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Advanced Quantum Mechanics | Lecture 6

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let's go way we ain't going today into the world of quantum field theory quantum field theory is of course the description of nature as we know know it will be exceptional gravity and sold practically everything that we know with the exception of gravity in principle we think is explainable if only we had computational power to be able to solve quantum field theories with accuracy that would be appropriate that problem is of course that problems tend to be much too complicated to solve except for some very special problems you can I understand the hydrogen atom but by the time you get a the boron Adam it too complicated and so forth and studying human beings as well beyond the capacity of any quantum field theories that I know end but nevertheless apart from gravity quantum field theory seems to be all areas and we're going to enter into the study of the very very very very elementary way who studied idea of what is called 2nd quantization before we do in 2nd quantization is about quantum fields but will come to where I want to review something 1st hand army may be generalized that's something a little bit the thing that I want to study or to remind you of all go through with you is again the harmonic oscillator but no aspect of the harmonic oscillator cup of nor aspect of the harmonic oscillator just so that we have in front of us because the harmonic oscillator is essential ingredient the central mathematical structure that goes in the quantum field theory so we need time the standard make sure we understand it well but when I worry about bringing all systems oscillating when I speak about the harmonic oscillator later I'm really speaking about algebra of those little operators a plus and minus and the product of a plus turns a marinas which is what remember it was a wall it's the number operators the number of the expectation Charlotte Mighty Mary overthrew victim after you work to facts that may not have told you about whether a particularly what's just right down rules for a single Hamas from our harmonic oscillator 8 classes they act on state to elevate them of the next level there a minuses which they give them at a bottle on level ground state a this gives you nothing just violates the state so a minus a lowering operator and a plus is raising operator will the call them creation operators and annihilation operators what is it that they create an entirely as will see particles Our but let's just the user same note that the same conventions it would use of American and there was also a plus kinds any miners that was the thing if you remember that went into the Hamiltonian feared the Hamiltonian was just basically a pluses and minuses was plus it is X plus vii Omega he divided by the square root of 2 Omega a minus the same sort of thing with a minus sign in there and you multiply them together you get things with X Quayle our X squared plus black the right people there are wrong Pete X if you multiply the 2 of them together you get peace squared for filmmakers Quebec squared plus the sexual the terms of the ground-state energy our and so that was that they that winter the Hamiltonian a plus times in my guess we called that the number operator who was the thing was I didn't values whose observable value would be integer level spacing of the integer levels of the harmonic oscillator so I believe we call this capital in the fair that I did values we called little in the observable values of it 1 more often the commutation relations of a plus a main mine that's what made a plus and minus Our permission conjugates of each other that's important for mathematical fact but the other mathematical fact is that the commutator minus with a plus is equal 1 battled made everything work battle used to define the properties of what happens when we act with a plus and minus we use those commutation relations that was basically it that's all that went into the harmonic oscillator now promoters I have many harmonic oscillators you could just think of them as a collection of Springs Khartoum springs all you could think of them as these various oscillations of a violin strained or what ever for whatever reason there happened to be many independent systems that are undergoing oscillation can undergo oscillations possibly with different frequencies possibly with different frequencies wears a frequency coming toward us the frequency will comes in here but by the time we get to that point over there of the place where are we on the frequency came was just the fact that energy is people H. barred turns the frequency times end here it is also plus a have on only about half the ground state it's not important for us so that's where the frequency came in Keynesian just as telling us what energy each increased the number operator the frequency of together with 8 are determined each time we go I can't level how much energy costs jumped oscillator up 1 level or how much energy we might get back in some other form of from the oscillator jumps that jumped up for some reason that much energy we have to pay for Omega gems down from merger you we get endeavor has said that a single harmonic oscillator but no we have many former cast orders we label them we could label them with an index values the index i.e. but I does not stand for the square root of reminds warned Mr. tho label which goes all the possible oscillators that I'm interested in the maybe a finite number maybe this the number In the examples I would be interested in there are infinite number now we get the label of the oscillators just by putting in well OK we have to be will care for Whittier but each oscillator think of it as an independent system as a system which is just an independent the degree of freedom you have independent degrees of freedom operators 1 subsystem you with the operators for the others for any other subsystem wrong we went through it her 2 years ago or you're going over was the operators describing system AD and the operators describing system be if any and be a truly independent systems they commute with each other wine because not commute would tell you that you cannot measure them independently if you can measure them independently they're not independent system presumably if you had to springs you could take 1 halfway over there a long off over there is no reason why they should influence each other there operators everything about them commutes and silicon right the following that lets tried out all the commutation relations because their important To get this for the moment let's
