Lecture 5 | New Revolutions in Particle Physics: Basic Concepts

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[MUSIC PLAYING] Stanford University. Group velocity and phase velocity-- OK, good. All right, let's talk about it a little bit. Generally, the phase velocity of a wave doesn't have anything to do with measurable properties of the wave of the measurable things in quantum mechanics. It's the group velocity, which is really the velocity at which signals move, the velocity at which particles move and so forth. So let me just show you what's at stake and what's going on. The first thing is if you have a plane wave-- a sine or cosine or an e to the ikx-- and that's all-- just e to the ikx. Let's write to e to the ikx minus omega t. Or just sines-- the sine of kx minus omega t will also do. In fact, let's just write that. Sine of kx minus omega t-- looks like that. And how does that wave move? How does the top of each crest move? The top of each crest moves in such a way that this thing inside the sine is constant. In other words, for example, right over here at time t equals 0. And at x equals 0, the sine wave is not a maximum at that point. But x equals 0 and k equals 0-- the sine wave happens to be over here. OK, but where is the sine wave after a certain amount of time t? That point here has moved. It's moved to the place where kx minus omega t is equal to 0. Where is that? That's at the position x is equal to omega over k times t. So it's dividing by k. Now that's not too surprising. If you think about it for a minute, omega is a frequency. So it's inverse to the period of oscillation, to the time of oscillation. It's 1 over the time of oscillation. There's a 2 pi in there, but the 2 pis appear in both things. So omega is like 1 over the period of the wave-- 1 over t, the amount of time that it takes for the wave to oscillate-- and k is like 1 over lambda. Actually, there are two pis in these equations-- two pis in the numerator. Let's put them in. And so omega over k-- what's omega over k? Omega over k is lambda over t. It's the wavelength divided by the period. The wavelength is the distance it moves in one oscillation. The period is the time that it takes. The ratio is the velocity. So this is the velocity of the peak of each wave. OK, for a plane wave that's called a phase velocity. Now let's think about Schrodinger waves for a minute. Schrodinger waves have a connection between omega and k. To remember it, all you have to remember is that omega is really secretly energy, and k is really secretly momentum. If we set h bar equal to one and so using the formula that energy is equal to momentum squared divided by mass, that just becomes omega squared is equal to k squared divided by 2m. Did I say momentum squared divided by mass over twice the mass? This is, as usual, with h bar equals c equals 1. In fact, there's no c's in this formula, but there are some h bars. OK, so that will tell you, for example, let's compute the phase velocity-- the phase velocity of a Schrodinger wave. For a Schrodinger wave, Omega-- sorry. Omega is equal to k squared over 2m, not omega squared. Omega is k squared over 2m So what is the group velocity? Its omega divided by k-- omega divided by k is equal to k over 2m. That's the phase velocity-- k divided by 2m. What's the velocity of a particle in terms of its momentum and its mass? Momentum divided by m. There's an extra factor of two here. What's that doing there? Peculiar thing-- the phase velocity is not the velocity of the motion of a classical particle. OK, let's just keep that in mind. That's true, but let's do something else now. Supposing I add a constant to the frequency of every wave. I simply take the frequency of every wave-- all of them for all k-- and I add the same constant. What should I call that constant? You got a name for it? It's mc squared, but let's not belabor. It's not important. Let's just add a constant-- an overall constant to the frequency of all waves. What does that do for the phase velocity? Well, it changes it omega divided by k will now have an extra term in it. It'll have an extra term in it, and it won't be the same. Whatever It does, it will be c of ak. It'll have an extra term in it. The phase velocity will have changed. But I tell you right now, there is no physics in just adding a constant to the frequency of every Schrodinger wave what does it correspond to? It corresponds to adding a constant for the energy of a particle and adding a numerical constant to the energy-- doesn't ever change anything. Only energy differences are ever important. Can we see what's going on in terms of the Schrodinger wave? Well, the Schrodinger field, psi, is some kind of sum or integral-- either a sum or an integral-- of creation operators, rather annihilation operators, in this case, e to the ikx, e to the minus i omega t, where omega is the omega, which is related to k. So let's write omega of k. Now supposing I add to omega just this constant c. What does it do? It adds in or multiplies in each one of these waves-- sorry, e to the minus ict. All of the waves in here that make up the Schrodinger field all get multiplied by the same time-dependent phase here. So what it does is if you had a solution of the Schrodinger equation and you added a constant to the phase, it would just change the solution. Let's say psi of x and t. It would just multiply it by e to the minus ict. That's all it would do. It would multiply the wave-- sorry-- by psi of x and t. It would change the wave by multiplying it by a time-dependent phase. All of the quantities of physical interest that are made out of the Schrodinger field involve psi times its complex conjugate. The probability density, for example, is psi times psi star. All possible expectation values-- quantum mechanical expectation values always involve psi times its complex conjugate. So what happens to psi times its complex conjugate when you perform this operation? It cancels out. It cancels out because psi star gets multiplied by e to the plus ict. And when you multiply them, they cancel out. As a consequence, there is no physical measurability to adding a constant to all of the phases. And nevertheless, it changes the velocity of the Schrodinger waves. So there must be something about this change of velocity-- which is, well, it's change of frequency. Let's say, for the phase velocity, the phase velocity is not something which is ever measured or ever appears when you multiply psi time psi star. So it's an artifact of a particular mathematical set of conventions. On the other hand, the group velocity really does mean something. It is the motion of the wave. If you have a wave packet, which is centered in some place-- in other words, if you have a bunch of waves which add up, a bunch of waves, bunch of plane waves, which add up to something that looks like that. Maybe it has some oscillations in it. But what you would call like a lump, the whole lump moves with a velocity that's called a group velocity. But let's see if we can see just go roughly why. If you take a whole bunch of plane waves-- every plane wave looks like this-- goes on and on forever. How is it possible to add them up so that you get a concentrated lump someplace? The answer is destructive interference. Destructive interference-- so that the waves are in phase and add up over here, and they're out of phase and cancel over here. So the question then comes up, if we're interested in the motion of waves and how waves of slightly different frequency reinforce each other and make high points and low points, let's try to follow how that works. Let's take two waves, not a single wave but let's take two waves of very neighboring close momentum and try to follow where the reinforcing constructive interference takes place. That'll be good enough. Let's just follow where the constructive interference takes place between the two waves. So in particular, take e to the ikx-- we could just take sine. Sine is good enough. Sine kx minus omega of kt and let's add to it-- this is just a way of moving along with momentum k, energy omega. And let's add to it another wave, which is-- let's put parentheses around this-- sine k prime where k prime and k are very close to each other. X minus omega of k prime times t. Where are they in phase and where are they out of phase? In phase means they reinforce each other. Out of phase-- and in particular, it will mean that the two terms here have the same sine-- and there are other places where they'll have opposite sine. Well, one easy way to find out where they reinforce each other is to ask when is the argument-- argument meaning the thing in the bracket here-- when does the argument of this wave the same as the argument of this wave? Those are the places in that case, in that particular place where the argument of this wave and the argument of this wave of the same. It's clear the waves will reinforce each other. They'll reinforce each other, and they will be constructive interference. Where is that? That's at the place where kx minus omega of kt is equal to k prime x minus omega-- let's just call it omega prime. Omega prime means omega of k prime t. But let's write this in the following way-- k minus k prime x is equal to omega minus mega prime t. Two neighboring waves will reinforce each other along a trajectory where x and t are related by this equation. OK, let's just divide it by k minus k prime. And they'll assume that k and k prime are very close to each other. Just for simplicity, let's imagine that they only differ by a very small amount-- a small fractional amount. Then what is omega minus omega prime over k minus k prime? That's the derivative of omega with respect to k. That's all it is a derivative of omega with respect to k if omega and omega prime are near each other. And so what it says is the place where they reinforce each other will travel along an orbit, which is x is equal to the derivative of omega with respect to k. d omega by decay times t. This is also a velocity. It's the velocity of the place where the waves reinforce each other, and it's called the group velocity. It's the group velocity. Let's calculate it. If omega is k squared over 2m plus an arbitrary constant-- this is the phase velocity, v phase. What about the group velocity? That's the derivative of omega with respect to k. Derivative of omega with