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So yesterday we had an awkwardness about this formula here because what overage arrived disagreed in the ordering here from what's in the book.

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And it just should reassure you that what I derived was correct and what's in the book is correct.

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And it's quite instructive, too, because these are important formulae, so it's good to have them in your mind.

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So let's just let's just understand how it comes about.

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The both of these are right. If you if so, what's happened between these two is that these things have been swapped in their order.

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But crucially, also, these things have been swapped.

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So if I go from here, I pick up a minus sign.

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If I simply invert the two operators, j and vs is just an arbitrary vector.

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Okay. It could be x, it could be P, it could be j, it could be whatever.

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Right? So if I swap these two operators, of course I pick up a, I pick up a minus sign,

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but then I can pick up a matching minus sign on the right side by swapping these two indices of the Epsilon symbol.

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So that's minus i Sigma K, Epsilon, j i k v k.

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And now I can say to myself, well, so, so, so I now have this formula from this formula of to to derive this formula except the what here is called I,

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which is the index on J and the index on the first index on epsilon is here called Little J and what here

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is called J being the middle index on Epsilon and the index on the vector operator is here called II.

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So it's a mere relabelling of what appears at the bottom. So these two familiar both?

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Correct. Okay.

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So. We just need to pull together.

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We really have everything now. We just need to pull together the results that we have and just calmly understand the physical significance of them.

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So we've discovered that these operators, like the momentum and the angular momentum, are associated with with displacements.

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They generate they're the generators of displacements. The momentum generates.

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You have a which shoves your system. It doesn't literally shove your system.

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It makes a new system that's translated that's the same as the old one,

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except its location is being incremented by a the incremental operators make a new system which is the same as your old system,

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except they've been rotated around the origin by some angle.

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And we have already seen that when the that when the momentum operator commits with the Hamiltonian,

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when one of these observables commits with a Hamiltonian, we have a conserved quantity and we have good quantum numbers and things like that.

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So what's the connection? Okay, so if so, let's say if P commits with H, what does that mean?

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That very easily implies that U of A the translation operator is going to compute with H because this

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operator right is e to the minus i a dot p pon h bar is a function of p and therefore it commutes.

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If h commits with p commits then the function of p. So if we have sorry.

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If if this equals nought, then this equals nought.

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Now, what does that, uh. What does that tell us?

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To see what it tells. What it tells us.

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We we have to think about the unitary operator associated with H because H is an observable and it's associated with the unitary operator.

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Each of the minus i h. Upon H.

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Bar t. Which we're going to call U of T.

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What's this, operator? Well, we know.

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We already know that IXI at time t is equal to the sum and each of the minus i e and over h for t times n that way.

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This is our standard expression for solving our standard means of solving the time dependent Schrodinger equation

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is to decompose the given state into a linear superposition of energy ion states as any particular time,

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for example, at t equals zero and then evolves by multiplying each term in the series by this exponential here.

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But we can see that this could also be written as e to the minus i h t upon h bar times sum and.

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Ian Wright because when this linear operator looks at this sum here, it passes.

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It's a linear operator, so it can be distributed, pass through these ends and look at each one of these things than H looks at it.

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I can get in and it says, Aha, that's my, I can get e n it returns,

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it returns e and times the number n and then you go and there you go, you have that.

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So this operator u of T is is very is a crucial operator.

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It's the thing that evolves you forward in time any state.

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This is nothing new. This is just a repackaging of old results.

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So the unitary operator associated with the observable time moves you forward in time.

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If it carries you from today into tomorrow or whatever.

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Right. So. If so, let's just repeat this.

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If P comma h equals nought, then that implies and indeed is implied by that the unitary operates a u of a commutes with the unitary operator u of t.

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Right. This is the thing that moves you forward in time. This is the thing that makes you a new system shoved along the bench by a.

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So what does that tell you? That tells you that take take the state of your system and evolve it in time and then shove

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it along the desk and you will have exactly the same state as if you take your system.

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Shove it along the bench and then evolve it in time.

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It says that whether you let it evolve here and then move it to its point where you want to have it,

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that is this that will give you the same results as if you move it now and let it evolve over there.

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So this this what is the physical implication of this simple equation is that.

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Physics is the same. Here is their.

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Now, that's not always the case. If. If that were a clock.

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That were a clock. And we let it evolve on the floor. Until until tomorrow.

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And then read it well and then moved it up here.

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Well, somewhere higher up here. We wouldn't have the same situation as if we let it evolve until tomorrow up there.

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Because. Because clocks the gravitational potential down on the floor is lower than it is up there.

