Lecture 2 | New Revolutions in Particle Physics: Basic Concepts

Stanford University alright um 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 I'm going to teach you a very elementary version of what a quantum field is I think tonight but I do want to get up on the blackboard some basic mathematical facts just to review and to remind you this is too elementary to be a serious part of this class at this point but I do want to get it up there the first mathematical set of facts does have to do with waves waves are of course an example of fields field configurations fields or things which vary throughout space waves a particular kind of configurations in which the fields are moving in a wave-like motion and waves and particles are closely connected through the link of quantum mechanics I am NOT going to teach a quantum mechanics course this clod of this quarter so if your quantum mechanics is fuzzy uncertain haha go back to the lectures on quantum mechanics the two sets there was a class called quantum entanglement which we'll get up there hopefully soon and another class just called quantum mechanics and that is the basic starting point for what the particle physics is also special relativity but let's just review very very quickly one mathematical fact which comes up over and over and over and over again just what the relation between sines cosines and Exponential's everybody probably most of the people here know this but I just want to get it up on the blackboard for future reference okay let's remember what an exponential is an exponential is an exponential function it has the form an e e is a certain number to the Alpha X X alpha is a number it could be any number real or complex or pure imaginary and e to the 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 separate little interval of X it grows by an amount proportional to itself for example a population of a petri dish the increase in the population is proportional to the population itself and so the hallmark of an exponential let's call this f of X is that the derivative of f of X with respect to X is proportional in this case with the proportional a proportionality factor alpha times f itself the rate of change is proportional to F itself and such a function is called an exponential and it's denoted e to the Alpha X or something it can be a number of times e to the Alpha X okay that's the hallmark of an exponential now let me remind you of some facts about sines and cosines and then put them together very quickly sines and cosines you know what they are they're functions which oscillate like this that's sine of X and cosine of X is exactly the same function except shifted over by 90 degrees and it looks like that and I'm sure you all know this I won't bother going into detail but the important thing about sine of let's say KX let's call it sine of KX sine of K where K is a number any number real number in this case perms X could be positive could be net could be negative and the other function of interest is cosine of K at alright sine of KX and cosine of KX can be differentiated both of them and I'll just remind you what the derivative is now you'll remind me what's the derivative of sine of KX with respect to X the sine of KX with respect to X is equal to K times cosine KX and the derivative of cosine with respect to X is minus K sine KX so neither one of these is an exponential neither one has the property that when you differentiate it you get something back proportion it's proportional to itself they have the odd property that when you differentiate 1 you get something proportional to the other whether we're in one case with a minus sign cosine KX thank you cosine KX so you differentiate this one you get the other one you differentiate the other one you get minus the first one alright so out of these two you can now build a exponential but it's an exponential of an imaginary thing so let's just let's just do it once and be done with it if you take the combination cosine KX plus I sine KX I being the square root of -1 and you differentiate it with respect to X what do you get you get from cosine you get minus sine KX plus I I I'm sorry minus K sine KX plus I K cosine KX did I do it correctly now I think I did which is just spend a couple of minutes doing the Algebra I times K T times the original thing back again cosine KX plus I sine of KX in other words to make a long story short cosine KX plus I sine KX is the function e to the I KX when you differentiate it with respect to X you get I K times the same thing all right so that's that's an important mathematical fact that comes up over and over again sines and cosines of course the functions which describe wave-like oscillations either oscillations with respect to space if X is a space coordinate there's an oscillation the La Salette toolspace if X happens to be time it corresponds to an oscillation with respect to time I will come now let's uh let's consider in a little more detail well not so much detail but just quick review all right we have a wave the wave is moving past us it has a certain frequency and it has a certain wavelength let me just remind you of the connections because we use them we can call f the frequency it's the number of cycles that pass you every second frequency frequency and it's measured in cycles per second C stands for cycles per second a cycle is one full oscillation of the wave another definition now this is definition Omega is another form of the frequency it's the angular frequency measured in radians per second and it's just equal to pi times F so f and Omega are the same thing except for a factor 2 pi ah and I may go back and forth between them try to keep equations as simple as possible with as few factors of Pi so sometimes I may use Omega sometimes I may use F okay lambda is of course the wavelength and now for a light wave a light wave moves at the speed of light because it's moving past you with the speed of light the wavelength and the frequency are related and they're related by the formula the wavelength is the distance traveled in one full cycle and if you multiply that by the frequency that gives you the speed of light that's the velocity the frequency is how long well the inverse of the time that it takes to go one full cycle so this is a distance divided by the time that it takes to go one for the full cycle and for a light ray this is see if it's a sound wave then you put the speed of sound there if it's a water wave you put the ball speed of water waves there or whatever it happens to be but will be of course particularly interested in in light waves now that's one set of equations the other set of equations that we discussed last time just quick briefly view review of last time is the equations of quantum mechanics which relate wave-like motions the particles and we're not going to go in heavy depth into the explanation of these equations for that you go back and open up the the lectures on quantum mechanics but nevertheless I'll try to be as complete as possible or at least as self-contained as possible or the important thing when going to quantum mechanics is a new number Planck's constant it comes in two varieties two separate notations there's the old Planck's constant which was plunks original notation and then there's the new Planck's constant which is the same as which is this which is the same as the old Planck's constant except for a factor of 2 pi and that 2 pi is put there so to save 2 PI's in somebody else's equation but if they always come up and there's just no escape from 2 PI's they're always you're always going to have them but the other definition is H bar and that's the old planets a new Planck constant that's the old clock constant divided by 2 pi I will simply save 2 PI's by sometimes using H and sometimes using H bar try to keep the number of 2 PI's to a minimum now armed lambda times let's say okay yeah that was that's not an equation that's just a definition and what is H H is a number which is a tiny tiny tiny number anybody remember what it is 6 times 10 to the minus something 34 or something in some units and some units are another oh that was an interesting question it sure my email it showed up somebody questioning whether the origin of the unit meter pointing out let's see what does the meter the meter is one forty millionth of the way of the circumference of the earth and said no I was wrong it didn't have anything to do with this it's 140 million 30 do you think that somebody just came along and said let's define a unit which is 140 millionth of the circumference of the earth no of course not that unit had been in use I don't know what it was called a meter or not in England it was called the yard but why 140 million why not 140 trillionth or 140 well because you wanted some unit that was easy to deal with for a human being good to measure rope with good to measure cloth with whatever and I do believe that the origin of a unit which is about the length of a human beings arm predated knowledge about what the circumference of the earth was anyway which in fact was known for a long time but there brother yeah in any case in standard units Meteor seconds and kilograms h-bar is a very small number do either write something like 10 to the minus 34 yeah one point that's little H bar that's H bar H is 6 times 10 to the minus 34 or whatever and C of course is 3 times 