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# The Theoretical Minimum | Lecture 4

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let's that's gone fact Watson let's forget for the moment the spins system will come back to it the short on probably tonight fact definitely tonight but let's remind ourselves I think I wrote down war principles of quantum mechanics less time and I wanna right now again and the band move on to a 5th principle In 2nd not all independent power it's easier for me to explain them 1 of a kind that the tribe arrived from the minimal absolute minimum number of them and of exhibit particular advantage is doing or storm was right there all the principles of wrote and the first one is that observables observables other things that you measure that could be called Miserables and if I had it was up to me I would call the measurable but about so-called observables but observables are represented on the issues equal sign indicate that observables are represented by permission operators are used a symbol sales for the moment for permission operated by operators now all linear operators are permission and it will not always be the case that the symbol Ill will stand forever permission operator but right now Els stems from her mission operate any commission operator every observable is identified with permission operator that the question of whether every her mission operator is a thing which can be measured is it generally hard questions after their home what can and can't be measured depends on what kind of materials you have available all kinds of things generally the abstract ruled that for the full ruled is it any her mission operator is identified with observable and someday somebody will figure out how to measure it OK but the the other way and any observable identified with permission operated that's for sure I didn't values of mission operate would call them Melendez Lander is eigenvalue values represent the possible numerical values that il can exhibit when is measured pace cell eigenvalue valueless hiding in value I love L which equal land they represent possible observed tour possible outcome of experiments possible outcome of an experiment to measure L solved L. measured and OK I think we have a 3rd 1 long line he young
physically distinguishable state such as see what physically distinct distinguishable state me on certainly opt in barriers are physically distinguishable and the meaning of saying 2 states are physically distinguishable is that exists a measure of some kind and you couldn't do that can tell you the difference between an upstate and downstate another words somebody hands you a spin from our pocket and say I created this spin prepared In the state which is you but or down can you do an experiment on ambiguously County which 1 yes you took your apparatus and towed pointing along with axis and you make the measures if the state was up you'll get plus 1 if the state is Baron you get minus 1 no ambiguity the same is true for a left and right if unless different thing that you would measure your measure the x component in Europe took juror apparatus along the x-axis but saying the same deal on the other hand supposing somebody that same person came along and said Hey I'm going to give you a spin derivatives and that's been I I even prepared but or I prepared it right I am not the tell you which way prepared at you which way I killed the mining apparatus when I did the expand when I did the preparation I'm just going to tell you it's I left sorry it's either up all right can you do 0 experiment which will uniquely tell you the difference the answer is no 0 for example supposing you decide to measure the spin along the z-axis won't stop you get plus 1 it is rife with 50 percent probability you get plus 1 so you can be sure whether you are what you know whether you are alright what still you would have a probability the idea School of Young who would not be ambiguous right OK so the next Osterberg USC deaths was there yes we heard what excesses then he should rarely accessed 1st said the new ahead if you measure the X-axis 1st you'll find something out it but you could have found that exactly the same answer it was along with case there is some probability I'm getting the same answer whether it was up or right is no experiment could do which unambiguously Willow will determine which awards that leads to the notion of gonna try to give a more precise definition of it that leads to the notion of physically distinguishable states by which is meant that there exists an experiment or a set of experiments that can unambiguously determine which of the 2 states you're talking about the Mets postulated that physically distinct physically distinguishable states are represented by orthogonal vectors physically distinct at distinct distinguishable center believes you correct distinct physically distinct states and plight orthogonality but I say observables are permission operators I could just said linear operators with flavored linear operators for the moment but let's and a postulated that the result of every experiment every simple primitive experiment was a real number was a real number it's a complex number you really means the 2 things to independent things were measured actual results come out of your apparatus the commander or your were the needles under apparatus the real numbers so a land that passed you with the results of experiments always real numbers and therefore the the values of observables Our real numbers that does not prove that the apparatus of permission enough none of the other candidates thing is that the various eigenvectors with different Eigen values physically distinguishable states orthogonal their posture and that tells you that observables are her mission operators that's the her mission operators call was I didn't values were always Rio and who was eigenvectors or orthogonal that's necessary and sufficient our store this
1 said not all completely are independent of each other 1 1 2 3 and let's right number for now our forced out hit it well at the behest of arms dealer was for work they're not the probability the probability principal objectives probability principled followers also called Bourne's rule that Max Born warns rule which is born born rule but for probabilities so far None of this has much content be edit content will has content but there are the real bite here is our for the prediction of probabilities for various experiments the board rule is a rule for Qatar calculate probabilities of Borel says if you will if your system has been prepared and state 80 per assistance and 80 end humor measure L the observable AOL 9 using interchangeably police physical idea of the observable with the operator which represents I'm not going to make a special language where they operator will always be call the operator and observable with Ill saying they'd be becoming pairs fuel and the system is prepared in state and you measure fails you measure L B outcome is going to be 1 of the Eigen value was so the only question you is what is the probability that you get the answer land for 1 of the specific eigenvalues against areas the probability of probability that you get result the a probability that you get out of the various Oregon values that you get land is equal the squalor of the inner product of the state vector of the system was in with eigenvectors corresponding to normalize eigenvectors all victims of normalized now our the inner product a with land the square and a little and under the enterprise sometimes incidentally in a product is called overlap it's a measure if we send an orthogonal orthogonality is complete distinguish ability a unique and a complete distinguish ability and a lack of orthogonality represents to some extent the inability to make a clear distinction between 2 states yacht ripe services this is not a question it will honor us by CUC let as that no and are right notation years every Eigen value is associated with the eigenvectors of the Freddie vector notation is that the vector labeled by