WEBVTT 00:02.610 --> 00:05.450 Prof: Okay, so we just glimpsed this at the 00:05.452 --> 00:06.382 end last time. 00:06.380 --> 00:10.320 This is a crystal structure of a complicated molecule that was 00:10.320 --> 00:14.390 performed by these same Swiss folks that we've talked about, 00:14.390 --> 00:16.840 and notice how very precise it is. 00:16.840 --> 00:21.260 The bond distances between atoms are reported to 00:24.450 --> 00:27.070 The bonds are like one and a half angstroms. 00:27.070 --> 00:30.740 So it's like a part in a thousand, is the precision to 00:30.736 --> 00:33.776 which the positions of the atoms is known. 00:33.780 --> 00:34.810 Okay? 00:34.810 --> 00:39.200 But those positions are average positions, because the atoms are 00:39.200 --> 00:41.570 constantly in motion, vibrating. 00:41.570 --> 00:44.970 In fact, the typical vibration amplitude, 00:44.970 --> 00:48.030 which depends on -- an atom that's off on the end of 00:48.026 --> 00:51.556 something is more floppy than one that's held in by a lot of 00:51.562 --> 00:55.592 bonds in various directions, in a cage sort of thing. 00:55.590 --> 01:00.160 But typically they're vibrating by about 0.05 angstroms, 01:00.158 --> 01:04.378 which is twenty-five times as big as the precision to which 01:04.375 --> 01:07.205 the position of the average is known. 01:07.209 --> 01:08.659 Okay? 01:08.659 --> 01:11.189 So no molecule looks like that at an instant, 01:11.192 --> 01:13.672 the atoms are all displaced a little bit. 01:13.670 --> 01:15.070 Now how big is that? 01:15.069 --> 01:17.309 Here, if you look at that yellow thing, 01:17.313 --> 01:19.793 when it shrinks down, that's how big it is, 01:19.793 --> 01:21.923 that's how big the vibration is. 01:21.920 --> 01:23.320 It's very small. 01:23.319 --> 01:25.469 But these are very precise measurements. 01:25.468 --> 01:25.908 Right? 01:25.909 --> 01:28.569 Now why did they do so precise measurements? 01:28.569 --> 01:32.499 Did they really care to know bond distances to that accuracy? 01:32.500 --> 01:35.530 Maybe for some purposes they did, but that wasn't the main 01:35.534 --> 01:37.774 reason they did the work very carefully. 01:37.769 --> 01:41.659 They did it carefully in order to get really precise positions 01:41.659 --> 01:44.629 for the average atom, so they could subtract 01:44.632 --> 01:47.982 spherical atoms and see the difference accurately. 01:47.980 --> 01:48.970 Okay? 01:48.970 --> 01:52.490 Because if you have the wrong position for the atom that 01:52.492 --> 01:54.992 you're subtracting, you get nonsense. 01:54.989 --> 01:59.629 Okay, and what this is going to reveal is some pathologies of 01:59.625 --> 02:04.025 bonding, from the point of view of Lewis concept of shared 02:04.028 --> 02:05.108 electrons. 02:05.108 --> 02:08.948 Okay, so here's a picture of this molecule. 02:08.949 --> 02:10.979 And remember, we had -- rubofusarin, 02:10.984 --> 02:14.304 which we looked at last time, had the great virtue that it 02:14.298 --> 02:15.168 was planar. 02:15.169 --> 02:17.939 So you could cut a slice that went through all the atoms. 02:17.938 --> 02:21.308 This molecule's definitely not planar, so you have to cut 02:21.305 --> 02:23.765 various slices to see different things. 02:23.770 --> 02:27.760 So first we'll cut a slice that goes through those ten atoms. 02:27.759 --> 02:28.729 Okay? 02:28.729 --> 02:32.179 And here is the difference electron density. 02:32.180 --> 02:39.010 What does the difference density show? 02:39.009 --> 02:40.429 Somebody? 02:40.430 --> 02:42.590 Yes Alex? 02:42.590 --> 02:43.690 Student: It's the electron density minus the 02:43.693 --> 02:45.163 spherical -- Prof: It's the total 02:45.161 --> 02:46.601 electron density minus the atoms; 02:46.598 --> 02:49.818 that is, how the electron density shifted when the 02:49.818 --> 02:52.248 molecule was formed from the atoms. 02:52.250 --> 02:56.190 Okay, and here we see just exactly what we expected to see, 02:56.190 --> 03:00.000 that the electrons shifted in between the carbon atoms, 03:00.000 --> 03:02.880 the benzene ring, and other pairs of carbon atoms 03:02.878 --> 03:03.478 as well. 03:03.479 --> 03:07.019 It also shows the C-H bonds, because in this case the 03:07.021 --> 03:09.271 hydrogen atoms were subtracted. 03:09.270 --> 03:12.490 We showed one last time where the hydrogen atoms weren't 03:12.489 --> 03:13.249 subtracted. 03:13.250 --> 03:15.730 Okay, so this is -- there's nothing special here, 03:15.729 --> 03:18.159 everything looks the way you expect it to be; 03:18.158 --> 03:22.638 although it's really beautiful, as you would expect from these 03:22.644 --> 03:24.854 guys who do such great work. 03:24.849 --> 03:26.179 Now we'll do a different slice. 03:26.180 --> 03:29.600 This is sort of the plane of the screen which divides the 03:29.602 --> 03:32.172 molecule symmetrically down the middle, 03:32.169 --> 03:35.949 cuts through some bonds, cuts through some atoms and so 03:35.949 --> 03:36.299 on. 03:36.300 --> 03:40.890 So here's the difference density map that appears on that 03:40.893 --> 03:41.553 slice. 03:41.550 --> 03:44.380 Now the colored atoms, on the right, 03:44.381 --> 03:49.321 are the positions of the atoms through which the plane slices, 03:49.316 --> 03:52.306 but the atoms are subtracted out; 03:52.310 --> 03:55.910 so what you see is the bonding in that plane. 03:55.910 --> 03:59.070 So you see those bonds, because both ends of the bond 03:59.073 --> 04:01.933 are in the plane, so the bonds are in the plane, 04:01.932 --> 04:04.612 and you see just what you expect to see. 04:04.610 --> 04:06.530 But there are other things you see as well. 04:06.530 --> 04:10.660 You see the C-H bonds, although they don't have nearly 04:10.657 --> 04:14.157 as much electron density as the C-C bonds did. 04:14.161 --> 04:14.941 Right? 04:14.938 --> 04:20.128 You also see that lump, which is the unshared pair on 04:20.132 --> 04:21.032 nitrogen. 04:21.031 --> 04:21.931 Right? 04:21.930 --> 04:25.160 But you see these two things, which are bonds, 04:25.160 --> 04:28.530 but they're cross-sections of bonds, 04:28.528 --> 04:31.778 because this particular plane cuts through the middle of those 04:31.783 --> 04:32.213 bonds. 04:32.209 --> 04:36.029 Everybody see that? 04:36.029 --> 04:38.309 Okay, so again that's nothing surprising. 04:38.310 --> 04:40.340 But here is something surprising. 04:40.339 --> 04:43.689 There's another bond through which that plane cuts, 04:43.692 --> 04:47.582 which is the one on the right, through those three-membered 04:47.579 --> 04:48.719 rings, right? 04:48.720 --> 04:54.420 And what do you notice about that bond? 04:54.420 --> 04:54.520 Student: It isn't there. 04:54.519 --> 04:56.609 Prof: It isn't there. 04:56.610 --> 05:00.080 There isn't any electron density for that bond. 05:00.079 --> 05:02.639 So it's a missing bond. 05:02.639 --> 05:06.459 This is what we'll refer to as pathological bonding, 05:06.459 --> 05:07.059 right? 05:07.060 --> 05:11.130 It's not what Lewis would have expected, maybe; 05:11.129 --> 05:14.339 we don't have Lewis to talk to, so we don't know what he 05:14.339 --> 05:17.259 would've thought about this particular molecule. 05:17.259 --> 05:21.069 So here's a third plane to slice, which goes through those 05:21.074 --> 05:24.024 three atoms, and here's the picture of it. 05:24.019 --> 05:29.579 And again, that bond is missing, that we saw before. 05:29.579 --> 05:31.309 Previously we looked at a cross-section. 05:31.310 --> 05:34.720 Here we're looking at a plane that contains the bond, 05:34.716 --> 05:37.006 and again there's no there there. 05:37.009 --> 05:37.599 Okay? 05:37.600 --> 05:40.550 But there's something else that's funny about this slice. 05:40.550 --> 05:44.190 Do you see what? 05:44.190 --> 05:51.200 What's funny about the bonds that you do see? 05:51.199 --> 05:55.169 Corey? 05:55.170 --> 05:56.450 Speak up so I can hear you. 05:56.449 --> 05:59.049 Student: They're connected; they're not totally 05:59.050 --> 05:59.580 separate. 