1 00:00:00,090 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,690 continue to offer high quality educational resources for free. 5 00:00:10,690 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:17,250 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,250 --> 00:00:18,210 at ocw.mit.edu. 8 00:00:21,490 --> 00:00:26,310 ROBERT FIELD: This is one of my favorite lectures for 5.61. 9 00:00:26,310 --> 00:00:29,190 And I hope you like it too. 10 00:00:29,190 --> 00:00:34,540 OK, last time, we talked about the hydrogen atom. 11 00:00:34,540 --> 00:00:37,830 And it's the last exactly solved problem. 12 00:00:37,830 --> 00:00:40,680 And you've probably noticed that like the way 13 00:00:40,680 --> 00:00:44,400 I treated each of the exactly solved problems, 14 00:00:44,400 --> 00:00:51,150 I avoided looking at the exact solutions. 15 00:00:51,150 --> 00:00:58,860 I said we should be able to build our own picture based 16 00:00:58,860 --> 00:01:00,990 on simplifications. 17 00:01:00,990 --> 00:01:03,090 And this is something that I really 18 00:01:03,090 --> 00:01:08,670 want to stress that, yes, the exact solutions are 19 00:01:08,670 --> 00:01:09,870 in textbooks. 20 00:01:09,870 --> 00:01:13,380 You can program them into your computers 21 00:01:13,380 --> 00:01:15,960 and ask your computer to calculate anything. 22 00:01:15,960 --> 00:01:18,150 But sometimes you want to know what 23 00:01:18,150 --> 00:01:20,730 it is you want to calculate and what 24 00:01:20,730 --> 00:01:25,620 is the right answer or approximately the right answer 25 00:01:25,620 --> 00:01:26,550 to build insights. 26 00:01:26,550 --> 00:01:34,410 Now, in 5.112 and possibly 5.111 and certainly not 3.091, 27 00:01:34,410 --> 00:01:38,790 you learn something about the periodic table. 28 00:01:38,790 --> 00:01:40,830 You learn how to make predictions 29 00:01:40,830 --> 00:01:45,769 about all properties of atoms based on some simple ideas. 30 00:01:45,769 --> 00:01:46,935 What are those simple ideas? 31 00:01:50,927 --> 00:01:52,385 I'm looking for three simple ideas. 32 00:01:58,430 --> 00:01:59,327 Yes. 33 00:01:59,327 --> 00:02:01,452 STUDENT: Things with similar valence configurations 34 00:02:01,452 --> 00:02:03,140 behave similarly. 35 00:02:03,140 --> 00:02:09,350 ROBERT FIELD: Yes, that's sort of a second order simple idea, 36 00:02:09,350 --> 00:02:13,640 that because-- 37 00:02:13,640 --> 00:02:16,070 I mean, it's using the idea that if you 38 00:02:16,070 --> 00:02:19,850 know that quantum numbers of the electrons and what orbitals 39 00:02:19,850 --> 00:02:27,620 are occupied, you can make connections between atomic 40 00:02:27,620 --> 00:02:29,180 and some molecular properties. 41 00:02:33,690 --> 00:02:36,330 Yes. 42 00:02:36,330 --> 00:02:38,830 STUDENT: If you understand the interplay 43 00:02:38,830 --> 00:02:41,070 between nuclear charge and electric charge, 44 00:02:41,070 --> 00:02:44,085 you can derive large physical magnitudes. 45 00:02:44,085 --> 00:02:44,835 ROBERT FIELD: Yes. 46 00:02:47,430 --> 00:02:53,440 I mean, there are things like electronegativity, shielding. 47 00:02:53,440 --> 00:02:58,510 And these are things where if we have some intuitive 48 00:02:58,510 --> 00:03:04,010 sense for how big orbitals are and what 49 00:03:04,010 --> 00:03:09,620 is the relative attractiveness to different nuclei 50 00:03:09,620 --> 00:03:13,370 of various orbitals, you build up everything. 51 00:03:13,370 --> 00:03:16,310 And most of it comes from this lecture. 52 00:03:19,150 --> 00:03:24,210 And so I'm going to talk about the hydrogen atom 53 00:03:24,210 --> 00:03:27,600 as a model for electronic structure. 54 00:03:27,600 --> 00:03:34,165 And in particular, I'm to talk about quantum number scaling 55 00:03:34,165 --> 00:03:34,665 effects. 56 00:03:38,140 --> 00:03:41,780 And I'm going to be using this semi-classical method a lot. 57 00:03:45,410 --> 00:03:51,170 So we know that for the hydrogen atom, 58 00:03:51,170 --> 00:03:54,290 you can factor the wave function into an angular part, which 59 00:03:54,290 --> 00:03:56,260 is universal. 60 00:03:56,260 --> 00:03:59,830 So if you understand it once for any problem, 61 00:03:59,830 --> 00:04:01,490 you understand it all the time. 62 00:04:01,490 --> 00:04:02,960 And so we can just put that aside, 63 00:04:02,960 --> 00:04:05,650 because that's a done deal. 64 00:04:05,650 --> 00:04:08,530 And then there's the radial part. 65 00:04:08,530 --> 00:04:11,740 And the radial part is different for every central force 66 00:04:11,740 --> 00:04:13,120 problem. 67 00:04:13,120 --> 00:04:14,680 And so we want to be able to know 68 00:04:14,680 --> 00:04:17,620 what goes into the radial part and how 69 00:04:17,620 --> 00:04:22,840 to use those ideas to be able to understand almost 70 00:04:22,840 --> 00:04:25,732 any problem, even problems where there 71 00:04:25,732 --> 00:04:26,815 is more than one electron. 72 00:04:30,850 --> 00:04:36,440 So I'm going to be making use of the classical, 73 00:04:36,440 --> 00:04:39,270 the semiclassical approach, where 74 00:04:39,270 --> 00:04:46,630 the classical momentum that's conjugate to the radius, 75 00:04:46,630 --> 00:04:49,480 in other words, the radial momentum, 76 00:04:49,480 --> 00:04:53,130 that we know what that is. 77 00:04:53,130 --> 00:04:56,790 It's related to the difference between the energy 78 00:04:56,790 --> 00:04:59,240 and the potential energy. 79 00:04:59,240 --> 00:05:04,530 And we get the momentum from that, the classical momentum 80 00:05:04,530 --> 00:05:05,430 function. 81 00:05:05,430 --> 00:05:08,310 And that gives us everything, because we 82 00:05:08,310 --> 00:05:12,210 use the DeBroglie relationship between the wavelength 83 00:05:12,210 --> 00:05:15,020 and the momentum. 84 00:05:15,020 --> 00:05:19,730 And I hope you believe what I'm going 85 00:05:19,730 --> 00:05:24,260 to try to do in the time we have for this. 86 00:05:24,260 --> 00:05:27,890 So there are several important things 87 00:05:27,890 --> 00:05:33,020 that I didn't get to have time to do in lecture. 88 00:05:33,020 --> 00:05:36,410 And I want to mention them, because I probably 89 00:05:36,410 --> 00:05:40,190 will write a question on the final connected to spin 90 00:05:40,190 --> 00:05:43,667 orbit and Stern-Gerlach. 91 00:05:43,667 --> 00:05:45,000 So you want to understand those. 92 00:05:48,240 --> 00:05:53,010 So the schedule for today is suppose 93 00:05:53,010 --> 00:05:56,760 you record a spectrum of the hydrogen atom. 94 00:05:59,520 --> 00:06:03,420 Now, that spectrum is going to reveal something 95 00:06:03,420 --> 00:06:09,880 that I regard as electronic structure, structure 96 00:06:09,880 --> 00:06:17,020 in this sense, not of bricks and mortar 97 00:06:17,020 --> 00:06:19,640 as you would talk about the structure of building, 98 00:06:19,640 --> 00:06:22,150 but what the architect had in mind 99 00:06:22,150 --> 00:06:24,580 and what the function of the building was. 100 00:06:24,580 --> 00:06:28,900 And so the structure is kind of a mystical thing, 101 00:06:28,900 --> 00:06:31,540 in which if you look at a little bit of it, 102 00:06:31,540 --> 00:06:34,550 you sort of get the idea of all of the rest. 103 00:06:34,550 --> 00:06:37,270 And so here, I'm going to show you 104 00:06:37,270 --> 00:06:41,880 how you can approach a spectrum and to know 105 00:06:41,880 --> 00:06:44,340 where you are, because the spectrum can 106 00:06:44,340 --> 00:06:45,960 be really complicated. 