1 00:00:00,060 --> 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 00:00:22,410 --> 00:00:24,010 PROFESSOR: Begin with a new topic, 9 00:00:24,010 --> 00:00:28,260 which is breakdown of classical statistical mechanics. 10 00:00:43,750 --> 00:00:47,990 So we developed a formalism to probabilistically describe 11 00:00:47,990 --> 00:00:50,590 collections of large particles. 12 00:00:50,590 --> 00:00:56,560 And once we have that, from that formalism calculate properties 13 00:00:56,560 --> 00:01:00,060 of matter that have to do with heat, temperature, 14 00:01:00,060 --> 00:01:02,210 et cetera, and things coming to equilibrium. 15 00:01:05,870 --> 00:01:14,190 So question is, is this formalism always successful? 16 00:01:14,190 --> 00:01:21,110 And by the time you come to the end of the 19th Century, 17 00:01:21,110 --> 00:01:25,240 there were several things that were hanging around that 18 00:01:25,240 --> 00:01:28,090 had to do with thermal properties of the matter 19 00:01:28,090 --> 00:01:33,810 where this formalism was having difficulties. 20 00:01:33,810 --> 00:01:36,500 And the difficulties ultimately pointed out 21 00:01:36,500 --> 00:01:39,220 to emergence of quantum mechanics. 22 00:01:39,220 --> 00:01:42,210 So essentially, understanding the relationship 23 00:01:42,210 --> 00:01:46,030 between thermodynamics, statistical mechanics, 24 00:01:46,030 --> 00:01:50,700 and properties of matter was very important to development 25 00:01:50,700 --> 00:01:52,020 of quantum mechanics. 26 00:01:52,020 --> 00:01:58,180 And in particular, I will mention three difficulties. 27 00:01:58,180 --> 00:02:03,820 The most important one that really originally set 28 00:02:03,820 --> 00:02:06,490 the first stone for quantum mechanics 29 00:02:06,490 --> 00:02:08,834 is the spectrum of black body radiation. 30 00:02:21,010 --> 00:02:23,320 And it's basically the observation 31 00:02:23,320 --> 00:02:25,840 that you heat something. 32 00:02:25,840 --> 00:02:30,110 And when it becomes hot, it starts to radiate. 33 00:02:30,110 --> 00:02:34,720 And typically, the color of the radiation that you get 34 00:02:34,720 --> 00:02:36,960 is a function of temperature, but does not 35 00:02:36,960 --> 00:02:40,850 depend on the properties of the material that you are heating. 36 00:02:40,850 --> 00:02:43,020 So that has to do with heat. 37 00:02:43,020 --> 00:02:45,770 And you should be able to explain 38 00:02:45,770 --> 00:02:50,570 that using statistical mechanics. 39 00:02:50,570 --> 00:02:53,650 Another thing that we have already mentioned 40 00:02:53,650 --> 00:02:57,450 has to do with the third law of thermodynamics. 41 00:02:57,450 --> 00:03:04,270 And let's say the heat capacity of materials such as solids. 42 00:03:08,960 --> 00:03:14,780 We mentioned this Nernst theorem that 43 00:03:14,780 --> 00:03:20,160 was the third law of thermodynamics 44 00:03:20,160 --> 00:03:22,200 based on observation. 45 00:03:22,200 --> 00:03:26,150 Consequence of it was that heat capacity of most things 46 00:03:26,150 --> 00:03:31,650 that you can measure go to 0 as you go to 0 temperature. 47 00:03:31,650 --> 00:03:33,770 We should be able to explain that again, 48 00:03:33,770 --> 00:03:39,040 based on the phenomena of statistical-- the phenomenology 49 00:03:39,040 --> 00:03:44,830 of thermodynamics and the rules of statistical mechanics. 50 00:03:44,830 --> 00:03:47,400 Now, a third thing that is less often mentioned 51 00:03:47,400 --> 00:03:54,910 but is also important has to do with heat capacity 52 00:03:54,910 --> 00:04:06,380 of the atomic gases such as the air in this room, which 53 00:04:06,380 --> 00:04:09,260 is composed of, say, oxygen and nitrogen that 54 00:04:09,260 --> 00:04:12,620 are diatomic gases. 55 00:04:12,620 --> 00:04:20,754 So probably, historically they were answered and discussed 56 00:04:20,754 --> 00:04:25,570 and resolving the order that I have drawn for you. 57 00:04:25,570 --> 00:04:27,480 But I will go backwards. 58 00:04:27,480 --> 00:04:30,590 so we will first talk about this one, 59 00:04:30,590 --> 00:04:35,070 then about number two-- heat capacity of solids-- 60 00:04:35,070 --> 00:04:39,500 and number three about black body radiation. 61 00:04:39,500 --> 00:04:41,650 OK. 62 00:04:41,650 --> 00:04:45,520 Part of the reason is that throughout the course, 63 00:04:45,520 --> 00:04:49,230 we have been using our understanding of the gas 64 00:04:49,230 --> 00:04:55,420 as the sort of measure of how well we understand 65 00:04:55,420 --> 00:04:57,870 thermal properties of the matter. 66 00:04:57,870 --> 00:05:02,230 And so let's stick with the gas and ask, what do I 67 00:05:02,230 --> 00:05:06,520 know about the heat capacity of the gas in this room? 68 00:05:06,520 --> 00:05:11,497 So let's think about heat capacity 69 00:05:11,497 --> 00:05:16,495 of dilute diatomic gas. 70 00:05:22,290 --> 00:05:28,440 It is a gas that is sufficiently dilute that it is practically 71 00:05:28,440 --> 00:05:30,290 having ideal gas law. 72 00:05:30,290 --> 00:05:34,720 So PV is roughly proportional to temperature. 73 00:05:34,720 --> 00:05:37,160 But rather than thinking about its pressure, 74 00:05:37,160 --> 00:05:39,040 I want to make sure I understand something 75 00:05:39,040 --> 00:05:42,810 about the heat capacity, another quantity that I can measure. 76 00:05:42,810 --> 00:05:44,270 So what's going on here? 77 00:05:44,270 --> 00:05:46,990 I have, let's say, a box. 78 00:05:46,990 --> 00:05:49,640 And within this box, we have a whole bunch 79 00:05:49,640 --> 00:05:51,910 of these diatomic molecules. 80 00:05:56,140 --> 00:06:01,910 Let's stick to the canonical ensemble. 81 00:06:01,910 --> 00:06:05,660 So I tell you the volume of this gas, the number 82 00:06:05,660 --> 00:06:07,690 of diatomic molecules, and the temperature. 83 00:06:10,380 --> 00:06:16,260 And in this formalism, I would calculate the partition 84 00:06:16,260 --> 00:06:16,780 function. 85 00:06:16,780 --> 00:06:18,800 Out of that, I should be able to calculate 86 00:06:18,800 --> 00:06:22,220 the energy, heat capacity, et cetera. 87 00:06:22,220 --> 00:06:24,470 So what do I have to do? 88 00:06:24,470 --> 00:06:30,980 I have to integrate over all possible coordinates that 89 00:06:30,980 --> 00:06:33,340 occur in this system. 90 00:06:33,340 --> 00:06:37,060 To all intents and purposes, the different molecules 91 00:06:37,060 --> 00:06:38,890 are identical. 92 00:06:38,890 --> 00:06:42,840 So I divide by the phase space that 93 00:06:42,840 --> 00:06:45,930 is assigned to each one of them. 94 00:06:45,930 --> 00:06:51,210 And I said it is dilute enough that for all intents 95 00:06:51,210 --> 00:06:55,880 and purposes, the pressure is proportional to temperature. 96 00:06:55,880 --> 00:06:58,420 And that occur, I know, when I can 97 00:06:58,420 --> 00:07:01,720 ignore the interactions between particles. 98 00:07:01,720 --> 00:07:05,490 So if I can ignore the interactions between particles, 99 00:07:05,490 --> 00:07:08,450 then the partition function for the entire system 100 00:07:08,450 --> 00:07:10,820 would be the product of the partition functions 101 00:07:10,820 --> 00:07:13,970 that I would write for the individual molecules, 102 00:07:13,970 --> 00:07:16,270 or one of them raised to the N power. 103 00:07:16,270 --> 00:07:20,230 So what's the Z1 that I have to calculate? 104 00:07:20,230 --> 00:07:23,880 Z1 is obtained by integrating over 105 00:07:23,880 --> 00:07:28,270 the coordinates and momenta of a single diatomic particle. 106 00:07:28,270 --> 00:07:30,530 So I have a factor of d cubed p. 107 00:07:30,530 --> 00:07:33,600 I have a factor of d cubed q. 108 00:07:33,600 --> 00:07:38,480 But I have two particles, so I have d cubed p1, d cubed q1, 109 00:07:38,480 --> 00:07:43,850 d cubed p2, d cubed q 2, and I have 110 00:07:43,850 --> 00:07:48,170 six pairs of coordinate momenta. 111 00:07:48,170 --> 00:07:50,800 So I divide it by h cubed. 112 00:07:50,800 --> 00:07:57,820 I have e to the minus beta times the energy of this system, 113 00:07:57,820 --> 00:08:04,230 which is p1 squared over 2m p2 squared over 2m. 114 00:08:04,230 --> 00:08:08,160 And some potential of interaction 115 00:08:08,160 --> 00:08:12,220 that is responsible for bringing and binding these things 116 00:08:12,220 --> 00:08:13,720 together. 117 00:08:13,720 --> 00:08:18,110 So there is some V that is function of q1 and q2 118 00:08:18,110 --> 00:08:21,760 that binds the two particles together and does not 119 00:08:21,760 --> 00:08:24,290 allow them to become separate. 120 00:08:24,290 --> 00:08:26,810 All right, so what do we do here? 121 00:08:26,810 --> 00:08:30,275 We realize that immediately for one of these particle,s there 122 00:08:30,275 --> 00:08:34,039 is a center of mass that can go all over the place. 123 00:08:34,039 --> 00:08:36,750 So we change coordinates to, let's 124 00:08:36,750 --> 00:08:42,049 say, Q, which is q1 plus q2 over 2. 125 00:08:42,049 --> 00:08:45,910 And corresponding to the center of mass position, 126 00:08:45,910 --> 00:08:51,660 there is also a center of mass momentum, P, 127 00:08:51,660 --> 00:08:59,320 which is related to p1 minus p2. 128 00:08:59,320 --> 00:09:03,430 But when I make the change of variables from these 129 00:09:03,430 --> 00:09:06,840 coordinates to these coordinates, what I will get 130 00:09:06,840 --> 00:09:10,450 is that I will have a simple integral 131 00:09:10,450 --> 00:09:12,380 over the relative coordinates. 132 00:09:12,380 --> 00:09:17,230 So I have d cubed Q d cubed big P h cubed. 133 00:09:17,230 --> 00:09:20,580 And the only thing that I have over there is 134 00:09:20,580 --> 00:09:27,640 e to the minus beta p squared divided by 2 big M. Big M 135 00:09:27,640 --> 00:09:30,430 being the sum total of the two masses. 136 00:09:30,430 --> 00:09:33,670 If the two masses are identical, it would be 2M. 137 00:09:33,670 --> 00:09:36,316 Otherwise, it would be M1 plus M2. 138 00:09:36,316 --> 00:09:42,040 And then I have an integration over the relative coordinate. 139 00:09:42,040 --> 00:09:46,642 Let's call that q relative momentum p 140 00:09:46,642 --> 00:09:52,330 h cubed e to the minus beta p squared over 2 times 141 00:09:52,330 --> 00:09:56,000 the reduced mass here. 142 00:09:56,000 --> 00:09:59,280 And then, the potential which only 143 00:09:59,280 --> 00:10:01,250 is a function of the relative coordinate. 144 00:10:06,470 --> 00:10:12,070 Point is that what I have done is I have separated out this 6 145 00:10:12,070 --> 00:10:15,270 degrees of freedom that make up-- or actually, 146 00:10:15,270 --> 00:10:18,850 the 3 degrees of freedom and their conjugate momenta that 147 00:10:18,850 --> 00:10:23,650 make up a single molecule into some degrees of freedom that 148 00:10:23,650 --> 00:10:27,210 correspond to the center of mass and some degrees of freedom 149 00:10:27,210 --> 00:10:29,910 that correspond to the relative motion. 