respect to k-- that's equal to 2k divided by 2m, otherwise known as k over m. And what do we get from here? Nothing, zilch. This is the group velocity, v group. Notice two things-- the group velocity does not depend on this extra additive thing that I put into each frequency-- number one. And number two, it's exactly the velocity of a non-relativistic particle momentum divided by mass. So the 2 went away. Yeah, that's right. The two went away-- exactly. When I differentiated k, I got 2k divided by 2m, and 2 went away. So the group velocity is the thing which is most closely associated with the classical motion of a particle. It's also the only thing that's associated with propagation of signals and so forth, not the phase velocity. We could go ahead a little more with this. Let's see, that we want to? If the speed of light is c, is that phase or group velocity? Both. So let's go through that. Let's go through that and see why that's true. The c here is not the speed of light, incidentally. This is not the speed of light, but let's go through that. Let's take a relativistic wave and calculate both the group and phase velocity. For that, we need to know a factor too. We need to start with the relationship between energy and momentum. Sometimes they talk about light goes faster than the speed of light. No, no, it doesn't. It doesn't. It doesn't. It doesn't. It doesn't. It is a good question, but it doesn't have to do with that. It's a separate issue. Could he be thinking of the classic [INAUDIBLE] waveguide for the phase velocity is always greater than the speed of light? The group velocity is what was left. The product is the speed of light. Yeah, so let me give you an example of that happening right now, not for a waveguide, not a waveguide, just a wave equation for a particle with a mass. The wave equation is nothing but the relation between omega and k. To hell with the wave equation-- just the relation between omega and k for a relativistic particle-- particle moving close to but not at the speed of light. It's got a mass. It's got a mass. If it didn't have a mass, it would move with the speed of light. OK, what's the connection? Omega is energy. Let's write this first in the form that we might remember best. What's the relation between energy and momentum for a particle, which is relativistic? [INAUDIBLE] What's that? Momentum equals [INAUDIBLE]? The relation with momentum, p, square root. OK, square root of p squared plus m squared. But do we want to put the speed of light into here? I think there's a c to the fourth and a squared. But let's set c equal to 1. Yeah, we've already set c equal to 1. Later on, we can put back the c's if we want to. It's not necessary. Here's the energy. And if we now set h bar equal to, 1 this is exactly the same as writing omega as equal to square root of k squared plus m squared. So for a relativistic wave, a relativistic particle, the frequency is the square root of k squared plus squared. Now if k happens to be equal to 0, then it's omega equals the magnitude of k. Apart from a speed of light, which I set equal to 1, that's the relationship for a wave moving with the speed of light that omega is equal to k. What's the group velocity? First of all, what's the phase velocity? The phrase velocity is omega over k. The phase is omega over k and that's 1. What does that mean-- 1? Speed of light-- right. What about the group velocity v group? That's the derivative over omega with respect to k. What is that? 1. The group velocity and the phase velocity are the same for a wave with mass equal to 0. Forget that this is mass. This is just a mass. Of course, it's mass. Energy equals square root of momentum squared plus mass squared. But now let's calculate the group velocity when the mass is not equal to 0-- well, both of them. Let's calculate both of them. Omega over k-- that's the phase velocity. That's the square root of k squared plus m squared over k, but let's put it all inside the square root. Let's put it all inside the square root. Then it becomes k squared percent squared over k squared. Is this is bigger than or smaller than one? Bigger than one-- k squared plus m squared over k squared is bigger than 1. So the phase velocity omega over k is bigger than 1-- bigger than the speed of light. OK, oh we get super scared? Waves moving faster than the speed of light. But what about the phase velocity. Let's work that out. Group velocity-- thank you. The group velocity is the derivative of omega with respect to k. What's the derivative of omega with respect to K? Whoops, not t-- d omega dk. We have to take the derivative of a square root. That gives us 1 over twice the square root of k squared plus m squared. And then we have to differentiate the argument here with respect to k, So That just gives us k. This is k over this. This is the derivative, which is also equal to the square root of k squared over k squared plus m squared. Isn't that nice? The group velocity is squared of k squared over k squared plus m squared. The phase velocity is square root of k squared plus m squared over k squared. They're inverse to each other. This one is greater than 1. This one is always less than 1. This one is less than 1. In fact, this is the velocity of a relativistic particle with momentum k. k squared-- thank you. Yeah, they're just inverse to each other. And as it happens-- as was pointed out by somebody-- the product of the group velocity, the group velocity, and the phase velocity are equal to 1. So they're just inverses of each other. Now that's somewhat accidental. That's not a very general fact. Buried in there somewhere is that the wave speed must be a function of frequency. What? Wave speed must be a function of frequency. If the wave speed is constant to the frequency, you get no dispersion. That's right. That's this case here where omega is equal or proportional to k. That's the case where all waves have the same velocity. So it's only in the case where all the waves of different frequency have the same velocity that the group and phase velocity are the same. That's a good approximation, not only for light. It also happens to be a good approximation for sound and other things-- to at least lower frequency sound waves. They all move with pretty much the same velocity and so they-- Then if it's not true, then the [INAUDIBLE].. That's right. That's right. So that's right. That's right. So the difference of these two kinds of velocities is connected with wave dispersion with the fact that waves change shape absolutely, absolutely. OK, so that's the story about waves, about group velocity, and phase velocity. And as I said, in quantum mechanics, for example, here the effect of an additive shift in omega is no physical effect whatever. It's an example of the fact that most times, the phase velocity is irrelevant to real physical propagation of any signals or any energy flow or anything like that. Probability flow. You said that you can't measure the phase velocity because these complex-- At least for the Schrodinger wave-- yeah. Well, for a photon, they're the same. They're the same. Now it gets a little more complicated when you have various kinds of dispersal of materials that the photon can move through. But that's for these squalid state physicists. Is all of this waves that don't spread out over time? Waves that don't spread out over time are the ones for which all waves have the same velocity. If different components of the wave have different velocities, then it's going to spread out. So that's the case where omega is a non-linear function of k, basically. If omega is a linear function of k, then they don't spread out, and that's the case of always having one universal velocity. At some point, maybe if somebody comes back and asks me this question about light cones and things, we can come back to it. But it was a good question, but it's not quite related to this. Let's see was there anything else I wanted to fill in before getting to fermions? Yeah, before getting the fermions, let's discuss the issue of momentum conservation. We discussed the issue of energy conservation. And I showed you an example. It was really just an example of how the time translation and variance of the amplitudes that go into a scattering process in quantum field theory conserve energy. I'll just remind you very quickly and then show you how the momentum conservation works-- very, very similar. We will come back to these things, but then I want to get onto fermions. That's what I want to get to tonight. Where were we? I was going to talk about momentum conservation just really quickly. Remember how energy conservation works? Energy conservation we had, for example, some process. It doesn't matter what it is-- a bunch of psi annihilation operators, a bunch of psi creation operators act on a state with a bunch of particles in it with momenta. And then we calculate the overlap of the inner product or the probability amplitude for a final state with a bunch of other particles, k prime, blah, blah, blah, blah, blah. OK, these fields act on the initial states. These fields act on the final states. And I'll just remind you a little bit how it worked. We said supposing all the processes take place at the same point of space and time at the same point or the example that I gave was one in which one particle came in. One particle went out. And they came in and scattered off a point in space, but we integrated over all possible times. The particle was there, not this particle, but the target was there at all times. And so a particle could have scattered at any time-- whenever it gets there. And that's represented by integrating this over time. If you remember, each one of these size or each one of these sizes made up out of a bunch of things involving a's, e to the ikx's, and e to the minus i omega t's-- omega as being the energies, k's being the momenta. Now if we said all of this took place of the origin of spatial coordinates, space coordinates, then this is just equal to 1. When the operators act and we're finished calculating everything, the only time dependence in this amplitude comes from these e to the i omega t's. Let me remind you what it looked like. It looked like e to the minus i omega of each momentum k1, t, e to the minus 1 omega of k2 t dot, dot, dot, dot. That's for the initial states. And then for the final