107 00:06:45,960 --> 00:06:48,480 And there are certain patterns that knowing 108 00:06:48,480 --> 00:06:52,200 about electronic structure enables you to say, yeah, 109 00:06:52,200 --> 00:06:53,190 I've got a Google map. 110 00:06:53,190 --> 00:06:55,790 I know exactly where I am. 111 00:06:55,790 --> 00:06:59,080 And it's based on some simple ideas. 112 00:06:59,080 --> 00:07:03,330 So this is first illustration of structure. 113 00:07:03,330 --> 00:07:07,320 Then I'm going to go and deal with the semiclassical methods 114 00:07:07,320 --> 00:07:10,830 for calculating all electronic properties of the hydrogen 115 00:07:10,830 --> 00:07:12,540 atom. 116 00:07:12,540 --> 00:07:15,500 And we get this business of scaling 117 00:07:15,500 --> 00:07:17,940 of properties with the principal quantum number. 118 00:07:21,130 --> 00:07:24,620 And that extends to Rydberg states of everything-- 119 00:07:24,620 --> 00:07:28,750 atoms and molecules-- not just the one electron spectrum. 120 00:07:28,750 --> 00:07:33,040 And the Rydberg states are special because one electron 121 00:07:33,040 --> 00:07:33,740 is special. 122 00:07:33,740 --> 00:07:36,040 It's outside of all of the others. 123 00:07:36,040 --> 00:07:39,370 And what we're going to learn about next time when 124 00:07:39,370 --> 00:07:42,370 we have many electrons or more than one, 125 00:07:42,370 --> 00:07:45,910 we don't have to think about antisymmetrization. 126 00:07:45,910 --> 00:07:47,980 We don't have-- we can still take 127 00:07:47,980 --> 00:07:52,090 our simple minded picture of an electron interacting 128 00:07:52,090 --> 00:07:53,770 with something. 129 00:07:53,770 --> 00:07:58,960 And so we can then take what we knew from hydrogen 130 00:07:58,960 --> 00:08:05,140 and describe everything about Rydberg states of molecules. 131 00:08:05,140 --> 00:08:08,180 And that's kind of exciting. 132 00:08:08,180 --> 00:08:12,300 And this will be followed by the bad news 133 00:08:12,300 --> 00:08:15,650 that when you have more than one electron, 134 00:08:15,650 --> 00:08:18,560 you have to do something else. 135 00:08:18,560 --> 00:08:24,180 You have to write antisymmetric wave functions, 136 00:08:24,180 --> 00:08:28,167 antisymmetric with respect to the exchange of all electrons. 137 00:08:28,167 --> 00:08:29,750 And that looks like it's going to lead 138 00:08:29,750 --> 00:08:32,760 to a tremendous headache. 139 00:08:32,760 --> 00:08:35,130 But it doesn't. 140 00:08:35,130 --> 00:08:38,380 But you do have to learn a new algebra. 141 00:08:38,380 --> 00:08:41,126 OK, so let's go on to this. 142 00:08:48,070 --> 00:08:54,390 OK, so for the hydrogen atom, it's 143 00:08:54,390 --> 00:09:00,290 easy to solve the time independent Schrodinger 144 00:09:00,290 --> 00:09:01,330 equation. 145 00:09:01,330 --> 00:09:06,720 And we get a complete set of these Rnl radial functions, 146 00:09:06,720 --> 00:09:08,520 complete set, an infinite number. 147 00:09:11,720 --> 00:09:13,870 We don't care. 148 00:09:13,870 --> 00:09:21,960 You can take this-- it's a simple one-dimensional 149 00:09:21,960 --> 00:09:23,065 differential equation. 150 00:09:23,065 --> 00:09:23,940 And you can solve it. 151 00:09:23,940 --> 00:09:25,564 You can tell your computer to solve it. 152 00:09:25,564 --> 00:09:27,540 You don't have to have any tricks at all. 153 00:09:27,540 --> 00:09:30,630 You can just use whatever numerical method 154 00:09:30,630 --> 00:09:34,260 to find the wave functions and calculate whatever you want. 155 00:09:34,260 --> 00:09:35,880 There's no insight there. 156 00:09:35,880 --> 00:09:39,270 It's the same thing as recording a spectrum, where 157 00:09:39,270 --> 00:09:41,520 you have tables of observed lines 158 00:09:41,520 --> 00:09:43,320 and maybe observed intensities. 159 00:09:43,320 --> 00:09:45,950 You don't know anything. 160 00:09:45,950 --> 00:09:47,570 It's a description. 161 00:09:47,570 --> 00:09:49,940 And you can have a mathematical description, 162 00:09:49,940 --> 00:09:53,030 or you can have an experimental description. 163 00:09:53,030 --> 00:09:55,010 You don't know anything. 164 00:09:55,010 --> 00:10:00,110 You want to have your insight telling you what's important 165 00:10:00,110 --> 00:10:03,620 and how to use what you know about part of the spectrum 166 00:10:03,620 --> 00:10:08,850 to determine other things. 167 00:10:08,850 --> 00:10:13,800 For example, we have two energy levels. 168 00:10:13,800 --> 00:10:18,640 Let's say n equals 1 and n equals 2. 169 00:10:18,640 --> 00:10:24,200 And there are several things that you 170 00:10:24,200 --> 00:10:28,100 would need to do in order to know what the selection 171 00:10:28,100 --> 00:10:33,530 rule for l is from this, because the l, the orbital angular 172 00:10:33,530 --> 00:10:36,991 momentum states of n equals 2 are degenerate. 173 00:10:39,880 --> 00:10:46,370 And so you could use the Zeeman effect. 174 00:10:46,370 --> 00:10:49,920 You can use radiative lifetimes. 175 00:10:49,920 --> 00:10:54,710 And what you would find is we have a p state and an s state. 176 00:10:54,710 --> 00:10:59,890 And the p state fluoresces to the s state 177 00:10:59,890 --> 00:11:03,450 and the s state doesn't, unless you apply an electric field 178 00:11:03,450 --> 00:11:07,330 and it mixes with a p state and it then fluoresces. 179 00:11:07,330 --> 00:11:10,570 So there are all sorts of things that will alert you to the fact 180 00:11:10,570 --> 00:11:13,040 that there's more going on here than just this simple level 181 00:11:13,040 --> 00:11:13,539 diagram. 182 00:11:18,580 --> 00:11:22,190 OK, so let's do some elementary stuff 183 00:11:22,190 --> 00:11:29,410 and then get to the Google map for the spectrum. 184 00:11:32,990 --> 00:11:37,940 We have this thing that makes the hydrogen atom interesting. 185 00:11:37,940 --> 00:11:41,090 The potential, the radial potential 186 00:11:41,090 --> 00:11:44,750 is a function of the orbital angular momentum. 187 00:11:44,750 --> 00:11:50,930 And the effective potential is given 188 00:11:50,930 --> 00:11:58,910 by h bar squared l, l plus 1, over 2 mu r squared-- 189 00:11:58,910 --> 00:12:01,550 and that's with a plus sign. 190 00:12:01,550 --> 00:12:08,910 And we have z e squared over 4 pi epsilon 0-- 191 00:12:08,910 --> 00:12:17,160 that's just stuff from the MKS units for the Coulomb 192 00:12:17,160 --> 00:12:20,480 equation-- and 1 over r. 193 00:12:20,480 --> 00:12:23,330 So we have an attractive thing that 194 00:12:23,330 --> 00:12:25,760 pulls the electron towards the nucleus. 195 00:12:25,760 --> 00:12:30,140 And we have a repulsion part, which keeps the electron away 196 00:12:30,140 --> 00:12:33,790 as long as l is not equal to 0. 