1st take me commutation relations of EU Loring operators plywood Jake the 1st oscillator with 14th oscillator well 1st of all I is not equal to J. talking about different oscillators talking about different oscillators they must commute but we're talking about the same sort then the commutator should be exactly the same as it was for the single our case namely 1 so only right that we write that but putting the climate of Delta over here all that means is also not 0 sorry 0 annihilation operators or commute with each other even for a single oscillator annihilation operator commutes with itself because everything commutes with itself seems they for the creation operators on raising operators now here's what I want to say amounted to grow if I take the annihilation operators and take its kind UK with the creation operator that if I is not J. we're talking about 2 different oscillators don't know about each other they don't talk to to each other our we stay we talk to each other but they are independent of each other and the country must be 0 on the other hand it's the same answer there were talking about then the commutator must be 1 so this is equal to dealt J. This is the algebra of the operators of mini remark oscillators that and 1 was saying What's the Togo energy total energy is just some of the energy of what we are orders so let's try that total energy which we can take to be the Hamiltonian is some or all IT all oscillators 1st onslaught the 2nd mouse we're all on nigger I'll include pH bought the moment probably forget about it before it was over on May double with major each oscillator may have a different frequency sold each oscillator the contribution on ice it is proportional to the frequency of the oscillator times and really mean to now we'll call In by ended with a call an occupation numbers now occupation like what's your occupation by occupation review are sitting near occupying some space and these are the occupation numbers just mean for the moment what is it doing the occupying will find out but this shows each oscillator has an energy proportional to its occupation number times Omega terms frequency will be are being there is extra half from ground state energy it's it's of no interest plays no role it's just an a bit of constant additive constants in energy or make any difference OK that's the whole theory if you like well not quite not quite not only label states that we have many oscillators once every have 1 oscillator label a complete basis of states just by go level for the the occupation numbers of the oscillator member will only states and equals 0 which is the ground state which we sometimes just called over and equals 1 which we call 1 and equals tool and so forth and so on we label the states by the occupation rats state where we have many oscillators and now imagine that I lay them out there are many oscillators use the first one over here said born here 1 Olivier 4th here I equals 1 was too soulful then in order to specify state I have specified the occupation number or go 1 way of specifying complete basis of states 1 way of specifying basis of states that label was right down the occupation number of each afterward How many quanta harmony of units of energy on service in the 1st hour or so later an article 1st isolated and not and warned In 1 end tool and free as a finite number of oscillators it terminates an infinite number doesn't only situations where are least mathematically and the number of oscillators what's at well if it had to be out on an example would be and I'm very idealized mathematical violin strings a violin string any number of harmonics everyone cannot solely with different frequency that mathematically idealized of course see real real violin strings made Adams and there are there really that many possibilities of but violin strength would be mathematically ideal idealized violence strain would have it infinite number of oscillators but of course very very high in the very high I would correspond to extremely rapid frequencies and really our be costs d of the atomic structure of the year of Ireland's playing it really doesn't have that many oscillators but that's an example mathematically of infinitely many are gonna be interested in that case OK let's talk about knowledge Komodo the blackboard over here With a little more a little more oscillation Mosher later of mathematics I told you Europe originally that if your that would creation operated a quality come recalled could slip in the calling the creation operator With the creation operated you increased by 1 year the well that's not quite right not if we normalize the states in the usual way the usual way to normalize a family of states there are thought to each other the different while father to each other because they have a different observably different number of In Seoul states with different values of the slide in value or orthogonal so the usual normalization would be the inner product In with end his daughter NATO or another world's each state and physical the 1 end and musical 2 0 and not equal to end their orthogonal equal pay and we normally would take them to be normalized state as standard quantum mechanical arm outpatient procedure with notation Millard each 1 of these states is of unit Lane each 1 of these states His of unit length but with notation a plus some in the news not give Air Force One gives a numerical coefficient from and numerical coefficient is the square root of and plus 1 but it proved that for you but see rigor Probert how about renewed of him OK let's not put the incident let's just call it some numerical coefficient and we don't know what it is yet we don't know what is yet it's summer sea and tried a figure out what aren't tool is the commutation relations for the offers a single oscillators multiple oscillators OK let's try this is this
is a cat the equation let's like the corresponding Bravo equation when you right when you go from I can't sector equation toe Abra vector equation what you do the operators to cats go abroad what about the operators mission country permission conjured permission country it at mission conjugate Plus is a minus so what another write the same fact is banishing conjugate on the bronze at the and is equal to CNN assuming real weakened below the complex case it doesn't matter we can OK the think of C and Israel I don't it doesn't it doesn't matter but what's Let it Be real if wasn't real then we would want the complex conjugate but we can use it to be real and plus 1 that's the exact same equation and accept kept got bronze operators go to their mission conjugate and numbers go to the complex conjugate taking CUB real OK now is really this equation next to baby in a product of this equation with pleasure was not among the top 1 was right about In a minus 8 Class as happens if you take me in a product of the left side view of the right side and that's going to equal C and squalor and privacy In a product of this vector with this that the boss in apart from this what 1 every 1 of these sectors is normalized the 1 which is 1 so now I have told the figure out what's CN as I always have to do is juggling a pluses and minuses isn't a clever way wanna where we do I think Irish the equation here should be our lips what's right down the guiding equation over here for over a year and funny instead is belittle him and funny and statewide definition of being a item nectar busy eigenvectors begin begin taxonomy Eigen state will I in value in to give land that's a definition of that yes or a lot 1 because I used the rule not know now assume the normalized I'm assuming now would be in of normalized and having said when eyes effect put the ambiguity in heavy seas I made sure that the normalized and then seek altered dust OK go 5 so maybe end is nothing but a plus a minus so let's put that here is a plus a minus that's almost what I have here a plus I don't have a plus a minus I have in mind is a plot but what I know is what a passing what I do will set operations commute commute and here is our commutator becoming caterers 1 so what that says is this same thing is equal to an end a plus minus plus 1 that's a curry favor Turner name OK now make good shape because I know what a passing might stars were back and it just multiplies the land so all of this just gives me In plus warm fuzzy In a product of with itself which is what 1 so we can CN square Quebec and blouse warm and therefore see and is the square root of canned plus 1 we now know what the creation operated doors In orchids glory here at increases the occupation number by warned and in multiplied by the number square root of plus 1 please a river restless ammoniated anymore next was about 80 disarm in well you go through essentially the same will give you could do it yourself it doesn't take a rather hard this time of year have villainy commutator lined up in the right order and would you find out if this is equal to the square root of