 43 00:02:49,180 --> 00:02:52,860 And as you've seen in one of your problems, 44 00:02:52,860 --> 00:02:55,360 orbital angular momentum has to do 45 00:02:55,360 --> 00:03:01,620 with values of j, which are integers. 46 00:03:01,620 --> 00:03:09,670 So half integers values of j cannot be realized for orbital 47 00:03:09,670 --> 00:03:10,820 angular momentum. 48 00:03:10,820 --> 00:03:12,460 It's a very interesting thing. 49 00:03:12,460 --> 00:03:18,840 So spin states don't have wave functions in the usual way. 50 00:03:18,840 --> 00:03:23,490 It's only states of integer angular momentum 51 00:03:23,490 --> 00:03:26,260 that have wave functions. 52 00:03:26,260 --> 00:03:27,905 And those are the spherical harmonics. 53 00:03:27,905 --> 00:03:32,330 So I will talk about lm, and l, as usual, 54 00:03:32,330 --> 00:03:36,040 will go from 0 to infinity. 55 00:03:36,040 --> 00:03:40,910 And m goes from l to minus l. 56 00:03:43,420 --> 00:03:48,660 And you had these states, and we said that algebraically you 57 00:03:48,660 --> 00:03:56,440 would have L squared equals h squared l times l plus 1 lm. 58 00:03:56,440 --> 00:04:03,960 And Lz lm equal hm lm. 59 00:04:07,610 --> 00:04:10,440 Now basically, the spherical harmonics 60 00:04:10,440 --> 00:04:17,339 are going to be wave functions for these states. 61 00:04:17,339 --> 00:04:20,019 And the way we can approach it is 62 00:04:20,019 --> 00:04:24,790 that we did a little bit of work already with constructing the l 63 00:04:24,790 --> 00:04:27,120 squared operator. 64 00:04:27,120 --> 00:04:34,050 And in last lecture we derived, starting from the fact that L 65 00:04:34,050 --> 00:04:39,720 is r cross p and using x,y, and z, px, py, pz, 66 00:04:39,720 --> 00:04:44,270 and passing through spherical coordinates that L squared is 67 00:04:44,270 --> 00:04:52,230 the operator minus h squared 1 over sine theta d d theta sine 68 00:04:52,230 --> 00:04:59,295 theta d d theta again plus 1 over sine squared theta d 69 00:04:59,295 --> 00:05:00,866 second d phi squared. 70 00:05:03,830 --> 00:05:10,220 And we didn't do it, but Lz, which you know 71 00:05:10,220 --> 00:05:23,790 is h bar over i x d dy minus y d dx can also 72 00:05:23,790 --> 00:05:28,320 be translated into angular variables. 73 00:05:28,320 --> 00:05:31,820 And it has a very simple form. 74 00:05:31,820 --> 00:05:33,740 Also purely angular. 75 00:05:33,740 --> 00:05:38,550 And you can interpret it Lz is rotations around the z-axis, 76 00:05:38,550 --> 00:05:40,420 so they change phi. 77 00:05:40,420 --> 00:05:42,530 So it will not surprise you, if you 78 00:05:42,530 --> 00:05:48,840 do this exercise, that this is h over i d d phi. 79 00:05:48,840 --> 00:05:52,870 And you should really check it. 80 00:05:52,870 --> 00:05:57,020 There's another one that is a bit more laborious. 81 00:05:57,020 --> 00:06:03,630 L plus minus, remember, is Lx plus minus i Lz. 82 00:06:03,630 --> 00:06:06,010 We have a big attendance today. 83 00:06:10,250 --> 00:06:13,900 Is equal to-- more people. 84 00:06:13,900 --> 00:06:30,630 h bar e to the plus minus i phi i cosine theta over sine theta 85 00:06:30,630 --> 00:06:36,130 d d phi plus minus d d theta. 86 00:06:41,000 --> 00:06:44,010 And that takes a bit of algebra. 87 00:06:44,010 --> 00:06:45,300 You could do it. 88 00:06:45,300 --> 00:06:47,520 It's done in many books. 89 00:06:47,520 --> 00:06:50,040 It's probably there in Griffith's. 90 00:06:50,040 --> 00:06:52,590 And these are the representations 91 00:06:52,590 --> 00:06:56,840 of these operators as differential operators that 92 00:06:56,840 --> 00:06:59,690 act and function on theta and phi 93 00:06:59,690 --> 00:07:03,940 and don't care about radius. 94 00:07:03,940 --> 00:07:08,910 So in mathematical physics, people study these things 95 00:07:08,910 --> 00:07:14,460 and invent these things called spherical harmonics 96 00:07:14,460 --> 00:07:19,210 Ylm's of theta and phi. 97 00:07:19,210 --> 00:07:22,940 And the way you could see their done 98 00:07:22,940 --> 00:07:28,230 is in fact, such that this L squared viewed 99 00:07:28,230 --> 00:07:34,870 as this operator, differential operator, acting on Ylm 100 00:07:34,870 --> 00:07:43,690 is indeed equal to h squared l times l plus 1 Ylm. 101 00:07:43,690 --> 00:07:49,890 And Lz thought also as a differential operator, the one 102 00:07:49,890 --> 00:07:51,930 that we've written there. 103 00:07:51,930 --> 00:07:58,960 On the Ylm is h bar m Ylm. 104 00:08:01,700 --> 00:08:06,430 So they are constructed in this way, 105 00:08:06,430 --> 00:08:10,100 satisfying these equations. 106 00:08:10,100 --> 00:08:12,980 These are important equations in mathematical physics, 107 00:08:12,980 --> 00:08:14,930 and these functions were invented 108 00:08:14,930 --> 00:08:18,380 to satisfy those equations. 109 00:08:18,380 --> 00:08:24,130 Well, these are the properties of those states over there. 110 00:08:24,130 --> 00:08:30,620 So we can think of these functions 111 00:08:30,620 --> 00:08:34,429 as the wave functions associated with those states. 112 00:08:34,429 --> 00:08:39,289 So that's interpretation that is natural in quantum mechanics. 113 00:08:39,289 --> 00:08:42,210 And we want to think of them like that. 114 00:08:42,210 --> 00:08:50,780 We want to think of the Ylm's as the wave functions associated 115 00:08:50,780 --> 00:08:52,900 to the states lm. 116 00:08:52,900 --> 00:08:56,760 So lm. 117 00:08:56,760 --> 00:09:02,575 And here you would put a position state theta phi. 118 00:09:08,610 --> 00:09:11,360 This is analogous to the thing that we usually 119 00:09:11,360 --> 00:09:17,090 call the wave function being a position state times the state 120 00:09:17,090 --> 00:09:18,590 side. 121 00:09:18,590 --> 00:09:25,440 So we want to think of the Ylm's in this way 122 00:09:25,440 --> 00:09:29,850 as pretty much the wave functions associated 123 00:09:29,850 --> 00:09:32,550 to those states. 124 00:09:32,550 --> 00:09:42,160 Now there is a little bit of identities that come once you 125 00:09:42,160 --> 00:09:45,150 accept that this is what you think of the Ylm's. 126 00:09:45,150 --> 00:09:50,480 And then the compatibility of these equations. 127 00:09:50,480 --> 00:09:57,390 Top here with these ones makes in this identification natural. 128 00:09:57,390 --> 00:10:01,920 Now in order to manipulate and learn things 129 00:10:01,920 --> 00:10:03,950 about those spherical harmonics the way 130 00:10:03,950 --> 00:10:06,220 we do things in quantum mechanics, 131 00:10:06,220 --> 00:10:10,190 we think of the completeness relation. 132 00:10:10,190 --> 00:10:17,450 If we have d cube x x x, this is a completeness relation 133 00:10:17,450 --> 00:10:19,770 for position states. 134 00:10:24,500 --> 00:10:31,320 And I want to derive or suggest a completeness relation 135 00:10:31,320 --> 00:10:33,760 for these theta phi states. 136 00:10:33,760 --> 00:10:38,900 For that, I would pass this integral 137 00:10:38,900 --> 00:10:41,170 to do it in spherical coordinates. 138 00:10:41,170 --> 00:10:52,290 So I would do dr rd theta r sine theta d phi. 139 00:10:52,290 --> 00:10:59,100 And I would put r theta phi position states 140 00:10:59,100 --> 00:11:01,050 for these things. 141 00:11:01,050 --> 00:11:06,180 And position states r theta phi. 142 00:11:06,180 --> 00:11:07,810 Still being equal to 1. 143 00:11:13,510 --> 00:11:21,920 And we can try to split this thing. 144 00:11:21,920 --> 00:11:25,360 It's natural for us to think of just theta phi, 145 00:11:25,360 --> 00:11:30,390 because these wave functions have nothing to do with r, 146 00:11:30,390 --> 00:11:34,940 so I will simply do the integrals this way. 147 00:11:34,940 --> 00:11:39,780 d theta sine theta d phi. 148 00:11:39,780 --> 00:11:45,360 And think just like a position state in x, y, z. 149 00:11:45,360 --> 00:11:48,770 It's a position state in x, in y, and in z multiplied. 150 00:11:48,770 --> 00:11:52,280 We'll just split these things without trying 151 00:11:52,280 --> 00:11:55,790 to be too rigorous about it. 152 00:11:55,790 --> 00:12:00,700 Theta and phi like this. 153 00:12:00,700 --> 00:12:11,845 And you would have the integral dr r squared r r equal 1. 154 00:12:15,800 --> 00:12:21,410 And at this point, I want to think 155 00:12:21,410 --> 00:12:29,580 of this as the natural way of setting a completeness relation 156 00:12:29,580 --> 00:12:32,060 for theta and phi. 157 00:12:32,060 --> 00:12:35,360 And this doesn't talk to this one, 158 00:12:35,360 --> 00:12:40,260 so I will think of this that in the space of theta and phi, 159 00:12:40,260 --> 00:12:42,740 objects that just depend on theta and phi, 160 00:12:42,740 --> 00:12:45,570 this acts as a complete thing. 161 00:12:45,570 --> 00:12:48,220 And if objects depend also in r, this 162 00:12:48,220 --> 00:12:50,580 will act as a complete thing. 163 00:12:50,580 --> 00:12:53,140 So I will-- I don't know. 164 00:12:53,140 --> 00:12:56,130 Maybe the right way to say is postulate 165 00:12:56,130 --> 00:12:59,960 that we'll have a completeness relation of this form. 166 00:12:59,960 --> 00:13:12,975 d theta sine theta d phi theta phi theta phi equals 1. 167 00:13:23,260 --> 00:13:31,070 And then with this we can do all kinds of things. 168 00:13:31,070 --> 00:13:34,230 First, this integral is better written. 169 00:13:34,230 --> 00:13:42,520 This integral really represents 0 to pi d theta sine theta 0 170 00:13:42,520 --> 00:13:46,690 to 2 pi d phi. 171 00:13:46,690 --> 00:13:49,785 Now this is minus d cosine theta. 172 00:13:57,530 --> 00:14:02,020 And when theta is equal to 0, cosine theta 173 00:14:02,020 --> 00:14:10,280 is 1 to minus 1 integral d phi 0 to 2 pi. 174 00:14:10,280 --> 00:14:18,550 So this integral, really d theta sine theta d 175 00:14:18,550 --> 00:14:25,090 phi this is really the integral from minus 1 to 1. 176 00:14:25,090 --> 00:14:32,955 Change that order of d cos theta integral d phi from 0 to 2 pi. 177 00:14:37,650 --> 00:14:43,227 And this is called the integral over solid angle. 178 00:14:43,227 --> 00:14:44,060 That's a definition. 179 00:14:47,710 --> 00:14:51,930 So we could write the completeness relation 180 00:14:51,930 --> 00:14:57,350 in the space theta phi as integral over solid angle theta 181 00:14:57,350 --> 00:15:02,800 phi theta phi equals 1. 