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So a clock up there will evolve faster than a clock down there.

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So it is by no means obvious.

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It's not necessarily the case that that it doesn't matter where you conduct your experiments,

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that evolving it in one place and then shoving it and then moving it somewhere else

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is going to give you the same results as shoving it somewhere else now and then, allowing it to evolve with that other place.

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This being the same here and there is is is a statement about the the homogeneity.

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So when this is the case, it's a statement about the homogeneity of space.

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It's it's a right. And we would say and physicists are of the view that that.

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Ultimately physics has to be the same here and there.

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And the reason and the reason that the clock evolves on the floor in a different

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way from on the table is because not because of any homogeneity of space,

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but the fact that is a dirty, great planet here.

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We're 8000 miles or whatever it is from the centre of the earth,

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and it's the relative movement of the earth and the clock which has changed the circumstances, not the homogeneity of space.

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So we're completely wedded to the concept that fundamentally space is the same everywhere,

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and therefore, fundamentally, this should be the case should be the case.

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If, if, if if your system is isolated.

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We say that. In other words, we say that when this principle is not observed, it's the reason it's not observed is your system.

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Johnny isn't isolated.

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In the case of the of the clock on the floor there, it's obvious what the court what they're not isolated in this is it's the dirty group planet.

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But in other circumstances it might be more subtle.

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But we would we conjecture that you will be able to find something which is violating the isolate, you know, which is which is affecting it.

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Which is which is violating its isolation.

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Okay. So where all this commuting of operators is associated with something being conserved, that something is momentum.

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It's also associated with a statement about.

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Invariance of physics on translations. So so we have a sort of set of ideas like this commuting.

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Well. P with h well, so p comber h is connected to conserved.

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Momentum. Which is itself connected to uniformity.

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Of space. Which is the same thing as symmetry under translation.

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And this is a set of three sort of separate things which are tightly connected by mathematics and basic principles of physics.

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We similarly if we if it's the case that say Jay Z comma H.

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So this is the generator of rotations around the Z axis, if that's equal to nought.

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So I need to have a nought equals here then that is associated with conservation in classical physics,

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that's associated with the conservation of angular momentum.

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Which is why we want to call this the angular momentum operator, which is associated with the result of space.

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So it's clearly the case that. A compass.

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A compass behaves differently.

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If you oriented east, west or north south. Right.

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It's. Of. Because because on the surface of the earth, on account of the Earth's magnetic field,

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the physics of space is not isotropic from from the perspective of a compass needle.

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And it's associated with. And as a consequence of that it's angular momentum.

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Operator will not compute with the Hamiltonian of the compass model and its angular momentum will not be conserved.

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That's why it swings to and fro around the North Pole when you let it go.

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Magnetic north. And he is on the momentum thing.

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Just remember what Newton said about bodies moving in a straight line, etc. He fundamentally said isolated bodies have conserved momentum.

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And so there already he was he was in fact, connecting the conservation momentum to the to the eyes of space.

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He had that concept of an isolated body. So so in general, we're always interested in finding these operators,

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these observables which commute with the Hamiltonian and in general it's hard to find.

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So in general, it's hard to find. The operators that we don't have a system.

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Unfortunately, for finding operators that can meet with the Hamiltonian,

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the best system we've in fact got is to look at the at the uniformity of the physics,

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to say to ourselves, can I see any reason why the system should be different?

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The behaviour of systems should be different. If I rotated where if I translate it or do some other thing to, to it.

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So observables. Commuting with age.

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But here's an example. When you when you can spot one, if you have any particles that interact with each other, nothing else.

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Oops into egg. Then you have the Hamiltonian of this system.

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Is the sum API squared over to him.

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I summed over particles. Hi.

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So this index here enumerates the particles plus.

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So that's the kinetic energy of each particle. Some of the particles makes the Hamiltonian to the whole system.

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And then that's time. That's, that's going to that.

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We have to add the potential energy of interaction between the particles,

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which will be the sum of a pairs of particles which we can get by saying that J is less than I.

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These are the vector positions of the particles. So there's some, there's some interaction potential between these particles.

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They interact in pairs and this, in this, in this, in this picture.

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So this would be the Hamiltonian of these particles which are interacting in some arbitrary way with each other, flows that interact in pairs.

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What we can say is that is that h is its invariant.

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This expression is manifestly the same. If x I goes to x, i plus a, if you simply add a vector A to all the locations of the particles,

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long as you shove the whole system along by a vector a then the arguments of

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all of these interactions stay the same and you don't affect the momentum.