150 00:10:29,910 --> 00:10:31,870 Furthermore, for the relative motion 151 00:10:31,870 --> 00:10:34,240 I expect that the form of this potential 152 00:10:34,240 --> 00:10:43,980 as a function of the separation has a form that is a minimum. 153 00:10:43,980 --> 00:10:48,270 Basically, the particles at 0 temperature 154 00:10:48,270 --> 00:10:52,110 would be sitting where this minimum is. 155 00:10:52,110 --> 00:10:55,900 So essentially, the shape of this diatomic molecule 156 00:10:55,900 --> 00:11:00,580 would be something like this if I find its minimum energy 157 00:11:00,580 --> 00:11:03,270 configuration. 158 00:11:03,270 --> 00:11:08,940 But then, I can allow it to move with respect 159 00:11:08,940 --> 00:11:11,130 to, say, the minimum energy. 160 00:11:11,130 --> 00:11:13,590 Let's say it occurs at some distance d. 161 00:11:16,350 --> 00:11:20,560 It can oscillate around this minimum value. 162 00:11:20,560 --> 00:11:22,800 If it oscillates around this minimum value, 163 00:11:22,800 --> 00:11:28,100 it basically will explore the bottom of this potential. 164 00:11:28,100 --> 00:11:35,350 So I can basically think of this center of mass contribution 165 00:11:35,350 --> 00:11:37,220 to the partition function. 166 00:11:37,220 --> 00:11:41,440 And this contribution has a part that 167 00:11:41,440 --> 00:11:46,390 comes from these oscillations around the center of mass. 168 00:11:46,390 --> 00:11:51,150 Let's call that u. 169 00:11:51,150 --> 00:11:53,430 Then, there is the corresponding momentum. 170 00:11:53,430 --> 00:11:56,220 I don't know, let's call it pi. 171 00:11:56,220 --> 00:12:02,490 I divide by h and I have e to the minus beta 172 00:12:02,490 --> 00:12:06,070 pi squared over 2 mu. 173 00:12:06,070 --> 00:12:10,000 And then I have minus beta. 174 00:12:10,000 --> 00:12:11,900 Well, to the lowest order, I have 175 00:12:11,900 --> 00:12:14,650 v of d, which is a constant. 176 00:12:14,650 --> 00:12:22,170 And then I have some frequency, some curvature 177 00:12:22,170 --> 00:12:23,890 at the bottom of this potential that I 178 00:12:23,890 --> 00:12:26,660 choose to write as mu omega squared 179 00:12:26,660 --> 00:12:30,160 over 2 multiplying by u squared. 180 00:12:30,160 --> 00:12:32,800 Essentially, what I want to do is 181 00:12:32,800 --> 00:12:38,120 to say that really, there is a vibrational degree of freedom 182 00:12:38,120 --> 00:12:41,480 and there is a harmonic oscillator that describes that. 183 00:12:41,480 --> 00:12:44,117 The frequency of that is related to the curvature 184 00:12:44,117 --> 00:12:45,950 that I have at the bottom of this potential. 185 00:12:49,530 --> 00:12:52,700 So this degree of freedom corresponds to vibrations. 186 00:12:59,590 --> 00:13:05,490 But that's not the end of this story because here I had three 187 00:13:05,490 --> 00:13:06,910 q's. 188 00:13:06,910 --> 00:13:10,000 One of them became the amplitude of this oscillation. 189 00:13:10,000 --> 00:13:16,240 So basically, the relative coordinate is a vector. 190 00:13:16,240 --> 00:13:19,730 One degree of freedom corresponds to stretching, 191 00:13:19,730 --> 00:13:22,800 but there are two other components of it. 192 00:13:22,800 --> 00:13:24,830 those two other components correspond 193 00:13:24,830 --> 00:13:28,320 to essentially keeping the length of this fixed 194 00:13:28,320 --> 00:13:31,340 but moving in the other directions. 195 00:13:31,340 --> 00:13:32,470 What do they correspond to? 196 00:13:32,470 --> 00:13:34,790 They correspond to rotations. 197 00:13:34,790 --> 00:13:38,280 So then there is essentially another partition function 198 00:13:38,280 --> 00:13:39,920 that I want right here. 199 00:13:39,920 --> 00:13:43,334 That corresponds to the rotational degrees of freedom. 200 00:13:46,020 --> 00:13:49,640 Now, the rotational degrees of freedom 201 00:13:49,640 --> 00:13:54,500 have a momentum contribution because this p is also 202 00:13:54,500 --> 00:13:55,960 three components. 203 00:13:55,960 --> 00:13:58,850 One component went in to the vibrations. 204 00:13:58,850 --> 00:14:01,360 There are two more components that 205 00:14:01,360 --> 00:14:04,620 really combine to tell you about the angular 206 00:14:04,620 --> 00:14:08,200 momentum and the energy that is proportional to the square 207 00:14:08,200 --> 00:14:09,750 of the angular momentum. 208 00:14:09,750 --> 00:14:12,090 But there is no restoring force for them. 209 00:14:12,090 --> 00:14:15,120 There is no corresponding term that is like this. 210 00:14:15,120 --> 00:14:20,090 So maybe I will just write that as an integral over angles 211 00:14:20,090 --> 00:14:22,270 that I can rotate this thing. 212 00:14:22,270 --> 00:14:24,860 An integral over the two components 213 00:14:24,860 --> 00:14:28,610 of the angular momentum divided by h squared. 214 00:14:28,610 --> 00:14:31,050 There's actually two angles. 215 00:14:31,050 --> 00:14:35,970 And the contribution is e to the minus beta angular 216 00:14:35,970 --> 00:14:39,350 momentum squared over 2I. 217 00:14:39,350 --> 00:14:42,990 So I wrote the entire thing. 218 00:14:42,990 --> 00:14:49,550 So essentially, all I have done is 219 00:14:49,550 --> 00:14:53,060 I have taken the Hamiltonian that 220 00:14:53,060 --> 00:14:55,070 corresponds to two particles that 221 00:14:55,070 --> 00:14:59,460 are bound together and broken it into three pieces 222 00:14:59,460 --> 00:15:03,640 corresponding to the center of mass, to the vibrations, 223 00:15:03,640 --> 00:15:04,515 and to the rotations. 224 00:15:08,300 --> 00:15:12,920 Now, the thing is that if I now ask, 225 00:15:12,920 --> 00:15:18,500 what is the energy that I would get for this one particle-- 226 00:15:18,500 --> 00:15:24,660 I guess I'll call this Z1-- what is the contribution of the one 227 00:15:24,660 --> 00:15:29,440 particular to the energy of the entire system? 228 00:15:29,440 --> 00:15:34,380 I have minus the log Z1 with respect to beta. 229 00:15:34,380 --> 00:15:37,610 That's the usual formula to calculate energies. 230 00:15:40,170 --> 00:15:44,360 So I go and look at this entire thing. 231 00:15:44,360 --> 00:15:48,070 And where do the beta dependencies come from? 232 00:15:48,070 --> 00:15:49,280 Well, let's see. 233 00:15:49,280 --> 00:16:00,220 So my Z1 has a part that comes from this center of mass. 234 00:16:00,220 --> 00:16:05,720 It gives me a V. We expect that. 235 00:16:05,720 --> 00:16:09,570 And then from the integration over the momenta, 236 00:16:09,570 --> 00:16:20,821 I will get something like 2 pi m over beta h squared to the 3/2 237 00:16:20,821 --> 00:16:21,321 power. 238 00:16:25,090 --> 00:16:31,200 From the vibrations-- OK, what do I have? 239 00:16:31,200 --> 00:16:36,340 I have e to the minus beta V of d, which is the constant. 240 00:16:36,340 --> 00:16:39,570 We really don't care. 241 00:16:39,570 --> 00:16:43,240 But there are these two components 242 00:16:43,240 --> 00:16:52,775 that give me root 2 pi mu divided by beta. 243 00:16:57,960 --> 00:17:02,910 There is a corresponding thing that 244 00:17:02,910 --> 00:17:06,990 comes from the variance that goes with this object, which 245 00:17:06,990 --> 00:17:16,119 is square root of 2 pi divided by beta mu omega squared. 246 00:17:16,119 --> 00:17:18,225 The entire thing has a factor of 1/h. 247 00:17:22,483 --> 00:17:23,566 So this is the vibrations. 248 00:17:26,930 --> 00:17:29,570 And for the rotations, what do I get? 249 00:17:29,570 --> 00:17:33,780 I will get a 4 pi from integrating 250 00:17:33,780 --> 00:17:37,200 over all orientations. 251 00:17:37,200 --> 00:17:40,300 Divided by h squared. 252 00:17:40,300 --> 00:17:46,970 I have essentially the two components of angular momentum. 253 00:17:46,970 --> 00:17:55,806 So I get essentially, the square of 2 pi I divided by beta. 254 00:17:58,602 --> 00:18:03,240 So this is rotations. 255 00:18:03,240 --> 00:18:05,220 And this is center of mass. 256 00:18:12,010 --> 00:18:14,520 We can see that if I take that formula, 257 00:18:14,520 --> 00:18:22,410 take its log divide by-- take a derivative with respect 258 00:18:22,410 --> 00:18:23,710 to beta. 259 00:18:23,710 --> 00:18:26,360 First of all, I will get this constant 260 00:18:26,360 --> 00:18:30,030 that is the energy of the bond state at 0 temperature. 261 00:18:30,030 --> 00:18:32,630 But the more interesting things are the things 262 00:18:32,630 --> 00:18:38,060 that I take from the derivatives of the various factors of beta. 263 00:18:38,060 --> 00:18:41,470 Essentially, for each factor of beta in the denominator, 264 00:18:41,470 --> 00:18:44,550 log Z will have a minus log of beta. 265 00:18:44,550 --> 00:18:48,850 I take a derivative, I will get a factor of 1 over beta. 266 00:18:48,850 --> 00:18:55,410 So from here, I will get 3/2 1 over beta, which is 3/2 kT. 267 00:18:55,410 --> 00:18:56,905 So this is the center of mass. 268 00:19:02,770 --> 00:19:09,180 From here, I have two factors of beta to the 1/2. 269 00:19:09,180 --> 00:19:12,990 So they combine to give me one factor of kT. 270 00:19:12,990 --> 00:19:13,960 This is for vibrations. 271 00:19:17,850 --> 00:19:21,360 And similarly, I have two factors of beta 272 00:19:21,360 --> 00:19:28,790 to the 1/2, which correspond to 1 kT for rotations. 273 00:19:35,660 --> 00:19:40,440 So then I say that the heat capacity at constant volume 274 00:19:40,440 --> 00:19:46,710 is simply-- per particle is related to d e1 by dT. 275 00:19:46,710 --> 00:19:54,960 And I see that that amounts to kb times 3/2 plus 1 plus 1, 276 00:19:54,960 --> 00:19:59,160 or I should get 7/2 kb. 277 00:19:59,160 --> 00:20:05,350 Per particle, which says that if you go and calculate the heat 278 00:20:05,350 --> 00:20:09,440 capacity of the gas in this room, 279 00:20:09,440 --> 00:20:11,430 divide by the number of molecules 280 00:20:11,430 --> 00:20:14,980 that we have-- doesn't matter whether they 281 00:20:14,980 --> 00:20:16,410 are oxygens or nitrogen. 282 00:20:16,410 --> 00:20:18,890 They would basically give the same contribution 283 00:20:18,890 --> 00:20:21,070 because you can see that the masses and all 284 00:20:21,070 --> 00:20:25,020 the other properties of the molecule 285 00:20:25,020 --> 00:20:27,890 do not appear in the heat capacity. 286 00:20:27,890 --> 00:20:32,080 That as a function of temperature, 287 00:20:32,080 --> 00:20:35,730 I should get a value of 7/2. 288 00:20:35,730 --> 00:20:38,890 So basically, C in units of kb. 289 00:20:38,890 --> 00:20:40,140 So I divide by kb. 290 00:20:40,140 --> 00:20:42,970 And my predictions is that I should see 7/2. 291 00:20:49,030 --> 00:20:53,680 So you go and do a measurement and what do you get? 292 00:20:53,680 --> 00:20:55,240 What you get is actually 5/2. 