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Okay. Good morning. So. The next item on the synopsis on the website says.

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Motion of a Particle and Magnetic Field. But I think it's better that we postpone that.

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We don't need to handle it now.

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And now open this new topic, this new chapter, which is chapter four in the book, The Relationship Between Transformations and Observables.

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We'll come back to the magnetic field later. But we we have this week and this is a I'd like to make sure we do this thing properly.

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Now, this topic is off syllabus, right?

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But it is actually very important is that the core of quantum mechanics and it's the core of 20th century physics.

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And I think you'll find it illuminating because we now on we it should should

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explain why the time dependent Schrodinger equation takes the form that it does.

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It should explain why the momentum operator takes the form that it does, why the canonical computation relations take the form that they do.

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So I think it explains many things, but because of historical reasons, it's not actually on the syllabus.

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Okay. So we have a we know that this thing is a function.

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This thing is a function of X where X is not going to be a position vector.

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This is the being the amplitude to find your system, your particle, whatever, at the location x right.

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So it's, it's because it's an amplitude which depends on X.

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It's a, it's a complex valued function of X.

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So we can take this here is expand this. Physicists always assume you can tell the theories, expand everything.

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So we take the series, expand this, and we say that if we evaluate this at X minus A, then that is going to be essentially.

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Well, to keep the notation simple. We, we call this up save x, right?

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So this is going to be obsessive x.

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Minus a dot D by the x of a CI plus a dot d by the x of CI squared over two factorial minus blah, blah, blah, blah, blah.

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This is just the here series expansion in three variables.

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It's covered in some prelims course. Right? And we've, we've, uh.

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Yeah. So this is just the Territory expansion and we now make an observation that this can be written with our now.

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Now that we understand how to take a function of an operator and we realise that DPD is an operator,

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we can write this as e of the exponential of minus a dot d by the x operating on ACI

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where we defining this exponential to mean this thing raised to the north power,

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namely one plus this thing raised to the first power plus this thing raised the second power on two factorial and so on and so on and so forth.

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That's what we mean by the exponential this. Operator Okay. But we, we notice that this can also be written as X.

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This is working on five X. This can be written by the rules of of operations and the definition of P as the exponential

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of minus i p upon h bar a sorry dot p over h bar operating operating on the catsup side.

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Just let me just remind you what my authority for that is.

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My authority for that is the observation that or the definition of P which was that x p sign was by definition minus i h bar d by d x.

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Of X cy.

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And this animal here could be rewritten as x upside.

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Okay. So. I can just make a change of notation here.

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Because of this. And where I've replaced P now by the function of P that you see there.

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And then we have a function of this operator. So what have we discovered?

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What we've discovered is that x minus A,

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this is the bottom line on this little piece of calculation which is really only Telesur is expanding into ABC is equal to X on.

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You have a website where you have a is a new notation for it simply means the exponential minus I a dot P over four.

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So that's what you really means. This could also be written as x minus A on upside is equal to x on it.

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So I primed where it was I primed.

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The cat is by definition use the operator operating on a sign.

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Right. We've just called this thing so primed.

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So let's think about let's just mathematics and it's nothing.

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But Telesur is expanding and a little bit of sophistication in taking functions of operators.

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But we've begun to do that. We understand that that comes with the territory.

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What does this physically say? It says that the sender is if you use this operator on an ABC, you get a new state.

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ABC Prime's. What's the point about this new states?

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Well, if your system is in this state, the new state, then the amplitude to be at X is the same as it was when we were in our old state.

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Somewhere behind our current location, back at x minus a.

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So here's his here's here's a visualiser. His meant to be a picture of this.

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We can get it to come back. Yes, we can get a come back. And if I could find a pointer, which I probably can't.

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Never mind. But. So. So. If upside with that.

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Sort of.

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If the probability density associated with sci, with that spherical blob on the left, the lower left and a is that vector displacement up there.

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Then the amplitude to be at some point take any point X in the sphere of upside primed.

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If you move back by A,

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you come to the corresponding point on the spherical density associated with cy and the amplitude in upside matches the amplitude in upside primed.

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That's through visualisation. This statement of what this is telling us.

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It's telling us that if cy primed the amplitude to be up, cy primed is the same as the amplitude over here at a point back,

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which means if cy primed is the state that our system would be in if we were able to just shove it down the vector a to translate it by a,

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then we would get a new state with with these properties.

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So what have we done? We have discovered what the operator you have a does you have a shoves the system by a displacement a.

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Now. Hey is just an ordinary, boring vector.

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This is an operator, but this is an ordinary, boring vector.

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It's a set of three real numbers. And we can differentiate.

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We can do d of upside primed. So the place that you.

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The state that you get is a function of a right.

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So we can do D by deep psi primed of a sub I well sub a sub j.

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Shall we say right. To avoid confusion between the index I and the square of two minus one.