Internet kind and minus 1 another words it takes you down level of multiplied by the square root and not just 1 of the architects that's what you find wants to know this formula here you don't have to worry about the fact what happens when a it's the bottom level when a team minus hits the bottom level the bottom level is an equal 0 you get 0 so this equation as it stands now colds the fact that these lowering operator annihilate kills or give 0 when it acts on the ground state just square squared event here so this is more oscillator stuff that I assume that Barak was was responsible for now we know I think everything we need to know about former at least for for the moment and that was as the mathematical interlude if you like go to so that we have from us we need it put up on top of questions about harmonic oscillators or multiple harm our across Florida's bowl 1 thing what happens here is a particular vector for a particular state which has a certain set of occupation numbers let's here with the creation operator for the RIT oscillate Florida I thought so and applying 1 unit of creation it warned you about celebration what does it do well it goes all but I thought if this were 3 if I was really go over to the slot and it increases the occupation number by 1 unit but only for the particular on that we wrote them so we got to the 7th isolated only apply the creation operator it increases the 7th oscillate level it does not increase the 6th 1 doesn't increase the 5th 1 was an oscillator over there it's over there that's not to be confused with anybody else and salt what this does is it leaves all the other ones alone but ideas are here somewhere and 1 in 2 Dr. until we get up to to me I'd and then it gives us an eye plus 1 than Beyond that nothing but that is not quite right this is not quite right because it fails to take into account the square root in postwar solo choreographed actually have the right when writing and writing a equals in 1 of about 5 . in I plus 1 . 5 but we have to put
it in the square root of the my plus 1 Sono is works exactly the same as as a single oscillator with all the other ones just been bystanders which played no role when you hit it would be Iike oscillator if you hit it with the GAO's oscillator another some other occupation number would give increased likewise for the lowering operators of the annihilation operators seemed kind of understanding sources but that's what we need to know about oscillators know come the quantum fields real genuine quantum fields but a very simple context nonrelativistic quantum mechanics would require mechanics not relativity yet for not doing special relativity just ordinary quantum mechanics who under a quantum mechanics theirs some kind of funny connection between particles and fields particles of described by away from the functions of functions of position they sort of look like fields well feels feels things degrees of freedom which depend on position and on time in particular but let's just say position for the moment what freeze hiring field are functions of position the wave function of a particle is a function of position so you might think that's all there at the connection between particles and fields but it's not it's not for me the story least it's only a very kind part of it and fact not real idea sigh of X trial X is the wave function of 1 particle where X is the position of particle X could stand for ex-wife z of course it was necessary stand for just text but sigh is a function of X but what is it it's not observable quantities it's the same there it's the state vector in the position representation it's not observable you don't Bill experiments to measure the wave function of particle you do experiments to measure the position of apart for the momentum of a particle of the angular momentum of a particle but not the state of particle you don't measure that's not a measurable thing that our thing which tells you a great deal about the probabilities of various measurements but it itself is not an observable so that's the 1st thing not an observable but next factor if we have several particles that sigh is not a function of 1 position as a function of all position I want to arm exaggerated no-show notation a little bit I want to make this all parties Oh Lord case side and it's a wave function if I have to particles scholar XMY again not be x coordinate them white court space but just a quarter of 2 particles the wave function is a function of 2 positions are functions our several positions affected the function of the of the position of although particles a 3rd factor was a 3rd fact a 3rd factor is given away functions which may depend on X 1 throws some number of wherever x 15 15 particles it's a description of a fixed number of particles we describe 15 particles would the 15 particle wave function fixed number of particles that is not the idea quantum field by contrast now by contrast work quantum field is in many respects it everything that this is not although we can use the same location for it and closely related 1st of all quantum field this is awaited a wave functions and they represent state vectors capital sorry is different number 1 it's an observable it's a thing you can measure let's just say from moment was a quantum field really going to be in there when we get to the real thing real Quiet more complicated there serious quantum field theory at things like electromagnetic field that's gonna be the quantum mechanical electromagnetic field electromagnetic field is most certainly an observable but With job and so press for whatever side is it is an observer No. 2 0 is a function of all only 1 chord always Bentley the fact that observable means that an operator means of operator and the spacer states it's not suffer state it is operator that must be if it's gotta be observable number till it's a function of only 1 quarter net of 1 position functional want free vector sorrow speak finally it describes systems of any number of particles and any number of particles no words it's capable of describing systems where particle numbers can change even with particle numbers could change happen particle numbers well we thinking about electromagnetic field and we remember electromagnetic fields fault times every time electromagnetic fields radiated it changes the number of photons sold whatever site is it's a description observables operator at a functional 1 position and it represents the quantum mechanics of a number particles not a fixed number of particles have some award why don't go too fast because this is a tricky stuff what is this that would at 2nd vary but defunct and a bomb we want extra board room the only 1 x but that doesn't mean it's a function of X would exercise that means it's a function of X X be anything but is not a function of X 1 X to an expert what are you gonna why the may White yet took a functional born quarter 1 coordinate In real world means three-quarters excellence that wanted the particles 1 is named whose name White Chris Who's on 1st that's what he said the 1st is that at quarter side when I get in my life not that a much what effect of faxes X 1 and 1 label 2 different articles Burke has of what it now that but ahold that's the whole point by late outfield big contracts and now we want to know what's a connection between them and could take a lot of now say could seemed kind of particles and in particular both are will kind the fermions later From the from later yes this is the theory of HBO's arms everything we're doing tonight will be both but that was what can with the is the all is
