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# Advanced Quantum Mechanics | Lecture 1

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Latin letters representing observables and they are all are for me she operators that act on the vectors permission for our purposes translates or is he quantum mechanical equivalent of real real as in real or complex or imaginary our they are the real operators which are equal to their own permission conjugate but every such operator in the space states is called an observer and presumably there is really a way to observe it we II get values and eigenvectors of these permission operators play a special role Saunders just remind you what it and I give value and eigenvectors if you'll have a Harnisch and operate a Yukon search for sectors let's label the vector what's label it little aII or a big better yet Alpha to run away but with the Greek index what's label it with an alpha In a meeting in operator can act on alpha and alpha is eigenvectors of 80 if the action of aII is just multiply the vet the Alpha my number in this case on user redundant notation saying patience for the Eigen value and the eigenvectors this as the Eigen value the eigenvectors Alpha is an eigenvectors of 80 with the Eigen value out and the significance of that equation is that 1st of all the set of ideas values of an operator of an observable other set of possible results of measures if you have a quantity which would you measure it can take on the value of 1 3 and 7 then the Eigen values of that operator all 1 3 and 7 the eigenvectors of the state vectors of the system for which if you make a measurement we answer is a definite not statistical determined determined stick and the answer measurement is out for a precise Sumitomo remember this summer doing is mostly just refresh your memory now be particularly systems were going to be discussing mostly in this quarter a systems of particles particles or largely characterized by saying things which have a locations based on their relocation space for example of spaces one-dimensional then a particle is the thing which has a coordinated X located some exit basis three-dimensional the particles move around they have x y z himself what but to write formulas for the case of 1 dimension but you would 2 4 3 dimensions a moment old started that's going back there were 1 point that I need to emphasize that every Braun and every cat but called the brine much codified it was not necessarily the particular brought goes together with siding with the kids sorry that every player like that there isn't in a proper was a number that numbers in general a complex number and it is represented as the inner product in a proper Edison number it's a number for every pair of vectors after it represents the kind of product of the 2 vectors its is similar in its mathematical structure the dot product between 2 vectors of ordinary space but it's a more abstract object at in a park between 2 vectors and plays an important role in the logic of quantum mechanics in particular if 2 states are distinguishable uniquely by the properties that somewhat observable that you could measure which would be different in the 2 of them but supposes some quantity that you could measure and it is definite but they answered in the state side is different than the
state fiery then owed them would say those 2 states of distinctly different there's no chance of confusing them is a measurement you could do to distinguish them in those states are said to be orthogonal orthogonal means physically identifiably different end mathematically it's the a statement that the 2 and inner product equal to 0 so orthogonality is a fundamental property of relationships between vectors which says that they are different you can't confuse them but you can confused but you shouldn't confuse them OK that when it comes to particles the most important observable is the position of a particle so let's just discussed particles moving or white wine is the X-axis the location of the particle is just the value of X an obviously X you should be thought of as an observable in particular there are states which are labeled by for our value of position but court action north this is the point x North over here I'm just using anorchia they indicate of particular position there are states are which labeled by X which have the property that if you measure the particle particles position you will definitely find it at text want next concept Oh and of course state vectors for different values let's call them X next prime of exit next prime are 2 different position X and X prime clearly those 2 states Our distinguishable operationally and by measurement distinguishable of eastern states father now take any state any state whatever of this particle citing its inner product with the state representing the particle at X is called the wave function of the wave function of particle and threatened sigh of X it's a sign that goes into the Schroedinger equation strutting sire believe Schroeder was the 1st for it to work the court sorry sorry and the meaning of the wave function is closely related to but not the same as the probability that the particle is at position x is an arbitrary function well a completely arbitrary but some functions What does it represents sigh of X Siletz is closely