1 00:00:10,330 --> 00:00:13,820 STUDENT: So you're probably all familiar with the field 2 00:00:13,820 --> 00:00:16,280 measured in this paper here by Shrodinger, 3 00:00:16,280 --> 00:00:17,770 which is a quantum theory. 4 00:00:17,770 --> 00:00:19,660 And you surely also know that the students 5 00:00:19,660 --> 00:00:22,460 have a lot of problems visualizing 6 00:00:22,460 --> 00:00:25,840 it, and understanding its behavior in some intuitive way. 7 00:00:25,840 --> 00:00:27,680 For example, you know the tunneling effect. 8 00:00:27,680 --> 00:00:31,240 And you won't expect a classical cat to just tunnel 9 00:00:31,240 --> 00:00:32,840 through potential wall. 10 00:00:32,840 --> 00:00:36,160 What I am going to show you in this presentation 11 00:00:36,160 --> 00:00:39,310 is an algorithm called the split operator Fourier transform 12 00:00:39,310 --> 00:00:42,880 algorithm, which can be used to describe quantum tunnel 13 00:00:42,880 --> 00:00:45,682 evolution, and therefore, for example, the tunneling effect. 14 00:00:45,682 --> 00:00:47,890 And I think that it also might be interesting for you 15 00:00:47,890 --> 00:00:51,220 as material scientists, on the one hand, because you might 16 00:00:51,220 --> 00:00:53,900 be interested in time evolution of your properties, 17 00:00:53,900 --> 00:00:56,530 especially for nanosystems, quantum systems. 18 00:00:56,530 --> 00:00:58,660 But on the other hand, I will show you 19 00:00:58,660 --> 00:01:00,580 iteration of this algorithm which can also 20 00:01:00,580 --> 00:01:02,320 be used to derive other algorithms, 21 00:01:02,320 --> 00:01:04,660 like the [INAUDIBLE] algorithm. 22 00:01:04,660 --> 00:01:08,590 And you also see some methods to judge you 23 00:01:08,590 --> 00:01:10,570 how good a simulation is. 24 00:01:10,570 --> 00:01:13,710 And I think that might be useful for you if you [INAUDIBLE] 25 00:01:13,710 --> 00:01:17,430 want to do some simulations on other systems. 26 00:01:17,430 --> 00:01:20,890 So in doing this, we are going to start at the foundations, 27 00:01:20,890 --> 00:01:22,475 which is Schrodinger equation. 28 00:01:22,475 --> 00:01:24,600 In it's time-dependent version, it looks like this. 29 00:01:24,600 --> 00:01:26,110 So you have your state. 30 00:01:26,110 --> 00:01:28,632 And then you derive it after t. 31 00:01:28,632 --> 00:01:29,590 And then it's the same. 32 00:01:29,590 --> 00:01:31,631 Like, if you apply your Hamiltonian to a state t, 33 00:01:31,631 --> 00:01:33,630 and if you integrate this equation here 34 00:01:33,630 --> 00:01:36,100 for time-independent Hamiltonian, 35 00:01:36,100 --> 00:01:38,050 you get this equation here which tells you 36 00:01:38,050 --> 00:01:41,380 that, if you apply this thing here with a Hamiltonian 37 00:01:41,380 --> 00:01:44,110 exponential to your state at time zero, 38 00:01:44,110 --> 00:01:46,580 you get out your state at time t. 39 00:01:46,580 --> 00:01:49,150 And you do this behavior as it propagates a wave 40 00:01:49,150 --> 00:01:51,550 function from time 0 to time t. 41 00:01:51,550 --> 00:01:53,540 We call this object here propagator. 42 00:01:53,540 --> 00:01:58,920 And this object will be the main focus of this review. 43 00:01:58,920 --> 00:02:00,990 And if you don't know, this Hamiltonian 44 00:02:00,990 --> 00:02:03,380 here gives us the total energy and is [INAUDIBLE] 45 00:02:03,380 --> 00:02:07,060 out of two parts, mainly kinetic part and the potential part. 46 00:02:07,060 --> 00:02:09,590 And in general, you write this kinetic part 47 00:02:09,590 --> 00:02:11,510 here with the momentum, so it tells you 48 00:02:11,510 --> 00:02:14,320 how fast the cat moves. 