Quantum Mechanics
sri
An (almost) self-contained introduction to the basic ideas of quantum mechanics. The theory and important experimental results will be discussed.
 And so he studied physics and and thinking we just all need a lot better understanding of quantum mechanics because he sees the theory theory being misused a lot by some weird esoteric theories kind of abusing it to just justify everything and anything so he wants to change that and he wants to have people with some understanding of this very important theory and so he will start today with all of us here and try to explain to us the wonders of quantum mechanics. Have a go. Well thank you for the warm welcome it will be about quantum mechanics we will see whether the gentleman introduction will be a lie depending on how good you can follow me so at first there will be a short introduction a bit meta discussion about physical theories and what is the aim of this talk and then we will discuss the experiments most of this is high school physics you you've probably seen it before and then it will get ugly because we'll do the theory and we'll really do the theory we'll write down the equations of quantum mechanics and try to make them plausible and hopefully understandable to a lot of people and finally some applications will be discussed so what is the concept of this talk the key experiments will be reviewed as said and but we will not do it in historical fashion we'll look at the experiments as physical facts and derive the theory from them and since quantum mechanics is rather abstract and not as a said in German and in science theory unschaulich we will need mathematics and most of this will be linear algebra so a lot of quantum mechanics is just linear algebra and steroids that means in infinite dimensions and in doing so we'll try to find a certain post classical unschaulish kite or lividness to understand the theory since there'll be a lot of math as the allergy advice said there will be crash courses tuned in to explain mathematical facts sorry for the mathematicians that are here they probably suffer because I lie a lot so at first how do scientific theories work to understand quantum mechanics we must understand the setting and setting where it was created and how scientific theories are created in general a scientific theory is a net of interdependent propositions so we have one proposition for example f equals m m times a in classical mechanics and we have another proposition that the gravitational force equals is proportional to the product of the masses divided by the distance between the masses squared so something like this and when we go around make experiments look into nature develop theories calculate we test those we test hypotheses different hypotheses and try to determine which one describes our experimental results best and if the hypothesis stand the experimental tests they're added to the theory but what happens if there's a experimental result that totally contradicts what we've seen before and that happened in the late 19th and early 20th century there were new results that could not be explained so such inconsistent results are for I found then our old theory has been falsified this is this term is due to popper who said that theory is scientific as long as it can be falsified that is as long as we can prove that it's not true and we can never prove a theory true but only approve it wrong and all that we have not yet proven wrong at least some approximation to truth and if this happens we have to amend our old theory and we have to use care there and find a minimal amendment this principle as Occam's razor or one could also say the principle of least surprise from software engineering and then we try that our theories again consistent with the experimental results and of course the new theory must explain why the hell that for example Newtonian mechanics work for 200 years if it's absolutely wrong and so the old theory must in some limit contain the new one and now how does it begin with quantum mechanics as already said the time frame is the late 19th and early 20th century and there were three or four fundamental theories of physics known then classical mechanics which is just governed by the single equation the force equals the mass times the acceleration was given forces and two known force laws the immediately the immediate distance action Newtonian gravitation and the Maxwell electrodynamics this funny equation here whoop this funny equation here's a way of writing down the Maxwell equations that basically contain all the known electromagnetic effects and finally there was a word of beginnings of the Maxwell Boltzmann statistical physics but classical statistical physics is is a pain doesn't really work so several experimental results as said could not be explained by classical theories for example the photoelectric effect the discovered by Hudson Talvex in 1887 or the discrete spectral lines of atoms first shown by Fraunhofer and the spectrum of the Sun and then and then studied by Bunsen and Kirchhoff with the so-called Bunsen Brenner you all know it from your chemistry classes and further radioactive rays were really a mystery nobody understood how can it happen that's something just decays at random intervals it was unclear and then the people looked into the atom rather forward using alpha particles to bombard the gold foil and saw there must be positively charged nuclear and they already knew that they were negatively charged what we now call electrons particles in the atom so this and this was really strange that atoms are stable at compost like this and I will explain why a bit later but now to more detail to the experiments the really big breakthrough in this time experimentally speaking were vacuum tubes so you took a piece of glass and pumped the air out and closed it off and put all all sorts of devices in there and now one thing is this nice cathodray experiment we have here a so-called electron gun and this is a heated electrode so he flows the current that heats it so that the electrons get energy and seep out into the vacuum then we have an electrode that goes around and the plate in front that is positively charged so we accelerate our electrons towards the plate there's a pinhole on the plate and we get a beam of electrons and now we had those evacuated two tubes and those electron guns so we put a electron gun in the evacuated tube perhaps left a bit of gas in because then it glowed when it was when the atoms and the gas were hit by the electron so we could see the cathodray and then we play around we take magnetic fields and see how does it react to magnetic fields we take electric fields how does it react to electric fields and so on and what we find out is we somehow must have negatively charged particles that flow nicely around in our almost vacuum and because atoms are neutral which is just known macroscopically there must be a positively charged component in the atom as well and this positively charged component was first thought to be kind of a plump pudding or so with the electrons sitting in there but the rather third geiger experiment so it was a rather thought invented the idea and must and geiger actually performed the experimental work showed that if you have a really thin gold foil really only a few hundred layers of atoms that's nothing about gold you can just hammer it out to really really thin sheets if you had that and then shot alpha particles that is helium not nuclear that are created by the radioactive decay of many heavy elements for example most uranium isotopes decay by alpha decay then they were deflected strongly if the charge would have been spaced throughout the atoms then this could not have happened you can calculate you can you can estimate the possible deflections with an extended charge and with a concentrated charge and you see the only explanation for this is that there is a massive and really really small positive thing in those atoms so atoms are small positively charged nucleus as rather foot called it and around it there's a there's a cloud of electrons or he thought orbiting electrons but orbiting electrons atoms are stable this doesn't really make sense in classical physics because in classical physics all accelerated charges must radiate energy and be slowed by this process and this means atoms that are stable and composed of some strange electrons and heavy nuclei they're just not possible it's it's a no-go so at least at this moment it was completely clear classical physics as they knew it up until then is wrong and a next experiment in this direction was the photoelectric effect what's shown there is a schematic of a photo tube and the photo tube is again a vacuum tube out of glass and there's a for example a caesium layer in in the tube at one side and there's a ring electrode removed from it and if we shine light on this they close the current but the peculiar thing is that if we use a bias a bias voltage across the two terminals of this tube to stop the electrons we see that the bias voltage that completely stops the flow is not proportional to the intensity of the light that is incident onto the tube but it's proportional to the frequency of the light that's incident on the photo tube and that was again really weird for the people of the time because the frequency shouldn't shouldn't make any difference for the energy and this was when Einstein derived the thought of that there must be some kind of energy portions in the electric field from this simple experiment which is often done in physics classes even even at the high school level so it's from out today's view it's not a complicated experiment and to go even further those weird stable atoms had discrete had discrete lines of emission and absorption of light and here we have again a very simplified experimental setup of a so-called discharge tube where we have high voltage between the terminals and the thin gas and then a current will flow will excite the atoms the atoms will relax and emit light and this light will have a specific spectrum with sharp frequencies that are that have strong emission and we can see this with a diffraction creating that sorts the light out according to its wavelength and then look on the screen or use some more fancy optical instrument to do precision measurements as a bundlen köche oft did so what we knew up up until now was that something was really weird and our physical theories didn't make sense and then it got worse someone took an electron gun and pointed it at the at the monocrystalline surface and such a monocrystalline surface is just like a diffraction creating grating a periodically arranged thing and of periodically arranged things there does happen regular interference pattern creation so they saw interference pattern with electrons but electrons are the particles how can particles so what was thought of then since the times of Newton as a little heart ball how can a little heart ball flowing around create interference patterns it was really weird and there's even more and as already mentioned radioactivity with the random decay of nucleus this doesn't make sense in classical physics so it was it was really really bad and here I've added some modern facts that we'll need later on namely that that if we measure but if we try to measure the position of a particle and use different position position sensors to do so only one of them so at only at one position will the single particle register but it will nevertheless show an interference pattern if I do this experiment with many many electrons so there must somehow be a strange divide between the free space propagation of particles and measuring the particles and you can do really weird stuff and record the information through which slit the particle went and if you do this the interference pattern vanishes and then you can even destroy this information in a coherent manner and the coherence and the interference pattern appears again so what we know up until now is that quantum mechanics is really really weird and really different from classical mechanics and now that we've talked about those experiments we'll begin with the theory and the theory will begin with a lot of mathematics the first one is simple complex numbers okay who who doesn't who doesn't know complex numbers okay sorry I'll have to ignore you for the sake of getting to the next points so I'll just say complex numbers are two components of real at two component objects with real numbers and one of them is multiplied by an imaginary number I and if we square the number I it gets minus one and this makes many things really beautiful for example all algebraic equations have exactly the number of degree solutions in complex numbers and if you count them correctly and if you work with complex functions it's really beautiful a function that once differentiable is infinitely many times differentiable and it's it's nice so now we had complex numbers you've all said you know them so we go on to vector spaces which probably also a lot of you know just to revisit it a vector space is a space of objects called vectors above some scalars that must be a field and here we only use complex numbers as the underlying fields there's a null vector we can add to the vectors we can invert vectors and we can multiply vectors by real numbers so we can say three that five times this vector and just scale the the arrow and the these operations interact nicely so that we have the those distributive laws and now it gets interesting even more maths L2 spaces L2 spaces are in a way a infinite dimensional or one form of an infinite dimensional extension of vector spaces instead of having just three directions xyz we have directions at each point of a function so we have an analogy here we have vectors which have three discrete components given by x index i on the right side and we have this function and we have this function and each component is the value of the function at one point along the x is x and then we can just as for vectors define a norm on those L2 functions which is just the integral over the absolute value squared of this function f and a nice thing about this choice of a norm there are other choices of the norm this norm is induced by a scalar product and this little asterisk for the that is there at the f denotes the complex conjugate so flipping i to minus i in all complex values and if you just if you just plug in f and f into the scalar product you will see that it's in the integral over the squared absolute value and this space this L2 space is a Hilbert space and the Hilbert space is a complete space vector space with a scalar product where complete means that it's mathematical nonsense forget it so but the nice surprise is that most things carry over from finite dimensional spaces what we know from finite dimensional spaces we can always diagonalize matrices with certain properties and this all more or less works at the mathematicians really really really do a lot of work for this but for physicists we just know when to be careful and how and don't care about it otherwise so it just works for us and that's nice and now that we have those complex numbers we can begin to discuss how particles are modeled in quantum mechanics and as we know from the Davis and German experiments there's diffraction of electrons but there's nothing in electrons that corresponds to a positive electric field in some direction or so some other periodicity has so periodicity of electrons during propagation has never been directly observed and the broi said particles have a wavelength that's related to the momentum and he was motivated primarily by the Bohr theory of the atom to do so and he was shown right by the Davis and German experiments so his relation for the wavelength of a particles older than the experiments showing this which is impressive I think and now the idea is to have a complex wave function and let the squared absolute value of the wave function describe the probability density of a particle so we make particles extended but probability measured object so there isn't no longer the position of the particle as long as we don't measure but we have just some description of a probability where the particle is and by making it complex we have a face and this face can allow still allow interference effects which we need for explaining the interference peaks in the Davis and German experiment and now a lot of textbooks say here there's a wave particle you listen blah blah blah distinct nonsense blah the point is it doesn't get you far to think about quantum objects as either wave or particle they're just quantum neither wave nor particle doesn't help you either but it doesn't confuse you as much as when you try to think about particles as waves or particles about quantum particles as ways of particles and now