1 00:00:00,000 --> 00:00:01,529 NICOLA MARZARI: Wonderful. 2 00:00:01,529 --> 00:00:02,029 OK. 3 00:00:02,029 --> 00:00:04,900 So welcome to class 5. 4 00:00:04,900 --> 00:00:09,520 We are actually counting the lab as class 4 in the numbering. 5 00:00:09,520 --> 00:00:13,330 What we'll see today is really a wrap-up 6 00:00:13,330 --> 00:00:16,810 on some of these ideas on the behavior of electrons 7 00:00:16,810 --> 00:00:18,520 as quantum particles. 8 00:00:18,520 --> 00:00:23,800 And we'll introduce really the first electronic structure 9 00:00:23,800 --> 00:00:25,840 method in the modern sense-- 10 00:00:25,840 --> 00:00:27,970 that is Hartree-Fock theory-- 11 00:00:27,970 --> 00:00:32,200 that you'll see as a very simple, conceptual framework. 12 00:00:32,200 --> 00:00:36,340 And it really derives directly from the Schrodinger equation 13 00:00:36,340 --> 00:00:38,980 and from the reformulation of the Schrodinger equation 14 00:00:38,980 --> 00:00:42,010 as a variational principle. 15 00:00:42,010 --> 00:00:44,800 Next class, we'll start with density functional theory. 16 00:00:44,800 --> 00:00:47,830 That is a completely different approach 17 00:00:47,830 --> 00:00:49,900 to the problem than Hartree-Fock. 18 00:00:49,900 --> 00:00:52,600 Although, in practice, it results 19 00:00:52,600 --> 00:00:55,900 in equations for the wave function of the electron 20 00:00:55,900 --> 00:00:58,660 that are formerly fairly simple. 21 00:00:58,660 --> 00:01:01,240 Historically, Hartree-Fock has been the method 22 00:01:01,240 --> 00:01:05,230 that has developed first, especially from the quantum 23 00:01:05,230 --> 00:01:06,580 chemistry community. 24 00:01:06,580 --> 00:01:09,790 And because it derives from the variational principle, 25 00:01:09,790 --> 00:01:12,920 it can actually be augmented with perturbation theory. 26 00:01:12,920 --> 00:01:16,090 So whenever you need very accurate calculations, 27 00:01:16,090 --> 00:01:18,520 things like molar [INAUDIBLE] approaches 28 00:01:18,520 --> 00:01:21,340 and other perturbation theory approaches 29 00:01:21,340 --> 00:01:24,730 on top of Hartree-Fock can actually help you out. 30 00:01:24,730 --> 00:01:28,010 But they tend to be very expensive. 31 00:01:28,010 --> 00:01:28,510 OK. 32 00:01:28,510 --> 00:01:33,320 Let me, in one slide, remind you of the most fundamental concept 33 00:01:33,320 --> 00:01:35,020 from last class that we have seen. 34 00:01:35,020 --> 00:01:37,300 That is the de Broglie relation. 35 00:01:37,300 --> 00:01:41,740 This is something that tells us that if a particle has 36 00:01:41,740 --> 00:01:45,460 a momentum mass times velocity p, well, 37 00:01:45,460 --> 00:01:48,250 then there is a fundamental constant in nature that is 38 00:01:48,250 --> 00:01:50,980 called the Planck constant-- that's written here-- 39 00:01:50,980 --> 00:01:53,470 such that this relation is satisfied 40 00:01:53,470 --> 00:01:57,530 where lambda is the wavelength of that particle. 41 00:01:57,530 --> 00:02:01,390 So basically, particles have wavelength properties. 42 00:02:01,390 --> 00:02:05,920 But the real issue is that only electrons are light enough 43 00:02:05,920 --> 00:02:10,570 to have wavelength comparable to the inter-atomic distance, 44 00:02:10,570 --> 00:02:13,402 so to give rise to diffraction phenomena. 45 00:02:13,402 --> 00:02:14,860 And so this is why we actually need 46 00:02:14,860 --> 00:02:18,340 to treat the electrons with the Schrodinger equation 47 00:02:18,340 --> 00:02:20,440 due to their wavelike properties. 48 00:02:20,440 --> 00:02:23,710 And in general, we don't really need to nuclei 49 00:02:23,710 --> 00:02:26,020 as a wave-like particle. 50 00:02:26,020 --> 00:02:27,980 So this was one thing. 51 00:02:27,980 --> 00:02:31,090 The other important thing that I'll repeat over and over again 52 00:02:31,090 --> 00:02:35,950 is that when we move from a particle-like description, 53 00:02:35,950 --> 00:02:37,990 where the only important quantity are 54 00:02:37,990 --> 00:02:42,190 the position and the momentum, to a wave-like quantity, well, 55 00:02:42,190 --> 00:02:45,520 we have a completely different computational content. 56 00:02:45,520 --> 00:02:47,920 That is, a wave function is actually 57 00:02:47,920 --> 00:02:51,850 defined by assigning the amplitude of the wave 58 00:02:51,850 --> 00:02:55,520 at every point in space and at every time. 59 00:02:55,520 --> 00:02:58,930 So in practice, it's a continuum field that in every, say, 60 00:02:58,930 --> 00:03:02,080 computational calculation needs actually 61 00:03:02,080 --> 00:03:07,300 to be discretized and approximated in different ways. 62 00:03:07,300 --> 00:03:11,890 And then I introduced what is the equivalent 63 00:03:11,890 --> 00:03:16,780 of the Newton's second law of dynamics for actually quantum 64 00:03:16,780 --> 00:03:17,350 objects. 65 00:03:17,350 --> 00:03:22,390 So classical particles evolve according to the law force 66 00:03:22,390 --> 00:03:24,840 equals mass times acceleration, where 67 00:03:24,840 --> 00:03:27,250 the acceleration is really the secondary derivative 68 00:03:27,250 --> 00:03:28,640 of the trajectory. 69 00:03:28,640 --> 00:03:33,040 Quantum particles obey a different fundamental law 70 00:03:33,040 --> 00:03:38,740 that's basically been discovered by Erwin Schrodinger in 1925. 71 00:03:38,740 --> 00:03:42,880 And the other major player where Werner Heisenberg, Paul Dirac, 72 00:03:42,880 --> 00:03:44,200 and many others. 73 00:03:44,200 --> 00:03:47,800 And I've written it here in the most general form 74 00:03:47,800 --> 00:03:52,270 for the case of a single electron, so a particle 75 00:03:52,270 --> 00:03:54,910 that is described by a wave function 76 00:03:54,910 --> 00:03:59,900 psi, a function of space and time. 77 00:04:08,130 --> 00:04:08,872 Very good. 78 00:04:08,872 --> 00:04:10,080 The machine [INAUDIBLE] beep. 79 00:04:38,670 --> 00:04:39,170 OK. 80 00:04:39,170 --> 00:04:42,680 Professor Ceder will take care of this. 81 00:04:42,680 --> 00:04:43,950 And let me continue. 82 00:04:43,950 --> 00:04:46,760 So we have the descriptor that is the wave function. 83 00:04:46,760 --> 00:04:48,911 And we have a potential. 84 00:04:48,911 --> 00:04:50,900 A potential is the general concept. 85 00:04:50,900 --> 00:04:54,980 It could be, say, the electromagnetic potential 86 00:04:54,980 --> 00:04:57,980 that is coming from the nucleus of an atom. 87 00:04:57,980 --> 00:05:00,920 So suppose that you are studying a hydrogen atom. 88 00:05:00,920 --> 00:05:04,070 Well, the nucleus has an attractive potential 89 00:05:04,070 --> 00:05:07,340 for the electron that is just 1 over the distance 90 00:05:07,340 --> 00:05:09,380 between the electron and the nucleus. 91 00:05:09,380 --> 00:05:12,530 So V here is really the same potential 92 00:05:12,530 --> 00:05:16,070 that you have learned in classical dynamics 93 00:05:16,070 --> 00:05:17,690 and electromagnetism. 94 00:05:17,690 --> 00:05:21,395 But now, instead of acting on a particle, 95 00:05:21,395 --> 00:05:23,520 it acts on the wave function. 96 00:05:23,520 --> 00:05:27,598 And the way it does, it just multiplies the wave function. 97 00:05:27,598 --> 00:05:29,390 So what the Schrodinger equation is telling 98 00:05:29,390 --> 00:05:32,420 me is that whenever I have a potential V, 99 00:05:32,420 --> 00:05:35,600 the solution for the dynamics of the electron 100 00:05:35,600 --> 00:05:38,720 will be given by the solution of this second-order differential 101 00:05:38,720 --> 00:05:41,570 equation, where here this is a Laplacian. 102 00:05:41,570 --> 00:05:44,630 We have a second derivative with respect to space. 103 00:05:44,630 --> 00:05:49,072 And so the wave function needs to satisfy this equation. 104 00:05:49,072 --> 00:05:53,600 In most cases, to actually deal with problems 105 00:05:53,600 --> 00:05:57,380 in which the potential does not depend on time-- 106 00:05:57,380 --> 00:05:59,900 suppose that you are studying a molecule. 107 00:05:59,900 --> 00:06:01,010 What is your potential? 108 00:06:01,010 --> 00:06:03,260 Think of, say, the benzene molecule. 109 00:06:03,260 --> 00:06:08,240 Well, you have six carbon atoms in a ring and six hydrogen 110 00:06:08,240 --> 00:06:10,040 atoms also in a ring. 111 00:06:10,040 --> 00:06:13,700 And in each nucleus, there is a Z/r 112 00:06:13,700 --> 00:06:16,730 that is an attractive potential for the electron. 113 00:06:16,730 --> 00:06:19,460 And that we can either think of at a potential 114 00:06:19,460 --> 00:06:21,950 that doesn't change in time-- 115 00:06:21,950 --> 00:06:25,160 or even if it changes, because the molecule is 116 00:06:25,160 --> 00:06:29,060 a certain temperature and so atoms move around, what 117 00:06:29,060 --> 00:06:30,920 it turns out will happen in practice 118 00:06:30,920 --> 00:06:35,180 is that electrons are so much lighter than nuclei 119 00:06:35,180 --> 00:06:38,660 that for all practical purposes they 120 00:06:38,660 --> 00:06:42,560 move so much faster that they see the nuclei basically 121 00:06:42,560 --> 00:06:44,760 as fixed in time. 122 00:06:44,760 --> 00:06:47,360 So if you think of a particle like a nucleus that 123 00:06:47,360 --> 00:06:50,150 has a huge mass and a particle like an electron that 124 00:06:50,150 --> 00:06:54,440 has a very light mass, well, even if the nuclei move, 125 00:06:54,440 --> 00:06:57,110 the electrons always are so much faster 126 00:06:57,110 --> 00:07:00,320 to rearrange themselves to be in the minimum energy 127 00:07:00,320 --> 00:07:01,280 configuration. 128 00:07:01,280 --> 00:07:03,560 We'll actually see more of what is 129 00:07:03,560 --> 00:07:06,770 called adiabatic approximation. 130 00:07:06,770 --> 00:07:08,930 So the important thing here is that, 131 00:07:08,930 --> 00:07:14,660 for all practical purposes, we consider in most problems 132 00:07:14,660 --> 00:07:18,320 the potential as not really dependent on time 133 00:07:18,320 --> 00:07:20,390 either because the atoms are not moving 134 00:07:20,390 --> 00:07:23,330 or because even if they move, they move so much more 135 00:07:23,330 --> 00:07:25,130 slowly than the electrons. 136 00:07:25,130 --> 00:07:27,902 Then the electrons can always [INAUDIBLE].. 137 00:07:31,780 --> 00:07:35,050 The form that the wave function can have. 138 00:07:35,050 --> 00:07:39,370 And I'll briefly sketch out this derivation here. 139 00:07:39,370 --> 00:07:43,210 So whenever the potential doesn't depend on time, 140 00:07:43,210 --> 00:07:45,010 this is how we proceed. 141 00:07:45,010 --> 00:07:47,620 We actually make an hypothesis-- often, 142 00:07:47,620 --> 00:07:49,270 this is called an ansatz-- 143 00:07:49,270 --> 00:07:51,820 in which our wave function can actually 144 00:07:51,820 --> 00:07:57,070 be separated into the product of two terms-- 145 00:07:57,070 --> 00:08:00,070 a term that depends only on space, 146 00:08:00,070 --> 00:08:06,010 we'll call it phi; and a term that depends only on time. 147 00:08:06,010 --> 00:08:08,600 So we are making this hypothesis. 148 00:08:08,600 --> 00:08:11,290 We have decided that the potential does not 149 00:08:11,290 --> 00:08:12,880 depend on time. 150 00:08:12,880 --> 00:08:15,970 And we see what [AUDIO OUT]. 151 00:08:20,240 --> 00:08:22,820 What we have now is that whenever 152 00:08:22,820 --> 00:08:25,190 we apply, say, the Laplacian secondary derivative 153 00:08:25,190 --> 00:08:28,970 with respect to space, well, [INAUDIBLE] 154 00:08:28,970 --> 00:08:33,000 the f component of the function, the time-dependent component. 155 00:08:33,000 --> 00:08:35,780 So the Laplacian of psi can be written actually 156 00:08:35,780 --> 00:08:40,460 as f times the Laplacian of phi. 157 00:08:40,460 --> 00:08:43,429 And then I'm not writing the explicit dependence 158 00:08:43,429 --> 00:08:44,540 on the variables. 159 00:08:44,540 --> 00:08:49,070 But we have the potential times the wave function, phi f. 160 00:08:49,070 --> 00:08:51,890 And then we have i h bar. 161 00:08:51,890 --> 00:08:54,840 And now, the derivative with respect to time of phi-- 162 00:08:54,840 --> 00:08:58,520 well, a phase-dependent part is not affected. 163 00:08:58,520 --> 00:09:01,640 So we write it as pih times df/dt. 164 00:09:04,080 --> 00:09:04,580 OK. 165 00:09:04,580 --> 00:09:06,480 So this is our first step. 166 00:09:06,480 --> 00:09:09,140 And now, we take this expression. 167 00:09:09,140 --> 00:09:15,890 And we just divide it by the product of phi times f. 168 00:09:15,890 --> 00:09:18,710 And so when I do that, what I obtain 169 00:09:18,710 --> 00:09:25,710 is minus h bar squared divided by 2m Laplacian of phi divided 170 00:09:25,710 --> 00:09:34,530 by phi plus V equal to i h bar 1/f df/dt. 171 00:09:37,500 --> 00:09:39,510 And this really is the end of the derivation 172 00:09:39,510 --> 00:09:42,990 because something very important happens at this point. 173 00:09:42,990 --> 00:09:45,810 Look for a moment at the right-hand side 174 00:09:45,810 --> 00:09:47,430 of this equation. 