197 00:12:33,790 --> 00:12:39,390 So this is the actual effective potential. 198 00:12:39,390 --> 00:12:43,010 This is the thing we would use for solving the Schrodinger 199 00:12:43,010 --> 00:12:48,320 equation for the Rnl of R functions 200 00:12:48,320 --> 00:12:52,520 exactly if you wanted to. 201 00:12:52,520 --> 00:12:57,950 And first surprise is that the energy levels come out to be-- 202 00:13:05,650 --> 00:13:10,700 for n equals 1, 2, 3, etc. 203 00:13:10,700 --> 00:13:14,600 So why are the energy levels not dependent on l? 204 00:13:14,600 --> 00:13:19,360 Because it takes more ink to write this than that. 205 00:13:19,360 --> 00:13:22,190 And this can be really important. 206 00:13:22,190 --> 00:13:24,270 But there is no l dependence. 207 00:13:24,270 --> 00:13:26,040 That's a surprise. 208 00:13:26,040 --> 00:13:30,860 It's not true for anything other than one-electron spectra. 209 00:13:30,860 --> 00:13:35,570 But if it's true and it has to be, 210 00:13:35,570 --> 00:13:38,930 you have to have something that says, OK, for something that's 211 00:13:38,930 --> 00:13:42,430 not hydrogen. Maybe there will be some l splitting. 212 00:13:42,430 --> 00:13:43,040 Why? 213 00:13:43,040 --> 00:13:43,800 Why is that? 214 00:13:43,800 --> 00:13:48,070 How does this effective potential 215 00:13:48,070 --> 00:13:50,600 have an effect for something other than hydrogen? 216 00:13:53,230 --> 00:13:57,960 And we'll understand that in a second. 217 00:13:57,960 --> 00:14:02,280 OK, now there's the question of degeneracy. 218 00:14:07,830 --> 00:14:15,750 So if we pick a value of n, we can have l equals 0, 1, 219 00:14:15,750 --> 00:14:16,920 up to n minus 1. 220 00:14:20,240 --> 00:14:21,860 That comes out of the mathematics. 221 00:14:21,860 --> 00:14:24,950 And I'm just assuming that you can accept that as a fact. 222 00:14:27,580 --> 00:14:34,180 We know from our study of the rigid rotor for the Ylm 223 00:14:34,180 --> 00:14:40,930 functions that the degeneracy of the l part of the wave function 224 00:14:40,930 --> 00:14:45,840 goes as 2l plus 1. 225 00:14:45,840 --> 00:14:47,790 If we have an orbital angular momentum of 0, 226 00:14:47,790 --> 00:14:50,260 there's a degeneracy of 1. 227 00:14:50,260 --> 00:14:52,930 Angular momentum of 1, it has a degeneracy of 3. 228 00:14:52,930 --> 00:14:53,590 We know that. 229 00:14:57,120 --> 00:15:02,230 So if we do a little game. 230 00:15:02,230 --> 00:15:06,790 And we look at the degeneracies for l equals 0, 1, 2, 231 00:15:06,790 --> 00:15:10,290 3, the degeneracies-- 232 00:15:10,290 --> 00:15:16,970 so g sub l, is 1, 3, 5, 7. 233 00:15:16,970 --> 00:15:25,620 OK, now, the degeneracy for n is going to be all of the possible 234 00:15:25,620 --> 00:15:26,520 l's. 235 00:15:26,520 --> 00:15:30,090 And so if we have n equals 1, we only have one l. 236 00:15:30,090 --> 00:15:31,770 And so the degeneracy is 1. 237 00:15:34,760 --> 00:15:40,100 If we have n equals 2, we have p and s. 238 00:15:40,100 --> 00:15:42,320 And so we have 4. 239 00:15:42,320 --> 00:15:47,600 And for n equals 2, we have-- 240 00:15:52,900 --> 00:16:05,960 I'm sorry, for n equals 1, we have only 1. 241 00:16:05,960 --> 00:16:08,230 And then we have for n equals 2, 4. 242 00:16:08,230 --> 00:16:12,040 And for n equals 3, 9. 243 00:16:12,040 --> 00:16:14,380 So you see 1 plus 3 is 4. 244 00:16:14,380 --> 00:16:17,620 1 plus 3 plus 5 is 9. 245 00:16:17,620 --> 00:16:22,150 And so the degeneracy for n is n squared. 246 00:16:25,870 --> 00:16:28,770 So as you go really high end, you get lots of states. 247 00:16:28,770 --> 00:16:33,820 And for hydrogen, they're all degenerate. 248 00:16:33,820 --> 00:16:37,930 But if they're not split, maybe you don't care. 249 00:16:37,930 --> 00:16:38,680 Of course, you do. 250 00:16:41,500 --> 00:16:43,050 I better leave this down. 251 00:16:47,180 --> 00:16:50,210 Now, here we are for-- 252 00:16:50,210 --> 00:16:51,440 suppose you take a spectrum. 253 00:16:54,230 --> 00:16:56,630 And suppose it's an emission spectrum. 254 00:16:56,630 --> 00:17:00,050 You run a discharge through gas of something 255 00:17:00,050 --> 00:17:04,839 that contains hydrogen. And you'll see a bunch of lines. 256 00:17:04,839 --> 00:17:08,740 And so one of the things you know 257 00:17:08,740 --> 00:17:15,069 is that the energy levels are separated-- 258 00:17:15,069 --> 00:17:17,349 they go as 1 over n squared. 259 00:17:17,349 --> 00:17:21,354 And so the energy levels-- 260 00:17:23,945 --> 00:17:26,430 well, actually, they're converging. 261 00:17:26,430 --> 00:17:29,470 And so let's just indicate that like this. 262 00:17:29,470 --> 00:17:35,677 So suppose you ask, well, suppose I'm at this level, 263 00:17:35,677 --> 00:17:38,010 there are going to be a series of transitions converging 264 00:17:38,010 --> 00:17:42,220 to a common limit from it. 265 00:17:42,220 --> 00:17:46,520 And that's a marker, a pattern in the spectrum. 266 00:17:46,520 --> 00:17:49,090 If you have essentially an infinite number 267 00:17:49,090 --> 00:17:53,560 of levels converging to a fixed point, that's easy to see. 268 00:17:53,560 --> 00:17:55,800 And so for each level, there will 269 00:17:55,800 --> 00:17:59,440 be convergence to the same fixed point. 270 00:17:59,440 --> 00:18:03,990 And so that tells you there's a kind of pattern recognition 271 00:18:03,990 --> 00:18:05,670 you would do. 272 00:18:05,670 --> 00:18:08,940 And you would begin to build up the energy level diagram 273 00:18:08,940 --> 00:18:13,410 and to know which levels you're looking at from these existence 274 00:18:13,410 --> 00:18:15,210 of convergent series. 275 00:18:18,410 --> 00:18:25,240 So if we're interested in the spectrum, 276 00:18:25,240 --> 00:18:30,040 we have the energy level differences, z 277 00:18:30,040 --> 00:18:38,279 squared Rydberg 1 over n squared minus 1 over n prime squared. 278 00:18:38,279 --> 00:18:39,445 That's the Rydberg equation. 279 00:18:42,120 --> 00:18:45,520 And so this also tells you something. 280 00:18:45,520 --> 00:18:49,670 So suppose you're in the nth level, well, you know this. 281 00:18:49,670 --> 00:18:54,010 And now this then converges to the ionization limit. 282 00:18:54,010 --> 00:18:55,710 But there's something else. 283 00:18:55,710 --> 00:19:01,990 And that is suppose we observe two consecutive members, 284 00:19:01,990 --> 00:19:03,610 n an plus 1. 285 00:19:03,610 --> 00:19:05,800 You don't know what n is, but because you 286 00:19:05,800 --> 00:19:08,230 see the pattern of this convergent series, 287 00:19:08,230 --> 00:19:12,010 you can see two levels that are obviously consecutive members 288 00:19:12,010 --> 00:19:13,900 of a series. 289 00:19:13,900 --> 00:19:17,750 And the question is what is n? 290 00:19:17,750 --> 00:19:26,040 And, again, you can solve that by knowing that the energy 291 00:19:26,040 --> 00:19:29,080 levels go as 1 over n squared. 292 00:19:29,080 --> 00:19:32,010 So the difference between energy levels 293 00:19:32,010 --> 00:19:43,030 goes as 2 z squared r over n cubed. 294 00:19:43,030 --> 00:19:46,590 This is just taking the derivative of 1 over n squared. 