that we say we have a complex wave function what about simply using a plane wave with constant probability as the states of definite momentum because we somehow have to describe a particle that is that that has a certain momentum and we do this those have the little problem that they are not in the Hilbert space because they're not normalizable the absolute value of psi is 1 1 over 2 pi everywhere so that's bad but we can write the superposition of any state by Fourier transformation those e to the I k dot r states are just the basis states of Fourier transformation we can write any function in terms of this basis and we can conclude that by Fourier transformation of the state psi of r to some state till the psi of k we describe the same information because we know we can invert the Fourier transformation and also this implies the uncertainty relation and because this is simply property of Fourier transformations that either the either the function can be very concentrated in position space or in momentum space and now that we have states of definite momentum and the other big ingredient in quantum mechanics are operators next to the state description and operators are just like matrices linear operators on the state space just as we can apply a matrilinear operator in the form of a matrix to a vector we can apply linear operators to two functions and when we measure an observable it will be that it's one of the eigenvalues of this operator that's the measurement value you know so eigenvalues are those values where a matrix does just if matrix just scales a vector by a certain amount that is an eigenvector of the eigenvalue of the matrix and in the same sense we can define eigenvalues and eigenvectors for two functions and there are some facts such as that non-commuting operators have eigenstates that are not common so we can't have a description of the of all the of the basis of the state space in terms of functions that are both eigen functions of both operators and some examples of operators are the momentum operator which is just minus i h bar napler which is the derivation operator in three dimensions so in the x component we have derivation in the direction of x and the y component in the direction of y and so on and the position operator which is just the operator that multiplies by the position x in the position space representation of the wave function and as for the non- commutativity of operators we can already show that those p and x operators that do not commute but fulfill a certain commutation relation and the commutation relation is just a measure for how much two operators do not commute and the commute commutators a b minus b a for the objects a b so if they commute if a b equals b a the commutator simply vanishes and there's more operators just to make it clear linear just means that we can split the argument if it's a sound if it's just linear just some linear combinations of vectors and apply the operator to the individual vectors occurring we can define multiplication of operators and this just exactly follows the template that is laid down by finite-dimensional linear algebra there's not there's nothing new here and they're inverse operators for some operators not for all of them that give the identity operator if it's multiplied with the original operator and further there's the so-called a joint our scalar product that had this little asterisk and this means that it's not linear in the first component if I scale the first component by some complex number alpha the total scalar product is not scaled by alpha but by the complex conjugate of alpha this kind of not quite by linearity is called sometimes called sask wheel in year sask wheel in the air at you a seldom used word and they are commonly defined classes of operators in terms of how the how the adjoint that is defined their acts and how some their operators for example where the adjoint is the inverse which is a generalization from the fact that for rotation operators in normal Euclidean space the transpose is the inverse and now that we have operators we can define expectation values just by some formula for now we we don't know what those expectation values are but we can assume it has something to do with the measurement values of the operator because why else would I tell you about it and later on we will show that this is actually the expectation value of the quantity if we prepare a system always in the same fashion and then do measurements on it we get random results each time but the expectation value will be this combination and now again a bit of mathematics eigenvalue problems well known you can diagonalize the matrix and you can diagonalize linear operators you have some equation a psi equals lambda psi where lambda is just a scalar so and if such an equation holds for some vector psi then it's an eigenvector and any linear scale if we scale the vector linearly this will again be an eigenvector and what can happen is that to one eigen eigenvalue there are several eigenvectors not only one ray of eigenvectors but a higher-dimensional subspace and important to know is that so-called Hermitian operators that is those that equal their joint which again means that the eigenvalues equal the complex conjugate of the eigenvalues have real eigenvalues because of a complex number equals its complex conjugate then it's a real number and the nice thing about those diagonalized matrices and all is we can develop any any vector in terms of the eigen basis of the operator again just like in linear algebra where when you diagonalize a matrix you get a new basis for your vector space and now you can express all vectors in that new basis and if the operator is a mission the eigenvectors have a nice property namely they are orthogonal if the eigenvalues are different and this is good because because this guarantees us that we can choose a orthonormal that is a basis in the vector space where two basis vectors always have vanishing scalar product are orthogonal and are normal that is we scale them to length one because we want our probability interpretation and in our probability interpretation we need to have normalized vectors so now we have that and now we want to know how does this strange function psi that describes the state of the system evolve in time and for this we can have several requirements that it must fulfill so again we're close to software engineering and one requirement is there's that if it's a sharp wave packet so if we have a localized state that is not smeared around the whole space then it should follow the classical equation of motion because we want that our new theory contains our old theory and the time evolution must conserve the total probability of finding the particle because otherwise we couldn't do probability interpretation of our wave function if the total probability of the particle wouldn't remain one further we wish the equation to be first order in time and to be linear because for example the Maxwell equations are linear and show nice show nice interference effects so we want that because then simply some of solutions is again a solution it's it's a good property to have and if it works that way why not and the third and the fourth requirement together already give us more or less the form of the Schrödinger equation because linearity just says that the right-hand side is some linear operator applied to psi and the time first order in time just means that there must be a time derivative a single time derivative in the equation on the left-hand side and this IH bar is just and we just want the debt there no particular reason we could have done this differently but it's convention now with this equation we can look what must happen for the probability to be conserved and by a simple calculation we can show that it must be her mission operator and there's even more than this global argument there's local conservation of probability that is a particle can simply vanish here and appear there but it must flow from one point to the other with local operations this can be shown when you consider this in more detail now we know how this equation of motion looks like but we don't know what this mysterious object H might be and this mysterious object H is the operator of the energy of the system which is from known from classical mechanics as the Hamilton function and which we here upgrade to a Hamilton operator by using the formula for the classical Hamilton function and inserting our P and R operators and we can also extend this to a magnetic field and by doing so we can show that our theory is more or less consistent with Newtonian mechanics we can show the Irnfest theorem that's the first equation and then those equations are almost Newton's equation of motion for the centers of mass of the particle because this is the expectation value of the momentum this is the expectation value of the position of the particle this just looks exactly like the classical equation the velocity is the momentum divided by the mass but this is weird here we ever we average over the force so the gradient of the potential is the force we average over the force and do not take the force at the center position so we can't in general solve this equation but again if we have a sharply defined wave packet we recover the classical equations of motion which is nice so we have shown our new theory does indeed explain why our old theory worked we only still have to explain why the centers of mass of massive particles are usually well localized and that's the question we're still having trouble with today but since the since it otherwise works don't worry too much about it and now you probably want to know how to solve the Schrödinger equation or you don't want to know anything more about quantum mechanics and to do this we make a so called separation ansatz where we say we have a form stable we have a form stable part of our wave function multiplied by some time-dependent part and if we do this we can write down the general solution for the Schrödinger equation because we already know that the one equation that we get is the is an eigenvalue equation or an eigenvector equation for the energy eigenvalues that is the eigenvalues of the Hamilton operator and we know that we can develop any function in terms of those and so the general solution must be of the form shown here and those states of this of specific energy have a simple evolution because their form is constant and only their face changes and depends on the energy and now this thing with the measurement in quantum mechanics is bad you probably know Schrödinger's cat and the point is that you don't know whether the cat's dead or alive while you don't look inside the box as long as you don't measure it's in a superposition or something so you measure you measure your cat and then it's dead it isn't dead before only by measuring it you kill it and that's really not nice to kill cats we like cats so and the the important part here is the TLDR quantum measurement is probabilistic and inherently changes the system state so I'll skip the multi particle things we can describe multiple particles and just show the axioms of quantum mechanics shortly don't read them too detailed but this is just a summary of what we've discussed so far and the the thing about the multiple particles is the axiom 7 which says that the sign of the wave function must change if we exchange the coordinates of identical fermions and this makes atoms stable by the way without this atoms as we know them would not exist and finally there's a notational convention in quantum mechanics called bracket notation and in bracket notation you label states by their eigenvalues and just think about such a cat as an abstract vector such as x with an with a vector arrow over it or a fat set x is an abstract vector and we can either represent it by its coordinates x1 x2 x3 or we can work with the abstract vector and this cat is such an abstract vector for the l2 function psi of r and then we can also define the adjoint of this which gives us if we multiply the joint at the function the scalar product so this is a really nice and compact notation for many physics problems and the last equation there just looks like component wise like working with components of matrices which is because it's it's nothing else this is just matrix calculus in a and a new and new clothes now for the applications the first one is quite funny oh there's a slight missing okay the first one is a quantum eraser at home because if you encode the which way information in to a double slit experiment you lose your you lose your interference pattern and we do this by using a vertical and a horizontal polar polarization filter and you know from classical physics then it won't it won't make a it won't make an interference pattern and if we add then at the diagonal polarization filter then the interference pattern will appear again so now just so you've seen it the harmonic oscillator can be exactly solved in quantum mechanics if you can solve the harmonic oscillator in any kind of physics then you're good then you'll get through the exams in the physics when you study physics so the harmonic oscillators is solved by introducing so-called destroyers and creation and destroyer operators and then we can determine the ground state function with the size in a much simpler manner than if we had to solve the Schrödinger equation explicitly for all those cases and we can we can determine the ground state the ground state function so the function of lowest energy this can all be done and then can from it by applying the creation operator create the higher the higher state eigenstates of the system and get all of them then there's this effect of tunneling that you've probably heard about and this just means that in quantum mechanics a potential barrier that is too high for the particle to penetrate does not mean the particle doesn't penetrate at all but just that the the probability of finding the particle inside a barrier decays exponentially and this can for example be understood in terms of this uncertainty relation because if we try to compress the particle to a smaller part of the boundary layer then its momentum has to be high so it can reach farther in because then it has more energy and there's this myth that tunneling makes particles travel to travel instantaneously from A to B and even some real physicists believe it but sorry it's not true the particle status extended anyway and to defining what how fast the particle travels is actually not a well-defined thing and deep quantum regimes and also the Schrödinger equation is not relativistic so there's nothing nothing really nothing stopping your particle from flying around with 30 times the speed of light it's just not in the theory another important consequence of quantum mechanics is so-called entanglement and this is a really weird one because it shows that our the universe that we live in is in a way non-local inherently non-local because we can create some state of if you can create a state for some internal degrees of freedom of two atoms and move them apart then measure the one system and the measurement result and the one system will determine the measurement result and the other system no matter how far removed they are from each other and this was first discovered in a paper by Einstein Podolsky and Rosen and they thought it was an argument that quantum mechanics is absurd this can't be true but sorry it is true so this works and this kind of state that we've written there that is such an entangled state of two particles but important to remark is that there are no hidden variables that means the measurement result is not determined beforehand it is only when we measure that is actually known what the result will be this is utterly weird but one can prove this experimentally is those are belt tests there's a bell inequality that's a limit for theories where they are hidden variables and it's by real experiments they violate this inequality and they may show that there are no hidden variables and there's a myth surrounding entanglement namely that you can transfer information with it between two sides instantaneously but again there's nothing hindering you in non relativistic quantum mechanics to distribute information arbitrarily fast it doesn't have a speed limit but you can also can't communicate with those entangled pairs of particles you can just create correlated noise at two ends which is what quantum cryptography is using so now because this is the hackers congress some short remarks and probably in intelligible due to their strong compression about quantum information a qubit the fundamental unit of quantum information is a system with two states zero and one so just like a bit but now we allow arbitrary superposition so arbitrary super positions of