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So we can take we can see the rate at which this thing changes when we change the parameters that appear in here,

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when we differentiate this exponential, as everybody knows, when you differentiate an exponential, you get the exponential back.

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That's just by the magic of uh, of the, that, that particular power series that defines the exponential.

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And then we need the differential. So that's going to be you. So differentiating you, we're going to get back you,

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but we're also going to get the derivative of this with respect to a sub j, which is going to be minus I a.

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J Sorry, minus PJ over HBO.

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And then, of course, Abassi will stick around because website is not a function of a.

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So if we would now set well, so now we can just recall that this thing is ACI prime.

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So I now have the deep CI primed by the ace of J is equal to minus I.

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Let's multiply through by bar and then we have that.

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This is equal to PGA of ACI primes.

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Whoops. So this this now answers a question which I forgot to ask at the beginning of the lecture, which is what actually does the operator P do?

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An operator associated with an observable so with each observable observable.

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We have associated an operator.

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We did it originally by saying that.

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Q The observable associated with Q was by definition.

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Q. J. Q.

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J. Q. J. And this operator, we're taking advantage of the fact that our mission operator is uniquely characterised by its iGen kits.

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And I can values. So if you specify these, you specify these.

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If you specify this, you specify this. There's a there's a relationship here which we found useful.

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We've we've discovered that the expectation value of Q, for example, is equal to this mathematical animal.

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And other things and other useful things.

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We found the rate of change of expectation values depends on the commentator of Q with the Hamiltonian operator,

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which is the operator associated with the energy, etc.,

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etc. But we haven't actually addressed or answered the question of what this observe what these operators

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that we're introducing actually do to states because an operator turns a state into a new state.

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So for example, so, so the operator Q turns up PSI if we expand its PSI in its iGen states, right.

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So if we, if we write it like this.

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So we know we can expand any upside. Thus in the Asian states of this operator.

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And then we know how to use this on this. So this is equal to the sum.

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Q. J. Q. J.

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UPS oops. So when we use the operator.

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Q one of PSI, we get this stuff here, which is some long gobbledegook.

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But if we measure Q. Then upside goes to Hugh K for some K, it doesn't go to this long list of stuff.

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It goes to one of these things on the one of these things is chosen at random somehow by nature, not discussed by theory,

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no answer offered by theory, merely probability distribution under which we get one of these things is predicted.

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But we know that the state of PSI on making a measurement collapses into one of these states here.

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So the operator Q is not doing measuring. That's the point.

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And we have discovered, apropos of the operator pee, what is it doing?

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What he does is give you the rate of change of your state when you shove something along.

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So this gives you the rate of change subside, gives you the rate of change of your state if you shove it down the x axis.

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So we're learning what the operator does. And what it does is not measure, but displace.

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Let's let's for a for a piece of practice.

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Let's check this out on, uh, uh, let, let's check this out on this state.

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Let's for fun, apply you a to this state,

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which is a state of definitely being at X and make sure that we can produce X plus a the state of being at X plus eight.

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Because if it's true, if you take the state X and you displace it you by a, you must have the state, right?

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Let's make sure that this is the case. So what we want to do is use you A on x.

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Now, this operator here is a function of the momentum operators, right?

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It's it's that exponential a dot P so the noisy way to do this is to decompose this into a linear combination of of states of.

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Of well-defined P. So we write This is d cube p of p x.

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P. So basically I've subbed to an identity operator in front of the ex.

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This is a boring, complex number. What is it?

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Is the complex conjugate of the wave function two of the wave function associated with being a having well-defined momentum.

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So we know what it is. It's e to the minus P upon h bar dot x over h bar to the three halves power.

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We discussed that when we talked about generalisation to three dimensions.

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That's what this complex number is.

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This operator ignores that complex number because it's a linear operator and goes straight to the to its target which is this.

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Then all the operators in here meet their Oregon State P and and get transformed simply into their eigenvalues.

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So this becomes when this thing hits this which this in the PS in here operates this but

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when they meet that because that's it's I can state they simply become eigenvalues.

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So we get an E to the minus I a dot P overage bar times the cat, the eigen cat left behind.

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And still we have to do a d q p integration.

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All right. So this is no longer an operator because it already worked on that and produced its eigenvalue.

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So we can rearrange this. We can put those two exponentials whose arguments are mere complex numbers.

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We can gather them together. And this becomes the integral due p of over each bar.

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Sorry, that isn't bar that's on board. Excuse me. Three horsepower h planck's naked constant e.

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To the minus i. P Well, it's right, it doesn't matter what order we write these in, you see, because this is a number and that's a number.

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So I'm going to write this as a plus x dot p on each bar u p.

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But if I now ask myself what is X in this in this notation, I probably should have written this down.

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Originally it was d, q p over h three halves power e to the minus i x dot p over h bar p this.

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This is just the standard expression which I've essentially used above for decomposing a

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state of well defined position is the superposition of states of well-defined momentum.

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Where this is, is this this thing here is nothing but p x.

