Lecture 7 | New Revolutions in Particle Physics: Basic Concepts

[Music] Stanford University we're going to study tonight is angular momentum we're after is one of the most important interesting unintuitive and yet very very simple aspects of elementary particles or anything else for that matter their spin their aim their spin angular momentum let's first talk a little bit about what angular momentum is well let me let me pretend for the moment that you knew what angular momentum was which you probably do but you get you get a rough idea it's got to do with how fast the system is rotating and how massive it is and the you know that sort of thing angular momentum is first of all a vector quantity a point center Direction the mathematical definition of the angular momentum is that its direction as a vector points along the axis of rotation so that's kind of obvious um there's a right-hand rule if the system is rotating about an axis you don't know offhand without a definition whether the angular momentum is pointing that way or that way all I told you it was along the axis so you need a rule the rule is the right-hand rule if it's rotating that way you wrap your fingers around the direction of rotation and your thumb points along the direction of the angular momentum that's not something you prove that's something you define alright the direction of angular momentum it's built up out of the mass and the speed of rotation and that sort of thing the moment of inertia to be exact which is a combination of a mass and the size of the system and the velocity the angular velocity and any system of any ordinary system composite system this cup here made up at a lots of atoms it actually can have two kinds of angular momentum what is called orbital angular momentum and the orbital angular momentum is a consequence of the motion of its center of mass it could be angular momentum angular momentum first of all is relative to a particular axis if an object is moving around an axis it has angular momentum relative to that axis even if the thing is it's not itself not spinning it's a Google Earth spinning spinning is rotating about some internal axis it has angular momentum by virtue of the fact that it's moving let's say in a circular orbit around my head I see it that has angular momentum that's not the angular momentum we're interested in that orbital angular momentum the angular momentum were interested in is the angular momentum of spin so what does the spin angular momentum the spin angular momentum is what ever angular momentum would be there in a frame of reference where the system was at rest so if the system is at rest the only angle where the center of mass of the system is at rest I don't mean that it's not rotating I mean that its center of mass is standing still in the frame of reference where its momentum is zero ordinary momentum any leftover angular momentum is called spin basically ah now I wish I had a basketball or something that I could spin but just to illustrate ordinarily in ordinary thinking about things you can take a basketball and you can start it rotating and you'll say that's the same thing that's the same object as the original basketball which wasn't rotating except it's now rotating if we also keep track of the fact that in quantum mechanics the amount of angular momentum is discrete you can't interpolate continuously between different angular momenta then you could ask the question it becomes a def question of definition if you start an object with a little bit with no angular momentum and then you rotate it up you spin it up and give it some angular momentum is it the same object rotating or is it a new object now that's obviously a matter of definition but in practice the real issue is how much energy does it take to set it into rotation now you say I can set it into the smallest amount of rotation arbitrarily small amount of rotation and so it takes arbitrarily small amount of energy to start rotating but that's true in classical mechanics you have a continuous interpolation between the thing not rotating and the thing rotating but in quantum mechanics you don't have a continuous interpolation and so I would say it's a matter of definition whether you want to think of a rotating nucleus as the same object or a discretely different object but the real question as I said is how much energy does it take the starter system in rotation given that the rotational States are discrete it makes sense how much task how much energy does it take to set it into the first excited state the first lowest amount of rotation that you can get if it's very small it only takes a little bit of energy then the object is probably pretty recognizable as the same object as when it was rest but if it takes an enormous amount of energy well it could take so much amount of so much energy that it would just blow the system apart so for example an atom if you try to set an atom into rotation with too much angular momentum the electrons will just fly off and that atom won't even be there so you certainly make things which are distinguished ibly different when you set them into rotation and discreetly different what about an electron can you set an electron forget for the moment that the electron has intrinsic spin for a minute let's forget that for a minute can you set an electron into rotation so that it resembles the same object except rotating about some axis or is the electron somehow so small and so point like that it doesn't make any sense to set it into rotation a thing which is infinitely small a simple point a simple mathematical point it's hard to conceive of at least with our usual mental pictures of setting that thing into rotation but we don't know that the electron is infinitely small maybe the electron is not infinitely small ah if it's not infinitely small maybe we have a chance of seeing a rotating electron so the question becomes how much energy does it take to set an electron into rotation now the answer tends to depend on the size of an object for a given mass it depends on the size of an object surprisingly maybe it's surprising maybe it's not surprising the smaller the object the more energy it takes to set it into rotation big objects you can set into very very small angular velocities small objects take larger amounts of energy to excite them to a given amount of rotations or given amount of angular momentum the electron may be so small and it probably is it's known to be so small that the energy that it would take to give it one more unit of angular momentum would be astronomical would be maybe the Planck energy or some humongously large energy and so we don't see in the laboratory rotating electrons in fact the rotating electron may be so different from an ordinary electron that we wouldn't even call it electrons are the upside of this is that when we talk about objects in quantum mechanics we are or particles in particular are nuclei are any simple systems relatively simple systems they have a amount of angular momentum which characterizes them and which once and for all is fixed you don't talk about electrons with different angular it's different spin angular momentum the angular momentum of the electron is always the same if it has any angular momentum at all you say why is that well just because if it had more angular momentum you call it a different object now angular momentum as I said is a vector we're going to define exactly what I have defined at the moment but it's a vector it has a length the length of the vector is proportional to the speed of rotation and so forth it can point in any direction or at least classically it can point in any direction shall we think of pointing the angular momentum of an object in different directions as corresponding to different objects let's suppose we now do have an object which has some angular momentum we call it what should we call it I don't want to call it an electron it's a what spin transparent okay spintronics pintron we identified as a spin Tron and its angular momentum is pointing that way can the angular momentum point in another direction well yes it better be able to because the laws of physics are rotationally invariant there's nothing special about one axis than another axis so yes the same object can be made to rotate in a different direction even if it can't be made to rotate with more total angular momentum so pointing it in different directions we would say corresponds to the same object if an electron does have angular momentum anthem we should be able to think of that angular momentum in any direction on the other hand the amount of angular momentum the magnitude of it is quantized and so as I said the gap to the first excited state