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So H is invariant. And what does that tell you?

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That tells you that the generator of this.

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Of this. Of this transformation is going to be a conserve quantity.

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So this so this transformation. Well, it implies.

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Conservation. Of the generator.

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Which is going to be the the total momentum being the sum of the momenta of the individual particles.

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Which of course we recognise as is the total momentum of this system is going to be

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conserved because action and reaction or equal and opposite back to back to what?

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Isaac Newton. These points you haven't seen probably made in this way before.

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But it is. I would like to make the point that. They are actually very fundamental points of physics.

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They are not peculiar. They're not special to quantum mechanics in classical physics.

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All of these statements remain true.

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It's just that when you do elementary mechanics, you don't have the machinery at hand to see the connection between symmetry and conserved quantities.

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These are really very basic points which are true in quantum mechanics, but they're also true in classical physics.

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But we have now the apparatus, so we can see these things rather more clearly than we can in classical physics.

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So we now have time to cycle back to something to what I skipped, which was.

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Which is motion in a magnetic field. This is a particularly important topic because an awful lot of quantum mechanics was developed.

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Historically, it was developed by sticking atoms into magnetic fields.

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It's obviously also an important topic in the sense that.

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We use magnetic fields in an awful lot of in an awful lot of devices.

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And people also now stick their crystals in magnetic fields to see what happens.

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So it's still a it's still an important way of probing systems when you're trying

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to understand systems which whose physics is is based on quantum mechanics.

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And it's very important to understand how this happens.

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And there's a fundamental difficulty we have to address up front, which is what we need to know is how to modify the Hamiltonian.

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Right. Because in quantum mechanics you put the physics into the Hamiltonian.

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The Hamiltonian tells you what forces are acting, what the system consists of.

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It encodes what the physical laws are.

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For your system. So if you switch on a magnetic field, it must be that it's changing the Hamiltonian somehow.

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So the question is, so how is it changing the Hamiltonian?

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And if you take the view that H is equal to P squared over two M plus V because you've got some particle,

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then you're in trouble because there's no magnetic contribution in the potential energy.

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Because the Lorentz force never does any work.

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The Lorenz force, the cross b is perpendicular to v so v dot v cross be identically vanishes, and the Lorenz force never does any work.

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So it can never contribute to the potential energy of your system. And therefore you can't look for you can't look for magnetic contributions in here.

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So it turns out that. And you need to do it because magnetism is a relativistic correction to electric statics, right?

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Fundamentally, that's what it is and I think it's one of them.

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I'm always amazed and I don't I don't think I really understand why it is that our electoral devices overwhelmingly use this,

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you know, your vacuum cleaners, your disk drive, your.

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I mean, we make the electricity, in fact, using a relativistic correction to electrostatic.

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We do almost nothing with electrostatic. There are a few electric a few scientific instruments use like electrostatic drives but

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it's it's it's almost an unused you know Coulomb so is almost unused except to get

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the electrons to go down the wires in order to generate this relativistic correction

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because they're moving at a slightly different speed from the ions in the wires anyway.

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So, so but it is a relativistic correction to electric statics.

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And so in order to, to find out how to change your Hamiltonian, you really need to do relativity.

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That's the proper place to look. And I'm not going to be able to derive this for you.

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I'm going to be able to tell you what it is, which is what we need to do is replace that P by P minus the charge on the particle.

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So when we have a magnetic field, we put it in by replacing P in our original Hamiltonian by P minus Q.

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What is the charge on your particle?

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And of course, B is the curl of A, so a is the magnetic back to potential that generates the magnetic field.

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So I'm not really able to justify this because to, to, to explain why this should be so we need to do some relativity,

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which is way out of scope for, for the quantum mechanics.

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But what we what I should do with this is use Ehrenfest theorem to convince you.

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This gives you that this gives us the classical equations of motion in the magnetic field and ultimately.

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You know, only experiment can tell you whether this is right or wrong. So let's let's let's use Aaron first.

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To recover classical physics out of this. So what we have is i h bar dvd t oops dvd t of the expectation value of x i.

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What's that? That's equal to the expectation value of x, comma h.

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We're going to will we'll drop this because we're not really interested for the moment.

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We're not interested in in V, which would contain, for example,

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the electrostatic interactions, the interactions with the electric field, if there were any.

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Let's just let's not worry about it. Let's just take it that what we have is a particle moving in a magnetic field.

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So I want to take this to be the Hamiltonian, to keep life simple.

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So what is this? This is one over two m expectation value of x i comma p minus q a squared comitato.

