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New Revolutions in Particle Physics: Basic Concepts  Lecture 3
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Title  New Revolutions in Particle Physics: Basic Concepts  Lecture 3 
Title of Series  Lecture Collection  Particle Physics: Basic Concepts 
Part Number  3 
Number of Parts  10 
Author 
Susskind, Leonard

License 
CC Attribution 3.0 Germany: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/15072 
Publisher  Stanford University 
Release Date  2010 
Language  English 
Production Year  2009 
Content Metadata
Subject Area  Physics 
Abstract  (October 19, 2009) Leonard Susskind gives the third lecture of a threequarter sequence of courses that will explore the new revolutions in particle physics. In this lecture he talks about what a quantum field is and how it is related to particles. 
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00:05
Steiger university flights
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of less time we will be due Sergei the idea of a quantum field and that's what I want it today what is a quantum field I was related the particles we spent over time talking about harmonic oscillators in the basic some mathematical tools for quantum field kind of of quantum fields a kind quantum feels is the harmonic oscillator and putting oscillators together to make good on your soul we talked about our fields on a periodic space occur to draw the periodic space a periodic space is 1 which FT Villa distant Aiello comes back to itself to useful trekked Eds and the reason useful trick better than just taking a fine line in the world is because when you have a periodic space you could have waves which go in 1 direction always which go in the direction without reflecting a that is analog momentum cars that they have target is momentum conservation of you will retain momentum conservation good idea if your maintain momentum conservation and temporarily put their system finite made box so that you don't have an infant volume to worry about then the right way to do it is periodic boundary conditions of those circumstances a wave a plane wave of being signed cosine will be of the form he DIE Kane X Our K can be positive or negative K being positive corresponds always moving the right around loop cabling negative Katie corresponds to a moment going the other direction just to remind you Kerry it is not enough arbitrary number it has to be could cast a come in integer multiples something around in order that the way when it comes back to itself after going around the distant L the way you come back on original value so we say that X you go always around so that would draw but gave this concern was l then eat I can l must be white Was go to war and tell you want everyone at Mexico's you're always exceed 0 0 over here this function is 1 Mexico 0 any exponential of acts it is 1 of the girls always around better come back to itself so that means that Ito the IKL must equal eating units AIK 0 0 Mexico 0 x equals L K 0 is just 1 more of that means he could be on a L equals 1 Oravetz very that's Casey times els must be an integer multiple of 2 pie he took Is it with the 1 I I know that while we could just work it out remember would be exponential means it means call sign our to Part II Plus Tupaiidae plus ii times assigned to party a cosigner took applied as 1 New Scientist took 5 0 so indeed this is equal to 1 and factor you put any multiple of 2
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pipe is also won so as long as L is an integer multiple of 2 Pyay This is a periodic function Exs periodic function of Cape Times l must be equal to 5 times and integer and let's call at integer and this In the measures should be distinguished From another in which will come later is seeking which counts the number of course want them reached tight like this is not counting the number of quantum particles is just camping allowable wave numbers is is called wave number and the dresser Cape Times l is still party must be took pride in or else that's the that's the conditions on Cheney that when you go around function comes back to itself so that's what I want think incidentally this is exactly the same thing as the quantization of the waiver Lane saying the wavelength past the figure in integer multiple times by the time you come back what's the connection between Kane Waverly promised the wavelength is inverse to Katie as took pie in the relationship L arm returned this all over its 1 K E equals L divided by 2 pie in yeah Our want graverobber but your this is this is the condition on case is equal on the condition that the wavelengths be quantized just fits into the circle integer number time is the kind awaits you could have Korea Korea on if we are talking about the waves that are associated with some quanta the case related to the moment that I guess we better worker is to work out the relation between and the wavering on what the wavelength associated with a wave white X while it's the distance you have to move before I eat me I go OK egos the 1 full cycle sold if I want to find that the wavelength of an object right there's a way of this kind of way I just say that that came 8 times X I love 8 times the wavelength has to be paid by the wavelength wavelength lambda as to be Tupaiidae all R. Kelly is the same thing and twoparty divided by the way for stress the relation between Kanye and wavelength oneway big cases small wavelength a small case is big and took part relating but
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there closely related but that's the relation between K and waiver as anybody remember the relationship between the Waverly Our photon already feels and momentum fresh momentum our particle making a budget connections momentum of a particle prescribed by giving corner mechanical weight the momentum of particle is equal also total what for so either by land that is our right the ayatollah the bike expressed in terms of age bar and then it would be to Pyay bomber overland so all you could see that at that P and Katie are essentially the same thing except for a factor of Kate they are sorry except for effective age from particular momentum in age ah times case so if I have a wave described this is called a wave number is called wave number if a wave number and 3 positive or negative waves will be the less willing to the right corresponding the moment they're going to the left the right that's a basic that's a basic waved on a circle of life a waiver on a periodic space now through each such kind of way they could be plant that quanta corresponding to particles only described as as those that what we do is just described the number of quanta With momentum P fluently the number of plant there With a given wave number case on a corn dog there would be given wave number case since Keyes itself and integer we have the specified and in the jury for each integer 1 of those integers issues related to to a wave number other integer is called the occupation number the number of quanta off DuPont that particular weight if you like on that particularly for right back so there is which correspond to the quantization of momentum but there also in which correspond to be on the number of quanta With each moment of those called occupation numbers so I put right and I will write a case was different and now KK is itself quantized but in this case is the number of particles of the number of quanta with wave number
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K which in effect means with momentum p but just remember there is a related P K I happened to be related by it of us might choose a quarter units in which cable cases eagled 1 for each farcical 1 thing came of the same thing so we often don't even bother to distinguish between wave number momentum we speak of Katie as the momentum of the particle not there but that all key he not that is theirs I got to have a collection of occupation numbers every value of including negative and positive numbers Dr. Michael if we wave numbers not only integer that's correct used to ponder of L but this did the jury is not the occupation number we could die I don't know what I don't know any of the good notation integer except in fortunate and so everything which is quantized lines of being called in except L but are closing out this is not our was angular momentum which is also quantized so that did track are 2 different kinds of integers 1 of them the discrete momentum and the other with discrete number of quanta with each momentum and the way they keep track of it it is occupation numbers I will call in case meaning the number of quanta with momentum cake Katie itself is also planned but still occupation number number of particles will give momentum cake next we introduce operators quantum mechanical operators for those who know a little bit of quantum mechanics you will remember the state vector are Lee near what's which call state vectors I will shoot anybody calls of back there state vectors because we'll get them confused with Ben directions Bass Lilly elements such just coal and linear elements elements that are elements of a vector space but they forget forget that the state vectors of a system are linear elements that we can and so forth but we can also operate on them with operators can't do this at least twice in this class and a series of questions and on the observables a quantum mechanical systems are the her mission or reveal valued operators which operate on the state's operate means they are operations which changes state to another state they were as well as a convenient Waco labeled the possible states of a system of quanta will lead such a circle Mrs. specified all the end of KU so you can think of it as In 1 and 1 and 2 0 0 0 1 2 and so forth refer to the values of K and terrain all I shows I also have to put in a minus 1 9 2 0 0 9 4 Thionnet do this I interceptor imagine that this strain the string goes out the left and the right Our and represents a state with in oneweek Kwan then it 1st value momentum until quanta with the 2nd value momentum would not specify what these values of are just first one sec ond of minus 1 . 