1 00:00:00,070 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a Creative Commons license. 3 00:00:04,019 --> 00:00:06,870 Your support will help MIT OpenCourseWare continue 4 00:00:06,870 --> 00:00:10,730 to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation, or view additional materials 6 00:00:13,330 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:21,510 --> 00:00:26,000 PROFESSOR: So last lecture, we laid out 9 00:00:26,000 --> 00:00:28,700 the foundations of quantum stat mech. 10 00:00:36,565 --> 00:00:39,700 And to remind you, in quantum mechanics 11 00:00:39,700 --> 00:00:45,830 we said we could very formally describe a state 12 00:00:45,830 --> 00:00:49,950 through a vector in Hilbert space 13 00:00:49,950 --> 00:00:53,080 that has complex components in terms of some basis 14 00:00:53,080 --> 00:00:55,570 that we haven't specified. 15 00:00:55,570 --> 00:00:58,850 And quantum mechanics describes, essentially 16 00:00:58,850 --> 00:01:01,170 what happens to one of these states, 17 00:01:01,170 --> 00:01:05,000 how to interpret it, et cetera. 18 00:01:05,000 --> 00:01:08,700 But this was, in our language, a pure state. 19 00:01:08,700 --> 00:01:13,480 We were interested in cases where we have an ensemble 20 00:01:13,480 --> 00:01:16,620 and there are many members of the ensemble that 21 00:01:16,620 --> 00:01:19,790 correspond to the same macroscopic state. 22 00:01:19,790 --> 00:01:24,050 And we can distinguish them macroscopically. 23 00:01:24,050 --> 00:01:27,510 If we think about each one of those states 24 00:01:27,510 --> 00:01:32,070 being one particular member, psi alpha, 25 00:01:32,070 --> 00:01:35,760 some vector in this Hilbert space occurring 26 00:01:35,760 --> 00:01:41,060 with some probability P alpha, then that's 27 00:01:41,060 --> 00:01:44,210 kind of a more probabilistic representation 28 00:01:44,210 --> 00:01:49,470 of what's going on here for the mech state. 29 00:01:49,470 --> 00:01:52,460 And the question was, well, how do we manipulate things 30 00:01:52,460 --> 00:01:58,020 when we have a collection of states in this mixed form? 31 00:01:58,020 --> 00:02:01,060 And we saw that in quantum mechanics, 32 00:02:01,060 --> 00:02:03,255 most of the things that are observable 33 00:02:03,255 --> 00:02:06,420 are expressed in terms of matrices. 34 00:02:06,420 --> 00:02:14,820 And one way to convert a state vector into a matrix 35 00:02:14,820 --> 00:02:23,190 is to conjugate it and construct the matrix whose elements are 36 00:02:23,190 --> 00:02:27,470 obtained-- a particular rho vector by taking element 37 00:02:27,470 --> 00:02:31,260 and complex conjugate elements of the vector. 38 00:02:31,260 --> 00:02:38,360 And if we were to then sum over all members of this ensemble, 39 00:02:38,360 --> 00:02:42,410 this would give us an object that we call the density 40 00:02:42,410 --> 00:02:45,220 matrix, rho. 41 00:02:45,220 --> 00:02:48,340 And the property of this density matrix 42 00:02:48,340 --> 00:02:53,970 was that if I had some observable for a pure state 43 00:02:53,970 --> 00:02:56,640 I would be able to calculate an expectation 44 00:02:56,640 --> 00:03:00,010 value for this observable by sandwiching it 45 00:03:00,010 --> 00:03:03,910 between the state in the way that quantum mechanic has 46 00:03:03,910 --> 00:03:05,460 taught us. 47 00:03:05,460 --> 00:03:10,450 In our mech state, we would get an ensemble average 48 00:03:10,450 --> 00:03:16,420 of this quantity by multiplying this density matrix 49 00:03:16,420 --> 00:03:21,100 with the matrix that represents the operator whose average we 50 00:03:21,100 --> 00:03:24,630 are trying to calculate, and then taking 51 00:03:24,630 --> 00:03:27,155 the trace of the product of these two matrices. 52 00:03:32,290 --> 00:03:35,980 Now, the next statement was that in the same way 53 00:03:35,980 --> 00:03:40,100 that the micro-state classically changes as a function of time, 54 00:03:40,100 --> 00:03:42,630 the vector that we have in quantum mechanics 55 00:03:42,630 --> 00:03:46,300 is also changing as a function of time. 56 00:03:46,300 --> 00:03:49,570 And so at a different instant of time, 57 00:03:49,570 --> 00:03:52,230 our state vectors have changed. 58 00:03:52,230 --> 00:03:56,970 And in principle, our density has changed. 59 00:03:56,970 --> 00:04:03,085 And that would imply a change potentially in the average 60 00:04:03,085 --> 00:04:04,320 that we have over here. 61 00:04:06,870 --> 00:04:10,410 We said, OK, let's look at this time dependence. 62 00:04:10,410 --> 00:04:15,540 And we know that these psis obey Schrodinger equation. 63 00:04:15,540 --> 00:04:19,480 I h bar d psi by dt is h psi. 64 00:04:19,480 --> 00:04:22,630 So we did the same operation, I h bar d 65 00:04:22,630 --> 00:04:26,850 by dt of this density matrix. 66 00:04:26,850 --> 00:04:29,290 And we found that essentially, what we 67 00:04:29,290 --> 00:04:33,410 would get on the other side is the commutator 68 00:04:33,410 --> 00:04:35,840 of the density matrix and the matrix 69 00:04:35,840 --> 00:04:39,690 corresponding to the Hamiltonian, 70 00:04:39,690 --> 00:04:47,100 which was reminiscent of the Liouville statement 71 00:04:47,100 --> 00:04:49,020 that we had classically. 72 00:04:49,020 --> 00:04:53,090 Classically, we had that d rho by dt 73 00:04:53,090 --> 00:04:59,391 was the Poisson bracket of a rho with H versus this quantum 74 00:04:59,391 --> 00:04:59,890 formulation. 75 00:05:03,300 --> 00:05:07,410 Now, in both cases we are looking 76 00:05:07,410 --> 00:05:10,940 for some kind of a density that represents an equilibrium 77 00:05:10,940 --> 00:05:13,180 ensemble. 78 00:05:13,180 --> 00:05:14,920 And presumably, the characteristic 79 00:05:14,920 --> 00:05:17,650 of the equilibrium ensemble is that various measurements 80 00:05:17,650 --> 00:05:21,500 that you make at one time and another time are the same. 81 00:05:21,500 --> 00:05:24,390 And hence, we want this rho equilibrium 82 00:05:24,390 --> 00:05:27,920 to be something that does not change as a function of time, 83 00:05:27,920 --> 00:05:31,600 which means that if we put it in either one of these equations, 84 00:05:31,600 --> 00:05:35,560 the right-hand side for d rho by dt should be 0. 85 00:05:35,560 --> 00:05:39,000 And clearly, a simple way to do that 86 00:05:39,000 --> 00:05:45,040 is to make rho equilibrium be a function of H. 87 00:05:45,040 --> 00:05:50,930 In the classical context, H was a function in phase space. 88 00:05:50,930 --> 00:05:54,000 Rho equilibrium was, therefore, made a function 89 00:05:54,000 --> 00:05:57,000 of various points in phase space implicitly 90 00:05:57,000 --> 00:06:01,450 through its dependence on H. In the current case, 91 00:06:01,450 --> 00:06:04,450 H is a matrix in Hilbert space. 92 00:06:04,450 --> 00:06:07,070 A function of a matrix is another function, 93 00:06:07,070 --> 00:06:09,610 and that's how we construct rho equilibrium. 94 00:06:09,610 --> 00:06:12,510 And that function will also commute with H 95 00:06:12,510 --> 00:06:15,740 because clearly H with H is 0. 96 00:06:15,740 --> 00:06:19,350 Any function of H with H will be 0. 97 00:06:19,350 --> 00:06:23,400 So the prescription that we have in order now 98 00:06:23,400 --> 00:06:27,460 to construct the quantum statistical mechanics 99 00:06:27,460 --> 00:06:33,960 is to follow what we had done already for the classical case. 100 00:06:33,960 --> 00:06:36,485 And classically following this equation, 101 00:06:36,485 --> 00:06:40,800 we made postulates relating rho-- 102 00:06:40,800 --> 00:06:42,710 what its functional dependence was 103 00:06:42,710 --> 00:06:47,930 on H. We can now do the same thing in the quantum context. 104 00:06:47,930 --> 00:06:51,360 So let's do that following the procedure 105 00:06:51,360 --> 00:06:55,480 that we followed for the classical case. 106 00:06:55,480 --> 00:06:57,820 So classically, we said, well, let's look 107 00:06:57,820 --> 00:07:00,090 at an ensemble that was micro-canonical. 108 00:07:05,940 --> 00:07:09,500 So in the micro-canonical ensemble, 109 00:07:09,500 --> 00:07:13,630 we specified what the energy of the system was, 110 00:07:13,630 --> 00:07:16,880 but we didn't allow either work to be performed 111 00:07:16,880 --> 00:07:21,440 on it mechanically or chemically so that the number of elements 112 00:07:21,440 --> 00:07:22,200 was fixed. 113 00:07:22,200 --> 00:07:25,060 The coordinates, such as volume, length of the system, 114 00:07:25,060 --> 00:07:26,810 et cetera, were fixed. 115 00:07:26,810 --> 00:07:30,620 And in this ensemble, classically the statement 116 00:07:30,620 --> 00:07:39,170 was that rho indexed by E is a function of H. 117 00:07:39,170 --> 00:07:42,600 And clearly, what we want is to only allow 118 00:07:42,600 --> 00:07:46,370 H's that correspond to the right energy. 119 00:07:46,370 --> 00:07:50,070 So I will use a kind of shorthand notation 120 00:07:50,070 --> 00:07:52,150 as kind of a delta H E. 121 00:07:52,150 --> 00:07:57,130 Although, when we were doing it last time around, we allowed 122 00:07:57,130 --> 00:08:00,660 this function to allow a range of possible E-values. 123 00:08:00,660 --> 00:08:03,850 I can do the same thing, it's just that writing that out 124 00:08:03,850 --> 00:08:05,690 is slightly less convenient. 125 00:08:05,690 --> 00:08:08,630 So let's stick with the convenient form. 126 00:08:08,630 --> 00:08:12,550 Essentially, it says within some range delta A 127 00:08:12,550 --> 00:08:15,600 allow the Hamiltonians. 128 00:08:15,600 --> 00:08:18,240 Allow states, micro-states, whose Hamiltonian 129 00:08:18,240 --> 00:08:22,530 would give the right energy that is comparable to the energy 130 00:08:22,530 --> 00:08:27,280 that we have said exists for the macro-state. 131 00:08:27,280 --> 00:08:30,040 Now clearly also, this definition 132 00:08:30,040 --> 00:08:34,190 tells me that the trace of rho has to be 1. 133 00:08:34,190 --> 00:08:38,679 Or classically, the integral of rho E over the entire phase 134 00:08:38,679 --> 00:08:40,179 space has to be 1. 135 00:08:40,179 --> 00:08:43,210 So there is a normalization condition. 136 00:08:43,210 --> 00:08:47,890 And this normalization condition gave us the quantity omega 137 00:08:47,890 --> 00:08:50,290 of E, the number of states of energy E, 138 00:08:50,290 --> 00:08:53,110 out of which we then constructed the entropy 139 00:08:53,110 --> 00:08:56,870 and we were running away calculating 140 00:08:56,870 --> 00:09:01,050 various thermodynamic quantities. 141 00:09:01,050 --> 00:09:07,400 So now let's see how we evaluate this in the quantum case. 142 00:09:07,400 --> 00:09:10,330 We will use the same expression, but now me 143 00:09:10,330 --> 00:09:14,570 realize that rho is a matrix. 