that exact effort by now there by might yet that would be I could be labeling states of North China and now you have a cattle farm that electrons a interact more complicated procedures just had not interacting electrons and that I would be labeling the state the OK but we're Comparex doubts Kumba right now but go back the single-particle quantum mechanics misses the period of many particles period no particles go with theory 1 particle theory to particles variable number particles but go back to the fury of just 1 particle and simplicity I'm going to take a particular example of particle in a box of potential energy is such that the particle kid get out of a box just 1 direction of space Road although nothing I'm saying is depend on that some bows on at any moment at for the moment doesn't matter it is only 1 of em it does matter it as 1 of them is no question of whether it is an exclusion principle not are so 1 man doesn't matter let's think about me energy Eigen states of particles in a box their wave functions for example fullest energy wave functions will be the smoothest function the satisfies assuring her equation and don't typically look something like that it'll be assigned function that just barely fits in the box that fits in the box from 1 end to the other end battling function a wave function in this sense Of this 1 already let's give it be less call
signed 1 of X this should be a little the Lakeshire little sorry it's just the way function 1 particle system in a box so I 1 of what they want to represent just that it's the 1st energy level the lowest energy level what about the next energy level the next energy level is another wave function and as you will open energy for the wave functions tend to be more variable the next 1 would have been no that ring quartzite of acts amusing funny notation the index here is not the number of no that's 1 more than the number of nodes number of nodes in the box so the next 1 will be cycle of X would have a higher energy and so forth and so on 1 of the pillars he do not represent Rivera maligned sigh buying of X gravesite is will be the same as the sire eventually but at the moment that 2 different thing in this case the labeling oscillators in this case it's labeling state vectors of 1 particle in a box is 1 particle each 1 has an energy bar right now or energy is at the center remind you that each feature of these wave functions her energy How area at all right with that 1 particle wave functions that let's imagine now that we have many particles where many particles of an undetermined number but some number of them with in-state warned some number of particles in the box occupying state won some number of particles occupying state to some number particles occupying state 3 and that is a pretty good description of a white class or a basis of states in fact if I know that is a basis of states saying how many particles occupied each energy level so attractive and his a family of states No. particles In state 1 number particles state to number particles in the state's rainy homeboy disheveled where is our earlier absolutely identical to the way I labeled the states of former multiple harmonic oscillators now of course the young there are infinitely many possible states in here arbitrarily wavelength arbitrarily small wavelength and that's the reason why we need to to study any number of oscillators up to infinity so here is apparel between you wouldn't wanna do this you couldn't do this affirming affirming aren't you canina have no particle in there with a particular state or 1 particle never to take temperature songwriting and the idea that you can have any number of particles in the 1st state any number and the 2nd state when the 3rd state resuming for the begin with that we're talking about on the notices complete parallel here because is a complete parallel in means we can in vain sisters and events and that of the Hague operators which do a exactly the same thing as the creation and annihilation operators to now they literally create particles for example a plus 1 on in 1 that is a square root of plus 1 and 1 plus 1 in 1 plus 1 and 2 and 3 so for increases the number of particles in the 1st state where I was that everyone and multiplied by the standard square root White multiplied square root better definition but the definition which is very cleverly chosen among other things so we can use all the apparatus that we have about oscillators which is very helpful rules likewise would be annihilation operators now creation and annihilation operators they create and annihilate remove particles from the system OK that's the basic idea of creation and annihilation operators yet we thought 1 In 1 in the within 3 of the number of particles in that state 1 2 1 3 are 1 2 and 3 of starlight lament past so this is tricky Luanda patient with the notation is it's not hard to do a bit to remember our so going slow for that reason the repeating myself physical meet this inquiry Square Ste achieved at the top but
8 plus a miners on in is end kind James right White tosses off what I gave my best so better be that plus in their this was somehow square roots event but we can do better than that let's see how we are you but a connotation relations don't it but suppose we know that's but oppose written know this but we did not about the square a today they did not know about square tape but 1st let's use are commutation relations the write this they a mine they are in a minus 1 I think yes they minus a pluses equal plus a minus I just want plus for right and 1 minus life the verdict a plus 8 9 is a plus minus a passing minus is equal to 1 of course now you would what I was what's what but right who does work that will change it today now was a plus Armando according to assumption it gives you square root a plus 1 times and plus 1 that's only a voice laughter multiply by a mine and then we have a number of miners but have him right OK this is just the number square event war which he squared course 1 what does any money list when it acts on square plus 1 plus 1 no think yards gives us in the Watson America Lumumba square and perform right so we get to square would plus 1 which is what and plus 1 firms and minus 1 times and which is just good In the in another words those square at the end of it To ensure that this vectors really and eigenvectors of a plus and minus an roughly speaking barely roughly a plus and a minus are of the same order of magnitude wonders at P Plus axioms P minus IX saying motor magnitude and since altogether they end up other multiply together to give you in each 1 is roughly the square root of a job look at plus must get 5 5 . young so we don't get it yet others that make a lot of it with define defined by making randomly but with findings and they want see how they work at earlier flight plus read that they had saying I did
understand what does it's clear had said and that's nothing but fault or maybe up from other letters of reprimand for the conference slated for closure what that will last played of Barco said but where is it what is what would you do if you that the definition of you can ask why about definition right finally emerges from where it decreases the number part of mathematics we Richter Ferguson's may refer consequences of so the question is why youthful that position well useful definition of floor systems of various numbers of particles in that we want to explore a systems with a number of particles could change for example with my radiator photon how would we go about mathematically indicating that we created a photon we take states we hit it with a Korea photon creation operator or photon might be absorbed a Adam might absorb on Adderley right the fact the highway express the fact that a full time is the photon has disappeared with expressed by saying annihilation operators acted so the purpose of all of this is not only to be able to study systems of arbitrary number of particles but even to be able to study systems of variable number particles and so here's what