1 00:00:00,000 --> 00:00:18,595 *35C3 preroll music* 2 00:00:18,595 --> 00:00:26,930 Herald Angel: And, so he studied physics and I'm thinking we just all need a lot 3 00:00:26,930 --> 00:00:33,550 better understanding of quantum mechanics, because he sees this theory being misused 4 00:00:33,550 --> 00:00:42,329 a lot by some weird esoteric theories, kind of abusing it to just justify 5 00:00:42,329 --> 00:00:48,250 everything and anything. So he wants to change that and he wants to have people 6 00:00:48,250 --> 00:00:53,809 with some understanding of this very important theory and so he will start 7 00:00:53,809 --> 00:00:58,570 today with all of us here and try to explain to us the wonders of quantum 8 00:00:58,570 --> 00:01:08,724 mechanics. Have a go. *applause* 9 00:01:08,724 --> 00:01:11,700 Sebastian Riese: Well thank you for a warm welcome. It will be about quantum 10 00:01:11,700 --> 00:01:18,110 mechanics. We will see whether the gentle introduction will be a lie depending on 11 00:01:18,110 --> 00:01:23,380 how good you can follow me. So at first there will be a short introduction, a bit 12 00:01:23,380 --> 00:01:29,560 meta discussion about physical theories and what is the aim of this talk. And then 13 00:01:29,560 --> 00:01:35,241 we will discuss the experiments. Most of this is high school physics, you've 14 00:01:35,241 --> 00:01:40,360 probably seen it before. And then it will get ugly because we'll do the theory and 15 00:01:40,360 --> 00:01:45,080 we'll really do the theory, we'll write down the equations of quantum mechanics 16 00:01:45,080 --> 00:01:50,450 and try to make them plausible and hopefully understandable to a lot of 17 00:01:50,450 --> 00:01:56,860 people. And finally some applications will be discussed. So what is the concept of 18 00:01:56,860 --> 00:02:02,470 this talk. The key experiments will be reviewed as said, and but we will not do 19 00:02:02,470 --> 00:02:07,110 it in historical fashion. We will look at the experiments as physical facts and 20 00:02:07,110 --> 00:02:13,450 derive the theory from them. And since quantum mechanics is rather abstract and 21 00:02:13,450 --> 00:02:19,140 not, as I said in German and in science theory "anschaulich", we will need 22 00:02:19,140 --> 00:02:23,780 mathematics and most of this will be linear algebra. So a lot of quantum 23 00:02:23,780 --> 00:02:28,600 mechanics is just linear algebra on steroids, that means in infinite 24 00:02:28,600 --> 00:02:35,550 dimensions. And in doing so we'll try to find a certain post classical 25 00:02:35,550 --> 00:02:40,871 "Anschaulichkeit" or lividness to understand the theory. Since there'll be a 26 00:02:40,871 --> 00:02:48,610 lot of math as the allergy advice said, there will be crash courses driven in to 27 00:02:48,610 --> 00:02:53,350 explain mathematical facts. Sorry for the mathematicians that are here they probably 28 00:02:53,350 --> 00:03:03,520 suffer because I lie a lot. So at first: How do scientific theories work? To really 29 00:03:03,520 --> 00:03:08,500 understand quantum mechanics we must understand the setting and setting where 30 00:03:08,500 --> 00:03:14,990 it was created and how scientific theories are created in general. A scientific 31 00:03:14,990 --> 00:03:19,920 theory is a net of interdependent propositions so we have one proposition 32 00:03:19,920 --> 00:03:27,100 for example "F = M times a" in classical mechanics and we have another proposition 33 00:03:27,100 --> 00:03:32,010 that the gravitational force equals is proportional to the product of the masses 34 00:03:32,010 --> 00:03:38,140 divided by the distance between the masses squared, so something like this. And when 35 00:03:38,140 --> 00:03:44,540 we go around, make experiments, look into nature, develop theories, calculate, we 36 00:03:44,540 --> 00:03:51,300 test those we test hypotheses, different hypotheses and try to determine which one 37 00:03:51,300 --> 00:03:57,030 describes our experimental results best. And if the hypothesis stands the 38 00:03:57,030 --> 00:04:02,790 experimental tests they're added to the theory. But what happens if there's an 39 00:04:02,790 --> 00:04:07,120 experimental result that totally contradicts what we've seen before? And 40 00:04:07,120 --> 00:04:12,520 that happened in the late 19th and early 20th century. There are new results that 41 00:04:12,520 --> 00:04:18,690 could not be explained. So if such inconsistent results are found then our 42 00:04:18,690 --> 00:04:24,940 old theory has been falsified. This term is due to Popper who said that a theory is 43 00:04:24,940 --> 00:04:28,770 scientific as long as it can be falsified, that is at least as long as we can prove 44 00:04:28,770 --> 00:04:33,940 that it's not true and we can never prove a theory true but only prove it wrong. And 45 00:04:33,940 --> 00:04:39,350 all that we have not yet proven wrong are at least some approximation to truth. And 46 00:04:39,350 --> 00:04:45,460 if this happens we have to amend our old theory and we have to use care there and 47 00:04:45,460 --> 00:04:51,810 find a minimal amendment. This principle is Occam's Razor. One could also say the 48 00:04:51,810 --> 00:04:58,660 principle of least surprise from software engineering. And then we try that our 49 00:04:58,660 --> 00:05:03,940 theory is again consistent with the experimental results. And of course the 50 00:05:03,940 --> 00:05:09,030 new theory must explain why the hell that, for example Newtonian mechanics work for 51 00:05:09,030 --> 00:05:14,520 two hundred years if it's absolutely wrong. And so the old theory must in some 52 00:05:14,520 --> 00:05:20,910 limit contain the new one. And now how does it begin with quantum mechanics. As 53 00:05:20,910 --> 00:05:25,820 already said the time frame is the late 19th and early 20th century. And there 54 00:05:25,820 --> 00:05:32,100 were three or four fundamental theories of physics known then: Classical mechanics, 55 00:05:32,100 --> 00:05:36,290 which is just governed by the single equation the force equals mass times the 56 00:05:36,290 --> 00:05:44,780 acceleration with given forces. And two known force laws: The immediate distance 57 00:05:44,780 --> 00:05:51,410 action Newtonian gravitation and the Maxwell electro dynamics, this funny 58 00:05:51,410 --> 00:05:56,320 equation here. This funny equation here is a way of writing down the Maxwell 59 00:05:56,320 --> 00:06:04,960 equations that basically contain all the known electromagnetic effects. And finally 60 00:06:04,960 --> 00:06:09,580 there were the beginnings of the Maxwell Boltzmann statistical physics, but 61 00:06:09,580 --> 00:06:17,250 classical statistical physics is a pain, doesn't really work. So several 62 00:06:17,250 --> 00:06:22,620 experimental results I said could not be explained by classical theories. For 63 00:06:22,620 --> 00:06:27,890 example the photoelectric effect discovered by Hertz and Hallwachs in 1887, 64 00:06:27,890 --> 00:06:32,130 or the discrete spectral lines of atoms first shown by Fraunhofer in the spectrum 65 00:06:32,130 --> 00:06:38,110 of the sun and then studied by Bunsen and Kirchhoff with the so-called 66 00:06:38,110 --> 00:06:43,430 "Bunsenbrenner", you all know it from the chemistry classes. And further, 67 00:06:43,430 --> 00:06:47,900 radioactive rays were really a mystery nobody understood: How can it happen that 68 00:06:47,900 --> 00:06:54,770 something just decays at random intervals? It was unclear. And then the people looked 69 00:06:54,770 --> 00:07:00,810 into the atom, Rutherford using alpha particles to bombard a gold foil and saw 70 00:07:00,810 --> 00:07:05,259 there must be positively charged nucleii and they already knew that they were 71 00:07:05,259 --> 00:07:13,490 negatively charged, what we now call electrons, particles in the atom. So this 72 00:07:13,490 --> 00:07:18,150 was really strange that atoms are stable at composed like this and I will explain 73 00:07:18,150 --> 00:07:23,660 why a bit later. But now to more detail to the experiments. The really big 74 