1 00:00:01,530 --> 00:00:04,151 Hello everybody. My name is Victor Galitski and I gave the 2 00:00:04,151 --> 00:00:08,000 first 8 lectures in this course Exploring Quantum Physics. 3 00:00:09,160 --> 00:00:13,360 Today it is my pleasure to welcome those of you who survived this course to the 4 00:00:13,360 --> 00:00:17,888 last concluding lecture of Exploring Quantum Physics. 5 00:00:17,888 --> 00:00:21,650 So this lecutre is going to be on the very interesting subject of time 6 00:00:21,650 --> 00:00:26,138 dependent quantum mechanics and we're going to talk about it a little later but 7 00:00:26,138 --> 00:00:30,098 first I would like to take a couple of minutes to talk about the final exam 8 00:00:30,098 --> 00:00:35,926 coming up. So the format for the final exam is going 9 00:00:35,926 --> 00:00:41,140 to be slightly different than that of our regular homeworks. 10 00:00:41,140 --> 00:00:46,080 So, there going to be, sixteen problems. Roughly one problem for every lecture and 11 00:00:46,080 --> 00:00:50,496 these problems intend to cover all the material, sort of cover, that we have 12 00:00:50,496 --> 00:00:54,720 studied in this course including this lecture and starting from the very first 13 00:00:54,720 --> 00:00:59,940 lecture. So the final exam will be due, two weeks 14 00:00:59,940 --> 00:01:03,710 from now. Now being the time, when we posted these, 15 00:01:03,710 --> 00:01:07,812 last lectures. and the deadline for the final is 16 00:01:07,812 --> 00:01:12,045 going to be the same as the hard deadline for the last posted homework. 17 00:01:12,045 --> 00:01:16,013 So you will notice that there is no homework for this last week, no regular 18 00:01:16,013 --> 00:01:21,007 homework but there is a final exam. Also you will have just one attempt to 19 00:01:21,007 --> 00:01:25,790 take the final unlike the homeworks where you have multiple attempts. 20 00:01:25,790 --> 00:01:29,170 And once you start there will be a time limit of six hours for you to complete 21 00:01:29,170 --> 00:01:33,552 the final. But you shouldn't worry too much about 22 00:01:33,552 --> 00:01:37,173 it, because the final exam should be a little bit on the easier side as compared 23 00:01:37,173 --> 00:01:42,030 to the last homeworks you've seen in this course. 24 00:01:42,030 --> 00:01:44,531 So if you've got this far, you certainly should take the final and get your 25 00:01:44,531 --> 00:01:47,868 certificate. Or probably certificate of distinction 26 00:01:47,868 --> 00:01:52,092 I'm sure many of you will get those. So in other words if you have so far 27 00:01:52,092 --> 00:01:55,628 completed most homeworks, all the homeworks, you definitely should take the 28 00:01:55,628 --> 00:01:59,676 final, don't even think about it. It's going to be pretty straightforward 29 00:01:59,676 --> 00:02:03,500 and you will do well. Now going to the scientific part, well to 30 00:02:03,500 --> 00:02:07,399 the lecture. So I've been sort of thinking A lot about 31 00:02:07,399 --> 00:02:11,280 the possible subject for the last lecture. 32 00:02:11,280 --> 00:02:14,238 And of course there's are many possibilities because there's a lot of 33 00:02:14,238 --> 00:02:17,298 material we haven't had the time to study unfortunately, due to the time 34 00:02:17,298 --> 00:02:22,055 constraints and some constraints on the format of these lectures. 35 00:02:22,055 --> 00:02:27,095 But I decided to talk about this subject, time-dependent quantum mechanics for two 36 00:02:27,095 --> 00:02:31,770 reasons. So the first reason is that this eh, 37 00:02:31,770 --> 00:02:37,320 topic of time-dependent quantum mechanics is among a very few topics in theoretical 38 00:02:37,320 --> 00:02:44,165 physics that is still being developed. So it's still a very active subject, or a 39 00:02:44,165 --> 00:02:48,220 research topic in theoretical physics. 