1 00:00:03,350 --> 00:00:08,580 Hello, bonjour, sdravo, ia orana, 2 00:00:09,620 --> 00:00:15,220 welcome to the fifth tutorial of Statistical Mechanics: Algorithms and Computations, 3 00:00:15,220 --> 00:00:18,210 from the Physics Department of Ecole Normale supérieure. 4 00:00:19,380 --> 00:00:25,210 In this week's lecture, we made our first move into quantum mechanics 5 00:00:25,210 --> 00:00:28,500 and quantum statistical mechanics. 6 00:00:28,500 --> 00:00:35,030 We used two languages. The first language is the one of wavefunctions 7 00:00:35,030 --> 00:00:40,930 and of quantized energy levels, both solutions of the Schroedinger equation. 8 00:00:42,490 --> 00:00:49,240 For the harmonic potential, this is what we studied in harmonic_wavefunctions.py. 9 00:00:50,500 --> 00:00:59,710 Besides this traditional approach to quantum mechanics, we also employed the language of density matrices. 10 00:01:00,950 --> 00:01:05,300 This lead us directly to matrix squaring 11 00:01:05,920 --> 00:01:12,120 to the Feynman path integral and to path integral Monte Carlo simulations. 12 00:01:13,550 --> 00:01:18,390 To make the connection between wavefunctions on the one hand 13 00:01:18,390 --> 00:01:24,390 and path integrals on the other, we must provide the missing link 14 00:01:25,070 --> 00:01:31,240 the density matrix at small beta, that holds the path together. 15 00:01:32,760 --> 00:01:35,620 This is like a spring constant 16 00:01:35,620 --> 00:01:39,620 of our quantum mechanical elastic object 17 00:01:41,230 --> 00:01:48,570 We will arrive at this connection between Schroedinger and Feynman in two steps 18 00:01:49,300 --> 00:01:51,820 First, in what follows 19 00:01:51,820 --> 00:01:56,740 Michael will actually compute the density matrix for a free particle 20 00:01:58,380 --> 00:02:04,040 and derive and illustrate a result that was already presented in the lecture. 21 00:02:05,460 --> 00:02:10,460 He will then extend this simplest of all calculations 22 00:02:10,460 --> 00:02:13,820 using the path integral representation. 23 00:02:15,330 --> 00:02:20,890 Then, Alberto will derive and explain 24 00:02:20,890 --> 00:02:23,770 the Trotter decomposition. 25 00:02:24,700 --> 00:02:30,000 Remember: among the three properties of the density matrix 26 00:02:30,000 --> 00:02:34,000 this was the one which allowed to incorporate 27 00:02:34,000 --> 00:02:38,000 arbitrary interaction at high temperature. 28 00:02:38,000 --> 00:02:42,000 This simulation knows 29 00:02:42,000 --> 00:02:46,390 about the free density matrix and the Trotter decomposition 30 00:02:46,390 --> 00:02:49,050 but not about much else. 31 00:02:49,580 --> 00:02:54,550 And certainly not about the solutions of the Schroedinger equation. 32 00:02:56,640 --> 00:02:59,460 So, now in these simulations 33 00:02:59,460 --> 00:03:03,460 let us spend a moment to speak about the axis. 34 00:03:04,950 --> 00:03:11,090 In the wavefunction picture, the axis are the position x and the energy level E. 35 00:03:11,090 --> 00:03:13,310 So things are clear. 36 00:03:15,440 --> 00:03:22,300 But in the path integral picture, the axes here is again x, the position 37 00:03:22,300 --> 00:03:30,510 but the y axis refers an "inverse temperature like" variable, tau. 38 00:03:32,320 --> 00:03:38,640 In quantum statistical mechanics, we call this variable the imaginary time, 39 00:03:39,850 --> 00:03:45,420 and this mysterious name adds to the fascination of path integrals. 40 00:03:47,070 --> 00:03:50,630 Vivien, in the last part of this tutorial, 41 00:03:50,630 --> 00:03:55,570 will tell us about the relationship between temperature and time. 42 00:03:57,180 --> 00:04:02,660 He will convince us that the Trotter decomposition at high temperature 43 00:04:02,660 --> 00:04:06,430 can also be used at small times. 