05:59.579 --> 06:02.449 Prof: What do you mean they're connected separately? 06:02.449 --> 06:08.379 Student: Usually you see separate electron densities, 06:08.384 --> 06:10.804 but they're connected. 06:10.800 --> 06:12.820 Prof: Somebody say it in different words. 06:12.819 --> 06:14.739 I think you got the idea but I'm not sure everybody 06:14.735 --> 06:15.345 understood it. 06:15.350 --> 06:19.080 John, do you have an idea? 06:19.079 --> 06:21.699 Student: The top one seems to be more dense than the 06:21.702 --> 06:22.282 bottom one. 06:22.278 --> 06:23.908 Prof: One, two, three, four; 06:23.910 --> 06:25.520 one, two, three four five. 06:25.519 --> 06:27.559 That's true, it is a little more dense. 06:27.560 --> 06:30.540 That kind of thing could be experimental error, 06:30.540 --> 06:33.160 because even though this was done so precisely, 06:33.160 --> 06:37.440 you're subtracting two numbers that are very large, 06:37.440 --> 06:40.450 so that any error you make in the experimental one, 06:40.449 --> 06:43.489 or in positioning things for the theoretical position of the 06:43.485 --> 06:45.785 atoms, any error you make will really 06:45.788 --> 06:47.808 be amplified in a map like this. 06:47.810 --> 06:49.350 But it's true, you noticed. 06:49.350 --> 06:52.080 But there's something I think more interesting about those -- 06:52.079 --> 06:52.489 that... 06:52.490 --> 06:53.630 Yes John? 06:53.629 --> 06:56.529 Student: The contour lines, they're connected, 06:56.526 --> 06:59.636 the contour lines between the top and the bottom bonds are 06:59.644 --> 07:00.414 connected. 07:00.410 --> 07:02.960 So maybe the electrons, maybe -- I don't know if 07:02.956 --> 07:04.036 they're connected. 07:04.040 --> 07:05.870 Prof: Yes, they sort of overlap one 07:05.872 --> 07:06.322 another. 07:06.319 --> 07:09.529 But of course if they're sort of close to one another, 07:09.528 --> 07:13.518 that doesn't surprise you too much because as you go out and 07:13.524 --> 07:15.674 out and out, ultimately you'll get rings 07:15.665 --> 07:19.225 that do meet, if you go far enough down. 07:19.230 --> 07:21.120 Yes Chris? 07:21.120 --> 07:24.700 Student: The center of density on the bonds doesn't 07:24.697 --> 07:27.347 intercept the lines connecting the atoms. 07:27.350 --> 07:28.150 Prof: Ah. 07:28.149 --> 07:32.939 The bonds are not centered on the line that connects the 07:32.935 --> 07:33.715 nuclei. 07:33.720 --> 07:40.300 These bonds are bent. 07:40.300 --> 07:42.120 Okay? 07:42.120 --> 07:46.270 So again, pathological bonding; and in three weeks you'll 07:46.274 --> 07:49.534 understand this, from first principles, 07:49.533 --> 07:52.283 but you've got to be patient. 07:52.279 --> 07:56.369 Okay, so Lewis pairs and octets provide a pretty good 07:56.369 --> 08:00.379 bookkeeping device for keeping track of valence, 08:00.379 --> 08:03.379 but they're hopelessly crude when it comes to describing the 08:03.380 --> 08:06.480 actual electron distribution, which you can see 08:06.480 --> 08:08.300 experimentally here. 08:08.300 --> 08:10.610 There is electron sharing. 08:10.610 --> 08:14.750 There's a distortion of the spheres of electron density that 08:14.747 --> 08:17.887 are the atoms, but it's only about 5% as big 08:17.886 --> 08:21.306 as Lewis would've predicted, had he predicted that two 08:21.310 --> 08:23.230 electrons would be right between. 08:23.230 --> 08:24.330 Okay? 08:24.329 --> 08:27.969 And there are unshared pairs, as Lewis predicted. 08:27.970 --> 08:31.830 And again they're less -- but in this case they're even less 08:31.827 --> 08:34.637 than 5% of what Lewis would've predicted. 08:34.639 --> 08:36.009 But you can see them. 08:36.009 --> 08:39.779 Now this raises the question, is there a better bond theory 08:39.783 --> 08:42.333 than Lewis theory, maybe even one that's 08:42.327 --> 08:44.537 quantitative, that would give you numbers for 08:44.543 --> 08:46.933 these things, rather than just say there are 08:46.928 --> 08:48.128 pairs here and there. 08:48.130 --> 08:48.590 Right? 08:48.590 --> 08:50.430 And the answer, thank goodness, 08:50.428 --> 08:53.858 is yes, there is a great theory for this, and what it is, 08:53.864 --> 08:55.954 is chemical quantum mechanics. 08:55.950 --> 08:58.330 Now you can study quantum mechanics in this department, 08:58.327 --> 09:00.927 you can study quantum mechanics in physics, you can probably 09:00.926 --> 09:02.376 study it other places, right? 09:02.379 --> 09:06.639 And different people use the same quantum mechanics but apply 09:06.636 --> 09:08.406 it to different problems. 09:08.410 --> 09:09.050 Right? 09:09.048 --> 09:14.408 So what we're going to discuss in this course is quantum 09:14.407 --> 09:17.717 mechanics as applied to bonding. 09:17.720 --> 09:20.170 So it'll be somewhat different in its flavor -- 09:20.168 --> 09:22.428 in fact, a lot different in its flavor -- 09:22.428 --> 09:26.468 from what you do in physics, or even what you do in physical 09:26.469 --> 09:29.219 chemistry, because we're more interested 09:29.217 --> 09:32.737 -- we're not so interested in getting numbers or solving 09:32.743 --> 09:36.063 mathematical problems, we're interested in getting 09:36.059 --> 09:39.239 insight to what's really involved in forming bonds. 09:39.240 --> 09:42.700 We want it to be rigorous but we don't need it to be 09:42.703 --> 09:43.523 numerical. 09:43.519 --> 09:44.479 Okay? 09:44.480 --> 09:47.390 So it'll be much more pictorial than numerical. 09:50.953 --> 09:54.133 that was discovered in, or invented perhaps we should 09:54.133 --> 09:56.243 say, in -- I don't know whether -- it's 09:56.235 --> 09:59.085 hard to know whether to say discovered or invented; 09:59.090 --> 10:03.460 I think invented is probably better -- in 1926. 10:06.904 --> 10:10.804 the sort of ninety-seven-pound weakling on the beach. 10:10.799 --> 10:11.549 Right? 10:11.548 --> 10:14.848 He's this guy back here with the glasses on. 10:14.850 --> 10:15.390 Okay? 10:15.389 --> 10:19.139 He was actually a well-known physicist but he hadn't done 10:19.143 --> 10:21.693 anything really earthshaking at all. 10:21.690 --> 10:25.480 He was at the University of Zurich. 10:25.480 --> 10:28.120 And Felix Bloch, who was a student then -- 10:28.120 --> 10:32.380 two years before he had come as an undergraduate to the 10:32.384 --> 10:35.944 University of Zurich to study engineering, 10:35.940 --> 10:38.330 and after a year and a half he decided he would do physics, 10:38.330 --> 10:41.470 which was completely impractical and not to the taste 10:41.465 --> 10:42.485 of his parents. 10:42.490 --> 10:46.490 But anyhow, as an undergraduate he went to these colloquia that 10:46.493 --> 10:50.603 the Physics Department had, and he wrote fifty years later 10:50.596 --> 10:54.406 -- see this was 1976, so it's the 50th anniversary of 10:54.410 --> 10:56.690 the discovery of, or invention, 10:56.692 --> 10:58.322 of quantum mechanics. 10:58.320 --> 11:00.240 So he said: "At the end of a 11:00.240 --> 11:02.600 colloquium I heard Debye"(there's a picture of 11:05.515 --> 11:10.015 you're not working right now on very important problems anyway. 11:10.019 --> 11:14.119 Why don't you tell us something about that thesis of de 11:14.115 --> 11:14.945 Broglie?' 11:17.842 --> 11:21.232 a beautifully clear account of how de Broglie associated a wave 11:21.227 --> 11:22.317 with a particle. 11:22.320 --> 11:25.690 When he had finished, Debye casually remarked that he 11:25.693 --> 11:28.943 thought this way of talking was rather childish. 11:28.940 --> 11:32.230 He had learned that, to deal properly with waves, 11:32.229 --> 11:34.559 one had to have a wave equation. 11:34.558 --> 11:37.118 It sounded rather trivial and did not seem to make a great 11:39.292 --> 11:41.712 thought a bit more about the idea afterwards, 11:41.710 --> 11:44.240 and just a few weeks later he gave another talk in the 11:44.240 --> 11:47.010 colloquium, which he started by saying, 11:47.006 --> 11:51.356 'My colleague Debye suggested that one should have a wave 11:51.356 --> 11:52.286 equation. 