is eigenvectors associated with the ideal value land that to the point during the your on the problem with trying to be a purist about patients is that the patients get very complicated and make it so complicated and difficult to read To many indices too many different letters you have to use so yup sometimes we use a slightly excuse the word bastardize notation where around where we fully assembled for a victory with a symbol in this case ID value that OK home yes that's the probability principle rule and that's just look at from it you must Hawaii the square why the square of the absolute value well in general these overlaps are neither positive not even real in general overlaps between 2 vectors and not necessarily real and sank in general and not become we're talking about complex vector space and of the components of the vectors are complex and generally in a product will be real little alone positive on the other hand the squalor of the art or are absolute value of the Square Square the absolute value of be in a product that Israel and it's not real it's also positive so this has a chance to be a candidate for probability whereas In the product itself doesn't feel the product itself is called the probability amplitude the probability amplitude of things that you square In the existence of absolute value to compute a probability the a justification for these principles and end his experiment on the other hand you could ask how much kind and still make physical sense Out of the predictions the instead is nobody has ever found a way to change the rules of quantum mechanics and still preserve a reasonable reasonably logical structure so we'll eventually get familiar with these principles there that principles several possible that 1 actually well I'll take yes L it is an an ideal of lot are over sales if is if they the starting state of the system happens to be and eigenvectors but it will only have product with 1 of the eigenvectors erupts again is a good question sir they say they all kept land judge show yes yes the chart this but when there is I lot still the rule in they are but what your right arm give you 2 answers the 1st since it is don't worry about the case where the Eigen values may be the same because it's very special but that soccer enough for me to say would we be write as rioters if you want the probability forgiven value of lambda and more than 1 eigenvectors with the same value of land that knew add them you simply a head the sum of the squares of the year the probability edition on Serbia the questions some the river god yacht please the that too yes they knew it and you take the sums of squares of the some of Scripture us several different eigenvectors all the same I get value 1st all you make them perpendicular to each of you can always make a perpendicular to each other no not even if they were same Mike in value our visit the beer market let's go back to your the eigenvectors with different ideas values are orthogonal it doesn't tell you anything about the eigenvectors for the same idea value prices opposing you'll have to 0 eigenvectors with the same ideas about you that it's also true any really a combination of those 2 eigenvectors is also an eigenvectors of the same value for example let's suppose land then lender 1 and lender to have not to different values of land is the same value when the but 2 different eigenvectors which arrived like lender the 1 and lender to 2 different eigenvectors both exhibiting the same Eigen value you can't take any what what that means is that ill on 1 and 1 equals land times Lembo 1 to go on land 2 eagles Lambda Lambda to multiply this by a complex number Alpha this 1 by any complex number beta and edit to equation what you find is L on the linear combination of linear some powerful and the 1 was beta landed 2 is equal on this side there you have when the time the same thing as in the bracket he other words if you have 2 0 eigenvectors value could get them with arbitrary coefficients and they remained eigenvectors no again until distinct vectors 2 distinct sectors and all
possible linear combinations of them you can always find perpendicular to this when you have this 1 Marogot always end them or subtract them with coefficients to make the perpendicular so the rest of the pyramid is that if the colliding values odds you will have the freedom to choose 2 0 eigenvectors which are perpendicular which are orthogonal at the end of the day the sun rises theorem is that given permission operator it's eigenvectors can be chosen so that about so that form the basis so that there also normal perpendicular to each other and normalized OK so willowy on that was an interruption where remember where wives that's the problem with corruption from that is over you yes there was talk about board ruled by boat yet OK so if you happened the several 0 eigenvectors but say land the 1 and when both the same eigenvectors you then you construct overlap on the state vector would land the 1 you construct overlap of the of the state vector would win the tool used square this 1 you square this 1 a them other words you simply and the probabilities are for the 2 possible ways you could get the same amount of land Justice summer probably think it and you're ready to let area Oreos but my wife would say this is probably healthy is already ROC at 1st the you go far enough after fixed do make the but stupid young but allowed acquitted these days also said the reason why way regardless of whether complex so big there's no right we noticed some of the squares of these things which up Oedipal warned from normalized state the normal vectors Dutch I'm not sure which which 1 implies which but clearly they fit together pronounced like that OK but if possible to get a result theirs start I'm not sure what unprepared experiment means but I think are I think what you're asking is supposing somebody gives you a spin and they don't tell you anything about prepared at just certain here it is I give it to you or nothing but then it is not possible defined experiment which has an unambiguous answers Her would point OK now now what we need to come 2 new idea and no question and classical mechanics all of this whole story here was arise our when I showed you a heads and tails that's as those of the 2 possible states of a simple system I didn't we talk about measuring it was also obvious and we have to try because we know the measuring it is doesn't do very much to a system measure but your work because she's system in any way here we come so much more complicated story but the story up till now is really quantum mechanical analog of just specifying that you have a collection of states the next question we ask was how the states change with time that we called him the classical cases dynamical laws of motion or dynamical laws of nature whatever the rule for how was big changes remember was examples of our calling a simple example would be good going down and rulers nothing happened just days the same way the Pre trivial another possible is in each interval of time and Scrooge rubber mascarpone time that we can imagine each interval of time it flipped up heads tails have scales such tales from the law Our we talked about the concept the same concept in the context of a buyer with 6 faces or any number of places including continuous infinity saying basic idea the laws of physics or updating from instant instead of a configuration that have a configuration changes that have good laws in classical physics the determine stick Italian have state changes are also under the rule and the other rule we called reversibility reversibility was basically just the idea the states don't run into each other states which are distinguished mainly different heads and tails of distinguishable shot to states which are different will evolve mathematically and state different art that was your idea reversibility comes down to is exactly the fact that if you know the stated the instant you not only will know what comes next but you also know what came before because no 1 chance that states run together our endeavor to different states giving you the same outcome than you wouldn't be able account where you came from so reversibility is the idea we also called information conservation and I suspect they refer to it as the miners 1st loss might 1st because comes before everything else quite mechanics also has rule out both reversibility and information conservation and autumn miners 1st long are but it comes down to the same states which are distinguishable in the sense that there exists an experiment that uniquely distinguishes them state distinguishable in our world talk about how States evolve with time this regular come back and discuss this in the context of the the simplest system of quantum mechanics and good idea the goal bound for that but let's talk about it the more general OK here is a posture In the past is that you have this if you have a system at time at some specific time let's call it a time 0 and I'm going to start changing my notation the states instead of calling in BU I'm going to start using notation which is moreover
