1 00:00:00,070 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,980 --> 00:00:21,890 PROFESSOR: OK. 9 00:00:21,890 --> 00:00:24,330 Let's start. 10 00:00:24,330 --> 00:00:30,470 So last lecture, what we talked about was limitations 11 00:00:30,470 --> 00:00:33,280 of classical statistical mechanics, 12 00:00:33,280 --> 00:00:37,180 and what I will contrast with what 13 00:00:37,180 --> 00:00:41,860 I will talk about today, which is new version. 14 00:00:41,860 --> 00:00:45,320 The old version of quantum mechanics, 15 00:00:45,320 --> 00:00:48,770 which was based on the observation originally from 16 00:00:48,770 --> 00:00:51,680 Planck, and then expanded by Einstein, 17 00:00:51,680 --> 00:01:01,220 that for a harmonic oscillator, a frequency omega, 18 00:01:01,220 --> 00:01:05,850 the energies cannot take all values. 19 00:01:05,850 --> 00:01:11,010 But values that are multiples of the frequency of the oscillator 20 00:01:11,010 --> 00:01:13,220 and then some integer n. 21 00:01:17,094 --> 00:01:19,890 What we did with this observation 22 00:01:19,890 --> 00:01:23,820 to extract thermal properties was to simply say 23 00:01:23,820 --> 00:01:25,600 that we will construct a partition 24 00:01:25,600 --> 00:01:28,870 function for the harmonic oscillator 25 00:01:28,870 --> 00:01:34,382 by summing over all of the states of e to the minus beta e 26 00:01:34,382 --> 00:01:38,860 n according to the formula given above. 27 00:01:38,860 --> 00:01:44,100 Just thinking that these are allowed states 28 00:01:44,100 --> 00:01:50,600 of this oscillator-- and this you can very easily do. 29 00:01:50,600 --> 00:01:53,660 It starts with the first term and then 30 00:01:53,660 --> 00:02:01,930 it's a algebraic series, which will give you that formula. 31 00:02:01,930 --> 00:02:06,720 Now, if you are sitting at some temperature t, 32 00:02:06,720 --> 00:02:10,990 you say that the average energy that you 33 00:02:10,990 --> 00:02:16,090 have in your system, well, the formula that you have is 34 00:02:16,090 --> 00:02:22,170 minus the log z by the beta, which if I apply to the z 35 00:02:22,170 --> 00:02:27,010 that I have about, essentially weighs each one of these 36 00:02:27,010 --> 00:02:30,170 by these Boltzmann weights by the corresponding energy 37 00:02:30,170 --> 00:02:31,910 and sums them. 38 00:02:31,910 --> 00:02:36,740 What we do is we get, essentially, the contribution 39 00:02:36,740 --> 00:02:38,660 of the ground state. 40 00:02:38,660 --> 00:02:44,390 Actually, for all intents and purposes, we can ignore this, 41 00:02:44,390 --> 00:02:45,870 and, hence, this. 42 00:02:45,870 --> 00:02:50,090 But for completion, let's have them around. 43 00:02:50,090 --> 00:02:52,960 And then from what is in the denominator, 44 00:02:52,960 --> 00:02:56,520 if you take derivative with respect to beta, 45 00:02:56,520 --> 00:02:59,330 you will get a factor of h bar omega. 46 00:02:59,330 --> 00:03:04,010 And then this factor of 1 minus e to the minus beta h bar. 47 00:03:06,950 --> 00:03:10,190 Actually, we then additional e to the minus beta 48 00:03:10,190 --> 00:03:11,642 h bar omega in here. 49 00:03:17,430 --> 00:03:20,860 Now, the thing that we really compare to 50 00:03:20,860 --> 00:03:25,540 was what happens if we were to take one more derivative to see 51 00:03:25,540 --> 00:03:28,870 how much the heat capacity is that we 52 00:03:28,870 --> 00:03:31,760 have in the harmonic oscillator. 53 00:03:31,760 --> 00:03:36,070 So basically taking an average of the raw formula with respect 54 00:03:36,070 --> 00:03:39,680 to temperature, realizing that these betas are 55 00:03:39,680 --> 00:03:42,970 inverse temperatures. 56 00:03:42,970 --> 00:03:46,896 So derivatives with respect to t will 57 00:03:46,896 --> 00:03:48,437 be related to derivative with respect 58 00:03:48,437 --> 00:03:52,370 to beta, except that I've will get an additional factor of 1 59 00:03:52,370 --> 00:03:54,770 over k bt squared. 60 00:03:54,770 --> 00:03:58,190 So the whole thing I could write that as kb 61 00:03:58,190 --> 00:04:07,500 and then I had the h bar omega over kt squared. 62 00:04:07,500 --> 00:04:11,570 And then from these factors, I had something like e 63 00:04:11,570 --> 00:04:20,275 to the minus e to the h bar omega over kt. 64 00:04:20,275 --> 00:04:25,860 E to the h bar omega over kt minus 1 squared. 65 00:04:29,510 --> 00:04:36,830 So if we were to plug this function, the heat 66 00:04:36,830 --> 00:04:44,570 capacity in its natural unit that are this kb, then 67 00:04:44,570 --> 00:04:51,390 as a function of temperature, we get behavior 68 00:04:51,390 --> 00:04:54,740 that we can actually express correctly 69 00:04:54,740 --> 00:04:56,160 in terms of the combination. 70 00:04:56,160 --> 00:04:58,870 You can see always we get temperature 71 00:04:58,870 --> 00:05:02,110 in units of kb over h bar omega. 72 00:05:02,110 --> 00:05:06,260 So I can really plug this in the form of, say, 73 00:05:06,260 --> 00:05:10,897 kt over h bar omega, which we call t 74 00:05:10,897 --> 00:05:12,355 to some characteristic temperature. 75 00:05:14,870 --> 00:05:22,090 And the behavior that we have is that close to 0 temperatures, 76 00:05:22,090 --> 00:05:24,450 you go to 0 exponentially, because 77 00:05:24,450 --> 00:05:27,690 of essentially the ratio of these exponentials. 78 00:05:27,690 --> 00:05:31,010 We leave one exponential in the denominator. 79 00:05:31,010 --> 00:05:34,668 So the gaps that you have between n equals to 0 80 00:05:34,668 --> 00:05:39,410 and n equals to 1 translates to behavior 81 00:05:39,410 --> 00:05:42,730 of that at low temperatures is exponentially 82 00:05:42,730 --> 00:05:45,230 decaying to leading order. 83 00:05:51,170 --> 00:05:55,040 Then, eventually, at high temperatures, 84 00:05:55,040 --> 00:06:01,260 you get the classical result where you saturate to 1. 85 00:06:01,260 --> 00:06:05,380 And so you will have a curve that 86 00:06:05,380 --> 00:06:10,350 has a shift from one behavior to another behavior. 87 00:06:10,350 --> 00:06:15,320 And the place where this transition occurs 88 00:06:15,320 --> 00:06:19,220 is when this combination is of the order of 1. 89 00:06:19,220 --> 00:06:23,030 I'm not saying it's precisely 1, but it's of the order of 1. 90 00:06:23,030 --> 00:06:25,324 So, basically, you have this kind of behavior. 91 00:06:28,782 --> 00:06:31,260 OK? 92 00:06:31,260 --> 00:06:34,740 So we use this curve to explain the heat 93 00:06:34,740 --> 00:06:39,160 capacity of diatomic gas, such as the gas in this room, 94 00:06:39,160 --> 00:06:44,800 and why at room temperature, we see a heat capacity in which it 95 00:06:44,800 --> 00:06:48,380 appears that the vibrational degrees of freedom are frozen, 96 00:06:48,380 --> 00:06:50,640 are not contributing anything. 97 00:06:50,640 --> 00:06:54,500 While at temperatures above the characteristic vibrational 98 00:06:54,500 --> 00:06:57,392 frequency, which for a gas is of the order of 10 99 00:06:57,392 --> 00:07:00,940 to the 3 degrees k, you really get 100 00:07:00,940 --> 00:07:02,680 energy in the harmonic oscillator 101 00:07:02,680 --> 00:07:04,120 also in the vibrations. 102 00:07:04,120 --> 00:07:06,530 And the heat capacity jumps, because you 103 00:07:06,530 --> 00:07:08,240 have another way of storing energy. 104 00:07:11,010 --> 00:07:27,510 So the next thing that we asked was whether this describes also 105 00:07:27,510 --> 00:07:31,990 heat capacity of a solid. 106 00:07:36,790 --> 00:07:38,860 So basically, for the diatomic gas, 107 00:07:38,860 --> 00:07:42,870 you have two atoms that are bonded together 108 00:07:42,870 --> 00:07:44,360 into a molecule. 109 00:07:44,360 --> 00:07:47,520 And you consider the vibrations of that. 110 00:07:47,520 --> 00:07:51,830 You can regard the solid as a huge molecule 111 00:07:51,830 --> 00:07:54,560 with lots of atoms joined together. 112 00:07:54,560 --> 00:07:56,720 And they have vibrations. 113 00:07:56,720 --> 00:08:00,830 And if you think about all of those vibrations giving you 114 00:08:00,830 --> 00:08:03,550 something that is similar to this, 115 00:08:03,550 --> 00:08:06,240 you would conclude that the heat capacity of a solid 116 00:08:06,240 --> 00:08:09,690 should also have this kind of behavior. 117 00:08:09,690 --> 00:08:12,440 Whereas we noted that in actuality, 118 00:08:12,440 --> 00:08:20,820 the heat capacity of a solid vanishes much more slowly 119 00:08:20,820 --> 00:08:23,230 at low temperatures. 120 00:08:23,230 --> 00:08:26,740 And the dependence at low temperatures 121 00:08:26,740 --> 00:08:28,560 is proportional to t cubed. 122 00:08:33,150 --> 00:08:36,590 So at the end of last lecture, we 123 00:08:36,590 --> 00:08:42,820 gave an explanation for this, which I will repeat. 124 00:08:42,820 --> 00:08:48,000 Again, the picture is that the solid, like a huge molecule, 125 00:08:48,000 --> 00:08:50,910 has vibrational modes. 126 00:08:50,910 --> 00:08:53,560 But these vibrational modes cover a whole range 127 00:08:53,560 --> 00:08:56,150 of different frequencies. 128 00:08:56,150 --> 00:09:04,630 And so if you ask, what are the frequencies omega 129 00:09:04,630 --> 00:09:20,510 alpha of vibrations of a solid, the most natural way 130 00:09:20,510 --> 00:09:23,640 to characterize them is, in fact, 131 00:09:23,640 --> 00:09:29,970 in terms of a wave vector k that indicates a direction 132 00:09:29,970 --> 00:09:35,270 for the oscillatory wave that you set up in the material. 133 00:09:35,270 --> 00:09:47,420 And depending on k, you'll have different frequencies. 134 00:09:47,420 --> 00:09:54,150 And I said that, essentially, the longest wave length 135 00:09:54,150 --> 00:09:58,180 corresponding to k equals to 0 is taking the whole solid 136 00:09:58,180 --> 00:09:59,340 and translating it. 137 00:09:59,340 --> 00:10:03,570 Again, thinking back about the oxygen molecule, the oxygen 138 00:10:03,570 --> 00:10:06,000 molecule, you have two coordinates. 139 00:10:06,000 --> 00:10:09,560 It's the relative coordinate that has the vibration. 140 00:10:09,560 --> 00:10:13,500 And you have a center of mass coordinate that has no energy. 141 00:10:13,500 --> 00:10:16,030 If you make a molecule more and more complicated, 142 00:10:16,030 --> 00:10:18,480 you will have more modes, but you will always 143 00:10:18,480 --> 00:10:22,480 have the 0 mode that corresponds to the translation. 144 00:10:22,480 --> 00:10:25,010 And that carries over all the way to the solid. 