00:07:23,660 --> 00:07:29,860 breakthrough in this time, experimentally speaking, were vacuum tubes, so you took a 75 00:07:29,860 --> 00:07:37,580 piece of glass and pumped the air out and closed it off and put all sorts of devices 76 00:07:37,580 --> 00:07:45,229 in there. And now one thing is this nice cathode ray experiment. We have here a so- 77 00:07:45,229 --> 00:07:53,960 called electron gun and this is a heated electrode, so here flows the current that 78 00:07:53,960 --> 00:08:00,210 heats it, so that the electrons get energy and seep out into the vacuum. Then we have 79 00:08:00,210 --> 00:08:08,240 an electrode that goes around and a plate in front that is positively charged. So we 80 00:08:08,240 --> 00:08:11,961 accelerate our electrons towards the plate. There's a pinhole in the plate and 81 00:08:11,961 --> 00:08:18,850 we get a beam of electrons. And now we had those evacuated tubes and those electron 82 00:08:18,850 --> 00:08:24,539 guns. So we put the electron gun in the evacuated tube, perhaps left a bit of gas 83 00:08:24,539 --> 00:08:29,050 in because then it glowed when it when the atoms in the gas were hit by the electrons 84 00:08:29,050 --> 00:08:33,940 so we could see the cathode ray, and then we play around. We take magnetic fields 85 00:08:33,940 --> 00:08:38,130 and see how does it react to magnetic fields. We take electric fields. How does 86 00:08:38,130 --> 00:08:42,879 it react to electric fields and so on. And what we find out is we somehow must have 87 00:08:42,879 --> 00:08:53,129 negatively charged particles that flow nicely around in our almost vacuum. And 88 00:08:53,129 --> 00:09:00,519 because atoms are neutral which is just known macroscopically there must be a 89 00:09:00,519 --> 00:09:08,339 positively charged component in the atom as well. And this positively charged 90 00:09:08,339 --> 00:09:14,079 component was first thought to be kind of a plum pudding or so with the electrons 91 00:09:14,079 --> 00:09:21,040 sitting in there. But the Rutherford- Marsden-Geiger experiment, so it was 92 00:09:21,040 --> 00:09:25,630 Rutherford invented the idea and Marsden and Geiger actually performed the 93 00:09:25,630 --> 00:09:30,819 experimental work, showed that if you had a really thin gold foil, really only a few 94 00:09:30,819 --> 00:09:35,139 hundred layers of atoms, that's the nice thing about gold, you can just hammer it 95 00:09:35,139 --> 00:09:41,529 out to really, really thin sheets, if you had that and then shot alpha particles 96 00:09:41,529 --> 00:09:47,540 that is helium nuclei that are created by the radioactive decay of many heavy 97 00:09:47,540 --> 00:09:54,900 elements for example, most uranium isotopes decay by alpha decay, then they 98 00:09:54,900 --> 00:10:00,050 were deflected strongly. If the charge would have been spaced throughout the 99 00:10:00,050 --> 00:10:04,009 atoms then this could not have happened. You can calculate, you can 100 00:10:04,009 --> 00:10:09,649 estimate the possible deflections with an extended charge and with a concentrated 101 00:10:09,649 --> 00:10:14,189 charge, and you see the only explanation for this is that there is a massive and 102 00:10:14,189 --> 00:10:22,050 really, really small positive thing in those atoms. So atoms are small, 103 00:10:22,050 --> 00:10:27,550 positively charged nucleus as Rutherford called it and around it there's a cloud of 104 00:10:27,550 --> 00:10:33,820 electrons or, he thought, orbiting electrons. But orbiting electrons atoms 105 00:10:33,820 --> 00:10:37,679 are stable, this doesn't really make sense in classical physics, because in classical 106 00:10:37,679 --> 00:10:43,029 physics all accelerator charges must radiate energy and be slowed by this 107 00:10:43,029 --> 00:10:51,920 process. And this means atoms that are stable and composed of some strange 108 00:10:51,920 --> 00:10:58,749 electrons and having nuclei they're just not possible. It's a no go, so at least at 109 00:10:58,749 --> 00:11:02,930 this moment it was completely clear classical physics as they knew it up until 110 00:11:02,930 --> 00:11:09,869 then is wrong. And the next experiment in this direction was the photoelectric 111 00:11:09,869 --> 00:11:15,350 effect. What's shown there is a schematic of a phototube. And a phototube is again a 112 00:11:15,350 --> 00:11:22,550 vacuum tube out of glass and there is a for example cesium layer in in the tube at 113 00:11:22,550 --> 00:11:27,399 one side and there is a ring electrode removed from it. And if we shine light on 114 00:11:27,399 --> 00:11:34,559 this there flows a current. But the peculiar thing is that if we do the bias 115 00:11:34,559 --> 00:11:44,129 voltage across the two terminals of this tube to stop the electrons, we see that 116 00:11:44,129 --> 00:11:49,040 the bias voltage that completely stops the flow is not proportional to the intensity 117 00:11:49,040 --> 00:11:53,689 of the light that is incident onto the tube, but it's proportional to the 118 00:11:53,689 --> 00:11:59,540 frequency of the light that's incident on the phototube. And that was again really 119 00:11:59,540 --> 00:12:05,179 weird for the people of the time because the frequency shouldn't make any 120 00:12:05,179 --> 00:12:11,929 difference for the energy. And this was when Einstein derived that, or thought of 121 00:12:11,929 --> 00:12:18,149 that there must be some kind of energy portions in the electric field, from this 122 00:12:18,149 --> 00:12:23,480 simple experiment, which is often done in physics classes even at the high school 123 00:12:23,480 --> 00:12:31,040 level. So it's, from today's view it's not a complicated experiment. And to go even 124 00:12:31,040 --> 00:12:37,149 further those weird stable atoms had discrete, had discrete lines of emission 125 00:12:37,149 --> 00:12:43,660 and absorption of light. And here we have again a very simplified experimental set 126 00:12:43,660 --> 00:12:48,010 up of a so-called discharge tube, where we have high voltage between the terminals 127 00:12:48,010 --> 00:12:53,449 and a thin gas and then a current will flow, will excite the atoms. The atoms 128 00:12:53,449 --> 00:12:58,560 will relax and emit light and this light will have a specific spectrum with sharp 129 00:12:58,560 --> 00:13:03,670 frequencies that are, that have strong emission and we can see this with a 130 00:13:03,670 --> 00:13:08,839 diffraction grating that sorts light out according to its wavelength and then look 131 00:13:08,839 --> 00:13:13,869 on the screen or view some more fancy optical instrument to do precision 132 00:13:13,869 --> 00:13:22,860 measurements as Bunsen and Kirchhoff did. So what we knew up until now was that 133 00:13:22,860 --> 00:13:28,730 something was really weird and our physical theories didn't make sense. And 134 00:13:28,730 --> 00:13:34,149 then it got worse. Someone took an electron gun and pointed it at a 135 00:13:34,149 --> 00:13:38,430 monocrystalline surface. And such a monocrystalline surface is just like a 136 00:13:38,430 --> 00:13:45,670 diffraction grating: A periodically arranged thing. And off periodically 137 00:13:45,670 --> 00:13:53,019 arranged things there does happen regular interference pattern creation. So they saw 138 00:13:53,019 --> 00:13:58,100 interference pattern with electrons. But electrons aren't that particles? How can 139 00:13:58,100 --> 00:14:03,450 particles, so what was thought of then, since the times of Newton as a little hard 140 00:14:03,450 --> 00:14:08,839 ball, how can a little hard ball flowing around create interference patterns? It 141 00:14:08,839 --> 00:14:18,369 was really weird. And there's even more and as already mentioned radioactivity 142 00:14:18,369 --> 00:14:24,420 with the random decay of a nucleus. This doesn't make sense in classical physics, 143 00:14:24,420 --> 00:14:30,040 so it was really, really bad. And here I've added some modern facts that we'll 144 00:14:30,040 --> 00:14:38,100 need later on. Namely that if we measure, if we try to measure the position of a 145 00:14:38,100 --> 00:14:44,410 particle and use different position sensors to do so, only one of them, so at 146 00:14:44,410 --> 00:14:48,449 only at one position will the single particle register, but it will 147 00:14:48,449 --> 00:14:53,500 nevertheless show an interference pattern if I do this experiment with many many 148 00:14:53,500 --> 00:14:59,470 electrons. So there must somehow be a strange divide between the free space 149 00:14:59,470 --> 00:15:05,819 propagation of particles and measuring the particles. And you can do really weird 150 00:15:05,819 --> 00:15:11,319 stuff and record the information through which slit the particle went. And if you 151 00:15:11,319 --> 00:15:16,449 do this, the interference pattern vanishes. And then you can even destroy 152 00:15:16,449 --> 00:15:24,639 this information in a coherent manner and the interference pattern appears again. So 153 00:15:24,639 --> 00:15:28,769 what we know up until now is that quantum mechanics is really, really weird and 154 00:15:28,769 --> 00:15:37,540 really different from classical mechanics. And now that we've talked about those 155 00:15:37,540 --> 00:15:41,480 experiments, we'll begin with the theory, and the theory will begin with a lot of 156 00:15:41,480 --> 00:15:49,439 mathematics. The first one is simple. Complex numbers. Who doesn't know complex 157 00:15:49,439 --> 00:15:59,220 numbers? Okay. Sorry I'll have to ignore you for the sake of getting to the next 158 00:15:59,220 --> 00:16:05,089 points. *laughter* So I'll just say complex numbers are two components of, two 159 00:16:05,089 --> 00:16:10,240 componented objects with real numbers. And one of them is multiplied by an imaginary 160 00:16:10,240 --> 00:16:16,069 number i. And if we square the number i it gets -1. And this makes many things really 161 00:16:16,069 --> 00:16:22,149 beautiful. For example all algebraic equations have exactly the number of 162 00:16:22,149 --> 00:16:29,369 degrees solutions in complex numbers, and if you count them correctly. And if you 163 00:16:29,369 --> 00:16:33,989 work with complex functions it's really beautiful. A function that once 164 00:16:33,989 --> 00:16:39,850 differentiable is infinitely many times differentiable and it's, it's nice. So now 165 00:16:39,850 --> 00:16:46,499 we had complex numbers. You've all said you know them. *laughter* So we go onto 166 00:16:46,499 --> 00:16:53,329 vector spaces, which probably also a lot of you know. Just to revisit it, a vector 167 00:16:53,329 --> 00:16:58,430 space is a space of objects called vectors, above some scalars that must be a 168 00:16:58,430 --> 00:17:03,369 field. And here we only use complex numbers as the underlying fields. There is 169 00:17:03,369 --> 00:17:07,760 a null vector, we can add vectors, we can invert vectors and we can multiply vectors 170 00:17:07,760 --> 00:17:13,990 by real numbers. So we can say three that five times this vector and just scale the 171 00:17:13,990 --> 00:17:24,690 arrow and these operations interact nicely so that we have those distributive laws. 172 00:17:24,690 --> 00:17:33,830 And now it gets interesting. Even more maths: L2 spaces. L2 spaces are in a way 173 00:17:33,830 --> 00:17:40,210 an infinite dimensional or one form of an infinite dimensional extension of vector 174 00:17:40,210 --> 00:17:47,080 spaces. Instead of having just three directions x, y, z, we have directions at 175 00:17:47,080 --> 00:17:53,420 each point of a function. So we have an analogy here. We have vectors which have 176 00:17:53,420 --> 00:18:01,240 three discrete components given by x index i on the right side and we have this 177 00:18:01,240 --> 00:18:06,790 function and each component is the value of the function at one point along the 178 00:18:06,790 --> 00:18:13,750 axis x. And then we can just as for vectors define a norm on those L2 179 00:18:13,750 --> 00:18:18,690 functions which is just the integral over the absolute value squared of this 180 00:18:18,690 --> 00:18:23,610 function f. And the nice thing about this choice of norm, there are other choices of 181 00:18:23,610 --> 00:18:33,400 the norm. This norm is induced by a scalar product and this little asterisk that is 182 00:18:33,400 --> 00:18:39,160 there at the f denotes the complex conjugate, so flipping i to minus i in 183 00:18:39,160 --> 00:18:46,860 all complex values. And if you just plug in f and f into the scalar product you 184 00:18:46,860 --> 00:18:53,410 will see that it's the integral over the squared absolute value. And this space, 185 00:18:53,410 --> 00:18:59,180 this L2 space is a Hilbert space and the Hilbert Space is a complete vector space 186 00:18:59,180 --> 00:19:04,810 with a scalar product where complete means that - It's mathematical nonsense. 187 00:19:04,810 --> 00:19:10,050 Forget it. So but the nice surprise is that most things carry over from finite 188 00:19:10,050 --> 00:19:13,430 dimensional space. What we know from finite dimensional space is we can always 189 00:19:13,430 --> 00:19:19,180 diagonalize matrices with certain properties and this more or less works. 190 00:19:19,180 --> 00:19:23,620 And the mathematicians really, really, really do a lot of work for this but for 191 00:19:23,620 --> 00:19:30,500 physicists we just know when to be careful and how and don't care about it otherwise. 192 00:19:30,500 --> 00:19:38,360 So just works for us and that's nice. And now that we have those complex numbers we 193 00:19:38,360 --> 00:19:44,530 can begin to discuss how particles are modeled in quantum mechanics. And as we 194 00:19:44,530 --> 00:19:48,560 know from the Davisson-Germer experiments there's diffraction of electrons but 195 00:19:48,560 --> 00:19:54,130 there's nothing in electrons that corresponds to an electric field in some 196 00:19:54,130 --> 00:20:00,050 direction or so. Some other periodicity has, so periodicity of electrons during 197 00:20:00,050 --> 00:20:07,770 propagation has never been directly observed. And De Broglie said particles 198 00:20:07,770 --> 00:20:12,490 have a wavelength that's related to their momentum. And he was motivated primarily 199 00:20:12,490 --> 00:20:19,430 by the Bohr theory of the atom to do so. And he was shown right by the Davisson- 200 00:20:19,430 --> 00:20:24,300 Germer experiments so his relation for the wavelength of a particle is older than the 201 00:20:24,300 --> 00:20:29,650 experiments showing this, which is impressive I think. And now the idea is 202 00:20:29,650 --> 00:20:33,450 they have a complex wave function and let the squared absolute value of the wave 203 00:20:33,450 --> 00:20:39,880 function describe the probability density of a particle. So we make particles 204 00:20:39,880 --> 00:20:46,400 extended but probability measured objects so there isn't no longer the position of 205 00:20:46,400 --> 00:20:50,500 the particle as long as we don't measure. But we have just some description of a 206 00:20:50,500 --> 00:20:56,860 probability where the particle is. And by making it complex we have a phase and this 207 00:20:56,860 --> 00:21:01,420 phase can allow, still allow, interference effects which we need for explaining the 208 00:21:01,420 --> 00:21:07,130 interference peaks in the Davisson-Germer experiment. And now a lot of textbooks say 209 00:21:07,130 --> 00:21:13,250 here there's a wave particle dualism, blah blah blah. Distinct nonsense, blah. 210 00:21:13,250 --> 00:21:19,760 The point is it doesn't get you far to think about quantum objects as either wave 211 00:21:19,760 --> 00:21:25,850 or particle, they're just quantum. Neither wave nor particle. Doesn't help you either 212 00:21:25,850 --> 00:21:30,050 but it doesn't confuse you as much as when you tried to think about particles as 213 00:21:30,050 --> 00:21:37,930 waves or particles, or about quantum particles as waves or particles. And now 214 00:21:37,930 --> 00:21:43,550 that we say we have a complex wave function what about simply using a plain 215 00:21:43,550 --> 00:21:50,570 wave with constant probability as the states of definite momentum