175 00:09:47,430 --> 00:09:50,400 You see it's just a constant-- the Planck constant, i, 176 00:09:50,400 --> 00:09:52,020 is the imaginary unit-- 177 00:09:52,020 --> 00:09:55,230 times 1/f df/dt. 178 00:09:55,230 --> 00:10:00,390 So everything that's written on the right-hand side here 179 00:10:00,390 --> 00:10:03,145 depends only on time. 180 00:10:09,510 --> 00:10:12,280 The left-hand side, on the other hand, 181 00:10:12,280 --> 00:10:16,170 we have a term that depends only on position. 182 00:10:16,170 --> 00:10:19,770 Phi is a function of position. 183 00:10:19,770 --> 00:10:22,860 And V-- remember, our initial hypothesis-- 184 00:10:22,860 --> 00:10:24,340 does not depend on time. 185 00:10:24,340 --> 00:10:26,410 The potential does not depend on time. 186 00:10:26,410 --> 00:10:30,390 So the left-hand side [AUDIO OUT].. 187 00:10:30,390 --> 00:10:34,620 Now, the only way such a relation can be satisfied-- 188 00:10:34,620 --> 00:10:38,760 that is a term dependent solely on position can be equal 189 00:10:38,760 --> 00:10:41,070 to a term that depends only on time-- 190 00:10:41,070 --> 00:10:45,250 is if each term separately. 191 00:10:45,250 --> 00:10:49,252 That's the only way the overall function of r 192 00:10:49,252 --> 00:10:52,600 can be [AUDIO OUT] overall function of t, 193 00:10:52,600 --> 00:10:53,770 if they are constant. 194 00:10:53,770 --> 00:10:56,750 Basically, because r and t are independent variables, 195 00:10:56,750 --> 00:10:58,240 there is no relation. 196 00:10:58,240 --> 00:11:05,990 If you want, in this ansatz [AUDIO OUT] down 197 00:11:05,990 --> 00:11:09,450 the Schrodinger equation in two parts-- 198 00:11:09,450 --> 00:11:12,620 the left part being equal to a constant, 199 00:11:12,620 --> 00:11:15,730 and the right part being equal to a constant. 200 00:11:15,730 --> 00:11:17,660 So there is a simplification. 201 00:11:17,660 --> 00:11:20,420 We can deal with separate equations 202 00:11:20,420 --> 00:11:25,400 for the time-dependent part and for the space-dependent part. 203 00:11:25,400 --> 00:11:29,930 What is the left-hand term equal to a constant becomes 204 00:11:29,930 --> 00:11:33,122 what is called the stationary Schrodinger equation. 205 00:11:33,122 --> 00:11:34,580 That is the equation that we'll try 206 00:11:34,580 --> 00:11:40,950 to solve over and over again in our calculation. 207 00:11:40,950 --> 00:11:43,930 And it's the term that I've written here. 208 00:11:43,930 --> 00:11:47,370 So the stationery Schrodinger equation is written here. 209 00:11:47,370 --> 00:11:49,170 And this is the differential equation 210 00:11:49,170 --> 00:11:52,290 that, in principle, is complex to solve 211 00:11:52,290 --> 00:11:55,990 because the potential can have an arbitrary complex shape. 212 00:11:55,990 --> 00:11:57,720 But again, think of the potential 213 00:11:57,720 --> 00:11:59,490 as being the attractive potential 214 00:11:59,490 --> 00:12:04,200 for the electrons generated by all the nuclei in a molecule 215 00:12:04,200 --> 00:12:05,820 or in a solid. 216 00:12:05,820 --> 00:12:12,850 And the time [AUDIO OUT] has become trivial to integrate. 217 00:12:12,850 --> 00:12:17,620 And we see this is the time part of the Schrodinger equation. 218 00:12:17,620 --> 00:12:19,920 We see, actually, in the next slide 219 00:12:19,920 --> 00:12:22,050 why this is simple because this is just 220 00:12:22,050 --> 00:12:24,300 a first-order differential equation. 221 00:12:24,300 --> 00:12:27,210 And we are looking, in general, for a function 222 00:12:27,210 --> 00:12:30,390 whose first derivative with respect to time 223 00:12:30,390 --> 00:12:34,145 is equal to the function itself times a constant. 224 00:12:34,145 --> 00:12:36,550 [AUDIO OUT] equation to integrate 225 00:12:36,550 --> 00:12:40,720 because the exponential function satisfies this condition. 226 00:12:40,720 --> 00:12:42,700 When you take the exponential, you just 227 00:12:42,700 --> 00:12:45,340 get the exponential itself times a constant. 228 00:12:45,340 --> 00:12:52,250 And so really, there is no analysis [AUDIO OUT] 229 00:12:52,250 --> 00:12:55,520 The other part, the space-dependent part 230 00:12:55,520 --> 00:12:57,830 is the one that is complex to solve. 231 00:12:57,830 --> 00:13:04,346 And we'll see here just a few examples [AUDIO OUT] 232 00:13:04,346 --> 00:13:07,940 If we want the simplest example possible, 233 00:13:07,940 --> 00:13:10,160 that is the case of a free particle. 234 00:13:10,160 --> 00:13:15,090 What happens to an electron that doesn't feel any potential? 235 00:13:15,090 --> 00:13:17,870 And again, its wave function is going 236 00:13:17,870 --> 00:13:21,770 to be the product of a space-dependent part phi 237 00:13:21,770 --> 00:13:24,380 times a time-dependent part f. 238 00:13:24,380 --> 00:13:26,855 The f term [AUDIO OUT]. 239 00:13:29,696 --> 00:13:35,680 The space-dependent term here is the solution 240 00:13:35,680 --> 00:13:39,400 of the stationary Schrodinger equation in the hypothesis 241 00:13:39,400 --> 00:13:42,520 that the potential V is equal to 0. 242 00:13:42,520 --> 00:13:46,021 So I've removed from the stationary [AUDIO OUT].. 243 00:13:49,430 --> 00:13:53,060 Again, this is a fairly simple differential equation 244 00:13:53,060 --> 00:13:57,410 to integrate because what we are looking for is basically 245 00:13:57,410 --> 00:14:00,920 a function whose secondary derivative is 246 00:14:00,920 --> 00:14:04,520 equal to the function itself times a constant. 247 00:14:04,520 --> 00:14:09,410 And then again, the solution is just given by an exponential, 248 00:14:09,410 --> 00:14:10,670 OK? 249 00:14:10,670 --> 00:14:15,740 Now, I'm looking in particular at the problem in which I'm 250 00:14:15,740 --> 00:14:21,350 trying to find a solution phi for a value of this constant E 251 00:14:21,350 --> 00:14:23,040 that is positive. 252 00:14:23,040 --> 00:14:25,430 And so when you actually work out the algebra, 253 00:14:25,430 --> 00:14:29,180 you see that what you need as a term in front 254 00:14:29,180 --> 00:14:33,680 of the x in the exponential is the square root 255 00:14:33,680 --> 00:14:35,660 of a negative number, is the square root 256 00:14:35,660 --> 00:14:40,910 of twice the mass times E divided by h bar squared. 257 00:14:40,910 --> 00:14:46,010 And so that's why we get this imaginary unit here. 258 00:14:46,010 --> 00:14:49,640 It's basically because we have rewritten this equation 259 00:14:49,640 --> 00:14:58,220 as the square Laplacian of phi equal to minus 2mE divided 260 00:14:58,220 --> 00:15:01,670 by h bar squared f. 261 00:15:01,670 --> 00:15:04,790 And you see straightforwardly that if you 262 00:15:04,790 --> 00:15:09,020 take the secondary whatever of the functions that 263 00:15:09,020 --> 00:15:11,360 are written here, well, we'll get, 264 00:15:11,360 --> 00:15:13,250 by taking the first derivative, a term 265 00:15:13,250 --> 00:15:16,245 i square root of 20 divided by h bar. 266 00:15:16,245 --> 00:15:17,870 And when we take the second derivative, 267 00:15:17,870 --> 00:15:19,460 we get that term again. 268 00:15:19,460 --> 00:15:23,120 And the square of that term is a square of i 269 00:15:23,120 --> 00:15:26,500 that gives you my minus sign. 270 00:15:26,500 --> 00:15:27,200 OK. 271 00:15:27,200 --> 00:15:28,370 So this is the solution. 272 00:15:28,370 --> 00:15:33,560 We have actually found the wave function for our free particle. 273 00:15:33,560 --> 00:15:35,940 And the wave function is the product 274 00:15:35,940 --> 00:15:40,460 of [AUDIO OUT] times the time-dependent part. 275 00:15:40,460 --> 00:15:43,490 And in the next slide, I am actually 276 00:15:43,490 --> 00:15:50,160 plotting this wave function in space. 277 00:15:50,160 --> 00:15:54,400 Sorry, let me jump to this slide. 278 00:15:54,400 --> 00:15:55,170 This is it. 279 00:15:55,170 --> 00:15:57,480 So I am plotting it. 280 00:15:57,480 --> 00:16:00,150 For simplicity, I've called it a different coefficient. 281 00:16:00,150 --> 00:16:03,090 E over h bar, I've called it omega. 282 00:16:03,090 --> 00:16:07,890 And square root of 2mE divided by h bar, I've called it t. 283 00:16:07,890 --> 00:16:12,450 So this is the wave function for a free electron. 284 00:16:12,450 --> 00:16:17,140 And this is what we call a plane wave. 285 00:16:17,140 --> 00:16:19,920 And the reason why is that is simple 286 00:16:19,920 --> 00:16:23,460 because if you think of what this term is, 287 00:16:23,460 --> 00:16:26,070 the exponential of a complex number 288 00:16:26,070 --> 00:16:31,350 is really a combination of sines and cosines. 289 00:16:31,350 --> 00:16:36,330 And now, suppose that you look at the amplitude of this wave 290 00:16:36,330 --> 00:16:40,080 function at a certain point in space-- 291 00:16:40,080 --> 00:16:44,460 that is at a certain x equal to x0. 292 00:16:44,460 --> 00:16:47,640 What you see is that that amplitude 293 00:16:47,640 --> 00:16:51,000 will have a kx factor that doesn't change because we 294 00:16:51,000 --> 00:16:53,850 are sitting at a certain x0. 295 00:16:53,850 --> 00:16:56,550 But what is changing with respect to time 296 00:16:56,550 --> 00:16:59,100 is the second term, omega t. 297 00:16:59,100 --> 00:17:04,900 And so you see an oscillating amplitude with a period 298 00:17:04,900 --> 00:17:08,130 omega at any point in time. 299 00:17:08,130 --> 00:17:11,430 Or in a different way, if for a moment 300 00:17:11,430 --> 00:17:15,660 you think that what is this field of the wave function 301 00:17:15,660 --> 00:17:18,450 at a frozen instant in time-- 302 00:17:18,450 --> 00:17:22,020 that is for a certain time t equal to t0-- 303 00:17:22,020 --> 00:17:25,690 well, it's nothing else than a complex exponential. 304 00:17:25,690 --> 00:17:28,680 So it's a complex-valued function 305 00:17:28,680 --> 00:17:32,010 that has a cosine or sine projection, depending 306 00:17:32,010 --> 00:17:35,650 if we look at the real or complex axes. 307 00:17:35,650 --> 00:17:37,170 So what you need to think of when 308 00:17:37,170 --> 00:17:39,540 you think that at a free electron 309 00:17:39,540 --> 00:17:43,950 is nothing else than this infinite oscillatory 310 00:17:43,950 --> 00:17:47,550 field propagating in space. 311 00:17:47,550 --> 00:17:51,360 But it has really no beginning and no end. 312 00:17:51,360 --> 00:17:56,190 And one of the important things is that the wave function 313 00:17:56,190 --> 00:18:00,480 really contains all the information that we might 314 00:18:00,480 --> 00:18:03,780 want to know about an electron. 315 00:18:03,780 --> 00:18:06,210 And we will see in the next slides 316 00:18:06,210 --> 00:18:08,610 how we actually obtain the physical properties 317 00:18:08,610 --> 00:18:11,130 about an electron from the function. 318 00:18:11,130 --> 00:18:17,300 But the simplest property in [AUDIO OUT] 319 00:18:17,300 --> 00:18:20,660 of finding an electron in a certain position 320 00:18:20,660 --> 00:18:22,440 and a certain time. 321 00:18:22,440 --> 00:18:27,470 And so [INAUDIBLE] that wave function amplitude 322 00:18:27,470 --> 00:18:30,700 at a certain x and a certain t gives me 323 00:18:30,700 --> 00:18:32,900 actually the physical probability 324 00:18:32,900 --> 00:18:34,850 of finding an electron. 325 00:18:34,850 --> 00:18:38,720 That means that if I have the wave function of an electron, 326 00:18:38,720 --> 00:18:41,720 I can take at every point in space 327 00:18:41,720 --> 00:18:44,280 the square models of its amplitude. 328 00:18:44,280 --> 00:18:48,080 And so what I have now is a probability distribution. 329 00:18:48,080 --> 00:18:50,180 And what quantum mechanics tells me 330 00:18:50,180 --> 00:18:52,610 is that if I'm going to make a measurement 331 00:18:52,610 --> 00:18:55,700 and try to see where the electron is, 332 00:18:55,700 --> 00:18:58,190 well, I will find it in different points 333 00:18:58,190 --> 00:19:02,520 in space with a probability [AUDIO OUT] 334 00:19:02,520 --> 00:19:05,370 into these square models. 335 00:19:05,370 --> 00:19:09,480 And you see one of the first conclusions from our analysis 336 00:19:09,480 --> 00:19:13,170 of the Schrodinger equation in the case of a potential that 337 00:19:13,170 --> 00:19:18,000 does not depend on time is a Schrodinger equation in which 338 00:19:18,000 --> 00:19:20,670 the wave function can actually be written 339 00:19:20,670 --> 00:19:24,480 as the product of a space part times a time part 340 00:19:24,480 --> 00:19:29,010 is that the probability of finding the electron somewhere 341 00:19:29,010 --> 00:19:31,470 does not depend on time basically 342 00:19:31,470 --> 00:19:33,630 because [INAUDIBLE] square models 343 00:19:33,630 --> 00:19:41,330 of psi that can also be written as psi 344 00:19:41,330 --> 00:19:44,570 times its complex conjugate. 345 00:19:44,570 --> 00:19:49,310 What you have is that the term exponential of minus i/h 346 00:19:49,310 --> 00:19:51,560 gets multiplied by its complex conjugate. 