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New Revolutions in Particle Physics: Basic Concepts | Lecture 2

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as I said the basic tool of the trade Is quantum field theory and that is what we're going to spend a little bit of time learning going to teach you a very elementary version of what a quantum field is a thing tonight but I don't want to get up on the blackboard some basic mathematical fact just to review and to remind you novices to elementary through these serious part of this class at this point but I do want to give up the 1st mathematical set of facts does have to do with waves waves are course an example of Fields field configurations fields of things which vary throughout space waves a particular kind of configurations in which the EU are moving In a way like motion and waves and particles are closely connected through the length of quantum mechanics I am not going to peach aqua mechanics courses close-quarter so if your quantum mechanics His fuzzy on certain hot pot of gold back the lectures on quantum mechanics the coast that's was West called quantum entanglement which will get up there and hopefully soon and clusters called on mechanics and that is the basic starting point for work particle physics also special relativity but let's just review very very quickly 1 mathematical fact which comes up over and over and over and over again just what the relation between signs Co signs exponential everybody probably most of the people here know this but despite give up from a black arm for future reference OK let's I remember what an exponential an exponential exponential function as the form an e uses
number out of X at Athens isn't number it could be any number of real or complex pure imaginary and eagerly Alpha X is a function of X an exponential is a function which grows in this case with respect to x by an amount each separately with the voter exit grows by an amount proportional to itself for example population of a B tree dish the increase in population is proportional for the population itself and so are the hallmark of an exponential Of course at next is derivative X with respect X is proportional in this case with a proportional proportionality factor alpha Times f itself the rate of change is proportional through a self and such a function His called an exponential and the noted he could be out that X or something be number times the OK that's the hallmark of an exponential now remind you of some facts about signs in color signs in the put together very quickly signs in things you know what they are functions which oscillate at sign X and call sign of X is exactly the same function except shifted over by 90 degrees and it looks like that and I am sure you will not visceral go into detail but the important thing about signing of let's say KTX was caught sight of kayak signing of K is a number in a number of real number in this case projects could be positive could be met could be negative end the other function of interest is call sign Our X I sign of kayaks cosigner KX commute differentiated both of them and not just remind you were derivative as well you remind me what's the derivative of scientific kayaks with respect please sign of X with respect to X is equal to Katie course earned care acts and the derivative of call sign With respect X it is my next signed exit so neither 1 of these is an exponential neither 1 has the property that when you differentiated to get something back proportions proportional for itself they have got property that we differentiate 1 you get something proportional to the other where away in 1 case with minus sign please both sank KXI killed because came to differentiate this 1 you get the other 1 you differentiate the other 1 medium minus
the first one but so out of these 2 you cannot now build a exponential but it's an exponential of the imaginary saying so let's just as do it warrants and their return would take the combination cosigner OK X plus on a sign kayaks I'm being square reminders warned you differentiate it with respect to x or you get you get From cosine you get minus signed checks plus a body I I'm sorry might this case signed X plus I carry cosine KXO do it correctly now occurred which it is the spend a couple minutes linear algebra I can't carry times the original thing back again cosigned KTX plus Army signed came most make a long story short course I'm OK X plus I signed X Is the fortune each of the Audi K X when you differentiator with respect our X you get Pike a terms right so that's that's an important mathematical fact comes up over and over again signs Inco signs of course the functions which describe wavelike oscillations Yeutter oscillations with respect the space it acts as a space coordinator of oscillation mastery of space X happens to be kind to become responsible oscillation with respect to to come Our let's but consider more detail our customers detail just quick review but remember waved the waivers moving past us it has a certain frequency and has a certain wavelengths just remind you of the connections cause we use them think OF the the frequency it's the number of cycles of you every 2nd frequency frequency and measured and cycles per 2nd c stands cycles per 2nd cycle was 1 floor source of the another definition was definition on May Day it is another form of the frequency angular frequency measured and radiant per 2nd and just Quebec to planned f so F and
only got the same thing except for a factor too I love and I may go back and forth between them and try to keep equations as simple as possible with as as few factors of apartheid so sometimes made Omega sometimes use of their
land does of course the wavelength and now for a light of lightwave moves at the speed of light because is moving past you with the speed of light the Waverly and frequency are related and their elated by the formula the wavelengths is the distance traveled in 1 full cycle and if you multiply that by the frequency that gives you the speed of light that the velocity frequency is how long the inverse of the time that it takes to the wonderful cycle so this is a distant divided by the time that it takes going full cycle and for a light Brady Mississippi a sound wave you put the speed of sound there it's a waterway to put the speed of water waves there or whatever happens to be but there will be of course particularly interested in that's 1 set of equations the other set of equations that we discussed last foreign quick briefly to review last time it is the equations of quantum mechanics which relate wavelike motions particles were not going to go in heavy stepped into the explanation of these equations for that you go back In open up B a video lectures on corn mechanics but arrests of drug abuse complete possible or at least a self-contained as possible are the important thing we're going to Quanah mechanics is in number plant can't comes in tool variety variety to would separate notations all of Planck's constant which was plants original notation and then there's no plants constant which is the same Paris riches which is the same as the old plants constant except for a factor to Pyay and that 2 pies but they have so far received 2 pies in somebody else's equation that it they always come up in this is no escape to parties they're always euros but the other definitions age bar and that's old part and new plant constant that's only part Kirsten divided by pride I will simply say supplied by sometimes using some time using each for Turkey to member 2 parties Our to a minimum now farm lender funds that state case yacht Bowers as equation that just a definition and H h is a number which is a tiny tiny tiny number 5 anybody remember this 6 times and my some 34 something in some units and some units about all that was an interesting question about said only e-mail was shut up somebody questioning whether the origin of the unit meter pointing out that Stewart was needed media is 1 14 million of the circumference of the now I know was I was wrong I didn't have anything to do with this 140 million you think that somebody just came along and said let's define unit which is 14 million of the circumference of you're out Of course that you have had been in use I don't know those call the leader of not England was called the yard but why 140 million wine that 140 trillion won 40 well because you wanted some unit easy to deal with a human being good measure rope with good to measure cloth with rubber and I don't believe the origin of the unit which is about the length of a human beings are up tree 88 knowledge about what the circumference of truthful answer anyway which effect was known for a long time but