295 00:19:46,590 --> 00:19:49,290 We get 2 times 1 over n cubed. 296 00:19:52,290 --> 00:19:57,280 And so that kind of a pattern enables you to say, oh, yeah, 297 00:19:57,280 --> 00:19:58,210 I know what n is. 298 00:20:02,380 --> 00:20:05,440 So this is a kind of a hand waving, 299 00:20:05,440 --> 00:20:08,350 but this is what we do as spectroscopists. 300 00:20:08,350 --> 00:20:12,520 We're looking for a clue as to how to begin 301 00:20:12,520 --> 00:20:14,770 to put assignments on levels. 302 00:20:14,770 --> 00:20:17,710 And because there is this simple structure, which 303 00:20:17,710 --> 00:20:21,800 is represented by this Rydberg equation, 304 00:20:21,800 --> 00:20:23,675 we can say where we are. 305 00:20:23,675 --> 00:20:24,550 We know where we are. 306 00:20:24,550 --> 00:20:26,270 We have two choices. 307 00:20:26,270 --> 00:20:29,110 One is this 1 over n cubed scaling of the energy 308 00:20:29,110 --> 00:20:30,160 differences. 309 00:20:30,160 --> 00:20:33,130 And the other is the existence of a convergence 310 00:20:33,130 --> 00:20:37,480 of every initial level to a common final level. 311 00:20:37,480 --> 00:20:39,320 And that's very important. 312 00:20:39,320 --> 00:20:41,170 You can't assign a spectrum unless you 313 00:20:41,170 --> 00:20:42,820 know what the patterns are. 314 00:20:42,820 --> 00:20:45,130 And these are two things that come out of the fact 315 00:20:45,130 --> 00:20:48,070 that there is a structure associated with the hydrogen 316 00:20:48,070 --> 00:20:50,670 atom. 317 00:20:50,670 --> 00:20:55,830 OK, now, I'm going to have to put some numbers on the board. 318 00:20:55,830 --> 00:20:58,328 And it's another illustration of structure. 319 00:21:02,540 --> 00:21:08,320 Now, I'm just taking things from a table in McQuarrie. 320 00:21:08,320 --> 00:21:14,050 And this table is a table of expectation values 321 00:21:14,050 --> 00:21:17,900 of integer powers of r. 322 00:21:17,900 --> 00:21:18,670 And what is this? 323 00:21:18,670 --> 00:21:22,060 Every electronic property, anything you would measure 324 00:21:22,060 --> 00:21:26,370 is going to be a function of the-- 325 00:21:26,370 --> 00:21:29,680 is going to involve an integral involving a function of r. 326 00:21:32,270 --> 00:21:34,820 That's what we mean by electronic properties. 327 00:21:34,820 --> 00:21:40,010 And these electronic properties have a wonderful behavior. 328 00:21:40,010 --> 00:21:41,630 So let's make a little table. 329 00:21:41,630 --> 00:21:43,640 Here is the integer power. 330 00:21:43,640 --> 00:21:47,060 Here is the expectation value. 331 00:21:47,060 --> 00:21:48,140 And that's l. 332 00:21:51,330 --> 00:21:54,150 So we start with n equals 2. 333 00:21:58,580 --> 00:22:04,050 The result you get by actually accurately evaluating 334 00:22:04,050 --> 00:22:05,250 an integral-- 335 00:22:05,250 --> 00:22:06,655 and these are doable integrals. 336 00:22:06,655 --> 00:22:08,910 They are not numerically evaluated. 337 00:22:08,910 --> 00:22:11,260 And they have a simple formula-- 338 00:22:11,260 --> 00:22:20,390 a0 squared n to the fourth z squared times 1 339 00:22:20,390 --> 00:22:31,380 plus 3/2 times 1 minus l, l plus 1 minus 1/3. 340 00:22:31,380 --> 00:22:32,460 These are done integrals. 341 00:22:32,460 --> 00:22:33,390 I can't do them. 342 00:22:33,390 --> 00:22:36,300 I wouldn't care to do them. 343 00:22:36,300 --> 00:22:42,270 So that's how any electronic property that 344 00:22:42,270 --> 00:22:46,320 goes as the square of r behaves, we get this. 345 00:22:46,320 --> 00:22:48,120 This is the Bohr radius. 346 00:22:48,120 --> 00:22:51,120 This is approximately half an angstrom. 347 00:22:51,120 --> 00:22:55,830 And it's the radius of the n equals 1 Bohr orbit, 348 00:22:55,830 --> 00:22:58,020 which doesn't even exist. 349 00:22:58,020 --> 00:22:58,520 Right? 350 00:22:58,520 --> 00:23:01,040 There are no Bohr orbits. 351 00:23:01,040 --> 00:23:03,260 But we have this Bohr model, which 352 00:23:03,260 --> 00:23:05,040 explains these energy levels. 353 00:23:05,040 --> 00:23:07,130 And so we use that. 354 00:23:07,130 --> 00:23:09,560 And there's nothing else here. 355 00:23:09,560 --> 00:23:11,210 That's the charge on the nucleus. 356 00:23:11,210 --> 00:23:17,240 Well, it's 1 for hydrogen. But there's nothing empirical here. 357 00:23:17,240 --> 00:23:20,200 And this is a sort of a fundamental constant now, 358 00:23:20,200 --> 00:23:21,810 but it's sort of empirical. 359 00:23:24,390 --> 00:23:31,470 And then the next one, it has a very interesting initial term, 360 00:23:31,470 --> 00:23:37,560 a0 not squared n squared over z. 361 00:23:37,560 --> 00:23:40,570 We've lost a power here. 362 00:23:40,570 --> 00:23:46,250 And it again is one of these complicated looking functions. 363 00:23:58,450 --> 00:24:04,560 OK, this r to the 1, that's how big an atom 364 00:24:04,560 --> 00:24:06,510 is, the radius of the atom. 365 00:24:06,510 --> 00:24:09,430 And again, we have a closed form. 366 00:24:09,430 --> 00:24:12,030 Notice that the l is present here, 367 00:24:12,030 --> 00:24:16,070 even though it's not present in the energies. 368 00:24:16,070 --> 00:24:20,020 0, that just goes n to the 0 power. 369 00:24:20,020 --> 00:24:26,902 It's just a constant minus 1, minus 2, minus 3. 370 00:24:26,902 --> 00:24:30,790 Well, minus 1, that's what you have 371 00:24:30,790 --> 00:24:35,530 for the Coulomb attraction. 372 00:24:35,530 --> 00:24:42,180 And that goes as z over a0 n squared. 373 00:24:42,180 --> 00:24:50,020 And for minus 2, that goes as z squared a0 n cubed. 374 00:24:53,580 --> 00:24:56,400 And all that plus 1/2. 375 00:24:56,400 --> 00:25:07,503 And for minus 3, we get z cubed a0 cubed n cubed l, 376 00:25:07,503 --> 00:25:14,270 l plus 1/2, l plus 1. 377 00:25:14,270 --> 00:25:17,890 So there are all these formulas. 378 00:25:17,890 --> 00:25:20,170 Well, the important thing is the leading term. 379 00:25:20,170 --> 00:25:26,510 And what you discover is that right here something happens. 380 00:25:26,510 --> 00:25:29,472 The leading term goes as 1 over n cubed. 381 00:25:29,472 --> 00:25:34,540 It doesn't matter what the negative power of r is. 382 00:25:34,540 --> 00:25:37,870 The leading term goes as 1 over n cubed. 383 00:25:41,530 --> 00:25:42,146 Why's that? 384 00:25:45,226 --> 00:25:47,100 Well, that's really important, because this 1 385 00:25:47,100 --> 00:25:51,080 over n cubed behavior is telling you something very important. 386 00:25:55,310 --> 00:25:58,230 So here's the nucleus. 387 00:25:58,230 --> 00:26:00,600 And here's the electron. 388 00:26:00,600 --> 00:26:02,120 We can think of it as a Bohr orbit. 389 00:26:05,220 --> 00:26:08,710 If we have a negative power of r, 390 00:26:08,710 --> 00:26:13,390 then as you get farther and farther away from r, 391 00:26:13,390 --> 00:26:16,740 the property gets small. 392 00:26:16,740 --> 00:26:22,740 And so what happens is for large enough negative powers of r, 393 00:26:22,740 --> 00:26:25,200 the only thing that matters is the first lobe, 394 00:26:25,200 --> 00:26:28,189 the innermost lobe of the wave function. 