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ABC. Now we know how to take combat cases.

203
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Why does x i not commute with this?

204
00:24:11,210 --> 00:24:17,360
It doesn't compute with this because it contains the momentum operators while in particular it contains the ith momentum operator.

205
00:24:19,730 --> 00:24:27,200
And this is going to be one over two m if we're pedantic, we could do it more quickly than this.

206
00:24:29,870 --> 00:24:34,580
This will commute. This is there are two of these couple together. It'll commute with the first one.

207
00:24:35,630 --> 00:24:39,530
We should do the commentator with the first one. Now X is going to commute with a.

208
00:24:39,950 --> 00:24:52,729
We're going to have that x comma a equals nought because the this vector potential is a function of x write it to the vector potential depends on x,

209
00:24:52,730 --> 00:24:55,820
it varies with x. Therefore it is a function of the operator x.

210
00:24:56,570 --> 00:25:02,750
So X is going to commute with it. So the reason it's going to hit with this bracket is because it contains p.

211
00:25:03,440 --> 00:25:13,790
So we're going to have x x i comma p, the vector p dotted in to p minus q a.

212
00:25:15,560 --> 00:25:18,590
Close brackets. That's one of the two terms.

213
00:25:18,590 --> 00:25:23,030
And then, unfortunately, there's there's another term which will in fact be identical.

214
00:25:23,120 --> 00:25:28,100
But just to be pedantic, let's get it. Let's keep it right. There's going to be P minus Q A.

215
00:25:30,820 --> 00:25:34,430
Dot x i. Com up.

216
00:25:36,770 --> 00:25:42,050
All right. So what we've done is regarded this as P minus times P dotted into P minus Q A,

217
00:25:42,500 --> 00:25:47,420
which is a product we've used the rule for doing the commentator on a product to the commentator on the first term,

218
00:25:47,420 --> 00:25:51,379
leave the second alone, that's that. Then leave the first time alone and do this.

219
00:25:51,380 --> 00:25:57,710
Commentator On the second one and the commentators on these brackets reduced media p because

220
00:25:58,250 --> 00:26:10,010
of that this so this product could be written as a sum over K of PK dot PK minus Q K Right.

221
00:26:10,520 --> 00:26:19,970
And this commentator is going to be nothing except when K equals I when this will be an h bar

222
00:26:20,090 --> 00:26:29,960
which will cancel that h bar and we'll discover that dvt t of x ii is equal to this h bar.

223
00:26:30,170 --> 00:26:36,049
Is is is a mere number. It'll commute with this. So we don't need to worry this this term is going to generate the same as that.

224
00:26:36,050 --> 00:26:45,140
So this gives me a one over M of P minus P-I minus q i.

225
00:26:49,980 --> 00:26:53,969
So this what this is telling us is that the classical velocity,

226
00:26:53,970 --> 00:26:59,879
because the rate of change of the expectation value, the position is what we would call the classical velocity.

227
00:26:59,880 --> 00:27:07,680
The expectation by the velocity, if you like, is not equal to the momentum over M It's equal to the momentum minus Q a over m.

228
00:27:13,880 --> 00:27:19,330
Or alternatively, it's telling you that pie is equal to m v.

229
00:27:19,330 --> 00:27:24,319
I always write these what their expectation values here this is.

230
00:27:24,320 --> 00:27:28,670
This is their expectation values here, right? This was always expectation value.

231
00:27:29,210 --> 00:27:36,890
So what we're discovering is that the expectation value of the momentum is equal to mass times,

232
00:27:36,890 --> 00:27:46,430
expectation of velocity plus Q Times expectation value of the magnetic vector potential at the location of the particle.

233
00:27:46,850 --> 00:27:53,060
And there's a problem on the problem sets that tries to convince you, this is all this is the momentum.

234
00:27:57,960 --> 00:28:04,130
Of the IMAG field. Point is that if you move a charge particle.

235
00:28:05,490 --> 00:28:09,030
You are moving. It's electromagnetic, you're moving.

236
00:28:09,030 --> 00:28:16,620
It's electrostatic field. The electrostatic field causes the magnetic.

237
00:28:16,890 --> 00:28:27,360
The combination of an electric field and a magnetic field endows the makes for a momentum flux in the now electro magnetic field.

238
00:28:27,690 --> 00:28:35,490
And there's a calculation which makes it look as if I think it probably is broadly true that that that.

239
00:28:37,950 --> 00:28:47,229
Yep. The removing charge to get a charge moving, you not only have to give it momentum, but you have to give the field some momentum.

