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The Theoretical Minimum | Lecture 8

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The Theoretical Minimum | Lecture 8
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(February 27, 2012) Leonard Susskind spends some time in the beginning of the lecture discussing some of the basic qualities of systems to lay a foundation for the rest of the lecture and the class.
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Stecher University we're
talking about what happened during from beginning from before measurement after measures talking about what happens to the evolution of an isolated black hole left to itself between the time that it was prepared and the time was measured OK nevertheless I could answer the questions it is entanglement inevitably irreversible if things which are not entangled come together interact with each other do whatever they do and become entangled is irreversible not necessarily anything can happen the opposite can happen and now we're not talking about an X-terminal apparatus measuring the system was source simply talking for example about to Spain to Spain's interact with each other come together go off and their entangled what happens is it possible to disentangle them by for example reflecting them letting them come together and interact again can they become distant yes yes no rule which chosen angle is irreversible on the other hand if entanglement happens and you take 1 of the electrons often roller away the other 1 is stuck left being entangled with the things that just those mounted a comeback will not get a comeback on west of course somebody out there throw it reflected back here so entanglement is River is reversible birth fires you are concerned the measurements you do are irreversible that's a very peculiar situation you have to think about it on the stairway means within a mile of the move off the question of these very simple systems spins sometimes called cube it's up tonight at some point but I want to just the go over just very briefly think probably discussed this class last time but nevertheless it's so important and that a better warmer arms Hammock 1st of all there was a motion of the density matrix ever have to systems Alice's system and Bob system analysis by Bob system of far away from each other so palace doesn't get to do any experiments on Bob system Bob get to do experiments on Alice's system but they're entangled leader they are or they are not entangled may be entangled is there a description that Alice by herself uses to describe just for a subsystem yes working description of her subsystems called the density matrix of emitters review that very very quickly and go back over some concepts if the wave function of the Alice Bob system that combined system of Alice's subsystem embodies the wave function is called sigh of any and be Beaver Bob a 4 hours then Alice's Alice's density matrix Alice's density matrix is defined to be star complex conjugate of the wave function North Salem role it role for the density matrix indicated by role it has subscript ANA prime because of the but it doesn't really mean the dual Bob sewage purely a matrix with respect palaces variables but indicate that by calling this Alice's density matrix and it's given by sigh star of a B prime be there was confused at a prime sigh of a B signed over Bob's variables it doesn't depend on Bob's variables dissidents some older when you some something it no longer depends on that variables so this is only a function of Alice's variables and it contains all possible all possible arms statistical information that Alice can ever know about her half of the system under less Bob comes and interacts with sister Gaza Bob comes from where areas and gives her system a shot of some sort that will change her density as long as Bob stays away from her system barbs in Palo Alto Alice's on office and Terry as long as Bob does not interact directly with systems are density matrix doesn't change no matter what Bob does Billy give a little proof of that love let's take Bob such subsystem not be so narrow was just be the little spin that is carrying around and also Alice's let's that Alice's subsystem consists of everything in-house neighborhood Everything that she could have any contact with everything in Bob system is everything in Bob's neighborhoods maybe even including Bob Imel Bob is going to do something he's going to do something to interact of course he can only interact direct interact means he may turn on a magnetic field he may go all sorts of things oddity to the evolution of his subsystem is gonna make subsystem evolve in some way 1 of the ways which Bob can cause his subsystem to involve many many ways but whatever they are there are always governed by unit matrices that's come over here and forget hours for a minute and ask what Bob computer with subsystem he has a wave function with Fleet Alice and picture altogether just but suppose Obama as a wave functions and he's good and manipulate the system somehow get manipulate the system somehow and send it to a new wave function the Ruelas the new wave function of his subsystem must be related to the old way function by a unitary operator unitary operators of the things which governs time evolution go back remember about time evolution unitary operators that means sign of of becomes replaced by some although be some unitary matrix b prime sigh of be Prime in terms of wave functions and
matrices this is the description of what happens the barbs matrix sorry Bob's column vector is basically column Bob's column back there describing his half of the world for his system and the find out what happens to with the matter what it does evenly doesn't do anything opposing does something does something it generates a unitary evolution of his where function now highly described the same thing in the composite language in the composite language of the wave function is volume a and B Alice's and Bob's variable 1 happens to it with Bob manipulates his hair for the world exactly the same thing with Alice's variables being just spectators just being passive spectators so sigh and they can be just becomes sigh of ropes were put in there b primed summer be prime but this is a function of beat and use some of would-be prime pay it together some Overby kind of your B & prime sigh I AT & be prime ages passed through Alice knew nothing whatever about about this whole process are subsystem was not in any way influenced her index part of the wave function for half of the dependence of the wave function was on influenced by Bob's manipulation so let's go back and take the manipulated wave function Is the manipulated wave function Court society Our manipulated manipulated way function of B N A of a and B that there's been modified by the action of barbs unitary operator OK now let's recalculate to recalculate Alice's density matrix after all Alice's density matrix contains all information that you can never know all about her hair for the world so let's see what happened to it while we 1st of all have to take the manipulated size star we still have some of the beat but now we take manipulated size Storrow we have criticized Starr sorry about that we have the right size stock manipulated of and we want to know about it with the index 8 prime prime and be mailing get just by complex conjugate this tells us how sigh changes let's just work out the complex conjugate that's a complex conjugate Of 8 trying there is a game we now this is a this is a plan to stay permanently but here since crying a summation variable we do not want to write prime he beat prime was summation variable used over here Assistant another 1 called be double priorities we best that can be a summation variable a moment and you complex conjugate of B & B double prime you'll complex conjugate and of course is a somewhat of a B double time here and the only reason I call the be double prime instead of be prime was so as not to confuse it with be prime is 1 son and he is a separate Phil it something where we know of a unitary operators you attend operators are operators that if we don't change be free interchange 2 indices and complex conjugate what happens that's called called the inverse operator or the Commission conjugate for Europe in general it's called her mission country specifically for unitary it becomes the inverse operator such coat accomplishing conjugated a fight on complex conjugate and interchange fees still will be double prime be double prime be this becomes permission conjured right so this is the usual story if operators operate the right on can't sectors the corresponding action on broad sectors is given by by mission conjugate I always have to do is plug in republics and left early for the manipulated size start that summation Overby be double prime our size star a prime and be doubled trying times you'll dagger Horace be double prior and be a citizens to many indices for me to be comfortable on the blackboard but I think it's worth doing now What outside outside the other half a comes into density matrix and that you will be b trauma sigh of sigh of a and B prime complicated the mess hall or early something our with something or all the bees basically yacht was summing over this b over here could be separation he we were something or be doubled trying to get the manipulated wait for the complex conjugate but we also have to some of would-be prime here are some of us all the bees inside each B is assimilation variable by the time we finished now look at what we have it looks complicated but what is this thing over here well not quite yet 1 is essentially correct but of course were to mean is viewing matrix this is the product matrix I love you dagger is a unitary matrix so you dagger is used in be double prime b b b prime this is just a crocodile b double prime be prime it just instructs you said b double prime equal to be prime