related to the probability the fine the particle the position next but it's not quite that the probability itself for its P E of X is the product of the wave function plans its complex conjugate wave function times its complex conjugate is positive for any number of any complex number at times its own conjugate is always positive probabilities are always positive sign Amex's are generally complex numbers they positively commune negative can be imaginary vacant B O every light sold it would not make sense to say that sign is a probability but it does make sense to say terms stars a probability that I Pepsi most important profitable particle that has a position it has eigenvectors which represent particles of known position and we can construct of course a position operated a a position operated just multiplies sigh of acts by exits itself ex of the idea position important observable for particle Romania certainly observables for particle but the other a particularly important 1 is called the you what the momentum the moment classical mechanics ending quantum mechanics positions and moment the come together all before we do that I should decide should discuss the EIB issue of a particle moving not just in
1 dimension 1 happens of a particle is moving in 3 dimensions Nova words are a real particle end it has the real observable positions the 3 components of the positioning the ex-wife if you like but you could rotate taxis our you could just think of the position as a point in three-dimensional space and that case who just think of X he was appointed three-dimensional space we could fill out we fell out the equations by saying there are state which represent particles at 9 locations In 3 dimension x y and z this represents a particle located at a point in three-dimensional space heater ex-wife as they enter the rules will be similar very similar the inner products of ex- prime why primacy crime that would be zeroing if is this not if X is not a Quebec's Prime and why not a good 1 but if the entire polling x y z is not the same as the point that far more prime G. prime then bizarre observably different positions for particle in the states are orthogonal to each other I'm not I'm going to suppress the wine industry but keep in mind that the position of a particle depends on the dimensionality of the space that would talking about and in the real world spaces three-dimensional OK for each component of positions there is also a component of momentum and classical physics but nonrelativistic classical physics the momentum is just romance times that component of velocity a quantum mechanics the momentum is also blew observable is also represented by an operator by mission operate permission operators be thought of In the abstract as objects which act on mathematical that vectors there's the space of the states although they can be thought of more concrete Lee as operations on wave functions he the 1 you can think of them concrete as operations that you watch what they know about X itself what is that represents be operated X well it's just taking the state vector sigh of taking the wave function and multiplying it by X if sigh have X as a function represent things beat and the probability amplitude sigh of Texas called probability amplitude away functional particle then if you want to apply the operator represent big positioned the size of eaters multiplies sigh of expects of our the eigenvectors of X on a wave functions which are highly P very very narrow Dirac delta function in the same way we momentum operator no I'm not going to explain the details of this you go back of lecture broached the world or whatever lectures themselves what lecture was it that we talked about momentum everybody remember part this case a yet A 8 8 8 it was the track and remember with great the grief Charita it OK so these momentum particle is represented by an operator called P a key also does something what does the size of actors it differentiates it it differentiates a with respect X but not quite the real thing is that multiplies it by minors all right the complex number there's a factor of monks constant age Bora and differentiate Beebe IDX that is being operator that acts on wave functions that represents these are the observable momentum Everybody remember this did we refer it OK now what about the eigenvectors of momentum the eigenvectors Of position a Dirac delta function their functions which are Zero everywhere
except look I mean value position another words the eigenvectors representing a particle known to be a X nor is a function which is 0 everywhere except the next marked with very high in a narrow but not worry about precise mathematical definition of scalded Iraq built a fortune 1 and it's not too hard to believe that the wave function particle is so concentrated at a particular point of the probability for finding it is 0 everywhere except at that point that of course makes sense that what I these states for which the momentum over what wave functions correspond to particles with definite momentum and also got by solving the eigenvalue equation the eigenvalue equation is that P whatever it is that we know what it is what's right performer what what's right beyond the abstract equations 1st appeal on society it is equal to Littlepine signed the adding value being wheeled not the