because we 216 00:21:50,570 --> 00:21:55,920 somehow have to describe a particle to say that has a certain momentum and we do 217 00:21:55,920 --> 00:22:00,110 this. Those have the little problem that they are not in the Hilbert space because 218 00:22:00,110 --> 00:22:07,420 they're not normalizable. The absolute value of psi is 1 over 2 pi everywhere, so 219 00:22:07,420 --> 00:22:13,630 that's bad. But we can write the superposition of any state by Fourier 220 00:22:13,630 --> 00:22:19,550 transformation those e to the i k dot r states are just the basis states of a 221 00:22:19,550 --> 00:22:25,550 Fourier transformation. We can write any function in terms of this basis. And we 222 00:22:25,550 --> 00:22:30,050 can conclude that by Fourier transformation of the state psi of r to 223 00:22:30,050 --> 00:22:35,500 some state till the psi of k, we describe the same information because we know we 224 00:22:35,500 --> 00:22:39,810 can invert the Fourier transformation and also this implies the uncertainty 225 00:22:39,810 --> 00:22:47,720 relation. And because this is simply property of Fourier transformations that 226 00:22:47,720 --> 00:22:52,130 either the function can be very concentrated in position space or in 227 00:22:52,130 --> 00:22:58,340 momentum space. And now that we have states of definite momentum. And 228 00:22:58,340 --> 00:23:04,530 the other big ingredient in quantum mechanics are operators, next to the state 229 00:23:04,530 --> 00:23:09,380 description. And operators are, just like matrices, linear operators on the state 230 00:23:09,380 --> 00:23:16,090 space. Just as we can apply a linear operator in the form of a matrix to a vector, 231 00:23:16,090 --> 00:23:24,710 we can apply linear operators to L2 functions. And when we measure an 232 00:23:24,710 --> 00:23:30,011 observable it will be that it's one of the eigenvalues of this operator that's the 233 00:23:30,011 --> 00:23:36,940 measurement value, you know. So eigenvalues are those values: If a matrix 234 00:23:36,940 --> 00:23:44,330 that just scales a vector by a certain amount that is an eigenvalue of the matrix 235 00:23:44,330 --> 00:23:49,410 and in the same sense we can define eigenvalues and eigenvectors for, L2 236 00:23:49,410 --> 00:23:57,250 functions. And there are some facts such as that non-commuting operators have 237 00:23:57,250 --> 00:24:05,830 eigenstates that are not common. So we can't have a description of the basis of 238 00:24:05,830 --> 00:24:12,160 the state space in terms of function that are both eigenfunctions of both operators 239 00:24:12,160 --> 00:24:16,760 and some examples of operators are the momentum operator which is just minus i 240 00:24:16,760 --> 00:24:23,810 h-bar Nabla which is the derivation operator in three dimensions. So in the x 241 00:24:23,810 --> 00:24:27,810 component we have derivation in the direction of x and in the y component in 242 00:24:27,810 --> 00:24:34,230 direction of y and so on. And the position operator which is just the operator that 243 00:24:34,230 --> 00:24:42,560 multiplies by the position x in the position space representation of the wave 244 00:24:42,560 --> 00:24:48,250 function. And as for the non- communtitivity of operators we can already 245 00:24:48,250 --> 00:24:54,710 show that those p and x operators that do not commute but fulfill a certain 246 00:24:54,710 --> 00:25:00,620 commutation relation. And a commutation relation is just a measure for how much 247 00:25:00,620 --> 00:25:07,250 two operators do not commute. And the commutator is AB minus BA for the objects 248 00:25:07,250 --> 00:25:15,180 AB, so if they commute, if AB equals BA the commutator simply vanishes. And 249 00:25:15,180 --> 00:25:20,870 there's more on operators just to make it clear: Linear just means that we can split 250 00:25:20,870 --> 00:25:26,660 the argument if it is just some linear combinations of vectors and apply the 251 00:25:26,660 --> 00:25:32,340 operator to the individual vectors occuring, we can define multiplication of 252 00:25:32,340 --> 00:25:38,240 operators and this just exactly follows the template that is laid down by finite 253 00:25:38,240 --> 00:25:44,110 dimensional linear algebra. There's nothing new here. And there are inverse 254 00:25:44,110 --> 00:25:49,380 operators for some operators, not for all of them, that give the identity operator 255 00:25:49,380 --> 00:25:54,450 if it's multiplied with the original operator. And further there's the so- 256 00:25:54,450 --> 00:26:00,400 called adjoint. Our scalar product had this little asterisk and this means that 257 00:26:00,400 --> 00:26:04,560 it's not linear in the first component. If I scale the first component by some 258 00:26:04,560 --> 00:26:10,600 complex number alpha the total scalar product is not scaled by alpha, but by the 259 00:26:10,600 --> 00:26:17,610 complex conjugate of alpha. This kind of not quite bi-linearity is sometimes called 260 00:26:17,610 --> 00:26:26,980 sesquilinearity, a seldomly used word, and they're commonly defined classes of 261 00:26:26,980 --> 00:26:35,990 operators in terms of how the adjoint that is defined there acts and how some other 262 00:26:35,990 --> 00:26:39,890 operators for example where the adjoint is the inverse which is a generalization from 263 00:26:39,890 --> 00:26:45,520 the fact that for rotation operators in normal Euclidean space, the transpose is 264 00:26:45,520 --> 00:26:53,950 the inverse. And now that we have operators we can define expectation values 265 00:26:53,950 --> 00:26:59,270 just by some formula. For now, we don't know what expectation values are, but we 266 00:26:59,270 --> 00:27:03,690 can assume, it has something to do with the measurement values of the operator 267 00:27:03,690 --> 00:27:09,510 because: why else would I tell you about it. And later on we will show that this is 268 00:27:09,510 --> 00:27:15,160 actually the expectation value of the quantity if we prepare a system always in 269 00:27:15,160 --> 00:27:20,780 the same fashion and then do measurements on it, we get random results each time, 270 00:27:20,780 --> 00:27:29,200 but the expectation value will be this combination. And now again: a bit of 271 00:27:29,200 --> 00:27:34,840 mathematics: eigenvalue problems. Well known: You can diagonalize a matrix and 272 00:27:34,840 --> 00:27:39,820 you can diagonalize linear operators. You have some equation A psi equals lambda 273 00:27:39,820 --> 00:27:47,080 psi, where lambda is just a scalar. And if such an equation holds for some vector psi 274 00:27:47,080 --> 00:27:52,770 then it's an eigenvector and if we scale the vector linearly, this will again be an 275 00:27:52,770 --> 00:28:01,380 eigenvector. And what can happen is that to one eigenvalue there are several 276 00:28:01,380 --> 00:28:05,350 eigenvectors, not only one ray of eigenvectors, but a higher dimensional 277 00:28:05,350 --> 00:28:11,940 subspace. And important to know is that so-called Hermitian operators, that is 278 00:28:11,940 --> 00:28:17,690 those that equal their adjoint, which again means that the eigenvalues equal the 279 00:28:17,690 --> 00:28:23,980 complex conjugate of the eigenvalues have a real eigenvalues. Because if a complex 280 00:28:23,980 --> 00:28:33,070 number equals its complex conjugate, then it's a real number. And the nice thing 281 00:28:33,070 --> 00:28:39,570 about those diagonalized matrices and all is: we can develop any vector in terms of 282 00:28:39,570 --> 00:28:46,890 the eigenbasis of the operator, again just like in linear algebra where when you 283 00:28:46,890 --> 00:28:51,420 diagonalize a matrix, you get a new basis for your vector space and now you can 284 00:28:51,420 --> 00:28:57,640 express all vectors in that new basis. And if the operator is Hermitian the 285 00:28:57,640 --> 00:29:05,050 eigenvectors have a nice property, namely they are orthogonal if the eigenvalues are 286 00:29:05,050 --> 00:29:11,380 different. And this is good because this guarantees us that we can choose an 287 00:29:11,380 --> 00:29:16,610 orthonormal, that is a basis in the vector space where to basis vectors always have 288 00:29:16,610 --> 00:29:24,090 vanishing scalar product are orthogonal and are normal, that is: we scale them to 289 00:29:24,090 --> 00:29:29,200 length one, because we want our probability interpretation, and in our 290 00:29:29,200 --> 00:29:37,220 probability interpretation we need to have normalized vectors. So now we have that 291 00:29:37,220 --> 00:29:41,890 and now we want to know: How does this strange function psi, that describes the 292 00:29:41,890 --> 00:29:49,580 state of the system, evolve in time. And for this we can have several requirements 293 00:29:49,580 --> 00:29:55,640 that it must fulfill. So again we are close to software engineering and one 294 00:29:55,640 --> 00:30:01,400 requirement is, that if it is a sharp wave packet, so if we have a localized state 295 00:30:01,400 --> 00:30:06,620 that is not smeared around the whole space, then it should follow the classical 296 00:30:06,620 --> 00:30:12,580 equation of motion because we want that our new theory contains our old theory. 