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 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:21,690 --> 00:00:24,680 PROFESSOR: So we've been developing the procedure 9 00:00:24,680 --> 00:00:27,610 to describe a quantum gas of identical particles. 10 00:00:38,335 --> 00:00:39,460 That's our non-interacting. 11 00:00:48,200 --> 00:00:50,390 So what does that mean? 12 00:00:50,390 --> 00:00:52,940 Is that we are thinking about the Hamiltonian 13 00:00:52,940 --> 00:00:55,860 for a system of N particles. 14 00:00:55,860 --> 00:00:58,240 If they are not interacting, it can 15 00:00:58,240 --> 00:01:00,990 be written as the sum of the contributions 16 00:01:00,990 --> 00:01:03,590 for the individual particles. 17 00:01:03,590 --> 00:01:05,750 And if they are identical, it's basically 18 00:01:05,750 --> 00:01:09,360 the same particle for each one of them. 19 00:01:09,360 --> 00:01:13,600 Same Hamiltonian for each one of them. 20 00:01:13,600 --> 00:01:14,840 OK? 21 00:01:14,840 --> 00:01:17,420 How do we handle these? 22 00:01:17,420 --> 00:01:20,870 Well, if you regard this as a quantum problem, 23 00:01:20,870 --> 00:01:23,400 for a single particle-- so for each one 24 00:01:23,400 --> 00:01:28,680 of these Hamiltonians-- we can find the eigenstates. 25 00:01:28,680 --> 00:01:33,790 So there's a bunch of states that say label by k because we 26 00:01:33,790 --> 00:01:36,300 are ultimately have in mind the plane waves that 27 00:01:36,300 --> 00:01:41,040 describe gas particles in a box. 28 00:01:41,040 --> 00:01:43,540 And there is corresponding energy levels 29 00:01:43,540 --> 00:01:46,020 for the single particles, epsilon k. 30 00:01:48,680 --> 00:01:53,180 And the single particle spectrum can therefore 31 00:01:53,180 --> 00:01:58,230 can be described by a bunch of energies, a lot of energy 32 00:01:58,230 --> 00:02:00,980 levels, with different values of k 33 00:02:00,980 --> 00:02:06,790 and different allowed discretized energies. 34 00:02:06,790 --> 00:02:11,390 Now, when we want to describe a case of N particles 35 00:02:11,390 --> 00:02:13,930 that are non-interacting, we said 36 00:02:13,930 --> 00:02:18,250 that we could, in principle, pick one k here, 37 00:02:18,250 --> 00:02:21,550 one k here, one k here. 38 00:02:21,550 --> 00:02:23,580 And if we are dealing with bosons, 39 00:02:23,580 --> 00:02:27,500 potentially more than one at each side. 40 00:02:27,500 --> 00:02:31,550 And so the state would then be described 41 00:02:31,550 --> 00:02:36,190 by a bunch of occupation numbers Nk. 42 00:02:36,190 --> 00:02:40,890 And this Nk will be 0 or 1 if you are talking about fermions, 43 00:02:40,890 --> 00:02:44,280 but any integer if you are talking about bosons. 44 00:02:44,280 --> 00:02:46,900 So this eta being plus or minus was 45 00:02:46,900 --> 00:02:49,162 used to distinguish appropriate symmetrization 46 00:02:49,162 --> 00:02:50,120 or anti-symmetrization. 47 00:02:53,270 --> 00:02:57,820 Now, once we have a state such as this, 48 00:02:57,820 --> 00:03:02,110 clearly the N particle hamintonium 49 00:03:02,110 --> 00:03:05,385 acting on a state that is symmetrized 50 00:03:05,385 --> 00:03:09,180 with these bunch of occupation numbers 51 00:03:09,180 --> 00:03:11,900 will give us back this state. 52 00:03:11,900 --> 00:03:14,830 The energy that we would be getting 53 00:03:14,830 --> 00:03:19,890 is summing over all possible k values, epsilon k 54 00:03:19,890 --> 00:03:25,850 times the number of occurrance of that in this state. 55 00:03:25,850 --> 00:03:27,970 Again, multiplying this state back. 56 00:03:32,750 --> 00:03:39,400 Of course, we have a constraint that the number of particles 57 00:03:39,400 --> 00:03:42,070 matches the sum of the occupation numbers. 58 00:03:46,760 --> 00:03:49,600 So then we said, OK, this is just 59 00:03:49,600 --> 00:03:54,930 a description of the state that they want to describe, 60 00:03:54,930 --> 00:03:59,600 or the set of states that represent 61 00:03:59,600 --> 00:04:03,950 this system of identical non-interacting partners 62 00:04:03,950 --> 00:04:05,960 in quantum mechanics. 63 00:04:05,960 --> 00:04:09,700 So the next stage, we want to do some kind of first statistics 64 00:04:09,700 --> 00:04:11,100 for them. 65 00:04:11,100 --> 00:04:13,830 And we had seen that basically, in order 66 00:04:13,830 --> 00:04:19,320 to assign appropriate statistics in case of quantum systems, 67 00:04:19,320 --> 00:04:22,670 we have to deal with a density matrix. 68 00:04:22,670 --> 00:04:26,660 And we saw that in the canonical representation, 69 00:04:26,660 --> 00:04:29,480 the density matrix is complicated 70 00:04:29,480 --> 00:04:33,850 and amounts to some kind of interaction among particles. 71 00:04:33,850 --> 00:04:36,930 Although there is no interaction at this level, 72 00:04:36,930 --> 00:04:40,620 the symmetrization or anti-symmetrization 73 00:04:40,620 --> 00:04:43,960 create states that are intermixed in a manner that 74 00:04:43,960 --> 00:04:47,080 appears as if you have an interaction. 75 00:04:47,080 --> 00:04:49,850 We found that, however, we could get rid of that 76 00:04:49,850 --> 00:04:59,960 by going to the grand canonical ensemble where we would trade 77 00:04:59,960 --> 00:05:02,440 in the constraint that the number of particles 78 00:05:02,440 --> 00:05:04,820 is fixed with a chemical potential. 79 00:05:07,570 --> 00:05:14,150 And in this ensemble, we found that the density matrix 80 00:05:14,150 --> 00:05:17,320 is diagonal in this representation. 81 00:05:17,320 --> 00:05:20,090 So the diagonal elements can be thought 82 00:05:20,090 --> 00:05:25,620 of as the probability of one of these occupation number sets. 83 00:05:25,620 --> 00:05:32,400 And so the probability of a particular set of occupations 84 00:05:32,400 --> 00:05:35,690 in this grand canonical prescription 85 00:05:35,690 --> 00:05:40,680 is proportional to e to the beta-- of course 86 00:05:40,680 --> 00:05:46,860 we have mu if we are going to this grand canonical ensemble-- 87 00:05:46,860 --> 00:05:53,610 minus epsilon k, and then we had Nk. 88 00:05:53,610 --> 00:06:00,720 So the probability that a particular one of these 89 00:06:00,720 --> 00:06:06,360 is occupied Nk times in this grand canonical prescription 90 00:06:06,360 --> 00:06:09,180 is simply an exponential. 91 00:06:09,180 --> 00:06:12,280 And of course, the different k levels 92 00:06:12,280 --> 00:06:14,440 are completely independent of each other. 93 00:06:14,440 --> 00:06:17,900 So the probability factor arises in this fashion. 94 00:06:17,900 --> 00:06:22,200 Of course, as any probability, we have to normalize this. 95 00:06:22,200 --> 00:06:26,120 Normalization is appropriate. 96 00:06:26,120 --> 00:06:29,560 We call the grand canonical partition function 97 00:06:29,560 --> 00:06:31,640 and will be different whether we are dealing 98 00:06:31,640 --> 00:06:34,460 with boson statistics or fermion statistics. 99 00:06:37,960 --> 00:06:43,880 Actually, because we have a structure such as this, 100 00:06:43,880 --> 00:06:47,330 Q is going to be the product of the normalization 101 00:06:47,330 --> 00:06:50,290 of each individual one. 102 00:06:50,290 --> 00:06:55,705 And therefore, log of Q is going to be 103 00:06:55,705 --> 00:06:58,970 a sum of the contributions that I 104 00:06:58,970 --> 00:07:02,960 would get for the normalization of each individual ones. 105 00:07:02,960 --> 00:07:06,150 And of course, the log of whatever normalization 106 00:07:06,150 --> 00:07:09,280 I have for the individual one. 107 00:07:09,280 --> 00:07:16,780 If I have the case of a fermions where Nk gets 0 or 1 as the two 108 00:07:16,780 --> 00:07:19,700 possibilities, then the normalization 109 00:07:19,700 --> 00:07:24,510 will be simply 1 plus this exponential. 110 00:07:24,510 --> 00:07:28,773 And I choose to write that as z e to the minus beta 111 00:07:28,773 --> 00:07:34,566 epsilon of k where I have introduced z 112 00:07:34,566 --> 00:07:36,900 to be e to the beta mu. 113 00:07:36,900 --> 00:07:40,030 Because this combination occurs all the time, 114 00:07:40,030 --> 00:07:44,270 I don't want to keep repeating that. 115 00:07:44,270 --> 00:07:47,570 So this was for the case of fermions. 116 00:07:47,570 --> 00:07:52,630 For the case of bosons where N goes 0, 1, 2, 3, et cetera, 117 00:07:52,630 --> 00:07:55,180 this is a geometric series. 118 00:07:55,180 --> 00:07:59,250 For the geometric series, when I sum it, 119 00:07:59,250 --> 00:08:01,910 I would get 1 over this factor. 120 00:08:01,910 --> 00:08:06,910 So when I take the log I will have a minus sign here. 121 00:08:06,910 --> 00:08:10,290 And I can put the Boson and Fermi cases 122 00:08:10,290 --> 00:08:14,290 together by simply putting a factor of minus eta 123 00:08:14,290 --> 00:08:18,990 here so that I have the plus for the case of fermions 124 00:08:18,990 --> 00:08:22,724 and the minus for the case of bosons. 125 00:08:22,724 --> 00:08:25,610 OK? 126 00:08:25,610 --> 00:08:30,000 So this is one thing that we will be using. 127 00:08:30,000 --> 00:08:33,370 The other thing is that this is the probability 128 00:08:33,370 --> 00:08:38,299 for this level having a certain occupation number. 129 00:08:38,299 --> 00:08:43,441 In k we can always ask, what is the average of Nk? 130 00:08:46,390 --> 00:08:52,290 And from this probability it is easy to calculate the average. 131 00:08:52,290 --> 00:08:55,100 And the answer, again, can be collapsed 132 00:08:55,100 --> 00:08:59,160 into one expression for the case of bosons and fermions 133 00:08:59,160 --> 00:09:07,600 as 1 over z inverse e to the beta epsilon of k minus eta 134 00:09:07,600 --> 00:09:11,800 encompassing both fermions and bosons. 135 00:09:11,800 --> 00:09:23,760 So these results give some idea both at the microscopic level-- 136 00:09:23,760 --> 00:09:28,060 in terms of occupation numbers of these different levels-- 137 00:09:28,060 --> 00:09:31,090 and macroscopic level a grand partition function 138 00:09:31,090 --> 00:09:34,130 from which we can construct thermodynamics. 139 00:09:34,130 --> 00:09:36,030 Actually, full probability, again, 140 00:09:36,030 --> 00:09:39,560 in this ensemble which we can transform in principle 141 00:09:39,560 --> 00:09:43,410 and get density matrices, et cetera, in other basis. 142 00:09:43,410 --> 00:09:45,940 But these are the kinds of things 143 00:09:45,940 --> 00:09:51,390 that we can do for any interacting Hamiltonian. 144 00:09:51,390 --> 00:09:54,390 OK? 145 00:09:54,390 --> 00:09:58,370 Now, what I said over here was that I'm actually 146 00:09:58,370 --> 00:09:59,905 interested in the case of a gas. 147 00:10:03,240 --> 00:10:07,670 So let's again think about the type of gas 148 00:10:07,670 --> 00:10:10,660 such as the one in this room that we are interested. 