10 to the 8th meters per second 3 times 10 to the 8th meters per second right okay so this is just a definition of some number of definition between a relation between two numbers but the fact is that the energy of a single photon a photon associated with a wave of a given wavelength and therefore a given frequency notice this equation here tells you the relationship between frequency and wavelength frequency and wavelength are inverse or inversely related the bigger one is the smaller the other is the energy of a single quantum single quantum energy now this is not the energy of an electromagnetic wave this is the energy of one quantum of the electromagnetic wave that's equal either to h-bar Omega where Omega is the frequency of the wave the angular frequency of the wave incidentally the frequency of an electromagnetic optical an optical wave one that you can see with your eye is very very rapid something like about 10 to the 15 cycles per second they'll have that right about 10 to the 15 cycles per second so typically Omega is very very large into the 15 but a little over per second but h-bar is more than small enough to to make the energy of a single quantum very very small if you multiply these two in the units that we've been using you get the energy in joules and you find out that the energy in joules is very small 10 to the minus 19th or something like that so we take a huge number of optical photons to add up to any appreciable macroscopic conventional energy all right so the energy and this of course is also equal to the other Planck's constant times the other frequency so they're the same thing are just two different notations all right now let's talk about momentum let's remind ourselves about momentum energy incidentally and momentum are especially interesting because they're conserved quantities they're conserved our total amount of energy is conserved total amount of momentum is conserved and therefore the good bookkeeping device is good for writing balanced equations and useful among other things besides being deep momentum is a vector it points in a direction points typically in the direction of motion of an object it's defined by being conserved okay but having said that it's defined by being conserved by the fact that it's conserved we count it and so forth we can ask what is the momentum of various kinds of objects in nature and the first thing we can write down is the momentum of a non relativistic object non relativistic object just means one moving slower than the speed of light Oh a lot slower a lot slower than the speed of light a lot slower the momentum of such an object is proportional to its mass times velocity and its velocity is also a vector if I give you any pulse of radiation electromagnetic radiation and it's all collimated along given direction not a dispersed wave which is going out in a variety of different directions but in which everything is lined up and collimated along an axis then this is nonrelativistic the opposite extreme is super relativistic in other words moving with the speed of light now this formula is not right for moving with the speed of light you could plug in for V the speed of light but what do you plug in for M remember I told you M is always by definition the rest mass where we used to call the rest mass and a photon doesn't have a rest mass okay its rest mass zero try to bring it to rest has nothing but this is not the right formula nevertheless the right formula has a look which looks a bit like this if we were to have used inappropriately e equals mc-squared now mainly I'm using this for unix in order to get units right energy is mass times velocity squared then of course mass would be an energy divided by the speed of light squared all right now if I as I said inappropriately ascribed to a photon a mass equal to its energy divided by C squared the energy of the photon divided by C squared and then said that the velocity of the photon was C in fact more or less somewhat accidentally I would get the right answer that the magnitude of the momentum this should be read as the magnitude of the moment Yeah right but the magnitude of the momentum is the energy of the photon or the energy of the electromagnetic wave for any electromagnetic wave all moving in a constant direction the direction of the momentum is of course the direction of motion of the wave that that's easy but the magnitude of the momentum is the energy of the wave divided by the speed of light from that we can say supposing the wave consisted only of a single photon now we're using it legitimately yeah you get the right answer what is the momentum of a single photon and to get the momentum of a single photon you just plug the energy from here into the formula for the momentum so let's do that the momentum for a single photon well let's let's do it over here momentum of a single photon let's write photon over here cannot be read everybody read that the momentum of a photon and I'm concentrating now on its magnitude incidentally the double bar here means the magnitude of it the direction is along the direction of the beam of light or along the direction of the photon let's plug it in the energy let's see let's use this formula over here H times frequency energy is H frequency divided by C all right what's the frequency divided by C right or well 1 over the wavelength of 1 over the wavelength frequency divided by C I've divided by C is equal to 1 over lambda right did I get that right yes frequency divided by C is 1 over lambda so this is just the equation that the photon momentum photon momentum in magnitude is equal to Planck's constant that makes it small times the frequency divided by C which is divided by the wavelength so the smaller the wavelength the larger the momentum of the photon that's a pattern and it's in some sense the most important fact coming from quantum mechanics about particle physics that if you want short wavelength for the purposes of studying very small objects the price that you pay is you have to put a lot of momentum equals each h-bar Omega of God is that the fundamental or is that derived from some symmetry it derives from the quantum mechanics of a harmonic oscillator so the place to look it up is on lectures previous lectures about the harmonic oscillator I will make use of those lectures but that's where it comes from from a deeper point of view will come will come to the the right statements although not the demonstration of them young just attention this is we have ways photons are under a wave it has an amplitude and always have amplitudes but a photon itself we haven't talked about it talk to you giving it such a net let me answer the question now I would I I know it was there and I was waiting to get to the right point but why don't I just say it now right an electromagnetic wave or any other wave has several features one of which I have not written down here it's the amplitude of the wave now the amplitude of the wave well okay here are two waves with the same frequency the same every and every respect the same except that as a and here's another one one of them is just some small number times sine KX for example maybe the small one is a sine KX the other one is a great big numerical multiple which I'll call a times sine KX they're exactly the same except one is much stronger than the other it's got a larger amplitude a is the amplitude of the wave and now the question is in photon language what is the meaning of the amplitude or another question you could ask is what is the amplitude associated with one photon now the question is a little bit ambiguous because quantum mechanics always has uncertainties technically it has uncertainties and you can't measure everything simultaneously for example you can't measure both the amplitude and the phase of the wave the phase of the wave is where it's located but never mind that or there's a sensible order magnitude question order magnitude what is the amplitude that's associated are with a single photon for example okay the answer is the following the energy of a wave now this is a completely classical statement completely classical statement the energy of a wave is proportional k times the square of the amplitude for example if this was the electromagnetic field let's say the electric field or the magnetic field associated with an electromagnetic wave what's the energy of an electromagnetic field it's proportional to the square of the electromagnetic field so typically the energy of a wave is proportional to the square of the amplitude if you take that good now the energy of the wave is also proportional to the number of photons in that wave so here I've described for you some features of a single photon but a wave of light consists of many photons let's suppose the number of photons in some wave of light is in then the energy is proportional to n in fact the energy will be in the number of photons times H bar Omega so that means that there's a proportionality between the square of the amplitude on the one hand and the number of photons in that wave let's just call it little in the precise exact detailed connection I'll work out for you another time but the square of the amplitude of the wave is proportional to the number of photons now there's a factor of H bar in there because the square of the amplitude is the energy and we should have an H bar H bar Omega and the