11:52.289 --> 11:57.169 Well I have found one.'" And we write it now: 11:57.167 --> 11:58.867 HΨ=EΨ. 11:58.870 --> 12:01.290 He actually wrote it in different terms, 12:04.528 --> 12:08.338 The reason the one we write is a little different from his is 12:08.342 --> 12:10.952 he included time as a variable in his, 12:10.950 --> 12:13.340 whereas we're not interested in, for this purpose, 12:13.340 --> 12:15.000 in changes in time. 12:15.000 --> 12:18.330 We want to see how molecules are when they're just sitting 12:18.331 --> 12:18.801 there. 12:18.799 --> 12:20.889 We'll talk about time later. 12:20.889 --> 12:23.829 So within, what? 12:27.682 --> 12:29.472 a good deal sharper. 12:29.472 --> 12:30.282 Right? 12:30.279 --> 12:32.509 And where is he standing? 12:32.509 --> 12:35.659 He's standing at the tram stop in Stockholm, 12:35.663 --> 12:39.483 where he's going to pick up his Nobel Prize for this. 12:39.476 --> 12:40.206 Right? 12:40.210 --> 12:45.140 And he's standing with Dirac, with whom he shared the Nobel 12:45.144 --> 12:47.114 Prize, and with Heisenberg, 12:47.106 --> 12:50.156 who got the Nobel Prize the previous year but hadn't 12:50.162 --> 12:53.382 collected it yet, so he came at the same time. 12:53.379 --> 12:54.939 Okay? 12:58.167 --> 13:00.347 HΨ=EΨ. 13:00.350 --> 13:03.230 H and E you've seen, but Ψ may be new to 13:03.225 --> 13:03.565 you. 13:03.570 --> 13:04.470 It's a Greek letter. 13:04.470 --> 13:09.320 We can call it Sigh or P-sighi, some people call it Psee. 13:09.317 --> 13:10.007 Right? 13:10.009 --> 13:12.119 I'll probably call it Psi. 13:12.120 --> 13:13.010 Okay. 13:13.009 --> 13:16.859 And it's a wave function. 13:16.860 --> 13:20.930 Well what in the world is a wave function? 13:20.929 --> 13:21.599 Okay? 13:21.600 --> 13:26.250 So this is a stumbling block for people that come into the 13:26.254 --> 13:28.594 field, and it's not just a stumbling 13:28.594 --> 13:31.264 block for you, it was a stumbling block for 13:31.258 --> 13:33.988 the greatest minds there were at the time. 13:33.990 --> 13:36.750 So, for example, this is five years later in 13:36.753 --> 13:38.953 Leipzig, and it's the research group of 13:38.952 --> 13:40.792 Werner Heisenberg, who's sitting there in the 13:40.793 --> 13:43.273 front, the guy that -- this was about the time he was 13:43.269 --> 13:45.979 being nominated or selected for the Nobel Prize. 13:45.980 --> 13:46.500 Right? 13:46.500 --> 13:51.630 So he's there with his research group, and right behind him is 13:51.634 --> 13:55.854 seated Felix Bloch, who himself got the Nobel Prize 13:55.845 --> 13:58.535 for discovering NMR in 1952. 13:58.538 --> 14:01.908 So he's quite a young guy here, and he's with these other -- 14:01.908 --> 14:05.488 there's a guy who became famous at Oxford and another one who 14:05.493 --> 14:08.543 became the head of the Physics Department at MIT. 14:08.539 --> 14:10.519 Bloch was at Stanford. 14:10.519 --> 14:14.169 So these guys know they're pretty hot stuff, 14:14.168 --> 14:16.138 so they're looking right into the camera, 14:16.139 --> 14:18.939 to record themselves for posterity, 14:18.940 --> 14:20.570 as part of this distinguished group; 14:20.570 --> 14:22.130 except for Bloch. 14:22.129 --> 14:23.849 What's he thinking about? 14:23.850 --> 14:28.310 > 14:28.308 --> 14:30.798 What in the world is Ψ? 14:30.797 --> 14:31.397 Right? 14:31.399 --> 14:33.279 Now, in fact, in that same year, 14:36.438 --> 14:38.318 wave equation, and Ψ. 14:38.321 --> 14:39.051 Right? 14:39.048 --> 14:43.868 And that summer these smart guys, who were hanging around 14:43.870 --> 14:47.290 Zurich at that time, theoretical physicists, 14:47.289 --> 14:49.919 the young guys went out on an excursion, 14:49.918 --> 14:53.618 on the lake of Zurich, and they made up doggerel 14:53.621 --> 14:58.351 rhymes for fun about different things that were going on, 15:05.490 --> 15:07.400 whom we'll talk about next semester, 15:07.399 --> 15:08.939 was about Ψ. 15:08.940 --> 15:12.010 "Gar Manches rechnet Erwin schon, Mit seiner 15:12.006 --> 15:12.966 Wellenfuction. 15:16.779 --> 15:18.719 vorstell'n soll." 15:18.720 --> 15:22.960 Which means: "Erwin with his Psi can do 15:22.958 --> 15:25.718 calculations, quite a few. 15:25.720 --> 15:30.040 We only wish that we could glean an inkling of what Psi 15:30.038 --> 15:31.398 could mean." 15:31.399 --> 15:32.119 Right? 15:32.120 --> 15:35.560 You can do calculations with it, but what is it? 15:35.559 --> 15:36.929 -- was the question. 15:36.929 --> 15:37.959 Okay? 15:37.960 --> 15:42.010 And it wasn't just these young guys who were confused. 15:46.345 --> 15:48.675 Ψ really means. 15:48.678 --> 15:51.138 Now if we're lucky, this'll play this time. 15:54.000 --> 15:56.860 "What is Matter", from 1952. 15:56.860 --> 15:59.130 <> 16:02.256 --> 16:05.376 einmal den Sancho Panza, sein liebes Eselchen auf dem er 16:11.130 --> 16:13.640 vergessen und das gute Tier ist wieder da. 16:13.639 --> 16:18.169 Nun werden sie mich vielleicht zuletzt fragen, 16:18.174 --> 16:23.324 ja was sind denn nun aber wirklich diese Korpuskeln, 16:31.630 --> 16:35.250 wo Sancho Panzas zweites Eselchen hergekommen ist." 16:41.042 --> 16:43.712 really know what Ψ was. 16:43.710 --> 16:44.470 Okay? 16:44.470 --> 16:49.040 So don't be depressed when it seems a little curious what 16:49.041 --> 16:51.001 Ψ might be. 16:51.000 --> 16:51.730 Okay? 16:53.922 --> 16:56.932 other guys -- first we'll learn how to find 16:56.931 --> 17:01.211 Ψ and use it, and then later we'll learn what 17:01.210 --> 17:01.910 it means. 17:01.909 --> 17:04.529 Okay? 17:04.528 --> 17:07.778 So Ψ is a function, a wave function. 17:07.778 --> 17:12.368 What do you want to know, from what I've shown here? 17:12.369 --> 17:14.089 What is a function? 17:14.089 --> 17:16.029 Student: A relationship. 17:16.028 --> 17:17.588 Prof: Like sine is a function; 17:17.589 --> 17:19.909 what does that mean? 17:19.910 --> 17:21.430 Yes? 17:21.430 --> 17:24.020 I can't hear very well. 17:24.019 --> 17:26.399 Student: You put an input into a function and you 17:26.400 --> 17:27.080 get an output. 17:27.078 --> 17:29.308 Prof: Yes, it's like a little machine. 17:29.308 --> 17:32.448 You put a number in, or maybe several numbers, 17:32.453 --> 17:34.063 and a number comes out. 17:34.060 --> 17:34.690 Right? 17:34.690 --> 17:40.600 That's what the function does; okay, you put in ninety degrees 17:40.599 --> 17:42.679 and sin says one. 17:42.680 --> 17:43.830 Okay? 17:43.828 --> 17:47.258 So what do you want to know about Ψ 17:47.263 --> 17:48.273 first? 17:48.269 --> 17:50.909 Student: What does it do? 17:50.910 --> 17:53.270 Prof: What's it a function of? 17:53.269 --> 17:57.219 What are the things you have to put in, in order to get a number 17:57.222 --> 17:57.602 out? 17:57.599 --> 17:59.829 Okay? 17:59.829 --> 18:03.079 So it's different from the name. 18:03.079 --> 18:06.549 The wave functions have names. 18:06.548 --> 18:09.398 That's not what they're a function of. 18:09.403 --> 18:10.023 Right? 18:10.019 --> 18:13.689 You can have sine, sine^2, cosine. 18:13.690 --> 18:17.550 Those are different functions, but they can be functions of 18:17.548 --> 18:19.138 the same thing, an angle. 18:19.144 --> 18:19.814 Right? 18:19.808 --> 18:22.958 So we're interested in what's it a function of; 18:22.960 --> 18:26.210 not what the function is but what's it a function of? 18:26.210 --> 18:28.640 So you can have different Ψs. 18:28.640 --> 18:34.880 They have names and quantum numbers give them their names. 18:34.880 --> 18:36.360 For example, you can have n, 18:36.355 --> 18:37.435 l, and m. 18:37.440 --> 18:39.480 You've seen those before, n, l, 18:39.478 --> 18:41.098 m, to name wave functions. 18:41.099 --> 18:42.329 Those are just their names. 