Advanced Quantum Mechanics | Lecture 7


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Advanced Quantum Mechanics | Lecture 7
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Susskind, Leonard
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(November 4, 2013) Leonard Susskind extends the presentation of quantum field theory to multi-particle systems, and derives the particle creation and annihilation operators.
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go although again the basic construction of a simple quantum field were not exactly what I have read your vote no again Murray gambit may be slightly different young were OK so as I said we've of introducing fields From particles seem not to let you look away Ucon starlet particles and reconstruct wiII described by field for you start with field and then work wife fields are described by particles Whitefield waves fields have corn in way that will go both ways for we started by starting with particles we said Look that's fine could be talking about 1 particle or 2 particles free particles a fixed number of particles that he won outright study all possible numbers of particles by particles Merrill speaking about a single species of particle we Conway the tonight what happens if you have more room when species but Bob for the moment 1 kind of particle in only 1 kind of Article II but later theory in which weakens simultaneously in 1 fell swoop service Creek are talk about a number of particles of all the things of multiple particles including situations where the number of particles change now why would be allegedly led there at wealth In real life numbers of particles change the number of photons in this room was constantly changing the being emitted by the electric light bulbs are in other contexts particles became turning in other kinds of particles sorrow Be neutron indicate into a proton and electron and neutrino and the process number of neutrons changes the number of protons change member electrons neutrinos in a fact that poll lumber particles change however we have the think about particles we have the think not only about the quantum mechanics of a fixed number of that we have the think of that number is changeable so we need a flexible enough pool the be studying what happened last single quantum mechanics which embraces essentially any number of particles and even cheating number particles that's what they are baffled point there is a matter for the point of field bodily questions about that and all over again so the mathematics is that it is not hard but if it's not abstract abstract and place it would not Her with as young as we use the B for 1 they are morass reminder is not necessarily a tool instantly people orthogonal functions member and all that kind of thing so it's good goes well I Levy you learn something rather than nothing of care there were no easy the learn not to or something OK so we started with a single particle court mechanics single-particle the description of 1 single particle quantum McCarrick going that need that we have state vectors but they perform 1 particle we have wave functions for those particles if we have a stake of the single-particle scoreless sigh a little sigh a little sire will also stay in Ford's wave function if we have a state vector for a single particle than and the wave function is just be in a product of the state vector with a particle at position acts labeled X means of particle act point X the state's sunny meters some general space but of the single-particle and that's called sigh what you do all of that it is a formula crash sigh star vexed sigh of X is a wave functions that these are not quantum fields when I got a quantum field I will use large capitalized letters even if they happen to be the same it was object was at correspond look the probability the following the particle act X probability density the probability density could find particle next probability density problem bands you could through in the rest of the letters back about the probability density result when you integrated although position where you get you want that happens in this case that what happens to be the number of particles wasn't so eager ones but you could also say you get the number of particles were a summary interesting because we only have 1 particle begin with doing 1 particle quarter Kerik's lasted 2 true fact now we next introduced a collection of 1 particle states form a basis for film normal orthogonal racers will you do not them away 1st of all the exit state themselves of farm to each other and they form a base which will go back minute but there are many many ways of choosing Macy's take br eigenvectors of any her mission operate will form a basis in particular a foreign arrested may be especially interested In energy Eigen states for example of particle box that represents a box the horizontal axis is position vertical axis is the potential energy when you get to the edge of the box the potential energy
becomes Hewlett's him apart from his reflects also came through in that case the ground-state wave function looks like that the 1st example call out sigh what next the next energy ideas they looks like that caught 2 of X is basically signed Chinese are over filter 2 knowledge and characterized by a number of novel about peculiarly Astana was sigh 1 which has no site too has 1 Knollwood Soviet X and since these are supposed to be energy argued that orthogonal pooch over a former North normal basis so it's right the equation back the equation variety is wrapped in the role of size star on of Texas was side Jr Rex but see product the excess was actually remember show go project if defies equal the J strange state of product as 1 or more state is different the orthogonal corresponding with 2 different energies dealt by J. nonzero Prijedor tomorrow a fact Israeli another uh another little manipulation on of what is because get to have that ticker 4 cases was that goes back to be a posture with quantum mechanics the posture of quantum mechanics said about probabilities desire next the Tuesday in a product of site with state localized that exits call the probability amplitude the posture watts of quantum mechanics which are like to go back to book square about probability Seoul OK here's a little trick a little mathematical trick that Over and over again thank you off normal your favorite author normal