standard notation inquiry character call them applies Greek letter and size a commonly used letter to represent the state of the system notice Data systems do change with time so you have to think of these these vectors as being functions of kindly evolve so let's put that in here in the following way sigh sigh of tedious simply represents a state which changes with time which could be in a state that's not the same after a certain time is as worth to begin with but were following 1 specific state system has been prepared and then are allowed to involve by whatever means things evolve they passed to it is a sign of E can be obtained from of 0 0 means time 0 by an operation on sigh of 0 a unique definite operation which is governed by the laws by the quantum mechanical laws for that system so we have to do something of a back because further status so let's represent what we built by the operation of some kind Our operate now incidentally we didn't have to start with kind of 0 I would with kind something else and evolved might amount t so this year really represents the kind of evolution from initial state for final states not necessary final but their favored some time the basic 2 basic pastor the 1st is that is a linear operator you was a linear operator and the same play any state whatever the initial state years you put that here and you hit it would say you were the saying operator who is is maybe this is called the time development operator it's Olivia operator Arthur and involves state from 1 incident or not you could try and ask what would happen if you more general hypothesis could sample you might have a hypothesis that you would depends on the initial state or you might have the hypothesis that you was in operation but not linear operator some other kind of operation you could try you will very shortly with the big troubles from all kinds of things it's been tried a guarantee you tried and repeatedly every so often some physicist will try to invent something in and some other person will come along and say Look that has troubles with low-calorie era has troubles with the causality your has troubles with probability interpretation and your freed try I've given up try I actually never tried our right OK so you was a linear operator we all rules for linear operators but 1 other thing and that's idea of conservation of information while conservation of distinguished ability that you have to stay tuned different states Of the same system and you evolve and of score 5 t by capital flight that's given by the same you will see the same operator times fire at 0 Britain yes 580 to begin with is orthogonal aside of physically distinguishable to begin with and they will remain orthogonal for all kinds of disease that is case slips on left singing the Greek letters uppercase on left is on the rise this is not right right so orthogonality breeds orthogonality physically distinct states involved a physically distinct states of distinguish ability but if remains intact its median analog of the mightiest 1st loss of classical mechanics so let's see if we could find way has all incidentally it's very easy approval from backed rower was it say it says that if sigh and fire were initially orthogonal right now if they were initially orthogonal if they end don't ask me to make capital letters achieved much quicker to maker bookcase ones I need capital here I uppercase then this is cruel and remains true for time it's an easy exercise I'll leave it to you to prove that of orthogonality is preserved the Duke estate a stronger thing it's implied by this you will save you take vectors which may or may not be orthogonal In a product of between them stays constant with time as a muscle trek you take any 2 vectors you expand and bases vectors and then you assume that the bases vectors remain orthogonal with time and you could prove that in a product indeed 2 vectors this time independent source follows From the orthogonality beyond the evolution of orthogonality that we in the product between 2 states remains the same overlap between them remains the same as a center with the overlap between 2 states Of course a measure of this similarity and what it says is a degree of similarity remains a sec OK let's see if we can figure out what that means for you what says let's was rewrite PARIS by a Slovenian for firefighting inside of the product yield times 5 a time side but before we do what we have to remember that we have the flip this equation chocolate upon this job without a star with the flap from a cat vector toll brought vector equation sewers do sigh of relief equals you not times you'll look times what you'd dagger that mission conjured it but combustor related to a complex conjugate but we permission conjugate answer now we applaud this intimacy quarries here a while says let's write about Brewer Berber erasure father drifted
honored to be orthogonal where they are to they have similar product that equal size 0 you dagger of T U of C 5 0 foreign donors plugged in a fight you of to beside you dagger now prosecute dagger terms you is another operator was a product of operators we talk about marked the product of 2 operators very simple concept after you take a linear operator and you applied to a vector and then you take the result and you apply another operated to it the process of repeatedly applying operators gives you an operation which is called a product of the 2 up so that's what this 1st you hit 5 with you and then you hit the result you back but and what it says is the threat if he fi inside any fines at all that when used plant is here you dagger and you the product review dagger renew you get something which is exactly the same as had you not evolve the state a product stays the same who upset think I made a mistake we should be firing right so that the principle of conservation of overlap with 2 while it is the theory I don't think again very very easily approved by going toward basis very easily that if the matrix elements as global matrix elements or be a product like this if this equation is true that any pair of vectors overall record store here on the general if you'll have operator a court case for you value and that any pair of vectors society sigh K 5 is equal to the sci-fi and pair of vectors whatever it follows that case is a unit operator operator is the operator which was nothing just gives you back the same nectar source follows and they you terms you must be be the unit operator that the content of conservation of overlap you dagger but take a Freddy tee times you of tea is equal the unit operator means the operator which does nothing scares you back the same Becker is safe is the interest the others make itself is you can be but to examine the surreal but now in particular examine it very very small car in very small cars no you only all the system by a tiny little incremental kind let's call the incremental time Epsilon right that's right that you will Epsilon your 1st of all what view of 0 no time evolution at all this 1 just unit