40 00:02:48,220 --> 00:02:52,291 So this is different by the way from most other things we have seen in this course 41 00:02:52,291 --> 00:02:57,300 where pretty much everything we want to know we do know so 42 00:02:57,300 --> 00:03:01,260 of course people are still arguing about foundations of quantum physics, some 43 00:03:01,260 --> 00:03:04,670 philosophical interpretations of quantum theory and et cetera, so these 44 00:03:04,670 --> 00:03:08,660 discussions will probably continue forever. 45 00:03:08,660 --> 00:03:11,988 But once you accept the axioms of quantum mechanics and look at it from the 46 00:03:11,988 --> 00:03:16,652 particle point of view. So, we can pretty much explain or predict 47 00:03:16,652 --> 00:03:21,176 the result of any conceivable experiment that you know, that can be performed with 48 00:03:21,176 --> 00:03:25,810 quantum systems. So the problem won't say not with the. 49 00:03:25,810 --> 00:03:29,030 mysteries of quantum physics, but with lack of mysteries at this stage. 50 00:03:29,030 --> 00:03:31,590 We, we actually understand pretty well what we see. 51 00:03:31,590 --> 00:03:35,677 So, but, this subject of time-dependent quantum mechanics is a bit different in 52 00:03:35,677 --> 00:03:40,626 that, there's some open, open questions. I wouldn't call necessarily This 53 00:03:40,626 --> 00:03:44,266 question's grand mysteries but they're definitely open problems and frontiers 54 00:03:44,266 --> 00:03:48,290 where which we are still exploring. So this is one reason. 55 00:03:48,290 --> 00:03:52,322 So the second reason I have chosen this subject is because I myself have been 56 00:03:52,322 --> 00:03:56,738 working on it quite extensively for the last few years and this is among my sort 57 00:03:56,738 --> 00:04:03,910 of chief research interests. So, of course what I just told you sort 58 00:04:03,910 --> 00:04:08,290 of implies that well, I hardly can tell you all this interesting things about 59 00:04:08,290 --> 00:04:12,870 time-dependent quantum mechanics in one lecture. 60 00:04:12,870 --> 00:04:15,865 So it's a very active research topic. It could be the subject of a separate 61 00:04:15,865 --> 00:04:18,392 course. But at least I can sort of give you a 62 00:04:18,392 --> 00:04:22,199 flavor. What time-dependent quantum mechanics is 63 00:04:22,199 --> 00:04:27,236 about and also maybe one message of this lecture will be to, for you to understand 64 00:04:27,236 --> 00:04:33,996 why this subject is difficult. Why unlike in most other subfields of 65 00:04:33,996 --> 00:04:39,610 quantum physics, here we still sort of struggling to 66 00:04:39,610 --> 00:04:42,385 understanding the tehcnical law of what's going on. 67 00:04:42,385 --> 00:04:45,620 So, this will be another sort of message of this lecture. 68 00:04:45,620 --> 00:04:49,460 So, what we're actually going to be studying in today's lecture is truly time 69 00:04:49,460 --> 00:04:55,105 dependent showing your equation. So, we actually derived, while in quotes, 70 00:04:55,105 --> 00:04:59,190 this equation, in the very first lecture of the course. 71 00:04:59,190 --> 00:05:02,630 but we never truly used the time-dependent part. 72 00:05:02,630 --> 00:05:06,506 So we in some sense, we got rid of it the first moment we got the opportunity to do 73 00:05:06,506 --> 00:05:10,211 so, because in most cases, we have, we have looked at so the potential, well 74 00:05:10,211 --> 00:05:14,201 actually null cases we have looked at the potential was not time-dependent, the 75 00:05:14,201 --> 00:05:20,645 potential was constant. So what were going to be studying today 76 00:05:20,645 --> 00:05:28,280 is the situation where the Hamiltonian H is a function, explicit function of time. 77 00:05:28,280 --> 00:05:30,760 So it's not a constant it's a functions of time. 78 00:05:30,760 --> 00:05:35,316 So of course even if it were not to be a function of time, even if the Hamiltonian 79 00:05:35,316 --> 00:05:40,826 were a constant. So the wave function would still have 80 00:05:40,826 --> 00:05:46,090 been time-dependent and the wave function would be the. 81 