energy so the square of the amplitude is proportional to H bar and in now of course the point is that H bar is an extremely small number this is the energy of the wave another form another way of writing the energy of a wave with many photons not a single photon but many photons right now you say well look this is a terribly tiny number how can you get any appreciable amount of energy in an electromagnetic wave well the answer is by having a lot of photons so a typical electromagnetic wave that carries an appreciable energy has an enormous number of photons to overwhelm this factor of H bar but the amplitude if you like a is proportional to the square root of the number of photons so that's the missing that's the missing wave characteristic that I didn't write down here and as I said it is proportional to the confusion about the oscillation we used in a diagram of the MV field that are perpendicular and there the temptation to think of that as a spatial you know the height of the sine wave no no right okay right okay so here's a wave it's moving down the x-axis what's plotted vertically here is not the y-axis or the z-axis what's plotted vertically the electric field right so this is not a plot there's only one of the directions here corresponds to space the other direction of course corresponds to the electric or the magnetic field I think that's what you are asking me to clarify and properly so now of course the wave could be oriented in any direction that way that way that way and the momentum carried by the wave is along the axis that the wave is moving yeah say number of photons in the way oh you you mean number of photons within one period let's say you move in one third of it yeah yeah because a square this is like the powered energy that's right so that's why this area to say within one within one wavelength that's right well the proportionality I purposefully wrote proportional to here if we were talking about two wavelengths three wavelengths for wavelengths of course the answer but guess you got the out you got the right idea well order-of-magnitude it's given by this formula with n equals 1 but yes in some sense it is meaningless it says meaningless for an electromagnetic wave electromagnetic wave has both a magnetic field and an electric field now the magnetic field in the electric field are related to each other by a relation which is similar mathematically to the relationship between the position of a particle and the velocity of a particle and those are things which cannot be measured simultaneously they are the uncertainty principle prevents you from knowing both of them and in the same way there's an uncertainty principle between electric field and magnetic field that prevents you from knowing both of them so yes there is it there are limitations on how well you can know the amplitude but as an order of magnitude statement this is correct well we'll try to do better as we go along we're sort of getting into the subject of quantum field theory which is a difficult and abstract subject ok um I wrote down some equations here which are true for photons some of them are true for any particle others are not alright the definition of frequency that's the general definition Omega being 2 pi F that's perfectly good lambda equals a wavelength that's a definition this is true for waves which will move with the speed of light so it would not be true for a general wave let's let's 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 wave or particle whether or not it moves of the speed of light this is not so general this is not so general anything that has a seen it will not be very general when I say not general it'll be general for anything which moves with the speed of light ok this is general now energy equals H bar Omega that is also a very general fact this clearly neither of these this is specific to things which move with the speed of light this is specific to objects which move with much less than the speed of light there's an interpolation in-between which I won't get into now which is true for things which move with some fraction of the speed of light so I can um where were we yeah we were we were trying to figure out which equations are great generality which are not this one H F over C that contains C in it that is not terribly general oops where's my blue momentum equals H divided by lambda is general okay now where this comes from I'm not going to try to prove these things to you now I'm just telling you as facts go back to your quantum mechanics momentum is H over lambda that's general okay let's use these facts now here's the relationship for a photon between what is it arm between frequency and wavelength and as I said it's not very general what about a slowly moving electron or a slowly moving nucleus or for that matter a slowly moving bowling ball what's the relationship between frequency and wavelength can we figure it out is it something on the blackboard or something that we have to add something else question for an electron I did tell you last time that electrons are also described by waves did I not I told you that you can do interference experiments with electrons so there is an electron wave let's call it Sai and the question now is what is the relationship between frequency and wavelength can we figure it out yes we can but we need to know one fact about nonrelativistic systems it's the analog of knowing it is not the analog it's knowing the relation between momentum and energy okay but for an electron moving slowly the momentum is not the energy divided by the speed of light what is it okay energy is equal to now we can start at the beginning and right I mean kinetic energy one-half MV squared and the momentum MV P equals MV so we can solve for the velocity P over m everything is a vector but that's alright we don't know that's right and then the energy becomes one-half M times V squared and V squared is P squared over m squared if you don't know this equation it's an important equation it occurs over and over the energy in terms of the momentum is one-half P squared divided by M people get confused the M is upstairs when you multiply by the velocity squared its downstairs when you multiply by the momentum squared okay now we have what we need to find the relationship between frequency and wavelength energy is where is it is h-bar Omega or H times frequency let's use H times frequency of a single of a single electron now we're talking about a single electron H times the frequency of the wave describing that electron H times frequency that's energy on the right hand side we have one-half let's put the N downstairs and what about the momentum the momentum is H divided by lambda so that's H squared over lambda squared and we can divide this equation by H and we find that the frequency is 1 over 2m times plunks constant divided by the square this time of the of the wavelength notice how different it is are over here the frequency was proportional to 1 over the wavelength here it's 1 over the wavelength squared you don't have to remember this you have to memorize this you can always read the rive it the basic ingredients are the equation for the energy and the relationship between momentum wavelength frequency and all that kind of stuff all right this is just the illustrate the connections so that's a non relativistic particle when moving slowly for example electron in an atom the frequency of its Schrodinger wave would be 1 over 2m times Planck's constant divided by the wavelength squared let me stop me slow down now and pick some questions that is true for any particle as long as it's moving slowly compared to the speed of light maybe we should well we'll come back and we'll interpolate between the slow and the fast another time there is an interpolation the gino is well depends on how fast a neutrino is moving most neutrinos in nature are moving very fast with close to the speed of light very close to the speed of light so for neutrinos the speed of light is a good approximation for but in principle neutrinos can be slowed down and if they're moving with very much less than the speed of light it's the other formula a non relativistic formula yeah and so that's where electrons passing wait yeah welcome back to a relation between amplitudes and and and quanta soon enough and that's that's what the whole story is about but we haven't gotten in an electron accelerator let's say with 20 kV however accelerator with 20 kV very low energy exhaustion example it's a slow fest it's a lot to consider slow or fast damage for an electron slow so all right well you I could alright um if you want to know first of all let's work in units in which C is equal to 1 in such units momentum and energy have the same year exactly the same units where do we have it momentum and energy have exactly the same units of C is equal to 1 if you want to know if something is relativistic a non relativistic you want to compare its momentum to its mass if its momentum is large compared to its mass its rest mass photon has no rest mass and so its momentum is always much larger than its a that its mass and so it's highly relativistic its relativistic as it can be the figure of Merit the thing that you compare is the momentum to the rest mass so now you have to tell me what's the rest mass of an electron you gave me in ke V ok ke V killer electron volts is a unit of