1 00:00:00,000 --> 00:00:02,520 The following content is provided under a Creative 2 00:00:02,520 --> 00:00:03,970 Commons license. 3 00:00:03,970 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,660 continue to offer high-quality educational resources for free. 5 00:00:10,660 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,190 --> 00:00:18,326 at ocw.mit.edu. 8 00:00:26,010 --> 00:00:31,750 JEFFREY GROSSMAN: Now, this is how we left on Tuesday. 9 00:00:31,750 --> 00:00:36,820 We got to this point where we had put a particle in a box. 10 00:00:36,820 --> 00:00:39,860 We put a quantum mechanical particle in a box. 11 00:00:39,860 --> 00:00:42,400 And it was an infinite square well. 12 00:00:42,400 --> 00:00:44,404 And what's this called? 13 00:00:44,404 --> 00:00:45,568 AUDIENCE: Energy levels. 14 00:00:45,568 --> 00:00:47,360 JEFFREY GROSSMAN: Energy levels, and what's 15 00:00:47,360 --> 00:00:49,262 really weird about them? 16 00:00:49,262 --> 00:00:50,720 AUDIENCE: They're discrete. 17 00:00:50,720 --> 00:00:54,680 JEFFREY GROSSMAN: Yeah, meaning if you have this one here 18 00:00:54,680 --> 00:00:56,540 and then you have another energy here, 19 00:00:56,540 --> 00:00:58,670 can it have energy in between? 20 00:00:58,670 --> 00:00:59,493 No. 21 00:00:59,493 --> 00:01:01,160 Can that quantum particle be in between? 22 00:01:01,160 --> 00:01:02,880 No, in energy. 23 00:01:02,880 --> 00:01:04,970 And so the question was, can we connect that 24 00:01:04,970 --> 00:01:07,550 to things like what we see in the universe, 25 00:01:07,550 --> 00:01:09,140 like the spectral lines? 26 00:01:09,140 --> 00:01:10,795 So what I want to do-- 27 00:01:10,795 --> 00:01:12,170 there's a little bit more we need 28 00:01:12,170 --> 00:01:15,240 to do to do that connection. 29 00:01:15,240 --> 00:01:18,440 And so I want to spend some time today just going through that 30 00:01:18,440 --> 00:01:21,650 and going to the hydrogen atom, because this is not 31 00:01:21,650 --> 00:01:22,400 the hydrogen atom. 32 00:01:22,400 --> 00:01:23,990 It's not an infinite well. 33 00:01:23,990 --> 00:01:28,160 And then from the hydrogen atom, we'll 34 00:01:28,160 --> 00:01:31,820 talk about how just from that, just 35 00:01:31,820 --> 00:01:33,800 by solving for the hydrogen atom, 36 00:01:33,800 --> 00:01:37,430 we literally have this incredibly deep understanding 37 00:01:37,430 --> 00:01:40,870 of the entire periodic table and all of chemistry. 38 00:01:40,870 --> 00:01:42,930 So it's a pretty powerful thing to do. 39 00:01:42,930 --> 00:01:45,530 So we're going to start with a little bit of review, 40 00:01:45,530 --> 00:01:49,032 and then this real-world example. 41 00:01:49,032 --> 00:01:50,990 And then I've got to tell you a few things that 42 00:01:50,990 --> 00:01:54,230 are really important, like you have this thing called "spin." 43 00:01:54,230 --> 00:01:55,520 So we'll talk about that. 44 00:01:55,520 --> 00:02:00,290 We'll end with that, and talk about how from this solution, 45 00:02:00,290 --> 00:02:01,490 you get this-- 46 00:02:01,490 --> 00:02:04,870 pretty awesome, pretty amazing. 47 00:02:04,870 --> 00:02:09,340 So some review-- so we went through this on Tuesday. 48 00:02:09,340 --> 00:02:14,320 So why did people even need quantum mechanics? 49 00:02:14,320 --> 00:02:16,530 Well, because things were failing, and crumbling, 50 00:02:16,530 --> 00:02:20,070 and catastrophing all around them. 51 00:02:22,640 --> 00:02:24,110 And so how long-- 52 00:02:24,110 --> 00:02:28,850 so classical atom, oh, no, there's a classical atom. 53 00:02:28,850 --> 00:02:30,620 Photoelectric effect was one of them. 54 00:02:30,620 --> 00:02:35,050 Who can explain what happened here? 55 00:02:35,050 --> 00:02:37,450 These are some of the reasons why we needed 56 00:02:37,450 --> 00:02:41,913 a new theory after 400 years. 57 00:02:41,913 --> 00:02:43,830 What's going on with the photoelectric effect? 58 00:02:51,610 --> 00:02:52,780 Anyone remember? 59 00:02:52,780 --> 00:02:54,490 Yeah. 60 00:02:54,490 --> 00:02:59,013 AUDIENCE: A description of the phenomenon or why it happened? 61 00:02:59,013 --> 00:02:59,930 JEFFREY GROSSMAN: Yes. 62 00:02:59,930 --> 00:03:02,680 AUDIENCE: OK, so basically, you take some light. 63 00:03:02,680 --> 00:03:06,320 You shine it at a metal, and a bunch of electrons come off. 64 00:03:06,320 --> 00:03:08,560 And if you increase the intensity of the light, 65 00:03:08,560 --> 00:03:10,090 the number of electrons coming off 66 00:03:10,090 --> 00:03:11,710 increases, but not the energy, which 67 00:03:11,710 --> 00:03:13,490 was sort of counterintuitive. 68 00:03:13,490 --> 00:03:15,490 JEFFREY GROSSMAN: Why was that counterintuitive? 69 00:03:15,490 --> 00:03:18,670 AUDIENCE: Because if light's not a particle, 70 00:03:18,670 --> 00:03:21,670 it's just like a wave, then you expect 71 00:03:21,670 --> 00:03:24,260 if you're shooting more light at it, that's more energy. 72 00:03:24,260 --> 00:03:26,920 So if an electron gets hit by a bunch of light, 73 00:03:26,920 --> 00:03:29,230 you'd think it would get a bunch of energy, 74 00:03:29,230 --> 00:03:31,750 and you'd also get a higher energy, but it doesn't. 75 00:03:31,750 --> 00:03:34,840 And the solution was that the photons 76 00:03:34,840 --> 00:03:36,970 are produced by electrons changing energy levels 77 00:03:36,970 --> 00:03:38,330 or something like that. 78 00:03:38,330 --> 00:03:41,445 And so they come in discrete packets of energy. 79 00:03:41,445 --> 00:03:42,820 And so when they hit an electron, 80 00:03:42,820 --> 00:03:44,800 they shoot it off at some particular energy. 81 00:03:44,800 --> 00:03:45,470 JEFFREY GROSSMAN: And what's that? 82 00:03:45,470 --> 00:03:46,210 AUDIENCE: It's frequency dependent. 83 00:03:46,210 --> 00:03:47,950 JEFFREY GROSSMAN: It's just dependent on the frequency, 84 00:03:47,950 --> 00:03:48,545 exactly. 85 00:03:48,545 --> 00:03:49,420 That's exactly right. 86 00:03:49,420 --> 00:03:50,920 So does everybody see that? 87 00:03:50,920 --> 00:03:53,440 So this was called the "photoelectric effect." 88 00:03:53,440 --> 00:03:58,750 And it was the reason Einstein was given the Nobel Prize. 89 00:03:58,750 --> 00:04:01,840 It's a very important explanation 90 00:04:01,840 --> 00:04:03,832 of this very confusing effect, which 91 00:04:03,832 --> 00:04:05,290 is that you increase the intensity, 92 00:04:05,290 --> 00:04:07,075 but the energy of those electrons-- 93 00:04:07,075 --> 00:04:07,950 see what's happening? 94 00:04:07,950 --> 00:04:10,575 You're shining light on a piece of metal and electrons fly off. 95 00:04:10,575 --> 00:04:13,840 And you can measure how much energy those electrons have. 96 00:04:13,840 --> 00:04:15,520 And you increase the intensity of light, 97 00:04:15,520 --> 00:04:18,130 and they don't have any more energy, which was really 98 00:04:18,130 --> 00:04:20,410 bizarre if light is a wave. 99 00:04:20,410 --> 00:04:24,130 What Einstein said is, no, actually, oh, 100 00:04:24,130 --> 00:04:27,990 but it does depend on the frequency of the light. 101 00:04:27,990 --> 00:04:30,590 So that makes sense if light can be considered 102 00:04:30,590 --> 00:04:34,500 as particles of zero mass-- 103 00:04:34,500 --> 00:04:38,490 that's weird-- that have an energy associated 104 00:04:38,490 --> 00:04:39,960 with their frequency. 105 00:04:39,960 --> 00:04:41,240 So that was really important. 106 00:04:41,240 --> 00:04:44,160 The energy of something you thought 107 00:04:44,160 --> 00:04:47,850 was a wave, light, actually can be considered 108 00:04:47,850 --> 00:04:50,220 as a particle associated with the frequency, 109 00:04:50,220 --> 00:04:53,310 have the energy of a particle associated with that frequency. 110 00:04:53,310 --> 00:04:55,320 And then what was the classical atom? 111 00:04:55,320 --> 00:04:58,010 What's wrong with this one? 112 00:04:58,010 --> 00:05:02,800 How long do we live in a classical world? 113 00:05:02,800 --> 00:05:05,086 What's the most we could live in a classical world? 114 00:05:05,086 --> 00:05:06,790 AUDIENCE: 10 to the minus 12 seconds. 115 00:05:06,790 --> 00:05:09,460 JEFFREY GROSSMAN: 10 to the minus 12 seconds, it's game 116 00:05:09,460 --> 00:05:10,000 over. 117 00:05:10,000 --> 00:05:10,750 Why? 118 00:05:10,750 --> 00:05:13,735 What's happening? 119 00:05:13,735 --> 00:05:16,433 AUDIENCE: [INAUDIBLE]. 120 00:05:16,433 --> 00:05:17,350 JEFFREY GROSSMAN: Why? 121 00:05:17,350 --> 00:05:18,058 What's happening? 122 00:05:18,058 --> 00:05:19,940 So it's accelerating, and we know 123 00:05:19,940 --> 00:05:22,250 that an accelerating charge-- 124 00:05:22,250 --> 00:05:25,248 radiation, and it loses energy. 125 00:05:25,248 --> 00:05:27,290 And if it loses energy, it should just spiral in, 126 00:05:27,290 --> 00:05:30,570 and that takes about 10 to the minus 12 seconds. 127 00:05:30,570 --> 00:05:32,720 So we know that's not true, either. 128 00:05:32,720 --> 00:05:34,910 Actually, I wanted to put this out there. 129 00:05:34,910 --> 00:05:40,590 This is from Wikipedia, and it's a sort of explanation of that. 130 00:05:40,590 --> 00:05:42,770 And so if you want to understand more and read more 131 00:05:42,770 --> 00:05:45,380 about the classical atom, get a book 132 00:05:45,380 --> 00:05:48,245 on Lienard-Wiechert potentials. 133 00:05:48,245 --> 00:05:50,120 I'm sure just the name alone is inspiring you 134 00:05:50,120 --> 00:05:53,870 guys to want to dig deep. 135 00:05:53,870 --> 00:05:55,160 And those are the ones. 136 00:05:55,160 --> 00:05:58,310 That's what you can do to get the time-varying EM 137 00:05:58,310 --> 00:06:01,100 field for a point charge in arbitrary motion. 138 00:06:01,100 --> 00:06:02,840 But look at that-- they're not corrected 139 00:06:02,840 --> 00:06:04,132 for quantum mechanical effects. 140 00:06:06,550 --> 00:06:08,710 So that's what was such a problem. 141 00:06:08,710 --> 00:06:12,310 This was right around 1900. 142 00:06:12,310 --> 00:06:15,670 So they had it, but then they tried to apply it to an atom, 143 00:06:15,670 --> 00:06:17,300 and here's what happens. 144 00:06:17,300 --> 00:06:21,457 It's accurate, but breaks down at the quantum level. 145 00:06:21,457 --> 00:06:23,290 Quantum mechanics sets important constraints 146 00:06:23,290 --> 00:06:26,800 on the ability of a particle to emit radiation, i.e., 147 00:06:26,800 --> 00:06:29,040 it doesn't in an atom. 148 00:06:31,433 --> 00:06:33,850 Sometimes, I think this is kind of interesting to look at. 149 00:06:33,850 --> 00:06:37,060 Basically, we have this problem that an atom will 150 00:06:37,060 --> 00:06:39,888 sort of self-destruct, and the electrons just spiral down. 151 00:06:39,888 --> 00:06:41,680 The solution really, if you think about it, 152 00:06:41,680 --> 00:06:44,920 is it doesn't do that anymore. 153 00:06:44,920 --> 00:06:46,030 But now why? 154 00:06:46,030 --> 00:06:51,320 Well, because of the fact that particles are waves and waves 155 00:06:51,320 --> 00:06:52,070 are particles. 156 00:06:52,070 --> 00:06:54,900 And it comes from the quantization. 157 00:06:54,900 --> 00:06:59,930 So I want to make sure that we're all on that page 158 00:06:59,930 --> 00:07:02,120 by the end of today, that we all kind of see that. 159 00:07:02,120 --> 00:07:05,900 So that's the reason, is because particles can 160 00:07:05,900 --> 00:07:10,130 be both waves and particles. 161 00:07:10,130 --> 00:07:12,440 Waves can be waves and particles. 162 00:07:12,440 --> 00:07:16,610 Everything is one happy family. 163 00:07:16,610 --> 00:07:18,350 Can't we all just get along? 164 00:07:18,350 --> 00:07:20,750 Here are some particles going through slits. 165 00:07:20,750 --> 00:07:21,860 Now what's happening here? 166 00:07:25,700 --> 00:07:27,590 Actually, you can think of this as a particle 167 00:07:27,590 --> 00:07:31,540 or it kind of looks like a water wave, doesn't it? 168 00:07:31,540 --> 00:07:33,160 But see, that's the whole point. 169 00:07:33,160 --> 00:07:34,630 That's what particles are. 170 00:07:37,590 --> 00:07:39,360 So what's happening here? 171 00:07:39,360 --> 00:07:44,090 Can somebody explain why it's showing this? 172 00:07:44,090 --> 00:07:44,810 What's going on? 173 00:07:47,612 --> 00:07:51,876 We've got on the left, how many slits? 174 00:07:51,876 --> 00:07:52,630 AUDIENCE: One. 175 00:07:52,630 --> 00:07:52,990 JEFFREY GROSSMAN: And on the right? 176 00:07:52,990 --> 00:07:53,770 AUDIENCE: Two. 177 00:07:53,770 --> 00:07:56,170 JEFFREY GROSSMAN: What's the difference? 178 00:07:56,170 --> 00:07:57,280 AUDIENCE: Interference. 179 00:07:57,280 --> 00:07:58,480 JEFFREY GROSSMAN: Yes. 180 00:07:58,480 --> 00:08:01,300 What's interference? 181 00:08:01,300 --> 00:08:03,008 AUDIENCE: Two different waves, and you've 182 00:08:03,008 --> 00:08:03,925 got peaks and troughs. 183 00:08:03,925 --> 00:08:06,730 JEFFREY GROSSMAN: Yeah, so that you get two waves coming out 184 00:08:06,730 --> 00:08:12,190 of-- but look, the same particle creates no interference here 185 00:08:12,190 --> 00:08:14,030 because it just bubbles through. 186 00:08:14,030 --> 00:08:19,210 But if you have two slits, it can create two little waves. 