00:05:46,090 --> 00:05:50,920 We have talked little bit about this when we were driving the Feynman path integral 82 00:05:50,920 --> 00:05:55,540 so the wave function is a function of time in this case would have been the 83 00:05:55,540 --> 00:06:02,020 evolution operatior, e to the power minus i over h-bar HT. 84 00:06:02,020 --> 00:06:06,320 You're acting on an initial condition the question is no. 85 00:06:06,320 --> 00:06:12,130 Now if the hamiltonian is a function of time. 86 00:06:12,130 --> 00:06:18,060 So this of course doesnt work. So this in this case doesnt work. 87 00:06:18,060 --> 00:06:24,040 And basically we dont know our priority. What the wave function is now. 88 00:06:24,040 --> 00:06:29,308 But this is exactly what we want to know. So basically to solve the time dependant 89 00:06:29,308 --> 00:06:36,110 Schrodinger equation means either to find the wave function, Si of T explicitly. 90 00:06:36,110 --> 00:06:40,490 Or even better, to find the evolution operator U of T that would relay the 91 00:06:40,490 --> 00:06:46,230 initial condition to the to define a wave function. 92 00:06:46,230 --> 00:06:50,286 So again in the case of time-independent Hamiltonian, so this evolution operator 93 00:06:50,286 --> 00:06:53,198 is written here, so this is this exponential e to the power of minus i 94 00:06:53,198 --> 00:07:00,335 over h-bar h t. So but if the 95 00:07:00,335 --> 00:07:04,831 Hamiltonian's implicit function of time? Well there is no simple closed equation 96 00:07:04,831 --> 00:07:09,588 for this. One thing I should eh, reiterate about 97 00:07:09,588 --> 00:07:13,718 this evolution operator which is true both in the case of time-independent, 98 00:07:13,718 --> 00:07:18,350 Hamiltonian, and in the case of time-dependents. 99 00:07:18,350 --> 00:07:21,806 Is that actually an evolutionary operator is independent of, of the initial 100 00:07:21,806 --> 00:07:25,630 conditions. So it essentially describes a rotation of 101 00:07:25,630 --> 00:07:31,916 the wave function in the Hilbert space. So let us recall that well wave function 102 00:07:31,916 --> 00:07:35,745 describes well has probabilistic interpretation. 103 00:07:35,745 --> 00:07:39,290 >> Okay and so if we take, let's say if we just treat spectral. 104 00:07:39,290 --> 00:07:44,650 If we have let's say one particle in the system. 105 00:07:44,650 --> 00:07:48,700 So the wave function must be normalized to one. 106 00:07:48,700 --> 00:07:52,988 So and it's true both for the initial condition and for the time dependent 107 00:07:52,988 --> 00:07:55,855 part. And it should remain. 108 00:07:55,855 --> 00:07:59,320 >> one just because if we have a single particle in normal divistic quantum 109 00:07:59,320 --> 00:08:03,405 mechanics is just going to stay there. It's not going to be not going to 110 00:08:03,405 --> 00:08:07,270 disappear and a new particles are not going to appear out of the blue. 111 00:08:07,270 --> 00:08:10,834 Well actually in quantum electrodynamics it is possible but were doing normal 112 00:08:10,834 --> 00:08:13,950 divistic single particle quantum mechanics. 113 00:08:13,950 --> 00:08:17,505 So this normalization condition should be preserved. 114 00:08:17,505 --> 00:08:22,640 >> And so if you think about Hilbert space, again vetric space. 115 00:08:22,640 --> 00:08:26,556 Let me just raise this arrow here. So if you think about Hilbert space as a 116 00:08:26,556 --> 00:08:30,430 vetric space and which our wave funcions leave. 117 00:08:30,430 --> 00:08:36,805 So what this condition implies in some sense is that the wave function exists on 118 00:08:36,805 --> 00:08:41,170 a hyper sphere. Well, I cannot draw it in a, in a higher 119 00:08:41,170 --> 00:08:44,338 than three dimensions but of course we should, sort of, take this picture with a 120 00:08:44,338 --> 00:08:48,255 grain of salt. So some multidimential Hubert space, and 121 00:08:48,255 --> 00:08:52,100 there is some sphere. Oh, it did not work out very well. 122 00:08:52,100 --> 00:08:53,830 Let me just. We draw it. 123 00:08:53,830 --> 00:08:57,240 Imagine it. So if we have a sphere. 