birth yacht taste In standard units meters seconds and a kg h Boris a very small number the bladder right American minus 34 at 1 point but like that's a little each that it ages 6 times and a 34 or whatever and see of course it is 3 times and the 8 meters per 2nd 3 times and 8 meters per 2nd OK so this is just a definition of some number of definition between a relation between 2 numbers but the fact is that they energy of any single photon of photon associated With anyway of a given wavelength and therefore a given frequency notice this equation he tells you the relationship between frequency and wavelengths frequency and wave inverse true inversely related the bigger 1 is a small or the other is the energy of a single quantum single quantum energy this is not the energy of an electromagnetic wave this is the energy of 1 quanta of electromagnetic wave that's equal either each borrower Omega well maybe it is the frequency of the wave of the angular frequency of the radiative incidentally the frequency of an electromagnetic optical on optical way you want and you could see with your time is very very rapid something like about 10 of the 15 cycles per 2nd on that right and the 15th cycles per 2nd so that it will make it is very very large and the 15th Brewer per 2nd but makes policy more than small enough to work to to make the energy of a single point very very small if you multiply these tool in the units that we've been using you get the energy and Jules and you find out that the energy and jewels is very small and the 19th or something like that so would take a huge number of optical photons it 20 appreciable macroscopic conventional energy price of energy and this of course is also equal the other plants constant frenzy the frequency for the same hour just 2 different official partner let's talk about momentum but remind ourselves about momentum energy incidentally in momentum are especially interesting because they're conserved quantities their conserve our total amount of energy is conserved problem at the moment is concerned and therefore the good bookkeeping devices could refer writing balance equations and a useful among others things besides being date momentum as a yeah the point of their exit points typically direction emotional object by its defined by being considered but having said that it's defined by being considered by the fact that serve we counted himself forth on we can ask What is the momentum of various kinds of objects in nature and the 1st thing we can write down is a momentum of the non relativistic object nonrelativistic object just means 1 moving slower than the speed of light or lost the a lot slower than the speed of light Watts the momentum of such as such an object is proportional finds velocity and velocity is also backed by debut 80 pulse of radiation electromagnetic radiation at all column made a long given direction not disbursed wave which is going out and our a variety of different directions but which every thing is lined up culminated along the axis then Ardis's not relativistic the opposite extreme super relativistic and was moving with the speed of light now this formula is not right from with the speed of light Ukiah plug-in from V the speed of light but would you plug-and-play and remember I told you and that is all the ways by definition rest but we used the poll arrest the photon doesn't have a rest it's rest 0 brief arrest has nothing but
this is not the right formula nevertheless the right formula has a look which looks a bit like us if we would used inappropriately E equals M C squared mainly and using this unit sonority units right energy is mass times velocity squared then of course Mass would be an energy divided by the speed of light squares by now if I as a sit inappropriately ascribed for photon on equal boots energy divided by C squared energy of the photon the provided of icy squared and then said that the velocity the photon was C and that more or less someone accidentally I would get the right answer that the magnitude of the moment of should be read as the magnitude of the moment they are dot but but the magnitude of the momentum is Ste energy of the photon all the energy of the electromagnetic waves that any electromagnetic wave or moving in a constant direction direction of the momentum is of course the direction of motion of the way but that's easy but the magnitude of the moment and Izzy energy of the wave provided by the speed of light from that say surpassing the wave consisted of only of a single photon now 0 using it legitimately the right and what it is the momentum of a single photon and to get the momentum of a single photon you just plug energy from here into the formula for the momentum so let's do that the momentum for a single photon crowds of thoroughly here momentum of a single photon left-right photon only you cannot be ready really read the momentum a photon concentrating on its magnitude incidentally the double bar means the magnitude of it the direction Is along the direction of the beam of light along the direction of the let's put it in energy but let's use this formula over here time frequency energy is age frequency provided by sea right what's the frequency divided by right or what level away like the 1 on the frequency provided by sea as divided by city is equal to 1 Orlando right get that right yes frequency divided by sea is 1 of so this is just the equation of photon momentum photon momentum in magnitude is equal to plants can't stand that makes small the frequency divided by sea which is divided by the wavelength of the smaller Waverly the larger and the momentum of a photon at a pattern and in some sense the most important fact coming from quantum mechanics of a particle physics that if you want short Lane for the purposes Our of studying very small objects price that you pay you have a lot more the 1st yah yah Gloria he is worried that a of work if this is 5
from a quantum mechanics of a harmonic oscillator Charles placed the look it up his arm lectures previous lectures about the harmonic oscillator are to make use of those who lectures but that's where comes from off from a deeper point of view that will come work on the right statement alone might be dentures and now that wrote I got a
call from memory and the question now our guys I know it was there and I was waiting at the right point but to said arm but an electromagnetic wave or any other way has several features 1 of which I have not written down here it's the amplitude of the wave the amplitude of of wealth OK the waves with the same frequency the same hour every in every respect the same except that concern and here's another 1 1 of them it just some small number of times signed kayaks for example may be small assigned KXT other ones a great big numerical multiple which I call any kind signed a acts exactly the same someone much stronger the other it's got a larger Tampa to Eisley and with the way of our and now the question is in full-time language what is the meaning of the amplitude or another question is what is the amplitude associated with 1 photon now the question is a little bit ambiguous because quantum mechanics always has on certain he's technically has uncertainties and you can't measure everything simultaneously for example you can measure both the amplitude and the phase of the way the phase of the waivers where located but never mind that orders there's a sensible order of magnitude question order magnitude was the amplitude that's associated our with a single photon for example OK the answer the following week energy of a wave now this is a completely classical statement completely classical statement energy of a wave His proportional Katie times the squalor of the amplitude are for example if this was meow electromagnetic field say the electric field or the magnetic field associated with electromagnetic relate to what energy of an electromagnetic field and proportional to the square of electromagnetic magnetic field typically the energy Ave way is proportional to the square of the imports if you take back but now the energy of the way is also proportional to the number of photons in that way so he had described for use on features of the single photon but a way of life consists of many photons let's oppose a number of photons in some way of life is In the end we energy proportional and in fact be energy will be in the number of photons turns borrow maker so that means that of proportionality between square of the emperor on the 1 hand and the number of photons in that way which is called In the precise exact detailed connection all worked out for you another time but the square of the amplitude of the wave is proportional to the number of thought