2 4 that's a quantum state of the quite moving and now there are operators analogous to not analogous to really equivalent to a harmonic oscillator creation and annihilation operators raising and Lauren operators remember the harmonic oscillator is described by operators a pass in a mine which are multiplying arms writing him a plus in my mind as a plus increases energy levels of the oscillator increases takes you from 0 Quantico that warned quanta warned or Roberto 1 and so forth and so on an a minus picture back down so what these operators or and is 1 of them the each moment 1 8 plus and 1 making minus for each moment so they should be labeled as a class of kV E N a case so what does a plus of Katie do when it acts on a state vector like well it goes the case slot for example let's take a possible want because of the 1st slot here and increases the number of quanta by 1 unit that also does something else you with does it multiplies by something square rooted in the password and plus 1 unit yup now minuses square on remind you to keep track of those coefficients that I explain last time as a bookkeeping devices bookkeeping devices passes the b squarely XOA plus of 1 and general a plus of Katie goes to the cave momentum slot I really should call it a plus end of Katie with the number of that none of the 8 plus of K increases the number of quanta with momentum carried by warned you in a minus decreases it at the same time multiplying by the appropriate square those are Betsy but a set of mathematical objects that we have to deal left and heavily build the quantum field what is connection between the quantum field and a pluses and minuses well I'll tell you now I will see great with a pluses and minuses in another language that the quantum mechanical versions of the floor yet coefficients the Fourier coefficients Of the field of field periodic circle like a canoe represented let's go let's give the fuel the name let's call it society let's courtside and sigh fields a function of X and any such field periodic interval like that can be represented as infinite son over the
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ol allowable values of k but write some AUC what value of k by something over a summer would be allowable values of Katie of what call efficient slips Cohen 5 kb he could be I carry X as saying expanding in signs and cosine this signed and there is another object sigh Star Wars star the complex conjugate of obviously side is a cut in general a complex thing eagerly IKX is a complex sigh is a complex thing as a real and imaginary parts the real pot plus I imaginary part side pot is I can't imagine a report is called a complex conjugate match equal to the sum of 10 18 of the complex conjugate of our flat times or so complex conjugate Evita the IKX my so to specifying a classical will closer classical wave you specify all of these foyer coefficient these Fourier coefficients you have to specify separately the in stars if you know how a start but you do have to specify the real and the imaginary part of OK so a classical configurational wave specified forgive me after being quantum mechanical a pluses and minuses the quantum mechanical versions of the after these observables why they observables recourse if you have a wave you could measured the amplitude of each Fourier components the amplitude of Fourier components could you measure the sheer fun of observables are least a real imaginary part of our observables classically if you can measure the real part and the imaginary part the whole things observable but strictly speaking we do it observation you get a real number so an alpha really stands for toward observables a real imaginary part on big quantum mechanical version of the and back I write equals that wouldn't quite makes sense these of classical variables is a quantum mechanical operators but instead of writing equals let's right arm Paramount sure equals that means approximately equal if you have yet backandforth Farrell that means it quantum of aliens quantize in system you replaced Alpha by a so the double arrow means replaced by the same thing here Alpha Stalin who Irish right but I wonder if I have a lot I wonder if you follow I think right as we go along when they decide to change the definition so that this is called out for star that's Alpha is not important because you you have the freedom to yup from retained autumn Gavra standard definition and that this is a convention of course if Alpha stars cannot be a conduit of alfalfa declined star so as a matter of definition here could choose away but I think it's standard of remember are to make this identification again it's not terribly important on this day the quantum field the definition of the quantum field is simply replace the alpha's by the creation and annihilation operators a marinas of Katie a process not class of case that got us we would have to prove that in some sense a another classical after in some appropriate limit be acres must behave like the elephant in what sense I can do is to make his replacement are on the blackboard but you probably save yourself why is that the right replacement and what since these quantum mechanical objects behave anything like that Our like a classical Fourier coefficients well in many ways but there are no buts to say that we have to go a little more into these dynamics of quantum fields and we will Lowell but let's just take it as a matter of definition now that quantum field is represented as a Fourier some with the Fourier coefficients are operators that makes sense the Fourier coefficients are operators that makes sense but supported coefficients you expect to be observables and that the structure of those observables of relation between them is exactly the relation between creation and annihilation operators the field is now the fielders and offered right Chang yet Barnegat permission conjugated game last year it allowed that she added that doesn't mean that the dynamic will create entirely particles that we have to go into the EU make the dynamics of these fields muck well they change the number particles but it would be a passes in a I due they change the number of particles these things he represent the number particles with momentum 1 number particles of momentum to number particles of 3 what to to change Nova has to say that they aid is change the number of particles doesn't mean the nature goes around changing the number particles is a mathematical symbols which change the number of particles it doesn't mean that particle number changes that that a matter of a laws of nature where the laws of nature permit the number particles to change but if the number if the laws of nature dual allow the the