energy all right and remember energy and momentum have the same units so what's the mass of an electron half a million electron volts so 20 ke V is a lot less than 20 then half a million and so that's a slow that's a slow electron right any other cookies any other question frequency and wavelength for a real way or related yeah start participar clockwise ready so is that equation the left one you have did you rewrite a facebook scheme and get the right equation but electrons electron waves do not move with the speed of light so there's no C so the question is can you derive a velocity from this yes the answer is yes you can but it won't be lost ah maybe probability well oh there are two yeah yeah there are two velocities to go to the waves one of them is called the group velocity one of them is called a phase velocity I'll I'll be specific since you asked me if you take a general wave the wave let's assume it's a wave packet now a wave packet means that it looks like a wave except that it it's some confined to some packet like structure like that so it locally looks like a wave meaning to say a plane wave a sine or a cosine but it has an envelope that that sort of combined confines it to some region now in general in general depending on the specific nature of the wave you'll see that as that wave moves there are two separate velocities associated with it and they're different one of them is the overall velocity of the wave packet the other is the velocity of the crests and troughs or whatever you call them and what you would see if you looked at this wave let's say from some other perspective where you see waves and troughs moving along is you would see the packet moving with one velocity but the waves moving through the packet with a different velocity in fact typically the wave the the one is called the phase velocity that's the velocity of the troughs and peaks and valleys and the other was called group velocity the group velocity is the velocity of the whole packet guess which one you identify with the nonrelativistic velocity of the particle the group velocity and the group velocity is usually slower than the phase velocity the phase velocity is not got to do with the actual velocity of the particle now there is one special case the special case is when all waves of all frequencies move with the same velocity that would not be true for a none relativistic wave like this let's see the velocity is the frequency times the wavelength right so let's let's look at this the frequency times the wavelength is just frequency times wavelength as 1 over 2m h-bar over lambda so different lambdas move with different velocities different lambdas move with different velocities and when different wavelengths move with different velocities you have this situation where group and phase velocities are not the same for the special case where all wavelengths move with the same velocity then group and phase velocities are the same so for a light wave all waves move with the same velocity the group and the packet velocity the the group and the phase velocity are the same that's a special and simplify a simple case all right but we didn't I just wanted to illustrate some of the equations here and incidentally this equation over here is equivalent to another equation of physics anybody know what it is the Schrodinger equation the Schrodinger equation is an equation for a wave and this tells you the time dependence of the wave the frequency this tells you the space dependence of the wave the Schrodinger equation is exactly an equation for the relationship between the time dependence of a wave and the space dependence of the way we will come back to it but this is just the Schrodinger equation in a slightly disguised form for better yet if you knew the Schrodinger equation and you said supposing I had a wave of a given frequency that had a given wave length what would the equation tell me about the relation between wavelength and frequency and it would be exactly this okay so that's the Schrodinger equation in a nutshell I think the face velocity grow faster than life yes yes yes yes but the phase velocity as a rule carries no information with it you can't send information faster than the group velocity phase velocity doesn't really carry any any information with it so we'll come back to that it's it's a technical issue that we don't need to deal with right at this moment okay let's see let's go on a little bit more with waves there are one or two other concepts about waves that are necessary and you know there's a lot of technical details and I'm getting into but every single piece of it is really very central or important okay we will be interested in waves moving in infinite space but I use the word infinite space infinite infinity computers don't like infinity and if physics is the art of writing equations that computers can solve in principle it is necessary to always remove the infinities and write the equations for finite systems first and then take a limit all right so in studying quantum mechanics waves quantum field theory particle physics and so forth it's necessary in the back of your mind to get rid first around of everything that's 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 so one way of what I want you to think about now is a wave moving in one dimension three dimensions is really no different than one dimension except there are three of them so let's concentrate on one dimensional wave motion could be a violin string it could be a rope being wiggled back and forth or it could just be a wave in a mathematical universe of one dimension one way of getting rid of the infinity of space would be to make the wave only propagate on a region of length L for example by holding down the wave at this end holding down your string at this in just like a violin string is held down holding it down at this end you can have all sorts of waves and the waves move and they bounce off when they get to the end they bounce off and reflect and they go back and forth and back and forth that's a perfectly good way to describe things the problem with this kind of let's call it boundary condition this is called the boundary condition namely that waves when they get to the end reflect it's called a reflecting boundary condition the problem with it is it violates one of the important conservation laws of physics what would that conservation law be momentum after all if the momentum of a wave is it's a vector quantity it points in a direction if a wave is moving along this at has a little wave it's moving along this axis its momentum is pointing to the right when against to the end it bounces off and reflects so the momentum has changed sign okay the momentum has changed sign the momentum is not conserved that's okay there's nothing wrong with it you can study it but if you want to keep track of things like momentum conservation there's a smarter thing you can do it goes as follows imagine this line is such that we identify the beginning and the end now we can do that by replacing the line by a circle it doesn't have to be a legitimate geometric circle just mathematically we connect the beginning back to the a so we go on a cycle alright so as we move from here to here to here we can imagine we start here move along here and in this magic line when you get to the other end you come back to the beginning that's a closed circle I don't want to call it a circle because I don't mean to imply it's a circle in two dimensions I just want to imply that it's periodic periodic is the term for this periodic space now you can imagine a wave which is moving along moving along here and when it comes to the end over here what does it do it just keeps going right it just keeps going it doesn't care about that point that point is not a special point anymore and so the wave just keeps cycling around cycling around cycling around and the momentum is conserved it doesn't suddenly reflect into the direction in terms of this picture you can just imagine when the wave gets to this end it reappears at this end and it just keeps going to the right if it were going to the right it go to the left of it we're going to the left and just keeps going forever with this trick with this mathematical trick of replacing space by a periodic space we can first of all get rid of the Infinity in the size of space feed it into a computer and we can get rid of the problem of momentum of noma of non momentum conservation momentum is now conserved but there is a cost I don't know that it's a cost but it's a fact the fact is that in such a world momentum is quantized now I don't mean in the sense that well it is that momentum only comes in discrete multiples of a unit and it's useful to work in understanding quantum fields that relation to particles and so forth is useful to remember that the momentum comes in discrete multiples of something so let's see how that works well first of all it's obvious that the wavelength an integer number of waves must fill it let's let's give this distance around here a name let's call it L L for length if it were a circle it would be the circumference but let's just call it L the total distance around from beginning back to beginning again quote L all right if we're going to put a wave in there and let's say now that it's a way of a sine or a cosine wave assault whatever an integer number of wavelengths