those states because that is what quantum mechanics allows we can always super imposed states and quantum computers are really bad for most computing tasks because they they have to have even if they built quantum computers they'll never be as capable as the state-of-the-art silicon electrical computers so don't fear for your jobs because of quantum computers but the problem is they can compute some things faster for example factoring primes and working with some elliptic curve algorithms and so on and determining discrete logarithms so our public key crypto would be destroyed by them and this all works by using the superposition to construct some kind of weird parallelism so it's actually I think nobody really can imagine how it works but we can compute it which is often the case in quantum mechanics and then there's quantum cryptography and that's fundamentally solve the same problem as a Diffie Helman key exchange we can generate the shared key and we can check by the statistics of our measured values that there was no even dropper which is cool actually but it's also quite useless because we can't detect them and in the middle how should the quantum particle knows of the other side is the one we that we want to talk to we still need some shared secret or public key infrastructure whatever so it doesn't solve a problem that we don't have solved and then the fun fact about this is that all the commercial implementations of quantum cryptography were susceptible to side channel text for example you could just shine with the light and the fiber that was used read out the polarization filter state that they used and then you could you could mimic the other side so that's not good either so finally some references for further study the first one is really difficult only try this a few for at the other two but the second one sorry that they're in German the first and the last are also available in translation but the second one has a really really nice and accessible introduction and the first in the last few pages so it's just 20 pages and it's really good and understandable so we can you get your hands you can get your hands on the books and are really interested read it so thank you for your attention and I'll be answering your questions next thank you Sebastian do we have questions and don't be afraid to sound naive or anything I'm sure if you didn't understand something many other people thank you for a good question as to understanding things in quantum mechanics Feynman said you can't understand quantum mechanics you can just accept it there's nothing to understand there it's just too weird okay we found some questions so microphone one please measure a system it looks like you change the state of the system how is it defined where the system starts or no what how is it defined by the system ends and the measurement system begins or in other words why does the universe have a state is there somewhere out there who measures the universe no there's a at least the beginning of a solution by now which is called decoherence and which says that that this measurement structure that we observe is not inherent in quantum mechanics but comes from the interaction with the environment and we don't care for the states of the environment and if we do this the technical term is trace out the states of the environment then the remaining state of the measurement apparatus and the system we're interested in will be just classically randomized states so it's it's rather a consequence of the complex dynamics of a system state and environment in quantum mechanics but this this is really the burning question we we don't really know we have this we know the coherence makes some makes it nice and looks good but it also doesn't answer the question finally and this is what all those discussions about interpretations of quantum mechanics are about how shall we make sense of this weird measurement process okay microphone for in the back please could you comment on your point in the theory section I'm I don't understand what you were trying to do did you want to show that you cannot understand really quantum mechanics without the mathematics or well yes you can't understand quantum mechanics without the mathematics and my point to show was that the make matter or at least my hope to show was that the mathematics is half ways accessible so probably not understandable after just exposure of a short talk but just to give an introduction where to look okay so you were trying to combat esoterics to and and say they they don't really understand the theory because they don't understand the mathematics I understand the mathematics I'm just interested what what you you were trying to say I was just trying to present the theory that was my aim okay thank you okay microphone to please so I know the answer to this question is that and you a little bit closer to the microphone maybe move it up so I know the answer to this question is that atoms behave randomly but could you provide an argument why they behave randomly and it is not the case that we don't have a model that's so our atoms behaving randomly or is it the case that we don't have a model accurate enough to predict the way they behave radioactive decay is just as random as quantum measurement and since it since if if we were to look at the whole story and look at the coherent evolution of the whole system we would have to include the environment and the problem is that the state space that we have to consider gross exponentially that's the point of quantum mechanics if I have two particles I have a two-dimensional space of a ten particles I have a 1024 dimensional space and that's only talking about non-interacting particles so things explode in quantum mechanics and large systems and therefore I would go so far as to say that it's objectively impossible to determine radioactivity k although there are things that is I think one experimentally confirmed method of letting an atom decay on purpose that this involves a meter stable states of nuclei and then you can do something like spontaneous emission in a laser you shine a strong gamma source by it and this shortens the lifespan of the nucleus but other than that completely hypothetical case if you know all the the starting conditions and what happens afterwards you would be able to say it's deterministic I mean I know I'm playing with with heavy words here but is it is just that we we say it's random because it's very very complex right that's what I'm understanding maybe think about that question one more time and we have the signal angel in between and then you can come back hi there's one question from the internet which is the ground state of BEH2 has been just calculated using a quantum eigen solver so is there still some use of quantum computing in quantum mechanics yes definitely one of the main motivations for inventing quantum computers was quantum simulators Feynman Feynman invented this kind of quantum computing and he showed that with digital quantum computer you can efficiently simulate quantum systems while you can't simulate quantum systems with a classical computer because of this problem of the exploding dimensions of the hibbert space that you have to consider and for this quantum computers are really really useful and will be used once they work which is a question when it will be perhaps never beyond two or three qubits or 20 or a hundred qubits but you need scalability for a real quantum computer but simu quantum simulation is a real thing and it's a good thing and we need it okay then we have microphone one again so very beginning you said that the theory is a set of interdependent propositions right and then if a new hypothesis is made it can be confirmed by an experiment it can't be confirmed but it well it's a it's a it's a philosophical question but the common stance that it can be made probable but not be confirmed because we can never absolutely be sure that there won't be some new experiment that shows that the hypothesis is wrong because this light said that the experiment confirms yeah confirm in the sense that it doesn't disconfirm it so it makes probable that it's a good explanation of the reality and that's the point physics is just models we don't we do get nothing about the ontology that is about the actual being of the world out of physics we just get models describe the world but all what I say about this wave function and what we say about elementary particles we can't say they are in the sense that you and I are here and exist because we can't see them we can't access them directly we can only use them as description tools but this is my personal position on philosophy of science so there are people who disagree microphone to please or maybe superposition by the way so on the matter of the collapsing of the wave function so this was already treated on the interpretation of Copenhagen and then as you mentioned it was expanded by the concept of decoherence and is this so the decoherence is including also the Girardirimini-Veba interpretation or not could decoherence be used in computation or no no so for the Girardirimini-Veba interpretation of the collapsing of the wave function that's that's one that I don't know okay I'm not so much into interpretations I actually think that this interesting work done there but I think they're a bit irrelevant because in the end what I just said I don't think we can derive ontological value from our physical theories and in this belief I think that the interpretations are in a sense void just they just help us to rationalize what we're doing but they don't really add something to the theory as long as they don't change what can be measured sorry for being an extremist totally fine someone just left from microphone one I don't know they want to come back I don't see any more questions as the signal angel have anything else ah there's some more signal angel do you have something no okay then we have microphone for I wanted to ask a maybe a noob question I want to know are there probabilities of quantum mechanics an inherent part of nature or maybe in some future we'll have a science that will determine all these values exactly well if the coherence theory is true then quantum mechanics is absolutely deterministic but so if or it's let's say if the ever it interpreted as so ever it says that all those possible measurement outcomes do happen and the whole state of the system is in a superposition and by looking at our measurement device and seeing some value we in a way select one strand of those superpositions and live in this of the many worlds and in this sense everything happens deterministically but we just can't access any other values so it's I think it's rather matter of for now rather matter of philosophy than of science I see thanks anything else I don't see any people lined up with microphones so last chance to run up now I think well then I think we're closing this and have a nice applause again for Sebastian thanks thank you and I hope I didn't make create more fear of quantum mechanics than I disqualified
 And so he studied physics and and thinking we just all need a lot better understanding  of quantum mechanics because he sees the theory theory being misused a lot by  some weird esoteric theories kind of abusing it to just justify everything  and anything so he wants to change that and he wants to have people with some  understanding of this very important theory and so he will start today with  all of us here and try to explain to us the wonders of quantum mechanics.  Have a go.  Well thank you for the warm welcome it will be about quantum mechanics we will  see whether the gentleman introduction will be a lie depending on how good you  can follow me so at first there will be a short introduction a bit meta discussion  about physical theories and what is the aim of this talk and then we will discuss  the experiments most of this is high school physics you you've probably seen  it before and then it will get ugly because we'll do the theory and we'll  really do the theory we'll write down the equations of quantum mechanics and try  to make them plausible and hopefully understandable to a lot of people and  finally some applications will be discussed so what is the concept of  this talk the key experiments will be reviewed as said and but we will not do  it in historical fashion we'll look at the experiments as physical facts and  derive the theory from them and since quantum mechanics is rather abstract and  not as a said in German and in science theory unschaulich we will need  mathematics and most of this will be linear algebra so a lot of quantum  mechanics is just linear algebra and steroids that means in infinite dimensions  and in doing so we'll try to find a certain post classical unschaulish  kite or lividness to understand the theory since there'll be a lot of math  as the allergy advice said there will be crash courses tuned in to explain  mathematical facts sorry for the mathematicians that are here they  probably suffer because I lie a lot so at first how do scientific theories work  to understand quantum mechanics we must understand the setting and setting  where it was created and how scientific theories are created in general a  scientific theory is a net of interdependent propositions so we have  one proposition for example f equals m m times a in classical mechanics and we  have another proposition that the gravitational force equals is  proportional to the product of the masses divided by the distance between  the masses squared so something like this and when we go around make  experiments look into nature develop theories calculate we test those we  test hypotheses different hypotheses and try to determine which one describes  our experimental results best and if the hypothesis stand the experimental tests  they're added to the theory but what happens if there's a experimental result  that totally contradicts what we've seen before and that happened in the late  19th and early 20th century there were new results that could not be explained  so such inconsistent results are for I found then our old theory has been  falsified this is this term is due to popper who said that theory is scientific  as long as it can be falsified that is as long as we can prove that it's not  true and we can never prove a theory true but only approve it wrong and all  that we have not yet proven wrong at least some approximation to truth and if  this happens we have to amend our old theory and we have to use care there and  find a minimal amendment this principle as Occam's razor or one could also say  the principle of least surprise from software engineering and then we try  that our theories again consistent with the experimental results and of course  the new theory must explain why the hell that for example Newtonian mechanics  work for 200 years if it's absolutely wrong and so the old theory must in some  limit contain the new one and now how does it begin with quantum mechanics as  already said the time frame is the late 19th and early 20th century and there  were three or four fundamental theories of physics known then classical  mechanics which is just governed by the single equation the force equals the  mass times the acceleration was given forces and two known force laws the  immediately the immediate distance action Newtonian gravitation and the  Maxwell electrodynamics this funny equation here whoop this funny equation  here's a way of writing down the Maxwell equations that basically contain all the  known electromagnetic effects and finally there was a word of beginnings of  the Maxwell Boltzmann statistical physics but classical statistical physics is  is a pain doesn't really work so several experimental results as said could not  be explained by classical theories for example the photoelectric effect the  discovered by Hudson Talvex in 1887 or the discrete spectral lines of atoms  first shown by Fraunhofer and the spectrum of the Sun and then and then  studied by Bunsen and Kirchhoff with the so-called Bunsen Brenner you all know it  from your chemistry classes and further radioactive rays were really a mystery  nobody understood how can it happen that's something just decays at random  intervals it was unclear and then the people looked into the atom rather  forward using alpha particles to bombard the gold foil and saw there must be  positively charged nuclear and they already knew that they were negatively  charged what we now call electrons particles in the atom so this and this  was really strange that atoms are stable at compost like this and I will  explain why a bit later but now to more detail to the experiments the really  big breakthrough in this time experimentally speaking were vacuum tubes so  you took a piece of glass and pumped the air out and closed it off and put all  all