that was a factor of age barring near because the square the amplitude is energy and we should have H R at age 4 maker and the energy so the square of the amplitude it is proportional to age foreign it not plus a point is that H. bars an extremely small numbers so the energy of away another form another way of writing the energy of a wave with many photons not a single photon that many times and I say well look this is terribly number how could you get any appreciable amount of energy and electromagnetic wave Waverly answered by having a lot of photons so typical electromagnetic waves that carries an appreciable energy has an enormous number of all kinds overwhelm the factors which but the amplitude if you like 18 is proportional to the square root Of the number of boat so that's the missing go that's missing we've characteristic that I did write down here a said it is proportional to the pot 5 you can try on the temptation that you know I signed a theft is on America racked up pace throw it away it's moving down the X-axis which plot of vertically here is not the axis or the z-axis what vertically electric field right service is not a plot only 1 of the directions year corresponds to space the other direction of course corresponds to be electric or magnetic field that security has been clarified and properly so that caused a wave could be oriented in a new direction that way that way that way and the momentum carried by the waves along the axis of believes now they thought you may
not want here that there has faced with is like a car that that tracked that's when say would be noon With no 1 way of life better but proportionality I purposefully wrote proportional but we are talking about two-way lanes freeway lanes forward cost from gets you got there you have the right if if if we will all honor of magnitude it's given by this formula within equals 1 Our but yes in some sense it is meaningless says meaningless for electromagnetic waves the electromagnetic wave has both a mobile a magnetic field and an electric field of the magnetic field and electric field are related to each other by a relations which is similarly mathematically the relationship between the position of a particle and the velocity of a particle another things which cannot be measured simultaneously the uncertainty principle prevents you from both of them in the same way as in uncertainty principle between electric field and magnetic field that prevents you from both so yes there is there are limitations on how well you could not be amplitude was an order of magnitude statement is correct will try to better as we go along with sorry getting into the subject the quantum field theory which is difficult an abstract subject OK I wrote them some equations utility true of photons some of them are true any particle others are not acted definition a frequency that the general definition being to pie at that perfectly good land the Eagles a wavelength at definition this is true for Wales which will lower the speed of light so would not be true for a general wave Charlotte's Web put something around those equations which are not expected to be of great general generality by great generality I mean they would be true for almost any kind of waiver particle whether or not most of the speed of light This is not for general This is not general anything that has seen it will not theory General general regional for anything which moved the speed of light OK this is a general was age bomb maker that is also a very general fact this clearly neither of these this is PacifiCorp things which more with the speed of light this is specific to objects which will with much less than the speed of light is interpolation in between which I won't get into national which is true for things which move with some fraction of the Speaker right on where or Yahweh we were trying to figure out which equations are great generality which I'm not this 1 H f Over seat that contains see if that is not terribly Gerald Erps ways murder bloke momentum equals 8 divided by land that is general with this comes from a tried approve these things to you now just telling you what facts about quite mechanics momentum is each of our land that that's general I'll get it lets you lose these facts now here's the relationship where photons between on between frequency and way of life and as I said it's not very general What about slowly moving electron tourists slowly moving nucleus or for that matter a slowly moving bowling ball what's the relationship between frequency end wavering we figure it out something on the blackboard or something we have had something else question for electron than I did tell you last time that electrons are also described by waves that I'm not I tell you that you could do interference experiments with electrons so there is an electron wave let's call signing and the question now is what is the relationship between frequency and bravery we figure it out yes we can we need to know 1 fact about nonrelativistic systems analog of knowing it is not at are its knowing the relation between momentum in energy but from electron moving slowly the momentum is not the energy divided by the speed of light what is it I'm afraid energy but physical tour elegant start at the beginning in right I mean kinetic energy why have every squared and momentum and very peoples and so we can solve for the velocity peeled and use of vector but a surreal Anthony energy becomes 1 half and times square to be squared piece scored all over him squared if you don't know this equation it's important equations occurs
all and all that energy in terms of the momentum is 1 half peace quit divided by air people get confused the embers upstairs when you multiplied by the velocity squared downstairs William multiply by the momentum squared OK now we have what we need to define the relationship between the frequency and wavelength AIG is swears it is age bottle may go or age times frequency to times frequency a single single electronic about a single electron H times the frequency of the way of describing them it frequency that energy from allowing inside we have 1 half for the end downstairs and what about the momentum the momentum is divided by war and that's H cornered Over land the square we can divide this equation by H. refined that the frequency is 1 or 2 times plants can't the abided by the squalor of this time of the year of the waiver notice different are over here at the frequency was proportional told 1 over the wavelength here it's 1 all the squared Billy don't the members of the memorized as always read the right of the basic ingredients are the equation for the energy and the relationship between momentum Waverly frequency unlike stuff but this is just don't illustrate our movies connection so that's a nonrelativistic particle 1 moving slowly for example electronic Adam the frequency of its shredding the wave would be won over to and comes plants constant divided by the river square based on down our quick question how head if that is true for any particle as long as moving slowly compared to the speed of light arms maybe we should well come back and will interpolate between the slowing the fast amount time there is an interpolation is always a lot depends on how best the trio of most neutrinos in nature are moving very fast with close to the speed of light very close to the speed of light so the neutrinos the speed of light is a good approximation but in principle neutrinos can be slowed down and if they're moving with very much less than the speed of light formula a number of Mr. on are this is the best way a yacht in a welcome back to relation between amplitudes and learn them and Kwan there so not best the whole story is about a program that the listen but gave me I accelerator with 20 very low energy the reclusive there is for electronic slow slow will you Iker right arm did you want to know 1st of all it's units In which city is equal to 1 such units momentum and energy have the same you exactly where we have momentum and energy have exactly the same unit of musical toward the yuan or something is relativistic among our respect you wanna compare its momentum toward maps of its momentum as a large compared to its message rest photon has no rest mass sought momentum was always much larger than that a mass and so it's highly relativistic relativistic as it may be a figure of merit thing that you compare is the momentum for the rest mass now you have to tell me what's the rest mass of an electron you gave me in need a k k killer electron volts is a unit of energy and remember energy momentum of the same year I saw its massive an electron half a million electron volts 20 kV is a lot less than 20 than half