states, there was something similar e to the plus i omega k prime one blah, blah, blah, blah times t. Blah, blah, blah means the sum of all of the outgoing momentum, and here it's minus the sum of all the incoming momenta-- omega k incoming times t. The last step was to say if we integrate this over all times representing the fact that the scattering could happen at any times, the integral of this gives what-- integral over time? A delta function-- a delta function of the sum of the final momentum minus the initial momentum-- sorry, final energies minus the initial energies. When we integrate this over time, what we get is a Dirac delta function of let's call it omega final. That means the sum of all the outgoing energies minus the sum of all the incoming energies. In other words, the answer is 0 unless the incoming energies are equal to the outgoing energies. And this was just as a consequence of saying that the process could happen with equal probability at any time. Now what about momentum conservation? Momentum conservation is exactly the same sort of thing, except it has to do with space translation. If a process can happen at any point of space with equal probability, then it's momentum that's conserved. Let me give you just one example-- one very simple example. This is a process in which a particle comes in and splits into two particles. Can this really happen in nature? Yes, it can really happen in nature-- particles decay. Particles decay. And an incoming particle of one momentum can decay into particles of other momenta. But as you know, momentum is conserved, but let's see if we can see why that is. So the incoming particle is described-- well, of course, it's described by a particle coming in. The outgoing particles-- this is k-- outgoing particles k prime 1 and k prime 2. The operator, which eats the initial particle-- let's call that psi of x-- misrepresents the particle being absorbed at point X. and then two particles created-- also at point x. That's a simple example. One particle is eaten at point x, and then it spits out two particles also at the same point for simplicity. But it can happen at any place. It can happen in any place and that means we have to integrate this overall space. Let's see what we get. Again, these size have creation and annihilation operators. The sum annihilation operator in here-- which can annihilate or just work on this particle-- eat that particle. That's a minus of k, and it has an e to the ikx in it. Let's forget the omegas. We don't need the time dependence in here. So that's the operator, which eats this particle over here, and then there's these two operators, which put back the particles over here. And they give us some a plus of k1 prime, a plus of k2 prime, and then there's an e to the minus i k1 plus k2 prime x-- same x. These operators act. They annihilate the particles. We calculate the matrix element. That's all trivial. The only thing of interest that's left over is this function of position. This is the amplitude that the process happens at point x. But then since x is not a special place, it can happen anywhere as we integrate it over x. What do we get when we integrate it over x? A delta function of momentum conservation. All right, so if a process can happen with translation and variance in space, then it will conserve momentum. And in fact, it didn't matter how many particles were coming in and going out in this little demonstration here. It would work the same way-- translation and variance or the idea that the process could happen anywhere with equal amplitude. That's momentum conservation. The idea that it can happen with equal probability at any time-- that's energy conservation. There was another symmetry we talked about a little bit. The other symmetry that we talked about a little bit was a possible symmetry. If we multiply psi by e to the I times-- let's call it lambda-- in other words, just change the phase of psi-- what will that do to probability amplitudes? Well, the answer is it will do nothing to them as long as it's the same number of psis and psi daggers. If there's the same number of psis inside daggers, that's the same thing as saying that the number of particles coming in is the same as the number of particles going out. And so symmetry with respect to changing the phase of the wave function like that-- just multiplying it by an overall phase-- and multiplying its complex conjugate by the opposite phase, that's associated with certain conservation laws, such as conservation of electric charge. Conservation of electric charge tells us that is contained in symmetries of this type here. If these were charge carrying fields-- charge carrying fields mean fields for electrons, for example-- if they were fields for electrons, then this symmetry would be correct only if there were as many electrons in the initial state as in the final state. This is, of course, a law of nature that charge is conserved. So the law of nature that charge is conserved is the same as the symmetry under changing the phase of the wave function. These are some examples of symmetries but that was a quick review. Now I want to move on to the notion of a fermion. Yeah. When you were doing the integration over space, that's overall three spatial dimension? Yeah, [? say. ?] That would mean conservation of all three components of the momentum. Exactly. Yeah. Then if you have two positive k's and a negative k, the k's may have a distribution. In other words, you could have one is larger, one is smaller, the other one is-- Yes, yes, yes, yes. Is there anything in here that could point to-- Some relative probability for different ways that the momentum is shared. Yeah, not for this case here. For this case here is basically neutral with respect to the waves. The momentum is shared. [INAUDIBLE] Here, just the number. We'll come back to these things. We'll come back to it. We'll come back to it. I simply wanted to give some very, very simple illustrations of the connection between symmetries and conservation laws and how quantum fields are used to describe physical processes of creation or annihilation of power of particles. Yeah. A technical question. I have, in my notes, where we were doing a sum over k of annihilation operator times either the plus ikx times some state. And I wrote that it reduces to one term. The state has a product line in it. There's only one particle, There's one particle I'm annihilating by summing over k, and I wrote that it reduces to 1 term. And I wanted to ask whether that means that if the terms which are annihilating places where there are no particles-- 0-- gives you 0. Yeah, 0 vector-- 0 vector of 0. It's nothing at all. Let's now depart from the theory of bosons The characteristic of bosons is, first of all, you can have any number of them in any quantum state. And as we described a couple of lectures ago, they have a tendency to want to accumulate in the same state. An example of that is if a particle decays and produces a photon, for example, an atom decays and produces a photon it may produce a photon in any direction with any momentum with some probability distribution. But if at the same time as it decays there happens to be a large number of photons all in the same state-- in other words, all moving the same way, with the same energy, and the same frequency-- it will preferentially decay. So that the photon that comes out has the same momentum as the extra photons, which are already there. That's the square roots of n that occur in the boson operators. Fermions are very different. Fermions don't like each other. They don't like to congregate into the same state. In fact, they simply can't be in the same state. This, of course, was a discovery that was first made by Pauli, I guess. Was it Pauli? Yeah, sure, Pauli exclusion principle in the context of atoms that two electrons never occupy the same quantum state within an atom and that was generalized by Pauli himself-- Pauli Fermi, others-- to a powerful principle that fermions of particles that simply can never be in the same quantum state in atoms, out of atoms, anywheres very different than bosons. OK, let's work out some of the mathematics or some of the bookkeeping. The mathematics of creation and annihilation operators is really a form of bookkeeping-- useful bookkeeping. But let's try to work out a theory of quantum fields of fermions-- particles that can never be in the same state. Well, we start with the same kind of construction, as we did for bosons. Imagine a quantum field psi of x, which is, again, exactly the same formula as for bosons. Summation over all of the allowed momenta-- I guess I call it k, usually. All of the allowed k, if the allowed k are continuous, it becomes an integral. Annihilation operator for a particle of momentum k, e to the ikx. The states of the system are again labeled by how many particles there are in each quantum state, but the rule is never more than one. So the state of the system is a bunch of 0s and 1's, never more than one particle in the same state. That's the character of a fermion. Now, are there other kinds of things that can have three particles in the same state but not more or that must have more than seven particles in the same state or other odd varieties? The answer is no. But to prove that the mathematics of quantum mechanics together with special relativity dictate that the only possibilities are fermions and bosons-- would take us far beyond this class so we're not going to do that. But we're are going to describe what a fermion is. All right, so this formula is exactly the same as for bosons. Psi dagger of x-- same thing. Incidentally, this is all non-relativistic particles. We have not got to relativistic particles. Eventually, we will. Same thing with creation operators-- a plus of k t to the ikx. But now the rule is a little different than before. Yeah, there is a minus sign. All right, so now let's focus just on one possible momentum of the electron, or this could be an electron. Let's focus on one possible state, not worry about all of the states. But let's just say some particular quantum state is either filled with an electron, or it's empty. Those are the only two possibilities. Let's represent that. I guess we can represent that by 0-- that represents no electron in that state-- or one-- that represents one electron in that state. This isn't multiplying. This is just two possible states-- put a comma here. What is the algebra or the abstract mathematics of the creation and annihilation operators? Let's see if we can make some guesses. You'll make the guess. The creation operator on the state with no particle-- want to guess? One particle. What else could it be? What about a creation operator on a state with one particle? 