293 00:21:02,720 --> 00:21:07,000 So something is not quite right. 294 00:21:07,000 --> 00:21:11,700 We are not getting the 7/2 that we predicted. 295 00:21:11,700 --> 00:21:13,400 Except that I really mentioned that you 296 00:21:13,400 --> 00:21:18,500 are getting this measurement when you do measurements 297 00:21:18,500 --> 00:21:21,190 at room temperature, you get this value. 298 00:21:21,190 --> 00:21:23,720 So when we measure the heat capacity 299 00:21:23,720 --> 00:21:27,110 of the gas in this room, we will get 5/2. 300 00:21:27,110 --> 00:21:29,890 But if we heat it up, by the time 301 00:21:29,890 --> 00:21:35,420 we get to temperatures of a few thousand degrees Kelvin. 302 00:21:35,420 --> 00:21:39,680 So if you heat the room by a factor of 5 to 10, 303 00:21:39,680 --> 00:21:45,410 you will actually get the value of 7/2. 304 00:21:45,410 --> 00:21:48,100 And if you cool it, by the time you 305 00:21:48,100 --> 00:21:53,190 get to the order of 10 degrees or fewer, 306 00:21:53,190 --> 00:21:57,950 then you will find that the heat capacity actually 307 00:21:57,950 --> 00:21:58,910 goes even further. 308 00:21:58,910 --> 00:22:00,970 It goes all the way to 3/2. 309 00:22:04,030 --> 00:22:06,910 And the 3/2 is the thing that you 310 00:22:06,910 --> 00:22:09,320 would have predicted for a gas that 311 00:22:09,320 --> 00:22:13,320 had monatomic particles, no internal structure. 312 00:22:13,320 --> 00:22:16,340 Because then the only thing that you would have gotten 313 00:22:16,340 --> 00:22:20,440 is the center of mass contribution. 314 00:22:20,440 --> 00:22:25,320 So it seems like by going to low temperatures, 315 00:22:25,320 --> 00:22:28,580 you somehow freeze the degrees of freedom 316 00:22:28,580 --> 00:22:35,820 that correspond to vibrations and rotations of the gas. 317 00:22:35,820 --> 00:22:38,086 And by going to really high temperatures, 318 00:22:38,086 --> 00:22:41,750 you are able to liberate all of these degrees of freedom 319 00:22:41,750 --> 00:22:43,550 and store energy in them. 320 00:22:43,550 --> 00:22:45,920 Heat capacity is the measure of the ability 321 00:22:45,920 --> 00:22:50,905 to store heat and energy into these molecules. 322 00:22:50,905 --> 00:22:52,990 So what is happening? 323 00:22:58,370 --> 00:23:03,020 Well, by 1905, Planck Had already 324 00:23:03,020 --> 00:23:09,010 proposed that there is some underlying quantization 325 00:23:09,010 --> 00:23:13,770 for heat that you have in the black body case. 326 00:23:13,770 --> 00:23:18,250 And in 1905, Einstein said, well, maybe we 327 00:23:18,250 --> 00:23:22,000 should think about the vibrational degrees 328 00:23:22,000 --> 00:23:26,360 of the molecule also as being similarly quantized. 329 00:23:26,360 --> 00:23:32,450 So quantize vibrations. 330 00:23:36,950 --> 00:23:39,880 It's totally a phenomenological statement. 331 00:23:39,880 --> 00:23:42,290 We have to justify it later. 332 00:23:42,290 --> 00:23:52,120 But the statement is that for the case where classically we 333 00:23:52,120 --> 00:23:53,720 had a harmonic oscillator. 334 00:23:53,720 --> 00:23:55,180 And let's say in this case we would 335 00:23:55,180 --> 00:23:58,000 have said that its energy depends 336 00:23:58,000 --> 00:24:03,590 on its momentum and its position or displacement-- 337 00:24:03,590 --> 00:24:11,510 I guess I called it u-- through a formula such as this. 338 00:24:14,270 --> 00:24:18,240 Certainly, you can pick lots of values of u and p 339 00:24:18,240 --> 00:24:21,430 that are compatible with any value of the energy 340 00:24:21,430 --> 00:24:24,010 that you choose. 341 00:24:24,010 --> 00:24:27,720 But to get the black body spectrum 342 00:24:27,720 --> 00:24:33,740 to work, Planck had proposed that really what you should do 343 00:24:33,740 --> 00:24:37,760 is rather than thinking of this harmonic oscillator 344 00:24:37,760 --> 00:24:42,180 as being able to take all possible values, that somehow 345 00:24:42,180 --> 00:24:45,710 the values of energy that it can take are quantized. 346 00:24:45,710 --> 00:24:48,180 And furthermore, he had proposed that they 347 00:24:48,180 --> 00:24:52,050 are proportional to the frequency involved. 348 00:24:52,050 --> 00:24:53,930 And how did he guess that? 349 00:24:53,930 --> 00:24:55,830 Ultimately, it was related to what 350 00:24:55,830 --> 00:24:58,260 I said about black body radiation. 351 00:24:58,260 --> 00:25:01,880 That as you heat up the body, you 352 00:25:01,880 --> 00:25:04,720 will find that there's a light that comes out 353 00:25:04,720 --> 00:25:07,190 and the frequency of that light is somehow 354 00:25:07,190 --> 00:25:10,140 related to temperature and nothing else. 355 00:25:10,140 --> 00:25:11,950 And based on that, he had proposed 356 00:25:11,950 --> 00:25:16,715 that frequencies should come up in certain packages that 357 00:25:16,715 --> 00:25:20,200 are proportional to-- the energies 358 00:25:20,200 --> 00:25:23,470 of the particular frequencies should come in packages that 359 00:25:23,470 --> 00:25:26,210 are proportional to that frequency. 360 00:25:26,210 --> 00:25:30,100 So there is an integer here n that tells you 361 00:25:30,100 --> 00:25:33,930 about the number of these packets. 362 00:25:33,930 --> 00:25:39,220 And not that it really matters for what we are doing now, 363 00:25:39,220 --> 00:25:42,780 but just to be consistent with what we currently know 364 00:25:42,780 --> 00:25:46,830 with quantum mechanics, let me add the 0 point 365 00:25:46,830 --> 00:25:48,590 energy of the harmonic oscillator here. 366 00:25:51,830 --> 00:25:55,950 So then, to calculate the contribution 367 00:25:55,950 --> 00:26:00,270 of a system in which energy is in quantized packages, 368 00:26:00,270 --> 00:26:02,240 you would say, OK, I will calculate 369 00:26:02,240 --> 00:26:06,720 a Z1 for these vibrational levels, 370 00:26:06,720 --> 00:26:10,740 assuming this quantization of energy. 371 00:26:10,740 --> 00:26:14,460 And so that says that the possible states 372 00:26:14,460 --> 00:26:18,510 of my harmonic oscillator have energies 373 00:26:18,510 --> 00:26:24,210 that are in these units h bar omega n plus 1/2. 374 00:26:24,210 --> 00:26:28,960 And if I still continue to believe statistical mechanics, 375 00:26:28,960 --> 00:26:33,620 I would say that at a temperature t, 376 00:26:33,620 --> 00:26:36,860 the probability that I will be in a state that 377 00:26:36,860 --> 00:26:40,550 is characterized by integer n is e to the minus 378 00:26:40,550 --> 00:26:44,570 beta times the energy that corresponds to that integer n. 379 00:26:44,570 --> 00:26:50,590 And then I can go and sum over all possible energies 380 00:26:50,590 --> 00:26:52,925 and that would be the normalization 381 00:26:52,925 --> 00:26:57,510 of the probability that I'm in one of these states. 382 00:26:57,510 --> 00:27:04,195 So this is e to the minus beta h bar omega over 2 383 00:27:04,195 --> 00:27:09,020 from the ground state contribution. 384 00:27:09,020 --> 00:27:12,490 The rest of it is simply a geometric series. 385 00:27:12,490 --> 00:27:18,820 Geometric series, we can sum very easily to get 1 minus e 386 00:27:18,820 --> 00:27:21,780 to the minus beta h bar omega. 387 00:27:21,780 --> 00:27:28,820 And the interesting thing-- or a few interesting things 388 00:27:28,820 --> 00:27:31,570 about this expression is that if I 389 00:27:31,570 --> 00:27:35,261 evaluate this in the limit of low temperatures. 390 00:27:35,261 --> 00:27:37,510 Well, actually, let's go first to the high temperature 391 00:27:37,510 --> 00:27:39,060 where beta goes to 0. 392 00:27:39,060 --> 00:27:42,940 So t goes to become large, beta goes to 0. 393 00:27:42,940 --> 00:27:48,080 Numerator goes to 1, denominator I can expand the exponential. 394 00:27:48,080 --> 00:27:52,000 And to lowest order, I will get 1 over beta h bar omega. 395 00:27:56,320 --> 00:28:03,960 Now, compare this result with the classical result 396 00:28:03,960 --> 00:28:07,330 that we have over here for the vibration. 397 00:28:07,330 --> 00:28:10,440 Contribution of a harmonic oscillator to the partition 398 00:28:10,440 --> 00:28:11,680 function. 399 00:28:11,680 --> 00:28:16,880 You can see that the mu's cancel out. 400 00:28:16,880 --> 00:28:21,280 I will get 1 over beta. 401 00:28:21,280 --> 00:28:24,950 I will get h divided by 2 pi. 402 00:28:24,950 --> 00:28:29,090 So if I call h divided by 2 pi to be h bar, 403 00:28:29,090 --> 00:28:36,370 then I will get exactly this limit. 404 00:28:36,370 --> 00:28:43,250 So somehow this constant that we had introduced 405 00:28:43,250 --> 00:28:46,420 that had dimensions of action made 406 00:28:46,420 --> 00:28:50,960 to make our calculations of partition function 407 00:28:50,960 --> 00:28:54,870 to be dimensionless will be related 408 00:28:54,870 --> 00:28:58,220 to this h bar that quantizes the energy 409 00:28:58,220 --> 00:29:01,480 levels through the usual formula of h being 410 00:29:01,480 --> 00:29:05,640 h bar-- h bar being h over 2 pi. 411 00:29:05,640 --> 00:29:10,920 So basically, this quantization of energy 412 00:29:10,920 --> 00:29:14,310 clearly does not affect the high temperature limit. 413 00:29:14,310 --> 00:29:16,400 This oscillator at high temperature 414 00:29:16,400 --> 00:29:21,920 behaves exactly like what we had calculated classically. 415 00:29:21,920 --> 00:29:24,670 Yes? 416 00:29:24,670 --> 00:29:26,670 AUDIENCE: Is it h equals h bar over 2 pi? 417 00:29:26,670 --> 00:29:30,850 Or is it the other way, based on your definitions above? 418 00:29:30,850 --> 00:29:31,834 PROFESSOR: Thank you. 419 00:29:38,071 --> 00:29:38,570 Good. 420 00:29:44,170 --> 00:29:45,710 All right? 421 00:29:45,710 --> 00:29:48,740 So this is, I guess, the corresponding formula. 422 00:29:52,150 --> 00:29:54,830 Now, when you go to low temperature, what do you get? 423 00:29:54,830 --> 00:30:02,130 You essentially get the first few terms in the series. 424 00:30:02,130 --> 00:30:05,710 Because at the lowest temperature 425 00:30:05,710 --> 00:30:10,460 you get the term that corresponds to n equals to 0, 426 00:30:10,460 --> 00:30:12,850 and then you will get corrections 427 00:30:12,850 --> 00:30:13,910 from subsequent terms. 428 00:30:21,090 --> 00:30:28,350 Now, what this does is that it affects the heat capacity 429 00:30:28,350 --> 00:30:29,020 profoundly. 430 00:30:29,020 --> 00:30:31,180 So let's see how that happens. 431 00:30:31,180 --> 00:30:36,330 So the contribution of 1 degrees of freedom to the energy 432 00:30:36,330 --> 00:30:44,120 in this quantized fashion d log Z by d beta. 433 00:30:44,120 --> 00:30:48,700 So if I just take the log of this expression, 434 00:30:48,700 --> 00:30:52,920 log of this expression will get this factor of minus beta h bar 435 00:30:52,920 --> 00:30:55,530 omega over 2 from the numerator. 