145 00:10:25,010 --> 00:10:28,500 So there is a mode that corresponds to translations-- 146 00:10:28,500 --> 00:10:31,420 and, in fact, rotations-- that would carry 147 00:10:31,420 --> 00:10:37,940 no energy, and corresponds, therefore, to 0 frequency. 148 00:10:37,940 --> 00:10:42,690 And then if you start to make long wavelength oscillations, 149 00:10:42,690 --> 00:10:45,980 the frequency is going to be small. 150 00:10:45,980 --> 00:10:49,460 And, indeed, what we know is that we tap on the solid 151 00:10:49,460 --> 00:10:51,830 and you create sound waves, which 152 00:10:51,830 --> 00:10:56,360 means that the low-frequency long wavelength 153 00:10:56,360 --> 00:11:00,050 modes have a dispersion relation in which omega 154 00:11:00,050 --> 00:11:02,090 is proportional to k. 155 00:11:02,090 --> 00:11:06,340 We can write that as omega is v times k, where 156 00:11:06,340 --> 00:11:10,600 v is the velocity of the sound in the solid. 157 00:11:10,600 --> 00:11:16,230 Now, of course, the shortest wave length that you can have 158 00:11:16,230 --> 00:11:20,540 is related to the separation between the atoms in the solid. 159 00:11:20,540 --> 00:11:23,190 And so, basically, there's a limit 160 00:11:23,190 --> 00:11:27,300 to the range of k's that you can put in your system. 161 00:11:27,300 --> 00:11:29,570 And this linear behavior is going 162 00:11:29,570 --> 00:11:34,170 to get modified once you get towards the age of the solid. 163 00:11:34,170 --> 00:11:39,100 And the reason I have alpha here is 164 00:11:39,100 --> 00:11:42,390 because you can have different polarizations. 165 00:11:42,390 --> 00:11:45,480 There are three different possible polarizations. 166 00:11:45,480 --> 00:11:47,820 So in principle, you will have three 167 00:11:47,820 --> 00:11:51,080 of these curves in the system hard. 168 00:11:55,210 --> 00:11:59,720 And these curves could be very complicated 169 00:11:59,720 --> 00:12:01,980 when you get to the edge of [INAUDIBLE] zone 170 00:12:01,980 --> 00:12:05,790 and you have to solve a big dynamical matrix 171 00:12:05,790 --> 00:12:09,060 in order to extract what the frequencies are, 172 00:12:09,060 --> 00:12:13,370 if you want to have the complete spectrum. 173 00:12:13,370 --> 00:12:17,770 So the solid is a collection of these harmonic oscillators that 174 00:12:17,770 --> 00:12:20,180 are, in principle, very complicated. 175 00:12:22,730 --> 00:12:26,070 But we have the following. 176 00:12:26,070 --> 00:12:28,545 So I say, OK, I have all of these. 177 00:12:28,545 --> 00:12:31,400 And I want to calculate at a given temperature 178 00:12:31,400 --> 00:12:34,480 how much energy I have put in the solid. 179 00:12:34,480 --> 00:12:38,690 So this energy that I have put in the vibrations 180 00:12:38,690 --> 00:12:43,180 at some temperature t, assuming that these vibrations are 181 00:12:43,180 --> 00:12:47,610 really a collection of these oscillators. 182 00:12:47,610 --> 00:12:52,930 Well, what I have to do is to add up all of these terms. 183 00:12:52,930 --> 00:12:56,820 There's going to be adding up all of the h bar omega over 2s 184 00:12:56,820 --> 00:12:58,920 for all of these. 185 00:12:58,920 --> 00:12:59,420 OK? 186 00:12:59,420 --> 00:13:02,500 That will give me something that I will simply call e 0, 187 00:13:02,500 --> 00:13:04,330 because it doesn't depend on temperature. 188 00:13:04,330 --> 00:13:06,950 Presumably will exist at 0 temperature. 189 00:13:06,950 --> 00:13:09,380 And I can even fold into that whatever 190 00:13:09,380 --> 00:13:13,300 the value of the potential energy of the interactions 191 00:13:13,300 --> 00:13:16,670 between the particles is at 0 temperature. 192 00:13:16,670 --> 00:13:20,690 What I'm interested in really is the temperature dependence. 193 00:13:20,690 --> 00:13:24,500 So I basically take the formula that I have over there, 194 00:13:24,500 --> 00:13:28,060 and sum over all of these oscillators. 195 00:13:28,060 --> 00:13:31,220 These oscillators are characterized by polarization 196 00:13:31,220 --> 00:13:33,780 and by the wave vector k. 197 00:13:33,780 --> 00:13:39,560 And then I have, essentially, h bar omega alpha 198 00:13:39,560 --> 00:13:48,750 of k divided by e to the beta h bar omega alpha of k minus 1. 199 00:13:48,750 --> 00:13:53,440 So I have to apply that formula to this potentially very 200 00:13:53,440 --> 00:13:57,320 complicated set of frequencies. 201 00:13:57,320 --> 00:14:01,720 The thing is, that according to the picture that I have over 202 00:14:01,720 --> 00:14:06,920 here, to a 0 order approximation, 203 00:14:06,920 --> 00:14:10,540 you would say that the heat capacity is 1 204 00:14:10,540 --> 00:14:15,190 if you are on this side, 0 if you're on that side. 205 00:14:15,190 --> 00:14:18,790 What distinguishes those two sides 206 00:14:18,790 --> 00:14:22,860 is whether the frequency in combination with temperature 207 00:14:22,860 --> 00:14:25,670 is less than or larger than 1. 208 00:14:25,670 --> 00:14:31,150 Basically, low frequencies would end up being here. 209 00:14:31,150 --> 00:14:33,610 High frequencies would end up being here. 210 00:14:33,610 --> 00:14:36,040 And would not contribute. 211 00:14:36,040 --> 00:14:41,150 So for a given temperature, there is some borderline. 212 00:14:41,150 --> 00:14:45,310 That borderline would correspond to kt over h bar. 213 00:14:45,310 --> 00:14:49,630 So let me draw where that borderline is. 214 00:14:49,630 --> 00:14:52,600 Kt over h bar. 215 00:14:52,600 --> 00:14:55,770 For a particular temperature, all of these modes 216 00:14:55,770 --> 00:14:57,930 are not really contributing. 217 00:14:57,930 --> 00:15:00,550 All of these modes are contributing. 218 00:15:00,550 --> 00:15:03,660 If my temperature is high enough, 219 00:15:03,660 --> 00:15:05,880 everything is contributing. 220 00:15:05,880 --> 00:15:09,600 And the total number of oscillators is 3 n. 221 00:15:09,600 --> 00:15:10,810 It's the number of atoms. 222 00:15:10,810 --> 00:15:13,490 So essentially, I will get 3 n times 223 00:15:13,490 --> 00:15:15,790 whatever formula I have over there. 224 00:15:15,790 --> 00:15:18,010 As a come further and further down, 225 00:15:18,010 --> 00:15:20,400 there's some kind of complicated behavior 226 00:15:20,400 --> 00:15:23,600 as I go through this spaghetti of modes. 227 00:15:23,600 --> 00:15:27,140 But when I get to low enough structures, 228 00:15:27,140 --> 00:15:29,110 then, again, things become simple, 229 00:15:29,110 --> 00:15:33,020 because I will only be sensitive to the modes that 230 00:15:33,020 --> 00:15:37,260 are described by this a omega being vk. 231 00:15:37,260 --> 00:15:38,040 OK? 232 00:15:38,040 --> 00:15:44,980 So if I'm interested in t going to 0, 233 00:15:44,980 --> 00:15:48,055 means less than some characteristic temperature 234 00:15:48,055 --> 00:15:50,890 that we have to define shortly. 235 00:15:50,890 --> 00:15:56,270 So let's say, replace this with t less than some theta 236 00:15:56,270 --> 00:16:00,140 d that I have to get for you shortly, 237 00:16:00,140 --> 00:16:06,930 then I will replace this with e 0 plus sum over alpha 238 00:16:06,930 --> 00:16:16,520 and k of h bar v alpha k e to the beta h bar e 239 00:16:16,520 --> 00:16:21,890 alpha k minus 1. 240 00:16:21,890 --> 00:16:22,390 OK? 241 00:16:26,320 --> 00:16:32,080 Now, for simplicity, essentially I 242 00:16:32,080 --> 00:16:33,700 have to do three different sums. 243 00:16:33,700 --> 00:16:36,300 All of them are the same up to having 244 00:16:36,300 --> 00:16:40,700 to use different values of v. Let's just for simplicity 245 00:16:40,700 --> 00:16:45,850 assume that all of the v alphas are the same v, 246 00:16:45,850 --> 00:16:48,570 so that I really have only one velocity. 247 00:16:48,570 --> 00:16:51,200 There's really no difficulty in generalizing this. 248 00:16:51,200 --> 00:16:55,140 So let's do this for simplicity of algebra. 249 00:16:55,140 --> 00:16:58,360 So if I do that, then the sum over alpha 250 00:16:58,360 --> 00:17:00,420 will simply give me a factor of 3. 251 00:17:00,420 --> 00:17:03,390 There are three possible polarizations, so I put a 3 252 00:17:03,390 --> 00:17:05,720 there. 253 00:17:05,720 --> 00:17:08,500 And then I have to do the summation over k. 254 00:17:08,500 --> 00:17:11,640 Well, what does the summation over k mean? 255 00:17:11,640 --> 00:17:16,440 When I have a small molecule for the, let's say, three 256 00:17:16,440 --> 00:17:19,119 or four atoms, then I can enumerate 257 00:17:19,119 --> 00:17:21,980 what the different vibrational states are. 258 00:17:21,980 --> 00:17:27,520 As I go to a large solid, I essentially 259 00:17:27,520 --> 00:17:32,930 have modes that are at each value of k, 260 00:17:32,930 --> 00:17:35,040 but, in reality, they are discrete. 261 00:17:35,040 --> 00:17:38,400 They are very, very, very, very finely separated 262 00:17:38,400 --> 00:17:41,390 by a separation that is of the order of 2 263 00:17:41,390 --> 00:17:45,110 pi over the size of the system. 264 00:17:45,110 --> 00:17:49,580 So to ensure that eventually when you count all of the modes 265 00:17:49,580 --> 00:17:51,550 that you have here, you, again, end up 266 00:17:51,550 --> 00:17:56,010 to have of the order of n states. 267 00:17:56,010 --> 00:18:00,330 So if that's the case, this sum, really I 268 00:18:00,330 --> 00:18:04,930 can replace with an integral, because going from one point 269 00:18:04,930 --> 00:18:07,320 to the next point does not make much difference. 270 00:18:07,320 --> 00:18:12,070 So I will have an integral over k. 271 00:18:12,070 --> 00:18:15,850 But I have to know how densely these things are. 272 00:18:15,850 --> 00:18:19,120 And in one direction it is 2 pi over l. 273 00:18:19,120 --> 00:18:21,840 So the density would be l over 2 pi. 274 00:18:21,840 --> 00:18:23,850 If I look at all three directions, 275 00:18:23,850 --> 00:18:26,180 I have to multiply all of them. 276 00:18:26,180 --> 00:18:28,990 So I will get v divided by 2 pi cubed. 277 00:18:28,990 --> 00:18:32,670 So this is the usual density of states. 278 00:18:32,670 --> 00:18:39,250 And you go to description in terms of wave numbers, 279 00:18:39,250 --> 00:18:42,280 or, later on, in terms of momentums. 280 00:18:42,280 --> 00:18:49,071 And what we have here is this integral h bar v k e 281 00:18:49,071 --> 00:18:53,352 to the beta h bar v k minus 1. 282 00:18:56,520 --> 00:18:58,960 OK? 283 00:18:58,960 --> 00:19:03,100 So let's simplify this a little bit further. 