149 00:10:10,660 --> 00:10:14,070 And again, to make a contrast with something 150 00:10:14,070 --> 00:10:16,370 that we may want to do later, let's 151 00:10:16,370 --> 00:10:19,090 emphasize that it is a non-relativistic gas. 152 00:10:26,300 --> 00:10:35,530 So the Hamiltonian H is essentially free particle 153 00:10:35,530 --> 00:10:39,100 in a box of volume V, the thing that we 154 00:10:39,100 --> 00:10:41,090 have been using all the time, just 155 00:10:41,090 --> 00:10:44,160 the kinetic energy of a one particle state. 156 00:10:44,160 --> 00:10:48,170 And for that we know that the single particle energy 157 00:10:48,170 --> 00:10:53,090 levels are characterized by a vector k. 158 00:10:53,090 --> 00:10:55,330 As three components, in fact. 159 00:10:55,330 --> 00:10:59,600 And h bar squared over 2mk squared. 160 00:10:59,600 --> 00:11:02,590 And we've discussed what the discretization of k 161 00:11:02,590 --> 00:11:06,070 is appropriate to the size of the box. 162 00:11:06,070 --> 00:11:07,500 We won't go over that. 163 00:11:10,790 --> 00:11:13,590 So again, maybe it is important to mention 164 00:11:13,590 --> 00:11:17,630 that because of the discretizatoin of the k values, 165 00:11:17,630 --> 00:11:20,145 the spacing between possible values of k 166 00:11:20,145 --> 00:11:23,270 are 2 pi over l in each direction. 167 00:11:23,270 --> 00:11:25,510 And ultimately what we want to do 168 00:11:25,510 --> 00:11:33,280 is to replace these sums over k's in the limit of large boxes 169 00:11:33,280 --> 00:11:35,620 with integrals over k. 170 00:11:35,620 --> 00:11:41,790 So these go over to an integral over k times 171 00:11:41,790 --> 00:11:45,920 the density factor that you get in going from the sum 172 00:11:45,920 --> 00:11:50,490 to the integral, which is 2 pi cubed times the volume. 173 00:11:55,050 --> 00:12:01,020 I will introduce one other factor here, 174 00:12:01,020 --> 00:12:04,710 which is that when we are talking about quantum 175 00:12:04,710 --> 00:12:10,010 particles and bosons and fermions, 176 00:12:10,010 --> 00:12:16,050 they are typically assigned also another state-- 177 00:12:16,050 --> 00:12:19,790 another parameter-- that characterizes their spin. 178 00:12:19,790 --> 00:12:23,430 And depending on the possible values of spin, 179 00:12:23,430 --> 00:12:27,270 you will have g copies of the system. 180 00:12:27,270 --> 00:12:32,680 And this g will be related to the quantum spin-- 181 00:12:32,680 --> 00:12:38,020 to the formula 2s 1-- so that when you have fermions, 182 00:12:38,020 --> 00:12:42,440 typically s is half integer and this will be an even number. 183 00:12:42,440 --> 00:12:46,730 Whereas for bosons it will be an odd number. 184 00:12:46,730 --> 00:12:48,720 So that's, again, something that we've seen. 185 00:12:48,720 --> 00:12:51,610 You just, essentially says multiply everything 186 00:12:51,610 --> 00:12:53,021 by some factor of g. 187 00:12:55,970 --> 00:12:57,720 Now if I go through this procedure 188 00:12:57,720 --> 00:13:01,180 that I have outlined here, I will ultimately 189 00:13:01,180 --> 00:13:05,015 get the grand partition function of the gas-- which 190 00:13:05,015 --> 00:13:08,210 as we will see is related to its pressure-- in terms 191 00:13:08,210 --> 00:13:10,790 of the chemical potential. 192 00:13:10,790 --> 00:13:13,950 Throughout the course we've seen that the more interesting thing 193 00:13:13,950 --> 00:13:18,120 to look at is how things depend on density rather than chemical 194 00:13:18,120 --> 00:13:19,290 potential. 195 00:13:19,290 --> 00:13:25,180 So I'd better use this result to get what the density is. 196 00:13:25,180 --> 00:13:26,475 How do I do that? 197 00:13:26,475 --> 00:13:31,780 Well, once I have determined what the chemical potential is, 198 00:13:31,780 --> 00:13:34,150 the average number of particles will 199 00:13:34,150 --> 00:13:38,360 be the sum over and the average occupation numbers 200 00:13:38,360 --> 00:13:40,070 that I have in all of these states. 201 00:13:43,520 --> 00:13:47,630 And according to the prescription that we have over 202 00:13:47,630 --> 00:13:51,980 here, in the limit of a large box, this sum over k 203 00:13:51,980 --> 00:13:58,820 becomes gV intergral d cubed k 2 pi Q. Ultimately 204 00:13:58,820 --> 00:14:01,540 I will divide by V so I will have 205 00:14:01,540 --> 00:14:04,050 the formula for the density. 206 00:14:04,050 --> 00:14:06,600 And what is Nk? 207 00:14:06,600 --> 00:14:08,300 Well, I have the formula for Nk. 208 00:14:08,300 --> 00:14:13,590 It is 1 over z inverse e to the beta. 209 00:14:13,590 --> 00:14:17,700 Epsilon k is h bar squared k squared over 2m, 210 00:14:17,700 --> 00:14:19,480 and then I have a minus 8. 211 00:14:22,460 --> 00:14:22,960 OK? 212 00:14:26,200 --> 00:14:31,390 So if I divide through by V, really I will have the density. 213 00:14:36,130 --> 00:14:39,590 Get rid of this V. But I have to do that integral. 214 00:14:43,230 --> 00:14:46,070 Now for this and a lot of other things, 215 00:14:46,070 --> 00:14:50,680 you can see that always I will have sums or integrations 216 00:14:50,680 --> 00:14:53,560 over the combination that appears in the exponent 217 00:14:53,560 --> 00:14:58,390 and enhances dimension list which is beta epsilon of k. 218 00:14:58,390 --> 00:15:00,380 So let me introduce a new variable. 219 00:15:00,380 --> 00:15:06,410 Let me call x to be the combination beta h bar squared 220 00:15:06,410 --> 00:15:08,560 k squared over 2m. 221 00:15:08,560 --> 00:15:12,420 So I'm going to call this e to the x, if you like. 222 00:15:12,420 --> 00:15:16,975 So I have changed variables from k to x. 223 00:15:16,975 --> 00:15:24,540 k in terms of x is simply square root of 2m inverse of beta-- 224 00:15:24,540 --> 00:15:30,130 which is kt-- divided by h bar squared. 225 00:15:30,130 --> 00:15:32,910 But I took a square root so it's just h bar. 226 00:15:32,910 --> 00:15:38,190 Times x to the 1/2. 227 00:15:38,190 --> 00:15:40,900 OK? 228 00:15:40,900 --> 00:15:49,590 Now, I realize that h bar is in fact h divided by 2pi. 229 00:15:49,590 --> 00:15:52,740 So the 2pi I can put in the numerator. 230 00:15:55,930 --> 00:16:00,710 And then I remember that actually the combination h 231 00:16:00,710 --> 00:16:04,540 divided by square root of 2pi mkt, 232 00:16:04,540 --> 00:16:08,470 I was calling these thermal wavelength lambda. 233 00:16:08,470 --> 00:16:12,570 And k should have the dimensions of an inverse landscape, 234 00:16:12,570 --> 00:16:15,195 so that's the appropriate thing to choose. 235 00:16:15,195 --> 00:16:17,220 So what do I get? 236 00:16:17,220 --> 00:16:19,850 I take a square root of pi inside 237 00:16:19,850 --> 00:16:25,000 and I'm left with 2 square root of pi divided by this parameter 238 00:16:25,000 --> 00:16:29,280 lambda x to the 1/2. 239 00:16:29,280 --> 00:16:34,020 Again, for later clarity, let's make 240 00:16:34,020 --> 00:16:40,650 sure we remember this was our lambda. 241 00:16:40,650 --> 00:16:45,370 So this is the variable k that I need. 242 00:16:45,370 --> 00:16:49,660 In fact, for the-- for the integration 243 00:16:49,660 --> 00:16:53,640 I have to also make a change in dk. 244 00:16:53,640 --> 00:16:58,830 So I not that dk, if I take a derivative here, 245 00:16:58,830 --> 00:17:05,030 I will get x-- 1/2x to the minus 1/2 becomes root pi over lambda 246 00:17:05,030 --> 00:17:06,740 x to the minus 1/2 dx. 247 00:17:12,130 --> 00:17:15,700 Actually, the factor that I need over here 248 00:17:15,700 --> 00:17:20,710 is dk-- d cubed k over 2pi cubed. 249 00:17:23,520 --> 00:17:26,359 Now everything only depends on the magnitude of k. 250 00:17:26,359 --> 00:17:30,260 So I will take advantage of spherical symmetry, 251 00:17:30,260 --> 00:17:39,950 write d cubed k as 4pi k squared dk divided by 8pi cubed. 252 00:17:39,950 --> 00:17:43,700 And then I substitute from k squared and dk 253 00:17:43,700 --> 00:17:47,120 above in terms of this x. 254 00:17:47,120 --> 00:17:52,610 So I will have a 1 over 2pi squared from this division. 255 00:17:52,610 --> 00:18:00,600 k squared is simply 4pi over lambda squared x. 256 00:18:00,600 --> 00:18:02,870 This is k squared. 257 00:18:02,870 --> 00:18:10,860 dk is root pi over lambda x to the minus 1/2 dx. 258 00:18:14,310 --> 00:18:16,990 So you can see lots of things cancel out 259 00:18:16,990 --> 00:18:21,410 and the answer is going to be x to the 1/2 260 00:18:21,410 --> 00:18:25,930 dx divided by lambda cubed. 261 00:18:29,610 --> 00:18:33,230 And what is left from all of these pi's et cetera, 262 00:18:33,230 --> 00:18:38,130 is a factor of square root of pi over 2. 263 00:18:38,130 --> 00:18:41,160 Or 2 in the numerator or a square root of pi over 2 264 00:18:41,160 --> 00:18:43,940 in the denominator. 265 00:18:43,940 --> 00:18:46,160 OK? 266 00:18:46,160 --> 00:18:50,305 So if I do these set of changes that I 267 00:18:50,305 --> 00:18:52,460 have on the right part of the board, 268 00:18:52,460 --> 00:18:54,330 what do I have for density? 269 00:18:54,330 --> 00:18:58,470 The density becomes simply g. 270 00:18:58,470 --> 00:19:02,500 I have this integral over k that I 271 00:19:02,500 --> 00:19:11,930 have recast as an integral that is x to the 1/2 dx. 272 00:19:11,930 --> 00:19:16,950 Let me pull out the factor of lambda cubed on the outside. 273 00:19:16,950 --> 00:19:20,340 So I have g over lambda cubed. 274 00:19:20,340 --> 00:19:25,450 I will pull out to that factor of root pi over 2 275 00:19:25,450 --> 00:19:27,500 that I had before. 276 00:19:27,500 --> 00:19:29,670 And then I have the integral from 0 277 00:19:29,670 --> 00:19:32,910 to infinity dx x to the 1/2. 278 00:19:32,910 --> 00:19:36,820 Actually, let me write it in this fashion. 279 00:19:36,820 --> 00:19:39,840 dx x to the 1/2. 280 00:19:39,840 --> 00:19:43,500 And in the denominator, what do I have? 281 00:19:43,500 --> 00:19:48,390 I have z universe into the x minus 8. 282 00:19:52,980 --> 00:19:59,750 So we did all of these and we ended up with an integral 283 00:19:59,750 --> 00:20:02,430 that we don't recognize. 284 00:20:02,430 --> 00:20:06,600 So what we are going to do is to pretend that we recognize it 285 00:20:06,600 --> 00:20:08,450 and give it a name. 286 00:20:08,450 --> 00:20:09,240 All right? 287 00:20:09,240 --> 00:20:23,210 So we define a set of functions f sub m of variable z. 288 00:20:23,210 --> 00:20:29,320 Once I do the integration over x, I will get a function of z. 289 00:20:29,320 --> 00:20:34,120 So that function of z I will call fm of z. 290 00:20:34,120 --> 00:20:39,060 But clearly I have a different fm if I have eta minus 1 or eta 291 00:20:39,060 --> 00:20:41,930 plus 1 for bosons of fermions. 292 00:20:41,930 --> 00:20:44,080 So there is two classes of things. 293 00:20:44,080 --> 00:20:46,030 What's this m? 294 00:20:46,030 --> 00:20:47,180 OK. 295 00:20:47,180 --> 00:20:51,460 So here I had the integral x to the 1/2. 