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Advanced Quantum Mechanics | Lecture 5

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With fief Stanford University but just review from
moment tho year chemistry that went into these discovery of Spain and the fermionic character of electrons basically the fermionic can their character employs 2 things they're actually believed To be related to each other but they're different 1 of them has to do with a span of electron spin of electrons half spin Ender that means that the Spanish spin goes if we forget about the orbital motion of electrons for electron who's been states corresponding to spend a half bus have mile a half hour the other property that qualifies electron as a firmly on news that it satisfies the Paoli exclusion principle you can put more than 2 of them in the same state in talking about the state's Spink County sorrel here for example on we ignored the spin and example we talked about putting tools electrons into the ground state wave function of a hydrogen atom yes you could do that you could put 2 electrons into the the lowest energy level but only because you have the spin the soak up these media Powell exclusion principle so you put them in with opposite spend to try for a 3rd 1 and you can't put it into the ground state which I preferred 1 let's say because we're dealing with lithium our quest for helium Wickham put both electrons and the ground-state park approximation that we ignore the interactions between your lack words in the chemists bad approximation into every these days chemists are very sophisticated quantum physicist could really good chemists very sophisticated but cannot the older of the old picture of the Adam the shell model and all sorts of things that were simplistic which didn't work very well but they would go a bit you put 2 electrons into the ground state the helium atom of the helium atom behaves as if you had to electrons both In the lowest energy state Our and that's OK because they have opposite spit if orders up his ban Farmer's bounty of resort in fact those 2 electrons are entangled their entangled in exactly the way that the robbery tangled orders up IBM's down Wednesday the is up and that's a that's state went to Italy you can't put electron into the ground state now coming into the ground survived the helium atoms and of course in the ground state of the electron wave functions In the acidity of the of the lithium equates our growth nucleus also has a collector a charge and also has a collection of wave functions for electrons you put 1st went to the ground state in the 2nd 1 in the ground state but in the 3rd set the tone of the state's soup with no room for it by room I mean from the point of view of the exclusive principle so we start filling another shelf and how many how many energy levels are there at the next shell well we can draw our energy level picture that withdrew before 5 angular momentum 0 which is called the S wave at least for the simple approximations which the electrons don't interact as the ground state and then there are excited states which still have angular momentum 0 and so forth and so on and then there's a P waves the P waves are angular momentum 1 or orbital angular momentum 1 peace stands for something I they don't know what and the 1st P wave as more energy than the 1st wave that has 3 states the 3 possible states of orbital angular momentum 1 and then there are above that and by accident by the accidents or leave them out really accidents of course but by these funny mathematics of cool potential their 1st it's well is generous he's actually generous but indicated he was bad and so forth and so on S P B spectroscopic stationed at the next level of 1st started the next angular momentum this is L. equals 0 l equals 1 0 equals 2 at the next waived level illegals the lowest energy state is already up here 1 2 3 4 5 1 2 3 4 5 0 4 5 6 2 and as for the pattern of energy levels looks like OK so you go to the um you go to be lithium batteries North Sea a man you put into electrons into the lowest energy state both In these lowest energy as we stayed the nite the extra electron you have to do something will you could put BOX electron upstairs at some higher energy state but it would be unstable because it would be carried by call upon which we have gotten too but it would but would be take down to something more energy or to condone is but the next electron intake any 1 of these poor States any Kigali where comes after let him I can't remember Peru and get a new kind beryllium beryllium if flawless 1 and you feel 1 of these next levels but when it is for electrons now so you 2 of them that's so beryllium Watkins comes at the beryllium what OK and 1 at the bottom 1 problem that right and if wasn't speed at the next level which would the next batter which is where the nitrogen at the next nitrogen level you have to start filling up these up here but that's not the way it works the spectroscopy and everything we know about it is consistent with pulling the 5th as the electron warned to whatever the via a year and that's 1 fits in again
into 1 of these levels and that works show your gotten too young male case in neon era 8 electrons In this 1st XIV energy level here why are they eat well 1 for each orbital angular orbital state but up but you also have the spin degree of freedom so there are 8 states that the occupied and then beyond neon I gotta remember my chemistry with who sodium got back sodium was the next level and the right you have to put 1 of them up here any love story you the story as I do surveillance electrons and so forth sodium looks sort of like hydrogen and sodas lithium Bacall it has won an extra electron in a house of in the out of most of the electron orbit and of what had been wanted to chemistry tonight I just want to remind you that to move things seemed to go together the exclusion principle and spin ahead half now could you have here is an interesting question could you have particles which was half but didn't have an exclusion principle you could put a spell I can all alone before I up but there was about 4 times the photons have exclusion principle photons most certainly do not have an exclusion principle and in fact classical electromagnetic waves are typically ways in which many many many quanta photons are all in the same state that makes them classical that they have a very very large amplitude of very large number of photons all in the same state of photons most certainly don't have an exclusion principle and fact that's almost the opposite is tendency for them to want to be in the same state but a broker selected for cars don't want anything but they do tend to congregate in in the same state whereas firming up has been particles can be in the same state so could maybe a particle of half Spain which doesn't satisfy these Powell exclusion principle or lawyer could be a particle of integer spin photons an integer spin particles been 1 already has been 0 could be 1 of those which does satisfies the power with exclusion principle and the interesting fact is that they go together have integer spin that doesn't necessarily mean spin 1 half it could be spent 3 have spent 5 years 7 are those have been objects always satisfied be exclusion principle and and the in digit particles never satisfied the exclusion principle is that Phil Babb Sephira resorts of 5 it can be derived From the principles of quantum field theory we will get that bus show a little demonstration narrowed her tonight there or use a belt so understand some aspects of it there's some interesting topological aspects of which will try to get through this is you will not have this on no it's not a game that's correct that's correct our fear or not the 1 thing you can say that stroke their maximum button in the