49 00:02:14,320 --> 00:02:16,340 And the potential part is some function, 50 00:02:16,340 --> 00:02:19,779 in most cases of x, which is your coordinate 51 00:02:19,779 --> 00:02:21,695 in the classical Cartesian space, for example. 52 00:02:21,695 --> 00:02:24,260 And it tells you how your potential looks 53 00:02:24,260 --> 00:02:27,050 like in which the cat moves. 54 00:02:27,050 --> 00:02:28,592 So let's focus on the propagator. 55 00:02:28,592 --> 00:02:30,050 And the first thing you might write 56 00:02:30,050 --> 00:02:33,230 is that you don't do the huge step from time 0 57 00:02:33,230 --> 00:02:34,860 to time n in one step. 58 00:02:34,860 --> 00:02:38,180 So you might think, if your cat, your system, a number of small 59 00:02:38,180 --> 00:02:42,740 kicks to go in that way from time 0 to time t. 60 00:02:42,740 --> 00:02:45,815 And you can write this down by using just a small time 61 00:02:45,815 --> 00:02:48,650 step, epsilon, which is t, your whole time, 62 00:02:48,650 --> 00:02:49,940 divided by number of steps. 63 00:02:49,940 --> 00:02:53,240 And then you do this n times, one after another. 64 00:02:53,240 --> 00:02:55,590 So you might ask, does it help us? 65 00:02:55,590 --> 00:02:57,580 And the answer is, unfortunately, no, 66 00:02:57,580 --> 00:02:59,870 because we still don't know how to compute 67 00:02:59,870 --> 00:03:03,830 the application of this operator to some general stage. 68 00:03:03,830 --> 00:03:07,269 So a second thing you might ask, OK, I know my Hamiltonian 69 00:03:07,269 --> 00:03:07,935 looks like this. 70 00:03:07,935 --> 00:03:10,340 So I have a kinetic part, a potential part. 71 00:03:10,340 --> 00:03:13,070 Might I be able to just write this propagator down 72 00:03:13,070 --> 00:03:17,330 as a product of two propagators, so one for the kinetic part 73 00:03:17,330 --> 00:03:19,070 and one for the potential part? 74 00:03:19,070 --> 00:03:21,270 And this still isn't true and helpful 75 00:03:21,270 --> 00:03:24,200 because we don't know if this is really 76 00:03:24,200 --> 00:03:27,260 a true rewriting, because, in general, quantum mechanics 77 00:03:27,260 --> 00:03:29,840 operators don't [INAUDIBLE] place or a row 78 00:03:29,840 --> 00:03:32,290 which all of your [INAUDIBLE]. 79 00:03:32,290 --> 00:03:35,130 So but the world isn't that bad. 80 00:03:35,130 --> 00:03:36,910 Schrodinger gave us an equation which 81 00:03:36,910 --> 00:03:41,490 helps us to do this time evolution in quantum mechanics. 82 00:03:41,490 --> 00:03:43,594 And it's this split up equation here. 83 00:03:43,594 --> 00:03:45,760 And it tells us, if you use, for example, this split 84 00:03:45,760 --> 00:03:49,330 up here, so half a time step in potential, 85 00:03:49,330 --> 00:03:51,700 and a full time step in kinetics, 86 00:03:51,700 --> 00:03:53,950 and half a time step in the potential, 87 00:03:53,950 --> 00:03:56,950 and if you do this for an infinite number of times, 88 00:03:56,950 --> 00:03:59,950 this is really, really, really time steps, it 89 00:03:59,950 --> 00:04:03,590 is an exact rewriting of the original equation. 90 00:04:03,590 --> 00:04:05,400 So this is really helpful, because now we 91 00:04:05,400 --> 00:04:09,750 know that this is a controllable approximation which get better 92 00:04:09,750 --> 00:04:13,216 if we use more time steps. 93 00:04:13,216 --> 00:04:14,840 Up to here, we're all totally abstract. 