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So since this is the general formula, this state that we're producing you air on X is given by the same formula, but with X replaced by x plus a.

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Right. Because the only difference between this formula and this formula is it is that here we have an X and there therefore we have an X,

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and here we have an X plus A, so we should have an X plus. So this establishes indeed that X plus a is equal to U of A on x.

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So that's just a particular extra very vivid example of a basic principle.

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So what we want to do now is. Generalise this to any continuous transformation.

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We always require. Proper normalisation we require ACI.

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ACI is equal to one. Why?

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Because this tells us that the total probability to find, to get some measurement to find something is one.

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That's why we're completely wedded to that normalisation. So we're interested in transformations that preserve this property.

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This shoving it along transformation was one example.

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In a minute, we'll talk about the transformations associated with rotating our system around some axes.

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But there are many transformations we might make. So what we require what we're going to say is that if PSI goes to some newfangled state

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which is some operator you on our old state and the restrict because in light of this

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we're going to restrict ourselves to one is equal to ABC primed ABC prime if we take our

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new states they've got to be properly normalised which means that we are looking at ABC.

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You dagger you ABC.

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All right. So we require this is one. But this is by definition, you see.

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So if we take the mod square of this, we're looking at that. Where you. Is this as yet undetermined.

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Operator. And the thing is. So this has to be true for all for all of sci, for any any quantum state.

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This has to be true that this thing is one. And there's a technical detail about establishing that this is one.

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There's a box in chapter four of the book doing this, which I don't propose to go through.

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It's very straightforward and simple, but I don't want to take the time to do it because it's mere mathematics from this,

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from the fact that this has to be the has to be one. For any of sai, we can deduce that u dagger u is in fact the identity operator.

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Okay. From from this statement. This follows fairly straightforwardly, but I'm not actually proving it right now.

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So operators of this sort as I expect you know from professor excellent course.

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I called unitary. So usually operators are precisely those operators which leave the length of our states unchanged.

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And in the present case, for physical reasons, the length is one. Now let's.

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So we're dealing with one such earlier on.

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But let's. Let's suppose that you.

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Is a function of theatre. In that case you as a function of a theatre.

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Just be some parameter where. So theatre is a parameter.

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Which we can make small. Well, shall we say, which can go to zero.

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So the idea is that theta is the amount by which you transformed there a was the distance which we had displaced.

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So A is analogous to theatre here we theatre is just stands vaguely for the amount by which you can do something.

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And we want to be able to say but we can we can reduce this amount continuously down to nothing when we're doing absolutely nothing.

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So we're going to have the you of of nought is the identity operator because that's that's the operator that does nothing.

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So we want to have this parameter. And now we're going to argue that if theatre small.

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We should be able to tailor expand. I said, physicists assume you can tailor expand everything.

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So we're going to tailor expand this.

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So we're going to have that you of theto which is now small, is you for feature equals nought, which we've said is one the identity.

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And now we're going to write the first order term in a slightly funny way.

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We're going to write in minus I see to tell and then we'll have terms or to see two squared.

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So this is a Taylor series expansion only the first two terms that the zero term in the first

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derivative term and all the other terms we just got wrapped up under order theta squared,

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not saying what they are and this is an operator, it has to be an operator because this is an operator.

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There is this, of course, is an operator. That is a mere number. That is a mere number.

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A real number. So therefore, this has to be doing the operating.

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But we've just chosen a particular way of writing the first order, the first derivative term in a Taylor series.

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So this is a tailor series. It relies only on the idea that theatre,

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that there's a whole family of transformations which could be reduced to the

197
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identity transformation as theatre goes down to nothing when you don't do anything.

198
00:24:24,420 --> 00:24:31,710
Okay, now we want to look at this condition that we want to have a look at the condition that the identity is you, dagger you.

199
00:24:32,760 --> 00:24:35,760
So let's write the let's write you once you dagger.

200
00:24:35,760 --> 00:24:40,290
If this is you, you, dagger is going to be a dagger, which is I.

201
00:24:40,620 --> 00:24:50,160
And then we'll need the dagger of this, which is going to be plus I theatre tao dagger plus order theta squared, which we're going to ignore.

202
00:24:50,730 --> 00:24:59,310
And that has to be multiplied on I minus I see to Tao plus order C two squared which we're going to ignore.

203
00:24:59,970 --> 00:25:03,450
So when you, when you multiply these two brackets together,

204
00:25:04,440 --> 00:25:08,280
ginormous job in principle because they're all this infinite number of terms and this and that.

205
00:25:08,280 --> 00:25:11,880
But we won't need to bother with much algebra.

206
00:25:12,540 --> 00:25:18,509
We must get the identity operator and we must get the identity, operate it completely regardless of what theatre is.

207
00:25:18,510 --> 00:25:23,550
Right. Because this is meant to be this. This is. This is a unitary transformation, regardless of theatre.