144 00:09:14,570 --> 00:09:18,090 So I have to evaluate, maybe elements of that matrix 145 00:09:18,090 --> 00:09:22,390 to clarify what this matrix looks like in some basis. 146 00:09:22,390 --> 00:09:25,190 What's the most convenient basis? 147 00:09:25,190 --> 00:09:28,570 Since rho is expressed as a function of H, 148 00:09:28,570 --> 00:09:40,880 the most convenient basis is the energy basis, 149 00:09:40,880 --> 00:09:45,750 which is the basis that diagonalizes your Hamiltonian 150 00:09:45,750 --> 00:09:46,790 matrix. 151 00:09:46,790 --> 00:09:54,250 Basically, there's some vectors in this Hilbert space 152 00:09:54,250 --> 00:09:56,630 such that the action of H on this 153 00:09:56,630 --> 00:09:59,400 will give us some energy that I will 154 00:09:59,400 --> 00:10:01,885 call epsilon n, some eigenvalue. 155 00:10:01,885 --> 00:10:02,590 An n. 156 00:10:02,590 --> 00:10:06,730 So that's the definition of the energy basis. 157 00:10:06,730 --> 00:10:11,930 Again, as all basis, we can make these basis vectors 158 00:10:11,930 --> 00:10:17,660 n to be unit length and orthogonal to each other. 159 00:10:17,660 --> 00:10:20,660 There is an orthonormal basis. 160 00:10:20,660 --> 00:10:25,590 If I evaluate rho E in this basis, what do I find? 161 00:10:25,590 --> 00:10:32,075 I find that n rho m. 162 00:10:32,075 --> 00:10:34,310 Well, 1 over omega E is just the constant. 163 00:10:34,310 --> 00:10:35,285 It comes out front. 164 00:10:39,280 --> 00:10:43,930 And the meaning of this delta function becomes obvious. 165 00:10:43,930 --> 00:10:48,400 It is 1 or 0. 166 00:10:48,400 --> 00:11:06,840 It is 1 if, let's say, Em equals to the right energy. 167 00:11:06,840 --> 00:11:11,970 Em is the right energy for the ensemble E. 168 00:11:11,970 --> 00:11:15,150 And of course, there is a delta function. 169 00:11:15,150 --> 00:11:21,460 So there is also an m equals to n. 170 00:11:21,460 --> 00:11:26,560 And it is 0, clearly, for states that have the wrong energy. 171 00:11:32,040 --> 00:11:35,710 But there is an additional thing here 172 00:11:35,710 --> 00:11:39,795 that I will explain shortly for m not equal to n. 173 00:11:43,520 --> 00:11:45,860 So let's parse the two statements 174 00:11:45,860 --> 00:11:48,670 that I have made over here. 175 00:11:48,670 --> 00:11:54,070 The first one, it says that if I know 176 00:11:54,070 --> 00:11:57,410 the energy of my macro-state, I clearly 177 00:11:57,410 --> 00:12:02,290 have to find wave functions to construct 178 00:12:02,290 --> 00:12:05,230 possible states of what is in the box that 179 00:12:05,230 --> 00:12:08,060 have the right energy. 180 00:12:08,060 --> 00:12:12,700 States that don't have the right energy are not admitted. 181 00:12:12,700 --> 00:12:16,380 States that are the right energy, 182 00:12:16,380 --> 00:12:19,570 I have nothing against each one of them. 183 00:12:19,570 --> 00:12:23,410 So I give them all the same probability. 184 00:12:23,410 --> 00:12:36,740 So this is our whole assumption of equal equilibrium 185 00:12:36,740 --> 00:12:39,940 a priori probabilities. 186 00:12:46,490 --> 00:12:52,480 But now I have a quantum system and I'm looking at the matrix. 187 00:12:52,480 --> 00:12:58,750 And this matrix potentially has off-diagonal elements. 188 00:12:58,750 --> 00:13:02,370 You see, if I am looking at m equals to n, 189 00:13:02,370 --> 00:13:04,890 it means that I am looking at the diagonal elements 190 00:13:04,890 --> 00:13:07,280 of the matrix. 191 00:13:07,280 --> 00:13:11,720 What this says is that if the energies are degenerate-- let's 192 00:13:11,720 --> 00:13:16,610 say I have 100 states, all of them have the right energy, 193 00:13:16,610 --> 00:13:19,080 but they are orthonormal. 194 00:13:19,080 --> 00:13:23,670 These 100 states, when I look at the density matrix 195 00:13:23,670 --> 00:13:26,930 in the corresponding basis, the off-diagonal elements 196 00:13:26,930 --> 00:13:29,760 would be 0. 197 00:13:29,760 --> 00:13:31,310 So this is the "or." 198 00:13:31,310 --> 00:13:34,210 So even if this condition is satisfied, 199 00:13:34,210 --> 00:13:37,400 even if Em equals to E, but I am looking 200 00:13:37,400 --> 00:13:41,370 at off-diagonal elements, I have to put 0. 201 00:13:41,370 --> 00:13:44,255 And this is sometimes called the assumption of random phases. 202 00:14:00,260 --> 00:14:02,030 Before telling you why that's called 203 00:14:02,030 --> 00:14:05,380 an assumption of random phases, let's also 204 00:14:05,380 --> 00:14:09,340 characterize what this omega of E is. 205 00:14:09,340 --> 00:14:17,640 Because trace of rho has to be 1, clearly omega of E 206 00:14:17,640 --> 00:14:22,010 is the trace of this delta h E. Essentially 207 00:14:22,010 --> 00:14:26,510 as I scan all of my possible energy levels 208 00:14:26,510 --> 00:14:30,910 in this energy basis, I will get 1 209 00:14:30,910 --> 00:14:34,310 for those that have the right energy and 0 otherwise. 210 00:14:34,310 --> 00:14:43,540 So basically, this is simply number of states of energy E, 211 00:14:43,540 --> 00:14:45,960 potentially with minus plus some delta E 212 00:14:45,960 --> 00:14:47,203 if I want to include that. 213 00:14:50,990 --> 00:14:56,220 Now, what these assumptions mean are kind of like this. 214 00:14:56,220 --> 00:15:00,160 So I have my box and I have been told 215 00:15:00,160 --> 00:15:04,350 that I have energy E. What's a potential wave function 216 00:15:04,350 --> 00:15:07,060 that I can have in this box? 217 00:15:07,060 --> 00:15:12,270 What's the psi given that I know what the energy E is? 218 00:15:12,270 --> 00:15:17,310 Well, any superposition of these omega E states 219 00:15:17,310 --> 00:15:21,460 that have the right energy will work fine. 220 00:15:21,460 --> 00:15:26,370 So I have a sum over, let's say, mu 221 00:15:26,370 --> 00:15:32,090 that belongs to the state such that H-- such 222 00:15:32,090 --> 00:15:36,770 that this energy En is equal to the energy 223 00:15:36,770 --> 00:15:38,010 that I have specified. 224 00:15:38,010 --> 00:15:41,920 And there are omega sub E such states. 225 00:15:41,920 --> 00:15:46,880 I have to find some kind of an amplitude for these states, 226 00:15:46,880 --> 00:15:50,397 and then the corresponding state mu. 227 00:15:53,260 --> 00:15:55,561 I guess I should write here E mu. 228 00:15:59,980 --> 00:16:05,460 Now, the first statement over here 229 00:16:05,460 --> 00:16:08,540 is that as far as I'm concerned, I 230 00:16:08,540 --> 00:16:12,850 can put all of these a mu's in whatever proportion 231 00:16:12,850 --> 00:16:13,690 that I want. 232 00:16:13,690 --> 00:16:16,830 Any linear combination will work out fine. 233 00:16:16,830 --> 00:16:19,780 Because ultimately, psi has to be normalized. 234 00:16:19,780 --> 00:16:23,680 Presumably, the typical magnitude a m 235 00:16:23,680 --> 00:16:27,380 squared, if I average over all members of the ensemble, 236 00:16:27,380 --> 00:16:30,340 should be 1 over omega. 237 00:16:30,340 --> 00:16:32,400 So this is a superposition. 238 00:16:32,400 --> 00:16:37,930 Typically, all of them would contribute. 239 00:16:37,930 --> 00:16:41,800 But since we are thinking about quantum mechanics, 240 00:16:41,800 --> 00:16:44,040 these amplitudes can, in fact, be complex. 241 00:16:47,700 --> 00:16:51,440 And this statement of random phases 242 00:16:51,440 --> 00:16:54,030 is more or less equivalent to saying 243 00:16:54,030 --> 00:17:00,440 that the phases of the different elements 244 00:17:00,440 --> 00:17:03,680 would be typically uncorrelated when you average 245 00:17:03,680 --> 00:17:06,308 over all possible members of this ensemble. 246 00:17:11,790 --> 00:17:15,140 Just to emphasize this a little bit more, 247 00:17:15,140 --> 00:17:18,849 let's think about the very simplest case 248 00:17:18,849 --> 00:17:23,520 that we can have for thinking about probability. 249 00:17:23,520 --> 00:17:26,810 And you would think of, say, a coin that 250 00:17:26,810 --> 00:17:30,090 can have two possibilities, head or tail. 251 00:17:30,090 --> 00:17:33,290 So classically, you would say, head or tail, 252 00:17:33,290 --> 00:17:37,120 not knowing anything else, are equally likely. 253 00:17:37,120 --> 00:17:40,480 The quantum analog of that is a quantum bit. 254 00:17:43,270 --> 00:17:47,770 And the qubit can have, let's say, up or down states. 255 00:17:47,770 --> 00:17:52,540 It's a Hilbert space that is composed of two elements. 256 00:17:52,540 --> 00:17:58,870 So the corresponding matrix that I would have would be a 2 by 2. 257 00:17:58,870 --> 00:18:01,260 And according to the construction that I have, 258 00:18:01,260 --> 00:18:02,880 it will be something like this. 259 00:18:07,220 --> 00:18:11,530 What are the possible wave functions for this system? 260 00:18:15,110 --> 00:18:17,440 I can have any linear combination 261 00:18:17,440 --> 00:18:26,760 of, say, up and down, with any amplitude here. 262 00:18:26,760 --> 00:18:29,740 And so essentially, the amplitudes, presumably, 263 00:18:29,740 --> 00:18:35,980 are quantities that I will call alpha up and alpha down. 264 00:18:35,980 --> 00:18:40,820 That, on average, alpha squared up or down would be 1/2. 265 00:18:40,820 --> 00:18:43,470 That's what would appear here. 266 00:18:43,470 --> 00:18:45,560 The elements that appear here according 267 00:18:45,560 --> 00:18:48,440 to the construction that I have up there, 268 00:18:48,440 --> 00:18:51,020 I have to really take this element, 269 00:18:51,020 --> 00:18:54,940 average it against the complex conjugate of that element. 270 00:18:54,940 --> 00:18:58,310 So what will appear here would be something like e 271 00:18:58,310 --> 00:19:04,910 to the i phi of up minus phi of down, where, let's say, 272 00:19:04,910 --> 00:19:07,201 I put here phi of up. 273 00:19:10,640 --> 00:19:14,840 And what I'm saying is that there 274 00:19:14,840 --> 00:19:17,655 are a huge number of possibilities. 275 00:19:17,655 --> 00:19:23,810 Whereas, the classical coin has really two possibilities, head 276 00:19:23,810 --> 00:19:27,000 or tail, the quantum analog of this 277 00:19:27,000 --> 00:19:30,010 is a huge set of possibilities. 278 00:19:30,010 --> 00:19:33,270 These phis can be anything, 0 to 2 pi. 279 00:19:33,270 --> 00:19:35,580 Independent of each other, the amplitudes 280 00:19:35,580 --> 00:19:38,000 can be anything as long as the eventual normalization 281 00:19:38,000 --> 00:19:39,720 is satisfied. 282 00:19:39,720 --> 00:19:42,760 And as you sort of sum over all possibilities, 283 00:19:42,760 --> 00:19:44,737 you would get something like this also. 284 00:19:56,920 --> 00:20:05,580 Now, the more convenient ensemble for calculating things 285 00:20:05,580 --> 00:20:13,830 is the canonical one, where rather than specifying 286 00:20:13,830 --> 00:20:16,410 what the energy of the system is, 287 00:20:16,410 --> 00:20:18,510 I tell you what the temperature is. 288 00:20:18,510 --> 00:20:21,150 I still don't allow work to take place, 289 00:20:21,150 --> 00:20:24,750 so these other elements we kept fixed. 