94 00:04:14,840 --> 00:04:18,560 So we just use Dirac notation for our state, psi. 95 00:04:18,560 --> 00:04:20,360 And if you don't know this equation here 96 00:04:20,360 --> 00:04:23,380 or this form of writing this down, don't bother. 97 00:04:23,380 --> 00:04:25,810 It's just Dirac's notation, and just tells us 98 00:04:25,810 --> 00:04:29,180 that we now go from a general state, psi, 99 00:04:29,180 --> 00:04:30,807 to a more-- to a basis. 100 00:04:30,807 --> 00:04:32,640 We project into a coordinate representation, 101 00:04:32,640 --> 00:04:33,980 to a coordinate basis. 102 00:04:33,980 --> 00:04:39,110 And this gives us then our state psi at some position x0, 103 00:04:39,110 --> 00:04:41,870 if I predict onto x0, for example, here, [INAUDIBLE] 104 00:04:41,870 --> 00:04:43,430 here. 105 00:04:43,430 --> 00:04:45,949 And after some algebra, which is really, if we could, 106 00:04:45,949 --> 00:04:48,240 I will show you this in a note which I will provide you 107 00:04:48,240 --> 00:04:51,140 with, you see that you get this equation out here, which 108 00:04:51,140 --> 00:04:52,330 is the most important one. 109 00:04:52,330 --> 00:04:54,710 And this [INAUDIBLE] cross just gives us our algorithm. 110 00:04:54,710 --> 00:04:57,200 So we start with our wave function 111 00:04:57,200 --> 00:05:00,260 in some current representation at time zero. 112 00:05:00,260 --> 00:05:02,090 Then we apply our potential propagator, 113 00:05:02,090 --> 00:05:06,347 which looks similar to what we had here for half a time step. 114 00:05:06,347 --> 00:05:08,180 And then we have two interesting parts here, 115 00:05:08,180 --> 00:05:11,840 because we have integrals with some special form. 116 00:05:11,840 --> 00:05:15,120 And you hopefully recognize that these two integrals are just 117 00:05:15,120 --> 00:05:15,920 Fourier transforms. 118 00:05:15,920 --> 00:05:18,470 So we start here on coordinate representation, 119 00:05:18,470 --> 00:05:22,100 and then Fourier transform it to our momentum basis. 120 00:05:22,100 --> 00:05:23,630 And in the momentum basis, we then 121 00:05:23,630 --> 00:05:27,470 apply our kinetic propagation, which has a p in it. 122 00:05:27,470 --> 00:05:29,330 And then we go back from momentum basis 123 00:05:29,330 --> 00:05:32,780 to quantum basis using a Fourier transform again, 124 00:05:32,780 --> 00:05:35,090 and then can apply our potential propagation 125 00:05:35,090 --> 00:05:37,890 a second time to complete the time step. 126 00:05:37,890 --> 00:05:41,000 If you write it down in an algorithm, it looks like this. 127 00:05:41,000 --> 00:05:43,490 It's really simple, just a few lines. 128 00:05:43,490 --> 00:05:46,320 So we started from our state in some coordinate basis. 129 00:05:46,320 --> 00:05:49,250 Then we have to use a grid to discretize our system, 130 00:05:49,250 --> 00:05:52,490 and so we have some points on which we compute the time 131 00:05:52,490 --> 00:05:54,320 evolution on. 132 00:05:54,320 --> 00:05:57,520 And then we just apply this exponential operator here, 133 00:05:57,520 --> 00:06:00,120 this potential propagator for half a time step. 134 00:06:00,120 --> 00:06:02,960 Then we do this Fourier transform into our momentum 135 00:06:02,960 --> 00:06:05,089 basis in which a cat maybe doesn't 136 00:06:05,089 --> 00:06:07,130 look that familiar to us because we are more used 137 00:06:07,130 --> 00:06:09,290 to a coordinate representation. 138 00:06:09,290 --> 00:06:12,950 Here we can then apply our potential-- our momentum 139 00:06:12,950 --> 00:06:13,700 propagation. 