when you do this start when you do 1 of the 2
some of say the B double prime so others then becomes some over be pride state what have we got Fritz Weaver said Beat double prize now we've done this summer will be that's done tho some Overby which is dated and now we said be double prime equal to be prime sigh star of a prime be prime sigh of a and B pride I said that doesn't matter whether we call be trying be prime or whether we call it P because it's a summation variable would do that With code B where we have we have exactly I hope I pray the original density matrix so bond no matter what he does as long as what he does is described by a unitary matrix marble have absolutely no effect on Alice's density matrix does not matter whether entangled or not knowing that I specify that this was a product that is what physicists call locality the inability of Bob to influence any statistical prediction of Alice's by doing a manipulation on his owned degrees of freedom locality is not violated this form of locality this form of locality is not violated in anyway by the phenomenon of entanglement at the technical arguments notice that it does depend on you can so I knew what to do the Terry do with unitary had to do with conservation of probe of round of distinction is what I call the miners 1st lordly since it's quite mechanical version of the funny into an interesting interplay funny interesting interplay between the conservation of distinctions on the 1 hand and locality if Bob somehow had access tool changes which were not described by unitary operators this would not have become the unit matrix over here and the final density matrix would house of parliament matrix indeed would have been modified this is interesting the way the whole thing holds together this is why it's so hard to change course McCarrick you wanna play will be wet so that Bob but the barbs evolution not quite b unitary only approximately unitary well then immediately you start sending signals faster than the speed of light how bite instantly modifying Allison the density matrix sold please hold together tightly taught to change them on good so yes it is in America's a bomb a major user see in the back yard he says that it is things Alice palaces of these changes don't system by doing something in there doing something was described by this unitary matrix that effort but we amid is biology for example he might be turning a magnet he might be turning a magnet and such a way that has an influence on his woes that influence il cheetah say been example yacht if you have a spin in a given directions couple for magnetic field and turned the magnetic field in some worry that typically change the evolution about their so Bob could do that can no change of magnetic field acting certain spin the result will be to change the way the spin evolves that will change his all measurements of the spin of his if he does act by rotating magnetic field which is mathematically described by some sort of evolution does not want some sort of evolution described by a unitary operator then the then he does change the results of zone measurements presiding rotates the magnetic field but it doesn't rotate as apparatus result there's there's a change the probability distributions of his own spent but you will not change any of these probabilities of about OK silly data centers they at the reality is defined by the inability of Bob the influence Alice's measure that not quite so what's the big deal is arguing start at on yes what you may ask progress so say a single word Laotian movies littered goes off to give direction Tech had detected fasten the speed of light classical you did what he views as close enough to be laid off class 0 in case I don't what happened with density matrix of the other systems does not change Korean when we would have but if it was my head I know I know I know people look I'm doing my absolute best to tell it to you as a really it and cut rule that perhaps that's out there that cut through the crap that's out there because there is a huge America there is up Pyle mile fight we somehow wanna cut through them this is the mathematics of cutting through it now nevertheless it was wasn't pretty damn smart people who are uncomfortable about something something that they call 9 localities from Iran Einstein was aegypti was pretty smart and are being column Einstein for nothing if my favorite job John Bell was pretty smart it call him study but so is a legal on something going on young the a few now Oh I wanna get I'm review in last week's or where brickwork is real and that it is your idea of melodies that he stays days so that each those 1st the densities now an amnesty that due for 1 is that the your things get mixed up as a presenter of fuzzy but it's true so I will try I will try to tell you as I understand it worked b concrete operational facts are facts are I toggle Alaska although was a little bit tired last time was completely clear again because it's very very interesting of really is at the heart of all of the confusion about confusions and ANSA including my own internal confusions about a mechanics when I 1st learned about Earp Einstein Rosen the bells that was the discovery love was a Friday
was discovered think maybe Schroedinger said some things about entanglement but it was Einstein Rosen of Khodorkovsky who really called attention power peculiar was basically what they said is we have a very funny situation here where although we could know everything the compatibly be known about a Compass system we know nothing at all about its constituents that's crazy that violates my sense of reality that violates my sense what it means no water system is doing Pilates all sorts of things and my 1st heard about it factory before John Doe sometime in the sixties when student and I wasn't stood there was a young assistant professor and from start to think about that for about a lot the strategy euro what it means for long hours so inclined to try the think about it In terms of stimulating quantum mechanics of still and what are the limits if if Kwan mechanics a very strange and so very different in classical physics doesn't mean that you can't pull somebody in that thinking the system quantum mechanical if all you have your resources are entirely classical the insert it is yes and no no I know I talked about this last time but it really is at the center of things Alaska back over a quick because I can Willabella ourselves a quantum simulator I think somebody built on site came across about our that now because of the work that he is not here is that picture is a picture your picture whose bitch of Einstein Anheuser Bergen Einstein's I don't like this birds of what I said I feel like that's but all over I from back half at that but I guess that Russia is that a good captors if why go on but OK let's let's talk about the nature of quantum simulators 1st of all I just want simulator singles been I think they tell you how to do it but with right down the steps would have a computer screen is our computer screen on the screen we have a picture of the she acted an detector apparatus in the apparatus can be oriented in any direction in three-dimensional space we need to do some of three-dimensional graphics here has a window and the window shows wonderful really possible states what is called blank be that media that's the detector of the apparatus before its interactive with anything so it can be beat your commute plus or minus 1 B is not