436 00:30:55,530 --> 00:30:58,590 The derivative of that will give you this ground state 437 00:30:58,590 --> 00:31:01,880 energy, which is always there. 438 00:31:01,880 --> 00:31:05,570 And then you'll have to take the derivative 439 00:31:05,570 --> 00:31:08,370 of the log of what is coming out here. 440 00:31:08,370 --> 00:31:11,730 Taking a derivative with respect to beta, 441 00:31:11,730 --> 00:31:16,880 we'll always pick out a factor of h bar omega. 442 00:31:16,880 --> 00:31:19,980 Indeed, it will pick out a factor of h bar omega 443 00:31:19,980 --> 00:31:23,160 e to the minus beta h bar omega. 444 00:31:23,160 --> 00:31:26,880 And then in the denominator, because I took the log, 445 00:31:26,880 --> 00:31:28,595 I will get this expression back. 446 00:31:36,720 --> 00:31:40,740 So again, in this expression, if I 447 00:31:40,740 --> 00:31:50,870 take the limit where beta goes to 0, what do I get? 448 00:31:50,870 --> 00:31:53,370 I will get this h bar omega over 2. 449 00:31:53,370 --> 00:31:54,990 It's always there. 450 00:31:54,990 --> 00:31:59,750 Expanding these results here, I will have a beta h bar omega. 451 00:31:59,750 --> 00:32:03,610 It will cancel this and it will give me a 1 over beta. 452 00:32:03,610 --> 00:32:07,830 I will get this kT that I had before. 453 00:32:07,830 --> 00:32:12,440 Indeed, if I am correct to the right order, 454 00:32:12,440 --> 00:32:16,810 I will just simply get 1 over beta. 455 00:32:16,810 --> 00:32:24,470 Whereas, if I go to large beta, what I get 456 00:32:24,470 --> 00:32:28,710 is this h bar omega over 2 plus a correction 457 00:32:28,710 --> 00:32:34,446 from here, which is h bar omega e to the minus beta h bar. 458 00:32:39,640 --> 00:32:47,790 And that will be reflected in the heat capacity, which 459 00:32:47,790 --> 00:32:49,876 is dE by dT. 460 00:32:52,690 --> 00:32:56,760 This h bar omega over 2 does not continue to heat capacity, 461 00:32:56,760 --> 00:32:59,210 not surprisingly. 462 00:32:59,210 --> 00:33:03,620 From here, I have to take derivatives with temperatures. 463 00:33:03,620 --> 00:33:07,930 They appear in the combination h bar omega over kT. 464 00:33:07,930 --> 00:33:10,000 So what happens is I will get something 465 00:33:10,000 --> 00:33:12,990 that is of the order of h bar omega. 466 00:33:12,990 --> 00:33:15,500 And then from here, I will get another h bar 467 00:33:15,500 --> 00:33:20,290 omega divided by kb T squared. 468 00:33:20,290 --> 00:33:25,900 I will write it in this fashion and put the kb out here. 469 00:33:25,900 --> 00:33:29,510 And then the rest of these objects 470 00:33:29,510 --> 00:33:32,270 will give me a contribution that is minus 471 00:33:32,270 --> 00:33:37,800 h bar omega over kT divided by 1 minus e 472 00:33:37,800 --> 00:33:42,760 to the minus h bar omega over kT squared. 473 00:33:47,140 --> 00:33:49,260 The important thing is the following-- 474 00:33:52,110 --> 00:33:54,460 If I plot the heat capacity that I 475 00:33:54,460 --> 00:33:57,760 get from one of these oscillators-- 476 00:33:57,760 --> 00:34:00,480 and the natural units of all heat 477 00:34:00,480 --> 00:34:03,770 capacities are kb, essentially. 478 00:34:03,770 --> 00:34:08,860 Energy divided by temperature, as kb has that units. 479 00:34:08,860 --> 00:34:12,969 At high temperatures, what I can see 480 00:34:12,969 --> 00:34:17,360 is that the energy is proportional to kT. 481 00:34:17,360 --> 00:34:20,500 So heat capacity of the vibrational degree of freedom 482 00:34:20,500 --> 00:34:25,870 will be in these units going to 1. 483 00:34:25,870 --> 00:34:30,920 At low temperatures, however, it becomes this exponentially hard 484 00:34:30,920 --> 00:34:35,210 problem to create excitations. 485 00:34:35,210 --> 00:34:38,280 Because of that, you will get a contribution 486 00:34:38,280 --> 00:34:43,130 that as T goes to 0 will exponentially go to 0. 487 00:34:43,130 --> 00:34:47,160 So the shape of the heat capacity that you would get 488 00:34:47,160 --> 00:34:49,098 will be something like this. 489 00:34:53,662 --> 00:35:01,450 The natural way to draw this figure 490 00:35:01,450 --> 00:35:05,500 is actually what I made the vertical axis 491 00:35:05,500 --> 00:35:06,650 to be dimensionless. 492 00:35:06,650 --> 00:35:08,970 So it goes between 0 and 1. 493 00:35:08,970 --> 00:35:11,310 I can make the horizontal axis to be 494 00:35:11,310 --> 00:35:16,180 dimensionless by introducing a theta of vibrations, 495 00:35:16,180 --> 00:35:21,340 so that all of the exponential terms are of the form e 496 00:35:21,340 --> 00:35:25,970 to the minus T over this theta of vibrations, which 497 00:35:25,970 --> 00:35:29,750 means that this theta of vibration 498 00:35:29,750 --> 00:35:33,130 is h bar omega over kb. 499 00:35:33,130 --> 00:35:36,260 That is, you tell me what the frequency of your oscillator 500 00:35:36,260 --> 00:35:37,260 is. 501 00:35:37,260 --> 00:35:40,270 I can calculate the corresponding temperature, 502 00:35:40,270 --> 00:35:41,410 theta. 503 00:35:41,410 --> 00:35:45,490 And then the heat capacity of a harmonic oscillator 504 00:35:45,490 --> 00:35:49,140 is this universal function there, presumably 505 00:35:49,140 --> 00:35:53,160 at some value that is of the order of 1. 506 00:35:53,160 --> 00:35:55,980 It switches from being of the order 507 00:35:55,980 --> 00:35:58,810 of 1 to going exponentially to 0. 508 00:35:58,810 --> 00:36:03,060 So basically, the dependence down here to leading order 509 00:36:03,060 --> 00:36:05,404 is e to the minus T over theta vibration. 510 00:36:10,250 --> 00:36:10,750 OK. 511 00:36:14,020 --> 00:36:17,750 So you say, OK, Planck has given us 512 00:36:17,750 --> 00:36:20,720 some estimate of what this h bar is based 513 00:36:20,720 --> 00:36:25,250 on looking at the spectrum of black body radiation. 514 00:36:25,250 --> 00:36:29,450 We can, more or, less estimate the typical energies 515 00:36:29,450 --> 00:36:31,760 of interactions of molecules. 516 00:36:31,760 --> 00:36:35,120 And from that, we can estimate what 517 00:36:35,120 --> 00:36:37,960 this frequency of vibration is. 518 00:36:37,960 --> 00:36:40,720 So we should be able to get an order of magnitude 519 00:36:40,720 --> 00:36:43,380 estimate of what this theta y is. 520 00:36:43,380 --> 00:36:47,350 And what you find is that theta y is of the order of 10 521 00:36:47,350 --> 00:36:50,014 to the 3 degrees Kelvin. 522 00:36:50,014 --> 00:36:52,910 It depends, of course, on what gas you are looking at, 523 00:36:52,910 --> 00:36:53,830 et cetera. 524 00:36:53,830 --> 00:36:59,140 But as an order of magnitude, it is something like that. 525 00:36:59,140 --> 00:37:03,540 So we can now transport this curve that we have over here 526 00:37:03,540 --> 00:37:06,680 and more or less get this first part of the curve 527 00:37:06,680 --> 00:37:08,280 that we have over here. 528 00:37:08,280 --> 00:37:11,910 So essentially, in this picture what we have 529 00:37:11,910 --> 00:37:14,760 is that there is no vibrations. 530 00:37:14,760 --> 00:37:18,212 The vibrations have been frozen out. 531 00:37:18,212 --> 00:37:19,960 And here you have vibrations. 532 00:37:26,480 --> 00:37:28,220 Of course, in all of the cases, you 533 00:37:28,220 --> 00:37:30,640 have the kinetic energy of the center of mass. 534 00:37:36,450 --> 00:37:42,100 And presumably since we are getting the right answer 535 00:37:42,100 --> 00:37:45,790 at very high temperatures now, we also have the rotations. 536 00:37:51,150 --> 00:37:52,910 And it makes sense that essentially 537 00:37:52,910 --> 00:37:59,420 what happened as we go to very low temperatures 538 00:37:59,420 --> 00:38:02,130 is that the rotations are also frozen out. 539 00:38:08,590 --> 00:38:11,750 Now, that's part of the story-- actually, 540 00:38:11,750 --> 00:38:15,770 you would think that among all of the examples that I gave 541 00:38:15,770 --> 00:38:20,010 you, this last one should be the simplest thing because it's 542 00:38:20,010 --> 00:38:21,810 really a two-body problem. 543 00:38:21,810 --> 00:38:23,730 Whereas, solids you have many things. 544 00:38:23,730 --> 00:38:25,470 Radiation, you have to think about 545 00:38:25,470 --> 00:38:27,930 the electromagnetic waves, et cetera. 546 00:38:27,930 --> 00:38:29,840 That somehow, historically, this would 547 00:38:29,840 --> 00:38:33,080 be the one that is resolved first. 548 00:38:33,080 --> 00:38:36,800 And indeed, as I said in 1905, Einstein 549 00:38:36,800 --> 00:38:38,950 figured out something about this. 550 00:38:38,950 --> 00:38:43,780 But this part dealing with the rotational degrees of freedom 551 00:38:43,780 --> 00:38:46,170 and quantizing them appropriately 552 00:38:46,170 --> 00:38:50,420 had to really wait until you had developed quantum mechanics 553 00:38:50,420 --> 00:38:53,300 beyond the statement that harmonic oscillators 554 00:38:53,300 --> 00:38:54,830 are quantized in energy. 555 00:38:54,830 --> 00:38:57,440 You had to know something more. 556 00:38:57,440 --> 00:39:02,000 So since in retrospect we do know something more, 557 00:39:02,000 --> 00:39:04,410 let's finish and give that answer 558 00:39:04,410 --> 00:39:09,180 before going on to something else. 559 00:39:09,180 --> 00:39:10,970 OK? 560 00:39:10,970 --> 00:39:15,820 So the next part of the story of the diatomic gas 561 00:39:15,820 --> 00:39:19,630 is quantizing rotations. 562 00:39:24,630 --> 00:39:32,150 So currently what I have is that there is an energy classically 563 00:39:32,150 --> 00:39:38,960 for rotations that is simply the kinetic energy 564 00:39:38,960 --> 00:39:41,680 of rotational degrees of freedom. 565 00:39:41,680 --> 00:39:43,780 So there is an angular momentum L, 566 00:39:43,780 --> 00:39:46,150 and then there's L squared over 2I. 567 00:39:46,150 --> 00:39:49,880 It looks pretty much like P squared over 2M, 568 00:39:49,880 --> 00:39:54,360 except that the degrees of freedom 569 00:39:54,360 --> 00:39:57,950 for translation and motion are positions. 570 00:39:57,950 --> 00:40:00,030 They can be all over the place. 571 00:40:00,030 --> 00:40:01,750 Whereas, the degrees of freedom that you 572 00:40:01,750 --> 00:40:04,000 have to think in terms of rotations 573 00:40:04,000 --> 00:40:06,830 are angles that go between 0, 2 pi, 574 00:40:06,830 --> 00:40:11,190 or on the surface of a sphere, et cetera. 575 00:40:11,190 --> 00:40:16,810 So once we figure out how to do quantum mechanics, 576 00:40:16,810 --> 00:40:22,110 we find that the allowed values of this 577 00:40:22,110 --> 00:40:29,270 are of the form h bar squared over 2I l, l plus 1, where 578 00:40:29,270 --> 00:40:36,290 l now is the number that gives you the discrete values that 579 00:40:36,290 --> 00:40:41,070 are possible for the square of the angular momentum. 