182 00:15:08,280 --> 00:15:12,460 Then the key property of the spherical harmonics, 183 00:15:12,460 --> 00:15:20,220 or the lm states, is that they are orthogonal. 184 00:15:20,220 --> 00:15:25,870 So delta l, l prime, delta m, m prime. 185 00:15:25,870 --> 00:15:28,180 So the orthogonality are of this state 186 00:15:28,180 --> 00:15:31,860 is guaranteed because Hermitian operators, 187 00:15:31,860 --> 00:15:36,790 different eigenvalues, they have to be orthogonal. 188 00:15:36,790 --> 00:15:38,990 Eigenstates of Hermitian. 189 00:15:38,990 --> 00:15:41,400 Operators with different eigenvalues. 190 00:15:41,400 --> 00:15:46,670 Here, you introduce a complete set of states of theta phi. 191 00:15:46,670 --> 00:15:58,950 So you put l prime m prime theta phi theta phi lm. 192 00:16:02,340 --> 00:16:12,850 And this is the integral over solid angle of Yl prime m 193 00:16:12,850 --> 00:16:17,610 prime of theta phi star. 194 00:16:17,610 --> 00:16:21,400 This is in the wrong position. 195 00:16:21,400 --> 00:16:27,866 And here Ylm of theta phi being equal delta l 196 00:16:27,866 --> 00:16:31,440 l prime delta m m prime. 197 00:16:37,650 --> 00:16:47,630 So this is orthogonality of the spherical harmonics. 198 00:16:47,630 --> 00:16:50,570 And this is pretty much all we need. 199 00:16:50,570 --> 00:16:58,690 Now there's the standard ways of constructing these things 200 00:16:58,690 --> 00:17:02,860 from the quantum mechanical sort of intuition. 201 00:17:02,860 --> 00:17:07,450 Basically, you can try to first build 202 00:17:07,450 --> 00:17:15,425 Yll, which corresponds to the state ll. 203 00:17:20,050 --> 00:17:23,390 Now the kind of differential equations 204 00:17:23,390 --> 00:17:27,349 this Yll satisfies are kind of simple. 205 00:17:27,349 --> 00:17:30,100 But in particular, the most important one 206 00:17:30,100 --> 00:17:34,980 is that L plus kills this state. 207 00:17:34,980 --> 00:17:37,720 So basically you use the condition 208 00:17:37,720 --> 00:17:40,920 that L plus kills this state to find a differential 209 00:17:40,920 --> 00:17:46,050 equation for this, which can be solved easily. 210 00:17:46,050 --> 00:17:47,850 Not a hard differential equation. 211 00:17:47,850 --> 00:17:50,300 Then you find Yll. 212 00:17:50,300 --> 00:17:57,700 And then you can find Yll minus 1 and all the other ones 213 00:17:57,700 --> 00:18:01,400 by applying the operator L minus. 214 00:18:01,400 --> 00:18:03,960 The lowering operator of m. 215 00:18:03,960 --> 00:18:06,420 So in principle, if you have enough patience, 216 00:18:06,420 --> 00:18:10,130 you can calculate all the spherical harmonics that way. 217 00:18:10,130 --> 00:18:14,240 There's no obstruction. 218 00:18:14,240 --> 00:18:17,170 But the form is a little messy, and if you 219 00:18:17,170 --> 00:18:21,640 want to find the normalizations so that these things work out 220 00:18:21,640 --> 00:18:25,790 correctly, well, it takes some work at the end of the day. 221 00:18:25,790 --> 00:18:30,680 So we're not going to do that here. 222 00:18:30,680 --> 00:18:34,500 We'll just leave it at that, and if we ever 223 00:18:34,500 --> 00:18:40,260 need some special harmonics, we'll just hold the answers. 224 00:18:40,260 --> 00:18:44,250 And they are in most textbooks. 225 00:18:44,250 --> 00:18:46,940 So if you do need them, well, you'll 226 00:18:46,940 --> 00:18:49,620 have to do with complicated normalizations. 227 00:18:52,230 --> 00:18:56,190 So that's really all I wanted to say about spherical harmonics, 228 00:18:56,190 --> 00:19:00,210 and we can turn then to the real subject, which 229 00:19:00,210 --> 00:19:02,720 is the radial equation. 230 00:19:02,720 --> 00:19:03,870 So the radial equation. 231 00:19:12,440 --> 00:19:17,480 So we have a Hamiltonian H equals p squared vector 232 00:19:17,480 --> 00:19:21,160 over 2m plus v of r. 233 00:19:21,160 --> 00:19:23,960 And we've seen that this is equal to h over 234 00:19:23,960 --> 00:19:33,030 2m 1 over r d second dr squared r plus 1 over 2mr 235 00:19:33,030 --> 00:19:37,755 squared L squared plus v of r. 236 00:19:40,500 --> 00:19:42,695 So this is what we're trying to solve. 237 00:19:45,510 --> 00:19:48,070 And the way we attempt to solve this 238 00:19:48,070 --> 00:19:49,920 is by separation of variables. 239 00:19:49,920 --> 00:19:54,600 So we'll try to write the wave function, psi, characterized 240 00:19:54,600 --> 00:19:56,980 by three things. 241 00:19:56,980 --> 00:20:04,680 Its energy, the value of l, and the value of m. 242 00:20:04,680 --> 00:20:08,350 And it's a function of position, because we're 243 00:20:08,350 --> 00:20:12,380 trying to solve H psi equal E psi. 244 00:20:12,380 --> 00:20:16,580 And that's the energy that we want to consider. 245 00:20:16,580 --> 00:20:20,980 So I will write here to begin with something that will not 246 00:20:20,980 --> 00:20:25,840 turn out to be exactly right, but it's 247 00:20:25,840 --> 00:20:29,520 important to do it first this way. 248 00:20:29,520 --> 00:20:36,700 A function of art r that has labels E, l, and m. 249 00:20:36,700 --> 00:20:40,000 Because it certainly could depend on E, could depend on l, 250 00:20:40,000 --> 00:20:43,100 and could depend on m, that radial function. 251 00:20:43,100 --> 00:20:45,420 And then the angular function will 252 00:20:45,420 --> 00:20:50,530 be the Ylm's of theta and phi. 253 00:20:50,530 --> 00:20:53,060 So this is the [INAUDIBLE] sets for the equation. 254 00:20:57,870 --> 00:21:03,390 If we have that, we can plug into the Schrodinger equation, 255 00:21:03,390 --> 00:21:04,790 and see what we get. 256 00:21:04,790 --> 00:21:10,010 Well, this operator will act on this f. 257 00:21:10,010 --> 00:21:15,520 This will have the operator L squared, 258 00:21:15,520 --> 00:21:18,900 but L squared over Ylm, you know what it is. 259 00:21:18,900 --> 00:21:23,040 And v of r is multiplicative, so it's no big problem. 260 00:21:23,040 --> 00:21:24,230 So what do we have? 261 00:21:24,230 --> 00:21:30,070 We have minus h squared over 2m 1 over r. 262 00:21:30,070 --> 00:21:32,720 Now I can talk normal derivatives. 263 00:21:32,720 --> 00:21:46,900 d r squared r times fElm plus 1 over 2mr squared. 264 00:21:46,900 --> 00:21:49,690 And now have L squared acting on this, 265 00:21:49,690 --> 00:21:54,680 but L squared acting on the Ylm is just this factor. 266 00:21:54,680 --> 00:22:03,640 So we have h squared l times l plus 1 times the fElm. 267 00:22:08,850 --> 00:22:12,910 Now I didn't put the Ylm in the first term 268 00:22:12,910 --> 00:22:15,200 because I'm going to cancel it throughout. 269 00:22:15,200 --> 00:22:28,700 So we have this term here plus v of r fElm equals E fElm. 270 00:22:34,140 --> 00:22:39,670 That is substituting into the equation h psi equal E psi. 271 00:22:39,670 --> 00:22:42,490 So first term here. 272 00:22:42,490 --> 00:22:46,300 Second term, it acted on the spherical harmonic. 273 00:22:46,300 --> 00:22:47,840 v of r is multiplicative. 274 00:22:47,840 --> 00:22:50,030 E on that. 275 00:22:50,030 --> 00:22:52,500 But then what you see immediately 276 00:22:52,500 --> 00:22:55,960 is that this differential equation doesn't depend on m. 277 00:22:59,550 --> 00:23:04,200 It was L squared, but no Lz in the Hamiltonian. 278 00:23:04,200 --> 00:23:06,610 So no m dependent. 279 00:23:06,610 --> 00:23:12,290 So actually we were overly proven 280 00:23:12,290 --> 00:23:17,370 in thinking that f was a function of m. 281 00:23:17,370 --> 00:23:21,390 What we really have is that psi Elm 282 00:23:21,390 --> 00:23:28,530 is equal to a function of E and l or r Ylm of theta phi. 283 00:23:31,350 --> 00:23:33,760 And then the differential equation 284 00:23:33,760 --> 00:23:37,230 is minus h squared over 2m. 285 00:23:37,230 --> 00:23:40,220 Let's multiply all by r. 286 00:23:40,220 --> 00:23:44,942 d second dr squared of r fEl. 287 00:23:50,410 --> 00:23:51,540 Plus look here. 288 00:23:55,760 --> 00:24:00,070 The r that I'm multiplying is going to go into the f. 289 00:24:00,070 --> 00:24:01,700 Here it's going to go into the f. 290 00:24:01,700 --> 00:24:03,260 Here it's going to go into the f. 291 00:24:03,260 --> 00:24:04,680 It's an overall thing. 292 00:24:04,680 --> 00:24:09,100 But here we keep h squared l times l 293 00:24:09,100 --> 00:24:26,090 plus 1 over 2mr squared rfEl plus v of r fEl rfEl equal 294 00:24:26,090 --> 00:24:27,555 e times rfEl. 295 00:24:36,470 --> 00:24:42,825 So what you see here is that this function is quite natural. 296 00:24:46,390 --> 00:24:53,432 So it suggests the definition of uEl to be rfEl. 297 00:24:59,530 --> 00:25:03,060 So that the differential equation now finally becomes 298 00:25:03,060 --> 00:25:13,930 minus h squared over 2m d second dr squared of uEl plus 299 00:25:13,930 --> 00:25:18,035 there's the u here, the u here, and this potential 300 00:25:18,035 --> 00:25:20,310 that has two terms. 301 00:25:20,310 --> 00:25:26,920 So this will be v of r plus h squared l 302 00:25:26,920 --> 00:25:36,036 times l plus 1 over 2mr squared uEl equals E times eEl. 303 00:25:40,540 --> 00:25:42,920 And this is the famous radial equation. 304 00:25:47,100 --> 00:25:50,870 It's an equation for you. 305 00:25:50,870 --> 00:25:57,250 And here, this whole thing is sometimes 306 00:25:57,250 --> 00:25:58,940 called the effective potential. 307 00:26:04,670 --> 00:26:07,280 So look what we've got. 308 00:26:07,280 --> 00:26:16,056 This f, if you wish here, is now of the form uEl 309 00:26:16,056 --> 00:26:24,040 of r Ylm over r theta phi. 310 00:26:24,040 --> 00:26:25,960 f is u over r. 311 00:26:25,960 --> 00:26:30,520 So this is the way we've written the solution, 312 00:26:30,520 --> 00:26:35,010 and u satisfies this equation, which is a one dimensional 313 00:26:35,010 --> 00:26:38,856 Schrodinger equation for the radius r. 314 00:26:38,856 --> 00:26:42,990 One dimensional equation with an effective potential 315 00:26:42,990 --> 00:26:44,990 that depends on L. 316 00:26:44,990 --> 00:26:49,580 So actually the first thing you have to notice 317 00:26:49,580 --> 00:26:52,890 is that the central potential problem 318 00:26:52,890 --> 00:27:00,080 has turned into an infinite collection of one 319 00:27:00,080 --> 00:27:01,510 dimensional problems. 