1 00:00:00,060 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,030 under a Creative Commons license. 3 00:00:04,030 --> 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 --> 00:00:13,330 To make a donation, or view additional materials 6 00:00:13,330 --> 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 00:00:21,160 --> 00:00:23,250 PROFESSOR: Particles in quantum mechanics. 9 00:00:23,250 --> 00:00:27,470 In particular, the ones that are identical and non-interacting. 10 00:00:37,140 --> 00:00:42,430 So basically, we were focusing on a type of Hamiltonian 11 00:00:42,430 --> 00:00:45,780 for a system of N particles, which 12 00:00:45,780 --> 00:00:53,190 could be written as the sum of contributions that correspond 13 00:00:53,190 --> 00:00:57,460 respectively to particle 1, particle 2, particle N. 14 00:00:57,460 --> 00:01:03,090 So essentially, a sum of terms that are all the same. 15 00:01:03,090 --> 00:01:06,960 And one-particle terms is sufficient 16 00:01:06,960 --> 00:01:10,290 because we don't have interactions. 17 00:01:10,290 --> 00:01:15,750 So if we look at one of these H's-- so one of these 18 00:01:15,750 --> 00:01:19,673 one-particle Hamiltonians, we said that we could find some 19 00:01:19,673 --> 00:01:21,950 kind of basis for it. 20 00:01:21,950 --> 00:01:25,270 In particular, typically we were interested in particles 21 00:01:25,270 --> 00:01:26,140 in the box. 22 00:01:26,140 --> 00:01:31,430 We would label them with some wave number k. 23 00:01:31,430 --> 00:01:37,130 And there was an associated one-particle energy, 24 00:01:37,130 --> 00:01:39,900 which for the case of one-particle in box 25 00:01:39,900 --> 00:01:44,270 was h bar squared k squared over 2m. 26 00:01:44,270 --> 00:01:48,770 But in general, for the one-particle system, 27 00:01:48,770 --> 00:01:54,330 we can think of a ladder of possible values of energies. 28 00:01:54,330 --> 00:02:00,815 So there will be some k1, k2, k3, et cetera. 29 00:02:05,590 --> 00:02:08,830 They may be distributed in any particular way corresponding 30 00:02:08,830 --> 00:02:10,389 to different energies. 31 00:02:10,389 --> 00:02:12,760 Basically, you would have a number 32 00:02:12,760 --> 00:02:15,600 of possible states for one particle. 33 00:02:15,600 --> 00:02:27,730 So for the case of the particle in the box, the wave functions 34 00:02:27,730 --> 00:02:32,335 and coordinate space that we had x, k were of the form e 35 00:02:32,335 --> 00:02:37,020 to the i k dot x divided by square root of V. 36 00:02:37,020 --> 00:02:45,220 We allowed the energies where h bar squared k squared over 2m. 37 00:02:45,220 --> 00:02:50,520 And this discretization was because the values of k 38 00:02:50,520 --> 00:02:56,165 were multiples of 2 pi over l with integers in the three 39 00:02:56,165 --> 00:02:57,380 different directions. 40 00:02:57,380 --> 00:02:59,550 Assuming periodic boundary conditions 41 00:02:59,550 --> 00:03:03,670 or appropriate discretization for closed boundary conditions, 42 00:03:03,670 --> 00:03:07,010 or whatever you have. 43 00:03:07,010 --> 00:03:11,290 So that's the one-particle state. 44 00:03:11,290 --> 00:03:15,350 If the Hamiltonian is of this form, 45 00:03:15,350 --> 00:03:20,620 it is clear that we can multiply a bunch of these states 46 00:03:20,620 --> 00:03:25,800 and form another eigenstate for HN. 47 00:03:25,800 --> 00:03:27,805 Those we were calling product states. 48 00:03:32,945 --> 00:03:38,180 So you basically pick a bunch of these k's and you 49 00:03:38,180 --> 00:03:39,280 multiply them. 50 00:03:39,280 --> 00:03:43,685 So you have k1, k2, kN. 51 00:03:46,410 --> 00:03:49,050 And essentially, that would correspond, 52 00:03:49,050 --> 00:03:52,460 let's say, in coordinate representation 53 00:03:52,460 --> 00:03:57,910 to taking a bunch of x's and the corresponding k's and having 54 00:03:57,910 --> 00:04:03,500 a wave function of the form e to the i k alp-- sum over alpha k 55 00:04:03,500 --> 00:04:09,290 alpha x alpha, and then divided by V to the N over 2. 56 00:04:13,210 --> 00:04:16,779 So in this procedure, what did we do? 57 00:04:16,779 --> 00:04:22,340 We had a number of possibilities for the one-particle state. 58 00:04:22,340 --> 00:04:26,030 And in order to, let's say, make a two-particle state, 59 00:04:26,030 --> 00:04:31,450 we would pick two of these k's and multiply 60 00:04:31,450 --> 00:04:33,900 the corresponding wave functions. 61 00:04:33,900 --> 00:04:37,060 If you had three particles, we could pick another one. 62 00:04:37,060 --> 00:04:39,710 If you had four particles, we could potentially 63 00:04:39,710 --> 00:04:43,050 pick a second one twice, et cetera. 64 00:04:43,050 --> 00:04:44,890 So in general, basically we would 65 00:04:44,890 --> 00:04:48,900 put N of these crosses on these one-particle states 66 00:04:48,900 --> 00:04:51,570 that we've selected. 67 00:04:51,570 --> 00:04:54,690 Problem was that this was not allowed 68 00:04:54,690 --> 00:04:58,730 by quantum mechanics for identical particles. 69 00:04:58,730 --> 00:05:01,960 Because if we took one of these wave functions 70 00:05:01,960 --> 00:05:05,800 and exchanged two of the labels, x1 and x2, 71 00:05:05,800 --> 00:05:10,280 we could potentially get a different wave function. 72 00:05:10,280 --> 00:05:14,650 And in quantum mechanics, we said that the wave function 73 00:05:14,650 --> 00:05:18,850 has to be either symmetric or anti-symmetric with respect 74 00:05:18,850 --> 00:05:21,940 to exchange of a pair of particles. 75 00:05:21,940 --> 00:05:27,040 And also, whatever it implied for repeating this exchange 76 00:05:27,040 --> 00:05:31,340 many times to look at all possible permutations. 77 00:05:31,340 --> 00:05:36,770 So what we saw was that product states are good 78 00:05:36,770 --> 00:05:38,467 as long as you are thinking about 79 00:05:38,467 --> 00:05:39,550 distinguishable particles. 80 00:05:44,950 --> 00:05:53,710 But if you have identical particles, 81 00:05:53,710 --> 00:05:57,810 you had to appropriately symmetrize or anti-symmetrize 82 00:05:57,810 --> 00:06:00,060 these states. 83 00:06:00,060 --> 00:06:05,320 So what we ended up was a wabe of, for example, 84 00:06:05,320 --> 00:06:09,250 symmetrizing things for the case of fermions. 85 00:06:09,250 --> 00:06:12,400 So we could take a bunch of these k-values again. 86 00:06:15,360 --> 00:06:21,020 And a fermionic, or anti-symmetrized version, 87 00:06:21,020 --> 00:06:25,980 was then constructed by summing over all permutations. 88 00:06:25,980 --> 00:06:31,110 And for N particle, there would be N factorial permutations. 89 00:06:31,110 --> 00:06:34,320 Basically, doing a permutation of all of these 90 00:06:34,320 --> 00:06:40,200 indices k1, k2, et cetera, that we had selected for this. 91 00:06:40,200 --> 00:06:43,670 And for the case of fermions, we had 92 00:06:43,670 --> 00:06:49,720 to multiply each permutation with a sign that 93 00:06:49,720 --> 00:06:56,130 was plus for even permutations, minus for odd permutations. 94 00:06:56,130 --> 00:06:59,970 And this would give us N factorial terms. 95 00:06:59,970 --> 00:07:03,780 And the appropriate normalization 96 00:07:03,780 --> 00:07:08,490 was 1 over square root of N factorial. 97 00:07:08,490 --> 00:07:12,030 So this was for the case of fermions. 98 00:07:12,030 --> 00:07:13,445 And this was actually minus. 99 00:07:13,445 --> 00:07:15,410 I should have put in a minus here. 100 00:07:15,410 --> 00:07:21,220 And this would have been minus to the P. 101 00:07:21,220 --> 00:07:29,530 For the case of bosons, we basically 102 00:07:29,530 --> 00:07:33,360 dispensed with this factor of minus 1 to the P. 103 00:07:33,360 --> 00:07:35,145 So we had a Pk. 104 00:07:39,400 --> 00:07:44,480 Now, the corresponding normalization that we had here 105 00:07:44,480 --> 00:07:46,420 was slightly different. 106 00:07:46,420 --> 00:07:50,020 The point was that if we were doing 107 00:07:50,020 --> 00:07:55,570 this computation for the case of fermions, 108 00:07:55,570 --> 00:07:58,165 we could not allow a state where there 109 00:07:58,165 --> 00:08:02,030 is a double occupation of one of the one-particle state. 110 00:08:02,030 --> 00:08:06,060 Under exchange of the particles that 111 00:08:06,060 --> 00:08:08,470 would correspond to these k-values, 112 00:08:08,470 --> 00:08:11,600 I would get the same state back, but the exchange would give me 113 00:08:11,600 --> 00:08:14,250 a minus 1 and it would give me 0. 114 00:08:14,250 --> 00:08:17,370 So the fermionic wave function that I have constructed 115 00:08:17,370 --> 00:08:19,840 here, appropriately anti-symmetrized, 116 00:08:19,840 --> 00:08:24,060 exists only as long as there are no repeats. 117 00:08:24,060 --> 00:08:29,680 Whereas, for the case of bosons, I could put 2 over same place. 118 00:08:29,680 --> 00:08:34,820 I could put 3 somewhere else, any number that I liked, 119 00:08:34,820 --> 00:08:38,090 and there would be no problem with this. 120 00:08:38,090 --> 00:08:41,780 Except that the normalization would be more complicated. 121 00:08:41,780 --> 00:08:44,320 And we saw that appropriate normalization 122 00:08:44,320 --> 00:08:47,135 was a product over k nk factorial. 123 00:08:50,580 --> 00:08:55,380 So this was for fermions. 124 00:08:57,970 --> 00:08:58,820 This is for bosons. 125 00:09:03,380 --> 00:09:09,110 And the two I can submerge into one formula 126 00:09:09,110 --> 00:09:16,890 by writing a symmetrized or anti-symmetrized state, 127 00:09:16,890 --> 00:09:22,550 respectively indicated by eta, where we have eta is minus 1 128 00:09:22,550 --> 00:09:27,750 for fermions and eta is plus 1 for bosons, 129 00:09:27,750 --> 00:09:33,510 which is 1 over square root of N factorial product 130 00:09:33,510 --> 00:09:37,100 over k nk factorials. 131 00:09:37,100 --> 00:09:42,360 And then the sum over all N factorial permutations. 