basis of parks they could be signed as they could because signs they could be harmonic oscillator wave functions whatever some collection of size which from normal and if the author longwall former then if you take this object to what is a subject subject it is thief operate it operates on a state where she worked but just explorer pros were act on state was built other acts are stateside then the beauty of the Dirac notation you could just take the formula that you've written them when your act with this thing on the state sigh with is a member of a park of so I will I went that's were guards are now used a fact about awful normal basis if you Of all elements of the awful normal based on roads in this case all the energy eigenvectors up to Woods Cemetery what get now you have operated by the identity operated the operator that we act on any vectors you back saying that while those elements in the yards me in the note should go back to it don't take my word for it improve it by it's a useful thinkin out OK let's see use that now the following following way let's take up project at the with this thing white Oh would like to do was going to take over act you on would only that act on each day to a right which is localized position x then on the left wherever act well actually what we're doing is taking the inner product of a staple the particle at with at White anybody know what that is their mathematical symbol of the letters the Dirac delta function was conqueror for more votes are like that yet let's try to that is equal to the sum on head wiII fight on a X or plug-in you're operated here at the nothing but now I have a formula here we had a product of with white is sigh all right of what's yacht theater product of any state were eigenvectors position is the army is wave functional let's face it status I have with functions like water it over here so IRA of extra lawyers Horvath equal has Siegel Delta of X murders wife not some more white or yacht Miller's the 2nd 1 yes the 2nd 1 as soon as the U.S. stock begin Alannah Myles syrup if we take any low-level normal basis they could be the eigenvectors of any her mission after the button particularly energy and you multiply for each 1 of those states sigh start Prime Star and in some although them you always get a delta function with thinkin hours will say useful Rameses many many ways of writing the delta function just take any normal bases and the lists these could be signed and co-starring sorry are another way of writing the delta function but end this to the basic mathematics boom over America's plan think OK now works comeback worked let's go to the problem of study many particles or the same car now
we characterized as 1 set of states which is particularly useful as a set of states in which you say how many particles are With each 1 of these energies for example how many particles are there with state by 1 site to site 3 himself for and so we can take the Maltby particle systems and characterized by a basis of states which is just In 1 in 2 0 0 laptop not enough I thought about that it is only after patient numbers so the occupation numbers which tell you how many particles are with each energy level and overstate metastable particle states they don't belong in the same vectors faced with the single-particle wave functions the single-particle wave functions are themselves or whole vector space but when talking about a big there's a bigger systems so we should not confuse these single-particle state with the armed with a more general structures or talking about we use saying patient for them because nobody of the better notations Barack roars kept Barker there are OK next we would like To be able to change the number of particles maybe we do maybe we don't want to change the total number of particles that not clear yet but we like gala change them and so would like to be able to raise and lower the end solely introduce a oscillator face each might be some oscillate at the moment this is nothing to do with oscillation this is just the track adjustable the bookkeeping strict or it will be more than that later for each won't ideas we introduce a creation operate already are raising operator but scholar Claessen operators and now want to create a chart creation operator and annihilation operator and the creator of annihilation operators it's convenient that make them just work in exactly the same way as harmonic oscillators just a convenient way of dealing with plus I and a minus sign our study right now that there another convention for the same thing on drug Ray is the call was a dagger that looks like a plus sign not quite but nevertheless and this calling this 1 a you often see written that way it's a dagger creates a particle 80 does not work we have a correspondent is exactly this OK this basis states of a mini particles which was called Fox FOC case but was a Russian physicist who tomorrow those name attached to this I don't know her OK now find very can of very convenient to see how works to introduce the notion of a field operated the field operator has the same symbol is the wave functions sigh of acts but it's not wave function with Taowei function not a state vector for 1 particle it's an operator Anderson operate at that and on the basis of state of the many particles it itself contains creation annihilation operators like the soul of clutch right the notion a field operator the best way to motivated is to use it and see what values a opera Sean I'm a little bit afraid that go that you won't be withheld myopic cases from my lord cases that's what uppercase site right on its definition is the following Mr. some although RIA lot straight the annihilation operator for particle in state party times the wave functions of a particle if warned stayed and that's it some of all-white know somebody asked me last week Aziz had anything to do with Fourier transforms into bed at if the wave functions here where the wave functions of a free particle guidance states due a free particle is not a box just the infinitely big box or autos Oregon State was argued wave functions plane waves equally IKX Anita the miners sorry to the IPX Simitis IPX so in that case if we were using free particles infinitely big-box this would be an expansion in signs and co-starred in those aid would be the coefficient of science because they would be the Fourier coefficients before expected extract