240
00:28:47,230 --> 00:28:51,250
So this really is the total momentum. But the thing is, it's not the field.

241
00:28:51,550 --> 00:28:55,000
The particle is not on its own. It's not the only repository of momentum.

242
00:28:55,180 --> 00:28:59,799
The electromagnetic field is also a repository of momentum anyway.

243
00:28:59,800 --> 00:29:05,110
So so we have this non-trivial relationship now between momentum and velocity.

244
00:29:05,440 --> 00:29:08,709
And again, this is not something special to quantum mechanics.

245
00:29:08,710 --> 00:29:12,610
We've derived this in quantum mechanics, but it's a known result in Hamiltonian mechanics.

246
00:29:12,610 --> 00:29:17,139
Those people who've done A7 may have encountered this formula.

247
00:29:17,140 --> 00:29:24,070
I'm not sure whether it goes quite that far. Let's have a look at the other of the other equation of motion.

248
00:29:24,070 --> 00:29:29,800
We should have a look at which is harder duty of the expectation value of pi.

249
00:29:31,360 --> 00:29:36,910
So. Is going to be a sigh.

250
00:29:37,930 --> 00:29:42,710
P comma h but what's H its peak?

251
00:29:43,330 --> 00:29:54,910
I'm going to write it out now. P.K. minus Q. A K squared some over k commentator sticking another upside.

252
00:29:58,410 --> 00:30:05,820
So this is the commentator of the momentum with the Hamiltonian, where I've now written out the Hamiltonian in Gorey in reasonably gory detail.

253
00:30:09,150 --> 00:30:12,540
Over to him. Over to him. I'm missing a lot of it to him. I not one of it to him.

254
00:30:14,200 --> 00:30:17,290
In all that. That's one over two more times. This is the Hamiltonian.

255
00:30:19,150 --> 00:30:24,160
Why does PI not give you this? Well, obviously, P.E.I. views with Peak, that's a problem.

256
00:30:24,460 --> 00:30:28,360
P.E.I. doesn't commute with this because sitting inside, because this is a function of X.

257
00:30:29,440 --> 00:30:36,520
So when we work this out, we get a one over two M CI big bracket.

258
00:30:37,210 --> 00:30:41,340
We're going to have this thing commuting with that.

259
00:30:41,350 --> 00:30:48,190
So we'll have a minus q p P.E.I. comma a k.

260
00:30:49,390 --> 00:30:57,460
This is going to be some do over K of maybe it'll be better if we put a some over K here.

261
00:30:57,790 --> 00:31:06,790
Got to have one somewhere. Times peak minus Q a k.

262
00:31:07,730 --> 00:31:15,530
So that's that's p commuting with the first of these two of these two brackets and then we will have.

263
00:31:17,550 --> 00:31:27,540
We will have P.K. minus q ak of pi ak.

264
00:31:27,750 --> 00:31:34,950
Commentator And the factor of minus Q from here, close big brackets sticking out.

265
00:31:34,950 --> 00:31:39,150
Another sign. So this is a disgusting mess that we have.

266
00:31:40,580 --> 00:31:44,159
And now we have to address the question of, so what is this comitato?

267
00:31:44,160 --> 00:31:47,250
What is pi, comma, ak?

268
00:31:47,270 --> 00:31:50,760
We need it. We need it in two slots. We need it here and we need it here.

269
00:31:53,150 --> 00:31:59,540
Well, this is we now use our rule for doing the commentator of a function of X.

270
00:31:59,690 --> 00:32:02,840
We use previously almost the rules for a function of P.

271
00:32:04,280 --> 00:32:10,640
The rule was that this is equal to the a k by the x.

272
00:32:12,330 --> 00:32:16,520
High Times. The Communist commentator p comma exi.

273
00:32:20,450 --> 00:32:25,340
Right. The reason this isn't. This is a function of exi. That's why this computer fails to vanish.

274
00:32:26,270 --> 00:32:34,999
And we derived this rule quite early on that you can you can buy Taylor series expanding your function.

275
00:32:35,000 --> 00:32:38,690
You can convince yourself that this is true, that we just have a derivative times.

276
00:32:38,690 --> 00:32:45,200
The commentator with respect to whatever it is we taking the derivative with respect to this is minus h bar.

277
00:32:47,210 --> 00:32:51,820
So. By the I.

278
00:32:54,960 --> 00:32:59,190
So we're going to get some ice balls, which we can cancel over there.