187 00:08:19,210 --> 00:08:22,430 And those interfere and they cause complex behavior. 188 00:08:22,430 --> 00:08:25,658 And that is what blew people away. 189 00:08:25,658 --> 00:08:27,700 And oh, by the way, you can see that there's also 190 00:08:27,700 --> 00:08:29,283 some really cool interference going on 191 00:08:29,283 --> 00:08:31,282 in the backscattering from the edges, 192 00:08:31,282 --> 00:08:32,740 because you get a little scattering 193 00:08:32,740 --> 00:08:34,880 off the edges on both cases. 194 00:08:34,880 --> 00:08:37,669 But what we're looking at is what's happening over here. 195 00:08:37,669 --> 00:08:40,169 And so it's just really weird that you could throw particles 196 00:08:40,169 --> 00:08:42,600 through slits, and they were like water waves, 197 00:08:42,600 --> 00:08:43,890 not like particles. 198 00:08:43,890 --> 00:08:49,680 That was another really bizarre experiment. 199 00:08:49,680 --> 00:08:55,660 And again, one of the big, important stepping stones 200 00:08:55,660 --> 00:08:59,290 was the fact that you could describe the energy 201 00:08:59,290 --> 00:09:00,940 of a particle by its frequency. 202 00:09:00,940 --> 00:09:06,180 And then de Broglie came and said, actually, not just light. 203 00:09:06,180 --> 00:09:09,300 No, everything-- all matter-- 204 00:09:09,300 --> 00:09:12,550 can be considered as both a particle and a wave. 205 00:09:12,550 --> 00:09:14,760 This is stuff we talked about Tuesday. 206 00:09:14,760 --> 00:09:18,330 And so what that means is that I have a wavelength. 207 00:09:18,330 --> 00:09:21,560 What's my wavelength? 208 00:09:21,560 --> 00:09:24,290 35, 10 to the minus 35. 209 00:09:24,290 --> 00:09:25,940 But I'm OK with that. 210 00:09:25,940 --> 00:09:30,560 I feel that quantum is right here. 211 00:09:30,560 --> 00:09:35,030 I feel it somewhere within. 212 00:09:35,030 --> 00:09:39,710 But see, when things are really small like an electron, which 213 00:09:39,710 --> 00:09:42,320 is this part of the class-- we care about those electrons. 214 00:09:42,320 --> 00:09:46,040 And I motivated that on Tuesday, and we'll come back to that 215 00:09:46,040 --> 00:09:48,120 as we go through examples. 216 00:09:48,120 --> 00:09:50,570 So if I want to know about an electron, 217 00:09:50,570 --> 00:09:52,190 that's a really small thing. 218 00:09:52,190 --> 00:09:56,450 And so its wavelength-- and we talked about this Tuesday-- 219 00:09:56,450 --> 00:09:59,570 the wavelength associated with the particle 220 00:09:59,570 --> 00:10:03,500 is about the same size as the distances 221 00:10:03,500 --> 00:10:05,480 the particle sort of goes out for a walk 222 00:10:05,480 --> 00:10:10,010 on, on a daily basis, otherwise known as its "chemical bond." 223 00:10:10,010 --> 00:10:11,960 If you're an electron, you kind of walk 224 00:10:11,960 --> 00:10:14,090 along this chemical bond. 225 00:10:14,090 --> 00:10:20,390 That's sort of what you do, if you're in a bonding situation. 226 00:10:20,390 --> 00:10:24,170 And that distance of that walk is roughly the same 227 00:10:24,170 --> 00:10:27,800 as your wavelength, as your wave nature. 228 00:10:27,800 --> 00:10:33,370 Your wave nature-ness is important then. 229 00:10:33,370 --> 00:10:35,770 And so because of that, we can't describe it 230 00:10:35,770 --> 00:10:37,060 without quantum mechanics. 231 00:10:37,060 --> 00:10:40,590 You have to have a way of describing it. 232 00:10:40,590 --> 00:10:44,060 And it has to be something that is based on a wave. 233 00:10:44,060 --> 00:10:45,980 So that's the breakthrough. 234 00:10:45,980 --> 00:10:48,620 And I'll go through that again just quickly. 235 00:10:51,290 --> 00:10:58,550 But what we came out with is the fact that particles are waves. 236 00:10:58,550 --> 00:11:00,800 So they have wave functions. 237 00:11:00,800 --> 00:11:03,740 And this wave function is what we'll be solving for. 238 00:11:03,740 --> 00:11:06,050 This is what we want in this part of the class. 239 00:11:06,050 --> 00:11:06,920 We want that. 240 00:11:10,550 --> 00:11:12,560 And so it'll have some form. 241 00:11:12,560 --> 00:11:15,170 And we showed a few forms, and I'll show you more today. 242 00:11:15,170 --> 00:11:17,480 But we showed a few for a particle in a box. 243 00:11:17,480 --> 00:11:24,350 That had a cosine, and then it had one node in the middle. 244 00:11:24,350 --> 00:11:28,730 But what's amazing about this is that this has no meaning 245 00:11:28,730 --> 00:11:30,060 we know of. 246 00:11:30,060 --> 00:11:32,340 We do not understand what this means. 247 00:11:32,340 --> 00:11:38,000 So you have this intimate function 248 00:11:38,000 --> 00:11:40,800 that describes the particle, and we 249 00:11:40,800 --> 00:11:43,940 don't know how to interpret it. 250 00:11:43,940 --> 00:11:48,710 The big breakthrough came in making the connection 251 00:11:48,710 --> 00:11:54,080 that the square of that function is meaningful. 252 00:11:54,080 --> 00:11:55,310 But that's pretty cool. 253 00:11:55,310 --> 00:11:58,100 So think about if I had a position. 254 00:11:58,100 --> 00:11:59,430 Wait, you had this-- 255 00:11:59,430 --> 00:12:02,000 you did classical molecular dynamics, 256 00:12:02,000 --> 00:12:04,100 so you have the position of the particle. 257 00:12:04,100 --> 00:12:06,740 That's a function of time. 258 00:12:06,740 --> 00:12:08,560 Imagine saying that that has no meaning, 259 00:12:08,560 --> 00:12:12,100 or you don't know what it means, only the square of it. 260 00:12:12,100 --> 00:12:14,110 So something as fundamental as the wave 261 00:12:14,110 --> 00:12:18,810 itself that is the solution to the equation to our f 262 00:12:18,810 --> 00:12:22,290 equals ma, which is Schrodinger, something as fundamental 263 00:12:22,290 --> 00:12:24,720 as that, we don't know what it means. 264 00:12:24,720 --> 00:12:27,400 But we do know what its square means. 265 00:12:27,400 --> 00:12:28,760 And what does the square mean? 266 00:12:28,760 --> 00:12:29,960 AUDIENCE: Probability. 267 00:12:29,960 --> 00:12:31,210 JEFFREY GROSSMAN: Probability. 268 00:12:31,210 --> 00:12:34,416 So tell me a little more about that. 269 00:12:34,416 --> 00:12:36,840 AUDIENCE: [INAUDIBLE]. 270 00:12:36,840 --> 00:12:39,870 JEFFREY GROSSMAN: Yeah, so tell me-- this is a particle right 271 00:12:39,870 --> 00:12:40,370 here. 272 00:12:40,370 --> 00:12:42,920 You're starting to see it, I can tell. 273 00:12:42,920 --> 00:12:45,540 I can see that you're starting to see it. 274 00:12:45,540 --> 00:12:49,220 I can feel the excitement about this, too, in the room. 275 00:12:49,220 --> 00:12:50,570 It's tangible. 276 00:12:50,570 --> 00:12:52,220 This is a particle. 277 00:12:52,220 --> 00:12:55,640 You see this now not as a wave, not as a particle. 278 00:12:55,640 --> 00:12:57,350 You see it as both. 279 00:12:57,350 --> 00:12:58,942 Now, if this is a particle and that's 280 00:12:58,942 --> 00:13:01,400 why I'm going to show you a little video on it in a second. 281 00:13:01,400 --> 00:13:05,400 But if this is a particle, where is it? 282 00:13:05,400 --> 00:13:07,697 AUDIENCE: [INAUDIBLE] by chance [INAUDIBLE].. 283 00:13:07,697 --> 00:13:09,530 JEFFREY GROSSMAN: But I mean, I know that it 284 00:13:09,530 --> 00:13:11,860 can be a particle again. 285 00:13:11,860 --> 00:13:13,222 We it can go back and forth. 286 00:13:13,222 --> 00:13:15,680 That's what it did when you sent it through the two slits-- 287 00:13:15,680 --> 00:13:19,050 it became wave, then it became a particle. 288 00:13:19,050 --> 00:13:23,400 So can I say that it's spread out? 289 00:13:23,400 --> 00:13:24,210 Is it spread out? 290 00:13:28,370 --> 00:13:29,540 How do I interpret that? 291 00:13:33,060 --> 00:13:35,378 AUDIENCE: The probability of finding a particle is 292 00:13:35,378 --> 00:13:37,112 2 to the negative 3. 293 00:13:37,112 --> 00:13:38,070 JEFFREY GROSSMAN: Good. 294 00:13:38,070 --> 00:13:43,880 That's sort of the way that I like to look at it. 295 00:13:43,880 --> 00:13:45,500 I mean, the particle is-- 296 00:13:45,500 --> 00:13:48,830 in a way, you can think about it is spread out 297 00:13:48,830 --> 00:13:50,750 until you measure it. 298 00:13:50,750 --> 00:13:55,160 It's sort of everywhere in here, because there's no here here. 299 00:13:55,160 --> 00:14:00,430 There's only a certain amount of chance to be here here. 300 00:14:00,430 --> 00:14:03,520 So I have a kind of big chance, if I measure where it is, 301 00:14:03,520 --> 00:14:04,930 of finding it here. 302 00:14:04,930 --> 00:14:08,230 And then I have no chance of finding it here. 303 00:14:08,230 --> 00:14:10,660 And then it's sort of medium chance here, and a big chance 304 00:14:10,660 --> 00:14:13,370 there, if I measure it. 305 00:14:13,370 --> 00:14:16,720 So that's really what these functions mean. 306 00:14:16,720 --> 00:14:19,750 And those functions are the ones that we 307 00:14:19,750 --> 00:14:23,060 solve for to find out where electrons are. 308 00:14:23,060 --> 00:14:26,180 And where those electrons are is what 309 00:14:26,180 --> 00:14:29,750 is absolutely critical to doing a whole lot of material 310 00:14:29,750 --> 00:14:32,270 science, as I mentioned on Tuesday, 311 00:14:32,270 --> 00:14:38,570 and as we'll talk about when we apply this to problems. 312 00:14:38,570 --> 00:14:40,070 And that's what you get. 313 00:14:40,070 --> 00:14:45,560 You get bizarre distributions of probabilities. 314 00:14:45,560 --> 00:14:49,760 Here's an electron thrown into a wall that is infinite, 315 00:14:49,760 --> 00:14:54,110 and here is an electron thrown into a wall that's finite. 316 00:14:54,110 --> 00:14:56,240 So that's my quantum mechanical ball. 317 00:14:56,240 --> 00:14:57,620 Did I show this Tuesday? 318 00:14:57,620 --> 00:14:59,870 I don't think I did. 319 00:14:59,870 --> 00:15:01,550 So if it's thrown into an infinite wall, 320 00:15:01,550 --> 00:15:02,900 it actually can't go through. 321 00:15:02,900 --> 00:15:04,610 The probability-- we worked this out. 322 00:15:04,610 --> 00:15:06,920 If you have an infinite boundary here-- 323 00:15:06,920 --> 00:15:09,110 we talked about that Tuesday-- 324 00:15:09,110 --> 00:15:11,300 you can't have any function. 325 00:15:11,300 --> 00:15:12,860 The only solution is 0. 326 00:15:12,860 --> 00:15:14,117 So it can't penetrate at all. 327 00:15:14,117 --> 00:15:15,950 So it completely bounces back, although look 328 00:15:15,950 --> 00:15:18,950 at the weird things it does when it bounces back. 329 00:15:18,950 --> 00:15:22,910 That's the wave function of a ball thrown into a wall, 330 00:15:22,910 --> 00:15:30,110 or if you want, an electron thrown into a material. 331 00:15:30,110 --> 00:15:32,660 But over here, you see it's not infinite, 332 00:15:32,660 --> 00:15:35,540 and all kinds of interesting things happen. 333 00:15:35,540 --> 00:15:38,420 It does bounce back in a similar way, but some of it 334 00:15:38,420 --> 00:15:39,218 goes through. 335 00:15:39,218 --> 00:15:40,010 What's that called? 336 00:15:42,650 --> 00:15:43,500 AUDIENCE: Tunneling. 337 00:15:43,500 --> 00:15:45,672 JEFFREY GROSSMAN: Tunneling, right. 338 00:15:45,672 --> 00:15:47,130 And you better believe that this is 339 00:15:47,130 --> 00:15:52,180 as effect we use in technology all the time-- 340 00:15:52,180 --> 00:15:57,660 tunneling of charges, very important. 341 00:15:57,660 --> 00:16:01,050 If I take my macroscopic ball and I throw it, 342 00:16:01,050 --> 00:16:06,340 or if I just run myself with a helmet on into the wall, 343 00:16:06,340 --> 00:16:08,770 how much of me is going to tunnel? 344 00:16:11,662 --> 00:16:13,808 AUDIENCE: Not enough. 345 00:16:13,808 --> 00:16:15,850 JEFFREY GROSSMAN: What is enough in that example? 346 00:16:19,300 --> 00:16:21,840 So why aren't I tunneling if I run into that wall? 347 00:16:21,840 --> 00:16:23,990 AUDIENCE: [INAUDIBLE]. 348 00:16:23,990 --> 00:16:26,262 AUDIENCE: You're not running fast enough. 349 00:16:26,262 --> 00:16:27,720 JEFFREY GROSSMAN: I'm not running-- 350 00:16:27,720 --> 00:16:31,310 so actually, that's true. 351 00:16:31,310 --> 00:16:32,570 I'm pretty slow. 352 00:16:32,570 --> 00:16:40,210 But the thing is that there's another reason. 353 00:16:40,210 --> 00:16:44,890 What is it about me that is really different here? 354 00:16:44,890 --> 00:16:45,870 AUDIENCE: You're big. 355 00:16:45,870 --> 00:16:48,710 JEFFREY GROSSMAN: I'm big, and so what does that mean? 356 00:16:48,710 --> 00:16:49,490 What's small? 357 00:16:49,490 --> 00:16:51,020 AUDIENCE: Your wavelength. 358 00:16:51,020 --> 00:16:52,790 JEFFREY GROSSMAN: My wave function-- 359 00:16:52,790 --> 00:16:54,920 my wavelength, not my wave function. 360 00:16:54,920 --> 00:16:58,430 My wave function is everywhere, but it just doesn't really 361 00:16:58,430 --> 00:16:59,900 spread out that much. 362 00:16:59,900 --> 00:17:05,690 Because the wavelength of me is 10 to minus 35 meters, 363 00:17:05,690 --> 00:17:08,869 and remember, when quantum matters 364 00:17:08,869 --> 00:17:12,920 is when your wavelength is sort of on the same scale roughly 365 00:17:12,920 --> 00:17:15,920 as the stuff you're involved with. 366 00:17:15,920 --> 00:17:19,109 And I'm involved with a thing that's a meter over there. 