208
00:25:24,120 --> 00:25:27,810
So let's work this out. This is equal to the. So what do we have?

209
00:25:27,820 --> 00:25:35,250
We have the lowest order term. Is this on this? Then there are first order terms which you get this on this and this on this.

210
00:25:35,520 --> 00:25:48,629
So we're going to have plus I thetr Tao Dagger minus Tao, and then we will have terms like this on this, which will be all to see,

211
00:25:48,630 --> 00:25:53,010
to square this on this wall you will see two squared will have this on this allows you to see this squared.

212
00:25:53,310 --> 00:25:58,110
So plus order C2 squared. We've accounted for everything through linear order.

213
00:26:00,870 --> 00:26:05,040
So this is supposed to be true for all theatre.

214
00:26:05,680 --> 00:26:10,240
It doesn't matter what theatre we take. Should be true.

215
00:26:10,960 --> 00:26:16,020
If it's going to be true for all theatre, then we can equate powers of theatre on both sides.

216
00:26:16,030 --> 00:26:21,189
So the coefficient of theatre to the north, namely the identity, should be the same on both sides.

217
00:26:21,190 --> 00:26:26,380
Well, it is. That's a relief. The coefficient of theatre to the first power should be the same on both sides.

218
00:26:26,620 --> 00:26:30,820
On this side of the equation there is well, the coefficient of theatre to the first power is nothing.

219
00:26:31,150 --> 00:26:38,200
So it better be nothing on this side to. So this implies that Tao Dagger is equal to Tao.

220
00:26:38,530 --> 00:26:51,890
That is to say Tao is permission. Mission operators, we suspect are associated with observables.

221
00:26:53,090 --> 00:27:01,309
So the argument here is that every such transformation is going to be associated with the emission operator.

222
00:27:01,310 --> 00:27:05,060
And the reason this I was put in here, this was totally gratuitous.

223
00:27:07,300 --> 00:27:08,620
Sorry. Right up there.

224
00:27:08,740 --> 00:27:16,719
The reason that I was put in there, which was a totally gratuitous decoration, but it went in because that ensures looking forward,

225
00:27:16,720 --> 00:27:23,140
it ensures that Towle is a mission operator rather than an anti commission operator, which it would have been if I had not been put in.

226
00:27:23,770 --> 00:27:30,400
So there's a there's a suspicion that this TOWLE And it will always turn out to be the case that this Towle will be associated with an observable.

227
00:27:31,000 --> 00:27:37,510
This is how observables become associated with operators in both classical mechanics and quantum mechanics,

228
00:27:37,510 --> 00:27:40,749
or should have said in quantum mechanics and in classical mechanics.

229
00:27:40,750 --> 00:27:47,700
It turns out that it's true in any mechanics. Right.

230
00:27:47,700 --> 00:27:59,549
And if we if we write the equation upside primed is equal to you, a32 times upside is equal to one minus.

231
00:27:59,550 --> 00:28:03,060
I see two tao plus dot, dot, dot.

232
00:28:04,610 --> 00:28:11,130
ABC and we do d things are primed.

233
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Sorry. Deep sigh, primed by the theatre. We find that this is equal to minus I tao psi plus order delta squared.

234
00:28:26,360 --> 00:28:34,590
So if. If we put if we put the theatre equal to nought, then the Delta Square goes away.

235
00:28:34,890 --> 00:28:36,270
The sorry, the order theatre squared.

236
00:28:40,040 --> 00:28:51,070
And multiply this equation through by eye and we get a very important equation, which is that i d ci primed by d c to is equal to tao.

237
00:28:52,730 --> 00:29:00,290
So this observe the operator to the mission operator Tao, which we suspect is connected to some observable well, will turn out to be connected.

238
00:29:00,290 --> 00:29:04,160
Some observable in every case is the thing. What does it do?

239
00:29:04,190 --> 00:29:10,400
What it does is it measures the rate of change of your states when you change the parameter setter.

240
00:29:11,550 --> 00:29:18,080
So this is a generalisation of. Where are we?

241
00:29:18,170 --> 00:29:21,490
This equation. This equation here.

242
00:29:21,500 --> 00:29:25,100
Yeah. All right, so this is a concrete example of this.

243
00:29:25,640 --> 00:29:29,000
Now, this equation has a tiresome h bar here.

244
00:29:30,020 --> 00:29:35,870
Y is equal to task image bar here because in that exponential, there's a tiresome h bar on the bottom.

245
00:29:37,040 --> 00:29:42,529
Right. So here we had the exponential of minus.

246
00:29:42,530 --> 00:29:51,710
I don't pay over h bar. And if you do the Taylor series expansion of that you get one minus I ape over H bar

247
00:29:52,040 --> 00:29:57,800
so that the role of Tao in conceptual apparatus here is played by P over there.

248
00:29:58,220 --> 00:30:06,950
And it's an unfortunate historical accident that the momentum that this operator,

249
00:30:06,950 --> 00:30:10,580
which we call the momentum operator, has been defined with a rigid H bar.