347 00:19:51,560 --> 00:19:54,080 That is the exponential of plus i/h. 348 00:19:54,080 --> 00:19:56,550 And those terms cancel out. 349 00:19:56,550 --> 00:20:00,470 So in the probability distribution for the electron, 350 00:20:00,470 --> 00:20:04,490 we have only the square models of the space part. 351 00:20:04,490 --> 00:20:07,070 That is to say that the probability of finding 352 00:20:07,070 --> 00:20:10,670 an electron does not depend on time. 353 00:20:10,670 --> 00:20:14,570 It's just a function of space if the potential does not 354 00:20:14,570 --> 00:20:17,660 depend on time. 355 00:20:17,660 --> 00:20:21,440 You see, [INAUDIBLE] our plane wave. 356 00:20:21,440 --> 00:20:24,330 The potential does not depend on time. 357 00:20:24,330 --> 00:20:28,790 If we take the square models of this function, 358 00:20:28,790 --> 00:20:32,900 well, what we have is really a constant 359 00:20:32,900 --> 00:20:36,740 because we are just taking an imaginary exponential 360 00:20:36,740 --> 00:20:39,150 times its complex conjugate. 361 00:20:39,150 --> 00:20:41,610 And so that product is a constant. 362 00:20:41,610 --> 00:20:45,380 So what we discover really is that the electron 363 00:20:45,380 --> 00:20:50,630 is free as a probability distribution that is constant. 364 00:20:50,630 --> 00:20:54,770 That is, you can find a free electron anywhere in space. 365 00:20:54,770 --> 00:20:55,688 Yes? 366 00:20:55,688 --> 00:20:59,680 AUDIENCE: [INAUDIBLE]. 367 00:20:59,680 --> 00:21:01,930 NICOLA MARZARI: So this is actually the solution 368 00:21:01,930 --> 00:21:03,820 in three dimensions. 369 00:21:03,820 --> 00:21:09,580 So what the difference is, if you work out the algebra-- 370 00:21:09,580 --> 00:21:12,190 and we can look at it after class-- 371 00:21:12,190 --> 00:21:15,160 is that what you have is-- 372 00:21:15,160 --> 00:21:21,250 in the space term, you have really a scholar product 373 00:21:21,250 --> 00:21:22,840 in three dimensions. 374 00:21:22,840 --> 00:21:26,980 That is, what you have is that now 375 00:21:26,980 --> 00:21:29,860 a point in your three-dimensional space 376 00:21:29,860 --> 00:21:33,520 is described by a three-dimensional vector r. 377 00:21:33,520 --> 00:21:39,010 And now, you can have solutions that are classified-- 378 00:21:39,010 --> 00:21:43,990 if you want, this k is actually called the quantum number. 379 00:21:43,990 --> 00:21:47,680 It's what classifies all this possible continuum 380 00:21:47,680 --> 00:21:49,280 of solutions. 381 00:21:49,280 --> 00:21:52,690 And if you think, k is telling you 382 00:21:52,690 --> 00:21:55,630 what the wavelength of this plane wave is 383 00:21:55,630 --> 00:21:58,100 and what is its orientation. 384 00:21:58,100 --> 00:21:59,950 So really, what I am plotting here 385 00:21:59,950 --> 00:22:02,980 is specific plane waves that, if you want, 386 00:22:02,980 --> 00:22:07,840 is a k vector that is ordered in this direction. 387 00:22:07,840 --> 00:22:12,040 So if we were in one dimension, you have, if you want, 388 00:22:12,040 --> 00:22:14,410 the equivalent of a sine or a cosine. 389 00:22:14,410 --> 00:22:18,310 And your only degree of freedom is the wavelength. 390 00:22:18,310 --> 00:22:19,960 But if you are in three dimensions, 391 00:22:19,960 --> 00:22:22,600 your degrees of freedom are not only the wavelength-- 392 00:22:22,600 --> 00:22:24,610 that is the distance between these crests-- 393 00:22:24,610 --> 00:22:26,530 but also this direction. 394 00:22:26,530 --> 00:22:32,660 And any arbitrary k vector is actually valid. 395 00:22:32,660 --> 00:22:37,270 So you can have plane waves with all possible wavelengths 396 00:22:37,270 --> 00:22:40,360 and all possible orientations. 397 00:22:40,360 --> 00:22:42,910 And actually, this gives me a good chance 398 00:22:42,910 --> 00:22:47,530 to mention another sort of physical observable 399 00:22:47,530 --> 00:22:49,720 that comes from the wave function. 400 00:22:49,720 --> 00:22:52,690 The first one that we have seen is the probability 401 00:22:52,690 --> 00:22:53,920 distribution. 402 00:22:53,920 --> 00:22:57,580 And that's obtained by taking the square models. 403 00:22:57,580 --> 00:23:01,810 Another one is what is called the quantum kinetic energy 404 00:23:01,810 --> 00:23:03,260 of that particle. 405 00:23:03,260 --> 00:23:05,980 And the quantum kinetic energy is really just 406 00:23:05,980 --> 00:23:12,730 given by minus h bar squared divided by 2m. 407 00:23:12,730 --> 00:23:16,700 So the quantum kinetic energy is given-- well, 408 00:23:16,700 --> 00:23:20,420 I'll describe it in more detail in a moment. 409 00:23:20,420 --> 00:23:24,270 But it is obtained by really taking, 410 00:23:24,270 --> 00:23:28,000 in a slightly more complex way than what I am saying now, 411 00:23:28,000 --> 00:23:31,850 the second derivative of the wave function. 412 00:23:31,850 --> 00:23:33,770 So the second derivative here with respect 413 00:23:33,770 --> 00:23:35,740 to the space of this wave function 414 00:23:35,740 --> 00:23:39,070 would give me a k square term. 415 00:23:39,070 --> 00:23:43,250 That basically means that a plane wave with all wavelengths 416 00:23:43,250 --> 00:23:48,310 corresponds to electrons with all possible energies. 417 00:23:48,310 --> 00:23:50,050 But there is something other that 418 00:23:50,050 --> 00:23:53,290 is very subtle that is reminiscent of what 419 00:23:53,290 --> 00:23:57,220 in quantum mechanics people call the indetermination principle. 420 00:23:57,220 --> 00:24:00,220 So this free electron, as you see, 421 00:24:00,220 --> 00:24:05,390 is an electron that is distributed equally in space. 422 00:24:05,390 --> 00:24:07,420 So we can find it with equal probability 423 00:24:07,420 --> 00:24:10,000 here, here, or everywhere else. 424 00:24:10,000 --> 00:24:15,543 And it is actually a perfectly well-defined second derivative 425 00:24:15,543 --> 00:24:16,460 with respect to space. 426 00:24:16,460 --> 00:24:19,690 It has a perfectly well-defined kinetic energy, 427 00:24:19,690 --> 00:24:22,720 has a perfectly well-defined momentum. 428 00:24:22,720 --> 00:24:27,100 And that's really one example of the indetermination principle. 429 00:24:27,100 --> 00:24:29,170 That is, there is a fundamental rule 430 00:24:29,170 --> 00:24:31,630 that quantum particles satisfy. 431 00:24:31,630 --> 00:24:34,480 That is, you can't actually measure 432 00:24:34,480 --> 00:24:37,150 their momentum and their position 433 00:24:37,150 --> 00:24:41,590 simultaneously with arbitrary accuracy. 434 00:24:41,590 --> 00:24:44,050 And so a free electron, if you want, 435 00:24:44,050 --> 00:24:47,270 is an electron that has perfectly well-defined 436 00:24:47,270 --> 00:24:50,260 momentum, perfectly well-defined kinetic energy. 437 00:24:50,260 --> 00:24:55,150 And for this reason, it has perfectly undefined position. 438 00:24:55,150 --> 00:24:58,960 That is, the electron can be found anywhere. 439 00:24:58,960 --> 00:25:04,610 And so this is, if you want, an exotic result 440 00:25:04,610 --> 00:25:11,920 coming from the Heisenberg indetermination principle. 441 00:25:11,920 --> 00:25:17,050 I wanted to give you another physical example, 442 00:25:17,050 --> 00:25:20,530 getting closer to reality. 443 00:25:20,530 --> 00:25:24,710 Suppose that, for a moment, we are studying a metal surface. 444 00:25:24,710 --> 00:25:27,520 This wouldn't be very different from what you have studied 445 00:25:27,520 --> 00:25:29,020 in your first laboratory. 446 00:25:29,020 --> 00:25:31,720 What you are studying is basically a slab. 447 00:25:31,720 --> 00:25:33,430 And you have been finding quantities 448 00:25:33,430 --> 00:25:36,400 like what is the energy, what is the lattice parameter, what 449 00:25:36,400 --> 00:25:38,090 is the surface energy. 450 00:25:38,090 --> 00:25:42,140 Suppose that now you are using quantum mechanics. 451 00:25:42,140 --> 00:25:47,510 Well, what do the electrons in the metal look like? 452 00:25:47,510 --> 00:25:55,540 And we'll see in next classes that an electron inside a metal 453 00:25:55,540 --> 00:26:01,170 really feels and behaves very much like a free particle, 454 00:26:01,170 --> 00:26:03,840 especially in very simple metals, in metals 455 00:26:03,840 --> 00:26:08,620 like sodium, potassium, even aluminum to a certain point. 456 00:26:08,620 --> 00:26:11,130 There is a very delicate cancellation 457 00:26:11,130 --> 00:26:14,820 of attractive and repulsive terms 458 00:26:14,820 --> 00:26:16,710 for the electron in the metal. 459 00:26:16,710 --> 00:26:19,950 So the net result, to a very good approximation, 460 00:26:19,950 --> 00:26:22,560 is that the high energy electrons, 461 00:26:22,560 --> 00:26:26,740 the valence electrons, really feel almost free. 462 00:26:26,740 --> 00:26:29,790 I'm saying almost free because they are really 463 00:26:29,790 --> 00:26:34,920 confined inside the box that is your metal 464 00:26:34,920 --> 00:26:39,300 crystal inside the box that is given by your metal slab. 465 00:26:39,300 --> 00:26:42,900 And so to have a simple but very good approximation, 466 00:26:42,900 --> 00:26:49,110 we can think of the potential felt by an electron in a metal 467 00:26:49,110 --> 00:26:51,330 as a type of potential. 468 00:26:51,330 --> 00:26:57,420 I'm thinking of a case in which, on this side, on the left, 469 00:26:57,420 --> 00:27:00,180 I have really my semi-infinite slab. 470 00:27:00,180 --> 00:27:03,930 And so the potential that is drawn by this thick line 471 00:27:03,930 --> 00:27:08,550 is actually 0 for x smaller than 0. 472 00:27:08,550 --> 00:27:11,010 And then there is a confinement. 473 00:27:11,010 --> 00:27:12,780 So there is a step. 474 00:27:12,780 --> 00:27:14,970 This is a little bit like a well. 475 00:27:14,970 --> 00:27:18,360 There is a potential difference v0 476 00:27:18,360 --> 00:27:22,270 that contains the electron inside the slab. 477 00:27:22,270 --> 00:27:25,170 So you really need to pay an energy v0 478 00:27:25,170 --> 00:27:27,190 to get out of the slab. 479 00:27:27,190 --> 00:27:29,370 But this is in physical terms what people would 480 00:27:29,370 --> 00:27:31,080 think of as the work function. 481 00:27:31,080 --> 00:27:33,540 The work function is really the energy 482 00:27:33,540 --> 00:27:37,380 that you need to pay to pick up the highest energy 483 00:27:37,380 --> 00:27:40,920 electron in the metal and bring it out, bring it very far 484 00:27:40,920 --> 00:27:43,020 away from your slab. 485 00:27:43,020 --> 00:27:46,170 And so to a good approximation, the potential 486 00:27:46,170 --> 00:27:50,420 that these electrons close, as we say, 487 00:27:50,420 --> 00:27:54,620 to the Fermi energy feel is this. 488 00:27:54,620 --> 00:27:57,140 And what we are trying to find now 489 00:27:57,140 --> 00:28:01,280 is a solution for the Schrodinger equation 490 00:28:01,280 --> 00:28:04,970 for a case of an electron having an energy. 491 00:28:04,970 --> 00:28:09,320 Actually, the E, that constant in the Schrodinger equation, 492 00:28:09,320 --> 00:28:12,750 is actually the total energy of the particle. 493 00:28:12,750 --> 00:28:15,860 So we are looking for a solution for an electron having 494 00:28:15,860 --> 00:28:19,940 an energy that is basically between 0 and v0. 495 00:28:19,940 --> 00:28:22,580 That means that that electron classically 496 00:28:22,580 --> 00:28:27,630 is really confined to the left side of the potential. 497 00:28:27,630 --> 00:28:29,590 And so what do we have here? 498 00:28:29,590 --> 00:28:34,990 Well, what we have is that, on the left, we have again, 499 00:28:34,990 --> 00:28:37,350 a very simple Schrodinger equation 500 00:28:37,350 --> 00:28:42,360 that is minus h bar squared over 2m d 501 00:28:42,360 --> 00:28:49,170 squared over phi equal to E times phi. 502 00:28:49,170 --> 00:28:52,350 So this is, on the left side, the Schrodinger equation 503 00:28:52,350 --> 00:28:54,760 for the free particle. 504 00:28:54,760 --> 00:28:56,550 And we know the solution. 505 00:28:56,550 --> 00:29:00,780 And it's just being given by one of these plane waves. 506 00:29:00,780 --> 00:29:04,290 But now, the difference is that, on the right, 507 00:29:04,290 --> 00:29:07,330 we have all of a sudden a potential. 508 00:29:07,330 --> 00:29:09,990 So the equation on the right is going 509 00:29:09,990 --> 00:29:15,420 to be minus h bar squared over 2m times delta squared phi. 510 00:29:15,420 --> 00:29:18,420 Remember now, in the stationary Schrodinger equation, 511 00:29:18,420 --> 00:29:21,810 we have a term that is given by the potential that, 512 00:29:21,810 --> 00:29:26,380 in this case, is just a constant times the wave function. 513 00:29:26,380 --> 00:29:30,846 And this is going to be equal to d phi. 514 00:29:30,846 --> 00:29:34,830 What's the difference between these two equations? 515 00:29:34,830 --> 00:29:38,620 Well, let me actually rewrite the one on the right side 516 00:29:38,620 --> 00:29:43,530 as minus h bar squared over 2m delta 517 00:29:43,530 --> 00:29:50,550 squared phi equal to E minus v0 phi. 518 00:29:50,550 --> 00:29:52,590 Well, in both cases, we are trying 519 00:29:52,590 --> 00:29:56,970 to find a wave function whose second derivative 520 00:29:56,970 --> 00:30:00,300 is basically equal to the wave function itself 521 00:30:00,300 --> 00:30:02,310 times a constant. 522 00:30:02,310 --> 00:30:04,590 But the big difference is that we are actually 523 00:30:04,590 --> 00:30:07,950 looking for physical solutions for an electron that 524 00:30:07,950 --> 00:30:12,460 has an energy lying somewhere in between 0-- 525 00:30:12,460 --> 00:30:14,190 that is our conventional minimum-- 526 00:30:14,190 --> 00:30:15,210 and v0. 