367 00:17:19,109 --> 00:17:22,579 So if I'm 35 orders of magnitude smaller in my wavelength, 368 00:17:22,579 --> 00:17:25,280 there's not going to be any quantum effects. 369 00:17:25,280 --> 00:17:30,430 You won't tunnel very often, not enough to make it worth trying. 370 00:17:34,130 --> 00:17:37,800 Now, here's the goofy video, which is kind of fun. 371 00:17:37,800 --> 00:17:40,311 So I'll just play it, and then we'll move on. 372 00:17:40,311 --> 00:17:40,936 [VIDEO PLAYING] 373 00:17:40,936 --> 00:17:41,561 [MUSIC PLAYING] 374 00:17:41,561 --> 00:17:42,320 I love the music. 375 00:17:42,320 --> 00:17:44,045 - A true giant of quantum revolution 376 00:17:44,045 --> 00:17:48,070 was Werner Heisenberg. 377 00:17:48,070 --> 00:17:50,935 And from the vagueness of positions of electrons and de 378 00:17:50,935 --> 00:17:54,460 Beuys standing wave orbits, he drew a tremendous insight, 379 00:17:54,460 --> 00:17:58,260 which he stated like this-- 380 00:17:58,260 --> 00:18:01,890 "The more precisely the position is determined, 381 00:18:01,890 --> 00:18:08,100 the less precisely the momentum is known, and vice versa." 382 00:18:08,100 --> 00:18:10,740 And while it may seem like a bunch of gobbledygook, 383 00:18:10,740 --> 00:18:14,310 it was the insight necessary to understand how an object can 384 00:18:14,310 --> 00:18:16,350 be both a particle and a wave. 385 00:18:20,380 --> 00:18:22,840 This single statement led to the understanding 386 00:18:22,840 --> 00:18:26,320 that subatomic particles can disappear and reappear 387 00:18:26,320 --> 00:18:27,028 in another place. 388 00:18:27,028 --> 00:18:28,612 JEFFREY GROSSMAN: That's really weird. 389 00:18:28,612 --> 00:18:30,400 - And they can do this without existing 390 00:18:30,400 --> 00:18:31,960 in the intervening space. 391 00:18:31,960 --> 00:18:34,760 JEFFREY GROSSMAN: That's really weird. 392 00:18:34,760 --> 00:18:38,650 - When an electron is trapped inside an atom, 393 00:18:38,650 --> 00:18:41,080 the places it can disappear and reappear 394 00:18:41,080 --> 00:18:43,465 include specific locations around the nucleus. 395 00:18:47,160 --> 00:18:51,000 And that makes it look like a shell. 396 00:18:51,000 --> 00:18:54,990 Basic particles can also be in more than one place at a time, 397 00:18:54,990 --> 00:18:56,310 if the time is brief enough. 398 00:19:00,920 --> 00:19:03,740 An electron can travel from here to there 399 00:19:03,740 --> 00:19:06,930 along all possible paths simultaneously. 400 00:19:09,860 --> 00:19:12,800 And even more astounding, these particles 401 00:19:12,800 --> 00:19:16,510 can appear out of the nothingness of space, 402 00:19:16,510 --> 00:19:21,640 exist for an extremely brief instant, and then disappear. 403 00:19:21,640 --> 00:19:25,150 Scientists call them "virtual particles." 404 00:19:25,150 --> 00:19:31,240 They may be virtual, but they can have very real effects. 405 00:19:31,240 --> 00:19:33,640 All this jumping around makes particle descriptions 406 00:19:33,640 --> 00:19:34,210 so inexact. 407 00:19:38,300 --> 00:19:41,870 But then existence itself is inexact at these tiny scales. 408 00:19:45,620 --> 00:19:48,560 The question is often asked, what 409 00:19:48,560 --> 00:19:50,570 is it that is wave when a particle is 410 00:19:50,570 --> 00:19:51,620 described as a wave? 411 00:19:54,370 --> 00:19:57,430 And after a lot of hem-hawing around the issue, 412 00:19:57,430 --> 00:20:02,810 the right answer is, existence is waving. 413 00:20:02,810 --> 00:20:04,260 The particles are jumping around, 414 00:20:04,260 --> 00:20:08,840 coming into and out of existence. 415 00:20:08,840 --> 00:20:11,750 And those places where the wave crests are maximum, 416 00:20:11,750 --> 00:20:14,570 are the places where the particle materializes 417 00:20:14,570 --> 00:20:18,160 most of the time, while the places where 418 00:20:18,160 --> 00:20:24,090 the crests are minimum are the places the particle avoids. 419 00:20:24,090 --> 00:20:28,250 The wave is a map of the particles' existence. 420 00:20:28,250 --> 00:20:31,450 And as the wave changes, the particle changes as well. 421 00:20:34,607 --> 00:20:35,190 [END PLAYBACK] 422 00:20:35,190 --> 00:20:36,482 JEFFREY GROSSMAN: There you go. 423 00:20:36,482 --> 00:20:38,880 I love these kinds of videos, and not just 424 00:20:38,880 --> 00:20:41,730 for the soundtracks, because those are pretty cool. 425 00:20:41,730 --> 00:20:46,260 But I think that he got a little bit ahead 426 00:20:46,260 --> 00:20:50,520 of where I want to be, but does everybody see that weirdness? 427 00:20:50,520 --> 00:20:52,760 Is everybody getting a feel for it? 428 00:20:52,760 --> 00:20:57,410 OK, he said something that's mind blowing. 429 00:20:57,410 --> 00:21:00,710 What did he say that I said was really deep. 430 00:21:00,710 --> 00:21:02,078 Anybody remember? 431 00:21:02,078 --> 00:21:03,757 AUDIENCE: Existence is waving. 432 00:21:03,757 --> 00:21:05,090 JEFFREY GROSSMAN: Existence is-- 433 00:21:05,090 --> 00:21:05,720 well, yeah. 434 00:21:05,720 --> 00:21:08,450 Oh, actually, there were so many things. 435 00:21:08,450 --> 00:21:10,820 You're going to have to see this again. 436 00:21:10,820 --> 00:21:13,520 And again, this is one of those things-- you take this with Dr. 437 00:21:13,520 --> 00:21:15,620 Quantum, was it Dr. Quantum? 438 00:21:15,620 --> 00:21:17,420 You take this video with you, too-- 439 00:21:17,420 --> 00:21:21,620 to the bars, to the dorms, wherever you're going 440 00:21:21,620 --> 00:21:23,540 to do your next social thing. 441 00:21:23,540 --> 00:21:26,820 And you just see what the response is. 442 00:21:26,820 --> 00:21:31,580 You ask what their opinion is, and it's a great way 443 00:21:31,580 --> 00:21:33,630 to engage with people. 444 00:21:33,630 --> 00:21:34,130 Yeah. 445 00:21:34,130 --> 00:21:37,050 AUDIENCE: Does the wave function exist all the time? 446 00:21:37,050 --> 00:21:38,050 JEFFREY GROSSMAN: Sorry? 447 00:21:38,050 --> 00:21:41,600 AUDIENCE: Does the wave function exist all the time, too? 448 00:21:41,600 --> 00:21:44,010 JEFFREY GROSSMAN: That is a good question. 449 00:21:44,010 --> 00:21:45,240 And yes. 450 00:21:45,240 --> 00:21:47,420 I mean, it's changing all the time, 451 00:21:47,420 --> 00:21:49,857 but it exists all the time. 452 00:21:49,857 --> 00:21:51,440 Here's the thing, though-- did anybody 453 00:21:51,440 --> 00:21:53,313 catch the other thing that's going on here? 454 00:21:53,313 --> 00:21:55,230 I'm going to talk about this at the end today. 455 00:21:55,230 --> 00:21:55,872 But-- 456 00:21:55,872 --> 00:21:57,290 AUDIENCE: [INAUDIBLE]. 457 00:21:57,290 --> 00:21:59,870 JEFFREY GROSSMAN: OK, yeah, I can't even go there, 458 00:21:59,870 --> 00:22:01,280 that's so crazy. 459 00:22:01,280 --> 00:22:03,620 What else? 460 00:22:03,620 --> 00:22:04,930 Yeah. 461 00:22:04,930 --> 00:22:06,140 AUDIENCE: Virtual particles. 462 00:22:06,140 --> 00:22:09,030 JEFFREY GROSSMAN: Vertical particles, vacuum fluctuations, 463 00:22:09,030 --> 00:22:10,610 but we're not going there either-- 464 00:22:10,610 --> 00:22:12,930 mind blowing. 465 00:22:12,930 --> 00:22:15,680 I mean, in quantum mechanics, stuff actually appears out 466 00:22:15,680 --> 00:22:17,060 of nowhere. 467 00:22:17,060 --> 00:22:19,190 Out of nothing you get something, 468 00:22:19,190 --> 00:22:20,660 and then it disappears. 469 00:22:20,660 --> 00:22:22,610 Those are called "vacuum fluctuations." 470 00:22:22,610 --> 00:22:23,300 What else? 471 00:22:23,300 --> 00:22:24,920 But it's all related to this. 472 00:22:24,920 --> 00:22:28,220 Basically what he said is that particles 473 00:22:28,220 --> 00:22:31,830 can disappear and then reappear somewhere else. 474 00:22:31,830 --> 00:22:35,520 And actually, if you take field theory, 475 00:22:35,520 --> 00:22:39,110 which I really don't recommend for most people-- 476 00:22:39,110 --> 00:22:41,810 I still remember my final exam in field theory. 477 00:22:41,810 --> 00:22:43,950 It was 90 pages long. 478 00:22:43,950 --> 00:22:45,470 It was a 24-hour, take-home exam. 479 00:22:50,080 --> 00:22:53,110 Anyway, I still remember asking my field theory teacher 480 00:22:53,110 --> 00:22:57,580 in graduate school, how can that be? 481 00:22:57,580 --> 00:22:59,590 How can a particle disappear? 482 00:22:59,590 --> 00:23:01,840 Because basically when you get down to it, 483 00:23:01,840 --> 00:23:04,300 that's pretty much how things move 484 00:23:04,300 --> 00:23:06,340 at the very smallest scales. 485 00:23:06,340 --> 00:23:08,320 They disappear and then they reappear. 486 00:23:08,320 --> 00:23:10,570 That's what the math tells us, at least. 487 00:23:10,570 --> 00:23:12,537 How can that be? 488 00:23:12,537 --> 00:23:13,495 I didn't get an answer. 489 00:23:16,590 --> 00:23:18,072 This gets deep. 490 00:23:18,072 --> 00:23:19,780 It's really cool and really mind blowing. 491 00:23:19,780 --> 00:23:21,100 And we just don't really know. 492 00:23:21,100 --> 00:23:24,910 We don't connect it so much anymore in the classrooms 493 00:23:24,910 --> 00:23:28,670 to the mind-blowing awesomeness of it all. 494 00:23:28,670 --> 00:23:30,400 But I love it. 495 00:23:30,400 --> 00:23:32,650 Now, I'll get to that towards the end. 496 00:23:32,650 --> 00:23:34,230 I'll come back to that. 497 00:23:37,690 --> 00:23:39,970 Now, what we need-- so we have these waves. 498 00:23:39,970 --> 00:23:42,495 We know that particles are waves and waves are particles. 499 00:23:42,495 --> 00:23:43,870 And we know that the particles we 500 00:23:43,870 --> 00:23:45,400 care about in this part of the class 501 00:23:45,400 --> 00:23:47,950 are going to have to be described by a wave. 502 00:23:47,950 --> 00:23:51,640 How do we talk-- what's the F equals ma of that wave? 503 00:23:51,640 --> 00:23:54,640 It's dictated by-- it's the Schrodinger equation. 504 00:23:54,640 --> 00:23:55,990 That's it. 505 00:23:55,990 --> 00:23:57,610 That's our f equals ma. 506 00:23:57,610 --> 00:24:00,100 It's telling us how this wave is going 507 00:24:00,100 --> 00:24:03,770 to change in time and space. 508 00:24:03,770 --> 00:24:04,990 And so we talked about that. 509 00:24:04,990 --> 00:24:08,860 And we basically just wrote this down 510 00:24:08,860 --> 00:24:10,280 and said this is the equation. 511 00:24:10,280 --> 00:24:13,510 We're not going to go into the details of where it comes from, 512 00:24:13,510 --> 00:24:17,710 but there's so many places where you can read more about that. 513 00:24:17,710 --> 00:24:20,627 But this is our governing equation. 514 00:24:20,627 --> 00:24:22,210 And basically, what you have is a kind 515 00:24:22,210 --> 00:24:24,820 of kinetic energy of the wave. 516 00:24:24,820 --> 00:24:28,000 That's this del squared thing, kinetic energy operator, 517 00:24:28,000 --> 00:24:29,260 and a potential. 518 00:24:29,260 --> 00:24:33,220 And that together, we call h, we call the Hamiltonian. 519 00:24:33,220 --> 00:24:36,340 That's got its own special name because just 520 00:24:36,340 --> 00:24:40,450 like T plus V in classical physics is the energy, 521 00:24:40,450 --> 00:24:45,940 we consider this thing, h squared over 2m del squared 522 00:24:45,940 --> 00:24:47,770 plus V to be the energy. 523 00:24:47,770 --> 00:24:49,810 That's h. 524 00:24:49,810 --> 00:24:51,940 But it has to act on the wave function 525 00:24:51,940 --> 00:24:53,320 in order to get that energy. 526 00:24:53,320 --> 00:24:55,780 And that's basically what the Schrodinger equation is. 527 00:24:55,780 --> 00:24:57,370 Because we said, we're going to say 528 00:24:57,370 --> 00:25:00,040 that h doesn't depend on time usually, 529 00:25:00,040 --> 00:25:01,555 for the things we care about. 530 00:25:01,555 --> 00:25:02,930 And if it doesn't depend on time, 531 00:25:02,930 --> 00:25:06,010 then we can get rid of the t here, 532 00:25:06,010 --> 00:25:08,350 and we could separate the time dependence out, 533 00:25:08,350 --> 00:25:10,090 which is what we do. 534 00:25:10,090 --> 00:25:13,750 And we're left with H psi equals E 535 00:25:13,750 --> 00:25:16,660 psi, which is the equation we need to solve. 536 00:25:16,660 --> 00:25:18,440 And I'll come back to it in a second. 537 00:25:18,440 --> 00:25:20,440 Here's the bit on Schrodinger I mentioned 538 00:25:20,440 --> 00:25:21,980 that I think is amazing. 539 00:25:21,980 --> 00:25:24,970 This is, again, from Wikipedia. 540 00:25:24,970 --> 00:25:28,600 You see, he found the standing waves. 541 00:25:28,600 --> 00:25:29,360 See, there it is. 542 00:25:29,360 --> 00:25:32,530 He found the standing waves of this relativistic equation, 543 00:25:32,530 --> 00:25:34,390 but the relativistic corrections disagreed 544 00:25:34,390 --> 00:25:35,590 with Sommerfeld's formula. 545 00:25:35,590 --> 00:25:38,230 Discouraged, he put away his calculations 546 00:25:38,230 --> 00:25:40,960 and secluded himself in an isolated mountain 547 00:25:40,960 --> 00:25:43,930 cabin with a lover. 548 00:25:43,930 --> 00:25:45,220 I find it amazing-- 549 00:25:45,220 --> 00:25:50,770 not that part, but I find it amazing that seriously, he 550 00:25:50,770 --> 00:25:52,300 had to be convinced. 