580 00:40:41,070 --> 00:40:45,690 So you say OK, let's calculate a Z for the rotational degrees 581 00:40:45,690 --> 00:40:50,110 of freedom assuming this kind of quantization. 582 00:40:50,110 --> 00:40:54,160 So what I have to do, like I did for the harmonic oscillator, 583 00:40:54,160 --> 00:40:58,890 is I sum over all possible values of l that are allowed. 584 00:40:58,890 --> 00:41:04,220 The energy e to the minus beta h bar squared over 2I 585 00:41:04,220 --> 00:41:05,340 l, l plus 1. 586 00:41:07,860 --> 00:41:10,500 Except that there is one other thing, 587 00:41:10,500 --> 00:41:13,910 which is that these different values of l 588 00:41:13,910 --> 00:41:19,370 have degeneracy that is 2l plus 1. 589 00:41:19,370 --> 00:41:26,290 And so you have to multiply by the corresponding degeneracy. 590 00:41:30,820 --> 00:41:34,280 So what am I doing over here? 591 00:41:34,280 --> 00:41:41,110 I have to do a sum over different values of l, 592 00:41:41,110 --> 00:41:45,410 contributions that are really the probability that I 593 00:41:45,410 --> 00:41:51,329 am in these different values of the index l-- 0, 1, 2, 3, 4, 5, 594 00:41:51,329 --> 00:41:51,828 6. 595 00:41:54,790 --> 00:41:58,970 And I have to add all of these contributions. 596 00:42:02,570 --> 00:42:05,660 Now, the first thing that I will do 597 00:42:05,660 --> 00:42:09,320 is I ask whether the limit of high temperatures that I 598 00:42:09,320 --> 00:42:13,860 had calculated before is correctly reproduced or not. 599 00:42:13,860 --> 00:42:17,090 So I have to go to the limit where temperature is high 600 00:42:17,090 --> 00:42:19,340 or beta goes to 0. 601 00:42:19,340 --> 00:42:22,310 If beta goes to 0, you can see that going 602 00:42:22,310 --> 00:42:25,150 from one l to another l, it is multiply 603 00:42:25,150 --> 00:42:28,050 this exponent by a small number. 604 00:42:28,050 --> 00:42:29,290 So what does that mean? 605 00:42:29,290 --> 00:42:32,730 It means that the values from one point to another point 606 00:42:32,730 --> 00:42:37,350 of what am I summing over is not really that different. 607 00:42:37,350 --> 00:42:41,020 And I can think of a continuous curve that 608 00:42:41,020 --> 00:42:44,090 goes through all of these points. 609 00:42:44,090 --> 00:42:46,450 So if I do that, then I can essentially 610 00:42:46,450 --> 00:42:48,240 replace the sum with an integral. 611 00:43:00,970 --> 00:43:04,580 In fact, you can systematically calculate corrections 612 00:43:04,580 --> 00:43:08,330 to replacing the sum with an integral mathematically 613 00:43:08,330 --> 00:43:13,590 and you have a problem set that shows you how to do that. 614 00:43:13,590 --> 00:43:18,580 But now what I can do is I can call this combination l, l 615 00:43:18,580 --> 00:43:21,330 plus 1 x. 616 00:43:21,330 --> 00:43:26,940 And then dx will simply be 2l plus 1 dl. 617 00:43:29,550 --> 00:43:35,630 So essentially, the degeneracy works out precisely 618 00:43:35,630 --> 00:43:39,110 so that when I go to the continuum limit, whatever 619 00:43:39,110 --> 00:43:43,120 quantization I had for these angular momenta 620 00:43:43,120 --> 00:43:45,110 corresponds to the weight or measure 621 00:43:45,110 --> 00:43:48,770 that I would have in stepping around the l-directions. 622 00:43:48,770 --> 00:43:53,320 And then, this is something that I can easily do. 623 00:43:53,320 --> 00:43:57,280 It's just an integral dx e to the minus alpha x. 624 00:43:57,280 --> 00:44:01,830 The answer is going to be 1 over alpha, or the answer to this 625 00:44:01,830 --> 00:44:06,785 is simply 2I beta h bar squared. 626 00:44:14,680 --> 00:44:18,130 So this is the classical limit of the expression 627 00:44:18,130 --> 00:44:20,260 that we had over here. 628 00:44:20,260 --> 00:44:23,465 Let's go and see what we had when we did things classically. 629 00:44:27,570 --> 00:44:29,820 So when we did things classically, 630 00:44:29,820 --> 00:44:35,890 I had two factors of h and 2 pi and 4 pi. 631 00:44:35,890 --> 00:44:40,100 So I can write the whole thing as h bar squared and 2. 632 00:44:40,100 --> 00:44:43,902 I have I, and then I have beta. 633 00:44:43,902 --> 00:44:48,360 And you can see that this is exactly what we 634 00:44:48,360 --> 00:44:49,610 have over there. 635 00:44:49,610 --> 00:44:55,920 So once more, properly accounting for phase space, 636 00:44:55,920 --> 00:45:01,150 measure, productive p q's being dimension-- 637 00:45:01,150 --> 00:45:05,400 made dimensionless by this quantity h 638 00:45:05,400 --> 00:45:10,080 is equivalent to the high temperature limit 639 00:45:10,080 --> 00:45:11,990 that you would get in quantum mechanics 640 00:45:11,990 --> 00:45:13,230 where things are discretized. 641 00:45:13,230 --> 00:45:15,030 Yes. 642 00:45:15,030 --> 00:45:17,832 AUDIENCE: When you're talking about the quantum 643 00:45:17,832 --> 00:45:20,686 interpretations, then h bar is the precise value 644 00:45:20,686 --> 00:45:23,150 of Planck's constant, which can be an experimental measure. 645 00:45:23,150 --> 00:45:23,858 PROFESSOR: Right. 646 00:45:23,858 --> 00:45:25,260 AUDIENCE: But when you're talking 647 00:45:25,260 --> 00:45:29,082 about the classical derivations, h is just some factor 648 00:45:29,082 --> 00:45:30,540 that we mention of curve dimension. 649 00:45:30,540 --> 00:45:32,026 PROFESSOR: That's correct. 650 00:45:32,026 --> 00:45:35,802 AUDIENCE: So if you're comparing the limits 651 00:45:35,802 --> 00:45:40,710 of large temperatures, how can you 652 00:45:40,710 --> 00:45:45,263 be sure to establish the h bar in two places means 653 00:45:45,263 --> 00:45:46,555 the same thing? 654 00:45:46,555 --> 00:45:47,930 PROFESSOR: So far, I haven't told 655 00:45:47,930 --> 00:45:50,390 you anything to justify that. 656 00:45:50,390 --> 00:45:52,760 So when we were doing things classically, 657 00:45:52,760 --> 00:45:55,870 we said that just to make things dimensionless, 658 00:45:55,870 --> 00:46:00,380 let's introduce this quantity that we call h. 659 00:46:00,380 --> 00:46:03,800 Now, I have shown you two examples where 660 00:46:03,800 --> 00:46:07,710 if you do things quantum mechanically properly 661 00:46:07,710 --> 00:46:10,930 and take the limit of going to high temperatures, 662 00:46:10,930 --> 00:46:14,060 you will see that the h that you would get-- 663 00:46:14,060 --> 00:46:16,660 because the quantum mechanical partition functions 664 00:46:16,660 --> 00:46:19,500 are dimensionless quantities, right? 665 00:46:19,500 --> 00:46:21,650 So these are dimensionless quantities. 666 00:46:21,650 --> 00:46:24,140 They have to be made dimensionless by something. 667 00:46:24,140 --> 00:46:27,530 They're made dimensionless by Boltzmann's constant. 668 00:46:27,530 --> 00:46:30,880 By a Planck's constant, h bar. 669 00:46:30,880 --> 00:46:33,270 And we can see that as long as we 670 00:46:33,270 --> 00:46:36,330 are consistent with this measure of phase space, 671 00:46:36,330 --> 00:46:41,980 the same constant shows up both for the case of the vibrations, 672 00:46:41,980 --> 00:46:46,610 for the case of the rotations. 673 00:46:46,610 --> 00:46:49,100 And very soon, we will see that it will also 674 00:46:49,100 --> 00:46:51,780 arise in the case of the center of mass. 675 00:46:51,780 --> 00:47:00,010 And so there is certainly something in the transcriptions 676 00:47:00,010 --> 00:47:04,130 that we ultimately will make between quantum mechanics 677 00:47:04,130 --> 00:47:08,080 and classical mechanics that must account for this. 678 00:47:08,080 --> 00:47:12,640 And somehow in the limit where quantum mechanics is dealing 679 00:47:12,640 --> 00:47:16,510 with large energies, it is indistinguishable 680 00:47:16,510 --> 00:47:19,160 from classical mechanics. 681 00:47:19,160 --> 00:47:23,840 And quantum partition functions are-- 682 00:47:23,840 --> 00:47:26,990 all of the countings that we do in quantum mechanics are kind 683 00:47:26,990 --> 00:47:30,880 of unambiguous because we are dealing with discrete levels. 684 00:47:30,880 --> 00:47:34,960 So if you remember the original part of the difficulty 685 00:47:34,960 --> 00:47:39,150 was that we could define things like entropy only 686 00:47:39,150 --> 00:47:41,820 properly when we had discrete levels. 687 00:47:41,820 --> 00:47:44,380 If we had a continuum probability distribution 688 00:47:44,380 --> 00:47:46,650 and if we made a change of variable, 689 00:47:46,650 --> 00:47:49,240 then the entropy was changed. 690 00:47:49,240 --> 00:47:52,290 But in quantum mechanics, we don't have that problem. 691 00:47:52,290 --> 00:47:57,080 We have discretized values for the different states. 692 00:47:57,080 --> 00:48:00,910 Probabilities will be-- once we deal with them appropriately 693 00:48:00,910 --> 00:48:02,640 be discretized. 694 00:48:02,640 --> 00:48:06,190 And all of the things here are dimensionless. 695 00:48:06,190 --> 00:48:11,490 And somehow they reproduce the correct classical dynamics. 696 00:48:11,490 --> 00:48:13,830 Quantum mechanics goes to classical mechanics 697 00:48:13,830 --> 00:48:16,090 in the appropriate high-energy limit. 698 00:48:16,090 --> 00:48:22,440 And what we find is that what happens is that this shows up. 699 00:48:22,440 --> 00:48:28,720 If you like, another way of achieving-- 700 00:48:28,720 --> 00:48:32,630 why is there this correspondence? 701 00:48:32,630 --> 00:48:36,920 In classical statistical mechanics, 702 00:48:36,920 --> 00:48:43,390 I emphasize that I should really write h in units of p and q. 703 00:48:43,390 --> 00:48:47,870 And it was only when I calculated partition functions 704 00:48:47,870 --> 00:48:51,880 in coordinates p and q that were canonically conjugate that I 705 00:48:51,880 --> 00:48:54,350 was getting results that were meaningful. 706 00:48:58,260 --> 00:49:01,600 One way of constructing quantum mechanics 707 00:49:01,600 --> 00:49:04,720 is that you take the Hamiltonian and you change these 708 00:49:04,720 --> 00:49:06,200 into operators. 709 00:49:06,200 --> 00:49:14,950 And you have to impose these kinds of commutation relations. 710 00:49:14,950 --> 00:49:19,480 So you can see that somehow the same prescription in terms 711 00:49:19,480 --> 00:49:23,780 of phase space appears both in statistical mechanics, 712 00:49:23,780 --> 00:49:26,340 in calculating measures of partition function, 713 00:49:26,340 --> 00:49:27,950 in quantum mechanics. 714 00:49:27,950 --> 00:49:31,710 And not surprisingly, you have introduced in quantum mechanics 715 00:49:31,710 --> 00:49:35,260 some unit for phase space p, q. 716 00:49:35,260 --> 00:49:37,810 It shows up in classical mechanics 717 00:49:37,810 --> 00:49:39,020 as the quantity [INAUDIBLE]. 