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number particles to to change these operators would be very useful in describing those kind of changes that his other situations even with the number of particles doesn't change will but let's look at this with all we can do with these operators armed suppose include want to describe a reaction a particle gathers a particle scatters would scatters would say we have 1 particle now hold onto particles 1 particle of a given moment the Katie and that particle going scatter collide with something that is going to come off with another moment how we describe how old how might we describe such a process we might describe process like that by saying but start With the state with 1 particle where so I put that particle in here where the momentum of the particle happens to be particle happens that momentum Casey said then an hour make 0 0 on top of the 7th place for a 1 man and then 0 0 0 that's supposed to be the Sept . 3 so this describes a state with 1 particle moment the case 7 whatever case 7 happens to be valid described a process where the particle on given momentum scanners and comes off with some different momentum now so far we've only been talking about onedimensional motion even talking about emotional line OK we can do most of our line and we can even think of changes of momentum to a particle gets hit by something and its momentum changes we describe what we can describe that as saying we described by an operator which makes exactly those changes namely they get operator which removes this article what's that at 18 miners on 7 Cahill a minus the case upset but that that removes that article and then put back another particle In another momentum space for example of the scattering where particle goes for momentum case 70 K 9 men will put here 8 plus of nite canine maybe I should quote court case what's right that a minus Our Katie 7 classes are Kanye 9 What does is don't it acts on the state with the momentum with K 7 and switches it canine Soto scattering to kind of process for does what was down after that well here and here it doesn't matter the radio added that just multiplied by 1 year to see if there will you show that he was a case where a wire route of instead buttons Evans right then having Drive but do it as that it 1st removes this article of the coefficient war song gives a past of canine times 0 0 0 0 0 0 what's what what is called all you go back that's or I speak of the backfield it is also the groundstate also state lowest energy and this simple example we could call vacuum meaning not particles at the 1st operator Texas state mix of active and then the 2nd operator calms the creation operated multiplied by square relentless 1 but we go then canine canine is out here its occupation numbers 0 so the end plus 1 just a gives 1 again the rights of this operation and gives us 0 0 0 0 0 0 bucks an hour 9th place it gives us a white so what is done with coefficient 1 is switch the place where the particle From 1 momentum toward a moment of that there if there are situations where these square to the incoming will talk about our will talk about what they do they are important for this particular problem because it was never more than 1 particle and problem it Querida and on doing OK so here's an example and where a scattering process is described by a certain combination of not let's let's you go further let's suppose that the laws of physics allow for the possibility that the number of particles it is not considered supposing 1 particle concomitant into particles can go out felt this process of course we could have described another way we didn't need to go to this long and complicated story will occupation numbers we could just set a particle makes the transition from 1 moment to another but ballots dual financier and say the laws of physics permeate the look on a process where the number of particles Cheney for example 1 particle comes in my collide with something and 2 particles of the same kind go out but to particles With different moment particle of key 1 comes in and 2 particles 1 53 in 1 of K 5 go out Illinois Temirtau momentum conservation assuming the something to soak up the year the require some other object how will we discard that that would be difficult to describe in 1 particle quantum mechanics 1 particle quantum mechanics means the quantum mechanics of a singleparticle this was this is possible but was never more than 1 particle 1 particle onetoone particle particle goes to 2 particles that you can't describe singleparticle mechanics but you can't describe it here again you removed OK the the 7th particle and you put in the 9th particle you also put in the 11th part formal I'm thinking odd numbers even No. 8 plus our 16 of K 16 so this is a process where a particle of momentum case 7 comes in and make all collide something more might say that that's something a absorbs the particles the language this is language be target wars and so this article and needed to particles 1 of canine were 16 sorrows this gives you flexibility this is the existence of his operators and this is based of states in which the number of particles as variable gives you flexibility in studying the process to use with a particle number could change for cooking number
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there the chocolate chip operator Giant I've told a plus creates a particle from momentum shaky they plus of Katie What does sign next to sign X is also an operator to some of operatives or what sticks BAGHDAD daggers made about creation operators creates a particle a moment Katie not sorry they plus creates a particle momentum what this I do understand that you have to to remember but in quantum mechanics on like classical mechanics you get that had state states former a linear ominous of a linear vector space that only means that you could hear them subtract and multiply them but would marry numbers so let's start with the back others are not gonna write a string of zeros Oh stands for our old and there don't react with sigh dagger firstorder site to wait I tries annihilate particle but no particle later when irate soldiers gives you 0 no state or but let's say the creation out of their creation object site plus an act on their way to the Narrows follow Warren sigh plus of X where X is some particular caught the value of position some specific value a position but see what it is yachts well we just substitute for write some overcame follows a plus Our K all times eatery IKS X minus site here but subject yet this is an object which represents 1 particle with Lumumba if all I some of you here this is a 1 particle state with momentum Kate what happens if I superpower was 1 particle states with moment Casey With a set of coefficients Ito the miners IKX as anybody remember from quantum mechanics what happens if used to propose or end up state vectors corresponding gold particle of momentum KEE NUA them up with coefficient C minus IKX better remember not Keyes armament you got the right idea but you of these are momentum states this is a moment that what happens if again this is a moment state we a would don't want particle quantum mechanics we might just call this a particle momentum k 1 particles in this in this problem here I never have more than 1 particle actor this operator it gives me us superposition of 1 particle states each state that appears years a 1 particle state there's a 1 particle stay with momentum Katie and told they had them up with each of the miners strike a and supposition state it's a state what a definite position what's a position next to a head up momentum states with NATO the IKX that gives you a position is a state located at position acts as 1 particle quantum mechanics Saleh that tells us there and what website dagger of X sigh dagger of X Creek eat a particle at position acts it creates a sum of particles it creates a coherent quantum mechanical superposition of states with moment all the different momenta only allowed moment all of coefficient minus IKX incidentally what's the probability for different values of all the different values of K have the same probability because the coefficient here and square or do want so this is a linear superposition states with different moment where the probability remember what the probability is a probability would just be square of this the square of this at times its complex conjugate this would be just a state with a uniform probability in momentum space but it is a state with the definite particle a definite position OK this is getting pretty now sigh and be a pluses of K and a minuses of carried which other Florio which up to be interpreted as Fourier decomposition of sigh X Loza up creation and annihilation operators for particles with definite momentum sort of banks and sigh Geiger acts ah creation and annihilation operators for particles of definite position so if you want to create a particle and positions at makes you would not do it with a of candy but you would do it instead side diver of X which you could really write in terms of the case but it's just giving you the quantum mechanical superposition on momentum states that adds up to a position and that's the basic idea of of a field our area where once In relativity it's a little more complicated but if we don't worry about the complications of a special theory of relativity don't worry about me at the quantum field dagger of X is a creation operator where particle pollution at NASA yeah this case in this case there will be a lot fun it is spot all young disbarment circle nothing to lead right eventually we make the circle bigger and bigger and bigger What does that remember the allowable values of k work to Pyay integer over a repressive airspace this space by a spacing to pie male does space as El gets bigger and bigger the