bull's-eye now no I novel there there was a strong tendency for them to want to do the same state OK and will try to get that by this business it a tail off 2 minus signs 2 distinct different sirens our 1 having to do with spin and 1 having to do with what core change but before but before we talk a little more about the relationship between the have been and be exclusion principle let's talk about how we exclusion principle itself which has a deeper meaning just saying you can't put 2 particles of the same state as a statement about the nature of wave functions of Malta particle system if I have a multi particle system and now when I say wave functions lets for the moment forget the spinach and not come back the spin and play with do about it but by let's forget about them from moment and say that the Cold War admits the the particles that were talking about called X sub IEI does not stay x y z stands for particle 1 article 2 particles 3 and so X 1 represent sparkle ones and so forth a X could be x wider z memory Romero suppressed the year of the 3 components of positions Comex but body a represents a particle and then on any basis of state vectors is to give the positions of all of the particles let's write X 1 X 2 x 3 just as for 1 particle weakened label a complete set of states but just prescribing the position of the article for a system of particles we can just label all the positions and those are all normal family of states a complete set of states and just as for 1 particle we can define the wave function To be the inner product for member state vector happens to be this is an abstract state it's an abstract state of the system in particles some Member particles and the wave function itself is in product of the state vector if With a specific state vectors a squad X and batters called sigh X 1 through accent what you do when it is Querida well you square to multiply by its complex conjugate multiplied by complex conjugate the result is a probability it's a probability defying the 1st particle X 1 the 2nd particle next to the 3rd part of X-ray and so forth so this is not a particle quantum mechanics and much now let's
suppose that we're talking about His identical particles of the particles of exactly the same species electrons wherever they happen to be ignoring spin from Olmert father articles which are identical we have as the question we talked about it last time our we have to ask the question of whether a part particle 1 at position 1 and particle Toa position to the seems state is it a different state than particle true decision 1 and Article 1 at position to another words but cut that's the blow electron part at position x 1 as the red electron positions 2 electrons don't colors there identical indistinguishable and so it's an interesting question to ask is this should this be counted as exactly the same as the state with the red electrons at X 1 and a blue electron next as far as we know from all experiments as well as from all fury from quantum field theory relativistic quantum mechanics the rule is that these are the things so for example let's just take 2 particles that tells us for example that but state that I would label X 1 X 2 look particle position x 1 a red particle position next 0 is exactly the same actors red particle at position next to start now which 1 was No. 1 boy was low particle an extra 0 led park or X 1 another words there is no difference between a state With part 2 particles up to particles are safe that way but don't end in quantum mechanics on the physics of estate the physical properties of the state the physical properties of the state don't depend on 1 particular parameter they don't depend on the overall fees In front of the state vector in terms of wave functions the overall phase in front of the wave function never plays anything that is physically measurable as because all probabilities of all expectation values involves flight times start so it's conceivable that really the right relationship is not that X 1 X 2 was equal to X 2 X 1 but something a little bit Lewis X 1 X 2 0 0 is off heatedly on a 5 hour some numerical number which is a pure phase I need for the eyes 5 some fine just a fixed number as possible but when you make a change to particles that the wave function should pick up a face on the other hand supposing you do it twice supposing you interchange X for next to change its form next to the wave functions should get multiplied by the phasing we are 5 and then you do it again you interchange them again but at the same place you get back to the same place you back the things they do started where I know the words when you interchange twice you really do have to combat the same thing as no other possibility the general guests would be bad Ito the I 5 years ago on my this 1 yeah that's an operation when you do it twice you to get 1 but the I took possibilities sensible possibilities you might say only 1 sensible possibility but then you'd be leaving out half the particles of a world class of called Bozo particles which satisfy that property that when you interchange 80 2 of them and factor could be a multi particles state we just take a tool and interchange them if the state is exactly the same then the called bows so bows on satisfy the property of the wave function of the state vector would you interchangeably 2 of them stays the same very Young's file a case with the a minus sign OK so you switch to firmly honors identical frontier areas the state vector changes sign keeping in mind that the changes sign is itself not observable it's not observable think because the probability distributions expectation values everything of interest lies involves site times I start by so if the state vectors change in this way then they must be true of the wave functions for example but stake X 1 X 2 sign bets by definition sigh of X 1 next to watch the wave function of particle 1 part and particle tour with the other hand is X 2 X 1 that's equal sigh of X 2 X 1 another if the 2 wave functions are related to each other but just by switching the arguments but now is the state vectors themselves have this property that they to change signed of don't change signed than the wave functions must have the same property so this becomes up property of wave functions and that's right if you have a wave function for firmly on a system of firmly on 6 1 and now what while sister with the 2 particles and now are you with the many particles for friendly on it must be equal from minus what you would get if you change the particles details of particles the wave function changes signed mirrors if where Barbosa and Primos no
changes that's what I too Eagles plus OK let's let's think about some states of seawater Wickham cadavers exposing I have let's take a case of HBO's 1st kiss a bows who'll simpler knows no changes but suppose we have tool bows arms into a different states well once state could be wet again forgetting spend 1 state that correspond some particular way does have been adamant that we could just be any system our but take somebody's of comic states this wave function of a wave function associated with this state he could be called signs you call it Zero because it can 0 angular momentum and because it's the lowest energy state I sort of a particle warned that state saw the wave function for particle 1 would be size 0 of X 1 West but the next particle in some other state could be about the same angular momentum are different angular momentum Oberon let's make it a different state but make it different states well that's 1st the case would put 2 particles in the same state but put 2 particles in the same state then the state vector would be size 0 of extra article 1 in stateside 0 article in the states 0 does this satisfy the rule from wasn't satisfied the rule for well this structure