0. Now, 0 does not mean the state with no particles. It just means the 0 vector. I don't like that as a just 0-- It doesn't mean anything or there's not a vector at all. It's a non-state. Hmm, what's that? Louder. Is the operator still linear? Yeah, they're all linear operators. In quantum mechanics, all operators are linear operators. When one speaks about an operator-- especially in quantum mechanics-- it's always a linear operator. I think mathematical terminology is that operator usually means, maybe always means linear operator. Otherwise, it's operation, but I'm not sure. But yeah, in quantum mechanics, operators are always linear operators. OK, let's see if we can make some more guesses. What about a minus on o? Well, you can't take away that which is not there. But what about a minus on the state with one particle? No particles. No, no particles. All right, so these are operators. They do what they do. They do what they do. Just to clarify-- the 0 there-- that means that doesn't happen and you can't do that. Yeah. Yeah. Yeah. Let me give an example. If we described harmonic oscillators by wave functions-- ordinary wave functions-- then there would be a wave function for the ground state, and the wave function for the ground state. Well, let's just discuss this for a minute. The wave function for the ground state would be some Gaussian wave function like e to the minus x squared-- that would be the wave function. This would be identified as the ground state and therefore, the state with no excitations. On the other hand, the wave function psi of x, which is literally equal to 0, that's the state 0. No wave function at all. Probability is just 0. It's not a state. It's just a nothing. OK, this is the starting point for the algebra of fermionic creation and annihilation operators. There's nothing in the notation that helps us distinguish between fermions and bosons then? Well, yes, I suppose we should. Yeah, it's a good point. We should probably use a different letter. Let's not call fermionic operators a. Could we call them alpha? We could call them alpha. Sometimes C is used. Why don't I use C? OK. The problem is that there are many different, not inconsistent, but many different notations in the literature. And I guess, for us, we probably are wise to use a notation where bosonic and fermion operators have different letters associated with them. So let's call it C. This is a common notation also get that fermionic operators I'll label c. But they are the analog for fermions of the a's for bosons-- whoops. Right, let's see if we can figure out what C dagger C-- C+ times C- does. What does that operator do when it acts on anything? So let's figure out what it does. What happens when we act on O? Well, C-, when it acts on O, just gives 0. And C+ it is not going to be helpful in getting it back to being non-zero. 0 is 0. So this is 0. You simply can't take away what's not there. Even if you try to put it back afterwards, you can't take it away to begin with. What about C- times C+ on O. What does that give? C plus-- when it gives O gives what? 1 and then c minus? O. Let's call it O. Let's call this O, as opposed to 0. So this gives back the state. In other words, it gives back exactly the same state. Let's see it gives back exactly the same state. What about C+, C- plus C-, C+? If it acts on O-- let's see. What does it give when it acts on O? We're just adding those two equations. It just gives O. The ground state-- sometimes known as the vacuum. All right, let's now do the same thing acting on one. What is C-, C+ plus acting on one? Well, C- minus an act on one, takes the particle away, and leaves O, the vacuum, and then C+ puts it back. So this gives one back again. What does it give when it acts on-- One. Four. Acting on one. So minus-- yeah, that's correct, isn't it? Yeah. Yeah. OK. Now what happens when we do this at 0 because we can't add another particle to one that's already present if it's a fermion. So this is 0. What happens if we add these two equations? We find again C+, C-, C-, C+ on 1 gives 1. So those two are equal to i, including that one. Yeah. Both of these operators-- while this operator, this particular combination when it acts on any state gives back exactly the same state. So that's written in operator notation by saying C+, C- plus C-, C+ is equal to the unit operator. It just gives back the same state-- whenever it acts on. This combination-- an operator times another-- plus the interchange. What would you call if it there was a minus here? You would it the commutator. With a plus here, it's called the anti-commutator-- that's what it's called, and it's the symbol for it is a curly bracket. Sometimes they use a plus sign. Sometimes. Curly bracket of A and B is, by definition, AB plus BA. So the first thing we learn is the creation and annihilation operators. Anti-commute to give one on the right-hand side. This is incidentally the analog-- yeah, it doesn't matter which order you write them in since they're the same-- well, now, sorry-- they're not the same but doesn't matter which order you write them in-- C+, C-, C-, C+-- plus same thing. This is the analog for bosons of a commutator of A- with A+ is equal to 1. Analog is not the same thing, but it's the analog. In this case, it's commutated for bosons-- in this case, it's anti-commutator. Now what about the anti-commutator of C+ with C+? The anti-commutator of C+ with C+ is just C+, C+ plus C+, C+ is just twice C+ squared-- twice this. What does this give when it acts on anything? Well, if it acts on the vacuum, it can create a particle, but then the next one is going to try to create a particle on top of a particle which is already there. You can't do that. If it acts on a particle which is already there, can't do that. So whatever it acts on C+, C+ always gives 0. All right, so C+, C+ gives 0. What about C-, C- anti-commutator, C-, C-? That's just twice C- squared. If it acts on a state with a particle, the first operator removes a particle, and the second one looks for particle, can't find one, and says 0. If it acts on a state without a particle, it also gives 0. This is the analog for harmonic oscillators of commutators. Is that 0 or O? 0. No, I mean, on C-, C-. Is that the vector for O? This is an operator. It's not a vector. Right, it's the operator 0. Yeah, it's the operator 0. Right, it's the operator 0. The operator 0 is such about when it acts on any vector or any state, it gives 0. It really is literally 0. Whenever it acts, it just doesn't give anything. It gives 0. There's an operator, which is 0, and there's a vector, which is 0, not a vector but a state vector-- a state vector which is 0. Two different uses of 0 but they're both 0. And in fact, when the 0 operator acts on any vector, state vector, it gives the state equal to 0-- 0 vector. OK, so fermions and bosons have a parallel mathematics to each other where wherever you see commutator for bosons, you replace it by anti-commutator for fermions, and it is a consequence of this that you can't put two particles in the same state. If you try-- let's try to put C+, C+ on O-- and C+ times C+ is 0. That's what this equation is saying. C+, C+ plus C+, C+ is equal to 0. So this algebra incorporates and encompasses the idea that you cannot put two particles into the same state described by these creation and annihilation operators. I want to prove a little observation about this. These Cs here create and annihilate particles with given momentum. What we prove, what we've assumed that some combination approve and assume-- oh, incidentally, yeah, we should write this more completely. Supposing we're now talking about creation annihilation operators for different momenta, k and k prime, then I will just tell you this is delta kk prime. In other words, it's one if k equals k prime, 0 otherwise, and these things are always just 0. C-, C- minus I'll just equal to 0 for all k and k prime. That's the algebra of creation and annihilation operators, which if you like, by definition, this defines the algebra of creation and annihilation operators for fermions. The fact that it really does describe fermions in nature, of course, is a marvelous correspondence between mathematics, simple mathematics, and some physics. But here's an interesting question. As defined up till now, what this says is that you cannot put two particles into the same momentum state. You cannot make two particles carrying the same momentum. Two fermions, two fermions of the same kind, incidentally. Two fermions of the same species-- you cannot put them into the same state, momentum state. Can you put them into the same position state? Well, what this tells us is that if we find a particle with momentum k fermions an electron, we cannot find another one with exactly the same quantum state, same momentum. What about the same position? Can we find two particles at the same place in not momentum but in position? So let's try to find out. How do we create a particle at point x? We create a particle at point x using psi of x. Psi of x on the vacuum represents a particle at point x. Oh, sorry. I think we should have side dagger creation operators. So the question is what happens if you try to create two at the same point? Well, let's see what we have. That means we're going to have a sum over a k and a k prime of C+ of k, C+ of k prime. Let's do it. Let's do it at the origin. Let's just do it at position 0, then ez to the ik. x's aren't there on 0. What is this? Anybody tell what this is? All terms are 0, and k does not equal k prime. No, no, there's 0 in any case. You can rewrite this in the form-- C+ of k, C+ of k prime 1/2 C+ of k plus 1/2 C+ of k prime C+ of k because it's under the summation here. But this is just the anti-commutator of two C+s. The anti-commutator of C+s are all equal to 0, so this is 0. Why? Because the anti-commutator of C+ is 0. Should've asked you that before because I wrote it on the blackboard. Well, yeah. Yeah. No. Right. Right. All