290 00:20:24,750 --> 00:20:30,850 And our classical description for rho sub T 291 00:20:30,850 --> 00:20:35,880 was e to the minus beta H divided 292 00:20:35,880 --> 00:20:40,611 by some partition function, where beta was 1 over kT, 293 00:20:40,611 --> 00:20:41,110 of course. 294 00:20:51,000 --> 00:20:56,640 So again, in the energy basis, this would be diagonal. 295 00:20:56,640 --> 00:21:00,720 And the diagonal elements would be e to the minus beta epsilon 296 00:21:00,720 --> 00:21:02,960 n. 297 00:21:02,960 --> 00:21:06,760 I could calculate the normalization Z, which 298 00:21:06,760 --> 00:21:10,270 would be trace of e to the minus beta H. 299 00:21:10,270 --> 00:21:13,605 This trace is calculated most easily the basis 300 00:21:13,605 --> 00:21:15,980 in which H is diagonal. 301 00:21:15,980 --> 00:21:19,590 And then I just pick out all of the diagonal elements, sum 302 00:21:19,590 --> 00:21:21,670 over n e to the minus beta epsilon. 303 00:21:28,790 --> 00:21:32,140 Now if you recall, we already did something like this 304 00:21:32,140 --> 00:21:34,900 without justification where we said 305 00:21:34,900 --> 00:21:38,510 that the states of a harmonic oscillator, we postulated 306 00:21:38,510 --> 00:21:42,870 to be quantized h bar omega n plus 1/2. 307 00:21:42,870 --> 00:21:45,460 And then we calculated the partition function 308 00:21:45,460 --> 00:21:48,310 by summing over all states. 309 00:21:48,310 --> 00:21:54,460 You can see that this would provide 310 00:21:54,460 --> 00:21:56,120 the justification for that. 311 00:22:04,950 --> 00:22:11,200 Now, various things that we have in classical formulations also 312 00:22:11,200 --> 00:22:17,180 work, such that classically if we had Z, 313 00:22:17,180 --> 00:22:19,500 we could take the log of Z. We could 314 00:22:19,500 --> 00:22:23,210 take a derivative of log Z with respect to beta. 315 00:22:23,210 --> 00:22:26,180 It would bring down a factor of H. 316 00:22:26,180 --> 00:22:33,880 And then we show that this was equal to the average 317 00:22:33,880 --> 00:22:36,210 of the energy. 318 00:22:36,210 --> 00:22:38,770 It was the average of Hamiltonian, which we then 319 00:22:38,770 --> 00:22:42,850 would identify with the thermodynamic energy. 320 00:22:42,850 --> 00:22:49,030 It is easy to show that if you take the same set of operations 321 00:22:49,030 --> 00:22:56,960 over here, what you would get is trace of H rho, which 322 00:22:56,960 --> 00:23:00,888 is the definition of the average that you find. 323 00:23:15,070 --> 00:23:21,550 Now, let's do one example in the canonical ensemble, 324 00:23:21,550 --> 00:23:27,526 which is particle in box. 325 00:23:31,590 --> 00:23:36,530 Basically, this is the kind of thing that we did all the time. 326 00:23:36,530 --> 00:23:40,780 We assume that there is some box of volume v. 327 00:23:40,780 --> 00:23:46,120 And for the time being, I just put one particle in it. 328 00:23:46,120 --> 00:23:48,900 Assume that there is no potential. 329 00:23:48,900 --> 00:23:52,290 So the Hamiltonian for this one particle 330 00:23:52,290 --> 00:23:56,250 is just its kinetic energy p1 squared over 2m, 331 00:23:56,250 --> 00:23:58,160 maybe plus some kind of a box potential. 332 00:24:03,410 --> 00:24:06,430 Now, if I want to think about this quantum mechanically, 333 00:24:06,430 --> 00:24:09,000 I have to think about some kind of a basis 334 00:24:09,000 --> 00:24:11,950 and calculating things in some kind of basis. 335 00:24:11,950 --> 00:24:14,450 So for the time being, let's imagine 336 00:24:14,450 --> 00:24:17,215 that I do things in coordinate basis. 337 00:24:22,730 --> 00:24:27,930 So I want to have where the particle is. 338 00:24:27,930 --> 00:24:30,050 And let's call the coordinates x. 339 00:24:30,050 --> 00:24:33,140 Let's say, a vector x that has x, y, and z-components. 340 00:24:36,480 --> 00:24:40,020 Then, in the coordinate basis, p is 341 00:24:40,020 --> 00:24:44,940 h bar over i, the gradient vector. 342 00:24:44,940 --> 00:24:51,240 So this becomes minus h bar squared Laplacian over 2m. 343 00:24:54,900 --> 00:25:01,690 And if I want to look at what are 344 00:25:01,690 --> 00:25:07,860 the eigenstates of his Hamiltonian represented 345 00:25:07,860 --> 00:25:11,270 in coordinates basis, as usual I want 346 00:25:11,270 --> 00:25:17,725 to have something like H1 acting on some function of position 347 00:25:17,725 --> 00:25:20,370 giving me the energy. 348 00:25:20,370 --> 00:25:29,250 And so I will indicate that function of position as x k. 349 00:25:33,696 --> 00:25:37,740 And I want to know what that energy is. 350 00:25:41,780 --> 00:25:44,710 And the reason I do that is because you all 351 00:25:44,710 --> 00:25:51,510 know that the correct form of these eigenfunctions 352 00:25:51,510 --> 00:25:55,350 are things like e to the i k dot x. 353 00:25:59,560 --> 00:26:04,500 And since I want these to be normalized when I integrate 354 00:26:04,500 --> 00:26:14,950 over all coordinates, I have to divide by square root of V. 355 00:26:14,950 --> 00:26:18,340 So that's all nice and fine. 356 00:26:18,340 --> 00:26:23,420 What I want to do is to calculate the density matrix 357 00:26:23,420 --> 00:26:27,040 in this coordinate basis. 358 00:26:27,040 --> 00:26:33,175 So I pick some point x prime rho x. 359 00:26:35,740 --> 00:26:37,440 And it's again, a one particle. 360 00:26:37,440 --> 00:26:37,940 Yes? 361 00:26:37,940 --> 00:26:40,290 AUDIENCE: I was thinking the box to be infinite 362 00:26:40,290 --> 00:26:45,610 or-- I mean, how to treat a boundary because [INAUDIBLE]. 363 00:26:45,610 --> 00:26:48,810 PROFESSOR: So I wasn't careful about that. 364 00:26:48,810 --> 00:26:52,530 The allowed values of k that I have to pick here 365 00:26:52,530 --> 00:26:55,550 will reflect the boundary conditions. 366 00:26:55,550 --> 00:26:59,520 And indeed, I will be able to use these kinds of things. 367 00:26:59,520 --> 00:27:03,530 Let's say, by using periodic boundary condition. 368 00:27:07,390 --> 00:27:10,190 And if I choose periodic boundary conditions, 369 00:27:10,190 --> 00:27:15,470 the allowed values of k will be quantized. 370 00:27:15,470 --> 00:27:19,475 So that, say, kx would be multiples 371 00:27:19,475 --> 00:27:24,195 of 2 pi over Lx times some integer. 372 00:27:27,660 --> 00:27:32,000 If I use really box boundary conditions such as the wave 373 00:27:32,000 --> 00:27:34,970 function having to vanish at the two ends, 374 00:27:34,970 --> 00:27:38,270 in reality I should use the sine and cosine type 375 00:27:38,270 --> 00:27:42,990 of boundary wave functions, which are superpositions 376 00:27:42,990 --> 00:27:46,600 of these e to the i kx and e to the minus i kx. 377 00:27:46,600 --> 00:27:50,950 And they slightly change the normalization-- 378 00:27:50,950 --> 00:27:53,706 the discretization. 379 00:27:53,706 --> 00:27:57,210 But up to, again, these kinds of superpositions and things 380 00:27:57,210 --> 00:28:01,590 like that, this is a good enough statement. 381 00:28:01,590 --> 00:28:03,790 If you are really doing quantum mechanics, 382 00:28:03,790 --> 00:28:07,650 you have to clearly be more careful. 383 00:28:07,650 --> 00:28:12,550 Just make sure that I'm sufficiently careful, but not 384 00:28:12,550 --> 00:28:16,595 overly careful in what I'm doing next. 385 00:28:20,750 --> 00:28:23,590 So what do I need to do? 386 00:28:23,590 --> 00:28:31,350 Well, rho 1 is nicely diagonalized 387 00:28:31,350 --> 00:28:36,880 in the basis of these plane waves. 388 00:28:36,880 --> 00:28:41,110 In these eigenstates of the Hamiltonian. 389 00:28:41,110 --> 00:28:48,250 So it makes sense for me to switch from this representation 390 00:28:48,250 --> 00:28:52,670 that is characterized by x and x prime 391 00:28:52,670 --> 00:28:58,390 to a representation where I have k. 392 00:28:58,390 --> 00:29:03,770 And so I can do that by inserting a sum over k here. 393 00:29:03,770 --> 00:29:05,680 And then I have rho 1. 394 00:29:05,680 --> 00:29:08,755 Rho 1 is simply e to the minus beta. 395 00:29:08,755 --> 00:29:13,870 It is diagonal in this basis of k's. 396 00:29:13,870 --> 00:29:19,420 There is a normalization that we call Z1, partition function. 397 00:29:19,420 --> 00:29:22,190 And because it is diagonal, in fact 398 00:29:22,190 --> 00:29:26,044 I will use the same k on both sides of this. 399 00:29:28,950 --> 00:29:33,930 So the density matrix is, in fact, diagonal in momentum. 400 00:29:33,930 --> 00:29:36,400 This is none other than momentum, of course. 401 00:29:39,090 --> 00:29:43,175 But I don't want to calculate the density matrix in momentum 402 00:29:43,175 --> 00:29:47,000 where it is diagonal because that's the energy basis we 403 00:29:47,000 --> 00:29:51,360 already saw, but in coordinate basis. 404 00:29:51,360 --> 00:29:54,560 So now let's write all of these things. 405 00:29:54,560 --> 00:29:59,020 First of all, the sum over k when l becomes large, 406 00:29:59,020 --> 00:30:01,770 I'm going to replace with an integral over k. 407 00:30:05,230 --> 00:30:07,860 And then I have to worry about the spacing 408 00:30:07,860 --> 00:30:10,710 that I have between k-values. 409 00:30:10,710 --> 00:30:14,650 So that will multiply with a density of states, which 410 00:30:14,650 --> 00:30:18,740 is 2 pi cubed product of lx, ly, lz, 411 00:30:18,740 --> 00:30:20,680 which will give me the volume of the box. 412 00:30:23,390 --> 00:30:29,790 And then I have these factors of-- how did I define it? 413 00:30:29,790 --> 00:30:36,310 kx is e to the i k dot x. 414 00:30:36,310 --> 00:30:44,880 And this thing where k and x prime are interchanged 415 00:30:44,880 --> 00:30:46,480 is its complex conjugate. 416 00:30:46,480 --> 00:30:49,410 So it becomes x minus x prime. 417 00:30:49,410 --> 00:30:52,940 I have two factors of 1 over square root of V, 418 00:30:52,940 --> 00:30:55,650 so that will give me a factor of 1 over V 419 00:30:55,650 --> 00:30:57,760 from the product of these things. 420 00:30:57,760 --> 00:31:01,940 And then I still have minus beta h bar 421 00:31:01,940 --> 00:31:03,990 squared k squared over 2m. 422 00:31:16,282 --> 00:31:17,670 And then I have the Z1. 423 00:31:29,390 --> 00:31:32,130 Before proceeding, maybe it's worthwhile for me 424 00:31:32,130 --> 00:31:36,530 to calculate the Z1 in this case. 425 00:31:36,530 --> 00:31:45,160 So Z1 is the trace of rho 1. 426 00:31:45,160 --> 00:31:46,550 It's essentially the same thing. 427 00:31:46,550 --> 00:31:53,870 It's a sum over k e to the minus beta h 428 00:31:53,870 --> 00:31:55,450 bar squared k squared over 2m. 429 00:31:59,400 --> 00:32:01,920 I do the same thing that I did over there, 430 00:32:01,920 --> 00:32:08,010 replace this with V integral over k divided by 2 pi cubed 431 00:32:08,010 --> 00:32:12,150 e to the minus beta h bar squared k squared over 2m. 432 00:32:17,040 --> 00:32:20,465 I certainly recognize h bar k as the same thing as momentum. 