here please obviously unchanged if you want to change 1 the doesn't matter which or you multiply later function play functions just numerical values than operators a commuter OK so this wave function here to particles in the same state that's symmetric toward a symmetric symmetric on the interchange of the tool arguments of the wave function that does not change so that if such a wave function makes any sense it makes sense only for most for firmly on this is not possible because this does not satisfied the rule for fermion OK let's try another state Ms. Rago as saying it is you can't you could only put bows on some of the same state for bows aren't OK let's take 1 particle and put it in the size 0 and the other particle permanent or another state which I'll call sign 1 OK why happens if I interchange 1 until 5 not 0 wonder what happens interchange X 1 X chose this will become size 0 of X tool side 1 of X 1 In general the fees at 2 different wave functions the left side and the right side will neither be equal to each other nor equal to their native just to just talk of 2 functions sign of X 1 times are hyperbolic moose over a Baxter that has no symmetry whatever it doesn't change signed but doesn't stay the same so this is not a possible wave function for either from Iran's oil Mozart was at me mean you can't pull 1 particle and the 1 statement particle tone of the statement quite when they make a let me now Jacob state which is made by taking both of these combinations here in a adding that this is a wave function perfectly good wave function it's a superposition off particle 1 inside 0 particle tool being sigh 1 and the opposite particle Toby inside but both of them have a particle inside 0 Apodiformes I want the ball satisfy the criteria that there's a particle inside 1 apart so both states of old but about this what happens this defined the change 1 and do nothing stays the same cake so this is a perfectly good candidate for bows on system tho particles and the not a particle and state 1 of groups state 0 and the article state 1 and a belt was about firmly arms could you build could you create a state with a firm in this state and this state yes you have to do is put on my side OK you checked as that change sign when you interchange X 1 annexed to does this 1 goes over to hear the song goes over the years and the whole thing just changes so yes you could put in both cases you compose a particle interstate 1 and the particle interstate 0 what you can't do a Fermi answers put to particles in same state and 2 particles order for HBO's onto you can't OK in particular let's take a special case of exactly where sign 1 5 0 same state but a special case in which instead of taking 2 different states I take the same state as strong the possibilities sigh 0 next 1 side 1 of next to plus size 0 of X tool Warner on X 1 bows on carrier wave function now what cases I said we're both wave functions are the same well this just gives twice the original wave function RIN may not be normalized correctly may have the normalized appropriately but it gives us twice the original away from not 0 nothing wrong with it again it's still Boaz Island in the same state perfectly correct but the fermionic construction he gives us what 0 so game we can make it a state in which to Fermi arms are in the same state the same rule continues if you have multi particle state demoted particle states in any number of particles and the wave function is a function are many variables and has a property but for bows out and if you take told them and switch and the wave function does not change signed for Fermi answers to take any 2 of them and switch their 1st change again what happens if switch twice so that you come back to the same configuration queried in both cases nothing happens right so the process of double swap is known as nothing at was trivial it doesn't do anything as you would expect as you would expect but the odd Is that the process of single swap what you might have thought does nothing changes a sign of the wave function of from your case about that if you
like me basis the mathematical basis the mathematical set of rules that goes with Powell the exclusion principle put 2 particles the same state prefer now what about when this data well when this then the easiest way to think about it is just to and another fate each particle instead of saying the wave function only a function of position think of it as a function of position and Spain so in that way you can think of the wave function as a function of position end let's say it whether it's upward we could choose singers each year Z component of Spain this wave function has the following meaning it square sorry it its magnitude square magnitude is the probability defying the particle add position X with Spain either up or down depending on whether you put them simmers equals 1 Olson was equals my so it's a function of the continuous variable a discrete variable plus or minus 1 and it represents these amplitude the probability amplitude for a particle travel certain locations f with a spin either up or down once you know that they you say OK the wave function from particles but 1 he is stands for the particle about whether it's ex-wife Missouri text see two-room room and so forth and so on this representing the wave function for particle could be a position 1 would spin signal 1 position to with Spain's Roberto and forth so 1 another rulers exactly the same back if you take although they did although we stuff that's associated with particle 1 as both X and the stigma and you take off the things which are associated with particle tools and switch them bows on nothing happens spin change our search design changes but what it boils down to is exactly what we said in the 1st place you simply can't put 2 particles in the same state including state including spend if firm eons you can cables are looking questions about that so far Young what is clear is that exist you can get a change at 1st that think about that the has aged this is list representations of provocation that this guy got switching the particles as a permutation and what happens to them on the permutation opossum minus sign defines a representation of the permutation growth that person very definitely has the station so it now we remind you of it they hoped would be wrote that war is a spot for wrote that the problem a disarmed the sum of two-way functions for what is a function for 1 particle it's simply the superposition of arms could build a theory of 2 particles these state vectors are they kantor products are cancer products based on the work of the wave functions that depend on 2 variables and no you would want a interrogating him would correspond to putting a single-particle in superposition of states so right they know if you have to weigh function you could take a single particle and put it in the summer of the 2 of them all you can take the particles and put 1 1 and the other and 4 1 1 1 of the rule would be prop you could be buy spot into his work well so far we have so far have it but it certainly does give interference effects of off we never talk about the chip and make talk about Wilbur about years later the effects of these they they got out of a hat that Campbell Blair so there's no problem laugh that ahead in a field that threat you could say the politics right you could say the Paoli exclusion principle is a manifestation of a funny kind of interference where if you put that the articles in the same state bait and switch them they interfere but so yes so it is a kind of interference what is right yes this is a lot problems while yes there of Gaza Ali was doing it he wasn't deriving principle it was a empirical fat and he was if you like interpreting the principal finding a mathematical framework in which the principle