284 00:19:03,100 --> 00:19:04,880 I have e 0. 285 00:19:04,880 --> 00:19:05,730 I have 3v. 286 00:19:05,730 --> 00:19:12,650 The integrand only depends on the magnitude of k, 287 00:19:12,650 --> 00:19:15,610 so I can take advantage of that spherical symmetry 288 00:19:15,610 --> 00:19:22,140 and write this as 4 pi k squared v k divided by this 8 pi cubed. 289 00:19:25,660 --> 00:19:31,200 What I can do is I can also introduce 290 00:19:31,200 --> 00:19:36,270 a factor of beta here, multiplied by k t. 291 00:19:36,270 --> 00:19:39,530 Beta k t is 1. 292 00:19:39,530 --> 00:19:45,890 And if I call this combination to be x, then what 293 00:19:45,890 --> 00:19:54,560 I have is k b t x e to the x minus 1. 294 00:19:58,280 --> 00:20:02,995 Of course, k is simply related to x by k 295 00:20:02,995 --> 00:20:17,970 being x kt over h bar v. 296 00:20:17,970 --> 00:20:20,700 And so at the next level of approximation, 297 00:20:20,700 --> 00:20:25,210 this k squared v k I will write in terms of x squared v x. 298 00:20:25,210 --> 00:20:26,550 And so what do I have? 299 00:20:26,550 --> 00:20:29,080 I have e 0. 300 00:20:29,080 --> 00:20:33,965 I have 3v divided by 2 pi squared. 301 00:20:38,600 --> 00:20:42,040 Because of this factor of kt that I will take outside 302 00:20:42,040 --> 00:20:44,890 I have a kt. 303 00:20:44,890 --> 00:20:47,270 I have a k squared vk that I want 304 00:20:47,270 --> 00:20:49,350 to replace with x squared v x. 305 00:20:49,350 --> 00:20:51,640 And that will give me an additional factor 306 00:20:51,640 --> 00:20:56,970 of kv over h bar v cubed. 307 00:20:56,970 --> 00:21:01,020 And then I have an integral 0 to e 0 v 308 00:21:01,020 --> 00:21:06,930 x x cubed e to the x minus 1. 309 00:21:13,035 --> 00:21:17,470 Now, in principle, when I start with this integration, 310 00:21:17,470 --> 00:21:23,670 I have a finite range for k, which presumably 311 00:21:23,670 --> 00:21:27,680 would translate into a finite range for x. 312 00:21:27,680 --> 00:21:31,880 But in reality none of these modes is contributing, 313 00:21:31,880 --> 00:21:34,430 so I could extend the range of integration 314 00:21:34,430 --> 00:21:38,300 all the way to infinity, and make very small error 315 00:21:38,300 --> 00:21:40,460 at low temperatures. 316 00:21:40,460 --> 00:21:43,100 And the advantage of that is that then this 317 00:21:43,100 --> 00:21:47,160 becomes a definite integral. 318 00:21:47,160 --> 00:21:50,110 Something that you can look up in tables. 319 00:21:50,110 --> 00:21:54,083 And its value is in fact pi to the fourth over 15. 320 00:21:57,870 --> 00:22:02,090 So substituting that over there, what do we have? 321 00:22:02,090 --> 00:22:12,900 We have that the energy is e 0 plus 3 divided by 15, 322 00:22:12,900 --> 00:22:17,530 will give me 5, which turns the 2 into a 10. 323 00:22:20,480 --> 00:22:23,990 I have pi to the fourth divided by pi squared, 324 00:22:23,990 --> 00:22:28,450 so there's a pi squared that will survive out here. 325 00:22:28,450 --> 00:22:31,515 I have a kt. 326 00:22:31,515 --> 00:22:39,990 I have kt over h bar v cubed. 327 00:22:39,990 --> 00:22:41,980 And then I have a factor of volume. 328 00:22:41,980 --> 00:22:46,280 But volume is proportional to the number of particles 329 00:22:46,280 --> 00:22:50,180 that I have in the system times the size of my unit cell. 330 00:22:50,180 --> 00:22:52,760 Let's call that a cubed. 331 00:22:52,760 --> 00:22:57,230 So this I can write this as l a cubed. 332 00:22:57,230 --> 00:23:00,900 Why do I do that is because when I then take the derivative, 333 00:23:00,900 --> 00:23:05,250 I'd like to write the heat capacity per particle. 334 00:23:05,250 --> 00:23:12,726 So, indeed, if I now take the derivative, which is de by dt, 335 00:23:12,726 --> 00:23:18,210 the answer will be proportional to n and kv. 336 00:23:18,210 --> 00:23:22,000 The number of particles and this k v, which 337 00:23:22,000 --> 00:23:28,460 is the function, the unit of heat capacities. 338 00:23:28,460 --> 00:23:32,180 The overall dependence is t to the fourth. 339 00:23:32,180 --> 00:23:35,800 So when I take derivatives, I will get 4t cubed. 340 00:23:35,800 --> 00:23:43,830 That 4 will change the 1 over 10 to 2 pi squared over 5. 341 00:23:43,830 --> 00:23:49,360 And then I have the combination kvt h bar 342 00:23:49,360 --> 00:23:54,540 v, and that factor of a raised to the third power. 343 00:23:54,540 --> 00:23:57,950 So the whole thing is proportional to t cubed. 344 00:23:57,950 --> 00:24:04,090 And the coefficient I will call theta d for [INAUDIBLE]. 345 00:24:04,090 --> 00:24:11,695 And theta d I have calculated to be h bar v over a h 346 00:24:11,695 --> 00:24:14,790 bar v a over k t. 347 00:24:14,790 --> 00:24:17,276 No, h bar v over a k t. 348 00:24:30,940 --> 00:24:35,520 So the heat capacity of the solid 349 00:24:35,520 --> 00:24:38,960 is going to be proportional, of course, to n k b. 350 00:24:38,960 --> 00:24:42,620 But most importantly, is proportional to t cubed. 351 00:24:42,620 --> 00:24:46,480 And t cubed just came from this argument 352 00:24:46,480 --> 00:24:48,860 that I need low omegas. 353 00:24:48,860 --> 00:24:51,310 And how many things I have at the omega. 354 00:24:51,310 --> 00:24:55,180 How many frequencies do I have that are vibrating? 355 00:24:55,180 --> 00:24:57,550 The number of those frequencies is essentially 356 00:24:57,550 --> 00:25:01,760 the size of a cube in k space. 357 00:25:01,760 --> 00:25:07,040 So it goes like this-- maximum k cubed in three dimensions. 358 00:25:07,040 --> 00:25:09,970 In two dimensions, it will be squared and all of that. 359 00:25:09,970 --> 00:25:14,110 So it's very easy to figure out from this dispersion relation 360 00:25:14,110 --> 00:25:17,180 what the low temperature behavior of the heat capacity 361 00:25:17,180 --> 00:25:18,350 has to be. 362 00:25:18,350 --> 00:25:21,990 And you will see that this is, in fact, predictive, 363 00:25:21,990 --> 00:25:24,930 in that later on in the course, we 364 00:25:24,930 --> 00:25:29,300 will come an example of where the heat capacity of a liquid, 365 00:25:29,300 --> 00:25:34,610 which was helium, was observed to have this t cubed behavior 366 00:25:34,610 --> 00:25:38,635 based on that Landau immediately postulated that there should be 367 00:25:38,635 --> 00:25:42,290 a phonon-like dispersion inside that superfluid. 368 00:25:44,970 --> 00:25:45,470 OK. 369 00:25:48,000 --> 00:25:52,980 So that's the story of the heat capacity of the solid. 370 00:25:52,980 --> 00:25:55,770 So we started with a molecule. 371 00:25:55,770 --> 00:26:00,470 We went from a molecule into an entire solid. 372 00:26:00,470 --> 00:26:04,480 The next step that what I'm going to do is I'm 373 00:26:04,480 --> 00:26:08,140 going to remove the solid and just keep the box. 374 00:26:14,640 --> 00:26:16,720 So essentially, they calculation that I 375 00:26:16,720 --> 00:26:19,530 did, if you think about it, corresponded 376 00:26:19,530 --> 00:26:24,000 to having some kind of a box, and having 377 00:26:24,000 --> 00:26:27,820 vibrational modes inside the box. 378 00:26:27,820 --> 00:26:31,670 But let's imagine that it is an empty box. 379 00:26:31,670 --> 00:26:36,170 But we know that even in empty space we have light. 380 00:26:36,170 --> 00:26:39,960 So within an empty box, we can still 381 00:26:39,960 --> 00:26:43,420 have modes of the electromagnetic field. 382 00:26:47,430 --> 00:26:51,590 Modes of electromagnetic field, just like the modes 383 00:26:51,590 --> 00:26:54,930 of the solid, we can characterize 384 00:26:54,930 --> 00:26:58,790 by the direction along which oscillations travel. 385 00:27:01,490 --> 00:27:08,220 And whereas for the atoms in the solid, they have displacement 386 00:27:08,220 --> 00:27:13,170 and the corresponding momentum for the electromagnetic field, 387 00:27:13,170 --> 00:27:15,580 you have the electric field. 388 00:27:15,580 --> 00:27:19,130 And its conjugate is the magnetic field. 389 00:27:19,130 --> 00:27:21,720 And these things will be oscillating 390 00:27:21,720 --> 00:27:23,310 to create for you a wave. 391 00:27:27,030 --> 00:27:31,390 Except that, whereas for the solid, for each atom 392 00:27:31,390 --> 00:27:33,410 we had three possible directions, 393 00:27:33,410 --> 00:27:37,965 and therefore we had three branches, for this, since e 394 00:27:37,965 --> 00:27:40,690 and b have to be orthogonal to k, 395 00:27:40,690 --> 00:27:42,710 you really have only two polarizations. 396 00:27:51,440 --> 00:27:55,520 But apart from that, the frequency spectrum 397 00:27:55,520 --> 00:27:58,270 is exactly the same as we would have 398 00:27:58,270 --> 00:28:01,290 for the solids at low temperature replacing 399 00:28:01,290 --> 00:28:07,090 to v that we have with the speed of light. 400 00:28:07,090 --> 00:28:12,060 And so you would say, OK. 401 00:28:12,060 --> 00:28:16,570 If I were to calculate the energy content that 402 00:28:16,570 --> 00:28:21,740 is inside the box, what I have to do 403 00:28:21,740 --> 00:28:28,160 is to sum over all of the modes and polarizations. 404 00:28:31,150 --> 00:28:35,550 Regarding each one of these as a harmonic oscillator, 405 00:28:35,550 --> 00:28:38,430 going through the system of quantizing according 406 00:28:38,430 --> 00:28:41,860 to this old quantum mechanics, the harmonic oscillators, 407 00:28:41,860 --> 00:28:44,660 I have to add up the energy content of each oscillator. 408 00:28:44,660 --> 00:28:53,920 And so what I have is this h bar omega of k. 409 00:28:53,920 --> 00:28:58,470 And then I have 1/2 plus 1 over e 410 00:28:58,470 --> 00:29:02,910 to the beta h bar omega of k minus 1. 411 00:29:06,280 --> 00:29:08,890 And then I can do exactly the kinds of things 412 00:29:08,890 --> 00:29:12,880 that I had before, replacing the sum over k 413 00:29:12,880 --> 00:29:16,280 with a v times an integral. 414 00:29:16,280 --> 00:29:19,370 So the whole thing would be, first of all, 415 00:29:19,370 --> 00:29:22,960 proportional to v, going from the sum over k 416 00:29:22,960 --> 00:29:24,610 to the integration over k. 417 00:29:27,410 --> 00:29:33,570 I would have to add all of these h bar omega over 2s. 418 00:29:33,570 --> 00:29:37,290 Has no temperature dependence, so let me just, again, call it 419 00:29:37,290 --> 00:29:39,150 some e 0. 420 00:29:39,150 --> 00:29:40,800 Actually, let's call it epsilon 0, 421 00:29:40,800 --> 00:29:44,820 because it's more like an energy density. 