WEBVTT 00:00:00.060 --> 00:00:01.780 The following content is provided 00:00:01.780 --> 00:00:04.030 under a Creative Commons license. 00:00:04.030 --> 00:00:06.870 Your support will help MIT OpenCourseWare continue 00:00:06.870 --> 00:00:10.730 to offer high-quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation, or view additional materials 00:00:13.330 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:21.160 --> 00:00:23.250 PROFESSOR: Particles in quantum mechanics. 00:00:23.250 --> 00:00:27.470 In particular, the ones that are identical and non-interacting. 00:00:37.140 --> 00:00:42.430 So basically, we were focusing on a type of Hamiltonian 00:00:42.430 --> 00:00:45.780 for a system of N particles, which 00:00:45.780 --> 00:00:53.190 could be written as the sum of contributions that correspond 00:00:53.190 --> 00:00:57.460 respectively to particle 1, particle 2, particle N. 00:00:57.460 --> 00:01:03.090 So essentially, a sum of terms that are all the same. 00:01:03.090 --> 00:01:06.960 And one-particle terms is sufficient 00:01:06.960 --> 00:01:10.290 because we don't have interactions. 00:01:10.290 --> 00:01:15.750 So if we look at one of these H's-- so one of these 00:01:15.750 --> 00:01:19.673 one-particle Hamiltonians, we said that we could find some 00:01:19.673 --> 00:01:21.950 kind of basis for it. 00:01:21.950 --> 00:01:25.270 In particular, typically we were interested in particles 00:01:25.270 --> 00:01:26.140 in the box. 00:01:26.140 --> 00:01:31.430 We would label them with some wave number k. 00:01:31.430 --> 00:01:37.130 And there was an associated one-particle energy, 00:01:37.130 --> 00:01:39.900 which for the case of one-particle in box 00:01:39.900 --> 00:01:44.270 was h bar squared k squared over 2m. 00:01:44.270 --> 00:01:48.770 But in general, for the one-particle system, 00:01:48.770 --> 00:01:54.330 we can think of a ladder of possible values of energies. 00:01:54.330 --> 00:02:00.815 So there will be some k1, k2, k3, et cetera. 00:02:05.590 --> 00:02:08.830 They may be distributed in any particular way corresponding 00:02:08.830 --> 00:02:10.389 to different energies. 00:02:10.389 --> 00:02:12.760 Basically, you would have a number 00:02:12.760 --> 00:02:15.600 of possible states for one particle. 00:02:15.600 --> 00:02:27.730 So for the case of the particle in the box, the wave functions 00:02:27.730 --> 00:02:32.335 and coordinate space that we had x, k were of the form e 00:02:32.335 --> 00:02:37.020 to the i k dot x divided by square root of V. 00:02:37.020 --> 00:02:45.220 We allowed the energies where h bar squared k squared over 2m. 00:02:45.220 --> 00:02:50.520 And this discretization was because the values of k 00:02:50.520 --> 00:02:56.165 were multiples of 2 pi over l with integers in the three 00:02:56.165 --> 00:02:57.380 different directions. 00:02:57.380 --> 00:02:59.550 Assuming periodic boundary conditions 00:02:59.550 --> 00:03:03.670 or appropriate discretization for closed boundary conditions, 00:03:03.670 --> 00:03:07.010 or whatever you have. 00:03:07.010 --> 00:03:11.290 So that's the one-particle state. 00:03:11.290 --> 00:03:15.350 If the Hamiltonian is of this form, 00:03:15.350 --> 00:03:20.620 it is clear that we can multiply a bunch of these states 00:03:20.620 --> 00:03:25.800 and form another eigenstate for HN. 00:03:25.800 --> 00:03:27.805 Those we were calling product states. 00:03:32.945 --> 00:03:38.180 So you basically pick a bunch of these k's and you 00:03:38.180 --> 00:03:39.280 multiply them. 00:03:39.280 --> 00:03:43.685 So you have k1, k2, kN. 00:03:46.410 --> 00:03:49.050 And essentially, that would correspond, 00:03:49.050 --> 00:03:52.460 let's say, in coordinate representation 00:03:52.460 --> 00:03:57.910 to taking a bunch of x's and the corresponding k's and having 00:03:57.910 --> 00:04:03.500 a wave function of the form e to the i k alp-- sum over alpha k 00:04:03.500 --> 00:04:09.290 alpha x alpha, and then divided by V to the N over 2. 00:04:13.210 --> 00:04:16.779 So in this procedure, what did we do? 00:04:16.779 --> 00:04:22.340 We had a number of possibilities for the one-particle state. 00:04:22.340 --> 00:04:26.030 And in order to, let's say, make a two-particle state, 00:04:26.030 --> 00:04:31.450 we would pick two of these k's and multiply 00:04:31.450 --> 00:04:33.900 the corresponding wave functions. 00:04:33.900 --> 00:04:37.060 If you had three particles, we could pick another one. 00:04:37.060 --> 00:04:39.710 If you had four particles, we could potentially 00:04:39.710 --> 00:04:43.050 pick a second one twice, et cetera. 00:04:43.050 --> 00:04:44.890 So in general, basically we would 00:04:44.890 --> 00:04:48.900 put N of these crosses on these one-particle states 00:04:48.900 --> 00:04:51.570 that we've selected. 00:04:51.570 --> 00:04:54.690 Problem was that this was not allowed 00:04:54.690 --> 00:04:58.730 by quantum mechanics for identical particles. 00:04:58.730 --> 00:05:01.960 Because if we took one of these wave functions 00:05:01.960 --> 00:05:05.800 and exchanged two of the labels, x1 and x2, 00:05:05.800 --> 00:05:10.280 we could potentially get a different wave function. 00:05:10.280 --> 00:05:14.650 And in quantum mechanics, we said that the wave function 00:05:14.650 --> 00:05:18.850 has to be either symmetric or anti-symmetric with respect 00:05:18.850 --> 00:05:21.940 to exchange of a pair of particles. 00:05:21.940 --> 00:05:27.040 And also, whatever it implied for repeating this exchange 00:05:27.040 --> 00:05:31.340 many times to look at all possible permutations. 00:05:31.340 --> 00:05:36.770 So what we saw was that product states are good 00:05:36.770 --> 00:05:38.467 as long as you are thinking about 00:05:38.467 --> 00:05:39.550 distinguishable particles. 00:05:44.950 --> 00:05:53.710 But if you have identical particles, 00:05:53.710 --> 00:05:57.810 you had to appropriately symmetrize or anti-symmetrize 00:05:57.810 --> 00:06:00.060 these states. 00:06:00.060 --> 00:06:05.320 So what we ended up was a wabe of, for example, 00:06:05.320 --> 00:06:09.250 symmetrizing things for the case of fermions. 00:06:09.250 --> 00:06:12.400 So we could take a bunch of these k-values again. 00:06:15.360 --> 00:06:21.020 And a fermionic, or anti-symmetrized version, 00:06:21.020 --> 00:06:25.980 was then constructed by summing over all permutations. 00:06:25.980 --> 00:06:31.110 And for N particle, there would be N factorial permutations. 00:06:31.110 --> 00:06:34.320 Basically, doing a permutation of all of these 00:06:34.320 --> 00:06:40.200 indices k1, k2, et cetera, that we had selected for this. 00:06:40.200 --> 00:06:43.670 And for the case of fermions, we had 00:06:43.670 --> 00:06:49.720 to multiply each permutation with a sign that 00:06:49.720 --> 00:06:56.130 was plus for even permutations, minus for odd permutations. 00:06:56.130 --> 00:06:59.970 And this would give us N factorial terms. 00:06:59.970 --> 00:07:03.780 And the appropriate normalization 00:07:03.780 --> 00:07:08.490 was 1 over square root of N factorial. 00:07:08.490 --> 00:07:12.030 So this was for the case of fermions. 00:07:12.030 --> 00:07:13.445 And this was actually minus. 00:07:13.445 --> 00:07:15.410 I should have put in a minus here. 00:07:15.410 --> 00:07:21.220 And this would have been minus to the P. 00:07:21.220 --> 00:07:29.530 For the case of bosons, we basically 00:07:29.530 --> 00:07:33.360 dispensed with this factor of minus 1 to the P. 00:07:33.360 --> 00:07:35.145 So we had a Pk. 00:07:39.400 --> 00:07:44.480 Now, the corresponding normalization that we had here 00:07:44.480 --> 00:07:46.420 was slightly different. 00:07:46.420 --> 00:07:50.020 The point was that if we were doing 00:07:50.020 --> 00:07:55.570 this computation for the case of fermions, 00:07:55.570 --> 00:07:58.165 we could not allow a state where there 00:07:58.165 --> 00:08:02.030 is a double occupation of one of the one-particle state. 00:08:02.030 --> 00:08:06.060 Under exchange of the particles that 00:08:06.060 --> 00:08:08.470 would correspond to these k-values, 00:08:08.470 --> 00:08:11.600 I would get the same state back, but the exchange would give me 00:08:11.600 --> 00:08:14.250 a minus 1 and it would give me 0. 00:08:14.250 --> 00:08:17.370 So the fermionic wave function that I have constructed 00:08:17.370 --> 00:08:19.840 here, appropriately anti-symmetrized, 00:08:19.840 --> 00:08:24.060 exists only as long as there are no repeats. 00:08:24.060 --> 00:08:29.680 Whereas, for the case of bosons, I could put 2 over same place. 00:08:29.680 --> 00:08:34.820 I could put 3 somewhere else, any number that I liked, 00:08:34.820 --> 00:08:38.090 and there would be no problem with this. 00:08:38.090 --> 00:08:41.780 Except that the normalization would be more complicated. 00:08:41.780 --> 00:08:44.320 And we saw that appropriate normalization 00:08:44.320 --> 00:08:47.135 was a product over k nk factorial. 00:08:50.580 --> 00:08:55.380 So this was for fermions. 00:08:57.970 --> 00:08:58.820 This is for bosons. 00:09:03.380 --> 00:09:09.110 And the two I can submerge into one formula 00:09:09.110 --> 00:09:16.890 by writing a symmetrized or anti-symmetrized state, 00:09:16.890 --> 00:09:22.550 respectively indicated by eta, where we have eta is minus 1 00:09:22.550 --> 00:09:27.750 for fermions and eta is plus 1 for bosons, 00:09:27.750 --> 00:09:33.510 which is 1 over square root of N factorial product 00:09:33.510 --> 00:09:37.100 over k nk factorials. 