of would be correct later on of course once there was a relativistic theory of particles yeah this rule here became part of a larger inescapable logic then it was a derivation every year of Equal principal back 3 exclusion principle he was trying it 1st described purple factor cause was he realizes that it was such a nice mathematics to it undoubtedly he saw looking for deeper reasons for the mathematics and eventually found me and Barak interactive Dirac and Einstein whether you will the people who really built and bows and various support some of the pain of the life of me I fear me was a very very great physicist who did many things to me insult the Italians but areas but the home but I do think that the state had do with few days ago off Georgia your homework is defined who if the so work saying part said following to soon past here is there some connect between what and 1 of the exotic particles in the air and sea blockade well with 2 buildings is Europe's yard around you have to pay to particles this path were 1 is of this way another pair where the cross over as a warning that the same way and with bows islands they had with Fermi city they canceled so that it's it is connected with interference but I did not do that now there is an interesting interference experiment that you could do calm but works what's complete OK that's the pale that's that's 1 of my assigned now after this I will come back at us many many times but there's another minus sign that has to do where exchange this take minus sign that has to do with exchange of articles This is the minus sign best with Spain but
has to do with the rotation about an axis bite 2 party retention Michael piety is another process that you might expect does nothing like you might expect rotation by 2 party is a mathematical process that to Camara my junior hand but if you take system and imagine mathematically rotating it by Tupaiidae but not Gore ending in rotating it mathematically just performing a transformation which rotates a by to Hi you've done nothing be done nothing to it because the rotation by 2 pie is just be identity rotation is the same as the identity rapacious you think or I thought you may not think I did I did think it 1 time of my life so you might say OK let's now consider a wave function for particle 1 particle size of X and their let's rotate a frame of reference war and yet rotate the article more interest I'm only interested in the stood right now but go thorough but prices away function article it might the pain both on positions and we so we can win and put some still in here it depends on where on burned particularly depends on the spin and now we rotate by some angle what happens when you rotate a state with a given angular momentum by an angle well an idea about any forever an exterior axis right next to the particle can't be located at the article could be located at theology says although they not what ever angular momentum everything Everything right after everything including the spin rotation we are lost in it Spain's that there could be yes it could be rotation about the z-axis remember this bid doesn't have to be along the z-axis could belong XX Kabila y-axis Kabila Logania exodus the beaten book yes there is angular momentum was pointing in some direction in Europe paid about some other axis annual momentum could rotate about taxes but by the time his angular momentum axis were rotating about end we're take a bow time we come back but to party we come back to the same place so you might expect that if you rotate the wave functions and rotations Our unitary transformations but just call aura aura for a rotation we rotate the wave function by an angle theta we really anyway function by an angle of data we get something in general but if we rotate to piety a then presumably we should get the same thing back the probability should be unchanged beyond the probability for spin should be unchanged probability anything should be unchanged is all we did was an imaginary process where we in our heads rotated about party source and so should combatants and then await the same issue are occurs as occurred over here for exchanging particles of all always require a probabilities and expectation values don't change in it's possible that when you rotate by 2 5 you pick up a face again could be up plus a minus 1 arm but could be a plus or minus 1 arm might even be some we're phase but was the possibility that things could be in here now how does the phase that occurs there depend on the angular momentum of the state total angular momentum of the state the angular momentum would the angular momentum the angular momentum but that the total angular momentum another encloses Spain and the orbital angular momentum it usually called Jake Jane is a generator of rotation so that means acts almost daily but when acts on state differentiated with respect to an angle of particular website Jay Z component of Jake busy component of the angular momentum that effectively takes the state end just like momentum differentiates state with respect position angular momentum differentiates it will respect the angle so that's right that that that's that correct minors I the derivative of the state with respect to some attention angle of reimagined rotating state our little bit for or any amount then acting with J. does the same thing as differentiating with respect to that's what a momentum does that's when the angular momentum does with respect angle in this angle now would be about the c-axis rotating about the z-axis events generated by the generator Genesee OK let's suppose that the angular momentum about z-axis had a definite value the eigenvalue equation then we will write that this is equal pay and game would be the eigenvalue value of Genesee Z who just me and I'm sorry for him signed with of this Revere size of functions Oryza vector it doesn't matter what the solution of this equation ashamed is the size of the the awards be sigh that you get when you rotate angle faded what Ito the data sigh of Zero is justly equation that a derivative of something it is a Quebec and planned something that would be the solution remember what what will the eigenvalue used that we discovered what possible with flimsy component of angular momentum when we analyzed the commutation relations remember we do that we found 2 kinds of things 1 had integer spectrum and the other had half integer spectrum that was just the mathematical exercise using the commutation relations just abstract algebra it just fiddling with the following following their fires as all of his nose toast the plane with commentators we found that but could only be integers or have integers What if M Is here the budget and you rotate but to party opposing you rotate always around the axis so that caters to party then we get a fact I to Pyay and front