132 00:09:42,360 --> 00:09:49,360 This phase factor for fermions and nothing for bosons 133 00:09:49,360 --> 00:09:52,120 of the appropriately permuted set of k's. 134 00:09:57,630 --> 00:10:04,920 And in this way of noting things, 135 00:10:04,920 --> 00:10:13,840 I have to assign values nk which are either 0 or 1 for fermions. 136 00:10:13,840 --> 00:10:19,370 Because as we said, multiple occupations are not allowed. 137 00:10:19,370 --> 00:10:21,500 But there is no restriction for bosons. 138 00:10:28,040 --> 00:10:33,830 Except of course, that in this perspective, 139 00:10:33,830 --> 00:10:40,607 as I go along this k-axis, I have 0, 1, 0, 2, 1, 3, 0, 140 00:10:40,607 --> 00:10:44,910 0 for the occupations. 141 00:10:44,910 --> 00:10:47,060 Of course, what I need to construct 142 00:10:47,060 --> 00:10:50,040 whether I am dealing with bosons or fermions 143 00:10:50,040 --> 00:10:54,660 is that the sum over k nk is the total number of particles 144 00:10:54,660 --> 00:10:55,490 that I have. 145 00:11:01,800 --> 00:11:08,430 Now, the other thing to note is that once I have given you 146 00:11:08,430 --> 00:11:13,700 a picture such as this in terms of which one-particle states 147 00:11:13,700 --> 00:11:19,360 I want to look at, or which set of occupation numbers 148 00:11:19,360 --> 00:11:23,300 I have nk, then there is one and only 149 00:11:23,300 --> 00:11:27,400 one symmetrized or anti-symmetrized state. 150 00:11:27,400 --> 00:11:30,730 So over here, I could have permuted 151 00:11:30,730 --> 00:11:34,770 the k's in a number of possible ways. 152 00:11:34,770 --> 00:11:36,740 But as a result of symmetrization, 153 00:11:36,740 --> 00:11:39,940 anti-symmetrization, various ways 154 00:11:39,940 --> 00:11:44,200 of permuting the labels here ultimately 155 00:11:44,200 --> 00:11:48,210 come to the same set of occupation numbers. 156 00:11:48,210 --> 00:11:54,900 So it is possible to actually label the state rather than 157 00:11:54,900 --> 00:11:57,530 by the set of k's. 158 00:11:57,530 --> 00:11:59,600 By the set of nk's. 159 00:11:59,600 --> 00:12:04,085 It is kind of a more appropriate way of representing the system. 160 00:12:16,850 --> 00:12:21,770 So that's essentially the kinds of states 161 00:12:21,770 --> 00:12:24,320 that we are going to be using. 162 00:12:24,320 --> 00:12:27,200 Again, in talking about identical particles, 163 00:12:27,200 --> 00:12:31,650 which could be either bosons or fermions. 164 00:12:31,650 --> 00:12:35,970 Let's take a step back, remind you of something 165 00:12:35,970 --> 00:12:41,020 that we did before that had only one particle. 166 00:12:41,020 --> 00:12:43,750 Because I will soon go to many particles. 167 00:12:43,750 --> 00:12:46,940 But before that, let's remind you 168 00:12:46,940 --> 00:12:52,510 what the one particle in a box looked like. 169 00:12:52,510 --> 00:12:57,180 So indeed, in this case, the single-particle states 170 00:12:57,180 --> 00:13:00,030 were the ones that I told you before, 171 00:13:00,030 --> 00:13:03,360 2 pi over l some set of integers. 172 00:13:03,360 --> 00:13:08,260 Epsilon k, that was h bar squared k squared over 2m. 173 00:13:08,260 --> 00:13:13,260 If I want to calculate the partition function 174 00:13:13,260 --> 00:13:19,330 for one particle in the box, I have to do a trace of e 175 00:13:19,330 --> 00:13:24,185 to the minus beta h for one particle. 176 00:13:24,185 --> 00:13:26,820 The trace I can very easily calculate 177 00:13:26,820 --> 00:13:30,340 in the basis in which this is diagonal. 178 00:13:30,340 --> 00:13:32,330 That's the basis that is parameterized 179 00:13:32,330 --> 00:13:34,320 by these k-values. 180 00:13:34,320 --> 00:13:40,280 So I do a sum over k of e to the minus beta h 181 00:13:40,280 --> 00:13:42,305 bar squared k squared over 2m. 182 00:13:45,040 --> 00:13:47,940 And then in the limit of very large box, 183 00:13:47,940 --> 00:13:50,880 we saw that the sum over k I can replace 184 00:13:50,880 --> 00:13:55,700 with V integral over k 2 pi cubed. 185 00:13:55,700 --> 00:13:59,040 This was the density of states in k. 186 00:13:59,040 --> 00:14:05,910 e to the minus beta h bar squared k squared over 2m. 187 00:14:05,910 --> 00:14:09,820 And this was three Gaussian integrals 188 00:14:09,820 --> 00:14:15,100 that gave us the usual formula of V over lambda cubed, where 189 00:14:15,100 --> 00:14:18,830 lambda was this thermal [INAUDIBLE] wavelength 190 00:14:18,830 --> 00:14:20,665 h root 2 pi mk. 191 00:14:26,970 --> 00:14:31,560 But we said that the essence of statistical mechanics 192 00:14:31,560 --> 00:14:34,120 is to tell you about probabilities 193 00:14:34,120 --> 00:14:37,680 of various micro-states, various positions of the particle 194 00:14:37,680 --> 00:14:41,870 in the box, which in the quantum perspective 195 00:14:41,870 --> 00:14:45,210 is probability becomes a density matrix. 196 00:14:45,210 --> 00:14:48,560 And we evaluated this density matrix 197 00:14:48,560 --> 00:14:49,986 in the coordinate representation. 198 00:14:52,810 --> 00:14:57,030 And in the coordinate representation, 199 00:14:57,030 --> 00:15:00,600 essentially what we had to do was 200 00:15:00,600 --> 00:15:07,360 to go into the basis in which rho is diagonal. 201 00:15:07,360 --> 00:15:11,750 So we had x prime k. 202 00:15:11,750 --> 00:15:18,616 In the k basis, the density matrix is just this formula. 203 00:15:18,616 --> 00:15:23,030 It's the Boltzmann weight appropriately normalized by Z1. 204 00:15:25,540 --> 00:15:26,880 And then we go kx. 205 00:15:31,860 --> 00:15:35,300 And basically, again replacing this 206 00:15:35,300 --> 00:15:42,360 with V integral d cubed k 2 pi cubed e to the minus beta h bar 207 00:15:42,360 --> 00:15:45,510 squared k squared over 2m. 208 00:15:45,510 --> 00:15:51,430 These two factors of xk and x prime k gave us a factor of e 209 00:15:51,430 --> 00:16:05,130 to the kx, xk I have as ik dot x prime minus x. 210 00:16:05,130 --> 00:16:07,070 Completing the square. 211 00:16:07,070 --> 00:16:11,140 Actually, I had to divide by Z1. 212 00:16:11,140 --> 00:16:14,140 There is a factor of 1 over V from the normalization 213 00:16:14,140 --> 00:16:16,400 of these things. 214 00:16:16,400 --> 00:16:21,380 The two V's here cancel, but Z1 is proportional to V. 215 00:16:21,380 --> 00:16:23,510 The lambda cubes cancel and so what 216 00:16:23,510 --> 00:16:28,740 we have is 1 over V e to the minus x 217 00:16:28,740 --> 00:16:33,730 minus x prime squared pi over lambda squared. 218 00:16:37,730 --> 00:16:41,340 So basically, what you have here is 219 00:16:41,340 --> 00:16:47,040 that we have a box of volume V. There 220 00:16:47,040 --> 00:16:50,135 is a particle inside at some location x. 221 00:16:53,860 --> 00:16:56,450 And the probability to find it at location 222 00:16:56,450 --> 00:17:00,000 x is the diagonal element of this entity. 223 00:17:00,000 --> 00:17:03,400 It's just 1 over V. But this entity 224 00:17:03,400 --> 00:17:08,369 has off-diagonal elements reflecting the fact 225 00:17:08,369 --> 00:17:12,510 that the best that you can do to localize something in quantum 226 00:17:12,510 --> 00:17:15,990 mechanics is to make some kind of a wave packet. 227 00:17:24,890 --> 00:17:27,170 OK. 228 00:17:27,170 --> 00:17:31,740 So this we did last time. 229 00:17:31,740 --> 00:17:37,216 What we want to do now is to go from one particle 230 00:17:37,216 --> 00:17:40,570 to the case of N particles. 231 00:17:40,570 --> 00:17:44,020 So rather than having 1x prime, I 232 00:17:44,020 --> 00:17:49,420 will have a whole bunch of x primes labeled 1 through N. 233 00:17:49,420 --> 00:17:55,660 And I want to calculate the N particle density matrix that 234 00:17:55,660 --> 00:18:03,440 connects me from set of points x to another set of points 235 00:18:03,440 --> 00:18:04,080 x prime. 236 00:18:04,080 --> 00:18:06,880 So if you like in the previous picture, 237 00:18:06,880 --> 00:18:11,160 this would have been x1 and x1 prime, 238 00:18:11,160 --> 00:18:19,160 and then I now have x2 and x2 prime, x3 and x3 prime, xN and 239 00:18:19,160 --> 00:18:20,500 xN prime. 240 00:18:20,500 --> 00:18:23,520 I have a bunch of different coordinates 241 00:18:23,520 --> 00:18:27,140 and I'd like to calculate that. 242 00:18:27,140 --> 00:18:28,100 OK. 243 00:18:28,100 --> 00:18:37,620 Once more, we know that rho is diagonal in the basis that 244 00:18:37,620 --> 00:18:40,710 is represented by these occupations 245 00:18:40,710 --> 00:18:44,430 of one-particle states. 246 00:18:44,430 --> 00:18:49,480 And so what I can do is I can sum over 247 00:18:49,480 --> 00:18:52,360 a whole bunch of plane waves. 248 00:18:52,360 --> 00:18:58,240 And I have to pick N factors of k out of this list 249 00:18:58,240 --> 00:19:02,580 in order to make one of these symmetrized or anti-symmetrized 250 00:19:02,580 --> 00:19:04,740 wave functions. 251 00:19:04,740 --> 00:19:08,230 But then I have to remember, as I said, 252 00:19:08,230 --> 00:19:14,170 that I should not over-count distinct set of k-values 253 00:19:14,170 --> 00:19:17,690 because permutations of these list of k's 254 00:19:17,690 --> 00:19:20,750 that I have over here, because of symmetrization 255 00:19:20,750 --> 00:19:24,140 or anti-symmetrization, will give me the same state. 256 00:19:24,140 --> 00:19:26,670 So I have to be careful about that. 257 00:19:26,670 --> 00:19:31,650 Then, I go from x prime to k. 258 00:19:35,770 --> 00:19:39,510 Now, the density matrix in the k-basis I know. 259 00:19:39,510 --> 00:19:43,110 It is simply e to the minus beta, the energy which 260 00:19:43,110 --> 00:19:48,610 is sum over alpha h bar squared k alpha squared over 2m. 261 00:19:48,610 --> 00:19:52,690 So I sum over the list of k alphas 262 00:19:52,690 --> 00:19:54,510 that appear in this series. 