New Revolutions in Particle Physics: Basic Concepts | Lecture 8


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New Revolutions in Particle Physics: Basic Concepts | Lecture 8
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Susskind, Leonard
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(November 16, 2009) Leonard Susskind discusses the theory and mathematics of particle spin and half spin, the Dirac equation, and isotopic spin.
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question her question arose revised b paradox Oral here 2 particles both firmly arms make of HBO's can we have bows on arms which are what say in the same state and songs of perhaps a more general question the wave function of follows on the way function of is supposed to be so backtracked on the interchange of the tow bar that means for example if you have the laws and the wave function describing them was a function of let's say XMY this doesn't mean tho coordinates X and Y 6 more direction XMY are simply now standing for particle number 1 particle number 2 if their bowlers on and the rule is add that must equal sigh of wine X affirm our way of saying you can't distinguish between the 2 were particles is no difference between saying you have a 80 to bows on system in the states I XMY wiII and saying was on state why are so warn interchange symmetric between 5 in that case you could certainly have to particles the state for example a special case of this would be society is a product function Wrexham product I don't want to use the label sign again because of you cite label to particle wave functions but like cement and other thing described 1 particle wave functions was disqualified and sigh of X and Y could be far higher vexed times 5 white this would cast 5 to 8 bows on a with Tobosa arms with wave function fire with the same wave function and rubber words in the same court that's workers would mean to particle state and each particle is In the state fire this of course is obviously symmetric the same as party of wine next started 5 white times fire wrecks a wave functions are not operators they come your way fortunes and numbers not matrices dysfunctional of X and they satisfy all of the algebraic properties perfectly ordinary functions and so apparently I am I regret things so yes to both arms can be In the same state and the wave function America or about To firmly on the rule through firmly on this if every put them that we have to firmly on the wave function has to change sign that's a very odd thing but never West are that is the mathematical rule the abstract mathematical will performing eons were new interchange them the sign of a wave function changes now OK that's the difference between and then can you'll have to firmly on the same state not because 5 x and y is not equal the miners 5 wire used to and 5 vexed fire White is not equal to 0 5 wine and 5 vets were I'd now supposing I a wave function of foot to particles cannot may make and it's not symmetric and it's not entice America can I make something out of it which is symmetric or something which is and so I made sure you how you do that follows an I have some way function which is neither symmetric north west let's just call it a fire of X and Y when interchange X and Y neither changes signed north of the state source neither symmetric nor impressed by that you want to make something some which is an appropriate way function with 2 on you just a sigh of worry novices obviously cement if you interchange XMY this 1 becomes this 1 this 1 becomes this 1 and nothing happens throughout the promises a symmetric wave function this as a possible way function for and that any sign of exon wife and if we're talking about on we do the same thing except for minor signed there now what happens if we interchange Iksan white this 1 goes through this 1 when most of this 1 whereby will change sign the whole thing will change signed if we interchange X Y services away function symmetrise is called symmetrise in impression rising away function with 2 particle OK now suppose we have for particles altogether for knows what are the rules for 4 from and for simplicity let's suppose 2 different kinds of firmly on let's say there is electrons and protons and with talking about hydrogen atoms rule Castoridae where the wrong we have a form body wave function West now wet electrons be called X and protons recalled wife of a wave function as a function of 2 x deserves corner X 1 0 next you why want and white its or functional for variables water Our who roles on the royals are 1st of all but the wave function has Cheney signed ifyou interchange X 1 next to press factor electrons affirmed it must also change sign when you owe interchange why warm and White who incidentally are works with was right they must also change sign when you interchange why Warren and white off Sarah where can work From that that's a of those are the only rules sign change would you interchange X 1 next to it because electrons of runs and assigned change when you interchange why women White too because proconsul him so lets your Conaway functions we condone let's start where are our hydrogen and hydrogen atom located the origin of of a high Ian located on the orange and has a wave function which is a function it's electron and its Proton so 1 announced picks out a particular were 1 of the 2 hydrogen atoms when their assistance 1 of the 2 hydrogen bombs and this might be some way of functioning wave function governing 1 of the hydrogen atoms we're here but surpassing I want to shift its location however of the location well it's easier surface location you just a shift you just shift the origin of both X Y makers don't really have to do it on the ship and then I have
another pair parent and the other pair who is describing a hydrogen atom of some other location since it's a hydrogen atom Arellano location it has 8 different wave function where function may be the same except that it's been translated use different fine of X 2 and white general this is neither a symmetric with respect to work the axes or an asymmetric with respect to taxes or for that matter with respect to a wife so it's not a good way function from your supposing that I want to make it and isometric Monday interchange of electrons that's very easy we just subtract off here stride Our just interchange X 1 next to leading why 1 tour Price throughout whoever 1st minus sigh X 2 Y warned Friday horror X 1 1 2 is no art is isometric with respect to the extras referring to change the 2 axes of changes are designed as symmetrical despite what how I make it isometric with respect to a white so is take the whole thing I have here they did change it to wise and put a minus sign rights put minus so I R X 1 minute and white tool YEAR X 2 white that's exactly this with Assad what want this is the same except that white World I warned him and interchange and then I want to entice symmetrise this 1 with respect to reach in exchange why becomes plus Jesus are side of X tool White 2 0 fire of X 1 1 1 OK so this is now enticed symmetric with respect for both the X is and the wife of a few interchange in this expression X 1 next to you about this 1 change signed if you will to change X 1 next to hear this 1 herself for the whole thing is fully in place in that regard if you should take salt water that flows also I think was there are those there be forces between there may be a need a Polly principle does create a kind of force between and it it will make it very very in who make it essentially impossible to put 2 Adams right on Copper Beech adjust simply attribute that force but will you be able to put 2 Adams into the same momentum state momentum status spread out over a place they are spread out space so when you put the 2 Adams intimacy