sorts of devices in there and now one thing is this nice cathodray  experiment we have here a so-called electron gun and this is a heated  electrode so he flows the current that heats it so that the electrons get  energy and seep out into the vacuum then we have an electrode that goes around  and the plate in front that is positively charged so we accelerate our  electrons towards the plate there's a pinhole on the plate and we get a beam  of electrons and now we had those evacuated two tubes and those electron  guns so we put a electron gun in the evacuated tube perhaps left a bit of  gas in because then it glowed when it was when the atoms and the gas were hit  by the electron so we could see the cathodray and then we play around we  take magnetic fields and see how does it react to magnetic fields we take  electric fields how does it react to electric fields and so on and what we  find out is we somehow must have negatively charged particles that flow  nicely around in our almost vacuum and because atoms are neutral which is just  known macroscopically there must be a positively charged component in the atom  as well and this positively charged component was first thought to be kind  of a plump pudding or so with the electrons sitting in there but the  rather third geiger experiment so it was a rather thought invented the idea and  must and geiger actually performed the experimental work showed that if you  have a really thin gold foil really only a few hundred layers of atoms that's  nothing about gold you can just hammer it out to really really thin sheets if  you had that and then shot alpha particles that is helium not nuclear  that are created by the radioactive decay of many heavy elements for example  most uranium isotopes decay by alpha decay then they were deflected strongly  if the charge would have been spaced throughout the atoms then this could not  have happened you can calculate you can you can estimate the possible  deflections with an extended charge and with a concentrated charge and you see  the only explanation for this is that there is a massive and really really  small positive thing in those atoms so atoms are small positively charged  nucleus as rather foot called it and around it there's a there's a cloud of  electrons or he thought orbiting electrons but orbiting electrons atoms  are stable this doesn't really make sense in classical physics because in  classical physics all accelerated charges must radiate energy and be slowed  by this process and this means atoms that are stable and composed of some  strange electrons and heavy nuclei they're just not possible it's it's a  no-go so at least at this moment it was completely clear classical physics as  they knew it up until then is wrong and a next experiment in this direction was  the photoelectric effect what's shown there is a schematic of a photo tube and  the photo tube is again a vacuum tube out of glass and there's a for example  a caesium layer in in the tube at one side and there's a ring electrode removed  from it and if we shine light on this they close the current but the peculiar  thing is that if we use a bias a bias voltage across the two terminals of this  tube to stop the electrons we see that the bias voltage that completely stops  the flow is not proportional to the intensity of the light that is incident  onto the tube but it's proportional to the frequency of the light that's  incident on the photo tube and that was again really weird for the people of the  time because the frequency shouldn't shouldn't make any difference for the  energy and this was when Einstein derived the thought of that there must be  some kind of energy portions in the electric field from this simple  experiment which is often done in physics classes even even at the high  school level so it's from out today's view it's not a complicated experiment and  to go even further those weird stable atoms had discrete had discrete lines  of emission and absorption of light and here we have again a very simplified  experimental setup of a so-called discharge tube where we have high voltage  between the terminals and the thin gas and then a current will flow will excite  the atoms the atoms will relax and emit light and this light will have a  specific spectrum with sharp frequencies that are that have strong  emission and we can see this with a diffraction creating that sorts the light  out according to its wavelength and then look on the screen or use some more  fancy optical instrument to do precision measurements as a  bundlen köche oft did so what we knew up up until now was that something was  really weird and our physical theories didn't make sense and then it got worse  someone took an electron gun and pointed it at the at the monocrystalline  surface and such a monocrystalline surface is just like a diffraction  creating grating a periodically arranged thing and of periodically arranged  things there does happen regular interference pattern creation so they  saw interference pattern with electrons but electrons are the particles how can  particles so what was thought of then since the times of Newton as a little  heart ball how can a little heart ball flowing around create interference  patterns it was really weird and there's even more and as already mentioned  radioactivity with the random decay of nucleus this doesn't make sense in  classical physics so it was it was really really bad and here I've added  some modern facts that we'll need later on namely that that if we measure but if  we try to measure the position of a particle and use different position  position sensors to do so only one of them so at only at one position will the  single particle register but it will nevertheless show an interference pattern  if I do this experiment with many many electrons so there must somehow be a  strange divide between the free space propagation of particles and measuring  the particles and you can do really weird stuff and record the information  through which slit the particle went and if you do this the interference pattern  vanishes and then you can even destroy this information in a coherent manner  and the coherence and the interference pattern appears again so what we know up  until now is that quantum mechanics is really really weird and really different  from classical mechanics and now that we've talked about those experiments  we'll begin with the theory and the theory will begin with a lot of  mathematics the first one is simple complex numbers okay who who doesn't who  doesn't know complex numbers okay sorry I'll have to ignore you for the sake of  getting to the next points so I'll just say complex numbers are two components  of real at two component objects with real numbers and one of them is  multiplied by an imaginary number I and if we square the number I it gets minus  one and this makes many things really beautiful for example all algebraic  equations have exactly the number of degree solutions in complex numbers and  if you count them correctly and if you work with complex functions it's really  beautiful a function that once differentiable is infinitely many times  differentiable and it's it's nice so now we had complex numbers you've all said  you know them so we go on to vector spaces which probably also a lot of you  know just to revisit it a vector space is a space of objects called vectors above  some scalars that must be a field and here we only use complex numbers as the  underlying fields there's a null vector we can add to the vectors we can invert  vectors and we can multiply vectors by real numbers so we can say three that  five times this vector and just scale the the arrow and the these operations  interact nicely so that we have the those distributive laws and now it gets  interesting even more maths L2 spaces L2 spaces are in a way a infinite  dimensional or one form of an infinite dimensional extension of vector spaces  instead of having just three directions xyz we have directions at each point of  a function so we have an analogy here we have vectors which have three discrete  components given by x index i on the right side and we have this function  and we have this function and each component is the value of the function  at one point along the x is x and then we can just as for vectors define a norm  on those L2 functions which is just the integral over the absolute value  squared of this function f and a nice thing about this choice of a norm  there are other choices of the norm this norm is induced by a scalar product and  this little asterisk for the that is there at the f denotes the complex  conjugate so flipping i to minus i in all complex values and if you just if you  just plug in f and f into the scalar product you will see that it's in the  integral over the squared absolute value and this space this L2 space is a  Hilbert space and the Hilbert space is a complete space vector space with a  scalar product where complete means that it's mathematical nonsense forget it so  but the nice surprise is that most things carry over from finite dimensional  spaces what we know from finite dimensional spaces we can always  diagonalize matrices with certain properties and this all more or less  works at the mathematicians really really really do a lot of work for this  but for physicists we just know when to be careful and how and don't care about  it otherwise so it just works for us and that's nice and now that we have those  complex numbers we can begin to discuss how particles are modeled in quantum  mechanics and as we know from the Davis and German experiments there's  diffraction of electrons but there's nothing in electrons that corresponds to  a positive electric field in some direction or so some other periodicity  has so periodicity of electrons during propagation has never been directly  observed and the broi said particles have a wavelength that's related to the  momentum and he was motivated primarily by the Bohr theory of the atom to do so  and he was shown right by the Davis and German experiments so his relation for  the wavelength of a particles older than the experiments showing this which is  impressive I think and now the idea is to have a complex wave function and let  the squared absolute value of the wave function describe the probability  density of a particle so we make particles extended but probability  measured object so there isn't no longer the position of the particle as long as  we don't measure but we have just some description of a probability where the  particle is and by making it complex we have a face and this face can allow  still allow interference effects which we need for explaining the interference  peaks in the Davis and German experiment and now a lot of textbooks say here  there's a wave particle you listen blah blah blah distinct nonsense blah the  point is it doesn't get you far to think about quantum objects as either wave or  particle they're just quantum neither wave nor particle doesn't help you either  but it doesn't confuse you as much as when you try to think about particles as  waves or particles about quantum particles as ways of particles and now that  we say we have a complex wave function what about simply using a plane wave  with constant probability as the states of definite momentum because we  somehow have to describe a particle that is that that has a certain momentum and  we do this those have the little problem that they are not in the Hilbert space  because they're not normalizable the absolute value of psi is 1 1 over 2 pi  everywhere so that's bad but we can write the superposition of any state by  Fourier transformation those e to the I k dot r states are just the basis states  of Fourier transformation we can write any function in terms of this basis and  we can conclude that by Fourier transformation of the state psi of r to  some state till the psi of k we describe the same information because we  know we can invert the Fourier transformation and also this implies  the uncertainty relation and because this is simply property of Fourier  transformations that either the either the function can be very concentrated in  position space or in momentum space and now that we have states of definite  momentum and the other big ingredient in quantum mechanics are operators next to  the state description and operators are just like matrices linear operators on  the state space just as we can apply a matrilinear operator in the form of a  matrix to a vector we can apply linear operators to two functions and when we  measure an observable it will be that it's one of the eigenvalues of this  operator that's the measurement value you know so eigenvalues are those values  where a matrix does just if matrix just scales a vector by a certain amount  that is an eigenvector of the eigenvalue of the matrix and in the same  sense we can define eigenvalues and eigenvectors for two functions and  there are some facts such as that non-commuting operators have eigenstates  that are not common so we can't have a description of the of all the of the  basis of the state space in terms of functions that are both eigen functions  of both operators and some examples of operators are the momentum operator  which is just minus i h bar napler which is the derivation operator in three  dimensions so in the x component we have derivation in the direction of x and  the y component in the direction of y and so on and the position operator  which is just the operator that multiplies by the position x in the  position space representation of the wave function and as for the non-  commutativity of operators we can already show that those p and x operators  that do not commute but fulfill a certain commutation relation and the  commutation relation is just a measure for how much two operators do not  commute and the commute commutators a b minus b a for the objects a b so if they  commute if a b equals b a the commutator simply vanishes and there's more  operators just to make it clear linear just means that we can split the  argument if it's a sound if it's just linear just some linear combinations of  vectors and apply the operator to the individual vectors occurring we can  define multiplication of operators and this just exactly follows the template  that is laid down by finite-dimensional linear algebra there's  not there's nothing new here and they're inverse operators for some operators  not for all of them that give the identity operator if it's multiplied  with the original operator and further there's the so-called a joint our  scalar product that had this little asterisk and this means that it's not  linear in the first component if I scale the first component by some complex  number alpha the total scalar product is not scaled by alpha but by the complex  conjugate of alpha this kind of not quite by linearity is called sometimes  called sask wheel in year sask wheel in the air at you a seldom used word and  they are commonly defined classes of operators in terms of how the how the  adjoint that is defined their acts and how some their operators for example  where the adjoint is the inverse which is a generalization from the fact that  for rotation operators in normal Euclidean space the transpose is the inverse  and now that we have operators we can define expectation values just by some  formula for now we we don't know what those expectation values are but we can  assume it has something to do with the measurement values of the operator  because why else would I tell you about it and later on we will show that this  is actually the expectation value of the quantity if we prepare a system always  in the same fashion and then do measurements on it we get random results  each time but the expectation value will be this combination and now again a bit  of mathematics eigenvalue problems well known you can diagonalize the  matrix and you can diagonalize linear operators you have some equation a psi  equals lambda psi where lambda is just a scalar so and if such an equation holds  for some vector psi then it's an eigenvector and any linear scale if we  scale the vector linearly this will again be an eigenvector and what can  happen is that to one eigen eigenvalue there are several eigenvectors not  only one ray of eigenvectors but a higher-dimensional subspace and important  to know is that so-called Hermitian operators that is those that equal their  joint which again means that the eigenvalues equal the complex conjugate  of the eigenvalues have real eigenvalues because of a complex number equals its  complex conjugate then it's a real number and the nice thing about those  diagonalized matrices and all is we can develop any any vector in terms of the  eigen basis of the operator again just like in linear algebra where when you  diagonalize a matrix you get a new basis for your vector space and now you can  express