279
00:32:59,190 --> 00:33:07,530
So we're going to have the DVD t of PI, the rate of change of momentum, which should be equal to false all being.

280
00:33:07,530 --> 00:33:11,280
Well, this is turning out to be.

281
00:33:11,670 --> 00:33:19,440
Oh, yeah, sorry. Yeah. So we're going to have a one over two now we can take out a factor of Q of A to M This minus sorry

282
00:33:19,440 --> 00:33:27,089
is going to cancel that minus sign that Q I've taken outside the sum over K has not collapsed.

283
00:33:27,090 --> 00:33:30,090
No, it's the left. Yep.

284
00:33:31,740 --> 00:33:44,970
So we're going to have a summer of okay, what of we're going to have a K by TXI for this one here.

285
00:33:45,300 --> 00:33:54,810
We've got all the factors, peak minus Q arc and we're going to have essentially the same thing.

286
00:33:54,810 --> 00:34:03,299
But in the reverse order, peak minus q a K of the K by the x,

287
00:34:03,300 --> 00:34:09,930
i close the brick bracket and stick a matching cat upside on the outside to take the expectation value.

288
00:34:11,280 --> 00:34:20,070
So unfortunately, I cannot combine these two terms as they stand into one term because this is a function of x which refuses to compute with that p.

289
00:34:21,120 --> 00:34:28,200
Similarly this one. So so this thing is trapped on the left side of P and that one trapped on the right side of P and I can't combine them.

290
00:34:29,340 --> 00:34:33,120
And then in quantum mechanics, this is as far as I can as I can go.

291
00:34:33,480 --> 00:34:40,670
I now have to. So this is a this is a respectable, totally aboveboard quantum mechanical calculation to go further,

292
00:34:40,690 --> 00:34:44,459
I, I have to say, well, look, what am I trying to do?

293
00:34:44,460 --> 00:34:47,550
I'm trying to recover the Lorentz force for you.

294
00:34:47,550 --> 00:34:52,980
I'm trying to show that the classical in this is predicting the correct classical physics.

295
00:34:53,970 --> 00:34:57,780
If I'm predicting the correct classical physics, I can.

296
00:35:00,310 --> 00:35:06,070
If I talk about the classical physics, each of these operators can get replaced by its own expectation value.

297
00:35:06,080 --> 00:35:13,150
So the issue here is that here I have to take the expectation value of a product of one operator on another operator,

298
00:35:14,500 --> 00:35:20,829
and such an expectation value is not automatically the same as the product of the expectation value of this.

299
00:35:20,830 --> 00:35:22,570
On the expectation value of that,

300
00:35:23,110 --> 00:35:30,100
because fluctuations in this operator may be correlated with fluctuations in that in that sort of quantum fluctuations.

301
00:35:31,030 --> 00:35:34,989
But if we're in the classical limit, we we don't worry about these fluctuations.

302
00:35:34,990 --> 00:35:40,330
We assume that they all they all average wage zero like the interference pattern associated with the bullets.

303
00:35:40,330 --> 00:35:44,710
I mean, the fluctuations average away. We're just left with the mean value of the mean.

304
00:35:45,190 --> 00:35:52,540
So we now replace this product with a product of the expectation of the product, with a product of the expectation values.

305
00:35:52,780 --> 00:35:59,680
And then these become in numbers. So this becomes an expectation value of this operator, the expectation of the value of this number.

306
00:35:59,980 --> 00:36:03,940
And then of course, the numbers can be arranged in either order and I can stop fussing.

307
00:36:05,630 --> 00:36:13,160
I can stop fussing about about this so we can we now say in the classical limit.

308
00:36:14,470 --> 00:36:24,100
We've specialising now to the classic limit when we can neglect fluctuations, we can write this is Q over.

309
00:36:24,100 --> 00:36:27,100
M Because I'm going to combine these two terms,

310
00:36:27,550 --> 00:36:39,280
the sum over K of the expectation value of decay k by the Z times the expectation value of peak minus q a k.

311
00:36:42,200 --> 00:36:48,170
Now we can simplify again because this expectation value p, k minus Q.

312
00:36:49,610 --> 00:36:56,299
Remember we showed above. We honestly showed above without any fudging was equal to the mass times.

313
00:36:56,300 --> 00:37:00,290
The expectation of value of the velocity. Well. Well, that's. That's unjust.

314
00:37:00,410 --> 00:37:06,380
Yeah, right. So this thing here can be replaced by M the K.

315
00:37:08,600 --> 00:37:13,540
And the EMS cancel while we're about it.