Philippine nor is the value of momentum that because particle has end in this particular state leader the momentum of the particles known to be penalized this equation translates just before placing P minus age body by DX minus each broader side by X is equal to pee not the number the article number forums sigh of exercises from genomics diuretics and this is easy to solve this issue whether these equations which says that derivative of something is proportional to that same day derivative side apart from a numerical number the numerical number we can get mulled during the ice age promised side will reappear on this side after this the equation of the type that represents an exponential derivatives something is proportional to something itself and the solution of it is easy for the R P naught X divided by plant constant now I often give very tired of writing parks constant and often just equal to 1 was sometimes do that In fact from road while tried attracted for the time being but later on Monday just drop it end I think by the time I do drop it will done enough times it you know where the right there right place where question that in the 2 television disrupted the would With say that would say those are a message I did Victor got at the eigenvectors but that's ideas and eigenvectors of P argue that not the blue little holds any think anyway functions eatery IPA piano were here no mostly functions don't satisfy this equation don't exactly which refer basically this way from ROK that you notice of course that is a huge difference between the eigenvectors of position which of his narrow little spiky functions and the eigenvectors of momentum in particular let's take the probability that's associated with each the VIP nor X that's multiplied by its own complex conjugate what happens if you multiply this by its own complex country you want Ito the something firms minor size some things just plain 1 so this wave function in the 1 which represents a particle at it would definitely momentum its probability distribution smeared out over the entire wine but complete contrast the eigenvectors of the particles located the a definite position is highly concentrated in infinitely narrowly and this of course is a manifestation of the year of the uncertainty principle our if you know 0 rout if you know the momentum of a particle then its position is completely uncertain and likewise if you know the position of a particle its momentum is also completely I'm certain refer you back to lecture notes our for that purpose as basically quantum mechanics and that's and it is what this is just momentum them is our only if they expect fog delayed probability is no probably at the probability for position what given that the momentum is pinot are now an experimental is going to do an experiment not the check its momentum was going to do an experiment which measures its position different kinds of experiments In that 5 5 is the fact that probably aptitude for that Fletcher if he argued that right now there is also a motion of and the amplitude for finding different momenta and I'll just remind you that connected the Fourier transforms promote would do that now if we need Fourier Transform will come back to it at the moment our Fourier transforms and the momentum representation this is called a position representation representation of state vectors by sign of X is called a position representation it was also a momentum representation but I I would get to pick us too far afield right now yes all of the position of the outside of acts he said we side with what is that was there it was it looks like the square of dope function which is higher than the dealt but concentrated at the origin of the concentrate the origin in our we have not talked about normalizing wave functions making the area under here equal to 1 that's a thing that we wanted to end herb but as I said I hesitate to go into any given thing too deeply because who warned the doing the whole class again so he days that although the call you can you can represent them by column vectors words but there be a little awkward to represent expire combat the X takes on a continuous infinity of values where's the column vectors useful quantity which takes on some discreet the you could think of it the formerly head is some kind of continuous column right there that's the but for the cat sectors and the broad vectors you can think of as rogue sectors OK but turf works only need that our I'll remind you but did you will have gone back studied the lecture you won't be OK now saying how things change with time things do change retiring the evolution of the system with time as a special case of transformations that you couldn't do well on the state of a system that glasses also touring classical mechanics media over most of the system is represented as a transformation in phase space a transformation of the phase position in quantum mechanics me evolution of a system is a special case of a transformation that you couldn't do on the space states taking
