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Quantum Entanglements, Part 1 | Lecture 9

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Quantum Entanglements, Part 1 | Lecture 9
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Lecture 9 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded November 27, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.
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World Woodward talking about how systems change with part I go back over VAXes is partly I don't remember where we were exactly but 2nd bookstores worth going over last week's material it does not mean I don't make nor I always throwaway my notes to make absolutely sure that I don't use her sports car it's a very bad thing to do OK but I do like to go over the material which started last week and just make sure Everybody has a sword fights over slaying system is described by came back there we courtside up till now we just thought of the flight I think which doesn't depend on time in fact we haven't even talked about time we did last week but before last week we even talk about are we just go look at me in the relationship between quantum states observables measurements that sort of thing now want cover qualms state changes with time of war for a clear argument it simply other vector which moves around the vector space with pine role is the moves around on the influence other 80 linear operator and so we can ride that sigh of tea unit linear operator which depends on time but acting on side times Euro that's about the most general thing weaker right down so far I haven't told you what you is what's the most general thing that I could write down our under the assumption that state vectors change later that's something that I emphasised many times I can that I can give you a profile is basically a pasture with surpassed you with that's rooted in our experiment plus the fact that anybody lover tries to change it something different always warns up conceptual problems energies features of something else something bad cop probability is a well-defined or wherever so will take it as a postulated that the way systems change with time is through a linear operators we're operators just means basically matrices matrices which now kind he penned we can apply the same or the analogous of a related rule actually complex conjugate rule how brought this change with time on his arm of Bravo don't and business basically the complex conjugate of some can affect so if the complex conjugate army right-hand side fires but then so of you'll have to write a conjugate your back board Bagram meters from the point of your matrices you cooperation to complex conjugate all the elements and you reflect transport them about back us that said these 2 rules of basically the same 1 of them just the complex conjugate of the assumptions about how systems Cheney UK and expressed in various ways you could just expresses probability conservation of probability the problem the total probability is equal but 1 which is supposed to be that should remain that way for all time was a troubled probability the public probabilities in a product of the vector of itself of the state vector with itself and so we always right that physical Rio state vectors satisfies size equals 1 we're not going to allow that to change with time so that means that we have for time side aside your sorry conduit sorry taking the intifada surveillance change with time it's not hard to prove from back that theater product of 8 2 vectors has to be prime independent that's rather easy to prove our I a player you do it how would you do if ordinary vectors I want approval for nearly vectors that if you of every vectors respects the inner product between any pair of actors stays fixed the way you do it get pair of actors and if you look at the different sovereign there's a difference of 2 vectors now the assumption is that the length of every vector stays in the area is unchanged when you do some operations on linear operation in particular that means the size of each 1 of these triangles shouldn't change on their across operation of the size of a triangle brought change that means the angle between these 2 2 vectors as well a these vectors of change and therefore the inner pop-rock between sold from the assumption that the link of every vector doesn't change with time to be a park itself it's also possible to prove that beyond that
inner product between any pair of members change of part that's true we cannot write down more general state next forever if I have to to be in a product for society is exactly equal the firefighter 0 times 5 5 0 you're partisanship OK now 12 for 5 I have all related by this object called you so at becomes 5 0 0 but your backer of cheating your if you will see because that's what the I'm sigh of 0 however simple role that find 0 you that sandwiched between Pawar vectors whatever is just equal to me in a product of the same pair of actors without you better you pass a simple elementary hero that is the inner product reviewed matrix element of an operator equal the inner product 4 all vectors Neukom immediately conclude the you'd dagger you was just you operated the only way to make this happen is if you back you for every veteran point this is true of every veteran of actors the only way that ritual you diary you Gregory you'll physical operate that product operated with that product is called unitary it's analog see analog for numbers for complex numbers a complex number of having unit late in the complex plane see analog of a number having the form for the we are they the kinds of song complex Eagle for 1 that's not for unitary operator is but held on the back but the operator plans Commission conjugal as you go to your OK that way if we were I think we derive this last time I think we derive much more restrictive a bookstore suppose now together I'd be out of town what you wear supposing instead of allowing car ball for a large amount of time let's just allow it to evolve from a short amount of time this is something that the was always doing physics studying how things change with respect to some parameter are chopping up the change in the little in has more pieces have been studying how things change under an infinite has more reversed small change in the building up big changes by superimposing what's right were juxtaposing what's more changes salt would take the time interval chopper Labrador while pieces instead of asking what you wish for a finite amount of time and what your would be for a very very small Pioneer and they'll try the build where you should be for a large part of our if we know our system changes over every small cars the world that it should be possible to build up our changes over long time bubbles is exactly of course the same logic errors and the Tony and the mechanics way you owe study the motion of the system by breaking up the the timing the little segments arriving differential equations solids not no idea so don't take that kind be more equals Absolon means it's small and his son is epsilon is small enough that epsilon squared and be ignored standard boasts more parameter approximation of soreness small epsilon square is much more so ignore it right now is the time was equal to 0 0 with equal 0 0 then you must just be 1 why is that time Zero amount of time shouldn't change system little evolves no time at all should change their obviously you should just be you have operated the small collection small correction should be proportional epsilon right my nest and now this is completely arbitrary among only putting in these factors here because I know that eventually younger ones who were from want to run for mayor minors I have important factors epsilon a small change in kind should lead to a small change in your find that object which will call a each and there were put in a lot of factors a really but warned that would be there in the textbook in the textbook that would be an age bar points cars now after this it is and highly conventional say was not content and putting the idea or the Epsilon or the age far although be absorbed in a each and