important it's just to say before the detector interacts with everything is a button over he had called and and there is the operator of the system of the computer game basically Oliva gets to do was pressed him and then look at the result of the a & M stands for measure performs a measure now all they hear connected to the computer by wires are by cables because connected to the screen by cables is a computer and the computer stores information in its memory and the basic information that stores too complex numbers how far up was supposed to be the components of the wave function of the spin and and say the bills us being here but that's all little talk about how up and down the computer only could generate House's which satisfy office star offer up will star offered barons equaled 1 whenever whenever generates some of them I could do something else canoe to love thing it can update the alpha's by solving the Schroedinger equation for that there's some Hamiltonian for example it could be that the spins around the spin is the man imaginary magnetic field which Casey offers which change with time on the computer knows how to do that it knows how Schroeder Ingres equation computers consortia Ingres equations beautifully and updates sees as a function of time to update them according to the rules of war mechanics 1 other thing the cables are instantaneous they convey information instantly will have 1 more element the other element will be a random number generator let me just say some words about random number generators random number generators have agreed to loosen choir mechanics No we don't require mechanics are you couldn't simulate a very very high degree of confidence rent a sequence of random numbers are random number generator purely out of classical our classical considerations you know how do we take the digits of pi whatever there are you make something that looks were full antibodies including a sophisticated mathematician into believing a baby with generating a set random numbers as a random number generator inside the computer by the apparatus starts by initializing nervous system and initializing the system are simply means Starting out some orientation for these detect therefore the apparatus as work and sending me an offers some initial value Alacant plug-in muddy once a bloody and initial values to beat or the computer sets them according to wherever rule likes this system runs a period of time and indeterminate period of time updating the Alford until the operator decides depressed and a certain amount of time has elapsed operated decide how much time and presses image at that . be random number generator generates a plus 1 minus 1 but with probability alpha star power from up there plus 1 sorry I up a pop star plus 1 or else out star offer down minus 1 member of the some of these 2 opera star offers up to 1 so with with on Libya random number generator the computer centers a signal With the probability of their properly probability detector and detected detects or registers what the computer center that's 1 measurement now persona happens the computer work initializes the Alpha has a really initializing office is the detectors rejected plus 1 then it back a message doesn't really need to send back a message after all the message that he the computer over he knows whether cup plus warmer 1 if the detector plus 1 day and it sets Alpha you back 2 1 and sends out would be 0 another word it does what we call collapsing the wave function according to be result of according to the outcome of the measurement and really initializes the Alice's so whatever was measured you start with probability 1 for the outcome of probability 0 4 B which we call it the entire calm and then arises again saw the Schroedinger equation does it again be operated and make ceramics whatever decision he likes what makes him decide the treasure of depressed In experimental physicist what makes experimental physicists do what they do I don't really know about that but he does experimental physicist don't now and then he presses and there's a whole operation all over again they all start before he does he may decide to rotate the apparatus now the computer and he knows what the orientation of the
apparatus is so when it calculates it not only calculates updated Alfred but also calculates the probabilities for that rotated detector to detect wherever it's going to detect that although algebra and a little bit of a purely algebra the computer makes kind of calculations that you did not presumably did makes a kind of calculations to calculate what the probability would be a free detector was rotated some arbitrary direction then the watch the probability that it will get up or down now hallways it that decides to rotate the computers well it's say operator again the experiment 1st experimental is not only compressed him but he can also real audience detector any time you want this is about as much as an experimental physicist duel with that's been rotating apparatus measure rotating apparatus again measure and was the result of the outcome of this outcome of this is indistinguishable From what the experimental physicist examining the spin would solely simulator corner mechanics right now we won't know whether we can simulate quantum mechanics of Toulouse fans the 1 on Alice's side of the world over here and 1 Bob side of the world young but in our in hours of videos that if if I don't if that that it that the computer was up they all the time but what are these in words but we did talk about Hamiltonian we talked about Hamiltonian how Hamiltonian up to date the state of a system because you're right we didn't discuss how NATO wouldn't discuss much about Hamiltonian for Spain's we did yes we did I believe I described the spin and magnetic field and how the average imprecise we didn't really have a field that has not but then times she left for there's no real magnetic field just imaginary magnetic field the computer knows about and with that come into it comes into the solution of a nature of the exact Schering equation object don't we don't this summer here at precisely the same thing over here except instead of Alfred we have bases status for Bob do exactly the same set up the replicated twice Bob Aziz measurement Alice measurements that perfectly happy each 1 says I have a spin quite mechanical don't talk to each other and each 1 says my is in a state the language that we would describe this by Quan mechanically is they have simulated a product state the truck state in which Bob sorry Alice's wave function is bits and bobs wave functions as this and the combined system is the product of pulse now we want to simulate a any state more complete description fluids and tangled states well again do it this way let's start with the waves are guaranteed to work we take computer in a single computer we could think of a single computer for life as a pair of computers connected by wires but just the fact that they're communicating with each other and interacting with each other really makes a single computer and this single computer we know store for complex numbers before complex numbers that we store there are Our thought look how far down offered them offered their own down these complex numbers here are not necessarily products in fact of assume they're not products another is it this is not a product that it purchased at a highly entangled state even a highly entangled state is described by 4 complex numbers like Paris that's what's in the computer I was a computer doorway it evolves the Schroedinger equation 1st of all we initialized saying deal with initialize we start with somehow the offers are allowed to run according to work at a Schroedinger equation describes them and the end Eva Bob Aliso both without knowing what the other 1 is bring press or as the case may be don't press their measurement problems once the measurement button has been pressed and instantaneous signal goes back to the computer and the computer grinds out a random number generator and whichever side was measured sends a signal to the detector with appropriate probability Beaver plus 1 minus 1 and then really initializes itself Reisch aligns itself in a way that doesn't touch for example if Alister's the experiment then it doesn't touch bobs and the things that Bob does the experiments the computer read initializes his half of the wave function which sooner later saying it throws away the piece that you didn't measure throws away the pieces you didn't measure an renormalize is the wave functions of the total probability is warning exactly the same thing no no real difference with a single system you can simulate it and you could stimulate away which