00:09:37.100 --> 00:09:42.360 And then the sum over all N factorial permutations. 00:09:42.360 --> 00:09:49.360 This phase factor for fermions and nothing for bosons 00:09:49.360 --> 00:09:52.120 of the appropriately permuted set of k's. 00:09:57.630 --> 00:10:04.920 And in this way of noting things, 00:10:04.920 --> 00:10:13.840 I have to assign values nk which are either 0 or 1 for fermions. 00:10:13.840 --> 00:10:19.370 Because as we said, multiple occupations are not allowed. 00:10:19.370 --> 00:10:21.500 But there is no restriction for bosons. 00:10:28.040 --> 00:10:33.830 Except of course, that in this perspective, 00:10:33.830 --> 00:10:40.607 as I go along this k-axis, I have 0, 1, 0, 2, 1, 3, 0, 00:10:40.607 --> 00:10:44.910 0 for the occupations. 00:10:44.910 --> 00:10:47.060 Of course, what I need to construct 00:10:47.060 --> 00:10:50.040 whether I am dealing with bosons or fermions 00:10:50.040 --> 00:10:54.660 is that the sum over k nk is the total number of particles 00:10:54.660 --> 00:10:55.490 that I have. 00:11:01.800 --> 00:11:08.430 Now, the other thing to note is that once I have given you 00:11:08.430 --> 00:11:13.700 a picture such as this in terms of which one-particle states 00:11:13.700 --> 00:11:19.360 I want to look at, or which set of occupation numbers 00:11:19.360 --> 00:11:23.300 I have nk, then there is one and only 00:11:23.300 --> 00:11:27.400 one symmetrized or anti-symmetrized state. 00:11:27.400 --> 00:11:30.730 So over here, I could have permuted 00:11:30.730 --> 00:11:34.770 the k's in a number of possible ways. 00:11:34.770 --> 00:11:36.740 But as a result of symmetrization, 00:11:36.740 --> 00:11:39.940 anti-symmetrization, various ways 00:11:39.940 --> 00:11:44.200 of permuting the labels here ultimately 00:11:44.200 --> 00:11:48.210 come to the same set of occupation numbers. 00:11:48.210 --> 00:11:54.900 So it is possible to actually label the state rather than 00:11:54.900 --> 00:11:57.530 by the set of k's. 00:11:57.530 --> 00:11:59.600 By the set of nk's. 00:11:59.600 --> 00:12:04.085 It is kind of a more appropriate way of representing the system. 00:12:16.850 --> 00:12:21.770 So that's essentially the kinds of states 00:12:21.770 --> 00:12:24.320 that we are going to be using. 00:12:24.320 --> 00:12:27.200 Again, in talking about identical particles, 00:12:27.200 --> 00:12:31.650 which could be either bosons or fermions. 00:12:31.650 --> 00:12:35.970 Let's take a step back, remind you of something 00:12:35.970 --> 00:12:41.020 that we did before that had only one particle. 00:12:41.020 --> 00:12:43.750 Because I will soon go to many particles. 00:12:43.750 --> 00:12:46.940 But before that, let's remind you 00:12:46.940 --> 00:12:52.510 what the one particle in a box looked like. 00:12:52.510 --> 00:12:57.180 So indeed, in this case, the single-particle states 00:12:57.180 --> 00:13:00.030 were the ones that I told you before, 00:13:00.030 --> 00:13:03.360 2 pi over l some set of integers. 00:13:03.360 --> 00:13:08.260 Epsilon k, that was h bar squared k squared over 2m. 00:13:08.260 --> 00:13:13.260 If I want to calculate the partition function 00:13:13.260 --> 00:13:19.330 for one particle in the box, I have to do a trace of e 00:13:19.330 --> 00:13:24.185 to the minus beta h for one particle. 00:13:24.185 --> 00:13:26.820 The trace I can very easily calculate 00:13:26.820 --> 00:13:30.340 in the basis in which this is diagonal. 00:13:30.340 --> 00:13:32.330 That's the basis that is parameterized 00:13:32.330 --> 00:13:34.320 by these k-values. 00:13:34.320 --> 00:13:40.280 So I do a sum over k of e to the minus beta h 00:13:40.280 --> 00:13:42.305 bar squared k squared over 2m. 00:13:45.040 --> 00:13:47.940 And then in the limit of very large box, 00:13:47.940 --> 00:13:50.880 we saw that the sum over k I can replace 00:13:50.880 --> 00:13:55.700 with V integral over k 2 pi cubed. 00:13:55.700 --> 00:13:59.040 This was the density of states in k. 00:13:59.040 --> 00:14:05.910 e to the minus beta h bar squared k squared over 2m. 00:14:05.910 --> 00:14:09.820 And this was three Gaussian integrals 00:14:09.820 --> 00:14:15.100 that gave us the usual formula of V over lambda cubed, where 00:14:15.100 --> 00:14:18.830 lambda was this thermal [INAUDIBLE] wavelength 00:14:18.830 --> 00:14:20.665 h root 2 pi mk. 00:14:26.970 --> 00:14:31.560 But we said that the essence of statistical mechanics 00:14:31.560 --> 00:14:34.120 is to tell you about probabilities 00:14:34.120 --> 00:14:37.680 of various micro-states, various positions of the particle 00:14:37.680 --> 00:14:41.870 in the box, which in the quantum perspective 00:14:41.870 --> 00:14:45.210 is probability becomes a density matrix. 00:14:45.210 --> 00:14:48.560 And we evaluated this density matrix 00:14:48.560 --> 00:14:49.986 in the coordinate representation. 00:14:52.810 --> 00:14:57.030 And in the coordinate representation, 00:14:57.030 --> 00:15:00.600 essentially what we had to do was 00:15:00.600 --> 00:15:07.360 to go into the basis in which rho is diagonal. 00:15:07.360 --> 00:15:11.750 So we had x prime k. 00:15:11.750 --> 00:15:18.616 In the k basis, the density matrix is just this formula. 00:15:18.616 --> 00:15:23.030 It's the Boltzmann weight appropriately normalized by Z1. 00:15:25.540 --> 00:15:26.880 And then we go kx. 00:15:31.860 --> 00:15:35.300 And basically, again replacing this 00:15:35.300 --> 00:15:42.360 with V integral d cubed k 2 pi cubed e to the minus beta h bar 00:15:42.360 --> 00:15:45.510 squared k squared over 2m. 00:15:45.510 --> 00:15:51.430 These two factors of xk and x prime k gave us a factor of e 00:15:51.430 --> 00:16:05.130 to the kx, xk I have as ik dot x prime minus x. 00:16:05.130 --> 00:16:07.070 Completing the square. 00:16:07.070 --> 00:16:11.140 Actually, I had to divide by Z1. 00:16:11.140 --> 00:16:14.140 There is a factor of 1 over V from the normalization 00:16:14.140 --> 00:16:16.400 of these things. 00:16:16.400 --> 00:16:21.380 The two V's here cancel, but Z1 is proportional to V. 00:16:21.380 --> 00:16:23.510 The lambda cubes cancel and so what 00:16:23.510 --> 00:16:28.740 we have is 1 over V e to the minus x 00:16:28.740 --> 00:16:33.730 minus x prime squared pi over lambda squared. 00:16:37.730 --> 00:16:41.340 So basically, what you have here is 00:16:41.340 --> 00:16:47.040 that we have a box of volume V. There 00:16:47.040 --> 00:16:50.135 is a particle inside at some location x. 00:16:53.860 --> 00:16:56.450 And the probability to find it at location 00:16:56.450 --> 00:17:00.000 x is the diagonal element of this entity. 00:17:00.000 --> 00:17:03.400 It's just 1 over V. But this entity 00:17:03.400 --> 00:17:08.369 has off-diagonal elements reflecting the fact 00:17:08.369 --> 00:17:12.510 that the best that you can do to localize something in quantum 00:17:12.510 --> 00:17:15.990 mechanics is to make some kind of a wave packet. 00:17:24.890 --> 00:17:27.170 OK. 00:17:27.170 --> 00:17:31.740 So this we did last time. 00:17:31.740 --> 00:17:37.216 What we want to do now is to go from one particle 00:17:37.216 --> 00:17:40.570 to the case of N particles. 00:17:40.570 --> 00:17:44.020 So rather than having 1x prime, I 00:17:44.020 --> 00:17:49.420 will have a whole bunch of x primes labeled 1 through N. 00:17:49.420 --> 00:17:55.660 And I want to calculate the N particle density matrix that 00:17:55.660 --> 00:18:03.440 connects me from set of points x to another set of points 00:18:03.440 --> 00:18:04.080 x prime. 00:18:04.080 --> 00:18:06.880 So if you like in the previous picture, 00:18:06.880 --> 00:18:11.160 this would have been x1 and x1 prime, 00:18:11.160 --> 00:18:19.160 and then I now have x2 and x2 prime, x3 and x3 prime, xN and 00:18:19.160 --> 00:18:20.500 xN prime. 00:18:20.500 --> 00:18:23.520 I have a bunch of different coordinates 00:18:23.520 --> 00:18:27.140 and I'd like to calculate that. 00:18:27.140 --> 00:18:28.100 OK. 00:18:28.100 --> 00:18:37.620 Once more, we know that rho is diagonal in the basis that 00:18:37.620 --> 00:18:40.710 is represented by these occupations 00:18:40.710 --> 00:18:44.430 of one-particle states. 00:18:44.430 --> 00:18:49.480 And so what I can do is I can sum over 00:18:49.480 --> 00:18:52.360 a whole bunch of plane waves. 00:18:52.360 --> 00:18:58.240 And I have to pick N factors of k out of this list 00:18:58.240 --> 00:19:02.580 in order to make one of these symmetrized or anti-symmetrized 00:19:02.580 --> 00:19:04.740 wave functions. 00:19:04.740 --> 00:19:08.230 But then I have to remember, as I said, 00:19:08.230 --> 00:19:14.170 that I should not over-count distinct set of k-values 00:19:14.170 --> 00:19:17.690 because permutations of these list of k's 00:19:17.690 --> 00:19:20.750 that I have over here, because of symmetrization 00:19:20.750 --> 00:19:24.140 or anti-symmetrization, will give me the same state. 00:19:24.140 --> 00:19:26.670 So I have to be careful about that. 00:19:26.670 --> 00:19:31.650 Then, I go from x prime to k. 00:19:35.770 --> 00:19:39.510 Now, the density matrix in the k-basis I know. 00:19:39.510 --> 00:19:43.110 It is simply e to the minus beta, the energy which 00:19:43.110 --> 00:19:48.610 is sum over alpha h bar squared k alpha squared over 2m. 00:19:48.610 --> 00:19:52.690 So I sum over the list of k alphas 00:19:52.690 --> 00:19:54.510 that appear in this series. 