Particle Physics 3: Angular Momentum and Spin

hello today we're going to continue in our series on particle physics looking at angular momentum and spin first a health warning whenever we talk about angular momentum and spin the problem is that we tend to think of classical things like a spinning top or a spinning ball and to some extent that gives us an understanding of what we mean when we talk about things happening in quantum mechanics and angular momentum and spin but that analogy the classical analogy only goes so far and we shall see that when it comes to quantum mechanics all is not as it seems so let's start by thinking about a ball here's a ball and we can spin it we can spin a ball on any axis but it has to have some axis so let's have an axis there and we spin it with a rotational velocity Omega and if we say that that spinning ball then has an angular momentum and the angular momentum vector points along the axis of the spin and the direction of the axis is given by the corkscrew rule so you've you look at the way the ball is spinning and if it is spinning clockwise then rather like a corkscrew if that is spinning clockwise the a logos in the corkscrew goes in if it's spinning anti-clockwise then the arrow would be in this direction so that's an object which spins on its own axis it has angular momentum we sometimes call that spin a different type of orbital motion is for example when the earth goes around the Sun this is the Sun here is the earth which goes in almost circular motion around to be accurate around the center of mass between the Sun and the earth but the center of mass is inside the Sun itself so the earth goes around the Sun and that is orbital motion and that is orbital angular momentum and once the angular momentum vector will in this case come out of the page because I've got it that the earth is going around anti-clockwise now I just want to remind you of what we've shown a diagram drawn several times before and doubtless will be drawn again the problem of an electron orbiting a nucleus there's nucleus there's the electron and what we said was that if you think of that electron as a particle orbiting it is essentially accelerating as it orbits and an accelerating charged particle would radiate energy away and as a consequence it would spiral into the nucleus and the whole Latin would disintegrate and that was a major problem but it was quantum mechanics that comes to our rescue and says that you don't think of the electron circulating orbiting as a particle rather you think of it as a wave and so you've got this electron wave which is going around and the point about that way visit to join that way there is no disjoint between where it starts and where it ends and that means that there is a whole number of wavelengths in that orbit so that there are n wavelengths where n is a whole number and integer in the circumference of the orbit which is 2 pi R where R of course is the radius of the orbit and then we remember the debroglie formula that lambda equals H over P and so we put lambda into here we've now got that n H over P is equal to 2 PI R and from that you can see that P R is equal to n h-bar where H bar is H over 2 pi and P R is classically what we call angular momentum it is the orbital momentum or the linear momentum times the radius so if you've got the electron having momentum P going in that direction then you multiply P by R and that gives you the angular momentum which you will now see is in integer units of H bar because n is an integer what that means is that angular momentum cannot just take any old value it has to be in units of H bar so if you've got an axis that is like this we'll call that M HP where m is a number whole number integer then the if you move it on a bit if you give it a little bit more angular momentum then it will have M plus 1 H bar and if you give it less angular momentum then it will be M minus 1 H bar the key point being it cannot have any value between these they are step changes because angular momentum comes in units of H bar and then we just want to clarify what we are talking about when we refer to spin and angular momentum there is a spin which is essentially a spin on your own axis so the likely spinning ball that we talked about earlier so we can represent that as a spin that goes around like this it's just going on on your own axis and that's called spin and we give that the letter S then there is the orbital motion where let's say we've got a vector like this and it simply goes around some central point but you'll notice that the vector doesn't change direction so it's just going around but keeping in the same pointing in the same direction and that we call orbital angular momentum L and then there's finally an angular momentum where the vector itself spins on its own axis and it goes around some central point so now the vector is going around like this and that's obviously a combination of spin and angular momentum we call that J and J is clearly going to be equal to L plus s in vector terms because J is both a spin on your own axis plus an orbit around some central point now from a classical mechanics point of view the angular momentum vector L which is what we're going to look at first is equal to the radius vector cross product with the momentum vector and the radius vector of course can be given its coordinates XY and at Z and the P vector will have coordinates px py and PZ and now these are just definitions of the cross product this is nothing to do with quantum mechanics now we're simply doing a definition that the X component of the angular momentum is given as Y PZ minus Z P Y so it's the