would describe all the possible things that Allison Bob could measure whether or not the system was was entangled where is where do we get trouble we get in trouble if after initializing which writers separated computers and break the wires for example is all we might do given these our says we might put them into both computers Take a set of alpha so we generated this into this computer this this computer computers on after the talk to each other they have to have separate random number generators the random number generators will not talk to each other each 1 contains the full entangled state and then we go through the same operation what's going to happen the computer is simply going to forget for example for and highly entangled state puts it final status as a single state where the right would be that if Bob and Alice both measures they should get opposite value for any space if they measure the x component the spin they should get opposite values if measure components of spin they should get opposite value but if only separated them they have set but random number generators the random number generators on talk to each other guess what's gonna happen is gonna be no correlation of any kind between what Bob measures in what hours measures not Bob make that measure plus an Alice may also measure plus because random number generators are not connected to each other so you even know both computers may know know about the full state of the system if the random number generators don't talk to each other then the results of the experiments will not contain the correlation the correlation that powers could
later come together and report Bob would say my 1st experiments are a plus if things working right than Allison said my 1st experiments reminders and so forth and so on that correlation will not be there want to drive separated computers are related to a plan that could have we have a single random number generator that sends out signals the both computers it sends out a signal plus want this computer it sends out a signal minus 1 that computer while you use a single random number generator but it has to be able to talk to both computers so what does study wires in the world is gotta wires in the world connecting this side and decide if for no other reason than they both connect the same random number generator think about DAR yup yup but that's all right once to separated them then what happens next day after some measurements made doesn't preserve the relationship between them they become disconnected from each other Yasser think about Overbeck I mean even better simulated this is if I never tried but I don't think it should be that hard as you to do a little bit of computer programming simulated much harder things when my son was a little kid we have this thing called basic the basic enumerate designed a basketball game had friction and it had a moves were really much harder so don't buy West experience ever programming anything a scenario you will learn a lot by trying to stimulate I get a signal from allah fall ran that could be it where will you please speaker come quickly we get a single some on some random event was me you please be exact we were very exact getting check that's what may be the case that they can add value each day but it is really is I noticed distribution that gave signals a person chance of being a 1 0 these so they could be from yes as a pig are being correlation turns dead where a the other 1 is safe for us 1 that's in the real world I go they wrote yes Bureau worst Come on get information so I in information age was ruled by will be through and there is a chance that look at it but and that a bit of a wave of change that we these using that data is Bob analysis St it was a a boy hurry up because I never get the particles what is what is viewed as a 0 . weird Road what's weird is you can't simulated classically without having central processor connected by white Our from Twitter so those wires To here is a look at OK now we have a lot of wasn't meant for goodness sakes Shirley Bob and send the message to alliance that palace takes no time at all because instantaneous not true enough if this were a real computers set out but as bot is restricted not only do the things which quantum mechanics allows him to do if he is restricted by the rules of quantum mechanics then we find out amazingly that Bob can never send the wrong message to the wires that violates for cowardly wine because all Bob do His manipulate his own system with unitary operators and so that's what's curious arm does not mean that the world is filled up with wires we can see I don't think that's the message I think the right message but of course you you could hear a with his room for debate with the right message of all of this on the message I take away from it is basically quantum mechanics cannot be stimulated by a classical system the logic is 2 different structures of the 2 theories of 2 different and I don't take away the message that the world is for wires if only we could access them we could send messages back and forth other people do our but I think at least if we know how good the quantum mechanics will agree on the fact we may disagree on basically whether there are or are not wires they are they really wires there are they're really not wires well perhaps culprit the word really but there are but this is this is although there is 1 other element to it which is quite interesting armed with this assumption that Bob Dylan Alice get kind of randomly make decisions about Wayne Tula press the button about what orientations about what to decide the measure Bob 6 years and says Newgate Northwest didn't Southeast or whatever makes these random decisions according to the principle of free will also a speedy that's fine as long as Alice and bomb our outside the system and not part of it we don't need they asked us question of consciousness in the questions of free will just admit that Allison Bob could make random decisions about when to went to measure various things it's obviously a pretty good approximation but once we take analysts and Bob and put them into the system so that they are governed by the laws of physics then things get a little bit touching what do we mean by saying that Alice can decide to do this Bob could decide to do that that I think is where the possible loopholes in all of this could calm From that I don't want to get to deepen the heavy philosophical discussions of 1st of all sorts of psychological questions of consciousness and I don't know the answers that's it I don't know the answers beyond what I've told your after I certain of 2 things about I am certain that there are a lot of people who think they do know theaters rebels of it don't you can never tell us if you vote for a challenge to find what really as a bully really fervor entangled cancer ACS desire to hear your vote villages answered yeah we've got OK I have a lot of dollars so easily not Bucketty that sets a very interesting again so I'm not the 1st wanted allowed that I was the 1st word to think about this word of telling his people for 50 years a
rock opera was a good thing maybe boss thought about persuade numerals doesn't calm lets a we we we could spend the rest of eternity talking about the and of why because they really are are interesting but uh you know we want to get something else we want get the particles you wanna get a more complex systems the most complex system we discussed his spinners four-dimensional Hilbert space with that is about to the point where in classical physics we got through the 1st lecture the 1st lecture was about these simple discrete systems and only rather immediately moved on continuous motion of things like particles moving continuously with an infinite artist infinite but a continuously infinite the range of possibilities that's where we have to go now now fortunately we set up so much of the apparatus of quantum mechanics so we can do it fairly quickly the vote a little bit of subtlety about replacing some by integrals that's about it that's about the nature of the subtlety sums intervals we've expressed the general principles of quantum mechanics and then exhibited In the end the variety of simple spin system examples now would plunge right in quantum gravity not a lot of who plans right in the problem of a single particle moving on a one-dimensional access will take the axis to be infinite although we if we do our that's mostly because they don't have time to be more careful if you wanna get anywhere as organ after worms br it slightly less mathematically rigorous so we have the vote be rigorous you really have to contain the system of a finite volume and so forth but we don't