422 00:29:44,820 --> 00:29:48,850 And then I have the sum over all of the other modes. 423 00:29:48,850 --> 00:29:50,200 There's two polarisations. 424 00:29:50,200 --> 00:29:54,260 So as opposed to the three that I had before, I have two. 425 00:29:54,260 --> 00:30:00,150 I have, again, the integral over k of 4 pi k 426 00:30:00,150 --> 00:30:04,600 squared v k divided by 8 pi cubed, which 427 00:30:04,600 --> 00:30:08,370 is part of this density of state calculation. 428 00:30:08,370 --> 00:30:17,030 I have, again, a factor of h bar omega. 429 00:30:17,030 --> 00:30:21,500 Now, I realize that my omega is ck. 430 00:30:21,500 --> 00:30:25,130 So I simply write it as h bar ck. 431 00:30:25,130 --> 00:30:30,850 And then I have e to the beta h bar ck minus 1. 432 00:30:37,490 --> 00:30:44,710 So we will again allow this to go from 0 to infinity. 433 00:30:44,710 --> 00:30:47,010 And what do we get? 434 00:30:47,010 --> 00:30:55,890 We will get v epsilon 0 plus, well, the 8 and 8 cancel. 435 00:30:55,890 --> 00:30:59,570 I have pi over pi squared. 436 00:30:59,570 --> 00:31:01,640 So pi over pi cubed. 437 00:31:01,640 --> 00:31:03,445 So it will give me 1 over pi squared. 438 00:31:07,170 --> 00:31:10,175 I have one factor of kt. 439 00:31:13,310 --> 00:31:16,460 Again, when I introduce here a beta 440 00:31:16,460 --> 00:31:20,279 and then multiply by kt, so that this dimension, 441 00:31:20,279 --> 00:31:21,320 this combination appears. 442 00:31:25,330 --> 00:31:29,305 Then I have, if I were to change variable and call 443 00:31:29,305 --> 00:31:33,730 this the new variable, I have factor 444 00:31:33,730 --> 00:31:39,010 of k squared dk, which gives me, just as 445 00:31:39,010 --> 00:31:43,870 before over there, a factor of kt over h bar c cubed. 446 00:31:49,150 --> 00:31:53,710 And then I have this integral left, 447 00:31:53,710 --> 00:32:01,780 which is the 0 to infinity v x x cubed e to the x minus 1, 448 00:32:01,780 --> 00:32:06,194 which we stated is pi to the fourth over 15. 449 00:32:11,810 --> 00:32:19,130 So the part that is dependent on temperature, the energy 450 00:32:19,130 --> 00:32:24,450 content, just as in this case, scales as t to the fourth. 451 00:32:27,590 --> 00:32:31,540 There is one part that we have over here 452 00:32:31,540 --> 00:32:35,975 from all of the 0s, which is, in fact, an infinity. 453 00:32:39,540 --> 00:32:44,260 And maybe there is some degree of worry about that. 454 00:32:44,260 --> 00:32:49,940 We didn't have to worry about that infinity in this case, 455 00:32:49,940 --> 00:32:55,420 because the number of modes that we had was, in reality, finite. 456 00:32:55,420 --> 00:33:01,190 So once we were to add up properly all of these 0 point 457 00:33:01,190 --> 00:33:04,050 energies for this, we would have gotten a finite number. 458 00:33:04,050 --> 00:33:07,540 It would have been large, but it would have been finite. 459 00:33:07,540 --> 00:33:11,990 Whereas here, the difference is that there is really 460 00:33:11,990 --> 00:33:14,170 no upper cut-off. 461 00:33:14,170 --> 00:33:21,780 So this k here, for a solid, you have a minimum wavelength. 462 00:33:21,780 --> 00:33:25,710 You can't do things shorter than the separation of particles. 463 00:33:25,710 --> 00:33:30,860 But for light, you can have arbitrarily short wavelength, 464 00:33:30,860 --> 00:33:34,690 and that gives you this infinity over here. 465 00:33:34,690 --> 00:33:37,310 So typically, we ignore that. 466 00:33:37,310 --> 00:33:40,640 Maybe it is related to the cosmological constant, et 467 00:33:40,640 --> 00:33:41,540 cetera. 468 00:33:41,540 --> 00:33:46,090 But for our purposes, we are not going to focus on that at all. 469 00:33:46,090 --> 00:33:48,150 And the interesting part is this part, 470 00:33:48,150 --> 00:33:52,110 that proportional to t to the fourth. 471 00:33:52,110 --> 00:33:55,090 There are two SOP calculations to this 472 00:33:55,090 --> 00:33:58,280 that I will just give part of the answer, 473 00:33:58,280 --> 00:34:00,110 because another part of the answer 474 00:34:00,110 --> 00:34:05,730 is something that you do in problem sets. 475 00:34:05,730 --> 00:34:09,989 One of them is that what we have here is an energy density. 476 00:34:09,989 --> 00:34:12,260 It's proportional to volume. 477 00:34:12,260 --> 00:34:14,989 And we have seen that energy densities 478 00:34:14,989 --> 00:34:17,770 are related to pressures. 479 00:34:17,770 --> 00:34:20,134 So indeed, there is a corresponding pressure. 480 00:34:23,030 --> 00:34:26,139 That is, if you're at the temperature t, 481 00:34:26,139 --> 00:34:33,190 this collection of vibrating electromagnetic fields 482 00:34:33,190 --> 00:34:37,429 exerts a pressure on the walls of the container. 483 00:34:37,429 --> 00:34:44,120 This pressure is related to energy density. 484 00:34:44,120 --> 00:34:49,870 The factor of 1/3 comes because of the dispersion relation. 485 00:34:49,870 --> 00:34:52,620 And you can show that in one of the problem sets. 486 00:34:52,620 --> 00:34:54,719 You know that already. 487 00:34:54,719 --> 00:34:56,400 So that would say that you would have, 488 00:34:56,400 --> 00:35:00,200 essentially, something like some kind of p 0. 489 00:35:02,930 --> 00:35:07,860 And then something that is proportional to t 490 00:35:07,860 --> 00:35:09,190 to the fourth. 491 00:35:09,190 --> 00:35:12,760 So I guess the correspondent coefficient here 492 00:35:12,760 --> 00:35:28,200 would be p squared divided by 45 kt kt over h bar c cubed. 493 00:35:28,200 --> 00:35:31,620 So there is radiation pressure that 494 00:35:31,620 --> 00:35:33,790 is proportional to temperature. 495 00:35:33,790 --> 00:35:38,160 The hotter you make this box, the more pressure it will get 496 00:35:38,160 --> 00:35:40,560 exerted from it. 497 00:35:40,560 --> 00:35:42,730 There is, of course, again this infinity 498 00:35:42,730 --> 00:35:45,140 that you may worry about. 499 00:35:45,140 --> 00:35:48,640 But here the problem is less serious, 500 00:35:48,640 --> 00:35:51,280 because you would say that in reality, 501 00:35:51,280 --> 00:35:53,980 if I have the wall of the box, it 502 00:35:53,980 --> 00:35:57,740 is going to get pressure from both sides. 503 00:35:57,740 --> 00:36:00,437 And if there's an infinite pressure from both sides, 504 00:36:00,437 --> 00:36:01,145 they will cancel. 505 00:36:01,145 --> 00:36:04,390 So you don't have to worry about that. 506 00:36:04,390 --> 00:36:06,450 But it turns out that, actually, you 507 00:36:06,450 --> 00:36:10,470 can measure the consequences of this pressure. 508 00:36:10,470 --> 00:36:16,450 And that occurs when rather than having one plate, 509 00:36:16,450 --> 00:36:19,380 you have two plates that there are 510 00:36:19,380 --> 00:36:22,940 some small separation apart. 511 00:36:22,940 --> 00:36:25,040 Then the modes of radiation that you 512 00:36:25,040 --> 00:36:29,830 can fit in here because of the quantizations that you have, 513 00:36:29,830 --> 00:36:34,470 are different from the modes that you can have out here. 514 00:36:34,470 --> 00:36:39,160 So that difference, even from the 0 point fluctuations-- 515 00:36:39,160 --> 00:36:42,840 the h bar omega over 2s-- will give you 516 00:36:42,840 --> 00:36:46,700 a pressure that pushes these plates together. 517 00:36:46,700 --> 00:36:50,280 That's called a Casimir force, or Casimir pressure. 518 00:36:50,280 --> 00:36:54,520 And that was predicted by Casimir in 1950s, 519 00:36:54,520 --> 00:36:59,150 and was measured experimentally roughly 10 years ago 520 00:36:59,150 --> 00:37:03,070 to high precision, matching the formula that we had. 521 00:37:03,070 --> 00:37:06,050 So sometimes, these infinities have 522 00:37:06,050 --> 00:37:09,330 consequences that you have to worry about. 523 00:37:09,330 --> 00:37:12,580 But that's also to indicate that there's 524 00:37:12,580 --> 00:37:16,250 kind of modern physics to this. 525 00:37:16,250 --> 00:37:20,590 But really it was the origin of quantum mechanics, 526 00:37:20,590 --> 00:37:24,580 because of the other aspect of the physics, which 527 00:37:24,580 --> 00:37:29,200 is imagine that again you have this box. 528 00:37:29,200 --> 00:37:32,760 I draw it now as an irregular box. 529 00:37:32,760 --> 00:37:37,740 And I open a hole of size a inside the box. 530 00:37:37,740 --> 00:37:41,990 And then the radiation that was inside at temperatures t 531 00:37:41,990 --> 00:37:46,400 will start to go out. 532 00:37:46,400 --> 00:37:48,070 So you have a hot box. 533 00:37:48,070 --> 00:37:49,350 You open a hole in it. 534 00:37:49,350 --> 00:37:52,450 And then the radiation starts to come out. 535 00:37:52,450 --> 00:37:58,103 And so what you will have is a flux of radiation. 536 00:38:01,670 --> 00:38:08,390 Flux means that this it energy that is escaping per unit area 537 00:38:08,390 --> 00:38:11,320 and per unit time. 538 00:38:11,320 --> 00:38:18,010 So there's a flux, which is per area per time. 539 00:38:18,010 --> 00:38:21,610 It turns out that that flux-- and this 540 00:38:21,610 --> 00:38:25,750 is another factor, this factor of 1/3 541 00:38:25,750 --> 00:38:29,030 that I mentioned-- is related the energy 542 00:38:29,030 --> 00:38:34,500 density with a factor of 1 c over 4. 543 00:38:34,500 --> 00:38:39,730 Essentially, clearly the velocity with which energy 544 00:38:39,730 --> 00:38:43,210 escaping is proportional to c. 545 00:38:43,210 --> 00:38:47,760 So you will get more radiation flux the larger c. 546 00:38:47,760 --> 00:38:50,470 The answer has to be proportional to c. 547 00:38:50,470 --> 00:38:53,250 And it is what is inside that is escaping, 548 00:38:53,250 --> 00:38:56,350 so it has to be proportional to the energy density 549 00:38:56,350 --> 00:39:02,270 that you have inside, some kind of energy per unit volume. 550 00:39:02,270 --> 00:39:08,000 And the factor of 1/4 is one of these geometric factors. 551 00:39:08,000 --> 00:39:10,920 Essentially, there's two factors of cosine of theta. 552 00:39:10,920 --> 00:39:13,960 And you have to do an average of cosine squared theta. 553 00:39:13,960 --> 00:39:18,720 And that will give you the additional 1/4. 554 00:39:18,720 --> 00:39:19,770 OK? 555 00:39:19,770 --> 00:39:23,620 But rather than looking-- so this 556 00:39:23,620 --> 00:39:26,700 would tell you that there is an energy that is streaming out. 557 00:39:26,700 --> 00:39:29,510 That is, the net value is proportional to t 558 00:39:29,510 --> 00:39:30,820 to the fourth. 