44 00:04:07,020 --> 00:04:13,090 This will allow you to simulate time evolution on the computer, 45 00:04:13,560 --> 00:04:20,390 and to see some of the amazing effect of quantum physics, like the quantum tunneling effect. 46 00:04:21,740 --> 00:04:24,670 But now, let us listen to Michael 47 00:04:25,190 --> 00:04:30,360 explain to us the whence and the whither of the free density matrix 48 00:04:31,080 --> 00:04:35,720 and how it combines with the language of path integrals. 49 00:04:40,310 --> 00:04:43,530 As Werner mentioned in this week lecture 50 00:04:43,530 --> 00:04:48,590 we are able to explicitly write down the density matrix of a particle 51 00:04:48,590 --> 00:04:55,060 in an arbitrary potential V. We do this by use of the Trotter decomposition formula. 52 00:04:55,060 --> 00:05:00,650 Unfortunately, though, this approach is limited to the high-temperature regime. 53 00:05:01,570 --> 00:05:06,720 On the contrary, for a free particle (that is the case V=0) 54 00:05:06,720 --> 00:05:13,210 we are actually able to calculate an exact result which is valid at any temperature. 55 00:05:13,210 --> 00:05:16,100 So let us see how this is done. 56 00:05:16,930 --> 00:05:21,440 As in this week's lecture, we consider the Schroedinger's equation, 57 00:05:21,440 --> 00:05:28,360 and in the case of the free particle it reduces to the expression shown here. 58 00:05:29,160 --> 00:05:31,600 This is simply a wave equation 59 00:05:31,600 --> 00:05:37,340 and you can easily convince yourself that it is satisfied by any sine or cosine wave 60 00:05:37,340 --> 00:05:42,210 as sine and cosine simply reproduce themselves 61 00:05:42,210 --> 00:05:47,990 in the second derivative, up to a sign and a scaling factor. 62 00:05:48,790 --> 00:05:52,680 When we put the particle into a box of size L 63 00:05:52,680 --> 00:05:55,930 say with periodic boundary conditions 64 00:05:55,930 --> 00:05:59,930 then we can only use sine or cosine waves 65 00:05:59,930 --> 00:06:03,930 whose period is equal to the box length L. 66 00:06:04,520 --> 00:06:07,310 Other waves simply don't fit. 67 00:06:08,680 --> 00:06:12,840 The complete set of waves that do fit is shown here 68 00:06:13,470 --> 00:06:18,630 The superscripts denote symmetric and antisymmetric wavefunctions 69 00:06:18,630 --> 00:06:23,310 with respect to the center of the interval from 0 to L. 70 00:06:24,140 --> 00:06:26,430 You see one ground-state 71 00:06:27,490 --> 00:06:31,110 and the symmetric and antisymmetric solutions 72 00:06:31,110 --> 00:06:35,560 for each excited state, with the energies E_n 73 00:06:35,560 --> 00:06:40,430 = 2 pi^2 n^2 / L^2. 74 00:06:40,430 --> 00:06:44,430 In Python, this gives the following program 75 00:06:44,430 --> 00:06:49,050 it is analogous to the program harmonic_wavefunction.py 76 00:06:49,050 --> 00:06:51,860 that was introduced by Werner in this week's lecture 77 00:06:51,860 --> 00:06:56,710 and it computes the density matrix by summing over all states. 78 00:06:56,710 --> 00:06:59,820 So let us have a look at the density matrix 79 00:06:59,820 --> 00:07:03,820 by simply plotting the results of our little program. 80 00:07:05,530 --> 00:07:08,170 The stripe on the diagonal of the matrix 81 00:07:08,170 --> 00:07:13,810 indicated that the density matrix rho_free(x, x', beta) 82 00:07:13,810 --> 00:07:17,810 is in fact a function of x-x' only. 83 00:07:17,810 --> 00:07:22,470 This means that rho_free is translation invariant 84 00:07:22,470 --> 00:07:26,470 as the individual values of x and x' do not matter. 85 00:07:26,470 --> 00:07:30,470 only the distance x-x' counts. 