263 00:19:54,510 --> 00:19:56,350 There will be n of them. 264 00:19:56,350 --> 00:19:58,360 I have to appropriately normalize 265 00:19:58,360 --> 00:20:03,100 that by the N-particle partition function, which we have yet 266 00:20:03,100 --> 00:20:04,800 to calculate. 267 00:20:04,800 --> 00:20:09,780 And then I go back from k to x. 268 00:20:17,245 --> 00:20:20,230 Now, let's do this. 269 00:20:20,230 --> 00:20:23,390 The first thing that I mentioned last time 270 00:20:23,390 --> 00:20:26,430 is that I would, in principle, like 271 00:20:26,430 --> 00:20:30,580 to sum over k1 going over the entire list, 272 00:20:30,580 --> 00:20:34,570 k2 going the entire list, k3 going over the entire list. 273 00:20:34,570 --> 00:20:39,170 That is, I would like to make the sum over k's unrestricted. 274 00:20:42,100 --> 00:20:45,170 But then I have to take into account 275 00:20:45,170 --> 00:20:48,260 the over-counting that I have. 276 00:20:48,260 --> 00:20:51,490 If I am looking at the case where all of the k's are 277 00:20:51,490 --> 00:20:57,250 distinct-- they don't show any double occupancy-- then 278 00:20:57,250 --> 00:21:00,160 I have over-counted by the number of permutations. 279 00:21:00,160 --> 00:21:03,890 Because any permutation would have given me the same number. 280 00:21:03,890 --> 00:21:07,000 So I have to divide by the number of permutations 281 00:21:07,000 --> 00:21:12,710 to avoid the over-counting due to symmetrization here. 282 00:21:12,710 --> 00:21:14,680 Now, when I have something like this, 283 00:21:14,680 --> 00:21:22,580 which is a multiple occupancy, I have overdone this division. 284 00:21:22,580 --> 00:21:28,730 I have to multiply by this factor, 285 00:21:28,730 --> 00:21:36,850 and that's the correct number of over-countings that I have. 286 00:21:36,850 --> 00:21:40,880 And as I said, this was a good thing because the quantity 287 00:21:40,880 --> 00:21:43,980 that I had the hardest time for, and comes 288 00:21:43,980 --> 00:21:47,020 in the normalizations that occurs here, 289 00:21:47,020 --> 00:21:49,750 is this factor of 1 over nk factorial. 290 00:21:53,140 --> 00:21:56,680 Naturally, again, all of these things 291 00:21:56,680 --> 00:21:58,670 do depend on the symmetry. 292 00:21:58,670 --> 00:22:03,240 So I better make sure I indicate the index. 293 00:22:03,240 --> 00:22:06,400 Whether I'm calculating this density matrix for fermions 294 00:22:06,400 --> 00:22:09,070 or bosons, it is important. 295 00:22:09,070 --> 00:22:16,000 In either case-- well, what I need to do 296 00:22:16,000 --> 00:22:23,960 is to do a summation over P here for this one and P prime 297 00:22:23,960 --> 00:22:26,150 here or P prime here and P here. 298 00:22:26,150 --> 00:22:28,440 It doesn't matter, there's two sets of permutations 299 00:22:28,440 --> 00:22:30,580 that I have to do. 300 00:22:30,580 --> 00:22:36,240 In each case, I have to take care of this eta P, 301 00:22:36,240 --> 00:22:36,815 eta P prime. 302 00:22:42,390 --> 00:22:45,450 And then the normalization. 303 00:22:45,450 --> 00:22:50,040 So I divide by twice, or the square of the square root. 304 00:22:50,040 --> 00:22:56,350 I get the N factorial product over k nk factorial. 305 00:22:56,350 --> 00:23:00,190 And very nicely, the over-counting factor 306 00:23:00,190 --> 00:23:03,580 here cancels the normalization factor 307 00:23:03,580 --> 00:23:06,660 that I would have had here. 308 00:23:06,660 --> 00:23:08,820 So we got that. 309 00:23:08,820 --> 00:23:11,280 Now, what do we have? 310 00:23:11,280 --> 00:23:23,500 We have P prime permutation of these objects going to x, 311 00:23:23,500 --> 00:23:27,330 and then we have here P permutation 312 00:23:27,330 --> 00:23:31,346 of these k numbers going to x. 313 00:23:31,346 --> 00:23:36,270 I guess the first one I got wrong. 314 00:23:36,270 --> 00:23:37,860 I start with x prime. 315 00:23:40,460 --> 00:23:44,630 Go through P prime to k. 316 00:23:44,630 --> 00:23:50,810 And again, symmetries are already taken into account. 317 00:23:50,810 --> 00:23:52,730 I don't need to write that. 318 00:23:52,730 --> 00:23:57,490 And I have the factor of e to the minus beta h 319 00:23:57,490 --> 00:24:01,670 bar squared sum over alpha k alpha squared 320 00:24:01,670 --> 00:24:04,540 over 2m divided by ZN. 321 00:24:09,940 --> 00:24:14,880 OK, so let's bring all of the denominator factors out front. 322 00:24:14,880 --> 00:24:17,170 I have a ZN. 323 00:24:17,170 --> 00:24:20,970 I have an N factorial squared. 324 00:24:20,970 --> 00:24:24,620 Two factors of N factorial. 325 00:24:24,620 --> 00:24:31,620 I have a sum over two sets of permutations P and P prime. 326 00:24:34,350 --> 00:24:44,480 The product of the associated phase factor of their parities, 327 00:24:44,480 --> 00:24:48,830 and then I have this integration over k's. 328 00:24:48,830 --> 00:24:50,820 Now, unrestricted. 329 00:24:50,820 --> 00:24:55,730 Since it is unrestricted, I can integrate independently 330 00:24:55,730 --> 00:25:01,320 over each one of the k's, or sum over each one of them. 331 00:25:01,320 --> 00:25:04,000 When I sum, the sum becomes the integral 332 00:25:04,000 --> 00:25:12,700 over d cubed k alpha divided by 2 pi-- yeah, 2 pi cubed V. 333 00:25:12,700 --> 00:25:16,820 Basically, the density in replacing the sum over k 334 00:25:16,820 --> 00:25:20,350 alpha with the corresponding integration. 335 00:25:20,350 --> 00:25:25,430 So basically, this set of factors 336 00:25:25,430 --> 00:25:28,550 is what happened to that. 337 00:25:34,240 --> 00:25:37,210 OK, what do we have here? 338 00:25:37,210 --> 00:25:47,680 We have e to the i x-- well, let's be careful here. 339 00:25:47,680 --> 00:26:02,130 I have e to the i x prime alpha acting on k of p prime alpha 340 00:26:02,130 --> 00:26:07,850 because I permuted the k-label that 341 00:26:07,850 --> 00:26:15,580 went with, say, the alpha component here with p prime. 342 00:26:15,580 --> 00:26:18,350 From here, I would have minus because it's 343 00:26:18,350 --> 00:26:20,050 the complex conjugate. 344 00:26:20,050 --> 00:26:30,746 I have x alpha k p alpha, because I permuted this by k. 345 00:26:34,080 --> 00:26:40,010 I have one of these factors for each V. With each one of them, 346 00:26:40,010 --> 00:26:43,900 there is a normalization of square root of V. 347 00:26:43,900 --> 00:26:48,930 So the two of them together will give me V. But that's only one 348 00:26:48,930 --> 00:26:53,210 of the N-particle So there are N of them. 349 00:26:53,210 --> 00:26:57,265 So if I want, I can extend this product 350 00:26:57,265 --> 00:26:59,660 to also encompass this term. 351 00:27:02,520 --> 00:27:07,420 And then having done so, I can also write here e to the minus 352 00:27:07,420 --> 00:27:10,000 beta h bar squared k alpha squared 353 00:27:10,000 --> 00:27:12,680 over 2m within the product. 354 00:27:18,682 --> 00:27:23,040 AUDIENCE: [INAUDIBLE] after this-- is it quantity xk minus 355 00:27:23,040 --> 00:27:25,540 [INAUDIBLE]. 356 00:27:25,540 --> 00:27:27,200 PROFESSOR: I forgot an a here. 357 00:27:27,200 --> 00:27:28,740 What else did I miss out? 358 00:27:31,433 --> 00:27:32,724 AUDIENCE: [INAUDIBLE] quantity. 359 00:27:35,592 --> 00:27:37,210 PROFESSOR: So I forgot the i. 360 00:27:39,890 --> 00:27:41,340 OK, good? 361 00:27:41,340 --> 00:27:44,848 So the V's cancel out. 362 00:27:44,848 --> 00:27:48,200 All right, so that's fine. 363 00:27:48,200 --> 00:27:49,380 What do we have? 364 00:27:49,380 --> 00:27:57,530 We have 1 over ZN N factorial squared. 365 00:27:57,530 --> 00:28:02,070 Two sets of permutations summed over, p and p prime. 366 00:28:02,070 --> 00:28:07,170 Corresponding parities eta p eta of p prime. 367 00:28:07,170 --> 00:28:13,300 And then, I have a product of these integrations 368 00:28:13,300 --> 00:28:17,980 that I have to do that are three-dimensional Gaussians 369 00:28:17,980 --> 00:28:20,900 for each k alpha. 370 00:28:20,900 --> 00:28:22,700 What do I get? 371 00:28:22,700 --> 00:28:24,890 Well, first of all, if I didn't have this, 372 00:28:24,890 --> 00:28:27,730 if I just was doing the integration of e 373 00:28:27,730 --> 00:28:32,100 to the minus beta h bar squared k squared over 2m, 374 00:28:32,100 --> 00:28:34,120 I did that already. 375 00:28:34,120 --> 00:28:36,720 I get a 1 over lambda cubed. 376 00:28:36,720 --> 00:28:39,760 So basically, from each one of them I will get a 1 377 00:28:39,760 --> 00:28:40,645 over lambda cubed. 378 00:28:43,830 --> 00:28:47,470 But the integration is shifted by this amount. 379 00:28:47,470 --> 00:28:50,690 Actually, I already did the shifted integration here also 380 00:28:50,690 --> 00:28:53,060 for one particle. 381 00:28:53,060 --> 00:29:03,400 So I get the corresponding factor of e to the minus-- ah. 382 00:29:03,400 --> 00:29:07,590 I have to be a little bit careful over here 383 00:29:07,590 --> 00:29:14,450 because what I am integrating is over k alpha squared. 384 00:29:14,450 --> 00:29:18,910 Whereas, in the way that I have the list over here, 385 00:29:18,910 --> 00:29:22,830 I have x prime alpha and x alpha, 386 00:29:22,830 --> 00:29:27,450 but a different k playing around with each. 387 00:29:27,450 --> 00:29:28,360 What should I do? 388 00:29:28,360 --> 00:29:31,700 I really want this integration over k 389 00:29:31,700 --> 00:29:36,620 alpha to look like what I have over here. 390 00:29:36,620 --> 00:29:39,680 Well, as I sum over all possibilities 391 00:29:39,680 --> 00:29:44,680 in each one of these terms, I am bound to encounter k alpha. 392 00:29:44,680 --> 00:29:49,690 Essentially, I have permuted all of the k's that I originally 393 00:29:49,690 --> 00:29:50,810 had. 394 00:29:50,810 --> 00:29:55,340 So the k alpha has now been sent to some other location. 