433 00:32:20,465 --> 00:32:24,630 This is really p squared. 434 00:32:24,630 --> 00:32:28,900 So just let me write this after rescaling 435 00:32:28,900 --> 00:32:31,730 all of the k's by h bar. 436 00:32:31,730 --> 00:32:37,210 So then what I would have is an integral over momentum. 437 00:32:37,210 --> 00:32:41,120 And then for each k, I essentially 438 00:32:41,120 --> 00:32:44,110 bring down a factor of h bar. 439 00:32:44,110 --> 00:32:45,700 What is 2 pi h bar? 440 00:32:45,700 --> 00:32:48,050 2 pi h bar is h. 441 00:32:48,050 --> 00:32:51,960 So I would have a factor of h cubed. 442 00:32:51,960 --> 00:32:57,910 I have an integration over space that gave me volume. 443 00:32:57,910 --> 00:33:01,650 e to the minus beta h bar-- p squared over 2m. 444 00:33:07,570 --> 00:33:10,660 Why did I write it this way? 445 00:33:10,660 --> 00:33:15,580 Because it should remind you of how we made 446 00:33:15,580 --> 00:33:18,890 dimensionless the integrations that we 447 00:33:18,890 --> 00:33:22,150 had to do for classical calculations. 448 00:33:22,150 --> 00:33:24,000 Classical partition function, if I 449 00:33:24,000 --> 00:33:27,380 wanted to calculate within a single particle, 450 00:33:27,380 --> 00:33:29,500 for a single particle Hamiltonian, 451 00:33:29,500 --> 00:33:31,150 I would have this Boltzmann weight. 452 00:33:31,150 --> 00:33:35,450 I have to integrate over all coordinates, over all momenta. 453 00:33:35,450 --> 00:33:40,110 And then we divided by h to make it dimensionless. 454 00:33:40,110 --> 00:33:43,320 So you can see that that naturally appears here. 455 00:33:43,320 --> 00:33:47,290 And so the answer to this is V over lambda 456 00:33:47,290 --> 00:33:51,180 cubed with the lambda that we had defined 457 00:33:51,180 --> 00:33:54,590 before as h over root 2 pi m kT. 458 00:34:00,630 --> 00:34:12,969 So you can see that this traces that we write down 459 00:34:12,969 --> 00:34:17,090 in this quantum formulations are clearly 460 00:34:17,090 --> 00:34:19,420 dimensionless quantities. 461 00:34:19,420 --> 00:34:22,489 And they go over to the classical limit 462 00:34:22,489 --> 00:34:25,780 where we integrate over phase space appropriately 463 00:34:25,780 --> 00:34:29,040 dimensional-- made dimensionless by dividing 464 00:34:29,040 --> 00:34:33,590 by h per combination of p and q. 465 00:34:33,590 --> 00:34:39,260 So this V1 we already know is V-- Z1 466 00:34:39,260 --> 00:34:42,070 we already know is V over lambda cubed. 467 00:34:42,070 --> 00:34:44,080 The reason I got a little bit confused 468 00:34:44,080 --> 00:34:47,489 was because I saw that the V's were disappearing. 469 00:34:47,489 --> 00:34:49,739 And then I remembered, oh, Z1 is actually 470 00:34:49,739 --> 00:34:52,050 proportional to mu over lambda cubed. 471 00:34:52,050 --> 00:34:53,239 So that's good. 472 00:34:53,239 --> 00:34:58,690 So what we have here from the inverse of Z1 is a 1 473 00:34:58,690 --> 00:35:00,620 over V lambda cubed here. 474 00:35:04,500 --> 00:35:08,430 Now, then I have to complete this kind of integration, 475 00:35:08,430 --> 00:35:11,780 which are Gaussian integrations. 476 00:35:11,780 --> 00:35:15,700 If x was equal to x prime, this would 477 00:35:15,700 --> 00:35:20,380 be precisely the integration that I'm doing over here. 478 00:35:20,380 --> 00:35:23,910 So if x was equal to x prime, that integration, 479 00:35:23,910 --> 00:35:26,020 the Gaussian integration, would have given me 480 00:35:26,020 --> 00:35:29,570 this 1 over lambda cubed. 481 00:35:29,570 --> 00:35:34,350 But because I have this shift-- in some sense 482 00:35:34,350 --> 00:35:40,950 I have to shift k to remove that shift in x minus x prime. 483 00:35:40,950 --> 00:35:43,170 And when I square it, I will get here 484 00:35:43,170 --> 00:35:48,360 a factor, which is exponential, of minus x minus x prime 485 00:35:48,360 --> 00:35:51,700 squared-- the Square of that factor-- times 486 00:35:51,700 --> 00:35:58,070 the variance of k, which is m kT over h bar squared. 487 00:35:58,070 --> 00:35:59,440 So what do I get here? 488 00:35:59,440 --> 00:36:01,540 I will get 2. 489 00:36:01,540 --> 00:36:05,430 And then I would get essentially m kT. 490 00:36:08,780 --> 00:36:10,874 And then, h bar squared here. 491 00:36:19,550 --> 00:36:26,910 Now, this combination m kT divided by h, same thing 492 00:36:26,910 --> 00:36:29,010 as here, m kT and h. 493 00:36:29,010 --> 00:36:33,840 Clearly, what that is, is giving me some kind of a lambda. 494 00:36:33,840 --> 00:36:36,130 So let me write the answer. 495 00:36:36,130 --> 00:36:40,250 Eventually, what I find is that evaluating 496 00:36:40,250 --> 00:36:45,210 the one-particle density matrix between two points, x 497 00:36:45,210 --> 00:36:50,825 and x prime, will give me 1 over V. 498 00:36:50,825 --> 00:36:54,320 And then this exponential factor, 499 00:36:54,320 --> 00:36:58,050 which is x minus x prime squared. 500 00:36:58,050 --> 00:37:02,185 The coefficient here has to be proportional to lambda squared. 501 00:37:02,185 --> 00:37:04,205 And if you do the algebra right, you 502 00:37:04,205 --> 00:37:10,590 will find that the coefficient is pi over 2. 503 00:37:10,590 --> 00:37:11,370 Or is it pi? 504 00:37:11,370 --> 00:37:14,650 Let me double check. 505 00:37:14,650 --> 00:37:17,640 It's a numerical factor that is not that important. 506 00:37:17,640 --> 00:37:19,860 It is pi, pi over lambda squared. 507 00:37:28,900 --> 00:37:33,750 OK, so what does this mean? 508 00:37:33,750 --> 00:37:36,150 So I have a particle in the box. 509 00:37:36,150 --> 00:37:40,990 It's a quantum mechanical particle at some temperature T. 510 00:37:40,990 --> 00:37:44,150 And I'm trying to ask, where is it? 511 00:37:44,150 --> 00:37:45,970 So if I want to ask where is it, what's 512 00:37:45,970 --> 00:37:49,230 the probability it is at some point x? 513 00:37:49,230 --> 00:37:52,400 I refer to this density matrix that I 514 00:37:52,400 --> 00:37:54,920 have calculated in coordinate space. 515 00:37:54,920 --> 00:37:57,090 Put x prime and x to be the same. 516 00:37:57,090 --> 00:38:00,800 If x prime and x are the same, I get 1 over V. 517 00:38:00,800 --> 00:38:03,550 Essentially, it says that the probability 518 00:38:03,550 --> 00:38:08,530 that I find a particle is the same anywhere in the box. 519 00:38:08,530 --> 00:38:10,250 So that makes sense. 520 00:38:10,250 --> 00:38:12,581 It is the analog of throwing the coin. 521 00:38:12,581 --> 00:38:14,080 We don't know where the particle is. 522 00:38:14,080 --> 00:38:16,940 It's equally likely to be anywhere. 523 00:38:16,940 --> 00:38:20,560 Actually, remember that I used periodic boundary conditions. 524 00:38:20,560 --> 00:38:22,970 If I had used sine and cosines, there 525 00:38:22,970 --> 00:38:27,980 would have been some dependence on approaching the boundaries. 526 00:38:27,980 --> 00:38:31,150 But the matrix tells me more than that. 527 00:38:31,150 --> 00:38:33,630 Basically, the matrix says it is true 528 00:38:33,630 --> 00:38:37,250 that the diagonal elements are 1 over V. 529 00:38:37,250 --> 00:38:41,280 But if I go off-diagonal, there is this [INAUDIBLE] scale 530 00:38:41,280 --> 00:38:44,540 lambda that is telling me something. 531 00:38:44,540 --> 00:38:46,660 What is it telling you? 532 00:38:46,660 --> 00:38:49,460 Essentially, it is telling you that through the procedures 533 00:38:49,460 --> 00:38:55,460 that we have outlined here, if you are at the temperature T, 534 00:38:55,460 --> 00:39:01,620 then this factor tells you about the classical probability 535 00:39:01,620 --> 00:39:03,660 of finding a momentum p. 536 00:39:03,660 --> 00:39:07,700 There is a range of momenta because p squared over 2m, 537 00:39:07,700 --> 00:39:09,680 the excitation is controlled by kT. 538 00:39:09,680 --> 00:39:12,410 There is a range of momenta that is possible. 539 00:39:12,410 --> 00:39:14,680 And through a procedure such as this, 540 00:39:14,680 --> 00:39:19,820 you are making a superposition of plane waves 541 00:39:19,820 --> 00:39:22,450 that have this range of momentum. 542 00:39:22,450 --> 00:39:25,840 The superposition of plane waves, what does it give you? 543 00:39:25,840 --> 00:39:29,230 It essentially gives you a Guassian blob, 544 00:39:29,230 --> 00:39:32,650 which we can position anywhere in the space. 545 00:39:32,650 --> 00:39:35,290 But the width of that Gaussian blob 546 00:39:35,290 --> 00:39:38,180 is given by this factor of lambda, 547 00:39:38,180 --> 00:39:40,790 which is related to the uncertainty that 548 00:39:40,790 --> 00:39:43,010 is now quantum in character. 549 00:39:43,010 --> 00:39:46,321 That is, quantum mechanically if you know the momentum, 550 00:39:46,321 --> 00:39:47,695 then you don't know the position. 551 00:39:47,695 --> 00:39:50,870 There is the Heisenberg uncertainty principle. 552 00:39:50,870 --> 00:39:53,250 Now we have a classical uncertainty 553 00:39:53,250 --> 00:39:56,500 about the momentum because of these Boltzmann weights. 554 00:39:56,500 --> 00:40:00,590 That classical uncertainty that we have about the momentum 555 00:40:00,590 --> 00:40:07,280 translates to some kind of a wave packet size for these-- 556 00:40:07,280 --> 00:40:10,960 to how much you can localize a particle. 557 00:40:10,960 --> 00:40:13,360 So really, this particle you should 558 00:40:13,360 --> 00:40:17,800 think of quantum mechanically as being some kind of a wave 559 00:40:17,800 --> 00:40:20,873 packet that has some characteristic size lambda. 560 00:40:24,260 --> 00:40:25,680 Yes? 561 00:40:25,680 --> 00:40:29,880 AUDIENCE: So could you interpret the difference 562 00:40:29,880 --> 00:40:35,430 between x and x prime appearing in the [INAUDIBLE] 563 00:40:35,430 --> 00:40:41,040 as sort of being the probability of finding a particle at x 564 00:40:41,040 --> 00:40:44,270 having turned to x prime within the box? 565 00:40:49,250 --> 00:40:50,950 PROFESSOR: At this stage, you are 566 00:40:50,950 --> 00:40:54,510 sort of introducing something that is not there. 567 00:40:54,510 --> 00:40:57,900 But if I had two particles-- so very soon we 568 00:40:57,900 --> 00:41:00,330 are going to do multiple particles-- 569 00:41:00,330 --> 00:41:03,070 and this kind of structure will be maintained 570 00:41:03,070 --> 00:41:07,490 if I have multiple particles, then your statement is correct. 571 00:41:07,490 --> 00:41:10,310 That this factor will tell me something 572 00:41:10,310 --> 00:41:13,160 about the probability of these things being exchanged, 573 00:41:13,160 --> 00:41:17,650 one tunneling to the other place and one tunneling back. 574 00:41:17,650 --> 00:41:22,180 And indeed, if you went to the colloquium yesterday, 575 00:41:22,180 --> 00:41:25,180 Professor [? Block ?] showed pictures 576 00:41:25,180 --> 00:41:27,950 of these kinds of exchanges taking place. 