side and each other to apply is 1 Nissan is an integer minus 1 of them was a half integer so evidently these representations of these multiple that have half in heavy on property that when you rotate around by toe pie the wave function changes those are the only 2 possibilities for spectrum as you the integer Spain or half integer spend its true in 2 dimensions different but in 3 dimensions love the spin is either too tight or the half and digitize which means that when you rotate around awful to Tupaiidae either the wave function changes signed with us our weather changes signed or it doesn't simply depends on what kind of particle you talking about incidentally if you rotate by for a party it comes back toward self for to pi squared now buyers to pipe and yet no hazard demonstration that goes with this which really does reflect some of the mathematics is mathematics it seems that rotating by 2 pie is not the identity for firming it's not identity it doesn't change expectation values it doesn't change probabilities but it is not the identity operation would allow my house show your house actually wrote that the rotation by 2 party is observable effect it is observable effect and therefore I think we have to say that in quantum mechanics rotation by 2 party is not the same as normal rotation of all the something going on in these mathematics of rotations which says you have to rotate by for a party for the mathematical operator to give you just 1 of thing this trick I believe was blew my friend and I don't know 1st and that it was the Racal we 1st invented it owes duo who understood it but long after Dirac media circus performer but remember their article Tuesday a Dutch circus performer 8 trek could demonstrate the fact that rotations by took I am not the identity and rotation but for part are the identity I love to do this so much that we could do it and you're supposed to do a cup of coffee so here's a attract users armor rotate pipe cleaner Bellmore earmarked ago repeat this way and when a rotate this way that is not the identity let me tell you I'm quite certain that's not the Iraq take another 2 pot that's the event now there is no a matter Uribe are I think he did discover it now work there working was there was a little bit different from those of the same thing I think his picture was you pick us year and you put in a box and you connect the sphere of my ropes your firm business leaders is a box in your have ropes connecting it the strings collecting it the box at all possible directions and firstly 2 dimensionally what happens if you wrote the ball but to fight near the triangle right you a tangle became don't do that tangle rotator for quite so came from the within an hour in 2 dimensions In freedom mentions what is true is if you rotate 2 party you get tangled that angle it is basically a tangle of mine on historical took party but then if you rotate again for plight you tangled so this is is Ophelie about Robert apology over a patient group a theorem that no matter how complicated this set of connections with the walls of the box if you rotate by pine Dupuy sorry you cannot disengaged strings back the original configuration of you rotate by 4 par you can so really is a mathematical fact about rotations that rotations do not bring you back to the same configuration of repay price on less you specifically say I'm only going to study the kind of way functions which don't come back when you dealing with bows and the other possibilities the kind wave functions that change sign with rotate by twice say that that you couldn't will they're like this show wages but it looked that is the 1st leg the 4th meant that the but the living yacht it but had what you also working and expedite mentioned here It's time neural URU rotating with time but there prep but but for yacht you want working numerous your following up chaos In the wrote in the space of rotation and a champ in in the space of rotations could be unwound for flop can be more then the young antique apologist will recognize this as a major feature of OK so that is the difference between the half spin rep to replace our representations and integer spinner representations their behavior under 2 pirate patient firmly and get a minus sign on get a modest size so an interesting situation a set of tale of 2 minds are firmly on the wave functions to Fermi arms novices to ferry arms changes signed swap the wave function 1st single firmly on changes when Europe aided by 2 parts would be interesting if there was some mathematical connection between rotation by 2 piety and exchange and there is such a connection this even the theoretical physicists can not notice for some reason I don't know what mathematicians were stunned diplomat job and not with most theoretical physicist or nor was discovery of a friend of mine and by the name of David Finckel Stein who was aware of the unsung heroes of physics he's a he is a famous physicist on depends on who were more what crowd of people you're asking if you're asking an older generation are very very famous physicists like Roger and rose and people like that they certainly no David Finckel Star Wars is if you ask me in the street never heard of David Finckel spent said my Butcher was named David Finckel starring a butcher in the Bronx but the baby fickle storm was 1 of the great unsung heroes of of theoretical was he figured this out and which I would feel with fascinating love him about now I do something about the memory talk about what it is I did I hope I can do it I think I broad bubble burst of my belt paying the how this is done easily with a rubber band but of a dinner with a rubber band you could see it from here I notice about display you see an exchange
you see the toe bottoms of the belt yeah having had been exchanged on the other hand if I turned around you see this a rotation by 2 the belt has been twisted
by took part in a rotation but to kiII together with an exchange gives you nothing at all resume on twisted belt so somehow the mathematics a ripped out draw the picture that I think that drawer Her picture it just in case you couldn't see a my belt and is drawn four-story belt but say should look something odd wrist we we then led ever get to this if I yell at defied twist but belt we don't think so where I have to do it to make it the legal a twist my belt what I have to do twisted but it's a legal twist of adult that's what I should do about it twisting it then verify that's a legal twist of the belt meaning to say you could twist about that way and if you do but apart you just get a trivial topologically trivial belt like that of this is equivalent of best I want telling us is deep topological connection between the process of exchanging if I have this could be thought of in the following way but I have to particles and they do to processes on the 1st process is I take 1 of them and rotated by to and the other process from the other process is I switched to they cancel each other out so that means quite apart from whether the affirming eons of HBO's arms appreciate this is not obviously a direct telling you this because a lot of people think this really is at the root of the spin statistics bureau might do found that the deep topological connection between rotation and exchange may very well be what's going on Our end where says His at whatever you doing nothing is like building a rotation by 2 piety and the exchange both together OK that that the is serve is not the way Pauly proved the US Sprint statistics yacht but so this wouldn't have thought about With that How they got the particles they still you know that a time this committee thought of as a history with Prime going up this can be thought of as a history of history your magic that a pair of particles was created over here you have no particles in you create the same thing over here you know particles and you create so by the time you've got at this level over here you have particles swarmed through 3 4 right for particles each being thought of as as a cross-section through the belt now it doing is years taking 1 of the particles a rotating if pride and at the same time you taking the other 2 particles and switch in them are saying is the switching of these 2 particles whatever sighing may you may not get must cancel starring did you get from rotating 1 of them OK this is this is something deep that's never been completely explorers are young that is where you switch size yet connected with for they left arm but but I can't process a process the hour OK so as would processes the brasses is you start with nothing we see this kind of logic only makes sense In a theory with particles could be created who would not make sense in a period where particles were turned off see the argument by exchanging by adding new Vorel basically 18 yesterday In exchange an end the wire but I don't think so but only you don't want those whose won by a vote of all but the cat but that didn't give a fuck all of them all again the identity that Rook baby for you but yes it there is slot and this is a lesson fast and is rotated may be the same as that 1 of air well a little bit different because we want a study is a rotation by 2 . 