has to fit onto the circle that means that the wavelength has to be equal to the length around divided by an integer okay divided by an integer so the wavelength must be a total distance around divided by some integer I don't want to call the integer in let's see a calculate capital n Capital n little n will be reserved for something else so the wavelength is necessarily the total distance around divided by n now that means the wavelength comes in discrete possibilities it can be L it can be L over 2 it can be L over 3 it can be L over 4 but it can't be L times the square root of 2 or L divided by the square root of 2 an integer number of waves just won't fit in here in other words if you try to put a wavelength of the wrong wavelength in here by the time you got back the wave wouldn't match was what it was supposed to be at the beginning ok so that tells you that wavelengths come in discrete calm possibilities with the integer appearing the denominator now we had an equation here somewheres for momentum in terms of wavelength what was it momentum was equal h-bar yes H over lambda P is equal to H over lambda so now this is also telling us that momenta come in discrete possibilities the momentum of any quantum must satisfy the constraint that it's H divided by an allowable wavelength so the allowable momenta of quanta the allowable momenta of electrons the allowable momenta of photons the allowable momentum of anything in this one-dimensional world has to be H divided by okay let's work it out that's H divided by lambda that means n divided by L I turned lambda upside down and over L and what does it say it says the momenta come in discrete multiples of a basic unit and the basic unit is Planck's constant divided by the total distance around there's a neat fact if you put a particle and you put it in if you put a particle on a wire and the wire is a circular wire a bead a bead on a wire that bead can only have momenta which are discrete multiples of 1 over L naught n over Ln is the multiple 1 over L 1 over the length of the wire now of course H is a very small number and the L if L is a big macroscopic length of wire or something 1 over L is a very small number and so the quanta the separation between different mo values different allowed values of the momenta is very small you can get own if L is large and H is small you can get almost any number not almost any number but you can get a very dense collection of possible numbers all right but okay that's the that's the quantization of momentum in a world in which everything is periodic yeah total momentum not total momentum of the of a of the object total momentum of any quantum now if every quantum has to have a integer multiple of this basic unit of momentum then any number of quanta also have to have an integer multiple of it all right you might have each quantum is identified in terms of its value of in in terms of its momentum but there are allowed momentum and disallowed momentum the allowed momentum are discrete multiples of a basic unit and any number of photons electrons and so forth necessarily come in this discrete multiples this is we get to get stationary model when you come back to the same place what is that scene something well that's the assumption that that the space is really genuinely periodic that there's nothing special about any point and it is a fair question it is a good question but it's also the assumption that for example the electromagnetic wave electric field is a observable quantity and it doesn't have sudden jumps in it in fact a sudden jump you would have if it didn't come back to itself we would have to make a sudden jump at this point a sudden jump is very costly in energy and in momentum so sudden jumps and fields fields are continuous things they're not allowed to jump like that if you have what what can we package eyes oh that's alright any packet ah that's a good word any packetized wave that's a packetized wave is a sum of individual plane waves sines and cosines that's the Fourier analysis the statement the real statement is any allowable wave is a sum of waves each of which has this discreet this discretized the possible values of momentum so it's a statement about Fourier transforms that the Fourier analysis of any allowable wave only has these discrete multiples of a basic unit of P the center problem would help very very small yeah small as momentum it's very very very big ah how small very small you don't wanna get smaller than the plonk line that's it that's it that's it that's a bad thing to do up till that now there's no problem oh we could yeah what let's pursue this a moment something that's not in my notes but let's pursue it it's important let's really imagine now that we genuinely have a particle moving on a genuine circular wire a fun let's really now take a bead moving on a really circular wire and see if we can understand what this quantization of momentum means in another language let's take the circular wire to have a radius R let it have a radius R oh I erased the important equation what was the important equation the important equation was the allowable momentum is equal to an integer times H over lambda over sorry over L or L right okay now the radius of the orbit is R and that means that P is n H over 2 pi all right you can see by now that h-bar would have been a better choice in this equation right let's leave it let's leave it and later on we'll put back h-bar okay that's the momentum and the momentum is quantized every single electron photon or anything else has to have an integer multiple of this now what about the angular momentum what's the relation between momentum radius and angular momentum for a circular orbit for the simplest possible ah yes it is every R well for a non relativistic particle its MV R and we are L stands for angular momentum MV R but what is MV P so as P times R now NV r is not completely general for a relativistic particle the momentum is not n times V but angular momentum is equal to momentum times distance is very general that is for particle that is the definition of the momentum at least for a circular orbit for other orbits it's the cross product R cross P but it's always R times P and so now look at this what this says is that R times P multiplied by our angular momentum is an integer multiple of H divided by 2 pi or an integer multiple of plunks other constant alright so momentum is another quantity which is axial angular momentum is a quantity which is quantized but always in the same unit unlike the momentum here the momentum is quantized in a unit which depends on the angular on loops that I use alpha 2 things excuse me what should I call our angular momentum angular momentum am I will eventually change that I don't like that but but for the moment angular momentum alright is n times H bar momentum is quantized in this periodic world but the unit of momentum depends on the length around the circuit okay the angular momentum is quantized but always in units of h-bar okay all angular momenta come in units of h-bar except when they don't when they don't is when you have fermions but we'll come to that that's later not tonight okay there are exceptions to this rule okay um 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 is a quantum field how is the quantum field related to particles and to its quanta what is the connection between particles and fields with some precision and that's what we'll do next but let's take a let's take a five-minute break alright now we have to enter into the real world of quantum mechanics a little bit and for that if you're not our knowledgeable about it you have various alternatives one is to go to sleep another is to listen and try to get you know sort of a what's the right word for a very impressionistic what yeah you know get the oh there must be some word for very impressionistic sort of soft fuzzy feel for things third is to go back left what I hope to do better than that please and the third is to go back and learn a little bit about of quantum mechanics there will be nothing here that I haven't discussed in previous lectures on quantum mechanics in particular the most important thing for our purposes now is one of the simplest of all quantum systems the harmonic oscillator why does the harmonic oscillator come up the harmonic oscillator is important for a great many reasons but in our particular context it's because waves are oscillations if you just see a wave going past you obviously the amp with the field electric magnetic whatever it is oscillates in fact the fact that oscillates is equivalent to the statement that it has a frequency a wave maybe way made up out of a superposition of many different wavelengths for example light you can mix together different colors of light and create light of mixed different wavelengths but you can also separate it as did Newton into its component wavelengths and so every light wave can be thought of as being made up of individual components each with its own frequency its own amplitude and so forth therefore it be thought up as being made up out of a collection of quanta and 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 each energy and momentum is associated with a wavelength each wavelength is associated with a frequency and a frequency is by definition and oscillation so one can think of a wave as a collection of harmonic