that another numbers their operators operators of things you measure of things that the that play the role of operators inquiry here OK so that's 1 thing we complex called her Michigan card you get over it sigh dagger of X of X yeah was it mean to say we have an operation which depends on it so what this means we have a different operators have reacts that's all we have a lot of creation and annihilation operators have we can build a lot of independent operators the Vatican build operated at each point x by combining the eighties with signs of our X point decks so we don't this we make a different operator at every position so we have a operate a function of position operator valued functional position this complex or emission conjugate is same thing except creation operators here this is a team right I get so used to the OK a plus sigh dagger stock might of X now it doesn't matter that it's rather weird the remote page some people use a patient with a score of 5 star from people use worn with this call sign illegal up there the definitions and see what you want but he is a definition we the to operate a functions deal 1 of these as mission operator Our very creation operators annihilation operated on mission
conduits Our each other if you a had them will also be if you ate them you make something the Commission hearing emission operator any operator and machine card you get richer soullessly these till they become permission and her mission means are observable means something you could measure in a laboratory solve some of them for you measure and the differences in different emission weapons if you take a different from our number and its complex conjugate get something real trucks it's only pure magic review multiplied by I renewed something mission sorrel each the sullen difference of these defined mission narrow observable right now we're not observing them would is commuters removed bookkeeping devices OK now show you another way interpretation of Sysorex what at In particular let's ask what when acts on the vacuum vacuum means a state with no particles of overall so vacuum call vacuum which we usually do notice by Oh is really a FoxBASE state in which all the occupation numbers 0 0 FoxBASE state with which all the occupation numbers you sometimes just called the vacuum orders called the state 0 OK let's act but back yard work right the following formula this is only 1 particle state 1 particle in only 1 particle and let's right at the following former lets you lose this anybody northern need for this kind of formula is no it's not uh is it projected to trivial projection operators the more it's called a resolution of the identity of a way of writing your operated as a solemn overall a lot of raw fodder vectors lots of lots but plugged in the resolution of the identity someone why are a side I was I I is a state in which is want particle and only 1 particle energy level here what is this thing here this is sorry X X X R was as object called the top formula and you see that when new project that X you get away function this year it is some money starter all of X it's a complex conjugate of the wave function of high energy level examined will have a wave function as complex it were fortune staff would have the opposite of order where now this stadium that's a state with 1 particle in the energy level so I'm right this way overtake creation operator from making a particle in life energy multiplying it by the vacuum I've created the 1 particle state party by applying creation operator on vacuum and that's exactly the stayed here on the other hand was just compote and usually function my rule is signed but I erase the death not definition of field all case go to the 2nd floor I star the field operator so now what are you plus a body times the complex view wave function that's exactly what you see here right losers exactly equal poor field operator position X on vacuum that gives us a new interpretation of sorry if they are greater which creates a vacuum a particle position I think that's a fairly intuitive things back 2 quantum operator the position next start the injury ART create particles with certain energy levels the Sinai I create particles of location that's also what kind of creation operator the field operator itself as a kind of creation operator but instead of creating a particle With the energy with definite managing it creates a particle With definite position yeah yup should be stars to be dagger but easy you never use star for operating whose is that good that where all not all of will yards away function on numbers the number is an ally of the trivia operators members sometimes ordinary numbers are called scene numbers this was the mother of invention of Dirac ceased classical as classical numbers operators he called killed numbers for quite June numbers don't always commute see numbers always commerical themselves but they also commit with everything else just numbers OK so now I have a lovely new interpretation of the field operator it's the creation operated for particle at a point of space let's look at some interesting quantities think with her a public can put this up just to get both more experience with this looks but with a little bit if adjusted force if it is interesting inquiry into OBX the operator side Barak
of X sigh of sex now is this is same is off top of the blackboard up they'll be exercised star x 5 x not it's not the same thing it looks very similar but not the same thing in formulas on top of the size is just wave functions 1 particle wave functions and operators and the right-hand side is 1 by total probability this object here is an operator Sison operated by beggars and operated act on a state that does something which creates a particle was 1 mile it's a particle so voters don't West it has some old structural similarity with the thing up on top 1 wonders what is it doesn't have the meaning music producer of something that has a physical significance something that you could measure incidentally intermission because you take any operator and multiply by a submission conjugate the result permission of for multiplying ADR complex number by complex conjugate result always Rio OK facility we figure out what the saying its 1st that lets replaced the sign by the definition so this is it kinda will be X and now star that's a somber although or failure messy but get easy ARE dagger