a million so that's a swell slow electron the other cookies are you saw through that way you have realize if he if that electrons electron waves do not moves be no city questioners could you derive velocity from yes yes you can but it won't be losses not if they believe flower filled out there until the last musical waves 1 of them is called a group velocity warmer core phase velocity all be specific soon to ask from if you take a general way the way it let's assume it's a wave packet wave packet means that looks like a wave except that it its army confined to some Hackett like structure like that so locally looks like a wave meaning to say a plane wave China workers sign but it has an envelope that that combined confined some regions now in general in general depending on the specific nature of the way you'll see that has that wave moves there are 2 separate velocities associated with and a different 1 of them is the overall velocity of the wave packet the other Is the velocity of the Crescent from whatever you call it what you would see if you look at this way saved from prospective where you see waves and PROFS moving along as you would see the track it moving with 1 velocity but beat waves moving through the packet with a different velocity In fact typically the ways the birds what is called a phase velocity that's the velocity of these troughs peaks and valleys and the other was called the group velocity the group velocity is the velocity of the whole packet guess which 1 you identify with the nonrelativistic velocity of the particles the group velocity and the group velocity is usually
slower than the phase velocity the Faisal last it is not got to do with the actual loss park now there is 1 special case special case is when old waves of all frequencies move with the scene velocity that would not be true for nonrelativistic wave like there's a lessee velocity is the frequency times wavering right so let's let's look at the frequency times a line is just frequency times wavering is 1 over 2 and H far back so different when visible with different velocities different lenders will differ velocities and when different wavelengths move with different velocities you have the situation with group and our phase velocities are not the same for the special case where all wavelengths move with the same velocity then group and a phase velocities are the same so from lightweight always move the same Molossidae group and video and packet velocity of group and please velocity harvesting that's a special and simplify a simple case but but we I just want to illustrate some of the equations here parents incidentally this equation over here is equivalent to another equation of physics and the shorting the pleasure the Schroedinger equation is an equation for ways and missed tells you that kind dependence of the way the frequency this tells you the space dependence of the layer of the Schroedinger equation is exactly an equation for the relationship between the time dependence of away and the space and the way we will come back but this is just a Schroedinger equation in a year this slightly disguised form for better yet if you knew the Schroedinger equation and use its opposing I had a wave of a given frequency that had a given wavelength what would be equation tell me about the relation between wavelength frequency would be exactly showcase necessary pleasure in a nutshell belief that this little but the Faisal last as a rule carries no information with arms you can't send information fast in the group last of laser last year wasn't really carry any media information from back it it's a technical issue but we don't need to deal with red small all OK let's let's go on who that more with waves there are those who want to other concepts about waves necessary since you have that there's a lot of technical detail getting into that every single piece this is really very central portent OK we will be interested in waves moving in infinite space but I use the word infinite space infinite infinity um computers don't like infinity and physics is the art of writing equations that computers can solve a principle is necessary tho always remove these entities an rightly equations I systems 1st and didn't take over still in studying quantum mechanics waiting quantum field theory particle physics and so forth it's necessary in the back of your mind to get rid 1st around of all everything the infinite and state the problems as if everything were finite and in particular what I have in mind now is the extent of space itself sold 108 on where I want to think about now is a way of moving In 1 dimension 3 dimensions is really no different than when they mention except there 3 of them so let's concentrate on one-dimensional wave motion it could be a violin straying it could be a role being will back-and-forth could just be a wave mathematical universe one-dimensional 1 way of getting rid of the infinity of space would be to make the wave only propagate from a region of late Al for example by holding down the way this and holding them your string at this end Ingersoll Violin Strings held ban on it and this and you can have all sorts of waves and the waves more and they bounced off when they get to the end they bounced often reflect little back and forth and back and forth that's a perfectly good way to describe the problem with this kind of core boundary conditions misses call the boundary conditions namely that waves when they get to the end reflect the school reflecting boundary condition the problem with it is it violates 1 of the important conservation laws of physics what with conservation of momentum after all if the momentum of a way is it's a vector quantity of points in the direction all wave is moving along a sack as way moving along this axis is momentum is pointing to the right went against the end it bounces off and reflects show me momentum has changed sign the momentum as change signed the momentum is not concerned that's OK there's nothing wrong with it you could study it but if you want to keep track of things like momentum conservation is a smarter thing you could do it goes as follows a man team this line is such that we identified the beginning and the end now we can do that by replacing the by a circle with doesn't have to be a legitimate geometric circle just mathematically we connect the beginning but the cycle so as we move from here they had he we can imagine we start year move along here and in his magic aligned when you get to the other and you come back to the beginning at a closed circuit out or course circle because mean plots a circle in 2 dimensions urges former employer periodic periodic is the terms of periodic space now you could imagine a wave which is moving along more than along here when it comes to the end of year was it dealt it just keeps going it just keeps going it doesn't care about that point that point is not a special point anymore and so the way it just keeps cycling around cycling around cycling around and the momentum is considered it doesn't suddenly reflect new direction the terms of this picture you could just imagine when the way gets to this end it reappears at this and and it just keep going to the writer Phil going to the right because of a lack of will go from the left and keeps going forever with this trek with this mathematical trick of replacing space by a
periodic space begin 1st of all get rid of these infinity and the size of space into a computer and we can get rid I the problem of momentum nomo none momentum conservation of momentum is now sir but there is a cost caused but the the fact the fact is that in such a world momentum is quantized now I don't mean in the sense that the hour broke it is cooked that momentum not only comes in discrete multiples of a unit and useful for work In understanding quantum fields their relation the particles and so forth is useful to remember that the momentum cons indiscreet multiples of something so it see how that works well 1st of all it's obvious that the waiver Lane an 8 digit number of waves must feel that slips give this distance around here and name a squad L. L Lane or circle would be the circumference of which is quite L total distance around from beginning back beginning again Korvale ever put away Aaron let's say now that it's a waiver signed a cosine wave however and it did you numbers of wavelengths has defeat onto the circle that means that the wavelength has to be equal the link around divided by an integer provided by integer of the wavelength must be total distance around divided by some integer I don't call the integer In the city of account put capital capital and well and will be reserved for something else so the wavelength