140 00:06:13,700 --> 00:06:18,380 So you have here your momentum squared and then a time step, 141 00:06:18,380 --> 00:06:21,640 and then just go back to the coordinate representation 142 00:06:21,640 --> 00:06:23,930 via a inverse Fourier transform, and then 143 00:06:23,930 --> 00:06:26,750 can apply the last half time step 144 00:06:26,750 --> 00:06:28,660 for our potential propagation. 145 00:06:28,660 --> 00:06:32,354 And this is the algorithm which is still used today, 146 00:06:32,354 --> 00:06:33,770 although it's so simple it's often 147 00:06:33,770 --> 00:06:37,020 used as a benchmark for new approximations. 148 00:06:37,020 --> 00:06:40,040 So let's apply it to some real examples, for example, 149 00:06:40,040 --> 00:06:42,330 for a free wave packet. 150 00:06:42,330 --> 00:06:45,130 And what I show here is the wave packet of two different masses. 151 00:06:45,130 --> 00:06:47,495 So here I have a lighter one. 152 00:06:47,495 --> 00:06:49,100 And here I have a heavy one. 153 00:06:49,100 --> 00:06:52,370 And what you see in red is [INAUDIBLE] 154 00:06:52,370 --> 00:06:54,710 is the probability density in Bond's interpretation 155 00:06:54,710 --> 00:06:56,370 of quantum mechanics. 156 00:06:56,370 --> 00:06:59,215 And in blue you have the wave function. 157 00:06:59,215 --> 00:07:02,600 And what you see is that both wave functions oscillate, 158 00:07:02,600 --> 00:07:06,170 but this here remains localized, whereas this here 159 00:07:06,170 --> 00:07:09,120 tends to spread out in the space it has. 160 00:07:09,120 --> 00:07:12,950 And this is some general behavior of quantum systems. 161 00:07:12,950 --> 00:07:15,700 And it's closely linked to what you call in German, 162 00:07:15,700 --> 00:07:18,490 [SPEAKING GERMAN],, which is, in English, 163 00:07:18,490 --> 00:07:20,880 the Heisenberg Uncertainty Principle, 164 00:07:20,880 --> 00:07:24,020 and which tells you that you have problems with studying 165 00:07:24,020 --> 00:07:26,480 quantum systems at state [INAUDIBLE] localize. 166 00:07:26,480 --> 00:07:28,260 So they tend to spread out. 167 00:07:28,260 --> 00:07:30,570 And it's just what you see here, the classical system, 168 00:07:30,570 --> 00:07:33,440 the heavy one tends to be localized on this timescale 169 00:07:33,440 --> 00:07:36,430 which we study, whereas this light one tends to spread out 170 00:07:36,430 --> 00:07:38,005 in the space it has. 171 00:07:38,005 --> 00:07:39,680 And what you also see here is that we 172 00:07:39,680 --> 00:07:42,890 study two additional properties of our system, which 173 00:07:42,890 --> 00:07:45,620 are the norm and the energy. 174 00:07:45,620 --> 00:07:48,530 And we do this to check if our simulation is correct. 175 00:07:48,530 --> 00:07:51,590 So you know that psi's corrective [INAUDIBLE] 176 00:07:51,590 --> 00:07:54,230 and probability density and the probability 177 00:07:54,230 --> 00:07:56,120 to find your system somewhere should remain 178 00:07:56,120 --> 00:07:58,610 constant of the simulation. 179 00:07:58,610 --> 00:08:01,700 Similar holds for the energy, which should remain constant 180 00:08:01,700 --> 00:08:04,370 over the simulation, as we don't add or remove energy 181 00:08:04,370 --> 00:08:05,184 from our system. 182 00:08:05,184 --> 00:08:07,100 Therefore, this should always remain constant. 