ARE plus times sigh star party of X without With Applewhite without the witch Side now I'm right side as another separate not some over the same i.e. it's a somewhat a different bodies but I can just college 80 that's a bubble sort of bubbles from here and the bubbles has 8 J. society J of X I looked a bit complicated and In the grow over the goal over acts would double son of jail plus the fact that the integral to as all space and eye injury go from 1 to infinity so if you actually a calculated to put a new computer you computer would run forever and ever and ever and ever with Mel Hall our however we can do better flexibility it the goal over X 1st now I think we have a formula here about what line of X shot there there was right here sigh I Iive X was so J that's just the orthogonality relations salt were making the gray over X flight bag approach sorry I just get built by J mice also write a simpler way this is equal to the value of the world some all on my injury a plus RED AJC Collins the chronic adult words we get at Terminal 3 each foggy where Free giant accept only when I'm equal and it is equal to the sum are of a dagger or ii a it was a bad idea they are for harmonic oscillator this is the number operator with a hose are I didn't values are we in the little in this is the beginning of this can be written as the the sum of all Of the number operator for each 1 of these oscillators another word it's just the total number of particles member particles 1st mode for some fresh wave functions but number particles in a 2nd wave function was part of a 3rd wave function so what thing it took operated that represents the total number of particles for each 1 of these Fox states is just some of the middling now is an infinite number of the right in into 1203 3 is is not as it's going to be a finite number well remember that most of those bodies correspond very very hearty energy if the state that we're talking about not dominated by the ceiling high energy use and these must fall off quickly the number particles with energies of Brazilian billions billions must be a century 0 so in order that the problem of on energy be fine you certainly have a habit of Lumumba particles missing here typically find most of full most similarity with the start was also the problem of particles but only in the case with Member particles was known to be 1 that's just go to the probe a probability is 1 have a rapport with a number of particles as out of here exactly the same formal looking structure is not telling us total probabilities it's telling us how many particles we have those that there was a farce a probability that deal with the white but just before poverty for a share of lead you would vote for a year before war I because this is not a look at me and the question this thing here is a density of particles from like them Barak white visited the Rosenthal Lumumba particles are Michael right that's a probability density rho right density but the density is not necessarily definite number would in Quanah Kerik's thinkin Beahm observable but there are states generally capable states thing is not going have a definite value you could measure you could measure the density of particles strictly speaking you would measure the density of particles . what would you do it is the great over little volume has comedy particles are in volume of my integrators or a
small volume of wall space that we have particles were that's more volume I can measure I will always give the same answer I measures Wednesday that measure another state is so like measuring any of the quantum mechanical variable you get a variety of answers there gone but this will tell you what the number of particles is more what twirl sought that the kind events and is to be literally that he but what some operators a supplier it is she added all those they will receive the dollar is the single largest right now let's come back from moment this is not the most general state you could this is a you have also a basis of states Yukon policies just like in any vector space Yukon and vectors to command vectors with different numbers of particles you can add value if you have the same number particles you could add vectors which particles years of particles years ago and that your mother vector with some of Pfizer's changed by 12 you could together with vector which while particles will bless they find their many state by hand and superposition of states many many possibilities in any buyer whatever the state is the operator representing the number of particles more while you're at his side by side the thing EU measures to think only a measure when you in that's more while you're U.S. company particles solved although I haven't told you had a measure signed up Tri-Star side there is a familiar face just number particles of Bali whilst also but the point is that a fair jury problem is fine or be part of a widowed desert you don't look a love said his wallet and rational that all of us aboard supposing you have ah particles rebound just classically what particles you want to know what the probability that anyone particles while you wealth or high approximation of number particles rightly of those 1 measurement measure how particles of romance more volume why by the that's not the definition of the probability of get you could accidentally gives you are so we get too many most of the times within a margin of error you will get the number of particles being the probability for a particle to be No. particles provided by rights or it's not identity to save this thing when the vise appropriately divided by the total some identity to say that it is Beyond the probability for particle be in that region the probability convene half a tall readers space probability a half year and a half years have a million particles you could accidentally discovered the old particles on 1 side not very likely but you could discover pay you make a measurement and be answered is million on 1 side of the site not called for the probabilities but most of the time within the margin of error within about a square root of the number particles you get the probability of accidents in which I was a little waffling on what we work or probability of not this thing really stands for the measure for the hour for the observable which really corresponds to the number of particles that you actually get when you look at it however