395 00:29:55,340 --> 00:30:00,170 But as I sum over all possible alpha, I will hit that. 396 00:30:00,170 --> 00:30:03,540 When I hit that, I will find that the thing that 397 00:30:03,540 --> 00:30:15,030 was multiplying k alpha is the inverse permutation of alpha. 398 00:30:15,030 --> 00:30:18,880 And the thing that was multiplying k alpha here 399 00:30:18,880 --> 00:30:23,390 is the inverse permutation of p. 400 00:30:29,140 --> 00:30:34,140 So then I can do the integration over k alpha easily. 401 00:30:34,140 --> 00:30:35,570 And so what do I have? 402 00:30:35,570 --> 00:30:39,510 I have x prime of p prime inverse 403 00:30:39,510 --> 00:30:43,780 alpha-- the inverse permutation-- minus x 404 00:30:43,780 --> 00:30:49,180 of p inverse alpha squared pi over lambda squared. 405 00:31:04,610 --> 00:31:09,190 Now, this is still inconvenient because I 406 00:31:09,190 --> 00:31:14,860 am summing over two N factorial sets of permutations. 407 00:31:14,860 --> 00:31:19,070 And I expect that since the sum only involves 408 00:31:19,070 --> 00:31:24,610 comparison of things that are occurring N times, 409 00:31:24,610 --> 00:31:29,690 as I go over the list of N factorial permutation squared, 410 00:31:29,690 --> 00:31:33,240 I will get the same thing appearing twice. 411 00:31:33,240 --> 00:31:37,590 So it is very much like when we are doing an integration over x 412 00:31:37,590 --> 00:31:39,610 and x prime, but the function only 413 00:31:39,610 --> 00:31:41,370 depends on x minus x prime. 414 00:31:41,370 --> 00:31:43,470 We get a factor of volume. 415 00:31:43,470 --> 00:31:47,250 Here, it is easy to see that one of these 416 00:31:47,250 --> 00:31:54,470 sums I can very easily do because it is just repetition 417 00:31:54,470 --> 00:31:57,390 of all of the results that I have previously. 418 00:31:57,390 --> 00:32:01,400 And there will be N factorial such terms. 419 00:32:01,400 --> 00:32:07,810 So doing that, I can get rid of one of the N factorials. 420 00:32:07,810 --> 00:32:11,780 And I will have only one permutation left, 421 00:32:11,780 --> 00:32:19,150 Q. And what will appear here would be the parity of this Q 422 00:32:19,150 --> 00:32:21,520 that is the combination, or if you like, 423 00:32:21,520 --> 00:32:25,600 the relative of these two permutations. 424 00:32:25,600 --> 00:32:32,680 And I have an exponential of minus sum 425 00:32:32,680 --> 00:32:41,190 over alpha x alpha minus x prime Q alpha squared 426 00:32:41,190 --> 00:32:44,650 pi over lambda squared. 427 00:32:44,650 --> 00:32:50,173 And I think I forgot a factor of lambda to the 3 [INAUDIBLE]. 428 00:32:55,103 --> 00:32:57,075 This factor of lambda 3. 429 00:33:03,510 --> 00:33:05,970 So this is actually the final result. 430 00:33:14,480 --> 00:33:21,090 And let's see what that precisely 431 00:33:21,090 --> 00:33:22,510 means for two particles. 432 00:33:27,030 --> 00:33:28,560 So let's look at two particles. 433 00:33:32,810 --> 00:33:40,020 So for two particle,s I will have on one side coordinates 434 00:33:40,020 --> 00:33:42,885 of 1 prime and 2 prime. 435 00:33:45,440 --> 00:33:49,525 On the right-hand side, I have coordinates 1 and 2. 436 00:33:54,210 --> 00:33:57,702 And let's see what this density matrix tells us. 437 00:33:57,702 --> 00:34:04,470 It tells us that to go from x1 prime x2 prime, a two particle 438 00:34:04,470 --> 00:34:11,540 density matrix connecting to x1 x2 on the other side, 439 00:34:11,540 --> 00:34:15,670 I have 1 over the two-particle partition function 440 00:34:15,670 --> 00:34:18,560 that i haven't yet calculated. 441 00:34:18,560 --> 00:34:20,920 Lambda to the sixth. 442 00:34:20,920 --> 00:34:24,699 N factorial in this case is 2. 443 00:34:24,699 --> 00:34:32,320 And then for two things, there are two permutations. 444 00:34:32,320 --> 00:34:37,710 So the identity maps 1 to 1, 2 to 2. 445 00:34:37,710 --> 00:34:39,560 And therefore, what I will get here 446 00:34:39,560 --> 00:34:47,144 would be exponential of minus x1' minus x1 prime squared 447 00:34:47,144 --> 00:34:55,070 pi over lambda squared minus x2 minus x2 prime squared pi 448 00:34:55,070 --> 00:34:56,060 over lambda squared. 449 00:35:02,450 --> 00:35:07,610 So that's Q being identity and identity 450 00:35:07,610 --> 00:35:10,470 has essentially 0 parity. 451 00:35:10,470 --> 00:35:14,120 It's an even permutation. 452 00:35:14,120 --> 00:35:17,410 The next thing is when I exchange 1 and 2. 453 00:35:17,410 --> 00:35:19,650 That would have odd parity. 454 00:35:19,650 --> 00:35:23,570 So I would get minus 1 for fermions, plus for bosons. 455 00:35:23,570 --> 00:35:28,460 And what I would get here is exponential of minus x1' 456 00:35:28,460 --> 00:35:33,990 minus x2 prime squared pi over lambda squared minus x2 457 00:35:33,990 --> 00:35:37,650 minus x1 prime squared pi over lambda squared. 458 00:35:43,950 --> 00:35:49,020 So essentially, one of the terms-- the first term 459 00:35:49,020 --> 00:35:53,320 is just the square of what I had before for one particle. 460 00:35:53,320 --> 00:35:57,560 I take the one-particle result, going from 1 to 1 prime, 461 00:35:57,560 --> 00:36:02,310 going from 2 to 2 prime and multiply them together. 462 00:36:02,310 --> 00:36:07,700 But then you say, I can't tell apart 2 prime and 1 prime. 463 00:36:07,700 --> 00:36:09,860 Maybe the thing that you are calling 1 prime 464 00:36:09,860 --> 00:36:13,330 is really 2 prime and vice versa. 465 00:36:13,330 --> 00:36:16,280 So I have to allow for the possibility 466 00:36:16,280 --> 00:36:18,430 that rather than x1 prime here, I 467 00:36:18,430 --> 00:36:23,080 should put x2 prime and the other way around. 468 00:36:23,080 --> 00:36:27,560 And this also corresponds to a permutation 469 00:36:27,560 --> 00:36:28,630 that is an exchange. 470 00:36:28,630 --> 00:36:33,840 It's an odd parity and will give you something like that. 471 00:36:33,840 --> 00:36:38,780 Say OK, I have no idea what that means. 472 00:36:38,780 --> 00:36:43,720 I'll tell you, OK, you were happy when I put x prime and x 473 00:36:43,720 --> 00:36:45,640 here because that was the probability 474 00:36:45,640 --> 00:36:48,790 to find the particle somewhere. 475 00:36:48,790 --> 00:36:50,800 So let me look at the diagonal term 476 00:36:50,800 --> 00:36:52,650 here, which is a probability. 477 00:36:57,720 --> 00:36:59,520 This should give me the probability 478 00:36:59,520 --> 00:37:02,120 to find one particle at position x1, 479 00:37:02,120 --> 00:37:03,650 one particle at position x2. 480 00:37:06,990 --> 00:37:09,130 Because the particles were non-interacting, 481 00:37:09,130 --> 00:37:10,890 one particle-- it could be anywhere. 482 00:37:10,890 --> 00:37:15,010 I had the 1 over V. Is it 1 over V squared 483 00:37:15,010 --> 00:37:17,240 or something like that? 484 00:37:17,240 --> 00:37:19,350 Well, we find that there is factor 485 00:37:19,350 --> 00:37:21,520 out front that we haven't yet evaluated. 486 00:37:21,520 --> 00:37:24,460 It turns out that this factor will give me a 1 487 00:37:24,460 --> 00:37:27,070 over V squared. 488 00:37:27,070 --> 00:37:31,650 And if I set x1 prime to x1, x2 prime 489 00:37:31,650 --> 00:37:36,800 to x2, which is what I've done here, this factor becomes 1. 490 00:37:36,800 --> 00:37:41,510 But then the other factor will give me eta e to the minus 2 491 00:37:41,510 --> 00:37:46,590 pi over lambda squared x1 minus x2 squared. 492 00:37:53,070 --> 00:37:58,960 So the physical probability to find one particle-- or more 493 00:37:58,960 --> 00:38:05,240 correctly, a wave packet here and a wave package there 494 00:38:05,240 --> 00:38:07,580 is not 1 over V squared. 495 00:38:07,580 --> 00:38:10,796 It's some function of the separation between these two 496 00:38:10,796 --> 00:38:11,295 particles. 497 00:38:15,030 --> 00:38:18,410 So that separation is contained here. 498 00:38:18,410 --> 00:38:23,290 If I really call that separation to be r, 499 00:38:23,290 --> 00:38:27,210 this is an additional weight that depends on r. 500 00:38:27,210 --> 00:38:30,650 This is r squared. 501 00:38:30,650 --> 00:38:38,410 So you can think of this as an interaction, which 502 00:38:38,410 --> 00:38:41,140 is because solely of quantum statistics. 503 00:38:44,880 --> 00:38:47,310 And what is this interaction? 504 00:38:47,310 --> 00:38:56,600 This interaction V of r would be minus kT log of 1 505 00:38:56,600 --> 00:39:02,337 plus eta e to the minus 2 pi r squared over lambda squared. 506 00:39:07,110 --> 00:39:11,670 I will plot out for you what this V or r looks 507 00:39:11,670 --> 00:39:16,730 like as a function of how far apart the centers of these two 508 00:39:16,730 --> 00:39:19,390 wave packets are. 509 00:39:19,390 --> 00:39:23,240 You can see that the result depends on eta. 510 00:39:23,240 --> 00:39:27,850 If eta is minus 1, which is for the case of fermions, 511 00:39:27,850 --> 00:39:30,330 this is 1 minus something. 512 00:39:30,330 --> 00:39:32,600 It's something that is less than 1. 513 00:39:32,600 --> 00:39:34,960 [INAUDIBLE] would be negative. 514 00:39:34,960 --> 00:39:39,580 So the whole potential would be positive, or repulsive. 515 00:39:39,580 --> 00:39:44,210 At large distances, indeed it would be exponentially going 516 00:39:44,210 --> 00:39:48,840 to 0 because I can expand the log at large distances. 517 00:39:48,840 --> 00:39:51,650 So here I have a term that is minus 2 pi 518 00:39:51,650 --> 00:39:55,620 r squared over lambda squared. 519 00:39:55,620 --> 00:40:00,440 As I go towards r equals to 0, actually things 520 00:40:00,440 --> 00:40:05,610 become very bad because r goes to 0 I will get 1 minus 1 521 00:40:05,610 --> 00:40:07,310 and the log will diverge. 522 00:40:07,310 --> 00:40:14,290 So basically, there is, if you like, an effective potential 523 00:40:14,290 --> 00:40:15,950 that says you can't put these two 524 00:40:15,950 --> 00:40:19,460 fermions on top of each other. 