250
00:30:10,820 --> 00:30:15,139
So we have to divide through by H bar to get rid of what we shouldn't do put in in the first place.

251
00:30:15,140 --> 00:30:21,709
So it's one of these many cases in physics where history forces us into a bad notation and even a degree of intellectual muddle.

252
00:30:21,710 --> 00:30:25,250
That bar had better not would would have been better left out, but.

253
00:30:29,510 --> 00:30:33,940
The reason is that momentum came to Isaac Newton's attention before quantum mechanics.

254
00:30:33,950 --> 00:30:36,140
All this stuff was thought about and.

255
00:30:38,940 --> 00:30:48,089
So it came to mean something which is really a derivative thing, which is really something which follows on from Momentum's fundamental role,

256
00:30:48,090 --> 00:30:55,440
which is something which shoves your system in it, which spatially translates your system.

257
00:30:57,660 --> 00:31:03,010
Okay. And if we want to do so.

258
00:31:03,220 --> 00:31:11,320
So we've we've defined Tao. Tao came in here through a formula for you of theatre where theatre is small.

259
00:31:12,700 --> 00:31:16,719
We would like to know how to do you of three to even win seats as large.

260
00:31:16,720 --> 00:31:20,600
So for large theatre. Well.

261
00:31:23,480 --> 00:31:34,180
What we should say is take a transformation. Through large theatre.

262
00:31:42,050 --> 00:31:49,370
In steps. So if we if we are told to find out what you is for a large value of theatre.

263
00:31:49,730 --> 00:31:57,620
The way to go is to is to make many transformations one after another through small steps of length.

264
00:31:58,580 --> 00:32:06,910
Feature over and. Then if any, is big enough, no matter what the value of theatre we're given, we can write that.

265
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What we can do is we can set up CI primed, which is you.

266
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A theatre of SCI of course is equal to.

267
00:32:18,200 --> 00:32:32,390
You of theatre over in. You a theatre over and you a theatre over in end of these terms, all multiple or multiplied together, operating on its side.

268
00:32:33,080 --> 00:32:39,770
So we make a transformation by and by a split by an amount feature over end, and then another one feature of random left.

269
00:32:39,950 --> 00:32:54,080
So there are n terms. And each one of these use, we can use that Nazi formula up there because for each one of these, these were over any small.

270
00:32:54,410 --> 00:33:02,540
So this can be written as one minus. I see two over and Tao plus stuff which we're going to be able to neglect.

271
00:33:02,900 --> 00:33:11,240
This is raised to the ends power because the end of these terms on ACI and now we

272
00:33:11,240 --> 00:33:17,209
take the limit then goes to infinity to be completely sure that this plus dot,

273
00:33:17,210 --> 00:33:23,360
dot, dot stuff can be neglected. Write this plus dot dot dot stuff is order theta over n squared.

274
00:33:23,570 --> 00:33:27,050
So to be sure it can be neglected, we can go to the limit, end to infinity.

275
00:33:27,440 --> 00:33:32,839
And then we have a theorem of of calculus though that for what?

276
00:33:32,840 --> 00:33:40,040
This one plus a bit plus something of n race to the nth power goes to an exponential.

277
00:33:40,050 --> 00:33:49,760
So this mathematics now tells us that this is the exponential of minus i thi to tell operating on side.

278
00:33:51,820 --> 00:33:57,310
So we introduced Tao as the first order Taylor series term.

279
00:33:57,760 --> 00:34:06,670
But this apparatus tells us that that's all we need to know in order to find out what you of theatre is for any feature.

280
00:34:08,120 --> 00:34:11,780
She's, I think, slightly surprising. You don't need to know anything in the higher in the higher orders.

281
00:34:16,040 --> 00:34:22,250
What do we say? We say. We say that Tao is the generator.

282
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Of both the unitary transformations.

283
00:34:30,620 --> 00:34:34,640
Usually operator, rather. And the transformations.

284
00:34:43,560 --> 00:34:48,629
Sy goes to print, he saves the generator by saying it's the generator.

285
00:34:48,630 --> 00:34:56,310
What? This is the story. This is badly written town. But the generator is the operator you stuff in up here in the exponential.

286
00:34:56,580 --> 00:35:03,660
It's always times minus I for conventional reasons and then a parameter feature that tells you how much you've generated.

287
00:35:04,920 --> 00:35:13,350
So for example P overage bar, not p sadly, but p over bar is the generator.

288
00:35:16,670 --> 00:35:25,910
Of translations. That's just jargon.

289
00:35:31,170 --> 00:35:34,860
So now let's think about time to move to a new board. Think about rotations.

290
00:35:36,210 --> 00:35:46,710
This is where it becomes slightly more interesting because we will discover that in quantum mechanics, rotations seem well.

291
00:35:46,710 --> 00:35:47,760
They're rather more complicated.