320 00:27:01,510 --> 00:27:04,956 One for each value of l. 321 00:27:04,956 --> 00:27:09,940 For different values of l, you have a different potential. 322 00:27:09,940 --> 00:27:12,310 Now they're not all that different. 323 00:27:12,310 --> 00:27:15,530 They have different intensity of this term. 324 00:27:15,530 --> 00:27:20,200 For l equals 0, well you have some solutions. 325 00:27:20,200 --> 00:27:23,780 And for l equal 1, the answer could be quite different. 326 00:27:23,780 --> 00:27:26,410 For l equal 2, still different. 327 00:27:26,410 --> 00:27:30,210 And you have to solve an infinite number 328 00:27:30,210 --> 00:27:35,330 of one dimensional problems. 329 00:27:35,330 --> 00:27:39,380 That's what the Schrodinger equation has turned into. 330 00:27:39,380 --> 00:27:45,210 So we filled all these blackboards. 331 00:27:45,210 --> 00:27:48,420 Let's see, are there questions? 332 00:27:48,420 --> 00:27:50,470 Anything so far? 333 00:28:01,068 --> 00:28:01,567 Yes? 334 00:28:07,536 --> 00:28:09,326 AUDIENCE: You might get to this later, 335 00:28:09,326 --> 00:28:13,303 but what does it mean in our wave equations, 336 00:28:13,303 --> 00:28:18,780 in our wave function there, psi of Elm is equal to fEl, 337 00:28:18,780 --> 00:28:22,516 and the spherical harmonic of that one mean that one has 338 00:28:22,516 --> 00:28:24,320 an independence and the other doesn't. 339 00:28:24,320 --> 00:28:27,550 Can they be separated on the basis of m? 340 00:28:27,550 --> 00:28:33,520 PROFESSOR: So it is just a fact that the radial solution is 341 00:28:33,520 --> 00:28:37,010 independent of n, so it's an important property. 342 00:28:37,010 --> 00:28:39,900 n is fairly simple. 343 00:28:39,900 --> 00:28:43,500 The various state, the states with angular momentum 344 00:28:43,500 --> 00:28:48,180 l, but different m's just differ in their angular dependence, 345 00:28:48,180 --> 00:28:52,010 not in the radial dependence. 346 00:28:52,010 --> 00:28:54,230 And practically, it means that you 347 00:28:54,230 --> 00:28:59,960 have an infinite set of one dimensional problems labeled 348 00:28:59,960 --> 00:29:04,790 by l, and not labeled by m, which conceivably could have 349 00:29:04,790 --> 00:29:07,240 happened, but it doesn't happen. 350 00:29:07,240 --> 00:29:10,380 So just a major simplicity. 351 00:29:10,380 --> 00:29:11,614 Yes? 352 00:29:11,614 --> 00:29:13,030 AUDIENCE: Does the radial equation 353 00:29:13,030 --> 00:29:15,784 have all the same properties as a one dimensional Schrodinger 354 00:29:15,784 --> 00:29:16,284 equation? 355 00:29:16,284 --> 00:29:18,720 Or does the divergence in the effect [INAUDIBLE] 0 356 00:29:18,720 --> 00:29:19,220 change that? 357 00:29:19,220 --> 00:29:21,470 PROFESSOR: Well, it changes things, 358 00:29:21,470 --> 00:29:24,640 but the most serious change is the fact 359 00:29:24,640 --> 00:29:28,650 that, in one dimensional problems, 360 00:29:28,650 --> 00:29:32,640 x goes from minus infinity to infinity. 361 00:29:32,640 --> 00:29:35,100 And here it goes from 0 to infinity, 362 00:29:35,100 --> 00:29:38,510 so we need to worry about what happens at 0. 363 00:29:38,510 --> 00:29:41,590 Basically that's the main complication. 364 00:29:41,590 --> 00:29:46,670 One dimensional potential, but it really just 365 00:29:46,670 --> 00:29:48,370 can't go below 0. 366 00:29:48,370 --> 00:29:53,470 r is a radial variable, and we can't forget that. 367 00:29:53,470 --> 00:29:54,290 Yes? 368 00:29:54,290 --> 00:29:58,224 AUDIENCE: The potential v of r will depend on whatever problem 369 00:29:58,224 --> 00:29:59,140 you're solving, right? 370 00:29:59,140 --> 00:30:00,139 PROFESSOR: That's right. 371 00:30:00,139 --> 00:30:04,670 AUDIENCE: Could you find the v of r [INAUDIBLE]? 372 00:30:04,670 --> 00:30:07,090 PROFESSOR: Well that doesn't quite 373 00:30:07,090 --> 00:30:09,620 make sense as a Hamiltonian. 374 00:30:09,620 --> 00:30:13,270 You see, if you have a v of r, it's 375 00:30:13,270 --> 00:30:17,180 something that is supposed to be v of r for any wave function. 376 00:30:17,180 --> 00:30:18,380 That's the definition. 377 00:30:18,380 --> 00:30:20,900 So it can depend on some parameter, 378 00:30:20,900 --> 00:30:25,230 but that parameter cannot be the l of the particular wave 379 00:30:25,230 --> 00:30:28,092 function. 380 00:30:28,092 --> 00:30:32,370 AUDIENCE: [INAUDIBLE] or something that would interact 381 00:30:32,370 --> 00:30:32,870 with the-- 382 00:30:32,870 --> 00:30:34,790 PROFESSOR: If you have magnetic fields, 383 00:30:34,790 --> 00:30:38,860 things change, because then you can split levels 384 00:30:38,860 --> 00:30:40,900 with respect to m. 385 00:30:40,900 --> 00:30:43,640 Break degeneracies and things change indeed. 386 00:30:46,780 --> 00:30:49,960 We'll take care of those by using perturbation theory 387 00:30:49,960 --> 00:30:50,760 mostly. 388 00:30:50,760 --> 00:30:53,840 Use this solution and then perturbation theory. 389 00:30:53,840 --> 00:31:01,420 OK, so let's proceed a little more on this. 390 00:31:01,420 --> 00:31:07,710 So the first thing that we want to talk a little about 391 00:31:07,710 --> 00:31:11,440 is the normalization and some boundary conditions, 392 00:31:11,440 --> 00:31:14,950 because otherwise we can't really 393 00:31:14,950 --> 00:31:17,610 understand what's going on. 394 00:31:17,610 --> 00:31:22,630 And happily the discussion is not that complicated. 395 00:31:22,630 --> 00:31:24,450 So we want to normalize. 396 00:31:24,450 --> 00:31:25,600 So what do we want? 397 00:31:25,600 --> 00:31:39,230 Integral d cube x psi Elm of x squared equals 1. 398 00:31:39,230 --> 00:31:44,200 So clearly we want to go into angular variables. 399 00:31:44,200 --> 00:31:53,290 So again, this is r squared dr integral d solid angle, 400 00:31:53,290 --> 00:31:54,530 r squared Er. 401 00:31:54,530 --> 00:32:10,800 And this thing is now uEl squared absolute value 402 00:32:10,800 --> 00:32:12,250 over r squared. 403 00:32:12,250 --> 00:32:14,230 Look at the right most blackboard. 404 00:32:14,230 --> 00:32:19,570 uEl of r, I must square it because the wave function 405 00:32:19,570 --> 00:32:20,770 is squared. 406 00:32:20,770 --> 00:32:22,720 Over r squared. 407 00:32:22,720 --> 00:32:31,565 And then I have Ylm star of theta phi Ylm of theta phi. 408 00:32:34,560 --> 00:32:37,330 And if this is supposed to be normalized, 409 00:32:37,330 --> 00:32:39,245 this is supposed to be the number 1. 410 00:32:42,280 --> 00:32:46,220 Well happily, this part, this is why 411 00:32:46,220 --> 00:32:50,570 we needed to talk a little about spherical harmonics. 412 00:32:50,570 --> 00:32:55,630 This integral is 1, because it corresponds precisely 413 00:32:55,630 --> 00:32:59,650 to l equal l prime m equal m prime. 414 00:32:59,650 --> 00:33:04,610 And look how lucky or nice this is. 415 00:33:04,610 --> 00:33:07,020 r squared cancels with r squared, 416 00:33:07,020 --> 00:33:10,120 so the final condition is the integral from 0 417 00:33:10,120 --> 00:33:23,000 to infinity dr uEl of r squared is equal to 1, 418 00:33:23,000 --> 00:33:26,770 which shows that kind of the u really 419 00:33:26,770 --> 00:33:30,120 plays a role for wave function and a line. 420 00:33:30,120 --> 00:33:32,760 And even though it was a little complicated, 421 00:33:32,760 --> 00:33:35,520 there was the r here, and angular dependence, 422 00:33:35,520 --> 00:33:38,530 and everything, a good wave function 423 00:33:38,530 --> 00:33:43,240 is one that is just think of psi as being u. 424 00:33:43,240 --> 00:33:46,630 A one dimensional wave function psi being u, 425 00:33:46,630 --> 00:33:49,784 and if you can integrate it square, you've got it. 426 00:33:49,784 --> 00:33:50,700 AUDIENCE: [INAUDIBLE]. 427 00:33:53,890 --> 00:33:57,900 PROFESSOR: Because I had to square this, 428 00:33:57,900 --> 00:33:59,662 so there was u over r. 429 00:33:59,662 --> 00:34:03,280 AUDIENCE: But that's [INAUDIBLE]. 430 00:34:03,280 --> 00:34:05,000 PROFESSOR: Oh, I'm sorry. 431 00:34:05,000 --> 00:34:09,449 That parenthesis is a remnant. 432 00:34:09,449 --> 00:34:11,284 I tried to erase it a little. 433 00:34:14,600 --> 00:34:16,830 It's not squared anymore. 434 00:34:16,830 --> 00:34:20,500 The square is on the absolute value is r squared. 435 00:34:24,980 --> 00:34:27,276 So this is good news for our interpretation. 436 00:34:34,230 --> 00:34:39,050 So now before I discuss the peculiarities of the boundary 437 00:34:39,050 --> 00:34:45,352 conditions, I want to introduce really the main point 438 00:34:45,352 --> 00:34:47,310 that we're going to illustrate in this lecture. 439 00:34:47,310 --> 00:34:51,029 This is the thing that should remain in your heads. 440 00:34:54,900 --> 00:35:00,390 It's a picture, but it's an important one. 441 00:35:03,470 --> 00:35:06,330 When you want to organize the spectrum, 442 00:35:06,330 --> 00:35:08,465 you'll draw the following diagram. 443 00:35:12,740 --> 00:35:17,820 Energy is here and l here. 444 00:35:17,820 --> 00:35:19,590 And it's a funny kind of diagram. 445 00:35:19,590 --> 00:35:21,660 It's not like a curve or a plot. 446 00:35:21,660 --> 00:35:26,760 It's like a histogram or kind of thing like that. 447 00:35:26,760 --> 00:35:33,656 So what will happen is that you have a one dimensional problem. 448 00:35:37,480 --> 00:35:44,200 If these potentials are normal, there will be bound states. 449 00:35:44,200 --> 00:35:46,940 And let's consider the case of bound states 450 00:35:46,940 --> 00:35:50,840 for the purposes of this graph, just bound states. 451 00:35:50,840 --> 00:35:53,670 Now you look at this, and you say OK, 452 00:35:53,670 --> 00:35:55,110 what am I supposed to do? 453 00:35:55,110 --> 00:36:00,335 I'm going to have states for all values of l, 454 00:36:00,335 --> 00:36:03,560 and m, and probably some energies. 455 00:36:03,560 --> 00:36:06,780 So m doesn't affect the radial equation. 456 00:36:06,780 --> 00:36:08,160 That's very important. 