551 00:25:52,300 --> 00:25:54,670 See, he came back from the cabin and he 552 00:25:54,670 --> 00:25:57,790 decided that his earlier calculations were novel enough 553 00:25:57,790 --> 00:25:59,140 to publish. 554 00:25:59,140 --> 00:26:01,900 It's a good thing, because it's pretty much 555 00:26:01,900 --> 00:26:06,070 one of the most important equations of the last century. 556 00:26:06,070 --> 00:26:10,360 And it's the equation we will be solving. 557 00:26:10,360 --> 00:26:13,720 That's why I think this history is interesting. 558 00:26:13,720 --> 00:26:14,660 OK, so there it is. 559 00:26:14,660 --> 00:26:18,100 So we talked about how you can take the f out, 560 00:26:18,100 --> 00:26:21,430 because if H doesn't depend on time, then 561 00:26:21,430 --> 00:26:24,780 you can just set them equal to a constant. 562 00:26:24,780 --> 00:26:27,630 You can set each side having its own function 563 00:26:27,630 --> 00:26:30,870 dependent on its own variable, r and t in this case, 564 00:26:30,870 --> 00:26:32,070 and they equal a constant. 565 00:26:32,070 --> 00:26:34,918 And then you get the Schrodinger equation we care about, 566 00:26:34,918 --> 00:26:36,960 which is the one we'll be solving, which is H psi 567 00:26:36,960 --> 00:26:37,800 equals E psi. 568 00:26:41,320 --> 00:26:43,520 And then we did this, and we said, well, 569 00:26:43,520 --> 00:26:45,250 how do you solve it? 570 00:26:45,250 --> 00:26:48,740 What if you have a very simple particle in a box. 571 00:26:48,740 --> 00:26:52,720 And the box's walls go up to infinity. 572 00:26:52,720 --> 00:26:54,040 Then you can solve it. 573 00:26:54,040 --> 00:26:55,810 And it's because you have basically 574 00:26:55,810 --> 00:27:00,030 the particle is either between 0 and L, in which case 575 00:27:00,030 --> 00:27:05,740 here, then v is 0, and so you get this equation, 576 00:27:05,740 --> 00:27:11,290 or it's out here in this region, in which case 577 00:27:11,290 --> 00:27:14,110 V is infinity, which means psi has to be 0. 578 00:27:14,110 --> 00:27:16,550 That's the only solution. 579 00:27:16,550 --> 00:27:18,970 And then this is the interesting part. 580 00:27:18,970 --> 00:27:21,470 And when you look at this, you see that the general solution 581 00:27:21,470 --> 00:27:23,630 is sines and cosines. 582 00:27:23,630 --> 00:27:26,540 So we did that, but then there was the very important thing 583 00:27:26,540 --> 00:27:28,210 that happened. 584 00:27:28,210 --> 00:27:29,110 What is it? 585 00:27:31,630 --> 00:27:34,095 And you get a solution for E. Once you 586 00:27:34,095 --> 00:27:36,220 say the general solution, you get a solution for E. 587 00:27:36,220 --> 00:27:38,320 So this is just a standing wave, and you 588 00:27:38,320 --> 00:27:41,470 get a solution for E, which depends on k squared. 589 00:27:41,470 --> 00:27:43,916 But what's really important that we did next? 590 00:27:43,916 --> 00:27:45,245 AUDIENCE: [INAUDIBLE]. 591 00:27:45,245 --> 00:27:46,620 JEFFREY GROSSMAN: And that does-- 592 00:27:46,620 --> 00:27:48,360 yes, and what did that do? 593 00:27:48,360 --> 00:27:49,320 AUDIENCE: It quantizes. 594 00:27:49,320 --> 00:27:51,780 JEFFREY GROSSMAN: Quantizes. 595 00:27:51,780 --> 00:27:54,690 Once you have the boundary conditions, 596 00:27:54,690 --> 00:27:58,330 it limits what this can be. 597 00:27:58,330 --> 00:28:01,000 We didn't limit it until we applied the boundary 598 00:28:01,000 --> 00:28:01,500 conditions. 599 00:28:01,500 --> 00:28:04,170 Then we applied the boundary conditions and boom, 600 00:28:04,170 --> 00:28:07,020 this thing is quantized, and out from that 601 00:28:07,020 --> 00:28:09,690 drops such a relief for the world, 602 00:28:09,690 --> 00:28:13,670 which can exist for more than 10 to the minus 12 seconds. 603 00:28:13,670 --> 00:28:18,140 Because that quantization is, in fact, the answer. 604 00:28:18,140 --> 00:28:21,905 It's that those electrons can only be at certain energies. 605 00:28:24,410 --> 00:28:29,480 They cannot radiate energy off of the atom and spiral 606 00:28:29,480 --> 00:28:33,080 to the middle because they can't be at another energy except 607 00:28:33,080 --> 00:28:34,820 the one it's in, and then the next one. 608 00:28:37,820 --> 00:28:38,780 And it's real. 609 00:28:38,780 --> 00:28:41,360 I think I may have shown this on Tuesday-- 610 00:28:41,360 --> 00:28:45,680 that you can actually visualize the spatial extent 611 00:28:45,680 --> 00:28:46,440 of the orbitals. 612 00:28:46,440 --> 00:28:48,380 And I put some more-- 613 00:28:48,380 --> 00:28:50,120 this is actually cutting-edge research. 614 00:28:50,120 --> 00:28:53,370 Visualizing the shape of electrons is not easy, 615 00:28:53,370 --> 00:28:54,450 but it can be done. 616 00:28:54,450 --> 00:28:58,160 There's been some really phenomenal breakthrough 617 00:28:58,160 --> 00:28:58,700 experiments. 618 00:28:58,700 --> 00:29:02,900 This was in Scientific American a decade ago. 619 00:29:02,900 --> 00:29:09,010 Now I want to move from particle in a box to real stuff. 620 00:29:09,010 --> 00:29:11,770 And the first real stuff that's relevant 621 00:29:11,770 --> 00:29:15,490 is hydrogen. And it's only one element, 622 00:29:15,490 --> 00:29:19,570 and I mentioned last time that it's not necessarily 623 00:29:19,570 --> 00:29:24,190 useful enough to do, say, materials engineering. 624 00:29:24,190 --> 00:29:26,110 And so you'd like to have a little bit more 625 00:29:26,110 --> 00:29:27,520 of the periodic table. 626 00:29:27,520 --> 00:29:31,630 That's true, but you'll see that with hydrogen alone, 627 00:29:31,630 --> 00:29:33,788 we get our answers to so many questions 628 00:29:33,788 --> 00:29:35,080 about the whole periodic table. 629 00:29:37,640 --> 00:29:39,503 And it is a real useful material. 630 00:29:39,503 --> 00:29:42,170 How much of the universe is made of hydrogen, does anybody know? 631 00:29:44,876 --> 00:29:47,425 AUDIENCE: 70? 632 00:29:47,425 --> 00:29:49,300 JEFFREY GROSSMAN: That's a really good guess. 633 00:29:49,300 --> 00:29:50,580 It's about 75%. 634 00:29:50,580 --> 00:29:54,030 That's a lot of hydrogen. Now, how much of that 635 00:29:54,030 --> 00:29:58,613 is in usable form on Earth for the hydrogen economy? 636 00:30:05,720 --> 00:30:11,920 How much hydrogen is usable for hydrogen-powered cars 637 00:30:11,920 --> 00:30:15,550 on this planet right now, in an immediate form? 638 00:30:18,640 --> 00:30:19,140 Zero. 639 00:30:21,800 --> 00:30:24,920 That's part one of the hydrogen economy problem. 640 00:30:24,920 --> 00:30:26,900 You've got to make the hydrogen. 641 00:30:26,900 --> 00:30:28,070 There isn't any more. 642 00:30:28,070 --> 00:30:29,090 It's too light. 643 00:30:29,090 --> 00:30:32,370 This planet's gravity couldn't hold it. 644 00:30:32,370 --> 00:30:36,590 So anything that was free, left or bonded to something else. 645 00:30:36,590 --> 00:30:39,500 You've got to make the hydrogen. What's 646 00:30:39,500 --> 00:30:41,510 part two of the hydrogen economy problem? 647 00:30:44,120 --> 00:30:45,410 Storing it. 648 00:30:45,410 --> 00:30:48,470 So you've got two big problems with the hydrogen economy. 649 00:30:48,470 --> 00:30:51,290 I'm not saying it's not worth looking at. 650 00:30:51,290 --> 00:30:56,780 We will not be driving around this way, 651 00:30:56,780 --> 00:31:00,800 but I'll talk a little bit more about hydrogen. I just 652 00:31:00,800 --> 00:31:04,160 love motivating the topic we're about to go into. 653 00:31:04,160 --> 00:31:07,460 And since I want to talk about the hydrogen atom, 654 00:31:07,460 --> 00:31:10,625 I will talk a little more about hydrogen storage materials. 655 00:31:10,625 --> 00:31:12,500 And we will calculate some properties of them 656 00:31:12,500 --> 00:31:15,410 using quantum mechanics. 657 00:31:15,410 --> 00:31:16,430 But I love this. 658 00:31:16,430 --> 00:31:18,660 The history of hydrogen is a very interesting one, 659 00:31:18,660 --> 00:31:20,930 but I love going back to Jules Verne. 660 00:31:20,930 --> 00:31:25,130 In 1874, Jules Verne said, "I believe 661 00:31:25,130 --> 00:31:27,170 that one day hydrogen and oxygen, which together 662 00:31:27,170 --> 00:31:28,760 form water, will be used either alone 663 00:31:28,760 --> 00:31:32,750 or together as an inexhaustible source of heat and light." 664 00:31:32,750 --> 00:31:34,010 I love that. 665 00:31:34,010 --> 00:31:37,590 Anyway, we're not there, not even close. 666 00:31:37,590 --> 00:31:41,560 I think a hydrogen economy is a possibility. 667 00:31:41,560 --> 00:31:44,700 And I think there's some great research going on. 668 00:31:44,700 --> 00:31:49,240 And it should stay as research for at least another 30, 669 00:31:49,240 --> 00:31:49,860 40 years. 670 00:31:49,860 --> 00:31:51,910 I'll come back to-- 671 00:31:51,910 --> 00:31:54,040 it's going to need about that much research 672 00:31:54,040 --> 00:31:58,390 to really make an impact in the energy scene. 673 00:31:58,390 --> 00:32:01,520 I'll come back to that later. 674 00:32:01,520 --> 00:32:06,890 So that's a little motivation for hydrogen. Now here it is. 675 00:32:06,890 --> 00:32:10,270 Hydrogen has a proton, an electron, 676 00:32:10,270 --> 00:32:14,830 and it's got some distance that that electron sits away 677 00:32:14,830 --> 00:32:17,070 from the proton. 678 00:32:17,070 --> 00:32:18,690 What do I need to know? 679 00:32:18,690 --> 00:32:25,300 Well, I need to know the potential. 680 00:32:25,300 --> 00:32:28,420 So what's the potential here between these two charges? 681 00:32:28,420 --> 00:32:29,920 One's negative, one's positive. 682 00:32:29,920 --> 00:32:32,880 They feel an attraction. 683 00:32:32,880 --> 00:32:36,600 How does it go with r? 684 00:32:36,600 --> 00:32:37,770 Yeah. 685 00:32:37,770 --> 00:32:39,960 It goes as 1 over r. 686 00:32:39,960 --> 00:32:46,140 So now that-- I got my V. It's the potential 687 00:32:46,140 --> 00:32:47,550 that charges feel. 688 00:32:47,550 --> 00:32:50,940 And that's got to go now into the Schrodinger equation, which 689 00:32:50,940 --> 00:32:51,990 I then need to solve. 690 00:32:51,990 --> 00:32:57,870 And that'll tell me what the possible wave functions are. 691 00:32:57,870 --> 00:32:59,550 And so here's what it looks like, see? 692 00:32:59,550 --> 00:33:01,410 We had H psi equals E psi. 693 00:33:01,410 --> 00:33:03,570 That's our master equation. 694 00:33:03,570 --> 00:33:06,690 Then we said, that's the kinetic plus potential. 695 00:33:06,690 --> 00:33:09,900 And then we said, that's the momentum term, remember, 696 00:33:09,900 --> 00:33:13,630 we talked about Tuesday, which is just a second derivative. 697 00:33:13,630 --> 00:33:15,100 It's a second partial derivative. 698 00:33:15,100 --> 00:33:18,390 It's a del squared, and plus this potential term, which 699 00:33:18,390 --> 00:33:23,970 I now put in here explicitly for the hydrogen atom, E 700 00:33:23,970 --> 00:33:26,780 squared over 4 pi epsilon 0 r. 701 00:33:26,780 --> 00:33:30,090 And that's the question I need to solve. 702 00:33:30,090 --> 00:33:35,260 And it turns out that you can solve this exactly. 703 00:33:35,260 --> 00:33:37,170 You can solve this exactly, just like you 704 00:33:37,170 --> 00:33:41,170 could the particle in the box with the infinite boundaries. 705 00:33:41,170 --> 00:33:43,620 And that's really cool. 706 00:33:43,620 --> 00:33:45,520 So you've got an electron. 707 00:33:45,520 --> 00:33:49,370 You've got one particle, just one electron. 708 00:33:49,370 --> 00:33:51,410 These are its spatial coordinates. 709 00:33:51,410 --> 00:33:53,510 And you can solve for it in this potential, 710 00:33:53,510 --> 00:33:56,810 in this 1 over r potential exactly. 711 00:33:56,810 --> 00:33:59,690 And the easy way to do that is to switch it to spherical 712 00:33:59,690 --> 00:34:00,700 coordinates. 713 00:34:00,700 --> 00:34:03,620 So I won't go through the math in detail. 714 00:34:03,620 --> 00:34:07,790 There's literally like 100 sites and 50 Wolfram Alpha apps 715 00:34:07,790 --> 00:34:11,090 you can look at that go through this, 716 00:34:11,090 --> 00:34:14,960 and really, really wonderful material on this. 717 00:34:14,960 --> 00:34:18,800 But basically, the key point is that when you write it 718 00:34:18,800 --> 00:34:23,239 in spherical coordinates, you switch from XYZ Cartesian 719 00:34:23,239 --> 00:34:27,080 to spherical coordinates, you can actually separate this 720 00:34:27,080 --> 00:34:35,929 into essentially three quantized problems-- theta, r, and phi. 721 00:34:35,929 --> 00:34:40,130 And so you can write psi now as a function of the r 722 00:34:40,130 --> 00:34:42,830 and a function of the theta and a function of the phi. 723 00:34:42,830 --> 00:34:46,175 And what ends up happening-- and then so you put that in. 724 00:34:46,175 --> 00:34:47,550 And there's a little bit of math. 725 00:34:47,550 --> 00:34:48,949 It's actually not that much. 726 00:34:48,949 --> 00:34:51,031 I'm not going to go through it. 727 00:34:51,031 --> 00:34:52,489 But you put that in, and basically, 728 00:34:52,489 --> 00:34:55,850 just like that particle in the box, 729 00:34:55,850 --> 00:35:01,420 just like that, each one of these variables gets quantized. 