577 00:41:32,630 --> 00:41:35,250 OK? 578 00:41:35,250 --> 00:41:38,490 But essentially, really at this stage 579 00:41:38,490 --> 00:41:42,290 for the single particle with no potential, 580 00:41:42,290 --> 00:41:45,630 the statement is that the particle is, in reality, 581 00:41:45,630 --> 00:41:47,602 some kind of a wave packet. 582 00:41:51,320 --> 00:41:53,460 OK? 583 00:41:53,460 --> 00:41:54,400 Fine. 584 00:41:54,400 --> 00:41:56,750 So any other questions? 585 00:41:59,950 --> 00:42:02,190 The next thing is, of course, what I said. 586 00:42:02,190 --> 00:42:05,760 Let's put two particles in the potential. 587 00:42:05,760 --> 00:42:19,180 So let's go to two particles. 588 00:42:23,460 --> 00:42:26,390 Actually, not necessarily even in the box. 589 00:42:26,390 --> 00:42:28,970 But let's say what kind of a Hamiltonian 590 00:42:28,970 --> 00:42:31,265 would I write for two particles? 591 00:42:34,010 --> 00:42:36,770 I would write something that has a kinetic energy 592 00:42:36,770 --> 00:42:42,580 for the first particle, kinetic energy for the second particle, 593 00:42:42,580 --> 00:42:48,020 and then some kind of a potential of interaction. 594 00:42:48,020 --> 00:42:50,500 Let's say, as a function of x1 minus x2. 595 00:42:59,290 --> 00:43:02,930 Now, I could have done many things for two particles. 596 00:43:02,930 --> 00:43:05,600 I could, for example, have had particles 597 00:43:05,600 --> 00:43:07,780 that are different masses, but I clearly 598 00:43:07,780 --> 00:43:10,910 wanted to write things that were similar. 599 00:43:10,910 --> 00:43:13,090 So this is Hamiltonian that describes 600 00:43:13,090 --> 00:43:16,820 particles labeled by 1 and 2. 601 00:43:16,820 --> 00:43:19,340 And if they have the same mass and the potential 602 00:43:19,340 --> 00:43:21,870 is a function of separation, it's 603 00:43:21,870 --> 00:43:27,540 certainly h that is for 2 and 1 exchange. 604 00:43:27,540 --> 00:43:30,590 So this Hamiltonian has a certain symmetry 605 00:43:30,590 --> 00:43:33,660 under the exchange of the labels. 606 00:43:33,660 --> 00:43:37,300 Now typically, what you should remember 607 00:43:37,300 --> 00:43:41,240 is that any type of symmetry that your Hamiltonian has 608 00:43:41,240 --> 00:43:46,120 or any matrix has will be reflected ultimately 609 00:43:46,120 --> 00:43:49,130 in the form of the eigenvectors that you 610 00:43:49,130 --> 00:43:51,250 would construct for that. 611 00:43:51,250 --> 00:43:54,370 So indeed, I already know that for this, I 612 00:43:54,370 --> 00:43:57,535 should be able to construct wave functions that 613 00:43:57,535 --> 00:44:01,590 are either symmetrical or anti-symmetric under exchange. 614 00:44:01,590 --> 00:44:04,170 But that statement aside, let's think 615 00:44:04,170 --> 00:44:08,200 about the meaning of the wave function 616 00:44:08,200 --> 00:44:10,630 that we have for the two particles. 617 00:44:10,630 --> 00:44:16,230 So presumably, there is a psi as a function of x1 and x2 618 00:44:16,230 --> 00:44:19,050 in the coordinate space. 619 00:44:19,050 --> 00:44:22,870 And for a particular quantum state, 620 00:44:22,870 --> 00:44:26,610 you would say that the square of this quantity 621 00:44:26,610 --> 00:44:36,840 is related to quantum probabilities of finding 622 00:44:36,840 --> 00:44:41,540 particles at x1 and x2. 623 00:44:47,810 --> 00:44:53,370 So this could, for example, be the case of two particles. 624 00:44:53,370 --> 00:44:57,780 Let's say oxygen and nitrogen. 625 00:44:57,780 --> 00:45:00,180 They have almost the same mass. 626 00:45:00,180 --> 00:45:02,760 This statement would be true. 627 00:45:02,760 --> 00:45:05,530 But the statement becomes more interesting 628 00:45:05,530 --> 00:45:10,120 if we say that the particles are identical. 629 00:45:10,120 --> 00:45:24,070 So for identical particles, the statement 630 00:45:24,070 --> 00:45:30,050 of identity in quantum mechanics is damped at this stage. 631 00:45:30,050 --> 00:45:33,660 You would say that I can't tell apart the probability 632 00:45:33,660 --> 00:45:36,610 that one particle, number 1, is at x1, 633 00:45:36,610 --> 00:45:40,220 particle 2 is at x2 or vice versa. 634 00:45:44,540 --> 00:45:48,080 The labels 1 and 2 are just thinks 635 00:45:48,080 --> 00:45:50,750 that I assign for convenience, but they really 636 00:45:50,750 --> 00:45:52,400 don't have any meaning. 637 00:45:52,400 --> 00:45:53,650 There are two particles. 638 00:45:53,650 --> 00:45:56,650 To all intents and purposes, they are identical. 639 00:45:56,650 --> 00:46:02,420 And there is some probability for seeing the events 640 00:46:02,420 --> 00:46:04,250 where one particle-- I don't know 641 00:46:04,250 --> 00:46:06,950 what its label is-- at this location. 642 00:46:06,950 --> 00:46:09,180 The other particle-- I don't know what its label is-- 643 00:46:09,180 --> 00:46:10,790 is at that location. 644 00:46:10,790 --> 00:46:12,110 Labels don't have any meaning. 645 00:46:15,260 --> 00:46:16,800 So this is different. 646 00:46:16,800 --> 00:46:19,940 This does not have a classic analog. 647 00:46:19,940 --> 00:46:23,080 Classically, if I put something on the computer, 648 00:46:23,080 --> 00:46:26,820 I would say that particle 1 is here at x1, 649 00:46:26,820 --> 00:46:29,280 particle 2 is here at x2. 650 00:46:29,280 --> 00:46:32,630 I have to write some index on the computer. 651 00:46:32,630 --> 00:46:36,790 But if I want to construct a wave function, 652 00:46:36,790 --> 00:46:38,280 I wouldn't know what to do. 653 00:46:38,280 --> 00:46:41,050 I would just essentially have a function of x1 and x2 654 00:46:41,050 --> 00:46:44,680 that is peaked here and there. 655 00:46:44,680 --> 00:46:52,740 And we also know that somehow quantum mechanics tells us 656 00:46:52,740 --> 00:46:55,130 that there is actually a stronger 657 00:46:55,130 --> 00:47:01,470 version of this statement, which is that psi of x1, x2 x1 658 00:47:01,470 --> 00:47:07,400 is either plus psi of x1, x2 or minus psi 659 00:47:07,400 --> 00:47:13,320 of x1, x2 for identical particles 660 00:47:13,320 --> 00:47:16,235 depending on whether they are boson or fermions. 661 00:47:31,900 --> 00:47:43,910 So let's kind of build upon that a little bit more 662 00:47:43,910 --> 00:47:45,730 and go to many particles. 663 00:48:00,680 --> 00:48:14,150 So forth N particles that are identical, 664 00:48:14,150 --> 00:48:22,770 I would have some kind of a state psi 665 00:48:22,770 --> 00:48:24,580 that depends on the coordinates. 666 00:48:24,580 --> 00:48:26,840 For example, in the coordinate representation, 667 00:48:26,840 --> 00:48:30,140 but I could choose any other representation. 668 00:48:30,140 --> 00:48:32,080 Coordinate kind of makes more sense. 669 00:48:32,080 --> 00:48:34,420 We can visualize it. 670 00:48:34,420 --> 00:48:38,430 And if I can't tell them apart, previously I 671 00:48:38,430 --> 00:48:41,800 was just exchanging two of the labels 672 00:48:41,800 --> 00:48:45,480 but now I can permute them in any possible way. 673 00:48:45,480 --> 00:48:56,690 So I can add the permutation P. And of course, there 674 00:48:56,690 --> 00:48:58,570 are N factorial in number. 675 00:49:06,130 --> 00:49:13,050 And the generalization of the statement that I had before 676 00:49:13,050 --> 00:49:15,820 was that for the case of bosons, I 677 00:49:15,820 --> 00:49:18,529 will always get the same thing back. 678 00:49:27,160 --> 00:49:31,380 And for the case of fermions, I will get a number back 679 00:49:31,380 --> 00:49:35,910 that is either minus or plus depending 680 00:49:35,910 --> 00:49:40,549 on the type of permutation that I apply of what I started with. 681 00:49:48,380 --> 00:49:55,010 So I have introduced here something that I call minus 1 682 00:49:55,010 --> 00:49:58,580 to the power of P, which is called 683 00:49:58,580 --> 00:50:00,300 a parity of the permutation. 684 00:50:06,220 --> 00:50:10,420 A permutation is either even-- in which case, this minus 1 685 00:50:10,420 --> 00:50:14,620 to the power of P is plus-- or is odd-- in which case, 686 00:50:14,620 --> 00:50:18,520 this minus 1 to P is minus 1. 687 00:50:18,520 --> 00:50:22,710 How do I determine the parity of permutation? 688 00:50:22,710 --> 00:50:27,250 Parity of the permutation is the number 689 00:50:27,250 --> 00:50:34,020 of exchanges that lead to this permutation. 690 00:50:42,110 --> 00:50:44,765 Basically, take any permutation. 691 00:50:44,765 --> 00:50:48,770 Let's say we stick with four particles. 692 00:50:48,770 --> 00:50:53,010 And I go from 1, 2, 3, 4, which was, let's say, 693 00:50:53,010 --> 00:50:56,920 some regular ordering, to some other ordering. 694 00:50:56,920 --> 00:51:00,082 Let's say 4, 2, 1, 2. 695 00:51:00,082 --> 00:51:06,000 And my claim is that I can achieve this transformation 696 00:51:06,000 --> 00:51:08,140 through a series of exchanges. 697 00:51:08,140 --> 00:51:11,192 So I can get here as follows. 698 00:51:11,192 --> 00:51:15,390 I want 4 to come all the way back to here, 699 00:51:15,390 --> 00:51:19,560 so I do an exchange of 1 and 4. 700 00:51:19,560 --> 00:51:22,050 I call the exchange in this fashion. 701 00:51:22,050 --> 00:51:23,970 I do exchange of 1 and 4. 702 00:51:23,970 --> 00:51:30,640 The exchange of 1 and 4 will make for me 4, 2, 3, 1. 703 00:51:30,640 --> 00:51:31,140 OK. 704 00:51:31,140 --> 00:51:33,620 I compare this with this. 705 00:51:33,620 --> 00:51:37,560 I see that all I need to do is to switch 3 and 1. 706 00:51:37,560 --> 00:51:41,290 So I do an exchange of 1 and 3. 707 00:51:41,290 --> 00:51:45,480 And what I will get here is 4, 2, 1, 3. 708 00:51:45,480 --> 00:51:49,340 So I could get to that permutation 709 00:51:49,340 --> 00:51:50,870 through two exchanges. 710 00:51:50,870 --> 00:51:52,123 Therefore, this is even. 711 00:51:55,830 --> 00:51:57,670 Now, this is not the only way that I 712 00:51:57,670 --> 00:51:59,810 can get from one to the other. 713 00:51:59,810 --> 00:52:04,470 I can, for example, sit and do multiple exchanges of this 4, 714 00:52:04,470 --> 00:52:08,980 2 a hundred times, but not ninety nine times. 715 00:52:08,980 --> 00:52:11,040 As long as I do an even number, I 716 00:52:11,040 --> 00:52:12,540 will get back to the same thing. 717 00:52:12,540 --> 00:52:15,510 The parity will be conserved. 718 00:52:15,510 --> 00:52:18,540 And there's another way of calculating parity. 719 00:52:18,540 --> 00:52:22,570 You just start with this original configuration 720 00:52:22,570 --> 00:52:25,550 and you want to get to that final configuration. 721 00:52:25,550 --> 00:52:27,670 You just draw lines. 