of this is very clear if we're talking about only one k. It gets some will impact nontrivial impact when we generalize it in this fashion here to all k and k prime. So there's basically some physics that's going in here. And what is this physics? It's the physics that not only can't you put two particles into the same momentum state, but you can't put them into the same position state, for example. I'm showing you is that with this algebra. If this algebra is correct, then it's not just that you can't put two particles into the same momentum state. You can't put them into the same position state. In fact, you can't put them into the same state altogether. But this was just an example. With this algebra, it says that you cannot put particles either into the same momentum state or the same position state or any other state, for that matter. So these are fermions. This is the pattern for fermions. Yeah. I've always maybe misunderstood that you could have two electrons at the same momentum if there were four and not one. Yeah. Yeah. The correct statements is you can't have two electrons in the same state. Now, what constitutes a state? What constitutes a state may not just be its momentum. It may be its momentum and its spin, which we haven't come to yet. This is one of the things we're going to come to shortly is the concept the spin-- very important concept. We haven't come to it yet. So at this stage, we're imagining that a particle is characterized by a momentum. And if it's a fermions, you can't have two in the same momentum. Now the correct statement is, of course, you can't have them in the same state. And if a state requires other ingredients besides its momentum, for example, its spin, then you can't have two particles with the same momentum with the same spin. But spin-- there's only really two parallel, anti-parallel. But what about two electrons and two hydrogen atoms that are separated some distance away each other. Could they possibly have the same momentum? Momentum? Momentum-- I'm just going to stick with momentum. You want the atoms. The hybrid-- just, well, just an electron. I maybe used a bad example of that. Well, electrons in the hydrogen atom are not in momentum states. They're in atomic energy levels. Now the atomic energy levels over here and over here for two different atoms are two quite different wave functions. So if you have an atom over here, an electron might occupy a wave function that looks something like this. Another atom over here might have an electron, which looks like this. This wave function and that wave function are not the same. They're quite distinguishable by the fact that one of them is over here, and one of them is over here. So yes, you can have two electrons in two different atoms both, for example, in the ground state-- in some particular state. Yes? But you can't have two electrons in the same atom with the same quantum numbers. To be more generic-- What's that? To be more generic, instead of specifically talking about atoms, could you just use the term systems? Is that used in this language? States is the right statement. The right nomenclature is you can't have two electrons in the same quantum state, in the same quantum state. A system means a collection of degrees of freedom, but it doesn't imply a particular state. So an electron is a system. An electron can be in a particular state. So nomenclature is electrons' system. The state of the electron is perhaps some particular orbital and spin state in a particular atom. Can I just carry that one further? Let's suppose you have an electron beam in a picture tube, cathode ray tube. You have a stream of electrons. So that means that no two of those in that little glass jar-- Yeah. Yeah. I really like this-- a packet over here and another packet over here and another packet over here and another packet over here and then not in the same state. If they're not in the same momentum, [INAUDIBLE]?? They're not. They're not in the same momentum. They're not in the same position. And in big contrast to the idea of photons, photons march down the axis all, for example, in the same plane wave. That's why you can't build up a coherent beam of electrons in the same sense that you can have a coherent beam. You just can't put more than one of them in the same state. So a formula is, in many respects, a bigger abstraction. Than a boson? Can you have two electrons in the same place with different momenta? No. No, no, no, no, no, no, no. No, no. Remember you either describe a thing by a position or a momentum, and not both. The position and momentum are intertwined. Mutually incompatible [INAUDIBLE].. Right. So it's only like spin position then. That's because spin commutes with momentum and position, but position and momentum don't commute. So you can't speak about an electron at a position with a momentum, but you can speak about an electron at a position with a particular spin. Is there some sense in which this found electron over here and that found the electron are, therefore, in that sense, in a position representation or something? No, they're not position eigenstates because the wave function isn't that concentrated, but they approximately are. They're approximately located at the position of the atom. You can't put two of them into the same atom in the same way, but you can have two different atoms, which is like having an electron over here and an electron over here-- that's fine. Yes? So what is the definition of a state? What is the definition of a state? I think I would have to refer you either to my own lectures or a book on quantum mechanics. But it just means it's the quantum mechanical analog of the configuration of a system. A system is a collection of degrees of freedom-- a collection of coordinates and momentum and things like that. A state is a particular value, in classical mechanics, of the positions and momenta. In quantum mechanics, the state of an electron is everything you have to specify about the electron to state all the things that can be measured about the electron. [INAUDIBLE] It's a wave function, actually. Can an electron here and an electron, let's say, in the Andromeda galaxy-- a free electron-- can they be in the same state? No, obviously, not-- one of them's here, and one of them is in the Andromeda Galaxy. That's pretty different. Suppose you have one in Nevada and one in Connecticut. Can they be in the same state? [LAUGHTER] Can they have the same state as momentum? [INTERPOSING VOICES] What's that? They can have the same momentum. They can both be going east at-- Well, if you know where they are, it means you don't really know their momentum with infinite precision. So if you know anything about where they are, it means that you don't completely know the momentum. But yes, to within the extent that an electron in Nevada can have a pretty well-defined momentum but still not so well-defined that it might be in Connecticut, right. Well-defined, but not so well-defined that it might be in Connecticut, then, yes, you can have an electron in Nevada, an electron in Connecticut, which have pretty much the same momentum. But if you really tried to make that momentum infinitely sharp, that electron in Nevada would be equally likely to be in Connecticut. And the one in Connecticut would be equally likely to be in Nevada, and you simply can't-- --can't put them both into the same state. Just multiply the uncertainty together and you get [INAUDIBLE]. OK. So we now know what a fermion is, more or less. We use the algebra of these operators in a very similar way. We write down expressions involving electron operators, which annihilate electrons, which take electrons out of the initial state, creation operators which put electrons into the final state. We can even multiply them by operators for photons-- photons incidentally being bosons, and we can write mathematical expressions for the process of some number of electrons coming in, some number of electrons going out plus a bunch of photons. And we can write similar kinds of bookkeeping as we use to describe just the simple processes involving bosons, and we're not going to go into that now. I mean it is the story of particle physics-- how you describe those reactions. But before we get to that, I want to talk a little more about the differences between fermions and bosons. Question. Yeah. From the two-state example is a poly-exclusion principle a locally communicated principle? [? What does it mean ?] to you? There's a globally enforced instantly entanglement. In other words, if you had to do a few different processes going on in Nevada and Connecticut, they're both about creating a particle. But if one is known to be in Nevada and the other is known to be in Connecticut, then they're not in the same state, period. Yeah, but let's say it's not known. Let's just say that there's a slight probability difference. So there is a certain amount of fuzziness. Here's the situation. Let's imagine that Nevada is a point, and Connecticut is a point, instead of a big state. Here's a point. Here's Nevada, and here's Connecticut. An electron located right at the center of Nevada, I'll call N-- N for Nevada not n for neutron, not n for anything else. Let's call it capital N incidentally. N is not an integer. It's just Nevada. All right, an electron in Connecticut, I will call C. Now there is a quantum state of an electron, which is a superposition of being in Nevada and Connecticut. It has some probability of being in Nevada and some probability of being in Connecticut Nevada plus Connecticut. This is one electron with equal probability. You should put a one over square root of 2 here to make the probability come out to be one. This is a state of an electron, which has equal probability of being in Nevada or Connecticut. It's a quantum state. And if you look for that electron, you'll either find it at the center of Nevada or the center of Connecticut with an equal probability. Incidentally, there is another state like this, which is orthogonal to it, which has a minus sign here. What you can't have is two electrons both in this quantum state. You can have one electron in Nevada, one electron in Connecticut, but you can't have two electrons, which are in this state of both being equally shared between Nevada and Connecticut. Now that has implications that probably