525 00:40:19,460 --> 00:40:21,860 So there is a statistical potential. 526 00:40:21,860 --> 00:40:26,670 So this is for eta minus 1, or fermions. 527 00:40:30,430 --> 00:40:34,490 For the case of bosons, eta plus 1. 528 00:40:34,490 --> 00:40:36,910 It is log of 1 plus something, so it's 529 00:40:36,910 --> 00:40:39,385 a positive number inside the log. 530 00:40:39,385 --> 00:40:43,520 The potential will be attractive. 531 00:40:43,520 --> 00:40:49,660 And it will actually saturate to a value of kT log 2 532 00:40:49,660 --> 00:40:51,360 when r goes to 0. 533 00:40:51,360 --> 00:40:57,850 So this is again, eta of plus 1 for the case of bosons. 534 00:41:38,320 --> 00:41:42,840 So the one thing that this formula does not have yet 535 00:41:42,840 --> 00:41:46,480 is the value for this partition function ZN. 536 00:41:46,480 --> 00:41:50,590 It gives you the qualitative behavior in either case. 537 00:41:50,590 --> 00:41:53,390 And let's calculate what ZN is. 538 00:41:53,390 --> 00:41:56,320 Well, basically, that would come from noting 539 00:41:56,320 --> 00:42:00,000 that the trace of rho has to be 1. 540 00:42:00,000 --> 00:42:12,050 So ZN is trace of e to the minus beta H. 541 00:42:12,050 --> 00:42:22,340 And essentially, I can take this ZN to the other side 542 00:42:22,340 --> 00:42:35,110 and evaluate this as x e to the minus beta H x. 543 00:42:35,110 --> 00:42:44,770 That is, I can calculate the diagonal elements 544 00:42:44,770 --> 00:42:46,600 of this matrix that I have calculated-- 545 00:42:46,600 --> 00:42:48,830 that I have over there. 546 00:42:48,830 --> 00:42:52,260 So there is an overall factor of 1 547 00:42:52,260 --> 00:42:59,430 over lambda cubed to the power of N. I have N factorial. 548 00:43:02,250 --> 00:43:12,800 And then I have a sum over permutations Q eta of Q. 549 00:43:12,800 --> 00:43:15,580 The diagonal element is obtained by putting 550 00:43:15,580 --> 00:43:18,610 x prime to be the same as x. 551 00:43:18,610 --> 00:43:30,290 So I have exponential of minus x-- sum over alpha x alpha 552 00:43:30,290 --> 00:43:33,130 minus x of Q alpha. 553 00:43:33,130 --> 00:43:36,620 I set x prime to be the same as x. 554 00:43:36,620 --> 00:43:37,560 Squared. 555 00:43:37,560 --> 00:43:41,530 And then there's an overall pi over lambda squared. 556 00:43:44,200 --> 00:43:50,030 And if I am taking the trace, it means that I have to do 557 00:43:50,030 --> 00:43:52,580 integration over all x's. 558 00:44:03,640 --> 00:44:08,560 So I'm evaluating this trace in coordinate basis, which 559 00:44:08,560 --> 00:44:11,040 means that I should put x and x prime to be 560 00:44:11,040 --> 00:44:13,560 the same for the trace, and then I 561 00:44:13,560 --> 00:44:19,970 have to sum or integrate over all possible values of x. 562 00:44:19,970 --> 00:44:21,190 So let's do this. 563 00:44:21,190 --> 00:44:31,070 I have 1 over N factorial lambda cubed raised to the power of N. 564 00:44:31,070 --> 00:44:31,900 OK. 565 00:44:31,900 --> 00:44:36,290 Now I have to make a choice because I 566 00:44:36,290 --> 00:44:40,660 have a whole bunch of terms because of these permutations. 567 00:44:43,180 --> 00:44:46,520 Let's do them one by one. 568 00:44:46,520 --> 00:44:51,120 Let's first do the case where Q is identity. 569 00:44:51,120 --> 00:44:55,530 That is, I map everybody to themselves. 570 00:44:55,530 --> 00:45:00,580 Actually, let me write down the integrations first. 571 00:45:00,580 --> 00:45:06,330 I will do the integrations over all pairs 572 00:45:06,330 --> 00:45:10,780 of coordinates of these Gaussians. 573 00:45:10,780 --> 00:45:17,730 These Gaussians I will evaluate for different permutations. 574 00:45:17,730 --> 00:45:22,160 Let's look at the case where Q is identity. 575 00:45:22,160 --> 00:45:27,540 When Q is identity, essentially I will put all of the x prime 576 00:45:27,540 --> 00:45:29,040 to be the same as x. 577 00:45:29,040 --> 00:45:33,600 It is like what I did here for two particles and I got 1. 578 00:45:33,600 --> 00:45:36,960 I do the same thing for more than one particle. 579 00:45:36,960 --> 00:45:38,210 I will still get 1. 580 00:45:42,740 --> 00:45:46,420 Then, I will do the same thing that I did over here. 581 00:45:46,420 --> 00:45:52,370 Here, the next term that I did was to exchange 1 and 2. 582 00:45:52,370 --> 00:45:56,070 So this became x1 minus x2. 583 00:45:56,070 --> 00:45:58,330 I'll do the same thing here. 584 00:45:58,330 --> 00:46:01,570 I look at the case where Q corresponds 585 00:46:01,570 --> 00:46:05,340 to exchange of particles 1 and 2. 586 00:46:05,340 --> 00:46:07,690 And then that will give me a factor which 587 00:46:07,690 --> 00:46:15,180 is e to the minus pi over lambda squared x1 minus x2 squared. 588 00:46:15,180 --> 00:46:19,110 There are two of these making together 2 pi over lambda 589 00:46:19,110 --> 00:46:21,340 squared, which I hope I had there, too. 590 00:46:25,420 --> 00:46:29,030 But then there was a whole bunch of other terms that I can do. 591 00:46:29,030 --> 00:46:35,360 I can exchange, let's say, 7 and 9. 592 00:46:35,360 --> 00:46:38,720 And then I will get here 2 pi over lambda 593 00:46:38,720 --> 00:46:43,340 squared x7 minus x9 squared. 594 00:46:43,340 --> 00:46:46,230 And there's a whole bunch of such exchanges 595 00:46:46,230 --> 00:46:50,220 that I can make in which I just switch 596 00:46:50,220 --> 00:46:55,950 between two particles in this whole story. 597 00:46:55,950 --> 00:47:00,840 And clearly, the number of exchanges that I can make 598 00:47:00,840 --> 00:47:05,060 is the number of pairs, N N minus 1 over 2. 599 00:47:08,830 --> 00:47:12,750 Once I am done with all of the exchanges, 600 00:47:12,750 --> 00:47:14,780 then I have to go to the next thing that 601 00:47:14,780 --> 00:47:18,120 doesn't have an analog here for two particles. 602 00:47:18,120 --> 00:47:20,850 But if I take three particles, I can permute them 603 00:47:20,850 --> 00:47:22,980 like a triangle. 604 00:47:22,980 --> 00:47:25,770 So presumably there would be next set 605 00:47:25,770 --> 00:47:32,190 of terms, which is a permutation that is like 1, 2, 3, 2, 3, 1. 606 00:47:32,190 --> 00:47:36,830 There's a bunch of things that involve two permutations, four 607 00:47:36,830 --> 00:47:40,190 permutations, and so forth. 608 00:47:40,190 --> 00:47:42,600 So there is a whole list of things 609 00:47:42,600 --> 00:47:47,710 that would go to here where these two-particle exchanges 610 00:47:47,710 --> 00:47:49,190 are the simplest class. 611 00:47:52,970 --> 00:47:57,590 Now, as we shall see, there is a systematic way 612 00:47:57,590 --> 00:48:02,540 of looking at things where the two-particle exchanges are 613 00:48:02,540 --> 00:48:05,570 the first correction due to quantum effects. 614 00:48:05,570 --> 00:48:09,300 Three-particle exchanges would be higher-order corrections. 615 00:48:09,300 --> 00:48:14,880 And we can systematically do them in order. 616 00:48:14,880 --> 00:48:19,685 So let's see what happens if we compare the case where there 617 00:48:19,685 --> 00:48:24,210 is no exchange and the case where there is one exchange. 618 00:48:24,210 --> 00:48:26,540 When there is no exchange, I am essentially 619 00:48:26,540 --> 00:48:31,500 integrating over each position over the volume. 620 00:48:31,500 --> 00:48:38,880 So what I would get is V raised to the power of N. 621 00:48:38,880 --> 00:48:41,880 The next term? 622 00:48:41,880 --> 00:48:45,420 Well, I have to do the integrations. 623 00:48:45,420 --> 00:48:51,120 The integrations over x3, x4, x5, all the way to x to the N, 624 00:48:51,120 --> 00:48:52,270 there is no factors. 625 00:48:52,270 --> 00:48:56,790 So they will give me factors of V. And there are N minus 2 626 00:48:56,790 --> 00:48:57,290 of them. 627 00:49:01,060 --> 00:49:04,600 And then I have to do the integration over x1 and x2 628 00:49:04,600 --> 00:49:08,840 of this factor, but it's only a function 629 00:49:08,840 --> 00:49:10,690 of the relative coordinate. 630 00:49:10,690 --> 00:49:12,650 So there is one other integration 631 00:49:12,650 --> 00:49:17,690 that I can trivially do, which is the center of mass gives me 632 00:49:17,690 --> 00:49:20,760 a factor of V. And then I am left 633 00:49:20,760 --> 00:49:24,973 with the integral over the relative coordinate of e 634 00:49:24,973 --> 00:49:30,170 to the minus 2 pi r squared over lambda squared. 635 00:49:30,170 --> 00:49:34,070 And I forgot-- it's very important. 636 00:49:34,070 --> 00:49:40,200 This will carry a factor of eta because any exchange is odd. 637 00:49:40,200 --> 00:49:42,520 And so there will be a factor of eta here. 638 00:49:45,950 --> 00:49:52,310 And I said that I would get the same expression 639 00:49:52,310 --> 00:49:55,510 for any of my N N minus 1 over 2 exchanges. 640 00:49:59,230 --> 00:50:05,480 So the result of all of these exchange calculations 641 00:50:05,480 --> 00:50:08,480 would be the same thing. 642 00:50:08,480 --> 00:50:10,580 And then there would be the contribution 643 00:50:10,580 --> 00:50:12,520 from three-body exchange and so forth. 644 00:50:19,460 --> 00:50:21,950 So let's re-organize this. 645 00:50:21,950 --> 00:50:27,630 I can pull out the factor of V to the N outside. 646 00:50:27,630 --> 00:50:32,620 So I would have V over lambda cubed to the power of N. 647 00:50:32,620 --> 00:50:35,710 So the first term is 1. 648 00:50:35,710 --> 00:50:38,830 The next term has the parity factor 649 00:50:38,830 --> 00:50:42,460 that distinguishes bosons and fermions, goes 650 00:50:42,460 --> 00:50:47,920 with a multiplicity of pairs which is N N minus 1 over 2. 651 00:50:47,920 --> 00:50:51,230 Since I already pulled out a factor of V to the N 652 00:50:51,230 --> 00:50:54,440 and I really had V to the N minus 1 here, 653 00:50:54,440 --> 00:51:00,210 I better put a factor of 1 over V here. 654 00:51:00,210 --> 00:51:02,420 And then I just am left with having 655 00:51:02,420 --> 00:51:06,420 to evaluate these Gaussian integrals. 