then changed the name of a quiet W saw what there's no content not whatever putting is in Sept forms and this age bar into the expression it's just
let them for purposes of comparison with other things would other things already pre established by History basically by history Mora less to do with physics and has do with history our we put in his factors the epsilon of course really belong there that you can get rid of because just telling you the time interval small and hour Rios was true convention whatever less using conventional let's see what you that you would hold 1 said so we democracy doors you dagger you dagger as 1 class i Epsilon aged dagger provided by wasn't barge was consequence of course where your plants cast their votes his Vite is such as ridiculous number ahead of visit but what the Yad why what what the world would their purse and our throws it the use of war were what history who would recur so small a ridiculous and president of viewing is creating vow has would had urinary Villa 1 um come up so ridiculously small that way I spoke that docket broker will it that was a similar question was a speed of white so big nor the are ones that want why has our driver number so big but yes this requires chemists EU only that they could manipulate with their hands are they have moved beat gurus and other things a paper fluids and gasses and how big the flow the containers they were big enough for holding your hand to be off tip over our do things worth and so was natural but the macho unit of measure of substance this because his gasses should be the amount of gas withheld the container about as big as a human being OK maybe 50 times more of a human being but also the question then is a question not of physics nothing until physics has to do with human biology white as it take take the 27th Adams were from World War II the 30th Adams or whatever the maker human being here sir that with amend many fewer Adams and that you couldn't make complicated enough circuitry and so forth the make and college and things can ask questions so it's the people's Tropic principal better that if you want this big you simply couldn't simply couldn't ask the questions or so you better be that that's the reason that that's the reason the plant constant was more direct it's the reason the that the electron mass is so small and so it really doesn't mean anything fundamental from the point of of view of physics is just the choice of units choice of units which goes back historically put kind people know about microscopic physics so we worked with large large numbers of electrons large large numbers of Adams because they themselves will launch a case so about a spy was just there will big H Is that as you have something rare dagger is there because you have to take the Commission conjugate the Epsilon was there because we're talking about a small deviations from T equals 0 and the eye is there because there was saved me writing on later on OK you'll see in a moment the IRA's there I what does your that EU will we have to say you value it will want to solve X 1 was REA Epsilon age that each bar finds 1 minors i Epsilon each of our bar Siew dagger you know we ignore everything of oil epsilon squares of your finger appears as 1 class i Epsilon H. bar of age bar bets that the terms and the other was might almost say the only difference between them is 1 resonates bagger and the other has made by borrowers that has to be or 1 that's being equal 1 cervical cancel out the 1 on the left hand side step the result Cooper 0 another way to write the same things simpler there's just have a staggering equals age that's the conditions that you'll be unitary that age Bihar mission permission means equal pose complex frontier now we've already argued you accept bits of loving don't let observables the things that you measure in a laboratory correspond mission operators emission operators are observables are observables are mission operators solve what is on Jake is age it's an ensemble according formal definition in quantum mechanics are observables Mission operator must cast onto something you can measure moreover it's got to be something very general we had our precisely what form readers may not be general precisely what operator but every system has 1 of these otherwise it could change with time so the existence of some kind of thing called age is very very general all systems have H is of course the Hamiltonian the Hamiltonian means these energy thought an operator Escobar Hamiltonian thought was a quantity that you measure called the energy so redundant location recall a each could also called E for energy sustained things we usually reserved and the label easy for the Eigen values of the things you actually measured and ages the operator OK Mr. White why is that the energy well what other definitions so far you have energy anybody guarantee of the definition of views about how his father voice a OK what's definition classical now for doing that were more not less suffered definition of energy we have that's most certainly not that that that is the energy of a wrote rip article were so we're not job the do was aware the Israeli air force as it that
way we could we could work at a from that from that and we will have the word with that's it that's a very special situation where you have nonrelativistic objects we're nonrelativistic forces on the structure of energy with that would that help you understand for example the early energy-storing electromagnetic field saying that amassed on velocity of mass velocity squared whatever now Mr. some much more general concept is a much more general concept In 1 way of characterizing it is consumer that's the single most important thing about it but doesn't change with time and that certain familiar systems it takes a certain form for certain familiar systems it takes a certain form and that the statement of the whole shebang all kinds of energies don't change with time some of their errands that's where energy so of the most important thing is that if we if we just taking an abstract system and we don't know how related to any the secular nonrelativistic classical system the 1st thing we would look for in a definition of energy would be there concern doesn't change with time or combat what will come back with they are white is the subject of object that doesn't change with time what sense of change would come back OK so now I have an object which will give it its name call the Hamiltonian and we now know that the evolution of the system for a small amount of time is characterized by this