297 00:30:12,580 --> 00:30:18,450 And the time evolution must conserve the total probability of finding the particle 298 00:30:18,450 --> 00:30:21,990 because otherwise we couldn't do probability interpretation of our wave 299 00:30:21,990 --> 00:30:29,010 function, if the total probability of the particle wouldn't remain one. Further we 300 00:30:29,010 --> 00:30:35,110 wish the equation to be first order in time and to be linear because for example 301 00:30:35,110 --> 00:30:41,950 the Maxwell equations are linear and show nice interference effects, so we want that 302 00:30:41,950 --> 00:30:46,370 because then simply a sum of solutions is again a solution, it's a good property to 303 00:30:46,370 --> 00:30:51,840 have and if it works that way: Why not? And the third and the fourth requirement 304 00:30:51,840 --> 00:31:00,600 together already give us more or less the form of the Schroedinger equation. Because 305 00:31:00,600 --> 00:31:04,830 linearity just says that the right-hand side of some linear operator applied to 306 00:31:04,830 --> 00:31:12,340 psi and the first order in time just means that there must be a single time 307 00:31:12,340 --> 00:31:19,840 derivative in the equation on the left- hand side. And this i-h bar: we just wanted 308 00:31:19,840 --> 00:31:23,550 that there, no particular reason we could have done this differently, but it's 309 00:31:23,550 --> 00:31:31,710 convention. Now with this equation we can look: What must happen for the probability 310 00:31:31,710 --> 00:31:42,290 to be conserved and by a simple calculation we can show that it must be a 311 00:31:42,290 --> 00:31:47,870 Hermitian operator. And there is even more than this global argument. There's local 312 00:31:47,870 --> 00:31:52,020 conservation of probability, that is, a particle can't simply vanish here and 313 00:31:52,020 --> 00:31:59,250 appear there, but it must flow from one point to the other with local operations. 314 00:31:59,250 --> 00:32:05,360 This can be shown when you consider this in more detail. Now we know how this 315 00:32:05,360 --> 00:32:09,460 equation of motion looks like, but we don't know what this mysterious object H 316 00:32:09,460 --> 00:32:16,500 might be. And this mysterious object H is the operator of the energy of the system 317 00:32:16,500 --> 00:32:21,770 which is known from classical mechanics as the Hamilton function and which we here 318 00:32:21,770 --> 00:32:26,610 upgrade to the Hamilton operator by using the formula for the classical Hamilton 319 00:32:26,610 --> 00:32:33,480 function and inserting our p into our operators. And we can also extend this to 320 00:32:33,480 --> 00:32:39,710 a magnetic field. And by doing so we can show that our theory is more or less 321 00:32:39,710 --> 00:32:45,960 consistent with Newtonian mechanics. We can show the Ehrenfest theorem, that's the 322 00:32:45,960 --> 00:32:55,160 first equation. And then those equations are almost Newton's equation of motion for 323 00:32:55,160 --> 00:33:05,690 the centers of mass of the particle because this is the expectation value of 324 00:33:05,690 --> 00:33:10,670 the momentum, this is the expectation value of the position of the particle. 325 00:33:10,670 --> 00:33:15,740 This just looks exactly like the classical equation. The velocity is the momentum 326 00:33:15,740 --> 00:33:23,140 divided by the mass. But this is weird: Here we average over the force, so the 327 00:33:23,140 --> 00:33:29,920 gradient of the potential is the force, we average over the force and do not take the 328 00:33:29,920 --> 00:33:34,980 force at the center position, so we can't in general solve this equation. But again 329 00:33:34,980 --> 00:33:38,620 if we have a sharply defined wave packet we recover the classical equations of 330 00:33:38,620 --> 00:33:44,640 motion, which is nice. So we have shown our new theory does indeed explain why our 331 00:33:44,640 --> 00:33:51,250 old theory worked. We only still have to explain why the centers of mass of massive 332 00:33:51,250 --> 00:33:55,520 particles are usually well localized and that's a question we're still having 333 00:33:55,520 --> 00:34:08,010 trouble with today. But since it otherwise works: don't worry too much about it. And 334 00:34:08,010 --> 00:34:12,319 now you probably want to know how to solve the Schroedinger equation. Or you don't 335 00:34:12,319 --> 00:34:18,201 want to know anything more about quantum mechanics. And to do this we make a so- 336 00:34:18,201 --> 00:34:24,559 called separation ansatz, where we say, we have a form stable part of our wave 337 00:34:24,559 --> 00:34:30,609 function multiplied by some time dependent part. And if we do this we can write down 338 00:34:30,609 --> 00:34:35,589 the general solution for the Schroedinger equation. Because we already know that the 339 00:34:35,589 --> 00:34:41,009 one equation that we get is an eigenvalue equation or an eigenvector equation for 340 00:34:41,009 --> 00:34:45,849 the energy eigenvalues, that is the eigenvalues of the Hamilton operator. And 341 00:34:45,849 --> 00:34:50,519 we know that we can develop any function in terms of those and so the general 342 00:34:50,519 --> 00:34:58,170 solution must be of the form shown here. And those states of specific energy have a 343 00:34:58,170 --> 00:35:02,220 simple evolution because their form is constant and only their phase changes and 344 00:35:02,220 --> 00:35:08,960 depends on the energy. And now this thing with the measurement in quantum mechanics 345 00:35:08,960 --> 00:35:13,200 is bad. You probably know Schroedinger's cat and the point is: there you don't know 346 00:35:13,200 --> 00:35:16,033 whether the cat is dead or alive while you don't look inside the box. While you don't 347 00:35:16,033 --> 00:35:19,480 look inside the box as long as you don't measure it's in a superposition or 348 00:35:19,480 --> 00:35:23,930 something. So You measure your cat and then it's dead. It isn't dead 349 00:35:23,930 --> 00:35:29,089 before only by measuring it you kill it. And that's really not nice to kill cats. 350 00:35:29,089 --> 00:35:36,970 We like cats. The important part here is, the TL;DR, quantum measurement is 351 00:35:36,970 --> 00:35:42,970 probabilistic and inherently changes the system state. So I'll skip the multi 352 00:35:42,970 --> 00:35:52,770 particle things. We can't describe multiple particles. And just show the 353 00:35:52,770 --> 00:35:58,700 axioms of quantum mechanics shortly. Don't don't read them too detailed, but this is 354 00:35:58,700 --> 00:36:04,009 just a summary of what we've discussed so far. And the thing about the multiple 355 00:36:04,009 --> 00:36:09,730 particles is the axiom 7 which says that the sign of the wave function must change 356 00:36:09,730 --> 00:36:15,109 if we exchange the coordinates of identical fermions. And this makes atom 357 00:36:15,109 --> 00:36:20,739 stable by the way. Without this atoms as we know them would not exist. And finally 358 00:36:20,739 --> 00:36:26,809 there is a notational convention in quantum mechanics called Bra-Ket-notation. 