656 00:51:06,420 --> 00:51:09,192 Each Gaussian integral will give me 657 00:51:09,192 --> 00:51:11,750 2 pi times the variance, which is 658 00:51:11,750 --> 00:51:16,620 lambda squared divided by 2 pi. 659 00:51:16,620 --> 00:51:21,040 And then there's actually a factor of 2. 660 00:51:21,040 --> 00:51:25,260 And there are three of them, so I will have 3/2. 661 00:51:25,260 --> 00:51:30,870 So what I get here is lambda cubed divided by 2 to the 3/2. 662 00:51:39,580 --> 00:51:43,690 Now, you can see that any time I go further 663 00:51:43,690 --> 00:51:47,240 in this series of exchanges, I will 664 00:51:47,240 --> 00:51:50,440 have more of these Gaussian factors. 665 00:51:50,440 --> 00:51:52,920 And whenever I have a Gaussian factor, 666 00:51:52,920 --> 00:51:54,990 I have an additional integration to do 667 00:51:54,990 --> 00:51:57,770 that has an x minus something squared in it. 668 00:51:57,770 --> 00:52:01,990 I will lose a factor of V. I don't have that factor of V. 669 00:52:01,990 --> 00:52:07,470 And so subsequent terms will be even smaller in powers of V. 670 00:52:07,470 --> 00:52:12,240 And presumably, compensated by corresponding factors of lambda 671 00:52:12,240 --> 00:52:13,550 squared-- lambda cubed. 672 00:52:27,610 --> 00:52:33,630 Now, first thing to note is that in the very, very high 673 00:52:33,630 --> 00:52:37,540 temperature limit, lambda goes to 0. 674 00:52:37,540 --> 00:52:40,680 So I can forget even this correction. 675 00:52:40,680 --> 00:52:42,170 What do I get? 676 00:52:42,170 --> 00:52:47,770 I get 1 over N factorial V over lambda cubed to the power of N. 677 00:52:47,770 --> 00:52:52,000 Remember that many, many lectures back we introduced 678 00:52:52,000 --> 00:52:55,610 by hand the factor of 1 over N factorial for measuring 679 00:52:55,610 --> 00:52:58,780 phase spaces of identical particles. 680 00:52:58,780 --> 00:53:01,130 And I promise to you that we would get it 681 00:53:01,130 --> 00:53:03,880 when we did identical particles in quantum mechanics, 682 00:53:03,880 --> 00:53:05,940 so here it is. 683 00:53:05,940 --> 00:53:09,390 So automatically, we did the calculation, 684 00:53:09,390 --> 00:53:12,030 keeping track of identity of particles 685 00:53:12,030 --> 00:53:14,300 at the level of quantum states. 686 00:53:14,300 --> 00:53:17,400 Went through the calculation and in the high temperature limit, 687 00:53:17,400 --> 00:53:22,410 we get this 1 over N factorial emerging. 688 00:53:22,410 --> 00:53:26,770 Secondly, we see that the corrections to ideal gas 689 00:53:26,770 --> 00:53:34,470 behavior emerge as a series in powers of lambda cubed over V. 690 00:53:34,470 --> 00:53:40,120 And for example, if I were to take the log of the partition 691 00:53:40,120 --> 00:53:45,550 function, I would get log of what 692 00:53:45,550 --> 00:53:50,250 I would have had classically, which is this V over lambda 693 00:53:50,250 --> 00:53:53,770 cubed to the power of N divided by N factorial. 694 00:53:57,110 --> 00:54:01,750 And then the log of this expression. 695 00:54:01,750 --> 00:54:05,070 And this I'm going to replace with N squared. 696 00:54:05,070 --> 00:54:07,720 There is not that much difference. 697 00:54:07,720 --> 00:54:13,630 And since I'm regarding this as a correction, log of 1 698 00:54:13,630 --> 00:54:18,620 plus something, I will replace with the something 699 00:54:18,620 --> 00:54:27,670 eta N squared 2V lambda cubed 2 to the 3/2 plus higher order. 700 00:54:30,580 --> 00:54:32,170 What does this mean? 701 00:54:32,170 --> 00:54:34,130 Once we have the partition function, 702 00:54:34,130 --> 00:54:36,740 we can calculate pressure. 703 00:54:36,740 --> 00:54:42,885 Reminding you that beta p was the log Z by dV. 704 00:54:45,615 --> 00:54:49,850 The first part is the ideal gas that we 705 00:54:49,850 --> 00:54:52,040 had looked at classically. 706 00:54:52,040 --> 00:54:57,480 So once I go to the appropriate large-end limit of this, 707 00:54:57,480 --> 00:55:03,330 what this gives me is the density n over V. 708 00:55:03,330 --> 00:55:06,150 And then when I look at the derivative here, 709 00:55:06,150 --> 00:55:11,400 the derivative of 1/V will give me a minus 1 over V squared. 710 00:55:11,400 --> 00:55:15,630 So I will get minus eta. 711 00:55:15,630 --> 00:55:18,500 N over V, the whole thing squared. 712 00:55:18,500 --> 00:55:25,180 So I will have n squared lambda cubed 2 to the 5/2, 713 00:55:25,180 --> 00:55:26,600 and so forth. 714 00:55:30,730 --> 00:55:35,600 So I see that the pressure of this ideal gas 715 00:55:35,600 --> 00:55:40,140 with no interactions is already different 716 00:55:40,140 --> 00:55:43,525 from the classical result that we had calculated 717 00:55:43,525 --> 00:55:47,800 by a factor that actually reflects the statistics. 718 00:55:47,800 --> 00:55:52,660 For fermions eta of minus 1, you get an additional pressure 719 00:55:52,660 --> 00:55:56,170 because of the kind of repulsion that we have over here. 720 00:55:56,170 --> 00:55:59,640 Whereas, for bosons you get an attraction. 721 00:55:59,640 --> 00:56:02,740 You can see that also the thing that determines this-- 722 00:56:02,740 --> 00:56:06,420 so basically, this corresponds to a second Virial 723 00:56:06,420 --> 00:56:16,830 coefficient, which is minus eta lambda cubed 2 to the 5/2, 724 00:56:16,830 --> 00:56:23,820 is the volume of these wave packets. 725 00:56:23,820 --> 00:56:27,310 So essentially, the corrections are 726 00:56:27,310 --> 00:56:32,720 of the order of n lambda cubed that 727 00:56:32,720 --> 00:56:37,750 is within one of these wave packets how many particles 728 00:56:37,750 --> 00:56:39,970 you will encounter. 729 00:56:39,970 --> 00:56:43,780 As you go to high temperature, the wave packets shrink. 730 00:56:43,780 --> 00:56:48,550 As you go to low temperature, the wave packets expand. 731 00:56:48,550 --> 00:56:52,010 If you like, the interactions become more important 732 00:56:52,010 --> 00:56:55,147 and you get corrections to ideal gas wave. 733 00:56:58,471 --> 00:57:01,393 AUDIENCE: You assume that we can use perturbation, 734 00:57:01,393 --> 00:57:05,289 but the higher terms actually had a factor [INAUDIBLE]. 735 00:57:09,185 --> 00:57:13,110 And you can't really use perturbation in that. 736 00:57:13,110 --> 00:57:14,570 PROFESSOR: OK. 737 00:57:14,570 --> 00:57:19,370 So what you are worried about is the story here, 738 00:57:19,370 --> 00:57:27,410 that I took log of 1 plus something here 739 00:57:27,410 --> 00:57:31,700 and I'm interested in the limit of n going to infinity, 740 00:57:31,700 --> 00:57:36,530 that finite density n over V. So already in that limit, 741 00:57:36,530 --> 00:57:38,800 you would say that this factor really 742 00:57:38,800 --> 00:57:42,080 is overwhelmingly larger than that. 743 00:57:42,080 --> 00:57:46,220 And as you say, the next factor will be even larger. 744 00:57:46,220 --> 00:57:50,490 So what is the justification in all of this? 745 00:57:50,490 --> 00:57:54,990 We have already encountered this same problem 746 00:57:54,990 --> 00:57:57,850 when we were doing these perturbations 747 00:57:57,850 --> 00:57:59,710 due to interactions. 748 00:57:59,710 --> 00:58:08,932 And the answer is that what you really want to ensure 749 00:58:08,932 --> 00:58:20,710 is that not log Z, but Z has a form 750 00:58:20,710 --> 00:58:23,245 that is e to the N something. 751 00:58:25,985 --> 00:58:28,375 And that something will have corrections, 752 00:58:28,375 --> 00:58:34,330 potentially that are powers of N, the density, which 753 00:58:34,330 --> 00:58:39,490 is N over V. And if you try to force it into a perturbation 754 00:58:39,490 --> 00:58:44,830 series such as this, naturally things like this happen. 755 00:58:44,830 --> 00:58:46,820 What does that really mean? 756 00:58:46,820 --> 00:58:51,630 That really means that the correct thing that you should 757 00:58:51,630 --> 00:58:55,860 be expanding is, indeed, log Z. If you 758 00:58:55,860 --> 00:58:59,350 were to do the kind of hand-waving that I did here 759 00:58:59,350 --> 00:59:03,950 and do the expansion for Z, if you also try to do it over here 760 00:59:03,950 --> 00:59:06,360 you will generate terms that look 761 00:59:06,360 --> 00:59:08,780 kind of at the wrong order. 762 00:59:08,780 --> 00:59:12,480 But higher order terms that you would get 763 00:59:12,480 --> 00:59:17,120 would naturally conspire so that when you evaluate log Z, 764 00:59:17,120 --> 00:59:19,870 they come out right. 765 00:59:19,870 --> 00:59:22,420 You have to do this correctly. 766 00:59:22,420 --> 00:59:24,120 And once you have done it correctly, 767 00:59:24,120 --> 00:59:26,120 then you can rely on the calculation 768 00:59:26,120 --> 00:59:28,430 that you did before as an example. 769 00:59:28,430 --> 00:59:33,300 And we did it correctly when we were doing these cluster 770 00:59:33,300 --> 00:59:38,440 expansions and the corresponding calculation we 771 00:59:38,440 --> 00:59:41,500 did for Q. We saw how the different diagrams were 772 00:59:41,500 --> 00:59:44,400 appearing in both Q and the log Q, 773 00:59:44,400 --> 00:59:47,910 and how they could be summed over in log Q. 774 00:59:47,910 --> 00:59:51,485 But indeed, this mathematically looks awkward 775 00:59:51,485 --> 00:59:55,250 and I kind of jumped a step in writing log of 1 776 00:59:55,250 --> 01:00:00,492 plus something that is huge as if it was a small number. 777 01:00:14,550 --> 01:00:17,300 All right. 778 01:00:17,300 --> 01:00:20,870 So we have a problem. 779 01:00:20,870 --> 01:00:23,410 We want to calculate the simplest 780 01:00:23,410 --> 01:00:26,940 system, which is the ideal gas. 781 01:00:26,940 --> 01:00:30,680 So classically, we did all of our calculations 782 01:00:30,680 --> 01:00:32,210 first for the ideal gas. 783 01:00:32,210 --> 01:00:34,030 We had exact results. 