527 00:30:15,210 --> 00:30:18,610 That is the energy cost to get that electron out. 528 00:30:18,610 --> 00:30:24,030 So what we have is that, for the left side equation, what 529 00:30:24,030 --> 00:30:27,720 we are trying to find is a second derivative 530 00:30:27,720 --> 00:30:29,700 in which, you see, the constant is 531 00:30:29,700 --> 00:30:33,420 going to be minus 2mE over h bar squared. 532 00:30:33,420 --> 00:30:34,870 So it's a negative number. 533 00:30:34,870 --> 00:30:36,930 So we need to take a second derivative 534 00:30:36,930 --> 00:30:40,740 and get a function times a negative number. 535 00:30:40,740 --> 00:30:43,860 In the other case, remember, what we are looking 536 00:30:43,860 --> 00:30:49,170 is for a solution in which the energy is between 0 and V0. 537 00:30:49,170 --> 00:30:54,390 So this the sign of the coefficient here has changed. 538 00:30:54,390 --> 00:30:56,500 Here, before, it was positive. 539 00:30:56,500 --> 00:30:57,700 Now, it's negative. 540 00:30:57,700 --> 00:31:00,790 And with the negative sign on the other side, 541 00:31:00,790 --> 00:31:02,910 it means that what we are looking 542 00:31:02,910 --> 00:31:06,240 is for a function whose secondary what 543 00:31:06,240 --> 00:31:10,020 is equal to a positive constant times the function itself. 544 00:31:10,020 --> 00:31:11,700 And all this long story is just to say 545 00:31:11,700 --> 00:31:14,460 that the solution on the left will 546 00:31:14,460 --> 00:31:19,920 be given by an exponential that has an i term in there. 547 00:31:19,920 --> 00:31:22,080 So it's a complex exponential. 548 00:31:22,080 --> 00:31:24,480 Well, the solution on the right-hand side 549 00:31:24,480 --> 00:31:27,390 will be given by an exponential that 550 00:31:27,390 --> 00:31:30,090 has a real term coefficient. 551 00:31:30,090 --> 00:31:34,140 And so what we have is on the left side 552 00:31:34,140 --> 00:31:38,800 a periodic oscillatory solution typical of the free electron. 553 00:31:38,800 --> 00:31:41,310 And what we have on the right side 554 00:31:41,310 --> 00:31:44,730 is actually a decaying exponential. 555 00:31:44,730 --> 00:31:47,910 And so what we have is that our free electron is 556 00:31:47,910 --> 00:31:51,930 going to have a wave function that really looks like this-- 557 00:31:51,930 --> 00:31:57,150 free particle like inside the oscillatory and exponentially 558 00:31:57,150 --> 00:31:59,820 decaying outside the surface. 559 00:31:59,820 --> 00:32:04,020 And this is how truly electrons in a metal-- 560 00:32:04,020 --> 00:32:07,770 and in most metals, it looks like the wave function 561 00:32:07,770 --> 00:32:09,690 decays exponentially. 562 00:32:09,690 --> 00:32:12,360 That's actually why people, say in 1981, 563 00:32:12,360 --> 00:32:16,200 could invent something as fancy as the scanning tunneling 564 00:32:16,200 --> 00:32:16,920 microscope. 565 00:32:16,920 --> 00:32:20,400 You see, the STM is one of the fanciest instruments 566 00:32:20,400 --> 00:32:24,910 that we have that can actually be used to see atoms. 567 00:32:24,910 --> 00:32:27,120 And what does an STM do? 568 00:32:27,120 --> 00:32:32,970 Well, an STM comes with really a nanometer-sized tip. 569 00:32:32,970 --> 00:32:39,100 And what the STM measures is the current between, say, 570 00:32:39,100 --> 00:32:41,130 the metal and the tip. 571 00:32:41,130 --> 00:32:43,020 And it's very sensitive because, you 572 00:32:43,020 --> 00:32:47,220 see, since the wave function decays exponentially, 573 00:32:47,220 --> 00:32:52,230 moving the tip by a small amount will change exponentially 574 00:32:52,230 --> 00:32:55,950 the current going through the scanning tunneling microscope. 575 00:32:55,950 --> 00:32:57,990 So we can actually move this paper 576 00:32:57,990 --> 00:33:00,870 and have an enormous response in the current 577 00:33:00,870 --> 00:33:02,740 to a very small displacement. 578 00:33:02,740 --> 00:33:05,700 So we can actually measure the topography 579 00:33:05,700 --> 00:33:09,120 on the top of a metal surface with great accuracy 580 00:33:09,120 --> 00:33:11,070 because basically what the scanning tunneling 581 00:33:11,070 --> 00:33:14,220 microscopy is doing is really filling up 582 00:33:14,220 --> 00:33:17,850 these exponentially decaying tails coming out 583 00:33:17,850 --> 00:33:22,230 from the metal, coming out from the metal surface. 584 00:33:22,230 --> 00:33:22,740 OK. 585 00:33:22,740 --> 00:33:24,750 So I'll show you just a few examples so 586 00:33:24,750 --> 00:33:28,380 that we get more and more the feeling of how 587 00:33:28,380 --> 00:33:32,250 wave functions look, getting closer and closer 588 00:33:32,250 --> 00:33:34,470 to physical problems. 589 00:33:34,470 --> 00:33:36,330 Keep in mind this general idea-- 590 00:33:36,330 --> 00:33:40,020 if we have got zero potential, we 591 00:33:40,020 --> 00:33:42,190 have this wave-like solution. 592 00:33:42,190 --> 00:33:45,420 So a typical problem that is very simple to actually solve 593 00:33:45,420 --> 00:33:48,330 analytically would be the one in which we 594 00:33:48,330 --> 00:33:50,160 have an infinite square well. 595 00:33:50,160 --> 00:33:52,800 That is, we have a particle that feels 596 00:33:52,800 --> 00:33:57,270 zero potential between, say, minus a and a 597 00:33:57,270 --> 00:34:00,420 and infinite potential outside. 598 00:34:00,420 --> 00:34:04,770 And this qualitatively starts to be very similar to what 599 00:34:04,770 --> 00:34:06,825 an electron in an atom feels. 600 00:34:06,825 --> 00:34:08,670 And electron in an atom doesn't really 601 00:34:08,670 --> 00:34:12,360 feel a potential that is 0 in a region and infinite outside. 602 00:34:12,360 --> 00:34:16,440 It really feels an attractive potential that is 1/r. 603 00:34:16,440 --> 00:34:19,770 So the potential is actually shaped very differently. 604 00:34:19,770 --> 00:34:23,760 It's going to be a Coulombic cusp ending up in 0. 605 00:34:23,760 --> 00:34:26,070 But you see, the solutions are very similar. 606 00:34:26,070 --> 00:34:27,840 And we can understand them better 607 00:34:27,840 --> 00:34:31,739 if we actually look at the solution for this case. 608 00:34:31,739 --> 00:34:35,070 And again, what we have is a problem in which, 609 00:34:35,070 --> 00:34:39,370 between minus a and a, we have 0 potential. 610 00:34:39,370 --> 00:34:45,179 So the solutions are going to be free electron-like, wave-like. 611 00:34:45,179 --> 00:34:48,389 But now, we have a new condition. 612 00:34:48,389 --> 00:34:51,825 That is, what we have is that being the potential 613 00:34:51,825 --> 00:34:56,350 is infinite, say from this point onwards on the right 614 00:34:56,350 --> 00:34:59,410 and from this point downwards on the left, 615 00:34:59,410 --> 00:35:02,040 what we need to have is that the wave 616 00:35:02,040 --> 00:35:06,990 function needs to have 0 amplitude outside this region. 617 00:35:06,990 --> 00:35:08,460 If the potential is infinite, you 618 00:35:08,460 --> 00:35:13,140 can think physically that there is 0 probability of finding 619 00:35:13,140 --> 00:35:13,950 the electron. 620 00:35:13,950 --> 00:35:14,805 It's truly infinite. 621 00:35:14,805 --> 00:35:17,710 So it's fully constraining the electron. 622 00:35:17,710 --> 00:35:20,010 So the wave function needs to be 0 623 00:35:20,010 --> 00:35:23,220 from here onwards and from here onwards. 624 00:35:23,220 --> 00:35:27,390 And so another great discovery of quantum mechanics 625 00:35:27,390 --> 00:35:32,550 is that the only sine or cosine-like solutions 626 00:35:32,550 --> 00:35:34,560 of the Schrodinger equation that are 627 00:35:34,560 --> 00:35:40,030 such that they have 0 amplitude at this point and at this point 628 00:35:40,030 --> 00:35:45,000 there are sines or cosines with a wavelength that is 629 00:35:45,000 --> 00:35:48,100 compatible with this distance. 630 00:35:48,100 --> 00:35:50,820 So all of a sudden, our solutions that, 631 00:35:50,820 --> 00:35:52,680 in the case of the free electron, 632 00:35:52,680 --> 00:35:55,060 could have any wavelength in the world 633 00:35:55,060 --> 00:35:57,360 have now become quantized. 634 00:35:57,360 --> 00:36:01,080 There is only a discrete set of solutions with wavelengths 635 00:36:01,080 --> 00:36:05,250 that are compatible with the distance between the two 636 00:36:05,250 --> 00:36:08,360 infinite, confining sides. 637 00:36:08,360 --> 00:36:13,950 And this is the same concept of an organ pipe and the music 638 00:36:13,950 --> 00:36:15,120 that it plays. 639 00:36:15,120 --> 00:36:19,590 When you play music in an organ pipe, you are really exciting 640 00:36:19,590 --> 00:36:23,520 either a sound wave with a wavelength corresponding 641 00:36:23,520 --> 00:36:25,410 to the fundamental harmonica. 642 00:36:25,410 --> 00:36:30,390 Or the only other notes that your organ pipe can produce 643 00:36:30,390 --> 00:36:33,250 are actually higher harmonics. 644 00:36:33,250 --> 00:36:35,970 And so they are, again, sound waves 645 00:36:35,970 --> 00:36:39,210 that will have 0 amplitudes at the borders. 646 00:36:39,210 --> 00:36:42,600 Or same thing, say, with the chord of a guitar. 647 00:36:42,600 --> 00:36:46,980 And so the wave functions for a confined potential 648 00:36:46,980 --> 00:36:49,260 are actually quantized. 649 00:36:49,260 --> 00:36:52,470 And they have corresponding energies-- 650 00:36:52,470 --> 00:36:56,340 that is, in this specific case, go actually as the square. 651 00:36:56,340 --> 00:36:59,170 Although, this is less relevant for our case. 652 00:36:59,170 --> 00:37:01,500 So you will see that, in most cases, 653 00:37:01,500 --> 00:37:03,810 like when we look at the solution 654 00:37:03,810 --> 00:37:08,820 for an atom in a confining Coulombic potential, 655 00:37:08,820 --> 00:37:13,620 we'll actually find only a discrete set of wave functions 656 00:37:13,620 --> 00:37:16,840 with discrete energy levels. 657 00:37:16,840 --> 00:37:20,880 And again, we can get some hints on how 658 00:37:20,880 --> 00:37:23,070 this solution would look like. 659 00:37:23,070 --> 00:37:24,990 If you think for a moment at what 660 00:37:24,990 --> 00:37:29,280 would happen if our square well, instead of having 661 00:37:29,280 --> 00:37:34,380 infinite confining walls, would have only finite confining 662 00:37:34,380 --> 00:37:37,440 walls with a finite height, it's really 663 00:37:37,440 --> 00:37:40,290 the same case as the metal surface 664 00:37:40,290 --> 00:37:43,050 because now you have a finite wall. 665 00:37:43,050 --> 00:37:49,020 You have a finite wave function decaying exponentially 666 00:37:49,020 --> 00:37:51,550 outside in that region. 667 00:37:51,550 --> 00:37:55,920 And so what was, you remember, our fundamental harmonic-- 668 00:37:55,920 --> 00:37:58,770 the second harmonic, third harmonic, fourth harmonic, 669 00:37:58,770 --> 00:38:02,670 what were those oscillatory solutions inside the box 670 00:38:02,670 --> 00:38:08,100 now start getting these exponential tails decaying 671 00:38:08,100 --> 00:38:09,520 in the vacuum. 672 00:38:09,520 --> 00:38:14,490 And this is really how solutions for the Schrodinger 673 00:38:14,490 --> 00:38:17,560 equation of an atom look like. 674 00:38:17,560 --> 00:38:20,560 And I don't want really to bore you with the algebra. 675 00:38:20,560 --> 00:38:23,520 Some of you might actually have seen this over and over again, 676 00:38:23,520 --> 00:38:25,500 if you have taken a quantum class. 677 00:38:25,500 --> 00:38:31,050 I'll just give you one slide with the conceptual framework 678 00:38:31,050 --> 00:38:34,810 of what we should do if we were to solve the Schrodinger 679 00:38:34,810 --> 00:38:39,100 equation for a single electron in what we 680 00:38:39,100 --> 00:38:40,850 call a central potential. 681 00:38:40,850 --> 00:38:42,370 This is a general concept. 682 00:38:42,370 --> 00:38:44,830 Central potential is a potential that 683 00:38:44,830 --> 00:38:48,880 depends only on the distance from a center, 684 00:38:48,880 --> 00:38:52,390 but not on the angle of distribution. 685 00:38:52,390 --> 00:38:54,220 And so central potential would be 686 00:38:54,220 --> 00:38:59,440 written as V of r, where r is the modulus of our vector, 687 00:38:59,440 --> 00:39:01,030 is just the distance. 688 00:39:01,030 --> 00:39:03,580 And the algebra is fairly complex. 689 00:39:03,580 --> 00:39:07,630 But the general idea is that this is now 690 00:39:07,630 --> 00:39:09,250 our Schrodinger equation. 691 00:39:09,250 --> 00:39:12,130 We need to find a wave function such 692 00:39:12,130 --> 00:39:15,400 that the application of the second derivative plus 693 00:39:15,400 --> 00:39:17,650 this potential is equal to the wave 694 00:39:17,650 --> 00:39:19,900 function itself times a constant. 695 00:39:19,900 --> 00:39:21,640 And because somehow we are dealing 696 00:39:21,640 --> 00:39:23,860 with the geometry of a central potential, 697 00:39:23,860 --> 00:39:27,850 it's convenient to write the Laplacian, 698 00:39:27,850 --> 00:39:31,600 the second derivative, not in Cartesian coordinate, 699 00:39:31,600 --> 00:39:34,820 but in spherical coordinates-- that is, in coordinates which, 700 00:39:34,820 --> 00:39:39,410 instead of giving the x, y, and z components of a vector, 701 00:39:39,410 --> 00:39:42,790 we give the distance from the origin and the two 702 00:39:42,790 --> 00:39:45,610 angles, theta and phi, that that vector makes 703 00:39:45,610 --> 00:39:47,340 with respect to different axes. 704 00:39:47,340 --> 00:39:49,480 Again, the solution is fairly complicated. 