all vectors in that new basis and if the operator is a mission the  eigenvectors have a nice property namely they are orthogonal if the eigenvalues  are different and this is good because because this guarantees us that we can  choose a orthonormal that is a basis in the vector space where two basis vectors  always have vanishing scalar product are orthogonal and are normal that is we  scale them to length one because we want our probability interpretation and in  our probability interpretation we need to have normalized vectors so now we have  that and now we want to know how does this strange function psi that describes  the state of the system evolve in time and for this we can have several  requirements that it must fulfill so again we're close to software engineering  and one requirement is there's that if it's a sharp wave packet so if we have a  localized state that is not smeared around the whole space then it should  follow the classical equation of motion because we want that our new theory  contains our old theory and the time evolution must conserve the total  probability of finding the particle because otherwise we couldn't do  probability interpretation of our wave function if the total probability of the  particle wouldn't remain one further we wish the equation to be first order in  time and to be linear because for example the Maxwell equations are linear  and show nice show nice interference effects so we want that because then  simply some of solutions is again a solution it's it's a good property to  have and if it works that way why not and the third and the fourth requirement  together already give us more or less the form of the Schrödinger equation  because linearity just says that the right-hand side is some linear operator  applied to psi and the time first order in time just means that there must be a  time derivative a single time derivative in the equation on the left-hand side  and this IH bar is just and we just want the debt there no particular reason we  could have done this differently but it's convention now with this equation we  can look what must happen for the probability to be conserved and by a  simple calculation we can show that it must be her mission operator and there's  even more than this global argument there's local conservation of  probability that is a particle can simply vanish here and appear there but it  must flow from one point to the other with local operations this can be shown  when you consider this in more detail now we know how this equation of motion  looks like but we don't know what this mysterious object H might be and this  mysterious object H is the operator of the energy of the system which is from  known from classical mechanics as the Hamilton function and which we here  upgrade to a Hamilton operator by using the formula for the classical Hamilton  function and inserting our P and R operators and we can also extend this  to a magnetic field and by doing so we can show that our theory is more or less  consistent with Newtonian mechanics we can show the Irnfest theorem that's the  first equation and then those equations are almost Newton's equation of motion  for the centers of mass of the particle because this is the expectation  value of the momentum this is the expectation value of the position of  the particle this just looks exactly like the classical equation the velocity  is the momentum divided by the mass but this is weird here we ever we average  over the force so the gradient of the potential is the force we average over  the force and do not take the force at the center position so we can't in  general solve this equation but again if we have a sharply defined wave packet  we recover the classical equations of motion which is nice so we have shown  our new theory does indeed explain why our old theory worked we only still have  to explain why the centers of mass of massive particles are usually well  localized and that's the question we're still having trouble with today but  since the since it otherwise works don't worry too much about it and now you  probably want to know how to solve the Schrödinger equation or you don't want  to know anything more about quantum mechanics and to do this we make a so  called separation ansatz where we say we have a form stable we have a form  stable part of our wave function multiplied by some time-dependent part  and if we do this we can write down the general solution for the Schrödinger  equation because we already know that the one equation that we get is the is an  eigenvalue equation or an eigenvector equation for the energy  eigenvalues that is the eigenvalues of the Hamilton operator and we know that  we can develop any function in terms of those and so the general solution must  be of the form shown here and those states of this of specific energy have a  simple evolution because their form is constant and only their face changes  and depends on the energy and now this thing with the measurement in quantum  mechanics is bad you probably know Schrödinger's cat and the point is that  you don't know whether the cat's dead or alive while you don't look inside the  box as long as you don't measure it's in a superposition or something so you  measure you measure your cat and then it's dead it isn't dead before only by  measuring it you kill it and that's really not nice to kill cats we like cats  so and the the important part here is the TLDR quantum measurement is  probabilistic and inherently changes the system state so I'll skip the  multi particle things we can describe multiple particles and just show the  axioms of quantum mechanics shortly don't read them too detailed but this is  just a summary of what we've discussed so far and the the thing about the  multiple particles is the axiom 7 which says that the sign of the wave function  must change if we exchange the coordinates of identical fermions and  this makes atoms stable by the way without this atoms as we know them would  not exist and finally there's a notational convention in quantum  mechanics called bracket notation and in bracket notation you label states by  their eigenvalues and just think about such a cat as an abstract vector such  as x with an with a vector arrow over it or a fat set x is an abstract vector and  we can either represent it by its coordinates x1 x2 x3 or we can work  with the abstract vector and this cat is such an abstract vector for the l2  function psi of r and then we can also define the adjoint of this which gives  us if we multiply the joint at the function the scalar product so this is  a really nice and compact notation for many physics problems and the last  equation there just looks like component wise like working with  components of matrices which is because it's it's nothing else this is just  matrix calculus in a and a new and new clothes now for the applications the  first one is quite funny oh there's a slight missing okay the first one is a  quantum eraser at home because if you encode the which way information in  to a double slit experiment you lose your you lose your interference pattern  and we do this by using a vertical and a horizontal polar polarization filter and  you know from classical physics then it won't it won't make a it won't make an  interference pattern and if we add then at the diagonal polarization filter then  the interference pattern will appear again so now just so you've seen it the  harmonic oscillator can be exactly solved in quantum mechanics if you can  solve the harmonic oscillator in any kind of physics then you're good then  you'll get through the exams in the physics when you study physics so the  harmonic oscillators is solved by introducing so-called destroyers and  creation and destroyer operators and then we can determine the ground state  function with the size in a much simpler manner than if we had to solve the  Schrödinger equation explicitly for all those cases and we can we can determine  the ground state the ground state function so the function of lowest  energy this can all be done and then can from it by applying the creation  operator create the higher the higher state eigenstates of the system and get  all of them then there's this effect of tunneling that you've probably heard  about and this just means that in quantum mechanics a potential barrier that  is too high for the particle to penetrate does not mean the particle  doesn't penetrate at all but just that the the probability of finding the  particle inside a barrier decays exponentially and this can for example  be understood in terms of this uncertainty relation because if we try  to compress the particle to a smaller part of the boundary layer then its  momentum has to be high so it can reach farther in because then it has more  energy and there's this myth that tunneling makes particles travel to  travel instantaneously from A to B and even some real physicists believe it  but sorry it's not true the particle status extended anyway and to defining  what how fast the particle travels is actually not a well-defined thing and  deep quantum regimes and also the Schrödinger equation is not relativistic  so there's nothing nothing really nothing stopping your particle from flying  around with 30 times the speed of light it's just not in the theory another  important consequence of quantum mechanics is so-called entanglement and  this is a really weird one because it shows that our the universe that we live  in is in a way non-local inherently non-local because we can create some  state of if you can create a state for some internal degrees of freedom of two  atoms and move them apart then measure the one system and the measurement result  and the one system will determine the measurement result and the other system  no matter how far removed they are from each other and this was first discovered  in a paper by Einstein Podolsky and Rosen and they thought it was an argument  that quantum mechanics is absurd this can't be true but sorry it is true so  this works and this kind of state that we've written there that is such an  entangled state of two particles but important to remark is that there are  no hidden variables that means the measurement result is not determined  beforehand it is only when we measure that is actually known what the result  will be this is utterly weird but one can prove this experimentally is those are  belt tests there's a bell inequality that's a limit for theories where they  are hidden variables and it's by real experiments they violate this inequality  and they may show that there are no hidden variables and there's a myth  surrounding entanglement namely that you can transfer information with it  between two sides instantaneously but again there's nothing hindering you in  non relativistic quantum mechanics to distribute information arbitrarily fast  it doesn't have a speed limit but you can also can't communicate with those  entangled pairs of particles you can just create correlated noise at two ends  which is what quantum cryptography is using so now because this is the hackers  congress some short remarks and probably in intelligible due to their strong  compression about quantum information a qubit the fundamental unit of quantum  information is a system with two states zero and one so just like a bit but now  we allow arbitrary superposition so arbitrary super positions of those  states because that is what quantum mechanics allows we can always super  imposed states and quantum computers are really bad for most computing tasks  because they they have to have even if they built quantum computers they'll  never be as capable as the state-of-the-art silicon electrical computers so  don't fear for your jobs because of quantum computers but the problem is  they can compute some things faster for example factoring primes and working  with some elliptic curve algorithms and so on and determining discrete logarithms  so our public key crypto would be destroyed by them and this all works by  using the superposition to construct some kind of weird parallelism so it's  actually I think nobody really can imagine how it works but we can compute  it which is often the case in quantum mechanics and then there's quantum  cryptography and that's fundamentally solve the same problem as a Diffie  Helman key exchange we can generate the shared key and we can check by the  statistics of our measured values that there was no even dropper which is cool  actually but it's also quite useless because we can't detect them and in the  middle how should the quantum particle knows of the other side is the one we  that we want to talk to we still need some shared secret or public key  infrastructure whatever so it doesn't solve a problem that we don't have  solved and then the fun fact about this is that all the commercial  implementations of quantum cryptography were susceptible to side channel text  for example you could just shine with the light and the fiber that was used  read out the polarization filter state that they used and then you could you  could mimic the other side so that's not good either so finally some references  for further study the first one is really difficult only try this a few  for at the other two but the second one sorry that they're in German the first  and the last are also available in translation but the second one has a  really really nice and accessible introduction and the first in the last  few pages so it's just 20 pages and it's really good and understandable so we  can you get your hands you can get your hands on the books and are really  interested read it so thank you for your attention and I'll be answering your  questions next  thank you Sebastian do we have questions and don't be afraid to sound naive or  anything I'm sure if you didn't understand something many other people  thank you for a good question as to understanding things in quantum mechanics  Feynman said you can't understand quantum mechanics you can just accept it  there's nothing to understand there it's just too weird okay we found some  questions so microphone one please  measure a system it looks like you change the state of the system how is it  defined where the system starts or no what how is it defined by the system  ends and the measurement system begins or in other words why does the universe  have a state is there somewhere out there who measures the universe no there's a  at least the beginning of a solution by now which is called decoherence and  which says that that this measurement structure that we observe is not  inherent in quantum mechanics but comes from the interaction with the  environment and we don't care for the states of the environment and if we do  this the technical term is trace out the states of the environment then the  remaining state of the measurement apparatus and the system we're interested  in will be just classically randomized states so it's it's rather a consequence  of the complex dynamics of a system state and environment in quantum mechanics  but this this is really the burning question we we don't really know we  have this we know the coherence makes some makes it nice and looks good but it  also doesn't answer the question finally and this is what all those discussions  about interpretations of quantum mechanics are about how shall we make  sense of this weird measurement process okay microphone for in the back please  could you comment on your point in the theory section I'm I don't understand  what you were trying to do did you want to show that you cannot understand  really quantum mechanics without the mathematics or well yes you can't  understand quantum mechanics without the mathematics and my point to show was  that the make matter or at least my hope to show was that the mathematics is  half ways accessible so probably not understandable after just exposure of a  short talk but just to give an introduction where to look okay so you  were trying to combat esoterics to and and say they they don't really understand  the theory because they don't understand the mathematics I understand the  mathematics I'm just interested what what you you were trying to say I was just  trying to present the theory that was my aim okay thank you okay microphone to  please so I know the answer to this question is that and you a little bit  closer to the microphone maybe move it up so I know the answer to this question  is that atoms behave randomly but could you provide an argument why they behave  randomly and it is not the case that we don't have a model that's so our atoms  behaving randomly or is it the case that we don't have a model accurate enough to  predict the way they behave radioactive decay is just as random as quantum  measurement and since it since if if we were to look at the whole story and look  at the coherent evolution of the whole system we would have to include the  