minus i Epsilon H was the general equation epsilon was not small incidentally of epsilon is not small we might want to quality and might give up calling at epsilon epsilon is usually reserved for a very small quantity our we might try writing you was of tea but this wouldn't be right a large time it wouldn't be lodged were from large firms but so let's see if we can improve it a little bit well you know I think I won't spend time now I'll just tell you the answer the answer the fist really should be written and E I H T now what the media for me just say a word or 2 about exponential if you have an exponential he some small number and let's just call the small number of Epsilon where epsilon small kind an ordinary number and nickel ordinary number little H epsilon times H is a small number of very small epsilon eagerly epsilon age it is 1 plus epsilon H. that that formal Taylor series expansion of keeping only the 1st assuming his 2nd term higher terms or negligible cake now suppose you want to take you wanna make epsilon little bigger instead of just Epsilon I wanted each of the 2 epsilon times in which each of the 2 epsilon times age it is just the square Evita the Epsilon H. Ito the tomb time something square so it's equal to 1 plus epsilon each square supposing you do this Over and over and all we which incidentally is equal to 1 plus tool epsilon H plus epsilon squared each quarter toe but this is still an approximation this was an approximation of this just below approximation of it is a theory that if if you do this all marine over enough enough times let's say in that time he epsilon times In The times such that epsilon times in his mind that no words epsilon times In epsilon isn't infinite personally small number but I do enough times so the 18 times epsilon is a finite number mainly T dues over and over what have excuse me your back but still it this way 1 plus Absolon H and I do this a number of times a large number of turned in such that in times epsilon equals fit well I get going that's a binomial it's a binomial expansion but also a construction of the exponential function Our it's a rigorous theorem that as epsilon gets smaller in a gets large situated in times of storms kept fix this just become each 30 team so Major but G for me 1 person thought that I think it's right abuse right right and the way you could just understand it is by saying that the small Absolon arms 1 plus epsilon H is the same as eatonii epsilon age and take Italy and power the Epsilon just multiplies the end and becomes OK so I will tell you right now these generalization of this for a finite kinds 1 my assigned here 1 minus H T for our side he really becomes Ito the miners HP and this is an important factor but a small intervals small epsilon we could just be consistent and work toward a epsilon will find out everything we need to know OK so there's a concept now of a Hamiltonian let's see if we can understand this equation a little bit better Armed let's say take the wave function society of state vector state vector size time times keep plus Epsilon I am starting at time t with a certain wave function and I'm allowing it to change over kindly time interval What is the answer to this to this In that is he called me for the H Epsilon in times of society at time t see what I did minus plus sorry my legacy Starting at time t let up to date by a small little time Epsilon the role it is we multiplied by you are but are you off epsilon word updating by an amount of time Epsilon and that's equivalent the multiplying by Ito the miners died at age epsilon website so this is the rule for going from a time when neighboring card for now let's right that down as 1 minus I epsilon times 5 teams and that's right beside America forgettable OK but and sunny areas society he plus so what now multiply out 1 time sigh of sigh of key but let's take them all with the left hand side I don't give a sign of plus Epsilon my sigh of key just term coming from 1 of the term coming from 1 will give us the difference between sigh of T plus Epsilon Miners 5 we still adorned with building a differential equation for signed Sciotti plus epsilon might decide if is just equal minors I H Epsilon and finally the bye bye Epsilon OK what's the left inside left-hand side is the kind derivative of the state vector site and the right inside His just might decide each side focus of that leads us finally to assuring a equation the Schroedinger equation says that side by side by sees this could be a wave function or it could be the state vector I will leave the notation a little bit abused I won't bother
be the same for the rotated or the on rotated another way of saying it is I couldn't tell whether I was rotated out Europe they mean just go away I can't however was rotated because the equations that govern my metabolism my internal structure and so forth are exactly the same as have and that rotate rotated OK so replaces symmetry is 1 example another example as translations symmetry to take me in outer space and you sit back and say how I feel find you take the same state but you translate me over by a meter U.S. me how I feel I say exactly the same as before 2 symmetry on the other hand there are situations where translations may not be a cemetery if there was a year of furnace over there fire fire and you put me over here and said have you feel if you'll find translate Leonia field fine so the presence of an object may break translations of trees on the other hand if you think about a minute later what if I really translate Everything and defenders then we restore the translations of tree so symmetries our operations