expression here OK let's see if we can make a differential equation for the changeover system with time why differential equations wealth not system changes over very small kind Kabul's it's natural to think of a differential equation evolving from 1 the next so instead of looking at society let's look however declined the river sorry but chief of sorry was well that's equal essentially by definition side of Epsilon Miners sigh of 0 as an athlete any kind could be taken to be the Zero of prices nothing special about 0 Justice small shift Siegel's Absolon the starting point was arbitrary the important thing is we change the car will bet I will change the time I will and now we have to abide by epsilon MST derivative the time derivative of the OK we have we have what sigh of epsilon is wary of of epsilon has gotten by applying 1 Myers I age old age 5 Zero is just gotten by applying 1 so what does this year I 1 my years now 1 of my I I ever saw H Over each bar when I get that right nor did sigh of epsilon this minus 5 0 that just cancels out the 2nd just cancel out 1 of the difference in the numerator there's nothing but I plans I saw 1 each over each bar prime sire 0 that's the difference between sigh of epsilon size 0 now divided by epsilon just cancels out the Epsilon From it that's not always think about 8 which is basically apart from the fact Iowa each bar if the operator which tells you how sigh changes differentially with time how it changes the prime derivative Robert by if of sigh equal progress no that tells you how sorry changes with tired from instant instead because a small change any more lapse of or basically it's a differential equation of the state vector size squares remained called Troy equation following are used to seeing you have form anybody have a favorite form was running required in want because there was his love more was at added old walked from votes short of that Margaret Hassan a yes vote out of the war there are good for me editors draft was that if there get is a lot of more yes that's it that's or quantum mechanics nutshell or somebody tells you what's fires 0 you always using the Schroedinger equation you find that at later time you the state delay currently makes a measurements inside gives you the probabilities for measurements that's it that's all started with nothing else but on character restores out things change we're ready the we're ready figured out of the state vector how to calculate expectation values of probabilities so what and I said OK but road or a or just stop there but Stratus Alves was simple situations but 1st of all on you know there is another definition of energy that I'm sure you've seen before it doesn't come from classical physics comes from Einstein and the book on freaking out and what now what's a former equals know right a little for H bar Omega Omega New Wave is frequency in cycles per 2nd or make years frequency in radiance per 2nd hour each page bar is a thing because of Omega Europe
plants constant which goes frequency frequency frequency plans Ponce constant is equal Quebec angular frequency and look for scraps Wal-Mart Stores arms but still I can understand why this h he might have something to do with this energy here to review our let's solve the showing operations but what I wanna it is a support and 80 is on a mission operator it certainly has eigenvectors the affect every her mission operate has a complete set of eigenvectors so there's some basis some basis of state off all normal base which are the eigenvectors of let's imagine working in that let's pick 1 of the eigenvectors would eigenvalue you he is the ideal values each on sigh is equal to eat palm-size now what happens if you start with a eigenvectors of each Y has said that it will study and eigenvectors each with time because if you wait a little bit of time you'll be any iand basically be teens the you airing in what you a heading into work the site irritating and my epsilon each time sigh but since each time society is proportional to site itself what you're indeed and will kind of ball is just something proportional to the original vector started with is sigh is an eigenvectors of the Hamiltonian if signs of eigenvectors here it will sustain eigenvectors Hamel from for all time because so little things to Eddie iand are pointing in the same direction this so if you ever of eigenvectors if you where you you just multiplying the eigenvectors by a number so let's look for a solution of the equations of the form is ours always this final is by definition Miss it is a cruel yet to indicate that his be eigenvectors with energy he West look for a solution of former sigh of decisive each of tea is equal to 0 what society is a precise 17 of 0 the redundant we don't need that just from eigenvectors of the energy as a function of time a similar solve the equation that way OK it was viewed by BT of sigh of sizes just as 50 times a constant back there well that is a constant vector you just differentiate yet 50 of left-hand side is what's called ever thought ever by means here that standard notation of dark or something means differentiating with respect party for ever thought firms side fee by GTE progress a quote Kuo minus RIA H. Paul H. Barr fans I've 0 strike but chance sorry he was just keep the misery out what I would at my best on DOH far what I identity as half of 3 very this suggests or not suggested that we can find a solution we just by the solution of the equation have got physical for Martinez RED over each bar concept 4th the plan to remove F is just proportional death self Price this is not such a difficult differential equations it says the derivative is proportional to the thing itself when I was equation like back social quarters in my Kravis exponential if you have any quite says the F by beauty is equal to some cast their vote Mr. constant over here the constant find F the social that equation is F is equal to ease privacy seat if you differentiate an exponential it just pulls down the coefficient of the kind in the exponential saw solution of any equation of the form derivatives proportional of the thing itself as an expert American off what the solution is a constant f is equal to each of miners over age bar Times t they could be a constant finds that but apart from a constant solution of actors beers it will now solve assuring required and destroying our cars solution the time dependence of the of the reassuring equation eigenvectors of the Hamiltonian just a multiplied by each of martyrs I E over age followed 3 times I've 0 well is this fear he might DOH products combination of 1st sign of signed call sign of IT overage party boss I signed I Heritage party sewers S that was a homemaker soros we see that F His Ito minus RIA McGerty where omega is equal to over each bar is now I suppose I hadn't told you that he was the energy was there have called you before their features a Hamiltonian easy energy at this point we might look at Paris and old I feel that equation before equals HO gage for maker this left-hand side must be the energy from a quantum point of view and from the point of view of our of signed and photons of plant and all that stuff we would recognize the left-hand side of the energy not so clear that this has anything to do with us with the energy from Tommy and mechanics for but motor study that relationship we would