183 00:08:07,100 --> 00:08:10,850 So these are two useful things to study of your simulation 184 00:08:10,850 --> 00:08:15,780 to see if it really shows you real behavior or not. 185 00:08:15,780 --> 00:08:18,150 So a more complicated example might 186 00:08:18,150 --> 00:08:20,184 be using one potential wall. 187 00:08:20,184 --> 00:08:22,350 And you probably have seen in your quantum mechanics 188 00:08:22,350 --> 00:08:26,280 introduction course that there is this tunneling effect, 189 00:08:26,280 --> 00:08:27,810 and that it shows some dependence 190 00:08:27,810 --> 00:08:29,800 on the height of the barrier. 191 00:08:29,800 --> 00:08:31,830 So this is what you can see here. 192 00:08:31,830 --> 00:08:34,049 Maybe it's the first time you see this 193 00:08:34,049 --> 00:08:35,190 really in some animation. 194 00:08:35,190 --> 00:08:39,929 So you see that for height area, which is shown here on top, 195 00:08:39,929 --> 00:08:42,390 some part of-- or most of the wave function 196 00:08:42,390 --> 00:08:45,480 just gets reflected and can't tunnel through the wall, 197 00:08:45,480 --> 00:08:54,360 whereas for the small potential wall, most of the packet 198 00:08:54,360 --> 00:08:56,310 can just travel through the wall. 199 00:08:56,310 --> 00:08:58,895 And therefore, you have now these more complicated 200 00:08:58,895 --> 00:09:01,420 interference patterns. 201 00:09:01,420 --> 00:09:03,620 Of course, you have this transmittance and also 202 00:09:03,620 --> 00:09:06,161 periodicity of the system, which then gives this interference 203 00:09:06,161 --> 00:09:07,431 pattern here. 204 00:09:07,431 --> 00:09:09,210 A even more complicated example would 205 00:09:09,210 --> 00:09:12,346 be if you applied two potential walls. 206 00:09:12,346 --> 00:09:14,720 And again, [INAUDIBLE] wave packet in the middle and then 207 00:09:14,720 --> 00:09:16,005 let it evolve in time. 208 00:09:16,005 --> 00:09:18,970 We just apply our split operator algorithm. 209 00:09:18,970 --> 00:09:21,450 And then we see that it sometimes 210 00:09:21,450 --> 00:09:23,260 encounters the first wall. 211 00:09:23,260 --> 00:09:26,080 And here we see, again, a part gets reflected 212 00:09:26,080 --> 00:09:28,070 and a part passes through wall. 213 00:09:28,070 --> 00:09:31,850 But what you also can see here, as was in the previous example, 214 00:09:31,850 --> 00:09:35,010 we have some problems in energy conservation. 215 00:09:35,010 --> 00:09:37,430 So with these settings here, the simulation 216 00:09:37,430 --> 00:09:39,170 is probably not the best one. 217 00:09:39,170 --> 00:09:40,950 So you might ask, how can I improve this? 218 00:09:40,950 --> 00:09:45,000 And the first thing one might try is using another time step. 219 00:09:45,000 --> 00:09:48,920 So in total, you can show that the time step-- 220 00:09:48,920 --> 00:09:51,490 [INAUDIBLE] time step is connected to your error 221 00:09:51,490 --> 00:09:53,030 with the order of three. 222 00:09:53,030 --> 00:09:55,730 So you might try a small time step, which I did here. 223 00:09:55,730 --> 00:09:59,030 And it takes much longer to compute this evolution then. 224 00:09:59,030 --> 00:10:01,610 And unfortunately, you see that you basically 225 00:10:01,610 --> 00:10:04,850 get for this system here basically the same evolution, 226 00:10:04,850 --> 00:10:07,580 which tells us that this time step was appropriate 227 00:10:07,580 --> 00:10:10,070 or good enough to study the system. 228 00:10:10,070 --> 00:10:12,430 So the second thing one might try, OK, 229 00:10:12,430 --> 00:10:14,120 one might use a finer grid. 230 00:10:14,120 --> 00:10:16,851 And then it gets really expensive, which is also 231 00:10:16,851 --> 00:10:18,350 the main drawback of this algorithm, 232 00:10:18,350 --> 00:10:21,470 that it really gets expensive as you go to more dimensions. 