2 rotations but high but the 1st sounds like they might be connected on but what the process that reflected here you start with nothing now in the few wear particles convict created out of nothing quantum field theory is through a particles can be created but there are other kinds of theories were particles can also be created just creating a certain kinds of expectations and condensed matter physics Creek and create articles but enough particles could be created you solid no particles starting point is not particles of white blackboard and that only here and only here basically you can you're creating a particle in that particle called here and over here creating a particle that part of these particles were charged and you create them out of nothing obviously they must have opposite charge but that's not the important thing here you whatever it takes a new theory to create a pair of particles you have to put in energy for sure but you do it and now you're left with 4 particles 1 2 3 4 but column particle and that particle particle and that particle and with time as time goes on at the same time that you rotate 1 of these like to party at the same time you interchange fees so I suppose this is plus charge misses a minus charge this enemy minus charge Mr. B plus charge because you're making the change to minus charge you get something he's from rotating this particle and you get some phase from means changing these 2 but as you concede topologically this is completely equivalent to just creating a single pair of particles and annihilating them and that there is no phase whatever phase you got from exchanging must cancel against the phase it you got from rotation I like this argument a lot I think this can be turned into a real argument about but I've never actually seen or where it had to make its way quote very surprising is how few physicists actually know anyway it
does show it does show that there is a connection between rotations and exchanges and a real relational the right kind of cocaine so that was it Taylor the this is a suggestive argument that I've never seen turned into a rigorous art is the closest I can do to proof of the spin statistics theorem without going through the the complexities of what what some of us noticed what kind of girl and that's when David whether the framers paper but firm not famous enough what people noticed is that this statistics was more general than just relativistic Lorentz invariant quantum field theory which is what Pauly proved it for you could prove it for all kinds of particles that you could make things called Saleh kinds of things called solitary waves solitary waves just alum along come along the field alone a field that doesn't have to be in a relativistic field theory any kind of Newry where you can manipulate their fields and create lumps this will apply to post a notice by a few people particularly Finkel Stein and Rubin starring and a few others those physical style has to look that's the criteria but you could create articles if you could create annihilate particles then they must satisfy this rule whether or not relativistic quantum field theory so that's why it's even true truly about our lumps of stuff in condensed matter physics certain kinds of particles which are just expectations true for magnetic monopoles true for all kinds of things where Powell these arguments were really doesn't apply directly so presumably this is work was at the root what's it is not yet known no young dot if you have a field theory let's say some sort of scalar field it doesn't matter where it is any kind of many clever of fields and laps suppose our case reviews his work you can do let's takes on applied field
configuration the field 0 everywhere except in some will want regional he and the slump region unless they have feel the could be a set of fields does not a single for you but in original here fields has a non-zero value after game also has a shape and structure so let's that puts on blue lumps over here and there put some black lumps on the other side who now take another configuration of exactly the same kind Of absolutely was all kinds of physical realization of lumps yet nuclei along and some theories nuclei uh lumps of this type theirs what courts Skyrme on theories of nucleus the others as walked and my magnetic monopoles selectors and in modern quantum field theories almost all particles Committee expressed persuade so yacht Rice's come OK here's another similar particle want Samoa Mulheren over here and the couple black lumps over here over here now are chosen to draw Tools want configurations which happened to be mirror images of each other and into that I to make to mirror images call a follow-up article calls for a nearby park or through right now start bringing them together a notice that when you bring them together the fields will exactly match wine because the field over here is the same as a field he the field in a little bit over here the same imagined bringing them together and what happens is you bring them together as you bring together you can get a situation was where'd you get to a situation for example you will get to a situation carefully where the black lumps annihilated each other and the blue alone Foster lamb well then you keep squeezing in squeezing and because the field always match on from the year on the run you could just bring these field curve fields together and do it again to see an example of this with a belt but half somebody come up novel beer and the bill for Tokyo might do live over a little closer to me already until being a mirror image of me OK crude oil but data right up there's a lump over here and there's a lot of arts and now it's push the
lumps together there habit of the bill would do this because but you could see what happens they go on to each other because they match in the price of the demo by the time we brought the almost none hold the holder for the birth of the toasted as a lump over here as a lump over here but I keep pushing him together with just unwind that's what's going on here is a three-dimensional two-dimensional version of the same thing so given the theory lumps along and mirror image can always annihilate each other so if you start With nothing and of course to Begonia the direction you configure empty space I am close full free of air-launched right so you could create Carillon chicken create another pair of wants over here and you can rotate 1 of them but to interchange the others end of be certain From beat topological fury that go there to say was doing nothing I saw the space statistics theorem says that lumps come in 2 types bows which when you take them away function changes signed and those who function doesn't change signed over affirming Mossad have been together always goes together with the properties the integer vs. half integer spin always goes together with the properties under interchange of of the particles that's the theory of bows on the Fermi on not a or seen a hat fueled by leading to use the term ended in a legal manner of half a mile what not sure would erode Michael Simitis work yup and easily the work I him yet not exactly this that's exactly this over here if you rotate by a little bit you pick up Anita may I say 8 in M an integer you Apple that exactly that's correct and that saw him the spin me that a vote rotates with half the Phillies so that a full rotation bring you back the same OK question and