y coordinate of the radius times the z coordinate of the momentum minus the Z coordinate of the radius minus the Y coordinates of the momentum and that's the only thing you need to learn x y&z and then you spin these two round the other way for here and if you want the other two all you have to do now is cycle so L X next one is y L Z equals equals and now cycled from Y so that's going to be Y Z X here you're going to cycle from Z so that's going to be P X P Y the minus terms are always the same cycle from 0 you get X Y you're once you get to Zed you go back to X again here you cycle from Y so it's going to be P Zed px and those are the three terms which identify the X the Y and the Z coordinates of the angular momentum in terms of the XYZ coordinates of the radius and the pxpypz coordinates of the momentum now a little remote reminder about what we called the commutator of X P the commutator position and momentum that's Pietra we showed in the videos on quantum mechanics concepts that that equals IH bar which of course means it does not equal zero and the significance of that is that if you want to measure two objects at the same time if you want to make two measurements at the same time their commutator must equal zero if it doesn't equal zero you can't measure them simultaneously and that's how we show the Heisenberg uncertainty principle that you cannot measure position and momentum at the same time now I should be a little more precise about this because I should give this the subscript X you cannot measure the ket x-coordinate-- and the p x coordinate of momentum at the same time you can measure the X component of position and the Y component of momentum at the same time that commutator does equal zero so it's only when you want to measure the position and the momentum in the same coordinate that you come unstuck and just to be clear of course there's nothing to stop you measuring the X and the y component of position or indeed the Z component at the same time and there's nothing that stops you from measuring the PX and the py or indeed please if component at the same time so you can measure any Kenney to coordinates or indeed any three coordinates of position any three coordinates of momentum you can even measure the x coordinate of position and the y coordinate of momentum the only thing you can't do simultaneously is to measure the x coordinate of position and the x coordinate of momentum with that in mind in two therefore I now want to ask the question can you measure the x coordinate of angular momentum and the y coordinate angular momentum at the same time if you can then the commutator must be equal to zero so we need to find out what this commutator is well the definition of a commutator you will recall from the quantum mechanics concepts videos is that LX ly the commutator of LX ly is simply defined as LX ly - ly LX and I remind you that if those were simply numbers they would be bound to equal 0 because 2 times 3 minus 3 times 2 is 0 but what I'm going to do is to take these terms and substitute for them for the values that we've got here and here so I'm going to substitute these two terms for LX and ly so that's going to give me Y PZ minus z py times z PX minus x pz - now we've got this term so that's LX that's where ly now I'm going to put minus ly which is Z px minus xpz x LX which is why PZ minus z py now although that looks rather horrifying you'll agree won't you that essentially you're going to get from this product here you're going to get four terms and from this product here you're going to get four similar terms except that this at this term these two terms here have been swapped around so the numbers are just going to be in a different different position so let's have a look at let's say this term multiplied by this term here so you're going to get a y PX PZ Z and then here you're going to get a similar term Z P X Y PZ so what you're going to get is y and px which of course commute so you're not going to have any problem with that because there's no problem about measuring Y and px but then you're going to get into P Zed z minus z pz I Y P X P Zed's egg will come from that product YP x z pz comes from this product similar the similar product here and the point is that p za minus ZB zed is the commutator of P Z with Z and that is not 0 because you can't measure them both in the same at the same time and that comes out to be minus IH bar insanely I've just spotted an error further up the page here and the commutator of X px is not minus IH bar it's plus I hope I will have put an annotation to that effect but as soon as I made the mistake but actually plus IH bar if you do X with P X so consequently if you do P Z with Z because this is actually the commutator P Zaid and Zed you get minus IH bar okay now let's do these two so that will be those two terms there that will give you X py so minus times minus will give you a plus so that's plus XP y into Z PZ minus P Z Z alright so you've got you're going to get an X py for solid from these two terms you're going to get an X P P Y Z times P Zed plus an x-pipe py PZ times Z so you're going to get that term there and that again is the commutator of Z and P ed and that is plus IH bar when you look at the other two terms you will find that there is nothing in them that it's going to cause you a problem in of computation they are all going to equal zero here you've got Y X P Z squared well that's no problem because when you get the same term from this multiple multiple impire and there is no commutation problem and so it will the two will just simply vanish one minus the other will equal zero and similarly for