00:19:54.510 --> 00:19:56.350 There will be n of them. 00:19:56.350 --> 00:19:58.360 I have to appropriately normalize 00:19:58.360 --> 00:20:03.100 that by the N-particle partition function, which we have yet 00:20:03.100 --> 00:20:04.800 to calculate. 00:20:04.800 --> 00:20:09.780 And then I go back from k to x. 00:20:17.245 --> 00:20:20.230 Now, let's do this. 00:20:20.230 --> 00:20:23.390 The first thing that I mentioned last time 00:20:23.390 --> 00:20:26.430 is that I would, in principle, like 00:20:26.430 --> 00:20:30.580 to sum over k1 going over the entire list, 00:20:30.580 --> 00:20:34.570 k2 going the entire list, k3 going over the entire list. 00:20:34.570 --> 00:20:39.170 That is, I would like to make the sum over k's unrestricted. 00:20:42.100 --> 00:20:45.170 But then I have to take into account 00:20:45.170 --> 00:20:48.260 the over-counting that I have. 00:20:48.260 --> 00:20:51.490 If I am looking at the case where all of the k's are 00:20:51.490 --> 00:20:57.250 distinct-- they don't show any double occupancy-- then 00:20:57.250 --> 00:21:00.160 I have over-counted by the number of permutations. 00:21:00.160 --> 00:21:03.890 Because any permutation would have given me the same number. 00:21:03.890 --> 00:21:07.000 So I have to divide by the number of permutations 00:21:07.000 --> 00:21:12.710 to avoid the over-counting due to symmetrization here. 00:21:12.710 --> 00:21:14.680 Now, when I have something like this, 00:21:14.680 --> 00:21:22.580 which is a multiple occupancy, I have overdone this division. 00:21:22.580 --> 00:21:28.730 I have to multiply by this factor, 00:21:28.730 --> 00:21:36.850 and that's the correct number of over-countings that I have. 00:21:36.850 --> 00:21:40.880 And as I said, this was a good thing because the quantity 00:21:40.880 --> 00:21:43.980 that I had the hardest time for, and comes 00:21:43.980 --> 00:21:47.020 in the normalizations that occurs here, 00:21:47.020 --> 00:21:49.750 is this factor of 1 over nk factorial. 00:21:53.140 --> 00:21:56.680 Naturally, again, all of these things 00:21:56.680 --> 00:21:58.670 do depend on the symmetry. 00:21:58.670 --> 00:22:03.240 So I better make sure I indicate the index. 00:22:03.240 --> 00:22:06.400 Whether I'm calculating this density matrix for fermions 00:22:06.400 --> 00:22:09.070 or bosons, it is important. 00:22:09.070 --> 00:22:16.000 In either case-- well, what I need to do 00:22:16.000 --> 00:22:23.960 is to do a summation over P here for this one and P prime 00:22:23.960 --> 00:22:26.150 here or P prime here and P here. 00:22:26.150 --> 00:22:28.440 It doesn't matter, there's two sets of permutations 00:22:28.440 --> 00:22:30.580 that I have to do. 00:22:30.580 --> 00:22:36.240 In each case, I have to take care of this eta P, 00:22:36.240 --> 00:22:36.815 eta P prime. 00:22:42.390 --> 00:22:45.450 And then the normalization. 00:22:45.450 --> 00:22:50.040 So I divide by twice, or the square of the square root. 00:22:50.040 --> 00:22:56.350 I get the N factorial product over k nk factorial. 00:22:56.350 --> 00:23:00.190 And very nicely, the over-counting factor 00:23:00.190 --> 00:23:03.580 here cancels the normalization factor 00:23:03.580 --> 00:23:06.660 that I would have had here. 00:23:06.660 --> 00:23:08.820 So we got that. 00:23:08.820 --> 00:23:11.280 Now, what do we have? 00:23:11.280 --> 00:23:23.500 We have P prime permutation of these objects going to x, 00:23:23.500 --> 00:23:27.330 and then we have here P permutation 00:23:27.330 --> 00:23:31.346 of these k numbers going to x. 00:23:31.346 --> 00:23:36.270 I guess the first one I got wrong. 00:23:36.270 --> 00:23:37.860 I start with x prime. 00:23:40.460 --> 00:23:44.630 Go through P prime to k. 00:23:44.630 --> 00:23:50.810 And again, symmetries are already taken into account. 00:23:50.810 --> 00:23:52.730 I don't need to write that. 00:23:52.730 --> 00:23:57.490 And I have the factor of e to the minus beta h 00:23:57.490 --> 00:24:01.670 bar squared sum over alpha k alpha squared 00:24:01.670 --> 00:24:04.540 over 2m divided by ZN. 00:24:09.940 --> 00:24:14.880 OK, so let's bring all of the denominator factors out front. 00:24:14.880 --> 00:24:17.170 I have a ZN. 00:24:17.170 --> 00:24:20.970 I have an N factorial squared. 00:24:20.970 --> 00:24:24.620 Two factors of N factorial. 00:24:24.620 --> 00:24:31.620 I have a sum over two sets of permutations P and P prime. 00:24:34.350 --> 00:24:44.480 The product of the associated phase factor of their parities, 00:24:44.480 --> 00:24:48.830 and then I have this integration over k's. 00:24:48.830 --> 00:24:50.820 Now, unrestricted. 00:24:50.820 --> 00:24:55.730 Since it is unrestricted, I can integrate independently 00:24:55.730 --> 00:25:01.320 over each one of the k's, or sum over each one of them. 00:25:01.320 --> 00:25:04.000 When I sum, the sum becomes the integral 00:25:04.000 --> 00:25:12.700 over d cubed k alpha divided by 2 pi-- yeah, 2 pi cubed V. 00:25:12.700 --> 00:25:16.820 Basically, the density in replacing the sum over k 00:25:16.820 --> 00:25:20.350 alpha with the corresponding integration. 00:25:20.350 --> 00:25:25.430 So basically, this set of factors 00:25:25.430 --> 00:25:28.550 is what happened to that. 00:25:34.240 --> 00:25:37.210 OK, what do we have here? 00:25:37.210 --> 00:25:47.680 We have e to the i x-- well, let's be careful here. 00:25:47.680 --> 00:26:02.130 I have e to the i x prime alpha acting on k of p prime alpha 00:26:02.130 --> 00:26:07.850 because I permuted the k-label that 00:26:07.850 --> 00:26:15.580 went with, say, the alpha component here with p prime. 00:26:15.580 --> 00:26:18.350 From here, I would have minus because it's 00:26:18.350 --> 00:26:20.050 the complex conjugate. 00:26:20.050 --> 00:26:30.746 I have x alpha k p alpha, because I permuted this by k. 00:26:34.080 --> 00:26:40.010 I have one of these factors for each V. With each one of them, 00:26:40.010 --> 00:26:43.900 there is a normalization of square root of V. 00:26:43.900 --> 00:26:48.930 So the two of them together will give me V. But that's only one 00:26:48.930 --> 00:26:53.210 of the N-particle So there are N of them. 00:26:53.210 --> 00:26:57.265 So if I want, I can extend this product 00:26:57.265 --> 00:26:59.660 to also encompass this term. 00:27:02.520 --> 00:27:07.420 And then having done so, I can also write here e to the minus 00:27:07.420 --> 00:27:10.000 beta h bar squared k alpha squared 00:27:10.000 --> 00:27:12.680 over 2m within the product. 00:27:18.682 --> 00:27:23.040 AUDIENCE: [INAUDIBLE] after this-- is it quantity xk minus 00:27:23.040 --> 00:27:25.540 [INAUDIBLE]. 00:27:25.540 --> 00:27:27.200 PROFESSOR: I forgot an a here. 00:27:27.200 --> 00:27:28.740 What else did I miss out? 00:27:31.433 --> 00:27:32.724 AUDIENCE: [INAUDIBLE] quantity. 00:27:35.592 --> 00:27:37.210 PROFESSOR: So I forgot the i. 00:27:39.890 --> 00:27:41.340 OK, good? 00:27:41.340 --> 00:27:44.848 So the V's cancel out. 00:27:44.848 --> 00:27:48.200 All right, so that's fine. 00:27:48.200 --> 00:27:49.380 What do we have? 00:27:49.380 --> 00:27:57.530 We have 1 over ZN N factorial squared. 00:27:57.530 --> 00:28:02.070 Two sets of permutations summed over, p and p prime. 00:28:02.070 --> 00:28:07.170 Corresponding parities eta p eta of p prime. 00:28:07.170 --> 00:28:13.300 And then, I have a product of these integrations 00:28:13.300 --> 00:28:17.980 that I have to do that are three-dimensional Gaussians 00:28:17.980 --> 00:28:20.900 for each k alpha. 00:28:20.900 --> 00:28:22.700 What do I get? 00:28:22.700 --> 00:28:24.890 Well, first of all, if I didn't have this, 00:28:24.890 --> 00:28:27.730 if I just was doing the integration of e 00:28:27.730 --> 00:28:32.100 to the minus beta h bar squared k squared over 2m, 00:28:32.100 --> 00:28:34.120 I did that already. 00:28:34.120 --> 00:28:36.720 I get a 1 over lambda cubed. 00:28:36.720 --> 00:28:39.760 So basically, from each one of them I will get a 1 00:28:39.760 --> 00:28:40.645 over lambda cubed. 00:28:43.830 --> 00:28:47.470 But the integration is shifted by this amount. 00:28:47.470 --> 00:28:50.690 Actually, I already did the shifted integration here also 00:28:50.690 --> 00:28:53.060 for one particle. 00:28:53.060 --> 00:29:03.400 So I get the corresponding factor of e to the minus-- ah. 00:29:03.400 --> 00:29:07.590 I have to be a little bit careful over here 00:29:07.590 --> 00:29:14.450 because what I am integrating is over k alpha squared. 00:29:14.450 --> 00:29:18.910 Whereas, in the way that I have the list over here, 00:29:18.910 --> 00:29:22.830 I have x prime alpha and x alpha, 00:29:22.830 --> 00:29:27.450 but a different k playing around with each. 00:29:27.450 --> 00:29:28.360 What should I do? 00:29:28.360 --> 00:29:31.700 I really want this integration over k 00:29:31.700 --> 00:29:36.620 alpha to look like what I have over here. 00:29:36.620 --> 00:29:39.680 Well, as I sum over all possibilities 00:29:39.680 --> 00:29:44.680 in each one of these terms, I am bound to encounter k alpha. 00:29:44.680 --> 00:29:49.690 Essentially, I have permuted all of the k's that I originally 00:29:49.690 --> 00:29:50.810 had. 00:29:50.810 --> 00:29:55.340 So the k alpha has now been sent to some other location. 