86 00:07:31,880 --> 00:07:34,860 rho_free is largest on the diagonal 87 00:07:34,860 --> 00:07:38,500 In fact at high temperature (that is small beta) 88 00:07:38,500 --> 00:07:43,410 it is all concentrated on the diagonal, where x=x'. 89 00:07:44,160 --> 00:07:48,550 As we lower the temperature, we see that the off-diagonal part 90 00:07:48,550 --> 00:07:52,550 become larger and larger with increasing beta. 91 00:07:52,550 --> 00:07:58,910 This is the famous coherence, that is governed by the thermal De Broglie wavelength. 92 00:07:59,500 --> 00:08:01,980 Now let us do a small calculation 93 00:08:01,980 --> 00:08:06,900 but for simplicity, we will not use the sines and cosines 94 00:08:06,900 --> 00:08:11,540 but rather the equivalent complex exponential wavefunctions. 95 00:08:12,980 --> 00:08:18,280 You can actually check this equivalence by writing a small Python program. 96 00:08:21,290 --> 00:08:26,850 When you compare the output of this program to the output of the previous program, you will see 97 00:08:26,850 --> 00:08:31,070 that it arrives exactly at the same results as before. 98 00:08:32,370 --> 00:08:36,340 However, we promised an exact expression 99 00:08:36,340 --> 00:08:40,340 for the free density matrix, so let us now calculate it. 100 00:08:41,020 --> 00:08:44,500 Using the complex exponential wavefunctions 101 00:08:44,500 --> 00:08:48,500 the density matrix is given by this expression 102 00:08:48,500 --> 00:08:53,860 where you notice that each term is in fact a complex number. 103 00:08:54,350 --> 00:09:00,480 At first glance this seems a bit strange as we know that the finite result will be real. 104 00:09:01,420 --> 00:09:05,600 This is because for each complex number in the sum 105 00:09:05,600 --> 00:09:10,810 there's a complex conjugate counterpart and together they become real. 106 00:09:12,530 --> 00:09:15,810 Having the particle in a box of size L 107 00:09:15,810 --> 00:09:20,450 is no true limitation, as we can let L go to infinity 108 00:09:20,450 --> 00:09:24,870 and replace the sum over n by an integral. 109 00:09:25,670 --> 00:09:30,450 This may sound difficult, but it can be easily understood 110 00:09:30,450 --> 00:09:35,820 when we introduce a dummy parameter delta_n = 1, 111 00:09:35,820 --> 00:09:40,410 difference between two successive n values. 112 00:09:41,120 --> 00:09:45,360 We then change the variables from n to lambda 113 00:09:45,360 --> 00:09:49,360 with lambda = 2 pi n / L 114 00:09:49,360 --> 00:09:56,760 Thus we have delta_lambda = 2 pi delta_n / L 115 00:09:58,030 --> 00:10:02,880 This leads to this term by term equivalent sum 116 00:10:04,280 --> 00:10:06,990 where we use this result 117 00:10:08,260 --> 00:10:16,270 which follows when we take ctilde = c / sigma^2, from the Gaussian integral. 118 00:10:17,930 --> 00:10:23,890 This Gaussian integral was already evaluated in Lecture 4, last week's lecture. 119 00:10:24,820 --> 00:10:28,630 We finally arrive at the free density matrix 120 00:10:28,630 --> 00:10:32,630 where we have reinserted Planck constant hbar 121 00:10:32,630 --> 00:10:35,580 and the particle mass m. 122 00:10:36,600 --> 00:10:42,030 The free density matrix is indeed a function of x-x' 123 00:10:42,400 --> 00:10:47,140 and see how at large temperatures (that is small beta) 124 00:10:47,570 --> 00:10:50,500 rho_free becomes infinitely peaked 125 00:10:50,500 --> 00:10:55,410 at the diagonal x=x', as we noticed earlier. 126 00:10:56,280 --> 00:11:04,350 Note in particular that the density matrix is positive, not only on the diagonal but everywhere. 127 00:11:04,350 --> 00:11:07,070 For us Monte Carlo specialists 128 00:11:07,070 --> 00:11:12,660 this is important as you can interpret the density matrix as a probability. 