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spacing gets smaller and smaller than eventually you can have any value of K or at least a value of
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Kano state arbitrarily close bite so making for making the rain bigger and bigger and bigger is equivalent to replacing the discrete values the moment by continuous values and what does that entail an equation like this right means that integrated over overcame instead of summer property but it's good the 1st time around the think about it discreetly once you know when you understand that you could replace it by integral became but not that yet OK next step and next question what if somebody it creates a particle opposition X obviously site dagger normal I think at that site dagger creates a particle position X what ozone sigh do general it's part cortex right OK let's think let's think now about the possible kinds of reaction reactions and now mean particle reactions that we could think about but in a language Scitex let's but he is a possible reaction you tell me if this if you think this is a likely particle comes in and when it hits a target with a target happens to be localized very precisely localized act position acts that particle
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disappears and then often Alpha Centauri somewhere in the particle goes off here and around somewhere designer Betelgeuse among the here at some likely will we we called on local violates the rules that a but signals could propagate the arbitrary kind crazy velocities what's what's the right they think of right thing think about if you want to preserve new series of very precise notion of locality is that follow up of particle is absorbed right over here and 2 particles are emitted be emitted particles will be absorbed will be invaded on the same spot now you might wanna loosen that up a little bit for 1 reason or another and say Well maybe only means that there is admitted within a certain region of that but in fact the precise notion of localities in quantum field theory is that when a reaction happens it happens very much at a spot like so how Lumley describe this reaction 1 particle comes in is absorbed that next 2 particles go out remediate from the same point we could use these apparatus to describe more nonlocal process both we want as local process where we write we write particle but the 1st thing that happens as a particle in removed from the system society at point X and then to particles are put back Iain position X that would be signed veteran of acts site wrecks that might act on state with some sort of Beckham state but estate was someone port where the particle in it it was some particle let's try to figure out he is the exercise a pot of gold With more than the case comes I want to know if that particle with momentum is absorbed that position acts What is the final state What is the final state and what Is the various probabilities for the final state have this for that momentum so will we have to do is very simple fact that 1st of all say there's a particles with momentum Kiki that means 0 0 0 0 and the Kate slot is 1 part as a 1st step right down the initial state that more 1 step No. 2 will substitute for these local musical local field operators because they're localized position substitute their expression before trying to find out information about momentum substitute in their expression in terms of momentum creation and annihilation operators so do that at OK sigh of 1st of all ears some overcame and again some overcame means some over the allowable values of k miners Katie eagerly I K X Sysorex what X I put in here the exit which these reactions have 3 what can a what kind of correction could we imagine you give me an example that would make some sense for what's up talk abstract concretely yet we could have a case so then we will have the have operators describing different kinds of particles right now but I mean you you've got the right idea but which is photons now this is not quite the right description a photons but let's ignore photons a relativistic particles is more appropriate number of particles but the 2 North my soul a photon comes in it's an Adam localized appalling point and from that point to photons go off so the 1st photon it is the 1st the photon which we put into the system to begin with what work comes next actually taken this field operator as field operator which absorbs a particle position X and we allow a strike this initial state 0 blah blah blah 1 0 0 0 in the case of the cake place over here this is some of this case is different in this case centers escape different escape he was summing this is the dummy index summation this is some definite Caleb's court Katie Mitchell Mitchell momentum of part of paint but that that's a 1st step may once we work that our will have to hit it with side dagger of extra side figure of ex ante aside Edronax so it's right at an army Keyes a summation index we should use a different letter we shouldn't use Katie over again with he's a summation variables doesn't matter what you call already I just wanna repeat the use of a summation variable over here and call called same thing again so we have summation for the 1st operator summation over a L 8 plus mail he be by L X this will are each of these might decide elections what about the 2nd application of sorry where does that direct the same thing again but again we better use of different summation index fears were not allowed to repeat the use of a summation next twice that would make sense me so we have to repeat the same thing recall the are new summations XKL M RAM doesn't mean Messier wave number and a plus about L minus on I know I am sorry and men and acts are what kind of state does is create let's see what kind of state creates 1st of all he is a big some weights
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terms of the and give something which is not equal to 0 case a abide only if this Katie here is not the same as this case for example of is case of 13 that comes from the 13th slot then what happens when I applied K 1 each of the minus side 1 watch eyes to absorb the 1st particle but there is no 1st particle saying the sector once and only the 3rd team slot is occupied sold only case of 13 will survive or 80 of 13 will survive where hits the state the rule is annihilation operator has defined something for hour or so this really reduces the 1 turn some Asian arts heads not submission warned terms the turn where 18 miners Our case 13 OK initial just carry image can firms he we are a case in this X acting enormous state warm waters is do we went what does the annihilation operated to work its the state it is removed that article with coefficient of 1 song Bittersweet Ray said that In erase the particle and bullets left his estate with no particles times Vitaly I. K initial X but now we have applied we have to create particles summation L. not just exactly the summation that we have here this part of finished with it and just gave us and eat RIA K. initial exit times the vacuum OK Airways where this 1st it puts in a particle of momentum hailed so it puts in a particle of momentum L that's what each of us what a plus a does and what is this wonder but it puts in a particle on momentum and so the whole thing in the end is we we keep that each of ERIC K. initial X he minors I an X minus L X rooms 0 0 0 but now where 1 0 0 0 war hero ever some which 1 to which a tour of sites are occupied William and so it's given me as some only Ellen and now must just replaces look just call us to particle state with a particle of momentum halal a particle momentum him the notation is now I've used a different location and labeling this now 2 particles 1 with momentum L and 1 more momentum him that's a sum over such states the sum has not gone away but it's a sum of Ellen where the coefficient widow widower of functional dependence Iike 1 next month explains RX altogether some money and they'll eat I Katie initial minus M. minus no X kind L. A. but such yet not to creation operators creation operators Comey OK that's a very good question the rule is all creation operators commute with each other all annihilation operators commute with each other and creation and annihilation operators continued if they refer to different moment if they refer to it they refer to the same moment that then they don't commit as it arm this but good the point good good good perfect right this is correct In does not equal a L what is this this what is the probability for different values of L & and comment from here no no no no no consolation momentum he because egg Adam which absorbed photons could require a writer at the head the could absorb any amount of momentum right so there's no conservation of momentum what this is up here is change in momentum initial momentum in final momentum it is of course he and the momentum carried away by the Adam but this function here which is a function of is just a its magnitude is 1 so that means they're equal probabilities for all the values of Ellen and New square it the way you think you would think of this as the wave function for to particle state by the wave function is you're phase and all values Of the moment have equal probability of cake except not quite what about Ala equals him that I made a mistake and the reason I made a mistake is when I'm so let's go to that particular case let's take 1 particular case where Ellen and mother saying course there are many many such cases l could be an end could be anything but I 1 of focus on those contributions where the 2 would delay the L and M of the same there's going to be the center of the factory he said To hand X on minus 2 IMAX that's just because Ellen M. of the scene of be the same factor of the same crime factory but 1 thing will be no where's it what happens when a plus acts the 1st time it made it gives me a particle and and site what happens if a plus act again no it has sustained momentum the it creates another particle but it his me a square root of 2 right it gives me a square at 2 0 after the 1st hack it now