is necessarily public distance around divided by now that means the wavelength comes in indiscreet possibilities it can be held me over to me over 3 B L 4 4 but it can't be L. Times Square root of tool or L divided by the square root of an integer number waves just won't fit in here and other words if you try to put away leg of the wrong wavering here by the time you got back the way you would match with little was supposed to be at the beginning of a case that tells you that wavelengths come indiscreet calm possibilities with the integer appearing the nominator now an equation here somewhere for momentum In terms of wavelength that comment on force equal each bar but yes a Orlando P is eagled HOME so now this is also telling us that momentum indiscreet possibilities the momentum of quantum must satisfy the constraints that H divided by an allowable wavelengths of allowable moment of quiet be allowable moment electron Villalobos moment of photons the be allowable momentum of anything in this one-dimensional world it has to be divided by OK let's work out each divided by land the means and divided by L I turned member upside down Noel I was it say it as the moment that come in discrete multiples basic unit and the basic unit is plants can't stand divided by the total distance around there's a need that if you put a particle put it ahead if you put a particle on sign wire and the wires a circular 1 be you'd be a wire that indeed can only have a moment which a discrete multiples of 1 over much in over L and multiple 1 over moreover we leave of water now of course H it is a very small number and be able Aiello is a big macroscopically length from wire or something 1 over L is a very small number and so quantum a separation between different values different allowed values the moment it is very small you could get only Elise larger nature small you could get almost any number of almost any number but every events collection of possible but I'll take that's idea that's the quantization on momentum is a world in which everything is Korea yacht what follows a woman way look total momentum the momentum of the early part of the object call momentum of any quantum now if every Quantum has integer multiples of his basic unit of momentum in a number of also have to have an integer multiple you might have each quantum is identified in terms of its value of did this momentum but there are allowed the momentum and disallowed momentum allowed momentum are discrete multiples of basic unit and any number of photons electrons and so forth necessarily come in his this is a lot of this is some rocks assumption that these spaces really genuinely periodic there's nothing special about 14 hour and it is a fair question it is a good question but it's also the assumption that for example the electromagnetic waves electric field is a observable quantities doesn't have jumps In fact the jump you have if it didn't come back to itself would have to make a sudden jump at this point the sum jump is very costly in energy and storm sudden jumps in fields feels a continuous things now to jump crap if you have what what can wait packet old aside any packet Tosh grew word packet ties of the is a sum individual plane waves signs Inco signs of the Orient House the statement the real statement and allowable weight is a sum of waves each of which has the discrete described by the possible moment so to stable enough for it transforms the Fourier analysis of any allowable weight only has these discrete multiples over the basic unit L P a toe this
part got more than 2 how small a year he had farm nobody is smaller than the plant point that's that's a that's a that's a bad thing up to let our problem only kid got much but procedures moments of it's out my note a that pursue it it's important let's really imagine that we genuinely have a particle moving Virginia wine circular wire from what's really never think of moving on really circular wire and see if we can understand what this quantization a moment means another language let's take the circular while have a radius aura but I have a radius R all all I erased important equation what was important equation important equation was the allowable momentum is equal to an integer times 8 Orlando posada available for rail right OK that the radius of the orbit it is our and that means that he is an H overnight tool piety aura you could see by now that each bar with a better choice in this equation right that's it Leslie later I will put package are OK that's the momentum and the momentum is quantized every single electron photon or anything else as they have an integer multiple of there's know about the angular momentum What's the relationship between momentum radius in angular momentum for a circular back new task at well for nonrelativistic particle it's and they are and they are stands angular momentum and they are but what isn't it date the peak times are now NVR is not completely general relativistic particle momentum is not entirely but angular momentum as equal the momentum times distance it is very general that is a particle that Is the definition of a momentum at least for a circular orbit Our other orbits the cross product or crossed P but always hard times P and so delicate that's what this says Is that times pity multiply by are angular momentum is an individual multiple of age divided by 2 for an integer multiple of Clark's other Kirsten but sold momentum is another quantity which is acts angular momentum as a quantity which is quantized but always in the same unit on alike the momentum to hear the momentum was quantizer unit which depends on angular on get hopes that I use Alpha 2 things excuse me but like all of our angular momentum that angular momentum stadium and with the wound will eventually change that I don't like that but for the moment angular momentum is In times each far momentum is quantized and the periodic world but be unit of momentum depends on the around the circuit of the angular momentum is quantized but always in units of age bar all angular momenta common units of bar except when they don't we don't Israel have firm eons but will come that that's later that nite there are exceptions to this rule book a cap on now we want in the next hour to invent the idea of a quantum field and what is a quantum field in the simplest possible case the simplest possible situation what exactly quantum field how was the quantum field related particles and would quite there was a connection between particles you would through with some precision and canceled next let's take 5 star now we have to enter into the real world of quantum mechanics set the rap if you're not harm knowledgeable about it you have various alternatives 1 is to go slate gray mothers the list and then try to get you know so what's the right word for a very impressionistic uh however if I get a year of the most some word varied In state saw off the a fuzzy feel of things the reduced to go back but at all the path I hope to do better and I fully support and a 3rd is to go back and learn a little bit about a quantum mechanics still be nothing here but I haven't discussed previous lectures on quantum mechanics in particular the most important thing for purposes is 1 of the simplest of all quantum systems the harmonic oscillator wine as a harmonic oscillator come up the harmonic oscillators importer Vera great many reasons but particular context it's because waves are oscillations if you just see a wavering past you obviously the and would field electric magnetic murders oscillate In fact the fact of us awaits is equivalent to a statement that has a frequency on a wave maybe we made up however superposition of many different wavelengths for example light that you could mix together different colors of light to create light were mixed different wavelengths but you could also separated as did note and do its component away rights and so every light wave commit was being made up of individual components each with its own frequency Anton amplitude and so forth their Florida commit love is being made up of a collection of quanta man the quanta are not all of the same energy and momentum but still it can be separated into quanta of different energy and momentum and that each energy and momentum is associated with a wavelength each wavelength is associated with frequency and frequency is by definition oscillation sorrow 1 can think of a wave as a collection of harmonic