705 00:39:49,480 --> 00:39:51,610 So I won't really go into this. 706 00:39:51,610 --> 00:39:54,550 But this is how our Hamiltonian-- 707 00:39:54,550 --> 00:39:58,840 this is how the left part of the stationary equation 708 00:39:58,840 --> 00:40:01,870 would look for this central potential, 709 00:40:01,870 --> 00:40:06,340 having just rewritten basically this second derivative 710 00:40:06,340 --> 00:40:08,560 in Cartesian coordinates. 711 00:40:08,560 --> 00:40:13,150 And when one goes through all the algebra, 712 00:40:13,150 --> 00:40:16,270 one finds out that the function-- 713 00:40:16,270 --> 00:40:19,420 the space part of the wave function 714 00:40:19,420 --> 00:40:23,770 can be written as a term that depends only 715 00:40:23,770 --> 00:40:27,820 on the distance from the center, from the place where 716 00:40:27,820 --> 00:40:32,200 the nucleus is, times an angular part. 717 00:40:32,200 --> 00:40:34,090 And if you are familiar with this, 718 00:40:34,090 --> 00:40:37,930 we call this angular part spherical harmonics. 719 00:40:37,930 --> 00:40:41,110 And this would be the radial part of the wave function. 720 00:40:41,110 --> 00:40:46,210 That is a solution of a slightly different, one-dimensional 721 00:40:46,210 --> 00:40:47,320 Schrodinger equation. 722 00:40:47,320 --> 00:40:49,810 Again, this is just the start, if you 723 00:40:49,810 --> 00:40:51,460 are interested to take this. 724 00:40:51,460 --> 00:40:57,580 But somehow, this is all the algebra that gives rise 725 00:40:57,580 --> 00:40:58,810 to the periodic table. 726 00:40:58,810 --> 00:41:00,730 That is, when we study actually what 727 00:41:00,730 --> 00:41:02,680 are the solutions of the Schrodinger 728 00:41:02,680 --> 00:41:05,080 equation for a central potential, 729 00:41:05,080 --> 00:41:09,130 well, we find that we can classify the solution as we 730 00:41:09,130 --> 00:41:13,150 were classifying the solution in the particle in a box 731 00:41:13,150 --> 00:41:17,080 depending on, if you want, the fundamental harmonic 732 00:41:17,080 --> 00:41:18,490 and the higher harmonics. 733 00:41:18,490 --> 00:41:21,640 Well, we can still classify the solution depending 734 00:41:21,640 --> 00:41:23,840 on the angular distribution-- 735 00:41:23,840 --> 00:41:27,100 so what are actually the L and M indices 736 00:41:27,100 --> 00:41:29,110 of the spherical harmonics. 737 00:41:29,110 --> 00:41:33,190 And we can also classify the radial distribution. 738 00:41:33,190 --> 00:41:35,500 And so all these possible solutions 739 00:41:35,500 --> 00:41:42,070 give us all the possible states of an electron in a hydrogen 740 00:41:42,070 --> 00:41:42,910 atom. 741 00:41:42,910 --> 00:41:45,860 And you probably have seen this over and over again. 742 00:41:45,860 --> 00:41:49,450 But it's really the fundamental alphabet that gives rise 743 00:41:49,450 --> 00:41:50,930 to the periodic table. 744 00:41:50,930 --> 00:41:54,820 So for the hydrogen atom, the fundamental solution 745 00:41:54,820 --> 00:41:58,630 is one in which the electron is distributed 746 00:41:58,630 --> 00:42:02,210 with spherical symmetry around the nucleus. 747 00:42:02,210 --> 00:42:05,500 And if we look at the probability density 748 00:42:05,500 --> 00:42:08,230 as a function of the distance from the center, 749 00:42:08,230 --> 00:42:12,620 it's something that really looks a little bit like this. 750 00:42:12,620 --> 00:42:19,710 So the probability of finding the electron in the hydrogen 751 00:42:19,710 --> 00:42:22,110 atom at a certain distance really 752 00:42:22,110 --> 00:42:30,270 has a maximum at a distance that is, I believe, at 1.5 bohrs. 753 00:42:30,270 --> 00:42:33,000 I need to check actually this number. 754 00:42:33,000 --> 00:42:39,780 The classical radius of the hydrogen atom is 1 bohr. 755 00:42:39,780 --> 00:42:42,570 It's 0.529 Angstrom. 756 00:42:42,570 --> 00:42:45,090 And the maximum, I think, is at 1.5. 757 00:42:45,090 --> 00:42:47,790 But basically, there is a maximum of probability. 758 00:42:47,790 --> 00:42:50,230 And then it decays again. 759 00:42:50,230 --> 00:42:52,290 But then there is a whole zoology 760 00:42:52,290 --> 00:42:56,550 of excited states that can have, say, a more 761 00:42:56,550 --> 00:42:58,980 complex radial part. 762 00:42:58,980 --> 00:43:01,200 And we classify these-- you probably remember 763 00:43:01,200 --> 00:43:02,520 this from your chemistry. 764 00:43:02,520 --> 00:43:06,150 We call this state a 1s state where 765 00:43:06,150 --> 00:43:09,030 1 really refers to the radial part 766 00:43:09,030 --> 00:43:11,730 and s refers to the angular part. 767 00:43:11,730 --> 00:43:14,670 And the next state would be a 2s-like state, 768 00:43:14,670 --> 00:43:18,840 in which, being s, the angular part is still the same. 769 00:43:18,840 --> 00:43:20,700 It's spherically symmetric. 770 00:43:20,700 --> 00:43:22,820 But now, the radial part has changed. 771 00:43:22,820 --> 00:43:25,380 And it actually changed sign. 772 00:43:25,380 --> 00:43:27,930 So it goes inside. 773 00:43:27,930 --> 00:43:29,160 Then it goes back to 0. 774 00:43:29,160 --> 00:43:30,150 Then it goes again. 775 00:43:30,150 --> 00:43:31,980 And it goes back to 0. 776 00:43:31,980 --> 00:43:34,800 And then there are states like the 2p states 777 00:43:34,800 --> 00:43:39,340 in which the angular part starts to play a role. 778 00:43:39,340 --> 00:43:43,000 And so this would be a specific p state that, if you want, 779 00:43:43,000 --> 00:43:45,690 has two lobes with opposite signs 780 00:43:45,690 --> 00:43:47,580 across a plane of symmetry. 781 00:43:47,580 --> 00:43:50,520 And then there are more complex states like the d states 782 00:43:50,520 --> 00:43:52,570 and so on and so forth. 783 00:43:52,570 --> 00:43:56,160 And a very important thing is that really this angular 784 00:43:56,160 --> 00:44:02,370 symmetry comes from just having considered a central potential. 785 00:44:02,370 --> 00:44:04,710 That is a potential that doesn't depend 786 00:44:04,710 --> 00:44:08,230 from the angular variable, but just from the distance. 787 00:44:08,230 --> 00:44:14,040 And so these solutions have the same qualitative difference 788 00:44:14,040 --> 00:44:18,840 not only for a single electron in a hydrogen atom, 789 00:44:18,840 --> 00:44:24,360 but, say, in general for an electron in an atom where 790 00:44:24,360 --> 00:44:27,600 really that electron will feel not 791 00:44:27,600 --> 00:44:30,070 only the attractive potential of the nucleus, 792 00:44:30,070 --> 00:44:34,440 but also the repulsive potential from all the other electrons. 793 00:44:34,440 --> 00:44:38,880 But to a great extent, that repulsive potential 794 00:44:38,880 --> 00:44:41,520 of all the other electrons is going 795 00:44:41,520 --> 00:44:46,290 to be a fairly spherical symmetry kind of potential 796 00:44:46,290 --> 00:44:49,620 because all the other electrons, when you take their charge 797 00:44:49,620 --> 00:44:53,140 density, are going to be roughly spherically 798 00:44:53,140 --> 00:44:54,930 symmetrically distributed. 799 00:44:54,930 --> 00:44:58,590 And so the solution really, for one electron 800 00:44:58,590 --> 00:45:00,990 in a central potential, does not only apply 801 00:45:00,990 --> 00:45:03,360 to the hydrogen atom, but applies 802 00:45:03,360 --> 00:45:06,990 to really any atom in the periodic table 803 00:45:06,990 --> 00:45:08,160 to some approximation. 804 00:45:08,160 --> 00:45:10,440 That is to the approximation that we consider. 805 00:45:10,440 --> 00:45:14,490 The repulsive potential that all the other electrons 806 00:45:14,490 --> 00:45:19,410 have for this, say, top valance electron in a generic atom, say 807 00:45:19,410 --> 00:45:21,870 iron, is that the only approximation 808 00:45:21,870 --> 00:45:24,870 is that these repulsive potential is also 809 00:45:24,870 --> 00:45:26,520 spherically symmetric. 810 00:45:26,520 --> 00:45:33,060 That is not true but is true to a very large extent. 811 00:45:33,060 --> 00:45:33,810 OK. 812 00:45:33,810 --> 00:45:38,070 Now, this concludes, if you want, our general panorama 813 00:45:38,070 --> 00:45:40,440 on what is quantum mechanics. 814 00:45:40,440 --> 00:45:44,250 And now, really, we want to go into electronic structure 815 00:45:44,250 --> 00:45:44,970 methods. 816 00:45:44,970 --> 00:45:48,570 That is, we want to arrive at what is called the Hartree-Fock 817 00:45:48,570 --> 00:45:49,410 solution. 818 00:45:49,410 --> 00:45:52,140 That is, again, one of the fundamental techniques, 819 00:45:52,140 --> 00:45:54,510 especially in quantum chemistry. 820 00:45:54,510 --> 00:45:57,210 And to do this, I need to give you 821 00:45:57,210 --> 00:45:58,890 a little bit of nomenclature. 822 00:45:58,890 --> 00:46:01,710 Again, if you have never seen a set of quantum mechanics 823 00:46:01,710 --> 00:46:04,410 or if you have never taken a quantum mechanics class, 824 00:46:04,410 --> 00:46:07,800 it would be very useful to have a look at some of the books 825 00:46:07,800 --> 00:46:10,200 that I've given you in the bibliography, either 826 00:46:10,200 --> 00:46:12,390 some fundamental books like Quantum Mechanics 827 00:46:12,390 --> 00:46:14,340 or Physics of Atoms and Molecules, 828 00:46:14,340 --> 00:46:17,850 both by Bransden and Joachain, that are very good quantum 829 00:46:17,850 --> 00:46:20,970 mechanics books, or some of the electronic structure books, 830 00:46:20,970 --> 00:46:24,420 like the Phoenix books on inter-atomic forces 831 00:46:24,420 --> 00:46:28,440 or the Richard Martin book, or the Kaxiras book. 832 00:46:28,440 --> 00:46:33,660 But the nomenclature that we'll use over and over again 833 00:46:33,660 --> 00:46:37,980 is what is called the bracket nomenclature of the Dirac. 834 00:46:37,980 --> 00:46:41,310 And so in general, a wave function 835 00:46:41,310 --> 00:46:44,700 is actually represented by this sign here. 836 00:46:44,700 --> 00:46:46,020 This is called [INAUDIBLE]. 837 00:46:46,020 --> 00:46:50,070 Again, that means nothing else than our function. 838 00:46:50,070 --> 00:46:53,850 That is an amplitude defined everywhere 839 00:46:53,850 --> 00:46:57,340 in space or everywhere in space and time. 840 00:46:57,340 --> 00:47:01,470 And in general, it's a complex amplitude. 841 00:47:01,470 --> 00:47:06,450 And now, one of the fundamentals of the rules of quantum 842 00:47:06,450 --> 00:47:10,770 mechanics is that, say, a wave function 843 00:47:10,770 --> 00:47:14,390 corresponding to different states of an electron-- 844 00:47:14,390 --> 00:47:17,690 say you are considering the function for the ground state 845 00:47:17,690 --> 00:47:20,120 of the hydrogen atom, the 1s. 846 00:47:20,120 --> 00:47:23,600 And then you are considering the function for an excited state, 847 00:47:23,600 --> 00:47:24,290 like the 2s. 848 00:47:24,290 --> 00:47:27,470 So the wave function for different states 849 00:47:27,470 --> 00:47:30,020 are going to be orthonormal. 850 00:47:30,020 --> 00:47:31,380 What does it mean? 851 00:47:31,380 --> 00:47:34,880 It means that when we take the scalar product, 852 00:47:34,880 --> 00:47:38,960 that scalar product is going to be either 0 or 1, 853 00:47:38,960 --> 00:47:42,245 depending if we are taking the scalar product with the wave 854 00:47:42,245 --> 00:47:46,380 function itself or with the wave function of a different state. 855 00:47:46,380 --> 00:47:50,660 So this delta ij is what is called usually a chronicle 856 00:47:50,660 --> 00:47:51,170 delta. 857 00:47:51,170 --> 00:47:54,590 It's a symbol that means this is going to be equal to 1 858 00:47:54,590 --> 00:47:58,160 if i is equal to j or is going to be equal to 0 859 00:47:58,160 --> 00:48:00,710 if i is different from j. 860 00:48:00,710 --> 00:48:04,760 And the scalar product between two wave functions 861 00:48:04,760 --> 00:48:08,390 is written in shorthand notation as this. 862 00:48:08,390 --> 00:48:12,710 It's written as a bra times a ket, a bracket. 863 00:48:12,710 --> 00:48:16,310 But again, that's nothing else than a shorthand 864 00:48:16,310 --> 00:48:21,470 form for the proper definition of the scalar product. 865 00:48:21,470 --> 00:48:25,760 Again, you are probably not heard of Hilbert spaces. 866 00:48:25,760 --> 00:48:28,580 But this is a vector space of wave function 867 00:48:28,580 --> 00:48:30,350 that has a metric defined. 868 00:48:30,350 --> 00:48:33,800 And so we can define something like a scalar product 869 00:48:33,800 --> 00:48:37,910 by taking the integral of the wave function 870 00:48:37,910 --> 00:48:40,820 on the left that has been complex 871 00:48:40,820 --> 00:48:44,510 conjugated times the wave function on the right. 872 00:48:44,510 --> 00:48:48,350 So all wave functions that are solutions of a Schrodinger 873 00:48:48,350 --> 00:48:51,110 equation satisfy this orthonormality normality 874 00:48:51,110 --> 00:48:52,130 property. 875 00:48:52,130 --> 00:48:56,330 If you take this integral, that is going to be equal to 1 876 00:48:56,330 --> 00:49:00,080 if you are just taking the square of the wave function. 