730 00:35:01,420 --> 00:35:05,620 It gets its own sort of E-type term, 731 00:35:05,620 --> 00:35:09,160 and it gets a quantization. 732 00:35:09,160 --> 00:35:12,190 So these are the equations that it 733 00:35:12,190 --> 00:35:15,350 leads to when you do the separation of variables. 734 00:35:15,350 --> 00:35:20,740 And I think I have-- yes, and the quantization is-- 735 00:35:20,740 --> 00:35:23,590 I think maybe on the next page I have the actual energies. 736 00:35:23,590 --> 00:35:26,140 But the solution-- remember, just 737 00:35:26,140 --> 00:35:28,720 like the quantization of a particle in a box, the same 738 00:35:28,720 --> 00:35:29,410 here. 739 00:35:29,410 --> 00:35:32,710 It only can happen, you can only have a solution, 740 00:35:32,710 --> 00:35:34,420 if in this case, we're going to have 741 00:35:34,420 --> 00:35:39,070 a quantum number called n is 1, 2, 3, or something else going 742 00:35:39,070 --> 00:35:39,670 up. 743 00:35:39,670 --> 00:35:42,370 Or in this case, we'll have a quantum number called l. 744 00:35:42,370 --> 00:35:44,800 That's also, by the way, the main quantum number, 745 00:35:44,800 --> 00:35:47,850 the orbital quantum number, and then we have this little m sub 746 00:35:47,850 --> 00:35:49,960 l, which is the magnetic one. 747 00:35:49,960 --> 00:35:53,790 How many of you have not seen these? 748 00:35:53,790 --> 00:35:55,020 So you've all seen these? 749 00:35:55,020 --> 00:35:57,050 How many of you have seen this? 750 00:35:57,050 --> 00:36:00,050 OK, so I'm glad to see all the hands go up. 751 00:36:03,050 --> 00:36:04,850 And so that's what you have. 752 00:36:04,850 --> 00:36:07,130 And it's a very special case. 753 00:36:07,130 --> 00:36:09,660 And how many of you remember that s stands for "sharp?" 754 00:36:09,660 --> 00:36:11,180 See, I forget that. 755 00:36:11,180 --> 00:36:15,510 Seriously, did you all remember this sharp principle 756 00:36:15,510 --> 00:36:16,740 and diffuse? 757 00:36:16,740 --> 00:36:20,440 And I love f, which is "fundamental." 758 00:36:20,440 --> 00:36:27,540 So I want to know what's not fundamental about silicon, 759 00:36:27,540 --> 00:36:31,030 which has no f electrons? 760 00:36:31,030 --> 00:36:34,180 Anyway, I thought that was an interesting label. 761 00:36:34,180 --> 00:36:39,160 So this is how we designate, from these quantum numbers, 762 00:36:39,160 --> 00:36:41,950 which come from just solving the Schrodinger equation for one 763 00:36:41,950 --> 00:36:44,830 electron in a 1 over r potential-- that's 764 00:36:44,830 --> 00:36:46,720 how we get all of these designations 765 00:36:46,720 --> 00:36:50,140 that you're used to-- s orbitals, p orbitals. 766 00:36:50,140 --> 00:36:53,770 And you see, you get different wave functions. 767 00:36:53,770 --> 00:36:59,630 You get these functions, these same wiggly functions 768 00:36:59,630 --> 00:37:01,720 that I've been showing you. 769 00:37:01,720 --> 00:37:05,155 You get four different values of these quantum numbers. 770 00:37:07,970 --> 00:37:11,930 And those functions can be plotted in space. 771 00:37:11,930 --> 00:37:16,580 This is just given those spherical coordinates 772 00:37:16,580 --> 00:37:19,490 and the functions of those, you can plot them. 773 00:37:19,490 --> 00:37:23,180 And you see that they make really funny shapes. 774 00:37:26,840 --> 00:37:31,740 Why is this so important? 775 00:37:31,740 --> 00:37:33,810 I said it on Tuesday. 776 00:37:33,810 --> 00:37:36,925 Why is the funniness of those shapes important? 777 00:37:40,725 --> 00:37:43,110 AUDIENCE: [INAUDIBLE]. 778 00:37:43,110 --> 00:37:45,732 JEFFREY GROSSMAN: Yeah, because what am I looking at? 779 00:37:45,732 --> 00:37:47,190 Actually, let's go to the next one. 780 00:37:47,190 --> 00:37:48,510 That's kind of fun. 781 00:37:48,510 --> 00:37:50,250 What am I looking at here? 782 00:37:50,250 --> 00:38:00,920 This is the 1s orbital of hydrogen, 2s orbital, 3s, 4s. 783 00:38:00,920 --> 00:38:03,020 Look at how they're all spheres. 784 00:38:03,020 --> 00:38:05,430 They're all spherically symmetric. 785 00:38:05,430 --> 00:38:10,560 And then you go 2p orbital and you see 786 00:38:10,560 --> 00:38:13,580 that it has a different shape. 787 00:38:13,580 --> 00:38:17,760 So its function, these are wave functions. 788 00:38:17,760 --> 00:38:22,830 Remember, the wave function is-- what's the wave function? 789 00:38:28,740 --> 00:38:31,050 Who knows? 790 00:38:31,050 --> 00:38:33,835 Does any of you know? 791 00:38:33,835 --> 00:38:38,170 If anybody has some ideas, this is like an open problem, 792 00:38:38,170 --> 00:38:41,010 an open discussion. 793 00:38:41,010 --> 00:38:42,510 But what is interesting about these? 794 00:38:42,510 --> 00:38:44,190 So these are the wave functions. 795 00:38:44,190 --> 00:38:46,176 From that, what can I get? 796 00:38:46,176 --> 00:38:47,490 AUDIENCE: [INAUDIBLE]. 797 00:38:47,490 --> 00:38:49,260 JEFFREY GROSSMAN: Yeah, if I do what? 798 00:38:49,260 --> 00:38:50,093 AUDIENCE: Square it. 799 00:38:50,093 --> 00:38:51,810 JEFFREY GROSSMAN: Square it. 800 00:38:51,810 --> 00:38:57,410 So these landscapes squared tell me what? 801 00:39:01,060 --> 00:39:03,370 Where the electron is. 802 00:39:03,370 --> 00:39:08,170 And that is incredibly important. 803 00:39:08,170 --> 00:39:11,680 That's it-- where is that electron? 804 00:39:11,680 --> 00:39:17,160 Well, if you are fundamental as opposed to diffuse, 805 00:39:17,160 --> 00:39:21,330 you have some really bizarre places you can be and not be. 806 00:39:24,360 --> 00:39:27,460 You can't be anywhere where this isn't. 807 00:39:27,460 --> 00:39:28,200 That's the point. 808 00:39:28,200 --> 00:39:31,290 It's a probability distribution. 809 00:39:31,290 --> 00:39:36,200 It dictates-- basically, all the behavior of materials 810 00:39:36,200 --> 00:39:39,590 comes from these, these distributions. 811 00:39:39,590 --> 00:39:44,330 They dictate what materials do and become. 812 00:39:44,330 --> 00:39:47,000 Now in the hydrogen atom-- 813 00:39:47,000 --> 00:39:48,750 so what happens? 814 00:39:48,750 --> 00:39:53,220 So you get these shapes of the size, 815 00:39:53,220 --> 00:39:55,710 and then you also get those E's. 816 00:39:55,710 --> 00:39:58,500 And remember, we solved for the E's and then 817 00:39:58,500 --> 00:40:03,340 we put the boundary conditions on, and then it was quantized. 818 00:40:03,340 --> 00:40:15,710 And the quantization in hydrogen can be calculated exactly. 819 00:40:15,710 --> 00:40:18,170 So the quantization of the energy levels in hydrogen 820 00:40:18,170 --> 00:40:21,513 is exactly minus 13.6 over n squared, 821 00:40:21,513 --> 00:40:23,180 where n is the principle quantum number. 822 00:40:23,180 --> 00:40:25,850 How many of you have seen this? 823 00:40:25,850 --> 00:40:28,900 Sort of half. 824 00:40:28,900 --> 00:40:32,140 So 13.6 eV over n squared is the quantization. 825 00:40:32,140 --> 00:40:38,960 That means the energy of an electron in this first orbital, 826 00:40:38,960 --> 00:40:42,590 that one up there, if I put an electron in that orbital, 827 00:40:42,590 --> 00:40:50,342 its energy is bound to the atom by 13.6 eV. 828 00:40:50,342 --> 00:40:51,300 So what does that mean? 829 00:40:53,920 --> 00:40:54,640 Yeah. 830 00:40:54,640 --> 00:40:59,112 AUDIENCE: If you give it 13.6 eV or greater, it'll become free. 831 00:40:59,112 --> 00:41:00,070 JEFFREY GROSSMAN: Yeah. 832 00:41:00,070 --> 00:41:03,850 That's the ionization energy of that electron. 833 00:41:03,850 --> 00:41:07,630 But hydrogen has all of these other orbitals 834 00:41:07,630 --> 00:41:08,830 accessible to it. 835 00:41:08,830 --> 00:41:13,000 Maybe I could put an electron up to another level, 836 00:41:13,000 --> 00:41:16,060 like the 2s level. 837 00:41:16,060 --> 00:41:20,560 And if I do that, it's all the way up to 3.4. 838 00:41:20,560 --> 00:41:24,260 So now it's only bound by 3.4 eV in the atom. 839 00:41:24,260 --> 00:41:25,960 And what does that make it? 840 00:41:31,650 --> 00:41:35,410 Is it sort of less stable, more stable? 841 00:41:35,410 --> 00:41:36,327 AUDIENCE: Less stable. 842 00:41:36,327 --> 00:41:37,577 JEFFREY GROSSMAN: Less stable. 843 00:41:37,577 --> 00:41:39,408 So it'd be much more easy to pull it off. 844 00:41:39,408 --> 00:41:41,950 Now, do you think that that has something to do with bonding? 845 00:41:45,430 --> 00:41:47,950 You better believe it. 846 00:41:47,950 --> 00:41:55,060 Where electrons sit in energy relative to other electrons 847 00:41:55,060 --> 00:41:57,760 in other atoms, or in their own atom, 848 00:41:57,760 --> 00:42:02,850 has everything to do with how things bond together. 849 00:42:02,850 --> 00:42:04,800 And that bonding is the one that you've 850 00:42:04,800 --> 00:42:09,140 been just fitting to like a Lennard-Jones potential. 851 00:42:09,140 --> 00:42:10,400 You've just been fitting it. 852 00:42:10,400 --> 00:42:13,400 All that bonding is what you just-- in classical physics, 853 00:42:13,400 --> 00:42:18,650 in classical modeling, you have to have some potential form. 854 00:42:18,650 --> 00:42:20,930 And all of the intricacies of the bonding 855 00:42:20,930 --> 00:42:22,370 get swept up into that. 856 00:42:22,370 --> 00:42:24,410 But you can see why that's so much more 857 00:42:24,410 --> 00:42:29,090 complicated than that, just by looking at these. 858 00:42:29,090 --> 00:42:32,840 OK, but now, that series that we just 859 00:42:32,840 --> 00:42:37,340 got from a simple particle in 1 over r potential 860 00:42:37,340 --> 00:42:43,550 explains all of these levels that are observed. 861 00:42:43,550 --> 00:42:45,950 So does everybody remember this? 862 00:42:45,950 --> 00:42:50,570 And poor number six who didn't get named? 863 00:42:50,570 --> 00:42:51,540 What are these? 864 00:42:54,730 --> 00:42:58,630 When you see these, what does it make you think? 865 00:42:58,630 --> 00:43:06,110 Spectral-- now explain what's happening in the Lymann series. 866 00:43:06,110 --> 00:43:07,140 What's happening? 867 00:43:07,140 --> 00:43:09,765 AUDIENCE: You're moving from one orbital down to the ground one 868 00:43:09,765 --> 00:43:12,075 and it's emitting photons at those wavelengths. 869 00:43:12,075 --> 00:43:13,950 JEFFREY GROSSMAN: And why are they different? 870 00:43:13,950 --> 00:43:17,270 Why do I have all these different ones in one series? 871 00:43:17,270 --> 00:43:19,974 AUDIENCE: The amount of energy it gives off as it goes from n 872 00:43:19,974 --> 00:43:22,980 equals 7 to n equals 1 is different from n equals 2 to n 873 00:43:22,980 --> 00:43:23,480 equals 1. 874 00:43:23,480 --> 00:43:25,020 JEFFREY GROSSMAN: Yeah, exactly. 875 00:43:25,020 --> 00:43:27,590 And if you go up to 2 and you stop at 2, 876 00:43:27,590 --> 00:43:29,720 then you have other energies. 877 00:43:29,720 --> 00:43:33,620 Because it all comes down to, then, what the differences are. 878 00:43:33,620 --> 00:43:36,680 To get an electron to go from here to here, 879 00:43:36,680 --> 00:43:38,870 you need the difference in these energies. 880 00:43:38,870 --> 00:43:41,270 And to go from here to here, you need that difference. 881 00:43:41,270 --> 00:43:44,520 No more and no less will do. 882 00:43:44,520 --> 00:43:48,420 Now, when the electron comes back down, what happens? 883 00:43:52,040 --> 00:43:55,438 Anybody know what happens? 884 00:43:55,438 --> 00:43:56,480 AUDIENCE: It emits light. 885 00:43:56,480 --> 00:43:57,855 JEFFREY GROSSMAN: It emits light. 886 00:43:57,855 --> 00:43:59,870 It gives that energy back out as light. 887 00:43:59,870 --> 00:44:01,940 And that's what you see when you look out 888 00:44:01,940 --> 00:44:03,500 at these spectral lines. 889 00:44:03,500 --> 00:44:07,360 So it emits light at exactly this energy. 890 00:44:07,360 --> 00:44:09,460 Can it emit light at something else, 891 00:44:09,460 --> 00:44:12,430 at a different amount of energy, or a different frequency? 892 00:44:12,430 --> 00:44:15,640 Because we know for light, the energy 893 00:44:15,640 --> 00:44:20,350 is equal to the frequency times a constant. 894 00:44:20,350 --> 00:44:23,435 Can it emit light at this frequency, 895 00:44:23,435 --> 00:44:24,310 something in between? 896 00:44:24,310 --> 00:44:24,810 Why not? 897 00:44:28,224 --> 00:44:29,210 AUDIENCE: Quantized. 898 00:44:29,210 --> 00:44:32,660 JEFFREY GROSSMAN: Yeah, it just can't 899 00:44:32,660 --> 00:44:38,600 because it's a wave problem that has 900 00:44:38,600 --> 00:44:42,540 as its solution quantization. 901 00:44:42,540 --> 00:44:46,320 And now, I come to the question of, well then how does-- 902 00:44:46,320 --> 00:44:47,730 so that's what's happening. 903 00:44:47,730 --> 00:44:50,425 That's the picture of these electrons coming back down, 904 00:44:50,425 --> 00:44:52,050 and emitting different colors of light. 905 00:44:52,050 --> 00:44:55,620 It explains what people saw and couldn't 906 00:44:55,620 --> 00:44:57,660 understand for so long. 907 00:44:57,660 --> 00:45:00,900 This simple solution, mathematical solution, 908 00:45:00,900 --> 00:45:01,800 does it all. 909 00:45:01,800 --> 00:45:06,660 It really crystallized what was going on, 910 00:45:06,660 --> 00:45:08,760 and it really confirmed that this is the right way 911 00:45:08,760 --> 00:45:10,920 to picture things-- 912 00:45:10,920 --> 00:45:14,790 the wave equation and the Schrodinger equation. 