296 00:20:51,460 --> 00:20:55,120 It turns out that you will get a whole bunch of these integrals 297 00:20:55,120 --> 00:20:57,980 where the [INAUDIBLE] that is out front in the numerator 298 00:20:57,980 --> 00:20:59,700 is different. 299 00:20:59,700 --> 00:21:04,680 So I will generalize this x to the 1/2, this 1/2 to something 300 00:21:04,680 --> 00:21:07,400 that I'll call it m minus 1. 301 00:21:07,400 --> 00:21:13,440 And then I have z inverse e to the x minus eta. 302 00:21:13,440 --> 00:21:17,470 And if these normalized so that what is in front 303 00:21:17,470 --> 00:21:19,855 is 1 over m minus 1 factorial. 304 00:21:26,604 --> 00:21:27,600 OK? 305 00:21:27,600 --> 00:21:30,350 So these are very important functions. 306 00:21:30,350 --> 00:21:32,250 Everything we will do later on we 307 00:21:32,250 --> 00:21:34,390 will be expressing in terms of these functions. 308 00:21:37,300 --> 00:21:42,060 Now, clearly what is happening here corresponds to m minus 1 309 00:21:42,060 --> 00:21:48,580 being equal to 1/2 or m being 3/2. 310 00:21:48,580 --> 00:21:51,426 What is this normalization? 311 00:21:51,426 --> 00:21:56,230 Well, for the normalization, if this is 1/2 312 00:21:56,230 --> 00:21:59,450 I would need 1/2 factorial. 313 00:21:59,450 --> 00:22:03,070 And if you go and look at the various definitions 314 00:22:03,070 --> 00:22:07,430 that we have for the gamma function or the factorial, 315 00:22:07,430 --> 00:22:11,580 you can check that 1/2 factorial is root pi over 2. 316 00:22:11,580 --> 00:22:14,560 And, indeed, we saw this already when 317 00:22:14,560 --> 00:22:17,910 we wrote the expression for the surface area 318 00:22:17,910 --> 00:22:21,246 in general dimensions that involved 319 00:22:21,246 --> 00:22:24,470 factorial function appearing as fractions. 320 00:22:24,470 --> 00:22:27,000 And in order to get the right result in two or three 321 00:22:27,000 --> 00:22:30,330 dimensions, you can check that this root 322 00:22:30,330 --> 00:22:33,830 pi over 2 for 1/2 factorial is correct. 323 00:22:33,830 --> 00:22:38,670 So, indeed, I already have this root pi over 2 over here. 324 00:22:38,670 --> 00:22:40,840 It was properly normalized. 325 00:22:40,840 --> 00:22:43,490 So the whole thing here can be written 326 00:22:43,490 --> 00:22:51,270 as n is g over lambda cubed f 3/2 eta of z. 327 00:22:51,270 --> 00:22:53,560 That is the expression that relates 328 00:22:53,560 --> 00:22:58,720 the density to the chemical potential or this communication 329 00:22:58,720 --> 00:23:03,350 e to the beta mu, which is z or the fugacity 330 00:23:03,350 --> 00:23:05,930 is simply this nice formula. 331 00:23:10,240 --> 00:23:10,740 OK? 332 00:23:19,750 --> 00:23:22,540 But what we want is to calculate something 333 00:23:22,540 --> 00:23:25,020 like the pressure for the gas. 334 00:23:25,020 --> 00:23:29,880 Well, let me remind you that log Q, essentially, quite 335 00:23:29,880 --> 00:23:35,430 generally, log Q comes from normalizing something 336 00:23:35,430 --> 00:23:41,580 that is like mu times the total number of particles 337 00:23:41,580 --> 00:23:45,100 minus the energy-- you can kind of read that from here-- 338 00:23:45,100 --> 00:23:50,820 and then there would be TS from the number of states. 339 00:23:50,820 --> 00:23:53,810 This was the general formula that we had. 340 00:23:53,810 --> 00:23:59,660 And when we substitute for a system that is extensive, 341 00:23:59,660 --> 00:24:08,410 where E for a gas is mu n plus TS minus PV, 342 00:24:08,410 --> 00:24:11,980 we see that the log of the partition function 343 00:24:11,980 --> 00:24:14,250 is simply the combination beta PV. 344 00:24:18,350 --> 00:24:20,785 Of course, again, this pressure as a function 345 00:24:20,785 --> 00:24:23,860 of density or chemical potential would be different 346 00:24:23,860 --> 00:24:26,070 whether we are thinking about bosons of fermions. 347 00:24:28,760 --> 00:24:32,440 But it can be calculated from the kind of sum 348 00:24:32,440 --> 00:24:34,920 that we have over here. 349 00:24:34,920 --> 00:24:37,060 So let's do that. 350 00:24:37,060 --> 00:24:42,150 So I have beta P which is something 351 00:24:42,150 --> 00:24:47,290 like log Q divided by V. Is obtained 352 00:24:47,290 --> 00:24:52,620 by doing this sum over k over V. And so what do I get? 353 00:24:52,620 --> 00:24:56,780 I will get-- first of all, I have the minus eta sum 354 00:24:56,780 --> 00:25:01,130 over k [INAUDIBLE] with g integral 355 00:25:01,130 --> 00:25:06,900 over k divided by 2pi cubed. 356 00:25:06,900 --> 00:25:11,310 The volume factor cancels because I divided by volume. 357 00:25:11,310 --> 00:25:20,020 I have log of 1 minus eta z e to the minus beta-- again, 358 00:25:20,020 --> 00:25:23,820 h bar squared k squared over 2m-- 359 00:25:23,820 --> 00:25:27,950 are the energy that I'm dealing with. 360 00:25:27,950 --> 00:25:28,621 OK? 361 00:25:28,621 --> 00:25:29,120 Fine. 362 00:25:29,120 --> 00:25:32,276 What do we do with this? 363 00:25:32,276 --> 00:25:35,310 We make the same change of variables 364 00:25:35,310 --> 00:25:39,450 that we have over here from k to x. 365 00:25:39,450 --> 00:25:47,040 So then I would have minus eta d cubed k over 2pi cubed 366 00:25:47,040 --> 00:25:53,590 gets replaced with this 2 over root pi lambda cubed. 367 00:25:56,750 --> 00:26:02,272 And then I have the integral 0 to infinity dx 368 00:26:02,272 --> 00:26:11,827 x to the 1/2 log of 1 minus eta z e to the minus x. 369 00:26:16,210 --> 00:26:18,160 Huh. 370 00:26:18,160 --> 00:26:20,314 It seems like-- yes? 371 00:26:20,314 --> 00:26:22,409 AUDIENCE: What happened to your factor g? 372 00:26:22,409 --> 00:26:22,950 PROFESSOR: G? 373 00:26:22,950 --> 00:26:23,810 Good question. 374 00:26:26,570 --> 00:26:27,590 Anything else? 375 00:26:32,520 --> 00:26:35,590 So I was hoping to have integrals 376 00:26:35,590 --> 00:26:39,270 that are all expressible in terms of this-- these set 377 00:26:39,270 --> 00:26:42,620 of functions that I defined. 378 00:26:42,620 --> 00:26:45,530 This log does not fall into this category. 379 00:26:45,530 --> 00:26:48,050 Should I define something else? 380 00:26:48,050 --> 00:26:50,010 Well, the answer is no because I can 381 00:26:50,010 --> 00:26:53,300 make that integral look like this 382 00:26:53,300 --> 00:26:57,590 by simbly doing an integration by parts. 383 00:26:57,590 --> 00:27:09,810 So we do integrate by parts in which case what I do 384 00:27:09,810 --> 00:27:14,890 is I take a derivative of this and I 385 00:27:14,890 --> 00:27:18,350 take an integral of x to the 1/2. 386 00:27:18,350 --> 00:27:23,910 Integral of x to the 1/2 becomes x to the 3/2 over 3/2. 387 00:27:23,910 --> 00:27:27,351 There is, of course, a minus sign involved. 388 00:27:27,351 --> 00:27:32,132 So this minus becomes plus eta. 389 00:27:32,132 --> 00:27:35,210 Make sure I have the g this time. 390 00:27:35,210 --> 00:27:41,010 2 over root pi lambda cubed. 391 00:27:41,010 --> 00:27:46,810 I have the integral from 0 to infinity dx. 392 00:27:46,810 --> 00:27:50,540 x to the 1/2 became x to the 3/2. 393 00:27:50,540 --> 00:27:52,990 The 3/2 I will bring out front. 394 00:27:52,990 --> 00:27:55,300 So I have 1 over 3/2 here. 395 00:27:58,030 --> 00:28:01,480 And then I have to take the derivative of this function. 396 00:28:01,480 --> 00:28:04,180 What's the derivative of the log? 397 00:28:04,180 --> 00:28:10,810 It is 1 over the argument of the log times the derivative 398 00:28:10,810 --> 00:28:14,820 of the argument of the log which is minus 399 00:28:14,820 --> 00:28:17,770 eta z, the derivative of e to the minus x, 400 00:28:17,770 --> 00:28:19,320 which is e to the minus x. 401 00:28:19,320 --> 00:28:21,720 And this minus becomes plus. 402 00:28:25,310 --> 00:28:26,670 OK? 403 00:28:26,670 --> 00:28:30,540 So let's arrange things a little bit. 404 00:28:30,540 --> 00:28:37,350 I can get rid of these etas because eta squared will be 1. 405 00:28:37,350 --> 00:28:38,850 What is the answer, therefore? 406 00:28:38,850 --> 00:28:44,900 It is g over lambda cubed. 407 00:28:44,900 --> 00:28:47,420 That's fine. 408 00:28:47,420 --> 00:28:52,270 If root pi over 2 is 1/2 factorial, 409 00:28:52,270 --> 00:28:57,000 and I multiply 1/2 factorial with 3/2, what I will get 410 00:28:57,000 --> 00:29:00,670 is 1 over 3/2 factorial. 411 00:29:00,670 --> 00:29:06,040 Because n factorial is n times n minus 1 factorial. 412 00:29:06,040 --> 00:29:07,700 And then, what do I have? 413 00:29:07,700 --> 00:29:10,546 I have 0 to infinity dx. 414 00:29:13,710 --> 00:29:20,710 x to the 3/2, divide the numerator and denominator by z 415 00:29:20,710 --> 00:29:28,060 e to the minus x and you get z inverse e to the x minus eta. 416 00:29:28,060 --> 00:29:32,670 And lo and behold, we have a function of the variety 417 00:29:32,670 --> 00:29:37,310 that we had defined on the board corresponding to n minus 1 418 00:29:37,310 --> 00:29:41,180 being 3/2 or n being 5/2. 419 00:29:41,180 --> 00:29:46,790 So what we have here is that beta P, depending 420 00:29:46,790 --> 00:29:49,500 on whatever statistics we are dealing with, 421 00:29:49,500 --> 00:29:57,432 is g over lambda cubed f 5/2 eta of z. 422 00:30:04,390 --> 00:30:05,258 Yes? 423 00:30:05,258 --> 00:30:08,130 AUDIENCE: [INAUDIBLE] 424 00:30:08,130 --> 00:30:09,090 PROFESSOR: OK. 425 00:30:09,090 --> 00:30:11,510 So what you have to do is always ensure 426 00:30:11,510 --> 00:30:14,080 what is happening to the boundary terms. 427 00:30:14,080 --> 00:30:16,630 Boundary terms will be a product of x 428 00:30:16,630 --> 00:30:21,290 to the 3/2 and this factor. 429 00:30:21,290 --> 00:30:27,270 At x equals to 0, x to the 3/s will ensure that you are 0. 430 00:30:27,270 --> 00:30:29,960 At x goes to infinity, the exponential 431 00:30:29,960 --> 00:30:32,640 will ensure that you're to 0. 432 00:30:32,640 --> 00:30:34,830 So the boundary terms are important 433 00:30:34,830 --> 00:30:36,880 and indeed you have to check them. 434 00:30:36,880 --> 00:30:39,440 And you can check that they are 0. 435 00:30:43,390 --> 00:30:44,616 OK? 436 00:30:44,616 --> 00:30:48,350 AUDIENCE: I think the factor that multiples x to the 3/2 437 00:30:48,350 --> 00:30:50,590 is [INAUDIBLE]. 438 00:30:50,590 --> 00:30:52,222 But it's-- the argument still holds. 439 00:30:52,222 --> 00:30:53,680 PROFESSOR: The argument still holds 440 00:30:53,680 --> 00:30:58,370 because as x goes to infinity, e to the minus x goes to 0. 441 00:30:58,370 --> 00:31:00,320 Log of 1 is 0. 442 00:31:00,320 --> 00:31:02,600 In fact, you can ask, how does it go to 0? 443 00:31:02,600 --> 00:31:04,690 It goes to 0 exponentially. 444 00:31:04,690 --> 00:31:08,300 Exponentially it will [INAUDIBLE]. 445 00:31:08,300 --> 00:31:08,800 Yes. 446 00:31:08,800 --> 00:31:10,008 Thank you for the correction. 