think so I have lying in line along line 1 particle can exist we want to particles 1 particle can exist in 1 dimension and that is why it is labeled by coordinate X not right down court X but also Miss Bates Of the particle be states of the particles are also labeled by the position of a particle now he only a departing from classical logic and classical logic the states of a particle labeled by a position any momentum if we were talking about the state of the spin we would save the state of spin is described by all of the Colombia deal that a least tool of the components of spin bowler can't 11 car mechanics who can't specify two-component simultaneously saying classical mechanics specifying that you sure you not specifying the position the momentum is too much Our and will see how that works will see how that works were the moment with figures a postulated that a complete description of the state of a particle you can't get any more out our is for example the knowledge that is located at a position X is located the position next will describe that I can say that they can't vector x that's sustainable particle was located at a position acts now of course we can consider more complicated more complicated states linear superposition of bees in quantum mechanics we always allowed to make linear superposition and so the space of states is bigger and just the base of particles located definite positions let's call the general state society is the general state of particle doesn't mean anything now just an abstract symbol as a sector in some sectors space the vector space now has a continuously infinite number of bases vectors strictly speaking mathematically bad for but Wigan a play that game if you want to understand more deeply of the mathematics of quantum mechanics you go when you learn about Hilbert space born items that are so the general state of particles called side remember what the wave function of the system that way wave function of the system in a particular basis is the projection of the state on the bases vectors and recall that side of it now we come to a particle the basis of states Our position states and was the wave function the wave function busy in a product of the state vector With the eigenvectors of position and sigh of acts that's the way function the classical rock classical but that's a standard quantum mechanical wave function of a particle plays the same role as in the simple systems we discussed in a product of the state vector with 8 it's a little more In this context the probability that the system whatever tears exhibits property AU is given by this thing square in fact given by such star of of any as the probability that the system is the outcome of an experiment to determine whether the system was had property a probability likewise exactly parallel the probability defined the particle X is equal to size star of X times sigh I really haven't done anything that we haven't done before now in quantum mechanics like in any of the sensible statistical theory the sum of all should add up to 1 so the son of size I should add up to to 1 Bowen probabilities refer to a continuous variable we don't talk about the probability that the particles located exactly a point acts as to start the probability that thing is located exactly it given x 0 we talk about the problem eludes located within a range of X or better yet we talk about the density of probability we talk about the probability density and we integrate this over little range of X defined the probability that do in that range of X we do something like that here if we were not whether the system is definitely specified aid but we wonder whether it's a range of a we some over there but that he always have to do it total probability equal want that's a condition that integral of sigh star of next sigh of X X is equal to 1 photo probability of the system has 1 and the only as said the only new thing here is that we have to replace sums by integrals that doesn't make mathematical subtlety that we have to worry about but it's so that the mathematical or details OK where about the inner product between 2 state vectors 1 X and 1 its prime well here we go back for inspiration right back to love right back to 0 this case 0 off their is not equal to a prime and won a fate is equal to a prime no not would quite get
away with that he we're not going quite be able to get away with this statement that X X Prime is equal to 0 or 1 in we are going to say that affects is not equal to X's prime that is 0 both secular say that with equal to X's prime it's 1 that would not lead us anywhere what we're going to say is that a function which vanishes on less
equals articles at times but has a real 1 not under it will see how that works out will see what that does us the moment what it really comes down to it is a different normalization here but again the basic idea that states are orthogonal if x is not equal tax prime that will retain but the numerical value of what we get what next is equal to tax Prime will take to be slightly different and the fact we're going to invent a function or that we are directed the Dirac delta function of X minus X Prize the Dirac delta function is not a real mathematical function although by now a mathematician the finding the deaths over is a real mathematical entity our but for other purposes we can think of it in a very simple white bits of function which is 0 what what's at stake Delta of acts not belt of X-Men Prime is basically next month 0 X Prize function Delta of acts as a function which is 0 unless sex is equal to 0 to as saying this thing 0 on busses Exs Eagle text prime so this is X 0 everywhere except x equals Prime and x equals times and some very high narrows how high a narrow high enough narrow enough the area under it is when you that means if it's approximated by unnatural interval of size epsilon that has high 1 over epsilon video a limit we take epsilon smaller and smaller 1 of epsilon higher and higher always keeping the area under equal to 1 that's the Dirac delta function their act of the functions 0 everywhere except for Mexico's prime if in some sense when Exs eagled the ex- prime but in such a way that the area under it is equal to 1 right now let's use this Turner further elaborating on what the meaning of sigh of exit on a guest and with a check on a guest at the meaning of sigh of X is the following I want a ride a general state vector side but I can think of it as a sum over the bases vectors in Assam and the world of course the role because of the bases vectors of continuous BX of some coefficients some coefficients Ross not threat as were society equal to Indigo exit of Somerset of coefficient guess what they are guess at this Pyrex finds the idea turns 80 on the state the quantum state where the particle as located X with them again a mop for different axes was exactly the same thing we did for ordinary quantum systems except this was replaced by right now I'm giving you tone distinct definitions of 5 x won his Senate seat in a product of the state vector would be eigenvectors Axel called the eigenvectors backs the other is that expansion coefficients in writing society that as a sum over sectors now if Hulu uncomfortable with the idea of a continuous basis of vectors to integrate over you're in good company the mathematics of this it as I said by now and maybe rigorous by 0 2 some that took pages and pages and pages and pages and books and books and box on something called distribution theory to make sense with precision but and toward it's pretty intuitive and will will stick with that right so let's calculate or let's put Prime this is a son and the role it doesn't matter how doesn't matter to call the integration variable prime and melody calculate in a product of extra side as I said they were 2 different definitions of sigh of X which checked into the same so here we have it until the X Prize next time enough product of X would next prime what seal a product of actual its prime I think I wrote someplace where is it I think I erased tough but it's a call to the inner productivity into OBX Prize sigh of ex- prime Delta of X minus X prime fit was sacked 1st old Delta is a function which is Zero everywhere except the X equals X prime so that means they're doing this to the girl here you'll only get a contribution when X primers equal acts that easy