124 00:08:57,240 --> 00:08:59,449 Well this is not much better. Well anyways you understand what I'm 125 00:08:59,449 --> 00:09:01,540 saying. So this sphere is essentially 126 00:09:01,540 --> 00:09:06,296 manifistation of this condition. Basically means that my wave functions 127 00:09:06,296 --> 00:09:09,886 exists inner most condition is presevered. 128 00:09:09,886 --> 00:09:14,310 So this essentiatlly mean phi squared is equal to 1. 129 00:09:14,310 --> 00:09:18,470 And if I start from some initial condition on this here. 130 00:09:18,470 --> 00:09:24,322 So let's say this is my psi node. And the only thing which can happen with 131 00:09:24,322 --> 00:09:28,162 this psi node is that it can sort of rotate in this Hilber space to a new psi 132 00:09:28,162 --> 00:09:35,599 of t, somewhere else on this sphere. And the operator U essentially enforces 133 00:09:35,599 --> 00:09:40,170 this rotation from Psi nought to Psi of T. 134 00:09:40,170 --> 00:09:43,446 So we already discussed it, as a matter of fact, during the derivation of the 135 00:09:43,446 --> 00:09:46,670 Feynman Path Integral, but it was a month, it was in the middle of, you know, 136 00:09:46,670 --> 00:09:51,514 this optional technical derivation. And I'm not sure if all of you have 137 00:09:51,514 --> 00:09:53,983 looked at it. But I think it's a very imporatnt 138 00:09:53,983 --> 00:09:57,217 message, that actually U of T rotates the intial condition essentually by some 139 00:09:57,217 --> 00:10:00,440 angle in this mulidimensional hilbert space. 140 00:10:00,440 --> 00:10:04,990 The new time dependant wave functions were somewhere on this hypersphere whihc 141 00:10:04,990 --> 00:10:08,680 is, which exits you to this normalization. 142 00:10:08,680 --> 00:10:11,120 And the important thing is that if we were to choose. 143 00:10:11,120 --> 00:10:15,584 Let's see a different initial condition somewhere well, let's say here on this 144 00:10:15,584 --> 00:10:19,255 sphere. So, the and if we were to apply this same 145 00:10:19,255 --> 00:10:23,930 time-dependant Hamiltonian or time-independent Hamiltonian. 146 00:10:23,930 --> 00:10:27,680 So, in both cases, so the rotation would be sorted by end laws. 147 00:10:27,680 --> 00:10:30,535 So, there would, it would, it would be different let's say Psi nought. 148 00:10:30,535 --> 00:10:35,868 Prime and upside prime of t, okay? But the evolution operator, this guy 149 00:10:35,868 --> 00:10:38,300 would be exactly the same. Okay? 150 00:10:38,300 --> 00:10:41,576 And so this is that's why I say even better you know, so of course we want to 151 00:10:41,576 --> 00:10:46,540 find in a particular case when we're dealing with a wave function. 152 00:10:46,540 --> 00:10:49,312 So, we want to find the wave function is a function of time because it will 153 00:10:49,312 --> 00:10:51,996 essentially describe everything we want to know about this system or the 154 00:10:51,996 --> 00:10:56,770 probabilities or the transitions. Within various states and amplitudes. 155 00:10:56,770 --> 00:11:03,445 But if we know the evolution operator. We in some sense can solve all possible 156 00:11:03,445 --> 00:11:07,740 initial conditions in one shot. Just because the Evolution operator has 157 00:11:07,740 --> 00:11:09,970 been enforced. This quantum of Evolution judges by 158 00:11:09,970 --> 00:11:13,880 acting on an initial condition. But, well the problem. 159 00:11:13,880 --> 00:11:17,905 With this I mentioned is that its very hard to find the evolution operator. 160 00:11:17,905 --> 00:11:21,760 Now why is that? Why is it so difficult? 161 00:11:21,760 --> 00:11:25,640 So let me sort of give you a very brief explanation why its difficult. 