now that they got shot what boat got a bad do that but would you could do not to that that normally doesn't occur but would you could do was have a clamor superposition of states in which this some number of electrons distributed throughout different states and then a some number of rigid distributed different ways number electron photons however because they don't have charged you have no problem to proposing different numbers of a four-part now when electrons you could uh don't forget your other species of particles in the world and particularly positron so you could superpose states With different number of electrons and part of positrons as long as the total charges of the Member OK but right now we just invent across all the concept of a quantum field are the 1 more little manipulation to assure you have our example oral form of things it you could build worried I thought I was going to start their working there the expressions of operator made up of a size which stands for the energy of the horses from harm we probably won't get quite there tonight but look what's what's would begin 2 foot is our we found that integral X upside Assad was seen as some of the number of particles of each hour goes with manipulation poll of fish His name is now let's slightly different and stock With the formula for the fog energy now I'm interested now only in systems with independent particles articles on directly with each other that's typically an approximation it pays the work when the particles of pretty loop even cards if you may convince enough will start to to interact with each other many AG will not just be the song of the of folk art that have been make them plenty events much of the new American elaborate our but still In principle in principal energy is not equal to the sum of the energies of particles system but I think later particles on not tool been aren't told many of them feel volume which generally good approximation so let's take that approximation and there are think malaise very principle we particles free particles particles of interact with each other was Energy Energy is sum all the modes mode means worries single-particle where single-particle state terminology mode means particles from of the number of particles in
that state Khalil the energy associated with that state could recall the last time or make the we could call age bar alleviates barred from Quebec moral from a guy in energy are the same thing I just use Omega because working well oscillators brokers where this just energy the on right OK this is the energy of a of we also right back as a summer wiII of a dagger Ottey AIT times on God votes are lot flight this summer here the only difference each is that each are is what we did with a factor of Omega are worldwide do what I won't do it tonight is to go from expression like this back to an expression like this except for his beast over here will again is something involving the field operators and that thing that involves a field operations is going to to be the energy of the field of energy the sum of all the images of all particles but could be something that's expressed in terms of field operators there will just call these the energy on the field but something reasonably familiar Our Burke who won't do it tonight tonight he said I thought I was the did go for our I was but the pleasure it it at the time was not but with the yes OK the post Wu about thousand draw Bullmore young not just be there it is Of the ice state wave functions of functions of position after a function of position with the energy that's associated aware this is the energy of the 1st of the ground state of the city of Iran's airborne particles that are not there may be is I'm not functions of position their functions on the functions of which energy level with talking about half hour OK so what are the only guy may eyes the energy levels which are the solutions of the Schroedinger equation time independent your equation defying the energy levels of system you solve the Schroedinger equations and you find out which I give values on the right hand side of the prime independent trotting our equations you've used solution has worked right at the time and the pen Schroedinger equation now although equation which determines a seal may get who who who who we are not not Norway directions what your interest rate particles loggers tech a yet Guard we're not no we're not assuming interactions yes we are assuming your nor a guy who's 4 at least that were a situation where directions or barium report that article school far off each other OK I've so all of our go may get is a solution of the time-dependent Schroedinger equation with strikebound down Greater destroying our equations the planned the assuring equation of course it's just that we're talking about the war particle Schroedinger equation which for only 1 particle quite well it's the 1 particle Hamiltonian acting based on a wave function is equal some Shiite and equal the ideal value or may go that's the energy level times Iife what Semel colony in the Hamiltonian works take the simplest thing P square over to end plus a potential energy the particles may be interacting with the potential energy but we're not going with them interact with each other so it could be electrons orbiting the sun would not treating the quantum mechanically we are treating electrons core McCarrick going forward thinking approximation of electrons don't interact with each other the only interactive the sun so on where we write down H H is potential and kinetic energy be square to wear because rear where Reid is potential energy the rise oil quira but with With with how multiplying x equals 0 may I am this is I sigh of ex wife energy level is Omega sigh X now novices on thing he or meet peace quit over to where we don't reveal next time on sigh of X means such as means multiply sigh of X by some potential energy for the thing increase Cuero where now we use P E equals mind I by DX Ucon has homework put in BH bars the quota components each dimensional space so it becomes PX wept was BY square was busy square this thing here becomes well for our right and three-dimensional language it becomes Dole square which means 2nd derivative respect X were a minus sign because of the ice squares times I signed divided by 2 now this if they do we know what to do with on-site just differentiates Iike twice it shows some of the 2nd derivatives in the various directions space plus of X directs glass rear axles put all in 1 hour and warned bracket plus BOX sigh he whores major so I saw make it depends on which and level with pot about well aware that there were so many showing or