1:04:25
I take the 2nd 1 it looks for a particle momentum in the end finds 1 banner defines 1 so 1 particle present already remember the rule is that if finds in particles you give it a square root of it's good plus 1 squared and was 1 so what's Izquierdo rentals brutal entity or elderly or not tolerate what they lost all of us something part of number with momentum number but that cake and so on right so for the special case where L and him with the 2 particles combat with the same momentum does it it could be that moment the Macoby got momentum could that momentum of the come with the same momentum was a factor of square root of 2 what is it about probabilities down to square square these where the coefficient of state vectors find abilities and not here 2 particles go off in arbitrary directions not the same with different moment there different lament that differ magnitudes of mom and that is what value of the probability but if you produce took particles with the same momentum and the probability is twice as large these particles like the B in the same state these are in all these are sorry these are This is the description of indistinguishable particles right is very much the the description of particles which are not distinguishable from each other so the production after 2 photons in a collision twice is probable if the 2 photons Comeaux from the scene they became more of a different order is if a guy you if if you go there is obvious you no it's not obvious were ordered that With different waters you could always commute them until like amount in order it which all the annihilation operators the right and all the creation operatives does called normal ordering and that ordering the 1st thing you do is you let the apparatus eat particles in the end spit out some new ones so normal order and will complement the saying 0 OK that Baghdad is a the question before that I ask you to keep in mind these example in which the moment quantize for example the movies could be photons in a year and a cavity where the moment the really are quantized so if funded if particle in a cavity with the a moment that are quantized of a particle of comes in hits and Adam and creates 2 photons you will be twice as probable for the 2 photons to be in the same discrete energy level as to be different levels there's only 1 case where to only 1 based on side where down then will be where was l and 2 other were a ALIR White saw focus work work work armed they write so what's your bets but see L & M on different Turkey so let's do that case I that's gonna give you any what kind of factors write your your rights TSUSA work were simply have 80 are too would be a better off twoway dagger of 3 we have that it said that plus a dagger of 3 a dagger or 2 true looks like this 1 more probable but it's not as as much more probable as it would be if that's where root so it's obvious that we give you another example we really see this is on a yacht had actually think you're right that want a different moment that because there are 2 different ways to do it maybe more probable that we give another example where you could see me square roots of end an action on supposing we have a particle which can decay so we have some sort of particle which convict Katie and made of photons described that we did legally described that by just 1 hour onsite Rex but can they want for policy we don't want full time 1 time as photon or do is create a photon point X let's create a photon point text present SSI Gagarin of X acting on the this would be the decay of an object producing a photo or just the emission of a photon from a point acts what does is give this gives some somewhat over moment that Kerry creation operator of momentum Kerry eatery mind 830 plus minus I K X acting on the back of vacuum means the state would know for times for example gift this gives some overcame each of the miners side IKX Kansas State with a particle momentum Katie obviously equal probability for possible moment the case in this particular case equal probability former mentor Kate do something else let presupposes that it was Free existing a particle momentum and Eyal was preexisting is a cavity who was a particle momentum L and hour Adam decays was advocate a nifty photon but there's already a
1:12:50
preexisting photon a moment of our 1st calculation told us that the simple marble situation we have equal probabilities for the emission of all possible moment don't worry about conservation forced to abandon them a wire back our tells us that there's nothing distinctive about any of the moment that they all have equal probability of being but not already of preexisting particle momentum they end that means we have to operate on this year where the particle that already has momentum and as 1 here someplace and locate supposing Katie is not this is place over here supposing case is not equal to aerial this just creates a particle momentum With no square roots of anything because it had looked up at creating a particle of Kate where there was no particle Katie so the square root of just square is it do it creates a state within Ito the miners IKX firms Katie In a else particle is still there and new particle case but now supposing we look for the church in this summer where K is equal to tell them where we get and we get a square root of 2 of them against squared to so this is correct as long as Cheney is not equal to L. we get a state with 2 particles with a wave function to the miners IKX original health particle particles still there was the bystanders a particle of momentum k was made it was mated with an amplitude Ito minus IKX acquirer chemical Umberto Eco emerges here but what about the possibility that OK here is equal to than what happened by a factor of 2 by a factor of 2 because now this aII of Katie is AOL veteran is just a avail already up particle layer momentum and so you get square root of tool that's what these operators billed as these things they give you factors the square root Of the number of particles Of the type each time things act so if there was a particles all over the place here then you would have an inherent probability for producing particles of a kind which already there yeah that's Corbett Court Justices quotes stimulated emission is part of a deal said he has 4th In if a well really particles come out and leave package that really come out and the speaker Mahmoud the states but they In a cavity they come out with definite momentum and and this is a real effect this is a real effect that if there are particles president over filling cavity and cable preferentially happened Inter photons rich were to have the moment that the particles in the cavity more particles you have in the cavity of a given moment the more probable that is that the makes particle will come out with moment our so if instead of having 1 particles In his L slot where hundreds particles in the cell slot that we would get a factor of square root of a hundred more I guess our got square a 101 and the probability would be a hundred more times that the Adam decays into the state photon with the same momentum is at 101 particles so you couldn't figure out more what generally what and who will be as squat stimulated emission it means is that Mr. President of particles of a given will increase the probability that a decay takes place Burke only when the decay takes place with the final moment the D. a slug it to bar for young mothers were not as much as if that were true what we flew back the uh it's not a real legitimate attraction of any kind that's a statistical effect it's the effect of the identity of particles on not particles I'm not all particles have this same incidentally particles common to kind their affirming