18:42.328 --> 18:44.058 It's not what they're a function of. 18:44.058 --> 18:46.908 Or you can have 1s or 3d_xy, 18:46.906 --> 18:49.126 or σ, or π, or π*. 18:49.130 --> 18:51.460 Those are all names of functions. 18:51.458 --> 18:52.038 Right? 18:52.038 --> 18:55.568 But they're not what it's a function of. 18:55.568 --> 19:01.538 What it's a function of is the position of a particle, 19:01.544 --> 19:04.254 or a set of particles. 19:04.250 --> 19:09.120 It's a function of position, and it's also a function of 19:09.122 --> 19:13.522 time, and sometimes of spin; some particles have spin and it 19:13.522 --> 19:15.382 could be a function of that too. 19:15.380 --> 19:18.890 But you'll be happy to know that for purposes of this course 19:18.891 --> 19:21.451 we're not so interested in time and spin. 19:21.450 --> 19:26.940 So for our purposes it's just a function of position. 19:26.940 --> 19:29.980 So if you have N particles, how many positions do 19:29.980 --> 19:32.690 you have to specify to know where they all are? 19:32.690 --> 19:34.970 How many numbers do you need? 19:34.970 --> 19:37.370 You need x, y, z for every 19:37.369 --> 19:37.839 particle. 19:37.839 --> 19:38.309 Right? 19:38.308 --> 19:42.808 So you need 3N arguments for Ψ. 19:42.808 --> 19:47.338 So Ψ is a function that when you tell it where all 19:47.340 --> 19:50.740 these positions are, it gives you a number. 19:50.740 --> 19:53.830 Now curiously enough, the number can be positive, 19:53.828 --> 19:56.188 it can be zero, it can be negative, 19:56.190 --> 20:00.170 it can even be complex, right, although we won't talk 20:00.165 --> 20:02.685 about cases where it's complex. 20:02.690 --> 20:06.680 The physicists will tell you about those, or physical 20:06.683 --> 20:07.533 chemists. 20:07.529 --> 20:08.129 Okay? 20:08.130 --> 20:11.880 And sometimes it can be as many as 4N+1 arguments. 20:11.880 --> 20:14.060 How could it be 4N+1? 20:14.058 --> 20:16.028 Student: > 20:16.028 --> 20:18.258 Prof: Because if each particle also had a spin, 20:18.259 --> 20:20.659 then it would be x, y, z and spin; 20:20.660 --> 20:22.030 that'd be four. 20:22.028 --> 20:25.888 And if time is also included, it's plus one. 20:25.890 --> 20:29.050 Okay, so how are we going to go through this? 20:29.048 --> 20:32.228 First we'll try -- this is unfamiliar territory, 20:32.233 --> 20:34.203 I bet, to every one of you. 20:34.200 --> 20:35.090 Okay? 20:35.088 --> 20:39.048 So first we're going to talk about just one particle and one 20:39.045 --> 20:42.125 dimension, so the function is fairly simple. 20:42.130 --> 20:42.990 Okay? 20:42.990 --> 20:46.810 And then we'll go on to three dimensions, but still one 20:46.806 --> 20:49.346 particle, the electron in an atom; 20:49.348 --> 20:52.538 so a one-electron atom, but now three dimensions, 20:52.542 --> 20:54.342 so it's more complicated. 20:54.338 --> 20:58.488 Then we'll go on to atoms that have several electrons. 20:58.490 --> 21:01.890 So you have now more than three variables, because you have at 21:01.892 --> 21:04.702 least two electrons; that would be six variables 21:04.704 --> 21:08.234 that you have to put into the function to get a number out. 21:08.230 --> 21:12.650 Then we'll go into molecules -- that is, more than one atom -- 21:12.645 --> 21:14.235 and what bonding is. 21:14.240 --> 21:17.980 And then finally we get to the payoff for organic chemistry, 21:17.980 --> 21:21.130 which is talking about what makes a group a functional group 21:21.134 --> 21:23.384 and what does it mean to be functional, 21:23.380 --> 21:24.550 what makes it reactive? 21:24.549 --> 21:26.019 That's where we're heading. 21:26.019 --> 21:29.739 But first we have to understand what quantum mechanics is. 21:31.380 --> 21:34.590 ΗΨ=EΨ, and we're talking about the 21:36.710 --> 21:41.500 so time is not a variable, and that means what we're 21:41.501 --> 21:44.981 talking about is stationary states. 21:44.980 --> 21:47.750 We don't mean that the atoms aren't moving, 21:47.748 --> 21:51.368 but just that they're in a cloud and we're going to find 21:51.374 --> 21:53.554 how is the cloud distributed. 21:53.548 --> 21:57.488 If a molecule reacts, the electrons shift their 21:57.491 --> 22:00.151 clouds and so on; it changes. 22:00.150 --> 22:02.340 We're not interested in reaction now, 22:02.339 --> 22:05.629 we're just interested in understanding the cloud that's 22:05.625 --> 22:08.055 sitting there, not changing in time. 22:08.058 --> 22:12.358 Okay, now the right part of the equation is E times 22:12.359 --> 22:14.019 Ψ, right? 22:14.019 --> 22:16.969 And E will turn out to be the energy of the system; 22:16.970 --> 22:19.050 maybe you won't be surprised at that. 22:19.049 --> 22:20.679 So that's quite simple. 22:20.680 --> 22:22.710 What's the left? 22:22.710 --> 22:25.510 It looks like H times Ψ. 22:25.509 --> 22:28.299 If that were true, what could you do to simplify 22:28.304 --> 22:28.844 things? 22:28.839 --> 22:31.149 Knock out Ψ. 22:31.150 --> 22:36.680 But HΨ is not H times Ψ. 22:36.680 --> 22:39.430 H is sort of recipe for doing something with 22:39.425 --> 22:41.795 Ψ; we'll see that right away. 22:41.798 --> 22:45.858 So you can't just cancel out the Ψ, 22:45.864 --> 22:47.314 unfortunately. 22:47.308 --> 22:50.008 Okay, so ΗΨ=EΨ. 22:50.009 --> 22:52.919 Oops sorry, what did I do, there we go. 22:52.920 --> 22:56.070 Now we can divide, you can divide the right by 22:56.068 --> 22:58.178 Ψ, and since it was E times 22:58.180 --> 22:59.970 Ψ, the Ψ goes away. 22:59.970 --> 23:04.980 But when you divide on the left, you don't cancel the 23:04.976 --> 23:09.016 Ψs, because the top doesn't mean 23:09.019 --> 23:10.849 multiplication. 23:10.848 --> 23:14.758 Now I already told you the right side of this equation is 23:14.761 --> 23:16.091 the total energy. 23:16.088 --> 23:19.568 So when you see a system, what does the total energy 23:19.571 --> 23:20.461 consist of? 23:20.460 --> 23:25.450 Potential energy and kinetic energy. 23:25.450 --> 23:29.230 So somehow this part on the left, ΗΨ/Ψ, 23:29.226 --> 23:32.336 must be kinetic energy plus potential energy. 23:32.338 --> 23:35.588 That recipe, H, must somehow tell you 23:35.592 --> 23:40.512 how to work with Ψ in order to get something which, 23:40.509 --> 23:43.669 divided by Ψ, gives kinetic energy plus 23:43.673 --> 23:44.903 potential energy. 23:44.900 --> 23:46.190 So there are two parts of it. 23:46.190 --> 23:48.640 There's the part that's potential energy, 23:48.636 --> 23:51.146 of the recipe, and there's the part that's 23:51.146 --> 23:52.306 kinetic energy. 23:52.308 --> 23:58.548 Now, the potential energy part is in fact easy because it's 23:58.551 --> 23:59.951 given to you. 23:59.951 --> 24:00.921 Right? 24:00.920 --> 24:04.270 What's Ψ a function of? 24:04.269 --> 24:05.289 Students: Position. 24:05.288 --> 24:06.748 Prof: Position of the particles. 24:06.750 --> 24:08.930 Now if you know the charges of the particles, 24:08.930 --> 24:12.280 and their positions, and know Couloumb's Law, 24:12.278 --> 24:14.378 then you know the potential energy, 24:14.380 --> 24:16.360 if Couloumb's Law is right. 24:16.359 --> 24:17.559 Is everybody with me on that? 24:17.558 --> 24:19.688 If you know there's a unit positive charge here, 24:19.690 --> 24:22.390 a unit negative charge here, another unit positive charge 24:22.390 --> 24:24.320 here and a unit negative charge here, 24:24.318 --> 24:27.678 or something like that, you -- it might be complicated, 24:27.680 --> 24:30.110 you might have to write an Excel program or something to do 24:30.106 --> 24:32.156 it -- but you could calculate the 24:32.156 --> 24:34.626 distances and the charges and so on, 24:34.630 --> 24:36.870 and what the energy is, due to that. 24:36.868 --> 24:40.558 So that part is really given to you, once you know what system 24:40.557 --> 24:43.337 you're dealing with, the recipe for finding the 24:43.339 --> 24:44.609 potential energy. 