457 00:36:08,160 --> 00:36:11,310 But l does, so I have a different problem 458 00:36:11,310 --> 00:36:12,890 to solve for different l. 459 00:36:12,890 --> 00:36:18,790 So I will make my histogram here and put here l 460 00:36:18,790 --> 00:36:22,570 equals 0 at this region. 461 00:36:22,570 --> 00:36:27,890 l equals 1, l equals 2, l equals 3, and go on. 462 00:36:30,490 --> 00:36:35,200 Now suppose I fix an l. 463 00:36:35,200 --> 00:36:36,520 l is fixed. 464 00:36:36,520 --> 00:36:41,010 Now it's a Schrodinger equation for a one dimensional problem. 465 00:36:41,010 --> 00:36:47,390 You would expect that if the potential suitably grows, which 466 00:36:47,390 --> 00:36:51,790 is a typical case, E will be quantized. 467 00:36:51,790 --> 00:36:54,830 And there will not be degeneracies, 468 00:36:54,830 --> 00:36:57,740 because the bound state spectrum in one dimension 469 00:36:57,740 --> 00:36:59,530 is not degenerate. 470 00:36:59,530 --> 00:37:03,730 So I should expect that for each l there 471 00:37:03,730 --> 00:37:08,500 are going to be energy values that are going to appear. 472 00:37:08,500 --> 00:37:14,780 So for l equals 0, I expect that there will be some energy here 473 00:37:14,780 --> 00:37:16,600 for which I've got a state. 474 00:37:16,600 --> 00:37:19,490 And that line means I got a state. 475 00:37:19,490 --> 00:37:23,385 And there's some energy here that could be called E1, 476 00:37:23,385 --> 00:37:35,720 0 is the first energy that is allowed with l equals 0. 477 00:37:35,720 --> 00:37:39,570 Then there will be another one here maybe. 478 00:37:39,570 --> 00:37:44,480 E-- I'll write it down-- 2,0. 479 00:37:44,480 --> 00:37:53,620 So basically I'm labeling the energies with En,l which means 480 00:37:53,620 --> 00:37:56,970 the first solution with l equals 0, 481 00:37:56,970 --> 00:37:58,980 the second solution with l equals 0, 482 00:37:58,980 --> 00:38:04,430 the third solution E 3,0. 483 00:38:04,430 --> 00:38:08,460 Then you come to l equals 1, and you 484 00:38:08,460 --> 00:38:11,640 must solve the equation again. 485 00:38:11,640 --> 00:38:15,530 And then for l equal 1, there will be the lowest energy, 486 00:38:15,530 --> 00:38:19,290 the ground state energy of the l equal 1 potential, 487 00:38:19,290 --> 00:38:21,240 and then higher and higher. 488 00:38:21,240 --> 00:38:25,130 Since the l equal 1 potential is higher 489 00:38:25,130 --> 00:38:30,270 than the l equals 0 potential, it's higher up. 490 00:38:30,270 --> 00:38:33,010 The energies should be higher up, 491 00:38:33,010 --> 00:38:35,600 at least the first one should be. 492 00:38:35,600 --> 00:38:38,980 And therefore the first one could 493 00:38:38,980 --> 00:38:44,190 be a little higher than this, or maybe by some accident 494 00:38:44,190 --> 00:38:49,180 it just fits here, or maybe it should fit here. 495 00:38:49,180 --> 00:38:51,540 Well, we don't know but know, but there's 496 00:38:51,540 --> 00:38:55,760 no obvious reason why it should, so I'll put it here. 497 00:38:55,760 --> 00:38:56,750 l equals 1. 498 00:38:56,750 --> 00:39:04,150 And this would be E1,1. 499 00:39:04,150 --> 00:39:07,460 The first state with l equals 1. 500 00:39:07,460 --> 00:39:11,150 Then here it could be E2,1. 501 00:39:11,150 --> 00:39:16,040 The second state with l equal 1 and higher up. 502 00:39:16,040 --> 00:39:21,570 And then for l equal-- my diagram is a little too big. 503 00:39:21,570 --> 00:39:25,200 E1,1. 504 00:39:25,200 --> 00:39:25,930 E2,1. 505 00:39:25,930 --> 00:39:30,230 And then you have states here, so maybe this one, l equals 2, 506 00:39:30,230 --> 00:39:31,590 I don't know where it goes. 507 00:39:31,590 --> 00:39:36,350 It just has to be higher than this one, so I'll put it here. 508 00:39:36,350 --> 00:39:39,760 And this will be E1,2. 509 00:39:39,760 --> 00:39:43,230 Maybe there's an E2,2. 510 00:39:43,230 --> 00:39:49,980 And here an E1,3. 511 00:39:49,980 --> 00:39:54,210 But this is the answer to your problem. 512 00:39:54,210 --> 00:39:59,520 That's the energy levels of a central potential. 513 00:39:59,520 --> 00:40:02,510 So it's a good, nice little diagram 514 00:40:02,510 --> 00:40:07,210 in which you put the states, you put the little line wherever 515 00:40:07,210 --> 00:40:08,590 you find the state. 516 00:40:08,590 --> 00:40:12,140 And for l equals 0, you have those states. 517 00:40:12,140 --> 00:40:17,760 Now because there's no degeneracies in the bound 518 00:40:17,760 --> 00:40:21,800 states of a one dimensional potential, 519 00:40:21,800 --> 00:40:27,000 I don't have two lines here that coincide, because there's 520 00:40:27,000 --> 00:40:30,180 no two states with the same energy here. 521 00:40:30,180 --> 00:40:34,030 It's just one state. 522 00:40:34,030 --> 00:40:35,200 And this one here. 523 00:40:35,200 --> 00:40:38,460 I cannot have two things there. 524 00:40:38,460 --> 00:40:41,500 That's pretty important to. 525 00:40:41,500 --> 00:40:46,230 So you have a list of states here. 526 00:40:46,230 --> 00:40:50,260 And just one state here, one state, but as you can see, 527 00:40:50,260 --> 00:40:54,500 you're probably are catching me in a little wrong play 528 00:40:54,500 --> 00:40:58,420 of words, because I say there's one state here. 529 00:40:58,420 --> 00:41:01,130 Yes, it's one state, because it's l equals 0. 530 00:41:01,130 --> 00:41:02,570 One state, one state. 531 00:41:02,570 --> 00:41:05,890 But this state, which is one single-- 532 00:41:05,890 --> 00:41:11,410 this should be called one single l equal 1 multiplet. 533 00:41:11,410 --> 00:41:15,660 So this is not really one state at the end of the day. 534 00:41:15,660 --> 00:41:19,940 It's one state of the one dimensional radial equation, 535 00:41:19,940 --> 00:41:24,600 but you know that l equals 1 comes accompanied 536 00:41:24,600 --> 00:41:28,250 with three values of m. 537 00:41:28,250 --> 00:41:31,750 So there's three states that are degenerate, 538 00:41:31,750 --> 00:41:33,560 because they have the same energy. 539 00:41:33,560 --> 00:41:37,430 The energy doesn't depend on l. 540 00:41:37,430 --> 00:41:43,440 So this thing is an l equal 1 multiplet, 541 00:41:43,440 --> 00:41:47,730 which means really three states. 542 00:41:47,730 --> 00:41:50,090 And this is three states. 543 00:41:50,090 --> 00:41:52,420 And this is three states. 544 00:41:52,420 --> 00:41:57,860 And this is 1 l equal 2 multiplet, which 545 00:41:57,860 --> 00:42:03,160 has possibility of m equals 2, 1, 0 minus 1 and minus 2. 546 00:42:03,160 --> 00:42:06,960 So in this state is just one l equal 2 multiplet, 547 00:42:06,960 --> 00:42:12,180 but it really means five states of the central potential. 548 00:42:12,180 --> 00:42:15,860 Five degenerate states, because the m 549 00:42:15,860 --> 00:42:17,810 doesn't change the energy. 550 00:42:17,810 --> 00:42:19,480 And this is five states. 551 00:42:19,480 --> 00:42:21,850 And this is seven states. 552 00:42:21,850 --> 00:42:28,250 One l equal 3 multiplet, which contains seven states. 553 00:42:28,250 --> 00:42:30,870 OK, so questions? 554 00:42:30,870 --> 00:42:35,240 This is the most important graph. 555 00:42:35,240 --> 00:42:37,330 If you have that picture in your head, 556 00:42:37,330 --> 00:42:39,900 then you can understand really where 557 00:42:39,900 --> 00:42:41,850 you're going with any potential. 558 00:42:41,850 --> 00:42:44,334 Any confusion here above the notation? 559 00:42:44,334 --> 00:42:44,834 Yes? 560 00:42:44,834 --> 00:42:46,929 AUDIENCE: So normally when we think about a one 561 00:42:46,929 --> 00:42:49,220 dimensional problem, we say that there's no degeneracy. 562 00:42:49,220 --> 00:42:51,770 Not really. 563 00:42:51,770 --> 00:42:53,287 No multiple degeneracy, so should we 564 00:42:53,287 --> 00:42:56,816 think of the radial equation as having copies for each m value 565 00:42:56,816 --> 00:42:59,760 and each having the same eigenvalue? 566 00:42:59,760 --> 00:43:02,450 PROFESSOR: I don't think it's necessary. 567 00:43:02,450 --> 00:43:05,930 You see, you've got your uEl. 568 00:43:05,930 --> 00:43:07,660 And you have here you solutions. 569 00:43:07,660 --> 00:43:10,700 Once the uEl is good, you're supposed 570 00:43:10,700 --> 00:43:13,340 to be able to put any Ylm. 571 00:43:13,340 --> 00:43:18,060 So put l, and now the m's that are allowed are solutions. 572 00:43:18,060 --> 00:43:19,560 You're solving the problem. 573 00:43:19,560 --> 00:43:26,100 So think of a master radial function as good for a fixed l, 574 00:43:26,100 --> 00:43:30,030 and therefore it works for all values of m. 575 00:43:30,030 --> 00:43:33,310 But don't try to think of many copies of this equation. 576 00:43:33,310 --> 00:43:37,010 I don't think it would help you. 577 00:43:37,010 --> 00:43:38,060 Any other questions? 578 00:43:42,956 --> 00:43:44,932 Yes? 579 00:43:44,932 --> 00:43:47,155 AUDIENCE: Sorry to ask, but if you could just 580 00:43:47,155 --> 00:43:49,790 review how is degeneracy built one more time? 581 00:43:49,790 --> 00:43:50,670 PROFESSOR: Yeah. 582 00:43:50,670 --> 00:43:53,390 Remember last time we were talking about, 583 00:43:53,390 --> 00:44:02,020 for example, what is a j equal to multiplet. 584 00:44:02,020 --> 00:44:05,860 Well, these were a collection of states jm with j 585 00:44:05,860 --> 00:44:09,310 equals 2 an m sum value. 586 00:44:09,310 --> 00:44:14,320 And they are all obtained by acting with angular momentum 587 00:44:14,320 --> 00:44:15,990 operators in each other. 588 00:44:15,990 --> 00:44:17,260 And there are five states. 589 00:44:17,260 --> 00:44:28,310 The 2,2, the 2,1, the 2,0, the 2, minus 1, and the 2, minus 2. 590 00:44:28,310 --> 00:44:30,240 And all these states are obtained 591 00:44:30,240 --> 00:44:35,630 by acting with, say, lowering operators l minus and this. 592 00:44:35,630 --> 00:44:38,930 Now all these angular momentum operators, 593 00:44:38,930 --> 00:44:43,560 all of the Li's commute with the Hamiltonian. 594 00:44:43,560 --> 00:44:46,200 Therefore all of these states are 595 00:44:46,200 --> 00:44:49,720 obtained by acting with Li must have the same energy. 596 00:44:49,720 --> 00:44:52,390 That's why we say that this comes in a multiplet. 597 00:44:52,390 --> 00:44:59,100 So when you get j-- in this case we'll call it l-- l equals 2. 598 00:44:59,100 --> 00:45:01,880 You get five states. 