have to be talking about systems which have a natural a nonrelativistic mechanical interpretation and were not His point so this is because we can get to a derivation award you are for an explanation of why we call E the energy used a finger Einstein would recognize our so we and reduce accord is a leader who was shot dead in his young men who are are you a that's right co-starring R P Myers are sorry but or make is equal to your way of vast would erode hourly wages of a value of of violence that they are allows visitors about that because that's because well I sprinkle me or so that the solution would be mission now and that's gloomy are were found each bag equals when stage was OK that's an n-type mission operator the score by provisional but I know what to do about it but I can't buy her mission operate on a mobile are quite high becomes a mission operator so those not currently I just put it there because I want to give a day I wanted the defined the thing which would be her mission Lewis's arm foresight or hindsight must show which but I knew Viet not yet yet the reason has to be Rio is because the unit parity we don't want a from APEC the change with time is you know we see coming out that await the vector boson shoulder and that was assured by having EB real vessels Valenzvela most factors there so it would have to work here with them here are brought to to a vote there is danger that race was
RAS vague setting where there is a vote as early as possible His visit as well as a way to get there really disease column no no are woes not rude that's next right we're right it is a principal of course there is but there are worth nothing up course point the point the point being that Eagles H-bomb is more general thing just show for palms of your frenetic radiation more general interpretation of the errant photons being worn special case of OK let's say or if I suppose and we have a vector which is a sum of eigenvectors of we a Hamiltonian with different ideas about in general a different eigenvectors will have different bargain values corresponding to the different values of the energy can have worker to be measured so supposing we take multiplying a vector by a phase like this is not very interesting it doesn't change it and interesting way which is more applies a buyer phase and know phases on port or mechanics but supposing we have a vector which has some superposition of energy ideas now a Hamiltonian he is emission operator it has a complete family a complete bases of eigenvectors soul in principle any vector any state of the system could be expanded linearly in eigenvectors of the Hamiltonian knows how are we solve the solemn it is simply labels different ideas about use of us let's just right hand coefficient alpha and pions sigh where scientists f eigenvectors of ammo from what happens to that with time well the point is that since the evolution is with a linear operator we can figure out what happens to each side a them up that's what that's the nature of linear operators that's major linear operators but you could bring them past the coefficients here just figure out what they do to individual vectors and ended up in the air so what happens to this evolves into a war so summation end after an from now we just have to figure out what's on indoors voice ready what work each eigenvectors energy does adjusted multiplied by Ito the RIA he's so the end when assigned Ahtisaari east of em over age 40 firings science each 1 of the eigenvectors is evolving with different frequency this is quite different than or overall Freddy's each phase of each component series involving differently but that's a general social assuring turquoise each eigenvectors energy just evolves with its own frequency a high energy was volary fast frequency goes around Arafat's philology states evolves slowly but if you break via the system up and 0 eigenvectors of the energy and the solution of any quantum mechanics problem is just the right car that each vector Van Van Horn says Shanghai view handling vote is His were losers of 4 users cars your Saleh said he connection that idea for faster more it's not normal you do apply richer ruptured vertical also brought forward for yard portables makes sigh effort is a disease then you need him as his unique about video saying shortages have vast wealth I'm not sure I'm not sure exactly how picked questions on the lots of situations where high frequency corresponds to bring energy but Kwan can be usually traced back the somehow 1 with Robert require mechanics but hurry we have to work at a case-by-case question right now on a yacht for what do an example but before I do the example OK that's that's that's all basic he's staff everything is easy inquiry 1 up observables observables not and that a Hamiltonian itself but other observables other quantities that we can measure How they change with time nor I mean I have change with time for the moment let's ask how their expectation values change with time the expectation value the average values yummy average value of a of an ovary observable was score we observable AT the average value of 80 sigh a-Si now is the size which change with time might be a BA is awesome very definite matrices but described very very particular observables it's the states with change with time giving you the rule from the states change of heart and that will allow us to figure out how the expectation value of observables change with part of states which so let's figure out what's just ask What's the kind river of 1st let's call this AD average average and now I'm interested in that time derivative on the stand time derivative Powell the average of any changes with time we have a product of 3 things here of stayed on the left a stain on the right a bunch of Oh coefficients corresponding to a matrix 80 in the middle order we differentiate with respect it's just the product of 3 things call them of time-dependent is really a has not signed the pact sewage is treated as a product of 3 things so product 3 things Hattie differentiator product well it's a sum of terms in each terms you will differentiate 1 of the factors and where more this that is so I got a sorry cause sigh eh sigh . no I don't have Bobbitt differentiating a vessel thinks matrix corresponding saw some very very definite off observable here by Victor want some flaws study some particular component of the spin or whatever happens to be a changing your with time average only changes because the state the changes Wellwood next and later served as a high versus as does shorter equation user strutting their equation I got so what is this 1 What's the sort of this factor here turned sigh AT now side not I put in my ears I H 0 8 bar times I arrived got sister and hear what this for this I put in sight but now the time derivative of the above gets up plus are at the complex conjugate that me a plus each of age kind I'm side most differences in the 2 1 case I have a minus IRA in India the case of a plot sorry that because I've complex conjugated when studied the remove yeah the complex can't plan side that the other differences as a form of operation a age is different 2 firms here I have a year I have 8 8 which was a little different age Eddie