are not at all obviously what those implications are. But supposing we tried to make two electrons in this state, you can't do it. You can't have two electrons, which are equally shared in this way between the Nevada and Connecticut. Can one have plus sign, the other have a minus sign? Yes, yes, one can of a plus sign and the other have a minus sign. Yes, yes, yes, yes, yes. One electron can be in the state with a plus sign. The other one In the state with the minus sign. And we're talking about at the same time here. we're talking about at the same time. So the ambiguity in that is Nevada and Connecticut are spatially separated, so there's a question of what do you mean by at the same time, simultaneity. But when you think about it in momentum space, you don't have the spatial separation. Remember we're doing non-relativistic physics, so there's no limitations of the speed of light in making experiments and so forth. At the same time, I mean, the same time. Somebody can jump from here to there in no time at all. Check So what happens when you throw in relativistic physics? Not much. Not much. There is an additional fuzziness because of relativity about how particles are located. But for the moment, I wanted to deal with the non-relativistic ideas. Let's come to the ground state of a system of boson and the ground state of a system of electrons to see how different they are. Let's imagine the three-dimensional world. And since I can never draw three dimensions, I will only draw two. I want to plot all of the possible allowable momenta-- two-dimensional momentum. This is the x component of momentum. This is the y component I really want three dimensions, but as I said, I can't draw it. Electrons confined to be in a certain finite volume. The finite volume is very much like when we studied one dimension, we studied electrons on an interval of length L with periodic boundary conditions, but that's not so important here. If electrons are confined or particles are confined to be within a given volume, then the momenta are discrete. Remember when the electron was moving on the periodic interval, it had momenta, which had the n over l. And there's a 2 pi in there-- 2 pi n over L is equal to k. So what do we say? We said if we drew the momentum axis, the possible values of momenta were discrete and that was a consequence of the space being finite-- space available to the particle being finite. The bigger L is, the closer these levels get. All right, let's continue to think about things confining than a finite volume. In that case, both the x and y and z components of the momentum would again be discrete so the allowable values of the momentum vector would form a lattice. This is not space. This is momentum space-- the values of the possible momentum vectors. Let's put 0 right over here at the center, and then they go on and on forever. It's not bounded-- just goes on and on forever, just like this axis goes on and on forever. This n can be anything. So these are possible allowable values of the momentum of a particle in a periodic three-dimensional box. OK, let's start with boson. Let's take a collection of in bosons all identical to each other. They're all the same kind of particle, and let's put them into our box. And let's look for the ground state-- the state of lowest energy. Well, let's put the first bosons in. Where will it go if it's supposed to go into the state of lowest energy? 0-- 0 what? 0 momentum, 0 momentum. In other words, we'll put the bosons in. That 0 is kx, ky, kx is equal to 0 right at that point. That's the single-particle state of lowest energy. Remember the energy is k squared over 2 m. The lowest possible energy for that boson is to carry 0 momentum. If we want to find the ground state, we just put in the particle with 0 momentum. Now supposing we want to add another particle to it-- another boson of the same kind. Where do we put it to keep the energy as low as possible? Same place. Right, just the same place. We put two particles in the same place. In fact, we put all end particles in the same place. That would be the state of lowest energy. So that is called a Bose condensate the idea of a Bose condensate is a large number of particles all in the same state. But in this case, all in the lowest energy state available. Incidentally, the lowest energy state available, since it has the lowest momentum, fills up. It's a wave function which fills up the box. It's a wave function which fills up the box. And so the wave function of each one of these particles fills up the box, and we simply put a lot of particles in, all of them very uncertain about where they are in the box-- a lot of them in. That's, first of all, called the Bose condensate, and it's also the ground state of a system of bosons. The large number of particles builds up a kind of classical field strength so that the shrouding wave function is pretty classical. OK, that's very, very different. Than what would happen if we had fermions. Let's forget spin. We're not going to be interested in spin now. Where shall we put the first one? We got put them in one at a time. We want to keep the energy as low as possible. OK, obviously, the first one to go in, we want to put into the lowest state. What about the next one? Whatever the next energy level is over here, the next one would maybe go over here. They happen to be four states of the same energy. So we could put it here, here, here, or here. This is in two dimensions. In three dimensions, there would be six states altogether. So by the time we put in five electrons, they will fill up these states here. Now let's put in some more electrons. We can't put them in the same state-- forbidden. So the next one will go in the next lowest energy level, which will be over here. Incidentally, the energy being proportional to k squared is proportional to the distance squared from the origin. So what we want to do to keep the energy as low as possible is crowd the electrons in momentum space into as small a sphere as we can. In other words, when we have a lot of electrons or we put them in, they will fill up in momentum space, fill up a sphere. There might be a little confusion about the jagged of the spherical edge here, but that's not important. This, of course, will have a lot more energy than the corresponding boson system. The corresponding boson system had all of its particles in the lowest energy. If you look for a particle, all you will find is a particle of 0 momentum. In the ground state of the mini fermion system, you will find the more formulas you put in, the higher the maximum momentum will be. There will be lots and lots and lots of particles of large momentum And of course, only a handful at low momentum because there aren't that many states of low momentum. There are clearly more states within a given shell at large momentum. So first of all, the energy of the system will be much higher, putting fermions into a box costs more energy, one. Number two, there's this pattern that they fill up a sphere out to some boundary where the boundary is determined by how many electrons you have. Depending on the number of electrons, it will fill a sphere and that sphere is called the Fermi sphere. It's called the Fermi sphere. And so the history of this is a little bit confused. It was Dirac who figured out fermions and that's why they're called fermions. It was Einstein who figured out bosons, and that's why they're called bosons. They all contributed very heavily to the subject. Anyway, this is the ground state of a collection of fermions fills the Fermi sphere. Let's draw over here the corresponding thing in momentum space is. This is in momentum space. I don't need to draw these lines-- just all the particles condensed at the center of the momentum space for bosons-- bosons, fermions. Let's put a little bit of this lattice in, so we can see where the next states are. OK, what's the first excited state? We have in bosons? They're all at the center. What's the first energy level above the ground state? Well, that's easy. We take one boson and move it to the next energy level. We take out one boson in the ground state and put it into the first excited state. Which boson did we do? Doesn't matter-- they're all the same. So what we have n minus 1 bosons in the ground state and one boson in the first excited state there are four first excited states. That's not so important. The second excited state-- it might correspond depending on the details. It might correspond to putting one electron in the second excited state or two electrons in the first excited state. Not electrons, bosons-- a variety of different things. But you can see the way the pattern to excite the system, to give it more energy is to take electrons from the ground state and put them into higher energy states. OK, let's come to the fermion system. Well, it's also very similar, but let's try to figure out exactly what would correspond to the lowest-- question? OK. What would correspond maybe not to the exact lowest energy state but to the very, very low energy states above the ground state? We want to give it just a little bit of energy. What's the cheapest way to give it energy? Should we take an electron from the center here and move it outside? We can't move an electron to someplace inside the Fermi sphere because there's already an electron there. I think as you were about to say, it wouldn't be too smart to take one deep inside the Fermi sphere and bring it to the outside because that would take a lot of energy. It takes how much energy? You're going to take one out of low energy and put it back at relatively high energy. That's pretty costly. What's the cheapest thing? Right, take one from very near the surface of the sphere, inside the sphere, and move it to the outside. How much energy does that cost? Not much because you've taken a pretty high energy electron and just displaced it a little bit in energy. So that creates an electron out here with a little bit of energy above the edge of the Fermi sphere. But it also leaves something-- it leaves a hole in the Fermi sphere behind it. Let's suppose these are really electrons. OK, let's say let's consider a real-world situation. We have a box. And that box is full of positive charge. The positive charge is protons, of course, or nuclei. But for simplicity, let's just imagine the positive