129 00:11:13,640 --> 00:11:16,040 Now in this calculation 130 00:11:16,040 --> 00:11:20,040 we made a transformation from a sum to an integral 131 00:11:20,040 --> 00:11:23,040 when taking the limit L to infinity. 132 00:11:23,850 --> 00:11:32,240 But what is the density matrix of a finite box with periodic boundary conditions, that is for finite L? 133 00:11:33,090 --> 00:11:38,520 The above sum can be evaluated using the famous Poisson's sum rule 134 00:11:38,520 --> 00:11:41,330 and the result is really simple. 135 00:11:42,780 --> 00:11:49,770 The parameter w is called the winding number and you will see in a moment where this name comes from. 136 00:11:50,260 --> 00:11:56,240 The density matrix rho(x,x',beta) is made up 137 00:11:56,240 --> 00:11:58,000 of all the paths 138 00:11:58,000 --> 00:12:04,480 (in Feynman's path integral picture) that go from x to x' directly 139 00:12:05,460 --> 00:12:09,860 plus all the paths that wind around the box 140 00:12:09,860 --> 00:12:14,860 once, that is they have the winding number w=1 141 00:12:15,500 --> 00:12:23,970 plus all the paths that wind around the box twice and so on and so on for higher winding numbers. 142 00:12:23,970 --> 00:12:32,220 Of course, we can also wind a path to the left, corresponding to negative winding numbers. 143 00:12:33,890 --> 00:12:38,580 This concept of winding number will come in handy two weeks from now. 144 00:12:38,580 --> 00:12:43,600 It is key to understand Bose-Einstein condensation. 145 00:12:44,430 --> 00:12:52,340 So we have now obtained the free density matrix at all temperatures and it is a Gaussian of x-x' 146 00:12:52,340 --> 00:12:56,470 with a normalization 1 / sqrt(2 pi beta) 147 00:12:56,470 --> 00:13:00,660 and a variance of beta. 148 00:13:00,660 --> 00:13:04,660 Alberto will determine a general expression 149 00:13:04,660 --> 00:13:09,910 for all potentials at high temperatures, using the Trotter formula. 150 00:13:11,840 --> 00:13:19,950 But before you go on, please take a moment or three to download, run and modify the programs discussed in this section. 151 00:13:19,950 --> 00:13:28,210 On the Coursera website, you will find the program free_periodic_sine_cosine.py and its cousin 152 00:13:28,210 --> 00:13:31,520 free_periodic_complex_exp.py 153 00:13:31,520 --> 00:13:37,640 Take these programs apart to understand the inner workings of complex arithmetics 154 00:13:37,640 --> 00:13:41,840 that lead to a final real-valued and positive result. 155 00:13:46,310 --> 00:13:53,310 As explained by Michael, for a free particle we can compute exactly the density matrix 156 00:13:53,310 --> 00:13:57,310 this is of great importance 157 00:13:57,310 --> 00:14:06,460 as we can compute the general density matrix - so to speak - which grows from this seed, as we will discuss 158 00:14:07,340 --> 00:14:09,840 For what follows 159 00:14:09,840 --> 00:14:17,430 you must understand that rho(x,x',beta) is a matrix of two indices 160 00:14:17,430 --> 00:14:24,450 x and x', but it is also an operator: exp(- beta H) 161 00:14:24,450 --> 00:14:30,800 The matrix elements of the Hamiltonian H can be written in an arbitrary basis 162 00:14:30,800 --> 00:14:36,080 and in this basis the matrix elements of rho write like this 163 00:14:38,450 --> 00:14:44,300 This expression can be familiar to those of you who have studied a little bit of quantum mechanics 164 00:14:44,670 --> 00:14:49,530 but for all of you, please retain that the density matrix 165 00:14:49,530 --> 00:14:53,670 is an operator, the exponential of minus beta H 166 00:14:53,670 --> 00:14:57,670 it is the direct generalization of the Boltzmann distribution 167 00:14:59,080 --> 00:15:05,510 Let us now consider the density matrix of a system with the following Hamiltonian 168 00:15:05,510 --> 00:15:09,510 H = Hfree + V 169 00:15:09,510 --> 00:15:13,510 where Hfree is the Hamiltonian discussed by Michael 170 00:15:13,510 --> 00:15:20,640 and V is the interaction potential, that for the harmonic oscillator is (1/2) x^2 171 00:15:20,640 --> 00:15:26,350 Now, the Trotter decomposition corresponds to the following expression. 