913 00:45:14,790 --> 00:45:17,160 Now, what's happening when an electron goes from here 914 00:45:17,160 --> 00:45:18,750 to here? 915 00:45:18,750 --> 00:45:21,980 How does it do that? 916 00:45:21,980 --> 00:45:28,400 It cannot go in between, but we know it does do this all 917 00:45:28,400 --> 00:45:29,310 the time. 918 00:45:29,310 --> 00:45:31,935 We just look out in space and we see it happening all the time, 919 00:45:31,935 --> 00:45:33,240 but we can do it here. 920 00:45:33,240 --> 00:45:37,910 We can build materials that do this for a living. 921 00:45:37,910 --> 00:45:43,760 Where are those electrons going when they go from the 2p 922 00:45:43,760 --> 00:45:46,690 to the 1s? 923 00:45:46,690 --> 00:45:47,940 Where are they going? 924 00:45:51,720 --> 00:45:54,640 Who can tell me? 925 00:45:54,640 --> 00:45:55,722 Yeah. 926 00:45:55,722 --> 00:45:58,180 AUDIENCE: They're all sort of described by a wave function, 927 00:45:58,180 --> 00:45:58,680 right? 928 00:45:58,680 --> 00:46:00,700 So it has some non-zero probability 929 00:46:00,700 --> 00:46:03,432 to be in one place and a non-zero probability 930 00:46:03,432 --> 00:46:04,390 to be in another place. 931 00:46:04,390 --> 00:46:06,700 And if you hit it with a photon or something, 932 00:46:06,700 --> 00:46:09,223 or have some perturbation to your hydrogen atom, 933 00:46:09,223 --> 00:46:11,140 then it's going to have a non-zero probability 934 00:46:11,140 --> 00:46:13,700 to be in one electron energy level than another. 935 00:46:13,700 --> 00:46:17,087 And so it just sort of ends up falling into one of them. 936 00:46:17,087 --> 00:46:18,670 JEFFREY GROSSMAN: That actually works. 937 00:46:18,670 --> 00:46:23,142 That is a great interpretation of what's happening. 938 00:46:23,142 --> 00:46:25,600 And that actually works, and it even works all the way down 939 00:46:25,600 --> 00:46:26,260 to the nucleus. 940 00:46:26,260 --> 00:46:30,250 By the way, decay to the nucleus is possible 941 00:46:30,250 --> 00:46:33,400 because those s orbitals-- 942 00:46:33,400 --> 00:46:36,670 I was just showing you a slice of them, an isosurface. 943 00:46:36,670 --> 00:46:40,420 But they have a non-zero magnitude all the way down 944 00:46:40,420 --> 00:46:41,290 to the nucleus. 945 00:46:41,290 --> 00:46:46,450 It just won't happen very often, because the probability is 946 00:46:46,450 --> 00:46:52,840 so low that it's why we have a universe. 947 00:46:52,840 --> 00:46:57,190 If that overlap was more, if the wave function 948 00:46:57,190 --> 00:46:58,390 going into the middle-- 949 00:47:02,840 --> 00:47:08,270 if this is the nucleus and this is the electron, 950 00:47:08,270 --> 00:47:13,340 and the wave function of the s orbital looks like this, 951 00:47:13,340 --> 00:47:21,830 and it's almost 0, that's why we are alive today. 952 00:47:21,830 --> 00:47:28,090 But if this looks more like this, no, we wouldn't make it. 953 00:47:28,090 --> 00:47:30,610 There's just too much probability for the electron 954 00:47:30,610 --> 00:47:35,140 to decay into the nucleus, and give off that amount of energy, 955 00:47:35,140 --> 00:47:37,150 and the atom collapses. 956 00:47:39,850 --> 00:47:44,375 So we should be very thankful to quantum mechanics. 957 00:47:44,375 --> 00:47:47,660 Now, there are situations, though, 958 00:47:47,660 --> 00:47:52,300 when you actually have nodes in between. 959 00:47:52,300 --> 00:47:54,340 And that's a tougher one. 960 00:47:54,340 --> 00:47:56,550 The node means the wave function goes exactly to 0. 961 00:47:56,550 --> 00:48:00,240 Now what's happening? 962 00:48:00,240 --> 00:48:02,820 And here, we go back to this world 963 00:48:02,820 --> 00:48:06,480 of particles, this field theory world, basically, of how 964 00:48:06,480 --> 00:48:10,260 particles move, by essentially annihilating themselves 965 00:48:10,260 --> 00:48:12,590 and reappearing. 966 00:48:12,590 --> 00:48:15,910 That's how electrons really move. 967 00:48:15,910 --> 00:48:19,030 Now any questions about any of that? 968 00:48:25,560 --> 00:48:26,970 No questions at all? 969 00:48:26,970 --> 00:48:31,920 OK, we're going to start taking this 970 00:48:31,920 --> 00:48:35,280 to the next level, which is obviously going 971 00:48:35,280 --> 00:48:38,713 to have to be more complexity. 972 00:48:38,713 --> 00:48:40,880 There are some constants that I'm just putting here. 973 00:48:40,880 --> 00:48:42,470 I'm not going to spend any time on it, 974 00:48:42,470 --> 00:48:44,120 but these are the constants you'll 975 00:48:44,120 --> 00:48:46,580 be using in the quantum world. 976 00:48:46,580 --> 00:48:49,310 We have an electron volt. We have a Rydberg, 977 00:48:49,310 --> 00:48:51,620 which is 13.6 electron volts. 978 00:48:51,620 --> 00:48:53,370 And just to make it interesting, there's 979 00:48:53,370 --> 00:48:57,200 a Hartree, which is a half of that. 980 00:48:57,200 --> 00:49:01,090 And then the same with angstroms and Bohr. 981 00:49:01,090 --> 00:49:02,680 These are the kinds of unit-- 982 00:49:02,680 --> 00:49:05,110 do you say angstrom, do you say Bohr-- 983 00:49:05,110 --> 00:49:09,190 to-may-to, to-mah-to-- you can choose, but just 984 00:49:09,190 --> 00:49:11,700 keep track of it all. 985 00:49:11,700 --> 00:49:13,720 Now, as we go to more complex-- 986 00:49:13,720 --> 00:49:17,670 so I've given you a particle in a box, 987 00:49:17,670 --> 00:49:20,490 and then I gave you a particle, actually, on Tuesday, 988 00:49:20,490 --> 00:49:22,140 in an oscillator. 989 00:49:22,140 --> 00:49:25,270 And then I give you a particle in 1 over r. 990 00:49:25,270 --> 00:49:26,800 And that got us a long way. 991 00:49:26,800 --> 00:49:28,060 We explained spectral lines. 992 00:49:28,060 --> 00:49:30,520 We got hydrogen right. 993 00:49:30,520 --> 00:49:32,770 But if I just add a little bit more complexity, 994 00:49:32,770 --> 00:49:36,060 you see things get really messy. 995 00:49:36,060 --> 00:49:38,280 Now, this was a problem. 996 00:49:38,280 --> 00:49:41,550 I wanted to show how messy things get 997 00:49:41,550 --> 00:49:47,340 with analytic solutions as a kind of motivational exercise. 998 00:49:47,340 --> 00:49:48,870 So I gave this on a homework problem 999 00:49:48,870 --> 00:49:51,610 in this class two years ago. 1000 00:49:51,610 --> 00:49:57,600 And I'll admit that was probably a mistake. 1001 00:49:57,600 --> 00:50:04,260 I had a lot of people at office hours, because you see, 1002 00:50:04,260 --> 00:50:05,220 this is the same thing. 1003 00:50:05,220 --> 00:50:07,050 I'm throwing my quantum mechanical ball 1004 00:50:07,050 --> 00:50:09,090 into a wall that has another wall behind it. 1005 00:50:11,800 --> 00:50:13,840 So can't you just set it up? 1006 00:50:13,840 --> 00:50:15,370 Well, sure, you can. 1007 00:50:15,370 --> 00:50:17,860 And this is what you get. 1008 00:50:17,860 --> 00:50:21,130 And you put that into Mathematica, 1009 00:50:21,130 --> 00:50:24,370 which every single student did, and it chokes-- 1010 00:50:24,370 --> 00:50:25,960 can't do it. 1011 00:50:25,960 --> 00:50:27,190 Too many for Matlab. 1012 00:50:30,280 --> 00:50:32,440 I guess too many boundary conditions, or too many 1013 00:50:32,440 --> 00:50:34,540 variables, or something, but nobody 1014 00:50:34,540 --> 00:50:37,000 could get Mathematica to do this without 1015 00:50:37,000 --> 00:50:39,340 some intelligent substitution of variables first, 1016 00:50:39,340 --> 00:50:40,660 which was sort of the trick. 1017 00:50:40,660 --> 00:50:44,380 But my point in this was not to make students suffer. 1018 00:50:44,380 --> 00:50:46,510 I don't like to do that, but it was just 1019 00:50:46,510 --> 00:50:48,430 to convey how hard this equation gets 1020 00:50:48,430 --> 00:50:53,050 to solve with just a little bit more complexity than one 1021 00:50:53,050 --> 00:50:57,400 particle in a well or in 1 over r, or something like that. 1022 00:50:57,400 --> 00:51:03,510 It gets really hard, and that's why we need computers. 1023 00:51:03,510 --> 00:51:06,690 And that's what computation in the last 50 years has done. 1024 00:51:06,690 --> 00:51:09,780 It has made this, together with some algorithms, 1025 00:51:09,780 --> 00:51:13,080 some simplifications that you'll hear about-- 1026 00:51:13,080 --> 00:51:17,250 it has made this equation solvable 1027 00:51:17,250 --> 00:51:20,970 for much more complex systems. 1028 00:51:20,970 --> 00:51:23,760 So I can also say, well, let's not 1029 00:51:23,760 --> 00:51:26,050 make the potential more complicated. 1030 00:51:26,050 --> 00:51:31,800 Let's just add another electron. 1031 00:51:31,800 --> 00:51:35,880 So now, I'm still with a 1 over r potential, 1032 00:51:35,880 --> 00:51:38,740 but I've got two of them. 1033 00:51:38,740 --> 00:51:41,950 So now, all of a sudden, my potential term 1034 00:51:41,950 --> 00:51:45,970 is these electrons interacting with the proton, 1035 00:51:45,970 --> 00:51:50,050 but also those electrons interacting with each other. 1036 00:51:50,050 --> 00:51:54,600 That's r12, the distance between the two electrons. 1037 00:51:54,600 --> 00:51:56,760 And boy, does that make this equation 1038 00:51:56,760 --> 00:52:00,090 hard to solve analytically. 1039 00:52:00,090 --> 00:52:05,650 You cannot use separation of variables. 1040 00:52:05,650 --> 00:52:12,450 So even just one more electron, it's a deal breaker, basically. 1041 00:52:12,450 --> 00:52:15,960 And now you think we want to do big molecules. 1042 00:52:15,960 --> 00:52:17,160 We want to do fullerenes. 1043 00:52:17,160 --> 00:52:19,530 We want to do silicon. 1044 00:52:19,530 --> 00:52:22,450 We want to do thousands of electrons. 1045 00:52:22,450 --> 00:52:24,270 Analytic approaches won't work. 1046 00:52:26,810 --> 00:52:28,380 So what can you do? 1047 00:52:28,380 --> 00:52:32,990 Well, there are two different ways to go. 1048 00:52:32,990 --> 00:52:34,760 You've got to do something. 1049 00:52:34,760 --> 00:52:39,470 You've got to make it easier, because as you'll see, 1050 00:52:39,470 --> 00:52:43,880 an exact solution, even with the most powerful computers 1051 00:52:43,880 --> 00:52:48,950 in the world times squared are not going to do it. 1052 00:52:48,950 --> 00:52:52,010 You need to make approximations to the actual equation. 1053 00:52:52,010 --> 00:52:55,530 And you'll hear about those next week. 1054 00:52:55,530 --> 00:52:58,980 And one thing you can do is you can use perturbation theory 1055 00:52:58,980 --> 00:53:03,360 to say that this Hamiltonian is actually some simpler 1056 00:53:03,360 --> 00:53:05,430 Hamiltonian. 1057 00:53:05,430 --> 00:53:08,190 That Hamiltonian is some simpler Hamiltonian 1058 00:53:08,190 --> 00:53:12,300 that you can solve easily plus some perturbation to it. 1059 00:53:12,300 --> 00:53:14,640 We're not going to really do that. 1060 00:53:14,640 --> 00:53:18,070 Has anybody used perturbation theory? 1061 00:53:18,070 --> 00:53:20,280 So we're not going to do it much but, 1062 00:53:20,280 --> 00:53:22,360 I'm just using it for reference. 1063 00:53:22,360 --> 00:53:25,530 And then, so you do something for something. 1064 00:53:25,530 --> 00:53:28,710 Perturbation theory, what you do is you say, well, that's hard, 1065 00:53:28,710 --> 00:53:30,130 so I'm not going to do that. 1066 00:53:30,130 --> 00:53:32,490 But this is actually kind of related, and it's easy, 1067 00:53:32,490 --> 00:53:33,960 and I know how to do this. 1068 00:53:33,960 --> 00:53:37,500 So I'll do this and then add something to it. 1069 00:53:37,500 --> 00:53:40,710 That's basically perturbation theory. 1070 00:53:40,710 --> 00:53:43,470 But what we're going to do is we're 1071 00:53:43,470 --> 00:53:48,630 going to try to solve it using more of a matrix eigenvalue 1072 00:53:48,630 --> 00:53:49,330 approach. 1073 00:53:49,330 --> 00:53:51,960 And again, we are not going to go deep into the math here. 1074 00:53:51,960 --> 00:53:55,830 And you will not be tested on the math. 1075 00:53:55,830 --> 00:53:59,250 As I said, my goal here is to have you guys 1076 00:53:59,250 --> 00:54:04,410 applying these equations to real materials problems. 1077 00:54:04,410 --> 00:54:08,160 But I do think that it's useful to know what's under the hood, 1078 00:54:08,160 --> 00:54:10,440 to know what equations are being solved 1079 00:54:10,440 --> 00:54:12,900 when you press "Simulate." 1080 00:54:12,900 --> 00:54:16,440 So you'll hear more about this, and how 1081 00:54:16,440 --> 00:54:19,740 you go about actually making these approximations, 1082 00:54:19,740 --> 00:54:21,210 making the approximations you need 1083 00:54:21,210 --> 00:54:27,690 to make in order to get an eigenvalue problem that 1084 00:54:27,690 --> 00:54:29,870 can be solved on a computer. 1085 00:54:29,870 --> 00:54:31,870 There are a couple of different ways to do that. 1086 00:54:31,870 --> 00:54:33,995 The way that we're going to settle on in this class 1087 00:54:33,995 --> 00:54:35,827 is density functional theory. 