722 00:52:27,670 --> 00:52:37,370 So 1 goes to 1, 2 goes to 2, 3 goes to 3, 4 goes to 4. 723 00:52:37,370 --> 00:52:41,160 And you count how many crossings you have, 724 00:52:41,160 --> 00:52:42,757 and the parity of those crossings 725 00:52:42,757 --> 00:52:44,590 will give you the parity of the permutation. 726 00:52:50,350 --> 00:52:56,480 So somehow within quantum mechanics, 727 00:52:56,480 --> 00:53:01,720 the idea of what is identical particle is stamped 728 00:53:01,720 --> 00:53:04,710 into the nature of the wave vectors, 729 00:53:04,710 --> 00:53:07,460 in the structure of the Hilbert space that you can construct. 730 00:53:10,940 --> 00:53:16,550 So let's see how that leads to this simple example of particle 731 00:53:16,550 --> 00:53:20,610 in the box, if we continue to add particles into the box. 732 00:53:36,780 --> 00:53:39,485 So we want to now put N particles in box. 733 00:53:43,910 --> 00:53:48,340 Otherwise, no interaction completely free. 734 00:53:48,340 --> 00:53:56,170 So the N particle Hamiltonian is some alpha running from 1 to N 735 00:53:56,170 --> 00:53:59,560 p alpha squared over 2m kinetic energies 736 00:53:59,560 --> 00:54:00,630 of all of these things. 737 00:54:04,760 --> 00:54:05,850 fine. 738 00:54:05,850 --> 00:54:11,830 Now, note that this Hamiltonian, since it is in some sense 739 00:54:11,830 --> 00:54:17,230 built up of lots of non-interacting pieces. 740 00:54:17,230 --> 00:54:19,940 And we saw already classically, that things are not 741 00:54:19,940 --> 00:54:23,400 interacting-- calculating probabilities, 742 00:54:23,400 --> 00:54:26,650 partition functions, et cetera, is very easy. 743 00:54:26,650 --> 00:54:28,520 This has that same structure. 744 00:54:28,520 --> 00:54:29,910 It's the ideal gas. 745 00:54:29,910 --> 00:54:31,690 Now, quantum mechanically. 746 00:54:31,690 --> 00:54:34,170 So it should be sufficiently easy. 747 00:54:34,170 --> 00:54:39,110 And indeed, we can immediately construct the eigenstates 748 00:54:39,110 --> 00:54:39,940 for this. 749 00:54:39,940 --> 00:54:42,250 So we can construct the basis, and then 750 00:54:42,250 --> 00:54:44,790 do the calculations in that basis. 751 00:54:44,790 --> 00:54:49,670 So let's look at something that I will call product state. 752 00:54:56,530 --> 00:55:01,780 And let's say that I had this structure 753 00:55:01,780 --> 00:55:05,890 that I described on the board above where I have a plane 754 00:55:05,890 --> 00:55:09,320 wave that is characterized by some wave number 755 00:55:09,320 --> 00:55:11,950 k for one particle. 756 00:55:11,950 --> 00:55:15,010 For N particles, I pick the first one to be k1, 757 00:55:15,010 --> 00:55:18,476 the second one to be k2, the last one to be kN. 758 00:55:21,740 --> 00:55:25,890 And I will call this a product state in the sense 759 00:55:25,890 --> 00:55:31,940 that if I look at the positional representation of this product 760 00:55:31,940 --> 00:55:38,380 state, it is simply the product of one-particle states. 761 00:55:38,380 --> 00:55:43,892 So this is a product over alpha of x alpha 762 00:55:43,892 --> 00:55:49,110 k alpha, which is this e to the i kx 763 00:55:49,110 --> 00:55:51,480 k dot x over square root of it. 764 00:55:51,480 --> 00:55:55,460 So this is perfectly well normalized 765 00:55:55,460 --> 00:55:58,860 if I were to integrate over x1, x2, x3. 766 00:55:58,860 --> 00:56:01,335 Each one of the integrals is individually normalized. 767 00:56:01,335 --> 00:56:03,560 The overall thing is normalized. 768 00:56:03,560 --> 00:56:08,030 It's an eigenstate because if I act Hn on this product 769 00:56:08,030 --> 00:56:16,290 state, what I will get is a sum over alpha h bar squared 770 00:56:16,290 --> 00:56:20,050 k alpha squared over 2m, and then the product state back. 771 00:56:26,480 --> 00:56:30,510 So it's an eigenstate. 772 00:56:30,510 --> 00:56:33,100 And in fact, it's a perfectly good eigenstate 773 00:56:33,100 --> 00:56:36,550 as long as I want to describe a set of particles 774 00:56:36,550 --> 00:56:37,620 that are not identical. 775 00:56:40,660 --> 00:56:43,200 I can't use this state to describe particles 776 00:56:43,200 --> 00:56:46,280 that are identical because it does not 777 00:56:46,280 --> 00:56:50,160 satisfy the symmetries that I set quantum mechanics 778 00:56:50,160 --> 00:56:53,420 stamps into identical particles. 779 00:56:53,420 --> 00:56:59,025 And it's clearly the case that, for example-- so this is not 780 00:56:59,025 --> 00:57:11,210 symmetrized since clearly, if I look at k1, k2, 781 00:57:11,210 --> 00:57:15,390 k1 goes with x1, k2 goes with x2. 782 00:57:15,390 --> 00:57:19,270 And it is not the same thing as k2, k1, 783 00:57:19,270 --> 00:57:24,970 where k2 goes with x1 and k1 goes with x2. 784 00:57:24,970 --> 00:57:31,090 Essentially, the two particles can be distinguished. 785 00:57:31,090 --> 00:57:33,530 One of them has momentum h bar k1. 786 00:57:33,530 --> 00:57:35,880 The other has momentum h bar k 2. 787 00:57:35,880 --> 00:57:40,370 I can tell them apart because of this unsymmetrized nature 788 00:57:40,370 --> 00:57:43,360 of this wave function. 789 00:57:43,360 --> 00:57:46,480 But you know how to make things that are symmetric. 790 00:57:46,480 --> 00:57:52,770 You basically add k1 k2 plus k2 k1 or k1 k2 minus k2 k1 791 00:57:52,770 --> 00:57:54,390 to make it anti-symmetrized. 792 00:57:54,390 --> 00:57:57,820 Divide by square root of 2 and you are done. 793 00:57:57,820 --> 00:58:00,490 So now, let's generalize that to the case 794 00:58:00,490 --> 00:58:03,220 of the N-particle system. 795 00:58:03,220 --> 00:58:09,360 So I will call a-- let's start with the case 796 00:58:09,360 --> 00:58:16,322 of the fermionic state. 797 00:58:16,322 --> 00:58:19,310 In fermionic state, I will indicate 798 00:58:19,310 --> 00:58:29,030 by k1, k2, kN with a minus index because 799 00:58:29,030 --> 00:58:31,360 of the asymmetry or the minus signs 800 00:58:31,360 --> 00:58:33,540 that we have for fermions. 801 00:58:33,540 --> 00:58:38,530 The way I do that is I sum over all N factorial 802 00:58:38,530 --> 00:58:41,860 permutations that I have. 803 00:58:41,860 --> 00:58:46,480 I let p act on the product state. 804 00:58:50,800 --> 00:58:54,950 And again, for two particles, you have the k1 k2, 805 00:58:54,950 --> 00:58:57,400 then you do minus k2 k1. 806 00:58:57,400 --> 00:59:02,800 For general particles, I do this minus 1 to the power of p. 807 00:59:02,800 --> 00:59:06,800 So all the even permutations are added with plus. 808 00:59:06,800 --> 00:59:09,630 All the odd permutations are added with minus. 809 00:59:13,110 --> 00:59:17,210 Except that this is a whole bunch of different terms 810 00:59:17,210 --> 00:59:18,770 that are being added. 811 00:59:18,770 --> 00:59:20,580 Again, for two particles, you know 812 00:59:20,580 --> 00:59:22,780 that you have to divide by a square root of 2 813 00:59:22,780 --> 00:59:25,120 because you add 2 vectors. 814 00:59:25,120 --> 00:59:27,715 And the length of the overall vector 815 00:59:27,715 --> 00:59:30,140 is increased by a square root of 2. 816 00:59:30,140 --> 00:59:34,330 Here, you have to divide in general by the number of terms 817 00:59:34,330 --> 00:59:36,930 that you have, square root of N factorial. 818 00:59:42,090 --> 00:59:47,310 The only thing that you have to be careful with is that you 819 00:59:47,310 --> 00:59:51,370 cannot have any two of these k's to be the same. 820 00:59:51,370 --> 00:59:54,820 Because let's say these two are the same, then along the list 821 00:59:54,820 --> 00:59:56,830 here I have the exchange of these two. 822 00:59:56,830 --> 00:59:59,800 And when I exchange them, I go from even to odd. 823 00:59:59,800 --> 01:00:03,170 I put a minus sign and I have a subtraction. 824 01:00:03,170 --> 01:00:07,590 So here, I have to make sure that all k 825 01:00:07,590 --> 01:00:09,744 alpha must be distinct. 826 01:00:20,876 --> 01:00:27,770 Now, say the bosonic one is simpler. 827 01:00:27,770 --> 01:00:36,890 I basically construct it, k1, k2, kN with a plus. 828 01:00:36,890 --> 01:00:38,740 By simply adding things together, 829 01:00:38,740 --> 01:00:42,450 so I will have a sum over p. 830 01:00:42,450 --> 01:00:44,400 No sign here. 831 01:00:44,400 --> 01:00:49,955 Permutation k1 through kN. 832 01:00:49,955 --> 01:00:52,940 And then I have to divide by some normalization. 833 01:00:59,930 --> 01:01:03,420 Now, the only tricky thing about this 834 01:01:03,420 --> 01:01:11,460 is that the normalization is not N factorial. 835 01:01:11,460 --> 01:01:15,860 So to give you an example, let's imagine 836 01:01:15,860 --> 01:01:20,460 that I choose to start with a product state 837 01:01:20,460 --> 01:01:23,720 where two of the k's are alpha and one of them is beta. 838 01:01:26,320 --> 01:01:33,265 So let's sort of put here 1, 1, 1, 2, 3 for my k1, k2, k3. 839 01:01:33,265 --> 01:01:37,730 I have chosen that k1 and k2 are the same. 840 01:01:37,730 --> 01:01:42,940 And what I have to do is to sum over all possible permutations. 841 01:01:42,940 --> 01:01:46,920 So there is a permutation here that is 1, 3, 2. 842 01:01:46,920 --> 01:01:50,410 So I get here alpha, beta, alpha. 843 01:01:50,410 --> 01:01:59,230 Then I will have 2, 1, 3, 2, 3, 1, 3, 1, 2, 3, 2, 1, 844 01:01:59,230 --> 01:02:05,440 which basically would be alpha, alpha, beta, alpha, beta, 845 01:02:05,440 --> 01:02:10,580 alpha, beta, alpha, alpha, beta, alpha, alpha. 846 01:02:13,930 --> 01:02:20,270 So there are for three things, 3 factorial or 6 permutations. 847 01:02:20,270 --> 01:02:26,700 But this entity is clearly twice alpha, 848 01:02:26,700 --> 01:02:32,670 alpha, beta, plus alpha, beta, alpha, plus beta, alpha, alpha. 849 01:02:35,370 --> 01:02:41,270 And the norm of this is going to be 4 times 1 plus 1 plus 1. 850 01:02:41,270 --> 01:02:44,410 4 times 3, which is 12. 851 01:02:44,410 --> 01:02:46,540 So in order to normalize this, I have 852 01:02:46,540 --> 01:02:50,300 to divide not by 1/6, but 1/12. 853 01:02:53,170 --> 01:02:55,560 So the appropriate normalization here 854 01:02:55,560 --> 01:02:57,110 then becomes 1 over root 3. 855 01:03:02,190 --> 01:03:05,335 Now, in general what would be this N plus? 856 01:03:07,990 --> 01:03:15,460 To calculate N plus, I have to make sure 857 01:03:15,460 --> 01:03:18,230 that the norm of this entity is 1. 858 01:03:18,230 --> 01:03:23,270 Or, N plus is the square of this quantity. 859 01:03:23,270 --> 01:03:26,320 And if I were to square this quantity, 860 01:03:26,320 --> 01:03:28,205 I will get two sets of permutations. 861 01:03:28,205 --> 01:03:31,910 I will call them p and p prime. 862 01:03:31,910 --> 01:03:39,522 And on one side, I would have the permutation p of k1 863 01:03:39,522 --> 01:03:41,420 through kN. 864 01:03:41,420 --> 01:03:45,720 On the other side, I would have a permutation k prime of k1 865 01:03:45,720 --> 01:03:46,220 through kN. 866 01:03:54,320 --> 01:03:59,260 Now, this is clearly N factorial square terms. 