environment and the problem is that the state space that we have to consider  gross exponentially that's the point of quantum mechanics if I have two  particles I have a two-dimensional space of a ten particles I have a 1024  dimensional space and that's only talking about non-interacting particles so  things explode in quantum mechanics and large systems and therefore I would go  so far as to say that it's objectively impossible to determine radioactivity  k although there are things that is I think one experimentally confirmed  method of letting an atom decay on purpose that this involves a meter stable  states of nuclei and then you can do something like spontaneous emission in  a laser you shine a strong gamma source by it and this shortens the lifespan of  the nucleus but other than that completely hypothetical case if you know  all the the starting conditions and what happens afterwards you would be able  to say it's deterministic I mean I know I'm playing with with heavy words here  but is it is just that we we say it's random because it's very very complex  right that's what I'm understanding maybe think about that question one more  time and we have the signal angel in between and then you can come back  hi there's one question from the internet  which is the ground state of BEH2 has been just calculated using a quantum  eigen solver so is there still some use of quantum computing in quantum  mechanics yes definitely one of the main motivations for inventing quantum  computers was quantum simulators Feynman Feynman invented this kind of  quantum computing and he showed that with digital quantum computer you can  efficiently simulate quantum systems while you can't simulate quantum systems  with a classical computer because of this problem of the exploding dimensions  of the hibbert space that you have to consider and for this quantum computers  are really really useful and will be used once they work which is a question  when it will be perhaps never beyond two or three qubits or 20 or a hundred  qubits but you need scalability for a real quantum computer but simu quantum  simulation is a real thing and it's a good thing and we need it  okay then we have microphone one again so very beginning you said that the  theory is a set of interdependent propositions right and then if a new  hypothesis is made it can be confirmed by an experiment it can't be confirmed  but it well it's a it's a it's a philosophical question but the common  stance that it can be made probable but not be confirmed because we can never  absolutely be sure that there won't be some new experiment that shows that the  hypothesis is wrong because this light said that the experiment confirms yeah  confirm in the sense that it doesn't disconfirm it so it makes probable that  it's a good explanation of the reality and that's the point physics is just  models we don't we do get nothing about the ontology that is about the actual  being of the world out of physics we just get models describe the world but  all what I say about this wave function and what we say about elementary  particles we can't say they are in the sense that you and I are here and exist  because we can't see them we can't access them directly we can only use them  as description tools but this is my personal position on philosophy of science  so there are people who disagree  microphone to please or maybe superposition by the way so on the  matter of the collapsing of the wave function so this was already treated on  the interpretation of Copenhagen and then as you mentioned it was expanded by  the concept of decoherence and is this so the decoherence is including also the  Girardirimini-Veba interpretation or not could decoherence be used in  computation or no no so for the Girardirimini-Veba interpretation of the  collapsing of the wave function that's that's one that I don't know okay I'm not  so much into interpretations I actually think that this interesting work done  there but I think they're a bit irrelevant because in the end what I just  said I don't think we can derive ontological value from our physical  theories and in this belief I think that the interpretations are in a sense  void just they just help us to rationalize what we're doing but they  don't really add something to the theory as long as they don't change what can be  measured sorry for being an extremist  totally fine someone just left from microphone one I don't know they want  to come back I don't see any more questions as the signal angel have  anything else ah there's some more signal angel do you have something no okay  then we have microphone for I wanted to ask a maybe a noob question I want to  know are there probabilities of quantum mechanics an inherent part of nature or  maybe in some future we'll have a science that will determine all these  values exactly well if the coherence theory is true then quantum mechanics is  absolutely deterministic but so if or it's let's say if the ever it  interpreted as so ever it says that all those possible measurement outcomes do  happen and the whole state of the system is in a superposition and by looking at  our measurement device and seeing some value we in a way select one strand of  those superpositions and live in this of the many worlds and in this sense  everything happens deterministically but we just can't access any other values so  it's I think it's rather matter of for now rather matter of philosophy than of  science I see thanks  anything else I don't see any people lined up with microphones so last chance  to run up now I think well then I think we're closing this and have a nice  applause again for Sebastian thanks  thank you and I hope I didn't make create more fear of quantum mechanics than I  disqualified
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And so he studied physics and and thinking we just all need a lot better understanding of quantum mechanics because he sees the theory theory being misused a lot by some weird esoteric theories kind of abusing it to just justify everything and anything so he wants to change that and he wants to have people with some understanding of this very important theory and so he will start today with all of us here and try to explain to us the wonders of quantum mechanics. Have a go. Well thank you for the warm welcome it will be about quantum mechanics we will see whether the gentleman introduction will be a lie depending on how good you can follow me so at first there will be a short introduction a bit meta discussion about physical theories and what is the aim of this talk and then we will discuss the experiments most of this is high school physics you you've probably seen it before and then it will get ugly because we'll do the theory and we'll really do the theory we'll write down the equations of quantum mechanics and try to make them plausible and hopefully understandable to a lot of people and finally some applications will be discussed so what is the concept of this talk the key experiments will be reviewed as said and but we will not do it in historical fashion we'll look at the experiments as physical facts and derive the theory from them and since quantum mechanics is rather abstract and not as a said in German and in science theory unschaulich we will need mathematics and most of this will be linear algebra so a lot of quantum mechanics is just linear algebra and steroids that means in infinite dimensions and in doing so we'll try to find a certain post classical unschaulish kite or lividness to understand the theory since there'll be a lot of math as the allergy advice said there will be crash courses tuned in to explain mathematical facts sorry for the mathematicians that are here they probably suffer because I lie a lot so at first how do scientific theories work to understand quantum mechanics we must understand the setting and setting where it was created and how scientific theories are created in general a scientific theory is a net of interdependent propositions so we have one proposition for example f equals m m times a in classical mechanics and we have another proposition that the gravitational force equals is proportional to the product of the masses divided by the distance between the masses squared so something like this and when we go around make experiments look into nature develop theories calculate we test those we test hypotheses different hypotheses and try to determine which one describes our experimental results best and if the hypothesis stand the experimental tests they're added to the theory but what happens if there's a experimental result that totally contradicts what we've seen before and that happened in the late 19th and early 20th century there were new results that could not be explained so such inconsistent results are for I found then our old theory has been falsified this is this term is due to popper who said that theory is scientific as long as it can be falsified that is as long as we can prove that it's not true and we can never prove a theory true but only approve it wrong and all that we have not yet proven wrong at least some approximation to truth and if this happens we have to amend our old theory and we have to use care there and find a minimal amendment this principle as Occam's razor or one could also say the principle of least surprise from software engineering and then we try that our theories again consistent with the experimental results and of course the new theory must explain why the hell that for example Newtonian mechanics work for 200 years if it's absolutely wrong and so the old theory must in some limit contain the new one and now how does it begin with quantum mechanics as already said the time frame is the late 19th and early 20th century and there were three or four fundamental theories of physics known then classical mechanics which is just governed by the single equation the force equals the mass times the acceleration was given forces and two known force laws the immediately the immediate distance action Newtonian gravitation and the Maxwell electrodynamics this funny equation here whoop this funny equation here's a way of writing down the Maxwell equations that basically contain all the known electromagnetic effects and finally there was a word of beginnings of the Maxwell Boltzmann statistical physics but classical statistical physics is is a pain doesn't really work so several experimental results as said could not be explained by classical theories for example the photoelectric effect the discovered by Hudson Talvex in 1887 or the discrete spectral lines of atoms first shown by Fraunhofer and the spectrum of the Sun and then and then studied by Bunsen and Kirchhoff with the so-called Bunsen Brenner you all know it from your chemistry classes and further radioactive rays were really a mystery nobody understood how can it happen that's something just decays at random intervals it was unclear and then the people looked into the atom rather forward using alpha particles to bombard the gold foil and saw there must be positively charged nuclear and they already knew that they were negatively charged what we now call electrons particles in the atom so this and this was really strange that atoms are stable at compost like this and I will explain why a bit later but now to more detail to the experiments the really big breakthrough in this time experimentally speaking were vacuum tubes so you took a piece of glass and pumped the air out and closed it off and put all all sorts of devices in there and now one thing is this nice cathodray experiment we have here a so-called electron gun and this is a heated electrode so he flows the current that heats it so that the electrons get energy and seep out into the vacuum then we have an electrode that goes around and the plate in front that is positively charged so we accelerate our electrons towards the plate there's a pinhole on the plate and we get a beam of electrons and now we had those evacuated two tubes and those electron guns so we put a electron gun in the evacuated tube perhaps left a bit of gas in because then it glowed when it was when the atoms and the gas were hit by the electron so we could see the cathodray and then we play around we take magnetic fields and see how does it react to magnetic fields we take electric fields how does it react to electric fields and so on and what we find out is we somehow must have negatively charged particles that flow nicely around in our almost vacuum and because atoms are neutral which is just known macroscopically there must be a positively charged component in the atom as well and this positively charged component was first thought to be kind of a plump pudding or so with the electrons sitting in there but the rather third geiger experiment so it was a rather thought invented the idea and must and geiger actually performed the experimental work showed that if you have a really thin gold foil really only a few hundred layers of atoms that's nothing about gold you can just hammer it out to really really thin sheets if you had that and then shot alpha particles that is helium not nuclear that are created by the radioactive decay of many heavy elements for example most uranium isotopes decay by alpha decay then they were deflected strongly if the charge would have been spaced throughout the atoms then this could not have happened you can calculate you can you can estimate the possible deflections with an extended charge and with a concentrated charge and you see the only explanation for this is that there is a massive and really really small positive thing in those atoms so atoms are small positively charged nucleus as rather foot called it and around it there's a there's a cloud of electrons or he thought orbiting electrons but orbiting electrons atoms are stable this doesn't really make sense in classical physics because in classical physics all accelerated charges must radiate energy and be slowed by this process and this means atoms that are stable and composed of some strange electrons and heavy nuclei they're just not possible it's it's a no-go so at least at this moment it was completely clear classical physics as they knew it up until then is wrong and a next experiment in this direction was the photoelectric effect what's shown there is a schematic of a photo tube and the photo tube is again a vacuum tube out of glass and there's a for example a caesium layer in in the tube at one side and there's a ring electrode removed from it and if we shine light on this they close the current but the peculiar thing is that if we use a bias a bias voltage across the two terminals of this tube to stop the electrons we see that the bias voltage that completely stops the flow is not proportional to the intensity of the light that is incident onto the tube but it's proportional to the frequency of the light that's incident on the photo tube and that was again really weird for the people of the time because the frequency shouldn't shouldn't make any difference for the energy and this was when Einstein derived the thought of that there must be some kind of energy portions in the electric field from this simple experiment which is often done in physics classes even even at the high school level so it's from out today's view it's not a complicated experiment and to go even further those weird stable atoms had discrete had discrete lines of emission and absorption of light and here we have again a very simplified experimental setup of a so-called discharge tube where we have high voltage between the terminals and the thin gas and then a current will flow will excite the atoms the atoms will relax and emit light and this light will have a specific spectrum with sharp frequencies that are that have strong emission and we can see this with a diffraction creating that sorts the light out according to its wavelength and then look on the screen or use some more fancy optical instrument to do precision measurements as a bundlen köche oft did so what we knew up up until now was that something was really weird and our physical theories didn't make sense and then it got worse someone took an electron gun and pointed it at the at the monocrystalline surface and such a monocrystalline surface is just like a diffraction creating grating a periodically arranged thing and of periodically arranged things there does happen regular interference pattern creation so they saw interference pattern with electrons but electrons are the particles how can particles so what was thought of then since the times of Newton as a little heart ball how can a little heart ball flowing around create interference patterns it was really weird and there's even more and as already mentioned radioactivity with the random decay of nucleus this doesn't make sense in classical physics so it was it was really really bad and here I've added