to do about how they had no desire to for basis so that other at all right OK now that they must we catch short book 1 of the 1 show that you had not Asia relationship that would these definitions it will follow that these operators have seen commutation right but other words this system of a variable number particles here can be represented just as a system of harmonic oscillators and oscillator the each state each single-particle states treat single-particle wave an oscillator and occupation number of about oscillator is the number of particles are occupying particular state server terminology is well chosen occupation now we know what it means is that you want to find a seat it's a particle occupying the state on a particular these creates a violation operators for particles how you label with the vacuum in this language the director 1st of all it's the state which is annihilated by all of the annihilation operate it's a state which renewal try the Lord any occupation number is nothing nothing for short of a state becomes 0 but a vacuum is simply a state with 0 occupation number in every state it has a property that all the occupation numbers 0 and has the property that any 1 of the minuses due 0 quarterback the ground-state state is Red Derby was all but over your occupy or they don't react with is not deprive can occupied the makes a dresses that I'm very sorry work let's while we're at it let's write an expression which we can do would be energy of the system not the right expression for the energy of the system we need to know energy of each 1 of these wave functions each wave function has associated with an energy ideas about what we call the energies we could call them he could only under a clonal may go for it to borrow maybe but let's just call the energy level of the I but tillage bar forget each barred because of a ready for that and just say energy of a particle in the 1st state would be 1 the energy of a particle in the 2nd state with the Omega too and so forth the energy of a particle in the ice state would be only got 5 less strident expressions in terms of creation of annihilation operators for the energy of the whole system Ms. is easy the energy of the whole system is 1st of all the energy are also particles In the 1st state the number of particles in the 1st state in 1 times the energy that's associated with particle her 1st day plus the number of particles in the 2nd state times Jamaica pluses and so forth and so so on sources some of and some by Omega survive that's a true statement but I want to write as an operator state these Enzo eigenvalue but their ideal values of the number operator so can really write this operate formed by saying it's the son of all states single-particle states of or maybe Guy Heinze the occupation number of the state was the occupation of state a dagger I notice that each 1 of these is just basically the oscillator formula is just a oscillate a formula that would be an age foreign here but if you like this is just a oscillate a formula but now it's some all the possible our states here and you configure the 2 ways you could think of it somehow this system of particles is equivalent oscillators and those just a oscillator energies or you could say it's just some of the energy of the particles number particles in the 1st mode 1st state transient you 1st plus a 2nd state from the energy the 2nd fell for the comes out of the same thing now what's more time minors Our sorry that were
it is also are not but the many context on Sharif remaining context once you go to quantum field theory which is what we're doing now a plus just becomes a dagger a minus just becomes well I'll try to stick with the same notation I want change notations now it is OK let's just the thing for where we're doing we're inventing some kind of object which is a heart which is a bunch of oscillators on the other hand the Violin Strings Barger ocellated the violin strings a field if you look at radiation in a cavity radiation bouncing between mirrors it's a bunch of oscillators people speak and this goes back from long before quantum mechanics speak about radiation as a collection of oscillators radiation oscillators long before quantum mechanics and so were beginning the build From thinking about what particles were beginning to build up an idea of collections of particles has harmonic oscillators and soon enough we see the connection with fields OK if there are no more questions middle of pick a five-minute break because so I stop for a few minutes and organizing thoughts both field a few questions before we go on at the so it Is West so that is that I will target a with 1 of the very elegant a writer but does herbal I guess what they're happy at half or how but they but Hi how you I am OK near a learned for Cape North freeware wanna be remember where where where they left off said at Wheat do with new per miner north of river but so what Europe 1st summit but much of the year before but was a beer But ever remember Her uh uh but predicted that call if get that at it that I have the remove the ignorant of what the problem the Fed the Robles if word I didn't see that a man who for Sept here today not that has nothing to do with no we had G levels it's nothing energy levels Of the single particles in the boxes which are which are evenly spaced It's state energy levels of no particle 1 particles who particle we particle Vercher evenly spaced reach oscillator back yet now at no no we're not talking about particles which are moving in a potential which is a mouse so it could you could think of a lot of articles in the potential of an oscillator the you 2 kinds of oscillators the articles of the oscillators the creation of highways and operators of possibly but now we're just talking about any collection of single-particle state single-particle states is just what you would have thought about where you just thinking about 1 particle but now since we can talk about any number particles we can we can general riders this way we had bit of this you all of that all of us Africa at and have actually told us us OK let's let's get back by some questions because I don't want to go to bat I know that there's a wild season little pieces in the nuts and bolts which if we go fast those slip out of your head this about might get from time to time yet at the back her for sure there's because the courtroom the eighties the aid of can't of Fourier coefficients lookup so deeply related with Fourier Fourier analysis absolutely still little but was still it's a managing the theory 1st state they have energy 1 but all particles which are the 1st state at the same level we are all particles in the 2nd have of the same energy an opposite who 1 of the 1st soap not played well look at the moment we're talking about articles at the moment with talking about articles which simply don't interact with each other at the start of the of later we have to ask what kind of