233 00:10:21,470 --> 00:10:24,320 So I had to run the simulation here over night. 234 00:10:24,320 --> 00:10:28,190 And I also only used half of the total time. 235 00:10:28,190 --> 00:10:30,650 And I also didn't save this animation here, 236 00:10:30,650 --> 00:10:32,900 because it would just take too much memory. 237 00:10:32,900 --> 00:10:35,810 And I don't know the computer could do this. 238 00:10:35,810 --> 00:10:39,300 So what I will show you here is the last step of the simulation 239 00:10:39,300 --> 00:10:39,800 here. 240 00:10:39,800 --> 00:10:42,890 And then we will compare it to the old simulation 241 00:10:42,890 --> 00:10:46,560 and see whether it's at the same state here 242 00:10:46,560 --> 00:10:49,310 and just compare the energy convergence of both cases. 243 00:10:49,310 --> 00:10:52,160 And I think it's now at the same stage. 244 00:10:52,160 --> 00:10:55,430 And you see that here we have a much better energy 245 00:10:55,430 --> 00:10:58,350 difference compared to an initial state than here. 246 00:10:58,350 --> 00:11:01,790 So this might be a way to improve our simulation, 247 00:11:01,790 --> 00:11:04,370 although you always have this tradeoff 248 00:11:04,370 --> 00:11:08,450 between time-step size, grid size, grid 249 00:11:08,450 --> 00:11:11,370 spacing, and simulation time. 250 00:11:11,370 --> 00:11:13,580 So you always have to consider if you 251 00:11:13,580 --> 00:11:15,530 want to do this heavy calculations 252 00:11:15,530 --> 00:11:17,900 or if you could use fine-- 253 00:11:17,900 --> 00:11:21,357 less fine settings to get the same results. 254 00:11:21,357 --> 00:11:22,940 So what I have shown you in this video 255 00:11:22,940 --> 00:11:25,625 is how to go to an algorithm which 256 00:11:25,625 --> 00:11:28,384 you can use to study quantum dynamics 257 00:11:28,384 --> 00:11:30,050 with control [INAUDIBLE] approximations, 258 00:11:30,050 --> 00:11:33,110 because we exactly know where we introduce our approximation. 259 00:11:33,110 --> 00:11:34,550 And we introduce our approximation 260 00:11:34,550 --> 00:11:35,720 at the Trotter split up. 261 00:11:35,720 --> 00:11:39,530 And we exactly know how large this error is at the step. 262 00:11:39,530 --> 00:11:42,080 We then saw two applications in quantum mechanics, which 263 00:11:42,080 --> 00:11:45,000 are to the uncertainty principle and to the tunneling effect. 264 00:11:45,000 --> 00:11:47,000 And then we also saw two methods which 265 00:11:47,000 --> 00:11:50,420 you can use to judge the convergence of your simulation. 266 00:11:50,420 --> 00:11:53,180 And then we got a bit of a feeling of the influence 267 00:11:53,180 --> 00:11:55,380 of grids, and time steps, and the tradeoff 268 00:11:55,380 --> 00:12:00,030 between simulation time and how accurate your results will be. 269 00:12:00,030 --> 00:12:02,060 I hope that it might be helpful for you 270 00:12:02,060 --> 00:12:06,807 and you might remember part of this presentation. 271 00:12:06,807 --> 00:12:08,640 If you study molecular dynamics for example, 272 00:12:08,640 --> 00:12:11,870 you might see some similarities between the [INAUDIBLE] 273 00:12:11,870 --> 00:12:15,199 operator formalism and this formalism I showed you here. 274 00:12:15,199 --> 00:12:16,490 I hope also this was enjoyable. 275 00:12:16,490 --> 00:12:18,340 I have a bit of a cold, so I hope 276 00:12:18,340 --> 00:12:21,170 you understood everything, although my voice isn't 277 00:12:21,170 --> 00:12:22,890 that clear.