172 00:15:27,100 --> 00:15:32,740 To see that this is true, we will expand both terms up to the order beta^2 173 00:15:33,550 --> 00:15:38,450 but attention, Hfree and V are operators 174 00:15:38,450 --> 00:15:46,290 the product V Hfree can be different from Hfree V. 175 00:15:46,290 --> 00:15:51,250 Operators, matrices in general do not commute. 176 00:15:52,450 --> 00:15:57,010 So, let's start from the second term of this equation 177 00:15:57,010 --> 00:16:04,140 We can first expand exp( (-beta/2) V(x)) 178 00:16:06,400 --> 00:16:12,780 'a' corresponds to the linear term in beta, and 'b' to the second order term 179 00:16:13,870 --> 00:16:16,780 Then we expand rho_free 180 00:16:17,890 --> 00:16:23,130 'c' is the linear term and 'd' is the second order term, 181 00:16:23,130 --> 00:16:30,890 finally we can also expand exp(-(beta/2) V(x')) 182 00:16:33,040 --> 00:16:38,490 So let's reorganize these terms order by order in beta. 183 00:16:38,490 --> 00:16:42,490 At the zero order we have one. 184 00:16:43,470 --> 00:16:50,510 At the first order in beta, we have a+c+e 185 00:16:51,360 --> 00:16:56,370 and then we need a little bit of work to write the second order term 186 00:16:57,590 --> 00:17:05,560 This expression fully agrees with the direct expansion of the full density matrix 187 00:17:06,180 --> 00:17:09,280 These two equations prove 188 00:17:09,280 --> 00:17:13,900 that the Trotter expansion is correct up to beta^2. 189 00:17:15,620 --> 00:17:18,520 Please note that this manipulations 190 00:17:18,520 --> 00:17:22,520 are valid on operators and wavefunctions 191 00:17:22,520 --> 00:17:24,440 in a finite box. 192 00:17:25,210 --> 00:17:31,700 In Python we can implement the Trotter's decomposition using Michael's approach, 193 00:17:31,700 --> 00:17:36,380 and plot the entire density matrix at high temperature. 194 00:17:36,380 --> 00:17:40,380 Remember: the free density matrix 195 00:17:40,380 --> 00:17:43,630 rho_free(x,x',beta) 196 00:17:43,630 --> 00:17:47,630 is a function of x-x' 197 00:17:47,630 --> 00:17:51,630 and this gives the nice stripe in this diagram. 198 00:17:52,830 --> 00:17:59,550 Now let us add the harmonic potential V to the Trotter decomposition. 199 00:18:01,250 --> 00:18:07,880 You see that for large x and x' the density matrix is suppressed 200 00:18:07,880 --> 00:18:14,370 and this is what keeps close to the origin the quantum harmonic oscillator. 201 00:18:14,370 --> 00:18:18,370 In this week's homework, you will use 202 00:18:18,370 --> 00:18:21,420 both the Trotter decomposition 203 00:18:21,420 --> 00:18:24,000 and the convolution properties 204 00:18:24,000 --> 00:18:28,860 to compute the density matrix up to very low temperatures. 205 00:18:30,360 --> 00:18:36,920 As you must square the matrix many times, the approximation errors will grow 206 00:18:37,900 --> 00:18:46,050 However, the quality of this approximation (correct up to beta^2) is so high that it will survive 207 00:18:46,050 --> 00:18:47,900 up to low temperatures. 208 00:18:47,900 --> 00:18:51,900 The Trotter break-up is completely general: 209 00:18:51,900 --> 00:18:55,150 it applies to an arbitrary potential 210 00:18:55,150 --> 00:18:59,150 and (as will be shown by Vivien in a moment) 211 00:18:59,150 --> 00:19:05,810 it can be even turned around to describe the real time evolution of the system. 