559 00:39:30,820 --> 00:39:35,040 But more interestingly, we can ask 560 00:39:35,040 --> 00:39:39,880 what is the flux per wavelength? 561 00:39:39,880 --> 00:39:42,980 And so for that, I can just go back to the formula 562 00:39:42,980 --> 00:39:46,260 before I integrated over k, and ask 563 00:39:46,260 --> 00:39:51,440 what is the energy density in each interval of k? 564 00:39:51,440 --> 00:39:56,350 And so what I have to do is to just go and look at the formula 565 00:39:56,350 --> 00:40:00,190 that I have prior to doing the integration over k. 566 00:40:00,190 --> 00:40:02,720 Multiply it by c over 4. 567 00:40:02,720 --> 00:40:04,130 What do I have? 568 00:40:04,130 --> 00:40:09,460 I have 8 pi divided by 8 pi cubed. 569 00:40:09,460 --> 00:40:14,550 I have a factor of k squared from the density of states. 570 00:40:14,550 --> 00:40:23,270 I have this factor of h bar c k divided by e to the beta h 571 00:40:23,270 --> 00:40:25,350 bar c k minus 1. 572 00:40:31,920 --> 00:40:38,190 So there's no analogue of this, because I am not doing 573 00:40:38,190 --> 00:40:41,220 the integration over k. 574 00:40:41,220 --> 00:40:45,470 So we can simplify some of these factors up front. 575 00:40:45,470 --> 00:40:49,000 But really, the story is how does this quantity 576 00:40:49,000 --> 00:40:53,970 look as the function of wave number, which 577 00:40:53,970 --> 00:40:57,500 is the inverse of wave length, if you like. 578 00:40:57,500 --> 00:41:02,480 And what we see is that when k goes to 0, essentially, 579 00:41:02,480 --> 00:41:05,740 this factor into the beta h bar ck 580 00:41:05,740 --> 00:41:08,210 I have to expand to lowest order. 581 00:41:08,210 --> 00:41:12,430 I will get beta h bar c k, because the 1 disappears. 582 00:41:12,430 --> 00:41:15,160 H bar ck is cancelled, so the answer is going to be 583 00:41:15,160 --> 00:41:17,070 proportional to inverse beta. 584 00:41:17,070 --> 00:41:20,840 It's going to be proportional to kt and k squared. 585 00:41:20,840 --> 00:41:25,830 So, essentially, the low k behavior part of this 586 00:41:25,830 --> 00:41:32,650 is proportional to k squared c, of course, and kt. 587 00:41:38,780 --> 00:41:43,300 However, when I go to the high k numbers, 588 00:41:43,300 --> 00:41:46,190 the exponential will kill things off. 589 00:41:46,190 --> 00:41:49,740 So the large k part of this is going 590 00:41:49,740 --> 00:41:51,370 to be exponentially small. 591 00:41:55,570 --> 00:41:59,580 And, actually, the curve will look something like this, 592 00:41:59,580 --> 00:42:02,400 therefore. 593 00:42:02,400 --> 00:42:08,770 It will have a maximum around the k, which presumably 594 00:42:08,770 --> 00:42:17,100 is of the order of kt over h bar c. 595 00:42:20,080 --> 00:42:26,410 So basically, the hotter you have, 596 00:42:26,410 --> 00:42:29,990 this will move to the right. 597 00:42:29,990 --> 00:42:34,060 The wavelengths will become shorter. 598 00:42:34,060 --> 00:42:39,890 And, essentially, that's the origin of the fact that can you 599 00:42:39,890 --> 00:42:45,200 heat some kind of material, it will start to emit radiation. 600 00:42:45,200 --> 00:42:48,720 And the radiation will be peaked at some frequency that 601 00:42:48,720 --> 00:42:52,210 is related to its temperature. 602 00:42:52,210 --> 00:42:56,910 Now, if we didn't have this quantization effect, 603 00:42:56,910 --> 00:43:01,960 if h bar went to 0, then what would happen 604 00:43:01,960 --> 00:43:07,720 is that this k squared kt would continue forever. 605 00:43:07,720 --> 00:43:08,380 OK? 606 00:43:08,380 --> 00:43:10,590 Essentially, you would have in each one 607 00:43:10,590 --> 00:43:13,470 of these modes of radiation, classically, 608 00:43:13,470 --> 00:43:15,610 you would put a kt of energy. 609 00:43:15,610 --> 00:43:19,520 And since you could have arbitrarily short wavelengths, 610 00:43:19,520 --> 00:43:23,170 you would have infinite energy at shorter and shorter 611 00:43:23,170 --> 00:43:24,080 wavelengths. 612 00:43:24,080 --> 00:43:26,370 And you would have this ultraviolet catastrophe. 613 00:43:31,500 --> 00:43:33,750 Of course, the shape of this curve 614 00:43:33,750 --> 00:43:38,490 was experimentally known towards the end of the 19th century. 615 00:43:38,490 --> 00:43:42,020 And so that was the basis of thinking about it, 616 00:43:42,020 --> 00:43:43,980 and fitting an exponential to the end, 617 00:43:43,980 --> 00:43:47,980 and eventually deducing that this quantization 618 00:43:47,980 --> 00:43:50,900 of the oscillators would potentially give you 619 00:43:50,900 --> 00:43:52,375 the reason for this to happen. 620 00:43:57,610 --> 00:44:01,060 Now, the way that I have described it, 621 00:44:01,060 --> 00:44:05,430 I focused on having a cavity and opening the cavity, 622 00:44:05,430 --> 00:44:08,720 and having the energy go out. 623 00:44:08,720 --> 00:44:12,260 Of course, the experiments for black body 624 00:44:12,260 --> 00:44:13,870 are not done on cavities. 625 00:44:13,870 --> 00:44:17,430 They're done on some piece of metal or some other thing 626 00:44:17,430 --> 00:44:18,580 that you heat up. 627 00:44:18,580 --> 00:44:21,730 And then you can look at the spectrum of the radiation. 628 00:44:21,730 --> 00:44:24,590 And so, again, there is some universality 629 00:44:24,590 --> 00:44:29,070 in this, that it is not so sensitive to the properties 630 00:44:29,070 --> 00:44:31,650 of the material, although there are 631 00:44:31,650 --> 00:44:34,010 some emissivity and other factors that 632 00:44:34,010 --> 00:44:36,900 multiply the final result. 633 00:44:36,900 --> 00:44:38,820 So the final result, in fact, would 634 00:44:38,820 --> 00:44:42,860 say that if I were to integrate over frequencies, 635 00:44:42,860 --> 00:44:53,340 the total radiation flux, which would be c over 4 times 636 00:44:53,340 --> 00:44:57,190 the energy density total, is going 637 00:44:57,190 --> 00:44:59,070 to be proportional to temperature 638 00:44:59,070 --> 00:45:01,730 to the fourth power. 639 00:45:01,730 --> 00:45:16,010 And this constant in front is the Stefan-Boltzmann, 640 00:45:16,010 --> 00:45:23,570 which has some particular value that you can look up, 641 00:45:23,570 --> 00:45:28,066 units of watts per area per degrees Kelvin. 642 00:45:33,870 --> 00:45:40,300 So this perspective is rather macroscopic. 643 00:45:40,300 --> 00:45:44,970 The radiated energy is proportional to the surface 644 00:45:44,970 --> 00:45:46,720 area. 645 00:45:46,720 --> 00:45:50,130 If you make things that are small, 646 00:45:50,130 --> 00:45:53,610 and the wavelengths that you're looking at over here become 647 00:45:53,610 --> 00:45:56,160 compatible to the size of the object, 648 00:45:56,160 --> 00:45:59,160 these formulas break down. 649 00:45:59,160 --> 00:46:04,000 And again, go forward about 150 years or so, 650 00:46:04,000 --> 00:46:08,120 there is ongoing research-- I guess more 200 years-- 651 00:46:08,120 --> 00:46:11,430 ongoing research on-- no, 100 and something-- 652 00:46:11,430 --> 00:46:18,550 ongoing research on how these classical laws of radiation 653 00:46:18,550 --> 00:46:22,340 are modified when you're dealing with objects that are small 654 00:46:22,340 --> 00:46:25,011 compared to the wavelengths that are emitted, etc. 655 00:46:35,002 --> 00:46:35,585 Any questions? 656 00:46:38,950 --> 00:46:47,810 So the next part of the story is why did you do all of this? 657 00:46:47,810 --> 00:46:51,400 It works, but what is the justification? 658 00:46:54,960 --> 00:47:00,960 In that I said there was the old quantum mechanics. 659 00:47:00,960 --> 00:47:06,840 But really, we want to have statements about quantum 660 00:47:06,840 --> 00:47:11,050 systems that are not harmonic oscillators. 661 00:47:11,050 --> 00:47:14,890 And we want to be able to understand actually 662 00:47:14,890 --> 00:47:18,540 what the underlying basis is in the same way 663 00:47:18,540 --> 00:47:21,190 that they understand how we were doing things 664 00:47:21,190 --> 00:47:25,420 for classical statistical mechanics. 665 00:47:25,420 --> 00:47:28,660 And so really, we want to look at how 666 00:47:28,660 --> 00:47:32,564 to make the transition from classical to quantum 667 00:47:32,564 --> 00:47:33,480 statistical mechanics. 668 00:47:39,850 --> 00:47:44,910 So for that, let's go and remind us. 669 00:47:44,910 --> 00:47:47,010 Actually, so basically the question 670 00:47:47,010 --> 00:47:49,920 is something like this-- what does 671 00:47:49,920 --> 00:47:53,810 this partition function mean? 672 00:47:53,810 --> 00:47:57,110 I'm calculating things as if I have 673 00:47:57,110 --> 00:47:59,590 these states that are the energy levels. 674 00:47:59,590 --> 00:48:05,080 And the probabilities are e to the minus beta epsilon n. 675 00:48:05,080 --> 00:48:06,320 What does that mean? 676 00:48:06,320 --> 00:48:09,640 Classically, we knew the Boltzmann rates 677 00:48:09,640 --> 00:48:12,390 had something to do with the probability of finding 678 00:48:12,390 --> 00:48:17,500 a particle with a particular position and momentum. 679 00:48:17,500 --> 00:48:20,340 So what is the analogous thing here? 680 00:48:20,340 --> 00:48:23,570 And you know that in new quantum mechanics, 681 00:48:23,570 --> 00:48:26,830 the interpretation of many things is probabilistic. 682 00:48:29,350 --> 00:48:32,490 And in statistical mechanics, even classically 683 00:48:32,490 --> 00:48:35,610 we had a probabilistic interpretation. 684 00:48:35,610 --> 00:48:40,020 So presumably, we want to build a probabilistic theory 685 00:48:40,020 --> 00:48:43,580 on top of another probabilistic theory. 686 00:48:43,580 --> 00:48:46,480 So how do we go about understanding 687 00:48:46,480 --> 00:48:49,950 precisely what is happening over here? 688 00:48:49,950 --> 00:48:53,840 So let's kind of remind ourselves 689 00:48:53,840 --> 00:48:58,400 of what we were doing in the original classical statistical 690 00:48:58,400 --> 00:49:01,740 mechanics, and try to see how we can make 691 00:49:01,740 --> 00:49:03,930 the corresponding calculations when 692 00:49:03,930 --> 00:49:06,900 things are quantum mechanical. 693 00:49:06,900 --> 00:49:11,210 So, essentially, the probabilistic sense 694 00:49:11,210 --> 00:49:15,150 that we had in classical statistical mechanics 695 00:49:15,150 --> 00:49:18,790 was to assign probabilities for micro states, given 696 00:49:18,790 --> 00:49:21,950 that we had some knowledge of the macro state. 697 00:49:21,950 --> 00:49:39,350 So the classical microstate mu was a point 698 00:49:39,350 --> 00:49:44,368 which was a collection of p's and q's in phase space. 