599 00:45:01,880 --> 00:45:04,980 They correspond to the various values of m. 600 00:45:04,980 --> 00:45:07,320 So when you did that radial equation that 601 00:45:07,320 --> 00:45:10,820 has a solution for l equals 2, you're 602 00:45:10,820 --> 00:45:12,450 getting the full multiplet. 603 00:45:12,450 --> 00:45:14,460 You're getting five states. 604 00:45:14,460 --> 00:45:16,970 1 l equal 2 multiplet. 605 00:45:16,970 --> 00:45:19,280 That's why one line here. 606 00:45:19,280 --> 00:45:21,555 That is equivalent to five states. 607 00:45:24,740 --> 00:45:31,770 OK, so that diagram, of course, is really quite important. 608 00:45:31,770 --> 00:45:41,430 So now we want to understand the boundary conditions. 609 00:45:41,430 --> 00:45:43,050 So we have here this. 610 00:45:43,050 --> 00:45:46,890 So this probably shouldn't erase yet. 611 00:45:46,890 --> 00:45:48,480 Let's do the boundary conditions. 612 00:45:58,390 --> 00:46:03,315 So behavior here at r equals to 0. 613 00:46:08,720 --> 00:46:11,300 At r going to 0. 614 00:46:17,360 --> 00:46:21,780 The first claim is that surprisingly, you would think, 615 00:46:21,780 --> 00:46:24,560 well, normalization is king. 616 00:46:24,560 --> 00:46:26,760 If it's normalized, it's good. 617 00:46:26,760 --> 00:46:29,470 So just any number. 618 00:46:29,470 --> 00:46:33,910 Just don't let it diverge near 0, and that will be OK. 619 00:46:33,910 --> 00:46:36,940 But it turns out that that's not true. 620 00:46:36,940 --> 00:46:38,290 It's not right. 621 00:46:38,290 --> 00:46:56,100 And you need the limit as r goes to 0 of uEl of r be equal to 0. 622 00:46:56,100 --> 00:47:03,610 And we'll take this and explore the simplest case. 623 00:47:03,610 --> 00:47:08,860 That is corresponds to saying what if the limit of r 624 00:47:08,860 --> 00:47:15,270 goes to 0 or uEl of r was a constant? 625 00:47:15,270 --> 00:47:17,870 What goes wrong? 626 00:47:17,870 --> 00:47:21,250 Certainly normalization doesn't go wrong. 627 00:47:21,250 --> 00:47:23,880 It can be a constant. 628 00:47:23,880 --> 00:47:26,830 u could be like that, and it would be normalized, 629 00:47:26,830 --> 00:47:29,270 and that doesn't go wrong. 630 00:47:29,270 --> 00:47:31,670 So let's look at the wave function. 631 00:47:31,670 --> 00:47:35,440 What happens with this? 632 00:47:35,440 --> 00:47:38,160 I actually will take for simplicity, 633 00:47:38,160 --> 00:47:45,840 because we'll analyze it later, the example of l equals 0. 634 00:47:45,840 --> 00:47:48,720 So let's put even 0. 635 00:47:48,720 --> 00:47:49,750 l equals 0. 636 00:47:53,830 --> 00:48:04,006 Well, suppose you look at the wave function now, 637 00:48:04,006 --> 00:48:06,700 and how does it look? 638 00:48:06,700 --> 00:48:16,270 Psi of E0-- if l is equal to 0, m must be equal to 0-- 639 00:48:16,270 --> 00:48:21,990 would be this u over r times a constant. 640 00:48:21,990 --> 00:48:25,520 So a constant, because y 0, 0 is a constant. 641 00:48:25,520 --> 00:48:30,640 And then you uE0 of r over r. 642 00:48:30,640 --> 00:48:41,050 So when r approaches 0, psi goes like c prime over r, 643 00:48:41,050 --> 00:48:44,140 some other constant over r. 644 00:48:44,140 --> 00:48:46,240 So I'm doing something very simple. 645 00:48:46,240 --> 00:48:53,360 I'm saying if uE0 is approaching the constant at the origin, 646 00:48:53,360 --> 00:48:57,930 if it's uE0, well, this is a constant because it's 0,0. 647 00:48:57,930 --> 00:48:59,800 So this is going to constant. 648 00:48:59,800 --> 00:49:01,890 So at the end of the day, the wave function 649 00:49:01,890 --> 00:49:03,360 looks like 1 over r. 650 00:49:08,280 --> 00:49:16,820 But this is impossible, because the Schrodinger equation H 651 00:49:16,820 --> 00:49:26,070 psi has minus h squared over 2m Laplacian on psi plus dot dot 652 00:49:26,070 --> 00:49:26,570 dot. 653 00:49:30,150 --> 00:49:39,330 And the up Laplacian of 1 over r is minus 4 pi times a delta 654 00:49:39,330 --> 00:49:43,355 function at x equals 0. 655 00:49:43,355 --> 00:49:49,220 So this means that the Schrodinger equation, 656 00:49:49,220 --> 00:49:52,680 you think oh I put psi equals c over r. 657 00:49:52,680 --> 00:49:56,300 Well, if you calculate the Laplacian, it seems to be 0. 658 00:49:56,300 --> 00:49:59,680 But if you're more careful, as you know for [? emm ?] 659 00:49:59,680 --> 00:50:03,750 the Laplacian of 1 over r is minus 4 pi times the delta 660 00:50:03,750 --> 00:50:05,240 function. 661 00:50:05,240 --> 00:50:09,990 So in the Schrodinger equation, the kinetic term 662 00:50:09,990 --> 00:50:12,330 produces a delta function. 663 00:50:12,330 --> 00:50:14,720 There's no reason to believe there's 664 00:50:14,720 --> 00:50:17,600 a delta function in the potential. 665 00:50:17,600 --> 00:50:21,310 We'll not try such crazy potentials. 666 00:50:21,310 --> 00:50:25,460 A delta function in a one dimensional potential, 667 00:50:25,460 --> 00:50:27,800 you've got the solution. 668 00:50:27,800 --> 00:50:31,480 A delta function in a three dimensional potential 669 00:50:31,480 --> 00:50:35,235 is absolutely crazy. 670 00:50:35,235 --> 00:50:38,760 It has infinite number of bound states, 671 00:50:38,760 --> 00:50:40,840 and they just go all the way down 672 00:50:40,840 --> 00:50:43,030 to energies of minus infinity. 673 00:50:43,030 --> 00:50:46,190 It's a very horrendous thing, a delta function 674 00:50:46,190 --> 00:50:49,660 in three dimensions, for quantum mechanics. 675 00:50:49,660 --> 00:50:54,070 So this thing, there's no delta function in the potential. 676 00:50:54,070 --> 00:50:56,630 And you've got a delta function from the kinetic term. 677 00:50:56,630 --> 00:50:58,800 You're not going to be able to cancel it. 678 00:50:58,800 --> 00:51:01,840 This is not a solution. 679 00:51:07,570 --> 00:51:14,060 So you really cannot approach a constant there. 680 00:51:14,060 --> 00:51:16,190 It's quite bad. 681 00:51:16,190 --> 00:51:20,130 So the wave functions will have to vanish, 682 00:51:20,130 --> 00:51:25,990 and we can prove that, or at least under some circumstances 683 00:51:25,990 --> 00:51:26,920 prove it. 684 00:51:26,920 --> 00:51:30,310 And as all these things are, they all 685 00:51:30,310 --> 00:51:33,500 depend on how crazy potentials you want to accept. 686 00:51:33,500 --> 00:51:37,110 So we should say something. 687 00:51:37,110 --> 00:51:41,990 So I'll say something about these potentials, 688 00:51:41,990 --> 00:51:46,606 and we'll prove a result. 689 00:51:50,830 --> 00:52:09,460 So my statement will be the centrifugal barrier, which 690 00:52:09,460 --> 00:52:14,860 is a name for this part of the potential, 691 00:52:14,860 --> 00:52:23,370 dominates as r goes to 0. 692 00:52:23,370 --> 00:52:27,050 If this doesn't happen, all bets are off. 693 00:52:27,050 --> 00:52:33,720 So let's assume that v of r, maybe it's 1 over r, 694 00:52:33,720 --> 00:52:36,210 but it's not worse than 1 over r squared. 695 00:52:36,210 --> 00:52:39,370 It's 1 over r cubed, for example, 696 00:52:39,370 --> 00:52:41,430 or something like that. 697 00:52:41,430 --> 00:52:43,790 You would have to analyze it from scratch 698 00:52:43,790 --> 00:52:45,080 if it would be that bad. 699 00:52:45,080 --> 00:52:50,260 But I will assume that the centrifugal barrier dominates. 700 00:52:50,260 --> 00:52:54,270 And then look at the differential equation. 701 00:52:54,270 --> 00:52:56,430 Well, what differential equation do I have? 702 00:52:56,430 --> 00:53:04,475 Well, I have this and this. 703 00:53:04,475 --> 00:53:07,210 This thing is less important than that, 704 00:53:07,210 --> 00:53:10,970 and this is also less important, because this is u 705 00:53:10,970 --> 00:53:12,530 divided by r squared. 706 00:53:12,530 --> 00:53:14,250 And here is just u. 707 00:53:14,250 --> 00:53:17,380 So this is certainly less important than that, 708 00:53:17,380 --> 00:53:19,610 and this is less important than that, 709 00:53:19,610 --> 00:53:22,740 and if I want to have some variation of u, 710 00:53:22,740 --> 00:53:26,170 or understand how it varies, I must keep this. 711 00:53:26,170 --> 00:53:31,880 So at this order, I should keep just the kinetic term 712 00:53:31,880 --> 00:53:38,490 h squared over 2m d second dr squared u of El. 713 00:53:42,400 --> 00:53:49,840 And h squared l times l plus 1 over 2 mr squared. 714 00:53:49,840 --> 00:53:55,770 And I will try to cancel these two to explore how the wave 715 00:53:55,770 --> 00:53:59,360 function looks near or equal 0. 716 00:53:59,360 --> 00:54:02,300 These are the two most important terms of the differential 717 00:54:02,300 --> 00:54:06,050 equation, so I have the right to keep those, and try 718 00:54:06,050 --> 00:54:12,450 to balance them out to leading order, and see what I get. 719 00:54:12,450 --> 00:54:16,960 So all the h squared over 2m's go away. 720 00:54:16,960 --> 00:54:26,300 So this is equivalent to d second uEl dr squared is 721 00:54:26,300 --> 00:54:32,165 equal to l times l plus 1 uEl over r squared. 722 00:54:35,810 --> 00:54:39,400 And this is solved by a power uEl. 723 00:54:43,080 --> 00:54:50,900 You can try r to the s, some number s. 724 00:54:50,900 --> 00:55:01,390 And then this thing gives you s times s minus 1. 725 00:55:01,390 --> 00:55:06,195 Taking two derivatives is equal to l times l plus 1. 726 00:55:11,580 --> 00:55:14,830 As you take two derivatives, you lose two powers of r, 727 00:55:14,830 --> 00:55:17,960 so it will work out. 728 00:55:17,960 --> 00:55:21,190 And from here, you see that the possible solutions 729 00:55:21,190 --> 00:55:25,725 are s equals l plus 1. 730 00:55:25,725 --> 00:55:29,100 And s equals 2 minus l. 731 00:55:33,660 --> 00:55:39,310 So this corresponds to a uEl that 732 00:55:39,310 --> 00:55:44,890 goes like r to the l plus 1, or a uEl 733 00:55:44,890 --> 00:55:48,958 that goes like 1 over r to the l. 734 00:55:52,782 --> 00:55:56,660 This Is far too singular. 735 00:55:56,660 --> 00:56:01,020 For l equals 0, we argued that the wave function 736 00:56:01,020 --> 00:56:02,586 should go like a constant. 737 00:56:06,900 --> 00:56:09,920 I'm sorry, cannot go like a constant. 738 00:56:09,920 --> 00:56:11,860 Must vanish. 