212 00:19:07,230 --> 00:19:13,260 But before understanding why the inverse temperature beta is an imaginary time, 213 00:19:13,260 --> 00:19:17,550 please take a moment to download, run and modify 214 00:19:17,550 --> 00:19:21,990 the program harmonic_trotter_movie.py 215 00:19:21,990 --> 00:19:28,740 which implements the Trotter decomposition and provides a nice graphical output. 216 00:19:30,290 --> 00:19:32,740 You can check this approximation 217 00:19:32,740 --> 00:19:36,220 and actually in this week's homework 218 00:19:36,220 --> 00:19:44,250 you will test this approximation against the output of harmonic_wavefunctions.py 219 00:19:48,250 --> 00:19:51,990 In this section we examine a complementary problem 220 00:19:51,990 --> 00:19:55,510 which is the time evolution of a quantum system 221 00:19:56,140 --> 00:20:01,190 Werner, Michael and Alberto have explained to you 222 00:20:01,190 --> 00:20:03,420 in the previous sequences 223 00:20:03,420 --> 00:20:07,860 how to describe a quantum system at finite temperature 224 00:20:07,860 --> 00:20:13,200 in equilibrium, by the use of the density matrix 225 00:20:14,260 --> 00:20:19,170 However, in general, systems are described by 226 00:20:19,170 --> 00:20:24,310 a wavefunction psi(t) which evolves in time. 227 00:20:25,590 --> 00:20:31,720 So, if we fix this wavefunction psi(t=0) 228 00:20:31,720 --> 00:20:35,720 to be equal to psi0 at initial time 229 00:20:35,720 --> 00:20:41,000 it evolves in time according to the Schroedinger equation displayed here 230 00:20:41,500 --> 00:20:45,330 which is i times dpsi/dt 231 00:20:45,330 --> 00:20:48,320 = H psi 232 00:20:48,320 --> 00:20:52,810 and here H is the Hamiltonian of the system 233 00:20:52,810 --> 00:20:58,300 and we have also fixed the constant hbar to be equal to one. 234 00:21:00,880 --> 00:21:05,400 To solve this equation, one may be tempted to replace 235 00:21:05,400 --> 00:21:09,870 the time derivative by a first order approximation 236 00:21:10,870 --> 00:21:13,410 which is displayed there 237 00:21:13,410 --> 00:21:19,840 but this is not an option in our case because this would break the conservation of probability. 238 00:21:22,330 --> 00:21:27,430 A better choice is to consider the operator solution 239 00:21:27,430 --> 00:21:31,590 of this equation, which writes as follows 240 00:21:31,590 --> 00:21:39,370 It is psi(t) = exp(-i H t) 241 00:21:39,370 --> 00:21:43,370 times the initial vector psi(0) 242 00:21:44,650 --> 00:21:49,570 You may indeed check, by differentiating with respect to t 243 00:21:49,570 --> 00:21:54,100 that this expression is the solution of the Schroedinger equation 244 00:21:54,100 --> 00:22:00,450 with initial condition psi(t=0)=psi0. 245 00:22:02,090 --> 00:22:06,040 Now, let's have a closer look to this formula. 246 00:22:06,040 --> 00:22:10,040 You note that by formally replacing 247 00:22:10,040 --> 00:22:12,700 (i t) by beta 248 00:22:12,970 --> 00:22:16,900 that is to say, by working in imaginary time 249 00:22:17,390 --> 00:22:23,220 you obtain the very same expression as that of the density matrix 250 00:22:23,220 --> 00:22:29,730 that we know for a quantum system at equilibrium at inverse temperature beta. 251 00:22:31,810 --> 00:22:36,950 So this relation provides us a dictionary 252 00:22:36,950 --> 00:22:42,520 a mathematical dictionary between time evolution and thermal equilibrium. 