miners a Each the factory taken out it just Ohio H. bar fight over a bar but sigh H a miners 5 that's the time derivative of the expectation value of it a bar top score remained for the product of war a chase wireless age we feel before we talk about yet with public commentators is rooted in his voice E ahead in his vote but doesn't act because we could try to edit I drove the Theron for you but says that said that further operators will have a say simultaneous eigenvectors if they knew York proving that but doesn't matter because just to readjust the definition a H to operators of comedy and be a combination aid being miners BA classical classical mathematics if India numbers this differences always 0 obvious story but they're not numbers linear operators or matrices and northern matrices don't generally commute but you can't change the order will you multiply so this object baby mines VAT is called the commutator of a and B and threatened the stereo or square brackets for comma a being miners BAT equals bracket or practical cycle or bracket was confuse it with rather kept by our country paid symbol carnivorous of square brackets ISO now intifada we derived a fundamental identity prime derivative on any observable 80 3 right now a dark is REA over age yet Beaureguard use or IOH connery's nor was as this is the average of AJ Missouri swords the average of the commutator of Asian Eddie so the time derivative on average of an operator is related to the average of the commutator with the Hamiltonian that is of fundamental significance of a Hamiltonian physics from want me decrying the river of something derivative of an average value back it is just equal to the average of the commutator with a Hamiltonian very simple away come from Paul because it's which between site Westley anything that sandwiched between sigh like as it is an expectation value retail once expectation value our sandwich which it between sigh some particular the comet data was itself an operator and I can ask about what is the expectation value or become a very volatile Vong voters derivative of the average of 80 is the average of a crime figure of age with but the 1st application of verse with 2 1 simple application with 12 applications are the ones before we take a break in the morgue after we take a break the question wasn't trying remove the average of the Hamiltonian itself but suppose now we didn't know what a Hamiltonian wars we have no idea what it stood for it's some observable we know that because conditions but we don't know what it means years In the broader 0 and the broader context of so what is a timely be kind derivative of the average of the Hamiltonian you general 0 because the commutator of anything with itself is always 0 the mathematics of quantum mechanics is not so bizarre where Burma h is not equal database that resides not Beijing physical thinks so advice plug-in Fahey h itself I find out that the expectation value of the Hamiltonian doesn't change with time I like as a conservation law can't conservation What is a conservation of age every system has and each every system has a conservation law or what more likely thing is there and the energy is the energy of V of our case somewhat relieved as revision was almost as well that's a set of offering a but a Sobol things which could serve as energy that depends on the details of for example we have a particle moving in free space with no forces are opened your has however was a force on the a system that was not so it depends on the details system with a roar the conserved quantities from the guest of rude to request is the collection of all things which commute with Tamil party and that collection he is a set of conserved quantities of things such concern for the particular system you might be thinking about a set of 1 0 a point that idea but I forgot to to mention do what we no example amended Ranariddh to undertake or less abstract staff and work it out very very simplest case namely us being in a magnetic field the simplest system that we deal with although the shape the all the of the abstract things out the world of will with the detail part of concrete example arms the Hamiltonian H as a mission operator that means that has a complete family of eigenvalues and eigenvectors and if you work in the bases of state described by the eigenvectors of H & H is a diagonal matrix of page is Abagael matrix In the bases of eigenvectors of 8 fetched revealed nothing nothing better and Sohail energy I basis what's called leader have in the basis of states which are energy eigenvectors County is just 80 Bagwell matrix you want to 83 84 and so forth with zeros everywhere self now we also saw what happened where we also saw what happened individual eigenvectors prime evolve as tiny evolves each eigenvectors gets multiplied by each other like I each your each party that immediately tells us In the same basic basis of same eigenvectors it tells us we you waves your is a matrix whose entries are just these exponential each eigenvectors gets multiplied by this rotating phase by this exponential brand of different eigenvectors don't get mixed up each 1 just more quiet and sword tells us that the matrix on the same basis in the same basis is just eat having minus I even 1 over age 40 he Myers I 2 0 over each party and so forth everything else being 0 effectively other elements of 0 meters the the eigenvectors don't get mixed up
with each other but they don't rotated to each other each 1 simply gets these and so it's a bag of of books like there's lawyers usually written is just say this is reason is a matrix representations of Lanier operators in terms of abstract operators what warm rights Is that you'll is equal to eat are a minor sorry each for OH bar you go interpret that just in terms of the Eigen values that elements are you are simply be exponential for T he appropriate year Of the yard of the elements of a H that's imitation of your life we are here dazzled there was a yard wide a share of power highlight that Our follow it may be it it can be 0 eigenvectors I don't know what they thought was it is a garden banners libraries to yet but I won't here shortages let me out where they are but yeah yeah yeah he doesn't wary all yeah false alarms in Texas of a your of 0 . Hi how do another frequencies of a combined system of the some of the frequencies Yahoo yet the gap young well not a sub because linear on its This is a separate past if you like not interacting separated systems which interacting with each other Have Hamiltonian switch of the some of the 2 from that we could we could justify Our but it's a the question the answer is that the energy eigenvalues of songs the barriers of thus a surprising energy of course if it back in that's are quite sure interacting that may not be quite true reduced quite at Fernanda acting and you take the combined system be energies are simply the some of the 2 energies the Hamiltonian of the Sons of the Hamiltonian maybe we can even do an example we have time to our from are not sure we will not why are we example of what in some detail is the problem of a and magnetic field let's work out what's right arrow all the various equations that we that's a Silsbee now nor entanglement from armored probably will get to it by grow so I have spent person is ones signatory they represent the components or screamed and also proportional the components of the magnetic moment of electronic support from then we have a magnetic field B of Soria magnetic field point