momentum state are you not to to remain on top of each other and doesn't cost water and you saw what are other question whether you could stick to things on top of each other is a different question whether there for me arms embargoes on it could even happen harm that your billiard balls can be bows on you can't stick them on top of each other their heart hard cores nevertheless they can be bows aren't you could put them down our UK but the standpoint just cost too much energy but you can't put them into the same moment now the euro also about adding that of movement and you have to worry is at the same by momentum then you would have to offer it is if they do not know not to stick in you anyway function like 1st business it entice symmetric with exchanges to X its investor magic with respect to wiry What about if I interchange 1 and Wyant tours into changing to Adams there this is symmetric young store symmetric with respect to the change of the 2 Adams and white because when you interchange EX is you get a minus sign when you interchange wise you get a minus sign mind time plus so say arm yacht if I interchange wanting to what X 1 why 1 will become extra to wear the same size extra white will be X 1 y once again with the same signed service wanted these 2 worlds transform into each other by simultaneous changes on Everything will wonder everything to at a change the atoms likewise to have of X 2 Y 1 but here we have sigh of X 1 more tool both of them have minor starring used so these will go to each other so also use symmetric with respect will be interchange of our more remember through on that doesn't mean propose and this is quite this whole story here is independent of a question of energy it may be impossible for other reasons to preclude things on top of each other in space the reasons being that may be very strong repulsive force by a good example RN young we just broke down a wave and appropriate way functions with 2 atoms for 2 bows on Academy it is not this season may or may not be a moment space there may be anything review Wyoming and what what is are we saying that if we had to was Iike Adams said that what their state will be year of recoverable if we have a half where wave functions Siurek swarmed more 1 another Adam Air free didn't know anything about the 7 year we might write them away function for the electron proton Adam and now I want to write a wave function which is appropriate for the 2 Adams systems are away function for the 2 items from past to be enticed interchange NTX whose investor magic when you change any wiII so it is basically users will accept a user twice once to exchange the X is America's River Park and then take the whole thing change the wise and change sign the result is quite clearly if you work out carefully interchange symmetric under the exchange of the 2 Adams warrants to mourn America through our soul that's and that at the end of the day a repeat in the very fact is the rule about a azonic Adam mother bows on Academy when you interchange and the wave function has have seems OK now what kind of states could you put that tool Our Yom momentum
state you could put up 2 atoms into our missing way function doesn't cost a great deal of energy because they hardly ever on top of each other but it is true that he tries to Adams on top of each other of our area part the bill because of course very very costly and energy home are what happened In fact what would happen OK let's see what would happen if we made both Adams with the same weight functions not 1 that will wave function fine when with wave functions with 2 with the same hour what happens there I see armed with a peso watch yet with OK so there 2 Adams were the same way function we have uh subtract Rice's urged or Siurek swarm of wife To sorry we we shot reserves ear or rock carvings our stars era was calm Is it job X 1 1 0 1 4 0 season young no no no it's not 0 no this 1 and this 1 are the same OK so it just twice this week coolers so was irrelevant we occupy a factor tool and twice as visitor and this is not a 0 OK this is still Adams in the same state is non-zero some magic is going on here you can put electrons on the same state media say the electron wave function has to be an isometric proton wave function has to be an isometric also true here but yet to Adams earnings so Colorado Rupert right wave function work at her of course the point is among other things that in the electron was not in a particularly in a well-defined position it has a probability of being many places in the Adams are you put 2 atoms in the same place as quite true with 2 electrons can be in the same such 2 electrons cannot be used calm but see what happened I hear a functions would be to Adams perfectly good and isometric so for now supposing we try to stick to electrons in the same place that would mean setting X Forney Quebec stool there will be 0 I saw Adams are in the same blood in the same state it's still true with in that state to try to find food to a lecture on the same place was to see what would be the 2 electrons in the same place would be sigh of X 1 why 1 side or next 1 white to antiseptic away Exs equaled whichever Myers sigh a R X 1 why 1 so I R X 1 white to forced to electrons to be in the same place and guess what I think they cancel yes they do so are sold as a set under and a maneuver from New York and you see how I can possibly be that too From arms commuter on arm is as I mean it he added that there is a type of position this before I'd stood before I set Forney Quebec's to hear these 2 Arab yet Adams all in the same state but be electrons and protons were never state not bad residue when you when you did this operation event by symmetrise and are you basically projected out the pieces of a wave function where electrons with standpoint they the protons standpoint so nothing was left of rack piece of the wave function they still below fugitive leader so that state and yet not give any energetic electrons Hussein place for losing momentum Derby 0 but the matter spillover over it is more fun than we've dealt with stop are the reason I'm spending time with spend it is because the mathematics of Turkey really ethanoic tomorrow such things as mathematics of I mathematics of cholera and the mathematics of wool of the symmetries of particle physics mathematics is the same the physics as was pointed out here I R B analogies between these different kinds of conserved quantities formal mathematical and our duties but nevertheless but so study the mathematics of spin more a little bit more about the Dirac equation just a bit but and then I want to introduce the concept of isotopic spin Price which is Koncer protecting dates back to a thirties Okayama salt let's start with half a has been the electron was a and particles called particle an electron I could quote from Timor could call you want anyone of the hedge particles corn electron it has possible states if 2 possible values for the angular momentum all along the z-axis along the z-axis called L sub z of Corvette and we are not quite not quite not quite but quite army was a major right but from now on most of these recalled you know what OK so if the electron this is not the a full angular momentum incidentally of electron but I am using a really weird notation usually L was used for storing your momentum and is used force so let's River that U.S. forces as 1st base but the mathematics of it is exactly the same as the mathematics of tales that we write Obama over here what's right over here In mathematics of angular momentum but now I'm calling it s for Sweden instead of Arroyo stands for angular momentum but I saw what was a cover commutation relations with Obama last time were LX with a wife equals or part of the Napa comes as Banks with as wiII will by H.