253 00:22:44,540 --> 00:22:49,460 It is not only beautiful from the abstract point of view 254 00:22:49,460 --> 00:22:53,460 but also very useful in practice 255 00:22:54,030 --> 00:22:58,140 Indeed, you can apply what Alberto has told you 256 00:22:58,140 --> 00:23:00,900 about the Trotter expansion 257 00:23:00,900 --> 00:23:04,900 to devise an algorithm which allows you 258 00:23:04,900 --> 00:23:09,660 to study very efficiently the evolution of quantum systems 259 00:23:10,880 --> 00:23:15,510 Let us now focus our attention to a particle at position x 260 00:23:15,510 --> 00:23:18,430 in a potential V(x) in dimension 1 261 00:23:19,100 --> 00:23:24,740 If we set the mass of the particles to be m=1, its evolution 262 00:23:24,740 --> 00:23:27,840 is governed by the Schroedinger equation 263 00:23:27,840 --> 00:23:32,970 of Hamiltonian H = Hfree + V(x) 264 00:23:34,300 --> 00:23:38,180 The idea is to decompose the full time interval t 265 00:23:38,180 --> 00:23:44,630 into N time steps of duration delta_t = t / N 266 00:23:45,900 --> 00:23:51,320 The full operator of evolution is now expressed exactly as a product of N 267 00:23:51,320 --> 00:23:56,060 infinitesimal operator of evolution, as displayed here. 268 00:23:57,550 --> 00:24:05,760 Each of these infinitesimal operators of evolution writes exp(-i delta_t H) 269 00:24:06,730 --> 00:24:09,760 Now to understand the contribution 270 00:24:09,760 --> 00:24:14,960 of the components Hfree and V(x) of this Hamiltonian 271 00:24:14,960 --> 00:24:19,890 we can use Alberto's Trotter decomposition, which writes as follows 272 00:24:20,370 --> 00:24:27,650 You recognize the same formula, where beta has been replaced by (i delta_t) 273 00:24:28,140 --> 00:24:33,250 This approximation is valid up to order delta_t ^2 274 00:24:33,250 --> 00:24:39,740 that is to say, the first error term is of order delta_t^3. 275 00:24:41,670 --> 00:24:45,060 Let us examine each term of this product 276 00:24:45,060 --> 00:24:49,060 the first and the last term are easy to understand 277 00:24:49,060 --> 00:24:57,110 they consist in the multiplication by the same factor exp(-(i/2) delta_t V(x)) 278 00:24:57,110 --> 00:25:01,110 The free term is a bit enigmatic 279 00:25:01,110 --> 00:25:07,230 but the mystery is lifted by transforming the wavefunction to Fourier space 280 00:25:07,230 --> 00:25:09,730 for the spatial coordinate. 281 00:25:10,340 --> 00:25:15,290 This consists in going from the real space psi(x,t) 282 00:25:15,290 --> 00:25:19,860 to the momentum space psi_hat(p,t) 283 00:25:21,530 --> 00:25:29,460 You will easily convince yourself - for instance by reading the short fact-sheet associated to this tutorial 5 - 284 00:25:29,460 --> 00:25:33,460 that the action of the free operator 285 00:25:33,460 --> 00:25:40,500 consists in a multiplication by a factor exp(- (i/2) p^2) 286 00:25:40,500 --> 00:25:46,230 So, the action of this operator is just the multiplication by a Gaussian factor. 287 00:25:48,370 --> 00:25:54,000 This provides a possible scheme to devise an algorithm which is the following: 288 00:25:55,720 --> 00:26:00,610 first, define the initial wavefunction, then 289 00:26:00,610 --> 00:26:07,160 discretize the time interval t into N time slices of duration delta_t, 290 00:26:07,160 --> 00:26:10,590 define the Fourier procedure, 291 00:26:10,590 --> 00:26:14,700 and apply the Trotter formula many times. 292 00:26:14,700 --> 00:26:17,080 At each time step 293 00:26:17,080 --> 00:26:25,010 first in real space, multiply by a factor exp(-(i/2) delta_t V(x)), 294 00:26:25,010 --> 00:26:27,180 make a Fourier transformation, 295 00:26:27,180 --> 00:26:33,380 multiply by exp(- (i/2) p^2 delta_t), 296 00:26:33,480 --> 00:26:38,340 and go back to the real space multiplying by the same factor as previously. 297 00:26:38,340 --> 00:26:47,530 This is the only step, you just have to iterate and it gives the following beautiful algorithm 298 00:26:47,530 --> 00:26:51,530 quantum_time_evolution.py. 299 00:26:51,530 --> 00:26:54,060 This method is really powerful. 300 00:26:54,060 --> 00:27:00,150 To illustrate it, let's consider the fate of a quantum particle confined between two barriers: 301 00:27:00,150 --> 00:27:04,940 an infinite barrier on its left and a finite barrier on its right. 