292
00:35:47,760 --> 00:35:55,440
They seem a bit different from they are actually significantly different, but quite amazingly different from rotations in classical physics.

293
00:35:55,440 --> 00:35:59,490
And I think this is not fully understood even now.

294
00:36:02,190 --> 00:36:08,009
All right. So. To generate translations.

295
00:36:08,010 --> 00:36:11,910
We in fact need three operators, don't we? We need X, Y and Z.

296
00:36:13,410 --> 00:36:17,309
Why do we need three operators? Because to define a translation,

297
00:36:17,310 --> 00:36:23,610
we need to specify a vector because we have to say in what direction we're going to go and how far we're planning to go.

298
00:36:24,540 --> 00:36:28,290
And those three numbers define a vector.

299
00:36:28,580 --> 00:36:37,830
Alternatively, you can say, Well, okay, you know that. So then we we should expect that there are.

300
00:36:38,430 --> 00:36:46,350
There's more than one generator of rotations because in order to specify a rotation, we have to specify a rotation axis.

301
00:36:46,980 --> 00:36:50,639
And how far around that axis we're going to go. Right.

302
00:36:50,640 --> 00:36:55,050
If you know the axis around which. So. So here's here's a solid body.

303
00:36:55,860 --> 00:37:00,899
I can rotate this in a whole variety of ways to specify one rotation.

304
00:37:00,900 --> 00:37:07,890
I specify the axis I'm going to rotate around and I specify how far around that axis I'm going to rotate.

305
00:37:10,100 --> 00:37:15,440
So we expect. Three generators.

306
00:37:19,100 --> 00:37:23,590
Of rotation. Because.

307
00:37:28,070 --> 00:37:32,110
We specify. A rotation.

308
00:37:38,070 --> 00:37:41,220
With three numbers. Now, there are many ways.

309
00:37:41,430 --> 00:37:49,380
Just as there are many sets of three numbers, I can use to specify a translation because I can orient my x, y and Z axes in any which way I like.

310
00:37:49,890 --> 00:37:56,040
There are many ways in which I can specify three numbers that define a rotation.

311
00:37:56,160 --> 00:38:05,340
And those of you who've done a seven, the classical mechanics option will have heard of Euler angles, of which there are three Theta Phi and PSI.

312
00:38:06,450 --> 00:38:12,180
But the handiest way to specify three rotations is actually through a vector.

313
00:38:14,070 --> 00:38:23,610
We're going to use alpha. So that's the that's alpha x, comma, alpha y, comma, whoops, comma, alpha z.

314
00:38:24,600 --> 00:38:29,009
So and alpha has the unit vector. So this now doesn't mean an operator.

315
00:38:29,010 --> 00:38:32,310
It means a unit vector. Whoops.

316
00:38:32,420 --> 00:38:39,140
Unit vector parallel to alpha is axis of rotation.

317
00:38:43,630 --> 00:38:52,690
And Model Alpha. Oh. The modulus of the vector alpha is the angle through which we plan to rotate by.

318
00:38:52,770 --> 00:39:00,220
Okay, so these three numbers are a handy, convenient system for specifying which rotation you wish to refer to.

319
00:39:02,220 --> 00:39:06,800
And when I now say that there must be.

320
00:39:06,810 --> 00:39:15,660
So the rotations form a continuous set of transformations of my system because I can rotate my system by a little bit or a lot.

321
00:39:17,490 --> 00:39:26,670
And when I. So if I. So there must be a state of the system which differs from my previous state only in in being rotated.

322
00:39:27,920 --> 00:39:34,430
And this state must be reachable by some unitary operator u of alpha.

323
00:39:35,030 --> 00:39:49,600
And this apparatus here tells me that you of Alpha can be written as an exponential of minus I alpha dot j where this is playing the role of title.

324
00:39:49,610 --> 00:39:58,870
This is the operator. It's a set of three operations as promised, because this means alpha x, x, plus alpha y, jay y plus alpha z, jay z.

325
00:39:59,530 --> 00:40:04,000
So it's a set of three operators. Jay Jay Y, Jay Z.

326
00:40:05,290 --> 00:40:09,130
They must exist because and they. And it's going to be commission.

327
00:40:12,560 --> 00:40:15,980
It's going to be his mission because this operation is going to be unitary.

328
00:40:17,600 --> 00:40:22,280
And we've shown. The connection between emission operators and unitary operators.

329
00:40:23,180 --> 00:40:26,690
So I think this I hope that much is absolutely self-evident.

330
00:40:28,270 --> 00:40:33,110
We'll be when you think about this through again, there must be this operator.

331
00:40:33,130 --> 00:40:37,420
It's going to be a mission. So it's going to be a candidate for an observable.

332
00:40:38,260 --> 00:40:42,700
And the question arises, what observable is it going to be the operator of?

333
00:40:44,420 --> 00:40:48,709
The operator associated with translations which had to exist.