00:29:55.340 --> 00:30:00.170 But as I sum over all possible alpha, I will hit that. 00:30:00.170 --> 00:30:03.540 When I hit that, I will find that the thing that 00:30:03.540 --> 00:30:15.030 was multiplying k alpha is the inverse permutation of alpha. 00:30:15.030 --> 00:30:18.880 And the thing that was multiplying k alpha here 00:30:18.880 --> 00:30:23.390 is the inverse permutation of p. 00:30:29.140 --> 00:30:34.140 So then I can do the integration over k alpha easily. 00:30:34.140 --> 00:30:35.570 And so what do I have? 00:30:35.570 --> 00:30:39.510 I have x prime of p prime inverse 00:30:39.510 --> 00:30:43.780 alpha-- the inverse permutation-- minus x 00:30:43.780 --> 00:30:49.180 of p inverse alpha squared pi over lambda squared. 00:31:04.610 --> 00:31:09.190 Now, this is still inconvenient because I 00:31:09.190 --> 00:31:14.860 am summing over two N factorial sets of permutations. 00:31:14.860 --> 00:31:19.070 And I expect that since the sum only involves 00:31:19.070 --> 00:31:24.610 comparison of things that are occurring N times, 00:31:24.610 --> 00:31:29.690 as I go over the list of N factorial permutation squared, 00:31:29.690 --> 00:31:33.240 I will get the same thing appearing twice. 00:31:33.240 --> 00:31:37.590 So it is very much like when we are doing an integration over x 00:31:37.590 --> 00:31:39.610 and x prime, but the function only 00:31:39.610 --> 00:31:41.370 depends on x minus x prime. 00:31:41.370 --> 00:31:43.470 We get a factor of volume. 00:31:43.470 --> 00:31:47.250 Here, it is easy to see that one of these 00:31:47.250 --> 00:31:54.470 sums I can very easily do because it is just repetition 00:31:54.470 --> 00:31:57.390 of all of the results that I have previously. 00:31:57.390 --> 00:32:01.400 And there will be N factorial such terms. 00:32:01.400 --> 00:32:07.810 So doing that, I can get rid of one of the N factorials. 00:32:07.810 --> 00:32:11.780 And I will have only one permutation left, 00:32:11.780 --> 00:32:19.150 Q. And what will appear here would be the parity of this Q 00:32:19.150 --> 00:32:21.520 that is the combination, or if you like, 00:32:21.520 --> 00:32:25.600 the relative of these two permutations. 00:32:25.600 --> 00:32:32.680 And I have an exponential of minus sum 00:32:32.680 --> 00:32:41.190 over alpha x alpha minus x prime Q alpha squared 00:32:41.190 --> 00:32:44.650 pi over lambda squared. 00:32:44.650 --> 00:32:50.173 And I think I forgot a factor of lambda to the 3 [INAUDIBLE]. 00:32:55.103 --> 00:32:57.075 This factor of lambda 3. 00:33:03.510 --> 00:33:05.970 So this is actually the final result. 00:33:14.480 --> 00:33:21.090 And let's see what that precisely 00:33:21.090 --> 00:33:22.510 means for two particles. 00:33:27.030 --> 00:33:28.560 So let's look at two particles. 00:33:32.810 --> 00:33:40.020 So for two particle,s I will have on one side coordinates 00:33:40.020 --> 00:33:42.885 of 1 prime and 2 prime. 00:33:45.440 --> 00:33:49.525 On the right-hand side, I have coordinates 1 and 2. 00:33:54.210 --> 00:33:57.702 And let's see what this density matrix tells us. 00:33:57.702 --> 00:34:04.470 It tells us that to go from x1 prime x2 prime, a two particle 00:34:04.470 --> 00:34:11.540 density matrix connecting to x1 x2 on the other side, 00:34:11.540 --> 00:34:15.670 I have 1 over the two-particle partition function 00:34:15.670 --> 00:34:18.560 that i haven't yet calculated. 00:34:18.560 --> 00:34:20.920 Lambda to the sixth. 00:34:20.920 --> 00:34:24.699 N factorial in this case is 2. 00:34:24.699 --> 00:34:32.320 And then for two things, there are two permutations. 00:34:32.320 --> 00:34:37.710 So the identity maps 1 to 1, 2 to 2. 00:34:37.710 --> 00:34:39.560 And therefore, what I will get here 00:34:39.560 --> 00:34:47.144 would be exponential of minus x1' minus x1 prime squared 00:34:47.144 --> 00:34:55.070 pi over lambda squared minus x2 minus x2 prime squared pi 00:34:55.070 --> 00:34:56.060 over lambda squared. 00:35:02.450 --> 00:35:07.610 So that's Q being identity and identity 00:35:07.610 --> 00:35:10.470 has essentially 0 parity. 00:35:10.470 --> 00:35:14.120 It's an even permutation. 00:35:14.120 --> 00:35:17.410 The next thing is when I exchange 1 and 2. 00:35:17.410 --> 00:35:19.650 That would have odd parity. 00:35:19.650 --> 00:35:23.570 So I would get minus 1 for fermions, plus for bosons. 00:35:23.570 --> 00:35:28.460 And what I would get here is exponential of minus x1' 00:35:28.460 --> 00:35:33.990 minus x2 prime squared pi over lambda squared minus x2 00:35:33.990 --> 00:35:37.650 minus x1 prime squared pi over lambda squared. 00:35:43.950 --> 00:35:49.020 So essentially, one of the terms-- the first term 00:35:49.020 --> 00:35:53.320 is just the square of what I had before for one particle. 00:35:53.320 --> 00:35:57.560 I take the one-particle result, going from 1 to 1 prime, 00:35:57.560 --> 00:36:02.310 going from 2 to 2 prime and multiply them together. 00:36:02.310 --> 00:36:07.700 But then you say, I can't tell apart 2 prime and 1 prime. 00:36:07.700 --> 00:36:09.860 Maybe the thing that you are calling 1 prime 00:36:09.860 --> 00:36:13.330 is really 2 prime and vice versa. 00:36:13.330 --> 00:36:16.280 So I have to allow for the possibility 00:36:16.280 --> 00:36:18.430 that rather than x1 prime here, I 00:36:18.430 --> 00:36:23.080 should put x2 prime and the other way around. 00:36:23.080 --> 00:36:27.560 And this also corresponds to a permutation 00:36:27.560 --> 00:36:28.630 that is an exchange. 00:36:28.630 --> 00:36:33.840 It's an odd parity and will give you something like that. 00:36:33.840 --> 00:36:38.780 Say OK, I have no idea what that means. 00:36:38.780 --> 00:36:43.720 I'll tell you, OK, you were happy when I put x prime and x 00:36:43.720 --> 00:36:45.640 here because that was the probability 00:36:45.640 --> 00:36:48.790 to find the particle somewhere. 00:36:48.790 --> 00:36:50.800 So let me look at the diagonal term 00:36:50.800 --> 00:36:52.650 here, which is a probability. 00:36:57.720 --> 00:36:59.520 This should give me the probability 00:36:59.520 --> 00:37:02.120 to find one particle at position x1, 00:37:02.120 --> 00:37:03.650 one particle at position x2. 00:37:06.990 --> 00:37:09.130 Because the particles were non-interacting, 00:37:09.130 --> 00:37:10.890 one particle-- it could be anywhere. 00:37:10.890 --> 00:37:15.010 I had the 1 over V. Is it 1 over V squared 00:37:15.010 --> 00:37:17.240 or something like that? 00:37:17.240 --> 00:37:19.350 Well, we find that there is factor 00:37:19.350 --> 00:37:21.520 out front that we haven't yet evaluated. 00:37:21.520 --> 00:37:24.460 It turns out that this factor will give me a 1 00:37:24.460 --> 00:37:27.070 over V squared. 00:37:27.070 --> 00:37:31.650 And if I set x1 prime to x1, x2 prime 00:37:31.650 --> 00:37:36.800 to x2, which is what I've done here, this factor becomes 1. 00:37:36.800 --> 00:37:41.510 But then the other factor will give me eta e to the minus 2 00:37:41.510 --> 00:37:46.590 pi over lambda squared x1 minus x2 squared. 00:37:53.070 --> 00:37:58.960 So the physical probability to find one particle-- or more 00:37:58.960 --> 00:38:05.240 correctly, a wave packet here and a wave package there 00:38:05.240 --> 00:38:07.580 is not 1 over V squared. 00:38:07.580 --> 00:38:10.796 It's some function of the separation between these two 00:38:10.796 --> 00:38:11.295 particles. 00:38:15.030 --> 00:38:18.410 So that separation is contained here. 00:38:18.410 --> 00:38:23.290 If I really call that separation to be r, 00:38:23.290 --> 00:38:27.210 this is an additional weight that depends on r. 00:38:27.210 --> 00:38:30.650 This is r squared. 00:38:30.650 --> 00:38:38.410 So you can think of this as an interaction, which 00:38:38.410 --> 00:38:41.140 is because solely of quantum statistics. 00:38:44.880 --> 00:38:47.310 And what is this interaction? 00:38:47.310 --> 00:38:56.600 This interaction V of r would be minus kT log of 1 00:38:56.600 --> 00:39:02.337 plus eta e to the minus 2 pi r squared over lambda squared. 00:39:07.110 --> 00:39:11.670 I will plot out for you what this V or r looks 00:39:11.670 --> 00:39:16.730 like as a function of how far apart the centers of these two 00:39:16.730 --> 00:39:19.390 wave packets are. 00:39:19.390 --> 00:39:23.240 You can see that the result depends on eta. 00:39:23.240 --> 00:39:27.850 If eta is minus 1, which is for the case of fermions, 00:39:27.850 --> 00:39:30.330 this is 1 minus something. 00:39:30.330 --> 00:39:32.600 It's something that is less than 1. 00:39:32.600 --> 00:39:34.960 [INAUDIBLE] would be negative. 00:39:34.960 --> 00:39:39.580 So the whole potential would be positive, or repulsive. 00:39:39.580 --> 00:39:44.210 At large distances, indeed it would be exponentially going 00:39:44.210 --> 00:39:48.840 to 0 because I can expand the log at large distances. 00:39:48.840 --> 00:39:51.650 So here I have a term that is minus 2 pi 00:39:51.650 --> 00:39:55.620 r squared over lambda squared. 00:39:55.620 --> 00:40:00.440 As I go towards r equals to 0, actually things 00:40:00.440 --> 00:40:05.610 become very bad because r goes to 0 I will get 1 minus 1 00:40:05.610 --> 00:40:07.310 and the log will diverge. 00:40:07.310 --> 00:40:14.290 So basically, there is, if you like, an effective potential 00:40:14.290 --> 00:40:15.950 that says you can't put these two 00:40:15.950 --> 00:40:19.460 fermions on top of each other. 