the fourth term so the only two terms where you will have a commutator problem of the two that we've already identified so that is going to be minus IH bar YP X plus IH bar X py I take the IH bars outside so I've got a IH bar into X P Y minus y P X but X py minus y PX x py minus YP X is LZ so this is simply equal to IH bar LZ so we've shown that the commutator LX ly is equal to IH bar LZ which of course does not equal zero consequently you cannot measure the angular momentum along the x axis and the angular momentum along the y axis at the same time and just for completeness let's just write down what the commutator is our LX and y as we've just shown is equal to IH bar LZ ly LZ commutator is equal to IH bar L X all you're doing or notice is cycling X Y why is it as a X so LZ L X is equal to IH bar L Y so those are the three commutator 's and what they essentially say is that you cannot measure more than one Co ordinate of angular mentum at any one time by convention we choose to measure the co ordinate of angular momentum along the z axis because it doesn't really matter which one you choose but we by convention choose it and in doing so we abandon all hope of ever knowing what is going on angular momentum wise along the x and y coordinates because as we've just shown you can never know if you know what's happening if you know what eles it is the angular momentum along the z axis you can never know what's going on along the X and the y coordinates so here is at my spinning ball as it were there's the axis which we'll call em and what we're going to do is to measure that angular momentum but along the z axis although Zed co-ordinate you remember we said that it's quantized which means the next notch up will be n plus 1 and you measure that along the z axis one down would be a minus one and you measure that along the z axis the point being that having measured the angular momentum along that z axis you can never know what is happening along the X and the y axis now I want to create two new terms L Plus which I'm going to call L X plus i ly and L minus which I'm going to call L X minus ily I'm not going to tell you what they are l plus an L lines though you may be able to guess we'll see what they are in a moment and the first thing I just want to check is what is the commutator of L plus and L Z this will tell me of course if I can measure L plus whatever it is and LZ at the same time well that of course is just the same as the commutator L X plus ily because that's what L plus is with Elzy and we know what that is because the commutator of LX and LZ is minus I h-bar ly because it would have been the other way around if you wanted it to be plus IH bar ly and the commutator of ly and LZ we've got an eye here so let's put that hi first the commutator of ly and LZ is IH bar LX this time it is plus and so that's going to give me these two eyes multiplied together will give me a minus one so that means I've got two minuses and bring the minus and the H bar on the outside from here I'm just going to be left with LX from here I'm going to be method left with plus ily and those shouldn't be that that's not a commutator that should just be round brackets so I've got minus h-bar into L X plus ily but LX plus ily is l plus so that equals minus h-bar L plus so I've shown that the commutator L plus and L Z is equal to minus h-bar L Plus which does not equal zero which as it happens means you couldn't measure these two at the same time even if you wanted to and I can also tell you without going through all the maths again but you can do it check it if you wish that the commutator L minus L Z is equal to plus h-bar L minus so that's the other commutator here now let's just think about what LZ actually does LZ is an operator and LZ is going to act on the state of the system which is in for our purposes the state of angular momentum of the system and what hermitian operators which are measurables do we've learned this many times in quantum mechanics concepts is they act on the state of the system to provide an eigen value and the eigen value of course the actual value you're going to measure is the value of the angular momentum along the z axis and of course it is multiplied by the eigen vector M that's a form that we've had many times in quantum mechanics concepts if you don't recognize it you really do need to look at that series because everything I do in this series is based on assuming you know and the quantum mechanics concepts so the operator LZ acts on the eigen vector M which is the state of the system or particularly the state of the angular momentum of the system and it gives you back the same state but he gives you a measurable now which is the actual measurement of the angular momentum along z axis okay so what happens if you take L plus and you act on M that will give you some new eigen vector some new state we don't know what it is yet that's what I want to try and find out but that is now a new state so let's measure it and of course we measure it along the z-axis using the operator L Z so I want to know when I operate on the state M with L plus I get a new state and then I measure it with L Z well what's that going to give me what I just want to remind you that we showed you that the commutator of L plus with L Z is minus I minus h-bar L plus so I can just write down here that L plus L Z the commutator thereof which is of course the same as saying L plus L Z minus LZ l plus that's what a commutator is and that equals minus h-bar L Plus so rearranging this here I can say that L Z L Plus which is that term there is equal to L plus LZ that term there plus h-bar L plus what we