24:44.608 --> 24:48.058 So that part of HΨ/Ψ is no 24:48.061 --> 24:49.531 problem at all. 24:49.529 --> 24:52.439 But hold your breath on kinetic energy. 24:52.440 --> 24:52.970 Sam? 24:52.970 --> 24:56.330 Student: Didn't we just throw away an equation? 24:56.328 --> 25:00.278 There was an adjusted Couloumb's Law equation. 25:00.278 --> 25:01.758 Prof: Yes, that was wrong. 25:01.759 --> 25:04.109 That was three years earlier, remember? 25:04.109 --> 25:07.289 1923 Thomson proposed that. 25:07.289 --> 25:08.869 But it was wrong. 25:08.869 --> 25:11.079 This is what was right. 25:11.078 --> 25:12.598 Student: How did they prove it wrong? 25:12.599 --> 25:13.579 Prof: How did what? 25:13.578 --> 25:14.768 Student: Did they prove it wrong or just -- 25:14.769 --> 25:17.199 Prof: Yes, they proved this right, 25:17.204 --> 25:21.004 that Couloumb's Law held, because it agreed with a whole 25:20.997 --> 25:24.157 lot of spectroscopic evidence that had been collected about 25:24.162 --> 25:26.762 atomic spectra, and then everything else that's 25:26.760 --> 25:28.020 tried with it works too. 25:28.019 --> 25:30.839 So we believe it now. 25:30.838 --> 25:32.898 So how do you handle kinetic energy? 25:32.900 --> 25:35.050 Well that's an old one, you did that already in high 25:35.046 --> 25:35.716 school, right? 25:35.720 --> 25:37.580 Forget kinetic energy, here it is. 25:37.578 --> 25:42.158 It's some constant, which will get the units right, 25:42.160 --> 25:44.500 depending on what energy units you want, 25:44.500 --> 25:48.510 times the sum over all the particles of the kinetic energy 25:48.505 --> 25:49.835 of each particle. 25:49.838 --> 25:51.678 So if you know the kinetic energy of this particle, 25:51.682 --> 25:53.382 kinetic energy of this particle, this particle, 25:53.375 --> 25:55.275 this particle; you add them all up and you get 25:55.279 --> 25:56.319 the total kinetic energy. 25:56.319 --> 25:58.129 No problem there, right? 25:58.130 --> 26:01.260 Now what is the kinetic energy that you're summing up over each 26:01.255 --> 26:01.805 particle? 26:03.880 --> 26:07.100 Has everybody seen that before? 26:07.098 --> 26:11.458 Okay, so that's the sum of classical kinetic energy over 26:11.457 --> 26:15.257 all the particles of interest in the problem, 26:15.259 --> 26:18.139 and the constant is just some number you put in to get the 26:18.142 --> 26:21.402 right units for your energy, depending on whether you use 26:21.396 --> 26:24.346 feet per second or meters per year or whatever, 26:24.349 --> 26:26.139 for the velocity. 26:26.140 --> 26:31.160 Okay, but it turned out that although this was fine for our 26:31.159 --> 26:35.309 great-grandparents, it's not right when you start 26:35.314 --> 26:37.484 dealing with tiny things. 26:37.477 --> 26:38.427 Right? 26:38.430 --> 26:43.330 Here's what kinetic energy really is. 26:43.329 --> 26:44.579 It's a constant. 26:44.578 --> 26:48.418 This is the thing that gets it in the right units: 26:48.415 --> 26:53.265 (h^2)/8(π^2) times a sum over all the particles -- 26:53.269 --> 26:57.979 it's looking promising, right? 26:57.980 --> 27:02.450 -- of one over the mass -- not the mass mv^2, 27:02.450 --> 27:06.950 but one over the mass of each particle -- 27:06.950 --> 27:12.040 and here's where we get it -- > 27:12.038 --> 27:21.168 -- times second derivatives of a wave function. 27:21.170 --> 27:24.340 That's weird. 27:24.338 --> 27:27.348 I mean, at least it has twos in it, like v^2, 27:27.347 --> 27:27.817 right? 27:27.819 --> 27:29.099 > 27:29.099 --> 27:30.769 That's something. 27:30.769 --> 27:33.999 And in fact it's not completely coincidental that it has twos in 27:34.002 --> 27:34.262 it. 27:34.259 --> 27:38.279 There was an analogy that was being followed that allowed them 27:38.280 --> 27:39.600 to formulate this. 27:39.598 --> 27:44.698 And you divide it by the number Ψ. 27:44.700 --> 27:47.910 So that's a pretty complicated thing. 27:47.910 --> 27:51.490 So if we want to get our heads around it, we'd better simplify 27:51.486 --> 27:51.776 it. 27:51.779 --> 27:55.309 And oh also there's a minus sign; 27:55.308 --> 27:59.448 it's minus, the constant is negative that you use. 27:59.450 --> 28:03.360 Okay, now let's simplify it by using just one particle, 28:03.358 --> 28:05.818 so we don't have to sum over a bunch of particles, 28:05.818 --> 28:08.368 and we'll use just one dimension, x; 28:08.369 --> 28:10.099 forget y and z. 28:10.099 --> 28:11.539 Okay? 28:11.539 --> 28:13.429 So now we see something simpler. 28:13.430 --> 28:18.390 So it's a negative constant times one over the mass of the 28:18.385 --> 28:21.315 particle, times the second derivative of 28:21.316 --> 28:23.906 the function, the wave function, 28:23.909 --> 28:26.199 divided by Ψ. 28:31.539 --> 28:32.429 Okay? 28:32.430 --> 28:34.080 Or here it is, written just a little 28:34.078 --> 28:34.738 differently. 28:34.740 --> 28:37.720 So there's a constant, C, over the mass, 28:37.721 --> 28:38.241 right? 28:38.240 --> 28:42.120 And then we have the important part, is the second derivative. 28:42.118 --> 28:45.268 Does everybody know that the second derivative is a curvature 28:45.270 --> 28:46.010 of a function. 28:46.007 --> 28:46.477 Right? 28:46.480 --> 28:47.920 What's the first derivative? 28:47.920 --> 28:48.780 Students: Slope. 28:48.779 --> 28:50.249 Prof: Slope, and the second derivative is 28:50.246 --> 28:50.836 how curved it is. 28:50.838 --> 28:54.368 It can be curving down, that's negative curvature; 28:54.368 --> 28:57.128 or curving up, that's positive curvature. 28:57.130 --> 28:58.380 So it can be positive or negative; 28:58.380 --> 29:01.070 it can be zero if the line is straight. 29:01.069 --> 29:02.809 Okay. 29:02.808 --> 29:07.158 So note that the kinetic energy involves the shape of 29:07.159 --> 29:09.669 Ψ, how curved it is, 29:09.667 --> 29:13.597 not just what the value of Ψ is; 29:13.599 --> 29:18.439 although it involves that too. 29:18.440 --> 29:22.100 Maybe it's not too early to point out something interesting 29:22.098 --> 29:22.918 about this. 29:22.920 --> 29:26.300 So suppose that's the kinetic energy. 29:26.298 --> 29:32.518 What would happen if you multiplied Ψ by two? 29:32.519 --> 29:35.859 Obviously the denominator would get twice as large, 29:35.862 --> 29:38.672 if you made Ψ twice as large. 29:38.670 --> 29:40.410 What would happen to the curvature? 29:40.410 --> 29:42.210 What happens to the slope? 29:42.210 --> 29:44.930 Suppose you have a function and you make it twice as big and 29:44.932 --> 29:46.872 look at the slope at a particular point? 29:46.868 --> 29:50.248 How does the slope change if you've stretched the paper on 29:50.251 --> 29:51.441 which you drew it? 29:51.440 --> 29:52.780 Student: It's sharper. 29:52.779 --> 29:55.069 Prof: The slope will double, right, 29:55.074 --> 29:56.534 if you double the scale. 29:56.529 --> 30:01.719 How about the curvature, the second derivative? 30:01.720 --> 30:06.470 Does it go up by four times? 30:06.470 --> 30:11.420 No it doesn't go up by four times, it goes up by twice. 30:11.420 --> 30:16.000 So what would happen to the kinetic energy there if we 30:15.999 --> 30:20.059 doubled the size of Ψ every place? 30:20.059 --> 30:20.769 Student: Stay the same. 30:20.769 --> 30:22.179 Prof: It would stay the same. 30:22.180 --> 30:25.830 The kinetic energy doesn't depend on how you scale 30:25.827 --> 30:28.957 Ψ, it only depends on its shape, 30:28.955 --> 30:30.515 how curved it is. 30:30.519 --> 30:32.739 Everybody see the idea? 30:32.740 --> 30:36.340 Curvature divided by the value. 30:36.338 --> 30:39.358 Okay, now solving a quantum problem. 30:39.358 --> 30:42.748 So if you're in a course and you're studying quantum 30:42.753 --> 30:45.353 mechanics, you get problems to solve. 30:45.348 --> 30:50.678 A problem means you're given something, you have to find 30:50.676 --> 30:51.836 something. 