447 00:31:18,330 --> 00:31:18,830 OK? 448 00:31:21,620 --> 00:31:22,130 All right. 449 00:31:22,130 --> 00:31:26,720 There's one other thing that we can take a look at. 450 00:31:26,720 --> 00:31:28,770 The total energy. 451 00:31:28,770 --> 00:31:29,830 OK. 452 00:31:29,830 --> 00:31:33,240 So the energy in each one of these states 453 00:31:33,240 --> 00:31:35,680 is given by that formula. 454 00:31:35,680 --> 00:31:39,640 So if I ask, what is the total expectation value of energy? 455 00:31:39,640 --> 00:31:43,980 And again, in thermodynamics we expect 456 00:31:43,980 --> 00:31:48,040 the extensive quantities such as the energy should have, 457 00:31:48,040 --> 00:31:50,650 in an aerodynamic sense, well defined values 458 00:31:50,650 --> 00:31:53,440 so I won't write the expectation value. 459 00:31:53,440 --> 00:32:03,020 Is going to be a sum over k Nk epsilon k. 460 00:32:03,020 --> 00:32:07,320 And this will depend on the statistics that I'm looking at. 461 00:32:07,320 --> 00:32:11,670 And what I will have is that this will go over 462 00:32:11,670 --> 00:32:21,220 to an integral over k divided by 2pi cubed multiplied g 463 00:32:21,220 --> 00:32:28,160 and multiplied by V. I have epsilon 464 00:32:28,160 --> 00:32:33,730 of k which is h bar squared k squared over 2m. 465 00:32:33,730 --> 00:32:36,840 The expectation value of N, which 466 00:32:36,840 --> 00:32:43,900 is z inverse e to the minus beta-- sorry-- z inverse 467 00:32:43,900 --> 00:32:49,556 e to the plus beta h bar squared k squared over 2m minus 8. 468 00:32:56,230 --> 00:32:56,730 OK? 469 00:32:59,550 --> 00:33:08,090 So the simplest way to do this is to first of all divide by V. 470 00:33:08,090 --> 00:33:11,960 So I can define an energy density. 471 00:33:15,770 --> 00:33:20,030 Actually, since I'm going to change variables 472 00:33:20,030 --> 00:33:23,740 to this combination that is x, I notice 473 00:33:23,740 --> 00:33:27,690 that up to a factor of betas the numerator is x. 474 00:33:27,690 --> 00:33:33,070 So let me actually calculate the combination beta e over V. 475 00:33:33,070 --> 00:33:37,080 That only depends on this combination x. 476 00:33:37,080 --> 00:33:38,200 So what is it? 477 00:33:38,200 --> 00:33:43,360 It is g-- the integral over k, I know 478 00:33:43,360 --> 00:33:46,090 how to replace it with something that 479 00:33:46,090 --> 00:33:54,295 involves x-- is going to give me 2 over root pi lambda cubed. 480 00:33:54,295 --> 00:33:59,970 Integral 0 to infinity dx. 481 00:33:59,970 --> 00:34:04,440 From the change of variables, I have x to the 1/2. 482 00:34:04,440 --> 00:34:07,300 But I introduced one factor of x here 483 00:34:07,300 --> 00:34:12,590 because I have this energy factor in the numerator. 484 00:34:12,590 --> 00:34:16,010 So that changes this to x to the 3/2. 485 00:34:16,010 --> 00:34:19,214 And then again, I have z inverse e to the x minus 8. 486 00:34:22,179 --> 00:34:22,679 OK? 487 00:34:25,659 --> 00:34:29,230 So if is one of the integrals that we have defined. 488 00:34:29,230 --> 00:34:34,989 In fact, it is exactly what I have above for the pressure, 489 00:34:34,989 --> 00:34:39,630 except that I don't have the factor of 3/2 that appeared 490 00:34:39,630 --> 00:34:44,850 nicely to give the right normalization. 491 00:34:44,850 --> 00:34:51,920 So what I conclude is that this quantity is 492 00:34:51,920 --> 00:35:09,199 in fact g over lambda cubed times 3/2 of f 5/2 eta of z. 493 00:35:12,691 --> 00:35:13,191 OK? 494 00:35:18,190 --> 00:35:21,060 But they are exactly the same functions 495 00:35:21,060 --> 00:35:22,595 for energy and pressure. 496 00:35:25,560 --> 00:35:30,480 So you can see that, irrespective 497 00:35:30,480 --> 00:35:33,740 of what the chemical potential is doing, 498 00:35:33,740 --> 00:35:40,110 I can state that for the quantum system e is always 3/2 PV. 499 00:35:42,970 --> 00:35:49,110 Of course, we have seen this expression classically. 500 00:35:49,110 --> 00:35:50,950 Now what happens quantum mechanically 501 00:35:50,950 --> 00:35:52,930 is that the pressure starts to vary 502 00:35:52,930 --> 00:35:55,440 in some complicated way with density. 503 00:35:55,440 --> 00:35:58,720 But what we learn is that energy will also 504 00:35:58,720 --> 00:36:02,810 vary in the same complicated way with density, 505 00:36:02,810 --> 00:36:10,310 such that always this formula EV 3/2 PV is valid. 506 00:36:10,310 --> 00:36:12,900 There is some nice symmetry reason 507 00:36:12,900 --> 00:36:16,025 for why that is the case that you explore 508 00:36:16,025 --> 00:36:17,930 in one of the problems that you will see. 509 00:36:22,311 --> 00:36:22,810 OK? 510 00:36:25,830 --> 00:36:31,850 So these are the characteristics. 511 00:36:31,850 --> 00:36:34,790 In some sense we have solved the problem 512 00:36:34,790 --> 00:36:39,670 because we know both density and pressure in terms 513 00:36:39,670 --> 00:36:41,840 of chemical potential. 514 00:36:41,840 --> 00:36:44,770 So all we need to do is to invert this, get 515 00:36:44,770 --> 00:36:47,750 the chemical potential as a function of density, 516 00:36:47,750 --> 00:36:49,780 insert it in the formula for pressure 517 00:36:49,780 --> 00:36:52,420 and we have pressure as a function of density. 518 00:36:52,420 --> 00:36:55,610 But since these are kind of implicit 519 00:36:55,610 --> 00:37:00,190 through these f functions, we don't quite 520 00:37:00,190 --> 00:37:02,480 know what that means. 521 00:37:02,480 --> 00:37:05,900 So the next stage is to really study 522 00:37:05,900 --> 00:37:10,270 how these f functions behave and how 523 00:37:10,270 --> 00:37:15,920 we can relate their behaviors to dependence on density. 524 00:37:15,920 --> 00:37:20,110 And we will see that we have to distinguish between low density 525 00:37:20,110 --> 00:37:23,090 and high density regimes. 526 00:37:23,090 --> 00:37:25,230 What does it mean? 527 00:37:25,230 --> 00:37:27,780 Well, you can see that both have players on the right hand 528 00:37:27,780 --> 00:37:30,650 side is dimensionless. 529 00:37:30,650 --> 00:37:33,390 The only dimension full combination 530 00:37:33,390 --> 00:37:36,970 that we have here is N-- well, we can make it 531 00:37:36,970 --> 00:37:41,850 dimensionless by multiplying by lambda cubed over g. 532 00:37:41,850 --> 00:37:46,180 So it turns out that whether or not 533 00:37:46,180 --> 00:37:48,930 you see things that are classical or things that 534 00:37:48,930 --> 00:37:51,510 are very quantum mechanical depend 535 00:37:51,510 --> 00:37:53,270 on this degeneracy factor. 536 00:37:56,070 --> 00:38:00,780 Which, within our context of statistical physics of gases, 537 00:38:00,780 --> 00:38:03,400 we define to be the communication N 538 00:38:03,400 --> 00:38:04,810 lambda cubed over g. 539 00:38:07,480 --> 00:38:11,890 And we will regain classical behaviors 540 00:38:11,890 --> 00:38:15,040 when degeneracy is small, density 541 00:38:15,040 --> 00:38:18,680 is small enough such that d is much less than 1. 542 00:38:18,680 --> 00:38:21,320 We will see totally new behavior that 543 00:38:21,320 --> 00:38:24,990 is appropriate to quantum mechanics in other limit 544 00:38:24,990 --> 00:38:27,852 where [? d ?] becomes large. 545 00:38:27,852 --> 00:38:29,710 OK? 546 00:38:29,710 --> 00:38:42,770 So this distinguishes classical versus quantum behavior. 547 00:38:48,540 --> 00:38:53,700 Now, we expect to regain classical behavior 548 00:38:53,700 --> 00:38:56,360 in the limit of very low densities. 549 00:38:56,360 --> 00:38:58,460 That's what we've been kind of saying 550 00:38:58,460 --> 00:39:02,050 from the very first lecture that low density gas has 551 00:39:02,050 --> 00:39:06,440 this ideal behavior where PV is Nkt. 552 00:39:06,440 --> 00:39:09,980 And, therefore, let's focus on the limit 553 00:39:09,980 --> 00:39:14,950 where V is much less than 1, which is classical. 554 00:39:18,290 --> 00:39:31,530 And since d is f 3/2 eta of z, we 555 00:39:31,530 --> 00:39:37,420 need the behavior of the function when it is small-- 556 00:39:37,420 --> 00:39:39,830 and I'll show you [INAUDIBLE] consistently 557 00:39:39,830 --> 00:39:45,030 that the function is small when it's argument is small. 558 00:39:45,030 --> 00:39:49,420 So we'll see that d less than 1 implies 559 00:39:49,420 --> 00:39:53,035 or is implied by z being less than 1. 560 00:39:55,770 --> 00:39:57,400 OK? 561 00:39:57,400 --> 00:40:00,275 So let's take a look at one of these functions, 562 00:40:00,275 --> 00:40:07,400 fm eta z, 1 over m minus 1 factorial, 0 563 00:40:07,400 --> 00:40:11,980 to infinity dx, x to the m minus 1, 564 00:40:11,980 --> 00:40:15,110 z universe e to the x minus 1. 565 00:40:17,940 --> 00:40:24,730 And I'd like to explore the limit of this when z is small. 566 00:40:24,730 --> 00:40:28,140 So that's currently not very good because I have z inverse. 567 00:40:28,140 --> 00:40:32,300 So what I will do is I will multiply by z e to the minus 568 00:40:32,300 --> 00:40:35,790 x both numerator and denominator. 569 00:40:35,790 --> 00:40:41,480 So I will have m minus 1 factorial integral 0 570 00:40:41,480 --> 00:40:46,550 to infinity dx x to the n minus 1. 571 00:40:46,550 --> 00:40:50,300 I have z e to the minus x. 572 00:40:50,300 --> 00:40:54,560 I have-- oops, there was an eta here. 573 00:40:54,560 --> 00:40:59,981 1 minus eta z e to the minus x. 574 00:40:59,981 --> 00:41:00,480 OK? 575 00:41:02,990 --> 00:41:05,850 Then I say, well, the thing that is appearing 576 00:41:05,850 --> 00:41:10,350 in the denominator, I can write as a sum 577 00:41:10,350 --> 00:41:12,990 of a geometric series in powers of z. 578 00:41:12,990 --> 00:41:15,750 And that's, after all, what I'm attempting to do, 579 00:41:15,750 --> 00:41:18,390 make an expansion in powers of z. 580 00:41:23,000 --> 00:41:26,620 It starts with z to the minus x. 581 00:41:26,620 --> 00:41:31,180 And 1 over 1 minus eta z to the minus 582 00:41:31,180 --> 00:41:38,300 x I can write as the sum alpha going from 0 to infinity of eta 583 00:41:38,300 --> 00:41:41,980 z e to the minus x raised to the power of [INAUDIBLE]. 584 00:41:41,980 --> 00:41:46,390 This geometric series sums to what I have over there. 585 00:41:46,390 --> 00:41:48,510 OK? 586 00:41:48,510 --> 00:41:51,830 Let's rearrange this a little bit. 587 00:41:51,830 --> 00:41:57,090 I will take the sum over alpha outside. 588 00:41:57,090 --> 00:42:02,320 Well, I notice that actually I have here, this first term, 589 00:42:02,320 --> 00:42:04,810 which is different from this by a factor of eta. 590 00:42:04,810 --> 00:42:10,280 So let me multiply by an eta and get another eta. 591 00:42:10,280 --> 00:42:13,600 And then I have a sum that starts from 1 592 00:42:13,600 --> 00:42:23,280 to infinity of eta z e to the minus x to the power of alpha. 