typically do in this in this arrangement as we've done so many times before is we set H bar to one so you may not get to see that H bar equals one and what that does is it gets rid of all the constants so that we don't clutter up the formula so now what I can do is to say that here I've got LZ l plus acting on em but for LZ l plus I can write instead this term here which is L plus LZ acting on M plus H bars once I forget about that so that's going to be plus L plus acting on M and what I want to know is what is that well well when LZ acts on em it simply gives me back the eigen value m LZ on M gives me m on M so this is now just L plus times M on M plus L plus M M of course is just a number that can come on the outside so now we've got M plus 1 all acting on this term this new state of L plus M and so you've got the answer that LZ acting on L plus of M gives you n plus 1 acting on L plus of m m plus 1 is the eigen value in other words what L Plus has done is it's moved up the angular momentum from M to state M plus 1 L plus is there for the promotion operator if you like um it's the operator that shits the angular momentum by one notch by one amount of h-bar to get up to the next value and you won't be surprised to know that l+ is responsible for shifting the angular momentum up by one notch l- is responsible for shifting the angular momentum down by one notch now the question is can L plus and L minus act for an infinite number of occasions remember we are measuring along the z axis so here is my angular momentum we'll call that M and we measure it along the z axis along comes my operator L + + l plus simply notches the angular momentum up by one notch so that's now in plus 1 which again we measure along the z axis we keep applying L Plus how far can we go well we can go up to the point at which the angular momentum is wholly aligned along the z axis at which point we will have reached a maximum value of M and now if you try to apply L Plus again you can get no result it's just not applicable because once you've got to the maximum value which is when the spin or the angular momentum is aligned with the z axis which is the axis you're measuring against that's the maximum value of M and L plus can have no further effect similarly if we act with L - that will take us down a notch and we can keep acting with L - until the angular momentum is now aligned in the opposite direction along the z axis and at that point we measure a minimum value of M so all the way down this this LZ scale we have got quantized values of measurements of the angular momentum subject to the fact that there's maximum value when the axis when the angular momentum is wholly aligned with the z axis and a minimum value when as it were its anti-aligned it's aligned in the opposite direction with the z axis so if you look at that z axis you will find that there's a maximum value of m there is a minimum value of m there'll be a what one might call the zero that will be a plus in that will be a minus m there's a zero value which which effectively accords when the angular momentum is aligned along either the X or the Y Direction you don't know which but it's essentially perpendicular to the z axis and so there are M values in here and there are M values here plus the zero that means that there are a total of 2m plus one possible states of angular momentum and they are each quantized because the states come in units of h-bar L plus will shift the angular momentum up one notch subject the fact that once you've got to the maximum you can't L plus has no more effect L minus will move you down a notch subject to the fact that once you're at a minimum L minus can have no further effect now from Pythagoras we can simply say that the total angular momentum squared is going to be the coordinate along the x axis squared plus the coordinate along the y axis squared plus the coordinate along z axis where it matters not that you can't measure all these three three things simultaneously the fact is Pythagoras requires that that must be so that's just geometric so let's just do a little bit more manipulation I want to find out what is the product L minus L Plus well that by definition is going to be LX - ily which is l - times L plus plus ILY which is l plus and if you multiply those out you we'll get that LX x LX is LX squared minus ILY x plus ILY comes 2 plus ly squared and then these two terms together will give you plus i LX ly - i ly L X and this term here is simply plus I into LX ly - ly L X which is I into the commutator of LX with ly because that's what that is that's the commutator and we know what the commutator of LX and ly is it's IH bar LZ now we set H bar equal to 1 in our usual fashion so that gets rid of the constant that would otherwise be in the way I times I is minus 1 so this just becomes minus LZ so the L minus times L plus is LX squared plus ly squared minus LZ and that means that l squared which you'll remember we said was an x squared plus ly squared plus LZ squared well there's the LX squared plus ly squared so L squared is going to be equal to L minus times L plus which gives us LX squared plus ly squared minus LZ well the minus L said is no good so we better get rid of that by going plus LZ and that just gives us now LX squared plus ly squared but we also need an LZ squared so we've now got the l squared which is of course LX squared plus ly squared plus LZ squared is the same thing as L 1 L minus L plus plus LZ plus LZ squared so when we act with l squared on our state m and we'll make that the maximum value of M so call that M in Emacs that is the same as saying L minus L plus acting on m m + LZ acting