699 00:49:51,690 --> 00:50:01,000 So what is a quantum microstate? 700 00:50:01,000 --> 00:50:01,500 OK. 701 00:50:01,500 --> 00:50:13,160 So here, I'm just going to jump several decades ahead, and just 702 00:50:13,160 --> 00:50:14,570 write the answer. 703 00:50:14,570 --> 00:50:17,560 And I'm going to do it in somewhat 704 00:50:17,560 --> 00:50:21,369 of a more axiomatic way, because it's not up to me 705 00:50:21,369 --> 00:50:22,660 to introduce quantum mechanics. 706 00:50:22,660 --> 00:50:25,190 I assume that you know it already. 707 00:50:25,190 --> 00:50:28,010 Just a perspective that I'm going to take. 708 00:50:28,010 --> 00:50:40,020 So the quantum microstate is a complex unit vector 709 00:50:40,020 --> 00:50:40,880 in Hilbert space. 710 00:50:47,810 --> 00:50:49,440 OK? 711 00:50:49,440 --> 00:51:00,770 So for any vector space, we can choose a set of unit vectors 712 00:51:00,770 --> 00:51:02,710 that form an orthonormal basis. 713 00:51:09,028 --> 00:51:13,630 And I'm going to use this bra-ket notation. 714 00:51:13,630 --> 00:51:20,580 And so our psi, which is a vector in this space, 715 00:51:20,580 --> 00:51:25,680 can be written in terms of its components 716 00:51:25,680 --> 00:51:30,160 by pointing to the different directions in this space, 717 00:51:30,160 --> 00:51:37,790 and components that I will indicate by this psi n. 718 00:51:37,790 --> 00:51:40,650 And these are complex. 719 00:51:45,500 --> 00:51:51,810 And I will use the notation that psi n 720 00:51:51,810 --> 00:51:55,630 is the complex conjugate of n psi. 721 00:51:58,980 --> 00:52:06,480 And the norm of this I'm going to indicate by psi psi, which 722 00:52:06,480 --> 00:52:12,260 is obtained by summing over all n psi psi n star, which 723 00:52:12,260 --> 00:52:17,280 is essentially the magnitude of n psi squared. 724 00:52:17,280 --> 00:52:20,190 And these are unit vectors. 725 00:52:20,190 --> 00:52:23,240 So all of these states are normalized such 726 00:52:23,240 --> 00:52:25,960 that psi psi is equal to 1. 727 00:52:31,130 --> 00:52:32,260 Yes. 728 00:52:32,260 --> 00:52:35,397 AUDIENCE: You're not allowing particle numbers 729 00:52:35,397 --> 00:52:36,770 to vary, are you? 730 00:52:36,770 --> 00:52:38,150 PROFESSOR: At this stage, no. 731 00:52:38,150 --> 00:52:40,820 Later on, when we do the grand canonical, 732 00:52:40,820 --> 00:52:43,850 we will change our Hilbert space. 733 00:52:58,480 --> 00:53:00,220 OK? 734 00:53:00,220 --> 00:53:03,030 So that's one concept. 735 00:53:03,030 --> 00:53:06,820 The other concept, classically, we measure things. 736 00:53:06,820 --> 00:53:08,690 So we have classical observable. 737 00:53:17,100 --> 00:53:23,720 And these are functions all of which 738 00:53:23,720 --> 00:53:27,750 depend on this p and q in phase space. 739 00:53:33,510 --> 00:53:36,510 So basically, there's the phase space. 740 00:53:36,510 --> 00:53:39,270 We can have some particular function, 741 00:53:39,270 --> 00:53:42,670 such as the kinetic energy-- sum over i pi squared 742 00:53:42,670 --> 00:53:46,220 over 2 n-- that's an example of an observable. 743 00:53:46,220 --> 00:53:49,920 Kinetic energy, potential energy, anything that we like, 744 00:53:49,920 --> 00:53:52,440 you can classically write a sum function 745 00:53:52,440 --> 00:53:54,880 that you want to evaluate in phase space, 746 00:53:54,880 --> 00:53:57,740 given that you are at some particular point in phase 747 00:53:57,740 --> 00:54:02,720 space, the state of your system, you can evaluate what that is. 748 00:54:02,720 --> 00:54:10,850 Now in quantum mechanics, observables 749 00:54:10,850 --> 00:54:29,260 are operators, or matrices, if you like, in this vector space. 750 00:54:29,260 --> 00:54:30,650 OK? 751 00:54:30,650 --> 00:54:37,310 So among the various observables, 752 00:54:37,310 --> 00:54:39,210 certainly, are things like the position 753 00:54:39,210 --> 00:54:41,760 and the momentum of the particle. 754 00:54:41,760 --> 00:54:43,670 So there are presumably matrices that 755 00:54:43,670 --> 00:54:47,850 correspond to position and momentum. 756 00:54:47,850 --> 00:54:52,830 And for that, we look at some other properties 757 00:54:52,830 --> 00:54:56,650 that this classical systems have. 758 00:54:56,650 --> 00:55:03,210 We had defined classically a Poisson bracket, 759 00:55:03,210 --> 00:55:10,940 which was a sum over all alphas d a by d q alpha 760 00:55:10,940 --> 00:55:21,145 d b by d p alpha minus the a by the p alpha b d by the q f. 761 00:55:21,145 --> 00:55:21,645 OK? 762 00:55:24,500 --> 00:55:29,210 And this is an operation that you 763 00:55:29,210 --> 00:55:33,050 would like to, and happens to, carry over in some sense 764 00:55:33,050 --> 00:55:37,220 into quantum mechanics. 765 00:55:37,220 --> 00:55:40,000 But one of the consequence of this 766 00:55:40,000 --> 00:55:45,470 is you can check if I pick a particular momentum, key, 767 00:55:45,470 --> 00:55:52,840 and a particular coordinate, q, and put it over here, most 768 00:55:52,840 --> 00:55:57,140 of the time I will get 0, unless the alphas match 769 00:55:57,140 --> 00:56:00,540 exactly the p q's that I have up there. 770 00:56:00,540 --> 00:56:02,940 And if you go through this whole thing, 771 00:56:02,940 --> 00:56:08,610 I will get something like that is like a delta i j. 772 00:56:08,610 --> 00:56:10,610 OK. 773 00:56:10,610 --> 00:56:16,590 So this structure somehow continues in quantum mechanics, 774 00:56:16,590 --> 00:56:22,050 in this sense that the matrices that correspond to p and q 775 00:56:22,050 --> 00:56:33,240 satisfy the condition that p i and q j, 776 00:56:33,240 --> 00:56:38,200 thinking of two matrices, and this is the commutator, 777 00:56:38,200 --> 00:56:50,786 so this is p i q j minus q j p i is h bar over i delta h. 778 00:56:57,570 --> 00:57:04,710 So once you have the matrices that correspond to p and q, 779 00:57:04,710 --> 00:57:11,370 you can take any function of p and q that you had over here, 780 00:57:11,370 --> 00:57:16,510 and then replace the p's and q's that appear in, let's say, 781 00:57:16,510 --> 00:57:20,020 a series expansion, or an expansion of this o in powers 782 00:57:20,020 --> 00:57:24,840 of p and q, with corresponding matrices p hat and q hat. 783 00:57:24,840 --> 00:57:28,980 And that way, you will construct a corresponding operator. 784 00:57:28,980 --> 00:57:32,240 There is one subtlety that you've probably encountered, 785 00:57:32,240 --> 00:57:34,480 in that there is some symmetrization 786 00:57:34,480 --> 00:57:38,130 that you have to do before you can make this replacement. 787 00:57:40,770 --> 00:57:42,070 OK. 788 00:57:42,070 --> 00:57:45,386 So what does it mean? 789 00:57:45,386 --> 00:57:51,130 In classical theory, if something is observable 790 00:57:51,130 --> 00:57:52,720 the answer that you get is a number. 791 00:57:52,720 --> 00:57:53,220 Right? 792 00:57:53,220 --> 00:57:56,180 You can calculate what the kinetic energy is. 793 00:57:56,180 --> 00:57:59,030 In quantum mechanics, what does it 794 00:57:59,030 --> 00:58:03,600 mean that observable is a matrix? 795 00:58:03,600 --> 00:58:06,530 The statement is that observables 796 00:58:06,530 --> 00:58:15,210 don't have definite values, but the expectation value 797 00:58:15,210 --> 00:58:22,310 of a particular observable o in some state pi 798 00:58:22,310 --> 00:58:26,340 is given by psi o psi. 799 00:58:26,340 --> 00:58:31,010 Essentially, you take the vector that correspond to the state, 800 00:58:31,010 --> 00:58:34,520 multiply the matrix on it, and then sandwich it 801 00:58:34,520 --> 00:58:36,850 with the conjugate of the vector, 802 00:58:36,850 --> 00:58:38,365 and that will give you your state. 803 00:58:38,365 --> 00:58:43,880 So in terms of elements of some particular basis, 804 00:58:43,880 --> 00:58:47,610 you would write this as a sum over n and m. 805 00:58:47,610 --> 00:58:53,200 Psi n n o m m psi. 806 00:58:57,280 --> 00:59:02,820 And in that particular basis, your operator 807 00:59:02,820 --> 00:59:04,630 would have these matrix elements. 808 00:59:07,160 --> 00:59:10,970 Now again, another property, if you're 809 00:59:10,970 --> 00:59:13,630 measuring something that is observable, 810 00:59:13,630 --> 00:59:17,200 is presumably you will get a number that is real. 811 00:59:17,200 --> 00:59:21,450 That is, you expect this to be the same thing 812 00:59:21,450 --> 00:59:24,790 as its complex conjugate. 813 00:59:24,790 --> 00:59:27,940 And if you follow this condition, 814 00:59:27,940 --> 00:59:34,490 you will see that that reality implies that n o m should 815 00:59:34,490 --> 00:59:41,940 m o n complex conjugate, which is typically written 816 00:59:41,940 --> 00:59:46,039 as the matrix being its Hermitian conjugate, 817 00:59:46,039 --> 00:59:46,830 or being Hermitian. 818 00:59:51,190 --> 00:59:54,510 So all observables in quantum mechanics 819 00:59:54,510 --> 00:59:58,665 would correspond to Hermitian operators or matrices. 820 01:00:01,761 --> 01:00:02,260 OK. 821 01:00:04,810 --> 01:00:06,730 There's one other piece, and then we 822 01:00:06,730 --> 01:00:14,430 can forget about axioms. 823 01:00:14,430 --> 01:00:19,950 We have a classical time evolution. 824 01:00:26,930 --> 01:00:30,530 We know that the particular point in the classical phase 825 01:00:30,530 --> 01:00:33,410 space changes as a function of time, such 826 01:00:33,410 --> 01:00:40,440 that q i dot is plus d h by d p i. 827 01:00:40,440 --> 01:00:46,190 P i dot is minus d h by d q i. 828 01:00:46,190 --> 01:00:49,920 By the way, both of these can be written as q i 829 01:00:49,920 --> 01:00:55,630 and h Poisson bracket and p i and h Poisson bracket. 830 01:00:55,630 --> 01:01:01,720 But there is a particular function observable, h, 831 01:01:01,720 --> 01:01:07,660 the Hamiltonian, that derives the classical evolution. 832 01:01:07,660 --> 01:01:13,800 And when we go to quantum evolution, 833 01:01:13,800 --> 01:01:19,480 this vector that we have in Hilbert space evolves according 834 01:01:19,480 --> 01:01:26,550 to i h bar d by dt of the vector psi 835 01:01:26,550 --> 01:01:30,716 is the matrix that we have acting on psi. 836 01:01:34,900 --> 01:01:36,541 OK. 837 01:01:36,541 --> 01:01:37,040 Fine. 838 01:01:37,040 --> 01:01:40,584 So these are the basics that we need. 839 01:01:40,584 --> 01:01:45,880 Now we can go and do statistical descriptions. 