593 00:42:23,280 --> 00:42:27,400 So I have eta z to the power of alpha. 594 00:42:27,400 --> 00:42:29,410 I have one other eta here. 595 00:42:32,620 --> 00:42:42,080 And then I have the integration over x, x to the m minus 1. 596 00:42:46,140 --> 00:42:48,690 I didn't write e to the minus x to the alpha 597 00:42:48,690 --> 00:42:51,480 because I pulled it out the integration. 598 00:42:51,480 --> 00:42:54,514 I will write it now. e to the minus alpha x. 599 00:42:57,720 --> 00:43:01,438 And I didn't write 1 over n minus 1 factor. 600 00:43:09,260 --> 00:43:14,080 Now, once more, this is an integration that you have seen. 601 00:43:14,080 --> 00:43:17,240 The result of doing integral dx x to the m 602 00:43:17,240 --> 00:43:21,440 minus 1 e to the minus x is simply m minus 1 factorial. 603 00:43:26,010 --> 00:43:28,270 Except that it is not e to the minus 604 00:43:28,270 --> 00:43:31,160 x but it's e to the minus alpha x. 605 00:43:31,160 --> 00:43:34,280 I can rescale everything by alpha 606 00:43:34,280 --> 00:43:38,078 and I will pull out a factor of alpha to there. 607 00:43:38,078 --> 00:43:39,560 OK? 608 00:43:39,560 --> 00:43:41,700 So what do we have? 609 00:43:41,700 --> 00:43:48,100 We have that fm eta of z as a [? power series ?] 610 00:43:48,100 --> 00:43:55,340 representation, alpha running from 1 to infinity, 611 00:43:55,340 --> 00:44:00,040 z to the alpha divided by alpha to the m. 612 00:44:00,040 --> 00:44:04,690 It's an alternating series that starts with plus 613 00:44:04,690 --> 00:44:07,760 and then becomes a minus, et cetera. 614 00:44:07,760 --> 00:44:10,440 So if I were to explicitly write this, 615 00:44:10,440 --> 00:44:12,350 you will see how nice it is. 616 00:44:12,350 --> 00:44:19,860 It is z plus eta z squared 2 to the m plus z cubed 3 617 00:44:19,860 --> 00:44:24,800 to the m plus eta z to the 4 4 to the m. 618 00:44:24,800 --> 00:44:28,820 And you can see why it is such a nice thing that really deserves 619 00:44:28,820 --> 00:44:30,800 to have a function named after it. 620 00:44:30,800 --> 00:44:34,080 So we call that nice series [? to Vf. ?] 621 00:44:41,930 --> 00:44:46,250 Now, for future reference, note the following. 622 00:44:46,250 --> 00:44:51,340 If I take a derivative of this function with respect 623 00:44:51,340 --> 00:44:56,290 to z-- its argument-- then in this power series, say, 624 00:44:56,290 --> 00:45:00,020 z to the 4 becomes 4z cubed. 625 00:45:00,020 --> 00:45:02,880 4 will cancel one of the 4's that I have here 626 00:45:02,880 --> 00:45:05,290 and it becomes 4 to the m minus 1. 627 00:45:05,290 --> 00:45:07,840 This becomes 3 to the m minus 1. 628 00:45:07,840 --> 00:45:10,070 So you can see that the coefficients that 629 00:45:10,070 --> 00:45:15,230 were powers of m become powers of m minus 1. 630 00:45:15,230 --> 00:45:18,290 Numerator has gone down by one factor. 631 00:45:18,290 --> 00:45:21,160 So let's multiply by z. 632 00:45:21,160 --> 00:45:23,880 So the numerator is now what it was before. 633 00:45:23,880 --> 00:45:27,020 The denominators, rather than being raised to the power of m, 634 00:45:27,020 --> 00:45:29,520 are raised to the power of m minus 1. 635 00:45:29,520 --> 00:45:36,030 So clearly, this is the same thing as fm minus 1. 636 00:45:36,030 --> 00:45:41,110 So basically these functions have this nice character 637 00:45:41,110 --> 00:45:42,400 that they are like a ladder. 638 00:45:42,400 --> 00:45:44,860 You take a derivative of one and you 639 00:45:44,860 --> 00:45:48,623 get the one that is indexed by one less. 640 00:45:53,053 --> 00:45:53,553 OK? 641 00:46:03,413 --> 00:46:06,371 AUDIENCE: Is there a factor of eta that comes out to be 642 00:46:06,371 --> 00:46:09,225 [INAUDIBLE]. 643 00:46:09,225 --> 00:46:10,475 PROFESSOR: In the derivatives? 644 00:46:10,475 --> 00:46:15,692 AUDIENCE: I mean, it seems to me that there's an eta-- like, 645 00:46:15,692 --> 00:46:16,650 in that relation that-- 646 00:46:16,650 --> 00:46:17,816 PROFESSOR: In this relation? 647 00:46:17,816 --> 00:46:18,560 AUDIENCE: Yeah. 648 00:46:18,560 --> 00:46:19,143 PROFESSOR: OK. 649 00:46:19,143 --> 00:46:21,620 So let's do it by step. 650 00:46:21,620 --> 00:46:28,420 If I take a derivative, divide dz, 651 00:46:28,420 --> 00:46:35,700 I get 1 plus eta z squared z 2 to the m minus 1 plus 652 00:46:35,700 --> 00:46:40,460 z squared' 3 to the m minus 1. 653 00:46:40,460 --> 00:46:44,845 Then I multiply by z, this becomes z, 654 00:46:44,845 --> 00:46:47,500 this becomes z squared, this becomes z cubed. 655 00:46:47,500 --> 00:46:49,210 And you can see that the etas are 656 00:46:49,210 --> 00:46:52,969 exactly where they should be. 657 00:46:52,969 --> 00:47:03,749 AUDIENCE: Would you explain why the index of summation 658 00:47:03,749 --> 00:47:04,868 starts from-- 659 00:47:04,868 --> 00:47:05,493 PROFESSOR: One? 660 00:47:05,493 --> 00:47:06,200 AUDIENCE: --one. 661 00:47:06,200 --> 00:47:06,930 PROFESSOR: OK. 662 00:47:06,930 --> 00:47:11,220 So let's take a look at this function. 663 00:47:11,220 --> 00:47:15,060 You can see there is a z inverse in the denominator. 664 00:47:15,060 --> 00:47:19,290 So I made it into a z here. 665 00:47:19,290 --> 00:47:26,640 So you can see that if z is very small, if will start with z. 666 00:47:26,640 --> 00:47:27,140 All right? 667 00:47:27,140 --> 00:47:28,740 Now what happened here? 668 00:47:28,740 --> 00:47:32,090 I wrote this as a sum of geometric series. 669 00:47:32,090 --> 00:47:37,050 I pulled out this factor of z that was in the numerator, 670 00:47:37,050 --> 00:47:42,480 and the denominator I wrote as a series that starts with 0. 671 00:47:42,480 --> 00:47:45,190 But then there is this z out front. 672 00:47:45,190 --> 00:47:49,790 I pulled this z e to the minus x in front here 673 00:47:49,790 --> 00:47:52,600 and then it starts with 1. 674 00:47:52,600 --> 00:47:55,030 I could have written it-- I can put this in here 675 00:47:55,030 --> 00:47:57,740 and write it as alpha plus 1 with alpha 676 00:47:57,740 --> 00:47:59,235 starting from 0, et cetera. 677 00:47:59,235 --> 00:48:00,640 Or I can write it [INAUDIBLE]. 678 00:48:00,640 --> 00:48:04,360 Or, alternatively, you can say, let's put the eta here 679 00:48:04,360 --> 00:48:05,770 and eta here. 680 00:48:05,770 --> 00:48:08,540 And then I see that this is the geometric series where 681 00:48:08,540 --> 00:48:12,309 the first term is this z to the minus x [INAUDIBLE]. 682 00:48:19,294 --> 00:48:19,794 OK? 683 00:48:25,300 --> 00:48:31,230 So I want to erase this board but keep this formula. 684 00:48:31,230 --> 00:48:33,826 So let me erase it and then rewrite it. 685 00:48:46,410 --> 00:48:52,991 So this I will rewrite as N lambda q over g, which 686 00:48:52,991 --> 00:48:59,170 is my degeneracy factor, is f 3/2 eta of z. 687 00:49:02,800 --> 00:49:09,260 And in the limit where z goes to small, 688 00:49:09,260 --> 00:49:13,950 I can use the expansion that I just derived, 689 00:49:13,950 --> 00:49:20,840 which is z plus eta z squared 2 to the-- what is it-- 690 00:49:20,840 --> 00:49:25,620 3/2 plus z cubed. 691 00:49:25,620 --> 00:49:30,870 3 to the 3/2 and so forth. 692 00:49:30,870 --> 00:49:35,130 You can see that precisely because as z goes to 0, 693 00:49:35,130 --> 00:49:38,860 this series goes to 0, that smallness of z 694 00:49:38,860 --> 00:49:41,994 is the same thing or implied by the smallness of z. 695 00:49:44,720 --> 00:49:50,140 But that z is equal to d only at the lowest order. 696 00:49:50,140 --> 00:49:52,010 So what happens at higher orders, 697 00:49:52,010 --> 00:49:58,990 z-- inverting this-- is d minus eta z squared 2 to the 3/2 plus 698 00:49:58,990 --> 00:50:05,390 z cubed 3 to the 3/2, and so forth. 699 00:50:05,390 --> 00:50:10,370 And we saw this when I was doing the calculation 700 00:50:10,370 --> 00:50:14,660 for the interacting gas, that clearly to lowest order z 701 00:50:14,660 --> 00:50:19,200 equals to d, if I want to get the next order, 702 00:50:19,200 --> 00:50:22,150 I substitute the result of the previous order 703 00:50:22,150 --> 00:50:23,770 into this expression. 704 00:50:23,770 --> 00:50:27,680 So z I substitute for d, I will d 705 00:50:27,680 --> 00:50:35,150 minus eta d squared 2 to the 3/2 as the first contribution. 706 00:50:35,150 --> 00:50:37,470 And then something that is order of d cubed. 707 00:50:40,340 --> 00:50:44,120 And if I want the answer to the d cubed order, 708 00:50:44,120 --> 00:50:47,310 I take this whole thing that is that order of d squared 709 00:50:47,310 --> 00:50:49,830 and substitute in this series. 710 00:50:49,830 --> 00:50:51,310 And so what do I get? 711 00:50:51,310 --> 00:50:59,620 I will get d minus eta 2 to the 3/2, 712 00:50:59,620 --> 00:51:03,560 square of z-- which is the square of this quantity-- 713 00:51:03,560 --> 00:51:08,690 square of this quantity is d squared minus twice 714 00:51:08,690 --> 00:51:10,960 the product of these two terms. 715 00:51:10,960 --> 00:51:17,030 So I have twice eta d cubed 2 to the 3/2. 716 00:51:17,030 --> 00:51:21,310 And then square of this term, but that's order of e to the 4. 717 00:51:21,310 --> 00:51:23,180 I don't care for that. 718 00:51:23,180 --> 00:51:32,410 And then I have, from here, plus d cubed 3 to the 3/2 719 00:51:32,410 --> 00:51:34,880 ignoring terms of the order of d to the 4th. 720 00:51:37,700 --> 00:51:39,590 OK? 721 00:51:39,590 --> 00:51:43,320 So z is-- 722 00:51:43,320 --> 00:51:48,677 AUDIENCE: [INAUDIBLE] z equals d minus and the d [INAUDIBLE]. 723 00:51:48,677 --> 00:51:49,552 PROFESSOR: Ah, great. 724 00:51:49,552 --> 00:51:50,070 Thank you. 725 00:51:50,070 --> 00:51:50,570 Minus. 726 00:51:53,650 --> 00:51:54,501 Minus. 727 00:51:54,501 --> 00:51:55,001 Good. 728 00:51:58,370 --> 00:52:04,195 Saves me having to go back later when the answer doesn't match. 729 00:52:04,195 --> 00:52:04,695 OK. 730 00:52:07,520 --> 00:52:09,076 Fine. 731 00:52:09,076 --> 00:52:11,720 Now let's organize terms. 732 00:52:11,720 --> 00:52:14,050 The lowest order I have d. 733 00:52:14,050 --> 00:52:17,280 At order of d squared I have still 734 00:52:17,280 --> 00:52:25,470 just this one term, minus eta d squared over 2 to the 3/2. 735 00:52:25,470 --> 00:52:29,050 We knew that was the correct result at order of d squared. 736 00:52:29,050 --> 00:52:32,330 At order of d cubed I have two terms. 737 00:52:32,330 --> 00:52:35,240 One is the product of these two terms. 738 00:52:35,240 --> 00:52:39,500 The etas disappear, minus 2, and this 739 00:52:39,500 --> 00:52:42,870 will give me 2 over 9 which is 1/4. 740 00:52:47,000 --> 00:52:53,180 And from other part, I have 1 over 3 to the 3/1 d cubed. 