of the electron may be so large that we wouldn't even call it an electron okay so the angular momentum can point in any direction it is quantized and now we have to enter into the theory of angular momentum tonight what we're going to do is the mathematics of angular momentum and it's very magical it's magical and it seems totally abstract totally unintuitive and then pop out comes experimental facts and predictions and the entire properties of spin as an experimental observational fact about particles all right let's yeah question okay do you know what they know at the moment of inertia of an object is okay let me answer the question the simplest it is kind of intuitive a little bit well isn't it isn't the moment of inertia is a combination of the the mass of the object and the radius of the object we take a ball of some sort all right it's M R squared now it depends on the detailed shape of the object and it depends on how the matter is distributed sometimes it's three three-halves mr squared you know but it's a boarder mr squared and it's called the moment of inertia I okay the energy of a rotating object we're four that's one thing now then there is the angular momentum the angular momentum of the object is made up if the object let's say the outer boundary the object let's say is moving with velocity V and then roughly speaking order magnitude the mass times the velocity that's the ordinary momentum of a piece of it mass times velocity and if you multiply that by our that's that's the angular momentum momentum times distance is angular momentum so the momentum of of a little piece of it times the distance from the center of it is the angular momentum and if you write down the kinetic energy the kinetic energy is one-half MV squared what you'll discover is that it's the angular momentum squared divided by the moment of inertia twice the moment of inertia to be exact so that's the energy the kinetic energy of rotation is the angular momentum squared divided by twice the moment of inertia now for a given mass let's say the mass of the electron the smaller it is the smaller the moment of inertia but the moment of inertia goes in the denominator and that means the amount of energy that it takes to increase the angular momentum by one unit is inversely proportional to the square of the size of the object okay so it's a classical mechanical fact the only thing that comes new in quantum mechanics is that the angular momenta is discrete or discrete but the fact that for a given angular momentum a smaller object cost more energy that's a classical mechanical fact okay let's are let's do some of the mathematics and tonight we're really going to do the mathematics of angular momentum it's both totally unintuitive and simple all right simple enough and this is a great example of where you see extremely abstract mathematics which you can follow suddenly popping out very unintuitive answers but nevertheless experimental answers alright let's let's begin well first of all let's begin with a single particle orbiting a center circular orbit for simplicity the angular momentum is the momentum of the particle times the distance from the origin mass times velocity times distance but we'll just call it momentum that's the angular momentum in this very simple context R is it positive or is it negative that depends on the sense of rotation going this way I think I'll call positive going that way I'll call negative but it's not very important that's the angular momentum of a point object now this object has momentum has angular momentum because it has momentum and it has momentum because it has velocity so this is not yet spin but now supposing we had two particles at opposite ends of the diameter here both going with the same sense of rotation now the center of mass would be at rest the center of mass of the system would be at rest but the angular momentum would certainly not be zero it would be the sum of the two angular momenta and they're going in the same direction so here's an example where the center of mass of the system can be at rest but there's still angular momentum this is spin okay alright now let's get a little more refined in our precise definition of angular momentum I said before angular momentum is a vector and that means or that's denoted by putting a little arrow over the top of it momentum is a vector and also spatial location is the spatial location of an object it is relative to an origin relative to an origin we can call it a vector it's the radial vector let's talk about the components first before we get on to an angular momentum the components of the R vector are just the coordinates of the position of the particle so the components of R this R over here let's write it over here the components of our are just x y and z x y&z being the coordinates of the position of the particle and I'm also going to sometimes call them X 1 X 2 and X 3 X 3 being Z X to being Y X 1 being X ok so we have these two vectors and somehow the angular momentum is a product of the two vectors the angular momentum is itself a vector how do you make a vector out of two vectors by multiple by multiplication what kind of rule of multiplication is there that takes two vectors and combines them together by some rule of multiplication to make a new vector that is only that's right there's only one rule there's only one such rule and that's the cross product the cross product R and in fact there's an ambiguity is it R cross P or P cross R that's a question of whether you're right-handed or left-handed do you use the right hand rule or the left hand rule the standard convention is that it's R cross P I don't remember I think it's right here a bit it's left-handed I'm not sure I don't remember we could figure it out but I don't want to yeah it's R cross P oh we labeled the coordinates of the R vector is also the P vector the P vector also has coordinates and it's coordinates are px py and PZ or p1 p2 p3 okay those that's that's the notation we'll use now I can ask what are the components of the angular momentum so for that all you have to know is how to build a cross-product and I'll assume everybody knows that build a cross-product so let's just do it here's the rule the X component of the cross-product the X component of the angular momentum is the Y component of position times the Z component of momentum minus the Z component of position times the y component of momentum this is the only one that I ever remember L x equals y times PZ all right I read that off the other one here I know that I have to put them in the upper I have to change PZ to Z P y 2y and then I want to go down from there LX ly and LZ I never remember them but I know the rule the rule is you just cycle from X to Y to Z and back to X 2 y 2 Z think of XY and Z as being our points on a clock and you go from one to the next so when you go from X to Y Y goes to Z Z goes to X let's go down to the next one y Z X this is not I'm not finished with the formula Z X Y py minus xpz minus our Y P X the only reason I emphasize the cycling is to keep track of the sign how do I know that L Y isn't XP z minus z PX ok that's the rule you cycle through that way and right that's the component of angular momentum of a point particle moving in the vicinity of some origin of coordinates right it's the orbital angular momentum of that particle is not the spin angular momentum to make a spin angular momentum for a system of ordinary in classical mechanics to make a spin angular momentum you've got to have a lot of particles at least two anyway I showed you how to make a spin angular momentum with two of them you have to have several particles to being a possible all right how do you build the angular momentum or composite system you just add the angular momenta as vectors you add them as vectors you add the angular momentum of all the constituents once you add the angular momentum then you can have a spin where the total momentum is equal to zero but the angular momentum is not equal to zero but this is the basic formula for a single indivisible point particle so and as I said you just add them up for for a lot of particles next step we want to do the quantum mechanics of angular momentum this is the classical mechanics actually it's also the quantum mechanics but to get to the quantum mechanics the basic mathematics of quantum mechanics is the quantum mechanics of operators all quantities of physical significance meaning to say things that you can observe measure all those things are represented in quantum mechanics by what operators emission operators operators and there's no special exception about angular momenta the components of angular momenta are may are represented by operators but