farm but also cargo for 1 s GE and cycle are your Virex s warriors equals ISX and so forth cyclic permutation at all former referring we have as plus 2 minus just to remind you as plus or minus just as acts plus minors S why these operators raised the value on busy component all Lawrie depending on what was plus or minus sign of the river and Lauren operators for spin angular momentum all I've done is chief of station from Belfast interest expense and doesn't it could be any Spain I mean it could be a spin which is Due to a load factor the rotating basketball broker let's say intrinsic interest you're OK let's focus on have spear been it is a case where S zee or amount could pay Garner value might just ahead of plus hair and that's all but 2 states I'm although I'm also forgetting completely Everything else about their work were forgetting for example about where the electron is what momentum is in stuff just forget it for for the moment and would just concentrating on the spit protector right so our concentrating on a sphere of quantum mechanics of the quantum mechanics of a system with tool orthogonal states only 4 Cardinals it doesn't matter where it came from a quantum mechanics these spaces of states is just space of states on 2 component or to state system and you could always write it as a column vector scorer Alpha and Beta wet How Is the amplitude that spin along the z-axis His pot and beta busy and withdrew the back so who could always right state of the system
but kept vector of scored the strike which is on Of course abound along the z-axis you could just symbolically represented by constructing a wheel column vector and putting it out for them the Beta Alpha star offers the probability that the electrons are based on data was probably found in the same location operators become matrices store operators become matrices Stewart Camara about matrices off these components of the span have to be told by 2 matrices quite Dubai tool because the space of status two-dimensional so they have to be 2 by 2 matrices and they have the satisfied this algebra that's it if confined to matrices to satisfy this out overall 1 other thing we might want St Zita be diagonal yacht not the dimensions is faced with a minimum of 3 dimensions of spins based in space and space yacht not the kind of place not for us to do is to forget about forget for the moment the position of electrons begets momentum with thinking warm electron somebody is nailed down the road for a wall so it can move around and all you could do is whether it's measure it spin it has Cohen this state if you measure busy component of its active only gooseberry and so let's label them up a bound NTT quantum state of the despair is a linear superposition of them with complex coefficients and probabilities part of credibility for is our often the probability of bound his latest it's just a change of notation of common change of notation 3 place this clumsy thing he I just riding Our column Alpha Beta remembering that the law component stands for bound the upper components up In the same way operators become Dubai 2 matrices and Toronto 2 matrices just act armies who compartment sectors that yes jazz now I said What do we kill I would choose as to be there it doesn't matter you just you just search around and see if you confine matrices to Michael matrices which satisfy this world R yes operator is angular momentum observable is observable as an operator and becomes a all operators become matrices that act on vectors to give our records same thing we've done over the past 5 Celestial confined with our gives course to care for the river will be doing this Track 3 matrices which satisfy these commutation relations I would but but but but but but but but but but cross a price or public domain too late to make you are not unique they're not unique 5 and Burma choice the choice of them really corresponds to a choice of orientation of taxis but there is a choice of them which is particularly useful for us the role quota or Steger related just a rotation of the spatial spatial real space taxis are and it is not important what choice to use it because same answers but I will choose the the following Joyce and you can check that these matrices satisfy SX physical 2 0 0 1 1 0 0 s why musical from minus RIA who are 0 0 S. G G vocal 1 0 0 Myers what you'll never find 3 matrices which have these rule which no sorry one-half of 1 half of black or jacket just go through it are on blackboard takes 2 minutes to build 3 3 minutes find that these matrices do satisfy these commutation relations here they are how we spend matrices Polly matrices are these things without the 1 here the program but the real angular momentum matrices which satisfy these rules here are half of them were up a half after coal power we may choose to use with behalf that is called the matrices think there proper recalled matrices with ahead he saw this is they represent pace it would say this is a representation of the angular momentum matrices particularly tool component representation of new momentum the mayor of the of Caspian grandeur moment the matrices OK let's now see if we can find the state's that means the state means the column America's which correspond to being off consumers E. being down I or as soon up NSC being rows of the 2 possible orthogonal state Eigen states are as I visited very easy with Israel what what does it mean to say our state whose as EU is plus or minus means of eigenvectors of SC where the Eigen value plus 1 or minus 1 we expect incidentally that the components of Sweden have value plus and minus a half hour plus a month as a half what that means the acid should have value plus
1 minus 1 of course Astros these movies diagonal elements here the Eigen driver values rest so let's just be very pedantic being very pedantic were looking for vectors the score Alpha Beta whichever property where story this is illegal of plus or minus a plus 1st Alpha Beta this would be an iron value which singers E equals plus 1 for about 3 1 that would be 1 0 0 minus 1 from beta equals minus How data at the meeting of an eigenvalue when I get back if we find solutions of these the solutions will correspond to the states in which the spin is up and it is now much check with check that out OK sold Eds completely obvious why watched as work out our carpentry here by Multan matrix multiplications 1 times affable 0 times beta so far just multiplying upper matrices going 3 1 and alpha 0 0 times and the top and 0 times Alpha Miners later and bottom obviously this is not a quarter its on less walk but later 0 beta is is 0 then we found a solution for the Eigen value of the eigenvectors problem which corresponds Oregon value plus 1 and surprise surprise it corresponds to a vector which only has entry in the the opera place the opera place stood for spin up what shall I chose Alfreton I visited you wonder why 1 because I want the songs probabilities to be worn so offer star offer post bail beta should be 1 there could be choices just to choose our equal toward nor fireworks were put back what I erased violates through Zappa restraint firms offer beta woes are from minus data supposing I wanna get a minus sign and we're like Joseph R. fate equal to 0 F are physical 2 0 straw and that corresponds to a vector 0 1 against no big surprise it corresponds to the state it would be electrons definitely down and has no probably being up focused the father what about we've done this many times classes but just spend move to doing it again while about wave functions state vectors were of rural call and can vectors corresponding so the possibility that acts component of Spear Ronaldo different express River measure instead