interesting question is whether we fact that when electrons a rotated by Tupaiidae of these changes is observable is there a way to to observe that the any might think that's completely observable on observable because rotating irritating but Tupaiidae only changes assigned the sign of a wave function doesn't matter but not quite our West show you how warm and this is being carried out experimentally we give you a stroke purists version of an experiment where Chen may not quite your way actually was carried out successfully by West start with an expanded where you wouldn't see anything we would love to do not see an effect he take electron His electronic it has they were not aware of the world yellow don't worry we'll worry about him is electron it's gotta go citizens slips do some interesting interference experiment of some kind could offer some slips and before goes into until selects from this saga over here would put in the cavity the cavity is going down magnetic field the magnetic field was a magnetic field do it latches I want to be in the same way a magnetic field latches onto a R Torre composed needle watches onto would hold the skills and they would think that box and slowly rotated by 2 pi philatelic electron other box and create an interference pattern Maliki we compare that with the same experiment in which instead of slowly rotating electron would not go slowly that's all do exactly the same experiment in 1 case of very slow pick up time that the long the figure thousand years do this experiment on 2 million years after the 1st to enter the 15 seconds or whatever it will spend setting up might be the rotating this electron or not rotating and will look at interference pattern over here while you want do you drag a fascist warm alarm toward Repsol OK so that's the idea that's the experiment and the result will be key interference pattern in Norway remembers we weather was rotated by 2 part the now although different experiment take electorate electrons alike photons Ucon command beam splitters beam splitters will take electronic and Split the wave function as well a chance at to electron 1 electron end beam splitters but wave functions that a part of the wave function as 1 way part of wave function grossly way again this is not to electrons is just 2 parts of these probability the 1st thing we don't his recent electron flow Being Split and to wave packets come out I will say over and over again is not to elect terms this is superposition estates in which the electron went this way antielectron went this way OK now we're go slow motion for a while both wave packets Barr said into a box in the same enter through different boxes and in the box again is a magnet magnetic field OK so early have we a superposition of states in which an electron is here electron here and in this 1 the Spain is kept How or magnetic field and this 1 is kept out of the magnetic field but no I rotate 1 of them and not be other slowly slowly taking it 1 of them is rotated the other has not rotated if rotation did nothing and rotation but to ply that Pacific did nothing is saying nothing a mean guy we may have done the same thing but nothing goes nowhere effect of it patient by 2 pi brings it back to the same configuration but not rotation by Tupaiidae we have a wave function wave function society 1 arm
X plus acts sigh of the wave function in his box cycle of X is away from this box but if the rotating we get sigh 1 of X minus societal effects if this is the wave function that comes out of here were might decide that passes through these 2 slits the wave the earth the year of interference pattern will be quite different then if you have plus you could say White will come out of here will be a certain way function what comes out of he'll be a certain way function and issue a N them or subtract them is a quite differently the destructive interference or constructive interference so this experiment shows that under certain conditions you really can be had the rotation by 2 plied by interference express why electorate which is a superposition states 1 boxer and the other box you go away and don't look because of you look you the experiment had his leg has but if captured it and errors if captured for while you rental box now you have a cavity electrons in a box what but only 1 now it's is in both boxers resume 1 but or the other it's In a wave function which is a superposition of 2 of you open the boxes and look you will find only 1 electron your why the find a lecture 1 box or other box if you don't of course if you open the box and look fuel you affect the experiment Norway which destroys any of that money he is their only time will tell them an yet but you really you could really do just 1 experiment With the minus sign here you might get In the wave function and those means a 0 0 rats say right the cemetery point of that the village if the wave function comes on minus them for sure you will not detect away from here so if you do it detect an electron over here it means that it must plus so it's it's it's good it's quite observable this is not exactly the way the experiment was done for have experiment and expand was a little bit differently electron beam was splitting and and some cylinders with magnetic field in them along here that it was arranged so that after being splitting electrons will polarized in the direction perpendicular to the magnetic field you remember what happened if elected Spain is in the magnetic field and the Spillers perpendicular to a magnetic field it processes so if you make a cavity just exactly long enough and the magnetic fields is right and the velocity of electrons correct you can arrange so the electron executes 1 Tupaia rotation love all I guess in 1 of them 1 of them you don't put the magnetic field 1 of them you don't put that magnetic field alone you do but the magnetic field so 1 branch of the wave function executed 2 pi rotation the other branch of the airway function was not execute 2 pirate station depending on whether you turn on or not the magnetic fields what that right watch that I would you don't that's always a point of quantum mechanics anything you do to determine the path ruins experiment Joe Battle but what you do is you go over here and you look at the interference pattern of Hi Laura Diaz until tonight with appropriate electric and magnetic were sent here that yet when 1 time 1 overtime absolutely no I want to know what is going on where branch slim karzai to or so the branch of wave function means wave packet because 1 way or the other the saying the same way what is nonzero here and 0 he and the other 1 was non-zero organs your near of the year this is the 1 that is the way it is young but you're you're arranging a wave function initially wave function is set up by the beam splitter so that it is let's call 1 of X cycle of X and so I 1 of is non-zero in the vicinity of this cavity and Saito Exs nonzero facility of this scale what's that the ad at that size so size was its size decided 1 it 1 I know I wanted for example is summed functions which will play that site 2 with his son function which like that they're in different places if but I was not is slated so you have OK so you have all these here years you have electron coming in In its it had something some target over here with target as a particular kind of target which splits the took packets the splits the beaten and later are packets so wave coming in splits in 2 pieces there that charged particles so we can manipulate them with electric and magnetic fields we take this piece of the wave function and through here this piece of shit are ever wasn't the reason that magnets this would is be too soon experiment to essentially be the to experiment is just where some fancy splits have fancy selects the head the are equipped with some magnetic fields which may or may not be turned on if the if the if the magnetic field is turned around here are even lower does this take the state and is rotated by 2 Piney it takes you from constructive destructive interference or is it a lot look you destroy the interference with you look you destroy the interference anything after its here now just makes its makes a not absolve doesn't tell you which it's a sell this is a which which flow which 1 is don't you John it should sell for that back to social experiment go back to the Tuzla experiment because exactly the same as it was for experiment if you like you could think of social experiment you could do it for electrons and 4 4 times