to figure out what those operators are all we really need to know is what the operators which represent the position and momentum are in fact we don't even really have to know very much about the detailed property of the position and momentum operators all we have to know is their commutation relations with their commutation relations that's all we need to know in order to carry on and work out everything okay so what what does it mean what are the implications of two operators commuting let's suppose I have two operators which represent two observable quantities the anus and the Venus of a system and I know that a and B commute what does that tell me about observation yeah exactly you can simultaneously measure them what if they don't commute hmm then you can't simultaneously measure them the most famous example of two things that you cannot simultaneously measure of course is position and momentum the implication is that in quantum mechanics the operator is representing whether I erase them P 1 P 2 P 3 on the one hand x1 x2 and x3 don't commute among themselves so what are the write commutation relations alright now to some extent these are postulates of quantum mechanics you find them in the first chapter of Dirac's book and they're basically postulates if you like but there there are limitations on what you can write down but we're going to take them as postulates all right the first postulate is that you can simultaneously measure all three coordinates of position there's no limitation on how well you can determine the x y&z coordinates of position they're all simultaneously measurable and the implication of that is that every X commutes with every every other X we can write that down X I XJ equals 0 for all I and J for all I and J incidentally anything commutes with itself would be very weird or something didn't commute with itself that would say you can measure it but you couldn't measure it simultaneously with itself that idea okay so everything commutes with itself same thing for momenta there is no limitation on being able to simultaneously measure the different and some momentum so p i pj equals zero what about X's with P so that's exactly what the uncertainty principle is about it tells us that we cannot simultaneously measure coordinates in position but which coordinates in which positions can which components can we not measure simultaneously and the answer there's not a lot of freedom about this but nevertheless I'm going to just state it as a postulate that X 1 and P 2 can be simultaneously measured the x coordinate of a particle and the y component of its momentum are not limited by the uncertainty principle it's the X component of position in the X component of momentum which are not simultaneously measurable so we have that X with P X is not equal to 0 anybody remember what it is equal to IH bar it's small why because we certainly don't expect macroscopic big heavy objects to behave with such bizarre behaviors so it's small one unit of angular momentum sorry one unit of Planck's constant ah and so forth for y&z and so we can write down a general formula that the commutator of X I with PJ is IH bar if I equals J and it's 0 if I doesn't equal J is that clear ok so to represent that we put the Kronecker Delta here and that's a nice systematic way to describe the properties of position and momentum that turns out all to be all we really need to know in order to in order to understand the properties of angular momentum in order to understand the properties of spin let's say there's one other point that is worth emphasizing that's that angular momentum has units of Planck's constant how do I know that well angular momentum has units of distance times momentum here's distance times momentum on the right-hand side is Planck's constant it also occurs of course in the uncertainty principle Delta X Delta P is greater than or equal to whatever some number times Planck's constant so you see that Planck's constant has units of length times momentum and angular momentum has units of length times momentum so it's not completely surprising that angular momentum is quantized in units of Planck's constant are units of the basic quantum in nature which is Planck's constant okay but there really we haven't gotten ready yet I just pointed that out what I want to work out for you is the commutation relations of the components of angular momentum we're going to find out that once we work out the commutation relations of the angular momenta components of the angular momentum we can forget this we won't need it anymore it's just the components of the angular momentum whose commutation relations we want all right so let's do it over here let's first compute the commutator of LX with ly it's not very hard I will have to wing it a little bit because I haven't I would have to remind you what all the rules about the algebraic rules of commutator ZAR but it's pretty simple it's simple enough that the that you can see what happens all right so we want the commutator of LX with ly LX with ly is Y PZ is the commutator of ypz minus zpy that's the first our entry into the commutator we're going to commute that with that was that was LX with ly which is zpx minus xpz I think in the last 24 hours I've done this three times on the blackboard first the first two times are last night when the 30 people who fail the course showed up late yeah but you can actually just cycle from left to right so why start you see si starts with a good however you like yeah yeah that's fine right okay let's see if we can figure this out let's see Y PZ let's look at the commutator of this term of the first factor with this term of the second factor there's nothing to prevent Y you know when you think about commutator is you're thinking about pushing operators through each other can they freely pass through each other without a change in the value of the operator all right now Y commutes with X so it can go like past X it also compute the commutes with P Z PZ commutes with X and PZ commutes with P Z so you can push this one right through this one no obstruction so these two commute with each other contribute nothing to the commutator I think there's another combination let's see is it this one Z yeah these two what yeah these two also commute with each other the two it's don't commute with each other or two pairs listen this and this with this so let's see if we can work it out um we have y pz z PX the only thing which doesn't commute here is P Z with Z all right so I maintain that the answer is just y times I don't want to use red is just y times PX and it doesn't matter which order you write them down y times PX and then there's the commutator of P Z with Z what's the commutator of P Z with Z minus IH bar why is it minus IH bar because commentators in the opposite order change sign commutator is anti-symmetric meaning when you change the order of what you write in the bracket it changes sign a B minus B a if you interchange B it would just change sign all right so this this commutator of P Z with Z is minus IH bar all right then we have this guy over here with this one over here what is it that doesn't commute here P Y and X commute there's a plus minus times minus is plus a py times X but then we have the commutator of Z with PZ and that's IH bar so I h-bar of IH bar or in other words the whole thing oh sorry what I write that piece of Lackey very different piece of Y right X peaceful X piece of Y and again in each of these expressions here it doesn't matter which way you order the two constituent operators here ok the whole thing then is IH bar X py awareness wire Obata mumbo jumbo but that was very effective xq minus ypx is LZ so we've learned that the commutator of LX with ly is just sorry IH bar I h-bar LZ that's kind of pretty that when you commute two components of the angular momentum you get back another component of angular momentum you don't have to remember anything about peas and X's anymore the commutation relations of angular momenta closed among themselves and the algebra is going to call this the algebra of angular momentum is something abstract which doesn't even really remember where it came from meaning to say that doesn't contain the P is an exercise that's LX with ly then in the cycling vertically or horizontally doesn't matter to the write down next one a wide with LZ I h-bar LLC and so that's what x YC y z x IH r l TL y okay those are those are fundamental of relationships on which the theory of angular momentum is built now the next factor each I will tell you is that if you have many particles and you add up the angular momenta and then commute them you'll find exactly the same relationship it doesn't matter that this was a