718 00:49:43,220 --> 00:49:46,200 But there is, indeed, a little bit more work 719 00:49:46,200 --> 00:49:49,100 than I have shown you here that one can do. 720 00:49:49,100 --> 00:49:53,650 Once we have developed the appropriate formalism 721 00:49:53,650 --> 00:49:56,900 for quantum statistical mechanics, which 722 00:49:56,900 --> 00:50:01,230 is this [INAUDIBLE] performed and appropriate quantities 723 00:50:01,230 --> 00:50:04,500 defined for partition functions, et cetera, 724 00:50:04,500 --> 00:50:06,780 in quantum statistical mechanics that we 725 00:50:06,780 --> 00:50:08,620 will do in a couple of lectures. 726 00:50:08,620 --> 00:50:11,260 Then if you take the limit h bar goes to 0, 727 00:50:11,260 --> 00:50:15,100 you should get the classical integration over phase space 728 00:50:15,100 --> 00:50:18,412 with this factor of h showing up. 729 00:50:23,290 --> 00:50:25,400 But right now, we are just giving you 730 00:50:25,400 --> 00:50:29,910 some heuristic response. 731 00:50:29,910 --> 00:50:32,610 If I go, however, in the other limit, where 732 00:50:32,610 --> 00:50:40,370 beta is much larger than 1, what do I get? 733 00:50:40,370 --> 00:50:43,210 Basically, then all of the weight 734 00:50:43,210 --> 00:50:48,560 is going to be in the lowest energy level, 0, 1. 735 00:50:48,560 --> 00:50:52,400 And then the rest of them will be exponentially small. 736 00:50:52,400 --> 00:50:55,840 I cannot replace the sum with an integral, 737 00:50:55,840 --> 00:51:00,380 so basically I will get a contribution that starts with 1 738 00:51:00,380 --> 00:51:03,140 for l equals to 0. 739 00:51:03,140 --> 00:51:10,000 And then I will get 3e to the minus beta h 740 00:51:10,000 --> 00:51:13,630 bar squared divided by 2I. 741 00:51:13,630 --> 00:51:17,770 l being 1, this will give me 1 times 2. 742 00:51:17,770 --> 00:51:20,760 So I will have a 2 here. 743 00:51:20,760 --> 00:51:22,220 And then, higher-order terms. 744 00:51:28,360 --> 00:51:35,120 So once you have the partition function, 745 00:51:35,120 --> 00:51:36,730 you go through the same procedure 746 00:51:36,730 --> 00:51:39,230 as we described before. 747 00:51:39,230 --> 00:51:45,440 You calculate the energy, which is d log Z by d beta. 748 00:51:48,460 --> 00:51:51,230 What do you get? 749 00:51:51,230 --> 00:51:53,300 Again, in the high temperature limit 750 00:51:53,300 --> 00:51:55,610 you will get the same answer as before. 751 00:51:55,610 --> 00:51:59,390 So you will get beta goes to 0. 752 00:51:59,390 --> 00:52:01,320 You will get kT. 753 00:52:04,100 --> 00:52:06,770 If you go to the low temperature limit-- well, 754 00:52:06,770 --> 00:52:08,440 let's be more precise. 755 00:52:08,440 --> 00:52:10,390 What do I mean by low temperatures? 756 00:52:10,390 --> 00:52:13,740 Beta larger than what? 757 00:52:13,740 --> 00:52:17,050 Clearly, the unit that is appearing everywhere 758 00:52:17,050 --> 00:52:26,690 is this beta h bar squared over 2I, which has units of 1 759 00:52:26,690 --> 00:52:30,200 over temperature from beta. 760 00:52:30,200 --> 00:52:34,220 So I can introduce a theta for rotations 761 00:52:34,220 --> 00:52:37,920 to make this demonstrate that this is dimensionless. 762 00:52:37,920 --> 00:52:41,120 So the theta that goes with rotations 763 00:52:41,120 --> 00:52:48,650 is h bar squared over 2I kb. 764 00:52:48,650 --> 00:52:53,630 And so what I mean by going to the low temperatures 765 00:52:53,630 --> 00:52:55,610 is that I go for temperatures that 766 00:52:55,610 --> 00:53:00,910 are much less than the theta of these rotations. 767 00:53:00,910 --> 00:53:06,780 And then what happens is that essentially this state 768 00:53:06,780 --> 00:53:10,590 will occur with exponentially small probability 769 00:53:10,590 --> 00:53:14,320 and will contribute to the energy and amount that 770 00:53:14,320 --> 00:53:19,490 is of the order of h bar squared 2I times 2. 771 00:53:19,490 --> 00:53:22,780 That's the energy of the l equals to 1 state. 772 00:53:22,780 --> 00:53:26,570 There are three of them, and they occur with probability e 773 00:53:26,570 --> 00:53:31,400 to the minus theta rotation divided 774 00:53:31,400 --> 00:53:34,118 by T times a factor of 2. 775 00:53:37,870 --> 00:53:40,870 All of those factors is not particularly important. 776 00:53:40,870 --> 00:53:45,100 Really, the only thing that is important 777 00:53:45,100 --> 00:53:50,600 is that if I look now at the rotational heat capacity, which 778 00:53:50,600 --> 00:53:55,160 again should properly have units of kb, 779 00:53:55,160 --> 00:53:56,770 as a function of temperature. 780 00:53:56,770 --> 00:53:59,320 Well, temperatures I have to make 781 00:53:59,320 --> 00:54:04,380 dimensionless by dividing by this rotational heat capacity. 782 00:54:04,380 --> 00:54:06,280 I say that at high, temperature I 783 00:54:06,280 --> 00:54:08,810 get the classical result back. 784 00:54:08,810 --> 00:54:15,160 So basically, I will get to 1 at high temperatures. 785 00:54:15,160 --> 00:54:19,070 At low temperatures, again I have this situation 786 00:54:19,070 --> 00:54:22,830 that there is a gap in the allowed energies. 787 00:54:22,830 --> 00:54:27,160 So there is the lowest energy, which is 0. 788 00:54:27,160 --> 00:54:31,490 The next one, the first type of rotational mode that is allowed 789 00:54:31,490 --> 00:54:34,100 has a finite energy that is larger than that 790 00:54:34,100 --> 00:54:37,960 by an amount that is of the order of h bar squared over I. 791 00:54:37,960 --> 00:54:40,120 And if I am at these temperatures that 792 00:54:40,120 --> 00:54:42,840 are less than this theta of rotation, 793 00:54:42,840 --> 00:54:45,870 I simply don't have enough energy 794 00:54:45,870 --> 00:54:50,660 from thermal fluctuations to get to that level. 795 00:54:50,660 --> 00:54:54,820 So the occupation of that level will be exponentially small. 796 00:54:54,820 --> 00:54:57,940 And so I will have a curve that will, in fact, look something 797 00:54:57,940 --> 00:54:58,912 like this. 798 00:55:04,270 --> 00:55:09,270 So again, you basically go over at a temperature 799 00:55:09,270 --> 00:55:12,500 of the order of 1 from heat capacity 800 00:55:12,500 --> 00:55:17,000 that is order of 1 to heat capacity that is exponentially 801 00:55:17,000 --> 00:55:24,740 small when you get to temperatures that are lower 802 00:55:24,740 --> 00:55:28,740 than this rotational temperature. 803 00:55:28,740 --> 00:55:30,680 AUDIENCE: Is that over-shooting, or is that-- 804 00:55:30,680 --> 00:55:31,780 PROFESSOR: Yes. 805 00:55:31,780 --> 00:55:35,260 So you have a problem set where you 806 00:55:35,260 --> 00:55:37,700 calculate the next correction. 807 00:55:37,700 --> 00:55:41,100 So there is the summation replacing the sum 808 00:55:41,100 --> 00:55:43,290 with an integral. 809 00:55:43,290 --> 00:55:45,850 This gives you this to the first order, 810 00:55:45,850 --> 00:55:48,040 and then there's a correction. 811 00:55:48,040 --> 00:55:51,440 And you will show that the correction is such 812 00:55:51,440 --> 00:55:54,790 that there is actually the approach to one 813 00:55:54,790 --> 00:55:59,250 for the case of the rotational heat capacity is from above. 814 00:55:59,250 --> 00:56:01,940 Whereas, for the vibrational heat capacity, 815 00:56:01,940 --> 00:56:05,000 it is from below. 816 00:56:05,000 --> 00:56:07,308 So there is, indeed, a small bump. 817 00:56:12,676 --> 00:56:15,610 OK? 818 00:56:15,610 --> 00:56:24,050 So you can ask, well, I know the typical size of one 819 00:56:24,050 --> 00:56:26,610 of these oxygen molecules. 820 00:56:26,610 --> 00:56:28,020 I know the mass. 821 00:56:28,020 --> 00:56:33,120 I can figure out what the moment of inertia I is. 822 00:56:33,120 --> 00:56:37,100 I put it over here and I figure out what the theta of rotation 823 00:56:37,100 --> 00:56:37,840 is. 824 00:56:37,840 --> 00:56:41,880 And you find that, again, as a matter of order of magnitudes, 825 00:56:41,880 --> 00:56:46,220 theta of rotations is of the order of 10 degrees 826 00:56:46,220 --> 00:56:56,120 K. So this kind of accounts for why when you go to sufficiently 827 00:56:56,120 --> 00:56:58,820 low temperatures for the heat capacity of the gas 828 00:56:58,820 --> 00:57:01,770 in this room, we see that essentially 829 00:57:01,770 --> 00:57:05,220 the rotational degrees of freedom are also frozen out. 830 00:57:13,558 --> 00:57:14,058 OK. 831 00:57:18,990 --> 00:57:25,370 So now let's go to the second item that we have, 832 00:57:25,370 --> 00:57:27,780 which is the heat capacity of the solid. 833 00:57:27,780 --> 00:57:28,855 So what do I mean? 834 00:57:31,920 --> 00:57:35,610 So this is item 2, heat capacity of solid. 835 00:57:43,710 --> 00:57:51,020 And you measure heat capacities for some solid 836 00:57:51,020 --> 00:57:53,880 as a function of temperature. 837 00:57:53,880 --> 00:57:57,430 And what you find is that the heat capacity 838 00:57:57,430 --> 00:58:00,970 has a behavior such as this. 839 00:58:04,950 --> 00:58:07,810 So it seems to vanish as you to go 840 00:58:07,810 --> 00:58:10,320 to lower and lower temperatures. 841 00:58:10,320 --> 00:58:11,610 So what's going on here? 842 00:58:15,690 --> 00:58:21,060 Again, Einstein looked at this and said, well, it's 843 00:58:21,060 --> 00:58:25,270 another case of the story of vibrations and some things 844 00:58:25,270 --> 00:58:28,370 that we have looked at here. 845 00:58:28,370 --> 00:58:32,490 And in fact, I really don't have to do any calculation. 846 00:58:32,490 --> 00:58:34,340 I'll do the following. 847 00:58:34,340 --> 00:58:37,620 Let's imagine that this is what we have for the solid. 848 00:58:37,620 --> 00:58:41,290 It's some regular arrangement of atoms or molecules. 849 00:58:46,630 --> 00:58:50,910 And presumably, this is the situation 850 00:58:50,910 --> 00:58:52,450 that I have at 0 temperature. 851 00:58:52,450 --> 00:58:55,770 Everybody is sitting nicely where 852 00:58:55,770 --> 00:58:58,500 they should be to minimize the energy. 853 00:58:58,500 --> 00:59:02,420 If I go to finite temperature, then these atoms and molecules 854 00:59:02,420 --> 00:59:03,650 start to vibrate. 855 00:59:08,320 --> 00:59:13,480 And he said, well, basically, I can 856 00:59:13,480 --> 00:59:17,912 estimate the frequencies of vibrations. 857 00:59:17,912 --> 00:59:26,840 And what I will do is I will say that each atom is 858 00:59:26,840 --> 00:59:31,430 in a cage by its neighbors. 859 00:59:35,740 --> 00:59:43,200 That is, this particular atom here, if it wants to move, 860 00:59:43,200 --> 00:59:51,610 it find that its distance to the neighbors has been changed. 861 00:59:51,610 --> 00:59:53,370 And if I imagine that there are kind 862 00:59:53,370 --> 00:59:57,750 of springs that are connecting this atom only 863 00:59:57,750 --> 01:00:01,630 to its neighbors, moving around there 864 01:00:01,630 --> 01:00:05,610 will be some kind of a restoring force. 