316
00:37:13,550 --> 00:37:18,600
Why don't we replace this p I with M?

317
00:37:19,200 --> 00:37:24,530
Well, with what we get from up there, I've lost it. Here we go. It's in the VII minus.

318
00:37:24,770 --> 00:37:41,120
Sorry. Plus Kua. So this now comes down to DVD T of M VII plus q expectation of a I that's using that

319
00:37:41,240 --> 00:37:46,160
respectable formula up there for the relationship between velocity and momentum.

320
00:37:47,030 --> 00:37:50,390
Yes, that's correct. And that is going to be.

321
00:37:50,630 --> 00:38:02,270
Q Because the M's are going to cancel the sum over K of D a K by the X II times.

322
00:38:02,870 --> 00:38:13,309
What did we say was weak? And we ought to put an expectation value on around everything because we are dealing now with expectation values.

323
00:38:13,310 --> 00:38:16,310
We've explicitly gone to the classical regime.

324
00:38:21,600 --> 00:38:25,430
Okay. So we're nearly there. It doesn't probably look as if. What are we trying to get?

325
00:38:25,440 --> 00:38:28,920
I'm trying to get that mass times acceleration is equal to V.

326
00:38:30,040 --> 00:38:35,510
Crosby. And it may look as if I'm still some way from that, but it's not so bad, actually.

327
00:38:37,520 --> 00:38:42,650
Why? So what? On the left, what we have on the left here is the rate of change,

328
00:38:43,040 --> 00:38:52,160
of obviously the velocity and the and the vector potential evaluate is at the location of the particle,

329
00:38:52,400 --> 00:38:57,140
not just anywhere else, but at the location of the particle. So suppose we have a static field.

330
00:38:57,890 --> 00:39:09,310
Static B field. So that means that the partial derivative of a with respect to time can be taken to vanish.

331
00:39:10,630 --> 00:39:16,470
If the if this thing were nonzero, it would generate an electric field that would be quick.

332
00:39:17,020 --> 00:39:21,850
And then you know right of time varying magnetic field creates by Faraday's law

333
00:39:21,940 --> 00:39:27,339
creates a curly field and we that that leads to more complicated equation of motion.

334
00:39:27,340 --> 00:39:35,050
So we got we're just in static B fields so that the rate of change of the magnetic vector potential at any given point is zero.

335
00:39:35,230 --> 00:39:43,810
But this time derivative is not zero because the particle is moving and, and sensing the impact the, the vector potential at different locations.

336
00:39:44,560 --> 00:39:52,810
So what we have is that d by d t of a i is equal to what is it equal to?

337
00:39:52,810 --> 00:39:55,870
It's equal to the by the chain rule.

338
00:39:56,050 --> 00:40:01,330
It's equal to the x k by d t sorry.

339
00:40:01,330 --> 00:40:08,410
That should be a total total derivative d by t times d i by the x k.

340
00:40:09,420 --> 00:40:16,260
So the reason that this quantity is changing is because the place where we're making the measurements is changing at this rate,

341
00:40:16,260 --> 00:40:19,020
and this is the rate at which a change is with location.

342
00:40:22,130 --> 00:40:35,300
So what we now have is the m d vii by e t e mass times acceleration is equal to Q Q is going to be a factor Q on the right.

343
00:40:35,660 --> 00:40:43,010
I write down the terms. I've already got some of a k k by the x i.

344
00:40:45,360 --> 00:40:50,300
VK and then I'm transferring from the left side.

345
00:40:51,470 --> 00:41:04,850
This time's Q Right. So I have and it's going to come minus this is v k and this is the I by the x k.

346
00:41:08,860 --> 00:41:12,300
Street speaking. This bracket should be here because that summation sign is over.

347
00:41:12,690 --> 00:41:20,019
Is over. Both these both these signings have. Now this is actually equal to.

348
00:41:20,020 --> 00:41:26,770
Q The cross B ice component.

349
00:41:27,280 --> 00:41:38,350
Uh, well, because it's V cross. Ask yourself what what this would.

350
00:41:38,830 --> 00:41:47,350
I mean, just I claim that this is true. Let us see whether it is true, what we in order to expand this vector triple product,

351
00:41:47,350 --> 00:41:51,640
we would say it was this thing does it with this thing in the direction of that thing.

352
00:41:52,330 --> 00:42:11,150
So if I expand this, I get Q. This thing dotted with this thing that means a sum over k a VK VK K direction of this thing gives me a nebula.

353
00:42:11,320 --> 00:42:20,799
I because I'm trying to calculate the ice component and then minus this thing dotted with this thing, direction of that thing.