of the z component of the and were the X from part of the again the 2 possibilities are plus a half and my what are the eigenvectors which correspond to talk possible measurements for that case measuring stigma X not sex with men case we wanna ask 0 1 1 0 Alpha Beta Eagles are from data that's the eigenvectors through with bargain value plus 1 and the eigenvectors bargain value minus warned will have to satisfy 1 1 if such things exist How from beta label my Israeli the eigenvalue equations of plus and minus Iraq from Sigma exostosis in Missouri City Tex OK so a copy way slip through popcorn 1st was about equations saying now some legal bid for a 1st ever be prevented and 0 times after 1 beta so working out of the matrix prod heroes natives and record it gives us a death how far this matrix just interchanges top and bottom 1 Times Alpha asserted 0 times are 4 1 times bigger than a bomb role and salt is there a solution of this equation for yes baby was in fact the bottom equations Sophocles beta which is saying baby equals the offer joke we solve the equation by setting out legal debate at a convenient choice for convene choice to all across the ridge of a committee in charge is that we choose 1 or squarish tool when our square tool MY 1 over square tools so again the server proposal probability adds up to warn here yes yes yes yes yes if we have included the 1 half we would be looking fine very warm here so pull off in values of SX we just look for their twice the ideally placed the assert report I let's see what on let's see what the other possibilities the other possibility is 0 1 1 0 Alpha Beta which happens to be Beta Alpha that's by multiplication he batch of now equal minus Alpha Beta other idea now you take us away from the middle here is just becomes the eigenvalue equation the eigenvectors equation for eigenvectors bargain value like this 1 was required passes bagels minus out front we also said athlete with miners baby the baby might suffer an offer people's minds beta as all we have to do is put into the entries here Cherry Paris but now
from Bahia it is equal to 1 or a square of to minus 1 over square I thought was is very high it has to have we pick a linear superposition of states where the electron up along the z-axis and along Rosiak service we superpose those states in quantum superposition we make an electron which is what you were right word which has polarized along positive x-axis if we make the same kind of linear combination with a minus sign we get 1 which is polarized along a minus X axis I would not counted for electrons richer up along the z-axis down along the z-axis Opel only x-axis down along the x-axis finally busy issue or the white are electrons polarized along the y-axis are not going work out procedures minus RIA miners IAI of very rough solve this 1 and this worn by equals my and that those solutions are easy varies you don't do solutions are Alpha Beta equals 1 of my best plus sign I think that's what plus I would say for plus sign and the other 1 is 1 minor society but 1 hope my we are sorry excuse or square notice that there's more information is complex coefficients with just the fact that the probability of busy components of spin a half a half both well this has probability half a half so the previous case with electron was along the x-axis factor activism I year something new it tells us that electron was a longer y axes now we've accounted for electrons which a polarized whom also 6 directions What about a general direction and electron which is spinners pointing along a definite back hour-long video another acts as well so that you have a corner route through it now you take linear combinations of these we want would now will do another time it's not something you do this evening process quantum mechanics are we have seen that the minus sign those on the other arrived in our area north of thinking minus for which they believe because of the minor slide may be right girl figured Akure progress so let's see service should minus 7 switch a B plus I think that may be rare mountain or have notes where your bike less than a half at the hall theory of half hour old wave functions all state vectors corresponding through all possible directions of spin on a linear combinations Of the 2 states With the spinners up along with z-axis Obama or a flight that yet posture my remember are than 1 we've done this before we miss a lot before the storm made fast work it but don't do spin 1 spoon also interesting again if there are many many choices of matrices Quello or physically representing the angular momentum show you 1 spin disease component of angular momentum has 3 states been 1 is busy integer angular momentum corresponds to both arms and that's the case would be angular momentum can be 0 busy component Cami's 0 1 or minus 1 so the strip possibilities evidently the space of states it's three-dimensional not there not two-dimensional three-dimensional again only 1 defined Spain makers to use which has 3 by 3 matrices which satisfy exactly the same algebra the algebra means the set of commutation relations X X Y equals or IS z result through there exist I've left each bar because oracle toward go there exist 3 by 3 matrices whichever fibers that there because we saw a work that we worked out the existence of a state system by assuming these commutation rumors yes I will just show you our representation a new courthouse to you Our if you could Provo commutation relations 3 by 3 matrices but things sorry yes Z Ghazi but for spread along busy direction is equal top 0 0 1 0 minus 1 0 0 0 0 0 0 the sorrow books so familiar but not from area 3 by 3 but it kind of looks like SYO rare birds but doesn't have a factor cool heads different IS starts see what's next that's why he is equal off 0 0 1 0 0 0 0 0 my ears sworn 0 0
not us that this 1 has euros in the 3rd row and the 3rd
column this 1 has euros in the 2nd row and 2nd column and as acts as zeros in the 1st row and the 1st column 0 0 0 2 0 0 1 0 0 minus 1 0 with real symmetry between each 1 has a 1 2 minus 1 and isometric the and systematic each 1 has a 1 and minus 1 and it has zeros all along the diagonal that match the letter he gives each stands for free and the federal one-third Commerce Iran of why stands for the 2nd round 2nd column 0 or so of course so that easy remember the only trouble I ever have remembering the sign again you check those by checking the commutation and then I do not completely unique broker let's see if we can find the states other S zee this time Lizzie is not back no I have not chosen as E. B. diagonal but nevertheless we could certainly worked out for the year with the island states are not enough to warrant the guest on are lower solve SEE foreigners pro Norfolk this makers and now multiply amount by our friend Bader Our fire a beta and gamma comedy what what should the idea used 1 0 and Maya sworn foes of the three-state serves you component which drive for 1 1st job I'm down with no not armed yard if you not just as family visit this corner over here is very similar to the press why over here so it's gotta be Samara ideas warn miners on an 0 Of course so here's what I that I think this is a true relationship you could check it later by here is 0 eigenvectors where the Eigen value plus 1 that corresponds to the state where a subsidy world aware In that is equal plus war I didn't give it to USC in the simplest possible for my game doing very symmetric form where Exon wines look very similar to which other nevertheless the right defined state which corresponds Damocles plus 1 justifying the eigenvectors lighting very across 1 or about agonizing minus 1 crap turns out to be this war minors I woke up the Eigen value 0 always badly should grow while the the square root of 2 if you want the OK what about the idea value 0 you would be valued the wheel eigenvalues Euro 0 1 but the obvious take the top Raul multiplier pursued 0 because as nothing to match this Swanton likewise here this is just brain 0 big Fred 0 OK well these things mean is that the restates ME quote from Class 1 chemicals from minus 1 inimical to 0 0 corresponds to a vectors 1 minus 0 1 RI 0 0 0 1 these you might want square to assure To make up probability and warned fatso again we really go find and you'll find you and no other eigenvectors nor the right values that these are there'd be mathematical representations of a very abstract things we did last time was a marvelous things is square served to really do turn up in experimental predictions of 1 of those couple of arms but not with ordinary spin bowlers topics from Russia Carlos Queiroz turned up in various experimental quantities but not not now truly is a colors full of a job a for 1st and my flowers a through minus Iowa escorted to a 0 while route i.e. a rude toward 0 this