Iran's and bows out bows ions are particles described as described here creation and annihilation operators with the mathematics of the creation and annihilation operators being mathematics of harmonic oscillators particular square root ends with creation and annihilation operators having the algebraic commutation relations of harmonic Castle father then you have this tendency this is very general tendency that the ball particles you have the more likely it is we will decay or produce particles of the same momentum Our same thing happens scattering reactions scattering reactions them can make particles of different moment if you already have moment the the presence of a certain kind of tend to preferentially caused the products of collusion to come out with a moment of particles and said they I said it's called stimulated emission when particle when a Adam decays 5 in the presence of a ready preexisting photons that enhances the probability for decay our when they're on 0 4 Khan's president and just decays rode the probability that would the coefficient 1 that spontaneous emission for spontaneous emission is the emissions which is unaffected by additional particles or no additional particles spontaneous image of stimulated emission is the tendency for other particles to stimulate emission and make it happen with a higher probability there is a classical analog of this incidentally on if you have an enemy which is radiating he and his classic and you have an electromagnetic waves passing by electromagnetic where you will vibrate please will shake electrons and cause them to radiate with a higher probability than they would've radiated the fast radiate faster than they would have radiated without that electromagnetic waves or us and is that it will tend the make amid electromagnetic wave comin out with the same direction moment the same wave number so it really is a quantum mechanical version of something that was known classic namely just that preexisting way you will excite the system to work to radiate more just because of preexisting wave will vibrate twoyear electrons OK particles of this type which have the tendency rule come out in the same state particles that they don't really attract track that would be a wrong language but which have the statistical probability because of these Querida Van Gogh
1:22:10
like to be in the same state those called bows out they characterized by creation and annihilation operators let's right down the the mathematical rules but mathematical rules of bows on this for each moment there creation and annihilation operators the creation of annihilation operators have the property that all creation operators any moment that Camille but a place of Cape Times a plus those same thing as a positive l terms it carry stroke of Kate equals Al Qaeda RRK arcade doesn't people same thing with be annihilation operators but an annihilation operator where the creation operated a they do not memo always found for the a harmonic oscillator we just want our a miners with a plus we don't want to what what is that this is 1 if Katie equals and 0 otherwise rule is they always commute if we're talking about a different moment they were talking about the same moment that the annihilation operators and the creation operated not commute but have a Warner a righthand side we come right back in the former Delta hello when Delta Kayla's chronic delta of the symbol which is 1 if Kate equals L and 0 otherwise use this together but together with the definition of Siam X but is equal some are a minus of Kanye the plus Errico were forever room I think he is a minor side KEX correctly With that definition this basically defines the simplest quantum field theory side is a quantum field and its job is to create and annihilate particles 1 more 1 more thing about HBO's arms before we go on Fermi arms from Iran's our kind of particle different obviously of course it would have another name 1 more thing about let's look at the sight gag of acts sigh of X integrated all based what is that I know what that object is a would written down Noah was what's would get our butts pretend we do know what it was as if we were out signed off on mechanical operators This is an observable West work it out its equal From the definition of society into a solid see some of our figure 1 on the some of a class of Katie but it's it right I keep getting confused about the pluses and minuses are probably make mistakes bats sigh of aside dagger of X will about sigh of X and what's a Carmine it would make a difference for calculation at a time that if the black boxes I can't fly that it didn't periods it is a case so of this is somewhat another 1 which is some already know which is the annihilation operator minus L further plus I X and this gets integrated over X now let's see if we can examine what that corporate years this is integral BX Miller a X dependency years in these coefficient functions so what we really need to know is what the integral each on a L X Italy I L minus K X DX Betsy into it appears he the ILO X Ito reminders IKX threats this integrated access the integration of peers where we integrate we integrate all over all the space which means over this circle of Lane L so well again if we integrators over star who was on thought the OK he will assess but the bite was there so you get L If K is equal what appears not equal 0 White biggest aid because this will be assigned color signs and if you eat the greatest sign in the colors on and around the close around current integer multiple away you get 0 bore signs in call signs are positive as much as there are negative so what is this thing this it is dealt KAL Kimes knowing that times a leader of the interval that means when we do this integral it's going up take only the terms where Katie equal that's what this means it's gotta give us a factor Big Al 2 accusers failed firms the sum of IKB announces single Solomon not doubles biggest K has to be equal L I want to go be it's gonna be a class of Carey me Minus of Katie Beers member for the harmonic oscillator what a plus and a minus 1 day after patient a plus times ignoring is occupation numbers the number of quanta With momentum case Bakhtiar notes a plus times a Martinez who work at all square at civilians and things a plus Hyundai minus acting on any state just gives you the same thing as the occupation number so a plus and a minus is the occupation number the number of particles of momentum carried where his some Nova case full particles In that if just counts the number particles total number particles so this subject here if I divided by mail have provided by because of this show which appears be back is equal to the number of particles the total number of particles now there are conventions where you absorb this AL into society and there's a more normal could I have been following a particular set of conventions its led
1:31:29
me to have this 1 of the hell out here but that we could leave but basically sigh start time society is accounting quantity counts the number of particles observable its acquiring chemical operator but all it is is the number of particles No. particles may not be conserved so it In general this interval can change with time if particles are absorbed in a mail carrier absorb more created on the other hand if the number of particles was conserved this is a consent quantity so this is something which counts the number of particles apart from this would you given name a good because the number particles appoint expert that not such a good idea because a nor the density of particles at acts right so that's part from a factor of L which I will ignore them it will go away were replaced songs over case integrals sobriquet arm but apart from the fact that sigh star society but just TriStar society is proportional to equals the density of particles and at looks but as a Quantum Chemical Operator threats it that is the simplest of quantum field theories of HBO's army quantum field theory with a few of them made up creation and both annihilation operators sign a size star and you could work out incidentally workout out the commutation relations between side of X and so start X we know the commutation relations of the aid is the size of built of be a use it's interesting to work out the commutation relations of the sun army OK so that tho moral bit by a Garfield as the density of particles Rex now as I said this this whole machinery so far assumes there's 1 species of identical particles identical all the same crime there identical except for the fact that they might have different memento a different position but you wouldn't want to build it into a single quantum field both electrons and photons this might be the field for photon snarl because because of relativity we have to modify a little bit this would not be the field of electrons because electrons on not bows on me you may have to be dealt with differently