answers will come out to be proportional to sigh of X furthermore area on the belt of functions exactly 1 if reimagined that this belt the functions of pinched now a narrower high Speight so the areas under his 1 then this interval would 0 sigh of X the rule is if you have an integral over a delta function like this it's simply takes out the value of the integration variable where the delta function vanishes started with the argument of delta function vanishes where X minus X Prize vanishes at forever functions not 0 an the interval of a functional the delta function just gives you the value of the function at the point x but a rule we might as well write that down that was a rule dual Delta of X sex prime F of ex- prime is always equal to half of them picks out the value X Prize Quebec part of the definition of function so it's just people decide next refined consistency that the thing that we called sigh of X over here His saying his single recalled Silex over here all of this has its exact parallel for the year from ordinary systems did proper barbed-wire border along with each side as far as I can gave proposing which is the basis vectors private slipped compare that with is equal to some all that maybe a problem which quoted Prime sigh of a prime a prime for French before you was a factor that a year and a prime orthogonal and do exactly the same thing except replacing sums by integrals We will also find that the side appears here is the same as the inner product of siding with a basis so really does nothing that we haven't done before I appreciate little possibly Ooh unfamiliar after all it is is thing with them before offices aside from the the VAX prime plays the same role as a prime here yeah say roles aside the sum was replaced by the Indigo some a prime is replaced by into OBX primed and otherwise it's exactly same but want calculate B in the product question may be living inside a three-year duress already you've got a verbal wire between TX over here the Middle East there's no vertical I which them but is line that it got it right here right not absolutely comedy in a product of 2 vectors William product of 2 actors assortment of product but 1 way of calculating in product go back to ordinary ordinary course mechanics ordinary means for the discrete systems we don't need to strike calculated a product would stick to its inner a product of fight with 1st right side and fly as sums over the bases factors so this is equal to some of the 2 sets of bases that there is a prime and star of a prime times a prime as say brought back then we complex conjugate of classroom we deal with broad actors that's a sign affirmative and the final effort is a somewhat AU of fiery that fans a fire they mean the coefficient the 5 sector size star being the coefficients Of the side brought back by fire just a number sigh star of is just a number and In a product of a prime with a is just a chronic Delta so all of this adds up to a some of sigh star of a prime 580 and then I can't look at Delta of and a prime which is exactly the instruction to set a equaled a prime nothing to seen that before this is simply the rule about role vectors and size star of 80 is represented by a row that star of a 1 Barber and fire is represented by column vector and this is just a rule to take this time this but this time this for 1st time this but a stance that food services a common garden variety quantum mechanical in a product I don't know we don't need to go through the same exercise here we can immediately write down what the result is going to be just by by analogy is almost not even analogy it's just replacing wherever we see a put Exs wherever we see some work integral theater product of total state vectors for a particle low-lying is integral X sigh star of X 5 better His satisfies incidentally 1 of the basic rules of products what happens when a proper if you want to change the 2 vectors what's the relation between inner products sci-fi In a product fireside complex conjugation know what happened here if you interchange with 5 warning that writing
five-star here and so I hear that again would be via the problem of the year complex conjugate and for exactly the same reason I this it is equal for both discrete systems and for these continuous their products in a product of a vector with itself is the norm of the vector and just as in previous examples to normalize the vector to 1 who expressed the fact total probability up so how much difference just a difference over on notation and do what I think what gets to be killed confusing about it is you know both of these things you know all about these simple rules vectors and column vectors and you know about function but did not used to thinking of functions as vectors or maybe wash you are good but not used to using our forgive functions as vectors FOX News from outside of Texas but you have to get used to it but it's not but functions are vectors of a vector space you could head then you could multiply them by complex constants are and prepared them you could multiply by numbers and so very satisfied the axioms of vector space space of functions is a vector space now sometimes you may want to be out in last letters that I will but it is members of immigration minus just a good which incidentally says something it says that we should be thinking about functions which grow it infinity for example we should be only thinking about functions which are squaring the verbal now would ever have the organ after modified Red Square Manger Square them Of course squaring can that be integral exist function punctures which growing Infiniti won't do that novels a crazy the because it would say that all of the but if you have functions which grew it infinity where would the probability of being the probability with all located them off at infinity and that's what we want we want to be talking about probability distributions which the total finite probability and then sold our that means we should take perhaps with a grain of salt the idea that we can write down any function but for the moment where orders go Bulu but loose about that OK I I think slowed down here from few minutes virtue but just after the Oscar for questions young nor yeah nor add bit more I'm meant about this but appear as if he is likely to set a date was almost all of boat entangled with all sorts of things in these environs and we haven't talked about the importance of a complex environment our it's not so important in the very foundations of subject is important to why we see the world the way we are there were constantly being bombarded getting getting entangled with things as there was this is a vote for while yet all you need is well our at ISU means it anything else besides clean up if I don't know at I think it's wrong need rigorous a set of concepts of physicist use all the time Our hazard a relief not normal during the period distributions will be out they there would be a matter for a physicist is made them laughed argue less with mathematician was 1 of the many examples were mathematicians got dragged screaming kicking by physicists Our course the opposite also happens but the was an example where mathematicians at 1st looked at this thing so where you crazy that doesn't make any sense by In time they came to peace with it through the finding it the wrong way but no I don't think they mathematical structure of it has influenced physics as much as you might have expected Obama unfortunately because big problems of confusion related to it in quantum field theory and hasn't resolve those problems prepared OK next place we wanna go is observables war talk about observables and give some examples of observables observables a linear operators what are linear operators acting space of functions their operations on state on a function that gives another function Germany operator with quality In my for some reason I saved it L stand observables used previously and for general linear operator source continued to sell whatever I'm is it's a machine and thought as an abstract operator space it Texas state vector until another state sigh prime or a veteran vector space but it did also Comey thought of as a thing which acts on wave functions every state has a wave function that describes it and media these machine but that that was what we call it and yes I remember we call them from machine the machine can also be thought of acting on a wave functions to give another wave function so let's think about ways a cigarette in concrete terms as an action that