single particle could be many particles and it could be many particles in your own rest frame in the rest frame of the center of mass so this will also be the commutation relations for the components of spin angular momentum so these as I said are our starter point ah you might ask is there anything special about the choice of axes chose the x or y and z axis but i never told you what x y&z were it could be the the edges of the room or they could be as you wanted to use a diagonal direction two other funny diagonal directions as long as they're mutually perpendicular they form a perfectly good Cartesian coordinate system giving the components of a vector in one frame of reference you can commute compute them in any other frame of reference so in principle we can work out from this the commutation relations of the components of angular momentum in some other set of axes what we would find is that they have exactly the same form they will have exactly the same form of I chose prime Vaccines axes which are different I was just put some primer in other words the commutation relations of angular momentum are rotationally invariant they take the same form in every frame of reference and that guarantees that whatever theory were making of angular momentum it will it will be independent the observable facts will be independent of what axis we choose to do the mathematics the farm is it infinite of translation as well it is but it is it is the spin spin spin they'll spinning but certainly this much is quite independent of direction is voltage it's also yes it is also independent of rod translation okay next step now we're going to be now we're going to be doing the abstract algebra okay Magic Mike we need to borrow so far good so far we didn't get anything yet it's only because magic will you get alright you can do you know you can write down into a number of formulas you can get a computer you can plug in these rules into a computer and then have it grind out all possible theorems you can do that and mostly they'll grind out uninteresting theorem it becomes magic when all of a sudden you look back and you say wow that's an experimental fact that I can confirm or that's telling me something interesting about the physics of the operational process of measurement so we'll get there but we're still just doing abstract mathematics for the moment of course I know where I'm going you don't know where I'm going so you just have to follow out after you see it you'll know and then you could do it yourself some you know okay XYZ axes that you choose are spinning alright that's it that we don't want to do that we don't want to do we want to use inertial reference fighters and spinning reference frames I'm not a vertical nevertheless I will tell you it's still true but the we don't want to get into that at the moment certainly the angular momentum is not invariant under going to rotating frames of reference if we have a system which is spinning in a handsome spin wheel into a rotating frame of reference which come with it then it wouldn't look like it had any angular momentum so it's clear that that we don't want to go to rotating frames of reference or at least the angular momentum might be invariant to inertial frames of reference reference frames which most in tropical horses know Coriolis forces and so forth are now we're going to invent two new operators which are very simply related to these simple as possible they called L plus and minus or to it let's write it down L plus in R minus L plus is equal at LX plus i ly and L minus is equal to LX of - ily now today right now what these are these are raising and lowering operators analogous for the harmonic oscillator creation and annihilation operators they take an angular momentum along a given axis namely the z axis and they bulk you up a step and they bump you down the step what we're going to show that we will show that another definition is that immeasurable values of LZ we're going to be concentrating on LC there is nothing special about the z axis but the other hand is nothing to forbid me from focusing on the z axis nothing special about it nevertheless I wish to choose an axis to work with which is the z-axis I'm going to call the eigenvalues of LZ the measureable values that can take on I'm going to call them n times H bar H bar is just a number if we set H bar equal to one which I think I will do now because I will never on the blackboard in real time be able to keep track of all the H bars later on you can put it back LZ is just called in that's a historical notation M as I said is the angular momentum in units of H bar okay so it's but and it'll turn out to be a quantized variable okay called in historical reasons that I don't know and that's a rotational use what I want to calculate I'm not interested for our purposes in the commutator between LX l plus ml - I'll tell you if you want to work it out it's proportional to LZ but don't that's not our interest right now I want to work out a commutator of L plus with LZ you'll see why is we go along all right so we want L plus 1 LZ also it'll - what LZ but let's do L plus what else this is equal to commutator of LX plus iy with LZ really how old rules this is easy oh my god we're going to do with them here they are how many dator of LX with LZ can somebody read it off for me - my idol - al lime is sitting H bar equal to 1 from the right so LX with LZ - I believe you I do - i ly [Music] and what about what about ly with LC so there's plus I and now know why with LC maybe we did or 9ln ILX that's I squared L X and I squared is just minus 1 and this is just equal to minus L plus all right it's - LX plus ily okay so we have our first commutation relation I think I can get rid of this it's just minus L plus what about L - what LZ I won't drag you through it I'll just write down the answer oh I think the answer is plus L - when you compute when you commute L plus 1 LZ you get back our plus when you compute when you community now - what LZ you get back l - with a sign that depends on whether it's L plus RL - these are the basic commutation relations that we've invited ok the next step is I want to show that L plus + l - are bumping operators which bump you up and down like raising a lower like creation and annihilation operators for harmonic oscillators now when L + NL - act they change the Z component of angular momentum so we're going to focus measure we can only measure one component the angular momentum and at or if you say that did I say don't say right now how many of the components of angular momentum can be simultaneously measure which is one any two of them both community any two of them don't commute and so at most we can measure one component of angular momentum at a time we pick one direction we call it the quantization axis but there's nothing special about it focus on it and though we interested now in the possible values that LZ can take on in other words was the possible values of him and what we're going to do now is we're going to show that L plus bumps you up in M increases the eigenvalue m by one year so let's see if we can do that let's suppose we have a state let's call it m such that LZ acting on him is equal to what an environment in other words that little m is an eigen value of LZ with eigenvector ket vector M that's what it means to say that m is an eigenvalue we don't have been through this a number of times what we want to prove is that if you act with L Plus on and that's a new vector let's put a red box around it it's a new it's a new ket vector what I want to prove is if that's a ket vector with eigenvalue n plus 1 with eigenvalue of l z equal to n plus 1 so let's do it all we have to do is multiply this by LZ and see what we get what should we get if it really is an eigenvector get some more we should get n plus 1 in terms the same thing in the red box we should get I won't bother writing it down but we should get n plus 1 times of thing in the red box alright now this would be easy to compute if L plus if LZ commuted with L plus then we would just push the LZ through and use the fact that LZ on em just gives us M fudging okay that would be easy but it's not that easy we can't push LZ through L plus y because they don't commute but if we use the commutation relations we can so I'll try it out what this commutation relation means it means L plus LZ - LZ l plus equals of minus L plus written it out by hand in excruciating detail the thing that I'm interested in here thing that came up here was LZ x l+ so let's put that on one side of the equation and write that this is L plus LZ plus L plus of style 0 plus equals L ZL plus that's what I want LZ l plus with LZ on the left of plus on the right ok and