in some direction the point the core of vector because we go source confused our Our were thing reports some directions space and has magnitude to beat the energy all are they spin and magnetic field is given by the product of the spin and the magnetic field in fact about product revert to you have a spin pointing some direction of magnetic field some of the direction the energy stored is proportional per about product of the spin and magnetic fuel really is the bar product of magnetic moment and B & B magnetic field now that moment is proportional post if you have a electrons spilling magnetic moment proportional Spain's entity on what they call fish correct view of real few right magnetic moment so magnetic or magnetic moment magnetic moment which is back there His ISR is actually a spin vectors times numbers the number is called magnetic moment I think as a factor tool he in the definition of what's not let Starbright that just right is an interesting quantity time signal tool the tour was entirely conventional could be absorbed in the was quoted as saying this is the thing that you multiply by beat forget the energy obviously has a low coefficient here are that has to do with charge respect if stiffest suspend had no charge to would-be nor electric current going around and sold it wouldn't interact with magnetic field at all so the magnitude of electric charge of particles somehow buried in Newell year end this tool here I think is just a convention beleaguered members of 2 pairs of any case energy is said not your got beat but for magnetic field so must be the Hamiltonian let's take that be the Hamiltonian where now segment is for lovers quantum operated B is just those just magnetic field its another quantum mechanical just great big magnificently Mayor of 10 Tesla magnet these and slows or averted art and pointing in some definite direction provided by his big heavy stable magnetic we don't think about the quantum character of a magnet resources their broker now let's choose B in a particular direction what's the the most convenient direction the chose let's chose B In the 3 in the 3rd direction along the z-axis and then the Hamiltonian has a simple form it's just you all the tools plans the magnitude of the magnetic field just beyond about our over the magnitude finds stigma so this is a says energy or the Hamiltonian I'll and spin In a magnetic field KYY calculator 1st as I wanna calculate the kind dependence of the averages Of the 3 components of the spare this is the analog of the classical mechanical question of calculating how the spear does whatever it thought as it rotate wars farm so let's let's begin by calculating how the expectation values change with time the expectation values a signal ones with 2 wins in 3 us all about sigma 3 hours Friday equated the equation for the abolition of an expectation value is right here sought solace in the 3 sigma 3 . won't while riding expectation values I mean here's the expectation value of certain referring harm derivative that's equal vii or H. bar finds commutator Of the Hamiltonian with stigma Obama just proportional certain so the Hamiltonian except for a numerical factor is Sigma 3 and commutator of Cigna free with sulfur 0 so firstly it tells us is but the at the time derivative of the expectation value of sigma 3 equal 0 let's talk are the stories the X-Y plane z-axis X Y and now we've discovered that the component of the spin along the z-axis doesn't change with time How much freedom those acts give them emotion of Sigma while it means that the projectionist signal on the 3rd axis of change and of the angle of the spin relative to the yard of a vertical axis of chain so Ordonez is does the expectation value of 3 components and must have somehow more calm of fixed angle here otherwise the component along the
z-axis would would change with cars OK now about b X component and the white component of span of more interesting 1 boring basically this was just conservation of energy physical to 0 that doesn't see energy is proportional of a 3rd component of spin the conservation of energy must tell the 3rd pomp of spin doesn't change with time or the average ever the average of the year he should change with Mexico about sports you which for merger but still use it born . 31 . is I always each bar times a commutator are saying ready it is a threat that what they choose Page Is new all the food prices beat France in the rate of them you ROH bar new divided by 2 0 lead kind data of Sigma 3 Watson born now are moving its 1 . for our rights warned instead of a 1st entry years still simmer 3 by not commentators were for use in Warrensville and has had to broker was worker reported about for but send me as arrives in prison or warned voters soberly orders With reserve 1 that equal proved 1 minus 1 0 0 tie worn warned 0 0 2 1 is here With 3 here what's product usual rule 1st wrote transfers column 0 1st rd prime 2nd column 1 2nd rower Prime's 1st column Myers warned 0 this looks a lot like signal White 2 were Nadal will accept that pros lacking and I think this would be similar to what was a minus sign plus I hear so this is actually 2 or ii kind of sites as I consider the check grabbed my can to RIA pions minus III 0 0 minus 5 times eyes warned that the correct so Cyprus in the West over the other order now the other order we have 0 1 1 0 times 1 0 0 my use OK so now we have here sir cholesterol plans 1st column 0 again 1st rd plans 2nd column minus 1 2nd Rover cars 1st column has warned the right 1 0 this is the same as this except for minor side services might I Nadal foreign number when you calculate the commutator you subtract 2 subtracting minus icing with tool for my signal to itself is what twice said Nadal overcoming said free with 1 is just twice ii signature found suitable 1 . sequel trial H. bar you b commentator of signatory with 1 is a factor of tool i.e. were Davison stuff together the whose cancel think that's what broke through there the can broke through there but the figures were for Pecan by Karzai is my best here the memories as these are expectations about really expectation values and not operators we prepare our room bar above them and being expectation values the components are just numbers so this tells me that the x component of a 1 component of Sigma kind river is proportional the assume tool can only tell you what you get from their out of the labor reports just found the answer until . looks very similar instead of commutator of signatory was 1 will have come later said the privacy the tool and that of giving class you will be over each bar pricing warned your forever primed Arevalo said foreign is white there's some constant admit to rule and the time derivatives tool is last soon worn From there was viewed as a guy just 8 wrote just a point rotating a planed Mohite it is a very simple solution of various is chose sumitomo is equal the call of Bach you be over age bar tea and warned is plus sign plus a sign of newbie over each party have a right of dawn you might decide somewheres OH was was re-signed sorry the switch because I derivative of is coastline derivative of course is mind sigh when you take a derivative you have proposed coefficient when you take part derivative and give you Mubi overage saltwater is that means bad in the y plane now a component of spin the XY plane rotates around like