162 00:11:25,640 --> 00:11:29,987 So if you look at this expression, this equation and if for a second, so let me 163 00:11:29,987 --> 00:11:34,196 just write it here in quotes in some sense this initial conditioning for a 164 00:11:34,196 --> 00:11:38,612 second we assume that the Hamiltonian is not really an operator but just some 165 00:11:38,612 --> 00:11:45,698 function H of T without the hat. So then I can simply divide this 166 00:11:45,698 --> 00:11:50,100 equation, let me put this in quotes since this is not true. 167 00:11:50,100 --> 00:11:55,336 So if we divide by Si of T and integrate both sides of this equation over time, we 168 00:11:55,336 --> 00:11:59,956 can ride the solution to this sort of sharing your equation down the down 169 00:11:59,956 --> 00:12:05,446 version. Sharing your equation as E to the power 170 00:12:05,446 --> 00:12:10,610 minus I over H bar, and then take it all from zero to T. 171 00:12:10,610 --> 00:12:14,835 Age of well seven T prime and let's say DT prime on and here's going to be the 172 00:12:14,835 --> 00:12:19,303 initial condition of sign up. So this would be a solution to an 173 00:12:19,303 --> 00:12:22,740 ordinary differential equation. Time dependent differential equation to 174 00:12:22,740 --> 00:12:24,730 first order. So if you look at it that's what it is, 175 00:12:24,730 --> 00:12:28,112 so it seems very simple. So why don't we just write that and put a 176 00:12:28,112 --> 00:12:31,048 hat here. So it turns out that we cannot do that. 177 00:12:31,048 --> 00:12:35,224 And well, those of you who are interested should take the time and well, stare at 178 00:12:35,224 --> 00:12:39,110 this eh, expression for a little while and well, work with it little bit to see 179 00:12:39,110 --> 00:12:43,790 why it is the case. Well, why it is not the case, actually. 180 00:12:43,790 --> 00:12:49,940 Why this is not true, that in the case of a true operator Schrodinger equation. 181 00:12:49,940 --> 00:12:53,250 So this solution doesn't work. And the chief reason for that is because 182 00:12:53,250 --> 00:12:57,090 the Hamiltonian H of t does not commute with itself necessarily at different 183 00:12:57,090 --> 00:13:01,569 moments of time. So if we take two different moments of 184 00:13:01,569 --> 00:13:05,709 time with let's say H of t1 and H of t2 so the Hamiltonian is not going to be 185 00:13:05,709 --> 00:13:09,780 the, come one, commutator, the Hamiltonian with itself, at different 186 00:13:09,780 --> 00:13:16,782 moments of time is not zero. And this is major, major complication 187 00:13:16,782 --> 00:13:20,422 which makes all quantum mechanical, time-dependent quantum mechanical 188 00:13:20,422 --> 00:13:24,300 problems extremely difficult. And this even goes even the simplest 189 00:13:24,300 --> 00:13:27,216 versions of this problem. Let's say with spin one half, with two 190 00:13:27,216 --> 00:13:31,559 level systems. Even there, there is no generally no 191 00:13:31,559 --> 00:13:36,576 close solution to a time-dependent problem. 192 00:13:36,576 --> 00:13:40,580 So let me just leave it here. So now let me provide a couple of general 193 00:13:40,580 --> 00:13:44,260 examples of time dependent and quantum, problems. 194 00:13:44,260 --> 00:13:47,810 So the first example will be a the following. 195 00:13:47,810 --> 00:13:53,298 So let's imagine we have a particle. Which is moving in a potential v of r. 196 00:13:53,298 --> 00:13:56,422 So we just draw a v of x here. For the purpose of illustration let's 197 00:13:56,422 --> 00:13:59,444 just assume it's one dimension. And let's imagine that this potential 198 00:13:59,444 --> 00:14:03,630 itself is sort of breathing. So it's moving as a function of time. 199 00:14:03,630 --> 00:14:06,894 For example it's changing as a function of time, and as time goes by, let's say 200 00:14:06,894 --> 00:14:11,248 it may become different. So, the question that one may ask is what 201 00:14:11,248 --> 00:14:15,826 happens with the particle? With the quantum particle moving in this 202 00:14:15,826 --> 00:14:18,680 potential. So, how does this wave function change as 203 00:14:18,680 --> 00:14:23,054 a function of time? So even if the potential were constant, 204 00:14:23,054 --> 00:14:28,270 so we know that this is strictly speaking, a rather non trivial problem. 