here I have it on the left side of the equation we'll see is a useful we're going to find out what is useful so this is equal then 2l plus LZ acting on n plus L plus acting on it I'll stop here if there any questions I think I've been clear but if not though ok but now it's easy what is LZ with an X on it can just replaces the LZ by an M right so let's take an amount put the M over here pick out the cells it close it up a little bit and x l+ and here we just have L plus tangent what did I do with that I know we're going to do with it but we just up oh we can do it is put a red box around it l plus x m l+ arms in same red box same thing in the red box on each side so what's in the red box as the property that L 1 LZ hits it and multiplies it by n plus 1 in the red box is an eigenvector of LZ with eigenvalue m plus 1 that's a little bit of magic and what it tells us is that if we have a state with a given value of LZ there must be another state with one additional unit of LZ in other words you could bump up LZ by one unit um you can bump down LZ the same way except using the lowering operator so you can check that that L minus our M gives you an eigenvector with eigenvalue in - water let's another let's think about now we learned something rather non-trivial we've discovered the spectrum of angular momentum with Z component that IV lemon right good is spaced by integers here is the spectrum of angular momentum and whatever it is it's spaced by integers I didn't know it could be I don't know it's spectrum of angular momentum is yet expecting the possible values of it it could be pi PI plus 1 PI plus 2 pi plus 3 PI minus 1 PI minus 2 or be the square root 2 2 plus 1 squared 2 plus 2 whatever they are the gap by integers but we still don't know what they are anyone ah yeah we'll come to that okay now next question and it's all running on forever now remember we're talking about a particular particle we're talking about Carlos we're not talking about the whole world we're just talking about some particular elementary particle what's a neutron or not or even a nucleus how much can we bump up the angular momentum the Z component of angular momentum before it's not the same object anymore before we simply run out of possibilities so we're talking about an object let's think classically from an ad it's rotating about some axis and we're not allowed to increase the rate of rotation that would give us a new object at different objects but we are not allowed to rotate the angular momentum around and rotate the angular momentum but we're not allowed to increase its amount without changing the whole nature of the object okay so there's no way of saying it that is a given length or given value of l squared the magnitude of L is fixed the magnitude of L is fixed and it characterizes the particle this is a particle is characterized by mass it's also characterized by own length of the angular momentum vector of LEF square of the length of the angular momentum vector okay how do we maximize the Z component of angular momentum we simply point it though we simply point the object upward you point the spin axis okay and that's going to be the maximum angular momentum Z maximum Z component of the angular momentum so how many to the point of view there ought to be a maximum value for the angular momentum of a specific article accomplice and after that mother well how can that be how can it be we've already proved that by acting with L Plus we bump up the angular momentum how can they possibly be a ceiling for the to the rule the answer is exactly the same as the harmonic oscillator creation and annihilation operators remember what happens to the annihilation operator if you hit the bottom state the state of lowest energy it just gives you zero okay that's the one possibility we left out here that perhaps there are states or a state where L Plus will give zero that would have to be the case if there was a max manual momentum so for the maximum angular momentum if there is such a thing there is for every given object l+ on let's call em max must equal zero so the curl can't go past a certain point what about the minimum value of the angular momentum minimum means the one keep this down the summer Megan it better just be the negative of this why must it be the negative why couldn't it be something else that's just rotational invariance if you can build a system with an angular momentum of only given axis of a certain value you must be able by rotational invariance by every axis being equivalent every other axis you must be able to rotate that on your momentum and have exactly the same value except pointing in the opposite direction so the bottom must be of course in Max and min must be equal to minus M max it was in name and then of course being negative and from a maximum in we can bump up an integer or bump bearing integer and then that now fixes for us some properties of a spectrum it's got to be symmetrically located relative to 0 and it's got to be gapped or spaced by integers there's only two possibilities of think about for a few minutes there's only two possibilities well there's many possibilities but they come in two families the first in the first family zero is a possible value of M this is M equals zero and then M equals one everyone's minus one and so forth until you get the hem max and until you get the M near and max slap laptop welcome and thanks - I'm max how many such states are there all together - and Max + 1 and Max here and Max here and then one more at the origin twice and Max was born that's the number of possible states because I haven't told you what I'm excerpts but that's because M max could be anything depending on the specific particle or the specific object we're talking about so specific particles have their own value of M Max and that in Max is called the spin of the particle for example there are particles that have no spin at all then the spectrum of angular momentum is just M equals 0 and nothing else when you hit with a raising operator you get zero are you hit with a lower number and you get zero that's spin zero then there's the possibility of spin 1 spin 1 is a situation where M equals 1 is the ceiling and N equals minus 1 is the floor can't go below the floor can't Corbeau the ceiling that's spin one or the spin one particle you cannot split two particles have spin three particles and so for example the Higgs boson is a spin zero particle the photon has been one particle and Z bosons has been one particle are there's been two particles in nature well the graviton - particle about spin three particles no object that is ordinarily called an elementary particles spin three but there are certainly objects with spin free there are nuclei with spin three there are basketballs with spin three there are galaxies who spin three rather hard finding but yeah there are lots of objects with string three but among the elementary particles in nature the things that we ordinarily call the electric particles spin zero spin one instance Pig goes in now that's not completely clear because I missed the possibility the possibility was that N equals zero was not going to spectrum the only other situation which is symmetrical about the vertical and which is gapped by integers based by integers as / 0 over here is to start at N equals 1/2 and M equals minus 1/2 and then go upward in units of 1 so this one would be three halves is going to be minus three halves and so forth until you come to a max in this case only all the eigenvalues would be half integers the simplest example being just the half spin particle again you call you use the you describe the particle as having a spin equal to the maximum value so an object which only had spin 1/2 or Z component equal to 1/2 and minus 1/2 that's the simplest object with the half spin kind of spectrum that's called a half spin particle mmm the next one will be called spin free halves particle and spin 5 halves particle the particles come in these two types integer spin and heritage better requests don't know yes a bit HEPA the string and integer spender yeah yeah yeah but the only thing is system that the maximum is going to be a half integer it's going to be non integer it's been three halfs particle the maximum would be three halves it's been five halves particle for spending maximum 200 oh yeah in all speculations about elementary particles basically a combination of Lorentz invariants quantum mechanics and fusing other things tell you that there are no particles elementary particles has been higher than true that will allow 0 1 & 2 and it will allow a half and three halves so all the standard particles of elementary particle physics are 0 1/2 1 3 halves into but you can