354
00:42:20,800 --> 00:42:23,800
So that's a VK Nebula K.

355
00:42:25,670 --> 00:42:38,980
A I. It's a little bit of a complicated vector triple product because this is a differential operator and it is operating only on this, never on this.

356
00:42:39,000 --> 00:42:43,860
So that's why I've written them in that form. It's this thing does it, this thing, direction of that thing.

357
00:42:44,100 --> 00:42:50,880
But this is only working on that. And then it's this thing dotted with this thing, direction of that thing that's nice and easy.

358
00:42:51,720 --> 00:42:57,450
And I think you can see that this term. Is this term and this term?

359
00:42:57,990 --> 00:43:01,230
Is this term with. If you move that around in back. Right.

360
00:43:01,950 --> 00:43:07,229
These these two terms are the same. So we have indeed recovered mass times.

361
00:43:07,230 --> 00:43:12,020
Acceleration is equal to Lorentz force. In the classical limit.

362
00:43:14,580 --> 00:43:20,050
Well, I think that's really all that I want to do. Yeah, that's all I want to do.

363
00:43:20,670 --> 00:43:25,260
That justifies provisionally the use of.

364
00:43:25,450 --> 00:43:29,120
So this Hamiltonian. Where was it? P minus.

365
00:43:33,370 --> 00:43:36,820
P minus A all squared over to him being the Hamill.

366
00:43:37,030 --> 00:43:47,500
That change in the Hamiltonian introduces a magnetic field into the physics, and we will use that when discussing atoms.

367
00:43:47,890 --> 00:43:51,219
Down, down the track. And if you look at the back end of chapter three,

368
00:43:51,220 --> 00:43:58,090
you can see there are some quite entertaining things you can do with the with the motion of a particle in a uniform magnetic field.

369
00:43:59,230 --> 00:44:05,740
When it turns out that you can recycle, you can recycle the physics well,

370
00:44:06,250 --> 00:44:11,230
you can recycle the formalism in the mathematics that we did for the harmonic harmonic oscillator.

371
00:44:11,530 --> 00:44:14,830
You can recycle it for this uniform magnetic field case.

372
00:44:15,070 --> 00:44:20,800
The basic principle is that if you if you have a uniform B field.

373
00:44:27,320 --> 00:44:31,820
And a non relativistic particle moving in a uniform charged particle moving in a uniform B field.

374
00:44:33,170 --> 00:44:44,270
You can have the orbits of circles, the particle circles around in this uniform B field with some radius that depends on its speed.

375
00:44:44,900 --> 00:44:48,620
If you have a fast particle, it goes round. It goes round.

376
00:44:49,040 --> 00:44:52,170
You know, we have we have empty square of all.

377
00:44:56,190 --> 00:45:01,200
Is equal to Q the b b.

378
00:45:03,780 --> 00:45:07,050
So we have that that v.

379
00:45:09,090 --> 00:45:15,629
Over R is equal to cube of m is equal to than the more frequency.

380
00:45:15,630 --> 00:45:22,470
So the the the the the angular frequency of which the particle goes on its orbit

381
00:45:22,890 --> 00:45:26,340
is depends on the strength of the magnetic field and the charge in the mass,

382
00:45:26,340 --> 00:45:29,370
but not on the energy. It doesn't depend on how fast you're going.

383
00:45:29,670 --> 00:45:33,990
So fast particles go in big circles and take the same time to go around to slow particles.

384
00:45:34,560 --> 00:45:35,760
So you have a characteristic.

385
00:45:35,880 --> 00:45:42,920
All the motion is at some characteristic frequency and that is reminiscent of a harmonic oscillator or it allows allows us to reset that.

386
00:45:42,930 --> 00:45:47,610
The fundamental underlying physical reason why we can solve the problem of motion,

387
00:45:47,610 --> 00:45:53,010
the quantum mechanical problem of motion in a uniform magnetic field using the apparatus of the harmonic oscillator.

388
00:45:54,310 --> 00:45:59,469
So I think you should have I mean, I hope some people will have some fun looking at that in the vacation.

389
00:45:59,470 --> 00:46:02,800
It is very good quantum mechanics. It's it's very important physics.

390
00:46:03,040 --> 00:46:07,959
But unfortunately, we are not going to have time to cover it in the in the lectures.

391
00:46:07,960 --> 00:46:12,970
But magnetic field will be important in in the context of atomic physics.

392
00:46:13,840 --> 00:46:15,160
Okay, so that's it until next time.