205 00:14:28,270 --> 00:14:33,350 But, if you ground the energy levels in this quantum potential. 206 00:14:33,350 --> 00:14:37,620 And at any moment of time we can define a sort of equivalent time-independent 207 00:14:37,620 --> 00:14:42,200 problem and find the corresponding energy levels. 208 00:14:42,200 --> 00:14:47,430 But if the Hamiltonian is time-dependent the energy itself is not conserved. 209 00:14:47,430 --> 00:14:51,600 So the, the, this would be wrong question to ask in some sense. 210 00:14:51,600 --> 00:14:53,460 What are the energy levels in the problem? 211 00:14:53,460 --> 00:14:57,268 So we know that at any moment of time there is formally a time-independent sort 212 00:14:57,268 --> 00:15:00,908 of eigenvalue problem but unless the potential is changing time very slowly, 213 00:15:00,908 --> 00:15:04,660 these energy levels do not have a meaning because we're always sort of pumping in 214 00:15:04,660 --> 00:15:10,080 energy in the system and the energy is not conserved. 215 00:15:10,080 --> 00:15:13,710 So this is really one of the major complications. 216 00:15:13,710 --> 00:15:17,814 But, the probabilistic interpretation of the wavefunction is still valid, and we 217 00:15:17,814 --> 00:15:21,861 can still ask the questions about, well, the probability, let's say, of finding a 218 00:15:21,861 --> 00:15:27,580 particle in a certain region in space. And it's still going to be given So this 219 00:15:27,580 --> 00:15:31,882 probability is still going to be given by, so let me just write it here. 220 00:15:31,882 --> 00:15:35,522 As an example, probability let's say to find a particle between points x1, and 221 00:15:35,522 --> 00:15:38,882 x2, is going to be given, well it's a function of time, let me write it down 222 00:15:38,882 --> 00:15:44,642 here. Is going to be equal to the integral from 223 00:15:44,642 --> 00:15:52,630 x1 to x2 sin Of x, of t, everything squared integrated over dx. 224 00:15:52,630 --> 00:15:57,391 So this is an example of a physical observable property that we can we can 225 00:15:57,391 --> 00:16:01,738 calculate using a solution to the Schrininger equation if this solution is 226 00:16:01,738 --> 00:16:06,663 no. Now the second kind of problem, well it's 227 00:16:06,663 --> 00:16:10,268 actually the same. Time-dependent Shroedinger equation but 228 00:16:10,268 --> 00:16:13,989 for a different type of a system is a let's say spin in a time-dependent 229 00:16:13,989 --> 00:16:18,231 magnetic field. So this relates to the lecture of 230 00:16:18,231 --> 00:16:22,562 Professor Appelbaum who introduced spin, and in particular he was talking about 231 00:16:22,562 --> 00:16:27,551 nuclear Magnetic resonance. MRI imaging for example is where, 232 00:16:27,551 --> 00:16:31,577 actually what's being done is that the spin is being rotated by time-dependent 233 00:16:31,577 --> 00:16:35,766 field. So and in this case, the Hamiltonian well 234 00:16:35,766 --> 00:16:39,506 the wave function first of all is a two-component spinner, and the 235 00:16:39,506 --> 00:16:43,994 Hamiltonian here, so this is our H of t, so in this case the Hamiltonian is simply 236 00:16:43,994 --> 00:16:50,232 B Dot sigma, sigma being the vector of poly matrices. 237 00:16:50,232 --> 00:16:54,651 And this Hamiltonian, two by two matrix acts on these two components speed. 