oscillators harmonic oscillators were being the simplest kind of oscillation signs cosigner in order to proceed with what quantum field is we have to go back toward the harmonic oscillators let's let's begin with a harmonic oscillator corner characters a strange theory Paris things numbers in it which don't satisfy the usual rules of arithmetic technically they are operators representing me observable things that you can measure where they have some roles had them you could multiply them but they don't necessarily commuter was I mean that means you have to variables describing through different attributes of properties of a system of each of which by itself might be measurable might be something you could measure let's call them AT and be given names 8 could stand for example for electric field because have a magnetic field for a could stand for the position electron-beam could stand the author of the electric field for any too pair of things then ordinarily in ordinary mathematics of across the product of them doesn't matter which you multiply them 8 times B's is equal defeat Sunday just a property of numbers quantum mechanics of course that is not generally true it may or may not be true of toward observable quantities of natural which is right that the difference between needs is not equal to 0 now what stands means different measures electric field some numbers and as a magnetic field and some other number and multiply them what the possibly be that doesn't make sense to multiply them in the opposite order Mansour generally speaking when meet answer it depends on the order of multiplication were really means another was now 0 what it really means is that you can't measure both of these things simultaneously that they interfere with each of the experiment that measures 1 of them necessarily screws up the measurement of the other 1 and in that sense there is no contradiction when this is not 0 simply means you can measure both of them together when you can measure to things simultaneously and you measure them you get numbers for them when you multiply the numbers the numbers have to commute it doesn't matter which or you multiply the men and so the test for whether you could measure 2 things at the same time is whether this quantity which is also called the commutator a B B a score the commutator newcomer alike go back-and-forth got to work and back you interchange fewer of them are labeled bracket come on an AT & B the order of matters so typically is not equal to 0 all incidentally the Taylor with Haiti that's just exactly the same thing except in the opposite order and that is assigned different beast or same except for a sign OK that that's quite a mechanics a nutshell we just a quantum mechanics examples now I'm that proving son just telling you this now but nevertheless fact partly empirical fact mathematical fact a particle has different components of position x y and z for example you could measure the position particles of the components of position of a particle on candidates for things which are observables these things commute with each other you can measure the X and the wife of a particle same time you could measure the X the Wise Museum particle all 3 simultaneously what else can you measure about particle Yukon measured velocity or its momentum mass times velocity and those things of called the components of Amoco PXP wine PC components Our view vector momentum and they also commute with each other you could measure the 3 components or better yet magnet to and Nemo and the direction of Over momentum Paducah measure all 3 components simultaneously there's no obstruction they commute with each other so I X commuter why commutes with Zea PEX commutes PY commutes with PC but the components of positions do not commute with all components of momentum so for example you cannot measure the position X component of the position of a particle and the x component of the momentum every time you tried and measures x component position you will give the particle black and change its momentum and the result is you can measure both of them this is not equal to 0 it's equal High Times monks constant it's more so bleak reflecting the fact that dominant Rugby the uncertainties in position a moment of small and the border each likewise the white components a white position and the wine momentum don't commute with each other same from components how that X with the Y component of momentum could you measure X Y component of them simultaneously yes cement affected can prompt but that may not be obvious loose draw OK cap on so it another aspect of quantum mechanics now let's consider the harmonic Oregon I'm telling you facts for most part try but the mathematical facts into some kind of context these other facts who received orders to go why is this but those days this is squared minus 1 the not budging lodging K which are unit vectors along with reacts and the also among quaternion which also have an IDE Immokalee associated with ex-wife but not the decide square reminds 1 wears a square mile 20 until formula like this don't don't ask we we we always do that the fact is we really need it we just need to know that algebraic manipulations in quantum mechanics are a little bit funny and the funniness it is that you can't as a rule of interchange the orders about our operation but were not best that Be that further there are a dozen this isn't doesn't mean measure to get measuring our these simply mathematical manipulations which are young it doesn't
mean the 1st measure a in then measure be its simply 80 knots of breeze it is a mathematical manipulation that matrix multiplication operator multiplication but not going where we're not going through the principles of quantum mechanics era and is restating things that we've learned in the past and 1st you need to understand the better they do it this you go back to earlier lectures OK now the it doesn't really apply to a photon apply strictly speaking nonrelativistic particles the problem is that the position of a photon is not so well defined the positional 4 times harder to define the position of the nonrelativistic particle are sold Oh absolutely back that's a subtlety which is not terribly important OK the harmonic oscillator the harmonic oscillator is a system of classical system which has a quantum mechanical analog and simply means a system which oscillate weight money in the spring which up closer it's up and down all a sound wave that oscillate a lightwave oscillate anything that oscillate like a spring going back and forth we could think of thousands of examples and the mathematics of them is always the same every oscillation for every has a frequency that you'd and we call on all major the frequency again Tupaiidae times the number of cycles per 2nd cycle per 2nd a spring with weight on it is just how long it takes to both the 1 full swing in combat at the same place not just the same place the same velocity of gravity weight on it in the spring and ocellated or what frequency is not how long it takes to come back the same place but politics back saying place moving in the same direction a 1 oscillation it is period the number of cycles of number of oscillations per 2nd is the frequency f and as usual Omega is took part nothing rarer unusual the usual now what unusual about the quantum mechanical harmonic Is it energy is quantized not momentum is not over but energy is quantized something you look up or accept so what plot vertically the energy levels of such a harmonic oscillator classically of course there's a lowest energy which is 0 does the oscillator rest and then you give it a little poker starts soaring back and forth and you could give it any positive value of energy of course if you hit it too hard and try to give too much energy you might melt the spring aura of break this spring but in a mathematically idealized a version of the harmonic oscillator you can give it any positive value of energy not sold the quantum mechanical story the quantum mechanical oscillator has discreet integer units of energy if we call the lowest energy 0 sometimes it's appropriate to call the lowest energy not quite 0 but we can define begin defined the lowest energy to be 0 them what's the next energy level 8 ball Omega the same formula as the energy of a photon of frequency omega what selects 1 2 NH farming that could even be the energy of a single photon associated with a frequency twice as big as the first one or Baker correspond to photons both associated With a wave of frequency omega this is 2nd our idea that for blow on that this is the same energy as 2 photons so father an awful times is only an springs and balls of oscillations back and forth but nevertheless this does have these energy of 1 photon associated with frequency omega the energy of 2 2 photons of frequency omega and so forth 3 age forum again so for those of the possible energies of our Castle nothing in between we label with an integer right energy is an integer funds H bomb