thing is that we have to to do to get particles Barak Georgia was saying particles really interacted with each other the energies would really be a edited that would just be the energy of particle 1 Aussie image a particle to be the energy particle 1 energy particle any kind of potential energy between them for example so right that's correct for the moment we're talking about a free particles free particles means particles which don't interact with each other are OK now we come no good nor the questions yet that is why you at all yes but doesn't play because the winning it's just 0 . 0 energy drink the wine as a 0 . 71 is irrelevant for for everything but I think what this energy is Hamil what happens if you a constant number for Hamiltonian well and No. commutes with everything and remember what the Hamiltonian does for you is it provides a method of getting equations emotion the time dependence of something is related to the commutator of that thing with a Hamiltonian constants nothing they commute with everything way of saying it is in keeping track of energies your thing they really energy difference is not the the sole control way that Gramm state energy except for certain very special purposes that a their concern us right now OK quantum field the field as a function of space it's not a function of 2 . space of three-pointers base so it's not like a wave functions of a many particles it's a thing which only depends on 1 quart also operate a white because it's a notable you observe fuels imaginary it was a 3rd point about now I remember birth yard somehow have summer has to do with your ability the change the number of particles OK so I'm going to do is I'm going to give you the rule of the quantum field our this system of particles and they would explore how works U.S. prematurely that the definition will it won't work it won't work the best thing to do sometimes it's just best to follow the definition and while I promise you we won't look to far before you start to see why it works of interesting in warrior why that's a definition but it's better not to try to work to out from the beginning white this is particular definition ravigote used to it and familiar with that and in a short amount of time you'll understand why that's the thing to do the definitions the value of the definitions depends on the utility what you could do with them and I might say this To begin with quantum field theory at least in the way with thinking about here among other things as a bookkeeping device it's a bookkeeping device for keeping track of many particles for keeping track of the quantum mechanics of particles which Condon goat cheese their state even disappear and reappear whatever it's a kind of bookkeeping device went operators like this you don't bookkeeping still more than that because you could actually measures these fields that things go by you you feel they have an effect on you but for the moment we just think of it as bookkeeping and whether a definition is a useful definition or not that depends on what you try to do with neck tell with quantum feels it and locked yacht is on well we go way way back we can start with the question of who invented feel when they could feel the wind the field theory I suppose Saturday with electric and magnetic field so I think probably the idea could best be approved the original history started with traffic with Farand Maxwell invented field equations parity didn't have field equations to just have a feel for the fields feel for the fields there there was able to visualize them Maxwell wrote down as far as I know but it's probably not true the first one to write down a wave equations wave equations ways of water waves of other things but I think it can fairly be said Maxwell wrote that the 1st modern versions of field theory those classical field theory on me neat quantized field of the need to do something involving each goes back the plot plant he goes back before a plant our we have not talked we will eventually talk about called the ultraviolet catastrophe won't talk about it now but a plant realized that classical field theory would not be adequate for studying thermodynamics of radiation introduced plants constant end In the process of basically said some things about harmonic oscillators which were more or less correct he didn't realize that the radiation field itself was made out of oscillators or photons as weaker quanta of the field he was thinking about be oscillators in the walls of the cavity mainly the Adams which he thought of the envisioned as literally oscillators will oscillating charges his hand his application of plants constant than the others didn't have mathematics but the idea that the energy levels with integer he wrongly thought that had to do with the year with a oscillators in the walls of the cavity was Einstein who 1st understood that that these quantization of oscillations I I had to do with electromagnetic field Maxwell's electromagnetic fields so as with almost everything else was Einstein who worked at the 1st inkling of theirs but them later those Isenberg Pauly Dirac was 3rd was media person who really put together form were talking about our and that was basically around 1929 in 1929 a lot of these ideas 1st really surface but not as usual Einstein had really did have all the ideas and although he hadn't articulated them and in exactly the modern form after that the whole history but Africa might like his notion of particles all a variable number of particles as harmonic oscillators that's a good question hole was the 1st I'm guessing Dirac again but I'm not sure that he was the first
one to identify particle creation and annihilation as a form of a harmonic oscillator images was direct public around 1930 the 1st major problem was the problem of the ultraviolet catastrophe of or what basically wire radiation was quantized right saying no radiation was quantized he knew the radiation came and photons but that a mathematical way to think about our 1st major problems those woes a formal mathematical quantum mechanics was consistent with the rest of quantum mechanics that included the idea that radiation King in photons radiation came so folly have never even talked about a photon we're getting close we're getting close of photon has something to do with electromagnetic field and it's a unit of energy is like his occupation numbers saying things which occupy the occupation numbers quanta feed particles with talking about are the quantity of electromagnetic field the calls for times so that's what we're going we're really getting close the talking about were 4 were quantum field is in fact right now will introduce the notion of a quantum field the simplest version of it give it to you as a definition the shortest see does how works OK so sigh is a quantum field again because it's a quantum field it's an operate that's something you could measure From capital sorry it's an operated but operator which depends on position so as not single operators an operated each position for each position was separate operator that act cyberspace states place of state was a space of state space states where is a stint well states and 1 and 2 0 incidentally this is called FoxBASE OK for Park was a Russian physicist but I doubt they're much that he was the first one to use this doctors he wasn't Bogart named after him for the usual reasons that P. get things get named after people named no reason OK the quantum field sigh of X is just a very short definition equal to the sun all all of the single-particle wave fortunes some wiII contribution for each 1 of these states to summer it contains start II of X the single-particle wave functions things like signs co-star signs Inco signs of different of different wave number of different Waverly as goes then and what multiplies the coefficients coefficients are the creation of annihilation operators in this case the annihilation operators I'm now somebody asked me if any of this has to do with Fourier analysis is the sign these a moment let alone what the the size of the Mint the state users signs and in these operate as any money would look would resemble very closely the coefficients that occur in Fourier analysis that is they are the quantum version of the Fourier coefficients when you expand the thing in science because there might not be signed because signs whatever we do you also normal family of state of single-particle states so that's 8 that's the definition of a quantum field Mr. Bush the simplest example Guo said that the site it these numbers are attacked as a separate No. 3 position sigh of X are just classical just functions so if I want to know what the quantum field is an