build composite objects yo the your liner loves or neighbors and guys well it is things is just it just means rotated around the access just pointed down and we characterize particle by the maximum value that's been meaning the maximum positive value so let me he'll and a way out of here so okay i acquire there only say two orientations who said there to spin one zero particle or none okay to screen three particle there are three I [Applause] know what you're asking you're asking what happens to the directions in between your ass yeah okay we're going to talk about that that's where quantum that's quantum mechanics in your face yeah we will compare that's not what topic of all okay when we ask for more really after that and you get different uptick which is good I like corkscrew a corkscrew motion is a motion which is moving and rotating we an electromagnetic wave can be a corkscrew way yeah meaning the circular polarization circularly polarized wave has a heard many all are is all or now very classical yeah yeah yeah but nevertheless a circularly polarized photon is one with a spin angular momentum along the axis going exactly the direction of U vector a circularly polarized wave is composed of circularly polarized photons and a circularly polarized means one whose angular momentum along that axis is pointing along that axis but to talk about to talk about objects at rest which is what we're talking about now it doesn't make sense to talk about corkscrew motion because both crew really has to do with simultaneous rotation and motion along an axis so at the moment the object you're hunting for the mathematical term is he listening olicity is got to do with the orientation of the spinning relative to the direction of motion a little contact very very important cause your element a particle that we know this rehab stick ah depends on whether way to find the terminal no electronic measure or that we have areas or punches like a simple arm it's a matter of speculations populations to to roam work conjecture it's a matter of conjecture that there exists a supersymmetric particle partner of the graviton which is called the gravity know which has been perhaps that has not and will never be experiment we love this if you use that parity parity Hector directly at this point the elves are no longer acting like the original definition because if you just further or at their original definition in you wouldn't get these limits so there isn't much be a difference well it looks like there's no longer acting on the original space of the side functions that we had somehow there's an aside mentions of the original space of X no we didn't know what the original space of catch was we didn't know what you're right okay well if this what we're doing is the properties of that space okay that's what we're doing we didn't know what they were here thanks it's not the UH face that we usually take in Frank's L got your honor no no this is something new so nice in space you can tell us what the spices oh it is a space of these integers that's it for a particle of spin in the space is that the basis vectors or you specify what a state is by saying where the basis vectors are its basis vectors are simply labeled with the values and along this ackers that's it that's that's uh yeah if you took the absolute value about that vector now is that is that an observable quantity and would you get this it is so let's let's alright so let me tell you what's true [Music] okay instead of the absolute value let's take the sums of the squares of the components and see what we can find out about that I try to do with other notes let's take l squared you might think that l squared is just M max squared so that would make sense right you you you pick this spinning you point it directly upward the problem with that is that you can't know the Z component and the x and y component simultaneously so if you know the edge of the Z component is vertically upward and max there's bound to be some uncertainty in the x and y component all right so that uncertainty in X & Y component will tell you that LX squared plus a y squared plus LZ squared not quite be and mass squared will have a little extra so what's computing that's a fun thing to compute them that's that I was going to go without my notes because always one thing which I have to go through to remember yeah okay let's take L square l squared means LX squared plus ly squared plus LZ squared and of course the name of the game here is to rewrite things in terms of L plus L minus L see L plus L minus n LZ are so was L squared let's start with Elma all right so what would we do to get l squared we would add LZ squared and that would be it right classically but not quite true quantum mechanically because this is extra commutator term the extra commutator term else plus L minus is not equal exactly to LX square plus ly squared it's in fact LX squared plus a y squared plus a term coming from the commutator of LX with or without LX with ly so let me write down what that extra term is what's the commutator of LX with ly it's got to do with LZ there's an eye over here but the commutator also has an R in it so the eyes are going to cancel and if you work it out you find out that it's just plus L that else well let's see that Jen L plus L minus is LX square plus ly squared I think it's - LZ and that means that l squared has another term in it which is just plus LZ it's definitely plus over here because - just an extra term coming from that commutator the little mistake we made when we said that L owe that L plus times L minus is LX squared plus L R squared give a little mistake and a little mistake is proportional LZ and here's how it comes in okay now let's take L square and operate on a state which is M max which has the maximum eigen value of M what are we get L square root of that is equal to L minus L plus on in Max Plus what do I get with it when L Z gets in max incident in MX so here we get plus n max squared plus M max times the ket vector M max what about this I went through all this effort but still left with something out of the one bit but I don't know what it is because L plus when it hits and Max has nowhere to go all right we reached the ceiling this is zero so what have we found out we found out that the eigenvalue of l squared for the state up here is just in max squared plus M X in other words in max times M max plus one in next time in max plus one but a mask with M max comes in nights squared plus classically it would just be a max squared the maximum value but because of the fluctuation because of the uncertainty principle which says the things which don't commute can simultaneously be specified there's got to be some fluctuation in them and that fluctuation manifests itself by a little shift here okay so let's take the case for example of a spin zero particles what is the l squared for a spin zero particles wow that's not interesting and Max is zero in that case this is zero good what about a spin one particle [Music] turns out to be two what about spin 1/2 particle 1/2 times three-halves there's 3/4 so for a spin what about a spin thousand particle well yeah yes right if index is big enough the Schnoodle shift is not important yeah we take a spin of a hundred billion billion billion we're just going to get a square of it plus a tiny irrelevant change so classically it's a very good approximation that l squared is just an X now if you worked out you can do this you can work out l squared on each one of these states and you'll get the same value all of these states have the same value of l squared that is the quantum mechanical error of the statement that when you rotate the particle to change its Z component of angular momentum it just doesn't change the square it doesn't change the magnitude of it so that's a little exercise you could work out using again just the commutation relations of the angular momentum can you know the square of the angular momentum and the Z component simultaneously yes you can and you can check that by computing the commutator of l squared with the individual component to the angular momentum what you'll find is a commutator of l squared with any one of the components is equal to zero so that means although you cannot know two components of the angular momentum simultaneously you can know a component and the total squared value of the angular member every electron has a squared spirit the spin angular momentum equal to 3/4 every electron is the same value and it doesn't matter which play way the electron is oriented you said today would spin 1/2 L square root of e 0 crawls or justice your case no L squared is 0 for the zeros okay 0 only has one state with collegia movement