867 01:03:59,260 --> 01:04:04,090 But this is not N factorial squared distinct terms. 868 01:04:04,090 --> 01:04:08,530 Because essentially, over here I could 869 01:04:08,530 --> 01:04:12,080 get 1 of N factorial possibilities. 870 01:04:12,080 --> 01:04:15,220 And then here, I would permute over all the other N 871 01:04:15,220 --> 01:04:17,410 factorial possibilities. 872 01:04:17,410 --> 01:04:20,290 Then I would try the next one and the next one. 873 01:04:20,290 --> 01:04:23,910 And essentially, each one of those 874 01:04:23,910 --> 01:04:25,600 would give me the same one. 875 01:04:25,600 --> 01:04:29,780 So that is, once I have fixed what this one is, 876 01:04:29,780 --> 01:04:35,015 permuting over them will give me one member of N 877 01:04:35,015 --> 01:04:37,780 factorial identical terms. 878 01:04:37,780 --> 01:04:42,370 So I can write this as N factorial sum over Q. Let's say 879 01:04:42,370 --> 01:04:49,700 I start with the original k1 through kN, 880 01:04:49,700 --> 01:04:54,300 and then I go and do a permutation of k1 through kN. 881 01:04:59,250 --> 01:05:02,680 And the question is, how many times 882 01:05:02,680 --> 01:05:05,600 do I get something that is non-zero? 883 01:05:05,600 --> 01:05:09,390 If these two lists are completely distinct, 884 01:05:09,390 --> 01:05:12,310 except for the identity any transformation 885 01:05:12,310 --> 01:05:14,830 that I will make here will make me 886 01:05:14,830 --> 01:05:19,490 a vector that is orthogonal to this and I will get 0. 887 01:05:19,490 --> 01:05:23,200 But if I have two of them that are identical, 888 01:05:23,200 --> 01:05:25,140 then I do a permutation like this 889 01:05:25,140 --> 01:05:27,540 and I'll get the same thing back. 890 01:05:27,540 --> 01:05:32,800 And then I have two things that are giving the same result. 891 01:05:32,800 --> 01:05:39,080 And in general, this answer is a product 892 01:05:39,080 --> 01:05:44,600 over all multiple occurrences factorial. 893 01:05:44,600 --> 01:05:46,740 So let's say here there was something 894 01:05:46,740 --> 01:05:49,050 that was repeated twice. 895 01:05:49,050 --> 01:05:52,270 If it had been repeated three times, 896 01:05:52,270 --> 01:05:56,100 then all six possibilities would've been the same. 897 01:05:56,100 --> 01:05:58,620 So I would have had 3 factorial. 898 01:05:58,620 --> 01:06:01,851 So if I had 3 and then 2 other ones that were the same, 899 01:06:01,851 --> 01:06:03,600 then the answer would have been multiplied 900 01:06:03,600 --> 01:06:06,050 by 3 factorial 2 factorial. 901 01:06:06,050 --> 01:06:10,440 So the statement here is that this N plus 902 01:06:10,440 --> 01:06:16,210 that I have to put here is N factorial product over k nk 903 01:06:16,210 --> 01:06:20,190 factorial, which is essentially the multiplicity that 904 01:06:20,190 --> 01:06:21,617 is associated with repeats. 905 01:06:25,030 --> 01:06:30,340 Oh 906 01:06:30,340 --> 01:06:38,810 So we've figured out what the appropriate bosonic 907 01:06:38,810 --> 01:06:44,160 and fermionic eigenstates are for this situation of particles 908 01:06:44,160 --> 01:06:47,310 in the box where I put multiple particles. 909 01:06:47,310 --> 01:06:50,820 So now I have N particles in the box. 910 01:06:50,820 --> 01:06:54,680 I know what the appropriate basis functions are. 911 01:06:54,680 --> 01:07:00,640 And what I can now try to do is to, for example, 912 01:07:00,640 --> 01:07:04,220 calculate the analog of the partition function, 913 01:07:04,220 --> 01:07:07,780 the analog of what I did here for N particles, 914 01:07:07,780 --> 01:07:10,970 or the analog of the density matrix. 915 01:07:10,970 --> 01:07:13,581 So let's calculate the analog of the density matrix 916 01:07:13,581 --> 01:07:14,455 and see what happens. 917 01:07:33,395 --> 01:07:39,500 So I want to calculate an N particle density 918 01:07:39,500 --> 01:07:43,840 matrix, completely free particles 919 01:07:43,840 --> 01:07:47,010 in a box, no interactions. 920 01:07:47,010 --> 01:07:51,270 For one particle, I went from x to x prime. 921 01:07:51,270 --> 01:07:57,830 So here, I go from x1, x2, xN. 922 01:07:57,830 --> 01:08:03,665 To here, x1 prime, x2 prime, xN prime. 923 01:08:07,110 --> 01:08:10,150 Our answer ultimately will depend on 924 01:08:10,150 --> 01:08:13,980 whether I am dealing with fermions or bosons. 925 01:08:13,980 --> 01:08:19,630 So I introduce an index here eta. 926 01:08:19,630 --> 01:08:28,800 Let me put it here, eta is plus 1 for bosons and minus 1 927 01:08:28,800 --> 01:08:29,784 for fermions. 928 01:08:33,509 --> 01:08:38,149 So kind of goes with this thing over here 929 01:08:38,149 --> 01:08:43,689 whether or not I'm using bosonic or fermionic symmetrization 930 01:08:43,689 --> 01:08:46,069 in constructing the wave functions here. 931 01:08:49,090 --> 01:08:50,960 You say, well, what is rho? 932 01:08:50,960 --> 01:08:55,270 Rho is e to the minus this beta hn divided by some partition 933 01:08:55,270 --> 01:08:58,430 function ZN that I don't know. 934 01:08:58,430 --> 01:09:03,510 But what I certainly know about it is because I constructed, 935 01:09:03,510 --> 01:09:09,130 in fact, this basis so that I have the right energy, which 936 01:09:09,130 --> 01:09:12,479 is sum over alpha h bar squared alpha 937 01:09:12,479 --> 01:09:14,870 squared k alpha squared over 2m. 938 01:09:14,870 --> 01:09:21,960 That is, this rho is diagonal in the basis of the k's. 939 01:09:21,960 --> 01:09:26,319 So maybe what I should do is I should switch from this 940 01:09:26,319 --> 01:09:30,590 representation to the representation of k's. 941 01:09:30,590 --> 01:09:33,770 So the same way that for one particle 942 01:09:33,770 --> 01:09:37,380 I sandwiched this factor of k, I am 943 01:09:37,380 --> 01:09:39,930 going to do the same thing over here. 944 01:09:39,930 --> 01:09:42,595 Except that I will have a whole bunch of k's. 945 01:09:45,850 --> 01:09:50,484 Because I'm doing a multiple system. 946 01:09:55,730 --> 01:09:56,630 What do I have? 947 01:09:56,630 --> 01:10:06,510 I have x1 prime to xN prime k1 through kN, 948 01:10:06,510 --> 01:10:10,830 with whatever appropriate symmetry it is. 949 01:10:10,830 --> 01:10:17,410 I have e to the minus sum over alpha beta h 950 01:10:17,410 --> 01:10:21,400 bar squared k alpha squared over 2m. 951 01:10:21,400 --> 01:10:25,080 Essentially, generalizing this factor that I have here, 952 01:10:25,080 --> 01:10:28,480 except that I have to normalize it with some ZN 953 01:10:28,480 --> 01:10:32,180 that I have yet to calculate. 954 01:10:32,180 --> 01:10:41,760 And then I go back from k1, kN to x1, xN. 955 01:10:41,760 --> 01:10:44,580 Again, respecting whatever symmetry I 956 01:10:44,580 --> 01:10:46,038 want to have at the end of the day. 957 01:10:50,720 --> 01:10:55,090 Now, let's think a little bit about this summation 958 01:10:55,090 --> 01:10:57,820 that I have to do over here. 959 01:10:57,820 --> 01:11:00,170 And I'll put a prime up there to indicate 960 01:11:00,170 --> 01:11:03,370 that it is a restricted sum. 961 01:11:03,370 --> 01:11:05,040 I didn't have any restriction here. 962 01:11:05,040 --> 01:11:08,400 I said I integrate over-- or sum over all possible k's 963 01:11:08,400 --> 01:11:12,080 that were consistent with the choice of 2 pi over l, 964 01:11:12,080 --> 01:11:14,130 or whatever. 965 01:11:14,130 --> 01:11:18,970 Now, here I have to make sure that I don't do over-counting. 966 01:11:18,970 --> 01:11:20,840 What do I mean? 967 01:11:20,840 --> 01:11:24,110 I mean that once I do this symmetrization, 968 01:11:24,110 --> 01:11:31,940 let's say I start with two particles and I have k1, k2. 969 01:11:31,940 --> 01:11:33,560 Then, I do the symmetrization. 970 01:11:33,560 --> 01:11:37,110 I get something that is a symmetric version of k1, k2. 971 01:11:37,110 --> 01:11:39,030 I would have gotten exactly the same thing 972 01:11:39,030 --> 01:11:42,670 if I started with k2, k1. 973 01:11:42,670 --> 01:11:47,030 That is, a particular one of these states 974 01:11:47,030 --> 01:11:51,410 corresponds to one list that I have here. 975 01:11:51,410 --> 01:11:58,120 And so over here, I should not sum over k1, k2, k3, et 976 01:11:58,120 --> 01:12:00,520 cetera, independently. 977 01:12:00,520 --> 01:12:05,110 Because then I will be over-counting by N factorial. 978 01:12:05,110 --> 01:12:12,780 So I say, OK, let me sum over things independently 979 01:12:12,780 --> 01:12:15,570 and then divide by the over-counting. 980 01:12:18,600 --> 01:12:22,190 Because presumably, these will give me 981 01:12:22,190 --> 01:12:25,180 N factorial similar states. 982 01:12:25,180 --> 01:12:29,320 So let's just sum over all of them, forget about this. 983 01:12:29,320 --> 01:12:31,640 You say, well, almost. 984 01:12:31,640 --> 01:12:34,280 Not quite because you have to worry about these factors. 985 01:12:37,390 --> 01:12:40,320 Because now when you did the N factorial, 986 01:12:40,320 --> 01:12:42,750 you sort of did not take into account 987 01:12:42,750 --> 01:12:47,010 the kinds of exchanges that I mentioned that you should not 988 01:12:47,010 --> 01:12:53,510 include because essentially if you have k1, k1, and then k3, 989 01:12:53,510 --> 01:12:56,330 then you don't have 6 different permutations. 990 01:12:56,330 --> 01:12:59,510 You only have 3 different permutations. 991 01:12:59,510 --> 01:13:03,380 So actually, the correction that I have to make 992 01:13:03,380 --> 01:13:07,420 is to multiply here by this quantity. 993 01:13:07,420 --> 01:13:14,900 So now that sum be the restriction has gone into this. 994 01:13:14,900 --> 01:13:23,470 And this ensures that as I sum over all k's independently, 995 01:13:23,470 --> 01:13:27,920 the over-counting of states that I have made 996 01:13:27,920 --> 01:13:31,344 is taken into account by the appropriate factor. 997 01:13:34,960 --> 01:13:39,740 Now, this is actually very nice because when 998 01:13:39,740 --> 01:13:44,550 I look at these states, I have the normalization factors. 999 01:13:44,550 --> 01:13:48,610 The normalizations depend on the list of k's that I 1000 01:13:48,610 --> 01:13:49,360 have over here. 1001 01:13:53,540 --> 01:13:58,530 So the normalization of these objects will give me a 1 1002 01:13:58,530 --> 01:14:03,950 over N factorial product over k nk factorial. 1003 01:14:03,950 --> 01:14:08,390 And these nk factorials will cancel. 1004 01:14:08,390 --> 01:14:10,420 Now you say, hang on. 1005 01:14:10,420 --> 01:14:11,760 Going too fast. 1006 01:14:11,760 --> 01:14:16,350 This eta says you can do this both for fermions and bosons. 1007 01:14:16,350 --> 01:14:19,180 But this calculation that you did over here 1008 01:14:19,180 --> 01:14:22,142 applies only for the case of bosons. 