741 00:52:53,180 --> 00:52:57,210 And I haven't calculate things at order of d to the 4th. 742 00:53:01,410 --> 00:53:03,370 OK? 743 00:53:03,370 --> 00:53:04,600 Fine. 744 00:53:04,600 --> 00:53:08,120 So we have calculated in a power series 745 00:53:08,120 --> 00:53:11,990 that we can go to whatever order you like. 746 00:53:11,990 --> 00:53:14,790 This is z which is related to chemical potential 747 00:53:14,790 --> 00:53:16,990 as a function of this degeneracy that 748 00:53:16,990 --> 00:53:19,810 is simply related to density. 749 00:53:19,810 --> 00:53:23,760 Now, the formula for the pressure is up there. 750 00:53:23,760 --> 00:53:36,150 It is that eta P is g over lambda cubed f 5/2 eta of z. 751 00:53:39,130 --> 00:53:40,310 Right? 752 00:53:40,310 --> 00:53:43,940 So reorganizing this a little bit, 753 00:53:43,940 --> 00:53:47,470 the dimensionless form is beta P lambda 754 00:53:47,470 --> 00:53:52,812 cubed over g f 5/2 eta of z. 755 00:53:52,812 --> 00:53:56,440 f 5/2 we can make an expansion. 756 00:53:56,440 --> 00:54:04,610 It is z plus eta z squared 2 to the 5/2 plus z cubed 3 757 00:54:04,610 --> 00:54:09,130 to the 5/2 and so forth. 758 00:54:09,130 --> 00:54:13,500 For z I substitute from the line above. 759 00:54:13,500 --> 00:54:25,710 So I have d minus eta d squared 2 to the 3/2 plus 1/4 minus 1 760 00:54:25,710 --> 00:54:30,500 over 3 to the 3/2 d cubed. 761 00:54:30,500 --> 00:54:32,630 So that's the first term. 762 00:54:32,630 --> 00:54:40,250 The next term is eta 2 to the 5/2 z squared. 763 00:54:40,250 --> 00:54:44,610 So I have to square what is up here and here keeping terms 764 00:54:44,610 --> 00:54:47,450 to order of d cubed. 765 00:54:47,450 --> 00:54:48,460 So what do I have? 766 00:54:48,460 --> 00:54:56,630 I have d squared from squaring this term. 767 00:54:56,630 --> 00:54:59,590 And a term that is twice the product of these two terms. 768 00:54:59,590 --> 00:55:02,420 That's the only term that is order of d cubed. 769 00:55:02,420 --> 00:55:09,650 So I have minus 2 eta 2 to the 3/2 d cubed. 770 00:55:13,220 --> 00:55:16,120 And finally, z cubed over 3 to the 5/2 771 00:55:16,120 --> 00:55:21,590 would be d cubed over 3 to the 5/2. 772 00:55:21,590 --> 00:55:23,020 And I haven't calculated anything 773 00:55:23,020 --> 00:55:24,940 at all of d to the 4th. 774 00:55:28,300 --> 00:55:29,910 OK? 775 00:55:29,910 --> 00:55:30,715 So what do I have? 776 00:55:30,715 --> 00:55:34,675 I have that beta P lambda cubed over g. 777 00:55:37,240 --> 00:55:40,360 Well, you can see that everything 778 00:55:40,360 --> 00:55:44,590 is going to be proportional to d d squared d cubed. 779 00:55:44,590 --> 00:55:49,280 So let's put out the factor of d which is m lambda cubed over g. 780 00:55:49,280 --> 00:55:55,510 So this is just one factor of d that I have pulled out. 781 00:55:55,510 --> 00:56:03,780 So that then I start with 1, and then the next correction 782 00:56:03,780 --> 00:56:09,730 I have an eta d squared 2 to the 3/2 here, 783 00:56:09,730 --> 00:56:15,770 eta d squared 2 to the 5/2 here with opposite sign. 784 00:56:15,770 --> 00:56:19,080 So this is twice bigger than that. 785 00:56:19,080 --> 00:56:24,350 So the sum total of them would be minus eta divided by 2 786 00:56:24,350 --> 00:56:29,090 to the 5/2 times d squared-- one factor 787 00:56:29,090 --> 00:56:33,820 of d I have out front-- the other factor I will write here 788 00:56:33,820 --> 00:56:35,158 and lambda cubed over g. 789 00:56:38,370 --> 00:56:44,820 And then at order of d cubed, I have a bunch of terms. 790 00:56:44,820 --> 00:56:53,360 I have a 1/4 here, and from the product of these terms, 791 00:56:53,360 --> 00:56:59,740 I will have etas disappear, I have 2 divided by 2 792 00:56:59,740 --> 00:57:03,900 to the 8/2 which is 2 to the 4th which is 16. 793 00:57:03,900 --> 00:57:08,830 So the product of these things will give me a minus 1/8. 794 00:57:08,830 --> 00:57:13,130 I have 1/4 minus 1/8 I will get 1/8. 795 00:57:16,300 --> 00:57:18,605 AUDIENCE: [INAUDIBLE]. 796 00:57:18,605 --> 00:57:19,530 PROFESSOR: OK. 797 00:57:19,530 --> 00:57:26,150 And then we have minus 3-- 1 over 3 to the 1/2 798 00:57:26,150 --> 00:57:29,050 and this 3 to the 5/2. 799 00:57:29,050 --> 00:57:31,200 This is three times larger than this. 800 00:57:31,200 --> 00:57:34,050 This is 3 over 3 to the 5/2. 801 00:57:34,050 --> 00:57:36,350 Subtract one of them, I will be left 802 00:57:36,350 --> 00:57:41,836 minus 2 divided by 3 to the 5/2 [? N ?] 803 00:57:41,836 --> 00:57:45,440 lambda cubed over g squared. 804 00:57:45,440 --> 00:57:46,210 And so forth. 805 00:57:49,010 --> 00:57:54,310 So you can see that the form that we have is beta 806 00:57:54,310 --> 00:57:58,540 P-- the pressure-- starts with density. 807 00:57:58,540 --> 00:58:02,440 I essentially divide it through by lambda q over g. 808 00:58:02,440 --> 00:58:07,640 I start with the ideal gas result, n, 809 00:58:07,640 --> 00:58:11,960 and then I have a correction that is order of n squared. 810 00:58:11,960 --> 00:58:17,200 I can write it as B2 n squared. 811 00:58:17,200 --> 00:58:20,020 In view up our [INAUDIBLE] co-efficients. 812 00:58:20,020 --> 00:58:29,380 And we see that our B2 is minus eta times-- sorry. 813 00:58:29,380 --> 00:58:41,360 Minus eta 2 to the 5/2 times n lambda cubed over g. 814 00:58:41,360 --> 00:58:47,190 Remember that B2s, et cetera, had dimensions of volume. 815 00:58:47,190 --> 00:58:51,190 So if the volume is provided by lambda cubed, 816 00:58:51,190 --> 00:58:55,490 it is an additional factor. 817 00:58:55,490 --> 00:58:59,060 For fermions it's a positive pressure. 818 00:58:59,060 --> 00:59:02,340 So the pressure goes up because fermions repel each other. 819 00:59:02,340 --> 00:59:04,440 For bosons it is negative. 820 00:59:04,440 --> 00:59:06,940 Bosons attract each other. 821 00:59:06,940 --> 00:59:09,770 And we have actually calculated this. 822 00:59:09,770 --> 00:59:13,260 We did a calculation of the canonical form 823 00:59:13,260 --> 00:59:16,490 where we had the one exchange corrections. 824 00:59:16,490 --> 00:59:19,280 And if you go back to the notes or if you remember, 825 00:59:19,280 --> 00:59:22,500 we had calculated precisely this. 826 00:59:22,500 --> 00:59:25,380 But then I said that if I went and tried 827 00:59:25,380 --> 00:59:28,370 to calculate the next correction using 828 00:59:28,370 --> 00:59:30,570 the canonical formulation, I would 829 00:59:30,570 --> 00:59:32,900 have to keep track of diagrams that 830 00:59:32,900 --> 00:59:36,450 involve three exchanges and all kinds of things 831 00:59:36,450 --> 00:59:38,080 and it was complicated. 832 00:59:38,080 --> 00:59:40,030 But in this way of looking at things, 833 00:59:40,030 --> 00:59:45,520 you can see that we can very easily compute the [INAUDIBLE] 834 00:59:45,520 --> 00:59:47,880 co-efficients and higher orders. 835 00:59:47,880 --> 00:59:49,490 In particular, the third [INAUDIBLE] 836 00:59:49,490 --> 00:59:58,660 co-efficient we compute to be 1/8 minus 2 3 to the 5/2 lambda 837 00:59:58,660 --> 01:00:02,390 cubed over g squared. 838 01:00:02,390 --> 01:00:05,350 And you can keep going and calculate 839 01:00:05,350 --> 01:00:07,540 higher and higher order corrections. 840 01:00:10,869 --> 01:00:11,660 AUDIENCE: Question. 841 01:00:11,660 --> 01:00:12,620 PROFESSOR: Yes. 842 01:00:12,620 --> 01:00:16,225 AUDIENCE: So if you're doing this expansion for the z being 843 01:00:16,225 --> 01:00:16,745 close to 0-- 844 01:00:16,745 --> 01:00:17,370 PROFESSOR: Yes. 845 01:00:17,370 --> 01:00:19,286 AUDIENCE: --it appears that chemical potential 846 01:00:19,286 --> 01:00:20,490 is a large negative number. 847 01:00:20,490 --> 01:00:22,020 PROFESSOR: Yes. 848 01:00:22,020 --> 01:00:24,580 AUDIENCE: What kind of interaction 849 01:00:24,580 --> 01:00:28,240 does it imply between our grand canonical and [INAUDIBLE] 850 01:00:28,240 --> 01:00:30,240 and the system from which it takes particles? 851 01:00:34,740 --> 01:00:37,990 Because chemical potential represents energy threshold 852 01:00:37,990 --> 01:00:40,237 for the particle to move from there to-- 853 01:00:40,237 --> 01:00:40,820 PROFESSOR: OK. 854 01:00:40,820 --> 01:00:47,650 So as you say, it corresponds to being large and negative. 855 01:00:47,650 --> 01:00:52,940 Which means that, essentially, I think 856 01:00:52,940 --> 01:00:57,550 the more kind of physical way of looking at it is-- this z 857 01:00:57,550 --> 01:01:01,980 is e to the beta mu-- is the likelihood of grabbing 858 01:01:01,980 --> 01:01:04,800 and adding one particle to your system. 859 01:01:04,800 --> 01:01:09,470 And what we have is that this is much, much less than 1. 860 01:01:09,470 --> 01:01:11,610 And the way to achieve that is to say 861 01:01:11,610 --> 01:01:16,080 that it is very hard to extract and bring in other particle 862 01:01:16,080 --> 01:01:18,510 from whatever reservoir you have. 863 01:01:18,510 --> 01:01:20,840 And because it is so hard that you 864 01:01:20,840 --> 01:01:24,020 have such low density of particles in the system 865 01:01:24,020 --> 01:01:25,000 that you have. 866 01:01:25,000 --> 01:01:30,390 So for example, if you are in vacuum and you put a surface, 867 01:01:30,390 --> 01:01:33,360 and some atoms from the surface detach 868 01:01:33,360 --> 01:01:36,390 and form a gas in your container, 869 01:01:36,390 --> 01:01:39,140 the energetics of binding those atoms 870 01:01:39,140 --> 01:01:42,760 to the substrate particles was huge and few of them 871 01:01:42,760 --> 01:01:43,510 managed to escape. 872 01:01:50,980 --> 01:01:51,480 OK? 873 01:02:03,480 --> 01:02:04,250 All right. 874 01:02:04,250 --> 01:02:06,600 But this is only one limit. 875 01:02:06,600 --> 01:02:09,990 And this is the limit that we already knew. 876 01:02:09,990 --> 01:02:13,450 We are really interested in the other limit 877 01:02:13,450 --> 01:02:16,000 where we are quantum dominated. 878 01:02:16,000 --> 01:02:18,900 And clearly we want to go to the limit 879 01:02:18,900 --> 01:02:22,930 where the argument here is not small. 880 01:02:22,930 --> 01:02:26,650 I have to include a large number of terms in this series. 881 01:02:26,650 --> 01:02:28,500 And then I have to figure out what happens. 882 01:02:28,500 --> 01:02:32,960 I mean, how do I have to treat the cases where 883 01:02:32,960 --> 01:02:41,970 I want to solve this equation and the left hand side, d, is 884 01:02:41,970 --> 01:02:44,360 a large number? 885 01:02:44,360 --> 01:02:49,820 So how can I make the value of the function large? 886 01:02:49,820 --> 01:02:53,263 What does that imply about argument of the function? 887 01:02:53,263 --> 01:02:54,170 OK? 888 01:02:54,170 --> 01:02:57,070 So different functions will become large 889 01:02:57,070 --> 01:03:00,080 for different values of the argument. 890 01:03:00,080 --> 01:03:04,510 And as we will show for the case of bosons, 891 01:03:04,510 --> 01:03:08,485 these functions become large where the argument becomes 1. 