395 00:26:28,189 --> 00:26:29,980 And it turns out that's the only thing that 396 00:26:29,980 --> 00:26:31,524 matters for almost everything. 397 00:26:34,130 --> 00:26:37,500 And so one of the things I'm going to show you 398 00:26:37,500 --> 00:26:41,160 is why does the innermost lobe matter 399 00:26:41,160 --> 00:26:45,380 and how do we use that to understand everything, 400 00:26:45,380 --> 00:26:47,330 not just about hydrogen, but about Rydberg 401 00:26:47,330 --> 00:26:48,564 states of everything. 402 00:26:51,290 --> 00:26:55,480 So the scaling of electronic properties 403 00:26:55,480 --> 00:27:00,590 depending on the power of r is another example of structure. 404 00:27:00,590 --> 00:27:06,320 And if you know one thing about the hydrogen atom, 405 00:27:06,320 --> 00:27:09,590 if you make one measurement, if you know how to use it, 406 00:27:09,590 --> 00:27:12,010 you know everything. 407 00:27:12,010 --> 00:27:14,750 You don't have to do all this other stuff. 408 00:27:14,750 --> 00:27:19,320 That's a really beautiful example of structure. 409 00:27:19,320 --> 00:27:23,630 And this is insight too, because sometimes the structure 410 00:27:23,630 --> 00:27:27,020 that you have for hydrogen is only approximately 411 00:27:27,020 --> 00:27:28,700 valid for other things. 412 00:27:28,700 --> 00:27:32,030 And you're going to want to know what you can use and know 413 00:27:32,030 --> 00:27:35,070 how to modify it. 414 00:27:35,070 --> 00:27:37,550 And I guarantee you that nobody teaching a course 415 00:27:37,550 --> 00:27:40,550 like this would ever talk about that kind of stuff, 416 00:27:40,550 --> 00:27:44,150 because, you know, it's approximate. 417 00:27:44,150 --> 00:27:44,990 It's intuitive. 418 00:27:44,990 --> 00:27:48,680 And all the good stuff is tabulated in the textbooks 419 00:27:48,680 --> 00:27:50,030 and you have to memorize it. 420 00:27:50,030 --> 00:27:52,450 But you don't know how to use it. 421 00:27:52,450 --> 00:27:53,600 And this is how you use it. 422 00:27:53,600 --> 00:27:55,500 This is really important. 423 00:27:58,040 --> 00:28:01,550 So now let me draw some pictures. 424 00:28:06,320 --> 00:28:13,165 So this is hydrogen. This is sodium. 425 00:28:16,060 --> 00:28:17,390 This is carbon monoxide. 426 00:28:21,720 --> 00:28:29,020 So the electron sees a point charge in hydrogen. 427 00:28:29,020 --> 00:28:31,510 That's easy. 428 00:28:31,510 --> 00:28:35,380 Now in sodium-- we don't know yet why, 429 00:28:35,380 --> 00:28:38,660 but once I've talked about helium and many atoms, 430 00:28:38,660 --> 00:28:41,110 you will know why-- 431 00:28:41,110 --> 00:28:46,270 there are a whole bunch of electrons in the nucleus 432 00:28:46,270 --> 00:28:49,930 or in the core of sodium. 433 00:28:49,930 --> 00:28:52,540 And outside the core, you have something 434 00:28:52,540 --> 00:28:54,520 that looks hydrogenic. 435 00:28:54,520 --> 00:28:55,900 But there is this core. 436 00:28:55,900 --> 00:28:57,490 It's not a point. 437 00:28:57,490 --> 00:29:03,580 And that leads to the appearance of l dependence in the energy 438 00:29:03,580 --> 00:29:07,330 levels, because what you're going to find 439 00:29:07,330 --> 00:29:11,860 is that l equals 0 penetrates into the core. 440 00:29:11,860 --> 00:29:13,720 l equals 1 can't. 441 00:29:13,720 --> 00:29:16,360 And so as you have higher and higher l, 442 00:29:16,360 --> 00:29:18,860 you're seeing less and less of this. 443 00:29:18,860 --> 00:29:22,000 And so as a result, there is an l dependence. 444 00:29:22,000 --> 00:29:24,130 But it's an s emphatically decreasing one. 445 00:29:24,130 --> 00:29:27,520 As l increases, you get less and less sensitivity 446 00:29:27,520 --> 00:29:29,260 to this inner part. 447 00:29:29,260 --> 00:29:30,985 Now, CO isn't even round. 448 00:29:35,200 --> 00:29:38,250 But there are nuclei. 449 00:29:38,250 --> 00:29:43,000 But if you have an electron at a high enough end, 450 00:29:43,000 --> 00:29:44,940 it's outside of this core. 451 00:29:48,010 --> 00:29:51,460 But again, if you pick l-- 452 00:29:51,460 --> 00:29:54,540 now, you can't have l for something that's not round. 453 00:29:54,540 --> 00:29:56,460 But if you're far enough away from it, 454 00:29:56,460 --> 00:29:59,700 you can pretend it's round and use the not roundness 455 00:29:59,700 --> 00:30:01,830 as a perturbation. 456 00:30:01,830 --> 00:30:04,830 And so you can have Rydberg states of CO. 457 00:30:04,830 --> 00:30:07,470 And you can use the same ideas that we developed 458 00:30:07,470 --> 00:30:12,030 for the hydrogen atom to explain the Rydberg states of CO, 459 00:30:12,030 --> 00:30:12,870 anthracene-- 460 00:30:15,810 --> 00:30:19,220 I don't even know how to say the names of biological molecules. 461 00:30:19,220 --> 00:30:21,060 So just consider it. 462 00:30:21,060 --> 00:30:22,710 If you had it in the gas phase and you 463 00:30:22,710 --> 00:30:28,110 could make a high enough Rydberg, high enough end state, 464 00:30:28,110 --> 00:30:34,380 it would look following this kind of extrapolation 465 00:30:34,380 --> 00:30:39,280 with features common to hydrogen. 466 00:30:39,280 --> 00:30:42,780 OK, so now, semiclassical-- 467 00:30:58,190 --> 00:31:03,240 OK, remember, you can calculate this. 468 00:31:03,240 --> 00:31:06,570 If you want to use rigorous mathematics, 469 00:31:06,570 --> 00:31:08,640 you can find an analytic solution 470 00:31:08,640 --> 00:31:12,530 for this, for hydrogen. And that's good, 471 00:31:12,530 --> 00:31:14,720 but you haven't extracted any insight from it. 472 00:31:14,720 --> 00:31:15,440 You just have it. 473 00:31:18,910 --> 00:31:21,460 But the semiclassical theory extracts insight. 474 00:31:27,020 --> 00:31:29,600 So semiclassical theory starts out 475 00:31:29,600 --> 00:31:34,310 with writing the classical equation for the momentum. 476 00:31:34,310 --> 00:31:36,460 And the momentum as a function of r, that's 477 00:31:36,460 --> 00:31:38,720 a no-no from quantum mechanics. 478 00:31:38,720 --> 00:31:44,200 You can't ever say we know the momentum as a function of r. 479 00:31:44,200 --> 00:31:46,100 And it's also true in quantum mechanics, 480 00:31:46,100 --> 00:31:48,770 whenever you calculate an integral 481 00:31:48,770 --> 00:31:53,490 that you need to evaluate some property, 482 00:31:53,490 --> 00:31:56,560 you're evaluating it over all space. 483 00:31:56,560 --> 00:31:58,620 But what I'm going to show you is that many 484 00:31:58,620 --> 00:32:02,670 of these integrals accumulated a specific point 485 00:32:02,670 --> 00:32:09,370 in space, which is a reminder of the classical mechanics. 486 00:32:09,370 --> 00:32:12,760 So there are things where quantum mechanics says 487 00:32:12,760 --> 00:32:15,100 you have to do something, but there's insight 488 00:32:15,100 --> 00:32:20,590 waiting to be harvested because many of the things you have 489 00:32:20,590 --> 00:32:25,360 to calculate are trivially interpretable in terms 490 00:32:25,360 --> 00:32:27,280 of semiclassical ideas. 491 00:32:27,280 --> 00:32:30,880 And the semiclassical means classical. 