maker where it is just an integer which labels which 1 of these state the oscillator is oscillating and it's just a matter of notation for us here it's just a matter of notation mathematics of it we don't need to get into we label the various states of the oscillator by an integer in conventional just labeled the states of the oscillator pointy thing like that but called it can collect can't because it's the right thing inside of a bright care but we don't need the brash and there would only Duquette and that's it's our has said go back a study quota can all this is always labels This is a symbol representing if you put in equal 0 here it represents the state of the oscillator in the ground state ground-state means the lowest energy state if you put 1 here and represents the oscillator oscillating with 1 quantum of energy 1 unit of energy you put 2 here and represents the oscillator with 2 units of energy and so forth so just a notation just a convenient location the labeling the various possible configurations of quantum castle now let's inventor and operation Itzhak phenomena operations and operate pour the difference operator mess operator acts on state to change the state former state sore mathematical operations where which takes you from any state the neighboring 1 1 you acts on this state to give you this 1 from the state to give you this 1 dive from the state to give you this 1 from the state of this 1 so we can write down what about calling a plus in the
class indicates that it raises the energy of the state acts on these Interstate do you want what and last 1 but with a coefficient in front it the square root in plus 1 that is a mumbling jumble unless you know they know a bit about the mathematics of linear operators and so forth said you you know that you could follow it but you don't know if you could just get a hazy impression of what it means but this is the mathematical property of this operation here it multiplies any state that there by a square root event plus 1 and raises you to the next level but basically it's a shift of thing which shift you from 1 level of the neck of the operation which shift you from 1 level for the next regatta keep track of numerical coefficient of what a a minus picture down and minus 1 it it was a vision square square Rubin y happens if this form summer Bob state where take you the it doesn't think you anywhere it is give 0 but because of back some bomb state in a 0 0 and it just gives you is nothing mistake just the none state is nothing to get through so it just gives you 0 This is the
mathematics of creation and annihilation operators are raising and lowering operators now we just gotta get similar mumbo jumbo with show you some properties Of these operations very interesting entertaining again you couldn't do some there you cannot fun them just knowing some very simple set of rules and then this is the rules let's see what happens if we can't state 8 and multiply it 1st apply a miners and then might lost another words let's it with a minus and then a plus what we get always do is follow the rules of 1st of all the old 1 1 important thing some things are just numbers that everything is 1 of these peculiar things that de where the order of multiplication matters some things are really just numbers things like in Niger's numbers that means that the order to multiply doesn't matter something's the order of multiplication is perfectly ordinary in is always something good ordinary OK so I happened what happens What is the sequel to what it pieces let's 1st multiplied a miners and what is it gives a plus and a minus times it what squared and times in mind this 1 so let's take that outside and in mind this 1 now what happens every year with a plus it we had any vector already sorry any state with a plus it raises us Back raises the next level up but multiplied by 80 square force 1 square plus 1 here it is a squared event since I'm starting in minus 1 hitting it with a plus will give me square and another square and and and great times greater than in it just gives me the number and kinds in that's is on game playing plays the game with a set of logical rules here so we find our is if we multiplied by a plus and a minus with this set of logical rules its equivalent to multiplying by the numerical number in whatever stay weak but here if we put here the 1st 2 0 state if would put here the lowest a meager 0 but here the 1st day we get 1 the 2nd state we get to and so forth so whenever we do this combined set of operation as it picks out for us it picks out for us the numerical level and tells us which levels were at that we want to find out which level with talking about and they don't know what it is and it is called by some other name of Lopes the blogosphere
find out what in bloopers corresponds to what we multiplied by a plus stands a minus we find out with the numerical coefficient fears that the numerical coefficient tells us what what level of the oscillator we were playing with so it can write this another way we could simply write that multiplication by a plus times a nite is the same as multiplication by what is in it is simply the energy level or not quite energy level we need something else we need each borrower we multiplied by each borrow maker so we can write in each borrower Major is 8 follow Omega and a plus a minus and what does it tell us that poses the energy of the oscillator if we didn't know what the energy of the oscillator while we could find them mathematically reprimand but not as a matter of a verb burger Express but as a matter of our playing this logical game we could determine what level we would talking about by taking the state unknown state and multiplying it by a plus a minus 10 may now this is over Dean Mazur said doesn't correspond to near to experiment but in fact this it is the actual energy of the oscillator so that's a mathematical budget treks budget trickery let's multiply the mini opposite order as a rule we find out we found out that a plus and minus the sequel to end meaning just say that if you do this operation on steady-state it just multiplies and what about the opposite order but stridently opposite order using the same biological rules a minus planned a plaster on it was accurate but I do that a minor spongy plots the officer or so as a plus is scarce square root event plus 1 firms and plus 1 right now it's multiply that by mine a money this will bring you back end but it would give you another factor of squid and plus 1 whenever a minus X gives you the the numerical coefficient which is seen as being inside so that gives you Square Square and Times Square and pass wanting in Thailand and bring him back down but due to the fact the overall factor in plus 1 so what we find
we found that multiplying in the opposite order abuses different answers on don't sustained cancer but not quite in 1 order a plus a mind is Maine and the other order a minus 10 a and plus 1 so this little bit of logical gamesmanship euro mathematical symbology we find that the order of these 2 operations the same or 1 of them was a plus and minus the free with measure them and can and then take the values that we measure for them numerical values wherever they are whatever they correspond to measure them we multiplied their values by each other's data on the no don't commute but does make sense numbers always commute Multiplan what this means is that whatever a plus and a minus correspond to you can't measure both of them simultaneously what they correspond to was a combination of the input to load of this harmonic oscillator how big the oscillation in its babies phase that we would phase of oscillation but basically the time at which the oscillator passes through the origin the oscillator back-and-forth or you could delay a little bit of the coast the origin of the leader a really time you can't you can't whatever aid is you can't measure it is a plus and a minus simultaneously that's what this tells you at taught her that they don't commute and therefore they're not simultaneously measurable but this still it stinging mathematical objects where they do what they do they are mathematical operations which raise you up up and down his power where they do they add and subtract 1 of the energy of the ocelot not if we oscillate a happens correspond to an electromagnetic oscillations oscillation of electromagnetic field then what these operators do is they they add and subtract power again and subtract photons they are were called creation and annihilation operators who has set forth a full year that wave is the work of photons but you can't measure the plus in minors separately yard it's an interesting question a whether you could measure 8 plus or a minus