877 00:49:00,080 --> 00:49:03,650 And it's going to be equal to 0 if i is different to j. 878 00:49:03,650 --> 00:49:06,605 And you see why it's equal to 1 actually 879 00:49:06,605 --> 00:49:09,320 if i is equal to j because remember 880 00:49:09,320 --> 00:49:13,610 what is psi i star times psi i? 881 00:49:13,610 --> 00:49:17,540 Well, it's just a square modulus of psi i. 882 00:49:17,540 --> 00:49:23,270 So psi i star times psi i in a certain point r in space 883 00:49:23,270 --> 00:49:26,300 gives me the probability of finding the electron 884 00:49:26,300 --> 00:49:28,920 in point in space. 885 00:49:28,920 --> 00:49:33,020 And when I integrate that probability all over space, 886 00:49:33,020 --> 00:49:36,110 it becomes obvious that that number must be 1 887 00:49:36,110 --> 00:49:39,650 because what is the probability of finding an electron anywhere 888 00:49:39,650 --> 00:49:40,440 in space? 889 00:49:40,440 --> 00:49:42,150 Well, you'll find it somewhere. 890 00:49:42,150 --> 00:49:44,360 So the probability of finding it anywhere-- 891 00:49:44,360 --> 00:49:52,940 that is the integral of psi squared as a function of r-- 892 00:49:52,940 --> 00:49:54,740 must be equal to 1. 893 00:49:54,740 --> 00:49:58,610 So that's the physical connection. 894 00:50:09,570 --> 00:50:12,840 And the last definition that I need 895 00:50:12,840 --> 00:50:16,110 is that of expectation values. 896 00:50:16,110 --> 00:50:18,180 That is, once we have found a wave 897 00:50:18,180 --> 00:50:20,760 function for this electron-- 898 00:50:20,760 --> 00:50:23,880 as I said, we know everything that we need 899 00:50:23,880 --> 00:50:25,920 to know about that electron. 900 00:50:25,920 --> 00:50:30,420 And so we can extract all the physical properties 901 00:50:30,420 --> 00:50:32,070 that we want. 902 00:50:32,070 --> 00:50:34,350 It's like, in classical mechanics, 903 00:50:34,350 --> 00:50:37,050 if you have the trajectory of your electrons, 904 00:50:37,050 --> 00:50:42,360 you can calculate at any instant in time what is, say, 1/2 times 905 00:50:42,360 --> 00:50:44,220 the mass times the square velocity. 906 00:50:44,220 --> 00:50:46,830 That would be the kinetic energy of that particle. 907 00:50:46,830 --> 00:50:51,810 Similarly, if you know the wave function of that electron, 908 00:50:51,810 --> 00:50:57,060 you can calculate any kind of physical quantity. 909 00:50:57,060 --> 00:51:00,570 And one of the most important physical quantities, 910 00:51:00,570 --> 00:51:03,330 apart from the probability of finding it somewhere, 911 00:51:03,330 --> 00:51:07,290 is what is the energy of that electron. 912 00:51:07,290 --> 00:51:13,530 And this is the shorthand term in the bracket nomenclature 913 00:51:13,530 --> 00:51:17,460 of what would be the energy of an electron. 914 00:51:17,460 --> 00:51:20,920 We call this an expectation value. 915 00:51:20,920 --> 00:51:25,590 So in quantum mechanics, for every physical observable, 916 00:51:25,590 --> 00:51:30,600 like the energy, there is going to be an operator that 917 00:51:30,600 --> 00:51:35,040 acts on the wave function and is such that its expectation 918 00:51:35,040 --> 00:51:39,360 value gives me the physical quantity I'm interested in. 919 00:51:39,360 --> 00:51:43,680 So if I want to know the energy of an electron that has a wave 920 00:51:43,680 --> 00:51:47,820 function psi i, I need to calculate this expectation 921 00:51:47,820 --> 00:51:50,700 value, psi h psi i-- 922 00:51:50,700 --> 00:51:52,830 that will give me the energy. 923 00:51:52,830 --> 00:51:57,120 And the quantum mechanical operator 924 00:51:57,120 --> 00:52:01,260 that gives me the energy is what is called the Hamiltonian 925 00:52:01,260 --> 00:52:05,560 operator, h, that I've written here. 926 00:52:05,560 --> 00:52:10,290 So the Hamiltonian operator is nothing else. 927 00:52:10,290 --> 00:52:11,760 And I forgot it here. 928 00:52:11,760 --> 00:52:14,670 And sorry, there is a Laplacina. 929 00:52:14,670 --> 00:52:16,590 The Hamiltonian operator is nothing 930 00:52:16,590 --> 00:52:19,590 else what we have seen in the Schrodinger equation. 931 00:52:19,590 --> 00:52:24,180 It's minus h bar squared divided by 2m times 932 00:52:24,180 --> 00:52:26,860 the Laplacian plus the potential. 933 00:52:26,860 --> 00:52:28,710 So if you want to calculate what is 934 00:52:28,710 --> 00:52:32,430 the energy of this electron, what you need to do 935 00:52:32,430 --> 00:52:37,710 is to apply the energy operator, that is the Hamiltonian, 936 00:52:37,710 --> 00:52:39,520 to the wave function. 937 00:52:39,520 --> 00:52:42,450 So you need to take the second derivative with respect 938 00:52:42,450 --> 00:52:44,280 to the wave function. 939 00:52:44,280 --> 00:52:46,410 With the appropriate coefficient, 940 00:52:46,410 --> 00:52:51,100 you need to sum the potential times the wave function. 941 00:52:51,100 --> 00:52:54,150 And now, to this new quantity that 942 00:52:54,150 --> 00:52:57,840 is again a field that is a function of r, 943 00:52:57,840 --> 00:53:00,900 well, you need to take that multiplied 944 00:53:00,900 --> 00:53:04,500 by the complex conjugate of the wave function 945 00:53:04,500 --> 00:53:07,840 and integrate it all over space. 946 00:53:07,840 --> 00:53:10,590 So if you want, it's a much more complex operation. 947 00:53:10,590 --> 00:53:14,280 But that's exactly what are the rules for quantum objects. 948 00:53:14,280 --> 00:53:15,990 You have a Schrodinger equation. 949 00:53:15,990 --> 00:53:18,180 You find its solutions. 950 00:53:18,180 --> 00:53:22,170 And then the solutions are the ground state or the excited 951 00:53:22,170 --> 00:53:23,530 state wave function. 952 00:53:23,530 --> 00:53:25,110 And if you want to calculate what 953 00:53:25,110 --> 00:53:27,810 are the physical values of what we 954 00:53:27,810 --> 00:53:31,080 say are observable quantities, like the energy, 955 00:53:31,080 --> 00:53:33,780 you need to take an expectation value 956 00:53:33,780 --> 00:53:35,820 of the appropriate operator. 957 00:53:35,820 --> 00:53:38,490 And if you want the most fundamental operator, 958 00:53:38,490 --> 00:53:43,050 it's the Hamiltonian h that is just written here. 959 00:53:43,050 --> 00:53:46,710 And the expectation value that is written in shorthand form 960 00:53:46,710 --> 00:53:50,310 here is nothing else but the integral overall space 961 00:53:50,310 --> 00:53:52,120 of the complex conjugate of the wave 962 00:53:52,120 --> 00:53:58,200 function times what we obtain by applying to the wave 963 00:53:58,200 --> 00:54:01,560 function the Hamiltonian operator. 964 00:54:01,560 --> 00:54:04,410 And we integrate that in space. 965 00:54:04,410 --> 00:54:07,680 Now, it's actually simpler than it 966 00:54:07,680 --> 00:54:10,680 looks because remember for a moment 967 00:54:10,680 --> 00:54:14,550 that this psi is actually a solution of the Schrodinger 968 00:54:14,550 --> 00:54:15,630 equation. 969 00:54:15,630 --> 00:54:18,060 This psi is a solution of the stationary Schrodinger 970 00:54:18,060 --> 00:54:18,910 equation. 971 00:54:18,910 --> 00:54:22,800 So we know already, because the Hamiltonian is really-- 972 00:54:22,800 --> 00:54:24,990 remember, it's the left and term that 973 00:54:24,990 --> 00:54:26,920 is in this stationary equation. 974 00:54:26,920 --> 00:54:30,360 So we know already that, by the fact of this 975 00:54:30,360 --> 00:54:35,040 being the solution, applying this operator to this wave 976 00:54:35,040 --> 00:54:39,330 function is going to give me a constant times 977 00:54:39,330 --> 00:54:40,780 this wave function. 978 00:54:40,780 --> 00:54:44,940 And so the concept, we call it nothing else than E 979 00:54:44,940 --> 00:54:46,590 with a subscript i. 980 00:54:46,590 --> 00:54:50,700 So this times the wave function gives me a constant times 981 00:54:50,700 --> 00:54:51,960 the wave function itself. 982 00:54:51,960 --> 00:54:56,160 And I can take the constant out of the integral sine. 983 00:54:56,160 --> 00:55:02,970 And I'm left with constant E of phi times the integral of psi i 984 00:55:02,970 --> 00:55:04,740 star times Ci. 985 00:55:04,740 --> 00:55:06,660 And that's nothing else than 1. 986 00:55:06,660 --> 00:55:10,550 And so this is, if you want, the explicit solution 987 00:55:10,550 --> 00:55:13,760 of why this is epsilon i. 988 00:55:13,760 --> 00:55:16,080 Of course, if you have never seen this, 989 00:55:16,080 --> 00:55:17,450 it might look very arcane. 990 00:55:17,450 --> 00:55:22,790 And this is why I urge you to read some introductory quantum 991 00:55:22,790 --> 00:55:24,620 mechanics textbooks. 992 00:55:24,620 --> 00:55:30,530 If you have seen it already, of course, it might seem trivial. 993 00:55:30,530 --> 00:55:32,030 It might seem trivial to you. 994 00:55:37,140 --> 00:55:37,920 OK. 995 00:55:37,920 --> 00:55:41,850 And now, in the next three or four slides, 996 00:55:41,850 --> 00:55:47,310 we go really into the two fundamental concepts 997 00:55:47,310 --> 00:55:51,180 of computational electronic structure. 998 00:55:51,180 --> 00:55:55,600 That involves two things. 999 00:55:55,600 --> 00:56:01,230 One, it involves finding the solution of the Schrodinger 1000 00:56:01,230 --> 00:56:04,440 equation on a computer-- that is, on a system 1001 00:56:04,440 --> 00:56:07,500 that is really not able to do analysis, if you want, 1002 00:56:07,500 --> 00:56:09,900 is not able to do analytic derivatives 1003 00:56:09,900 --> 00:56:11,760 and analytic integrals. 1004 00:56:11,760 --> 00:56:15,390 And so we'll how we actually deal in practice 1005 00:56:15,390 --> 00:56:16,590 with this issue. 1006 00:56:16,590 --> 00:56:19,560 And then in the second set of slides, 1007 00:56:19,560 --> 00:56:22,720 we'll give you the fundamental concept 1008 00:56:22,720 --> 00:56:26,580 that is a reformulation of the differential equation, 1009 00:56:26,580 --> 00:56:30,360 of the Schrodinger equation, in a different form that 1010 00:56:30,360 --> 00:56:33,755 is actually computer-friendly and algorithmic-friendly. 1011 00:56:33,755 --> 00:56:35,130 So we have a Schrodinger equation 1012 00:56:35,130 --> 00:56:37,330 that we know we need to solve. 1013 00:56:37,330 --> 00:56:40,140 But we can actually rewrite it in a form 1014 00:56:40,140 --> 00:56:42,060 that is much more convenient. 1015 00:56:42,060 --> 00:56:45,780 And respectively, you can think of them 1016 00:56:45,780 --> 00:56:49,860 as the general concept of expanding wave function 1017 00:56:49,860 --> 00:56:56,100 in a basis that transforms all our analysis and integrals 1018 00:56:56,100 --> 00:56:58,700 and differentials in matrix algebra. 1019 00:56:58,700 --> 00:57:01,770 And I'll show you in a moment why that is the case. 1020 00:57:01,770 --> 00:57:03,690 And in the second part, we'll see 1021 00:57:03,690 --> 00:57:06,450 how the Schrodinger equation is transformed 1022 00:57:06,450 --> 00:57:08,790 into a variational principle. 1023 00:57:08,790 --> 00:57:11,340 So really, what you are seeing now is, I would say, 1024 00:57:11,340 --> 00:57:16,240 the central content of today's class. 1025 00:57:16,240 --> 00:57:16,740 OK. 1026 00:57:16,740 --> 00:57:20,340 Let me first go back to the Schrodinger equation 1027 00:57:20,340 --> 00:57:22,570 as you have seen it before. 1028 00:57:22,570 --> 00:57:25,770 And now, I'm using actually the shorthand term 1029 00:57:25,770 --> 00:57:28,920 H, this Hamiltonian operator, that is again nothing 1030 00:57:28,920 --> 00:57:32,080 less than minus H bar squared over 2m times 1031 00:57:32,080 --> 00:57:34,360 the Laplacian plus the potential. 1032 00:57:34,360 --> 00:57:36,210 And so the Schrodinger equation tells us 1033 00:57:36,210 --> 00:57:39,840 that, for a given potential, we need to find a wave function 1034 00:57:39,840 --> 00:57:43,680 psi such that when we apply on it 1035 00:57:43,680 --> 00:57:46,020 those derivatives and those multiplication 1036 00:57:46,020 --> 00:57:49,200 contained in the Hamiltonian operator, that 1037 00:57:49,200 --> 00:57:53,130 is H, what we obtain is nothing else 1038 00:57:53,130 --> 00:57:56,820 but the wave function itself times a constant. 1039 00:57:56,820 --> 00:58:01,020 So this is the Schrodinger equation in shorthand form. 1040 00:58:01,020 --> 00:58:04,830 And we can even be more compact and write it 1041 00:58:04,830 --> 00:58:09,180 in the bracket formulation on the right side 1042 00:58:09,180 --> 00:58:12,330 as H psi equal to epsilon psi. 1043 00:58:12,330 --> 00:58:16,260 So this is our Schrodinger equation. 1044 00:58:16,260 --> 00:58:19,860 And now, this is the fundamental step. 1045 00:58:19,860 --> 00:58:22,440 And this is what really makes all 1046 00:58:22,440 --> 00:58:25,620 of this solvable on a computer. 1047 00:58:25,620 --> 00:58:31,060 What we say now is that any generic wave function-- 1048 00:58:31,060 --> 00:58:31,740 OK. 1049 00:58:31,740 --> 00:58:35,730 So we are still considering one electron 1050 00:58:35,730 --> 00:58:37,870 in three dimensional space. 1051 00:58:37,870 --> 00:58:42,300 So what we have is really an amplitude field all over space. 