x equals 0 I plug in Mexico 0 here if I want to know what is it X equals a square the Pieper the square applied here so it's a the whole set of functions as a whole set of operators 1 for each positions based does operators 1 for each position positions space Weisel so rich why so many operators the reason is because we're trying to study a system where they could be any number of particles that's a lot of state a lot of states a lot of operators just because the number of particles as totally unbounded than this Her idea showcases an operator at each position this X in the sex of a same the size of different misses an operator this is just a function of function a position the operator is in here that's where the operators areas the complex conjugate order her mission conjured is an operator it's not a year a number that has an emission conjugate don't exactly the same thing except that would put creation operators there and so I started here these quantum fields are not permission and not her mission because the creation and annihilation operators armor furniture and other not really observables but if you enter society size stars side that daggers all you subtract them you can make permission combinations out of them and those who really are things that you could observe laboratory are just because of their mission operators with exploding we will explore these objects Lagos alone but the 1st thing I wanna show you is what the actions of sign is sigh of X capital of acts when it hits the vacuum what does it do when you apply it the vacuum of Let's take side that were excited that incidentally was sigh do when they get back in it kills it violates why because it only got annihilation operators dagger however has creation operators it creates a particle each can he creates a particle doesn't create a lot articles it creates a superposition of particles born 1 particle the superposition estates so what figure out what it does win it some backyard of case Sony is over and over let's take
single-particle quantum mechanics 1 particle corner mechanics to begin with another words forget all the fancy stuff or oscillators just the good old quantum mechanics every single particle there's a complete set of state the complete set of states other positions states sued her or write OK states which are located where the particles located at position next year a complete set of states that means complete basis states and that means that you could use some or all positions or integrate of all positions I'll use a storekeeper simple homages summing over positions but we may may have do integral if we somehow of positions were get on the right hand side won the unit operator at a general Rolph said any basis of states OK now the CIA just Jerry members which where did this hot that a true statement but it was actually wanted was in different states that is a Tuesday but what actually want wires Ow this some more white as usual but this time of nite I started making good mistakes and was just about to make a mistake but I caught myself these eyes remember the 1 particle states the ones which required sigh 1 side to side 3 sigh forms by himself they are a complete ban the energy Icahn's states they are a complete set of states I take the signs 2nd fit year for example sigh of exercise of 2 3 or 4 x that's a complete set of states each 1 of these I see is a state whose wave function is sigh I emphasize a complete set of states and so it follows that this expression here is just to you this is true for any complete basis states the back your quantum mechanics you remember what this means now that means there's a little bit trickery but does nothing to give foreign here some wiII applying money but some operator a seal of operator it's trivial to operator and what happens if I applied to a state in which the particle is known to be X were get I get a particle at and then nothing and just rewritten the article at position tactics and the audience was a subject over here up well that's a coefficient alright but a social wave function I have to yes away from the wave functions of the state might be in a product of a particle with the state vector is just a society was size star know I know epa's resource stocks Abbas besides so this could also be written some line sigh star of high X prices state that's a particle located point X number slow down here and give any question about this because some jurors this must be bit confusing this view that operator we apply it mistake located in excellent gives us estate located next Carnelian this in a product of a particle X would the wave functions sigh of state that the site is just sigh star of life is just a way function think bought about this about the state here that state is 0 1 particle is 1 particle with wave functions sigh I I can write it another way can write another way write as follows the piety is equal through a plus are on the vacuum rectum there's no particles if I created particles in the state leaders have a particle I'm an article in the state so I can substitute for this substitute 8 plus call on vacuum no I want vital 1 particle is 1 particle in the state the soldiers want particle 2 the gives me a particle act position x but wait a 2nd the thing that I wrote them here it is it exactly the quantum field side that so we can now write this and other former we can write a particle a position X use society at position next on the vacuum so now we know what side DataEase it said operate at which creates a particle position if we have no particles really want to create a particle Cheney number particles anyone it positioned X and the a way to do it is to apply side entered the vernacular creates a particle opposition acts what we want to create another particle position why he won't subtle hard a workout but what you expect to expect just hit it with field operated will call this a field operated now for the creation of leave the field operative we just hit it with another sigh dagger but position y so where this state is if you want a in a particle at a particular location that's the job of the quantum field as a function of X Y has known no 1 who is a function of X B callers it has nowhere to put the product it's an operator that puts a particle of exits not a part of the puts the particle Hawaii so it must be itself a function of X that's what it is simple situation the simple situation that were working on now that's the meaning of the quantum field it creates a particle apposition acts the creation and annihilation operators create particles in the energy items you will apply a dagger to evacuate and you get a particle not position I doesn't mean anything but the wave function i.e. signed their signs and co-stars so for us that's the meaning of a plaster on the