operator sofa very small epsilon you must be itself closer to the unit operator plus a small deviation presumably on order Epsilon and I'll be right there will be a broader epsilon so there will be something of a lawyer epsilon times another operating here I'm going to make use of freedom and farce but I have I'm going to put a minus sign he where but operate over here my sign has no content yet or the operator is over here the minus signs absorbed definitions and I'm also the poem I here again is no content yet because I haven't told you anything about h h is somewhat operator Absolon times H is just an operator and I Epsilon each is just an operator so I am not really said anything other than a small epsilon we operate EU is close to the identity how closed a water Epsilon and then there may be additional things where epsilon square Cuba but with a drop them and study this equation Washington wishing given here more specifically we should do talk operators which satisfied rule you daggers plans you is equal to 1 all are as was said you dagger permission conjugal view the inverse of your such operators are called unitary you'll is a unitary operator so this is again this is a restatement of the principle of conservation of World War II while less now apply this equation this condition to you of epsilon C will we find out about each OK so it's going over to the blackboard you of epsilon is equal 1 minus I epsilon age story what about you daggers you is permission conjugate permission can't 1 is just 1 but then we have a complex conjugate plus i Epsilon and then we have to permission conjugate program so here it is dagger and here's you know it's multiply them together and that the result because the 1 solar epsilon epsilon so where have we have viewed that is 1 plus i Epsilon and each times 1 minors i Epsilon each that has the equal 1 of the largest expand out toward epsilon 1 plus Lai Epsilon age that minus equals 1 as a sick food that the approximation of epsilon is so small that square Erkan because legally ignored compared Torah Epsilon right so the ones can part of the but basically what we find is that a dagger reminders which has to be 0 cancel out the ones 0 on the right side and or Epsilon after HST equally this is the
condition but each permission now I didn't call at age for her cold for something else call it Hamiltonian clear what it has to do with the Hamiltonian of classical mechanics will eventually become clear the Hamiltonian for classical mechanics also in Tunisian equations for systems evolved from what instantly to the next but Of course that's not a good enough reason call was Hamiltonian we want to see the enters equations away very very similar to the way the Hamiltonian and classical mechanics bars it is also the thing which this remission right that we use that some of their award could be an observer its list operate it could be an observable who is an observable it's a Hamiltonian energy Hamiltonian and energy of the same this it is what energy of the system is sleek Hamiltonian Woods generates the evolution of state vectors according to this rule here yet that Gobi 18 emission operator of the moment now what determines what her mission operate you put that were can't be Jessica BNE her mission operator associated with the system you study for selling system once and we want to make it up Out of operators that are associated with spin so yes to operator but it's a year experimental some of the same things which govern why you choose 1 Hamiltonian will grungy in classical mechanics either express some prejudice about the way things work of symmetry principal whatever whatever you have that might give you a clue as to what a at the end of the day you may have to resort for experiment the find out what their abroad whatever whatever OK so high that but watch let's rewrite our equation question Young said would you do that it states that they believed and each carts that's what would be the 1st visit there might so be each way Our abuse cut it's not type it never the situation with a possible explicit time-dependent of age is the same as in classical physics classical physics H can depend on time for example develop particle moving in a magnetic field and the strength of the magnets is being varied with time changing the current food electromagnet then b Hamiltonian for the particle moving in a magnetic fields time-dependent otherwise if you have the parameters of a problem not wearing what user the Hamiltonian is not it's exactly the same exact young young but that the tractor that that's right in principle could be a font time let's take a case what's not and what that corresponds to is a situation where the parameters Of the problems are not time-dependent where in classical physics we called the kind translation invariant remembering classical physics kind translation variants which means that every time is same as every other time could you do the experiment the same experiment at a later time will have the same output as this as identical experimental earlier we a statement about principal controversial invariants visit each does not have explosive depends just an operator which the part West said you could imagine situations where you're ramping up the kind that an electromagnet or the universe is expanding or God knows what there some explicit time-dependent then HK time because that what happens to energy conservation of the Hamiltonian kind depended their right sending here of the Hamiltonian this time depending on what we haven't yet seen wise where they might be a conservation of energy will contract but some are OK so let's come back this equation over here all of but again this is lost all along it yet yellow contract basically it is multiplied and together will come to that or you could think of about away updated by a small amount of time you just do it again and again and again Our to repeatedly but we can use this to derive a differential equation for the way the state changes with time withstood that on Wednesday that OK so let's look at the state of the system at a time Epsilon and this time epsilon could be thought of as a prime epsilon right after some arbitrary kind which I'm gonna call 0 but it doesn't matter sigh of epsilon is equal to 1 minors all eyes epsilon mines age epsilon times side say a 0 no later say it is a sign of the epsilon might sigh of 0 this is the small incremental change in the state vector that's equal to minors
i Epsilon
operation h on nor was I said this 0 here the these Eureka be any time as long as epsilon epsilon units after OK let's divide this equation by Epsilon you're allowed to do this because you could multiply vectors by numbers is multiplying both sides by 1 over epsilon 1 older epsilon kind this whole thing OK what system the left side 1 of the epsilon the difference of 2 things which you which at slightly different times that's just the kind derivative of this vector we haven't talked very much about taking derivatives of sectors but it's clear you could do that you could take the difference between 2 vectors of perfectly well defined are they will be close to each other if epsilon small divide by the small epsilon you get something which on the right-hand side is just the kind derivative after site time derivatives equals minus I H at any given time it doesn't doesn't have to be a prime 0 and indeed given time if you want to find out how the state of the system changes already incremental kind thinkable this is you're equation completely analogous to the way you update classical systems but b costs because knowing the state vector is not the same as knowing the values of V experimental output of experiments you could nor the state and still there can be ambiguity or uncertainty in the value of experiment this tells you how States change but it doesn't tell you have the results of experiments changed express results of experiments are still statistical but continuously updated state OK this is this equation has a name name of this equation it is generalized kind lead paint Schroedinger equation best