784 01:00:34,030 --> 01:00:36,220 Then, let's say we had interactions. 785 01:00:36,220 --> 01:00:39,100 We did perturbations around that and all of that. 786 01:00:39,100 --> 01:00:43,690 And we saw that having to do things for interacting systems 787 01:00:43,690 --> 01:00:46,540 is very difficult. 788 01:00:46,540 --> 01:00:49,960 Now, when we start to do calculations for the quantum 789 01:00:49,960 --> 01:00:53,810 problem, at least in the way that I set it up for you, 790 01:00:53,810 --> 01:00:57,250 it seems that quantum problems are inherently 791 01:00:57,250 --> 01:01:00,640 interacting problems. 792 01:01:00,640 --> 01:01:03,850 I showed you that even at the level of two particles, 793 01:01:03,850 --> 01:01:08,820 it is like having an interaction between bosons and fermions. 794 01:01:08,820 --> 01:01:11,070 For three particles, it becomes even worse 795 01:01:11,070 --> 01:01:14,890 because it's not only the two-particle interaction. 796 01:01:14,890 --> 01:01:16,920 Because of the three-particle exchanges, 797 01:01:16,920 --> 01:01:19,720 you would get an additional three-particle interaction, 798 01:01:19,720 --> 01:01:24,020 four-particle interaction, all of these things emerge. 799 01:01:24,020 --> 01:01:27,550 So really, if you want to look at this 800 01:01:27,550 --> 01:01:32,870 from the perspective of a partition function, 801 01:01:32,870 --> 01:01:36,520 we already see that the exchange term 802 01:01:36,520 --> 01:01:38,800 involved having to do a calculation that 803 01:01:38,800 --> 01:01:42,500 is equivalent to calculating the second Virial 804 01:01:42,500 --> 01:01:46,780 coefficient for an interacting system. 805 01:01:46,780 --> 01:01:49,570 The next one, for the third Virial coefficient, 806 01:01:49,570 --> 01:01:52,550 I would need to look at the three-body exchanges, 807 01:01:52,550 --> 01:01:56,570 kind of like the point clusters, four-point clusters, 808 01:01:56,570 --> 01:01:59,510 all kinds of other things are there. 809 01:01:59,510 --> 01:02:00,410 So is there any hope? 810 01:02:05,150 --> 01:02:11,300 And the answer is that it is all a matter of perspective. 811 01:02:11,300 --> 01:02:22,560 And somehow it is true that these particles 812 01:02:22,560 --> 01:02:26,890 in quantum mechanics because of the statistics 813 01:02:26,890 --> 01:02:31,310 are subject to all kinds of complicated interactions. 814 01:02:31,310 --> 01:02:34,250 But also, the underlying Hamiltonian 815 01:02:34,250 --> 01:02:37,000 is simple and non-interacting. 816 01:02:37,000 --> 01:02:39,870 We can enumerate all of the wave functions. 817 01:02:39,870 --> 01:02:41,460 Everything is simple. 818 01:02:41,460 --> 01:02:44,770 So by looking at things in the right basis, 819 01:02:44,770 --> 01:02:49,470 we should be able to calculate everything that we need. 820 01:02:49,470 --> 01:02:55,290 So here, I was kind of looking at calculating the partition 821 01:02:55,290 --> 01:03:00,590 function in the coordinate basis, which is the worst case 822 01:03:00,590 --> 01:03:06,390 scenario because the Hamiltonian is diagonal in the momentum 823 01:03:06,390 --> 01:03:07,860 basis. 824 01:03:07,860 --> 01:03:16,690 So let's calculate ZN trace of e to the minus beta H 825 01:03:16,690 --> 01:03:19,630 in the basis in which H is diagonal. 826 01:03:23,860 --> 01:03:28,240 So what are the eigenvalues and eigenfunctions? 827 01:03:28,240 --> 01:03:32,890 Well, the eigenfunctions are the symmetrized/anti-symmetrized 828 01:03:32,890 --> 01:03:34,000 quantities. 829 01:03:34,000 --> 01:03:37,170 The eigenvalues are simply e to the minus beta H 830 01:03:37,170 --> 01:03:40,890 bar squared k alpha squared over 2m. 831 01:03:40,890 --> 01:03:43,820 So this is basically the thing that I 832 01:03:43,820 --> 01:03:47,690 could write as the set of k's appropriately 833 01:03:47,690 --> 01:03:52,370 symmetrized or anti-symmetrized e to the minus beta 834 01:03:52,370 --> 01:04:00,060 sum over alpha H bar squared k alpha squared over 2m k eta. 835 01:04:06,230 --> 01:04:11,710 Actually, I'm going to-- rather than go through this procedure 836 01:04:11,710 --> 01:04:15,530 that we have up there in which I wrote 837 01:04:15,530 --> 01:04:22,860 these, what I need to do here is a sum over all k 838 01:04:22,860 --> 01:04:25,070 in order to evaluate the trace. 839 01:04:25,070 --> 01:04:31,200 So this is inherently a sum over all sets of k's. 840 01:04:31,200 --> 01:04:35,180 But this sum is restricted, just like what 841 01:04:35,180 --> 01:04:38,790 I had indicated for you before. 842 01:04:38,790 --> 01:04:42,190 Rather than trying to do it that way, 843 01:04:42,190 --> 01:04:48,180 I note that these k's I could also 844 01:04:48,180 --> 01:04:52,860 write in terms of these occupation numbers. 845 01:04:52,860 --> 01:04:57,200 So equivalently, my basis would be 846 01:04:57,200 --> 01:05:04,370 the set of occupation numbers times the energy. 847 01:05:04,370 --> 01:05:13,400 The energy is then e to the minus beta sum over k epsilon k 848 01:05:13,400 --> 01:05:20,830 nk, where epsilon k is this beta H 849 01:05:20,830 --> 01:05:23,650 bar squared k alpha squared over 2m. 850 01:05:23,650 --> 01:05:28,760 But I could do this in principle for any epsilon k 851 01:05:28,760 --> 01:05:30,800 that I have over here. 852 01:05:30,800 --> 01:05:35,230 So the result that I am writing for you is more general. 853 01:05:35,230 --> 01:05:37,120 Then I sandwich it again, since I'm 854 01:05:37,120 --> 01:05:40,365 calculating the trace, with the same state. 855 01:05:43,360 --> 01:05:46,900 Now, the states have this restriction 856 01:05:46,900 --> 01:05:47,970 that I have over there. 857 01:05:47,970 --> 01:05:54,070 That is, for the case of fermions, my nk can be 0 or 1. 858 01:05:54,070 --> 01:05:58,440 But there is no restriction for nk on the bosons. 859 01:05:58,440 --> 01:06:03,360 Except, of course, that there is this overall restriction 860 01:06:03,360 --> 01:06:11,670 that the sum over k nk has to be N 861 01:06:11,670 --> 01:06:14,865 because I am looking at N-particle states. 862 01:06:22,810 --> 01:06:29,610 Actually, I can remove this because in this basis, 863 01:06:29,610 --> 01:06:32,560 e to the minus beta H is diagonal. 864 01:06:32,560 --> 01:06:39,330 So I can, basically, remove these entities. 865 01:06:39,330 --> 01:06:41,715 And I'm just summing a bunch of exponentials. 866 01:06:47,860 --> 01:06:55,160 So that is good because I should be able to do for each nk 867 01:06:55,160 --> 01:06:57,580 a sum of e to something nk. 868 01:07:00,410 --> 01:07:03,150 Well, the problem is this that I can't sum over 869 01:07:03,150 --> 01:07:04,553 each nk independently. 870 01:07:08,990 --> 01:07:14,660 Essentially in the picture that I have over here, 871 01:07:14,660 --> 01:07:16,730 I have some n1 here. 872 01:07:16,730 --> 01:07:18,870 I have some n2 here. 873 01:07:18,870 --> 01:07:21,660 Some n3 here, which are the occupation numbers 874 01:07:21,660 --> 01:07:24,260 of these things. 875 01:07:24,260 --> 01:07:26,290 And for that partition function, I 876 01:07:26,290 --> 01:07:29,760 have to do the sum of these exponentials e 877 01:07:29,760 --> 01:07:33,760 to the minus epsilon 1 n1, e to the minus epsilon 2 n2. 878 01:07:33,760 --> 01:07:40,140 But the sum of all of these n's is kind of maxed out by N. 879 01:07:40,140 --> 01:07:46,092 I cannot independently sum over them going over the entire 880 01:07:46,092 --> 01:07:47,740 range. 881 01:07:47,740 --> 01:07:52,745 But we've seen previously how those constraints can 882 01:07:52,745 --> 01:07:55,400 be removed in statistical mechanics. 883 01:07:55,400 --> 01:07:56,790 So our usual trick. 884 01:07:56,790 --> 01:08:01,910 We go to the ensemble in which n can take any value. 885 01:08:01,910 --> 01:08:06,040 So we go to the grand canonical prescription. 886 01:08:12,880 --> 01:08:19,479 We remove this constraint on n by evaluating a grand partition 887 01:08:19,479 --> 01:08:30,800 function Q, which is a sum over all N of e to the beta mu N ZN. 888 01:08:30,800 --> 01:08:34,240 So we do, essentially, a Laplace transform. 889 01:08:34,240 --> 01:08:38,300 We exchange our n with the chemical potential mu. 890 01:08:41,850 --> 01:08:47,620 Then, this constraint no longer we need to worry about. 891 01:08:47,620 --> 01:08:55,390 So now I can sum over all of the nk's without worrying 892 01:08:55,390 --> 01:09:00,140 about any constraint, provided that I multiply with e 893 01:09:00,140 --> 01:09:03,560 to the beta mu n, which is a sum over k nk. 894 01:09:07,770 --> 01:09:10,660 And then, the factor that I have here, 895 01:09:10,660 --> 01:09:15,771 which is e to the minus beta sum over k epsilon of k nk. 896 01:09:22,180 --> 01:09:28,319 So essentially for each k, I can independently 897 01:09:28,319 --> 01:09:38,359 sum over its nk of e to the beta mu minus epsilon of k nk. 898 01:09:45,189 --> 01:09:49,069 Now, the symmetry issues remain. 899 01:09:49,069 --> 01:09:53,180 This answer still depends on whether or not 900 01:09:53,180 --> 01:09:57,870 I am calculating things for bosons or fermions 901 01:09:57,870 --> 01:10:03,566 because these sums are differently constrained 902 01:10:03,566 --> 01:10:06,200 whether I'm dealing with fermions. 903 01:10:06,200 --> 01:10:09,420 In which case, nk is only 0 or 1. 904 01:10:09,420 --> 01:10:12,075 Or bosons, in which case there is no constraint. 905 01:10:17,190 --> 01:10:20,020 So what do I get? 906 01:10:20,020 --> 01:10:28,540 For the case of fermions, I have a Q minus, 907 01:10:28,540 --> 01:10:33,020 which is product over all k. 908 01:10:33,020 --> 01:10:39,580 And for each k, the nk takes either 0 or 1. 