is I think M equals 1 forces M equals minus 1 and this is chemical 0 not said there are other representations there are other triplets of matrices which satisfy the same algebra but they are in fact physically completely equivalent much or going over reasons for about now these are good serve matrices toward where very elegant will use various museum Morgan what OK from another question supposing returned to their real-life charter which can move around the space it can offer the space so it has a position or a momentum and despair incidentally spin and position things which simultaneously measured spin momentum offerings which simultaneously demand new measures what other things which can simultaneously measures well position and momentum but also the different components of the spin there commute with each other speaking momentum who commute from each other but how we represent the wave function of the state director Ali a electron given the fact that also has a position as a position and a spin and is is we turn these 2 component that called spirit which could cost splinters what's the right hand side a 2 Component Object like Paris where charges represents ups and downs called a spinner we just turned these spinner compartments the functions of position so let's so what's the measurement now electrons freedom will moon-round befall state of electrons is described not just by a simple function of X but vital functions of Drexler score psyop inside now code of of we're like we can just because symbolic Europe Asia we put the McCollum was meaning numbers the meaning of this was a fallen back you don't just What's the probability that electron opt for Barak charms down US 2 more refined question more refined question is if electron found Our position X was the probability that that they depend on next to a probability that electron up might depend on where you look our way of saying it is classically discussed plans for a classical statement of the spin of electron could vary from place to place quite mechanically varies as the probability defined up probably would define a solid every point in space each point in space you'll have a spinner and the spinners Kazuo at that point in space what's the probability electron was up well sigh but but but but sigh off of acts as a probability that if you look at point decks discover electron there and then measure theaters Zeke comply and the delivery of the Standard busy component of the state the probability will be up and size star Obama Rex value of X is a but at each point x you have all of this apparatus exactly this form and that's the whole story of what electronic electron is a particle described by a wave functions which has told compartments withdrew components being be amplitude of electron is up or down what about a spin 1 particles exactly the same a spin 1 not exactly the same almost exactly the same but the spin 1 particle I'm not going to court society was his Germany 5 more common name from both on and it's been 1 particle always balls it becomes FIA 1 of X 5 2 5 3 of X where each point of space at each point of space fly 1 fight revive 3 replace alpha beta and gamma from before free of X she your yawns arms is reproducing exactly the same structure or were norms based their Lieberman that's better than his son is not known nor are OK so let's go back Let's go back good and that's a good question our let's find out what's at what is the probability is wave function what is the probability electron has Our speed plus along with z-axis who's been minor strong z-axis Ann's been 0 OK work out his or have to build on those eigenvectors the eigenvectors of spin for spin 1 of my 1st win 9 0 were Ave OK equals 1 that's been pop along the z-axis the eigenvectors rose 1 over squarish to minors ii over square 2 0 0 Mrs. the eigenvectors chemicals 1 now I'll remind you of what the rule for calculating the probability for this condition given this wave
function the rulers could be enough product all of this with the complex conjugate biggest became in a product of this with the brought back corresponding to his cat vector which is its complex conjugate has been square and it plans of complex conjugate so oleo wrong standard Kwan mechanical rule we take enough product Corliss Spadea foggy and let's call this M equals 1 with a better product and the square but tickets absolute value squared it kind of complex conjugate come out this project ever call them equals 1 brother actors are always complex conjugate of contractor's so that means the Brodrick there can be thought of as we rolled back as just an hour convention while escorted tour plus lie a or with square of 2 0 0 plus I because it's complex conjured a set of moisture farmer square to retain the inner product back with fire 1 fight 205 free for that square root of 2 fire 1 no 1 ever squirted to try 1 nite if I want more square tool plus on a fight tool over square to it 1 0 Mosquero to fly 1 fight to over square to and on and then we multiply this by its complex conjugate get what's a complex conjugate per worker resist standards complex conjugate here Malaysia's fire 1 square cross fire swept to square overthrew Her this transit complex conjugate Whoa here find warned squared plus fight to square over 2 so the probability that electron was up along the z-axis the appears to be fire 1 square fight to square over to what about the probability that down along with z-axis they say that was the wrapping or found northern winter camping figure when earliest reversed day the status setting off what about what about the probability that busy component of Spain 0 despite 3 squared In that case the chemical 0 state it is 1 1 0 and the inner product is just fight sorry 0 1 0 0 0 1 meal a product is just 5 3 so the probability that the electron is in these zeroing do momentum space here fights which square so here the probability sigh of fire 1 squared over passed by the fire 1 square fight to squared till saying things fire 1 squared plus 5 2 square all the tools and try 3 square From our off arose over 3 probabilities 1 fire 1 square fight to a square for strike 3 square are sorry Mr. sister None on violence square means firework firms far WordStar nor just hours regards as I think I'll value this to be worked out in detail how am I hate doing things on the blackboard wire Herman can make mistakes a problem we have made a mistake but you get the principle the principle taking in a product of these things with all I know what I did yacht the reason I get be equal is because I assumed that by 1 fight toward Rio they don't have to be real if they're not real you'll get different dancers used to check out prior to it again but the principle was always the same fate the eigenvectors take its in a product are with b actual wave functions and take square comp time complex of probability and not the most important in this case is that a function of position these supplies of functions of position vary from place to place so that's basic idea of it's an exercise to work it out for Sprint office and 3 hands Wehrenberg could significantly more complicated but not principal not principal are fares are modifying the year of the matrices blew apart a ride I'm very very quickly want to come back Dirac equations Watson how we now what about 3 hamstrung video for your birthday 3 of 3 has a case plus I have miners have 3 have 3 of the 4 states in our spirit who is white he knows there is a key law after a guy I was not armed this something like Mass testifying in the rest frame most easily defined in the rest frame throw in that sense