840 01:01:53,100 --> 01:02:04,540 So the main element that we had in constructing statistical 841 01:02:04,540 --> 01:02:07,666 descriptions was deal with a macrostate. 842 01:02:15,850 --> 01:02:18,440 We said that if I'm interested in thinking 843 01:02:18,440 --> 01:02:22,760 about the properties of one cubic meter of gas 844 01:02:22,760 --> 01:02:25,960 at standard temperature and pressure, 845 01:02:25,960 --> 01:02:29,340 I'm not thinking about a particular point in phase 846 01:02:29,340 --> 01:02:32,020 space, because different gases that 847 01:02:32,020 --> 01:02:35,300 have exactly the same macroscopic properties would 848 01:02:35,300 --> 01:02:38,100 correspond to many, many different possible points 849 01:02:38,100 --> 01:02:42,060 in this phase space that are changing as a function of time. 850 01:02:42,060 --> 01:02:47,850 So rather than thinking about a single microstate, 851 01:02:47,850 --> 01:02:50,740 we talked about an ensemble. 852 01:02:50,740 --> 01:03:00,280 And this ensemble had a whole bunch of possible microstates. 853 01:03:00,280 --> 01:03:02,340 In the simplest prescription, maybe we 854 01:03:02,340 --> 01:03:06,280 said they were all equally likely. 855 01:03:06,280 --> 01:03:09,560 But, actually, we could even assign some kind of probability 856 01:03:09,560 --> 01:03:10,060 to them. 857 01:03:13,740 --> 01:03:19,430 And we want to know what to do with this, because then what 858 01:03:19,430 --> 01:03:24,130 happened was from this description, 859 01:03:24,130 --> 01:03:31,150 we then constructed a density which was, again, 860 01:03:31,150 --> 01:03:36,170 some kind of a probability in phase space. 861 01:03:36,170 --> 01:03:38,690 And we looked at its time evolution. 862 01:03:38,690 --> 01:03:42,140 We looked at the averages and all kinds of things 863 01:03:42,140 --> 01:03:45,250 in terms of this density. 864 01:03:45,250 --> 01:03:50,340 So the question is, what happens to all of this 865 01:03:50,340 --> 01:03:55,070 when we go to quantum descriptions? 866 01:03:55,070 --> 01:03:55,930 OK. 867 01:03:55,930 --> 01:03:58,700 You can follow a lot of that. 868 01:03:58,700 --> 01:04:04,640 We can, again, take the example of the one cubic meter of gas 869 01:04:04,640 --> 01:04:07,430 at standard temperature and pressure. 870 01:04:07,430 --> 01:04:11,380 But the rather than describing the state of the system 871 01:04:11,380 --> 01:04:13,750 classically, I can try to describe it 872 01:04:13,750 --> 01:04:15,930 quantum mechanically. 873 01:04:15,930 --> 01:04:18,140 Presumably the quantum mechanical description 874 01:04:18,140 --> 01:04:21,930 at some limit becomes equivalent to the classical description. 875 01:04:21,930 --> 01:04:28,005 So I will have an ensemble of states. 876 01:04:30,820 --> 01:04:33,940 I don't know which one of them I am in. 877 01:04:33,940 --> 01:04:35,360 I have lots of boxes. 878 01:04:35,360 --> 01:04:39,710 They would correspond to different microstates, 879 01:04:39,710 --> 01:04:41,480 presumably. 880 01:04:41,480 --> 01:04:45,790 And this has, actually, a word that 881 01:04:45,790 --> 01:04:48,010 is used more in the quantum context. 882 01:04:48,010 --> 01:04:50,400 I guess one could use it in the classical context. 883 01:04:50,400 --> 01:04:53,570 It's called a mixed state. 884 01:04:53,570 --> 01:04:57,270 A pure state is one you know exactly. 885 01:04:57,270 --> 01:05:00,740 Mixed state is, well, like the gas I tell you. 886 01:05:00,740 --> 01:05:03,430 I tell you only the macroscopic information, 887 01:05:03,430 --> 01:05:06,340 you don't know much about microscopically what it is. 888 01:05:06,340 --> 01:05:09,470 If these are possibilities, and not knowing 889 01:05:09,470 --> 01:05:11,510 those possibilities, you can say that it's 890 01:05:11,510 --> 01:05:14,960 a mixture of all these states. 891 01:05:14,960 --> 01:05:16,830 OK. 892 01:05:16,830 --> 01:05:21,490 Now, what would I use, classically, a density for? 893 01:05:21,490 --> 01:05:24,700 What I could do is I could calculate 894 01:05:24,700 --> 01:05:34,530 the average of some observable, classically, in this ensemble. 895 01:05:34,530 --> 01:05:37,990 And what I would do is I would integrate 896 01:05:37,990 --> 01:05:43,470 over the entirety of the six n-dimensional phase space 897 01:05:43,470 --> 01:05:47,850 the o at some particular point in phase space 898 01:05:47,850 --> 01:05:51,540 and the density at that point in phase space. 899 01:05:51,540 --> 01:05:56,920 And this average I will indicate by a bar. 900 01:05:56,920 --> 01:06:03,770 So my bars stand for ensemble average, 901 01:06:03,770 --> 01:06:08,100 to make them distinct from these quantum averages 902 01:06:08,100 --> 01:06:11,880 that I will indicate with the Bra-Kets. 903 01:06:11,880 --> 01:06:14,180 OK? 904 01:06:14,180 --> 01:06:20,880 So let's try to do the analogue of that in quantum states. 905 01:06:20,880 --> 01:06:25,740 I would say that, OK, for a particular one of the members 906 01:06:25,740 --> 01:06:28,410 of this ensemble, I can calculate 907 01:06:28,410 --> 01:06:30,410 what the expectation value is. 908 01:06:33,260 --> 01:06:37,310 This is the expectation value that 909 01:06:37,310 --> 01:06:42,800 corresponds to this observable o, if I was in a pure state psi 910 01:06:42,800 --> 01:06:44,620 alpha. 911 01:06:44,620 --> 01:06:48,370 But I don't know that I am there. 912 01:06:48,370 --> 01:06:52,780 I have a probability, so I do a summation 913 01:06:52,780 --> 01:06:54,850 over all of these states. 914 01:06:54,850 --> 01:06:58,660 And I will call that expectation value ensemble average. 915 01:07:03,930 --> 01:07:07,770 So that's how things are defined. 916 01:07:07,770 --> 01:07:11,280 Let's look at this in some particular basis. 917 01:07:11,280 --> 01:07:19,780 I would write this as a sum over alpha m and n p alpha psi 918 01:07:19,780 --> 01:07:28,350 alpha m m o n n psi alpha. 919 01:07:28,350 --> 01:07:33,060 So, essentially, writing all of these psis 920 01:07:33,060 --> 01:07:38,290 in terms of their components, just as I had done above. 921 01:07:38,290 --> 01:07:39,920 OK. 922 01:07:39,920 --> 01:07:44,080 Now what I want to do is to reorder this. 923 01:07:44,080 --> 01:07:48,350 Do the summation over n and m first, 924 01:07:48,350 --> 01:07:50,380 the summation over alpha last. 925 01:07:50,380 --> 01:07:52,395 So what do I have? 926 01:07:52,395 --> 01:07:57,670 I have m o n. 927 01:07:57,670 --> 01:08:06,850 And then I have a sum over alpha of p alpha n psi alpha 928 01:08:06,850 --> 01:08:07,810 psi alpha n. 929 01:08:14,290 --> 01:08:23,359 So what I will do, this quantity that alpha is summed over-- 930 01:08:23,359 --> 01:08:26,939 so it depends on the two indices n and m. 931 01:08:26,939 --> 01:08:28,550 I can give it a name. 932 01:08:28,550 --> 01:08:31,430 I can call it n rho m. 933 01:08:35,550 --> 01:08:41,600 If I do that, then this o bar average becomes simply-- 934 01:08:41,600 --> 01:08:43,819 let's see. 935 01:08:43,819 --> 01:08:50,560 This summation over n gives me the matrix product o rho. 936 01:08:50,560 --> 01:08:54,790 And then summation over m gives me the trace of the product. 937 01:08:54,790 --> 01:08:57,221 So this is the trace of rho o. 938 01:08:59,987 --> 01:09:02,300 OK? 939 01:09:02,300 --> 01:09:09,850 So I constructed something that is 940 01:09:09,850 --> 01:09:16,260 kind of analogous to the classical use 941 01:09:16,260 --> 01:09:20,930 of the density in phase space. 942 01:09:20,930 --> 01:09:23,850 So you would multiply the density and the thing 943 01:09:23,850 --> 01:09:28,310 that you wanted to calculate the observable. 944 01:09:28,310 --> 01:09:31,290 And the ensemble average is obtained 945 01:09:31,290 --> 01:09:34,710 as a kind of summing over all possible what values 946 01:09:34,710 --> 01:09:37,550 of that product in phase space. 947 01:09:37,550 --> 01:09:40,729 So here, I'm doing something similar. 948 01:09:40,729 --> 01:09:45,109 I'm multiplying this o by some matrix row. 949 01:09:45,109 --> 01:09:52,370 So, again, this I can think of as having introduced 950 01:09:52,370 --> 01:09:55,630 a new matrix for an operator. 951 01:09:55,630 --> 01:09:57,510 And this is the density matrix. 952 01:10:07,520 --> 01:10:12,910 And if I basically ignore, or write it 953 01:10:12,910 --> 01:10:16,610 in basis-independent form, it is obtained 954 01:10:16,610 --> 01:10:20,620 by summing over all alphas the alphas, 955 01:10:20,620 --> 01:10:23,710 and, essentially, cutting off the n and m. 956 01:10:23,710 --> 01:10:31,710 I have the matrix that I would form out of state alpha by, 957 01:10:31,710 --> 01:10:35,370 essentially, taking the vector and its conjugate 958 01:10:35,370 --> 01:10:41,710 and multiplying rows and columns together to make a matrix. 959 01:10:41,710 --> 01:10:44,790 And then multiplying or averaging that matrix 960 01:10:44,790 --> 01:10:48,270 over all possible values of the ensemble-- 961 01:10:48,270 --> 01:10:52,640 elements of the ensemble-- would give me this density. 962 01:10:52,640 --> 01:10:59,060 So in the same way that any observable 963 01:10:59,060 --> 01:11:07,200 in classical mechanics goes over to an operator in quantum 964 01:11:07,200 --> 01:11:10,920 mechanics, we find that we have another function 965 01:11:10,920 --> 01:11:13,260 in phase space-- this density. 966 01:11:13,260 --> 01:11:17,320 This density goes over to the matrix or an operator that 967 01:11:17,320 --> 01:11:18,770 is given by this formula here. 968 01:11:21,330 --> 01:11:26,958 It is useful to enumerate some properties of the density 969 01:11:26,958 --> 01:11:27,458 matrix. 970 01:11:44,940 --> 01:11:55,150 First of all, the density matrix is positive definite. 971 01:11:55,150 --> 01:11:56,492 What does that mean? 972 01:11:56,492 --> 01:12:01,430 It means that if you take the density matrix, 973 01:12:01,430 --> 01:12:07,780 multiply it by any state on the right and the left 974 01:12:07,780 --> 01:12:12,780 to construct a number, this number will be positive, 975 01:12:12,780 --> 01:12:16,570 because if I apply it to the formula that I have over there, 976 01:12:16,570 --> 01:12:19,750 this is simply sum over alpha p alpha. 977 01:12:19,750 --> 01:12:25,910 Then I have phi psi psi alpha, and them 978 01:12:25,910 --> 01:12:29,440 psi alpha psi, which is its complex conjugate. 979 01:12:29,440 --> 01:12:34,690 So I get the norm of that product, which is positive. 980 01:12:34,690 --> 01:12:39,140 All of the p alphas are positive probabilities. 