oscillators harmonic oscillators being the simplest kind of oscillation sines and cosines in order to proceed with what a quantum field is we have to go back to what the harmonic oscillator is so let's let's begin with the harmonic oscillator quantum mechanics is a strange theory it has things numbers in it which don't satisfy the usual rules of arithmetic technically they are operators representing the observable things that you can measure and they have some rules you can add them you can multiply them but they don't necessarily commute what does that mean that means if you have two variables describing two different attributes or properties of a system each of which by itself might be measurable might be something you can measure let's call them a and B just to give them names a could stand for example for electric field B could stand for magnetic field or a could stand for the position of an electron B could stand for the electric field but any two pair of things then ordinarily in ordinary mathematics of course the product of them doesn't matter which order you multiply them a times B is equal to B times a that's just the property of numbers in quantum mechanics of course that is not generally true it may or may not be true of two observable quantities well it's not true we just write that the difference between these is not equal to zero now what nonsense I mean if I measure the electric field and it's some number and I measure the magnetic field and it's some other number and I multiply them what how can it possibly be that it doesn't make sense to multiply them in the opposite order and get the same answer generally speaking when the answer depends on the order of multiplication what it really means in other words when this is not zero what it really means is that you can't measure both of these things simultaneously that they interfere with each other the experiment that measures one of them necessarily screws up the measurement of the other one and in that sense there's no contradiction when this is not zero simply means you can't measure both of them together when you can measure two 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 order you multiply them in and so the test for whether you can measure two things at the same time is whether this quantity which is also called the commutator a B minus B a is called the commutator of a and B commute commutator you know like going back and forth go in to working back you interchange the order of them it's labeled by a bracket with a comma and a and B the order matters so typically this is not equal to zero incidentally the commutator of B with a e that's just exactly the same thing except in the opposite order and that's just a sign difference these two are the same except for a sign okay that the that's quantum mechanics in a nutshell we just did quantum mechanics examples now I'm not proving this I'm just telling you this now but nevertheless it's fact partly empirical fact partly mathematical fact a particle has different components of position x y&z for example you can measure the position of a particle so the components of position of a particle are candidates for things which are observables these things commute with each other you can measure the X and the y of a particle at the same time you can measure the X the Y and the Z of a particle all three simultaneously what else can you measure about a particle you can measure its velocity or its momentum mass times velocity and those things are called the components of them are called px py and PZ they're the components of the vector momentum and they also commute with each other you can measure the three components or better yet the magnitude and nima and the direction of of the momentum but you can measure all three components simultaneously there's no obstruction they commute with each other so X commutes with Y commutes with Z P X commutes with py commutes with PZ but the components of position do not commute with the polar components of momentum so for example you cannot measure the position the X component of the position of a particle and the X component of the momentum every time you try to measure the X component of position you will give the particle a whack and change its momentum and the result is you can't measure both of them this is not equal to zero it's equal to I times Planck's constant it's small simply reflecting the fact that well the uncertainties in position and momentum are small and of order H bar likewise for the Y components the Y position and the y momentum don't commute with each other same for the Z components how about X with the y component of momentum can you measure X and the y components of momentum simultaneously yes as a matter of fact you can that that may not be obvious but it is true okay so that's another aspect of quantum mechanics and now let's consider the harmonic oscillator now I'm telling you facts for the most part but I'll try to put the mathematical facts into some kind of context these are the facts that we went through previous quarters so I'm not going to derive it when you go from X to Y no this is the square root of -1 this is not the I J this is not our J and K which are unit vectors along the three axes and they're also not quaternions which also have an i J and a K associated with XY and Z but nope just I just the square root of -1 where does the square root of -1 get into a formula like this don't don't ask we we we don't we're not going to need it the fact is we won't really need it we just need to know that algebraic manipulations in quantum mechanics are a little bit funny and the funniness is that you can't as a rule interchange the orders of our operations sir yeah that's not a ap - beer that means first measure a it doesn't mean measure this is a it doesn't mean measure forget measuring are these are simply mathematical manipulations which are yeah it doesn't mean first measure a and then measure B it's simply a not simply it is a mathematical manipulation that the matrix multiplication operate a multiplication yes but we're not going to we're not going through the principles of quantum mechanics here I'm just restating things that we've learned in the past and if you need to understand them better the only way to do it is to go back to earlier lectures okay nice and sliced but close there is a it doesn't really apply to a photon it applies strictly speaking to nonrelativistic particles the problem is that the position of a photon is not so well defined the position of a photon is harder to define in the position of a non relativistic particle so but let's say we can come back to that that's a subtlety which is not terribly important to us now okay the harmonic oscillator the harmonic oscillator is a system a classical system which has a quantum mechanical analog and it simply means a system which oscillates a weight on the end of a spring which oscillates up and down on a sound wave that oscillates a light wave that oscillates anything that oscillates like a spring going back and forth we can think of thousands of examples and the mathematics of them is always the same every oscillation or every oscillator has a frequency it can either be taken and we can call it an Omega the frequency again is 2pi times the number of cycles per second alright cycle per second for a spring with a weight on it is just how long it takes to go through one full swing and come back to the same place not just the same place but the same velocity so if you have a weight on the end of a spring and it oscillates it's the the frequency is not how long it takes to come back to the same place but how long it takes to come back to the same place moving in the same direction so one full oscillation is the period the number of cycles or number of oscillations per second is the free qín C F and as usual Omega is 2 pi F nothing unusual the usual Omega now what is unusual about the quantum mechanical harmonic oscillator is that energy is quantized not momentum not the but energy is quantized this is something which you'll either look up or accept so let's plot vertically the energy levels of such a harmonic oscillator classically of course there was a lowest energy which is zero just the oscillator at rest and then you give it a little poke and it starts to swing back and forth and you can give it any positive value of energy of course if you hit it too hard and try to give it too much energy you might melt the spring or I'll break the spring but in a mathematically idealized version of the harmonic oscillator you can give it any positive value of energy not so the quantum mechanical oscillator the quantum mechanical oscillator has discrete integer units of energy if we call the lowest energy 0 sometimes it's appropriate to call the lowest energy not quite zero but we can define we can define the lowest energy to be zero then what's the next energy level h-bar Omega the same formula as the energy of a photon of frequency Omega what's the next one - h-bar Omega that could either be the energy of a single photon associated with a frequency twice as big as the first one or it could correspond to two Tan's both associated with a wave of frequency Omega it's the second idea that I want to dwell on that this has the same energy as two photons now so far there are no photons there's only Springs and balls and oscillations back and