that of motion of the spin vectors looks like a procession of procession about be vertical axis voters about the magnetic field is nothing special about using magnetic field belongs reactors we could use any acts our does our we wanna found that component of the spin-off along the magnetic field doesn't change with time that's energy conservation and then in these planes perpendicular to a magnetic field we were fined processing around magnetic field just like that are that of course is also known classical solutions of away on magnetic moment there would there would rotate around your magnetic Huber professor magnetic field 3rd gyroscope as workers for television viewers here is that there is a vibrant visit zone velocity is more a petite force yet the art that's right arm you could you can put it here this is a social required the phase factor simply corresponds look on the export right there's a solution would equal 0 he equals zeroed only said 1 is nonzero cross 1 so we found a solution where the spend at time equals 0 0 is along the axis army is nothing special about equals 0 Norway indicate that is to realize that we could start any kind with are Witherspoon pointing along the axis and then we would put in here somewhere in the state of mind they With data would just be we face factor or a or B kind which the spins was what expect from Val Verde Varley has there is a is vow of 1 man had very visit is very hot air and sea view van is valued views on there is no room at the point where doing so with doing some experiment we want do at the same experiment with we elect price so what's the experiment
experiment starts or why up old electrons or or also Max and doing it got so we pick axis politely electron Opel and then we will go it would join the same experimental electron means that we initially sought more and more our if we have a system leadership everywhere and we have a system that nobody's been careful to start exactly the same way we just out them right and we have made a mess with that we have in density where a mixed stake by the but we saw all the electrons identically that could mean that we start for example we start them all point to go along 1 axis apparently equal 0 no remains remained a report Nov chorus so 1 cynical answered with 3 or quantum mechanical variables Was at means for example supposing we start this so that we know that the pointing along 1 axis timetable 0 tell us that of we measure signaled what we we get plus 1 every measure Sigma tool we would get a half probability for it to be Upward Bound and free measures Sigma free will also get a hip problem but let's start with a system of honest and born axes and then let it golfer time after a little bit of time the system is processed into the food lecture that means if a little bit of time free with measure Sigma tool we would find it definitely being opt for a free measures similar ones we would find it would have probability of direction freeway mother have period on the quark career legacy Is Our we would find the spate of rumors along the x-axis who find assure that was my of measures signal again we will find it randomly distributed Allah that that's the basic solution of the problem of he said were hang up that had already paid for by ruse debated ADDS were was ordered last year so on were resource Europe for over a year to lose their last year then how Iowa of years we can't say what they do on us we have an expression for the energy Joe 1 possibility is that there's no magnet around but each 1 provides a magnet for the other 1 preparing if our case you could work out for me I would be a jagged various armed with the eigenvectors awfully the answer the answer is let us Over the United had very good and I had a view of 1 man 1 vote marked for a pay From a strong magnetic field and born them up but found laughter or we are on view this is shown up at wicket it's easy to answer the questions about electrons when they're not a magnetic field but when the providing of magnetic field for each other are they in the Hamiltonian is proportional the bar product of stigma and power still more wondered how 1 person with 2 outs to the recovery we're not a single 1 of the signal was in the freezer how 1 of power toward recovery we know how they act the states only to work out With the eigenvalues Europe till I values the odds are good mentality of the basis for a means of your for linearly independent vectors 2 times war but he you know what turned his wound when there are equal Eigen values for an operator for from as Colby General source Toa states have exactly the same energy and the our the energy spectrum is called generate save us our proposed more them once they would exact energy this Hamiltonian happens the have really be generous states and 1 state which is not be be Jarrett worker we therefore there are free there are 2 energy levels of value on the NC Yukon work this out the answer is both issues are we work In the basis with springs far along the z-axis if on both spinners are up then you have 1 particular value of the energy was just court equals evening as energy false spins are bad you also have the same energy and a list of possibilities of scored up their own plus some ideas down off 1 of them is the so-called singlet state that has a different energy the 3rd that's the more might start with them either sigh of course he warned you warned this time we get energy to 0 but with this side down up we get energy he won again at some even checked that's why this trip whether states are pop down and our family frost-bound up score Triplett was called so I have 3 different energy levels I want to know although 3 different energy levels and you know what the eigenvectors are you can then expand any vector that you like to start pick anything you want stock worth our which is some money a combination of use and then just evolve using while using a formula we have on blackboard born with showed gone by now formula said you're expanding to expand the state's an eigenvectors in each eigenvectors girls where the phases that's determined by the what toward the question while you are right down general strike because state because because of Vermont Views Van Gogh battle just tool but will have a tried stay that way and I AID any energy eigenvectors doesn't do anything terribly interesting it is rotates with strong faith so what'll happen to it Is it will pick up a phased eatery I need to always each bar or what's more interesting but supposing we just pick from vis a vis the vast and varied vats at the moment I don't have a magnetic field from magnetic field that provided each 1 season magnetic field for the of a wonderful book it so for example we could start with soot store problem let's suppose we start system as ot Baron that's 0 eigenvectors re-emerging IOC among these where are great among these is up up found their own and are banned Wednesday them up and up and forced so West that's actually work out what would happen if we started with without back that won't be trivial were stored of the 1st thing we do is we expanded you eigenvectors less right down what the eigenvectors are pop up bounce back end Bob downed plus stand up for some ideas found out provided by the square root of 2 vroom vroom obviously these toward Muckler comment game its these 2 widow common game the fact sector right up down equal their class down for bracket Arad British paused of their own Miners found rock provide despite tool and this by 2 0 but thanks Bob their plus down and down this down up the down or cancel majuscular Brown that but right now this area's square enough food and square root of 2 square through Times Square tool Let's give these 2 guys name of them forced them up and down my is down I have some names