OK that's that's that's basic theory of HBO's they are described by harmonic oscillator creation and annihilation operators which are the Fourier coefficients yield on they like some statistical they like to come out the same state already particles president that will enhance the rate of production of particles of the same momentum army and recreation annihilation operators have definite algebraic properties commutation relations which have the listed at the whole theory of a very simple quantum field again emphasized that you could use it described the accident as part of the onedimensional Babbitt that the bad news that could be immediately changes but putting in more exit right there nothing special about honesty people there are some people want to go and I don't want start subject poor people going but perhaps we can spend a little while talking about something where you want to talk about but not look at it were there is related to the local pub transparency transparency is simply interference you command thing which is transparent electrons which is trying OK so well that things in nature of flap a lot of things bows on as a borrows armed the Expos underwater robots is things are blows islands granite Burroughs arms on ice was used we say that sign deal aside daggers forth is observable that means that the field society is cannot electromagnetic field observatory could do an experiment that measure when does side behave like classical thing now is a problem about measuring size its components don't Camille with each other it's real imaginary part stone commute with each other and you can't simultaneously measure to none commuting thing signs of science I beg don't commute that means the real and imaginary parts of cyber your how could you measure sign the real imaginary parts community you can you cannot you also can measure Exon P. at the same time Wednesday is or for that matter the harmonic oscillator variables AT&T dagger array even a you can't measure because it has to none commuting a real imaginary part what is the situation let's take the harmonic oscillator 1st in which armed we harmonic oscillator variables behave classically an approximate sends you could measure a car both In a dagger or from ordinary young weather was the situation when the oscillator is highly excited when the oscillator is highly excited and that's the situation in which the number of expectations 8 times a dagger is much much larger than H. is much much larger than the right hand side of the commutation relations you kid precisely measure these things simultaneously which is in the same sense that you could measure excellent P approximately as long as you don't violate the uncertainty principle you could measures the components of fields approximately as long as you don't violate certain uncertainty principles wealth was at practice means in practice means that you committed these things as long as the the degree of expectations as long as the field strengths all large by comparison with the uncertainties large by comparison with the obstruction and measuring the both simultaneously on ways in these times of folk it means lots of photon right when it when you have a sufficiently large number of photons approximately you could measure moving the different components of the few simultaneously within certain areas but those areas may be a lot smaller than the magnitude of the U. themselves so the observable quantum fields but others things you could measure science for those behave like classical fields when the number of Qantas is very large it makes sense when the number of Qantas is very large the Fieldside behaves as if it were a classical wave field it's only when the number of quantity is small that you see discreteness and that's when side does not behave
1:40:47
like possible for you in face of Aceh set up acquire mechanical operator which does behave classically under certain conditions is any measurable quantity or at least parts and components of a measurable and it can be approximately measure all of its components simultaneously as long as the number of photons is much larger than the uncertainties dictated by the uncertainty principle this is an uncertainty principle it tells you that the components of AD in a bag of a real imaginary parts can be simultaneously measures is a poor 8 times a day a their 80 years the number of particles that is roughly speaking the magnitude of a year is like the square root of the number of particles as long as AU is much bigger than the right hand side as long as 8 times a dagger is much bigger than the right inside the right hand side can largely be ignored for many purposes and that's the situation where our are ADB haves approximately classically many of the questions he or is the that they have got a lot of on the left to the a correction Bush it for me I thought I use your day use 0 wait you get always still see from but at the state Zero is a state with no particles it is not the same as the state 0 Frank was this label this state is the ground state of a harmonic oscillator did its normal is 1 it's not the zeros 0 estate is minors Oh if it's it started this is just a vector which i've labeled as 0 is not this Euro sector and a vector space it's the vector describing no particles that is different than the 0 vector very different very different 0 vector is no victory will it doesn't describe anything our This is a perfectly good normalized vectors describing nor particles it sees a piece of what they are if we as state that's an atom operator feel bound state Bell In quantum field theory you over off of us and such yet the I said I guess that is about work about then Richard Moss and by like like it take debt from life that he would design a rack are a more common situations when you have the cavity with reflecting walls case come situation but you could magic john's moving around on a wire closed 1 that would be used for it each limit on video no it doesn't know it doesn't it does it doesn't go over McGovern member states what it says Is that KUB allowable values can you have to be to L kind and where does is in limits the number of states with momentum below a certain momentum it if you were to say the largest momentum however 1 think about his words of laws at the moment never want to allow more momentum than that in this would limit there a number of states of have arbitrarily high momentum so that they can be arbitrarily large but the number for days not all is not lost at Arroyo state test last year you really what they might have the exhibition right that the EU real and imaginary parts of the field another words are the Fieldside 1 piece of it has 8 passes peace Harris a minus I use is equal in order me if you waited outside there but if you and sidetoside dagger of course you get the real part of society and that's a plus plus a minus that's like X maybe that they be real and imaginary parts of the 2 conjugate variables and that the 2 variables the real imaginary part of 2 variables which have a distinct uncertainty relations between them so plus sigh dagger plus ii there plus or minus ii are 2 a distinct objects that the real and imaginary part of society and they correspond to the X of oscillator that's a good question that 3 that is slated calls for fresh yet no the 3 dimensional space but start with 2 dimensional space the dimensional space would you imagine the analog of the circle is the Taurus chorus at the Taurus usually think of Torres Obama off on surface of Burnett but thus tromethamine from mathematicians pointed to work is you take a rectangle spends identified the left edge with the right edge and the bottom edge with the top edge you could think of it as 1st cutting the Taurus imagine cutting the Taurus like that of course we're talking about the surface of the Taurus always you have to retake the surface of the Taurus and we lay out what happens if we cut the Taurus ruled and open up well we get a piece of a cylinder but and now we can't fathom you remember that this age is to be identified with that then we through here an open it up but remember that the top edge as to be identified with the bottom edge saw Torres is a rectangle in which we remember that this point is the same as this point this point is the same as pouring and for example of food talking about
1:50:06