238 00:16:54,651 --> 00:16:58,769 And essentially this is first order differential equation and the problem 239 00:16:58,769 --> 00:17:03,430 here would be lets say to find the you know, this side of t. 240 00:17:03,430 --> 00:17:11,370 And in this case for example the absolute values squared of this psy op of T. 241 00:17:11,370 --> 00:17:15,132 Squared this would give us the probability of finding the spin in the 242 00:17:15,132 --> 00:17:19,158 state up in the moment of time machine so this would be another observable 243 00:17:19,158 --> 00:17:26,013 characteristic we can talk about. Now an amazing thing actually and I have 244 00:17:26,013 --> 00:17:29,175 to admit I didn't know it until just a few years ago, is that this equation is 245 00:17:29,175 --> 00:17:34,025 actually, well it's two by two matrix. Time dependent of the first differential 246 00:17:34,025 --> 00:17:37,468 equation for a two by two matrix. So this equation appears not to be 247 00:17:37,468 --> 00:17:41,152 solvable in elementary functions. You cannot solve this equation in 248 00:17:41,152 --> 00:17:44,145 general, analytically at least. And so when I first looked at this 249 00:17:44,145 --> 00:17:47,800 equation, I thought well I was just solving during my lunch break. 250 00:17:47,800 --> 00:17:49,620 I didn't even want to look into the text books. 251 00:17:49,620 --> 00:17:53,710 I just thought It was just a toy model. But it turns out that even this equation, 252 00:17:53,710 --> 00:17:58,195 maybe the simplest quantum mechanical problem we can think of, you know, two by 253 00:17:58,195 --> 00:18:02,355 two matrix, or two component, two dimensional complex Hillber, Hilburt 254 00:18:02,355 --> 00:18:07,205 state. Even this problem is not exactly solvable 255 00:18:07,205 --> 00:18:08,970 for a general >> H of T. 256 00:18:08,970 --> 00:18:12,600 So this gives you something well some sort of insight into the complexity of 257 00:18:12,600 --> 00:18:16,441 the subject. But as usual in theoretical physics if we 258 00:18:16,441 --> 00:18:20,473 can't solve the problem exactly it doesn't stop us from solving it by some 259 00:18:20,473 --> 00:18:25,310 approximation methods. Actually I should mention one major 260 00:18:25,310 --> 00:18:29,290 mission in this course was Perturbation theory. 261 00:18:29,290 --> 00:18:35,218 Which is a general theory of solving approximately various stationary quantum 262 00:18:35,218 --> 00:18:38,628 problems. We didn't talk about it, but it's a very, 263 00:18:38,628 --> 00:18:41,480 very important piece of quantum mechanics and you, and those of you who are 264 00:18:41,480 --> 00:18:44,700 interested in professionally pursuing physics you should definitely, you should 265 00:18:44,700 --> 00:18:50,037 definitely look into this option. But in this context, in the context of 266 00:18:50,037 --> 00:18:53,515 non-equilibrium or time-dependent quantum mechanics there is a similar, there is, 267 00:18:53,515 --> 00:18:58,050 there is a similar math as various approximation schemes. 268 00:18:58,050 --> 00:19:03,783 And there are two clear situations. Two sort of asymptotically opposite cases 269 00:19:03,783 --> 00:19:08,540 of either very fast perturbation, or very slow perturbations or so-called adabatic 270 00:19:08,540 --> 00:19:12,480 perturbations. So let me just write it down then. 271 00:19:12,480 --> 00:19:18,920 so so we're going to study actually in in the next lecture, we're going to study 272 00:19:18,920 --> 00:19:26,372 fast or sudden perturbations, which means fast changes in the potential and in the 273 00:19:26,372 --> 00:19:34,426 third part of this lecture we're going to discuss slow. 274 00:19:34,426 --> 00:19:42,616 Or, adiabatic pertubations, which lead to very interesting, so-called topological 275 00:19:42,616 --> 00:19:53,820 quantum, phase that appears, in, in, in the solution to the Schrodinger Equation. 