what you do is you study in the rest frame and then you find the right way Torrance some of frames the duet narrow it did really doesn't change much when you look relativity and that's B costs the definition of spin is pretty much what you would get if you do all this in the rest frame of a particle but their Yao we weekend there we can simplify the story just say so far this spring pretty number or restrict it won't change much of our with 1 exception which won't to now OK let me come back to the Dirac equation turned to see what the Dirac equation has to do with gained Spain is best defined in the rest frame but let's over OK here's a Dirac equation had matrices 4 by 4 major cities and it looked like this I desire by beating signs for a component objects by 1 side suicide 3 strive for where I come from our where it came from a book equation greater physical reminders on alphasub Internet and runs from 1 the 3 with 3 directions of space 3 times the derivative of society with respect X and some older and serves as a powerful 1 beside RDX for suffered through sovereignty white cassava 3 decided by a busy plus beta times the mass of the particle site that was a Dirac equation that we wrote their lectures Agarwal and let me remind you what the conditions on the matrices Alpha and Beta war in order of what was what was a condition condition was that the frequency of America's Hawaii frequency it is equal from momentum our about Katie with emeritus wary of half and KM it is the same as our forewarned wander off XKX forced offered to Cato course after 3 K 3 which is the same as alpha XKX for YKY offers EKV plus data this a short was destroyed where our story and using a nite yacht is the index this is an act of particle pocket a howitzer views and fired precious quick you that say there's been some usual summation convention indexes repeated twice with always means you some over required that Omega square is you could to a square pass and where we did we took Omega squared by squaring the sorrow and required now we got a case for for term square the conditions for traffic conditions were conditions on the matrices Alpha and Beta remind you of war they required every Alpha are square adhere to be true I believe it is to be to know no 1 Warren beret office where might see each RID X squared equals awful Y squared equals offers you squared away but no is 1 matrix called is a matrix traders a matrix requires beta square to be equal 1 of these things are just so that you get a case squared Boston square but then annoyed that you don't get across in order that you don't get things like 8 times and when you square our 81 times K 2 and so forth we required things like Alpha X powerful wiII plus out Hawaii Texas equals era that would be for example across terms between KXM KYT James Van Ralph X offers we can Wigan I just need a notation and I commutator about facts out for y 0 and commutator Alpha Access offers equal 0 and and I can't figure out where zebra 4 white equals 0 whereby and iconic data about 4 x 4 x 4 sorry and 1 more of a Alpha With beta should be also Coke 0 what CAI commutator of data with itself too because as times oppose Beta Pi Beta OK right so I can summarized these commutation relations just by saying that everybody's commutator with himself told everybody is an iconic figure was somebody else's 0 it's a book I have come I'm just here and it's not hard to prove a food on rat go be smallest dimension the smallest number of components small makers matrices which consolidates are off former for you didn't have the biggest here you could find to by 2 matrices which satisfied In fact the stigma matrices the Polly matrices satisfied the Polly matrices and from commute with each other that's something you could check and this Queiroz equaled 1 so if you didn't have the extra Bader matrix here the Polly matrices would solve these equations but until by 2 matrices they wrongly matrices which aired commuter among themselves are wary of Europe 4th 1 just won't be 1 hour you have to go larger than matrices you try 3 by 3 matrices you are you drive for by 4 matrices and you find yes that there are our representations of this impact commutation algebra here I think I wrote down a representation of a write down another 1 tonight which is a little different basically yeah it is again took what or start give you an example of matrices which satisfy exactly this out now format for makers to use former for makers to use a different Alasdair in block for writing form
means to provide this off until tool by blocks job To my tool box each 1 of which is a approved by 2 matrix it is just a number of possible wherever they end for the Africa's feet each half a powerful 1 Daffodil after 3 In this to my toolbox over here a corresponding Pauly matrix seigniory visit to my to make direct source fills up for entries here now only put the same Polly matrix except support a minus sign an 0 COS as crazy I know I thought I started used different not a different representation West time they are EEPROM last time octopus ignores off your visit were the mathematically equivalent and Ross similarity relations officer Larry transformed which relate them but I like this from better like this will never know about data Vader it is just the same notation in to block rapacious and I just 1 what's toward warned that is toe but to make it 6 1 0 0 1 it really is 1 unit matrix even matrix down here but most of those matrices I have the desire property intraday incidentally when you multiply fall by 4 major seasons form you couldn't do it by by tricks we're really you multiply in the blocks to by 2 major issues so easy way to do this no harder than working with 2 by 2 matrices but only that you're devices to check these are the Dirac matrices the office of the beta or 1 particular realization of the officers were made in a place that has a lot of minus idea drugs left turn now what is different this is a different representation the last time I had these off by a former army I don't remember what I have this 1 I can remember apply beta bag Norwest I thought I had a very can remember offer young Out of it is who with different and you get a choice said that the physics of it will be exactly the same they just correspond to choosing linear combinations of for a entries in the area and was a real difference it's as physically interesting is rotating quartan what doing is rotating coordinates in the four-dimensional space which is not a four-dimensional space-time is for entries you do not have to do with our our work for Mac direct the same as before dimensions of space are are just the smallest representation of this algebraic structure Okayama that's what