forth but nevertheless this does have the energy of one photon associated with a frequency Omega this has the energy of two photons of frequency Omega and so forth three h-bar Omega and so forth those are the possible energies of the harmonic oscillator nothing in between we can label them with an integer and we can write that the energy is an integer times H bar Omega wherein is just an integer which labels which one of these states the oscillator is oscillating in 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 N and it's conventional to just label the states of the oscillator by a little pointy thing like that that's called a cat it's called a ket because it's the right hand side of a bracket but we don't need the brat in that we only need the ket end of it and it's a as I said go back and study quantum mechanics but all this is all this labels this is a symbol representing if you put N equals 0 here it represents the state of the oscillator where it's in its ground state ground state means the lowest energy state if you put a 1 here it represents the oscillator oscillating with one quantum of energy one unit of energy you put a 2 here it represents the oscillator with 2 units of energy and so forth right so it's just a note it's just a convenient notation for labeling the various possible configurations of the quantum harmonic oscillator now let's invent an operation it's actually not only an operation it's an operate or there's a difference an operator and this operator acts on a state to change the state to a new state so a mathematical operation which takes you from any state to the neighboring one one unit above it acts on this state to give you this one acts on this state to give you this one acts on this state to give you this one acts on this state to give you this one so we can write down what it does let's call it a plus and the plus indicates that it raises the energy of a state and then act on the end state to give one n plus one but with a coefficient in front of it square root of n plus one now this is mumbly jumbly unless you know the bit about the mathematics of linear operators and so forth so as I said either you know it then you can follow it or you don't know it then you can just get a hazy impression of what it means but this is the mathematical property of this operation here it multiplies any state vector by a square root of n plus 1 and raises you to the next level but basically it's a shift a thing which shifts you from one level to the next it's just an operation which shifts you from one level to the next and we got to keep track of a numerical coefficient in front of it what about a - what does it do takes you down and minus one what's the coefficient square root of n square root of n what happens if this one acts on the bottom state where does it take you it doesn't take you anywhere it just gives you zero that's because if it acts on the bottom state n is zero and it just gives you zero nothing no state they're just the non state there's nothing to get to so it just gives you zero all right this is the mathematics of creation and annihilation operators or raising and lowering operators now we're just going to we're going to do some little mumbo jumbo with it I'm going to show you some properties of these operations they're interesting they're entertaining and you can do some you can have say fun with them just knowing some very simple set of rules for it and this is the rules let's see what happens if we take the state N and multiply it first by a minus and then by a plus in other words let's hit it with a minus and then a plus what do we get okay all we do is follow the rules alright so first of all oh one one important thing some things are just numbers not everything is one of these peculiar things that where the order of multiplication matters some things are really just numbers things like in are just numbers that means that the order that you multiply them doesn't matter some things the order of multiplication is perfectly ordinary in in is always something rather ordinary okay so what happens what happens what is this equal to well let's take it in pieces let's first multiply a minus by n what does it give a plus and then a minus times n is what square root of n times n minus 1 so let's take that on the outside and n minus 1 now what happens if we hit it with a plus if we hit any vector or any sorry any state with a plus it raises us back it raises us to the next level up but multiplies by square root of n plus 1 but what is square root of n plus 1 here mean it just means square root of n since I'm now starting at n minus 1 hitting it with a plus will give me square root of n another square root of n + n square root of n times square root of n is in it just gives me the number n times n that's just a little game that we're playing we're playing a little game with a set of logical rules here and what we find out is if we multiply by a plus and a minus with this set of logical rules it's equivalent to multiplying by the numerical number in whatever state we put here if we put here the first a zero state therefore if we put here the lowest state we get zero if we put here the first state we get one we put it in the second state we get two and so forth so whenever we do this combined set of operations it picks out for us it picks out for us the numerical level and tells us which level we're at if we want to find out which level we're talking about I may not know what it is I may just call it by some other name of Lupus we just put Lucas here we want to find out what in blue this corresponds to well we multiply it by 8 plus time - and we find out what the numerical coefficient is that numerical coefficient tells us what what level of the oscillator we were playing with so we can write this in another way we can simply write that multiplication by a plus times a minus is the same as multiplication by n but what is in n is simply the energy level well not quite the energy level we need something else we need H bar Omega we need to multiply by H bar Omega so we can write in H bar Omega is H bar Omega times a plus a - and what does that tell us that tells us that the energy of the oscillator if we didn't know what the energy of the oscillator was we could find out mathematically mathematical experiment but as a matter of playing this logical game we could determine what level we were talking about by taking the state unknown state and multiplying it by a plus a - times H bar Omega now this is a little game as I said and doesn't correspond to any actual experiment but in fact this is the actual energy of the oscillator okay so that's a mathematical bunch of tricks bunch of trickery let's multiply them in the opposite order all right so what did we find out we found out that a plus times a minus is equal to n meaning to say that if you do this operation on any state it just multiplies by n what about the opposite order let's try it in the opposite order using the same little logical rules a minus times a plus on n what does that give did I do that right a minus times a plus the opposite order so what is a plus give this gives square root of n plus 1 times n plus 1 right and now let's multiply that by a - a - we'll bring you back down to n but it will give you another factor of square root of n plus 1 all right whenever a - axe it gives you the numerical coefficient which is the same as the thing inside here so that gives you square of square root of n plus 1 times square root of n plus 1 again times n brings you back down but gives you the factor overall factor of n plus 1 so what do we find we found that multiplying in the opposite order gives us different answers almost the same answer but not quite in one order a + a - is N and the other order a minus times a plus is n plus 1 so with this little bit of logical gamesmanship here or mathematical symbology we find that the order of these two operations isn't the singer well what does that mean does that mean that a plus and a minus that if we were to measure them and count and then take the values that we measure for them the numerical values whatever they are whatever they correspond to we measure them that when we multiply their values by each other they don't they don't commute that doesn't make sense numbers always commute when we multiply them what this means is that whatever a plus and a minus correspond to you can't measure both of them simultaneously now what they correspond to is a combination of the amplitude of this harmonic oscillator how big the oscillation is and it's phase it's phase does everybody know what the phase of an oscillation is it's basically the in which the oscillator passes through the origin the oscillator can oscillate back and forth or you can delay it a little bit so that it goes through the origin of the later or earlier time you can't you can't whatever a is you can't measure it a plus and a minus simultaneously that's what this tells you that's all it tells you that they don't commute and therefore they're not simultaneously measurable but they're still interesting mathematical little objects what do they do what do they do they are mathematical operations which raise you up and down this tower what do they do they add and subtract quanta of energy of the oscillator now if the oscillator happens to correspond to an electromagnetic oscillation