00:40:19.460 --> 00:40:21.860 So there is a statistical potential. 00:40:21.860 --> 00:40:26.670 So this is for eta minus 1, or fermions. 00:40:30.430 --> 00:40:34.490 For the case of bosons, eta plus 1. 00:40:34.490 --> 00:40:36.910 It is log of 1 plus something, so it's 00:40:36.910 --> 00:40:39.385 a positive number inside the log. 00:40:39.385 --> 00:40:43.520 The potential will be attractive. 00:40:43.520 --> 00:40:49.660 And it will actually saturate to a value of kT log 2 00:40:49.660 --> 00:40:51.360 when r goes to 0. 00:40:51.360 --> 00:40:57.850 So this is again, eta of plus 1 for the case of bosons. 00:41:38.320 --> 00:41:42.840 So the one thing that this formula does not have yet 00:41:42.840 --> 00:41:46.480 is the value for this partition function ZN. 00:41:46.480 --> 00:41:50.590 It gives you the qualitative behavior in either case. 00:41:50.590 --> 00:41:53.390 And let's calculate what ZN is. 00:41:53.390 --> 00:41:56.320 Well, basically, that would come from noting 00:41:56.320 --> 00:42:00.000 that the trace of rho has to be 1. 00:42:00.000 --> 00:42:12.050 So ZN is trace of e to the minus beta H. 00:42:12.050 --> 00:42:22.340 And essentially, I can take this ZN to the other side 00:42:22.340 --> 00:42:35.110 and evaluate this as x e to the minus beta H x. 00:42:35.110 --> 00:42:44.770 That is, I can calculate the diagonal elements 00:42:44.770 --> 00:42:46.600 of this matrix that I have calculated-- 00:42:46.600 --> 00:42:48.830 that I have over there. 00:42:48.830 --> 00:42:52.260 So there is an overall factor of 1 00:42:52.260 --> 00:42:59.430 over lambda cubed to the power of N. I have N factorial. 00:43:02.250 --> 00:43:12.800 And then I have a sum over permutations Q eta of Q. 00:43:12.800 --> 00:43:15.580 The diagonal element is obtained by putting 00:43:15.580 --> 00:43:18.610 x prime to be the same as x. 00:43:18.610 --> 00:43:30.290 So I have exponential of minus x-- sum over alpha x alpha 00:43:30.290 --> 00:43:33.130 minus x of Q alpha. 00:43:33.130 --> 00:43:36.620 I set x prime to be the same as x. 00:43:36.620 --> 00:43:37.560 Squared. 00:43:37.560 --> 00:43:41.530 And then there's an overall pi over lambda squared. 00:43:44.200 --> 00:43:50.030 And if I am taking the trace, it means that I have to do 00:43:50.030 --> 00:43:52.580 integration over all x's. 00:44:03.640 --> 00:44:08.560 So I'm evaluating this trace in coordinate basis, which 00:44:08.560 --> 00:44:11.040 means that I should put x and x prime to be 00:44:11.040 --> 00:44:13.560 the same for the trace, and then I 00:44:13.560 --> 00:44:19.970 have to sum or integrate over all possible values of x. 00:44:19.970 --> 00:44:21.190 So let's do this. 00:44:21.190 --> 00:44:31.070 I have 1 over N factorial lambda cubed raised to the power of N. 00:44:31.070 --> 00:44:31.900 OK. 00:44:31.900 --> 00:44:36.290 Now I have to make a choice because I 00:44:36.290 --> 00:44:40.660 have a whole bunch of terms because of these permutations. 00:44:43.180 --> 00:44:46.520 Let's do them one by one. 00:44:46.520 --> 00:44:51.120 Let's first do the case where Q is identity. 00:44:51.120 --> 00:44:55.530 That is, I map everybody to themselves. 00:44:55.530 --> 00:45:00.580 Actually, let me write down the integrations first. 00:45:00.580 --> 00:45:06.330 I will do the integrations over all pairs 00:45:06.330 --> 00:45:10.780 of coordinates of these Gaussians. 00:45:10.780 --> 00:45:17.730 These Gaussians I will evaluate for different permutations. 00:45:17.730 --> 00:45:22.160 Let's look at the case where Q is identity. 00:45:22.160 --> 00:45:27.540 When Q is identity, essentially I will put all of the x prime 00:45:27.540 --> 00:45:29.040 to be the same as x. 00:45:29.040 --> 00:45:33.600 It is like what I did here for two particles and I got 1. 00:45:33.600 --> 00:45:36.960 I do the same thing for more than one particle. 00:45:36.960 --> 00:45:38.210 I will still get 1. 00:45:42.740 --> 00:45:46.420 Then, I will do the same thing that I did over here. 00:45:46.420 --> 00:45:52.370 Here, the next term that I did was to exchange 1 and 2. 00:45:52.370 --> 00:45:56.070 So this became x1 minus x2. 00:45:56.070 --> 00:45:58.330 I'll do the same thing here. 00:45:58.330 --> 00:46:01.570 I look at the case where Q corresponds 00:46:01.570 --> 00:46:05.340 to exchange of particles 1 and 2. 00:46:05.340 --> 00:46:07.690 And then that will give me a factor which 00:46:07.690 --> 00:46:15.180 is e to the minus pi over lambda squared x1 minus x2 squared. 00:46:15.180 --> 00:46:19.110 There are two of these making together 2 pi over lambda 00:46:19.110 --> 00:46:21.340 squared, which I hope I had there, too. 00:46:25.420 --> 00:46:29.030 But then there was a whole bunch of other terms that I can do. 00:46:29.030 --> 00:46:35.360 I can exchange, let's say, 7 and 9. 00:46:35.360 --> 00:46:38.720 And then I will get here 2 pi over lambda 00:46:38.720 --> 00:46:43.340 squared x7 minus x9 squared. 00:46:43.340 --> 00:46:46.230 And there's a whole bunch of such exchanges 00:46:46.230 --> 00:46:50.220 that I can make in which I just switch 00:46:50.220 --> 00:46:55.950 between two particles in this whole story. 00:46:55.950 --> 00:47:00.840 And clearly, the number of exchanges that I can make 00:47:00.840 --> 00:47:05.060 is the number of pairs, N N minus 1 over 2. 00:47:08.830 --> 00:47:12.750 Once I am done with all of the exchanges, 00:47:12.750 --> 00:47:14.780 then I have to go to the next thing that 00:47:14.780 --> 00:47:18.120 doesn't have an analog here for two particles. 00:47:18.120 --> 00:47:20.850 But if I take three particles, I can permute them 00:47:20.850 --> 00:47:22.980 like a triangle. 00:47:22.980 --> 00:47:25.770 So presumably there would be next set 00:47:25.770 --> 00:47:32.190 of terms, which is a permutation that is like 1, 2, 3, 2, 3, 1. 00:47:32.190 --> 00:47:36.830 There's a bunch of things that involve two permutations, four 00:47:36.830 --> 00:47:40.190 permutations, and so forth. 00:47:40.190 --> 00:47:42.600 So there is a whole list of things 00:47:42.600 --> 00:47:47.710 that would go to here where these two-particle exchanges 00:47:47.710 --> 00:47:49.190 are the simplest class. 00:47:52.970 --> 00:47:57.590 Now, as we shall see, there is a systematic way 00:47:57.590 --> 00:48:02.540 of looking at things where the two-particle exchanges are 00:48:02.540 --> 00:48:05.570 the first correction due to quantum effects. 00:48:05.570 --> 00:48:09.300 Three-particle exchanges would be higher-order corrections. 00:48:09.300 --> 00:48:14.880 And we can systematically do them in order. 00:48:14.880 --> 00:48:19.685 So let's see what happens if we compare the case where there 00:48:19.685 --> 00:48:24.210 is no exchange and the case where there is one exchange. 00:48:24.210 --> 00:48:26.540 When there is no exchange, I am essentially 00:48:26.540 --> 00:48:31.500 integrating over each position over the volume. 00:48:31.500 --> 00:48:38.880 So what I would get is V raised to the power of N. 00:48:38.880 --> 00:48:41.880 The next term? 00:48:41.880 --> 00:48:45.420 Well, I have to do the integrations. 00:48:45.420 --> 00:48:51.120 The integrations over x3, x4, x5, all the way to x to the N, 00:48:51.120 --> 00:48:52.270 there is no factors. 00:48:52.270 --> 00:48:56.790 So they will give me factors of V. And there are N minus 2 00:48:56.790 --> 00:48:57.290 of them. 00:49:01.060 --> 00:49:04.600 And then I have to do the integration over x1 and x2 00:49:04.600 --> 00:49:08.840 of this factor, but it's only a function 00:49:08.840 --> 00:49:10.690 of the relative coordinate. 00:49:10.690 --> 00:49:12.650 So there is one other integration 00:49:12.650 --> 00:49:17.690 that I can trivially do, which is the center of mass gives me 00:49:17.690 --> 00:49:20.760 a factor of V. And then I am left 00:49:20.760 --> 00:49:24.973 with the integral over the relative coordinate of e 00:49:24.973 --> 00:49:30.170 to the minus 2 pi r squared over lambda squared. 00:49:30.170 --> 00:49:34.070 And I forgot-- it's very important. 00:49:34.070 --> 00:49:40.200 This will carry a factor of eta because any exchange is odd. 00:49:40.200 --> 00:49:42.520 And so there will be a factor of eta here. 00:49:45.950 --> 00:49:52.310 And I said that I would get the same expression 00:49:52.310 --> 00:49:55.510 for any of my N N minus 1 over 2 exchanges. 00:49:59.230 --> 00:50:05.480 So the result of all of these exchange calculations 00:50:05.480 --> 00:50:08.480 would be the same thing. 00:50:08.480 --> 00:50:10.580 And then there would be the contribution 00:50:10.580 --> 00:50:12.520 from three-body exchange and so forth. 00:50:19.460 --> 00:50:21.950 So let's re-organize this. 00:50:21.950 --> 00:50:27.630 I can pull out the factor of V to the N outside. 00:50:27.630 --> 00:50:32.620 So I would have V over lambda cubed to the power of N. 00:50:32.620 --> 00:50:35.710 So the first term is 1. 00:50:35.710 --> 00:50:38.830 The next term has the parity factor 00:50:38.830 --> 00:50:42.460 that distinguishes bosons and fermions, goes 00:50:42.460 --> 00:50:47.920 with a multiplicity of pairs which is N N minus 1 over 2. 