on m m plus LZ squared acting on mm okay because l squared is simply this and l squared acting on m is each of those terms acting on the maximum value of M but what does that actually do well let's start off by thinking about this term here when you've got a product of operators it either acts entirely or not at all here we've got an L plus acting on the aim of max what does L plus do when you are already at the maximum nothing it cannot do anything at all so that term there equal 0 because the L plus can have no effect what does LZ acting on M produce mmm that produces M times mmm it's the eigenvalue it's the maximum value of M so I can call that M M and what does il's N squared do well that of course just produces mm squared acting on mmm LLL say we always return the value of the angular momentum along the z axis so LZ will return the maximum value because we are at the maximum and LZ squared will simply return the square of that and that gives us therefore that l squared acting on the maximum value of M gives us M Squared maximum plus m maximum acting on all that acts on m m LZ squared acting on mmm would simply give you M acting on M right so if you have the maximum value of angular momentum which means the angular momentum is aligned with the z-axis let's go all the way back up here to the diagram I drew when the z axis is aligned sorry when the angular momentum is aligned with the z axis you get the maximum value of angular momentum along the z axis because that's the only thing we can measure against and when you measure that LZ which is the operator will simply return you the eigen value of the angular momentum which is the maximum value of n so that's what we've got down here that LZ squared returns M the maximum value of M Squared no problem but when you measure the total angular momentum what do you find the maximum value squared plus another bit another mm another in the max term now this is where quantum mechanics and classical mechanics part company if they hadn't already because if you measure the angular momentum along the z axis when the angular momentum is aligned with these n axis then the total angular momentum would be along the z axis in other words all of the angular momentum is along the z axis classically but quantum mechanically when you measure L you get what you would expect to get classically which is the M max squared plus a little bit extra and where does that little a little bit extra come from well that is what is the residual angular momentum which is somehow distributed between the X and the y coordinates of the angular momentum that you can never measure individually because you are forbidden we've shown that you cannot measure more than one coordinate at a time so what we're saying is that unlike the classical position when angular momentum is wholly aligned along the z axis quantum mechanically there remains and the mount equal to M max which is distributed between the X and the y coordinates and you can never measure it you know that between them there is m max but you don't know how its distributed and you never can and if you think about it that must be true because if the angular momentum were wholly assigned to the z axis and you measured it then you would know that the angular momentum measured along the z axis was ill was m max and that the angular momentum along the x and the y coordinates were zero and that would mean you would know the angular momentum along the z the x and the y coordinates and you're not allowed to do that and this is the way quantum mechanics and as it were undermines your ability to get round it it says yes when you measure along an axis which is aligned with the z axis yes you will indeed get a max squared which is what you'd expect to get when you measure LZ along the z axis but there's a little bit extra which is still being distributed and which can never be aligned wholly along the z axis a little bit extra is still distributed between x and y to provide an unknowable that you can never know precisely in terms of how its distributed and if we think back to the third video that I did in the quantum mechanics concept series when you remember I said that we could align the spin of an electron in any direction that we chose is simply by putting it to a magnetic field and I say let's suppose we align the spin of the electron so that it is wholly along the z axis and then we measure if you remember we measured it using a device with red and green lights which told us whether the spin was up or down and I said that if you aligned the spin entirely along the z axis you got a hundred percent of all those electrons would measure up what you may have inferred but what I was very careful not to ever say was that all the spin is holy along the z-axis and that is not of course true what we've just shown is that you can certainly align the spin along the z axis there's no problem at doing that and that will give you the maximum value of spin along the z axis but when you do that there is still a residual amount of spin that is somehow shared between the X and the y axis and you can never measure what that is you know what the total value is shared between the two but you don't know how its distributed so what I now want to know is can you measure l squared that is the total angular momentum squared and the angular momentum along the z axis at the same time if you can that commutator must equal zero well we showed that l squared is L minus L plus plus L Z squared plus LZ that was what L squared actually is and what I want to know is does that commute with LZ well this of course will need to act on a state so let's have