909 01:10:39,580 --> 01:10:45,500 So if it takes 0, I will write e to the 0, which is 1. 910 01:10:45,500 --> 01:10:46,465 Or it takes 1. 911 01:10:46,465 --> 01:10:50,562 It is e to the beta mu minus epsilon of k. 912 01:10:57,110 --> 01:11:04,640 For the case of bosons, I have a Q plus. 913 01:11:04,640 --> 01:11:10,370 Q plus is, again, a product over Q. In this case, 914 01:11:10,370 --> 01:11:15,240 nk going from 0 to infinity, I am summing a geometric series 915 01:11:15,240 --> 01:11:20,320 that starts as 1, and then the subsequent terms 916 01:11:20,320 --> 01:11:26,950 are smaller by a factor of beta mu minus epsilon of k. 917 01:11:26,950 --> 01:11:31,260 Actually, for future reference note 918 01:11:31,260 --> 01:11:35,350 that I would be able to do this geometric sum 919 01:11:35,350 --> 01:11:39,830 provided that this combination beta mu minus epsilon of k 920 01:11:39,830 --> 01:11:41,130 is negative. 921 01:11:41,130 --> 01:11:44,570 So that the subsequent terms in this series are decaying. 922 01:11:51,890 --> 01:11:54,680 Typically, we would be interested in things 923 01:11:54,680 --> 01:11:58,660 like partition functions, grand partition functions. 924 01:11:58,660 --> 01:12:05,620 So we have something like log of Q, which would be a sum over k. 925 01:12:09,760 --> 01:12:15,600 And I would have either the log of this quantity 926 01:12:15,600 --> 01:12:20,330 or the log of this quantity with a minus sign. 927 01:12:20,330 --> 01:12:23,180 I can combine the two results together 928 01:12:23,180 --> 01:12:28,910 by putting a factor of minus eta because in taking the log, 929 01:12:28,910 --> 01:12:32,794 over here for the bosons I would pick a factor of minus 1 930 01:12:32,794 --> 01:12:34,460 because the thing is in the denominator. 931 01:12:37,100 --> 01:12:42,310 And then I would write the log of 1. 932 01:12:42,310 --> 01:12:45,030 And then I have in both cases, a factor which 933 01:12:45,030 --> 01:12:49,810 is e to the beta mu minus epsilon of k. 934 01:12:49,810 --> 01:12:52,050 But occurring with different signs 935 01:12:52,050 --> 01:12:54,850 for the bosons and fermions, which again I 936 01:12:54,850 --> 01:12:58,890 can combine into a single expression by putting a minus 937 01:12:58,890 --> 01:13:01,920 eta here. 938 01:13:01,920 --> 01:13:10,890 So this is a general result for any Hamiltonian 939 01:13:10,890 --> 01:13:14,210 that has the characteristic that we wrote over here. 940 01:13:14,210 --> 01:13:16,960 So this does not have to be particles in a box. 941 01:13:16,960 --> 01:13:20,350 It could be particles in a harmonic oscillator. 942 01:13:20,350 --> 01:13:23,750 These could be energy levels of a harmonic oscillator. 943 01:13:23,750 --> 01:13:27,470 All you need to do is to make the appropriate sum 944 01:13:27,470 --> 01:13:30,520 over the one-particle levels harmonic oscillator, 945 01:13:30,520 --> 01:13:34,010 or whatever else you have, of these factors that 946 01:13:34,010 --> 01:13:37,650 depend on the individual energy levels 947 01:13:37,650 --> 01:13:39,120 of the one-particle system. 948 01:13:50,720 --> 01:13:52,790 Now, one of the things that we will encounter 949 01:13:52,790 --> 01:13:56,870 having made this transition from canonical, where we knew 950 01:13:56,870 --> 01:14:01,070 how many particles we had, to grand canonical, where we only 951 01:14:01,070 --> 01:14:04,510 know the chemical potential, is that we would ultimately 952 01:14:04,510 --> 01:14:09,360 want to express things in terms of the number of particles. 953 01:14:09,360 --> 01:14:16,070 So it makes sense to calculate how many particles 954 01:14:16,070 --> 01:14:21,060 you have given that you have fixed the chemical potential. 955 01:14:21,060 --> 01:14:42,985 So for that we note the following. 956 01:14:45,560 --> 01:14:55,410 That essentially, we were able to do this calculation for Q 957 01:14:55,410 --> 01:14:58,580 because it was a product of contributions 958 01:14:58,580 --> 01:15:04,560 that we had for the individual one-particle states. 959 01:15:04,560 --> 01:15:10,510 So clearly, as far as this normalization is concerned, 960 01:15:10,510 --> 01:15:15,680 the individual one-particle states are independent. 961 01:15:15,680 --> 01:15:20,752 And indeed, what we can say is that in this ensemble, 962 01:15:20,752 --> 01:15:27,610 there is a classical probability for a set of occupation numbers 963 01:15:27,610 --> 01:15:31,550 of one particle states, which is simply 964 01:15:31,550 --> 01:15:37,960 a product over the different one-particle states of e 965 01:15:37,960 --> 01:15:49,830 to the beta mu minus epsilon k nk appropriately normalized. 966 01:15:49,830 --> 01:15:54,010 And again, the restriction on n's 967 01:15:54,010 --> 01:15:57,200 being 0 or 1 for fermions or anything 968 01:15:57,200 --> 01:16:01,100 for bosons would be implicit in either case. 969 01:16:01,100 --> 01:16:05,830 But in either case, essentially the occupation numbers 970 01:16:05,830 --> 01:16:08,690 are independently taken from distributions 971 01:16:08,690 --> 01:16:11,810 that I've discussed [INAUDIBLE]. 972 01:16:11,810 --> 01:16:13,860 So you can, in fact, independently 973 01:16:13,860 --> 01:16:17,350 calculate the average occupation number 974 01:16:17,350 --> 01:16:22,460 that you have for each one of these single-particle states. 975 01:16:22,460 --> 01:16:29,790 And it's clear that you could get that by, for example, 976 01:16:29,790 --> 01:16:34,150 bringing down a factor of nk here. 977 01:16:34,150 --> 01:16:37,190 And you can bring down a factor of nk 978 01:16:37,190 --> 01:16:44,170 by taking a derivative of Q with respect to beta epsilon k 979 01:16:44,170 --> 01:16:48,420 with a minus sign and normalizing it, 980 01:16:48,420 --> 01:16:50,632 so you would have log. 981 01:16:50,632 --> 01:16:53,350 So you would have an expression such as this. 982 01:16:57,250 --> 01:16:59,890 So you basically would need to calculate, 983 01:16:59,890 --> 01:17:01,960 since you are taking derivative with respect 984 01:17:01,960 --> 01:17:08,740 to epsilon k the corresponding log for which epsilon 985 01:17:08,740 --> 01:17:10,500 k appears. 986 01:17:10,500 --> 01:17:12,890 Actually, for the case of fermions, 987 01:17:12,890 --> 01:17:14,890 really there are two possibilities. 988 01:17:14,890 --> 01:17:18,360 n is either 0 or 1. 989 01:17:18,360 --> 01:17:21,260 So you would say that the expectation value would 990 01:17:21,260 --> 01:17:27,440 be when it is 1, you have e to the beta epsilon of k minus mu. 991 01:17:27,440 --> 01:17:28,370 Oops. 992 01:17:28,370 --> 01:17:31,550 e to the beta mu minus epsilon of k. 993 01:17:34,330 --> 01:17:36,910 The two possibilities are 1 plus e 994 01:17:36,910 --> 01:17:40,450 to the beta mu minus epsilon of k. 995 01:17:40,450 --> 01:17:44,630 So when I look at some particular state, 996 01:17:44,630 --> 01:17:46,030 it is either empty. 997 01:17:46,030 --> 01:17:48,830 In which case, contributes 0. 998 01:17:48,830 --> 01:17:51,140 Or, it is occupied. 999 01:17:51,140 --> 01:17:53,362 In which case, it contributes this weight, 1000 01:17:53,362 --> 01:17:55,070 which has to be appropriately normalized. 1001 01:17:57,650 --> 01:18:01,460 If I do the same thing for the case of bosons, 1002 01:18:01,460 --> 01:18:04,400 it is a bit more complicated because I 1003 01:18:04,400 --> 01:18:07,490 have to look at this series rather than geometric 1 1004 01:18:07,490 --> 01:18:10,440 plus x plus x squared plus x cubed is 1005 01:18:10,440 --> 01:18:14,970 1 plus x plus 2x squared plus 3x cubed, which can be obtained 1006 01:18:14,970 --> 01:18:19,430 by taking the derivative of the appropriate log. 1007 01:18:19,430 --> 01:18:22,750 Or you can fall back on your calculations 1008 01:18:22,750 --> 01:18:27,160 of geometric series and convince yourself 1009 01:18:27,160 --> 01:18:29,880 that it is essentially the same thing with a factor of minus 1010 01:18:29,880 --> 01:18:30,380 here. 1011 01:18:34,850 --> 01:18:40,798 So this is fermions and this is bosons. 1012 01:18:45,480 --> 01:18:52,000 And indeed, I can put the two expressions together 1013 01:18:52,000 --> 01:18:57,390 by dividing through this factor in both of them 1014 01:18:57,390 --> 01:19:03,380 and write it as 1 over Z inverse e 1015 01:19:03,380 --> 01:19:08,700 to the beta epsilon of k minus eta, 1016 01:19:08,700 --> 01:19:11,560 where for convenience I have introduced 1017 01:19:11,560 --> 01:19:16,117 Z to be the contribution e to the beta. 1018 01:19:30,100 --> 01:19:36,530 So for this system of non-interacting particles 1019 01:19:36,530 --> 01:19:40,090 that are identical, we have expressions 1020 01:19:40,090 --> 01:19:45,890 for log of the grand partition function, the grand potential. 1021 01:19:45,890 --> 01:19:48,220 And for the average number of particles, 1022 01:19:48,220 --> 01:19:51,260 which is an appropriate derivative of this, 1023 01:19:51,260 --> 01:19:54,410 expressed in terms of the single-particle energy 1024 01:19:54,410 --> 01:19:58,440 levels and the chemical potential. 1025 01:19:58,440 --> 01:20:01,710 So next time, what we will do is we 1026 01:20:01,710 --> 01:20:03,880 will start this with this expression 1027 01:20:03,880 --> 01:20:06,576 for the case of the particles in a box 1028 01:20:06,576 --> 01:20:11,350 to get the pressure of the ideal quantum gas 1029 01:20:11,350 --> 01:20:13,510 as a function of mu. 1030 01:20:13,510 --> 01:20:15,090 But we want to write the pressure 1031 01:20:15,090 --> 01:20:19,260 as a function of density, so we will invert this expression 1032 01:20:19,260 --> 01:20:21,610 to get density as a function of-- chemical 1033 01:20:21,610 --> 01:20:24,620 potential as a function of density [INAUDIBLE] here. 1034 01:20:24,620 --> 01:20:27,580 And therefore, get the expression for pressure 1035 01:20:27,580 --> 01:20:30,620 as a function of density.