739 00:56:11,860 --> 00:56:13,480 This is not possible. 740 00:56:13,480 --> 00:56:14,510 It's not a solution. 741 00:56:14,510 --> 00:56:16,020 It must vanish. 742 00:56:16,020 --> 00:56:22,200 For l equals 0, uE0 goes like r and vanishes. 743 00:56:22,200 --> 00:56:25,650 So that's consistent, and this is good. 744 00:56:25,650 --> 00:56:29,650 For l equals 0, this would be like a constant as well 745 00:56:29,650 --> 00:56:30,400 and would be fine. 746 00:56:30,400 --> 00:56:34,790 But for l equals 1 already, this is 1 over r, 747 00:56:34,790 --> 00:56:36,035 and this is not normalizable. 748 00:56:38,900 --> 00:56:50,210 So this time this is not normalizable for l greater 749 00:56:50,210 --> 00:56:52,320 or equal than one. 750 00:56:52,320 --> 00:56:59,170 So this is the answer [INAUDIBLE] this assumption, 751 00:56:59,170 --> 00:57:01,880 which is a very reasonable assumption. 752 00:57:01,880 --> 00:57:05,480 But if you don't have that you have to beware. 753 00:57:08,010 --> 00:57:16,490 OK, this is our condition for u there. 754 00:57:16,490 --> 00:57:24,090 And so uEl goes like this as r goes to 0. 755 00:57:24,090 --> 00:57:27,840 It would be the whole answer. 756 00:57:27,840 --> 00:57:37,910 So f, if you care about f still, which is what appears here, 757 00:57:37,910 --> 00:57:40,970 goes like u divided by r. 758 00:57:40,970 --> 00:57:50,562 So fEl goes like cr to the l. 759 00:57:54,258 --> 00:58:02,460 And when l is equal to 0, f behaves like a constant. 760 00:58:02,460 --> 00:58:06,400 u vanishes for l equal to 0, but f 761 00:58:06,400 --> 00:58:08,760 goes like a constant, which means 762 00:58:08,760 --> 00:58:13,540 that if you take 0 orbital angular momentum, 763 00:58:13,540 --> 00:58:16,670 you may have some probability of finding 764 00:58:16,670 --> 00:58:21,660 the particle at the origin, because this f behaves 765 00:58:21,660 --> 00:58:25,520 like a constant for l equals 0. 766 00:58:25,520 --> 00:58:28,720 On the other hand, for any higher l, 767 00:58:28,720 --> 00:58:31,920 f will also vanish at the origin. 768 00:58:31,920 --> 00:58:36,840 And that is intuitively said that the centrifugal barrier 769 00:58:36,840 --> 00:58:39,990 prevents the particle from reaching the origin. 770 00:58:39,990 --> 00:58:42,840 There's a barrier, a potential barrier. 771 00:58:42,840 --> 00:58:47,020 This potential is 1 over r squared. 772 00:58:47,020 --> 00:58:49,420 Doesn't let you go to close to the origin. 773 00:58:49,420 --> 00:58:53,480 But that potential disappears for l equals 0, 774 00:58:53,480 --> 00:58:56,840 and therefore the particle can reach the origin. 775 00:58:56,840 --> 00:59:00,500 But only for l equals 0 it can reach the origin. 776 00:59:03,120 --> 00:59:08,710 OK, one more thing. 777 00:59:08,710 --> 00:59:13,320 Behavior near infinity is of interest as well. 778 00:59:20,130 --> 00:59:25,260 So what happens for r goes to infinity? 779 00:59:32,440 --> 00:59:34,850 Well, for r goes to infinity, you also 780 00:59:34,850 --> 00:59:39,770 have to be a little careful what you assume. 781 00:59:39,770 --> 00:59:44,000 I wish I could tell you it's always like this, but it's not. 782 00:59:44,000 --> 00:59:46,455 It's rich in all kinds of problems. 783 00:59:49,040 --> 00:59:51,050 So there's two cases where there's 784 00:59:51,050 --> 00:59:53,500 an analysis that is simple. 785 00:59:53,500 --> 00:59:59,890 Suppose v of r is equal to 0 for r greater than some r0. 786 01:00:02,520 --> 01:00:12,896 Or r times v of f goes to 0 as r goes to infinity. 787 01:00:12,896 --> 01:00:13,645 Two possibilities. 788 01:00:16,660 --> 01:00:22,440 The potential is plane 0 after some distance. 789 01:00:22,440 --> 01:00:30,140 Or the potential multiplied by r goes to 0 790 01:00:30,140 --> 01:00:31,310 as r goes to infinity. 791 01:00:31,310 --> 01:00:36,660 And you would say, look, you've missed the most important case. 792 01:00:36,660 --> 01:00:40,390 The hydrogen atom, the potential is 1 over r. 793 01:00:40,390 --> 01:00:43,640 r times v of r doesn't go to 0. 794 01:00:43,640 --> 01:00:45,850 And indeed, what I'm going to write here 795 01:00:45,850 --> 01:00:49,380 doesn't quite apply to the wave functions of the hydrogen atom. 796 01:00:49,380 --> 01:00:51,440 They're a little unusual. 797 01:00:51,440 --> 01:00:56,805 The potential of the hydrogen atom is felt quite far away. 798 01:00:59,480 --> 01:01:04,520 So never the less, if you have those conditions, 799 01:01:04,520 --> 01:01:13,000 we can ignore the potential as we go far away. 800 01:01:13,000 --> 01:01:17,595 And we'll consider the following situation. 801 01:01:33,100 --> 01:01:37,930 Look that the centrifugal barrier satisfies this as well. 802 01:01:37,930 --> 01:01:41,330 So the full effective potential satisfies. 803 01:01:41,330 --> 01:01:44,870 If v of r satisfies that, r times 1 804 01:01:44,870 --> 01:01:48,780 over r squared of effective potential also satisfies that. 805 01:01:48,780 --> 01:01:52,250 So we can ignore all the potential, 806 01:01:52,250 --> 01:01:58,170 and we're left ignore the effective. 807 01:01:58,170 --> 01:02:01,440 And therefore we're left with minus h squared 808 01:02:01,440 --> 01:02:09,793 over 2m d second uEl dr squared is equal to EuEl. 809 01:02:13,900 --> 01:02:17,920 And that's a very trivial equation. 810 01:02:17,920 --> 01:02:19,481 Yes, Matt? 811 01:02:19,481 --> 01:02:20,954 AUDIENCE: When you say v of r goes 812 01:02:20,954 --> 01:02:23,409 to 0 for r greater than [INAUDIBLE] 0. 813 01:02:23,409 --> 01:02:25,870 Are you effectively [INAUDIBLE] the potential? 814 01:02:25,870 --> 01:02:30,870 PROFESSOR: Right, there may be some potentials like this. 815 01:02:30,870 --> 01:02:34,340 A potential that is like that. 816 01:02:34,340 --> 01:02:38,180 An attractive potential, and it vanishes after some distance. 817 01:02:38,180 --> 01:02:41,940 Or a repulsive potential that vanishes after some distance. 818 01:02:41,940 --> 01:02:44,440 AUDIENCE: But say the potential was a [INAUDIBLE] potential. 819 01:02:44,440 --> 01:02:47,072 Are you just approximating it to 0 after it's [INAUDIBLE]? 820 01:02:47,072 --> 01:02:49,280 PROFESSOR: Well, if I'm in the [INAUDIBLE] potential, 821 01:02:49,280 --> 01:02:53,460 unfortunately I'm neither here nor here, 822 01:02:53,460 --> 01:02:55,830 so this doesn't apply. 823 01:02:55,830 --> 01:02:58,390 So the [INAUDIBLE] potential is an exception. 824 01:02:58,390 --> 01:02:59,994 The solutions are a little more-- 825 01:02:59,994 --> 01:03:02,160 AUDIENCE: The conditions you're saying. [INAUDIBLE]. 826 01:03:02,160 --> 01:03:05,230 PROFESSOR: So these are conditions 827 01:03:05,230 --> 01:03:08,490 that allow me to say something. 828 01:03:08,490 --> 01:03:10,900 If they're not satisfied, I sort of 829 01:03:10,900 --> 01:03:14,810 have to analyze them case by case. 830 01:03:14,810 --> 01:03:18,110 That's the price we have to pay. 831 01:03:18,110 --> 01:03:22,280 It's a little more complicated than you would think naively. 832 01:03:22,280 --> 01:03:27,000 Now here, it's interesting to consider two possibilities. 833 01:03:27,000 --> 01:03:30,800 The case when E is less than 0, or the case when 834 01:03:30,800 --> 01:03:33,660 E is greater than 0. 835 01:03:33,660 --> 01:03:37,910 So scattering solutions or bound state solutions. 836 01:03:37,910 --> 01:03:42,800 For these ones, if the energy is less than 0 837 01:03:42,800 --> 01:03:47,480 and there's no potential, you're in the forbidden zone far away, 838 01:03:47,480 --> 01:03:51,080 so you must have a decaying exponential. 839 01:03:51,080 --> 01:03:57,200 El goes like exponential of minus square root 840 01:03:57,200 --> 01:04:03,280 of 2m E over h squared r. 841 01:04:03,280 --> 01:04:04,845 That solves that equation. 842 01:04:07,962 --> 01:04:10,040 You see, the solution of these things 843 01:04:10,040 --> 01:04:15,810 are either exponential decays or exponential growths 844 01:04:15,810 --> 01:04:19,040 and oscillatory solutions, sines and cosines, 845 01:04:19,040 --> 01:04:21,790 or E to the i things. 846 01:04:21,790 --> 01:04:27,840 So here we have a decay, because with energy less than 0, 847 01:04:27,840 --> 01:04:29,080 the potential is 0. 848 01:04:29,080 --> 01:04:32,850 So you're in a forbidden region, so you must decay like that. 849 01:04:32,850 --> 01:04:35,520 In this hydrogen atom what happens 850 01:04:35,520 --> 01:04:39,090 is that there's a power of r multiplying here. 851 01:04:39,090 --> 01:04:43,710 Like r to the n, or r to the k or something like that. 852 01:04:43,710 --> 01:04:51,216 If E is less than 0, you have uE equal exponential 853 01:04:51,216 --> 01:05:04,561 of plus minus ikr, where k is square root of 2m E over h 854 01:05:04,561 --> 01:05:05,060 squared. 855 01:05:08,420 --> 01:05:11,670 And those, again, solve that equation. 856 01:05:11,670 --> 01:05:16,280 And they are sort of wave solutions far away. 857 01:05:22,350 --> 01:05:25,150 Now with this information, the behavior 858 01:05:25,150 --> 01:05:31,460 of the u's near the origin, the behavior of the u's far away, 859 01:05:31,460 --> 01:05:35,040 you can then make qualitative plots 860 01:05:35,040 --> 01:05:37,820 of how solutions would look at the origin. 861 01:05:37,820 --> 01:05:39,610 They grow up like r to the l. 862 01:05:39,610 --> 01:05:42,150 Then it's a one dimensional potential, 863 01:05:42,150 --> 01:05:46,310 so they oscillate maybe, but then decay exponentially. 864 01:05:46,310 --> 01:05:47,940 And the kind of thing you used to do 865 01:05:47,940 --> 01:05:51,170 in 804 of plotting how things look, 866 01:05:51,170 --> 01:05:55,550 it's feasible at this stage. 867 01:05:55,550 --> 01:05:59,580 So it's about time to do examples. 868 01:05:59,580 --> 01:06:01,260 I have three examples. 869 01:06:01,260 --> 01:06:04,900 Given time, maybe I'll get to two. 870 01:06:04,900 --> 01:06:06,030 That's OK. 871 01:06:06,030 --> 01:06:08,320 The last example is kind of the cutest, 872 01:06:08,320 --> 01:06:13,550 but maybe it's OK to leave it for Monday. 873 01:06:13,550 --> 01:06:17,100 So are there questions about this 874 01:06:17,100 --> 01:06:19,015 before we begin our examples? 875 01:06:24,840 --> 01:06:26,262 Andrew? 