1052 00:58:42,300 --> 00:58:45,330 Well, that amplitude field can actually 1053 00:58:45,330 --> 00:58:50,610 be written as a linear combination 1054 00:58:50,610 --> 00:58:58,120 of an infinite number of simple functions times a coefficient. 1055 00:58:58,120 --> 00:59:03,780 So we are saying, well, we can expand every possible field 1056 00:59:03,780 --> 00:59:09,390 into a linear combination of orthogonal functions 1057 00:59:09,390 --> 00:59:13,440 that would be our basis function. 1058 00:59:13,440 --> 00:59:16,470 Say, if you have seen Fourier analysis, 1059 00:59:16,470 --> 00:59:19,290 you have seen that, in most cases, 1060 00:59:19,290 --> 00:59:21,990 a well-behaved function can actually 1061 00:59:21,990 --> 00:59:26,370 be expanded as a linear combination of sines 1062 00:59:26,370 --> 00:59:27,780 and cosines. 1063 00:59:27,780 --> 00:59:30,330 Or in general, if you have seen more of analysis, 1064 00:59:30,330 --> 00:59:33,210 you have seen that, in a lot of these Hilbert spaces, 1065 00:59:33,210 --> 00:59:38,580 you can define what are called complete sets 1066 00:59:38,580 --> 00:59:40,890 of orthogonal functions. 1067 00:59:40,890 --> 00:59:45,370 Sines and cosines are a complete set of orthogonal functions. 1068 00:59:45,370 --> 00:59:48,630 If you take sines or cosines with different wavelengths 1069 00:59:48,630 --> 00:59:50,550 and you integrate them over a space, 1070 00:59:50,550 --> 00:59:53,370 you get 0 unless the wavelength is the same. 1071 00:59:53,370 --> 00:59:57,990 And by using an infinite number of sines and cosines 1072 00:59:57,990 --> 01:00:03,030 with different wavelengths, you can describe any function 1073 01:00:03,030 --> 01:00:07,020 that is, in some ways, smooth enough, that doesn't, say, 1074 01:00:07,020 --> 01:00:09,120 infinite discontinuities. 1075 01:00:09,120 --> 01:00:12,030 And so what we do really, in most 1076 01:00:12,030 --> 01:00:14,130 of our computational application, 1077 01:00:14,130 --> 01:00:17,760 is that we define a basis set. 1078 01:00:17,760 --> 01:00:21,120 Let's say, for the case of solids, 1079 01:00:21,120 --> 01:00:25,260 it's actually going exactly to be the basis set of plane 1080 01:00:25,260 --> 01:00:31,290 waves, those E to the i kr with all possible wavelength 1081 01:00:31,290 --> 01:00:33,930 and with all possible directions. 1082 01:00:33,930 --> 01:00:35,850 That is an orthogonal set. 1083 01:00:35,850 --> 01:00:37,590 And it's complete. 1084 01:00:37,590 --> 01:00:41,940 And so once we decide what is our basis set, 1085 01:00:41,940 --> 01:00:44,670 say sines and cosines, and plane waves-- 1086 01:00:44,670 --> 01:00:48,450 and there are a lot more that I'll describe in the following. 1087 01:00:48,450 --> 01:00:52,950 Well, at that point, our function 1088 01:00:52,950 --> 01:00:58,530 needs to be specified by a set of coefficients. 1089 01:00:58,530 --> 01:01:00,760 And in all practical implementation, 1090 01:01:00,760 --> 01:01:03,300 we won't use infinite sets. 1091 01:01:03,300 --> 01:01:06,390 If you have got a well-behaved function, 1092 01:01:06,390 --> 01:01:08,550 you don't need to describe-- 1093 01:01:08,550 --> 01:01:12,810 it probably won't have oscillation 1094 01:01:12,810 --> 01:01:15,850 with extremely thin wavelength. 1095 01:01:15,850 --> 01:01:19,470 So you can expand a wave function 1096 01:01:19,470 --> 01:01:22,830 into a combination of sines and cosines 1097 01:01:22,830 --> 01:01:26,580 that have progressively shorter and shorter wavelengths. 1098 01:01:26,580 --> 01:01:30,090 But at a certain point, you can say, well, I'll stop here. 1099 01:01:30,090 --> 01:01:35,010 I don't need a resolution that is thinner than what 1100 01:01:35,010 --> 01:01:36,540 is physically reasonable. 1101 01:01:36,540 --> 01:01:39,090 I mean, if you are looking at electrons, 1102 01:01:39,090 --> 01:01:41,400 well, they are going to have oscillations 1103 01:01:41,400 --> 01:01:44,430 that are of the order of the inter-atomic distance. 1104 01:01:44,430 --> 01:01:47,190 But they are not going to have oscillations 1105 01:01:47,190 --> 01:01:51,060 that are of the order, say, of the diameter of a nucleus. 1106 01:01:51,060 --> 01:01:53,320 It just doesn't make any physical sense. 1107 01:01:53,320 --> 01:01:58,410 So you can understand, for every physical problem at hand, 1108 01:01:58,410 --> 01:02:02,970 when you can actually stop this infinite expansion 1109 01:02:02,970 --> 01:02:04,710 in your orthogonal basis. 1110 01:02:04,710 --> 01:02:07,800 And at that point, your wave function 1111 01:02:07,800 --> 01:02:12,900 becomes defined just by a finite number of coefficients. 1112 01:02:12,900 --> 01:02:15,540 And so suppose that for a moment you 1113 01:02:15,540 --> 01:02:18,090 have only 10 basis functions. 1114 01:02:18,090 --> 01:02:23,940 It could be sine of x, sine of 2x, sine of 3x, sine of 4x, 1115 01:02:23,940 --> 01:02:26,670 up to sine of 10x in one dimension. 1116 01:02:26,670 --> 01:02:30,360 Well, once you have decided that those 10 are your good basis 1117 01:02:30,360 --> 01:02:36,000 function, your wave function is described by 10 numbers. 1118 01:02:36,000 --> 01:02:37,980 It's just a vector. 1119 01:02:37,980 --> 01:02:42,420 And so what you need to do is find those 10 numbers. 1120 01:02:42,420 --> 01:02:46,860 So if you want, what was an analysis problem 1121 01:02:46,860 --> 01:02:50,730 now becomes a bit of a linear algebra problem in the ways 1122 01:02:50,730 --> 01:02:52,860 you'll see in a moment, in which we really need 1123 01:02:52,860 --> 01:02:55,020 to find those 10 coefficients. 1124 01:02:55,020 --> 01:02:59,700 We don't need to find any more integrals 1125 01:02:59,700 --> 01:03:02,320 and derivatives of functions. 1126 01:03:05,270 --> 01:03:10,880 And so let's actually see how this develops. 1127 01:03:10,880 --> 01:03:15,860 Remember, what we need to solve is the Schrodinger equation 1128 01:03:15,860 --> 01:03:18,900 that I've written over here. 1129 01:03:18,900 --> 01:03:23,640 And now, I have decided that I've got an orthogonal basis. 1130 01:03:23,640 --> 01:03:26,400 And I am going to do some algebra. 1131 01:03:26,400 --> 01:03:31,650 In particular, I take this Schrodinger equation above. 1132 01:03:31,650 --> 01:03:34,980 Remember, it's operator applied on the wave 1133 01:03:34,980 --> 01:03:37,830 function equal to a constant times the wave function. 1134 01:03:37,830 --> 01:03:43,410 And what I do is I say I multiply this relation 1135 01:03:43,410 --> 01:03:51,510 on the left and on the right by the complex conjugate of phi m. 1136 01:03:51,510 --> 01:03:54,330 So in going from here to here, what I do 1137 01:03:54,330 --> 01:04:00,330 is I multiply this times the complex conjugate of phi m. 1138 01:04:00,330 --> 01:04:01,860 So it's phi m star. 1139 01:04:01,860 --> 01:04:08,320 And then I integrate everything over space. 1140 01:04:08,320 --> 01:04:10,840 And so you see, when I take H psi, 1141 01:04:10,840 --> 01:04:13,000 I multiply it by phi m star. 1142 01:04:13,000 --> 01:04:14,620 And I integrate. 1143 01:04:14,620 --> 01:04:18,160 What I obtain is something that, in shorthand bracket 1144 01:04:18,160 --> 01:04:22,420 formulation, can be written as phi m H psi 1145 01:04:22,420 --> 01:04:24,610 This is nothing less than a shorthand 1146 01:04:24,610 --> 01:04:30,010 for the integral of overall space of phi star m times 1147 01:04:30,010 --> 01:04:33,640 what results from applying H to psi. 1148 01:04:33,640 --> 01:04:36,160 Same to the right-hand side-- 1149 01:04:36,160 --> 01:04:40,810 I multiply E times by phi m star. 1150 01:04:40,810 --> 01:04:42,970 And I integrate over space. 1151 01:04:42,970 --> 01:04:44,990 Now, E is just a constant. 1152 01:04:44,990 --> 01:04:47,530 So it can come out of the integral sine. 1153 01:04:47,530 --> 01:04:49,870 And what I am left with is the integral 1154 01:04:49,870 --> 01:04:52,690 of phi m star times psi. 1155 01:04:52,690 --> 01:04:56,290 And again, it's very important that you go back just 1156 01:04:56,290 --> 01:05:00,040 after this class and you rewrite explicitly 1157 01:05:00,040 --> 01:05:03,100 all these algebraic terms to make sure that you 1158 01:05:03,100 --> 01:05:06,490 become comfortable with them. 1159 01:05:06,490 --> 01:05:08,440 So I've done this operation. 1160 01:05:12,850 --> 01:05:15,700 This is what I have obtained. 1161 01:05:15,700 --> 01:05:20,970 And now, what I do I exploit my hypothesis. 1162 01:05:20,970 --> 01:05:24,460 I've said that my wave function can 1163 01:05:24,460 --> 01:05:29,030 be written as a linear combination of my basis 1164 01:05:29,030 --> 01:05:32,090 function with certain coefficients. 1165 01:05:32,090 --> 01:05:36,910 So in this integral, now I substitute for psi 1166 01:05:36,910 --> 01:05:41,660 at sum of wave function with appropriate coefficients. 1167 01:05:41,660 --> 01:05:45,310 And as usual, I can take the sum and the coefficient 1168 01:05:45,310 --> 01:05:47,410 out of the integral sine. 1169 01:05:47,410 --> 01:05:51,100 So what I'm left with is a sum of the Cn that 1170 01:05:51,100 --> 01:05:56,920 multiplies the integral of phi star m H phi star n. 1171 01:05:56,920 --> 01:06:00,850 This is what I have on the left-hand side. 1172 01:06:00,850 --> 01:06:07,480 And now, on the right-hand side, this is likely trickier to see. 1173 01:06:07,480 --> 01:06:12,440 Again, I'm substituting for psi that sum. 1174 01:06:12,440 --> 01:06:15,310 So what I obtain here is really-- 1175 01:06:15,310 --> 01:06:18,130 let me write it as an integral for a moment-- 1176 01:06:18,130 --> 01:06:27,000 is the integral over all space of phi m star times 1177 01:06:27,000 --> 01:06:33,260 the sum over n of Cn phi m. 1178 01:06:37,340 --> 01:06:39,851 I can take a-- 1179 01:06:39,851 --> 01:06:41,100 sorry, it's a sum over-- 1180 01:06:46,400 --> 01:06:48,270 it's a bit difficult to see. 1181 01:06:48,270 --> 01:06:49,970 Let me write it more clearly. 1182 01:06:49,970 --> 01:06:56,550 It's a sum over n Cn phi star n. 1183 01:06:56,550 --> 01:06:58,440 OK. 1184 01:06:58,440 --> 01:06:59,790 These are just coefficients. 1185 01:06:59,790 --> 01:07:02,320 I can take them out of the integral sine. 1186 01:07:02,320 --> 01:07:07,290 So what I'm left with is sum over n Cn times 1187 01:07:07,290 --> 01:07:15,810 the integral over the r of phi n star times phi m. 1188 01:07:15,810 --> 01:07:19,030 For some reason, I wrote this down here. 1189 01:07:19,030 --> 01:07:21,450 So it's just a phi n. 1190 01:07:21,450 --> 01:07:25,380 And now, the integral of dr phi m star 1191 01:07:25,380 --> 01:07:31,350 times phi n is going to be equal either to 1 or to 0, 1192 01:07:31,350 --> 01:07:35,340 depending if m is equal to Min or m is different from n 1193 01:07:35,340 --> 01:07:39,630 because, in our hypothesis, we have said that these basis 1194 01:07:39,630 --> 01:07:41,220 functions are orthogonal. 1195 01:07:41,220 --> 01:07:45,630 And that orthogonal means we take the integral of them. 1196 01:07:45,630 --> 01:07:47,520 If they are identical, it gives 1. 1197 01:07:47,520 --> 01:07:50,130 If they are different, it gives 0. 1198 01:07:50,130 --> 01:07:53,220 So in all the sum over the n, there 1199 01:07:53,220 --> 01:07:57,250 is only one index that will be identical to this. 1200 01:07:57,250 --> 01:08:00,450 So for that case, the integral gives 1. 1201 01:08:00,450 --> 01:08:02,100 Otherwise, it gives 0. 1202 01:08:02,100 --> 01:08:05,550 So what is saved in all of this is 1203 01:08:05,550 --> 01:08:08,460 only the coefficient that corresponds 1204 01:08:08,460 --> 01:08:10,450 to this m coefficient here. 1205 01:08:10,450 --> 01:08:14,640 So what we obtain from this is the Cm times a constant. 1206 01:08:14,640 --> 01:08:15,570 OK. 1207 01:08:15,570 --> 01:08:19,680 So again, I've rewritten the Schrodinger equation 1208 01:08:19,680 --> 01:08:23,970 by multiplying it by phi m star, integrating it. 1209 01:08:23,970 --> 01:08:27,270 And what I obtain is a new formulation 1210 01:08:27,270 --> 01:08:33,840 of the Schrodinger equation that I have also in the next slide. 1211 01:08:33,840 --> 01:08:36,580 And that we are actually going to analyze. 1212 01:08:44,770 --> 01:08:50,950 So this is nothing else than your last slide. 1213 01:08:50,950 --> 01:08:56,890 And again, what we need to find are these coefficiencies. 1214 01:08:56,890 --> 01:09:01,180 Those have become now our solution. 1215 01:09:01,180 --> 01:09:06,279 And we can actually write this explicitly using 1216 01:09:06,279 --> 01:09:07,450 a bit of nomenclature. 1217 01:09:07,450 --> 01:09:11,979 That is, we can call that scalar, that expectation 1218 01:09:11,979 --> 01:09:12,899 value-- 1219 01:09:12,899 --> 01:09:19,029 phi m H phi n, we can call it that shorthand H m n. 1220 01:09:19,029 --> 01:09:21,520 We call it actually the matrix element 1221 01:09:21,520 --> 01:09:27,250 of the Hamiltonian over the two basis phi m and phi n. 1222 01:09:27,250 --> 01:09:33,310 And so again, our Schrodinger equation 1223 01:09:33,310 --> 01:09:37,899 has been rewritten in compact form as this. 1224 01:09:37,899 --> 01:09:41,410 That is, the physical problem tells 1225 01:09:41,410 --> 01:09:44,970 us what is the potential that the electron feels. 