charge is smeared over the box. And then we put electrons in on top of it. And we put enough electrons in to fill up a Fermi sphere like this. It's electrically neutral because there are as many electrons as there are protons in the box, so it's electrically neutral. In the ground state, the fermions fill up this Fermi circle, this Fermi sphere here. Now we take an electron out from just below the Fermi sphere and put it in just above the Fermi sphere. We have not changed the charge. It's still electrically neutral. But we have created one extra particle with a little bit of extra energy above the Fermi surface and left a hole in the Fermi surface and left a hole where a missing particle, a missing particle-- just think of it literally as a hole. It's an absence of a bit of negative charge-- a bit of negative charge with a certain momentum over here. It can be thought of as the presence of a positive charge-- whole. A whole can be thought of as a kind of particle. In this context, the whole behaves as a particle. Is it a particle with negative energy because it's below the Fermi sea? Now let's see how much energy it takes. How much energy does it take to move a particle from here to here? Got a square in it. Let's just say we took out a particle of energy. Let's call a particle that we took out. Let's call it energy epsilon. And we put in another particle of energy, epsilon prime. , When we took out the particle of momentum of energy. how much energy does that cost? Difference-- the difference. [INAUDIBLE] It's both. Is it the sum or the difference? I think the difference. You think the difference. I think the sum. OK. [LAUGHTER] I'm sorry. I'm sorry. Let me be more precise. It is the difference, but I'm getting tired. I'm reaching a point of-- the point is a custom energy-- a custom energy. I'll tell you how much energy later. Would you have to push energy into the system in order to pop an electron off for the next energy level? Well, the point is how much energy do you have to put in? You have to put in the amount of energy that it would take to remove this particle and bring it exactly to the Fermi surface and then an additional amount of energy to bring it from the Fermi surface off the Fermi surface. So here's the picture. Here's the Fermi sphere. The Fermi surface is the edge of the Fermi sphere. Let's take a particle just below the Fermi surface, bring it from here to here. How much energy does it take? Well, it takes two bits of positive energy. We can count it this way. The first bit of positive energy is the energy that it would take to remove this particle and bring it right to the Fermi surface. That's a little bit of positive energy to bring it to here and then a little more positive energy to bring it to here. So it's actually the sum of two positive terms. The two positive terms being the absolute value of the difference of the energies from the Fermi surface. You have to add two little bits of energy. One of them can be thought of as the energy of the electron over here measured relative to the Fermi surface and the other one can be thought of as the energy of the whole. And they're both positive. This one has a bit of positive charge because it's an electron. This one has a bit of negative charge because it's missing-- no sorry. This one has a bit of negative charge because it's a positron. No, I did that on purpose. I didn't. This one has a bit of negative charge because it's an electron, and this one has a bit of positive charge because it's a hole. Thank you. Now this electron could be moving around and then suddenly pop into that hole. The hole is available now. The hole is available for the electron to drop down into. Just as an electron can drop from an excited level of an atom down to an unoccupied quantum state-- if there's an unoccupied quantum state-- this electron can drop down into this hole. What will happen when an electron drops its energy and drops down into the hole? Well, energy has to be conserved. So the energy of the electron has decreased. A photon is emitted. You can say it another way. You can say an electron comes together with a hole and annihilates. The electron annihilates the hole and a photon or more than one photon. Some number of photons goes out. So you can speak about this in the language that a hole is a kind of particle because it's a fake particle. It's not a real particle in a real sense of the word. It's a kind of fake particle that has the opposite charge of the true electron. It can be counted as having some positive energy. And when the electron drops down into the hole, you can think of it as the electron hole coming together, annihilating, and producing some heat, radiation, whatever-- the process of annihilation. Question. Yeah. When you say you're moving these electrons, you're moving from one lattice point to another lattice point-- Lattice in momentum space. Yeah, no, no, no. You're moving them from one energy level to another. So it's the analog of having an atom. All right, let's speak about the analog of what this means in atomic physics. In atomic physics, the box could be not the momentum space box but the box that confines the electron could just be the atom. The proton at the center is some positive charge, and we fill it up with some negative charged electrons. Of course, in this case, the proton charge is not smeared over the atom, so it's concentrated in the center. It's a little bit different. But we take an atom, which has the same number of electrons as protons, and we start with all of the electrons in the lowest available states. The lowest available states if the ground state of the atom-- that's the analog of filling up the Fermi sphere here. Now a photon comes along and hits an electron and kicks it up into an excited state. Photon comes along, hits an electron just below the Fermi surface, and kicks it up just above the Fermi surface. It's a low energy electron, a low energy photon, so it doesn't have much energy. It can't take an electron from the center here and bring it to the outside. That would take a lot of energy. It's a very low energy photon, and so it comes in and all it can do is hit an electron but be absorbed by an electron near the Fermi surface and kick it just above the Fermi surface. Another language is the photon has annihilated and been replaced by a particle or an electron and a hole. So it's called particle, hole, creation by taking the energy from a photon. In the same way that a photon hitting an atom excites the atom, of course, when it excites the atom, it leaves a vacancy. The vacancy in this language is called a hole. Holes has gone So across the whole sphere, if the photon hits anything in that sphere, will it kick off something nearly-- Well, if it's a low-energy photon, let's say some radio wave photon-- very low energy-- and it tries to get absorbed by an electron near the center, it just can't give that electron enough energy to get out of the Fermi sphere. It can't jump to any place inside the Fermi sphere because it's already occupied. So the answer is a low energy photon can't be absorbed by an electron in the interior. So the cross section doesn't get bigger as the sphere gets bigger? It's only into the surface, not the volume-- It all comes off the surface. That's right-- from a high-energy photon. Of course, that can kick an electron from deep inside. The language is that things inside the Fermi sphere are in the Fermi Sea, S-E-A. This is deep in the Fermi Sea. This is near the surface of the Fermi Sea. Only a high energy photon or a collection of high-energy photons coming together can kick an electron from deep inside the Fermi Sea to the outside. A low-energy photon can take a shallow electron or one just beneath the surface and put it just above. Yeah? Is there anything that keeps the electrons from interacting with electrons directly without using photons? Nothing, except we're ignoring it. No, interactions between electrons make the story somewhat more complicated, but interactions between electrons are pretty weak. They're pretty weak. They're not very strong. And for the most part, they can be ignored inside of metal for example. If a photon has excess energy, it has enough to kick multiple electrons-- Say it again. If a photon has a lot of energy, enough to the excite multiple elections, then what happens? Does one electron go get extra excited and then fall back? Well, anything can happen as long as it's consistent with the conservation laws. But typically, it's a much lower probability process to create two electrons in two holes. If it kicked out two electrons, it would make two holes. So that would be a process in which four particles, four particles were created altogether, and that's much less likely. [INAUDIBLE] super excited one. Yeah, that's right. It's more likely to super excite one. Another question. Photo-electric effect here, is sort of what we're talking about. Is it ever the case two photons simultaneously can be absorbed by the electron? Yeah. One of them wouldn't do it, but both of them will. Yeah. OK. Often, you need both of them to conserve both energy and momentum. It's just a probability situation. It' not prohibition. If you have a low-energy photon that's doesn't have enough energy to excite any of the electrons, does it absorb-- Well, it just takes a tiny, tiny bit of energy to kick it from here to here. So yeah, there is some energy, which is so low that it can't do anything. It's true. Does it then not interact or does it absorb it? Yeah, pretty much just passes through the system. Well, that's not really true. Yeah, very low energy photon will find other ways to interact. It'll start the whole collection of electrons sloshing, and that's more complicated. You mean that they're actually moving on the grid back and forth? They actually would be moving on the grid back and forth-- yeah. Space. From space, well, that's a little bit late, but that's more complicated question. No. Why did you say electrons interact weakly with each other? Wait-- say it again. You said that electrons will not interact with each other very strongly. But don't they all have the same negative charges? Yeah, but they're also in a background sea of positive charges. So the whole thing is electrically neutral. The electrons are interacting with each other, but the electron interactions are pretty