some modern facts that we'll need later on namely that that if we measure but if we try to measure the position of a particle and use different position position sensors to do so only one of them so at only at one position will the single particle register but it will nevertheless show an interference pattern if I do this experiment with many many electrons so there must somehow be a strange divide between the free space propagation of particles and measuring the particles and you can do really weird stuff and record the information through which slit the particle went and if you do this the interference pattern vanishes and then you can even destroy this information in a coherent manner and the coherence and the interference pattern appears again so what we know up until now is that quantum mechanics is really really weird and really different from classical mechanics and now that we've talked about those experiments we'll begin with the theory and the theory will begin with a lot of mathematics the first one is simple complex numbers okay who who doesn't who doesn't know complex numbers okay sorry I'll have to ignore you for the sake of getting to the next points so I'll just say complex numbers are two components of real at two component objects with real numbers and one of them is multiplied by an imaginary number I and if we square the number I it gets minus one and this makes many things really beautiful for example all algebraic equations have exactly the number of degree solutions in complex numbers and if you count them correctly and if you work with complex functions it's really beautiful a function that once differentiable is infinitely many times differentiable and it's it's nice so now we had complex numbers you've all said you know them so we go on to vector spaces which probably also a lot of you know just to revisit it a vector space is a space of objects called vectors above some scalars that must be a field and here we only use complex numbers as the underlying fields there's a null vector we can add to the vectors we can invert vectors and we can multiply vectors by real numbers so we can say three that five times this vector and just scale the the arrow and the these operations interact nicely so that we have the those distributive laws and now it gets interesting even more maths L2 spaces L2 spaces are in a way a infinite dimensional or one form of an infinite dimensional extension of vector spaces instead of having just three directions xyz we have directions at each point of a function so we have an analogy here we have vectors which have three discrete components given by x index i on the right side and we have this function and we have this function and each component is the value of the function at one point along the x is x and then we can just as for vectors define a norm on those L2 functions which is just the integral over the absolute value squared of this function f and a nice thing about this choice of a norm there are other choices of the norm this norm is induced by a scalar product and this little asterisk for the that is there at the f denotes the complex conjugate so flipping i to minus i in all complex values and if you just if you just plug in f and f into the scalar product you will see that it's in the integral over the squared absolute value and this space this L2 space is a Hilbert space and the Hilbert space is a complete space vector space with a scalar product where complete means that it's mathematical nonsense forget it so but the nice surprise is that most things carry over from finite dimensional spaces what we know from finite dimensional spaces we can always diagonalize matrices with certain properties and this all more or less works at the mathematicians really really really do a lot of work for this but for physicists we just know when to be careful and how and don't care about it otherwise so it just works for us and that's nice and now that we have those complex numbers we can begin to discuss how particles are modeled in quantum mechanics and as we know from the Davis and German experiments there's diffraction of electrons but there's nothing in electrons that corresponds to a positive electric field in some direction or so some other periodicity has so periodicity of electrons during propagation has never been directly observed and the broi said particles have a wavelength that's related to the momentum and he was motivated primarily by the Bohr theory of the atom to do so and he was shown right by the Davis and German experiments so his relation for the wavelength of a particles older than the experiments showing this which is impressive I think and now the idea is to have a complex wave function and let the squared absolute value of the wave function describe the probability density of a particle so we make particles extended but probability measured object so there isn't no longer the position of the particle as long as we don't measure but we have just some description of a probability where the particle is and by making it complex we have a face and this face can allow still allow interference effects which we need for explaining the interference peaks in the Davis and German experiment and now a lot of textbooks say here there's a wave particle you listen blah blah blah distinct nonsense blah the point is it doesn't get you far to think about quantum objects as either wave or particle they're just quantum neither wave nor particle doesn't help you either but it doesn't confuse you as much as when you try to think about particles as waves or particles about quantum particles as ways of particles and now that we say we have a complex wave function what about simply using a plane wave with constant probability as the states of definite momentum because we somehow have to describe a particle that is that that has a certain momentum and we do this those have the little problem that they are not in the Hilbert space because they're not normalizable the absolute value of psi is 1 1 over 2 pi everywhere so that's bad but we can write the superposition of any state by Fourier transformation those e to the I k dot r states are just the basis states of Fourier transformation we can write any function in terms of this basis and we can conclude that by Fourier transformation of the state psi of r to some state till the psi of k we describe the same information because we know we can invert the Fourier transformation and also this implies the uncertainty relation and because this is simply property of Fourier transformations that either the either the function can be very concentrated in position space or in momentum space and now that we have states of definite momentum and the other big ingredient in quantum mechanics are operators next to the state description and operators are just like matrices linear operators on the state space just as we can apply a matrilinear operator in the form of a matrix to a vector we can apply linear operators to two functions and when we measure an observable it will be that it's one of the eigenvalues of this operator that's the measurement value you know so eigenvalues are those values where a matrix does just if matrix just scales a vector by a certain amount that is an eigenvector of the eigenvalue of the matrix and in the same sense we can define eigenvalues and eigenvectors for two functions and there are some facts such as that non-commuting operators have eigenstates that are not common so we can't have a description of the of all the of the basis of the state space in terms of functions that are both eigen functions of both operators and some examples of operators are the momentum operator which is just minus i h bar napler which is the derivation operator in three dimensions so in the x component we have derivation in the direction of x and the y component in the direction of y and so on and the position operator which is just the operator that multiplies by the position x in the position space representation of the wave function and as for the non- commutativity of operators we can already show that those p and x operators that do not commute but fulfill a certain commutation relation and the commutation relation is just a measure for how much two operators do not commute and the commute commutators a b minus b a for the objects a b so if they commute if a b equals b a the commutator simply vanishes and there's more operators just to make it clear linear just means that we can split the argument if it's a sound if it's just linear just some linear combinations of vectors and apply the operator to the individual vectors occurring we can define multiplication of operators and this just exactly follows the template that is laid down by finite-dimensional linear algebra there's not there's nothing new here and they're inverse operators for some operators not for all of them that give the identity operator if it's multiplied with the original operator and further there's the so-called a joint our scalar product that had this little asterisk and this means that it's not linear in the first component if I scale the first component by some complex number alpha the total scalar product is not scaled by alpha but by the complex conjugate of alpha this kind of not quite by linearity is called sometimes called sask wheel in year sask wheel in the air at you a seldom used word and they are commonly defined classes of operators in terms of how the how the adjoint that is defined their acts and how some their operators for example where the adjoint is the inverse which is a generalization from the fact that for rotation operators in normal Euclidean space the transpose is the inverse and now that we have operators we can define expectation values just by some formula for now we we don't know what those expectation values are but we can assume it has something to do with the measurement values of the operator because why else would I tell you about it and later on we will show that this is actually the expectation value of the quantity if we prepare a system always in the same fashion and then do measurements on it we get random results each time but the expectation value will be this combination and now again a bit of mathematics eigenvalue problems well known you can diagonalize the matrix and you can diagonalize linear operators you have some equation a psi equals lambda psi where lambda is just a scalar so and if such an equation holds for some vector psi then it's an eigenvector and any linear scale if we scale the vector linearly this will again be an eigenvector and what can happen is that to one eigen eigenvalue there are several eigenvectors not only one ray of eigenvectors but a higher-dimensional subspace and important to know is that so-called Hermitian operators that is those that equal their joint which again means that the eigenvalues equal the complex conjugate of the eigenvalues have real eigenvalues because of a complex number equals its complex conjugate then it's a real number and the nice thing about those diagonalized matrices and all is we can develop any any vector in terms of the eigen basis of the operator again just like in linear algebra where when you diagonalize a matrix you get a new basis for your vector space and now you can express all vectors in that new basis and if the operator is a mission the eigenvectors have a nice property namely they are orthogonal if the eigenvalues are different and this is good because because this guarantees us that we can choose a orthonormal that is a basis in the vector space where two basis vectors always have vanishing scalar product are orthogonal and are normal that is we scale them to length one because we want our probability interpretation and in our probability interpretation we need to have normalized vectors so now we have that and now we want to know how does this strange function psi that describes the state of the system evolve in time and for this we can have several requirements that it must fulfill so again we're close to software engineering and one requirement is there's that if it's a sharp wave packet so if we have a localized state that is not smeared around the whole space then it should follow the classical equation of motion because we want that our new theory contains our old theory and the time evolution must conserve the total probability of finding the particle because otherwise we couldn't do probability interpretation of our wave function if the total probability of the particle wouldn't remain one further we wish the equation to be first order in time and to be linear because for example the Maxwell equations are linear and show nice show nice interference effects so we want that because then simply some of solutions is again a solution it's it's a good property to have and if it works that way why not and the third and the fourth requirement together already give us more or less the form of the Schrödinger equation because linearity just says that the right-hand side is some linear operator applied to psi and the time first order in time just means that there must be a time derivative a single time derivative in the equation on the left-hand side and this IH bar is just and we just want the debt there no particular reason we could have done this differently but it's convention now with this equation we can look what must happen for the probability to be conserved and by a simple calculation we can show that it must be her mission operator and there's even more than this global argument there's local conservation of probability that is a particle can simply vanish here and appear there but it must flow from one point to the other with local operations this can be shown when you consider this in more detail now we know how this equation of motion looks like but we don't know what this mysterious object H might be and this mysterious object H is the operator of the energy of the system which is from known from classical mechanics as the Hamilton function and which we here upgrade to a Hamilton operator by using the formula for the classical Hamilton function and inserting our P and R operators and we can also extend this to a magnetic field and by doing so we can show that our theory is more or less consistent with Newtonian mechanics we can show the Irnfest theorem that's the first equation and then those equations are almost Newton's equation of motion for the centers of mass of the particle because this is the expectation value of the momentum this is the expectation value of the position of the particle this just looks exactly like the classical equation the velocity is the momentum divided by the mass but this is weird here we ever we average over the force so the gradient of the potential is the force we average over the force and do not take the force at the center position so we can't in general solve this equation but again if we have a sharply defined wave packet we recover the classical equations of motion which is nice so we have shown our new theory does indeed explain why our old theory worked we only still have to explain why the centers of mass of massive particles are usually well localized and that's the question we're still having trouble with today but since the since it otherwise works don't worry too much about it and now you probably want to know how to solve the Schrödinger equation or you don't want to know anything more about quantum mechanics and to do this we make a so called separation ansatz where we say we have a form stable we have a form stable part of our wave function multiplied by some time-dependent part and if we do this we can write down the general solution for the Schrödinger equation because we already know that the one equation that we get is the is an eigenvalue equation or an eigenvector equation for the energy eigenvalues that is the eigenvalues of the Hamilton operator and we know that we can develop any function in terms of those and so the general solution must be of the form shown here and those states of this of specific energy have a simple evolution because their form is constant and only their face changes and depends on the energy and now this thing with the measurement in quantum mechanics is bad you probably know Schrödinger's cat and the point is that you don't know whether the cat's dead or alive while you don't look inside the box as long as you don't measure it's in a superposition or something so you measure you measure your cat and then it's dead it isn't dead before only by measuring it you kill it and that's really not nice to kill cats we like cats so and the the important part here is the TLDR quantum measurement is probabilistic and inherently changes the system state so I'll skip the multi particle things we can describe multiple particles and just show the axioms of quantum mechanics shortly don't read them too detailed