the Dirac equation here it is that it can be summarized the perform like this where this really means of men on here are the major what I wanna do the only thing I wanna point I might was there 2 kinds of cool by of 1 of these blocks and the other was wounded within each block this what your core is doubling of the by to nearest corresponds to a doubling of the 2 by 2 news in physics between 1 of the tow but is positive and negative energy the other is Spain's sold there are particles of positive energy with all possible spins a positive particles with negative energy with all possible spent I would see their and that's why the 4 by 4 matrices if you like roughly speaking is an entry for positive energy with positive spree and positive energy with negative spin negative energy with positive spin negative energy would negatives are those of 4 possibilities that's why positive spin and they do Spearman spin-off z-axis half them along with VX that's why there are 4 components Dirac equation positive and negative energy exactly as an one-dimensional example what we describe and stand up spin bounced in that's why there are 4 entries in the Dirac spinor Dirac equation is perhaps the easiest way to what's going on is still look at Dirac particle with Zero momentum just take a particle with 0 momentum that means Katie is equal to 0 In that case we could just Raul Alfred's altogether we get a very simple equation where the momentum a 0 0 that means the strike is constant in space is what it means to have no derivatives with respect the space is absolutely 0 in that case although rack equations becomes a car decide by IDT is equal top height and baby I D cyborg outside that's a that's a Dirac equation for a particle at rest particle momentum equals 0 means of particle arrest that rests the wave function has no space dependents it just completely constant respect the space and this the whole thing that tells you what possible frequencies are the week arrival in different we can write Our maybe yacht why here nor there but Seurat says Tom is is a little about will take please Taiwan where foreign 3 1 2 3 4 provided the same kind of division in the box yeah each half of the opera here from Aurora of all put food there the film with entries will cost 5 classes and Siam minus the surplus miners being positive and negative and not they don't they just below mean anything right the plus a minus nothing special but each side plus was I plus is equal plus is inside upside down and the bottom when he arrived here is a summary of store others this corresponds to an electron OK we
just writing the column record In a notation which is similar to the way I see provided up matrices the violent matrices by destroying some red lines and doing so by 2 matrices here will divide up br spinners column vectors just providing them in half and in each place put a toe Component Object so sigh pluses I'm writing this on themselves to Component Object Toto Component Object maker for component it's just so that it's just a pretty way to organize the Dirac indices regret Barak grow structures William rightist equation very nicely this equation is pretty simple or charity data what is the Matrix 0 1 1 0 what this is matrix due when it hits Otto Component Object Class sigh might What does it swaps opera when this matrix hits like there's just interchanges plus we can also write this as a tow Component Object sigh plus really of 4 compartment object but to component objects and I see what the equations as the equations says is body buy fishing or are Syed pause he quotes in new sorry my years an IEEE Debye to be offside miners equals or side plots that's the Hoa are whole upshot of the Dirac equation of particles arrest is even simpler you care and and the next step they buy got a good idea what to do with this simplified up that's 1 thing you could do amnesty subtract from Newman's from so I have a few of them Debye IDT Aug 5 plus plus martyrs now with equals M 9 so I plus sigh arrest price of airing them and subtracting member
body be buddy did see our side plus minus sign minus equals minus him side plus minus sign the upshot here is that this really breaks up until on coupled equations with things 1 thing is I plus plus I this does not get mixed up with Cyprus miners farmers and sigh plus minus miners doesn't get mixed up with bypass plus a satisfy equations which are completely independent of each other how hot the frequencies of this 1 locally say about the frequency of this 1 ID by DT is just Omega right the f the wave function as a definite frequency than ID by DT just gives you it frequency so this subject here has a frequency this 1 has a frequency omega which is just equal this 1 has a frequency omega equal to minus and because of a minus sign here was a fair rugby energies of these 2 solutions 1 other positive energy solutions and 1 of them was a negative energy solutions solemn just exactly the same as the one-dimensional example we gave her a positive energy electrons and negative energy electrons I don't expect it to remember all of this proper and be able to deal with orders from one's sitting with there is worth going back now finding of a book on the Dirac equation going forward but basically told by told blocking this year has those positive and negative energy I plus plus I minus are positive energy things plus minus sign miners are negative energy but that still leaves the by told Of the individual entries here on each 1 of you use it buy 2 entries so let's forget sigh Parsons was taking care of that by hitting subtracting but now let's look at the entrance the entry of just 1 of them he also was rumored to think what is that Dubai to think it's spin spin which is described by these how we made our every completely forgot about the tho Tunis associated with the opera Maltese this year and is worried about the Dubai we're here and that's same as a towboat who knows what and Roberto matrices here that's a spin star Dirac equation predicts that electrons positive energy electrons common to different kinds of 2 different kinds have to do with the upper and lower components are of the up-and-down components with me and strike last year and are suspended it also predicts that as positive and energy of positive and negative energy story that from from here that has to do with the existence of positrons holes in the negative energy see a positron that's what the floor by floor is about you could even think about it as positive and negative energies been up and down or you could think about it as electrons and positrons each of which I have been particles Dr. GATT trade he was not yet you're asking whether you can now young absolutely can kapuka Berkey of a positive and negative energy this into a smaller matrices and the spinning they River bigger matrices yes you can you wait class your their kind a quota to which other images correspond reshuffling of he he's is the 1 with the club jack you could see from here but positive frequency and negative frequency solution things which have positive frequently on songs of these 2 0 and the things which are negative frequency on the difference between negative frequency means negative energy positive frequency means parser song after a positive energy solutions and positive energy electrons their negative energy solutions and means negative energy electrons what you do with a negative energy electrons you fill it up you you fill out Bergerac with all states negative energy you always lower energy by putting in a particle of negative energy if you want a law the energy down to the absolute bottom you feel every negative energy states among others you fill a negative energy states which have no moment