00:50:47.920 --> 00:50:51.230 Since I already pulled out a factor of V to the N 00:50:51.230 --> 00:50:54.440 and I really had V to the N minus 1 here, 00:50:54.440 --> 00:51:00.210 I better put a factor of 1 over V here. 00:51:00.210 --> 00:51:02.420 And then I just am left with having 00:51:02.420 --> 00:51:06.420 to evaluate these Gaussian integrals. 00:51:06.420 --> 00:51:09.192 Each Gaussian integral will give me 00:51:09.192 --> 00:51:11.750 2 pi times the variance, which is 00:51:11.750 --> 00:51:16.620 lambda squared divided by 2 pi. 00:51:16.620 --> 00:51:21.040 And then there's actually a factor of 2. 00:51:21.040 --> 00:51:25.260 And there are three of them, so I will have 3/2. 00:51:25.260 --> 00:51:30.870 So what I get here is lambda cubed divided by 2 to the 3/2. 00:51:39.580 --> 00:51:43.690 Now, you can see that any time I go further 00:51:43.690 --> 00:51:47.240 in this series of exchanges, I will 00:51:47.240 --> 00:51:50.440 have more of these Gaussian factors. 00:51:50.440 --> 00:51:52.920 And whenever I have a Gaussian factor, 00:51:52.920 --> 00:51:54.990 I have an additional integration to do 00:51:54.990 --> 00:51:57.770 that has an x minus something squared in it. 00:51:57.770 --> 00:52:01.990 I will lose a factor of V. I don't have that factor of V. 00:52:01.990 --> 00:52:07.470 And so subsequent terms will be even smaller in powers of V. 00:52:07.470 --> 00:52:12.240 And presumably, compensated by corresponding factors of lambda 00:52:12.240 --> 00:52:13.550 squared-- lambda cubed. 00:52:27.610 --> 00:52:33.630 Now, first thing to note is that in the very, very high 00:52:33.630 --> 00:52:37.540 temperature limit, lambda goes to 0. 00:52:37.540 --> 00:52:40.680 So I can forget even this correction. 00:52:40.680 --> 00:52:42.170 What do I get? 00:52:42.170 --> 00:52:47.770 I get 1 over N factorial V over lambda cubed to the power of N. 00:52:47.770 --> 00:52:52.000 Remember that many, many lectures back we introduced 00:52:52.000 --> 00:52:55.610 by hand the factor of 1 over N factorial for measuring 00:52:55.610 --> 00:52:58.780 phase spaces of identical particles. 00:52:58.780 --> 00:53:01.130 And I promise to you that we would get it 00:53:01.130 --> 00:53:03.880 when we did identical particles in quantum mechanics, 00:53:03.880 --> 00:53:05.940 so here it is. 00:53:05.940 --> 00:53:09.390 So automatically, we did the calculation, 00:53:09.390 --> 00:53:12.030 keeping track of identity of particles 00:53:12.030 --> 00:53:14.300 at the level of quantum states. 00:53:14.300 --> 00:53:17.400 Went through the calculation and in the high temperature limit, 00:53:17.400 --> 00:53:22.410 we get this 1 over N factorial emerging. 00:53:22.410 --> 00:53:26.770 Secondly, we see that the corrections to ideal gas 00:53:26.770 --> 00:53:34.470 behavior emerge as a series in powers of lambda cubed over V. 00:53:34.470 --> 00:53:40.120 And for example, if I were to take the log of the partition 00:53:40.120 --> 00:53:45.550 function, I would get log of what 00:53:45.550 --> 00:53:50.250 I would have had classically, which is this V over lambda 00:53:50.250 --> 00:53:53.770 cubed to the power of N divided by N factorial. 00:53:57.110 --> 00:54:01.750 And then the log of this expression. 00:54:01.750 --> 00:54:05.070 And this I'm going to replace with N squared. 00:54:05.070 --> 00:54:07.720 There is not that much difference. 00:54:07.720 --> 00:54:13.630 And since I'm regarding this as a correction, log of 1 00:54:13.630 --> 00:54:18.620 plus something, I will replace with the something 00:54:18.620 --> 00:54:27.670 eta N squared 2V lambda cubed 2 to the 3/2 plus higher order. 00:54:30.580 --> 00:54:32.170 What does this mean? 00:54:32.170 --> 00:54:34.130 Once we have the partition function, 00:54:34.130 --> 00:54:36.740 we can calculate pressure. 00:54:36.740 --> 00:54:42.885 Reminding you that beta p was the log Z by dV. 00:54:45.615 --> 00:54:49.850 The first part is the ideal gas that we 00:54:49.850 --> 00:54:52.040 had looked at classically. 00:54:52.040 --> 00:54:57.480 So once I go to the appropriate large-end limit of this, 00:54:57.480 --> 00:55:03.330 what this gives me is the density n over V. 00:55:03.330 --> 00:55:06.150 And then when I look at the derivative here, 00:55:06.150 --> 00:55:11.400 the derivative of 1/V will give me a minus 1 over V squared. 00:55:11.400 --> 00:55:15.630 So I will get minus eta. 00:55:15.630 --> 00:55:18.500 N over V, the whole thing squared. 00:55:18.500 --> 00:55:25.180 So I will have n squared lambda cubed 2 to the 5/2, 00:55:25.180 --> 00:55:26.600 and so forth. 00:55:30.730 --> 00:55:35.600 So I see that the pressure of this ideal gas 00:55:35.600 --> 00:55:40.140 with no interactions is already different 00:55:40.140 --> 00:55:43.525 from the classical result that we had calculated 00:55:43.525 --> 00:55:47.800 by a factor that actually reflects the statistics. 00:55:47.800 --> 00:55:52.660 For fermions eta of minus 1, you get an additional pressure 00:55:52.660 --> 00:55:56.170 because of the kind of repulsion that we have over here. 00:55:56.170 --> 00:55:59.640 Whereas, for bosons you get an attraction. 00:55:59.640 --> 00:56:02.740 You can see that also the thing that determines this-- 00:56:02.740 --> 00:56:06.420 so basically, this corresponds to a second Virial 00:56:06.420 --> 00:56:16.830 coefficient, which is minus eta lambda cubed 2 to the 5/2, 00:56:16.830 --> 00:56:23.820 is the volume of these wave packets. 00:56:23.820 --> 00:56:27.310 So essentially, the corrections are 00:56:27.310 --> 00:56:32.720 of the order of n lambda cubed that 00:56:32.720 --> 00:56:37.750 is within one of these wave packets how many particles 00:56:37.750 --> 00:56:39.970 you will encounter. 00:56:39.970 --> 00:56:43.780 As you go to high temperature, the wave packets shrink. 00:56:43.780 --> 00:56:48.550 As you go to low temperature, the wave packets expand. 00:56:48.550 --> 00:56:52.010 If you like, the interactions become more important 00:56:52.010 --> 00:56:55.147 and you get corrections to ideal gas wave. 00:56:58.471 --> 00:57:01.393 AUDIENCE: You assume that we can use perturbation, 00:57:01.393 --> 00:57:05.289 but the higher terms actually had a factor [INAUDIBLE]. 00:57:09.185 --> 00:57:13.110 And you can't really use perturbation in that. 00:57:13.110 --> 00:57:14.570 PROFESSOR: OK. 00:57:14.570 --> 00:57:19.370 So what you are worried about is the story here, 00:57:19.370 --> 00:57:27.410 that I took log of 1 plus something here 00:57:27.410 --> 00:57:31.700 and I'm interested in the limit of n going to infinity, 00:57:31.700 --> 00:57:36.530 that finite density n over V. So already in that limit, 00:57:36.530 --> 00:57:38.800 you would say that this factor really 00:57:38.800 --> 00:57:42.080 is overwhelmingly larger than that. 00:57:42.080 --> 00:57:46.220 And as you say, the next factor will be even larger. 00:57:46.220 --> 00:57:50.490 So what is the justification in all of this? 00:57:50.490 --> 00:57:54.990 We have already encountered this same problem 00:57:54.990 --> 00:57:57.850 when we were doing these perturbations 00:57:57.850 --> 00:57:59.710 due to interactions. 00:57:59.710 --> 00:58:08.932 And the answer is that what you really want to ensure 00:58:08.932 --> 00:58:20.710 is that not log Z, but Z has a form 00:58:20.710 --> 00:58:23.245 that is e to the N something. 00:58:25.985 --> 00:58:28.375 And that something will have corrections, 00:58:28.375 --> 00:58:34.330 potentially that are powers of N, the density, which 00:58:34.330 --> 00:58:39.490 is N over V. And if you try to force it into a perturbation 00:58:39.490 --> 00:58:44.830 series such as this, naturally things like this happen. 00:58:44.830 --> 00:58:46.820 What does that really mean? 00:58:46.820 --> 00:58:51.630 That really means that the correct thing that you should 00:58:51.630 --> 00:58:55.860 be expanding is, indeed, log Z. If you 00:58:55.860 --> 00:58:59.350 were to do the kind of hand-waving that I did here 00:58:59.350 --> 00:59:03.950 and do the expansion for Z, if you also try to do it over here 00:59:03.950 --> 00:59:06.360 you will generate terms that look 00:59:06.360 --> 00:59:08.780 kind of at the wrong order. 00:59:08.780 --> 00:59:12.480 But higher order terms that you would get 00:59:12.480 --> 00:59:17.120 would naturally conspire so that when you evaluate log Z, 00:59:17.120 --> 00:59:19.870 they come out right. 00:59:19.870 --> 00:59:22.420 You have to do this correctly. 00:59:22.420 --> 00:59:24.120 And once you have done it correctly, 00:59:24.120 --> 00:59:26.120 then you can rely on the calculation 00:59:26.120 --> 00:59:28.430 that you did before as an example. 00:59:28.430 --> 00:59:33.300 And we did it correctly when we were doing these cluster 00:59:33.300 --> 00:59:38.440 expansions and the corresponding calculation we 00:59:38.440 --> 00:59:41.500 did for Q. We saw how the different diagrams were 00:59:41.500 --> 00:59:44.400 appearing in both Q and the log Q, 00:59:44.400 --> 00:59:47.910 and how they could be summed over in log Q. 00:59:47.910 --> 00:59:51.485 But indeed, this mathematically looks awkward 00:59:51.485 --> 00:59:55.250 and I kind of jumped a step in writing log of 1 00:59:55.250 --> 01:00:00.492 plus something that is huge as if it was a small number. 01:00:14.550 --> 01:00:17.300 All right. 01:00:17.300 --> 01:00:20.870 So we have a problem. 