876 01:06:26,262 --> 01:06:30,472 AUDIENCE: What is consumption of [INAUDIBLE] [? barrier ?] 877 01:06:30,472 --> 01:06:30,972 dominates. 878 01:06:30,972 --> 01:06:33,340 But why is that a reasonable assumptions? 879 01:06:33,340 --> 01:06:35,720 PROFESSOR: Well, potentials that are just 880 01:06:35,720 --> 01:06:41,450 too singular at the origin are not common. 881 01:06:41,450 --> 01:06:45,820 Just doesn't happen. 882 01:06:45,820 --> 01:06:50,770 So mathematically you could try them, 883 01:06:50,770 --> 01:06:54,730 but I actually don't know of useful examples 884 01:06:54,730 --> 01:06:57,315 if a potential is very singular at the origin. 885 01:07:00,595 --> 01:07:03,976 AUDIENCE: [INAUDIBLE] in the potential [INAUDIBLE] 886 01:07:03,976 --> 01:07:05,425 the centrifugal barrier. 887 01:07:05,425 --> 01:07:08,323 That [INAUDIBLE]. 888 01:07:08,323 --> 01:07:09,310 PROFESSOR: Right. 889 01:07:09,310 --> 01:07:12,900 An effective potential, the potential doesn't blow up-- 890 01:07:12,900 --> 01:07:17,880 your potential doesn't blow up more than 1 over r squared 891 01:07:17,880 --> 01:07:19,090 or something like that. 892 01:07:19,090 --> 01:07:24,640 So we'll just take it like that. 893 01:07:24,640 --> 01:07:31,720 OK, our first example is the free particle. 894 01:07:31,720 --> 01:07:33,260 You would say come on. 895 01:07:33,260 --> 01:07:34,890 That's ridiculous. 896 01:07:34,890 --> 01:07:35,525 Too simple. 897 01:07:38,510 --> 01:07:42,980 But it's fairly non-trivial in spherical coordinates. 898 01:07:42,980 --> 01:07:45,720 And you say, well, so what. 899 01:07:45,720 --> 01:07:49,550 Free particles, you say what the momentum is. 900 01:07:49,550 --> 01:07:51,070 You know the energy. 901 01:07:51,070 --> 01:07:52,780 How do you label the states? 902 01:07:52,780 --> 01:07:55,250 You label them by three momenta. 903 01:07:55,250 --> 01:07:58,540 Or energy and direction. 904 01:07:58,540 --> 01:08:01,830 So momentum eigenstates, for example 905 01:08:01,830 --> 01:08:04,000 But in spherical coordinates, these will not 906 01:08:04,000 --> 01:08:06,080 be momentum eigenstates, and these 907 01:08:06,080 --> 01:08:08,220 are interesting because they allow 908 01:08:08,220 --> 01:08:12,450 us to solve for more complicated problems, in fact. 909 01:08:12,450 --> 01:08:15,390 And they allow you to understand scattering out 910 01:08:15,390 --> 01:08:16,810 of central potential. 911 01:08:16,810 --> 01:08:18,725 So these are actually pretty important. 912 01:08:22,410 --> 01:08:25,010 You can label these things by three numbers. 913 01:08:25,010 --> 01:08:27,729 p1, p2, p3. 914 01:08:27,729 --> 01:08:33,840 Or energy and theta and phi, the directions of the momenta. 915 01:08:33,840 --> 01:08:39,700 What we're going to label them are by energy l and m. 916 01:08:39,700 --> 01:08:45,399 So you might say how do we compare all these infinities, 917 01:08:45,399 --> 01:08:48,779 but it somehow works out. 918 01:08:48,779 --> 01:08:53,600 There's the same number of states really in either way. 919 01:08:53,600 --> 01:08:57,279 So what do we have? 920 01:08:57,279 --> 01:09:01,300 It's a potential that v is equal to 0. 921 01:09:01,300 --> 01:09:03,790 So let's write the differential equation. 922 01:09:03,790 --> 01:09:06,390 v is equal to 0. 923 01:09:06,390 --> 01:09:07,920 But not v effective. 924 01:09:07,920 --> 01:09:16,760 So you have minus h squared over 2m d second uEl 925 01:09:16,760 --> 01:09:23,950 dr squared plus h squared over 2m l times 926 01:09:23,950 --> 01:09:32,279 l plus 1 over r squared uEl equal EuEl. 927 01:09:32,279 --> 01:09:35,494 This is actually quite interesting. 928 01:09:35,494 --> 01:09:39,590 As you will see, it's a bit puzzling the first time. 929 01:09:39,590 --> 01:09:43,649 Well, let's cancel this h squared over 2m, 930 01:09:43,649 --> 01:09:47,270 because they're kind of annoying. 931 01:09:47,270 --> 01:09:54,090 So we'll put d second uEl over dr squared with a minus-- I'll 932 01:09:54,090 --> 01:10:03,350 keep that minus-- plus l times l plus 1 over r squared uEl. 933 01:10:03,350 --> 01:10:08,610 And here I'll put k squared times uEl. 934 01:10:08,610 --> 01:10:13,070 And k squared is the same k as before. 935 01:10:13,070 --> 01:10:16,880 And E is positive because you have a free particle. 936 01:10:16,880 --> 01:10:19,330 E is positive. 937 01:10:19,330 --> 01:10:25,940 And k squared is given by this, 2m E over h squared. 938 01:10:25,940 --> 01:10:28,145 So this is the equation we have to solve. 939 01:10:33,330 --> 01:10:40,990 And it's kind of interesting, because on the one hand, 940 01:10:40,990 --> 01:10:45,590 there is an energy on the right hand side. 941 01:10:45,590 --> 01:10:48,700 And then you would say, look, it looks like this just typical 942 01:10:48,700 --> 01:10:51,290 one dimensional Schrodinger equation. 943 01:10:51,290 --> 01:10:53,610 Therefore that energy probably is 944 01:10:53,610 --> 01:10:57,250 quantized because it shows in the right hand side. 945 01:10:57,250 --> 01:11:03,340 Why wouldn't it be quantized if it just shows this way? 946 01:11:03,340 --> 01:11:06,990 On the other hand, it shouldn't be quantized. 947 01:11:06,990 --> 01:11:12,550 So what is it about this differential equation that 948 01:11:12,550 --> 01:11:16,780 shows that the energy never gets quantized? 949 01:11:16,780 --> 01:11:20,330 Well, the fact is that the energy in some sense 950 01:11:20,330 --> 01:11:22,850 doesn't show up in this differential equation. 951 01:11:22,850 --> 01:11:27,630 You think it's here, but it's not really there. 952 01:11:27,630 --> 01:11:29,710 What does that mean? 953 01:11:29,710 --> 01:11:31,760 It actually means that you can define 954 01:11:31,760 --> 01:11:41,240 a new variable rho equal kr, scale r. 955 01:11:41,240 --> 01:11:47,870 And basically chain rule or your intuition, this k 956 01:11:47,870 --> 01:11:49,500 goes down here. 957 01:11:49,500 --> 01:11:53,810 k squared r squared k squared r squared, it's all rho. 958 01:11:53,810 --> 01:11:56,730 So chain rule or changing variables 959 01:11:56,730 --> 01:12:03,650 will turn this equation into a minus d second uEl d rho 960 01:12:03,650 --> 01:12:08,850 squared plus l times l plus 1 rho squared 961 01:12:08,850 --> 01:12:14,900 is equal to-- times uEl-- is equal to uEl here. 962 01:12:20,450 --> 01:12:23,270 And the energy has disappeared from the equation 963 01:12:23,270 --> 01:12:27,400 by rescaling, a trivial rescaling of coordinates. 964 01:12:27,400 --> 01:12:30,860 That doesn't mean that the energy is not there. 965 01:12:30,860 --> 01:12:35,510 It is there, because you will find solutions 966 01:12:35,510 --> 01:12:37,720 that depend on rho, and then you will 967 01:12:37,720 --> 01:12:40,830 put rho equal kr and the energies there. 968 01:12:40,830 --> 01:12:44,070 But there's no quantization of energy, 969 01:12:44,070 --> 01:12:49,490 because the energy doesn't show in this equation anymore. 970 01:12:49,490 --> 01:12:55,310 It's kind of a neat thing, or rather conceptually interesting 971 01:12:55,310 --> 01:12:59,350 thing that energy is not there anymore. 972 01:12:59,350 --> 01:13:04,560 And then you look at this differential equation, 973 01:13:04,560 --> 01:13:09,720 and you realize that it's a nasty one. 974 01:13:09,720 --> 01:13:15,280 So this equation is quite easy without this. 975 01:13:15,280 --> 01:13:18,130 It's a power solution. 976 01:13:18,130 --> 01:13:23,700 It's quite easy without this, it's exponentials are this. 977 01:13:23,700 --> 01:13:26,580 But whenever you have a differential equation that 978 01:13:26,580 --> 01:13:31,930 has two derivatives, a term with 1 979 01:13:31,930 --> 01:13:35,440 over x squared times the function, 980 01:13:35,440 --> 01:13:40,600 and a term with 1 times the function, 981 01:13:40,600 --> 01:13:42,255 you're in Bessel territory. 982 01:13:44,960 --> 01:13:47,300 All these functions have Bessel things. 983 01:13:49,820 --> 01:13:53,940 And then you have another term like 1 over x d dx. 984 01:13:53,940 --> 01:13:57,540 That is not a problem, but the presence of these two things, 985 01:13:57,540 --> 01:14:01,170 one with 1 over x squared and one with this, 986 01:14:01,170 --> 01:14:02,640 complicates this equation. 987 01:14:02,640 --> 01:14:05,270 So Bessel, without this, would be 988 01:14:05,270 --> 01:14:08,880 exponential solution without this would be powers. 989 01:14:08,880 --> 01:14:12,610 In the end, the fact is that this is spherical Bessel, 990 01:14:12,610 --> 01:14:15,200 and it's a little complicated. 991 01:14:15,200 --> 01:14:16,860 Not terribly complicated. 992 01:14:16,860 --> 01:14:20,280 The solutions are spherical Bessel functions, 993 01:14:20,280 --> 01:14:22,970 which are not all that bad. 994 01:14:22,970 --> 01:14:25,190 And let me say what they are. 995 01:14:38,550 --> 01:14:41,960 So what are the solutions to this thing? 996 01:14:41,960 --> 01:14:45,390 In fact, the solutions that are easier to find 997 01:14:45,390 --> 01:14:54,960 is that the uEl's are r times the Bessel function 998 01:14:54,960 --> 01:14:58,020 jl is called spherical Bessel functions. 999 01:14:58,020 --> 01:15:01,860 So it's not capital j that people 1000 01:15:01,860 --> 01:15:06,320 use for the normal Bessel, but lower case l. 1001 01:15:06,320 --> 01:15:08,640 Of kr. 1002 01:15:08,640 --> 01:15:13,430 As you know, you solve this, and the solutions for this 1003 01:15:13,430 --> 01:15:18,050 would be of the form rho jl for rho. 1004 01:15:18,050 --> 01:15:22,530 But rho is kr, so we don't care about the constant, 1005 01:15:22,530 --> 01:15:25,590 because this is a homogeneous linear equation. 1006 01:15:25,590 --> 01:15:26,890 So some number here. 1007 01:15:26,890 --> 01:15:29,320 You could put a constant if you wish. 1008 01:15:29,320 --> 01:15:32,130 But that's the solution. 1009 01:15:32,130 --> 01:15:34,820 Therefore your complete solutions 1010 01:15:34,820 --> 01:15:42,590 is like the psi's of Elm would be u divided 1011 01:15:42,590 --> 01:15:50,406 by r, which is jl of kr times Ylm's of theta phi. 1012 01:15:50,406 --> 01:15:51,780 These are the complete solutions. 1013 01:15:56,280 --> 01:15:59,400 This is a second order differential equation. 