other hand when you combine them together into the quantum field the quantum field creates a particle position next what if any our you think annihilation after leaving the other piece does that removes a particle X X is 1 that was there than I awaits state so sigh daggers creates a particle position next whether or not as 1 they signed annihilated particle Amex or gives 0 this is no but what kind of party they a should be in the state lost 1st of all we're talking about bars are get firm your number orders were right now is that the quantum field for each species of HBO's so what kind of bows on the modern world photons Gravatai Higgs boson islands of blue ones for each species Of those on there is a separate quantum field create that colleague of particle induced members of Our dealt a function of the huge vise I that's exactly right Greg so it didn't is know that side glowing reviews for the same thing that 1 when and annihilation operated a state with no particle and Garwin annihilation operating the ground state give Zero no Zero what it is but for the position you started with back in the vacuum is now 0 acting as a state it's a normalized state whose leaders 1 it a good status is not sure whether there a better has properties the properties of those of empty space where annihilation operator had literally gives 0 0 vector vector of 0 Lane but rule 0 is unique can only vector there if you edit any other vector you get the same back it's a veteran of vector space go back the vector spaces vector spaces for em In the end 0 vectors 0 vectors of vectors is relying on this not interesting it doesn't only properties just the describe minivans described physically anything it's just but if you at 0 0 state her back and say What if you add the back start you certainly not that U. S vacuum let's April 1 particle state then you get a state which has worked say you wait it would be cool coefficients just to be simple and you get a state that has a probability of a half of having a particle and a probability of half not the scene they found that U.S. and zeros stayed at the vacuum told 1 particle state users get 1 particles thank you and the vacuum of particle state you get something that has a statistical probability of what they have Have a particle and the statistical probability cannot have a particle you you go back to us 0 cannot be normalized is definitely not normalize 0 leads the 1 thing you can normalize vector 0 right the vacuum is not a vector of 0 it does for a little bit of time people generally get confused by the notation this looks like it 0 but it's not 0 assistant need 0 here does not mean that the vector of 0 length it does not mean that a few aided to another vector you get back the same vector it's just the name of the vector of represents a perfectly good physical state it happens to be a state with no particles in the study and displaced but to state staff you could do it could recall call them in a so-called George Bush everyone 4th doesn't have a him what a little talk about violations George Bush a class at LSU annihilated it can't be that might assist 15 games inches 0 literally 0 any operator that acts on 0 vector disease back 0 vectors of anything interesting was like 1 like this 1 0 the Getty vector and subtracted from the same vector 0 2 8 0 2 any number you get the same number it effect 0 2 now there is 0 record right has no length the vacuum on the other hand has only from 1 as vector and vector space normalized but easy Money is Inc not easy but to get used to it under what or some yet not that not only this is an operator any Point Texas is just an ordinary number was a matter which for a number of times operators are operators just sigh of acts as a collection of numbers 1 each X let me think it'll complex numbers Yum yum general become tourist numbers as that of course depends on the solutions of the Schroedinger equation with real complex they 1 the few fears another for that is the virus by applying a deal and another of your upper particle position y and top of this so far now list yet you want but another particle position worry OK let's make a toe particles state that the maker took particle stayed and make 1 more observation about visitor particle study this is a particle at X and particle why now alike could be the same point where they could be different points of the middle 1 thing there 1 interesting fact on the size are made out of annihilation operators are no creation operators in here every annihilation operator commutes with every other annihilation operator no I make a mistake yah yah yah are right side there is a clear sign of course on but still it's true that every creation after this is the important thing was Rose creation or annihilation each 1 commutes with every other 1 think every Cray every operator and he creates every opportunity here commutes with every operators here that means then-Gov wiII inside acts Camille let's call this the state with particle that wiII and acts while this
is also equal to 0 sigh dagger hex times sigh of NYT on home thank you are losing it and that's equal to the state with particle ad X and a particle Hawaii equal to each other the equal as a consequence of the fact that these commute the fact that these commute because built out oscillators in a certain way tells us that it doesn't matter which order we put the price at which or we write the particles and he and other worlds particle wonder why particle to annex is the same as particle 1 next particle told White that's the defining property of bows out so that it doesn't matter which order your expressed a particle particle here Jeffrey over here is the same as Jeffrey over here and Harry over there but the it's was fully equivalent systematization and tells you that the particles are most definitely bows so I didn't have I did nearly half the safe from the beginning well wilderness study bows on Zico with this construction and we would have to rise the particles studied this way on both so the interesting question is How the hell did you make from Iran's as more abstract why that while X and Y X Y position the press yeah yet have the same kind of bars on ride reversing kind of sewing giving quantum field describes 1 type of particle is a quantum field of photons will actually do it Till quantum field photons because of the 2 polarization is a quantum 3 field with of photons along some match is a quantum field for about 4 times so polarized Nova quantum field of photons polarized is a quantum field of the each different type of particle as a separate quantum field and if you want to make that kind of particle you apply that quantum field sufferer quantum field is horrible east in its simplest incarnation and assure you this is the supporters but the basic concept on the connection you see what we're doing is talking about the connection between particles and fields this year is a connection between particles and fields misses a field when you apply to a vacuum you create In defying etc. we defined it but once defined it's an operating if we add to its complex conjugate its position operated permission operators are observables all of a sudden the takes on a life as something that we can measure but I end of the wall the things with observables we don't feel that these provisions yes yes none of this would be interesting if we were talking about a situation where you could not create and annihilate particles hole point of well I think and take them to move a strong none of this would be interesting if we were only working in a context where they were 37 particles never 36 never 45 37 particles in the story would interested in studying the whole set of possibilities of any number of particles and this is a good tool but even better when was studying the possibility of the number of particles itself being ambiguous mean to say a statistical variable quantum mechanical variable because study states in which there is superposition of different numbers of particles not particle 1 particle to particles and movies have real meaning an electromagnetic waves a laser waves like that is a way of life is not a definite number of particles superposition 1 2 3 0 1 2 3 and were 1st of 4 more please
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Metadaten

Formale Metadaten

Titel Advanced Quantum Mechanics | Lecture 6
Serientitel Lecture Collection | Advanced Quantum Mechanics
Teil 6
Anzahl der Teile 7
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15100
Herausgeber Stanford University
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 28, 2013) Leonard Susskind introduces quantum field theory and its connection to quantum harmonic oscillators. Gravity aside, quantum field theory offers the most complete theoretical description of our universe.

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