Ray function condoms even go through this whole of his whole but you don't look at this Hollywood equipped with a little stronger here and this whole you equipped with wheels them and 1 of them you do or you don't put a magnetic field of electric coherent beam of electrons here and our for wavelike electrons comes just a wave comes in some on the way it goes so here some part of the way it goes so here and the way resembles itself Florida so we detect an electron over here if there's no magnetic field in the year here than the wave function is symmetric up and down and we expect that the word that these are the amplitude center years could be the sum of the 2 of them it's gonna be particularly large inside I saw a fractured get some overlap is evidently and then is feared that the this is an 18 year obstructed eating get Rudolph promised and they spread out you don't know set up what went on to note that relies on the Dennis C. P. standing orders away like that damage to its quarter of the way he has remember this FDA small Hey I got this is not really the way we do it but if you want to do it by a conventional who slipped yesterday small it is old hat road that just as ready to do the same for Halloween to rotate the polarization of the photon wasted away all it come out here would not be sensitive toward you rotated to pie probably America thought about what he said was slick she did he say that they sought cheated His lawyer on the purest version of it is you want to take that electron vary very slowly rotated so that you know nothing else is going on here it's not salt electron quickly but still it's the same amount of to say noise a the slowness this just make sure that there are other things going on jiggling environment it's the all the lecture that really all that we may feel that they should all they get into legitimate its measuring it's measures its measuring the effect a rotation by 2 pi arbitrary right goddesses is or a special case Our a phenomenon called a Berry phase Berry's phase strawberry phase raspberry phase Michael Berry phase at to expel those 1st proposed by over myself a long-term growth deaths the when the theory was proposed was considered a big deal everybody by that point enough of these kind of things a bit that everybody knew would have to honor but the euro out was considered a something surprising that there are people have thought that under no circumstances was there any empirical signature of rotation but Parks who was interesting got mixed the teams those regions get course doubt but way function the wave functions Missouri combines mean from jurors through and comes out spreads out here this 1 spreads out of here and it creates an interference pattern by classical interference From our mom what's set it is it is it really is just a Tuzla experiment except with was something inserted new 1 of the select do something like this said these things are always destroyed if there's any year you know little Max will be watching what's going on that recourse what goes on he can have a not on Saturday for a raft all around Dublin or was at therefore 1967 never forget are a bomb effect rose 52nd day on warm effect you hear visit is on testers give the says all the yeti on the right or them that's a very beautiful argued the theory of and a that gives you the idea through OK that he or his substances as they say if that's the problem is that the actually have did the but only 1 but we'll get that combining 2 from not yet but this is not a duet also chance Barry customers could you do the same thing exchanging or charms yeah you could do the same thing exchanging terms a suitable complicated hard to visualize you have you start with a pair of electrons and I literally start with a pair of electrons and what you do is you Duke pocket but a furious some possibilities you take these things a wonderfully anything you want them to be 2 electrons now what it would do what it is you're going to see Blake the beat of tool electrons that means you have created a state which is a superposition of 2 electrons over here and 2 electrons only only talked novel for electrons not 1 stuff for electrons it's to elective but even the electrons a here or the 2 electrons a here an if the 2 electrons in here there's an apparatus which exchanges and somebody a field grabs onto them and exchanges it's 0 down here no such thing happened old or perhaps we forget the switch on the thing which is which exchanges them so the other possibility is no exchange outside a branch so we Split the beam of 2 electrons so that we have unit to electrons healed oracle electrons if they hear exchanges if they here we don't exchange them and then we bring them back together again and the fear there and see what comes out of this thing Afterwards question is do you get the same thing or something different if you decide to exchange these 2 guys here or if you don't does the exchange which after all those nothing to them right mean that the state doesn't care about which electrons which course the answer is that the experiment for firmly
arms will be path the fact that you exchange the 2 particles experiment Phobos arms were not now happened is another experiment we Split the parent area for
Perahia the Paris here we interchange then it's a Perez them here we rotate to price the set Doyle guess what they were canceled to effect for that their best theater nobody at the knowledge nobody would as ever done that experiment Trevor feeling that news you look at the moment probably be on that English the idea that but what all know what the heck it is is indistinguishable means that if you change them changes such as what it means the quantum notion of arms are indistinguishable is more subtle and classical as usual and that sort of thing now Of course what's what's going on here had eaten in both of these cases in the case of these single electronic where you rotate 1 part of the world and you don't wrote other part of course that is not truly a rotation by 2 party with true rotation To part is the whole world has if the whole world rotates then you don't see anything but all experiments that you ever do compartment belies the toe of sorts it should still will not be a visible gap so the reason this
is detectable was because you've rotated 1 part of the universe relative to another but still you do would be cut the fact that rotations by two-party cause a change of sign and that was surprising OK the that the gadgets search for Argentine Robert orders were referred for more please visit us at
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Formale Metadaten

Titel Advanced Quantum Mechanics | Lecture 5
Serientitel Lecture Collection | Advanced Quantum Mechanics
Teil 5
Anzahl der Teile 7
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15099
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 21, 2013) Leonard Susskind introduces the spin statistics of Fermions and Bosons, and shows that a single complete rotation of a Fermion is not an identity operation, but rather induces a phase change that is detectable.

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