492 00:32:30,880 --> 00:32:32,650 You can do classical mechanics. 493 00:32:32,650 --> 00:32:34,720 You can calculate the probability 494 00:32:34,720 --> 00:32:37,307 of finding a particle at a particular place. 495 00:32:37,307 --> 00:32:38,890 And you can use that information here. 496 00:32:41,520 --> 00:32:48,780 And so always the relationship between the classical momentum 497 00:32:48,780 --> 00:32:54,790 function and the potential is this equation. 498 00:33:04,510 --> 00:33:07,300 p squared over 2 mu is the momentum. 499 00:33:07,300 --> 00:33:12,360 And energy minus potential is the momentum, 500 00:33:12,360 --> 00:33:14,590 is the kinetic energy. 501 00:33:14,590 --> 00:33:15,959 And this is how you get there. 502 00:33:15,959 --> 00:33:17,125 OK, so this is the function. 503 00:33:19,720 --> 00:33:22,150 And so that's one of the important pieces. 504 00:33:22,150 --> 00:33:25,810 And the other important piece is really unexpected, 505 00:33:25,810 --> 00:33:27,730 but wonderful. 506 00:33:27,730 --> 00:33:34,630 And so we have this Vl of r. 507 00:33:34,630 --> 00:33:37,780 If l is 0, the inner wall of the potential is vertical. 508 00:33:41,460 --> 00:33:44,880 If l is not 0, the inner wall of the potential 509 00:33:44,880 --> 00:33:48,000 isn't quite vertical, but it might as well 510 00:33:48,000 --> 00:34:01,650 be because when r gets small, even the l l plus 1 over r 511 00:34:01,650 --> 00:34:04,770 squared term gets small. 512 00:34:04,770 --> 00:34:08,969 And so we get this vertical potential. 513 00:34:08,969 --> 00:34:14,070 And so what happens is we can use the idea from DeBroglie 514 00:34:14,070 --> 00:34:19,400 and say, OK, where is the first lobe? 515 00:34:19,400 --> 00:34:20,400 Where is the first node? 516 00:34:20,400 --> 00:34:24,040 Well, it's half a wavelength from the turning point. 517 00:34:24,040 --> 00:34:26,159 So if we know where the turning point is we 518 00:34:26,159 --> 00:34:30,320 can draw something that looks like that. 519 00:34:30,320 --> 00:34:34,670 Now, it's a wave function, so it's going to be oscillating. 520 00:34:34,670 --> 00:34:39,860 So the important thing is that the first lobe, 521 00:34:39,860 --> 00:34:49,630 the innermost lobe, of all n and l lines up because this 522 00:34:49,630 --> 00:34:54,070 is nearly vertical and the energy distance from here 523 00:34:54,070 --> 00:34:58,060 to the bottom of the potential at r not equal to 0 524 00:34:58,060 --> 00:35:01,120 is really large. 525 00:35:01,120 --> 00:35:06,400 And so if you change n or l by 1 or 2, nothing much happens. 526 00:35:06,400 --> 00:35:09,670 And so if you just say, OK, for n greater than 527 00:35:09,670 --> 00:35:17,590 or equal to 6, 6, 36, the Rydberg constant divided by 36 528 00:35:17,590 --> 00:35:26,800 is not a big number, 3,000, 3,000 wave numbers. 529 00:35:26,800 --> 00:35:33,630 The vertical distance is more than 100,000 wave numbers. 530 00:35:33,630 --> 00:35:41,340 So the correction from n equals 6-- 531 00:35:41,340 --> 00:35:43,800 so if we have n equals 6, n equals 7, 532 00:35:43,800 --> 00:35:45,090 there isn't much change. 533 00:35:45,090 --> 00:35:51,070 And so the first lobe is going to always be at the same place. 534 00:35:51,070 --> 00:35:54,000 And so one thing that happens is they line up. 535 00:35:54,000 --> 00:36:00,310 The first node for all n and l is at the same place, 536 00:36:00,310 --> 00:36:02,520 to a good approximation. 537 00:36:02,520 --> 00:36:09,110 And the amplitude in this first lobe scales as n-- 538 00:36:09,110 --> 00:36:14,190 so the amplitude scales n to the minus 3/2. 539 00:36:14,190 --> 00:36:16,530 I'm going to prove that. 540 00:36:16,530 --> 00:36:19,630 This is where all the insight comes from. 541 00:36:19,630 --> 00:36:25,000 If you have a property, which is essentially determined 542 00:36:25,000 --> 00:36:29,750 close to the nucleus, because you 543 00:36:29,750 --> 00:36:38,150 have a negative power of the r in the electronic property, 544 00:36:38,150 --> 00:36:41,540 then the amplitude of the wave function 545 00:36:41,540 --> 00:36:44,420 will scale as n to the minus 3/2. 546 00:36:44,420 --> 00:36:51,790 And we can use this in evaluating all integrals, 547 00:36:51,790 --> 00:36:57,290 because everybody has a lobe here, at the same point. 548 00:36:57,290 --> 00:36:59,890 And the only thing that's different from one state, 549 00:36:59,890 --> 00:37:05,440 at one n l and another is the amplitude of this lobe. 550 00:37:05,440 --> 00:37:09,220 And that goes as 1 over n to 3/2 power. 551 00:37:13,960 --> 00:37:21,031 OK, that's what I was afraid of. 552 00:37:21,031 --> 00:37:21,530 OK. 553 00:37:31,420 --> 00:37:33,625 So we're interested in turning points. 554 00:37:44,040 --> 00:37:46,680 And what's the definition of a turning point? 555 00:37:46,680 --> 00:37:49,560 Or what is the mathematical equation that tells you 556 00:37:49,560 --> 00:37:54,240 how to get the inner nuclear dis-- the r 557 00:37:54,240 --> 00:37:59,510 value for the turning point. 558 00:37:59,510 --> 00:38:00,160 Yes. 559 00:38:00,160 --> 00:38:01,370 STUDENT: Derivative equals 0. 560 00:38:01,370 --> 00:38:02,370 ROBERT FIELD: I'm sorry. 561 00:38:02,370 --> 00:38:05,210 STUDENT: When the derivative of V l d r equals 0. 562 00:38:07,546 --> 00:38:09,170 ROBERT FIELD: I'm not looking for that. 563 00:38:09,170 --> 00:38:13,310 I'm looking for-- so when the energy 564 00:38:13,310 --> 00:38:20,910 is equal to V l of r, plus or minus. 565 00:38:20,910 --> 00:38:22,430 That's the equation that tells you 566 00:38:22,430 --> 00:38:24,020 where the turning points are. 567 00:38:24,020 --> 00:38:26,060 And these are simple equations. 568 00:38:26,060 --> 00:38:32,130 So it's child's play, adult child's, to calculate 569 00:38:32,130 --> 00:38:34,380 what the inner and outer turning point is 570 00:38:34,380 --> 00:38:38,320 as a function of n and l. 571 00:38:38,320 --> 00:38:43,370 And so we know the effective potential. 572 00:38:43,370 --> 00:38:44,300 We pick an energy. 573 00:38:44,300 --> 00:38:48,660 We know what it is because it's the Rydberg over n squared. 574 00:38:48,660 --> 00:38:55,950 So we have enough to calculate this function, which 575 00:38:55,950 --> 00:39:07,370 is a0 n squared 1 plus or minus 1 minus l l plus 1 576 00:39:07,370 --> 00:39:12,100 over n squared square root. 577 00:39:12,100 --> 00:39:13,300 Now I could derive this. 578 00:39:13,300 --> 00:39:15,850 You could derive this. 579 00:39:15,850 --> 00:39:18,080 It's a simple closed form equation. 580 00:39:18,080 --> 00:39:21,950 And that's the foundation of all of this. 581 00:39:21,950 --> 00:39:25,960 Now, the thing that's common for everything 582 00:39:25,960 --> 00:39:32,570 is the inner turning point, r minus, 583 00:39:32,570 --> 00:39:38,100 because what we're interested in is this. 584 00:39:38,100 --> 00:39:44,910 Where is the maximum or where is the first node 585 00:39:44,910 --> 00:39:46,817 and what is the amplitude? 