it acting on the maximum value of M and what happens now first we've got this term here acting on the maximum value of M what happens when L Plus acts on the maximum value of M you get nothing so that term goes what happens when L Z squared acts on the maximum value of M it returns the eigen value of M M Squared what happens when L Z acts on the maximum value of M you just get a min and similarly here these are all numbers these are eigen values real numbers all real numbers commute consequently this commutator equals zero and that means there's absolutely no problem about measuring the total angular momentum squared and the angular momentum along the z axis at the same time and that's why as I say if you measure the total angular momentum you will get M max squared plus M max whereas if you just measure the LZ component you will be per squared you will get n max squared it's that little bit of extra angular momentum embedded in L squared that is distributed along the x and the y coordinates just to maintain the uncertainty about exactly where the angular momentum is now I want to focus on spin as such and a lot of this has been covered in the first three videos that I did in the series on quantum mechanics concepts so I refer you to those but I remind you that one of the things we said was that if you measure remember if we put electrons through this device here and measure the spin you can only ever get two answers you can add this box aligned in any way you like but however you align the Box you can either get one or two answers either you will get an up spin or you will get a down spin and so those are the only two options up and down and what we do is we assign angular momentum values or spin values of 1/2 H bar and minus 1/2 H bar and why do we do that we do that because we said that the difference between two states comes in quantized values of H bar and the difference between plus h bar and minus H bar is of course H bar so that's the reason that we give it plus 1/2 and minus 1/2 because they each have symmetric values and and the difference between the two is a whole value of H bar and just as with angular momentum so it is true of spin that you cannot measure the spin along the x axis and the y axis at the same time the commutator is IH bar as said the commutator of sy is s ed is IH bar s X and the commutator of s said with SX is IH bar sy your that's exactly the same as what we've just shown for angular momentum just as you cannot measure angular momentum along more than one axis so you cannot measure spin along more than one axis we showed in the videos on ponte mechanics concepts that you could represent the spin state of a particle electron whatever as a combination of a probability amplitude for it being up and a probability amplitude for it being down this is the probability amplitude that you'll get an up this is the probability amplitude that you'll get it down to get the probability you have to square the probability amplitude or more accurately multiplied by its complex conjugate so the probability of getting an up will be alpha alpha star where alpha star is the complex conjugate and the probability of getting a down is Beta Beta star and since you must get either an up or down you never get a zero you always get one or the other that means that the probability of a plus the problem live down is equal to one so consequently alpha alpha star plus beta beta star always has to equal one that's called normalization and what are these spin operators SX sy and SZ well we've actually already done this in quantum mechanics concepts SX is nothing more than the power matrix Sigma X which you'll remember was 0 1 1 0 sy is the power matrix Sigma Y which is naught minus I I naught and s and s ed is the Pauli matrix Sigma Z which was 1 0 0 minus 1 so we have effectively ready covered the values or the measurables the operators SX sy and SZ and and that equates to remember our device if you place it so that you're measuring along the z-axis you will get an up-or-down if you place it so that you're measuring essentially along the x axis again you will still get up or down but up in this orientation is right and down is left and if you as it were have it going into the page I can't easily draw that but you have to imagine that that is pointing into the page then up will be in and down will be out so on the three dimensions you either get up or down left or right or in or out and we showed that the eigenvectors associated with those bearing in mind that you always want to get eigenvalues of +1 on plus 1 or minus 1 which represent up or down in the alignment that you've got whichever way you happen to align this you will always get a red light or a green light and up or down so you want the plus or minus 1 as the eigen values the eigenvectors that we identified and this remember this was fully identified in the in the quantum mechanics concept series which is why i'm going over it quickly now up was represented as 1 naught down was represented as naught 1 right was represented as 1 over root 2 1 over root 2 left was represented as 1 over root 2 minus 1 over root 2 ha in was represented as 1 over root 2 i over root 2 and out was represented as 1 over root 2 minus I over root 2 and those were the six and values the eigen value for up was +1 the eigen value for down was minus one eigen value for right was plus one eigen value for left was minus one eigen value for in was plus one eigenvalue for out was minus one and all of this fitted into the general formula that the operator SJ where SJ is either SX sy or s ed up here s exits like said the operator acting on the state sy which