it is the general form it is shown in Equation specific versions which Schroeder wrote down was a very specific version of that we can't evaporated and the course but this is the general idea the rate of Byron only to say it again it is both possible appeals it you have to know what ages yes it's salt on the reversibility yet so I ideas for the conservation of the rare that that but at the end of a long long time now and will made up for it apart but you have to remember when you do that you're intervening with an apparatus In all the follow the system under those circumstances you have to include the apparatus as part of the system you cannot interact with the system and not include the things which are interacting with it in classically not such a bad those but quite mechanically of some something's things interact with the system's strongly enough to to affected were strongly enough to measure it you have to include as part of the system now this is the way system evolves it is isolated and completely not in contact with anything else will have to come back they ask how we've always what happens when you put in contact with the measuring apparatus will have to have a model for how measurements take place but this is the way the system behaves as if nobody disturbs it during the course of the evolution from time 0 time he will have to come back to that important question look yacht religious 2 assumptions well maybe 1st is that time evolution takes place by a linear operator called you and that operators independent of the state that's at with me plus that's 1st and the next statement is that in products are considerable time and that tells you you was unitary from there we went on here and discovered that was small little incremental change is governed by mission operator so we get to the idea of a Hamiltonian the aim history wrote that book the small incremental change could be thought of as 1 over epsilon kinds the time derivative of the state vector of this misses Mr. the Schroeder equation it How do you solve the problem along time and that will not a short time at the it basically solve this equation will talk about how to solve this equation will boost for a lot of examples but Roma to born and explain what this has scheduled classical mechanics what has to do with the corresponding concept in classical mechanics we need a couple of ideas are up about how you relate classical ideas quite chemical years we need an idea of what is called the expectation value of observable it's a bad name it's a bad name because the expectation value as defined may have very little to do with what you expect the experiment to get past this doesn't have much to do with Cuomo mechanics is just an issue of probability theory and it was talked about before about just remind you if you have a probability distribution which looks like a fierce In the least likely Anthony the possibly get his right here it's also the expectation value so it's at a poor a poorly chosen terminology the right terminology should and sometimes summer of some fancy people it was named Maria ones but some fancy people like the court be expected value would Leslie expected value that is the expectation value at sea merger value the see average value and hammered valueless something could be of value which the thinking even take on the possible value that it could even have the average value for example if you assign assigned head plus 1 entails minus war and you flip the calling what the average yeah be detained get 0 or so it's got nothing to do with the expected value of experiment but nevertheless it's called the expectation value and we will call we will call the expectation of of the average let's talk about the average value of any observable Japan a particular state so our state is 80 said I was never change the side but sometimes I will sometimes look will just that that bad debt at that said it that have no no definitely not with funnier but never occurs never ever ever but even make a worse please as the probability X is some variable something that you measure this could be the probability of finding a particle at different locations of tank track it could be be that after some particular state the probability that slips make its symmetric about the origins of it could be that the probability distribution looks like there's and me and I meant to make a symmetric and is a really
very very 0 0 in a neighborhood of the origin here the nonsense in which x equals 0 0 is expected you could build the experiment forever and ever and ever and you will never get x equals 0 it will begin next summer in here or here the quality expect and expected of the expectation value Costa a misnomer it is the average value defining averages a very particular way that our from many many smooth probability distribution in particular probability distribution big holes in them something like this let's say it can very often be the case in the often is the case that the probability that B average value is the most likely value to get it like the peak there is a most likely our guest by definition proper beaches and thing many cases if you have a simple probability galaxy probability of bell-shaped curve was so for then the expectation value or the average value is usually quite close our average and narrowed peak value so you usually close to each other but certainly there can be violent exceptions that calling or the spin which could only be plus 1 minus 1 that Iran the expectation value and the average value a quite different OK but that that's just a little sermons and gave me a chance to bed now my friend Marie demand for that a where not said hurry billions of euros not now that we off but that yacht well OK it depends on what you mean by average was emotional of average which is strictly mathematical right down there are there is a unwritten law which I think it's safe to say nobody really understands but that's the law of large numbers that you do a thing enough times how many times not to do a thing enough time experiment enough arms about average got by averaging the results of your experiment will be equal to the average defined mathematically to within 1 the within the margin of error was a margin of error is specific numbers are really mean that every time you do it so will come out within the margin of error sorrow we're assuming stand the probability theory that you do if a finger not Web classes extra little ingredient extra physical assumption that you know what to call that if repeated experiments enough times the average of your data will be the mathematical average can anybody prove that no that's a that's a lot of large numbers there 1st the but no not nothing and that approved the real you can probably is because sooner or later somewhere in the Beavers somewhere is exception to it and how many times do a zillion times as user numbers salad 7 Sidoti zillions 1 with the logarithm of Brazilian 0 lifters half price if you don't expand if you do that and the isn't enough repetitions of it eventually somewhere someplace gonna be a violation of this is closely connected to learn of the 2nd lot thermodynamics ever was a principal with 2nd love thermodynamics entropy never decreases Baltimore and was never able to prove that finally realize that the right lawyer is the entropy probably never decreases except what it does which is rarely the except when about so I am not going to try to give you an explanation why probability theory works this is me very puzzling but we will assume probability theory works the statement that that it works is the same as the statement that the average of your data will be for big enough experiment will be mathematical average but so that's how well that's come over to the side of prestigious supposing we have a collection of Earth Hour we have a probability distribution for some variable except Lambda Lambda 1 of the measure 19 value of El supposing we have up probability land areas right recalled problem of Lambda Our after probability for Member end land that can take on a whole bunch of different