we do all our wave function to produce some of word not leader Rita that would act on the sum of 2 functions it gives back some of the results it means that act of function which is numerical prada numerical multiple of some function is back the same numerical multiple all the ideas of linear operator are continue to be exactly the same let's talk about somebody Rob 0 OK that's that's the notion however a linear operator know what about our mission linear operator I'll tell you the definition of her mission we will talk about it before but I'm going to give it a condensed form the condensed form is also true for all of the vector spaces we studied and you may recognize it Our if not go back and prove that follows From the previous definition her mission operator permission for operators is roughly speaking the analog of reels as opposed to a complex OK here's what's true about her mission operators you have a mission operator which I will now quote they'll end has a property that if you take if you are act on a wave function society or state vector side and take it in a product with body that that is equal to 0 L sigh but not quite what I have to do with complex conjugate now Hill 1 that lets go back a steps opposing ill was not permission would this be right but quite well we have to do we have to put dagger here another words we have roughly speaking complex conjugate OK fellas her mission In the role for that is that it to make the matrix elements In want sandwiched when sandwiched in the opposite order you don't have to complex conjugate because it's already the analog of real This is a condition that an operator His permission that's important because her mission operators of the class of observables so let's work at 102 her mission operators for Rabbit Vargas will me when they need this or work out a couple 1st of all the simplest things that I could do it away function I know what many simple things the conduit but a very simple thing Mr. multiplied by Texas and even simple thing would be to a multiplied by a simple numbers but let's take slightly more complicated multiplied by multiplying by excuse back a function so this is it at minimum an operation on a function as a linear sure it is X multiplies diabetic foot paper fire wrecks gives exercise of X plus X 5 Texas doors were summit gives us some of the results of the acts on a numerical multiple oversight is back as the numerical multiple linear operator question is it her mission is a demolition OK so let's check it is away the case have to take sorry it a product of X firms if this warned well sorry because that occurred work to do regularly fly xxx size top 5 x site and ask whether it this is true or not with 2nd said no X X is a function X we multiplied 1 function by another function that thanks functions the function number it's a variable any given value of his number by it's a machine that takes the functions sigh of X and replaces it by exercise that right throughout calculate the wave function of video that there is an ex Siurek acts the wave function of the brother His star of X the inner product is but brackets around Frank this is a cat vector inside the red bracket and this is a brought back then and the rule for taking me in a product whips interchange star suburb will taking me the product is integrate South River bracket here and just integrate DX the story DX right that in this order where about sort this summer over here in a product without the start without the start to will compare with it is the same thing except where sigh star x 5 x well look at it is this interval complex conjugate of this yes it is a complex and unify star far-right complex 100 of site sigh star and what about X X is a Tong complex conjugate so yes indeed X is an operator which satisfies the basic rule of permission operator that if you interchanges assigned the fine it complex conjugate what observable those acts correspond to position it is a position is the delta function aII eigenvectors position when they will come to that next time I think I want you to there will come back X is an observable was a correspond to it corresponds to position of particles will prove that by showing that the eigenvectors X that the vectors acts are 0 eigenvectors about operator but not right now will do better next time is easy trivial I you write checks implies that is really the best yes is a function of X at X it is an an effort X yes to replied that that X is a function of
X it has all but that's right itself as a function which is multiplying another function so it's an operation on a function which gives another function that's exactly right but said another operation or operator differentiation differentiated function may get a function of the 1st test of whether it's operator by differentiator some of the functions I get the sum of derivatives of differentiate numerical multiples of function I get the miracle multiple of the derivative it's a linear function is it a her mission operator source we have here we have to get ourselves a fresh blackboard and checked once and for all whether it's well check tradition of 1 happened let's call the derivative operation let's call a capital D D on society gives us some side Prime where about wave function deal worth acts on side backed by definition of this is the definition of hobby acts it simply gives beside by VX and they sometimes lapse in the partial derivative of respect exits find the matter as only 1 x so just differentiates it creates a no functioned as a linear operator Is it permission OK so that's right down the 2 sides of the intermission condition here and see a further sign that's all that's all there is to it the left side is integral Vice star that looks kind operation Dionne society which is just beside VX what's the other way of doing it on the right hand side the other way of doing it grow so start now not doing the same thing doing something different so I star X defined IDX and the question is whether this is a complex conjugate of that so it's right he also the question mark we don't know complex conjugate so what what do listened to grow over here hi but 1st of all cinema bracket here it is just equal Indigo sigh of X beef I start by DX just complex the complex conjugate of a product is just a product of complex conjures thing in the bracket of 65 star IDX Norway asking whether this is equal to this they look similar to set aside this as a side this is a five-star this as a five-star but unfortunately this 1 has signed differentiated and this 1 has sigh side and find how you go from this to this integration by parts and do look like this assuming quote assuming we say we can integrate car only that has far away the integral functions go to 0 but beat In point contributions from integration by parts don't give us anything well Of course the functions must go 0 far away because the integral of a couple things must be 1 so we can integrate by parts everybody integrating their products is not find out fast it's simply a royal play with the rulers you could shift the derivative From 1 if you have a problem the X DXC products sigh of extra a derivative of 5 star of X in Ucon simply shift the derivatives from 1 function to the other at cost cost a minus sign right there so this is equal minus the integral Tri-Star Rex desire by DX what we're measure this isn't the row five-star decide next This is my steam to a five-star decide DX they are not equal the fact worse than that they're absolutely the opposite each other opposition the center of the negative each other terrific said well not we can fix this not weaken redefined operator which is mission the redefined the operator which is her mission is not derivative it is I the minus sign doesn't matter you could I or my side Debye IDX so let's go through it again and check whether now which mission is the new thing that we're called away give a different name quality because of derivatives at my side of the river and have a good suggestion for a record peak brilliant but a B for momentum momentum and if I really wanted to be momentum with units of momentum 0 elsewhere but where right left right way but all Africa Hebrew better pray downstairs I don't like Olivetti SpA we use our usual conventional ignoring each far from home he is minus I D by DX see what happens Over here we have to replace this by an eye over here the are but sigh Thank you might decide what about this storm we've complex conjugated everything we better changes sign because the complex conjugate of side is plus side so but for photo display minus Ari we should get plus I hear if intermission White because in the process of taking the complex conjugate every eye gets converted to a minor side and every minor side