892 01:03:08,485 --> 01:03:12,450 Whereas, for fermions, they become large when the argument 893 01:03:12,450 --> 01:03:15,020 itself becomes very large. 894 01:03:15,020 --> 01:03:17,910 So the behavior of these functions is very different. 895 01:03:17,910 --> 01:03:19,610 When they become large they become 896 01:03:19,610 --> 01:03:21,765 large at different points when we 897 01:03:21,765 --> 01:03:24,550 are thinking about bosons and fermions. 898 01:03:24,550 --> 01:03:26,370 And because of that, at this stage 899 01:03:26,370 --> 01:03:29,600 I have to separately address the two problems. 900 01:03:29,600 --> 01:03:32,530 So up to here, the high temperature limit, 901 01:03:32,530 --> 01:03:35,160 we sort of started from the same point 902 01:03:35,160 --> 01:03:38,550 and branched off slightly by going into boson and fermion 903 01:03:38,550 --> 01:03:39,500 direction. 904 01:03:39,500 --> 01:03:41,090 But when I go to the other limit, 905 01:03:41,090 --> 01:03:44,050 I have to discuss the two cases separately. 906 01:03:44,050 --> 01:03:48,452 So let's start with the case of fermions. 907 01:04:12,860 --> 01:04:23,510 So degenerate-- degenerating a sense of d being larger 908 01:04:23,510 --> 01:04:27,820 than 1 rather than character flaws. 909 01:04:27,820 --> 01:04:31,380 So what do we do in this limit? 910 01:04:34,370 --> 01:04:39,450 So what I need to know is to look at one of these functions. 911 01:04:39,450 --> 01:04:49,460 So let's say d n lambda cubed over g is f 3/2 minus 912 01:04:49,460 --> 01:04:56,970 of z which is 1 over this 1/2 factorial integral 0 913 01:04:56,970 --> 01:05:03,590 to infinity dx x to the 1/2. 914 01:05:03,590 --> 01:05:05,840 And here I have a plus because I'm 915 01:05:05,840 --> 01:05:09,300 looking at the fermionic version of these. 916 01:05:09,300 --> 01:05:14,230 I have z inverse e to the x and I can write that as e 917 01:05:14,230 --> 01:05:18,652 to the x minus log z. 918 01:05:18,652 --> 01:05:19,152 OK? 919 01:05:24,350 --> 01:05:30,400 Now, essentially, if I look at this function-- actually, 920 01:05:30,400 --> 01:05:37,370 let's look at the function that is just 1 over e 921 01:05:37,370 --> 01:05:46,450 to the x minus log z plus 1 as a function of x. 922 01:05:46,450 --> 01:05:51,510 What does it do in the limit where z is large? 923 01:05:51,510 --> 01:05:56,370 In the limit where z is large, log z is also very large. 924 01:05:56,370 --> 01:06:00,990 And then what I have is something 925 01:06:00,990 --> 01:06:06,300 that if x is much less than log z-- essentially, 926 01:06:06,300 --> 01:06:11,600 this factor will be 0 and I have 1. 927 01:06:11,600 --> 01:06:15,540 Whereas if x is much greater than log z, 928 01:06:15,540 --> 01:06:19,880 then this is exponentially large quantity and I will have 0. 929 01:06:19,880 --> 01:06:24,540 So basically, at around log z, this 930 01:06:24,540 --> 01:06:28,480 switches from being very close to 0 to being 1. 931 01:06:28,480 --> 01:06:32,740 Indeed, that x equals 2 log z is exactly 1/2. 932 01:06:32,740 --> 01:06:34,960 So here is 1/2. 933 01:06:34,960 --> 01:06:37,440 It is switching very rapidly. 934 01:06:40,850 --> 01:06:43,770 And you can see that, essentially, 935 01:06:43,770 --> 01:06:48,210 the value of the function depends on the combination 936 01:06:48,210 --> 01:06:51,030 x minus log z. 937 01:06:51,030 --> 01:06:54,960 So if I were to make log z much larger, 938 01:06:54,960 --> 01:06:59,390 I'm essentially transporting this combination 939 01:06:59,390 --> 01:07:02,750 without changing its shape, either in one direction 940 01:07:02,750 --> 01:07:05,220 or the other direction. 941 01:07:05,220 --> 01:07:07,450 OK? 942 01:07:07,450 --> 01:07:11,430 And if I go to the limit, essentially that log z is 943 01:07:11,430 --> 01:07:16,290 very large, I will have 1 for a huge distance and then, 944 01:07:16,290 --> 01:07:20,080 at some point, I would switch to 0. 945 01:07:20,080 --> 01:07:25,300 And what I'm doing, therefore, is to essentially approximately 946 01:07:25,300 --> 01:07:32,680 integrating from 0 to log z x to the 1/2 dx. 947 01:07:32,680 --> 01:07:39,531 Which would give me x to the 3/2-- sorry, log z to the 3/2 948 01:07:39,531 --> 01:07:40,030 over 3/2. 949 01:07:45,190 --> 01:07:46,750 And you can see that I can repeat 950 01:07:46,750 --> 01:07:50,870 this for any other value of m here. 951 01:07:50,870 --> 01:07:54,690 I would be integrating x to the m minus 1. 952 01:07:54,690 --> 01:08:00,930 So you would conclude that the limit as z is much, much larger 953 01:08:00,930 --> 01:08:08,390 than 1, of one of these functions fm minus of z, 954 01:08:08,390 --> 01:08:12,890 is obtained by integrating 0 to log z dx x to the m. 955 01:08:12,890 --> 01:08:15,630 I will get another factor of log z. 956 01:08:15,630 --> 01:08:20,439 And then I will get log z to the power 957 01:08:20,439 --> 01:08:24,359 of m divided by 1 over m multiplying 958 01:08:24,359 --> 01:08:26,960 by 1 over m minus 1 factorial. 959 01:08:26,960 --> 01:08:28,390 It will give me m factorial. 960 01:08:35,020 --> 01:08:40,590 Now, of course there will be correction to this 961 01:08:40,590 --> 01:08:44,510 because this function would be equal to this 962 01:08:44,510 --> 01:08:49,029 if I was integrating a step function. 963 01:08:49,029 --> 01:08:50,630 And I don't have a step function. 964 01:08:50,630 --> 01:08:53,350 I have something that gradually vanishes. 965 01:08:53,350 --> 01:08:56,460 Although, the interval over which it vanishes 966 01:08:56,460 --> 01:09:00,180 is scaling much smaller than log z. 967 01:09:00,180 --> 01:09:04,510 So what you can do, is you can do some manipulations 968 01:09:04,510 --> 01:09:08,899 and calculate corrections to this formula that 969 01:09:08,899 --> 01:09:14,250 become significant as log z is not exactly infinite 970 01:09:14,250 --> 01:09:16,390 but becomes smaller and smaller. 971 01:09:16,390 --> 01:09:20,450 It turns out that that will generate for you a series that 972 01:09:20,450 --> 01:09:23,340 is inverse powers of log z. 973 01:09:23,340 --> 01:09:27,760 And in fact, inverse even powers of log z. 974 01:09:27,760 --> 01:09:31,760 So that the next term will fall off as 1 over log z squared. 975 01:09:31,760 --> 01:09:36,390 The next term will fall off as 1 over z to the fourth 976 01:09:36,390 --> 01:09:39,609 and so forth. 977 01:09:39,609 --> 01:09:43,350 And the coefficient of the first one-- and I 978 01:09:43,350 --> 01:09:46,550 show you this in the notes, how to calculate it-- 979 01:09:46,550 --> 01:09:50,800 is pi squared over 6 times m m minus 1. 980 01:09:50,800 --> 01:09:53,082 The coefficient of the next term is 981 01:09:53,082 --> 01:10:02,730 7 pi to the fourth 360 m m minus 1, m minus 2, m minus 3. 982 01:10:02,730 --> 01:10:05,740 And there's a formula for subsequent terms 983 01:10:05,740 --> 01:10:08,740 in the series, how they are generated. 984 01:10:08,740 --> 01:10:11,940 And this thing is known as the Sommerfeld expansion. 985 01:10:23,083 --> 01:10:23,583 OK? 986 01:10:28,030 --> 01:10:38,290 So, essentially, if you plot as a function of z one 987 01:10:38,290 --> 01:10:43,670 of these fm minuses of z's, what we have according 988 01:10:43,670 --> 01:10:48,110 to the previous expansion, as you go to z 989 01:10:48,110 --> 01:10:52,350 equals to 0 you have a linear function z. 990 01:10:52,350 --> 01:10:55,240 And then the next correction is minus z squared. 991 01:10:55,240 --> 01:10:57,920 You start to bend in this direction. 992 01:10:57,920 --> 01:11:00,540 And then you have higher order corrections. 993 01:11:00,540 --> 01:11:04,340 What we see, that at larger values, it keeps growing. 994 01:11:04,340 --> 01:11:07,850 And the growth at larger values is this. 995 01:11:07,850 --> 01:11:12,815 So here you have z minus z squared 2 to the power of m. 996 01:11:12,815 --> 01:11:18,150 Out here you have log z to the m over m factorial. 997 01:11:18,150 --> 01:11:22,980 And then this corrections from the Sommerfeld expansion. 998 01:11:22,980 --> 01:11:24,710 OK? 999 01:11:24,710 --> 01:11:28,700 And so once we have that function, 1000 01:11:28,700 --> 01:11:33,630 which is the right hand side of this equality, 1001 01:11:33,630 --> 01:11:39,440 for a particular density we are asking, what is the value of z? 1002 01:11:39,440 --> 01:11:45,210 And at low densities we are evaluating solutions here. 1003 01:11:45,210 --> 01:11:49,250 At high values of density we are picking up 1004 01:11:49,250 --> 01:11:52,250 solutions on this other. 1005 01:11:52,250 --> 01:11:54,270 OK? 1006 01:11:54,270 --> 01:11:58,810 So let's imagine that we are indeed in that limit. 1007 01:11:58,810 --> 01:12:01,250 So what do we have? 1008 01:12:01,250 --> 01:12:13,360 We have that n lambda cubed over g is log z 1009 01:12:13,360 --> 01:12:18,240 to the 3/2 over 3/2 factorial. 1010 01:12:18,240 --> 01:12:21,080 That's the leading term. 1011 01:12:21,080 --> 01:12:26,920 Sub-leading term is 1 plus pi squared over 6. 1012 01:12:26,920 --> 01:12:29,870 m, in our case, is 3/2. 1013 01:12:29,870 --> 01:12:35,480 m minus 1 would be 1/2 divided by log z squared. 1014 01:12:35,480 --> 01:12:36,850 I will stop at this order. 1015 01:12:36,850 --> 01:12:40,120 That's all we will need. 1016 01:12:40,120 --> 01:12:45,670 And so I will use this to calculate z as a function n. 1017 01:12:45,670 --> 01:12:48,090 Again, same procedure as before. 1018 01:12:48,090 --> 01:12:50,610 I can invert this and write this as log 1019 01:12:50,610 --> 01:12:58,905 z being 3/2 factorial n lambda cubed over g. 1020 01:12:58,905 --> 01:13:01,170 so I multiplied by this. 1021 01:13:01,170 --> 01:13:06,580 I have to raise to the 2/3 power-- the inverse of 3/2. 1022 01:13:06,580 --> 01:13:09,260 I take this combination to the other side. 1023 01:13:09,260 --> 01:13:11,250 So I have 1. 1024 01:13:11,250 --> 01:13:17,440 This becomes plus pi squared over 8 1 over log z 1025 01:13:17,440 --> 01:13:21,280 squared raised to the minus 2/3 power 1026 01:13:21,280 --> 01:13:24,157 because I had to take it to the other side. 1027 01:13:28,447 --> 01:13:28,947 OK. 1028 01:13:32,790 --> 01:13:40,270 So at the lowest order, what I need to do 1029 01:13:40,270 --> 01:13:46,100 is to calculate log z from the zeroed order term and then, 1030 01:13:46,100 --> 01:13:52,220 at the next order, substitute that value of log z in here 1031 01:13:52,220 --> 01:13:54,740 and get the next correction. 1032 01:13:54,740 --> 01:13:59,024 So the lowest order correction will tell me that log z--- 1033 01:13:59,024 --> 01:14:03,470 which, remind you, is simply beta mu-- 1034 01:14:03,470 --> 01:14:09,100 is this combination raised to the 2/3 power. 1035 01:14:11,980 --> 01:14:13,280 What is that communication? 1036 01:14:13,280 --> 01:14:14,580 3/2 factorial. 