individually in answer for technical reasons nor were there but our bar this doesn't tell you that but we have the idea Merrill of all operations a mathematical quantum mechanical operations which would make no sense class acquittal would we would multiply states by operators like this but quite mechanically we have the idea of creation and annihilation operators mathematical operations which they add and subtract quanta of energy for a fixed frequency now we have a system which has many of oscillators you could imagine we have many springs with many balls hanging from different frequencies all you could imagine electromagnetic waves in a cavity each different wavelength has a different frequency we can excite them independently weakened excite 1 way of Regan excited other ways excite them independently that we have a system which contains many different oscillations with many different frequencies that means to describe this may be a good place to slow down a little bit this is a this of course is a little bit and I just a bowler must have seen it before I make people seen it before but when a the 4 no not passing lecture other courts I think in many lectures on probably by I'm not paid and this course I have not talked about this yet the sequence of course yes we talk about many times I think but it certainly is In the lectures on how about we had to get court if is game for a long time so that they shed the harmonic oscillator the most basic or mechanical systems we've learned it if you want lectures and it is not you could you could see where to go and we do we describe particles as oscillations and we create and annihilate particles put them into the system take command of the system by mathematical operations that are these creation and annihilation operators Lee absolutely central be up to the understanding of God quantum field theory of particle physics just completely OK let's that's come back very quickly now and I think will have to discuss this again but let's come back to this will army world on a circle what we made the world a finite by making space to be a circle instead of an infinite wine and now it's very simple world of only 1 dimension but we have Line distant around l let's imagine photons are electromagnetic radiation or whatever kind of particles moving on this line and there are some allowable values on the way for number of allowable values of the wavering R L provided the body and weigh in any integer that means there are allowable values of momentum there are about allowable values of frequency and allowable values of momentum those allowable values of frequency we can call home may get in In goes from minus within NATO infinity whatever goes over our well in the past start from 0 to infinity all the possible that a frequent be waves moving on this system here are a collection of harmonic oscillators 1 harmonic oscillator for each value of the each frequency of oscillation
you can they had a energy independently to each frequency and you think any given frequency and an energy to oscillations in their frequency Ubon then a energy on top of that with some of the frequency sold a system of waves moving on a circle like this moving in anything for that matter it is really just a collection of harmonic oscillators with a certain set allowable frequency the question is how do you describe this system mathematically in quantum mechanics the interiors let's begin with the idea of a state when I can do this in detail Maryland actually getting get to our 1st example of a quantum field with the next time off but let's work around a little bit however we have labeled the states a system where there are many possible with mini oscillators you could even think about this is all oscillations of a waiver circle by just think of it as a bunch of balls and springs with different frequencies how would you label well each harmonic oscillator traders a quantized energy the quantized energy for each 1 of these answer raiders summit integer times for may gain times each book so what we have to stay there and is we are in danger X citations of each ocelot the number all of quanta With the lowest frequency boss frequency whatever it is more number of quanta with frequency into we offer 2nd from the number of quanta 3rd frequency and so forth and on but fringe AT&T forever this would correspond to a state in which Lois wave of longest wait along this wavelength oscillator has In 1 quantity in the next longest Waverly oscillations in 2 and 3 I'm getting a bit tired I suspect you are also but what I want to think about it the guitar strings of the violin strings and how you alone I you create harmonics would talking about us from our fixed prices overall mode of oscillation of strain were being used up and down like that if we forgot everything else and just thought of this as a simple harmonic oscillator oscillating back and forth with that the energy level the system is just 1 possibly know excitations 1 unit of excitation to you that expectation 3 of excitation how much would each unit of excitation call us it would cost an energy which was the frequency of that particularly no Omega times each times the number of oscillations With that frequency that I could add additional energy in the form of the 1st harmonic after this oscillation co-sponsor the Middle Sea this oscillation would be what you see 1 not above a little CEO whatever and some energy and that that has twice the frequency twice the frequency so it would be then a adding a number of quanta With the 2nd error frequency and then we get a oscillations on top of that there was trouble drawings next 1 became drive you normally Fredia With 2 nodes see to nodes but would have 3 times a frequency right 3 ways food to in house the oscillation like that has 3 times a frequency and harmony oscillators Haminoea quanta would we put it I wouldn't put any number in but I would be in the ring and the entire configuration of this oscillating strained quantum mechanically oscillating string would be characterized by a collection of integers which tells us how much quantized energy theories in each mode of oscillation basically what we have to to do mathematically Is repeat the mathematics of the harmonic oscillator infinitely many times 1 each mode of oscillation result is a quantum field that's quantum field is is collection a harmonic oscillators collection of creation and annihilation operators which they add and subtract energy law on in each mode of oscillation the mathematics recollection of oscillators which is called quantum field theory so if you wanna carry on with this subject may have to get used to this language OK from the scored here rapper say 1 more thing if we're interested in particle physics experiments opens particle physics experiments particles with given energy momentum coming in and be rearranged particles with other energy and momentum go out so another words we remove particles with nite we remove particles from the initial state and replaced them but particles in the final state and mathematical description of that is to remove particles with annihilation operators and replace them with particles of other the moment that energy involving other creation and annihilation operators that's what these operators do they remove particles per particles in mathematical description of the creation and annihilation of particles is through these kind of a mathematical symbols that what if did but DPP irregular it's also count if you are 1 integer has to do with the number of nodes here and the other has to do with a number of quanta House bill the savage if the are 2 Indistinct integers and this this example the number of nodes would sell you the frequency and then the number of quanta with each frequency for the 2 integers around that 1 of them is associated with the hot spot is not abstract formal please
visit and Epstein tracked down EDU
Kaltumformen
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Licht
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Werkzeug
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Metadaten

Formale Metadaten

Titel New Revolutions in Particle Physics: Basic Concepts | Lecture 2
Serientitel Lecture Collection | Particle Physics: Basic Concepts
Teil 2
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/15073
Herausgeber Stanford University
Erscheinungsjahr 2010
Sprache Englisch
Produktionsjahr 2009

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (October 12, 2009) Leonard Susskind gives the second lecture of a three-quarter sequence of courses that will explore the new revolutions in particle physics. In this lecture he explores quantum field theory.

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