865 01:00:05,610 --> 01:00:09,160 So it's like it is sitting in some kind 866 01:00:09,160 --> 01:00:11,690 of a harmonic potential. 867 01:00:11,690 --> 01:00:15,820 And if it tries to move, it will experience 868 01:00:15,820 --> 01:00:17,270 this restoring force. 869 01:00:17,270 --> 01:00:21,730 And so it will have some kind of a frequency. 870 01:00:21,730 --> 01:00:30,520 So each atom vibrates at some frequency. 871 01:00:35,499 --> 01:00:40,370 Let's call it omega E. 872 01:00:40,370 --> 01:00:44,210 Now, in principle, in this picture 873 01:00:44,210 --> 01:00:47,050 if this cage is not exactly symmetric, 874 01:00:47,050 --> 01:00:49,600 you may imagine that oscillations in the three 875 01:00:49,600 --> 01:00:52,820 different directions could give you different frequencies. 876 01:00:52,820 --> 01:00:56,140 But let's ignore that and let's imagine that the frequencies is 877 01:00:56,140 --> 01:00:58,790 the same in all of these. 878 01:00:58,790 --> 01:01:00,190 So what have we done? 879 01:01:00,190 --> 01:01:04,220 We have reduced the problem of the excitation energy 880 01:01:04,220 --> 01:01:08,340 that you can put in the atoms of the solid to be 881 01:01:08,340 --> 01:01:21,330 the same now as 3N harmonic oscillators of frequency omega. 882 01:01:24,150 --> 01:01:25,100 Why 3N? 883 01:01:25,100 --> 01:01:28,620 Because each atom essentially sees restoring 884 01:01:28,620 --> 01:01:30,380 force in three directions. 885 01:01:30,380 --> 01:01:32,790 And forgetting about boundary effects, 886 01:01:32,790 --> 01:01:35,240 it's basically three per particle. 887 01:01:35,240 --> 01:01:39,270 So you would have said that the heat capacity that I would 888 01:01:39,270 --> 01:01:47,200 calculate per particle in units of kb 889 01:01:47,200 --> 01:01:52,770 should essentially be exactly what we have over here, 890 01:01:52,770 --> 01:01:55,910 except that I multiply by 3 because each particular has 891 01:01:55,910 --> 01:01:57,830 3 possible degrees of freedom. 892 01:01:57,830 --> 01:02:01,620 So all I need to do is to take that green curve 893 01:02:01,620 --> 01:02:04,540 and multiply it by a factor of 3. 894 01:02:04,540 --> 01:02:11,080 And indeed, the limiting value that you get over here is 3. 895 01:02:11,080 --> 01:02:14,830 Except that if I just take that green curve 896 01:02:14,830 --> 01:02:18,420 and superpose it on this, what I will get 897 01:02:18,420 --> 01:02:19,665 is something like this. 898 01:02:24,980 --> 01:02:31,545 So this is 3 times harmonic oscillator. 899 01:02:39,530 --> 01:02:44,240 What do I mean by that, is I try to sort of do my best 900 01:02:44,240 --> 01:02:47,110 to match the temperature at which you 901 01:02:47,110 --> 01:02:49,210 go from one to the other. 902 01:02:49,210 --> 01:02:53,550 But then what I find is that as we had established before, 903 01:02:53,550 --> 01:02:57,300 the green curve goes to 0 exponentially. 904 01:02:57,300 --> 01:03:00,770 So there is going to be some theta associated 905 01:03:00,770 --> 01:03:02,950 with this frequency. 906 01:03:02,950 --> 01:03:06,400 Let's call it theta Einstein divided by T. 907 01:03:06,400 --> 01:03:10,440 And so the prediction of this model 908 01:03:10,440 --> 01:03:16,140 is that the heat capacities should vanish very rapidly 909 01:03:16,140 --> 01:03:18,255 as this form of exponential. 910 01:03:18,255 --> 01:03:22,050 Whereas, what is actually observed in the experiment 911 01:03:22,050 --> 01:03:25,182 is that it is going to 0 proportional 912 01:03:25,182 --> 01:03:33,200 to T cubed, which is a much slower type of decay. 913 01:03:33,200 --> 01:03:35,304 OK? 914 01:03:35,304 --> 01:03:38,650 AUDIENCE: That's negative [INAUDIBLE]? 915 01:03:38,650 --> 01:03:43,810 PROFESSOR: As T goes to 0, the heat capacity goes to 0. 916 01:03:43,810 --> 01:03:45,030 T to the third power. 917 01:03:48,920 --> 01:03:51,910 So it's the limit-- did I make a mistake somewhere else? 918 01:03:56,710 --> 01:03:57,670 All right. 919 01:03:57,670 --> 01:04:02,080 So what's happening here? 920 01:04:02,080 --> 01:04:05,370 OK, so what's happening is the following. 921 01:04:08,590 --> 01:04:12,010 In some average sense, it is correct 922 01:04:12,010 --> 01:04:18,430 that if you try to oscillate some atom in the crystal, 923 01:04:18,430 --> 01:04:22,660 it's going to have some characteristic restoring force. 924 01:04:22,660 --> 01:04:24,850 The characteristic restoring force 925 01:04:24,850 --> 01:04:28,660 will give you some corresponding typical scale 926 01:04:28,660 --> 01:04:30,460 for the frequencies of the vibrations. 927 01:04:32,845 --> 01:04:33,345 Yes? 928 01:04:36,076 --> 01:04:38,490 AUDIENCE: Is this the historical progression? 929 01:04:38,490 --> 01:04:39,115 PROFESSOR: Yes. 930 01:04:42,710 --> 01:04:44,970 AUDIENCE: I mean, it seems interesting that they 931 01:04:44,970 --> 01:04:48,570 would know that-- like this cage hypothesis is very good, 932 01:04:48,570 --> 01:04:52,707 considering where a quantum [INAUDIBLE] exists. 933 01:04:56,980 --> 01:04:59,170 I don't understand how that's the logic based-- 934 01:04:59,170 --> 01:05:01,690 if what we know is the top board over there, 935 01:05:01,690 --> 01:05:08,657 the logical progression is that you would have-- I don't know. 936 01:05:08,657 --> 01:05:09,240 PROFESSOR: No. 937 01:05:12,660 --> 01:05:18,120 At that time, the proposal was that essentially 938 01:05:18,120 --> 01:05:22,000 if you have oscillator of frequency omega, 939 01:05:22,000 --> 01:05:25,200 its energy is quantized in multiples of omega. 940 01:05:25,200 --> 01:05:27,800 So that's really the only aspect of quantum mechanics. 941 01:05:27,800 --> 01:05:31,310 So I actually jumped the historical development 942 01:05:31,310 --> 01:05:35,030 where I gave you the rotational degrees of freedom. 943 01:05:35,030 --> 01:05:42,761 So as I said, historically this was resolved last in this part 944 01:05:42,761 --> 01:05:44,885 because they didn't know what to do with rotations. 945 01:05:48,910 --> 01:05:52,040 But now I'm saying that you know about rotations, 946 01:05:52,040 --> 01:05:54,860 you know that the heat capacity goes to 0. 947 01:05:54,860 --> 01:05:58,910 You say, well, solid is composed. 948 01:05:58,910 --> 01:06:01,450 The way that you put heat into the system, 949 01:06:01,450 --> 01:06:05,240 enhance its heat capacity, is because there is kinetic energy 950 01:06:05,240 --> 01:06:07,620 that you put in the atoms of the solid. 951 01:06:07,620 --> 01:06:10,090 And as you try to put kinetic energy, 952 01:06:10,090 --> 01:06:14,080 there is this cage model and there's restoring force. 953 01:06:14,080 --> 01:06:16,610 The thing that is wrong about this model 954 01:06:16,610 --> 01:06:24,420 is that, basically, if you ask how easy it is to give energy 955 01:06:24,420 --> 01:06:28,530 to the system, if rather than having one frequency 956 01:06:28,530 --> 01:06:33,150 you have multiple frequencies, then at low temperatures 957 01:06:33,150 --> 01:06:36,840 you would put energy in the lower frequency. 958 01:06:36,840 --> 01:06:44,800 Because the typical scale we saw for connecting temperature 959 01:06:44,800 --> 01:06:47,850 and frequency, they are kind of proportional to each other. 960 01:06:47,850 --> 01:06:50,310 So if you want to go to low temperature, 961 01:06:50,310 --> 01:06:55,290 you are bound to excite things that have lower frequency. 962 01:06:55,290 --> 01:06:58,025 So the thing is that it is true that there 963 01:06:58,025 --> 01:07:00,310 is a typical frequency. 964 01:07:00,310 --> 01:07:03,990 But the typical frequency becomes less and less important 965 01:07:03,990 --> 01:07:06,010 as you go to low temperature. 966 01:07:06,010 --> 01:07:08,840 The issue is, what are the lowest frequencies 967 01:07:08,840 --> 01:07:11,160 of excitation? 968 01:07:11,160 --> 01:07:13,510 And basically, the correct picture 969 01:07:13,510 --> 01:07:17,890 of excitations of the solid is that you bang on something 970 01:07:17,890 --> 01:07:22,180 and you generate these sound waves. 971 01:07:22,180 --> 01:07:32,550 So what you have is that oscillations or vibrations 972 01:07:32,550 --> 01:07:50,240 of solid are characterized by wavelength and wave number 973 01:07:50,240 --> 01:08:01,290 k, 2 pi over lambda. 974 01:08:01,290 --> 01:08:19,450 So if I really take a better model of the solid 975 01:08:19,450 --> 01:08:27,069 in which I have springs that connect all of these things 976 01:08:27,069 --> 01:08:33,290 together and ask, what are the normal modes of vibration? 977 01:08:33,290 --> 01:08:36,970 I find that the normal modes can be characterized 978 01:08:36,970 --> 01:08:39,660 by some wave number k. 979 01:08:39,660 --> 01:08:43,399 As I said, it's the inverse of the wavelength. 980 01:08:43,399 --> 01:08:47,819 And frequency depends on wave number. 981 01:08:47,819 --> 01:08:56,069 In a manner that when you go to 0k, frequency goes to 0. 982 01:08:56,069 --> 01:08:57,340 And why is that? 983 01:08:57,340 --> 01:09:00,819 Essentially, what I'm saying is that if you 984 01:09:00,819 --> 01:09:05,420 look at particles that are along a line 985 01:09:05,420 --> 01:09:07,649 and may be connected by springs. 986 01:09:07,649 --> 01:09:11,859 So a kind of one-dimensional version of a solid. 987 01:09:11,859 --> 01:09:14,920 Then, the normal modes are characterized 988 01:09:14,920 --> 01:09:18,124 by distortions that have some particular wavelength. 989 01:09:23,160 --> 01:09:26,350 And in the limit where the wavelength 990 01:09:26,350 --> 01:09:30,439 goes to 0, essentially-- 991 01:09:30,439 --> 01:09:32,580 Sorry, in the limit where the wavelength goes 992 01:09:32,580 --> 01:09:36,729 to infinity or k goes to 0, it looks 993 01:09:36,729 --> 01:09:38,990 like I am taking all of the particles 994 01:09:38,990 --> 01:09:43,120 and translating them together. 995 01:09:43,120 --> 01:09:47,910 And if I take the entire solid here and translate it, 996 01:09:47,910 --> 01:09:50,124 there is no restoring force. 997 01:09:50,124 --> 01:09:55,370 So omega has to go to 0 as your k goes to 0, 998 01:09:55,370 --> 01:09:57,810 or wavelength goes to infinity. 999 01:09:57,810 --> 01:10:01,920 And there is a symmetry between k and minus k, 1000 01:10:01,920 --> 01:10:04,240 in fact, that forces the restoring force 1001 01:10:04,240 --> 01:10:06,560 to be proportional to k squared. 1002 01:10:06,560 --> 01:10:08,570 And when you take the square root of that, 1003 01:10:08,570 --> 01:10:10,170 you get the frequency. 1004 01:10:10,170 --> 01:10:15,160 You always get a linear behavior as k goes to 0. 1005 01:10:15,160 --> 01:10:19,810 So essentially, that's the observation 1006 01:10:19,810 --> 01:10:22,620 that whatever you do with your solid, 1007 01:10:22,620 --> 01:10:27,140 no matter how complicated, you have sound modes. 