values Landis abide would call them was becomes the probability of all life than the average the definition of the average is the sum of all of lender columns the probability for land that's the definition of the average and will represent the average by the symbol the brackets symbol will labeled by the brackets will come OK but that's a definition standard probability theory definition of the average of a quantity the quantity of probability that the quantity takes on that particular value some of our life an assistant but standard definition of average OK what is the average Interstate 80 of the observable Rambo so let's say that 80 explained in a basis which faces the basis of eigenvectors of led out of the other I'll remember how was mission operated its eigenvectors former bases could expand vectors in an awful from normal based so let's try this the former House possible I saw what I lenders of 5 Member of the lenders are awful normal basis what is a probability lender in the state are probability the land by his eagle alpha star 5 maintained Arbors's a probability handy average of Lambda the average of land that is some although I lenders a powerful I start out fight for let me prove that this is the same thing as taking that Eddie the Brava and say which in between the brought a and the kids that they be operated hailed the operator L is the operator who was eigenvalue use land but still you prove that it's actually quite easy boy is say let's just with composed of the blackboard plug-in and there are plans stupid prove that I so it can't do it this creature Arabia 80 is summation and let's use also that brought a is some additional body of lenders of Alpha star I still make a sandwich which is the son of injury wider 2 sons 1 for broad care what form from abroad and 1 for the kept Llandysul JT Alpha star Jerry Rice's abroad vector AT and that we put Delaware and then we put the candidate there are 5 land I just Purup the enology now what is ill with Exxon Landis a bite but they when the survivors are the eigenvectors avail so will act when the survive a slew of some I J and the J storage a authority and then they'll have slammed the RI images of Landis times land I know all this junk and here is just the
number for each are in jail for each eye injuries Justin Armour so that means we are called upon to take in a product of when the I remember Jerry that right it's 1 or 0 depending on what I was Geyer not given that this on the double some collapses to a single sum which led by a lender identified and this just becomes some land the might well but sir cut which cut go through some of the steps if J is not equal died you get nothing in days equal Tie me in a product which is 1 so this just becomes the son on how the start house on land the the which is exactly but we have over here so we produce this relationship B average of any observable is just the city in which the track kept the Brock had brought kept the market work at our sandwiching Ill be observable between a bracket because the prime etiquette but the same branch at the branch kept associated with the state of the system that's where this notation of were brought back from OK that's a good thing cannot that if you want average of a quantity just see in which it between the bracket of the rocket the representing the state we questions about that OK now detailed will go wanted to is going to try and find equations governing kind evolution of these average value the idea is that under suitable reasonable circumstances at the probability distributions on nicely shaped but they have a bell-shaped curve that that by calculating time evolution averages you doing something pretty close through work classical physics would instruct you to do about calculating the equations of motion of a classical the way averages tend to we've all the time as a under suitable circumstances follows equations which will vary very much like the corresponding classical Corey sold on the next goal is to try and find EU rules for that kind of evolution of these average value expectations and that's not so hard we have a we have all the equipment we need to do that let's see if we do know it quite so it any given time at any given time the average of Lander is given by society a state vector time free the operator L representing the observable times sigh of tea another words I just Colombian effect AT the actual crime to the system this is average of L as a function of time which we call or forward scorer her if the average of failed as a function of time again that kind of rope wrong notation it's not the average of L of T a C average of L has a function of time we could our wikibook the key outside the bracket that's standard somebody but only just goes standard notation standard notation and then remember well OK I had to do it emerges core average of L she as a function of time as a function is a function of time answering figures functions we do something else we could call it il Boerum other notation the averages to put a bar on top of something that's another notation for average stated notation caught L. of tea average as a function of time what what with worry well like to do with this would like defined equations of motion for it another letter that the find out how it changes with time by differentiating respect time want to take this quantity and differentiate it with respect to time so the court il got now we have a bars with thoughts on top of them 0 . of tea is time to remove its not L of assistant fixed operator it's some fixed definite operate at all the time dependence is in the state vectors stated that this change with time the operators those effects was that operators so let's see if we can take the time derivative of this year's saying the time derivatives act on the side of when you take the time derivative of a product in this could be a product of vectors economy a product number is a product of functions well functions
on the rulers always saying it's the time derivative of 1 20 other plus a time the removal the other times 1 that's very general sells this is going to equal then these signed by IDT the brought Rector L sigh class L decide ETA hopes erased of very important equation outside area under the year and the year blackboard arise over here these side DT physical reminders a at the site that's all it stake in over here so we get we begin to see that the Hamiltonian as tell us something about how average values change with time Hamiltonian of classical physics tells us how the corresponding quantities change with time Hamiltonian on mechanics would toss the averages change we're gaunt also needed the brawl version of this so let's try brought version of the robbers in orbit is deemed by C of the brought the size of tree it is equal Class II plus are side H h is her mission and so we don't have the permission conjugated his emissions but we do have to change sign of body where we go from brought to kept pace of Allegis got plug-in was just 40 will get L . yeah right elbow but commit to lose from 1st over here this will resume plus sign a Simitis sign Furnia minus teII sorry l h sigh and so we get from here about the other 1 the other 1 8 years plus RIA society H. L sigh that we are light frenzy expectation value we're Versailles here idea we're sigh year hearty kind H L minus a L H side at HL minus OH 0 when a challenge rural cares widow operators don't necessarily Camille commute means but you can freely interchange order of the general this is true of matrices products of matrices don't necessarily Camille some time until have particularly operator commutes with itself ill-timed l might sell times at that that's OK you could do it of times Omar missile times 0 0 OK age times each times age but In general L times is not equal each time out maybe but typically not was animal called H stands times called the commutator through prayer services 3 definition given until operators Forbes but they could be just a agent combination H. L minus L H is written with a square bracket H. Kahn a L and it's called the commutator commuting has something to do with passing near freely between each other exactly what has to do with taking the subway in New York and people remember Our was called the commutator so I have here is the equation the deed average L a given time average of kinder kind derivative of the average is equal society comments that H L but this subject only here is just the expectation value of commutator of each with we can think of this as an equation relating the time derivative of a certain average the average of another quantity wicket idea idea is a plus a plus I quite but for me right L . se