334
00:40:48,710 --> 00:40:57,830
It was a logical necessity that it existed. And we have shown that the operator is actually a.

335
00:41:00,580 --> 00:41:03,700
It's actually the momentum operator divided by age bar.

336
00:41:03,970 --> 00:41:09,220
So I hope it won't come now as a great surprise that this operator is going to be the angular momentum operator.

337
00:41:11,680 --> 00:41:15,850
We're not proving this. I'm saying it will turn out.

338
00:41:18,550 --> 00:41:23,140
To be. Angular.

339
00:41:24,010 --> 00:41:27,370
Angular right in this terrible angular.

340
00:41:36,030 --> 00:41:44,640
So the angular momentum operates the generators of rotations in the same way that the momentum operators are the generators of translations.

341
00:41:45,000 --> 00:41:53,180
But we will we will have to build confidence that that's the case as we go along.

342
00:41:53,190 --> 00:41:57,959
I'm saying that this will turn out to be the case, and I hope it will be clear at the moment.

343
00:41:57,960 --> 00:42:03,780
I just hope that that's a plausible conjecture, that it is the angular momentum that we're talking here about, the angle momentum operators.

344
00:42:04,080 --> 00:42:08,010
And of course, the reason there are three of them is that angular momentum itself is a vector.

345
00:42:08,190 --> 00:42:13,290
So you can have an angle centred around the X axis and Anglicanism around the Y axis and Anglicanism about the Z axis and those.

346
00:42:14,130 --> 00:42:22,080
That's because you have those three numbers, you have three operators, and we're going to have the analogue of, well,

347
00:42:22,080 --> 00:42:36,569
this formula here is going to be that D by the alpha, the modulus of this angle d by the alpha of of psi primed i times.

348
00:42:36,570 --> 00:42:44,310
This is going to be the unit vector alpha dot j on up sine.

349
00:42:47,110 --> 00:42:51,250
You might want to just check the algebra on this.

350
00:42:51,640 --> 00:42:55,390
If you do the derivative. So why is this a function of alpha?

351
00:42:55,630 --> 00:42:59,380
Only because this is u, which depends on alpha. On upside, which does not.

352
00:42:59,740 --> 00:43:04,930
So we're talking about the derivative of this exponential with respect to the modulus of the vector.

353
00:43:05,200 --> 00:43:11,559
But this exponential could clearly be written as model alpha times, the unit vector,

354
00:43:11,560 --> 00:43:14,680
alpha do the derivative, and you'll find this important relationship here.

355
00:43:16,690 --> 00:43:20,560
So what does the operator, Alpha?

356
00:43:20,590 --> 00:43:26,319
This is just a single operator. If you take the unit vector alpha and you dot it into the three operators J,

357
00:43:26,320 --> 00:43:29,980
you get a linear combination of these three operators, which is an operator.

358
00:43:30,280 --> 00:43:40,720
And what does this thing do for you? It measures the rate of change of your state when you rotate it around that axis that's specified by this.

359
00:43:40,930 --> 00:43:46,600
That's what it's physically doing for you. And this is this is an important relation.

360
00:43:46,900 --> 00:43:53,440
We'll come back to. Okay.

361
00:43:58,500 --> 00:44:07,050
We're not going to quite finish this, but let's let's get going. So we've talked about translations and rotations.

362
00:44:08,590 --> 00:44:14,829
And they have this in common that you they have a free for us parameter how much you do of them would you can

363
00:44:14,830 --> 00:44:19,660
turn right down to zero when you do nothing and they just become the the operators just become the identity.

364
00:44:20,260 --> 00:44:25,870
But we have to use in physics important we have important use to make of

365
00:44:25,870 --> 00:44:30,429
transformations which which in a discrete you cannot turn them down to nothing.

366
00:44:30,430 --> 00:44:40,850
You either do it or you don't do it. And the classic example is, is the parity or reflection operator.

367
00:44:40,860 --> 00:44:52,469
So if I have an ordinary vector, then it's p turns the position vector x into minus x.

368
00:44:52,470 --> 00:45:00,750
There's a transformation you can make where you start with a thing and you choose a point which you call the origin,

369
00:45:01,080 --> 00:45:07,739
and you move every, every part of your thing through that origin into another thing.

370
00:45:07,740 --> 00:45:15,330
So if if the origin is here and my lower hand is here, if I if I move every, every part of my lower hand.

371
00:45:16,680 --> 00:45:21,569
Through the origin by a certain amount to this equal distance opposite from the origin.

372
00:45:21,570 --> 00:45:24,680
It should become my up, my my upper hand, my right hand, right,

373
00:45:24,720 --> 00:45:28,950
left hands and right hands related in this way, through reflection, through the origin.

374
00:45:28,950 --> 00:45:34,930
If you put the origin symmetrically between the two. So this is a transformation you can make.

375
00:45:35,010 --> 00:45:39,809
This is a this is a mental transformation. It's not a real physical transformation, but it's a mental transformation.