mergers work was Roman are due for all the negative energy states and that's you take a negative energy electron stick even tho a positive energy state you made electron an all or electron and positron sold in exactly the same way is for the one-dimensional example positive and negative energy electrons become electrons and positrons the or other bubbling becomes a script there have been particles which comes in particle particle pairs and that was the prediction of the Dirac equation arts Julio when these marvelous examples of just fiddling with abstract mathematics a little little and boy outcomes very very dramatic incredible predictions positive and or negative energy particles with spin and remembered only discovered I was sort of political fact from Machado couple years before me just a few years before good there's shortly a short time before her 0 yard was if Theis is not saying the absence of a negative energy electrons apart from Willetta this before you feel a lot all the negative energy particles new Kuiper you and you can put more than 1 negative energy particle estate staff evacuated all the negative energy you less than 0 you greater 0 and fill them up you put an electron and every 1 of those states has lowest energy because all the negative energies of field that's a vacuum as empty space you removed an electron from here and put or hear novices earlier the same place what is at all now as a positive energy electrons and the absence of a negative energy From the absence of a negative energy electrons is a positive integer positron OK it's exact exactly the same as in the one-dimensional example what we did with the absence of negative energy electron or hold b Commons a particle of opposite charge but no just hired energy for them to do here your last fall yes your school unlike electron annihilation recreates photons what the 1st has to those who surrendered energy know how much energy does an electron or positron have both positron and electron positive energy why like was a positron positive energy because it's the absence of a major energy drink it's the absence of a major energy source as positive although this is a positive charge major when the positive
charge in negative charge come together you could find it described it as the annihilation of the positive charge a negative charge but energy has to be conserved so far pounds after go off so I think about it is as it had mom where electron excited electron dropping down below state what happens then put on school question of that really very clear each hour Missouri State added what they are you get is deserve it duty to give up the party ahead borrow cheered we are made more normally after Tom Rees voters rumors so is deserve healthy while 0 moment members the momentum positive energy statements a zoology positive Fowler correspond to a spinner Oliver form of courtside last start the same side Austria are we were set minus equal to 0 because we're going negative energy solutions that means what's well how offer for example offers not quite is not really are you asking me how you make me on the spin operators from the Dirac operators death out is ways they should all they do is they show how all flung discovered you've discovered that a positive energy particles calmed in 1 which you could often 1 what Yukon called back now you will receive pair really has a angular momentum young really has America see that howling matrices come into work but the really corresponds to a angular momentum you have 2 ways of going about 1 is through the connection between angular momentum and rotations of tree and the other is the rule refractive angular momentum is considered so what you would wanna Charlotte user conservation our definition of what you would want the show was there any kind of collusion where an electron comes and scatters or something else that if you assigned electron an angular momentum who was descriptions isn't terms of this tool for Ms. within here that angular momentum together with all the other angular momentum the problem would be conserved is a little too much for us to do too much more on what we can do more easily show the connection with rotations tree but I want I think we've all reached the limit by now there have been other around annihilate positron he didn't have the same and yes yeah and immediate number or a photons Camira were toward a player or happens OK supposing you have 2 electrons with the same energy electrons and serve as as far as he has for their and then another half public finances you have follower of America from word from work Trumbull remember which work but members yes young or you can this electrons out of here you can kick it up any energy doesn't necessarily have the same images a pass from Young where you had are you will necessarily kick the same energy is headed to hit nickel reserved for Samoa energy so there and then there's no memory of which I suppose it was another electron around Barak sees filled and now 4 a positron are part of ah yes a positron was formed and another electron then this electron conforming to hold all of this electrons from me the case you before I unclear if you have the data and a whole lot farther from each level native while like native Senator native state drop into that lot Matadi be all that I don't like it happened have services to good thing happened photons of our acts would see we stopped see what that says we start with negative 7 over here hole that means a positron 1 positron 1 plus positive charged electrons notation for positron plots of from electron beam minus artery Oriental 1 knee each class with energy equal minus our women energy equal to a 7 strife we took electron with minus 7 units and then removed it which means we have plus 7 units of positron and electron went someplace now was no worry about with electron warned electron became an electron appears someplace are you aware white can't 1 these for them to hear what I mean you Saleh positron innuendo apart but you and know positron different energy this whole positron now or electron from here columns and falls into this holders and negative energy electron falls into this whole as would arrest preparing to start the positron here was worthy have a whole new repository is just a positron a different energy saw positron has changed its energy where positron changes energy can only do so by emitting photons right How do you make of electron Ameritrade energy by emitting photons slamming it into something in having the solve a positron from here under the right conditions can start a an electron from here for he just another way of saying the positron scattered from 1 state to another and redhead wouldn't be always always higher energy electrons will need all call on CD-ROM on raising the reason that doesn't happen to say wiser positrons fabled right not to be answered is because you have to conserve energy and moment of an hour the conservation of energy and momentum doesn't allow it to happen to retort that our prepared again parts or as the beginning of next week I'll show you wiII why look you could ask you could just exactly the same thing about the original electron why can't electron always lower its energy and radiate photon why can't the energy here our Charges Dropped Baron energy coats lowest energy electron and the answer is because it key simultaneously conserve energy and momentum and photon just doesn't are sold so I will get Touch Review and give you the exact schedule we will definitely need Our uses quarter next quarter I have um obligations our won't be a superior and your next quarter thank you for your pressure for more
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