where we scored more and more than of his Seymour Teresa they and Triple-A as well as Teresa but services Triplette divided by the square root of 2 of them closed down provided by squared who was Ross Seymour Laura above square root through so that's a starting point busses or importing now let's Eve all this for some time to do it different energy ID value was 1 of them goes with a single it let's call it a was each report each report EC work each report so why happens this state as time goes on to the triplet state picks up a factory on our right down the straight after time to this will become each are each replied minus ii each of age bar See you are boys committed he's always better car caliber core Caresa capital Tripp Lite use capital this 1 this 1 picks up for phase heat from ISI East staying away bars view Villarreal I'll vision a village vigilante ADDS var var that depends on whether a plus sign on Marine assigned from a whole fares so if I have a plus sign think probably 1 of 4 main magnets warm decide on the energy is typically bigger for the triplet state that's what Ireland to maggots are life you have more energy with by warriors right because I like to be and alike North Pole likes to next southpaw so they want to and Pioline was and you can buy line and that means the singlet energy is less which but we could define the energy by saying What are the same those subtract constant from the energy that will make any difference we get set the industry with energy equal 0 then the trip energy would be possible nothing actually depends on constable which is well-paid we have now we have the evolution of the system introduced of the triplet and singlet state but we like to we could go back to this representation at the the raw I mean we or be off the up-and-down Spain's might b the finger were interested in whether along 3rd axis on our these both entangled states might want to write it in terms of Pangle states so then I would simply put back from a droplets state to what we would get your each other miners Ali Saeedi feel age bar graph poor all the square root of 2 now we would have replaced so that would be up down plus down up or square to yell or square toes so I would be down he would want cup 1 component and the other components were be something very similar each minus you we provided by each bar or the square of of downstairs or have up-and-down minus down up well I do what I now want to do is I wanna pick off the ob down and down separately from the 2 expressions and combined together expresses in terms of a particular states of famine bound not so let's do what I get what multiplies abound he a minors I see see over bar class I need TV he's single TV Over each borrower kind of barrel or 2 but now what about the other components of the component with down up it's not quite the same factory in fact has among the site right my assigned from here so we get down not is different than that of the apparent Paramount gets multiplied by 1 hair off nor was this year 1 here With the plans no not quite Chretien EDS different from each other him along von voter altered nor Syria's hyperbolic on it is our each week it's obvious we got do signs and what it is is it's he should be let's Weaver persuade voters will be collided terms science co-stars on my each other miners I see rage bar my anxiety he SOH bar so you see that the Arpent down the up down and down upscale different coefficients greater at time t equals 0 the coefficient about them and then up over saying that kind of sequel Zero is a right not a strong O'Leary sorry that time Keagle Zero we started with pure up down remember so what kind of he equal 0 0 Our these this was your plan equal 0 only up down but then we would a little bit this becomes nonzero this gets a little more the fact 1 was maximum currently equal 0 it gets a little smaller this 1 grows and grows in magnitude and after a little bit of time you have a bid of down a lot mixed in state vectors there for a little bit of time even tho you started with pure up down after a little bit of time you will have some probability of down now you wait for a large enough time after a while this 1 will cancel was 1 of a kind which this 1 cancels as what factory at that time this one's maximum so it goes back and forth between up down and down up back and forth between our paramount Upshaw of interpolating in between the 2 of us so what happened but you see the role the rowers 1st start with the bases that you're interested in which happens to be in this case the Oct on entangled based writer state spot reads precedent tons of eigenvectors of the Hamiltonian van they each eigenvectors Hamiltonian multiply it by the right and then this is Pandora back in video the original bases and that will tell you how the probabilities in the original bases change with time let's just eluded simplify was a little bit on let's say I think this is easy Our theory minors are E C plus he yes now no other . com what's at 9 wrong nine-under murals hum who or what Yellowstone or what the average yet so right our group and the average of theology is he people us over till us Our according to the RIA muddied the miners hi fee plus he s 2 meters thick America problems if he s had been equal to 0 0 0 then with how long we have this phase here soon wouldn't change all or that With before
so visual or along the walls of Harris view that he II this forgery buddy at Rich they are or area e-commerce I E T minus team cost Peterborough yes my 18 bt plus 4 boy yah yah OK this was done because I of he was it is God wire by twice each far-right French free the other ones looks exactly the same as exactly the same factor a phase but that it has signed for the same thing this is not for us all all phases on for overall phrase doesn't mean anything was change anything when you calculate the probabilities see overall phase goes away orders drop overall face to get job for forms right yesterday Roswell happens the probabilities only depend on the difference of energy as a good thing always depend on difference of energies the overall sum of energies doesn't make any difference from the although only the difference of energies counts and you can see what happens because of oscillations back and forth between Obama bound up with a computable frequency here was something I know most the frequency related to to the difference of energy use so the difference of energies of the singlet and triplet state tell you how things go back and forth between the 2 were states of barrel not that's an example of a quantum mechanical problem more about there From or better for the rules are always expand the backers of the eigenvectors hand on your left each 1 Eve all of its own and then put it back together again a former you want to compute where probability 1 so it's your Standard ritual you reduce the problem ritual which awards jury Walker Wolfram's promote
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