376
00:45:39,810 --> 00:45:48,330
You could ask yourself. Suppose I had a system which was obtained from my real system right here in the lab by this operation.

377
00:45:50,010 --> 00:45:57,450
Would I mean a question you can ask is would it would the dynamics of this system so is my real system around here is moving around, you know,

378
00:45:57,450 --> 00:46:04,380
imagine this thing is a solar system on my hand, wiggling with the system that you get above by mirroring each of these points through the origin.

379
00:46:04,590 --> 00:46:09,030
Would that behave like a real thing in the universe and in classical physics?

380
00:46:09,150 --> 00:46:17,930
That's the case. This what you see down here with the reflected through the origin, produces a wiggle up here which could happen all on its own there.

381
00:46:17,940 --> 00:46:20,310
It would be a dynamical system that would produce that wiggling.

382
00:46:21,270 --> 00:46:25,649
One of the amazing one of the great discoveries of the 20th century was that that's actually not true in all physics.

383
00:46:25,650 --> 00:46:29,730
Weak interactions mean when weak interactions are involved, things happen.

384
00:46:29,850 --> 00:46:35,310
If you make a model by by taking your real system and playing this silly game with it,

385
00:46:35,490 --> 00:46:41,520
you get a thing up here which you can distinguish from a real system because a real the real system up here couldn't behave in that way.

386
00:46:42,640 --> 00:46:48,340
So there is this there is this operation of taking your system and and mirroring it through the origin.

387
00:46:48,640 --> 00:46:57,280
And there's a classical operator, P which does this. It just changes the sign of all your of all your of all vectors of all components of a vector.

388
00:46:57,280 --> 00:47:01,390
It turns x it to minus x, it puts a point to the opposite point across there.

389
00:47:01,750 --> 00:47:07,300
So what's the quantum mechanical analogue of this? Uh, well, it's this animal, so.

390
00:47:07,390 --> 00:47:12,610
So I'm going to have a long tail on P to imply a classical operator, which simply changes the sign.

391
00:47:12,650 --> 00:47:18,300
Right. This is this is classical. And it's just a change. Of sign.

392
00:47:20,090 --> 00:47:23,930
Uh, of of of of all vectors.

393
00:47:28,330 --> 00:47:33,250
Okay. We want a quantum analogue and the quantum animal dog is the thing.

394
00:47:33,250 --> 00:47:38,120
This I'm going to define it thus. What's this?

395
00:47:38,660 --> 00:47:44,809
This is the amplitude to be an X if you're in the state of sci primed.

396
00:47:44,810 --> 00:47:56,070
I should have written this separately. This is the amplitude. So Pepsi is going to make a new state.

397
00:47:56,100 --> 00:47:59,910
What state is it going to make this state? What's the point about this state?

398
00:48:00,120 --> 00:48:07,109
The point about this state is if you're in this state, the amplitude to be at X is minus is this rise,

399
00:48:07,110 --> 00:48:10,920
the amplitude to be at minus X if you're in the original state.

400
00:48:12,560 --> 00:48:22,430
Right. So this is our original state and the amplitude to be at minus X in this state is equal to the amplitude to be plus X in this state,

401
00:48:22,430 --> 00:48:31,850
which we've gotten by using P on up side. So P takes my the state of my hand here and makes this this state.

402
00:48:32,510 --> 00:48:35,830
That's what it does. And.

403
00:48:39,090 --> 00:48:42,290
What can we say about what what interesting statements can we make about this?

404
00:48:42,300 --> 00:48:56,640
Well, one very obvious statement that we can make is that if we look at X squared CI, then that's x p p CI.

405
00:49:01,450 --> 00:49:06,549
By definition, obviously, we use this rule.

406
00:49:06,550 --> 00:49:14,020
We use this rule here on this state so that it's equal to minus X on.

407
00:49:15,730 --> 00:49:21,070
Right. Because this P and that X using that rule on this state gives me this.

408
00:49:21,250 --> 00:49:28,510
I'm replacing a sign, this formula by Pepsi, and now I can play the game all over again.

409
00:49:28,720 --> 00:49:39,700
So by using the same rule, I find that this is equal to minus minus X on its side, which of course is equal to X on its side.

410
00:49:41,050 --> 00:49:45,250
So if you use. What does this tell me? It tells me the amplitude to be an X when I'm in the state.

411
00:49:45,250 --> 00:49:49,590
P squared up ci is the same as the amplitude to be the x if I was an up sign.

412
00:49:49,610 --> 00:49:56,350
Other words, these two states are the same. So that implies that p squared is the identity operator.

413
00:49:57,840 --> 00:50:03,750
Which implies that p inverse is equal to P.

414
00:50:04,920 --> 00:50:13,739
P is its own inverse. And what I should do next is show that permission.

415
00:50:13,740 --> 00:50:20,580
But we'll have to do that tomorrow. And therefore he's is going to have the properties of an observable.