an oscillation of the electromagnetic field then what these operators do is they add and subtract photons they add and subtract photons they are what are called creation and annihilation operators all right so let's up on any physical experiment to talk about well you can measure the number of photons but you can't measure the a plus in the a minus separately you know it's an interesting question of whether you can measure a plus or a minus individually and the answer for technical reasons is no but but but this doesn't tell you that okay all right so we have the idea now of operations or mathematical quantum mechanical operations which would make no sense classically at all we wouldn't we wouldn't multiply States by the operators like this but quantum mechanically we have the idea of creation and annihilation operators mathematical operations which add and act quanta of energy for a fixed frequency now when we have a system which has many oscillators you can imagine we have many Springs with many balls hanging off from of different frequencies or you can imagine electromagnetic waves in a cavity each different wavelength has a different frequency we can excite them independently we can excite one way if we can excite another wave excite them independently then 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 this of course is a little bit indigestible unless you've seen it before how many people have seen it before when no no of course you give a lecture in another course yeah all right like a mini lectures on it probably by now in this I'm not in this course I have not talked about this yet oh the sequence of course yes we've talked about many times I think but the it certainly is in the lectures on wine Mari might have been recorded because we added loot to get me wrong it wasn't recorded it may not have been to the work in a different room not represent set it up about who the next room over yeah it's okay I hung on a casa later is the most basic quantum mechanical system please learn it if you want to follow these lectures and it is you know you can you can see where it's going we describe particles as oscillations and we create and annihilate particles put them into the system take them out of the system by mathematical operations that are these creation and annihilation operators so they are absolutely central to the to the understanding of quantum field theory and particle physics just completely ok let's let's come back very quickly now and I think we'll have to discuss this again but let's come back to this little model of a world on a circle where we made the world finite by making space to be a circle instead of an infinite line and now it's a very simple world of only one dimension but we have this line distance around it is L let's imagine photons or electromagnetic radiation or whatever kind of particles moving on this line there are some allowable values of the wavelength remember where the allowable values of the wavelength are L divided by n where n is any integer that means there are allowable values of momentum is there are allowable values of frequency and allowable values of momentum okay those allowable values of frequency we can call Omega in n goes from minus infinity to infinity whatever it goes over well angles in the puzzle sorry from zero to infinity all the possible values of frequencies all right the waves moving on the system here are a collection of harmonic oscillators one harmonic oscillator for each value of n the each frequency of oscillation you can add energy independently to each frequency and you can take any given frequency and add energy to oscillations in that frequency you can then add energy on top of that with some other frequency so a system of waves moving on a circle like this so moving in anything for that matter is really just a collection of harmonic oscillators with a certain set of allowable frequencies the question is how do you describe this system mathematically in quantum mechanics and the answer is let's begin with the idea of a state we're not going to do this in detail now we're not actually going to get to our first example of a quantum field we'll get there next time but let's work around it a little bit how would we label the states of a system where there are many possible where there are many oscillators around you can either think about this as all the oscillations of a wave on a circle or you can just think of it as a bunch of balls and Springs with different frequencies how would you label it well each harmonic oscillator has a quantized energy the quantized energy for each one of these oscillators is some integer times Omega n times H bar so what we have to state then is the integer excitations of each oscillator the number of quanta with the lowest frequency the lowest frequency whatever it is or in one number of quanta with frequency in two or the second the number of quanta with the third frequency and so forth and on and on for infinity forever this would correspond to a state in which the lowest wavelength or the longest wavelength the longest wavelength oscillator has in one quanta in it the next longest wavelength oscillation would have in two and three I'm getting a little bit tired and I suspect you are also but what I want you to think about is the guitar string or the violin string and how you how you create harmonics what we're talking about is harmonics all right so there's an overall mode of oscillation of the string where it behaves up and down like that if we forgot everything else and just thought of this as a simple harmonic oscillator oscillating back and forth we could say that the energy level of the system is just one possibly no excitations one unit of excitation two units of excitation three units of excitation and how much would each unit of excitation cost it would cost an energy which was the frequency of that particular note Omega times H bar times the number of oscillations with that frequency then we could add additional energy in the form of the first harmonic all right so if if this oscillation correspond with the middle C this oscillation would be what the C one knocked of above middle C or whatever add some energy in that that has twice the frequency twice the frequency so we would be then adding a number of quanta with the second frequency and then we could add oscillations on top of that I always have trouble drawing this next one I can't draw it you know what I mean three with two nodes yeah two nodes but it would have three times the frequency right three ways fit in that has an oscillation like that has three times of frequency and how many oscillators how many quanta would we put in or we can put any number in but that would be in three and the entire configuration of this oscillating string quantum mechanically oscillating string would be characterized by a collection of integers which tells us how much quantized energy there is in each mode of oscillation so basically what we have to do mathematically is repeat the mathematics of the harmonic oscillator infinitely many times one for each mode of oscillation the result is a quantum field that's what a quantum field is is it's a collection of harmonic oscillators a collection of creation and annihilation operators which add and subtract energy in each mode of oscillation and it's the mathematics of that collection of oscillators which is called quantum field theory so if we want to carry on with this subject they have to get used to this language okay let's quit here Wow let me just say one more thing if we're interested in particle physics experiments what happens in particle physics experiments particles with given energy momentum come in they rearrange particles with other energy and momentum go out so in other words we remove particles with an eye we remove particles from the initial state and replace them by particles in the final state the mathematical description of that is to remove particles with annihilation operators and replace them with particles of other momenta and energy involving other creation and annihilation operators so that's what these operators do they remove particles and put particles in and the mathematical description of the creation and annihilation of particles is through these kind of mathematical symbols that's why that's so important the other the N is used like indicate the energy level but then it's also used indicate the count of number of content is that linkage is going on here one integer has to do with the number of nodes here and the other has to do with the number of quanta I'll spell this out again there are two distinct integers in this class in this example the number of nodes would tell you the frequency and then the number of quanta with each frequency so there are two three two integers floating around Yeah right one of them is associated with you yeah this is hard stuff this is not easy it's abstract it's too abstract for more please visit us at stanford.edu

Channel: Stanford

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Keywords: science, physics, theoretical, particle, nature, matter, quantum field theory, light wave, Planck's constant, energy, photon, quanta, electromagnetic wave, momentum, frequency, magnitude, schrodinger equation, oscillation, level, Newton

Id: oT7bTr5YEO4

Length: 110min 34sec (6634 seconds)

Published: Thu Jan 21 2010