equations tonight we're just gonna take all major crimes IRA which notice what's it doesn't know it hasn't appeal yet which is going to you lose the formula without solving it will use the formula occurred they these use we get Mr. right down cancer and improved the answer was use when you get a flawed put this upon top Bush
Margarite Bellamy injury remembered a mushroom resisting Zach 2 OK before I do we point out something
about nearly scored Kerik's 1 particle or mechanics deposit at wrote their own the expression will be X sigh star of X anyway function whatever times my my squared off with 2 and a paucity of X Siurek so what what this thing Beery by now let's say expectation value of the energy saying this is each has basically cemetery age between Brian Kidd vectors the expectation value of the energy so his a formula for the expectation value the energy of 1 particle won't tell you the actual operator represent Hamiltonian of many particles system but gift from this would be answers for sex still well-known that but I'm looking for another expression for from the same thing I wanna get to this point and saying still they left with that found that but it but yeah I know it is not like that but I can make a guess what we build goal from me on what we do to go from probability defying particle could get the number of particles of small region indeed we replaced the little signs the big size and that so I gave us the sum of all the particles are so the same question Saleh let's try used his is a guest I guess from me it would be a guest for you accept further with of course is not a gash if and I remember learning about this with the little he probably thinking 60 warned there was kind of blown away service reading direct book redirect book Mrs. blew me away that this is the same as there's so much progress but go through the steps the 1st thing to do is to replace society by by definition of so replaced cyborg definitions this becomes a double the X down this 1 here it is some Hawaii AT that so I star of X right at formal review Ms. are here it was some more with J. R sorry Jr the X 80 J. remote happy with that reminders J I guess but all I did I formula this committee written as a summer what I NJT places 2nd summations signed it and no sandwich is my used elsewhere or touring because know what that this mean through this Back on the subject here what does it mean well it simply means that you do these operations on ordinary wave functions this is an ordinary wave function we know differentiated we know have multiplied by VRX solved we have an operational we know how do we might not be able umbilicus America too complicated but we define operation this differentiate side twice as multiplied by but we already know at least I hope we now somewhat of a black hole here but throw in place is a short hair equations Schroeder nickel equations tells us exactly what this ears Miners elsewhere total and was via VMX act signed just gives us made by time so we can arrange moderation because right this until the X some J a plus I scuba together with a miners jailed there now start II a star of there is this beast multiplying side had just Omega GA Cyrus J. books what's MXF How much a debate over X in debate over X when you into made acts star of acts I J. that's just chronic adult just quota to some OR injury a plus RIA a minus J dealt by J. but With 1 no factor that we didn't have room we did the probability or we did just account bigger particles major that's exactly the this Lobley expression this is the energy of a field and given in terms on something quadratic in the field will show differential operator Walker new compare that with the elected by memory electron dynamic very very similar kinds of things or any field theory you take a field you differentiate them a couple of times quadratic expression and integrate the space that the characteristic formula for the integral for the energy of a field all field theories have our property that the energy is equal typically want something quadratic in the field with a couple of derivatives acting on the field of derivatives of sight so this is an example of the energy of a field that's also the Hamiltonian these things are energy on these things are operators who left designers as an operator son number that's it's an operator 1 operator about Thing which has to do with the energy it's the Hamiltonian of the field theory the Hamiltonian of a field theory but it's also Rio as the what we would have to go to war for us to be EU South how how I went to every single step the fact that New Labour de steps Wells but I don't see why it's a function of position but it's a observable poorly Cyprus side daggers quantum mechanical observable measured it 1 place you could measurably of a place at the same time so what what you want it to come to the position of observable
quantities of the last causes vast all if bad about Yukon measure and opposition there is no other definitions now you could ask what he owes electromagnetic field not what's not electromagnetic field it stays in the state and the same relationship electromagnetic field as full-time would be in a certain relationship electromagnetic field when I look at it particles with talking about here is among relativistic particles that have amassed lady could be protons and neutrons moving slowly they could be electron moving slowly along with out relativistic along with the energy is given by square to win these this field is a field for those particles most particles our quanta about field in the same precise that folk corner of electromagnetic field electromagnetic field would have a expression for the energy it also has to derivatives field square were R they're quite similar although not identical look this Werner OK so we've who've invent their quantum field we vented to what observables 1 of them particle density there is now we have the energy of the energy is integral all the space of something what would you call a friend of spring integrated the energy density energy densities so what's in here collar Watson here the energy density the local energy density by