30:51.838 --> 30:56.168 You're given a set of particles, like a certain nuclei 30:56.172 --> 31:00.102 of given mass and charge, and a certain number of 31:00.095 --> 31:04.225 electrons; that's what you're given. 31:04.230 --> 31:07.580 Okay, the masses of the particles and the potential law. 31:07.578 --> 31:11.038 When you're given the charge, and you know Couloumb's Law, 31:11.040 --> 31:14.320 then you know how to calculate the potential energy; 31:14.319 --> 31:15.949 remember that's part of it. 31:15.950 --> 31:16.930 Okay? 31:16.930 --> 31:19.490 So that part's easy, okay? 31:19.490 --> 31:23.680 Now what do you need to find if you have a problem to solve? 31:23.680 --> 31:26.330 Oh, for example, you can have one particle in 31:26.334 --> 31:29.064 one dimension; so it could be one atomic mass 31:29.061 --> 31:32.351 unit is the weight of the particle, and Hooke's Law could 31:32.353 --> 31:33.943 be the potential, right? 31:33.940 --> 31:38.010 It doesn't have to be realistic, it could be Hooke's 31:38.010 --> 31:42.320 Law, it could be a particle held by a spring, to find a 31:42.321 --> 31:43.681 Ψ. 31:43.680 --> 31:47.000 You want to find the shape of this function, 31:46.996 --> 31:49.306 which is a function of what? 31:49.308 --> 31:50.668 Student: Positions of the particles. 31:50.670 --> 31:54.660 Prof: Positions of the particles, and if you're higher, 31:54.659 --> 31:57.209 further on than we are, time as well; 31:57.210 --> 31:58.760 maybe spin even. 31:58.759 --> 32:04.319 But that function has to be such that HΨ/Ψ is 32:04.323 --> 32:08.293 the total energy, and the total energy is the 32:08.292 --> 32:11.382 same, no matter where the particle is, 32:11.380 --> 32:14.390 right, because the potential energy and the kinetic energy 32:14.392 --> 32:15.082 cancel out. 32:15.078 --> 32:16.998 It's like a ball rolling back and forth. 32:17.000 --> 32:20.090 The total energy is constant but it goes back and forth 32:20.094 --> 32:22.734 between potential and kinetic energy, right? 32:22.730 --> 32:23.840 Same thing here. 32:23.838 --> 32:26.628 No matter where the particles are, you have to get the same 32:26.626 --> 32:27.056 energy. 32:27.058 --> 32:31.368 So Ψ has to be such that when you calculate the 32:31.368 --> 32:34.868 kinetic energy for it, changes in that kinetic energy, 32:34.869 --> 32:38.539 in different positions, exactly compensate for the 32:38.538 --> 32:41.178 changes in potential energy. 32:41.180 --> 32:45.950 When you've got that, then you've got a correct 32:45.954 --> 32:48.244 Ψ; maybe. 32:48.240 --> 32:51.900 It's also important that Ψ remain finite, 32:51.904 --> 32:53.994 that it not go to infinity. 32:53.990 --> 32:55.890 And if you're a real mathematician, 32:55.888 --> 32:59.048 it has to be single valued; you can't have two values for 32:59.048 --> 33:00.108 the same positions. 33:00.108 --> 33:03.248 It has to be continuous, you can't get a sudden break in 33:03.251 --> 33:04.111 Ψ. 33:04.108 --> 33:08.838 And Ψ^2^( )has to be integrable; 33:08.838 --> 33:12.628 you have to be able to tell how much area is under 33:12.634 --> 33:16.124 Ψ^2, and you'll see why shortly. 33:16.118 --> 33:20.388 But basically what you need is to find a Ψ such that 33:20.394 --> 33:24.534 the changes in kinetic energy compensate changes in potential 33:24.530 --> 33:25.290 energy. 33:25.289 --> 33:27.139 Now what's coming? 33:27.140 --> 33:29.740 Let's just rehearse what we did before. 33:29.740 --> 33:33.120 So first there'll be one particle in one dimension; 33:33.118 --> 33:37.328 then it'll be one-electron atoms, so one particle in three 33:37.328 --> 33:40.648 dimensions; then it will be many electrons 33:40.647 --> 33:43.397 and the idea of what orbitals are; 33:43.400 --> 33:45.440 and then it'll be molecules and bonds; 33:45.440 --> 33:49.050 and finally functional groups and reactivity. 33:49.048 --> 33:53.268 Okay, but you'll be happy to hear that by a week from Friday 33:53.270 --> 33:56.420 we'll only get through one-electron atoms. 33:56.420 --> 33:58.990 So don't worry about the rest of the stuff now. 33:58.990 --> 34:03.610 But do read the parts on the webpage that have to do with 34:03.607 --> 34:06.327 what's going to be on the exam. 34:06.328 --> 34:09.228 Okay, so normally you're given a problem, 34:09.230 --> 34:11.400 the mass and the charges -- that is, 34:11.400 --> 34:14.290 that potential energy as a function of position -- 34:14.289 --> 34:18.009 and you need to find Ψ. 34:18.010 --> 34:21.250 But at first we're going to try it a different way. 34:21.250 --> 34:25.810 We're going to play Jeopardy and we're going to start with 34:25.809 --> 34:29.649 the answer and find out what the question was. 34:29.650 --> 34:31.090 Okay? 34:31.090 --> 34:36.540 So suppose that Ψ is the sine of x; 34:36.539 --> 34:41.449 this is one particle in one dimension, the position of the 34:41.447 --> 34:46.007 particle, and the function of Ψ is sine. 34:46.010 --> 34:48.720 If you know Ψ, what can you figure out? 34:48.719 --> 34:50.639 We've just been talking about it. 34:50.639 --> 34:53.029 What can you use Ψ to find? 34:53.030 --> 34:54.180 Student: Kinetic energy. 34:54.179 --> 34:54.779 Prof: Kinetic energy. 34:54.780 --> 34:58.620 How do you find it? 34:58.619 --> 35:01.519 So we can get the kinetic energy, which is minus a 35:01.516 --> 35:04.886 constant over the mass times the curvature of Ψ 35:04.887 --> 35:07.547 divided by Ψ at any given position. 35:07.550 --> 35:08.260 Right? 35:08.260 --> 35:12.320 And once we know how the kinetic energy varies with 35:12.315 --> 35:14.845 position, then we know how the potential 35:14.849 --> 35:18.319 energy varies with position, because it's just the opposite, 35:18.320 --> 35:20.360 in order that the sum be constant. 35:20.355 --> 35:20.905 Right? 35:20.909 --> 35:23.729 So once we know the kinetic energy, then we know what the 35:23.726 --> 35:26.286 potential energy was, which was what the problem was 35:26.293 --> 35:27.353 at the beginning. 35:27.349 --> 35:29.059 Okay? 35:29.059 --> 35:33.789 So suppose our answer is sin(x). 35:33.789 --> 35:36.169 What is the curvature of sin(x), 35:36.173 --> 35:37.683 the second derivative? 35:37.679 --> 35:39.149 <> 35:40.713 --> 35:41.993 -sin(x). 35:41.989 --> 35:44.809 Okay, so what is the kinetic energy? 35:44.809 --> 35:46.239 Student: C/m. 35:46.239 --> 35:47.609 Prof: C/m. 35:47.610 --> 35:52.140 Does it depend on where you are, on the value of x? 35:52.139 --> 35:54.459 No, it's always C/m. 35:54.460 --> 35:57.200 So what was the potential energy? 35:57.199 --> 36:00.239 How did the potential energy vary with position? 36:00.239 --> 36:02.109 Student: It doesn't. 36:02.110 --> 36:04.820 Prof: The potential energy doesn't vary with 36:04.822 --> 36:05.422 position. 36:05.420 --> 36:10.490 So sin(x) is a solution for what? 36:10.489 --> 36:15.939 A particle that's not being influenced by anything else; 36:15.940 --> 36:19.170 so its potential energy doesn't change with the position, 36:19.172 --> 36:21.022 it's a particle in free space. 36:21.019 --> 36:25.539 Okay? 36:25.539 --> 36:28.329 So the potential energy is independent of x. 36:28.329 --> 36:31.779 Constant potential energy, it's a particle in free space. 36:31.780 --> 36:34.960 Now, suppose we take a different one, 36:34.958 --> 36:36.458 sin(ax). 36:36.460 --> 36:41.260 How does sin(ax) look different from sin(x)? 36:41.260 --> 36:44.220 Suppose it's sin(2x). 36:44.219 --> 36:47.419 Here's sin(x). 36:47.420 --> 36:52.450 How does sin(2x) look? 36:52.445 --> 36:53.655 Right? 36:53.659 --> 36:56.779 It's shorter wavelength. 36:56.780 --> 36:58.430 Okay? 36:58.429 --> 37:04.599 Now so we need to figure out -- so it's a shortened wave, 37:04.597 --> 37:06.577 if a*>*1. 37:06.579 --> 37:11.159 Okay, now what's the curvature? 37:11.159 --> 37:13.629 Russell? 37:13.630 --> 37:15.080 Student: It's -a^2 times sin(ax). 37:15.079 --> 37:18.359 Prof: It's -a^2 times sin(ax). 37:18.355 --> 37:18.875 Right? 