01:00:20.870 --> 01:00:23.410 We want to calculate the simplest 01:00:23.410 --> 01:00:26.940 system, which is the ideal gas. 01:00:26.940 --> 01:00:30.680 So classically, we did all of our calculations 01:00:30.680 --> 01:00:32.210 first for the ideal gas. 01:00:32.210 --> 01:00:34.030 We had exact results. 01:00:34.030 --> 01:00:36.220 Then, let's say we had interactions. 01:00:36.220 --> 01:00:39.100 We did perturbations around that and all of that. 01:00:39.100 --> 01:00:43.690 And we saw that having to do things for interacting systems 01:00:43.690 --> 01:00:46.540 is very difficult. 01:00:46.540 --> 01:00:49.960 Now, when we start to do calculations for the quantum 01:00:49.960 --> 01:00:53.810 problem, at least in the way that I set it up for you, 01:00:53.810 --> 01:00:57.250 it seems that quantum problems are inherently 01:00:57.250 --> 01:01:00.640 interacting problems. 01:01:00.640 --> 01:01:03.850 I showed you that even at the level of two particles, 01:01:03.850 --> 01:01:08.820 it is like having an interaction between bosons and fermions. 01:01:08.820 --> 01:01:11.070 For three particles, it becomes even worse 01:01:11.070 --> 01:01:14.890 because it's not only the two-particle interaction. 01:01:14.890 --> 01:01:16.920 Because of the three-particle exchanges, 01:01:16.920 --> 01:01:19.720 you would get an additional three-particle interaction, 01:01:19.720 --> 01:01:24.020 four-particle interaction, all of these things emerge. 01:01:24.020 --> 01:01:27.550 So really, if you want to look at this 01:01:27.550 --> 01:01:32.870 from the perspective of a partition function, 01:01:32.870 --> 01:01:36.520 we already see that the exchange term 01:01:36.520 --> 01:01:38.800 involved having to do a calculation that 01:01:38.800 --> 01:01:42.500 is equivalent to calculating the second Virial 01:01:42.500 --> 01:01:46.780 coefficient for an interacting system. 01:01:46.780 --> 01:01:49.570 The next one, for the third Virial coefficient, 01:01:49.570 --> 01:01:52.550 I would need to look at the three-body exchanges, 01:01:52.550 --> 01:01:56.570 kind of like the point clusters, four-point clusters, 01:01:56.570 --> 01:01:59.510 all kinds of other things are there. 01:01:59.510 --> 01:02:00.410 So is there any hope? 01:02:05.150 --> 01:02:11.300 And the answer is that it is all a matter of perspective. 01:02:11.300 --> 01:02:22.560 And somehow it is true that these particles 01:02:22.560 --> 01:02:26.890 in quantum mechanics because of the statistics 01:02:26.890 --> 01:02:31.310 are subject to all kinds of complicated interactions. 01:02:31.310 --> 01:02:34.250 But also, the underlying Hamiltonian 01:02:34.250 --> 01:02:37.000 is simple and non-interacting. 01:02:37.000 --> 01:02:39.870 We can enumerate all of the wave functions. 01:02:39.870 --> 01:02:41.460 Everything is simple. 01:02:41.460 --> 01:02:44.770 So by looking at things in the right basis, 01:02:44.770 --> 01:02:49.470 we should be able to calculate everything that we need. 01:02:49.470 --> 01:02:55.290 So here, I was kind of looking at calculating the partition 01:02:55.290 --> 01:03:00.590 function in the coordinate basis, which is the worst case 01:03:00.590 --> 01:03:06.390 scenario because the Hamiltonian is diagonal in the momentum 01:03:06.390 --> 01:03:07.860 basis. 01:03:07.860 --> 01:03:16.690 So let's calculate ZN trace of e to the minus beta H 01:03:16.690 --> 01:03:19.630 in the basis in which H is diagonal. 01:03:23.860 --> 01:03:28.240 So what are the eigenvalues and eigenfunctions? 01:03:28.240 --> 01:03:32.890 Well, the eigenfunctions are the symmetrized/anti-symmetrized 01:03:32.890 --> 01:03:34.000 quantities. 01:03:34.000 --> 01:03:37.170 The eigenvalues are simply e to the minus beta H 01:03:37.170 --> 01:03:40.890 bar squared k alpha squared over 2m. 01:03:40.890 --> 01:03:43.820 So this is basically the thing that I 01:03:43.820 --> 01:03:47.690 could write as the set of k's appropriately 01:03:47.690 --> 01:03:52.370 symmetrized or anti-symmetrized e to the minus beta 01:03:52.370 --> 01:04:00.060 sum over alpha H bar squared k alpha squared over 2m k eta. 01:04:06.230 --> 01:04:11.710 Actually, I'm going to-- rather than go through this procedure 01:04:11.710 --> 01:04:15.530 that we have up there in which I wrote 01:04:15.530 --> 01:04:22.860 these, what I need to do here is a sum over all k 01:04:22.860 --> 01:04:25.070 in order to evaluate the trace. 01:04:25.070 --> 01:04:31.200 So this is inherently a sum over all sets of k's. 01:04:31.200 --> 01:04:35.180 But this sum is restricted, just like what 01:04:35.180 --> 01:04:38.790 I had indicated for you before. 01:04:38.790 --> 01:04:42.190 Rather than trying to do it that way, 01:04:42.190 --> 01:04:48.180 I note that these k's I could also 01:04:48.180 --> 01:04:52.860 write in terms of these occupation numbers. 01:04:52.860 --> 01:04:57.200 So equivalently, my basis would be 01:04:57.200 --> 01:05:04.370 the set of occupation numbers times the energy. 01:05:04.370 --> 01:05:13.400 The energy is then e to the minus beta sum over k epsilon k 01:05:13.400 --> 01:05:20.830 nk, where epsilon k is this beta H 01:05:20.830 --> 01:05:23.650 bar squared k alpha squared over 2m. 01:05:23.650 --> 01:05:28.760 But I could do this in principle for any epsilon k 01:05:28.760 --> 01:05:30.800 that I have over here. 01:05:30.800 --> 01:05:35.230 So the result that I am writing for you is more general. 01:05:35.230 --> 01:05:37.120 Then I sandwich it again, since I'm 01:05:37.120 --> 01:05:40.365 calculating the trace, with the same state. 01:05:43.360 --> 01:05:46.900 Now, the states have this restriction 01:05:46.900 --> 01:05:47.970 that I have over there. 01:05:47.970 --> 01:05:54.070 That is, for the case of fermions, my nk can be 0 or 1. 01:05:54.070 --> 01:05:58.440 But there is no restriction for nk on the bosons. 01:05:58.440 --> 01:06:03.360 Except, of course, that there is this overall restriction 01:06:03.360 --> 01:06:11.670 that the sum over k nk has to be N 01:06:11.670 --> 01:06:14.865 because I am looking at N-particle states. 01:06:22.810 --> 01:06:29.610 Actually, I can remove this because in this basis, 01:06:29.610 --> 01:06:32.560 e to the minus beta H is diagonal. 01:06:32.560 --> 01:06:39.330 So I can, basically, remove these entities. 01:06:39.330 --> 01:06:41.715 And I'm just summing a bunch of exponentials. 01:06:47.860 --> 01:06:55.160 So that is good because I should be able to do for each nk 01:06:55.160 --> 01:06:57.580 a sum of e to something nk. 01:07:00.410 --> 01:07:03.150 Well, the problem is this that I can't sum over 01:07:03.150 --> 01:07:04.553 each nk independently. 01:07:08.990 --> 01:07:14.660 Essentially in the picture that I have over here, 01:07:14.660 --> 01:07:16.730 I have some n1 here. 01:07:16.730 --> 01:07:18.870 I have some n2 here. 01:07:18.870 --> 01:07:21.660 Some n3 here, which are the occupation numbers 01:07:21.660 --> 01:07:24.260 of these things. 01:07:24.260 --> 01:07:26.290 And for that partition function, I 01:07:26.290 --> 01:07:29.760 have to do the sum of these exponentials e 01:07:29.760 --> 01:07:33.760 to the minus epsilon 1 n1, e to the minus epsilon 2 n2. 01:07:33.760 --> 01:07:40.140 But the sum of all of these n's is kind of maxed out by N. 01:07:40.140 --> 01:07:46.092 I cannot independently sum over them going over the entire 01:07:46.092 --> 01:07:47.740 range. 01:07:47.740 --> 01:07:52.745 But we've seen previously how those constraints can 01:07:52.745 --> 01:07:55.400 be removed in statistical mechanics. 01:07:55.400 --> 01:07:56.790 So our usual trick. 01:07:56.790 --> 01:08:01.910 We go to the ensemble in which n can take any value. 01:08:01.910 --> 01:08:06.040 So we go to the grand canonical prescription. 01:08:12.880 --> 01:08:19.479 We remove this constraint on n by evaluating a grand partition 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. 01:08:30.800 --> 01:08:34.240 So we do, essentially, a Laplace transform. 01:08:34.240 --> 01:08:38.300 We exchange our n with the chemical potential mu. 01:08:41.850 --> 01:08:47.620 Then, this constraint no longer we need to worry about. 01:08:47.620 --> 01:08:55.390 So now I can sum over all of the nk's without worrying 01:08:55.390 --> 01:09:00.140 about any constraint, provided that I multiply with e 01:09:00.140 --> 01:09:03.560 to the beta mu n, which is a sum over k nk. 01:09:07.770 --> 01:09:10.660 And then, the factor that I have here, 01:09:10.660 --> 01:09:15.771 which is e to the minus beta sum over k epsilon of k nk. 01:09:22.180 --> 01:09:28.319 So essentially for each k, I can independently 01:09:28.319 --> 01:09:38.359 sum over its nk of e to the beta mu minus epsilon of k nk. 01:09:45.189 --> 01:09:49.069 Now, the symmetry issues remain. 01:09:49.069 --> 01:09:53.180 This answer still depends on whether or not 01:09:53.180 --> 01:09:57.870 I am calculating things for bosons or fermions 01:09:57.870 --> 01:10:03.566 because these sums are differently constrained 01:10:03.566 --> 01:10:06.200 whether I'm dealing with fermions. 01:10:06.200 --> 01:10:09.420 In which case, nk is only 0 or 1. 01:10:09.420 --> 01:10:12.075 Or bosons, in which case there is no constraint. 01:10:17.190 --> 01:10:20.020 So what do I get? 01:10:20.020 --> 01:10:28.540 For the case of fermions, I have a Q minus, 01:10:28.540 --> 01:10:33.020 which is product over all k. 01:10:33.020 --> 01:10:39.580 And for each k, the nk takes either 0 or 1. 