1009 01:14:22,142 --> 01:14:26,390 You say, never mind because for fermions, the allowed values 1010 01:14:26,390 --> 01:14:30,140 of nk are either 0 or 1. 1011 01:14:30,140 --> 01:14:32,710 I cannot put two of them. 1012 01:14:32,710 --> 01:14:37,740 So these nk factorials are either-- again, 1 or 1. 1013 01:14:37,740 --> 01:14:40,775 So even for fermions, this will work out fine. 1014 01:14:45,330 --> 01:14:48,720 There will be appropriate cancellation of wave functions 1015 01:14:48,720 --> 01:14:51,030 that should not be included when I 1016 01:14:51,030 --> 01:14:53,010 do the summation over permutations 1017 01:14:53,010 --> 01:14:58,240 with the corresponding factors of plus and minus. 1018 01:14:58,240 --> 01:15:01,530 So again, the minus signs-- everything will work. 1019 01:15:01,530 --> 01:15:04,690 And this kind of ugly factor will 1020 01:15:04,690 --> 01:15:07,412 disappear at the end of the day. 1021 01:15:07,412 --> 01:15:09,350 OK? 1022 01:15:09,350 --> 01:15:11,246 Fine. 1023 01:15:11,246 --> 01:15:15,440 Now, each one of these is a sum over permutations. 1024 01:15:15,440 --> 01:15:17,120 Actually, before I write those, let 1025 01:15:17,120 --> 01:15:22,240 me just write the factor of e to the minus sum over alpha beta h 1026 01:15:22,240 --> 01:15:26,094 bar squared k alpha squared over 2m. 1027 01:15:26,094 --> 01:15:27,810 And then I have ZN. 1028 01:15:27,810 --> 01:15:29,250 I don't know what ZN is yet. 1029 01:15:33,480 --> 01:15:40,130 Each one of these states is a sum over permutations. 1030 01:15:40,130 --> 01:15:44,970 I have taken care of the overall normalization. 1031 01:15:44,970 --> 01:15:47,270 I have to do the sum over the permutations. 1032 01:15:49,900 --> 01:15:56,150 And let's say I'm looking at this one. 1033 01:15:56,150 --> 01:15:59,400 Essentially, I have to sandwich that with some combination 1034 01:15:59,400 --> 01:16:00,060 of x's. 1035 01:16:00,060 --> 01:16:01,520 What do I get? 1036 01:16:01,520 --> 01:16:07,800 I will get a factor of eta to the p. 1037 01:16:07,800 --> 01:16:10,370 So actually, maybe I should have really 1038 01:16:10,370 --> 01:16:13,680 within the following statement at some point. 1039 01:16:30,270 --> 01:16:33,180 So we introduced these different states. 1040 01:16:33,180 --> 01:16:39,800 I can combine them together and write k1, k2, kN 1041 01:16:39,800 --> 01:16:41,890 with a symmetry that is representative 1042 01:16:41,890 --> 01:16:43,610 of bosons or fermions. 1043 01:16:43,610 --> 01:16:47,810 And that's where I introduced a factor of plus or minus, which 1044 01:16:47,810 --> 01:16:53,730 is obtained by summing over all permutations. 1045 01:16:53,730 --> 01:16:58,235 This factor eta, which is plus or minus, raised to the p. 1046 01:16:58,235 --> 01:17:01,700 So for bosons, I would be adding everything in phase. 1047 01:17:01,700 --> 01:17:04,370 For fermions, I would have minus 1 to the p. 1048 01:17:04,370 --> 01:17:09,615 I have the permutation of the set of indices k1 1049 01:17:09,615 --> 01:17:14,283 through kN in the product state. 1050 01:17:14,283 --> 01:17:16,010 Whoops. 1051 01:17:16,010 --> 01:17:17,447 This should be product. 1052 01:17:17,447 --> 01:17:20,820 This should be a product state. 1053 01:17:20,820 --> 01:17:26,160 And then I have to divide by square root of N 1054 01:17:26,160 --> 01:17:29,700 factorial product over k nk factorial. 1055 01:17:34,250 --> 01:17:43,080 And clearly, and we will see a useful way of looking at this. 1056 01:17:43,080 --> 01:17:49,810 I can also look at the set of states that are occupied among 1057 01:17:49,810 --> 01:17:52,030 the original list of k's. 1058 01:17:52,030 --> 01:17:54,060 Some k's do not occur in this list. 1059 01:17:54,060 --> 01:17:55,970 Some k's occurs once. 1060 01:17:55,970 --> 01:17:58,110 Some k's occur more than once. 1061 01:17:58,110 --> 01:18:00,225 And that's how I construct these factors. 1062 01:18:03,050 --> 01:18:06,130 And then the condition that I have is that, of course, 1063 01:18:06,130 --> 01:18:10,610 sum over k nK should be the total number 1064 01:18:10,610 --> 01:18:17,962 N. That nk is 0 or 1 for fermions. 1065 01:18:17,962 --> 01:18:23,905 That nk is any value up to that constraint for bosons. 1066 01:18:28,900 --> 01:18:36,020 Now, I take these functions and I substitute it here and here. 1067 01:18:36,020 --> 01:18:37,990 What do I get? 1068 01:18:37,990 --> 01:18:40,730 I get the product state. 1069 01:18:40,730 --> 01:18:42,870 So I have eta to the p. 1070 01:18:42,870 --> 01:18:48,580 The product state is e to the i sum over, 1071 01:18:48,580 --> 01:18:59,965 let's say, beta k of p beta x beta prime. 1072 01:19:03,260 --> 01:19:10,610 And then from the next one, I will get e to the minus i sum 1073 01:19:10,610 --> 01:19:13,240 over-- again, some other beta. 1074 01:19:13,240 --> 01:19:15,470 Or the same beta, it doesn't matter. 1075 01:19:15,470 --> 01:19:18,128 p prime beta x beta. 1076 01:19:22,290 --> 01:19:25,390 I think I included everything. 1077 01:19:25,390 --> 01:19:25,890 Yes? 1078 01:19:25,890 --> 01:19:26,858 AUDIENCE: Question. 1079 01:19:26,858 --> 01:19:31,214 Why do you only have one term of beta to the p? 1080 01:19:35,086 --> 01:19:36,070 PROFESSOR: OK. 1081 01:19:36,070 --> 01:19:38,980 So here I have a list. 1082 01:19:38,980 --> 01:19:45,740 It could be k1, k2, k2-- well, let's say k3, k4. 1083 01:19:45,740 --> 01:19:48,080 Let's say I have four things. 1084 01:19:48,080 --> 01:19:50,080 And the product means I essentially 1085 01:19:50,080 --> 01:19:52,430 have a product of these things. 1086 01:19:52,430 --> 01:19:54,940 When I multiply them with wave functions, 1087 01:19:54,940 --> 01:19:58,530 I will have e to the i, k1, x1 et cetera. 1088 01:19:58,530 --> 01:20:00,700 Now, I do some permutation. 1089 01:20:00,700 --> 01:20:04,450 Let's say I go from 1, 2, 3 to 3, 2, 1. 1090 01:20:04,450 --> 01:20:07,160 So there is a permutation that I do like that. 1091 01:20:07,160 --> 01:20:10,010 I leave the 4 to be the same. 1092 01:20:10,010 --> 01:20:14,310 This permutation has some parity. 1093 01:20:14,310 --> 01:20:18,584 So I have to put a plus or minus that depends on the parity. 1094 01:20:18,584 --> 01:20:19,392 AUDIENCE: Yeah. 1095 01:20:19,392 --> 01:20:24,230 But my question is why in that equation over there-- 1096 01:20:24,230 --> 01:20:24,980 PROFESSOR: Oh, OK. 1097 01:20:24,980 --> 01:20:28,445 AUDIENCE: --we have a term A to the p for the k. 1098 01:20:28,445 --> 01:20:29,070 PROFESSOR: Yes. 1099 01:20:29,070 --> 01:20:29,750 AUDIENCE: k factor and not for the bra. 1100 01:20:29,750 --> 01:20:30,580 PROFESSOR: Good. 1101 01:20:30,580 --> 01:20:34,822 You are telling me that I forgot to put p prime and eta of p 1102 01:20:34,822 --> 01:20:35,322 prime. 1103 01:20:35,322 --> 01:20:35,730 AUDIENCE: OK. 1104 01:20:35,730 --> 01:20:37,320 I wasn't sure if there was a reason. 1105 01:20:37,320 --> 01:20:37,920 PROFESSOR: No. 1106 01:20:37,920 --> 01:20:40,270 I was going step by step. 1107 01:20:40,270 --> 01:20:46,250 I had written the bra part and I was about to get to the k part. 1108 01:20:46,250 --> 01:20:48,000 I put that part of the ket, and I 1109 01:20:48,000 --> 01:20:49,850 was about to do the rest of it. 1110 01:20:49,850 --> 01:20:51,120 AUDIENCE: Oh, sorry. 1111 01:20:51,120 --> 01:20:54,110 PROFESSOR: It's OK. 1112 01:20:54,110 --> 01:20:56,390 OK. 1113 01:20:56,390 --> 01:21:03,760 So you can see that what happens at the end of the day here 1114 01:21:03,760 --> 01:21:07,670 is that a lot of these nk factorials disappear. 1115 01:21:07,670 --> 01:21:10,695 So those were not things that we would have liked. 1116 01:21:13,530 --> 01:21:19,230 There is a factor of N factorial from here 1117 01:21:19,230 --> 01:21:21,950 and a factor of N factorial from there that remains. 1118 01:21:21,950 --> 01:21:26,390 So the answer will be 1 over ZN N factorial squared. 1119 01:21:29,120 --> 01:21:34,530 I have a sum over two permutations, p and p prime, 1120 01:21:34,530 --> 01:21:36,260 of something. 1121 01:21:36,260 --> 01:21:40,260 I will do this more closing next time around, 1122 01:21:40,260 --> 01:21:42,720 but I wanted to give you the flavor. 1123 01:21:42,720 --> 01:21:45,880 This double sum at the end of the day, 1124 01:21:45,880 --> 01:21:50,640 just like what we did before, becomes one sum up 1125 01:21:50,640 --> 01:21:54,090 to repetition of N factorial. 1126 01:21:54,090 --> 01:21:58,430 So the N factorial will disappear. 1127 01:21:58,430 --> 01:22:02,060 But what we will find is that the answer here 1128 01:22:02,060 --> 01:22:07,550 is going to be something that is a bunch of Gaussians that 1129 01:22:07,550 --> 01:22:09,340 are very similar to the integration 1130 01:22:09,340 --> 01:22:11,800 that I did for one particle. 1131 01:22:11,800 --> 01:22:15,060 I have e to the minus k squared over 2m. 1132 01:22:15,060 --> 01:22:18,220 I have e to the i k x minus x prime. 1133 01:22:18,220 --> 01:22:21,120 Except that all of these k's and x's and x 1134 01:22:21,120 --> 01:22:25,310 primes have been permuted in all kinds of strange ways. 1135 01:22:25,310 --> 01:22:29,280 Once we untangle that, we find that the answer is going 1136 01:22:29,280 --> 01:22:34,400 to end up to be eta of Q x alpha minus x 1137 01:22:34,400 --> 01:22:40,910 prime Q alpha squared divided by pi 2 lambda squared. 1138 01:22:40,910 --> 01:22:45,200 And we have a sum over alpha, which 1139 01:22:45,200 --> 01:22:49,260 is really kind of like a sum of things 1140 01:22:49,260 --> 01:22:52,940 that we had for the case of one particle. 1141 01:22:52,940 --> 01:22:55,630 So for the case of one particle, we 1142 01:22:55,630 --> 01:22:59,390 found that the off-diagonal density matrix 1143 01:22:59,390 --> 01:23:03,760 had elements that's reflected this wave packet 1144 01:23:03,760 --> 01:23:05,600 nature of this. 1145 01:23:05,600 --> 01:23:09,946 If we have multiple particles that are identical, 1146 01:23:09,946 --> 01:23:15,370 then the thing is if I have 1, 2 and then 1147 01:23:15,370 --> 01:23:20,280 I go 1, 1 prime, 2, 2 prime for the two different locations 1148 01:23:20,280 --> 01:23:23,840 that I have x1 prime, x2 prime, et cetera. 1149 01:23:23,840 --> 01:23:27,065 And 1 and 2 are identical, then I could have really 1150 01:23:27,065 --> 01:23:30,860 also put here 2 prime, 1 prime. 1151 01:23:30,860 --> 01:23:33,360 And I wouldn't have known the difference. 1152 01:23:33,360 --> 01:23:36,960 And this kind of sum will take care of that, 1153 01:23:36,960 --> 01:23:39,000 includes those kinds of exchanges 1154 01:23:39,000 --> 01:23:42,140 that I mentioned earlier and is something 1155 01:23:42,140 --> 01:23:44,630 that we need to derive more carefully 1156 01:23:44,630 --> 01:23:47,580 and explain in more detail next time.