but this is just a summary of what we've discussed so far and the the thing about the multiple particles is the axiom 7 which says that the sign of the wave function must change if we exchange the coordinates of identical fermions and this makes atoms stable by the way without this atoms as we know them would not exist and finally there's a notational convention in quantum mechanics called bracket notation and in bracket notation you label states by their eigenvalues and just think about such a cat as an abstract vector such as x with an with a vector arrow over it or a fat set x is an abstract vector and we can either represent it by its coordinates x1 x2 x3 or we can work with the abstract vector and this cat is such an abstract vector for the l2 function psi of r and then we can also define the adjoint of this which gives us if we multiply the joint at the function the scalar product so this is a really nice and compact notation for many physics problems and the last equation there just looks like component wise like working with components of matrices which is because it's it's nothing else this is just matrix calculus in a and a new and new clothes now for the applications the first one is quite funny oh there's a slight missing okay the first one is a quantum eraser at home because if you encode the which way information in to a double slit experiment you lose your you lose your interference pattern and we do this by using a vertical and a horizontal polar polarization filter and you know from classical physics then it won't it won't make a it won't make an interference pattern and if we add then at the diagonal polarization filter then the interference pattern will appear again so now just so you've seen it the harmonic oscillator can be exactly solved in quantum mechanics if you can solve the harmonic oscillator in any kind of physics then you're good then you'll get through the exams in the physics when you study physics so the harmonic oscillators is solved by introducing so-called destroyers and creation and destroyer operators and then we can determine the ground state function with the size in a much simpler manner than if we had to solve the Schrödinger equation explicitly for all those cases and we can we can determine the ground state the ground state function so the function of lowest energy this can all be done and then can from it by applying the creation operator create the higher the higher state eigenstates of the system and get all of them then there's this effect of tunneling that you've probably heard about and this just means that in quantum mechanics a potential barrier that is too high for the particle to penetrate does not mean the particle doesn't penetrate at all but just that the the probability of finding the particle inside a barrier decays exponentially and this can for example be understood in terms of this uncertainty relation because if we try to compress the particle to a smaller part of the boundary layer then its momentum has to be high so it can reach farther in because then it has more energy and there's this myth that tunneling makes particles travel to travel instantaneously from A to B and even some real physicists believe it but sorry it's not true the particle status extended anyway and to defining what how fast the particle travels is actually not a well-defined thing and deep quantum regimes and also the Schrödinger equation is not relativistic so there's nothing nothing really nothing stopping your particle from flying around with 30 times the speed of light it's just not in the theory another important consequence of quantum mechanics is so-called entanglement and this is a really weird one because it shows that our the universe that we live in is in a way non-local inherently non-local because we can create some state of if you can create a state for some internal degrees of freedom of two atoms and move them apart then measure the one system and the measurement result and the one system will determine the measurement result and the other system no matter how far removed they are from each other and this was first discovered in a paper by Einstein Podolsky and Rosen and they thought it was an argument that quantum mechanics is absurd this can't be true but sorry it is true so this works and this kind of state that we've written there that is such an entangled state of two particles but important to remark is that there are no hidden variables that means the measurement result is not determined beforehand it is only when we measure that is actually known what the result will be this is utterly weird but one can prove this experimentally is those are belt tests there's a bell inequality that's a limit for theories where they are hidden variables and it's by real experiments they violate this inequality and they may show that there are no hidden variables and there's a myth surrounding entanglement namely that you can transfer information with it between two sides instantaneously but again there's nothing hindering you in non relativistic quantum mechanics to distribute information arbitrarily fast it doesn't have a speed limit but you can also can't communicate with those entangled pairs of particles you can just create correlated noise at two ends which is what quantum cryptography is using so now because this is the hackers congress some short remarks and probably in intelligible due to their strong compression about quantum information a qubit the fundamental unit of quantum information is a system with two states zero and one so just like a bit but now we allow arbitrary superposition so arbitrary super positions of those states because that is what quantum mechanics allows we can always super imposed states and quantum computers are really bad for most computing tasks because they they have to have even if they built quantum computers they'll never be as capable as the state-of-the-art silicon electrical computers so don't fear for your jobs because of quantum computers but the problem is they can compute some things faster for example factoring primes and working with some elliptic curve algorithms and so on and determining discrete logarithms so our public key crypto would be destroyed by them and this all works by using the superposition to construct some kind of weird parallelism so it's actually I think nobody really can imagine how it works but we can compute it which is often the case in quantum mechanics and then there's quantum cryptography and that's fundamentally solve the same problem as a Diffie Helman key exchange we can generate the shared key and we can check by the statistics of our measured values that there was no even dropper which is cool actually but it's also quite useless because we can't detect them and in the middle how should the quantum particle knows of the other side is the one we that we want to talk to we still need some shared secret or public key infrastructure whatever so it doesn't solve a problem that we don't have solved and then the fun fact about this is that all the commercial implementations of quantum cryptography were susceptible to side channel text for example you could just shine with the light and the fiber that was used read out the polarization filter state that they used and then you could you could mimic the other side so that's not good either so finally some references for further study the first one is really difficult only try this a few for at the other two but the second one sorry that they're in German the first and the last are also available in translation but the second one has a really really nice and accessible introduction and the first in the last few pages so it's just 20 pages and it's really good and understandable so we can you get your hands you can get your hands on the books and are really interested read it so thank you for your attention and I'll be answering your questions next thank you Sebastian do we have questions and don't be afraid to sound naive or anything I'm sure if you didn't understand something many other people thank you for a good question as to understanding things in quantum mechanics Feynman said you can't understand quantum mechanics you can just accept it there's nothing to understand there it's just too weird okay we found some questions so microphone one please measure a system it looks like you change the state of the system how is it defined where the system starts or no what how is it defined by the system ends and the measurement system begins or in other words why does the universe have a state is there somewhere out there who measures the universe no there's a at least the beginning of a solution by now which is called decoherence and which says that that this measurement structure that we observe is not inherent in quantum mechanics but comes from the interaction with the environment and we don't care for the states of the environment and if we do this the technical term is trace out the states of the environment then the remaining state of the measurement apparatus and the system we're interested in will be just classically randomized states so it's it's rather a consequence of the complex dynamics of a system state and environment in quantum mechanics but this this is really the burning question we we don't really know we have this we know the coherence makes some makes it nice and looks good but it also doesn't answer the question finally and this is what all those discussions about interpretations of quantum mechanics are about how shall we make sense of this weird measurement process okay microphone for in the back please could you comment on your point in the theory section I'm I don't understand what you were trying to do did you want to show that you cannot understand really quantum mechanics without the mathematics or well yes you can't understand quantum mechanics without the mathematics and my point to show was that the make matter or at least my hope to show was that the mathematics is half ways accessible so probably not understandable after just exposure of a short talk but just to give an introduction where to look okay so you were trying to combat esoterics to and and say they they don't really understand the theory because they don't understand the mathematics I understand the mathematics I'm just interested what what you you were trying to say I was just trying to present the theory that was my aim okay thank you okay microphone to please so I know the answer to this question is that and you a little bit closer to the microphone maybe move it up so I know the answer to this question is that atoms behave randomly but could you provide an argument why they behave randomly and it is not the case that we don't have a model that's so our atoms behaving randomly or is it the case that we don't have a model accurate enough to predict the way they behave radioactive decay is just as random as quantum measurement and since it since if if we were to look at the whole story and look at the coherent evolution of the whole system we would have to include the environment and the problem is that the state space that we have to consider gross exponentially that's the point of quantum mechanics if I have two particles I have a two-dimensional space of a ten particles I have a 1024 dimensional space and that's only talking about non-interacting particles so things explode in quantum mechanics and large systems and therefore I would go so far as to say that it's objectively impossible to determine radioactivity k although there are things that is I think one experimentally confirmed method of letting an atom decay on purpose that this involves a meter stable states of nuclei and then you can do something like spontaneous emission in a laser you shine a strong gamma source by it and this shortens the lifespan of the nucleus but other than that completely hypothetical case if you know all the the starting conditions and what happens afterwards you would be able to say it's deterministic I mean I know I'm playing with with heavy words here but is it is just that we we say it's random because it's very very complex right that's what I'm understanding maybe think about that question one more time and we have the signal angel in between and then you can come back hi there's one question from the internet which is the ground state of BEH2 has been just calculated using a quantum eigen solver so is there still some use of quantum computing in quantum mechanics yes definitely one of the main motivations for inventing quantum computers was quantum simulators Feynman Feynman invented this kind of quantum computing and he showed that with digital quantum computer you can efficiently simulate quantum systems while you can't simulate quantum systems with a classical computer because of this problem of the exploding dimensions of the hibbert space that you have to consider and for this quantum computers are really really useful and will be used once they work which is a question when it will be perhaps never beyond two or three qubits or 20 or a hundred qubits but you need scalability for a real quantum computer but simu quantum simulation is a real thing and it's a good thing and we need it okay then we have microphone one again so very beginning you said that the theory is a set of interdependent propositions right and then if a new hypothesis is made it can be confirmed by an experiment it can't be confirmed but it well it's a it's a it's a philosophical question but the common stance that it can be made probable but not be confirmed because we can never absolutely be sure that there won't be some new experiment that shows that the hypothesis is wrong because this light said that the experiment confirms yeah confirm in the sense that it doesn't disconfirm it so it makes probable that it's a good explanation of the reality and that's the point physics is just models we don't we do get nothing about the ontology that is about the actual being of the world out of physics we just get models describe the world but all what I say about this wave function and what we say about elementary particles we can't say they are in the sense that you and I are here and exist because we can't see them we can't access them directly we can only use them as description tools but this is my personal position on philosophy of science so there are people who disagree microphone to please or maybe superposition by the way so on the matter of the collapsing of the wave function so this was already treated on the interpretation of Copenhagen and then as you mentioned it was expanded by the concept of decoherence and is this so the decoherence is including also the Girardirimini-Veba interpretation or not could decoherence be used in computation or no no so for the Girardirimini-Veba interpretation of the collapsing of the wave function that's that's one that I don't know okay I'm not so much into interpretations I actually think that this interesting work done there but I think they're a bit irrelevant because in the end what I just said I don't think we can derive ontological value from our physical theories and in this belief I think that the interpretations are in a sense void just they just help us to rationalize what we're doing but they don't really add something to the theory as long as they don't change what can be measured sorry for being an extremist totally fine someone just left from microphone one I don't know they want to come back I don't see any more questions as the signal angel have anything else ah there's some more signal angel do you have something no okay then we have microphone for I wanted to ask a maybe a noob question I want to know are there probabilities of quantum mechanics an inherent part of nature or maybe in some future we'll have a science that will determine all these values exactly well if the coherence theory is true then quantum mechanics is absolutely deterministic but so if or it's let's say if the ever it interpreted as so ever it says that all those possible measurement outcomes do happen and the whole state of the system is in a superposition and by looking at our measurement device and seeing some value we in a way select one strand of those superpositions and live in this of the many worlds and in this sense everything happens deterministically but we just can't access any other values so it's I think it's rather matter of for now rather matter of philosophy than of science I see thanks anything else I don't see any people lined up with microphones so last chance to run up now I think well then I think we're closing this and have a nice applause again for Sebastian thanks thank you and I hope I didn't make create more fear of quantum mechanics than I disqualified
down the equations of quantum mechanics and try some applications will be discussed so what is experimental tests they're added to the theory