measures N equals 1 square its to it 1 times 1 plus 1 right right if I spin 3 what is it L times L plus 1 is 3 times 4 well he said that yeah I probably see how that it's always the same no matter which level you oh that's meant to work out so how do you work out your when you work it out by saying let me take em max and now hit it without - this gives me in that excuse me the next one and then hit it with l squared and use the tricks use the algebraic trick tricks and so that when you hit with l squared it just gives you the same idea value the same is our trivia algebraic statement is that l squared on the thing in the red box get your red box around the vector if you're asking that this is equal this is in that this is equal to M max prime n max plus one necklace I am axe and men's dress and that's plus one times the thing in the red box all right the l squared is exactly the same eigenvalue as it had firm x itself yeah you get that from the girl over there you get that from manipulating the operators you get that from manipulating operators you use yeah you use this formula here and manipulative and fiddle around with it darling and yup it takes it takes a few minutes work it also follows from the fact that l squared commutes with l plus an L minus commutes with all the components of the angular Momentum's of computers commutes with l fussing on - and it just follows through that so yeah they all have the same squared angular now let's talk about the rotational invariance not in detail not gonna talk about the details that would be great over time but we seem to have nothing but some discrete states of angular momentum along the z-axis what about rotating that ah if we rotated it wouldn't we get angular momenta which had fractional value along the z axis and maybe some components in the other direction so Aladdin let's let's concentrate for simplicity on the spinning up case from spin 1/2 case the spin 1/2 particle the spin angular momentum has only two states it's not true that an electron has only two states it has an orbital motion and it has a spin angular momentum let's just concentrate on the spin and nothing else then concentrating on the spin there's only two possibilities up and down half spin up has been down let's call them like using red for equations let's call them up plus and down lights these are the two basis States with spin angular momentum along the z axis but there are certainly many more states that I can write down these are not the only states I can write in fact the general quantum state of a system with two states like this is to add them with complex numbers alpha to beta or complex numbers what is the probability in such a state that the spin is our compass with alpha star alpha okay probability or up equals alpha star alpha and the probability for down is equal to beta star beta we add them together total probability has to be 1 and so one of the rules is alpha star alpha plus beta star beta equals one one other fact that if I were to make a phase rotation of each of these complex numbers in other words if I was to multiply alpha and beta by the same phase e to the I theta e to the I theta that does not change the physical character of the state and the reason is because all interesting quantities are things times complex conjugates these will cancel out of any interesting physical quantity and so there's one well the way to say it is there's one degree of freedom namely the overall phase of alpha and beta which is unphysical in which doesn't matter which is irrelevant okay let's count now the number of variables the number of it takes to specify a quantum state of a novice spin of an electron alpha is a complex number it has two real components beta is a complex number it has two real components so so far we have four real components but we have a constraint alpha star alpha plus beta star beta is 1 so that means only 3 real components the number of independent variables on the other hand one combination the overall phase is unphysical let's just ignore limiting in some one way or another that brings us down to only two independent variables specifying a quantum state two independent variables specify the quantum state now how many independent variables if an electron were a little arrow of a given length its angular momentum vector how many variables does it take to specify the orientation of that angular momentum to polar and as a mutant angle of the sphere describing the end of the vector they're the same thing pause you rotate around the spin of the electron by varying alpha and beta but no matter which axis you measure the angular momentum it's always an integer that is to say the actual measured value is an integer what about the average value the average value B now I'm sorry it's not an integer it's a half integer plus a half remind us area can you get anything but plus a half or minus a half for the average value the average price so what is the average value mean was it average value the average value means you take repeated identical experiments where in each experiment the electron was prepared with exactly the same quantum state and in each case you measure the Z component of the angular momentum sometimes you get plus 1/2 sometimes you get minus 1/2 the average is to take the ensemble of all of them together and up the Z component of angular momentum and divide by the number of experiments that you did just the average and euro member that can certainly be anything that does not have to be for example if we just chose alpha equals 1 and beta equals 1 what's the average Z component of the angular momentum zero zero in fact this happens to correspond to the spin of the electron along the x axis instead of along the z axis up along this x axis we rotated by 90 degrees nevertheless when you multiple you measure it along the z axis you'll get plus or minus one but the average is zero so it's the averages which behave like the classical variables if you take a spin up and you rotate it by 90 degrees the average spin along that vertical axis will be zero the averages which behave classically this is a spin along the x axis what's the spin along the y axis everybody remember no no no no okay good good good guess but the wrong guess this is a spin I visit the z axis is the z axis is the x axis this corresponds to a spin which is pointing along the x axis in this direction for a plus sign with a minus sign it corresponds to a spin pointing still along the x axis but in the office erection we still haven't found the one which points along the y-axis i- i correspond to orientation only y-axis and with other values of alpha and beta you can make spins which are pointing along an arbitrary axis pointing along an arbitrary axis means if you measure the component along that axis you'll always get the same answer so that's pretty much spinning in a nutshell and as i said it is the electric charge the mass the spin of a particle are its most important properties now there's a correlation which i should not only mention but emphasize very strongly because it's one of the prime facts of elementary particle physics it's actually a theorem of relativistic quantum mechanics but we're not going to try to prove it then the field is a correlation between the values of the spin of the particle and whether it's Primula nor a boson the fermions are the ones that you can't put two of them in the same state bosons they have a probably exclusion principle bosons are the ones that you can put into the same state to behave like photons and they make classical waves have spin particles are always fertilizers without exception by Heston or now meaning one half we have five have anything which is measuring 1/2 units where the structure is health units instead of our units those are all fermions and all particles which have integer spectrum like this of all times spin zero particle is always a boson without exception a spin-2 particle is always a pulsar now what happens if you take two half spin particles techno have spin particles which one of which happens to have one value of m and the other has another value of n each one having 1/2 unit it could be three halves of - 7 hams or whatever what happens if you add up their angular momentum oh you get an integer not a half integer that tells you that objects which are made of ours are an even number of fermions are always bosons any object made of an even number of fermions will have an integer spin because it hasn't it - just think it's emotional ah what happens if you take a boson and add it to a friend on oh let me give an example an example would be a hydrogen atom eight protons a half spin particle just like an electron half not three halves are 5 