physical from might average times average of the commutator each Ill now indicate that by a bar on top of it so enamored equations of motion now for averages In terms of averages if the
probability distributions for everything on the recent peak then we really can't just say approximately nice and peak so that the averages closer the of highest point on the curve the curve is reasonably narrows the reason the enormous amount of uncertainty then is basically says that the time derivative of classical worker time derivative of the approximate the military is equal to the approximate average of commutator which without someone writes the of course is not quite right but to write anyway got is equal top Times commutate H houses and this has to be taken will little low-grade assault what it means is an equation among averages means an equation among averages but often is written in this former 0 . is equally H OK now before we can finish I want to remind you only classical McCarrick equation of motion it's the equations of motion represented in terms of pass from bracket remember passed on brackets porcelain rackets fish bracket what were the plus only equations of motion you may remember who manner Sulek as right debate exactly but did see classical functions and this could be a classical function of queues and peas classical mechanics we accused peas but but but L incidentally is a standard notation for angular momentum this could see the angular momentum but there right now just using it to represent any observable after Debye IDT L equal to a certain parcel bracket and passed song bracket is pop bracket hello defamatory dilemma that probably summary not advise you to go and look it up Norwest next collapsed lecture of a classical mechanics or bracket Our and I will not dare go through it here I just want to remind you of it and I want you to notice the similarity with reserve really bearish similarity of this will be held by average value is on a minus force on most precise times the commutator of H With incidentally OK let's just point there were things plus 0 in brackets Have the property that if you in that change the tool expressions what happens the pass on bracket they just changes are commutator ATL miners L H if you interchange order of them Chisinau so we can also write this was money L with a commutator of the Lord the remarkable similarity and suggest suggests that we have identified the 4 stones bracket With aside times commutator no not down discussion was I just realized just completely forgot about what happened that each born what happened to each women working in units in which each bar is 1 I've been working units in which each pot plants constant as warned the question is if we want to reinstate monks constant and make it not equal toward where you would go in these equations and Lisette yes those enshrined as equation that correct but how the hell we traced heavily Tracy here's courage here I told you that each has the significance of the Hamiltonian and therefore his units of energy so let's look at this equation Villa cancels out on both sides as far as units go units of the size of the units of the site and so the right-hand side of the equation apart from the size as units of an energy the left-hand side of the equation as units of the 1st time in most time in energy don't have the same units in the 1st time in their 1st 2nd and energy is jewels or whatever so we have an equation here which doesn't make good the rituals that we need a constant in here and the constant we have to put constant in here Our is a constant goes a gourmet writing inside left side plant constant go riding inside or left last year the whole if it goes in 1 place and 1 thing goes in University of other place circuit here's all they to mail is plants constant has units of energy times time plant constant has units of energy times are so it energy terms some sort of plants that he and Izzy times time over time is equal to energy that's where Planck's constant goes into these equation and I apologize for having completely forgotten about it work goes into if we followed through the equations no doubt that appears over here I think it appears in the denominator here young I get everything right except where I have h I should have H HOA age bar reactors as we age age so although my equations wherever we wrote a check the Hamiltonian we really don't have a lot of keep unit straight each although plants constant and they go straight so it really the right equations including the constant is LDT it is 1 of 1 although H. bar he won all the plaudits constant plants can as a small number so this looks like a very big but the other hand this is L. times H minus each time jailed in classical physics that would be 0 so whatever this is classically Zero and quantum mechanically it must be a small correction of a classical physics to extend the choir mechanics is a small collection the classical physics and so the commutator is itself something which is typically small and a boy aged bar so that allows us to guests to guests and identification between commutator and porcelain
brackets now I this this is my onto with a moment that's just a year just looking at 2 equations and saying Look was given me any connection between quantum mechanics in classical mechanics then there must be a connection between the porcelain bracket and between the on rocket and the comet namely plus all bracket must be equal the minors I already each bar times commutator or is usually written the other way commutator is by for storm record commutator was a small thing because his major for air force on bracket classical physics is not small thing some number of some characteristic quantity which is marked by a small or big and the same units units that dual classical physics the standard unit meters seconds whatever commentators ovary small that's because they're almost 0 NYTimes beam-ends becomes a azo more 0 OK no I don't think so I think I am writing this on undecided desire rage bar multiply by age forum and no multiply both sides but I can use that decide terms plus sizes warned him Ryder was theoretical recognized this connection and built His quantum mechanics it builders quantum mechanics are over making an identification of plus on brackets with with her commentators because the foundations of the subject on character is a consistent subject without having to make any reference the classical physics the right way to think about things is the core of the Canucks from before a classical mixed calmly pages before apostle bracket you have to derive classical physics is an approximation through to work on mechanics so hilly see some formal similarity at this point here we have no idea why this commutator should have features in common with the pass on bracket be folly Gulf leaders say they write down some of the features of contemplated and for strong brackets to remind you are again although this should be at this stage to be a little mysterious while lowers is funny connection between us record but broke throats pursue OK I remind you about about writer 1st of all I have tool quantities I'll call may be now or maybe India use answers Carmela nature don't have to be the Hamiltonian and special just LMH 1st thing is the bracket L with age Is eagled miners surplus bracket of H with the commutator of Each is minus the coming of age with so that's a parallelism that we can begin with Now let's take some other 6 mother properties of