is one of these states here gives you back the eigen value which is either plus 1 or minus 1 times the same same state that was again all covered in quantum mechanics concepts well that was um fermions that was um particles that have a spin of 1/2 and you can either get plus 1/2 or minus 1/2 what about bosons like photons will they have a spin 1 which means that there are three possible measurements that you can make we call those plus 1 H bar 0 and minus 1 H bar and you'll notice that once again the gap between them is H bar because angular momentum comes in units of + H bar or 1 H bar and the eigenvalues of measuring these will be +1 naught and minus 1 so we're going to need spin operators which are the measurables and we're going to need I can vectors which will deliver us I gain values of plus 1 or a minus 1 to represent the three possible spin States of a boson which is of spin 1 and I can tell you that you can't do that with the 2 by 2 matrix so the spin States for um a boson along the z-axis turn out to be naught I naught minus I naught naught naught naught naught that's the dead spin state the Y spin state is naught naught I naught naught naught minus I naught naught and the said since pay spin state sorry the X spins state is naught naught naught naught naught I naught minus I naught those are the three spin operators for bosons which have spin 1 and the ideal is that you take the spin operator SJ which is X Y or Z act it on the spin state and that gives you a value lambda times sy that is the usual equation where we have an operator acting on a state gives us an eigenvalue times the same state so I've just given you the values of the operators X Y and Zed and I've told you that the eigenvalues must be plus 1 naught or minus 1 so now we can establish what the eigenvectors have to be and i'm not going to do this for all of them I'll just show the principle how it works you can do them for yourself so I'll take the operator s Z and I want us to get the eigenvalue plus 1 so this is when we measure along the z axis and we want a plus one h-bar measurable so essentially we're going to need to take the SZ operator which is going to be naught I naught minus I naught naught naught naught naught that's the operator and we're going to have to act on an eigen vector what is the mention of the eigenvector it must be three dimensions because we've got a three by three matrix so let's just call that alpha beta and gamma then oh what that is yet and that is going to produce an eigenvalue which i've said i want to be plus one for this illustration and that obviously has got to be equal to x the same eigen vector because that's that's the whole deal but what do you get if you multiply this matrix by this eigenvector well you're going to get naught times alpha plus i times beta plus naught times gamma so that's going to be the top term is going to be i times beta the next one is going to be minus I times alpha plus naught plus naught so that's minus I alpha and then you're going to get naught naught naught naught so this the guy we called this which means the three terms must be equal so let's see if I can write that out that means that alpha has got to equal I beta because that's the top term in each case beta has got to equal minus I alpha because that's the second term in each case and gamma equals zero because that's the third term in each case so we've identified gamma straight away that's zero and I can tell you that this there are several solutions but they are all the same kind of solution but it works if you have that alpha equals one beta equals minus I and gamma equals zero so consequently the actual eigen vector is one minus I and zero and that is the eigenvector of the spin operator measuring along the z axis that returns a value of plus 1/4 and the eigenvalue this is not quite right why is it not right because if you work out the probability these are all probability amplitudes remember you square them or more accurately you multiply by their complex conjugate if you and then add them all together and they have to total one if you do that you'll get the total two so that's no good so the way we solve that is you have to divide each of these terms by the square root of two you don't have to divide 0 by the square root of two because that is 0 so technically you need to divide by the square root 2/2 what's called normalize this term so 4s said and with the eigenvalue plus 1 we've just shown that the eigenvector is 1 over root 2 minus I over root 2 and naught I can tell you that if you wanted the eigen value minus 1 the eigenvector would turn out to be 1 over root 2 plus I over root 2 and naught and if you wanted the eigen value 0 which is the third possible spin state then the eigen vector would be naught naught and 1 so those are your three eigen vectors for your three separate eigen values plus 1 minus 1 and 0 all associated with the s Z eigen the SN miserable or operator and you'd need to do the same calculation again for SX and sy

Channel: DrPhysicsA

Views: 125,367

Rating: 4.9512892 out of 5

Keywords: Physics, Particle physics, quantum mechanics, quantum field theory, wave particle duality, Photoelectric effect, De Broglie, Einstein, quantised angular momentum, momentum, energy, hamiltonian, Schrodinger Equation, mass, Higgs, spin, creation operator, annihilation operator, fermions, bosons, SU3, SU2, U1, gauge invariance, spontaneous symmetry breaking, QED, QCD, Standard Model, Feynman diagrams, Supersymmetry, SUSY, Strong Nuclear Force, Weak Nuclear Force, Electromagnetic Force, QFD

Id: V7DcOXbVY70

Length: 56min 50sec (3410 seconds)

Published: Tue Oct 22 2013