has 1/2 proton has half spin just like the electron you take a hydrogen and spur me on you take an electron and you put it in orbit and you create a hydrogen atom the hydrogen atom becomes a hose out now this is not the height this is not the hydrogen isotope this is the hydrogen atom with just a single proton is goodliest guter on is a proton Valco Neutron Neutron also is a half-strength particle a deuteron is a boson a spin and another what about deuterium deuterium is a proton and a neutron that forms a nucleus with a electron in orbit around it okay what's the what are they quantum ability of the possible values of the Z component to spin going to be you take it's basically three half integers added up three half integers will always be a half integer again okay so a bond together with a fermion is Fermi on a boson it together with a BOS arms of boson and a fermion together with her me on as opposed so deuterium is Fermi on a hydrogen with only one proton what I say petunia think I got right incidentally particles and antiparticles always have the same spin same mass opposite charge what about positronium australium is a positron in orbit around that rough electron and a positron orbiting each other over that paper also both on set of property being able to go through things as I forget the O's arms have a property to what be in the same state either like super conductivity is that two pairs of electrons together and they act like that goes on there for but it's a transparent or something like that well okay before you can understand the superconductor you have to understand the superfluid a superfluid for example helium from superfluid let's talk about helium what though is helium a boson and a fermion wonder helium okay why is that it's not your even numbers that's got two protons two neutrons and two electrons two protons and two neutrons makes the helium nucleus an alpha particle and two electrons it's opposing so whatever bosons are helium is a good example of it helium atoms can all go into the same state and it's when basically a super can su perfluorinated is almost like a classical wave that you make by piling them up all in the same state that's not quite right but it's called another girl's they all move together in the same state now an electron the metal is a firm yarn P Kemper to electrons into the same state but how about pairs of electrons here's a paradox okay here's a paradox which we talked about last night but the total diving in tonight you take two fermions and you put together to make a boson how can it be that you can put two of those composites into the same state when you couldn't put the constituents into the same state does that make sense does it it does a good what no okay I'll tell you it does make sense okay what or exactly means you know that's to the next time I think I'm running running enzyme and remind me remind me about this this is so interesting that it's worth it's on how to fermions when they make a boson how you can put the bosons into the same state even though the firming ons can't be part of the same state we should get back to that I'm reaching that point in the evening where oh I'll get confused at the dryer yeah since protons and neutrons are composite why is it that their haps tin that this no no the composite of odd numbers of fermions probably brickworks a quark is a half spin particle so it's going to be a friendly on three fermions make a premiere all right just the last observation has to do with the Pauli exclusion principle or comes along discovered from staring at the periodic table long enough how we said that you can't put more than one electron into the same orbital orbital in an atom no he didn't say that he said you can't put more than one electron into the same quantum state he know very well that you can put two and no more than two electrons in the same orbital state in the atom you know that because he knew roughly speaking that the hydrogen atom you double the charge of nucleus and you put another electron in the two electrons go into the same state the same orbital spin so Pauli's exclusion principle did not apply to the orbital motion if it's it applies to the entire quantum state the entire quantum state includes the spin state you can put two electrons into the same orbital in an atom as long as the spins are in the opposite direction you can't put two electrons with spin in the same direction as long as the axis here you can't put them in the same direction but you can't put them in two spin States in the opposite direction as long as they are different as long as they are different two electrons and two of them can't be the same corporate state but quantum state includes everything about the particle so that means in every orbital in an atom you can put no more than two electrons and there are two electrons they have to be what's called spin singlets which mean this spin is canceled you can think of that like mesh gears if they're going in the same rotation they're going up a structure going the same direction you can serve the ears indeed I know I connect my car or below I simply mean the state of an electron if you ignore the spin orbit or am I using orbital in a way that chemists wouldn't use it Louisiana's yeah yeah okay I'm what I need by orbital motion is not what it chemists use by appointment what's an orbital again absolutely else PDF I mean a quantum display a solution of the Schrodinger equations a solution of the Schrodinger equation ignoring spirit it creates the chart the periodic traffic up there and so that's how come you have two electrons in here there are two things you have SPD Analects darkest blue orbital angular momentum but then this principal quantum number which is in and when I speak about the orbital motion I mean both in the orbital angular momentum it's a chemist's notation to call orbital SPD and all that stuff you know this stands for I don't want a stands for America yeah because we need it doesn't mean circle but it actually means a quantum state of zero angular momentum P means angular momentum warm this is the orbital motion of there D is angular momentum to F this angular momentum three and I know G comes after if I don't remember that that's the that's the orbital angular momentum of the electron but the other quantum number is the distant basically the distance of the electron from the from the proton and that's the principal quantum number so what I would call the orbital motion meaning this the actual motion of the electron is a calm positive principal quantum number and the orbital call that let's call that a state of orbital motion the state of orbiting both principal and alright so the rule is that Paola devised was that no electron can have all of its quantum numbers the same okay incidentally the orbital well yeah yep I really care me not how many are you can reduce the maximum yeah yeah [Applause] yeah we're not going to do that now [Music] see that I have intended to do it at all see that the principle quantum number is in the orbital quantum number than the spin what let's take let's talk about more battles in the sense that chemists use it orbital means s1b and i okay yeah the s and s particle is spin zero that means the orbital angular momentum is zero there's only one or one state of the orbital angular momentum what about P P is angular momentum or I'm talking about orbital angular momentum one but the rules of angular momentum of the same for the leg Euler momentum spin in your home so if you live in like the old method is one that uses three possibilities if the orbital angular momentum is 2 and it's a P wave or D by V state there are our five states so when you say one that actually needs up and down everything everything being specified there are two electrons or signals which all the things which can be simultaneously specified specified and then when there's three and there's to each of those young you get six more electrons for a total of 80 right exactly I hadn't intended to talk about atomic physics very interesting and some features of atomic physics show up again when you're thinking about quarks orbiting each other and so forth different okay are any questions at a brothel I have a date tonight [Music] for more please visit us at stanford.edu

Channel: Stanford

Views: 62,017

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Keywords: classical quantum physics, science, biology, engineering, abstract math, formula, angular momentum, elementary particles, spin, rotation, kinetic energy, propulsion, atom, vector, velocity, speed, direction, moment of inertia, particle, crossproduct, simu

Id: e22G9U7OlOc

Length: 102min 59sec (6179 seconds)

Published: Thu Feb 25 2010