586 00:39:46,817 --> 00:39:49,150 And so we're going to use this equation to get all that. 587 00:39:56,610 --> 00:40:03,030 All right, so the semiclassical theory lambda n l-- 588 00:40:03,030 --> 00:40:05,110 that's the wavelength-- 589 00:40:05,110 --> 00:40:07,906 is equal to h over Pr of r. 590 00:40:10,570 --> 00:40:12,400 That's DeBroglie. 591 00:40:12,400 --> 00:40:16,840 But it's generalized to a potential, which 592 00:40:16,840 --> 00:40:20,290 is dependent on r, or a momentum function, 593 00:40:20,290 --> 00:40:24,520 which is dependent on r. 594 00:40:24,520 --> 00:40:28,540 So now, we have to calculate several important things 595 00:40:28,540 --> 00:40:29,860 in order to build our model. 596 00:40:36,620 --> 00:40:42,420 One is the classical oscillation period. 597 00:40:48,170 --> 00:40:53,480 So if I told you we have a harmonic oscillator, 598 00:40:53,480 --> 00:41:00,780 we know the energy level spacing things are h bar omega. 599 00:41:00,780 --> 00:41:03,600 And they're constant. 600 00:41:03,600 --> 00:41:08,340 If we make a superposition state of a harmonic oscillator, 601 00:41:08,340 --> 00:41:13,490 there's going to be beat nodes at integer multiples of omega. 602 00:41:13,490 --> 00:41:20,610 So we can simply say, well, for Rydberg states or for anything, 603 00:41:20,610 --> 00:41:26,210 we can define the period of oscillation as h over, 604 00:41:26,210 --> 00:41:33,410 in this case, n plus 1/2 minus n Vn minus 1/2. 605 00:41:33,410 --> 00:41:38,660 This is the energetic separation between levels-- 606 00:41:38,660 --> 00:41:40,130 or even not levels. 607 00:41:40,130 --> 00:41:44,310 We have a formula that's giving the independence of the energy. 608 00:41:44,310 --> 00:41:47,810 So this is related to the period that we get 609 00:41:47,810 --> 00:41:50,580 from the harmonic oscillator. 610 00:41:50,580 --> 00:41:53,000 So this is a perfectly legitimate way 611 00:41:53,000 --> 00:41:54,840 of knowing the period. 612 00:41:58,750 --> 00:42:03,450 Well, when we do that, this is something I wrote down before, 613 00:42:03,450 --> 00:42:07,710 the energy separation of two levels differing in n 614 00:42:07,710 --> 00:42:17,890 by 1 centered at n is going to be the Rydberg divided by 2n-- 615 00:42:17,890 --> 00:42:19,400 well, I'll just write it down. 616 00:42:19,400 --> 00:42:20,720 So we have h. 617 00:42:20,720 --> 00:42:27,180 And now this hcR 2 over n cubed. 618 00:42:30,440 --> 00:42:34,840 And n cubed is really important. 619 00:42:34,840 --> 00:42:38,360 It came just from taking the derivative of the 1 620 00:42:38,360 --> 00:42:42,230 over n squared energy dependent. 621 00:42:42,230 --> 00:42:45,260 So this is a formula that's perfectly legitimate. 622 00:42:45,260 --> 00:42:48,990 What n tells us is the period is proportional to n cubed. 623 00:42:51,970 --> 00:42:53,680 The higher you go, the slower you go. 624 00:43:00,260 --> 00:43:10,972 Second, node to node probability. 625 00:43:13,630 --> 00:43:18,570 So how long does it take-- you have this well. 626 00:43:18,570 --> 00:43:22,980 And I don't mean to draw it like a harmonic oscillator, 627 00:43:22,980 --> 00:43:25,270 but it's a well. 628 00:43:25,270 --> 00:43:28,000 And so you have a node here, and you have a node here. 629 00:43:28,000 --> 00:43:31,086 And you have a particle that's moving, classically. 630 00:43:31,086 --> 00:43:33,210 How long did it take for the center of the particle 631 00:43:33,210 --> 00:43:35,610 to go from here to here? 632 00:43:35,610 --> 00:43:37,770 That's an easy thing to calculate too, 633 00:43:37,770 --> 00:43:40,110 because we know classically what the momentum is 634 00:43:40,110 --> 00:43:41,940 at any point in space. 635 00:43:41,940 --> 00:43:45,000 The momentum is related to the energy difference 636 00:43:45,000 --> 00:43:46,170 between here and here. 637 00:43:52,610 --> 00:43:56,920 So what we're going for is the ratio 638 00:43:56,920 --> 00:44:06,840 of the time node to node, or to next node, 639 00:44:06,840 --> 00:44:13,100 to delta t turning to turning point. 640 00:44:15,780 --> 00:44:20,420 OK, well, this is 1/2 the period, right? 641 00:44:20,420 --> 00:44:22,640 So we know what 1/2 the period is. 642 00:44:22,640 --> 00:44:28,700 And the period goes as n cubed. 643 00:44:28,700 --> 00:44:34,450 So this is the probability of finding the particle within one 644 00:44:34,450 --> 00:44:38,000 lobe of the wave function. 645 00:44:38,000 --> 00:44:44,530 So we can calculate the probability of finding 646 00:44:44,530 --> 00:44:47,880 the particle within one lobe. 647 00:44:47,880 --> 00:44:50,940 And it's clearly going as 1 over n cubed. 648 00:44:53,550 --> 00:44:55,544 Now, we want an amplitude. 649 00:44:55,544 --> 00:44:56,460 What is the amplitude? 650 00:44:56,460 --> 00:44:59,620 It's square root of the probability. 651 00:44:59,620 --> 00:45:04,240 All of a sudden, we start to see that the amplitude in whatever 652 00:45:04,240 --> 00:45:09,670 lobe we want goes as 1 over n to the 3/2. 653 00:45:09,670 --> 00:45:14,620 Now, because we have this lining up of nodes, the first node 654 00:45:14,620 --> 00:45:16,620 and the first lobe being identical, 655 00:45:16,620 --> 00:45:20,500 what we have now is the scaling of the probability 656 00:45:20,500 --> 00:45:23,440 or the amplitude in the first lobe of the wave function. 657 00:45:27,000 --> 00:45:34,110 And the next thing is again something 658 00:45:34,110 --> 00:45:36,790 that's almost never talked about in textbooks, 659 00:45:36,790 --> 00:45:39,930 but it's a really fantastic tool for understanding 660 00:45:39,930 --> 00:45:47,640 stuff is suppose we want to know the value of an integral. 661 00:45:51,430 --> 00:45:54,340 So we have an integral. 662 00:45:54,340 --> 00:46:04,430 And that will be, say, the electronic wave function 663 00:46:04,430 --> 00:46:06,490 in this chi representation, as opposed 664 00:46:06,490 --> 00:46:08,650 to the Rnl representation. 665 00:46:18,030 --> 00:46:21,060 OK, so this is the integral we want. 666 00:46:21,060 --> 00:46:24,450 But what we really want is a way of saying, 667 00:46:24,450 --> 00:46:26,790 I can convert this integral to a number 668 00:46:26,790 --> 00:46:30,800 that I can figure out on the back of a postage stamp. 669 00:46:30,800 --> 00:46:32,210 No methods for integration. 670 00:46:32,210 --> 00:46:33,650 No numerical methods. 671 00:46:33,650 --> 00:46:35,420 Just insight. 672 00:46:35,420 --> 00:46:45,450 So we have some electronic property, 673 00:46:45,450 --> 00:46:47,680 which comes from this integral. 674 00:46:47,680 --> 00:46:52,010 But now, this is a rapidly oscillating function. 675 00:46:52,010 --> 00:46:55,650 This is another rapidly isolating function. 676 00:46:55,650 --> 00:46:57,960 And so if we were to ask, well, how 677 00:46:57,960 --> 00:46:59,880 does the integral accumulate? 678 00:46:59,880 --> 00:47:01,770 In other words, let's say we're integrating 679 00:47:01,770 --> 00:47:12,810 from r minus to r prime chi n l of r r of k chi n prime l 680 00:47:12,810 --> 00:47:17,640 prime of r Vr. 681 00:47:17,640 --> 00:47:25,890 So if we plotted the integral, the value of the integral, 682 00:47:25,890 --> 00:47:28,550 as it accumulates-- 683 00:47:28,550 --> 00:47:31,555 so we're not evaluating the whole integral. 