1088 00:54:35,827 --> 00:54:37,410 So we'll talk a little bit about that. 1089 00:54:37,410 --> 00:54:40,050 And again, the only Nobel Prize in computation 1090 00:54:40,050 --> 00:54:42,720 was given for that, the development of that theory. 1091 00:54:45,640 --> 00:54:48,070 Now, there's one more thing I need to talk about 1092 00:54:48,070 --> 00:54:52,030 before I move to just a couple of examples 1093 00:54:52,030 --> 00:54:54,850 online that I'd like to show you. 1094 00:54:54,850 --> 00:54:58,340 And that is that there's something still missing here. 1095 00:54:58,340 --> 00:55:12,690 So we've got-- let's see, we've got this 1s. 1096 00:55:12,690 --> 00:55:15,270 So now we know where these things come from. 1097 00:55:15,270 --> 00:55:17,370 We really know where they come from, the solution 1098 00:55:17,370 --> 00:55:18,720 to the hydrogen atom-- 1099 00:55:21,250 --> 00:55:27,890 2p, 3s, and so on. 1100 00:55:27,890 --> 00:55:30,290 And then hydrogen would be what? 1101 00:55:30,290 --> 00:55:31,730 How would I show hydrogen here? 1102 00:55:35,170 --> 00:55:36,310 Put an arrow there. 1103 00:55:36,310 --> 00:55:39,160 What would helium be? 1104 00:55:39,160 --> 00:55:40,240 One more. 1105 00:55:40,240 --> 00:55:42,190 OK, tell me carbon-- 1106 00:55:42,190 --> 00:55:43,590 I like carbon. 1107 00:55:43,590 --> 00:55:44,590 AUDIENCE: [INAUDIBLE]. 1108 00:55:44,590 --> 00:55:46,215 JEFFREY GROSSMAN: Carbon is right here. 1109 00:55:46,215 --> 00:55:47,740 It hits me in the heart every time. 1110 00:55:47,740 --> 00:55:49,804 How many more? 1111 00:55:49,804 --> 00:55:51,707 AUDIENCE: Four. 1112 00:55:51,707 --> 00:55:53,040 JEFFREY GROSSMAN: And then what? 1113 00:55:53,040 --> 00:55:53,767 AUDIENCE: Two up. 1114 00:55:53,767 --> 00:55:54,850 JEFFREY GROSSMAN: Two up-- 1115 00:55:54,850 --> 00:55:57,550 oh, why not like this? 1116 00:55:57,550 --> 00:55:58,050 No? 1117 00:55:58,050 --> 00:56:01,640 It just feels wrong-- exclusion. 1118 00:56:01,640 --> 00:56:03,120 Now we're talking. 1119 00:56:03,120 --> 00:56:05,760 You guys already kind of have a feeling for this. 1120 00:56:05,760 --> 00:56:08,660 But actually, you can have materials 1121 00:56:08,660 --> 00:56:10,400 that violate someone's rule. 1122 00:56:10,400 --> 00:56:12,662 Anyone know what that rule is called? 1123 00:56:12,662 --> 00:56:13,787 AUDIENCE: Pauli exclusions. 1124 00:56:13,787 --> 00:56:15,370 JEFFREY GROSSMAN: Well, actually, this 1125 00:56:15,370 --> 00:56:16,760 doesn't violate Pauli exclusions. 1126 00:56:16,760 --> 00:56:17,570 AUDIENCE: Hund's rule. 1127 00:56:17,570 --> 00:56:19,320 JEFFREY GROSSMAN: Hund's rule-- very good, 1128 00:56:19,320 --> 00:56:21,800 whoever pulled that one out. 1129 00:56:21,800 --> 00:56:23,150 That was good. 1130 00:56:23,150 --> 00:56:26,870 But this is how carbon usually looks. 1131 00:56:26,870 --> 00:56:28,490 Now, something that we've done here 1132 00:56:28,490 --> 00:56:31,370 is we've already done what I'm about to tell you, which 1133 00:56:31,370 --> 00:56:33,290 is we've put spin in there. 1134 00:56:33,290 --> 00:56:36,510 We've put those electrons pointing up or down. 1135 00:56:36,510 --> 00:56:38,390 But we also have those quantum numbers. 1136 00:56:38,390 --> 00:56:40,960 This is n. 1137 00:56:40,960 --> 00:56:42,910 N equals 1. 1138 00:56:42,910 --> 00:56:48,970 This is n equals 2, and this is l equals 0, l equals 0. 1139 00:56:48,970 --> 00:56:50,710 And this is p. 1140 00:56:50,710 --> 00:56:55,720 So this is n equals 2, l equals 1. 1141 00:56:55,720 --> 00:56:59,860 And each one of these is an m. 1142 00:56:59,860 --> 00:57:04,710 m equals plus or minus 1 and 0. 1143 00:57:04,710 --> 00:57:07,103 So that's why I have three p's. 1144 00:57:07,103 --> 00:57:08,520 So those are those quantum numbers 1145 00:57:08,520 --> 00:57:10,920 you get from quantizing the solution to the Schrodinger 1146 00:57:10,920 --> 00:57:14,280 equation for an electron in the 1 over r potential. 1147 00:57:14,280 --> 00:57:17,340 But there's this one more thing, which was also very surprising 1148 00:57:17,340 --> 00:57:19,710 at the time, which is that if you 1149 00:57:19,710 --> 00:57:22,230 put a particle, in this case silver atoms, 1150 00:57:22,230 --> 00:57:25,440 if you think about electrons as charges spinning, 1151 00:57:25,440 --> 00:57:27,987 you should have a reaction to a magnetic field. 1152 00:57:27,987 --> 00:57:29,445 They should have a magnetic moment. 1153 00:57:32,100 --> 00:57:35,220 But classically, you see, if you have charged spinning, 1154 00:57:35,220 --> 00:57:36,780 you get a magnetic field. 1155 00:57:36,780 --> 00:57:41,040 Classically, if you shoot this through a really large magnet, 1156 00:57:41,040 --> 00:57:44,250 you should just get a whole range of lines of fields. 1157 00:57:44,250 --> 00:57:48,450 But just like the double slit, it didn't turn out that way. 1158 00:57:48,450 --> 00:57:53,360 You got two dots instead, only two dots. 1159 00:57:53,360 --> 00:57:54,460 And these are very famous. 1160 00:57:54,460 --> 00:57:56,290 These are the Stern-Gerlach experiments, 1161 00:57:56,290 --> 00:57:58,810 which were done all around the same time they 1162 00:57:58,810 --> 00:58:00,190 were figuring all this out. 1163 00:58:00,190 --> 00:58:04,000 And what it showed is that the spin can only be up or down. 1164 00:58:04,000 --> 00:58:06,850 It can only be two things. 1165 00:58:06,850 --> 00:58:12,040 There's only two values for the spin of an electron, the spin 1166 00:58:12,040 --> 00:58:14,200 being the thing that's causing this thing 1167 00:58:14,200 --> 00:58:17,830 to have a magnetic moment that's causing it to react 1168 00:58:17,830 --> 00:58:20,825 to the external magnetic field. 1169 00:58:20,825 --> 00:58:23,260 You should get a line. 1170 00:58:23,260 --> 00:58:25,090 You should get all this-- 1171 00:58:25,090 --> 00:58:28,900 classically, electrons should have any spin they want. 1172 00:58:28,900 --> 00:58:31,510 I don't know what that was, but it should be 1173 00:58:31,510 --> 00:58:33,400 able to point in any direction. 1174 00:58:33,400 --> 00:58:36,040 But no, it can only point this way or this way. 1175 00:58:36,040 --> 00:58:39,170 Now, is an electron really spinning? 1176 00:58:39,170 --> 00:58:43,810 If a charge is spinning, it has a magnetic moment, 1177 00:58:43,810 --> 00:58:45,445 but is an electron really spinning? 1178 00:58:49,770 --> 00:58:52,320 How does it have a magnetic moment? 1179 00:58:52,320 --> 00:58:54,420 Let's talk. 1180 00:58:54,420 --> 00:58:58,200 That's a really good one, too, but it does. 1181 00:58:58,200 --> 00:59:00,270 And you think about it as spinning 1182 00:59:00,270 --> 00:59:06,060 because that is our classical Maxwell sort of ENN framework. 1183 00:59:06,060 --> 00:59:07,680 We think about charged spinning. 1184 00:59:07,680 --> 00:59:08,310 You've done it. 1185 00:59:08,310 --> 00:59:12,930 You make current go through a wire, and you get a field. 1186 00:59:16,087 --> 00:59:17,670 And that's why we use the word "spin." 1187 00:59:17,670 --> 00:59:19,530 It's sort of like "orbital." 1188 00:59:19,530 --> 00:59:21,810 Should we really be using the word "orbital?" 1189 00:59:21,810 --> 00:59:25,110 Is this electron orbiting? 1190 00:59:25,110 --> 00:59:27,000 No. 1191 00:59:27,000 --> 00:59:29,950 And that's another ice-breaking question. 1192 00:59:29,950 --> 00:59:32,430 You go to your friend, and you just say, 1193 00:59:32,430 --> 00:59:36,150 do you think electrons are really orbiting? 1194 00:59:36,150 --> 00:59:37,440 Just like that. 1195 00:59:37,440 --> 00:59:40,290 And see how many people say yes. 1196 00:59:40,290 --> 00:59:43,180 They kind of are spinning in our minds. 1197 00:59:43,180 --> 00:59:48,040 But the spin is up or down because of those experiments. 1198 00:59:48,040 --> 00:59:50,740 Now then, the sort of end of the story-- 1199 00:59:50,740 --> 00:59:58,440 so they have a magnetic moment, and it gives them up and down. 1200 00:59:58,440 --> 01:00:01,770 And they can only be up and down, 1201 01:00:01,770 --> 01:00:04,690 only up and down, two values. 1202 01:00:04,690 --> 01:00:09,850 And so what you have is sort of a new quantum number. 1203 01:00:09,850 --> 01:00:15,040 You have spin, which is equal to plus or minus 1. 1204 01:00:15,040 --> 01:00:17,890 That's that and that. 1205 01:00:17,890 --> 01:00:19,360 So now each of these-- and that's 1206 01:00:19,360 --> 01:00:21,490 why with these collections of quantum numbers, 1207 01:00:21,490 --> 01:00:25,900 you can get that many electrons into those shells. 1208 01:00:25,900 --> 01:00:26,790 And I love this. 1209 01:00:26,790 --> 01:00:29,998 So somebody said "Pauli exclusion." 1210 01:00:29,998 --> 01:00:32,290 Somebody tell me what the Pauli exclusion principle is. 1211 01:00:34,800 --> 01:00:35,597 Yes. 1212 01:00:35,597 --> 01:00:37,680 AUDIENCE: Two fermions can't be in the same state. 1213 01:00:37,680 --> 01:00:41,400 JEFFREY GROSSMAN: OK, so two electrons, we'll just say, 1214 01:00:41,400 --> 01:00:44,470 which are fermions, cannot be in the same state. 1215 01:00:44,470 --> 01:00:47,370 And what do you mean by the "same state?" 1216 01:00:47,370 --> 01:00:53,970 AUDIENCE: That they can be in sort of the same energy shell 1217 01:00:53,970 --> 01:00:56,880 with the same spin at the same time. 1218 01:00:56,880 --> 01:00:58,290 JEFFREY GROSSMAN: OK, very good. 1219 01:00:58,290 --> 01:01:00,840 Basically what it means is they cannot have the exact same 1220 01:01:00,840 --> 01:01:01,530 quantum numbers. 1221 01:01:04,530 --> 01:01:09,290 Pauli exclusion says you cannot do this. 1222 01:01:09,290 --> 01:01:12,290 Now, I mentioned you can violate Hund's rule. 1223 01:01:12,290 --> 01:01:15,410 And you can violate Hund's rule by having 1224 01:01:15,410 --> 01:01:18,230 degenerate orbitals with the same energy, 1225 01:01:18,230 --> 01:01:20,570 and doing it there. 1226 01:01:20,570 --> 01:01:21,920 But usually, you don't. 1227 01:01:21,920 --> 01:01:25,640 Pauli exclusion is an incredibly important aspect 1228 01:01:25,640 --> 01:01:26,720 of quantum mechanics. 1229 01:01:26,720 --> 01:01:30,500 It says that no two electrons, fermions, 1230 01:01:30,500 --> 01:01:32,720 can occupy the same set of quantum numbers 1231 01:01:32,720 --> 01:01:33,830 when you include spin. 1232 01:01:36,480 --> 01:01:40,560 That is what gives us literally the periodic table and all 1233 01:01:40,560 --> 01:01:43,920 of chemistry when you add that to quantum mechanics. 1234 01:01:43,920 --> 01:01:49,370 That is what gives us the rest, that plus the hydrogen atom. 1235 01:01:49,370 --> 01:01:51,560 You don't need much more, except you do. 1236 01:01:51,560 --> 01:01:58,880 But I love this letter from Thomas to Goudsmit in 1926. 1237 01:01:58,880 --> 01:02:01,850 "I think you and Uhlenbeck"-- because they're 1238 01:02:01,850 --> 01:02:07,580 the ones that discovered this stuff in working out 1239 01:02:07,580 --> 01:02:08,600 the theory-- 1240 01:02:08,600 --> 01:02:11,030 "have been very lucky to get your spinning electron 1241 01:02:11,030 --> 01:02:14,360 published and talked about before Pauli heard of it. 1242 01:02:14,360 --> 01:02:17,240 It appears that more than a year ago, Kronig 1243 01:02:17,240 --> 01:02:19,820 believed in the spinning electron 1244 01:02:19,820 --> 01:02:21,320 and worked out something. 1245 01:02:21,320 --> 01:02:23,352 The first person he showed it to was Pauli. 1246 01:02:23,352 --> 01:02:26,450 Pauli ridiculed the whole thing so much 1247 01:02:26,450 --> 01:02:30,110 that the first person became also the last, and no one else 1248 01:02:30,110 --> 01:02:32,600 heard anything of it, which all goes 1249 01:02:32,600 --> 01:02:35,900 to show that the infallibility of the deity 1250 01:02:35,900 --> 01:02:39,770 does not extend to his self-styled vicar on Earth." 1251 01:02:43,200 --> 01:02:46,140 So Pauli had an interesting reputation. 1252 01:02:46,140 --> 01:02:50,340 But that is amazing to me, because this is it. 1253 01:02:50,340 --> 01:02:53,640 This is one of the foundations of the Pauli exclusion 1254 01:02:53,640 --> 01:02:57,690 principle, so Pauli built his whole reputation 1255 01:02:57,690 --> 01:03:00,300 and really made an important contribution 1256 01:03:00,300 --> 01:03:02,790 because of what apparently he didn't really 1257 01:03:02,790 --> 01:03:03,870 believe in at first. 1258 01:03:03,870 --> 01:03:06,060 So it goes to show-- keep an open mind. 1259 01:03:09,240 --> 01:03:12,660 So that's what we just said, can't have the same quantum 1260 01:03:12,660 --> 01:03:13,620 numbers. 1261 01:03:13,620 --> 01:03:16,710 Got to mix up the quantum numbers, and that's it. 1262 01:03:16,710 --> 01:03:18,090 And that is it. 1263 01:03:18,090 --> 01:03:22,290 You see, because now, we had all these levels from the hydrogen 1264 01:03:22,290 --> 01:03:24,180 atom solution-- 1265 01:03:24,180 --> 01:03:30,240 an electron a one electron in a simple 1 over r potential, 1266 01:03:30,240 --> 01:03:31,740 and we got all those levels. 1267 01:03:31,740 --> 01:03:35,560 And now all we need to do is fill them up. 1268 01:03:35,560 --> 01:03:39,960 And with the Pauli exclusion principle, we know the rules. 1269 01:03:39,960 --> 01:03:41,760 We know the rules. 