276 00:19:53,820 --> 00:19:58,112 So, just, sort of to wrap up this, course introductory part, will you let me first, 277 00:19:58,112 --> 00:20:02,056 eh, define what we actually mean by fast and slow, so as we discussed throughout 278 00:20:02,056 --> 00:20:07,162 the course actually. well, if you are talking about the 279 00:20:07,162 --> 00:20:11,192 physically quantity which has a non-trivial physical dimension be the 280 00:20:11,192 --> 00:20:15,823 length or time or whatever. So you cannot really say that it's small 281 00:20:15,823 --> 00:20:18,934 or large unless you compare it with a quantity which has the same physical 282 00:20:18,934 --> 00:20:23,720 dimension. So when we have a time scale, let me call 283 00:20:23,720 --> 00:20:28,480 it for instance tau of a certain time-dependent perturbation, let's say 284 00:20:28,480 --> 00:20:33,376 this a typical time at which a potential changes so we in order for us to see to 285 00:20:33,376 --> 00:20:38,204 define whether it's short time scale or long time scale, we have to get another 286 00:20:38,204 --> 00:20:46,471 quantity to compare with. Which has the same physical dimension and 287 00:20:46,471 --> 00:20:49,266 it turns out that in the case of quantum problems, if we have a spectrum, so lets 288 00:20:49,266 --> 00:20:54,315 say this is our energy. Okay, so let's say these are different 289 00:20:54,315 --> 00:21:00,705 energy, energies and there are some level spacing between the neighboring energy 290 00:21:00,705 --> 00:21:05,321 levels. It's in the harmonic oscillator case it 291 00:21:05,321 --> 00:21:09,165 would be H omega and the case of the potential well it would be something 292 00:21:09,165 --> 00:21:13,049 else. And they will in general change with time 293 00:21:13,049 --> 00:21:16,697 so, but let's say if we have a system which is quantum particle which is 294 00:21:16,697 --> 00:21:21,828 sitting on this certain quantum level, and neighboring levels. 295 00:21:21,828 --> 00:21:25,860 So the appropriate energy scale associated with it is delta, this level 296 00:21:25,860 --> 00:21:30,710 spacing. And, and h over delta is another 297 00:21:30,710 --> 00:21:34,806 timescale. So this is a timescale to compare with 298 00:21:34,806 --> 00:21:39,620 when we are talking about fast or slow perturbations. 299 00:21:39,620 --> 00:21:43,938 So if tau So if tau is much smaller than h over 300 00:21:43,938 --> 00:21:49,770 delta, we will say that this is a fast or some perturbation. 301 00:21:49,770 --> 00:21:53,910 If on the other hand if tau is much longer than h over level spacing, we're 302 00:21:53,910 --> 00:21:58,830 going to be talking about slow or adiabatic perturbations. 303 00:21:58,830 --> 00:22:01,885 And these are going to be the fast and slow perturbations again are going to be 304 00:22:01,885 --> 00:22:05,750 the subjects of the next two parts of this lecture. 305 00:22:05,750 --> 00:22:10,106 Now just the final thing I'm going to say here is that those of you who are 306 00:22:10,106 --> 00:22:15,320 interested in a research on this sort of quantum dynamical systems and in sort of 307 00:22:15,320 --> 00:22:21,890 exact non-perturbative methods of solving quantum problems. 308 00:22:21,890 --> 00:22:25,106 So I gave a lecture, well it's a research level presentation but I think it shoudl 309 00:22:25,106 --> 00:22:28,820 be understandable to most of you. So I gave a lecture on this subject at 310 00:22:28,820 --> 00:22:31,030 the Cavalier Institute for Theoretical Physics a couple pf years ago and so if 311 00:22:31,030 --> 00:22:33,780 you are interested, you may want to take a look at it. 312 00:22:33,780 --> 00:22:36,199 Of course, there are a lot of other people working on this and they have a 313 00:22:36,199 --> 00:22:39,735 number of amazing results. But I think there is another there is an 314 00:22:39,735 --> 00:22:42,920 interesting development in this field which allows us to construct exact 315 00:22:42,920 --> 00:22:46,671 solutions. But unfortunately, it is well beyond sort 316 00:22:46,671 --> 00:22:49,986 of this school, this course and the level of this course, so I just put a link 317 00:22:49,986 --> 00:22:52,970 here. So if you're interested take a look. 318 00:22:52,970 --> 00:22:57,986 Otherwise we, we'll talk now about the sudden and slow perturbations and see how 319 00:22:57,986 --> 00:23:02,474 quantum problems can be solved, in these cases, and what new phenomena appear 320 00:23:02,474 --> 00:23:04,940 there. [NOISE]