684 00:47:31,555 --> 00:47:33,530 We're evaluating the integral up to the r 685 00:47:33,530 --> 00:47:35,210 point, the r prime point. 686 00:47:39,380 --> 00:47:50,700 And what we see that there is a stationary phase point, 687 00:47:50,700 --> 00:47:55,650 where at some point in space, the two functions 688 00:47:55,650 --> 00:47:59,760 are oscillating spatially at the same frequency. 689 00:47:59,760 --> 00:48:02,670 And we already saw this for Fermi's golden rule, right. 690 00:48:02,670 --> 00:48:04,680 You have a time integral where you have rapidly 691 00:48:04,680 --> 00:48:06,060 oscillating functions. 692 00:48:06,060 --> 00:48:09,960 And then when the applied frequency 693 00:48:09,960 --> 00:48:14,310 is resonant with the difference in frequency, 694 00:48:14,310 --> 00:48:16,690 we get the integral accumulating. 695 00:48:16,690 --> 00:48:17,940 So same thing. 696 00:48:17,940 --> 00:48:19,830 It's easier. 697 00:48:19,830 --> 00:48:25,380 And so we have this idea that there is a stationary phase 698 00:48:25,380 --> 00:48:26,520 point. 699 00:48:26,520 --> 00:48:29,580 And the integral accumulates from 0 to its final value 700 00:48:29,580 --> 00:48:31,120 there. 701 00:48:31,120 --> 00:48:33,870 And then there's just little dithering 702 00:48:33,870 --> 00:48:36,140 as you go out the rest of the way. 703 00:48:36,140 --> 00:48:38,610 You get in complications when they're two stationary phase 704 00:48:38,610 --> 00:48:41,490 points, because then there's destructive or constructive 705 00:48:41,490 --> 00:48:42,820 interference between them. 706 00:48:42,820 --> 00:48:45,060 But it's very rare that you have two. 707 00:48:45,060 --> 00:48:46,330 You only have one. 708 00:48:46,330 --> 00:48:50,850 And so if you know the amplitude of the wave functions 709 00:48:50,850 --> 00:48:55,140 at a stationary phase point, you can calculate the integral 710 00:48:55,140 --> 00:48:56,790 as a product of three things. 711 00:48:59,460 --> 00:49:02,180 No integration, just one number. 712 00:49:02,180 --> 00:49:04,730 And the thing that we care about is 713 00:49:04,730 --> 00:49:10,070 that almost all electronic properties 714 00:49:10,070 --> 00:49:16,530 accumulate in the innermost lobe, because they're the same. 715 00:49:16,530 --> 00:49:18,650 The wave functions are the same. 716 00:49:18,650 --> 00:49:20,150 And they get more and more different 717 00:49:20,150 --> 00:49:23,310 as you go farther out. 718 00:49:23,310 --> 00:49:25,100 And as a result, since the amplitude 719 00:49:25,100 --> 00:49:31,090 in this innermost lobe goes as 1 over n to the 3/2 720 00:49:31,090 --> 00:49:32,700 and you've got two functions-- 721 00:49:32,700 --> 00:49:34,630 you've got the nl function and the n prime l 722 00:49:34,630 --> 00:49:37,310 prime function-- you have that the integral 723 00:49:37,310 --> 00:49:45,053 is proportional to 1 over n to the 3/2 1 over n 724 00:49:45,053 --> 00:49:48,310 prime to the 3/2. 725 00:49:48,310 --> 00:49:50,710 So not only do you get expectation values, 726 00:49:50,710 --> 00:49:54,030 you get off diagonal matrix elements. 727 00:49:54,030 --> 00:49:56,230 Now this is not exact. 728 00:49:56,230 --> 00:50:00,370 But it tells you this is the structure of the problem. 729 00:50:00,370 --> 00:50:03,400 And if I look at enough stuff, this 730 00:50:03,400 --> 00:50:09,156 is really important to say, how does everything scale? 731 00:50:09,156 --> 00:50:10,530 And if you know how things scale, 732 00:50:10,530 --> 00:50:13,020 you know that if you're combining 733 00:50:13,020 --> 00:50:15,300 things that aren't part of that scaling rule, 734 00:50:15,300 --> 00:50:19,030 that they're just not going to be relevant. 735 00:50:19,030 --> 00:50:22,670 This is a fantastic simplification, 736 00:50:22,670 --> 00:50:26,000 because, yes, you can do it exactly 737 00:50:26,000 --> 00:50:28,010 by programming your computer to calculate 738 00:50:28,010 --> 00:50:29,550 the integral numerically. 739 00:50:29,550 --> 00:50:31,010 And that's not a big deal. 740 00:50:31,010 --> 00:50:33,080 But you don't know anything. 741 00:50:33,080 --> 00:50:35,930 And here, you know what to expect. 742 00:50:35,930 --> 00:50:39,140 And this is a professional spectroscopist 743 00:50:39,140 --> 00:50:43,800 who survives by recognizing patterns and using patterns. 744 00:50:43,800 --> 00:50:46,760 This is beyond numerical integrals. 745 00:50:46,760 --> 00:50:50,360 This is actually understanding the structure of the problem. 746 00:50:50,360 --> 00:50:55,050 And the fact that the molecule gives you this beautiful-- 747 00:50:55,050 --> 00:50:57,770 or the atom gives you this beautiful lobe 748 00:50:57,770 --> 00:51:01,520 always at the same place, you know stationary phase. 749 00:51:01,520 --> 00:51:03,020 Now, there are other problems where 750 00:51:03,020 --> 00:51:05,900 you use the stationary phase approximation. 751 00:51:05,900 --> 00:51:09,080 You always want to be trying to use stationary phase. 752 00:51:09,080 --> 00:51:13,100 For example-- and I'm going to stop soon-- 753 00:51:13,100 --> 00:51:16,349 suppose you have two potential energy curves. 754 00:51:16,349 --> 00:51:18,140 Now, you don't know what a potential energy 755 00:51:18,140 --> 00:51:20,080 curve for a molecule is yet. 756 00:51:20,080 --> 00:51:22,100 But you can imagine what there is. 757 00:51:22,100 --> 00:51:24,200 And so they cross. 758 00:51:24,200 --> 00:51:32,120 And this is the place where at this energy, 759 00:51:32,120 --> 00:51:36,290 this curve is exactly the same as the momentum on that curve. 760 00:51:36,290 --> 00:51:40,310 And so the integral accumulates at this curve crossing point. 761 00:51:40,310 --> 00:51:42,246 Isn't that neat? 762 00:51:42,246 --> 00:51:44,120 If you know what the curve crossing point is, 763 00:51:44,120 --> 00:51:46,310 you know the amplitude of the wave function. 764 00:51:46,310 --> 00:51:49,930 And you can estimate any integral. 765 00:51:49,930 --> 00:51:54,250 So this is something you would use if you're actually 766 00:51:54,250 --> 00:51:59,080 creating new knowledge again and again, because it's deeper 767 00:51:59,080 --> 00:52:01,480 than the actual experimental observation. 768 00:52:01,480 --> 00:52:03,220 It's the explanation for it. 769 00:52:03,220 --> 00:52:05,680 And it's what you're looking for and how 770 00:52:05,680 --> 00:52:10,310 you use your knowledge of what you're looking for. 771 00:52:10,310 --> 00:52:13,640 That's why I like this lecture, because it really 772 00:52:13,640 --> 00:52:20,520 gives the approach that I take to all scientific problems. 773 00:52:20,520 --> 00:52:23,210 I look for an approximate way where 774 00:52:23,210 --> 00:52:26,690 I don't have to do any complicated mathematics 775 00:52:26,690 --> 00:52:28,040 or numerical integration. 776 00:52:28,040 --> 00:52:32,880 I can do that just sitting at my desk. 777 00:52:32,880 --> 00:52:37,420 OK, so on Friday, we'll hear about helium and why helium 778 00:52:37,420 --> 00:52:39,840 looks horrible, but isn't.