01:10:39.580 --> 01:10:45.500 So if it takes 0, I will write e to the 0, which is 1. 01:10:45.500 --> 01:10:46.465 Or it takes 1. 01:10:46.465 --> 01:10:50.562 It is e to the beta mu minus epsilon of k. 01:10:57.110 --> 01:11:04.640 For the case of bosons, I have a Q plus. 01:11:04.640 --> 01:11:10.370 Q plus is, again, a product over Q. In this case, 01:11:10.370 --> 01:11:15.240 nk going from 0 to infinity, I am summing a geometric series 01:11:15.240 --> 01:11:20.320 that starts as 1, and then the subsequent terms 01:11:20.320 --> 01:11:26.950 are smaller by a factor of beta mu minus epsilon of k. 01:11:26.950 --> 01:11:31.260 Actually, for future reference note 01:11:31.260 --> 01:11:35.350 that I would be able to do this geometric sum 01:11:35.350 --> 01:11:39.830 provided that this combination beta mu minus epsilon of k 01:11:39.830 --> 01:11:41.130 is negative. 01:11:41.130 --> 01:11:44.570 So that the subsequent terms in this series are decaying. 01:11:51.890 --> 01:11:54.680 Typically, we would be interested in things 01:11:54.680 --> 01:11:58.660 like partition functions, grand partition functions. 01:11:58.660 --> 01:12:05.620 So we have something like log of Q, which would be a sum over k. 01:12:09.760 --> 01:12:15.600 And I would have either the log of this quantity 01:12:15.600 --> 01:12:20.330 or the log of this quantity with a minus sign. 01:12:20.330 --> 01:12:23.180 I can combine the two results together 01:12:23.180 --> 01:12:28.910 by putting a factor of minus eta because in taking the log, 01:12:28.910 --> 01:12:32.794 over here for the bosons I would pick a factor of minus 1 01:12:32.794 --> 01:12:34.460 because the thing is in the denominator. 01:12:37.100 --> 01:12:42.310 And then I would write the log of 1. 01:12:42.310 --> 01:12:45.030 And then I have in both cases, a factor which 01:12:45.030 --> 01:12:49.810 is e to the beta mu minus epsilon of k. 01:12:49.810 --> 01:12:52.050 But occurring with different signs 01:12:52.050 --> 01:12:54.850 for the bosons and fermions, which again I 01:12:54.850 --> 01:12:58.890 can combine into a single expression by putting a minus 01:12:58.890 --> 01:13:01.920 eta here. 01:13:01.920 --> 01:13:10.890 So this is a general result for any Hamiltonian 01:13:10.890 --> 01:13:14.210 that has the characteristic that we wrote over here. 01:13:14.210 --> 01:13:16.960 So this does not have to be particles in a box. 01:13:16.960 --> 01:13:20.350 It could be particles in a harmonic oscillator. 01:13:20.350 --> 01:13:23.750 These could be energy levels of a harmonic oscillator. 01:13:23.750 --> 01:13:27.470 All you need to do is to make the appropriate sum 01:13:27.470 --> 01:13:30.520 over the one-particle levels harmonic oscillator, 01:13:30.520 --> 01:13:34.010 or whatever else you have, of these factors that 01:13:34.010 --> 01:13:37.650 depend on the individual energy levels 01:13:37.650 --> 01:13:39.120 of the one-particle system. 01:13:50.720 --> 01:13:52.790 Now, one of the things that we will encounter 01:13:52.790 --> 01:13:56.870 having made this transition from canonical, where we knew 01:13:56.870 --> 01:14:01.070 how many particles we had, to grand canonical, where we only 01:14:01.070 --> 01:14:04.510 know the chemical potential, is that we would ultimately 01:14:04.510 --> 01:14:09.360 want to express things in terms of the number of particles. 01:14:09.360 --> 01:14:16.070 So it makes sense to calculate how many particles 01:14:16.070 --> 01:14:21.060 you have given that you have fixed the chemical potential. 01:14:21.060 --> 01:14:42.985 So for that we note the following. 01:14:45.560 --> 01:14:55.410 That essentially, we were able to do this calculation for Q 01:14:55.410 --> 01:14:58.580 because it was a product of contributions 01:14:58.580 --> 01:15:04.560 that we had for the individual one-particle states. 01:15:04.560 --> 01:15:10.510 So clearly, as far as this normalization is concerned, 01:15:10.510 --> 01:15:15.680 the individual one-particle states are independent. 01:15:15.680 --> 01:15:20.752 And indeed, what we can say is that in this ensemble, 01:15:20.752 --> 01:15:27.610 there is a classical probability for a set of occupation numbers 01:15:27.610 --> 01:15:31.550 of one particle states, which is simply 01:15:31.550 --> 01:15:37.960 a product over the different one-particle states of e 01:15:37.960 --> 01:15:49.830 to the beta mu minus epsilon k nk appropriately normalized. 01:15:49.830 --> 01:15:54.010 And again, the restriction on n's 01:15:54.010 --> 01:15:57.200 being 0 or 1 for fermions or anything 01:15:57.200 --> 01:16:01.100 for bosons would be implicit in either case. 01:16:01.100 --> 01:16:05.830 But in either case, essentially the occupation numbers 01:16:05.830 --> 01:16:08.690 are independently taken from distributions 01:16:08.690 --> 01:16:11.810 that I've discussed [INAUDIBLE]. 01:16:11.810 --> 01:16:13.860 So you can, in fact, independently 01:16:13.860 --> 01:16:17.350 calculate the average occupation number 01:16:17.350 --> 01:16:22.460 that you have for each one of these single-particle states. 01:16:22.460 --> 01:16:29.790 And it's clear that you could get that by, for example, 01:16:29.790 --> 01:16:34.150 bringing down a factor of nk here. 01:16:34.150 --> 01:16:37.190 And you can bring down a factor of nk 01:16:37.190 --> 01:16:44.170 by taking a derivative of Q with respect to beta epsilon k 01:16:44.170 --> 01:16:48.420 with a minus sign and normalizing it, 01:16:48.420 --> 01:16:50.632 so you would have log. 01:16:50.632 --> 01:16:53.350 So you would have an expression such as this. 01:16:57.250 --> 01:16:59.890 So you basically would need to calculate, 01:16:59.890 --> 01:17:01.960 since you are taking derivative with respect 01:17:01.960 --> 01:17:08.740 to epsilon k the corresponding log for which epsilon 01:17:08.740 --> 01:17:10.500 k appears. 01:17:10.500 --> 01:17:12.890 Actually, for the case of fermions, 01:17:12.890 --> 01:17:14.890 really there are two possibilities. 01:17:14.890 --> 01:17:18.360 n is either 0 or 1. 01:17:18.360 --> 01:17:21.260 So you would say that the expectation value would 01:17:21.260 --> 01:17:27.440 be when it is 1, you have e to the beta epsilon of k minus mu. 01:17:27.440 --> 01:17:28.370 Oops. 01:17:28.370 --> 01:17:31.550 e to the beta mu minus epsilon of k. 01:17:34.330 --> 01:17:36.910 The two possibilities are 1 plus e 01:17:36.910 --> 01:17:40.450 to the beta mu minus epsilon of k. 01:17:40.450 --> 01:17:44.630 So when I look at some particular state, 01:17:44.630 --> 01:17:46.030 it is either empty. 01:17:46.030 --> 01:17:48.830 In which case, contributes 0. 01:17:48.830 --> 01:17:51.140 Or, it is occupied. 01:17:51.140 --> 01:17:53.362 In which case, it contributes this weight, 01:17:53.362 --> 01:17:55.070 which has to be appropriately normalized. 01:17:57.650 --> 01:18:01.460 If I do the same thing for the case of bosons, 01:18:01.460 --> 01:18:04.400 it is a bit more complicated because I 01:18:04.400 --> 01:18:07.490 have to look at this series rather than geometric 1 01:18:07.490 --> 01:18:10.440 plus x plus x squared plus x cubed is 01:18:10.440 --> 01:18:14.970 1 plus x plus 2x squared plus 3x cubed, which can be obtained 01:18:14.970 --> 01:18:19.430 by taking the derivative of the appropriate log. 01:18:19.430 --> 01:18:22.750 Or you can fall back on your calculations 01:18:22.750 --> 01:18:27.160 of geometric series and convince yourself 01:18:27.160 --> 01:18:29.880 that it is essentially the same thing with a factor of minus 01:18:29.880 --> 01:18:30.380 here. 01:18:34.850 --> 01:18:40.798 So this is fermions and this is bosons. 01:18:45.480 --> 01:18:52.000 And indeed, I can put the two expressions together 01:18:52.000 --> 01:18:57.390 by dividing through this factor in both of them 01:18:57.390 --> 01:19:03.380 and write it as 1 over Z inverse e 01:19:03.380 --> 01:19:08.700 to the beta epsilon of k minus eta, 01:19:08.700 --> 01:19:11.560 where for convenience I have introduced 01:19:11.560 --> 01:19:16.117 Z to be the contribution e to the beta. 01:19:30.100 --> 01:19:36.530 So for this system of non-interacting particles 01:19:36.530 --> 01:19:40.090 that are identical, we have expressions 01:19:40.090 --> 01:19:45.890 for log of the grand partition function, the grand potential. 01:19:45.890 --> 01:19:48.220 And for the average number of particles, 01:19:48.220 --> 01:19:51.260 which is an appropriate derivative of this, 01:19:51.260 --> 01:19:54.410 expressed in terms of the single-particle energy 01:19:54.410 --> 01:19:58.440 levels and the chemical potential. 01:19:58.440 --> 01:20:01.710 So next time, what we will do is we 01:20:01.710 --> 01:20:03.880 will start this with this expression 01:20:03.880 --> 01:20:06.576 for the case of the particles in a box 01:20:06.576 --> 01:20:11.350 to get the pressure of the ideal quantum gas 01:20:11.350 --> 01:20:13.510 as a function of mu. 01:20:13.510 --> 01:20:15.090 But we want to write the pressure 01:20:15.090 --> 01:20:19.260 as a function of density, so we will invert this expression 01:20:19.260 --> 01:20:21.610 to get density as a function of-- chemical 01:20:21.610 --> 01:20:24.620 potential as a function of density [INAUDIBLE] here. 01:20:24.620 --> 01:20:27.580 And therefore, get the expression for pressure 01:20:27.580 --> 01:20:30.620 as a function of density.