weak. Partly, they're weak because the electron charge is screened by the proton charges, but I really didn't want to get into the electron interaction. That's a separate issue that-- Will we talk about that in screening? Yeah, we can talk about it. No, I don't want to talk about that now. Yeah, screening is an important phenomenon in particle physics, and it has its analog here. But let's come back to it another time. What I really wanted to get to was the Dirac equation, and I have 15 minutes. To give you the very, very simplest version of the Dirac equation, which we will come back to repeatedly-- but the very, very simplest version of the Dirac equation just sort of complete a story here. You know what story I'm starting to tell you. I'm telling you the story about antiparticles. Holes are antiparticles. But what does this have to do with not a metal in a box but with relativistic quantum field theory and the vacuum and particles and antiparticles? So I want to spend just a couple of minutes giving you the very, very simplest version of Dirac's logic. And to keep it really simple, let's stay with one dimension of space-- one dimension of space on only one dimension, x. We'll come back to more dimensions next time. And let s write a wave equation. This wave equation is going to be for a field that I'm going to call psi. OK? Ah. This electron-- there's not an electron. It's a neutrino. I don't know what it is. It's a particle. It's a fermion. It's a fermion, but it happens to move with the speed of light. It moves with the speed of light and that means omega equals k. Omega equals k is the relationship between frequency and wave number or between energy and momentum-- C equal 1. Omega equals k. There is another possibility-- omega is equal to k. And what's the other possibility? Omega equals minus k, OK? But let's take omega equals k. What kind of wave equation does that correspond to? Well, that corresponds to the wave equation, d psi by dt is equal-- I think it's minus d psi by dx-- this is partial derivative. This is even simpler than the Schrodinger equation. Why do I say that this is omega equals k? Let's imagine the psi is equal to e to the ikx minus omega t. d psi by dt t brings down a factor minus i omega. d psi by dx brings down a factor, ik. If the relation between omega and k is just omega equals k corresponding to c equals 1, then that just says d psi by dt is equal to minus d psi by dx. So this is a very, very simple wave equation, and it describes waves which only move in one direction. They cannot move in both directions. They move only is it to the right or to the left? Yeah, this these waves move to the right. What would it take to make waves that move to the left? You want d psi by dt equals plus d psi by dx. With either of those two equations, you have particles which move in one direction-- let's say, in this case, to the right. And let's suppose these are fermions, and let's even imagine they carry electric charge. So they're electrically charged fermions. OK, now there's something a little bit crazy about this. Omega equals k-- energy equals momentum. Can k be negative? Sure, why not? I can write e to the ikx, and I can also write e to the minus ikx. I can write both possibilities. But there's something a little ugly-- not ugly, a little bit dangerous-- about negative values of k. They carry negative values of omega and negative values will make omega mean what? Negative energy-- right. So this equation describes electrons of positive and negative energy. Let's just draw a picture. Omega equals k. Here's omega equals k, omega versus k. It's just a straight line. For k positive, the energy is positive. For k negative, the energy is negative. This equation simply describes electrons, which can have either positive or negative energy. That sounds like trouble. Why is it trouble? Why is electrons with negative energy trouble? Well, if electrons can have arbitrarily low momentum-- remember what happens in an atom to an electron, which is excited. It radiates and drops down to a lower state. If there were unboundedly low energies arbitrarily negative, then let's say positive energy electrons could emit photons and drop down energy. But they wouldn't necessarily have to stop at 0 energy. They could keep dropping down and dropping down and dropping down and make arbitrarily negatively energy electrons while at the same time pumping up the energy of the electromagnetic field. This sounds very dangerous, and it is dangerous-- something wrong with this. You can't have arbitrarily low energies. Well, what is the lowest energy state of the multi-electron system? [INTERPOSING VOICES] What's that? The Fermi sphere? What is the Fermi sphere in this case? The Fermi sphere in this case is [INAUDIBLE]?? Let's suppose we have a zillion electrons-- as many as we like. And we put the first electron into the system. Where does it go? To the hole. It goes down into the lowest energy state, but the lowest energy state is way the hell down there somewhere. It's off at minus infinity somewhere. The next one off at minus infinity. This is very dangerous. This is a bad idea. This is the simplest version of Dirac equation. This is a Dirac equation in one dimension. What Dirac said is let's pretend that all of the negative energy states were filled. Just like filling the Fermi sphere, fill all the negative energy states all the ways up to 0 energy. So all of these states are filled. There's an electron in every negative energy state. You can't make a state of lower energy than that incidentally. That's the lowest possible energy you have. You filled up all the negative energy states. Every time you put a negative energy thing in, that lowers the energy. You've put as many negative energy states in as you can. Now if I start putting in positive energy electrons, that will increase the energy. So the very lowest energy state-- the thing that you could call a ground state-- is to fill it with as many negative energy electrons as there are negative energy states. And that's your starting point. Instead of a starting point with no electrons, your starting point is fill up completely the negative energy sea. That's the vacuum. That's empty space, according to Dirac's theory, Dirac's simple version. Start with that. Since it's the state of lowest energy, let's call it the vacuum. And then what can you do? You can take the only thing you can do is to take an electron out of the sea-- the negative energy sea-- and give it positive energy. So you can take an electron from over here and put it someplace else. what are the properties of a state with one electron over here with positive energy and positive momentum and one missing electron over here? It's a missing electron. How much momentum does it have [INAUDIBLE]?? [INTERPOSING VOICES] k is negative. k negative. Well, we removed the negative momentum electrons. Yeah, so it's 0. No, no no, no, no, no-- the hole. How much momentum does the hole have? Is it positive or negative? Positive, positive, positive. Why is it positive? Because it's a missing deficit of negative momentum. So what we've done is create a positive energy electron and positive momentum. We've created twp positive momentum objects both moving to the right-- one with minus charge, the electron with minus charge, and the missing electron having positive charge-- negative charge and positive charge. That in summary-- and we're going to do this more completely-- but that was Dirac's theory of the electron and the positron-- that the positron was simply a hole in the infinite negative energy sea. You wrote down this equation or the generalization of it, looked at it. And said I love that equation, but it's got this terrible disease of having negative energy states. No problem-- these are fermions. You can't put more than one electron in a negative energy state. So just fill it up completely and call that empty space. Having called that empty space, what can you do to it? You can remove one of the negative energy electrons and put it back as a positive energy electrons-- yes, put it back as positive energy electron. That leaves a hole. And Dirac side that hole is a particle of positive charge. Now at first, he was very excited because he thought it was the proton-- wasn't the proton. It's easy to prove that the mass of the particle here has to be the same as the mass of the-- these are particle and antiparticle. This was this discovery of particles and antiparticles. But this was the basic logic, and it was equations like this which led to it. Notice what a disaster you'd be in if you tried to describe a boson by this equation. That would be a madhouse. You would just keep putting particles into the negative energy lowest for as many as you like, and there would be no ground state. There would be no ground state. You could just keep putting in more and more and more negative energy bosons. So this wave equation only makes sense for fermions, would not make sense for bosons. And that was Dirac's great discovery-- that fermions can be described by Dirac equations like this that would not make sense for bosons. Why? Well, just the reasons I explained and the idea of antiparticles. We'll come to antiparticles for bosons next time, and we'll discuss the Dirac equation in more completeness-- the real Dirac equation. This is an elementary version of it, but we'll come to it next time. There's a number of things wrong with this electron. For one thing, it only moves to the right-- that's crazy. For another thing, it moves at the speed of light-- that's crazy. So is a number of things that are wrong with this electron. No, it's a neutrino. No, it's a neutrino. Neutrino is a boson. This neutrino is a boson. This is a neutrino, but it's not quite a neutrino because the neutrino also has mass. For more, please visit us at Stanford.edu.
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Channel: Stanford
Views: 84,723
Rating: 4.7960339 out of 5
Keywords: science, physics, particle physics, simple quantum field, vectors, field theory, particle, wave, momentum, energy, phase velocity, quantum mechanics, velocity, oscillation, Schrodinger, fermions, Fermi-Dirac statistics, harmonic oscillati
Id: TshXa3lGqQ8
Channel Id: undefined
Length: 118min 26sec (7106 seconds)
Published: Thu Feb 11 2010
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