37:18.880 --> 37:21.110 The a comes out, that constant, 37:21.105 --> 37:23.145 each time you take a derivative. 37:23.150 --> 37:30.250 So now what does the kinetic energy look like? 37:30.250 --> 37:34.140 It's a^2^( )times the same thing. 37:34.139 --> 37:34.959 Okay? 37:34.960 --> 37:39.120 So again, the potential energy is constant. 37:39.117 --> 37:39.907 Right? 37:39.909 --> 37:42.619 It doesn't change with position. 37:42.619 --> 37:46.809 But what is different? 37:46.809 --> 37:50.269 It has higher kinetic energy if it's a shorter wavelength. 37:50.268 --> 37:55.158 And notice that the kinetic energy is proportional to one 37:55.155 --> 37:58.465 over the wavelength squared, right?; 37:58.469 --> 38:01.989 a^2; a shortens the wave, 38:01.985 --> 38:05.035 it's proportional to a^2, one over the 38:05.041 --> 38:06.571 wavelength squared. 38:06.570 --> 38:07.830 Okay. 38:07.829 --> 38:10.949 Now let's take another function, exponential, 38:10.954 --> 38:12.094 so e^x. 38:12.090 --> 38:19.430 What's the second derivative of e^x? 38:19.429 --> 38:20.239 Pardon me? 38:20.239 --> 38:21.159 Student: e^x. 38:21.159 --> 38:21.889 Prof: e^x. 38:21.889 --> 38:24.249 What's the 18th derivative of e^x? 38:24.250 --> 38:25.080 Students: e^x. 38:25.079 --> 38:26.209 Prof: Okay, good. 38:26.210 --> 38:27.150 So it's e^x. 38:27.150 --> 38:31.210 So what's this situation, what's the kinetic energy? 38:31.210 --> 38:38.040 Student: -C/m. Prof: -C/m. 38:38.039 --> 38:40.369 Negative kinetic energy. 38:40.369 --> 38:44.569 Your great-grandparents didn't get that. 38:44.570 --> 38:49.740 You can have kinetic energy that's less than zero. 38:49.739 --> 38:51.219 What does that mean? 38:51.219 --> 39:00.489 It means the total energy is lower than the potential energy. 39:00.489 --> 39:04.279 Pause a minute just to let that sink in. 39:04.280 --> 39:10.910 The total energy is lower than the potential energy. 39:10.909 --> 39:13.689 The difference is negative. 39:13.690 --> 39:14.910 Okay? 39:14.909 --> 39:17.629 So the kinetic energy, if that's the difference, 39:17.625 --> 39:20.105 between potential and total, is negative. 39:22.949 --> 39:24.209 Yes? 39:24.210 --> 39:26.700 Student: Does this violate that Ψ has to 39:26.702 --> 39:27.392 remain finite? 39:27.389 --> 39:29.039 Prof: Does it violate what? 39:29.039 --> 39:30.909 Student: That Ψ has to remain 39:30.911 --> 39:31.271 finite? 39:31.269 --> 39:33.179 Prof: No. 39:33.179 --> 39:34.839 You'll see in a second. 39:34.840 --> 39:38.850 Okay, so anyhow the constant potential energy is greater than 39:38.849 --> 39:40.719 the total energy for that. 39:40.719 --> 39:46.569 Now, how about if it were minus exponential, e^-x? 39:46.570 --> 39:48.250 Now what would it be? 39:48.250 --> 39:51.970 It would be the same deal again, it would still be 39:51.972 --> 39:56.612 -C/m, and again it would be a constant potential energy 39:56.606 --> 39:59.186 greater than the total energy. 39:59.190 --> 40:05.750 This is not just a mathematical curiosity, it actually happens 40:05.751 --> 40:09.411 for every atom in you, or in me. 40:09.409 --> 40:13.909 Every atom has the electrons spend some of their time in 40:13.914 --> 40:18.014 regions where they have negative kinetic energy. 40:18.010 --> 40:23.110 It's not just something weird that never happens. 40:23.110 --> 40:27.650 And it happens at large distance from the nuclei where 40:27.648 --> 40:32.868 1/r -- that's Couloumb's Law -- where it stops changing 40:32.871 --> 40:34.071 very much. 40:34.070 --> 40:37.860 When you get far enough, 1/r gets really tiny and 40:37.860 --> 40:41.170 it's essentially zero, it doesn't change anymore. 40:41.170 --> 40:41.860 Right? 40:41.860 --> 40:46.500 Then you have this situation in any real atom. 40:46.500 --> 40:53.500 So let's look at getting the potential energy from the shape 40:53.503 --> 40:58.373 of Ψ via the kinetic energy. 40:58.369 --> 41:00.759 Okay, so here's a map of Ψ, 41:00.760 --> 41:02.730 or a plot of Ψ,it could be positive, 41:02.730 --> 41:05.860 negative, zero -- as a function of the one-dimension x, 41:05.860 --> 41:07.860 wherever the particle is. 41:07.860 --> 41:08.720 Okay? 41:08.719 --> 41:12.669 Now let's suppose that that is our wave function, 41:12.668 --> 41:15.468 sometimes positive, sometimes zero, 41:15.467 --> 41:17.357 sometimes negative. 41:17.360 --> 41:17.900 Okay? 41:17.900 --> 41:22.510 And let's look at different positions and see what the 41:22.512 --> 41:25.572 kinetic energy is, and then we'll be able to 41:25.570 --> 41:28.050 figure out, since the total will be 41:28.050 --> 41:31.250 constant, what the potential energy is. 41:31.250 --> 41:31.770 Okay? 41:31.768 --> 41:34.988 So we'll try to find out what was the potential energy that 41:34.987 --> 41:36.427 gave this as a solution? 41:36.429 --> 41:38.399 This is again the Jeopardy approach. 41:38.400 --> 41:40.840 Okay? 41:40.840 --> 41:43.930 Okay, so the curvature minus -- remember it's a negative 41:43.929 --> 41:46.019 constant -- minus the curvature over the 41:46.021 --> 41:49.141 amplitude could be positive -- that's going to be the kinetic 41:49.139 --> 41:50.639 energy; it could be positive, 41:50.639 --> 41:52.569 it could be zero, it could be negative, 41:52.574 --> 41:55.684 or it could be that we can't tell by looking at the graph. 41:55.679 --> 41:58.869 So let's look at different positions on the graph and see 41:58.865 --> 41:59.715 what it says. 41:59.719 --> 42:01.719 First look at that position. 42:01.719 --> 42:03.759 What is the kinetic energy there? 42:03.760 --> 42:16.760 Positive, negative, zero? 42:16.760 --> 42:18.940 Ryan, why don't you help me out? 42:18.940 --> 42:20.070 Student: No Prof: Well no, 42:20.070 --> 42:20.580 you can help me out. 42:20.579 --> 42:21.909 > 42:21.909 --> 42:22.949 Look, so what do you need to know? 42:22.949 --> 42:26.549 You need to know -- here's the complicated thing you have to 42:26.547 --> 42:27.337 figure out. 42:27.340 --> 42:33.260 What is minus the curvature divided by the amplitude at this 42:33.257 --> 42:34.057 point? 42:34.059 --> 42:36.839 Is it positive, negative or zero? 42:36.840 --> 42:40.050 So what's the curvature at that point? 42:40.050 --> 42:48.180 Is it curving up or down at that point? 42:48.179 --> 42:49.329 No idea. 42:49.329 --> 42:51.409 Anybody got an idea? 42:51.409 --> 42:52.229 Keith? 42:52.230 --> 42:53.790 Kevin? 42:53.789 --> 42:56.179 Student: It looks like a saddle point so it's probably 42:56.175 --> 42:56.445 zero. 42:56.449 --> 42:57.339 Prof: It's not a saddle point. 42:57.340 --> 42:58.170 What do you call it? 42:58.170 --> 42:59.140 Students: Inflection points. 42:59.139 --> 43:01.569 Prof: A saddle point's for three dimensions. 43:01.570 --> 43:03.440 In this it's what? 43:03.440 --> 43:04.590 Inflection point. 43:04.590 --> 43:05.740 It's flat there. 43:05.739 --> 43:08.139 It's curving one way on one side, the other way on the other 43:08.143 --> 43:08.433 side. 43:08.429 --> 43:10.869 So it's got zero curvature there; 43:10.869 --> 43:12.909 okay, zero curvature. 43:12.909 --> 43:15.569 Now Ryan, can you tell me anything about that, 43:15.565 --> 43:17.155 if the curvature is zero? 43:17.159 --> 43:18.019 Student: Zero. 43:18.019 --> 43:18.839 Prof: Ah ha. 43:18.840 --> 43:20.130 > 43:20.130 --> 43:20.820 Prof: Not bad. 43:20.820 --> 43:23.470 So that one we'll color grey for zero. 43:23.469 --> 43:25.649 The kinetic energy at that point is zero, 43:25.646 --> 43:27.276 if that's the wave function. 43:27.280 --> 43:28.670 Now let's take another point. 43:28.670 --> 43:31.250 Who's going to help me with this one? 43:31.250 --> 43:34.640 How about the curvature at this point right here? 43:34.639 --> 43:35.629 Student: Negative. 43:35.630 --> 43:36.990 <> 43:36.989 --> 43:39.849 Prof: It's actually -- I choose a point that's not 43:39.849 --> 43:40.309 curved. 43:40.309 --> 43:40.949 Student: Ah. 43:40.949 --> 43:42.559 Prof: It's straight right there. 