1037 01:14:14,580 --> 01:14:20,860 3/2 factorial we saw was 3/2 1/2 factorial-- 1038 01:14:20,860 --> 01:14:25,220 which was root pi over 2-- so this is 3/2 factorial. 1039 01:14:25,220 --> 01:14:32,990 I have n over g. 1040 01:14:32,990 --> 01:14:34,800 Oh, sorry. n lambda cubed. 1041 01:14:37,490 --> 01:14:43,170 So write it in this fashion. n over g. 1042 01:14:43,170 --> 01:14:46,905 And then I have lambda cubed. 1043 01:14:49,470 --> 01:14:55,450 Let me put this to the 2/3 power because lambda cubed 1044 01:14:55,450 --> 01:14:59,180 to the 2/3 power is simply lambda squared. 1045 01:14:59,180 --> 01:15:02,560 And lambda squared is going to be h squared 1046 01:15:02,560 --> 01:15:06,560 divided by 2 pi mkT. 1047 01:15:06,560 --> 01:15:07,848 So this is lambda cubed. 1048 01:15:11,520 --> 01:15:13,270 What I'm going to do is notice that one 1049 01:15:13,270 --> 01:15:17,890 of these h over 2pi's is in fact already an h bar. 1050 01:15:17,890 --> 01:15:20,340 I'd like to write the result in terms 1051 01:15:20,340 --> 01:15:25,510 of h bar squared divided by 2m. 1052 01:15:25,510 --> 01:15:29,380 The 1 over kT I can write as beta. 1053 01:15:29,380 --> 01:15:32,800 Ultimately, it is my intention to cancel that beta which 1054 01:15:32,800 --> 01:15:37,200 is the 1 over kT in the beta that I have out here. 1055 01:15:37,200 --> 01:15:41,180 And I get the value for mu. 1056 01:15:41,180 --> 01:15:42,600 OK? 1057 01:15:42,600 --> 01:15:50,340 To go from h squared over 2pi to h bar squared over 2m, 1058 01:15:50,340 --> 01:15:54,320 I need a factor of 4pi squared. 1059 01:15:54,320 --> 01:15:59,200 Those 4pi squared I will put out front in here. 1060 01:15:59,200 --> 01:16:05,600 And they will change the result to something that is like this. 1061 01:16:05,600 --> 01:16:12,270 It is n over g is going to be remaining there. 1062 01:16:12,270 --> 01:16:19,240 I have pi by the time it gets in front over here 1063 01:16:19,240 --> 01:16:20,950 becomes pi to the 3/2. 1064 01:16:20,950 --> 01:16:27,450 Pi to the 3/2 multiplied by pi to the 1/2 becomes pi squared. 1065 01:16:27,450 --> 01:16:30,490 You can convince yourself that the factors of 2 1066 01:16:30,490 --> 01:16:33,740 and all of those things manage to give you 1067 01:16:33,740 --> 01:16:41,090 something that is 6pi squared n over g to the 2/3 power. 1068 01:16:41,090 --> 01:16:42,230 Yes? 1069 01:16:42,230 --> 01:16:44,886 AUDIENCE: So when [INAUDIBLE] chemical potential 1070 01:16:44,886 --> 01:16:46,840 as no temperature [INAUDIBLE]. 1071 01:16:46,840 --> 01:16:47,780 PROFESSOR: Exactly. 1072 01:16:47,780 --> 01:16:52,183 When you go to 0 temperature, the chemical potential 1073 01:16:52,183 --> 01:16:55,750 of a Fermi gas goes to a constant 1074 01:16:55,750 --> 01:16:58,320 that you have actually seen-- and I just 1075 01:16:58,320 --> 01:17:02,580 was about to make it clear to you what that constant is. 1076 01:17:02,580 --> 01:17:08,490 So we get that limit as essentially this T goes 1077 01:17:08,490 --> 01:17:13,550 to 0, which is another way of achieving the combination n 1078 01:17:13,550 --> 01:17:16,020 lambda cubed over g goes to infinity. 1079 01:17:16,020 --> 01:17:18,620 When T goes to 0, lambda goes to infinity, 1080 01:17:18,620 --> 01:17:21,980 this combination becomes large. 1081 01:17:21,980 --> 01:17:25,530 Canceling the betas, the limit of mu 1082 01:17:25,530 --> 01:17:30,000 is something that potentially in solid state courses 1083 01:17:30,000 --> 01:17:32,750 you have seen as the fermion energy. 1084 01:17:32,750 --> 01:17:38,160 And the fermion energy is h bar squared over 2m. 1085 01:17:38,160 --> 01:17:42,340 Some Fermi momentum squared. 1086 01:17:42,340 --> 01:17:45,780 And hopefully the Fermi momentum is 1087 01:17:45,780 --> 01:17:51,092 going to come out to be 6pi squared n over g to the 1/3. 1088 01:17:51,092 --> 01:17:55,290 And one way that you usually get your Fermi momentum is 1089 01:17:55,290 --> 01:17:57,705 you say that, what I need to do is 1090 01:17:57,705 --> 01:18:04,460 to fill up all of this state up to some value of kf. 1091 01:18:04,460 --> 01:18:07,182 And the number of states that there are 1092 01:18:07,182 --> 01:18:10,680 is the volume of the sphere- 4pi over 3k 1093 01:18:10,680 --> 01:18:15,650 f cubed, of course times the volume times the degeneracy 1094 01:18:15,650 --> 01:18:19,310 coming from spins if I have multiple possibilities. 1095 01:18:19,310 --> 01:18:22,840 And this should give me the total number n. 1096 01:18:37,540 --> 01:18:39,970 Ah. 1097 01:18:39,970 --> 01:18:41,980 I made some big mistake here. 1098 01:18:41,980 --> 01:18:43,626 Can somebody point it out to me? 1099 01:18:49,340 --> 01:18:54,080 How do we calculate Fermi momentum and Fermi [INAUDIBLE]? 1100 01:18:54,080 --> 01:18:56,100 Not coming out the answer I want. 1101 01:18:56,100 --> 01:18:58,157 AUDIENCE: One [INAUDIBLE] is a sphere, right? 1102 01:18:58,157 --> 01:18:58,740 PROFESSOR: No. 1103 01:18:58,740 --> 01:19:00,219 It's a sphere. 1104 01:19:00,219 --> 01:19:01,698 AUDIENCE: [INAUDIBLE] 1105 01:19:01,698 --> 01:19:03,180 PROFESSOR: Thank you. 1106 01:19:03,180 --> 01:19:04,770 Divide by density of state. 1107 01:19:04,770 --> 01:19:09,880 I multiplied by the V but I forgot the 2pi cubed here. 1108 01:19:09,880 --> 01:19:11,120 OK? 1109 01:19:11,120 --> 01:19:16,100 So when you divide 4pi divided by 8pi cubed, 1110 01:19:16,100 --> 01:19:19,980 you will have 2pi squared times 3, 1111 01:19:19,980 --> 01:19:25,850 which will give you 6pi squared. 1112 01:19:25,850 --> 01:19:28,520 n over V is the density. 1113 01:19:28,520 --> 01:19:31,130 Divided by g. 1114 01:19:31,130 --> 01:19:38,680 So you can see that we get the right result. 1115 01:19:38,680 --> 01:19:39,180 Yes? 1116 01:19:39,180 --> 01:19:41,540 AUDIENCE: [INAUDIBLE] cubed. 1117 01:19:41,540 --> 01:19:43,110 PROFESSOR: f cubed. 1118 01:19:43,110 --> 01:19:44,670 Yes. 1119 01:19:44,670 --> 01:19:45,493 That's right. 1120 01:19:45,493 --> 01:19:48,934 So kf is 1/3 of that. 1121 01:19:48,934 --> 01:19:50,656 Let's go step by step. 1122 01:19:50,656 --> 01:19:55,870 A is 6pi squared n over g to the 1/3. 1123 01:19:55,870 --> 01:20:02,290 kf squared would be the 2/3 power [? of g. ?] Yes? 1124 01:20:02,290 --> 01:20:04,460 AUDIENCE: So you said that this is the lowest order 1125 01:20:04,460 --> 01:20:06,069 term for the chemical potential. 1126 01:20:06,069 --> 01:20:06,860 PROFESSOR: Exactly. 1127 01:20:06,860 --> 01:20:07,360 Yes. 1128 01:20:07,360 --> 01:20:10,370 AUDIENCE: But aren't there going to be maybe a second order 1129 01:20:10,370 --> 01:20:12,290 temperature dependent term? 1130 01:20:12,290 --> 01:20:13,160 PROFESSOR: Yes. 1131 01:20:13,160 --> 01:20:17,780 So when we substitute that formula in here, 1132 01:20:17,780 --> 01:20:21,140 we will find the value of the chemical potential. 1133 01:20:21,140 --> 01:20:23,440 So I will draw it for you and then 1134 01:20:23,440 --> 01:20:25,630 recalculate the result later. 1135 01:20:25,630 --> 01:20:27,680 So this is the chemical potential 1136 01:20:27,680 --> 01:20:32,960 of the Fermi system as a function of its temperature. 1137 01:20:32,960 --> 01:20:35,280 What did we establish so far? 1138 01:20:35,280 --> 01:20:37,195 We established that at zero temperature 1139 01:20:37,195 --> 01:20:40,480 it is in fact epsilon f. 1140 01:20:40,480 --> 01:20:42,790 We had established that at high temperature-- 1141 01:20:42,790 --> 01:20:44,740 this was the discussion we had before-- it's 1142 01:20:44,740 --> 01:20:45,990 large and negative. 1143 01:20:45,990 --> 01:20:48,830 It is down here. 1144 01:20:48,830 --> 01:20:52,090 So indeed the behavior of the chemical potential 1145 01:20:52,090 --> 01:20:55,830 of the Fermi gas as a function of temperature 1146 01:20:55,830 --> 01:20:58,200 is something like this. 1147 01:20:58,200 --> 01:21:03,420 And so the chemical potential at zero temperature is epsilon f. 1148 01:21:03,420 --> 01:21:08,290 As you go to finite temperature, it will get reduced by a factor 1149 01:21:08,290 --> 01:21:10,925 that I will just calculate. 1150 01:21:10,925 --> 01:21:12,830 OK? 1151 01:21:12,830 --> 01:21:19,050 So if I now substitute this result in here 1152 01:21:19,050 --> 01:21:21,350 to get the next correction, what do I have? 1153 01:21:21,350 --> 01:21:21,850 OK. 1154 01:21:21,850 --> 01:21:25,090 So this zero to order term we established 1155 01:21:25,090 --> 01:21:28,965 is the same thing as beta epsilon f. 1156 01:21:31,550 --> 01:21:38,480 And then the next order term will be 1 plus-- this log 1157 01:21:38,480 --> 01:21:42,850 z is going to be small-- minus 2/3 times this 1158 01:21:42,850 --> 01:21:46,455 is going to give me minus pi squared over 6. 1159 01:21:49,880 --> 01:21:50,860 Right? 1160 01:21:50,860 --> 01:21:53,970 And then I have 1 over log z squared. 1161 01:21:53,970 --> 01:21:59,830 One over log z is simply 1 over beta epsilon f 1162 01:21:59,830 --> 01:22:08,880 which is the same thing as kT over epsilon f squared So 1163 01:22:08,880 --> 01:22:16,780 what we see is that the chemical potential starts 1164 01:22:16,780 --> 01:22:22,310 to be epsilon f and then it starts to go down parabolically 1165 01:22:22,310 --> 01:22:25,180 in temperature. 1166 01:22:25,180 --> 01:22:30,300 And the first correction is of the order of kT over epsilon f. 1167 01:22:30,300 --> 01:22:34,110 Not surprisingly, it will go to 0 1168 01:22:34,110 --> 01:22:42,040 at the value T that is off the order of epsilon f over kT. 1169 01:22:42,040 --> 01:22:44,310 So the chemical potential of the Fermi gas 1170 01:22:44,310 --> 01:22:45,850 actually changes sign. 1171 01:22:45,850 --> 01:22:48,890 It's negative and large at high temperature, 1172 01:22:48,890 --> 01:22:50,810 it goes to the Fermi energy which 1173 01:22:50,810 --> 01:22:54,820 is positive at low temperatures. 1174 01:22:54,820 --> 01:23:00,270 So once we know this behavior or the chemical potential, 1175 01:23:00,270 --> 01:23:04,810 we can calculate what the energy and pressure on. 1176 01:23:04,810 --> 01:23:07,630 And again, at zero temperature you 1177 01:23:07,630 --> 01:23:10,360 will have some energy which comes from filling out 1178 01:23:10,360 --> 01:23:15,430 this Fermi C. And then we will get corrections to energy 1179 01:23:15,430 --> 01:23:19,030 and pressure that we can calculate 1180 01:23:19,030 --> 01:23:21,970 using the types of formulas that we have that 1181 01:23:21,970 --> 01:23:27,180 relate pressure and energy to f 5/2 of z. 1182 01:23:27,180 --> 01:23:30,420 And so we can start to calculate the corrections 1183 01:23:30,420 --> 01:23:33,940 that you have to completely Fermi sphere. 1184 01:23:33,940 --> 01:23:35,750 And how those corrections give you, 1185 01:23:35,750 --> 01:23:38,690 say, the linear heat capacity is something 1186 01:23:38,690 --> 01:23:41,046 that we will derive next time.