359 00:36:26,809 --> 00:36:35,380 And in Bra-Ket-notation you label states by their eigenvalues and just think about 360 00:36:35,380 --> 00:36:42,470 such a Ket as an abstract vector such as x with a vector arrow over it or a fat set x 361 00:36:42,470 --> 00:36:48,849 is an abstract vector and we can either represent it by its coordinates x1 x2 x3, 362 00:36:48,849 --> 00:36:53,059 or we can work with the abstract vector and this Ket is such an abstract vector 363 00:36:53,059 --> 00:37:02,089 for the L2 function psi of r. And then we can also define the adjoint of this which 364 00:37:02,089 --> 00:37:07,690 gives us, if we multiply the adjoint and a function, the scalar product. So this is a 365 00:37:07,690 --> 00:37:15,249 really nice and compact notation for many physics problems. And the last equation 366 00:37:15,249 --> 00:37:20,750 there just looks like component wise, like working with components of matrices, which 367 00:37:20,750 --> 00:37:31,579 is because it's nothing else. This is just matrix calculus in new clothes. Now for 368 00:37:31,579 --> 00:37:43,559 the applications. The first one is quite funny. There's a slide missing. Okay. Uh 369 00:37:43,559 --> 00:37:48,670 the first one is a quantum eraser at home. Because if you encode the "which way" 370 00:37:48,670 --> 00:37:56,050 information into a double slit experiment you lose your interference pattern. And we 371 00:37:56,050 --> 00:38:01,220 do this by using a vertical and horizontal polarisation filter. And you know from 372 00:38:01,220 --> 00:38:16,390 classical physics then it won't make an interference pattern. And if we then add a 373 00:38:16,390 --> 00:38:25,641 diagonal polarization filter then the interference pattern will appear again. So 374 00:38:25,641 --> 00:38:31,470 now, just so you've seen it, the harmonic oscillator can be exactly solved in 375 00:38:31,470 --> 00:38:36,499 quantum mechanics. If you can solve the harmonic oscillator in any kind of physics 376 00:38:36,499 --> 00:38:41,930 then you're good, then you'll get through the axioms when you study physics. So the 377 00:38:41,930 --> 00:38:47,771 harmonic oscillator is solved by introducing so-called creation and 378 00:38:47,771 --> 00:38:53,560 destroyer operators and then we can determine the ground state function, in a 379 00:38:53,560 --> 00:38:57,720 much simpler manner than if we had to solve the Schroedinger equation explicitly 380 00:38:57,720 --> 00:39:05,559 for all those cases. And we can determine the ground state function, so the function 381 00:39:05,559 --> 00:39:11,440 of lowest energy. This can all be done and then from it by applying the creation 382 00:39:11,440 --> 00:39:18,609 operator create the highest eigenstate of the system and get all of them. Then 383 00:39:18,609 --> 00:39:22,589 there's this effect of tunnelling that you've probably heard about and this just 384 00:39:22,589 --> 00:39:27,609 means that in quantum mechanics a potential barrier that is too high for the 385 00:39:27,609 --> 00:39:32,460 particle to penetrate does not mean that the particle doesn't penetrate at all but 386 00:39:32,460 --> 00:39:36,749 that the probability of finding the particle inside the barrier decays 387 00:39:36,749 --> 00:39:42,589 exponentially. And this can for example be understood in terms of this uncertainty 388 00:39:42,589 --> 00:39:47,569 relation because if we try to compress the particle to the smaller part of the 389 00:39:47,569 --> 00:39:52,009 boundary layer then its momentum has to be high so it can reach farther in because 390 00:39:52,009 --> 00:39:58,829 then it has more energy. And there's this myth that tunnelling makes particles 391 00:39:58,829 --> 00:40:04,530 traveling to travel instantaneously from A to B and even some real physicists believe 392 00:40:04,530 --> 00:40:12,700 it. But sorry it's not true. The particle states is extended anyway and to defining 393 00:40:12,700 --> 00:40:17,140 what how fast the particle travels is actually not a well-defined thing in deep 394 00:40:17,140 --> 00:40:23,079 quantum regimes, and also the Schroedinger equations is not relativistic. So there is 395 00:40:23,079 --> 00:40:27,630 nothing, really nothing stopping your particle from flying around with 30 times 396 00:40:27,630 --> 00:40:34,339 the speed of light. It's just not in the theory. Another important consequence of 397 00:40:34,339 --> 00:40:38,670 quantum mechanics is so-called entanglement and this is a really weird 398 00:40:38,670 --> 00:40:44,109 one, because it shows that the universe that we live in is in a way non-local, 399 00:40:44,109 --> 00:40:51,859 inherently non-local. Because we can create some states for some internal 400 00:40:51,859 --> 00:40:57,400 degrees of freedom of two atoms and move them apart then measure the one system and 401 00:40:57,400 --> 00:41:01,460 the measurement result in the one system will determine the measurement result in 402 00:41:01,460 --> 00:41:07,790 the other system, no matter how far removed they are from each other. And this 403 00:41:07,790 --> 00:41:12,099 was first discovered in a paper by Einstein, Podolski and Rosen and they 404 00:41:12,099 --> 00:41:17,619 thought it was an argument that quantum mechanics is absurd. This can't be true, 405 00:41:17,619 --> 00:41:23,359 but sorry it is true. So this works and this kind of state that we've written 406 00:41:23,359 --> 00:41:32,970 there that is such an entangled state of two particles. But important to remark is 407 00:41:32,970 --> 00:41:37,029 that there are no hidden variables, that means the measurement result is not 408 00:41:37,029 --> 00:41:42,250 determined beforehand. It is only when we measure that is actually known what the 409 00:41:42,250 --> 00:41:47,741 result will be. This is utterly weird but one can prove this experimentally. Those 410 00:41:47,741 --> 00:41:52,239 are Bell tests. There's a Bell-inequality that's the limit for theories where they 411 00:41:52,239 --> 00:41:57,410 are hidden variables and it's by real experiments they violate the inequality 412 00:41:57,410 --> 00:42:02,819 and thereby show that there are no hidden variables. And there's a myth surrounding 413 00:42:02,819 --> 00:42:07,250 entanglement, namely that you can transfer information with it between two sides 414 00:42:07,250 --> 00:42:12,910 instantaneously. But again there's nothing hindering you in non relativistic quantum 415 00:42:12,910 --> 00:42:18,309 mechanics to distribute information arbitrarily fast. It doesn't have a speed 416 00:42:18,309 --> 00:42:24,440 limit but you can't also count communicate with those entangled pairs of particles. 417 00:42:24,440 --> 00:42:28,319 You can just create correlated noise at two ends which is what quantum 418 00:42:28,319 --> 00:42:36,109 cryptography is using. So now because this is the hackers congress, some short 419 00:42:36,109 --> 00:42:41,549 remarks and probably unintelligible due to their strong compression about quantum 420 00:42:41,549 --> 00:42:47,190 information. A qubit, the fundamental unit of quantum information, is a system with 421 00:42:47,190 --> 00:42:53,852 two states zero and one. So just like a bit. But now we allow arbitrary super 422 00:42:53,852 --> 00:42:58,309 positions of those states because that is what quantum mechanics allows. We can 423 00:42:58,309 --> 00:43:03,000 always superimpose states and quantum computers are really bad for most 424 00:43:03,000 --> 00:43:08,940 computing tasks because they have to, even if they build quantum computers they 425 00:43:08,940 --> 00:43:14,171 will never be as capable as the state-of- the-art silicon electrical computer. So 426 00:43:14,171 --> 00:43:18,359 don't fear for your jobs because of quantum computers. But the problem is they 427 00:43:18,359 --> 00:43:23,599 can compute some things faster. For example factoring primes and working with 428 00:43:23,599 --> 