1014 01:15:59,400 --> 01:16:03,870 Therefore it has to have two solutions. 1015 01:16:03,870 --> 01:16:07,960 And this is what is called a regular solution at the origin. 1016 01:16:07,960 --> 01:16:12,250 The Bessel functions come in j and n type. 1017 01:16:12,250 --> 01:16:15,670 And the n type is singular at the origins, 1018 01:16:15,670 --> 01:16:17,340 so we won't care about it. 1019 01:16:20,810 --> 01:16:23,870 So what do we get from here? 1020 01:16:23,870 --> 01:16:27,780 Well, some behavior that is well known. 1021 01:16:27,780 --> 01:16:36,110 Rho jl of rho behaves like rho to the l plus 2 over 2l 1022 01:16:36,110 --> 01:16:42,690 plus 1 double factorial as rho goes to 0. 1023 01:16:42,690 --> 01:16:46,540 So that's a fact about these Bessel functions. 1024 01:16:46,540 --> 01:16:50,570 They behave that way, which is good, 1025 01:16:50,570 --> 01:16:56,640 because rho jl behaves like that, 1026 01:16:56,640 --> 01:17:00,770 so u behaves like r to the l plus 1, which 1027 01:17:00,770 --> 01:17:03,700 is what we derived a little time ago. 1028 01:17:03,700 --> 01:17:05,950 So this behavior of the Bessel function 1029 01:17:05,950 --> 01:17:09,370 is indeed consistent with our solution. 1030 01:17:09,370 --> 01:17:13,890 Moreover, there's another behavior that is interesting. 1031 01:17:13,890 --> 01:17:15,910 This Bessel function, by the time 1032 01:17:15,910 --> 01:17:19,580 it's written like that, when you go 1033 01:17:19,580 --> 01:17:25,390 far off to infinity jl of rho, it 1034 01:17:25,390 --> 01:17:31,950 behaves like sine of rho minus l pi over 2. 1035 01:17:35,900 --> 01:17:40,120 This is as rho goes to infinity. 1036 01:17:40,120 --> 01:17:45,080 So as rho goes to infinity, this is 1037 01:17:45,080 --> 01:17:47,550 behaving like a trigonometric function. 1038 01:17:47,550 --> 01:17:54,680 It's consistent with this, because rho-- this is rho jl 1039 01:17:54,680 --> 01:17:58,120 is what we call u essentially. 1040 01:17:58,120 --> 01:18:02,470 So u behaves like this with rho equal kr. 1041 01:18:02,470 --> 01:18:03,730 And that's consistent. 1042 01:18:03,730 --> 01:18:07,350 This superposition of a sine and a cosine. 1043 01:18:07,350 --> 01:18:10,570 But it's kind of interesting though that this l pi over 2 1044 01:18:10,570 --> 01:18:12,940 shows up here. 1045 01:18:12,940 --> 01:18:15,890 You see the fact that this function 1046 01:18:15,890 --> 01:18:17,940 has to vanish at the origin. 1047 01:18:17,940 --> 01:18:20,800 It vanishes at the origin and begins to vary. 1048 01:18:20,800 --> 01:18:23,990 And by the time you go far away, you contract. 1049 01:18:23,990 --> 01:18:28,970 And the way it behaves is this way. 1050 01:18:28,970 --> 01:18:32,580 The face is determined. 1051 01:18:32,580 --> 01:18:36,060 So that actually gives a lot of opportunity 1052 01:18:36,060 --> 01:18:43,190 to physicists because the free particle-- 1053 01:18:43,190 --> 01:18:51,090 so for the free particle, uEl behaves 1054 01:18:51,090 --> 01:18:59,285 like sine of kr minus l pi over 2 as r goes to infinity. 1055 01:19:05,560 --> 01:19:20,590 So from that people have asked the following question. 1056 01:19:20,590 --> 01:19:25,720 What if you have a potential that, for example 1057 01:19:25,720 --> 01:19:30,140 for simplicity, a potential that is localized. 1058 01:19:30,140 --> 01:19:32,650 Well, if this potential is localized, 1059 01:19:32,650 --> 01:19:35,850 the solution far away is supposed 1060 01:19:35,850 --> 01:19:39,280 to be a superposition of sines and cosines. 1061 01:19:39,280 --> 01:19:42,750 So if there is no potential, the solution 1062 01:19:42,750 --> 01:19:44,780 is supposed to be this. 1063 01:19:44,780 --> 01:19:48,270 Now another superposition of sines and cosines, 1064 01:19:48,270 --> 01:19:50,330 at the end of the day, can always 1065 01:19:50,330 --> 01:19:55,040 be written as some sine of this thing plus a change 1066 01:19:55,040 --> 01:20:01,230 in this phase So in general, uEl will 1067 01:20:01,230 --> 01:20:11,100 go like sine of kr minus l pi over 2 plus a shift, a phase 1068 01:20:11,100 --> 01:20:17,140 shift, delta l that can depend on the energy. 1069 01:20:21,490 --> 01:20:26,170 So if you haven't tried to find the radial solutions 1070 01:20:26,170 --> 01:20:28,600 of a problem with some potential, 1071 01:20:28,600 --> 01:20:32,400 if the potential is 0, there's no such term. 1072 01:20:32,400 --> 01:20:37,780 But if the potential is here, it will have an effect 1073 01:20:37,780 --> 01:20:41,330 and will give you a phase shift. 1074 01:20:41,330 --> 01:20:44,550 So if you're doing particle scattering experiments, 1075 01:20:44,550 --> 01:20:47,300 you're sending waves from far away 1076 01:20:47,300 --> 01:20:50,560 and you just see how the wave behaves far away, 1077 01:20:50,560 --> 01:20:54,190 you do have measurement information on this phase 1078 01:20:54,190 --> 01:20:55,140 shift. 1079 01:20:55,140 --> 01:20:58,310 And from this phase shift, you can learn something 1080 01:20:58,310 --> 01:21:00,900 about the potential. 1081 01:21:00,900 --> 01:21:05,850 So this is how this problem of free particle 1082 01:21:05,850 --> 01:21:10,130 suddenly becomes very important and very interesting. 1083 01:21:10,130 --> 01:21:13,370 For example, as a way through the behavior at 1084 01:21:13,370 --> 01:21:17,410 infinity learning something about the potential. 1085 01:21:17,410 --> 01:21:20,800 For example, if the potential is attractive, 1086 01:21:20,800 --> 01:21:26,020 it pulls the wave function in and produces some sign 1087 01:21:26,020 --> 01:21:30,520 of delta that the corresponds to a positive delta. 1088 01:21:30,520 --> 01:21:33,120 If the potential is repulsive, it 1089 01:21:33,120 --> 01:21:36,580 pushes the wave function out, repels it 1090 01:21:36,580 --> 01:21:39,120 and produces a delta that is negative. 1091 01:21:39,120 --> 01:21:42,740 You can track those signs thinking carefully. 1092 01:21:42,740 --> 01:21:48,100 But the potentials will teach you something about delta. 1093 01:21:48,100 --> 01:21:53,460 The other case that this is interesting-- I will just 1094 01:21:53,460 --> 01:21:58,400 introduce it and stop, because we might as well stop-- 1095 01:21:58,400 --> 01:22:03,880 is a very important case. 1096 01:22:03,880 --> 01:22:05,890 The square well. 1097 01:22:05,890 --> 01:22:09,250 Well, we've studied in one dimension 1098 01:22:09,250 --> 01:22:11,640 the infinite square well. 1099 01:22:11,640 --> 01:22:14,580 That's one potential that you now how to solve, 1100 01:22:14,580 --> 01:22:18,440 and sines and cosines is very easy. 1101 01:22:18,440 --> 01:22:21,740 Now imagine a spherical square well, 1102 01:22:21,740 --> 01:22:25,770 which is some sort of cavity in which a particle is 1103 01:22:25,770 --> 01:22:28,500 free to move here, but the potential becomes 1104 01:22:28,500 --> 01:22:30,900 infinite at the boundary. 1105 01:22:33,740 --> 01:22:38,940 It's a hollow sphere, so the potential v of r 1106 01:22:38,940 --> 01:22:42,180 is equal to 0 for r less than a. 1107 01:22:42,180 --> 01:22:46,090 And it's infinity for r greater than a. 1108 01:22:46,090 --> 01:22:51,590 So it's like a bag, a balloon with solid walls 1109 01:22:51,590 --> 01:22:53,460 impossible to penetrate. 1110 01:22:53,460 --> 01:22:58,590 So this is the most symmetric simple potential 1111 01:22:58,590 --> 01:23:00,200 you could imagine in the world. 1112 01:23:02,990 --> 01:23:05,520 And we're going to solve it. 1113 01:23:05,520 --> 01:23:07,190 How can we solve this? 1114 01:23:07,190 --> 01:23:12,900 Well, we did 2/3 of the work already in solving it. 1115 01:23:12,900 --> 01:23:14,050 Why? 1116 01:23:14,050 --> 01:23:18,890 Because inside here the potential is 0, 1117 01:23:18,890 --> 01:23:22,060 so the particle is free. 1118 01:23:22,060 --> 01:23:29,480 So inside here the solutions are of the form uEl 1119 01:23:29,480 --> 01:23:33,180 go like rjl of kr. 1120 01:23:36,130 --> 01:23:40,750 And the only thing you will need is that they vanish at the end. 1121 01:23:40,750 --> 01:23:44,410 So you will fix this by demanding 1122 01:23:44,410 --> 01:23:57,050 that ka is a number z such-- well, the jl of ka will be 0. 1123 01:23:57,050 --> 01:23:59,460 So that the wave function vanishes 1124 01:23:59,460 --> 01:24:03,290 at this point where the potential becomes infinite. 1125 01:24:03,290 --> 01:24:05,530 So you've solved most of the problem. 1126 01:24:05,530 --> 01:24:10,700 And we'll discuss it in detail, because it's an important one. 1127 01:24:10,700 --> 01:24:16,940 But this is the most symmetric potential, you may think. 1128 01:24:16,940 --> 01:24:23,380 This potential is very symmetric, very pretty, 1129 01:24:23,380 --> 01:24:27,520 but nothing to write home about. 1130 01:24:27,520 --> 01:24:30,250 If you tried to look-- and we're going 1131 01:24:30,250 --> 01:24:34,060 to calculate this diagram. 1132 01:24:34,060 --> 01:24:35,820 You would say well it's so symmetric 1133 01:24:35,820 --> 01:24:39,950 that something pretty is going to happen here. 1134 01:24:39,950 --> 01:24:41,720 Nothing happens. 1135 01:24:41,720 --> 01:24:44,290 These states will show up. 1136 01:24:44,290 --> 01:24:49,040 And these ones will show up, and no state ever 1137 01:24:49,040 --> 01:24:50,460 will match another one. 1138 01:24:50,460 --> 01:24:55,130 There's no pattern, or rhyme, or reason for it. 1139 01:24:55,130 --> 01:24:57,630 On the other hand, if you would have 1140 01:24:57,630 --> 01:25:07,550 taken a potential v of r of the form beta r squared, 1141 01:25:07,550 --> 01:25:10,960 that potential will exhibit enormous amounts 1142 01:25:10,960 --> 01:25:14,130 of degeneracies all over. 1143 01:25:14,130 --> 01:25:17,480 And we will have to understand why that happens. 1144 01:25:17,480 --> 01:25:19,500 So we'll see you next Monday. 1145 01:25:19,500 --> 01:25:21,390 Enjoy your break. 1146 01:25:21,390 --> 01:25:26,540 Homework will only happen late after Thanksgiving. 1147 01:25:26,540 --> 01:25:28,940 And just have a great time. 1148 01:25:28,940 --> 01:25:32,190 Thank you for coming today, and will see you soon. 1149 01:25:32,190 --> 01:25:33,740 [APPLAUSE]