43:42.559 --> 43:44.699 I assure you that's true. 43:44.699 --> 43:47.329 So I bet Ryan can help me again on that one. 43:47.329 --> 43:49.469 How about it? 43:49.469 --> 43:50.179 Student: Zero. 43:50.179 --> 43:51.479 Prof: Ah ha. 43:51.480 --> 43:53.190 So we'll make that one grey too. 43:53.190 --> 43:54.290 Now I'll go to someone else. 43:54.289 --> 43:55.859 How about there? 43:55.860 --> 43:58.240 What's the curvature at that point do you think? 43:58.239 --> 43:59.709 Shai? 43:59.710 --> 44:02.050 Student: It looks straight, zero curvature. 44:02.050 --> 44:03.410 Prof: It looks straight, zero curvature. 44:03.409 --> 44:05.659 So does that mean that this value is zero? 44:05.659 --> 44:06.639 Student: Not necessarily, because the 44:06.635 --> 44:08.115 amplitude -- Prof: Ah, 44:08.115 --> 44:10.475 the amplitude is zero there too. 44:10.480 --> 44:14.180 So really you can't be sure. 44:14.175 --> 44:15.095 Right? 44:15.099 --> 44:17.569 So that one we're going to have to leave questionable, 44:17.572 --> 44:18.742 that's a question mark. 44:18.739 --> 44:21.009 How about out here? 44:21.010 --> 44:23.260 Not curved. 44:23.260 --> 44:26.460 So what's the kinetic energy? 44:26.460 --> 44:28.930 Josh? 44:28.929 --> 44:29.849 Student: Questionable. 44:29.849 --> 44:31.089 Prof: Questionable, right? 44:31.090 --> 44:32.690 Because the amplitude is zero again; 44:32.690 --> 44:36.110 zero in the numerator, also zero in the denominator; 44:36.110 --> 44:37.970 we really don't know. 44:37.969 --> 44:44.119 Okay, how about here? 44:44.119 --> 44:46.049 Tyler, what do you say? 44:46.050 --> 44:47.660 Is it curved there? 44:47.659 --> 44:48.519 Student: Yes. 44:48.519 --> 44:49.369 Prof: Curving up or down? 44:49.369 --> 44:50.109 Student: Down. 44:50.110 --> 44:53.480 Prof: So negative, the curvature is negative. 44:53.480 --> 44:55.010 The value of Ψ? 44:55.010 --> 44:55.570 Student: Positive. 44:55.570 --> 44:56.240 Prof: Positive. 44:56.239 --> 44:57.859 The energy, kinetic energy? 44:57.860 --> 44:58.380 Student: Positive. 44:58.380 --> 44:59.270 Prof: Positive. 44:59.268 --> 45:02.578 Okay, so we can make that one green. 45:02.579 --> 45:05.079 Okay, here's another one. 45:05.079 --> 45:08.669 Who's going to help me here? 45:08.670 --> 45:09.950 Kate? 45:09.949 --> 45:11.379 Student: Yes. 45:11.380 --> 45:13.150 Prof: Okay, so how about the curvature; 45:13.150 --> 45:14.360 curving up, curving down? 45:14.360 --> 45:16.170 Student: It's curving down, that's negative. 45:16.170 --> 45:16.810 Prof: Yes. 45:16.809 --> 45:17.819 Amplitude? 45:17.820 --> 45:19.600 Student: Zero. 45:19.599 --> 45:20.049 So it should be green. 45:20.050 --> 45:20.840 Prof: Ah, green again. 45:20.840 --> 45:22.110 Okay. 45:22.110 --> 45:24.850 How about here? 45:24.849 --> 45:27.719 Ah, now how about the curvature? 45:27.719 --> 45:30.729 Seth? 45:30.730 --> 45:32.770 Student: I don't know. 45:32.768 --> 45:34.888 Prof: Which way is it curving at this point here? 45:34.889 --> 45:35.679 Student: Curving up. 45:35.679 --> 45:36.389 Prof: Curving up. 45:36.389 --> 45:38.729 So the curvature is -- Student: Positive. 45:38.730 --> 45:39.940 Prof: Positive. 45:39.940 --> 45:43.220 Student: The amplitude is negative, so it's positive. 45:43.219 --> 45:44.749 Prof: Yeah. 45:44.750 --> 45:46.360 So what color would we make it? 45:46.360 --> 45:48.480 Green again. 45:48.480 --> 45:52.720 Okay, so if you're -- you can have -- be curving down or 45:52.721 --> 45:55.421 curving up and still be positive; 45:55.420 --> 45:57.880 curving down if you're above the baseline, 45:57.884 --> 46:00.234 curving up if you're below the baseline. 46:00.228 --> 46:00.828 Right? 46:00.829 --> 46:04.759 So as long as you're curving toward the baseline, 46:04.760 --> 46:09.430 towards Ψ=0, the kinetic energy is positive. 46:09.429 --> 46:13.359 How about here? 46:13.360 --> 46:15.430 Zack? 46:15.429 --> 46:18.709 Which way is it curving? 46:18.710 --> 46:20.080 Curving up or curving down? 46:20.079 --> 46:20.989 Student: It should be curving up. 46:20.989 --> 46:22.959 Prof: Curving up, curvature is positive. 46:22.960 --> 46:24.840 The value? 46:24.840 --> 46:25.620 Student: Positive. 46:25.619 --> 46:26.219 Prof: Positive. 46:26.219 --> 46:27.309 Student: I guess it'll be negative. 46:27.309 --> 46:29.369 Prof: So it's negative kinetic energy there. 46:29.369 --> 46:33.779 Make that one whatever that pinkish color is. 46:33.780 --> 46:34.660 Okay? 46:34.659 --> 46:39.209 Here's another one, how about there? 46:39.210 --> 46:41.150 Alex? 46:41.150 --> 46:43.840 Which way is it curving at the new place? 46:43.840 --> 46:45.250 Here? 46:45.250 --> 46:46.150 Student: Curving down. 46:46.150 --> 46:48.970 Prof: Curving down, negative curvature. 46:48.969 --> 46:50.919 Student: Negative amplitude. 46:50.920 --> 46:51.430 Prof: Negative amplitude. 46:51.429 --> 46:52.679 Student: Negative kinetic energy. 46:52.679 --> 46:54.289 Prof: Negative kinetic energy; 46:54.289 --> 46:55.569 pink again. 46:55.570 --> 46:56.570 Is that enough? 46:56.570 --> 47:02.400 Oh, there's one more, here, the one right here. 47:02.400 --> 47:09.410 Okay? 47:09.409 --> 47:10.289 Student: Negative. 47:10.289 --> 47:10.919 Prof: Pardon me? 47:10.920 --> 47:11.720 Student: Negative. 47:11.719 --> 47:13.319 Prof: Negative, because it's -- how did you do 47:13.318 --> 47:13.748 it so quick? 47:13.750 --> 47:14.850 We didn't have to go through curvature. 47:14.849 --> 47:16.269 Student: It goes away from the line. 47:16.268 --> 47:19.448 Prof: Because it's curving away from the baseline, 47:19.454 --> 47:20.084 negative. 47:20.079 --> 47:22.369 Okay, pink. 47:22.369 --> 47:24.899 Okay, curving away from Ψ=0 means that the 47:24.898 --> 47:26.308 kinetic energy is negative. 47:26.309 --> 47:29.619 So now we know at all these positions whether the kinetic 47:29.615 --> 47:31.735 energy is positive, negative or zero, 47:31.742 --> 47:34.872 although there are a few that we aren't certain about. 47:34.871 --> 47:35.581 Right? 47:35.579 --> 47:38.559 So here's the potential energy that will do that. 47:38.559 --> 47:41.719 If you have this line for the total energy -- right? 47:41.719 --> 47:48.609 -- then here and here you have zero. 47:48.614 --> 47:50.194 Right? 47:50.190 --> 47:53.210 Also, incidentally, here and here, 47:53.206 --> 47:56.036 you have zero kinetic energy. 47:56.039 --> 47:56.989 With me? 47:56.989 --> 47:59.129 Okay. 47:59.130 --> 48:01.740 So no curvature, right? 48:01.739 --> 48:07.179 At these green places, the total energy is higher than 48:07.179 --> 48:09.539 the potential energy. 48:09.539 --> 48:13.209 So the kinetic energy is positive. 48:13.210 --> 48:15.550 Okay? 48:15.550 --> 48:19.370 At these places, the potential energy is higher 48:19.367 --> 48:21.357 than the total energy. 48:21.360 --> 48:24.900 So the kinetic energy is negative and the thing is 48:24.900 --> 48:27.140 curving away from the baseline. 48:27.139 --> 48:27.789 Right? 48:27.789 --> 48:31.329 And now we know something about this point. 48:31.329 --> 48:35.829 If the potential energy is a continuous kind of thing, 48:35.829 --> 48:39.149 then, although we couldn't tell by looking at the wave function, 48:39.150 --> 48:40.960 it's curving away from the baseline, 48:40.960 --> 48:43.970 but very slightly, right? 48:43.969 --> 48:47.989 It's negative kinetic energy there, and also on the right 48:47.989 --> 48:50.429 here is negative kinetic energy. 48:50.429 --> 48:55.909 And here we know, just by continuity, 48:55.909 --> 48:59.379 that at this point it must've been positive kinetic energy, 48:59.380 --> 49:01.770 even though we couldn't tell it by looking at the curve. 49:01.768 --> 49:06.818 There must be an inflection point when you go through zero, 49:06.820 --> 49:11.260 otherwise you'd get a discontinuity in the potential 49:11.262 --> 49:12.222 energy. 49:12.219 --> 49:14.209 Okay, so that one was green. 49:14.210 --> 49:17.500 Okay, now I have to stop. 49:17.500 --> 49:23.000