brackets on there's a properties of course brackets probably just a right there were 2 more if I have tool operators to not operators but to quantities of cold ill & M L kind and is itself a classical quantity and I can compute its past on bracket with each is only minimally answer remind you of tell you go look it up because this is a report its Ehrlich supports some bracket of NO with age plunged plus a all kinds of Prosser bracket on him with a each nite classical physics the order of multiplication doesn't matter so you could bring in the left or the right of doesn't matter you could bring the Eleanor left or right matter but this it is the relationship for the product of 2 variables the porcelain bracket back go back to rule next to last lecture a check this out of 1 of the defining finding 1 of the properties of course brackets you could do the same mucosus same question over here supposing you'll have a L and a new pickets commutator with what do do you get this is an exercise this is an exercise all you do is you write that this is equal to L N H minus a each bill em and then start juggling and ending sometimes subtracting the same terms and eventually you'll discover that this it is the same as those same relationship purposes equal to tell times Taylor of em with age plus are cut of filled with H & M the weighted Oh this is just right L & H minus H L and and then write out these guys here this is held MH minors L H & plus L H and Midas HBO and for what turns here total cancel pairs you just get the fact that it it's a toward settling Teresi OK so you begin to see some countering that plus own rockets area country pages must somehow be closely related the show you tonight is about energy conservation a mail we're not going consider the most general concept of energy conservation was simply going to ask is average Of the energy served with we're not even really got to what it would mean energy itself because served in a deeper way but we can ask we have all the ingredients which will allow us but Kobe average of our average of any quantity changes with time period is over here and we can now specifically asked how the average of the energy changes with time would do it the the energy I claim is the Hamilton art definition if you like what is energy Tamil so let's check and ask whether energy is conserved energy is a Hamiltonian this equation would read the time derivative Of the average of the Hamiltonian is equal to minors I only H kind the commutator of the
Hamiltonian with itself the Hamiltonian with itself h times my H times H is the same as 8 times 8 cut characters not that weird weird but 8 times H is the same as expenditure so this is 0 whatever they Hamiltonian 8 years its average average expectation value would kind is always year-old sought conserving maps and minimum and that is a stronger sense in which its concern come but even this week sense is kind of interesting and tells us want some kind of right track to be able to compare quantum mechanics and classical OK I was start there would pick a few questions and young at that time here get Was the probably I just mean average I mean members of mind for summation of probability of land that land so I don't know what's a standard symbol and statistics were an average in the sense Justin the when the user of the North 1st moment of the distribution the occurs let you're not OK bomb I'm usage is calling it a land that right now Hey bracket warmer and that's the expect the expected value even those up expected value that kept on you never seen they expect the average value of a bar on top of them gold they but the when I is a term average I will mean it's it's also called the expectation value and I will freely interchange the direct notation for the average with the modification of OK and that that's not the Standard according to statistics book bad gave a death of the place yes and then wrote down I'll be I'll have to go from the left and the right to complex that reached for missing conjugation but remember that his mission would prove that that wars that was the condition of territory each tradition but how you do if you do change seawater outside a lot DDT islam where where where where where on the left side play she said at 18 DDT the operator yet time derivative of abroad there was a broad time derivative can the time derivative is just another way of talking about differences the difference of till bra vectors of Bravo the difference of 2 can't sectors as a kid paint in the limit the difference becomes the time derivative of derivatives sells the derivative of a abroad vectors of brought the derivative cat vectors a vector if you want to take this kept equation What is the broth that goes with this kept branches was told kept time derivative of a correspondent Robert but the plan this year it just think of this as taking a difference of 2 vectors young it's just a difference of 2 vectors love right exactly when her again hot corner mechanics weird but work taking a difference of 2 vectors replacing it by a derivative when the difference a small batch of prefer Georgia Is that all you have to do a better don't agree of rock about that initiative to try to cool not sure why others but it did we welcome it is good 5th what we don't know that corresponding to energy and we do a lot more we don't know where we could find me energy Hamiltonian but that's not satisfied we want to see cases where we understand the classical physics we understand where energy use will want to see that Ricky quantum mechanics of same system classical system is an approximation require a chemicals the physics is really quite mechanical under certain conditions such as a system heavy are basically that heavy big a large masses the system behaves approximately classically which means that the average values move in in ways which are consistent with the classical equations of motion with wanna see that we know that when we study such big heavy systems that thing that we call the quantum mechanical Hamiltonian it is so essentially closely connected with the things that we call classical cars will complement for came to everything for 1 nite at the moment just showing use similarities nor better weather will actually get to do better than that I don't know we can't do everything but but for Wachter KBS radio instead what we are our right Harry had saved some time to do that but I learned to good enough for tonight on who was center the solution to a 3rd yeah yeah well-done right Charles are a brawl who will do it will do it they're just a run out of steam on Brad will come back comeback for that so fast that last part helping costly were collapsed last well really what we really want to derive could collapse pasture derivation is deeply connected with something else that we haven't come to you have to go to next turn entanglement our at the measurement process as really a processors of establishing entanglement between systems of apparatus and so is talked about the entanglement process I think until we do we simply have about the language of collapse but the next time they met some remind me about collapse horse to collapse the credit for more please
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### Metadaten

#### Formale Metadaten

 Titel The Theoretical Minimum | Lecture 4 Serientitel Lecture Collection | The Theoretical Minimum: Quantum Mechanics Teil 4 Anzahl der Teile 10 Autor Susskind, Leonard Lizenz CC-Namensnennung 3.0 Deutschland:Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. DOI 10.5446/15015 Herausgeber Stanford University Erscheinungsjahr 2012 Sprache Englisch

#### Inhaltliche Metadaten

 Fachgebiet Physik Abstract January 30, 2012 - In this course, world renowned physicist, Leonard Susskind, dives into the fundamentals of classical mechanics and quantum physics. He discovers the link between the two branches of physics and ultimately shows how quantum mechanics grew out of the classical structure. In this lecture, he continues his discussion on the vectors and operators that define the language of quantum physics.