1226 01:09:44,970 --> 01:09:49,590 The potential specifies what the Hamiltonian operator 1227 01:09:49,590 --> 01:09:52,620 H is because it's, again, just the Laplacian 1228 01:09:52,620 --> 01:09:54,520 plus the potential. 1229 01:09:54,520 --> 01:09:58,440 And so once we have decided our basis set 1230 01:09:58,440 --> 01:10:01,500 onto which we develop our wave function, what 1231 01:10:01,500 --> 01:10:05,550 we need to calculate are all these matrix elements. 1232 01:10:05,550 --> 01:10:09,030 Now, these H m n are explicit numbers. 1233 01:10:09,030 --> 01:10:11,430 Once my potential is well-defined, 1234 01:10:11,430 --> 01:10:17,100 once my bases sector is defined, those H are just numbers. 1235 01:10:17,100 --> 01:10:20,550 And they actually can be written out in a matrix. 1236 01:10:20,550 --> 01:10:23,940 And my Schrodinger equation has been rewritten in this form. 1237 01:10:23,940 --> 01:10:32,040 That is, I have that this matrix times the vector of coefficient 1238 01:10:32,040 --> 01:10:37,130 of my expansion needs to be equal to a number 1239 01:10:37,130 --> 01:10:39,460 times the same vector. 1240 01:10:39,460 --> 01:10:43,110 And if you want, I have two things to find out. 1241 01:10:43,110 --> 01:10:48,090 I need to find out what are the values of E 1242 01:10:48,090 --> 01:10:53,880 for which this linear algebra system is a solution at all. 1243 01:10:53,880 --> 01:10:57,780 And we know from linear algebra that there 1244 01:10:57,780 --> 01:11:02,100 aren't an infinite number of values of E 1245 01:11:02,100 --> 01:11:04,000 for which this system has a solution. 1246 01:11:04,000 --> 01:11:06,400 And I'll comment in that on a moment. 1247 01:11:06,400 --> 01:11:09,450 And once we have found those values of E 1248 01:11:09,450 --> 01:11:13,170 for which there is a solution, we need to find the solution. 1249 01:11:13,170 --> 01:11:16,560 That is, we need to find this vector. 1250 01:11:16,560 --> 01:11:19,950 And so again, our problem is matrix times 1251 01:11:19,950 --> 01:11:24,270 vector needs to be equal to a constant times vector. 1252 01:11:24,270 --> 01:11:27,090 And we actually-- again, I don't have the time 1253 01:11:27,090 --> 01:11:28,770 to go into linear algebra. 1254 01:11:28,770 --> 01:11:35,280 But we know this can actually be rewritten as a Hamiltonian 1255 01:11:35,280 --> 01:11:42,510 matrix, an H matrix minus an E on all the diagonal terms 1256 01:11:42,510 --> 01:11:44,940 times this vector equal to 0. 1257 01:11:44,940 --> 01:11:47,700 And we know that this linear algebra solution-- 1258 01:11:47,700 --> 01:11:50,700 this linear algebra problem has a solution 1259 01:11:50,700 --> 01:11:58,860 only for values of E such that the determinant of the matrix H 1260 01:11:58,860 --> 01:12:02,400 minus E times the identity is 0. 1261 01:12:02,400 --> 01:12:08,670 So our goal is first finding those values of E for which 1262 01:12:08,670 --> 01:12:10,320 this determinant is 0. 1263 01:12:10,320 --> 01:12:13,770 And we call those eigenvalues, are the only values 1264 01:12:13,770 --> 01:12:18,670 of the energy that are allowed in our quantized Schrodinger 1265 01:12:18,670 --> 01:12:19,450 equation. 1266 01:12:19,450 --> 01:12:23,220 And then for each of these eigenvalues, E, 1267 01:12:23,220 --> 01:12:25,260 there is going to be a vector. 1268 01:12:25,260 --> 01:12:29,310 There is going to be a set of coefficient from C1 and Ck 1269 01:12:29,310 --> 01:12:32,880 that satisfies this linear algebra problem. 1270 01:12:32,880 --> 01:12:38,430 And we call that specific vector of coefficients an eigenvector. 1271 01:12:38,430 --> 01:12:40,980 So this is an eigenvalue equation 1272 01:12:40,980 --> 01:12:43,410 that only certain values of phi which 1273 01:12:43,410 --> 01:12:45,270 this equation is a solution. 1274 01:12:45,270 --> 01:12:49,830 And those values of phi for which it has a solution 1275 01:12:49,830 --> 01:12:52,500 will give rise to the eigenvector 1276 01:12:52,500 --> 01:12:54,900 that satisfies this equation. 1277 01:12:54,900 --> 01:12:57,600 And in a computer, you see the only thing that we need to do 1278 01:12:57,600 --> 01:13:01,860 is calculate this matrix element of the Hamiltonian, 1279 01:13:01,860 --> 01:13:05,910 find its eigenvalue, and find its eigenvectors that 1280 01:13:05,910 --> 01:13:09,780 are all well-defined linear algebra problems. 1281 01:13:09,780 --> 01:13:14,190 And so this is the first fundamental conclusion. 1282 01:13:14,190 --> 01:13:18,630 We can always rewrite our quantum mechanical problem 1283 01:13:18,630 --> 01:13:22,950 into a linear algebra problem once we specify 1284 01:13:22,950 --> 01:13:25,470 a basis set for our function. 1285 01:13:25,470 --> 01:13:27,510 And depending on the problem at hand, 1286 01:13:27,510 --> 01:13:30,630 we can choose basis sets that are more appropriate or less 1287 01:13:30,630 --> 01:13:31,560 appropriate. 1288 01:13:31,560 --> 01:13:34,350 If we are studying, say, electrons in a metal 1289 01:13:34,350 --> 01:13:37,180 that would look like free electron-like, 1290 01:13:37,180 --> 01:13:40,440 a basis set of plane waves with different wavelengths 1291 01:13:40,440 --> 01:13:41,910 is a very good choice. 1292 01:13:41,910 --> 01:13:45,240 If we are studying solutions, say, in a molecule, 1293 01:13:45,240 --> 01:13:49,110 maybe a basis set in which each state looks more 1294 01:13:49,110 --> 01:13:52,600 like atomic orbitals would be a more appropriate choice. 1295 01:13:52,600 --> 01:13:56,100 And again, we'll see this in the next classes. 1296 01:13:56,100 --> 01:14:01,510 But now, the problem has become a linear algebra problem. 1297 01:14:01,510 --> 01:14:06,640 And there is another conceptual and last step for today's class 1298 01:14:06,640 --> 01:14:08,650 that I want to highlight. 1299 01:14:08,650 --> 01:14:13,360 That is very powerful because, especially when 1300 01:14:13,360 --> 01:14:18,340 we move from trying to find the solution for a single electron 1301 01:14:18,340 --> 01:14:21,750 to finding the solution for many interacting electrons, 1302 01:14:21,750 --> 01:14:25,510 it will give us a systematic handle 1303 01:14:25,510 --> 01:14:30,010 to move in a problem that becomes conceptually 1304 01:14:30,010 --> 01:14:31,250 more and more complex. 1305 01:14:31,250 --> 01:14:34,720 We have discussed how there is this exponentially exploding 1306 01:14:34,720 --> 01:14:36,460 complexity. 1307 01:14:36,460 --> 01:14:41,950 And there is what is called a variational principle. 1308 01:14:41,950 --> 01:14:44,530 That is nothing else but reformulation 1309 01:14:44,530 --> 01:14:46,570 of the Schrodinger equation. 1310 01:14:46,570 --> 01:14:47,920 And what is this? 1311 01:14:47,920 --> 01:14:50,530 Well, what I am saying is that there 1312 01:14:50,530 --> 01:14:56,410 is what we call a functional of any arbitrary function. 1313 01:14:56,410 --> 01:14:57,670 What is a functional? 1314 01:14:57,670 --> 01:15:01,600 Well, it's something that, given a function, 1315 01:15:01,600 --> 01:15:05,730 operates on that function and produces a number. 1316 01:15:05,730 --> 01:15:10,350 So we call our functional E of phi. 1317 01:15:10,350 --> 01:15:15,090 It's an algorithm that eats as input a function 1318 01:15:15,090 --> 01:15:17,970 and gives as output a number. 1319 01:15:17,970 --> 01:15:20,670 And the explicitly definition of this algorithm 1320 01:15:20,670 --> 01:15:24,100 is given here on the right-hand side. 1321 01:15:24,100 --> 01:15:25,030 And what is it? 1322 01:15:25,030 --> 01:15:28,380 Well, for any arbitrary function, what we need 1323 01:15:28,380 --> 01:15:32,770 is to calculate the expectation value of the Hamiltonian. 1324 01:15:32,770 --> 01:15:35,430 So we need to calculate again the integral 1325 01:15:35,430 --> 01:15:41,940 over space of phi star times the Hamiltonian operator applied 1326 01:15:41,940 --> 01:15:43,200 onto phi. 1327 01:15:43,200 --> 01:15:50,440 And we divide that this by the integral of phi star times phi. 1328 01:15:50,440 --> 01:15:50,940 OK. 1329 01:15:50,940 --> 01:15:52,620 This is an operative definition. 1330 01:15:52,620 --> 01:15:53,970 I gave you a function. 1331 01:15:53,970 --> 01:15:57,490 What you obtain back by calculating this integral 1332 01:15:57,490 --> 01:16:00,060 and taking the ratio is a number. 1333 01:16:00,060 --> 01:16:04,740 But now, the important thing is, what can we prove? 1334 01:16:04,740 --> 01:16:09,630 Well, we can prove this fundamental theorem that tells 1335 01:16:09,630 --> 01:16:12,210 us, for a given Hamiltonian-- that is, 1336 01:16:12,210 --> 01:16:14,280 for a given potential-- 1337 01:16:14,280 --> 01:16:19,770 you can throw into this functional any arbitrary 1338 01:16:19,770 --> 01:16:21,000 function. 1339 01:16:21,000 --> 01:16:26,430 You can try to calculate this ratio for any phi. 1340 01:16:26,430 --> 01:16:29,220 And what you'll find is that the number 1341 01:16:29,220 --> 01:16:32,880 that you obtain for any phi that you throw in 1342 01:16:32,880 --> 01:16:38,970 is going to be always greater or equal than the ground state 1343 01:16:38,970 --> 01:16:41,130 energy of your electron. 1344 01:16:41,130 --> 01:16:44,490 Not only that, but in particular this number 1345 01:16:44,490 --> 01:16:47,700 is equal to the ground state energy of your electron, 1346 01:16:47,700 --> 01:16:51,030 the function that you have used is really the ground state wave 1347 01:16:51,030 --> 01:16:53,640 function, is the wave function corresponding 1348 01:16:53,640 --> 01:16:57,220 to the lowest energy of the Schrodinger equation. 1349 01:16:57,220 --> 01:17:01,410 And so you see this gives us a sort of trial and error 1350 01:17:01,410 --> 01:17:07,530 practical recipe to try and find out what is the ground state 1351 01:17:07,530 --> 01:17:11,730 energy of an electron in very complex potential 1352 01:17:11,730 --> 01:17:14,190 because the only thing that we have to do 1353 01:17:14,190 --> 01:17:17,970 is try a lot of wave functions. 1354 01:17:17,970 --> 01:17:22,080 And what we will obtain as our best 1355 01:17:22,080 --> 01:17:26,940 guess is the wave function that has the minimum expectation 1356 01:17:26,940 --> 01:17:28,050 value. 1357 01:17:28,050 --> 01:17:31,320 It will never be an exact solution, 1358 01:17:31,320 --> 01:17:34,650 unless we examine all the possible wave 1359 01:17:34,650 --> 01:17:36,390 functions in the world. 1360 01:17:36,390 --> 01:17:39,810 But it's a solution that becomes better and better 1361 01:17:39,810 --> 01:17:42,060 the more systematic we become. 1362 01:17:42,060 --> 01:17:45,450 The more wave functions we try, the better 1363 01:17:45,450 --> 01:17:47,460 our solution will be, the closer it 1364 01:17:47,460 --> 01:17:49,990 will be to the ground state energy. 1365 01:17:49,990 --> 01:17:53,200 And I think I'll leave it as an exercise-- 1366 01:17:53,200 --> 01:17:54,900 and this is my last slide-- 1367 01:17:54,900 --> 01:17:57,760 to be proven by you for next time. 1368 01:17:57,760 --> 01:18:01,800 And that is, I'm asking you to go back and try to prove 1369 01:18:01,800 --> 01:18:03,870 this variational principle. 1370 01:18:03,870 --> 01:18:07,680 And the only operation that you really need to do 1371 01:18:07,680 --> 01:18:11,490 is, again, expand your wave function phi 1372 01:18:11,490 --> 01:18:16,710 into a linear combination now of states 1373 01:18:16,710 --> 01:18:21,240 that I can call chi of n that are actually 1374 01:18:21,240 --> 01:18:24,120 the eigenstates of the Hamiltonian. 1375 01:18:24,120 --> 01:18:27,880 That is, we have a well-defined Hamiltonian operator. 1376 01:18:27,880 --> 01:18:31,050 So in principle, at least formally, 1377 01:18:31,050 --> 01:18:34,980 we know that it will have a set of solutions 1378 01:18:34,980 --> 01:18:40,710 that we write as H applied to this eigenfunction 1379 01:18:40,710 --> 01:18:47,220 is going to be a number times the eigenfunction itself. 1380 01:18:47,220 --> 01:18:51,630 And the set of solutions of a Hamiltonian 1381 01:18:51,630 --> 01:18:56,400 are actually in themselves a complete orthonormal set. 1382 01:18:56,400 --> 01:18:59,890 So we use that complete orthonormal 1383 01:18:59,890 --> 01:19:05,550 set to expand our wave function as a linear combination 1384 01:19:05,550 --> 01:19:07,710 of these eigenvectors. 1385 01:19:07,710 --> 01:19:13,130 And with this ansatz, wave function expanded 1386 01:19:13,130 --> 01:19:15,920 in a combination of eigenfunctions 1387 01:19:15,920 --> 01:19:20,270 of the Hamiltonian, you can go back and prove by yourself 1388 01:19:20,270 --> 01:19:23,720 the variational principle in the previous slide. 1389 01:19:23,720 --> 01:19:27,110 And that, again, is probably the most powerful principle 1390 01:19:27,110 --> 01:19:32,900 that we have to develop all the many-body solutions 1391 01:19:32,900 --> 01:19:35,360 to the electronic structure problem. 1392 01:19:35,360 --> 01:19:39,290 And we'll see that in next class on Thursday. 1393 01:19:39,290 --> 01:19:42,140 And again, all of this, if you have never 1394 01:19:42,140 --> 01:19:45,620 seen quantum mechanics, might seem very Arcane. 1395 01:19:45,620 --> 01:19:48,620 So either read some of the readings 1396 01:19:48,620 --> 01:19:50,810 that I have given you or just come and see me. 1397 01:19:50,810 --> 01:19:53,500 Fix up an appointment.