measuring it for integrating their over small volume you find out how much energy and Wal-Mart you could more space in the global energy where we have something in the total energy is equal an integral Ave it stands to reason or we can call occurring Mike New year it's a definition of the energy density rugby or we definitions of this a little bit of a ambiguity but little is leaf is dead or alive I have a policy of and other items value for bigger begin as of operate and the numbers for requesting that the new express that he lifted the club Suffield equation there but but terms of but that that drug limitation because bright for travel book for which you expressed in need of a rep get Karen directs the whole Gaza rocket normal normal but these are not staying these are operators they don't that brought kept the brass and kept these other kept of sister brother but it is you abandon these days to be a little elaboration of a scar tissue what brought Rob rack price and was replaced a state vector of system that's to state that state state with 1 man 1 particles in the 1st state to air war the yup let seperate single-particle quantum mechanics is multipart is multi particle quantum mechanics who things John operators which Dhoni than a car in the single-particle B Armed right now this formalism August technology does allow you to recover Everything that we've ever said about single-particle quarter Kerik's you just go to the states which the phone number of particles as 1 new work map set of state all of this formalism just reproduces the physics of the single-particle but allows you to deal with a number particles and that they kill us there are states there are states with large numbers of particles typically observables now operators when they're a larger number of particles saying state then these societies tend to behave very much like classical field will talk about more houses can make the classical field theory what connected the classical fury like any quantum mechanics is connected to what's classical or its classical counterpart for example expectation values of things follow the same kind of equations of motion as a classical equations of motion you could ask for example what happens to the expectation values Of side sighs operator has an expectation value at any time and we know the Hamiltonian of a system we can use it to work out the plane dependence of the state and then we can use that to work out the car in the path of the expectation value of signed guess what essentially satisfies the classical equations emotional way feels with the same classical her career as fact satisfies the throwing of time-dependent Schroedinger equation for sure that this is a very beautiful formalism is bookkeeping devices but also the fundamental connection between particles ways your and resort a connection is it that folks the that's a correction will explore In a great deal over of exploration OK after playoff game struggle with the closure of of crucial solution is slow the disease's drew most is not 1 area Iowa Irene never switch them I never commuted fried commuter them switch that water of the only numbers they are numbers there on Saturday these signs on numbers liaison offered vote so on they the West where monitors all birds slot sludges goes develop it they operate on in 1 tool and threw himself he vanishes at a site operated by not because it covers all out write down NATO unless he is the only relation you wouldn't want to do freely without thinking America is to move an 18 miners past than a class commute with each other that's
where the operator character of the coming but as far as moving him past ordinary numbers that's fine with derivatives year bomb act on them derivatives act on the sign is so are there no cause to worry about any of the operations I it would be a cost or I tried to move my past but change yet reduced what is following the I've religious ethnic FOR America from yard but that's only these derivatives are there on ordinary folks daily but see you grow over at but 1st thought it back that the both following a legal billion the role exodus form what we did was say but this operation here just multiplies J. by maker pointedly became may go then we could into rollbacks now so I don't cheat you know I think that uh rapper the the number is 8 judges walls geologists sees full here the Wednesday in the age of you operators must be this fall just like you cannot can help you confused they were nearly before I Bearak mastered the role of put together they was very confused about the meaning of the wave functions some but some people thought it was a field particularly in all real field like electromagnetic field some people thought was just an auxiliary device for computing probabilities and and that's all it was them what does it have to do with the electromagnetic field and watch the connect with several different the ideas about what the connections between particles and leads were on the 1 hand the single-particle wave functions satisfied a wave equation the Schroedinger equation look suspiciously like Maxwell's equations was roughly speaking quite well the actual equation bachelors equations 1 not equations as far as anybody's fool forum for quantum amplitudes they were equations for measurable values of the field but still there was some connections some funny connection between these fields and particles namely the field was made out of Quantico folk art so there was a lot of confusion and this is so the resolution about computer how about this other that he said that Torres the that days is of not less than Charles To said did know job which is fortunate because heavy use the 10 notations that what they already know I thought quantum mechanics last year for them in the mathematics department which a lot of fun but it was a crazy point at which explaining direct pictures and I told them that every vectors Chloe's recalled vector space or our society and official trial we kept the official mathematical terms for the Bravo's is the Doole vector space needs complex conduits Blvd I would say Dole happy and I said this is in the product between every vector includes dual person not our huge can prevent cannot have been in a prior between veterans do you only have an inner product between the veteran itself I somewhat on telling you visit just don't talk about them in a problem which we are not not