00:43:29,599 some elliptic curve algorithms and so on and determining discrete logarithm so our 429 00:43:29,599 --> 00:43:35,339 public key crypto would be destroyed by them. And this all works by using the 430 00:43:35,339 --> 00:43:41,220 superposition to construct some kind of weird parallelism. So it's actually I 431 00:43:41,220 --> 00:43:47,359 think nobody really can imagine how it works but we can compute it which is often 432 00:43:47,359 --> 00:43:51,670 the case in quantum mechanics. And then there's quantum cryptography and that 433 00:43:51,670 --> 00:43:56,279 fundamentally solves the same problem as a Diffie-Hellman key exchange. We can 434 00:43:56,279 --> 00:44:01,150 generate the shared key and we can check by the statistics of our measured values 435 00:44:01,150 --> 00:44:07,349 that there is no eavesdropper, which is cool actually. But it's also quite useless 436 00:44:07,349 --> 00:44:10,201 because we can't detect a man in the middle. How should the quantum particle 437 00:44:10,201 --> 00:44:14,630 knows of the other side is the one with that we want to talk to. We still need 438 00:44:14,630 --> 00:44:18,839 some shared secret or public key infrastructure whatever. So it doesn't 439 00:44:18,839 --> 00:44:27,210 solve the problem that we don't have solved. And then the fun fact about this 440 00:44:27,210 --> 00:44:30,970 is that all the commercial implementations of quantum cryptography were susceptible 441 00:44:30,970 --> 00:44:35,150 to side channel text, for example you could just shine the light with a fiber 442 00:44:35,150 --> 00:44:40,920 that was used, read out the polarization filter state that they used and then you 443 00:44:40,920 --> 00:44:50,609 could mimic the other side. So that's not good either. So finally some references 444 00:44:50,609 --> 00:44:55,150 for further study. The first one is really difficult. Only try this if you've read the 445 00:44:55,150 --> 00:45:00,650 other two but the second one. Sorry that they're in German. The first and the last 446 00:45:00,650 --> 00:45:04,579 are also available in translation but the second one has a really really nice and 447 00:45:04,579 --> 00:45:09,650 accessible introduction in the last few pages so it's just 20 pages and it's 448 00:45:09,650 --> 00:45:14,660 really good and understandable. So if you can get your hands on the books and are 449 00:45:14,660 --> 00:45:22,079 really interested, read it. So thank you for the attention and I'll be answering 450 00:45:22,079 --> 00:45:24,259 your questions next. 451 00:45:24,259 --> 00:45:33,159 *Applause* 452 00:45:33,159 --> 00:45:41,240 Herald: Thank you Sebastian. Do we have questions? And don't be afraid to sound 453 00:45:41,240 --> 00:45:45,380 naive or anything. I'm sure if you didn't understand something many other people 454 00:45:45,380 --> 00:45:49,099 would thank you for a good question. Sebastian: As to understanding things in 455 00:45:49,099 --> 00:45:53,180 quantum mechanics, Fineman said "You can't understand quantum mechanics, you can just 456 00:45:53,180 --> 00:45:57,269 accept that there there's nothing to understand. That's just too weird." 457 00:45:57,269 --> 00:46:01,590 Herald: Ok,we've found some questions. So microphone one please. 458 00:46:01,590 --> 00:46:09,349 M1: Can you explain that, if you measure a system, it looks like you changed the 459 00:46:09,349 --> 00:46:15,130 state of the system. How is it defined where the system starts? No. How is it 460 00:46:15,130 --> 00:46:20,200 defined when the system ends and the measurement system begins. Or in other 461 00:46:20,200 --> 00:46:24,410 words why does the universe have a state? Is there somewhere out there who 462 00:46:24,410 --> 00:46:29,450 measures the universe? S: No. There's at least the beginning of a 463 00:46:29,450 --> 00:46:34,880 solution by now which is called "decoherence" which says that this 464 00:46:34,880 --> 00:46:39,970 measurement structure that we observe is not inherent in quantum mechanics but 465 00:46:39,970 --> 00:46:43,920 comes from the interaction with the environment. And we don't care for the 466 00:46:43,920 --> 00:46:48,460 states of the environment. And if we do this, the technical term is traced out the 467 00:46:48,460 --> 00:46:52,529 states of the environment. Then the remaining state of the measurement 468 00:46:52,529 --> 00:46:59,789 apparatus and the system we're interested in will be just classically a randomized 469 00:46:59,789 --> 00:47:05,700 states. So it's rather a consequence of the complex dynamics of a system state 470 00:47:05,700 --> 00:47:10,739 and environment in quantum mechanics. But this is really the burning question. We 471 00:47:10,739 --> 00:47:15,690 don't really know. We have this we know decoherence make some makes it nice and 472 00:47:15,690 --> 00:47:20,749 looks good. But it also doesn't answer the question finally. And this is what all 473 00:47:20,749 --> 00:47:25,430 those discussions about interpretations of quantum mechanics are about. How shall we 474 00:47:25,430 --> 00:47:28,940 make sense of this weird measurement process. 475 00:47:28,940 --> 00:47:37,150 Herald: Okay. Microphone 4 in the back please. M4: Could you comment on your point in the 476 00:47:37,150 --> 00:47:44,220 theory section. I don't understand what you were trying to do. Did you want to 477 00:47:44,220 --> 00:47:49,369 show that you cannot understand really quantum mechanics without the mathematics 478 00:47:49,369 --> 00:47:51,990 or? S: Well, yes you can't understand quantum 479 00:47:51,990 --> 00:47:56,010 mechanics without the mathematics and my point to show was that mathematics, or at 480 00:47:56,010 --> 00:48:02,380 least my hope to show was that mathematics is halfways accessible. Probably not 481 00:48:02,380 --> 00:48:07,799 understandable after just exposure of a short talk but just to give an 482 00:48:07,799 --> 00:48:12,849 introduction where to look M4: OK. So you are trying to combat the 483 00:48:12,849 --> 00:48:18,050 esoterics and say they don't really understand the theory because they don't 484 00:48:18,050 --> 00:48:29,380 understand the mathematics. I understand the mathematics. I'm just interested. What 485 00:48:29,380 --> 00:48:33,809 were you trying to say? S: I was just trying to present the 486 00:48:33,809 --> 00:48:39,339 theory. That was my aim. M4: Okay. Thank you. 487 00:48:39,339 --> 00:48:45,759 Herald: Okay, microphone 2 please. M2: I know the answer to this question is 488 00:48:45,759 --> 00:48:48,569 that ... Herald: Can you go a little bit closer to 489 00:48:48,569 --> 00:48:52,660 the microphone maybe move it up please. M2: So I know the answer to this question 490 00:48:52,660 --> 00:48:59,510 is that atoms behave randomly but could you provide an argument why they behave 491 00:48:59,510 --> 00:49:07,369 randomly and it is not the case that we don't have a model that's. So, are atoms 492 00:49:07,369 --> 00:49:12,019 behaving randomly? Or is it the case that we don't have a model accurate enough to 493 00:49:12,019 --> 00:49:17,890 predict the way they behave? S: Radioactive decay is just as random as 494 00:49:17,890 --> 00:49:24,089 quantum measurement and since if we were to look at the whole story and look 495 00:49:24,089 --> 00:49:28,219 at the coherent evolution of the whole system we would have to include the 496 00:49:28,219 --> 00:49:33,809 environment and the problem is that the state space that we have to consider grows 497 00:49:33,809 --> 00:49:37,809 exponentially. That's the point of quantum mechanics. If I have two particles I have 498 00:49:37,809 --> 00:49:42,900 a two dimensional space. I have 10 particles I have a 1024 dimensional space 499 00:49:42,900 --> 00:49:47,269 and that's only talking about non interacting particles. So things explode 500 00:49:47,269 --> 00:49:51,950 in quantum mechanics and large systems. And therefore I would go so far as to say 501 00:49:51,950 --> 00:49:57,140 that it's objectively impossible to determine a radioactive decay although 502 00:49:57,140 --> 00:50:03,670 there are things, there is I think one experimentally confirmed method of letting 503 00:50:03,670 --> 00:50:11,279 an atom decay on purpose. This involves meta stable states of nuclei and then you 504 00:50:11,279 --> 00:50:15,690 can do something like spontaneous emission in a laser. You shine a strong gamma 505 00:50:15,690 --> 00:50:21,710 source by it and this shortens the lifespan of the nucleus. But other than 506 00:50:21,710 --> 00:50:25,410 that. M4: So in a completely hypothetical case. If you 507 00:50:25,410 --> 00:50:29,839 know all the starting conditions and what happens afterwards,wouldn't it be able, 508 00:50:29,839 --> 00:50:37,220 we could say it's deterministic? I mean I'm playing with heavy words here. 509 00:50:37,220 --> 00:50:43,619 But is it just that we say it's randomised because it's very very complex right? 