gets converted to a plus is a true now yacht white because this is equal to minus those that would prove over here OK so we stay airline he had a Qantas I 18th assigned the complex conjugate this little operation here tells us that might I by DX is mission offers art isn't it that might I by DX should somehow be a real thing to represent the real thing and that Debye DX should not but let's let's check something out of sticks and something right now let's that's just so we'll finish up a moment we haven't talked about expectation values here tonight but I think you could get a think about them expectation values in the State Society of an observable El I just the sandwich composed of Siam both sides in the middle so let's take the expectation value of momentum size star X sigh of X and then according to the rule here he is minors I by DX it is real imaginary you're wrong Israel sought approval to Rio Albert's real battled to act so that integrate by parts this is equal 1st or set up a minor minus are into gold sighs Our X decided by the Yanks notes integrated by parts this is equal to 0 minus RIT and then we could switch the derivative wikis which the derivative the start variable and the wrong these start but DX time sigh of X but then knowingly integration by parts we have to change the size so here we have a little fear of the interval of this thing he side Is it the row of this thing here were plus I what's a complex conduit of this whole thing too complex conjugate it you replaced start by society or size over here you replace these side by side by the X by these start by IDX new replaced Miners I but plus or this here is a complex Condrieu next to everybody see that a show explain it for a vote from minus out here right that this is equal to decide but just gives us decide by mariticide he said but this is clearly the complex of the integrals more complex conjugate of each other sigh by DX is to use the complex conjugate of this Of the public's Pentagon this an ID and my desire complex conjugate of each other self given that this is a complex and of this equal to each other what say about a thing which is equal boots on complex conjugate Israel so through a rather startling fact that the integral of sigh star of X minus these I by DX Israel not yes I did right again it's because when you integrate by part you pick up an extra minus signs storms that this thing being minus its complex plus complex case or we learned we learned that this it is a mission operated at her mission operated and therefore observable corner checks the next time we explore a little more will discuss with this has to do with Fourier transforms will discuss questions about what the probability for finding the particle a different locations on the and will also talk about what probability for having different momentum are never move the shark or equation will move to the evolution of the of the wave function perceptual now is really in the classic sense of the word wave function will think about how it evolves with time all 1 other thing we were we could do it and I think I think would probably at the point of diminishing returns a review exhausted but could do 1 more thing we do more of it for let's calculate an interesting quantities and why it's an interesting quantity will come moment so let's take a quantity which seems very which is interesting calculated commutate ahead of ethics with Pete who anyway function where overdose sire Rex 81 a dodo things for 1st the multiplied by banks and then multiplied by miners ID or operate with like my his ideal IPX this could recalled he operating on strike Mrs. P this is X in water 1st X then a calculated the other lady X P side was that what's the other half of the blackboard over here a right kind miners RID by DX X and see what we get and factual those with a computer commutator which means calculate this 1 plus sign a calculator this the plus sign this summer Microsoft the following nite but before we subtract them with just calculate this is just
minors body and now the derivative of a product the derivative of a product is derivatives here now that act on not hold result of X acting on 5 x x x excuse in wave function and he acts on it so get when you hit the product 2 terms for the 1st time as a derivative of X time sigh River exit is 1 of the 1st 10 Josiah Rex and the 2nd areas plus X decide IDX theory real event derivative acts were signed in the derivative sign in left Texas pesticide over here about the size over here here the Debye DX doesn't act on exit attacked where we just acts on of X and gives us minors I by X side checks backs start X minor side Texas beside RDX now subtract them but they do this 1 with and this 1 with plots this corresponds to taking the commutator of X with acting on sigh acts committed acts of P another operator and on size of or we get always track but see this 1 now has to come with a plus sign since which objecting soros subtracted change signed here and this journal cancels mark yacht this cancels goes away and all were left with the size of X that's all left with this turn cancels just left tours of Isiah I size of acts said is always found any sign of its ever for any sign of the commutator of X with P acting on sigh of X just gives the plus side plus I Sysorex now if an operator acting on any wave function gives another operator if the operator XP acting on anyway function is the same as ii acting on that same way functions for any length function whatever it follows that this operator is equal to this operator another way of saying it is X P minors all eyes will be bracket when his anyway function give 0 then it means that this is 0 is only 1 operator which gives you when acts on every state and every that is no sector which escapes when this thing hit the implication is that this thing is 0 is only 1 operator which and annihilate which kills every Vector 0 0 XP minus 0 0 committed XP is equal top allies but I but that pH bars it would be an age bar here yeah I H bar Taylor we see things like this before you may have to go back to classical mechanics we've seen things before like this we think was sold brackets which had similar relationships with plus all bracket of X with PT physical earlier in this course we discussed the part the potential relationship we had reason to discuss why can't it has had to deal with the sure Ingres equations go back but we discussed the relationship might be a relationship between commutate isn't on brackets and particular the relation involved I each but this is actually with Dirac began his discussion of quantum mechanics he said there must be something analogous to a plus bracketing quantum mechanics and don't think of think of which had the right algebraic structure was and by submit predicted that would have a certain mathematical structure was accounted and he postulated that can't be and apostle brackets were related with each other tho an ice age bar and then proceeded to go exactly backwards from the way that we went he said If this is it if this is the parcel bracket and this must be the commutator and so there must be an operator P whose commutator with X is I and went back rule the operation and discovered that he was minors ID by DX so we went the opposite direction we discovered P by just playing a game said let's see how about this operated Beebe IDX was due found visitor that was was not her mission we had multiply by item maker explored a little bit found out some interesting properties of it but the most interesting property is that its commutator musical died each bar which us but suggests that connected with the pass on bracket the classical passed on brackets being everyone Of course the upshot of it is of course we need to do a lot more work to convince ourselves that the operator minus ID by DX really behaves like quantum mechanical momentum we can't just say you are looks like those opus on bracketed must be the former chemical momentum we have to shell that way if packets move with velocities that governed by abide by this the thing be that served has all the properties of momentum but we will the minute we operator of
size would best I want and what shall we say yes but would like to show that in some way that it is that's the best time verbosity so Sewell trial will eventually show is there if you have a wave packet of yours a sigh of X which is concentrated a distribution fears that wave packet moves along the x-axis with a velocity which is given by the expectation value of this momentum operator divided by the facts after we would like to sharper velocity with which this wave packet propagates along the axis that velocity Inco the group velocity choose recalled the group velocity velocity that the group will raise HMOs along kinds of mess of the particle is a quarterly expectations are more than once we do that and I think were rural roads and better serve level took us a long way to call from arm of some point we have to say quantum mechanics more fundamental classical mechanics and the definition of momentum should be on mechanics which ensure behaves in a certain way we want particles ahead you soulful for Moor please visit ash
Eckstein truck down