302 00:27:05,690 --> 00:27:09,760 The particle starts from some position between the two walls. 303 00:27:09,760 --> 00:27:16,050 If the particle were classical, it would bounce forever between the two walls. 304 00:27:17,680 --> 00:27:25,160 You observe that each time the wavefunction bounces on the wall on the right 305 00:27:25,160 --> 00:27:31,830 a finite but tangible proportion of the wave actually goes through the wall. 306 00:27:33,190 --> 00:27:40,270 This quantum tunneling effect is a genuine quantum effect and has no classical counterpart. 307 00:27:41,580 --> 00:27:45,630 We have illustrated how our mathematical dictionary between 308 00:27:45,630 --> 00:27:48,010 real time and imaginary time 309 00:27:48,010 --> 00:27:54,480 has allowed us to observe the quantum tunneling effect in a simple one-dimensional situation. 310 00:27:55,980 --> 00:27:59,320 However, have in mind that the very same effect 311 00:27:59,320 --> 00:28:01,840 has deep physical consequences 312 00:28:01,840 --> 00:28:09,710 from for instance the radioactive decay to the construction of the transistor in the CPUs of your computer, 313 00:28:09,710 --> 00:28:16,520 or in more complex situation to the devise of the scanning tunneling microscope, 314 00:28:16,520 --> 00:28:20,520 or even to the evaporation of black holes. 315 00:28:21,200 --> 00:28:26,110 It is now time for you to download, run and modify 316 00:28:26,110 --> 00:28:31,770 the beautiful program we have seen in this section: quantum_time_evolution.py. 317 00:28:31,770 --> 00:28:34,210 Check it out. 318 00:28:34,210 --> 00:28:38,210 See by yourself how a quantum particle can tunnel through a wall 319 00:28:38,210 --> 00:28:42,210 and program maybe some other exciting potentials 320 00:28:42,210 --> 00:28:49,420 Take also a look to the fact-sheet which explains to you how the Fourier transform from x 321 00:28:49,420 --> 00:28:53,420 to p is implemented in practice. 322 00:28:54,120 --> 00:28:56,470 In this tutorial 323 00:28:56,470 --> 00:29:02,230 we zoomed in to the density matrix and its three main properties 324 00:29:02,230 --> 00:29:08,790 the convolution, the free density matrix and the Trotter decomposition 325 00:29:10,410 --> 00:29:16,440 we already perceive the plot of these three stage characters 326 00:29:16,440 --> 00:29:23,390 that prepare for a great story and that will lead us to quite complicated situations 327 00:29:25,240 --> 00:29:29,300 In what Alberto and Michael told us 328 00:29:29,700 --> 00:29:34,160 the density matrix remains positive and real 329 00:29:34,470 --> 00:29:42,190 allowing us to interpret it as a probability, and to do quantum Monte Carlo. 330 00:29:42,190 --> 00:29:48,440 This positivity is preserved for many particles, 331 00:29:48,440 --> 00:29:53,280 arbitrary interactions, and even for bosons 332 00:29:53,280 --> 00:29:58,950 It is broken in the case of fermionic particles 333 00:30:00,080 --> 00:30:02,110 In what Vivien told us 334 00:30:02,440 --> 00:30:06,470 in the time-evolution all of this was broken also 335 00:30:07,080 --> 00:30:12,270 the density matrix was complex-valued and it could be negative 336 00:30:13,100 --> 00:30:18,300 The great algorithm produced only works in one-dimension 337 00:30:18,300 --> 00:30:23,280 because quantum time evolution is a really hard nut to crack 338 00:30:24,650 --> 00:30:27,280 So now it is time for you 339 00:30:27,280 --> 00:30:32,310 to look at how these three characters that we described 340 00:30:32,310 --> 00:30:34,330 intervene in practice 341 00:30:34,850 --> 00:30:41,710 and we wish you good luck and a lot of fun with the homework session 5.