1008 01:10:27,140 --> 01:10:29,520 And sound modes are things that happen 1009 01:10:29,520 --> 01:10:32,360 in the limit where you have long wavelengths 1010 01:10:32,360 --> 01:10:37,930 and there is a relationship between omega and k 1011 01:10:37,930 --> 01:10:39,785 through some kind of velocity of sound. 1012 01:10:43,400 --> 01:10:49,370 Now, to be precise there are really 1013 01:10:49,370 --> 01:10:51,910 three types of sound waves. 1014 01:10:51,910 --> 01:10:55,360 If I choose the direction k along which 1015 01:10:55,360 --> 01:10:58,650 I want to create an oscillation, the distortions 1016 01:10:58,650 --> 01:11:01,760 can be either along that direction or perpendicular 1017 01:11:01,760 --> 01:11:02,790 to that. 1018 01:11:02,790 --> 01:11:07,830 They can either be longitudinal or transfers. 1019 01:11:07,830 --> 01:11:12,340 So there could be one or two other branches. 1020 01:11:12,340 --> 01:11:15,470 So there could, in principle, be different straight lines 1021 01:11:15,470 --> 01:11:17,560 as k goes to 0. 1022 01:11:17,560 --> 01:11:19,920 And the other thing is that there 1023 01:11:19,920 --> 01:11:23,470 is a shortest wavelength that you can think about. 1024 01:11:23,470 --> 01:11:28,670 So if these particles are a distance a apart, 1025 01:11:28,670 --> 01:11:31,840 there is no sense in going to wave numbers that 1026 01:11:31,840 --> 01:11:34,390 are larger than pi over a. 1027 01:11:34,390 --> 01:11:38,400 So you have some limit to these curves. 1028 01:11:38,400 --> 01:11:41,010 And indeed, when you approach the boundary, 1029 01:11:41,010 --> 01:11:43,475 this linear dependence can shift and change 1030 01:11:43,475 --> 01:11:46,270 in all kinds of possible ways. 1031 01:11:46,270 --> 01:11:51,480 And calculating the frequency inside one 1032 01:11:51,480 --> 01:12:00,526 of these units that is called a Brillouin zone 1033 01:12:00,526 --> 01:12:05,980 is a nice thing to do for the case 1034 01:12:05,980 --> 01:12:09,180 of using methods of solid state. 1035 01:12:09,180 --> 01:12:11,660 And you've probably seen that. 1036 01:12:11,660 --> 01:12:18,390 And there is a whole spectrum of frequencies 1037 01:12:18,390 --> 01:12:21,570 as a function of wave number that 1038 01:12:21,570 --> 01:12:24,280 correctly characterize a solid. 1039 01:12:24,280 --> 01:12:27,370 So it may be that somewhere in the middle of this spectrum 1040 01:12:27,370 --> 01:12:32,300 is a typical frequency omega E. But the point is 1041 01:12:32,300 --> 01:12:35,900 that as you go to lower and lower temperatures, 1042 01:12:35,900 --> 01:12:42,830 because of these factors of e to the minus beta h bar omega, 1043 01:12:42,830 --> 01:12:45,970 you can see that as you go to lower and lower temperature, 1044 01:12:45,970 --> 01:12:50,370 the only things that get excited are omegas that are also 1045 01:12:50,370 --> 01:12:54,340 going to 0 proportionately to kT. 1046 01:12:54,340 --> 01:13:01,810 So I can draw a line here that corresponds to frequencies that 1047 01:13:01,810 --> 01:13:04,372 are of the order of kT over h bar. 1048 01:13:08,630 --> 01:13:10,680 All of the harmonic oscillators that 1049 01:13:10,680 --> 01:13:15,190 have these larger frequencies that occur at short wavelengths 1050 01:13:15,190 --> 01:13:16,440 are unimportant. 1051 01:13:16,440 --> 01:13:19,960 They're kind of frozen, just like the vibrations 1052 01:13:19,960 --> 01:13:22,410 of the oxygen molecules in this room are frozen. 1053 01:13:22,410 --> 01:13:24,710 You cannot put energy in them. 1054 01:13:24,710 --> 01:13:27,610 They don't contribute to heat capacity. 1055 01:13:27,610 --> 01:13:30,590 But all of these long wavelength modes 1056 01:13:30,590 --> 01:13:35,080 down here have frequencies that go to 0. 1057 01:13:35,080 --> 01:13:38,500 Their excitation possibility is large. 1058 01:13:38,500 --> 01:13:40,500 And it, indeed, these long wavelength 1059 01:13:40,500 --> 01:13:44,885 modes that are easy to excite and continue to heat capacity. 1060 01:13:48,090 --> 01:13:51,130 I'll do maybe the precise calculation next time, 1061 01:13:51,130 --> 01:13:53,610 but even within this picture we can figure out 1062 01:13:53,610 --> 01:13:56,190 why the answer should be proportional to T cubed. 1063 01:13:58,770 --> 01:14:02,460 So what I need to do, rather than counting 1064 01:14:02,460 --> 01:14:06,790 all harmonic oscillators-- the factor of 3n-- 1065 01:14:06,790 --> 01:14:10,380 I have to count how many oscillators have frequencies 1066 01:14:10,380 --> 01:14:13,600 that are less than this kT over h bar. 1067 01:14:16,140 --> 01:14:28,260 So I claim that number of modes with frequency less than kT 1068 01:14:28,260 --> 01:14:40,970 over h bar goes like kT over h bar cubed V. 1069 01:14:40,970 --> 01:14:43,657 Essentially, what I have to do is 1070 01:14:43,657 --> 01:14:50,710 to do a summation over all k that is less than some k max. 1071 01:14:50,710 --> 01:14:55,080 This k max is set by this condition 1072 01:14:55,080 --> 01:14:58,450 that Vk max is of the order of kT over h bar. 1073 01:14:58,450 --> 01:15:04,620 So this k max is of the order of kT over h bar V. 1074 01:15:04,620 --> 01:15:09,770 So actually, to be more precise I have to put a V here. 1075 01:15:09,770 --> 01:15:14,700 So I have to count all of the modes. 1076 01:15:14,700 --> 01:15:18,090 Now, this separation between these modes-- 1077 01:15:18,090 --> 01:15:21,230 if you have a box of size l is 2 pi over l. 1078 01:15:21,230 --> 01:15:24,870 So maybe we will discuss that later on. 1079 01:15:24,870 --> 01:15:27,770 But the summations over k you will always 1080 01:15:27,770 --> 01:15:31,730 replace with integrations over k times 1081 01:15:31,730 --> 01:15:37,900 the density of state, which is V divided by 2 pi cubed. 1082 01:15:37,900 --> 01:15:41,880 So this has to go between 0 and k max. 1083 01:15:41,880 --> 01:15:46,330 And so this is proportional to V k max 1084 01:15:46,330 --> 01:15:50,685 cubed, which is what I wrote over there. 1085 01:15:53,300 --> 01:15:56,000 So as I go to lower and lower temperature, 1086 01:15:56,000 --> 01:15:59,930 there are fewer and fewer oscillators. 1087 01:15:59,930 --> 01:16:04,620 The number of those oscillators grows like T cubed. 1088 01:16:04,620 --> 01:16:07,300 Each one of those oscillators is fully excited 1089 01:16:07,300 --> 01:16:11,550 as energy kT contributes 1 unit to heat capacity. 1090 01:16:11,550 --> 01:16:15,480 Since the number of oscillators goes to 0 as T cubed, 1091 01:16:15,480 --> 01:16:18,030 the heat capacity that they contribute also 1092 01:16:18,030 --> 01:16:22,450 goes to 0 as T cubed. 1093 01:16:22,450 --> 01:16:27,860 So you don't really need to know-- this is actually 1094 01:16:27,860 --> 01:16:29,570 an interesting thing to ponder. 1095 01:16:29,570 --> 01:16:31,420 So rather than doing the calculations, 1096 01:16:31,420 --> 01:16:34,480 maybe just think about this. 1097 01:16:34,480 --> 01:16:40,980 That somehow the solid could be arbitrarily complicated. 1098 01:16:40,980 --> 01:16:44,460 So it could be composed of molecules 1099 01:16:44,460 --> 01:16:46,270 that have some particular shape. 1100 01:16:46,270 --> 01:16:49,460 They are forming some strange lattice 1101 01:16:49,460 --> 01:16:50,970 of some form, et cetera. 1102 01:16:53,920 --> 01:16:58,180 And given the complicated nature of the molecules, 1103 01:16:58,180 --> 01:17:02,080 the spectrum that you have for potential frequencies 1104 01:17:02,080 --> 01:17:05,220 that a solid can take, because of all 1105 01:17:05,220 --> 01:17:07,680 of the different vibrations, et cetera, 1106 01:17:07,680 --> 01:17:09,560 could be arbitrary complicated. 1107 01:17:09,560 --> 01:17:13,430 You can have kinds of oscillations such as the ones 1108 01:17:13,430 --> 01:17:16,040 that I have indicated. 1109 01:17:16,040 --> 01:17:19,230 However, if you go to low temperature, 1110 01:17:19,230 --> 01:17:22,730 you are only interested in vibrations 1111 01:17:22,730 --> 01:17:25,470 that are very low in frequency. 1112 01:17:25,470 --> 01:17:28,090 Vibrations that are very low in frequencies 1113 01:17:28,090 --> 01:17:30,340 must correspond to the formations 1114 01:17:30,340 --> 01:17:32,920 that are very long wavelength. 1115 01:17:32,920 --> 01:17:34,340 And when you are looking at things 1116 01:17:34,340 --> 01:17:36,250 that are long wavelength, this is, again, 1117 01:17:36,250 --> 01:17:38,950 another thing that has statistical in character. 1118 01:17:38,950 --> 01:17:41,350 That is, rather you are here looking 1119 01:17:41,350 --> 01:17:47,510 at things that span thousands of atoms or molecules. 1120 01:17:47,510 --> 01:17:50,510 However, as you go to lower and lower temperature, 1121 01:17:50,510 --> 01:17:53,240 more and more atoms and molecules. 1122 01:17:53,240 --> 01:17:57,280 And so again, some kind of averaging is taking place. 1123 01:17:57,280 --> 01:18:00,840 All of the details, et cetera, wash out. 1124 01:18:00,840 --> 01:18:04,380 You really see some global characteristic. 1125 01:18:04,380 --> 01:18:06,370 The global characteristic that you see 1126 01:18:06,370 --> 01:18:07,650 is set by this symmetry. 1127 01:18:07,650 --> 01:18:12,580 Just the fact that when I go to exactly k equals to 0, 1128 01:18:12,580 --> 01:18:13,800 I am translating. 1129 01:18:13,800 --> 01:18:18,190 I have 0 frequency. 1130 01:18:18,190 --> 01:18:21,560 So when I'm doing something that is long wavelength, 1131 01:18:21,560 --> 01:18:24,710 the frequency should somehow be proportional 1132 01:18:24,710 --> 01:18:25,520 to that wavelength. 1133 01:18:25,520 --> 01:18:29,120 So that's just a statement of continuity if you like. 1134 01:18:29,120 --> 01:18:33,300 Once I have made that statement, then it's 1135 01:18:33,300 --> 01:18:37,170 just a calculation of how many modes are possible. 1136 01:18:37,170 --> 01:18:40,350 The number of modes will be proportional to T cubed. 1137 01:18:40,350 --> 01:18:43,885 And I will get this T cubed law irrespective 1138 01:18:43,885 --> 01:18:46,520 of how complicated the solid is. 1139 01:18:46,520 --> 01:18:50,570 All of the solids will have the same T cubed behavior. 1140 01:18:50,570 --> 01:18:55,120 The place where they come from the classical behavior 1141 01:18:55,120 --> 01:18:57,880 to this quantum behavior will depend 1142 01:18:57,880 --> 01:19:01,110 on the details of the solid, et cetera. 1143 01:19:01,110 --> 01:19:04,005 But the low temperature law, this T cubed law, 1144 01:19:04,005 --> 01:19:07,700 is something that is universal. 1145 01:19:07,700 --> 01:19:10,975 OK, so next time around, we will do this calculation 1146 01:19:10,975 --> 01:19:15,210 in more detail, and then see also its connection 1147 01:19:15,210 --> 01:19:17,590 to the blackbody radius.