a classical particle classical particle which left the Taurus over here would reappear over here who come here who were reappear here that's the same as a classical particle just moving around the torus it's got to directions of period this city and for this object both components of the momentum the X and the white component of momentum would be quantized so that's the way you think about Torres and rectangle with periodic identifications identifications across the EU because we have field we require that feel that this position to be the same as the field his position at that point to be the same as that a death where state workers space or the object space found now we're talking about space but the only reason for making a periodic is still avoid questions on infinity and and eventually global below make this space infinite by replacing sums by intervals incidentally this emotional 3 dimensional Torres threedimensional Torres is just a few wound when identifications his point is identified well you know you just defied the of this faces identify that face the opposite of the fire barber from the back so Torres's couldn't right can exist in the new dimensions and the notion of a periodic particle moving in periodic space is just the notion of a particle moving within the interior of the but in such a way that when it reaches the edge Of the torrents of red to the queue reappears at least there's flag area your shoes on what hear us back if have 1 ship but they were still were what is they may want to visit 1 of them require a quarter of the particles required was up for the continued they come in they go their creative Niland irradiated beef up that course it is a very very speculative question my own and would probably be no bar that figure as you're asking probably questions about length scales plunking size and so forth and our that kuwaiti beyond what we're talking about here soul most likely quantum field theory break it down by the time you get such scales just probably doesn't make sense Our so that's where we're talking about situations on same modestly scales a minus 13 centimeters and the momentous 20th sent them minors 28 cm but not can't minus 33 where a clerk for now of death a over a budget of almost all practical purposes could take space could be continuous from discrete are it said later its momentum which was discreet not space weather report things on a circle was a moment that decays widow discrete not positions particles position the particles were periodic the momentum with half of the year but we do we assume away from 2 of contract were younger question at a minimum fact use legitimate questions that Fed I thought that blocked it can't be sure that not changes a lot of that the cafeteria a few replaced commentators by Ian by car so the commutator Romanian any BNE sat down at a quarter commutator of will be at its end isometric and that a few interchange a and B it changes citing the in the icon Taylor is by definition is usually written this way be a NTV plus VAT say this for West because it's harder to swallow it a B doesn't have that maybe was BA equals 0 part of the swallows ADB might be physical 0 Italy have remained Healy almost everything is normal a plus and a minus is seen as a minus State Customs it passes it was a past times a pass in the opposite direction it's only a small subset of things which are all I OK if we do it to replace this by the intake country traders almost everything would be all Our a depots BA would equal 0 0 almost all over all the time so a country that but such mathematical manipulations defined Firmiana and firm Iran's are exactly the particles that have Powell the exclusion principle you simply can't but 2 of them and not only the not like to be in the same state they can't be in the same state media a complete opposite to the notion of stimulated emission and if you have a firm Iran a box and you try to create another 1 with the same momentum or saying everything we can to chronicle 1 of them in the state and that has to be reflected somehow by the algebra the creation and annihilation operators there are creation annihilation operators Jeremy but is such that the algebra instead tell you that they like to come out of the same state because they can I from out state her now and you can't put a lot of them in the same state never be his classes look that are made to edge weird if you picked of Ramirez you make above them you cannot put to Fernandez a mistake but if you make them to make bows out fermion to put it to you could put to both arms in the same state but that's not the the same as as putting the premier will or will learn Lewis the lot of next fees and did us right now Hill like next back you free external fields could be described as classical external Yorba Ricker be described as large numbers Kwan and that's a kind of boundary condition that these effects of large numbers of quanta should be in 1 To be essentially the same within small deviations from the effect classical as a cost a lot of this was reduced by saying well somehow of electromagnetic field has described frequent there but when there's a lot of energy reduced the right thing for Lula please visit us up
1:59:25
steam for that ETA
00:00
Girl (band)
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Order and disorder (physics)
Negativer Widerstand
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Dyeing
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Flight
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Roll forming
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Elektronenbeugung
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Spare part
Orbital period
Tool
Negation
TARGET2
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04:23
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Bomber
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Abrichtwerkzeug
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Steckverbinder
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White
Quantization (physics)
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Flugbahn
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10:55
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Last
28:18
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45:41
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47:52
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Fahrgeschwindigkeit
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Photonics
Hour
Miner
1:04:21
Ruler
Integrated circuit
Hot working
Photography
Emissionsvermögen
Year
Order and disorder (physics)
Annihilation
Fire apparatus
Energy level
Particle
Zeitdiskretes Signal
Rootstype supercharger
Apparent magnitude
Lecture/Conference
Spare part
Rail transport operations
Optical cavity
Magnetic moment
Water vapor
Homogeneous isotropic turbulence
Collision
Dolch
Photon
Vakuumphysik
White
Pair production
Yacht
Focus (optics)
Quantization (physics)
Excited state
Direct current
Cartridge (firearms)
Photonics
Hour
Radioactive decay
Miner
1:12:50
Weapon
Analog signal
Effects unit
Bjarke Ingels Group
Harmonic
Amplitude
Particle
Bird vocalization
Colorfulness
Continuous track
Cardinal direction
Tool
Magnetic moment
Energy conservation
Quantum
Speckle imaging
Reaction (physics)
Photon
Single (music)
Electronics
Vertical integration
Pair production
Excited state
Direct current
Cartridge (firearms)
Noise figure
Microwave
Ruler
Hyundai Motor Co.
Hot working
Day
Food packaging
Year
Emissionsvermögen
Supernova
Summer (George Winston album)
Force
Annihilation
Spontaneous fission
Access network
Interval (mathematics)
Cell (biology)
Rootstype supercharger
Lecture/Conference
Ship class
Spontaneous emission
Spare part
Radiation
Orbital period
Black box
Ion
Rail transport operations
Optical cavity
Mail (armour)
Star
Typesetting
Dolch
Ballpoint pen
Electromagnetic radiation
White
Loudspeaker
Chemical substance
Cable
Fulling
Abwasserverband Obere Lutter
Photonics
Bow (ship)
Marble sculpture
Radioactive decay
Wire
Miner
1:31:27
Theory of relativity
Weapon
Roman calendar
Analog signal
Bestrahlungsstärke
Order and disorder (physics)
Negativer Widerstand
Aircraft carrier
Particle
Variable star
Bird vocalization
Turning
Packaging and labeling
Limiter
Quantum
Weather
Magnetic moment
Mechanic
Photon
Single (music)
Observatory
Electronics
Vertical integration
Pair production
FACTS (newspaper)
Doorbell
Excited state
Flap (aircraft)
Density
Cartridge (firearms)
Ford Taurus
Microwave
Interference (wave propagation)
Hot working
Ground state
Sunlight
Schubvektorsteuerung
Day
Measurement
Year
Annihilation
Interval (mathematics)
Mitsubishi A6M Zero
Zeitdiskretes Signal
Gemstone
Apparent magnitude
Rack railway
Rootstype supercharger
Alcohol proof
Automobile
Kometenkern
Wednesday
Spare part
Video
Rail transport operations
Mail (armour)
Optical cavity
Station wagon
Star
Array data structure
Dolch
Field strength
Electronic component
Magnetspule
Machine
Suitcase
Ice
Chemical substance
Classical mechanics
Sizing
Forging
Schwache Lokalisation
Photonics
Bow (ship)
Wire
Last
1:50:05
Weapon
Effects unit
Scale (map)
Year
Annihilation
Horseshoe
Interval (mathematics)
Manipulator
Particle
Turning
Fermion
Automobile
Ship class
Cardinal direction
Spare part
Orbital period
Rail transport operations
Magnetic moment
Weather
Kontraktion
Ship breaking
Electronic component
White
Pair production
FACTS (newspaper)
Classical mechanics
Excited state
Ship
Direct current
Sizing
Forging
Computer animation
Ford Taurus
Electronic media
Bow (ship)