510 00:50:43,619 --> 00:50:48,470 That's what I'm understanding. Herald: Maybe think about that question 511 00:50:48,470 --> 00:50:53,099 one more time and we have the signal angel in between and then you can come back. 512 00:50:53,099 --> 00:50:57,690 Signal Angel do we have questions on the Internet? 513 00:50:57,690 --> 00:51:04,940 Angel: There's one question from the Internet which is the ground state of a BEH-2 has 514 00:51:04,940 --> 00:51:12,150 been just calculated using a quantum eigensolver. So is there still some use of 515 00:51:12,150 --> 00:51:16,849 quantum computing in quantum mechanics? S: Yes definitely. One of the main 516 00:51:16,849 --> 00:51:22,309 motivations for inventing quantum computers was quantum simulators. 517 00:51:22,309 --> 00:51:26,700 Feynman invented this kind of quantum computing and he showed that with 518 00:51:26,700 --> 00:51:32,009 digital quantum computer you can efficiently simulate quantum systems. While 519 00:51:32,009 --> 00:51:36,489 you can't simulate quantum systems with a classical computer because of this problem 520 00:51:36,489 --> 00:51:41,790 of the exploding dimensions of the Hilbert space that you have to consider. And for 521 00:51:41,790 --> 00:51:46,280 this quantum computers are really really useful and will be used once they work, 522 00:51:46,280 --> 00:51:52,760 which is the question when it will be. Perhaps never. Beyond two or three qubits 523 00:51:52,760 --> 00:51:59,180 or 20 or 100 qubits but you need scalability for a real quantum computer. But quantum 524 00:51:59,180 --> 00:52:03,349 simulation is a real thing and it's a good thing and we need it. 525 00:52:03,349 --> 00:52:07,960 Herald: Okay. Then we have microphone 1 again. 526 00:52:07,960 --> 00:52:13,779 M1: So very beginning, you said that the theory is a set of interdependent 527 00:52:13,779 --> 00:52:21,539 propositions. Right? And then if a new hypothesis is made it can be confirmed by 528 00:52:21,539 --> 00:52:28,219 an experiment. S: That can't be confirmed but, well it's 529 00:52:28,219 --> 00:52:33,559 a philosophical question about the common stance, it can be made probable but not 530 00:52:33,559 --> 00:52:37,210 be confirmed because we can never absolutely be sure that there won't be 531 00:52:37,210 --> 00:52:40,599 some new experiment that shows that the hypothesis is wrong. 532 00:52:40,599 --> 00:52:44,920 M1: Yeah. Because the slide said that the experiment confirms... 533 00:52:44,920 --> 00:52:51,150 S: Yeah, confirm in the sense that it doesn't disconfirm it. So it makes 534 00:52:51,150 --> 00:52:57,040 probable that it's a good explanation of the reality and that's the point. Physics 535 00:52:57,040 --> 00:53:01,710 is just models. We do get nothing about the ontology that is about 536 00:53:01,710 --> 00:53:06,349 the actual being of the world out of physics. We just get models to describe 537 00:53:06,349 --> 00:53:11,539 the world but all what I say about this wave function and what we say about 538 00:53:11,539 --> 00:53:18,150 elementary particles. We can't say they are in the sense that you and I are here 539 00:53:18,150 --> 00:53:22,749 and exist because we can't see them we can't access them directly. We can only 540 00:53:22,749 --> 00:53:28,960 use them as description tools. But this is my personal position on philosophy of 541 00:53:28,960 --> 00:53:33,380 science. So there are people who disagree. M1: Ok, thanks. 542 00:53:33,380 --> 00:53:39,960 Herald: Microphone 2 please. M2: Or maybe superposition. By the way, so 543 00:53:39,960 --> 00:53:47,550 on the matter of the collapsing of the wave function, so this was already treated 544 00:53:47,550 --> 00:53:52,589 on the interpretation of Copenhagen and then as you mentioned it was expanded by 545 00:53:52,589 --> 00:53:59,331 the concept of decoherence. And is this, so the decoherence is including also the 546 00:53:59,331 --> 00:54:03,319 Ghirardi–Rimini–Weber interpretation or not? 547 00:54:03,319 --> 00:54:06,880 S: Could decoherence be used in computation or? 548 00:54:06,880 --> 00:54:13,019 M2: No so for the Ghirardi–Rimini–Weber interpretation of the collapsing of the 549 00:54:13,019 --> 00:54:15,700 wave function. S:That's one that I don't know. 550 00:54:15,700 --> 00:54:24,269 I'm not so much into interpretations. I actually think that there's interesting 551 00:54:24,269 --> 00:54:29,630 work done there but I think they're a bit irrelevant because in the end what I just 552 00:54:29,630 --> 00:54:33,690 said I don't think you can derive ontological value from our physical 553 00:54:33,690 --> 00:54:40,519 theories and in this belief, I think that the interpretations are in a sense void, 554 00:54:40,519 --> 00:54:44,890 they just help us to rationalize what we're doing but they don't really add 555 00:54:44,890 --> 00:54:49,339 something to the theory as long as they don't change what can be measured. 556 00:54:49,339 --> 00:54:58,180 M2: Oh okay. Thanks. S: Sorry for being an extremist. 557 00:54:58,180 --> 00:55:03,580 M2: Totally fine. Herald: Someone just left from microphone 1 558 00:55:03,580 --> 00:55:07,810 I don't know if they want to come back. I don't see any more questions as to 559 00:55:07,810 --> 00:55:13,680 signal angel have anything else. There is some more. Signal angel, do you have 560 00:55:13,680 --> 00:55:16,519 something? Signal Angel: No. 561 00:55:16,519 --> 00:55:19,609 Herald: Okay. Then we have microphone 4. 562 00:55:19,609 --> 00:55:27,809 M4: I want to ask a maybe a noob question. I want to know, are the probabilities of 563 00:55:27,809 --> 00:55:32,770 quantum mechanics inherent part of nature or maybe in some future we'll have a 564 00:55:32,770 --> 00:55:37,490 science that will determine all these values exactly? 565 00:55:37,490 --> 00:55:44,799 S: Well if decoherency theory is true, then quantum mechanics is absolutely 566 00:55:44,799 --> 00:55:53,779 deterministic. But so let's say, Everett says that all those possible measurement 567 00:55:53,779 --> 00:55:58,869 outcomes do happen and the whole state of the system is in a superposition and by 568 00:55:58,869 --> 00:56:03,839 looking at our measurement device and seeing some value we in a way select one 569 00:56:03,839 --> 00:56:10,219 strand of those superpositions and live in this of the many worlds and in this sense 570 00:56:10,219 --> 00:56:21,349 everything happens deterministically, but we just can't access any other values. So 571 00:56:21,349 --> 00:56:27,999 I think it's for now rather a of philosophy than of science. 572 00:56:27,999 --> 00:56:32,900 M4: I see. Thanks. 573 00:56:32,900 --> 00:56:38,559 Herald: Anything else? I don't see any people lined up at microphones. So last 574 00:56:38,559 --> 00:56:46,709 chance to round up now, I think. Well then I think we're closing this and have a nice 575 00:56:46,709 --> 00:56:59,510 applause again for Sebastian. *applause* 576 00:56:59,510 --> 00:57:02,654 Sebastian: Thank you. And I hope I didn't create more fear of 577 00:57:02,654 --> 00:57:05,080 quantum mechanics than I dispersed. 578 00:57:05,080 --> 00:57:30,000 subtitles created by c3subtitles.de in the year 2020. Join, and help us!