are there all our operations that you could do want states of systems and if the operation doesn't change the equations doesn't change the properties of the system doesn't change the way you described the system then that transformations called asymmetry for a free particle fervor NEC ordinary system without outer space far from anything else and you translated that's a cemetery you rotated Baptist Street what his queasy change its dimensions by squeezing it that's generally not circuitry detector crystal over rock salt or something new squeeze down compressive by a factor of 2 doesn't behave the same way are so the all kinds of things which not trees nevertheless they still may be operations that you can define but they may not be cemeteries OK let's see let's talk about the condition but 1st of all private quantify idea we have state let's Janek typical state of the system that were interested in his core side will operation on that operation might correspond to rotating system and my correspond to translating the system or my correspond something Alderlea than that like squeezing stretching it but doing something through it and represent backed by a In the operator V. I use V because I've already use you but intended to be a unitary operate a white military apparatus well if I really have the symmetry and I have to states which are different in each other I apply the symmetry operation I expect them stay different if there are 2 states all of meet happy me and said the different from each other they're observably different there's a visit observable Duke measure the happiness of operator and the different you then wrote take the state out in outer space the distinction should remain so that means that orthogonal states should remain orthogonal when you do a symmetry operation and that says that they operate a V should be unitary Of these unitary represents vehicles a dagger a 1 and I'll use you will because I'm saving you for a special case Of kind translation of evolution Okatie now let's suppose Our job is Wilbur tricky soulful as you lose and his it's it's quite tricky picks me half an hour each time I tried to do this to get right but it's also very very simple wet suppose that a wave function or a state vector them interchangeably but scored sign 1 under kind evolution become this site to or if I thanks I won and I allow it to evolve with time that represent that will you hear your represents time evolution does not obvious something else and if I take so I won't allow it to revolve for a certain amount of time it will become size of the U. another way to say that over here is that you'll on sigh 1 is equal to site as evolves I won from time and you get site now let's imagine the transformed version of this transformed it could be by rotation it could be by translation by some symmetry operation What does this equation safer transformed things well let's call is what's called a side prior to many the keep in mind exactly what's going on here at 1 end to represent some initial state and some final state evolution from 1 to another than under Prime represent the action of the symmetry operations which could be rotation could be rotational could be translation OK here's what I maintain if there really is a symmetry and warned Prime if we were talking about rotations and sigh 1 Prime is just the rotated state of sigh Warren sigh 1 Prime will evolve through the rotated version of side too but the that says if I of all myself over a period of time and I'm in some state may get some other state I go from happy cannot so happy then when I wrote myself I should also find under the same evolution that I go from the rotated happy state of the role of the rotated not so happy state that really means that have a symmetry that found that she evolution of be transformed state behaves the same way as the usual Stitzel Buttrick OK let's see if we can make equation nervous now arm yup 1st the 1st here is just be equation site to use EU sigh 1 of Africa's 2nd equation let's go to the 2nd equation the goal from unkind the prime you multiplied by V so the 2nd equation here says that v it serves that I'm sorry is always confuses me wrong but but if says the flight 2 prices is equal to and sigh want go from
wanted to which is kind of erosion the girl from trying the prime is V uses symmetry operations are I time it he I want a story however you will site 2 0 is you times sigh 1 Prime and sigh warned Prime is NYTimes that's the right hand side the left-hand side is V on-site tool but site low His you'll I 1 so left a perpetual confusion that the that a is sums me every time I tried to do this on the blackboard what it says is that of a thing the symmetry another word preserves the way he ever evolution takes place the transformation which preserves be kind relationships between sectors then it says that he kinds you must be equal to you times read this of course must be true affect every state of its Toby a truce symmetry of nature the true symmetry of nature rotation symmetry for example is not something which only applies a firm stand vertically up bright should apply Frank any particular state of the system and so if Vita sends you on any sigh 1 is equal to you times V on any sigh wonder what it says is the symmetries of nature of their operations ease unitary operators which