1270 01:03:41,760 --> 01:03:44,140 We know we can only put two electrons here. 1271 01:03:44,140 --> 01:03:48,100 We know we can only put two electrons in s 1272 01:03:48,100 --> 01:03:51,130 and six electrons in p, so eight here. 1273 01:03:51,130 --> 01:03:54,480 We know we can only put two, six, and how many in d? 1274 01:03:59,520 --> 01:04:06,790 10, because n goes from minus 2 to plus 2, so 5. 1275 01:04:06,790 --> 01:04:08,380 I mean, d has that. 1276 01:04:08,380 --> 01:04:13,150 So that's starting to sound a whole lot like this. 1277 01:04:13,150 --> 01:04:14,190 That's it. 1278 01:04:14,190 --> 01:04:19,320 Essentially, with those fundamental principles 1279 01:04:19,320 --> 01:04:22,350 that were worked out in the '20s, and the solution 1280 01:04:22,350 --> 01:04:24,750 of the hydrogen atom, you get essentially all 1281 01:04:24,750 --> 01:04:26,900 of the periodic table. 1282 01:04:26,900 --> 01:04:28,870 It's really powerful. 1283 01:04:28,870 --> 01:04:31,310 It's pretty darn cool. 1284 01:04:31,310 --> 01:04:35,570 And you can connect, as I said, back to the spectral lines 1285 01:04:35,570 --> 01:04:36,990 and all of that. 1286 01:04:36,990 --> 01:04:41,990 Now, what I want to spend the last couple of minutes on 1287 01:04:41,990 --> 01:04:48,660 is I wanted to show you-- has anybody looked at the quantum 1288 01:04:48,660 --> 01:04:49,160 part? 1289 01:04:49,160 --> 01:04:50,830 You guys are still doing homework from the last part, 1290 01:04:50,830 --> 01:04:51,330 right? 1291 01:04:55,020 --> 01:04:58,590 So on the NanoHUB, there is-- 1292 01:04:58,590 --> 01:05:02,848 so you go to the NanoHUB, and you go to-- 1293 01:05:02,848 --> 01:05:04,140 sorry, I should check this out. 1294 01:05:04,140 --> 01:05:06,330 There I am. 1295 01:05:06,330 --> 01:05:07,560 That's amazing. 1296 01:05:07,560 --> 01:05:12,600 That's a picture from, like, I won't say how long ago. 1297 01:05:12,600 --> 01:05:16,140 That's embarrassing. 1298 01:05:16,140 --> 01:05:16,830 So you go. 1299 01:05:16,830 --> 01:05:18,790 This is my personal thing. 1300 01:05:18,790 --> 01:05:21,120 But anyway, the tool you want is this one, 1301 01:05:21,120 --> 01:05:22,900 and it has a little star next to it. 1302 01:05:22,900 --> 01:05:23,850 That's so exciting. 1303 01:05:23,850 --> 01:05:28,210 It's called the "MIT Atomic Scale Modeling Toolkit," 1304 01:05:28,210 --> 01:05:31,910 formerly known as the Berkeley Atomic Scale Modeling Toolkit, 1305 01:05:31,910 --> 01:05:35,290 because I developed it at Berkeley. 1306 01:05:35,290 --> 01:05:39,440 So much for that, but it's now at MIT. 1307 01:05:39,440 --> 01:05:42,170 And if you click that, these are actually 1308 01:05:42,170 --> 01:05:46,750 tools I developed for a class I made up when I was at Berkeley, 1309 01:05:46,750 --> 01:05:49,480 when I valued sunshine over science. 1310 01:05:49,480 --> 01:05:51,860 It doesn't say that. 1311 01:05:51,860 --> 01:05:55,030 And you can launch it, and you'll see that-- 1312 01:05:55,030 --> 01:05:56,810 and you can rate it a 10. 1313 01:05:56,810 --> 01:05:58,640 Don't rate it anything else. 1314 01:05:58,640 --> 01:06:03,260 But please rate it, or don't rate it. 1315 01:06:03,260 --> 01:06:07,800 Everybody is so star driven these days, number of stars. 1316 01:06:07,800 --> 01:06:10,270 What was that? 1317 01:06:10,270 --> 01:06:12,020 There was an NPR piece the other day about 1318 01:06:12,020 --> 01:06:15,500 how there's this whole business, opportunity sprouting up, 1319 01:06:15,500 --> 01:06:18,980 of people who have companies, where all they do 1320 01:06:18,980 --> 01:06:22,100 is they get your number of stars for your products high. 1321 01:06:22,100 --> 01:06:23,827 So they have people around the world, 1322 01:06:23,827 --> 01:06:25,910 because you've got to do it from different places, 1323 01:06:25,910 --> 01:06:29,570 and you've got to write fake reviews of products, 1324 01:06:29,570 --> 01:06:34,070 and have different fake Google email addresses-- 1325 01:06:34,070 --> 01:06:38,870 blowing me away, the so-called internet. 1326 01:06:38,870 --> 01:06:44,620 Now, you look here, and you see that actually in this MIT 1327 01:06:44,620 --> 01:06:47,410 Atomic Scale Modeling Toolkit, there are many things. 1328 01:06:47,410 --> 01:06:48,340 There are many things. 1329 01:06:48,340 --> 01:06:52,660 We will be using two tools from this, 1330 01:06:52,660 --> 01:06:55,130 and both are quantum mechanical tools. 1331 01:06:55,130 --> 01:06:59,350 One is going to be GAMESS, which is a quantum chemistry package. 1332 01:06:59,350 --> 01:06:59,870 It's free. 1333 01:06:59,870 --> 01:07:03,790 You can download it and run it on your laptop, actually. 1334 01:07:03,790 --> 01:07:05,710 And it's for molecules and atoms. 1335 01:07:05,710 --> 01:07:09,750 And another is going to be called SIESTA. 1336 01:07:09,750 --> 01:07:12,810 So you play games, you take a siesta. 1337 01:07:12,810 --> 01:07:17,040 And this one, we're going to use when we do solids. 1338 01:07:17,040 --> 01:07:19,733 Both of them are going to use an approximation called 1339 01:07:19,733 --> 01:07:21,900 "density functional theory" to solve the Schrodinger 1340 01:07:21,900 --> 01:07:24,010 equation for real materials. 1341 01:07:24,010 --> 01:07:26,640 So let's just take, in the last few minutes of class, 1342 01:07:26,640 --> 01:07:28,200 let's take a look at GAMESS. 1343 01:07:28,200 --> 01:07:31,030 And you can see-- 1344 01:07:31,030 --> 01:07:34,440 why isn't it letting me-- it's not letting me scroll down. 1345 01:07:34,440 --> 01:07:35,070 Seriously? 1346 01:07:38,720 --> 01:07:41,250 Oh, there's my scrollbar there. 1347 01:07:41,250 --> 01:07:45,158 OK, so the input is actually pretty straightforward. 1348 01:07:45,158 --> 01:07:47,450 So you can calculate the energy or you can optimize it. 1349 01:07:47,450 --> 01:07:50,840 The method is going to be DFT, Density Functional Theory. 1350 01:07:50,840 --> 01:07:53,540 The basis set is something you'll learn about on Tuesday, 1351 01:07:53,540 --> 01:07:57,930 what that is, but pick Medium. 1352 01:07:57,930 --> 01:08:00,450 And then I want to not have a solvent, 1353 01:08:00,450 --> 01:08:03,650 but I want to plot these orbitals. 1354 01:08:03,650 --> 01:08:09,830 And then, I think it disappeared again. 1355 01:08:09,830 --> 01:08:12,840 Don't you guys love the NanoHUB? 1356 01:08:12,840 --> 01:08:16,220 It's so bug free. 1357 01:08:16,220 --> 01:08:19,310 And now it's not dragging. 1358 01:08:19,310 --> 01:08:19,939 It's great. 1359 01:08:19,939 --> 01:08:22,430 So what you can do is you can do New, 1360 01:08:22,430 --> 01:08:26,520 and let's do the carbon atom, my favorite one. 1361 01:08:26,520 --> 01:08:27,569 And I'm going to-- 1362 01:08:27,569 --> 01:08:30,850 do you guys know about XYZ coordinates? 1363 01:08:30,850 --> 01:08:33,229 So you put the number of atoms in the top line. 1364 01:08:33,229 --> 01:08:37,540 You leave the next line blank or you write whatever you want. 1365 01:08:37,540 --> 01:08:40,720 And, well, don't let me see some of the things you might write. 1366 01:08:40,720 --> 01:08:41,529 I don't know. 1367 01:08:41,529 --> 01:08:44,795 And then you put the atoms with their XYZ positions. 1368 01:08:44,795 --> 01:08:46,420 So carbon is going to be at the origin, 1369 01:08:46,420 --> 01:08:47,830 and we click "Simulate." 1370 01:08:47,830 --> 01:08:48,880 And it's doing it. 1371 01:08:48,880 --> 01:08:49,600 That's it. 1372 01:08:49,600 --> 01:08:53,620 It just solved the Schrodinger equation for carbon, something 1373 01:08:53,620 --> 01:08:55,670 you couldn't do for a million years-- 1374 01:08:55,670 --> 01:08:57,010 no, for 50 years-- 1375 01:08:57,010 --> 01:08:58,540 analytically. 1376 01:08:58,540 --> 01:09:00,670 And it just did it. 1377 01:09:00,670 --> 01:09:07,819 And now it's still executing a script, 1378 01:09:07,819 --> 01:09:11,529 which is very frustrating. 1379 01:09:11,529 --> 01:09:14,370 Is it this slow for you guys? 1380 01:09:14,370 --> 01:09:17,170 This is one atom. 1381 01:09:17,170 --> 01:09:17,670 Man. 1382 01:09:22,220 --> 01:09:24,529 Well, so we go back to here. 1383 01:09:24,529 --> 01:09:26,540 One thing you got right there is you 1384 01:09:26,540 --> 01:09:30,800 got those energy levels, which I can't scroll down to look at. 1385 01:09:33,830 --> 01:09:40,220 This is so amazingly frustrating. 1386 01:09:40,220 --> 01:09:45,350 So now, what I'm going to do, since it's not working, 1387 01:09:45,350 --> 01:09:48,420 is I'm going to try it again. 1388 01:09:48,420 --> 01:09:53,560 And I'll pick a different basis, low. 1389 01:09:53,560 --> 01:09:54,860 See if that works. 1390 01:09:54,860 --> 01:09:59,110 And I just did the Schrodinger equation for carbon. 1391 01:09:59,110 --> 01:10:02,350 You've got to be kidding me. 1392 01:10:02,350 --> 01:10:05,860 You've got to be kidding me. 1393 01:10:05,860 --> 01:10:07,760 Really? 1394 01:10:07,760 --> 01:10:09,080 Seriously? 1395 01:10:09,080 --> 01:10:11,130 Let's try a different atom. 1396 01:10:11,130 --> 01:10:12,743 Let's try-- somebody give me an atom. 1397 01:10:12,743 --> 01:10:13,410 AUDIENCE: Boron. 1398 01:10:13,410 --> 01:10:15,702 JEFFREY GROSSMAN: All right, this is going to be lucky. 1399 01:10:19,142 --> 01:10:20,100 This is going to do it. 1400 01:10:23,200 --> 01:10:25,270 Oh, yeah, It's not able to do it-- 1401 01:10:28,890 --> 01:10:31,690 unbelievable. 1402 01:10:31,690 --> 01:10:34,530 Let me just try one more thing. 1403 01:10:34,530 --> 01:10:40,580 I'm going to try putting two atoms in. 1404 01:10:40,580 --> 01:10:41,740 Let's see if that helps. 1405 01:10:45,260 --> 01:10:47,810 And we'll optimize it. 1406 01:10:47,810 --> 01:10:50,120 Yeah, now it is a party. 1407 01:10:50,120 --> 01:10:58,590 Oh, OK, but did it give me the orbitals? 1408 01:10:58,590 --> 01:10:59,090 No. 1409 01:11:02,050 --> 01:11:03,870 Hang on. 1410 01:11:03,870 --> 01:11:05,890 Sorry, guys. 1411 01:11:05,890 --> 01:11:07,620 Oh, it's still here. 1412 01:11:15,660 --> 01:11:19,200 Let's just see if it gave me the orbitals. 1413 01:11:19,200 --> 01:11:20,970 OK, there they are. 1414 01:11:20,970 --> 01:11:24,660 OK, this is it. 1415 01:11:24,660 --> 01:11:28,470 It was Firefox, which I never use, but it came to the rescue 1416 01:11:28,470 --> 01:11:29,220 for this one. 1417 01:11:29,220 --> 01:11:31,200 So look at this, just really quickly, 1418 01:11:31,200 --> 01:11:32,330 because if you want to play with this, 1419 01:11:32,330 --> 01:11:33,955 this will just take one minute to show. 1420 01:11:33,955 --> 01:11:37,890 OK, there, I did the carbon dimer. 1421 01:11:37,890 --> 01:11:40,890 That's a lot of electrons, and it just solved it. 1422 01:11:40,890 --> 01:11:43,650 Because of what you're going to learn on Tuesday, 1423 01:11:43,650 --> 01:11:45,780 the approximations, it just solved it. 1424 01:11:45,780 --> 01:11:49,470 And look-- if you right click on top of the image, 1425 01:11:49,470 --> 01:11:51,690 you get surfaces, and then you get orbitals. 1426 01:11:51,690 --> 01:11:53,040 And look at that-- 1427 01:11:53,040 --> 01:11:56,976 that is actually-- what do you think those two are? 1428 01:11:56,976 --> 01:12:00,150 There are two carbon atoms here-- 1429 01:12:00,150 --> 01:12:01,230 the s's. 1430 01:12:01,230 --> 01:12:05,040 Those are those 1s's together in the dimer. 1431 01:12:05,040 --> 01:12:09,300 And these are now the 2s and p's, all bonding together. 1432 01:12:09,300 --> 01:12:11,100 And you want to know how they look? 1433 01:12:11,100 --> 01:12:16,350 Well, here's one of those s's. 1434 01:12:16,350 --> 01:12:17,500 There it is. 1435 01:12:17,500 --> 01:12:19,750 It's an S orbital in the dimer, though. 1436 01:12:19,750 --> 01:12:22,930 And it looks very much like the atomic s orbital, 1437 01:12:22,930 --> 01:12:24,020 which it should. 1438 01:12:24,020 --> 01:12:24,520 Why? 1439 01:12:28,020 --> 01:12:31,490 Remember, those are the ones we don't care about. 1440 01:12:31,490 --> 01:12:34,370 They don't really do anything when you bond. 1441 01:12:34,370 --> 01:12:39,130 It's all about the other ones, like for example, 1442 01:12:39,130 --> 01:12:42,120 one of these p's. 1443 01:12:42,120 --> 01:12:43,260 That's it. 1444 01:12:43,260 --> 01:12:45,000 That's the solution to the Schrodinger 1445 01:12:45,000 --> 01:12:46,620 equation for the carbon dimer. 1446 01:12:46,620 --> 01:12:50,160 And that's where that p-type orbital-- 1447 01:12:50,160 --> 01:12:53,070 that's where the electron in that level sits. 1448 01:12:53,070 --> 01:12:54,330 And this is no longer an atom. 1449 01:12:54,330 --> 01:12:55,830 You can do it for the carbon atom, 1450 01:12:55,830 --> 01:12:58,320 and see the p orbitals and the s orbitals, 1451 01:12:58,320 --> 01:13:00,780 but now when you make a molecule, 1452 01:13:00,780 --> 01:13:04,830 things get more complicated, and more and more complicated. 1453 01:13:04,830 --> 01:13:07,440 So that'll be the tool we use for our molecule things, 1454 01:13:07,440 --> 01:13:10,000 assuming it actually works. 1455 01:13:10,000 --> 01:13:14,370 And if anyone has any questions, email me. 1456 01:13:14,370 --> 01:13:18,830 And remember, class next Tuesday, but not next Thursday.