981 01:12:39,140 --> 01:12:43,890 So this is certainly something that is positive. 982 01:12:43,890 --> 01:12:50,780 We said that anything that makes sense in quantum mechanics 983 01:12:50,780 --> 01:12:51,850 should be Hermitian. 984 01:12:55,720 --> 01:12:58,630 And it is easy to check. 985 01:12:58,630 --> 01:13:02,350 That if I take this operator rho and do 986 01:13:02,350 --> 01:13:06,010 the complex conjugate, essentially what happens 987 01:13:06,010 --> 01:13:08,540 is that I have to take sum over alpha. 988 01:13:08,540 --> 01:13:11,520 Complex conjugate of p alpha is p alpha itself. 989 01:13:11,520 --> 01:13:14,270 Probabilities are real numbers. 990 01:13:14,270 --> 01:13:19,940 If I take psi alpha psi alpha and conjugate it, essentially 991 01:13:19,940 --> 01:13:21,510 I take this and put it here. 992 01:13:21,510 --> 01:13:23,450 And I take that and put it over there. 993 01:13:23,450 --> 01:13:24,985 And I get the same thing. 994 01:13:24,985 --> 01:13:26,620 So it's the same thing as rho. 995 01:13:29,920 --> 01:13:32,706 And, finally, there's a normalization. 996 01:13:41,720 --> 01:13:48,170 If, for my o over here in the last formula, I choose 1, 997 01:13:48,170 --> 01:13:52,790 then I get the expectation value of 1 998 01:13:52,790 --> 01:13:57,050 has to be the trace of rho. 999 01:13:57,050 --> 01:14:00,250 And we can check that the trace of rho, 1000 01:14:00,250 --> 01:14:02,760 essentially, is obtained by summing over 1001 01:14:02,760 --> 01:14:08,960 all alpha p alpha, and the dot product of the two psi alphas. 1002 01:14:08,960 --> 01:14:13,280 Since any state in quantum mechanics 1003 01:14:13,280 --> 01:14:17,660 corresponds to a unit vector, this is 1. 1004 01:14:17,660 --> 01:14:21,532 So I get a sum over alpha of p alphas. 1005 01:14:21,532 --> 01:14:24,280 And these are probabilities assigned 1006 01:14:24,280 --> 01:14:25,830 to the members of the ensemble. 1007 01:14:25,830 --> 01:14:28,060 They have to add up to 1. 1008 01:14:28,060 --> 01:14:30,010 And so this is like this. 1009 01:14:32,840 --> 01:14:40,960 So the quantity that we were looking at, 1010 01:14:40,960 --> 01:14:46,600 and built, essentially, all of our later classical statistical 1011 01:14:46,600 --> 01:14:50,500 mechanics, on is this density. 1012 01:14:50,500 --> 01:14:54,630 Density was a probability in phase space. 1013 01:14:54,630 --> 01:14:56,810 Now, when you go to quantum mechanics, 1014 01:14:56,810 --> 01:14:58,360 we don't have phase space. 1015 01:14:58,360 --> 01:15:00,650 We have Hilbert space. 1016 01:15:00,650 --> 01:15:03,680 We already have a probabilistic theory. 1017 01:15:03,680 --> 01:15:07,450 Turns out that this function, which 1018 01:15:07,450 --> 01:15:10,240 was the probability in phase space classically, 1019 01:15:10,240 --> 01:15:15,080 gets promoted to this matrix, the density matrix, that has, 1020 01:15:15,080 --> 01:15:17,990 once you take traces and do all kinds of things, 1021 01:15:17,990 --> 01:15:21,020 the kinds of properties that you would expect the probability 1022 01:15:21,020 --> 01:15:22,660 to have classically. 1023 01:15:22,660 --> 01:15:26,200 But it's not really probability in the usual sense. 1024 01:15:26,200 --> 01:15:28,810 It's a matrix. 1025 01:15:28,810 --> 01:15:31,060 OK. 1026 01:15:31,060 --> 01:15:37,350 There is one other thing element of this to go through, 1027 01:15:37,350 --> 01:15:44,450 which is that classically, we said 1028 01:15:44,450 --> 01:15:49,670 that, OK, I pick a set of states. 1029 01:15:49,670 --> 01:15:52,170 They correspond to some density. 1030 01:15:52,170 --> 01:15:55,960 But the microstates are changing as a function of time. 1031 01:15:55,960 --> 01:15:59,510 So the density was changing as a function of time. 1032 01:15:59,510 --> 01:16:07,190 And we had Liouville's theorem, which 1033 01:16:07,190 --> 01:16:15,625 stated that d rho by d t was the Poisson bracket 1034 01:16:15,625 --> 01:16:17,810 of the Hamiltonian with rho. 1035 01:16:20,860 --> 01:16:24,700 So we can quantum mechanically ask, 1036 01:16:24,700 --> 01:16:28,630 what happens to our density matrix? 1037 01:16:28,630 --> 01:16:33,290 So we have a matrix rho. 1038 01:16:33,290 --> 01:16:38,400 I can ask, what is the time derivative of that matrix? 1039 01:16:38,400 --> 01:16:44,870 And, actually, I will insert the i h bar here, 1040 01:16:44,870 --> 01:16:51,130 because I anticipate that, essentially, 1041 01:16:51,130 --> 01:16:56,820 rho having that form, what I will have is sum over alpha. 1042 01:16:56,820 --> 01:17:06,870 And then I have i h bar d by dt acting on these p alpha psi 1043 01:17:06,870 --> 01:17:08,560 alpha psi f. 1044 01:17:11,080 --> 01:17:15,860 So there rho is sum over alpha p alpha psi alpha psi alpha. 1045 01:17:15,860 --> 01:17:19,110 Sum over alpha p alpha I can take outside. 1046 01:17:19,110 --> 01:17:23,520 I h bar d by dt acts on these two psis 1047 01:17:23,520 --> 01:17:26,590 that are appearing a complex conjugates. 1048 01:17:26,590 --> 01:17:30,180 So it can either, d by dt, act on one or the other. 1049 01:17:30,180 --> 01:17:33,240 So I can write this as sum over alpha p 1050 01:17:33,240 --> 01:17:45,320 alpha i h bar d by dt acting on psi alpha psi alpha, or i, 1051 01:17:45,320 --> 01:17:56,466 or psi alpha, and then i h bar d by dt acting on this psi alpha. 1052 01:18:02,370 --> 01:18:08,060 Now, i h bar d by dt psi alpha, we 1053 01:18:08,060 --> 01:18:13,620 said that, essentially, the quantum rule for time evolution 1054 01:18:13,620 --> 01:18:16,490 is i h bar d by dt of the state will 1055 01:18:16,490 --> 01:18:21,656 give you is governed by h times acting on psi alpha. 1056 01:18:24,830 --> 01:18:31,220 If I were to take the complex conjugate of this expression, 1057 01:18:31,220 --> 01:18:35,690 what I would get is minus i h bar 1058 01:18:35,690 --> 01:18:39,020 d by dt acting on psi that is pointing 1059 01:18:39,020 --> 01:18:42,080 the other way of our complex conjugation 1060 01:18:42,080 --> 01:18:49,770 is h acting on the psi in the opposite way. 1061 01:18:49,770 --> 01:19:05,493 So this thing is minus psi alpha with h f acting on it. 1062 01:19:09,270 --> 01:19:10,520 OK. 1063 01:19:10,520 --> 01:19:22,020 So then I can write the whole thing as h-- for the first term 1064 01:19:22,020 --> 01:19:24,230 take the h out front. 1065 01:19:24,230 --> 01:19:28,140 I have a sum over alpha p alpha psi alpha 1066 01:19:28,140 --> 01:19:37,000 psi alpha, minus, from this complex conjugation-- here, h 1067 01:19:37,000 --> 01:19:40,740 is completely to the right-- I have a sum over alpha 1068 01:19:40,740 --> 01:19:45,380 p alpha psi alpha psi alpha. 1069 01:19:45,380 --> 01:19:46,969 And then we have h. 1070 01:19:49,690 --> 01:19:51,825 Now, these are again getting rho back. 1071 01:19:56,463 --> 01:20:04,490 So what I have established is that i h bar, the time 1072 01:20:04,490 --> 01:20:08,420 derivative of this density matrix, 1073 01:20:08,420 --> 01:20:15,800 is simply the commutator of the operators h and o. 1074 01:20:20,960 --> 01:20:31,140 So what we had up here was the classical Liouville theorem, 1075 01:20:31,140 --> 01:20:35,050 relating the time derivative of the density in phase space 1076 01:20:35,050 --> 01:20:37,860 to the Poisson bracket with h. 1077 01:20:37,860 --> 01:20:41,430 What we have here is the quantum version, where 1078 01:20:41,430 --> 01:20:45,220 the time derivative of this density matrix 1079 01:20:45,220 --> 01:20:47,580 is the commutator of rho with h. 1080 01:20:52,340 --> 01:20:59,550 Now we are done, because what did we use this Liouville for? 1081 01:20:59,550 --> 01:21:04,290 We used it to deduce that if I have things that are not 1082 01:21:04,290 --> 01:21:10,090 changing as a function of time, I have equilibrium systems, 1083 01:21:10,090 --> 01:21:12,590 where the density is invariant. 1084 01:21:12,590 --> 01:21:14,180 It's the same. 1085 01:21:14,180 --> 01:21:21,490 Then rho of equilibrium not changing as a function of time 1086 01:21:21,490 --> 01:21:27,440 can be achieved by simply making it a function of h. 1087 01:21:27,440 --> 01:21:32,940 And, more precisely, h and conserved quantities 1088 01:21:32,940 --> 01:21:36,100 that have 0 Poisson bracket with h. 1089 01:21:43,700 --> 01:21:47,330 How can I make the quantum density matrix 1090 01:21:47,330 --> 01:21:50,810 to be invariant of time? 1091 01:21:50,810 --> 01:21:54,440 All I need to do is to ensure that the Poisson bracket 1092 01:21:54,440 --> 01:21:58,350 of that density with the Hamiltonian is 0. 1093 01:21:58,350 --> 01:22:01,300 Not the Poisson bracket, the commutator. 1094 01:22:01,300 --> 01:22:04,560 Clearly, the commutator of h with itself is 0. 1095 01:22:04,560 --> 01:22:07,510 Hh minus hh is 0. 1096 01:22:07,510 --> 01:22:12,390 So this I can make a function of h, 1097 01:22:12,390 --> 01:22:15,270 and any other kind of quantity also 1098 01:22:15,270 --> 01:22:18,610 that has 0 commutator with h. 1099 01:22:18,610 --> 01:22:22,100 So, essentially, the quantum version also applies. 1100 01:22:22,100 --> 01:22:26,810 That is, the quantum version of rho equilibrium, 1101 01:22:26,810 --> 01:22:30,350 I can make it by constructing something 1102 01:22:30,350 --> 01:22:34,420 that depends on the Hilbert space 1103 01:22:34,420 --> 01:22:36,460 through the dependence of the Hamiltonian 1104 01:22:36,460 --> 01:22:41,730 and on the Hilbert space and any other conserved quantities that 1105 01:22:41,730 --> 01:22:45,436 have 0 commutator also with the Hamiltonian. 1106 01:22:49,440 --> 01:22:53,130 So now, what we will do next time 1107 01:22:53,130 --> 01:22:55,900 is we can pick and choose whatever rho equilibrium 1108 01:22:55,900 --> 01:22:56,680 we had before. 1109 01:22:56,680 --> 01:22:59,560 Canonical e to the minus beta h. 1110 01:22:59,560 --> 01:23:02,930 We make this matrix to be e to the minus beta h. 1111 01:23:02,930 --> 01:23:06,710 Uniform anything, we can just carry over 1112 01:23:06,710 --> 01:23:10,160 whatever functional dependence we had here to here. 1113 01:23:10,160 --> 01:23:13,240 And we are ensure to have something 1114 01:23:13,240 --> 01:23:16,280 that is quantum mechanically invariant. 1115 01:23:16,280 --> 01:23:21,120 And we will then interpret what the various quantities 1116 01:23:21,120 --> 01:23:25,620 calculated through that density matrix and the formulas 1117 01:23:25,620 --> 01:23:29,060 that we described actually. 1118 01:23:29,060 --> 01:23:30,575 OK?