continue with the time evolution operator world won't brawl yesterday to thank for all time if you will see its enough food to be true for 1 time and will be true time yes that's correct OK so what you Intel a operation things you could do to the state vectors which preserve orthogonality which commute with the time evolution operator aura symmetries that preserved the evolution of EU evolutionary relationships as it is today exactly the world's of very simple and they make perfect sense Endo yes us exactly words OK so how we identify symmetries we look for operations which commute with the kind of Aleutian operator now remembered that the time evolution operator is itself take the the limit of very small kind evolution just analyze a little bit says were 1st of all says that he can't 1 plus Epsilon H 1 miner said former each 1 9 aside for each is equal to 1 minus epsilon H. This is 1st order again Wednesday and just expanding about are the 1st daughter was good enough accuses everything we really want not 1st of all V times 1 is equal to 1 times was canceled and what it says Is that minus i Epsilon each equals Midas Epsilon agency are canceling out the minus side epsilon it says that he commutes with the Hamiltonian OK would cancel out these things might decide Epsilon a symmetry is a unitary operation which commute for the Hamiltonian any military operation which commute the Hamiltonian is a symmetry in symmetry is a unitary operation which commute but very abstract extremely abstract and ought to appreciate it we have to do a couple of examples will do some examples of a moment but let's just focus on that Terry operations on the space of states which commute with the Hamiltonian our symmetries how that will be limited but that it will require please help yacht a symmetry whatever symmetry is it preserves logical relationships between the vectors with the assumption that assumption that the symmetries of nature take orthogonal states for Cardinal states that bro seems plausible a symmetry should not take 2 states which are distinctly different from each other and act on them to give states which are not distinctly different from each other you wouldn't call rose tree not mutually exclusive thing should stay mutually exclusive OK so we don't have India abstract mathematical definitions of any symmetry operator of any cemetery but with this overcome much clearer when we do some examples cemetery try me a trader after each with D is equal to 0 we remember from 2 quarters ago what the commutator of they were the Hamiltonian is kind derivative your soul with this there is so whatever the operator it's conserved conservation of a certain quantity in this case V mean that a commuter the Hamiltonian is the seen as saying there's a cemetery is seems before classical mechanics what's the connection between symmetries in conservation laws called in classical mechanics 12 lithosphere right it's much simpler and court the mechanics much simpler in quantum mechanics are we don't need any fancy Lagrangian not anything else always need to know is that time evolution commutes with the symmetry operation itself and that to be ready in 2 different ways they can say that doing this symmetry operations well it says works can be red 2 different ways but the rebuttable saves and OK that's now let's go 1 step further there are different kinds of cemeteries for me give you some examples of 2 different kinds of secretaries discrete symmetries and there are continuous symmetry is a good example of a
X another words we can write translated wave function Vatican right this way whatever whatever is we have to get our views yet aware of is on sigh of X to give us sigh of X plus a as easy that that's what every year that he does in this case and act sigh acts and it translates the wave function by amount that so what now epsilon the small we can also write that this is equal to sigh of X plus a derivative of sided with respect to X Front Epsilon Saleh where he does Ford V onside drugs is it gives back the same size plus something proportional to the derivative that suggests suggested proves the Olivier's is the unit operator 1st turned here Exxon sigh of Exeter givebacks ibex plus a small change in the small changes epsilon tying the derivative operation I haven't written on extra Carrai on X V vii is an operator it's the unit operator plus at the 1 times the derivative operator we can act on any wave function where will give us is the wave function infinitesimal we displayed now about decide by X What about Beebe IDX Debye IDX is an operator it operates on wave functions a linear operator it's connected with the momentum operator remember relationship the momentum operator he was minus I age body by DX riots over where we were doing right multiplied by applying here and abide by each differentiation is I over age Barre Times momentum so what's right wait 1 plus Epsilon Phi Epsilon over each times the momentum we don't have an example of infinitesimal symmetry Operation Infinite testimony despair will be chef and we found that when generator is the GE where was we have to get is 1 letter-size epsilon g in this case I would have much preferred that I defined things of this came out we might this year but lost his that a results were right on the last strike and if it doesn't want tried so