only illegal alarm full of course it turned out but that they just have a patient in a product of 2 vectors take those sectors and you put next to each other this way and they put some funny symbol around the more they and that's the inner product between sigh fire was illegal till it's equal to 5 when they finally got the hang the mathematics professors often while this it is a powerful makes everything work simply for example where will again blown away by the formula that I is equal to 1 and all you have to do to multiply it by a state is just don't don't sign severe price never seen before that of course is so of course it's here they understood all completely but they had never seen this compact notation where a newly almost can make mistakes anything you do turns out to to be right now if so Barak was clever we'll study eat your if U.S. or alarms his mind ordinary everybody was not you probably ought to go request but dear so we saw things in different patterns of most of us Oh directors saying well stalwart of sort for just 4 normal could measure butter urges measuring properties collection yes but measuring resident part of properties that you might not have guessed now not measuring the properties of classical particles measuring properties of quantum prior particles and quantum particles have properties like interference a that the classical particles so yes you're only dealing with a system of particles of tall but because the quantum mechanical they have kinds of properties interference Pangle about things which ordinary particles repair classical particles and this account for our hope is that they as is columns and electrical field yet daiwa what has to be a little bit careful when you're dealing with relativistic particles careful marked the order or particles or a number dealing with relativistic particles the notion of the momentum of a particles
very well-defined notion of a position of particles not with the motion of the position of a particle was not defined then notions of the density of particles what is the fine is the density of energy and the density of energy has a similar are eligible for marked prior fair free don't put too many errors there may be a square of electric field is qualitatively related to be interviewed for sure How relativity there but the acquisition is the opposition's Margaret concert wrote the sticker system the idea could even container maker if you do if you defined this is not special useful but a fact if you define a vector a slice of electromagnetic field is defined as electric field plus I times a magnetic field quantity sigh star number mail this is 3 component of sought quite the same users 3 components electric field Miners icons of Mary magnetic field and sigh star sign is not quite the density of that's the that's the energy so what's the density of photons waited with the energy of each for that's a good concept for part of the density of photons of self was Margaret so I start time each square Class B squared if you nor your remember originally square post be squared his video museum and you get to the end sigh satisfies radioed for Schroeder like equation plant work workout from Maxwell's equations what so I was like that if you know Maxwell's equations you could work out equations for signing society and it's simple equation support with Maxwell's equations themselves OK but on energy equals the union role of Chapel fire that 5 you're not just being that that was that the differential operator where started slide that Will you not the Pearson who each other without taking care of the so it has a meaning but just above his work as it was in a corresponds to an extra pair of can't photographs they do you right let's see if we can make a guess a marker have approved this is easier than we here and you could try it on Kirk as for the single-particle quantum mechanics momentum he is able to minors Hi Dubai DX plant curves what's the expectation values supposing I have a single-particle wave functions Siurek so what average expert or the expectation value of the moment wild you take he'd take their kids that side you take the momentum operator and you sandwich usual sandwich to create the expectation value what that means is you did radio ext VX Tri-Star times my are these cyborg EX so I took my me outside Tri-Star firms we strive IBX a new integrated Robles space What about the momentum operator nothing expectation value of the momentum but the momentum operator for this field theory to feel some momentum feel some of them to take electromagnetic Dean from lasers shine on wall if laser beams powerful enough to blow a hole in the wall white because it carries our momentum exerts a force on the wall he'll have momentum classical field quantum fields of the quantum case the field momentum is just the moment the move or booked on cellular have a expression of average moment of a particle corn can you guess what expression for the colorful momentum of Wall particles as no work just replace the size but size of bigger signs a sly before have to if this is the operator who represents the momentum if you measure the momentum how you measure lament the museum hold you will in the wall by you measure the moment not the average of the moment the actual momentum in a wave going by your Harry measured you see how hard it to Humbert forces exert from bring this is the operator will be observable corresponds Croat is mathematical permission operators corresponds observable you could also calculate it expectation value this is an observable it is not the expectations are you if you give me a state I can calculate the expected this is something of this summer will go try with these operators they have expectations they are not the expectations you state your operator and state to construct an expectation value as a stake in this state wealthy that those in states up there in 1 and 2 and 3 and road calculate the expectation value more than OK on a day before the ship like the from rehearsal requesters because I have not remember things to talk about the actual breaking point next time we'll talk about passes to use and in particle physics and how they represent by these feels that have very simple car reversible class of about field operators implement processors used place where particles the Katie Wood particle scattering formal
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