almost say I substituted I think I got this right away are yeah that's right do you disagree with this contract that now wore on laughter I think shift the wave function variety which is still a function of the right going minors I'm here now the I made a mistake In defining a shift to the right the shift to both suck it should X plus epsilon but sigh of Expo society X minus 1 so this is actually a minus sign here if you track that rule will find our as a young a minus sign here OK so it identifies for us what these symmetry generated G for translations what is it it's some momentum divided by plant constant momentum the buyer but plants constant Wickham right the generator of translations use the language generators are generator extra lotion if this PT sub X P Sebec strictly speaking divided by party we drop Clark's constant there will drop plants constant and as a matter of fact the definition the official definition really doesn't have formed constant but calm as an example as an example he of X now is Felix is the momentum along the X axis conserved pens on the Hamiltonian it depends on the Hamiltonian so let's write down Hamiltonian Hamiltonian for a free particle particle moving in empty space with no forces acting on it H In that case it is just a squared all the away at the moment the square divided by twice the man if this is a one-dimensional problem than a really is just P extras the x component the momentum as the Expos beat Hamiltonian commute with PEX what becomes equivalent to the question of whether PEX commutes with PX squares yes every operator commutes with every function of itself POX commutes with POX squared POX commutes with any function of p of X sold for a free particle moving along the x-axis we 1st of all find that the momentum is concerned and of course we also have a translation salutatory the translation symmetry is just a fact of a free particle will behave the same way wherever you start started here and get sticky and then starting the starting point just everything would translate OK so that that's an example P is both the conserved quantity and generator and translation was more a space if there are more directions a space than there are several cemeteries you could translate along the x-axis you could translate along the y-axis you could translate along the z-axis in that case the Hamiltonian becomes PX square was PY school in order to land to land and the translations symmetry along the x-axis is still just PX translation along the Y axis generators P Y and so forth in each of them commute for the Hamiltonian services an example of the connection between symmetries in conservation laws which is very simple the next time you sucker about out of time the next time we will do rotations trees cases symmetry is much much more interesting it's much more involved does and it will bring us the subject of group theory some group theory but the group theory is completely trivial next time we will talk about rotations of space what's interesting about rotations in space is they don't commute with each other a rotation about X followed by a rotation by wine is not the same as a rotation my wife and that will give us the idea although none commuting cemeteries symmetries collections of symmetries which don't commute with each other and refined those they have power they have real power tell us about all kinds of physical properties of poker from external pressure for more
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Brennpunkt <Optik>
Waffentechnik
Eisenbahnbetrieb
Dielektrische Funktion
Niederspannungskabel
Rootsgebläse
Elektronik
Kardieren
Übungsmunition
Elektronische Medien
Steckdose
Werkzeug
Großtransformator
Zylinderkopf
Nissan Sunny
Jahr
Kit-Car
Absolute Datierung
Lineal
Wasserstoffatom
Generator
Schreibware
Messung
Hochofen
Tunnelstrom-stimulierte Lumineszenz
Magnetisches Dipolmoment
Mechanik
Sensor
Mechanikerin
Bergmann
Fahrzeug
Konfektionsgröße
Feuerwehrfahrzeug
Begrenzerschaltung
Jacht
Minute
Dolch
Teilchen
Schlauchkupplung
Kristallwachstum
Rotationszustand
Umlaufzeit
Elektronische Schaltung
Druckluftanlage
Kristallgitter
Schreibschrank
GIRL <Weltraumteleskop>
Abwrackwerft
Steckverbinder
Array
Stunde
Eisenkern
Kaltumformen
Waffentechnik
Eisenbahnbetrieb
Energieeinsparung
Übungsmunition
Werkzeug
Zeitdiskretes Signal
Großtransformator
Jahr
Halo <Atmosphärische Optik>
Mittwoch
Fliegen
Münztechnik
Manipulator
Direkte Messung
Kugelblitz
Mechanik
Mondphase
Irrlicht
Sturm
Feuerwehrfahrzeug
Leistungssteuerung
Rückspiegel
Teilchen
Rotationszustand
Vorlesung/Konferenz
Speckle-Interferometrie
Klappmechanismus
Raumfahrtprogramm
Brennpunkt <Optik>
Licht
Eisenbahnbetrieb
Bett
Dielektrische Funktion
Energieeinsparung
Proof <Graphische Technik>
Elektronik
Übungsmunition
Gleiskette
Elementarteilchen
Zeitdiskretes Signal
Reflexionskoeffizient
Großtransformator
Jahr
Zylinderkopf
Ersatzteil
Lineal
Anstellwinkel
Generator
Magnetisches Dipolmoment
Direkte Messung
Elektronisches Bauelement
Konfektionsgröße
Diesellokomotive Baureihe 219
Eisenbahnbetrieb
Erdefunkstelle
Rotverschiebung
Leistungssteuerung
Druckkraft
Feldeffekttransistor
Übungsmunition
Teilchen
Rotationszustand
Regelstrecke
Gleichstrom
Römischer Kalender
Druckfeld
Gasturbine
Jahr
Lineal
Schreibstift
Kontraktion
Familie <Elementarteilchenphysik>
Ausgleichsgetriebe
Generator
Steckverbinder