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And. So we arrived yesterday.
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Well, at a statement of the fundamental dynamical principle of quantum mechanics,
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the time dependent SchrÃ¶dinger equation here I didn't yet justify didn't offer any justification for this at this time.
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The justification or arguments which will make this seem plausible are about to come.
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But they have not yet come.
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However, we already, at the end of yesterday's lecture realised that if if Asi if the state of our system is such that the energy is well determined,
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is, is is the result of measuring energy is certain, then the time evolution of that stage is absolutely trivial.
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The state evolves only in that its phase increments at a rate with a frequency and angular frequency upon bar,
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which is typically very large because the bar is very small.
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And we went on from that result there to show that the solution to this equation for any state of PSI takes this form that the state,
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the the the state of the system at an arbitrary time is this is this some over
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the states of world are termed energy where an observer is and evaluated.
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So if you evaluate these coefficients that time t equals nought or indeed at any time at your convenience,
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then just by inserting into this sum, this ordinary sum, which we've seen few times of upside expanded in terms of a complete set of states.
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If we just insert these exponential factors, bingo, we evolve the state according to that equation.
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So that makes these states of well-defined energy, of crucial operational significance.
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And the result of that is that we spend a great deal of time solving the defining equation of these of these states,
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which is that and is equal to E and in the states of well-defined energy by construction.
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I can states of the Hamiltonian operator and this is the time independent Schrodinger equation
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to distinguish it from the time dependent up there so only states of well of definite energy.
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So if the time independent Schrodinger equation any state the state of our system always has to solve the time dependent Schrodinger equation.
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So what we want to do now, the next item on the agenda fundamentally is to provide some justification,
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make it seem plausible that that is the correct equation of motion.
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And a good way to do that is to link back to classical mechanics.
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So recall classical mechanics, classical physics is that limit of where we where the uncertainty in the values of dynamical variables
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is sufficiently small that isn't necessary to calculate the whole probability distribution.
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It's enough to know what the expectation value of the probability distribution is, because the true value will be very close to the expectation value.
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And we ordinarily don't distinguish between the expectation value and the true value when doing classical physics.
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So what we want to do is calculate divide e t of the expectation value of some observable.
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I think I think we calling observables Q.
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And let's put an ice bar in front of here just in order to clarify what's right, in order to simplify the algebra.
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So we're trying to calculate this the rate of change of the expectation value of something, this something might be position.
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So this might be X, this might be momentum, it might be energy, it might be whatever you want to know,
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might be angular momentum, whatever you want to know about the system.
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This would tell you the rate of change of the classical value of that variable, because the classical value is the expectation value.
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So what does that what is that? Well, this is just a this is just a product.
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So obviously it comes into three parts h bar the CI by e t times Q times of PSI plus.
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Let's take the last bit now. No. Yeah.
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Plus upside to Q to Q by d t sorry.
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These things probably should be with the. Whether we make these things partial derivatives or total derivatives, this is no significant.
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This is clearly a total derivative because this thing doesn't depend on anything except time.
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Whether we make these partial or total is is very unimportant.
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So here we have an Irish bar rate of change of size and this can immediately be replaced by H ABC.
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Here we have the bra upside and the original equation we can take.
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We can. We can take the Hamish now joint of the original equation when it becomes minus
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h bar d by d t of the bra psi because he's brought up psi equals upside h.
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Remember the rule when we take commisioner joints is we reverse the order of the symbols and we and we take the joints of the individual bits.
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H isn't h is an observable.
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So it's a high emission operator. So H equals h dagger.
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So I can pop those into here and here. So the first two terms give me including the bar and this one is going to be minus psi.
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H So that is bar divide T of the bra up psi times.
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Q Times of psi. And this one the I including the bar.
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This is going to be plus of psi cubed h.
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And then we have a trailing bit here, which will be plus H bar.
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Q Sorry, upside. Q By d.
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T So. These two can be handily combined together into a commentator because there's a minus sign here.
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Otherwise the order of the Q in the Asia swapped around. So this can be written as as upside the comitato Q comma h ci.
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And then this is plus h by. Oops.
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So this result goes by the name of Ehrenfest Theorem.
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And it's one of the more important results of quantum mechanics in most of our applications.
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We can forget about this because what is this last term? Here is the expectation value of the rate of change of your observable.
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So if the observable was, for example, x, it would have no rate of change.
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Xs, Xs X It's always the same operator if you're observable.
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P it would have no rate of change because the momentum.
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Momentum is momentum. It's momentum. It doesn't change the our understanding of what momentum isn't.
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It doesn't change from moment to moment. If it were angular momentum, it doesn't change from moment to moment.
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So this this term here usually falls away.
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And so so if the upside size de Q by d t equals nought, which is the normal state of affairs, then oh,
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we have that bar divided t of the expectation value of Q which is often written like this in shorthand notion notation.
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Right. We've left out the upside either side. Just leave the angle brackets,
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which means the expectation value in any state whatsoever is equal to the expectation value of the commentator cubed comma h.
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Okay. What does that tell us? It tells us immediately that if an observable commutes with the Hamiltonian.
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If Q commutes.
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But with age that was being the same thing as Q, comma h equals nought.
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Then clearly the expectation value of Q is always constant.
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And physicists are always very excited by quantities which are constant.
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So we call it what do we call it in classical physics, we call it a constant of motion.
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Famously, Newton said that if you didn't go from messing with particles, the momentum was constant.
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Or maybe he said the velocity was constant, right? So velocity is a constant of motion.
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We now would reinterpret that as momentum is constant.
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We often know that angular momentum is of the angle.
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Momentum with the earth is almost constant insofar as it's not acted on too much by the moon and so on and so forth.
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So. So. Physics is full of constants of motion. They're very important in quantum mechanics.
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There's a special notation around here, which is to say that we say that.
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Q The eigenvalue of this is a good quantum number.
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So if somebody says that angular momentum is a good quantum number, they're simply saying angular momentum is a constant to the motion.
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We can go further than that, though, because so we've shown that the expectation value of Q is constant.
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We could show easily that the expectation value of Q squared is constant too.
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It's very straightforward, but it's obviously the case because Q squared is an observable.
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If Q commits with a Hamiltonian, then Q squared has to compute with the Hamiltonian.
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So Q squared is also a constant to the motion. That means that if you start.
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So this implies that if if initially if t equals nought, abassi is one of the states of well-defined Q So.
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Q are one of the eigen states of the operator. Q So we know for certain the result of a measurement will be.
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Q Are then what does that mean? That means that T equals nought.
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The expectation value of Q is obviously equal to Q R and the expectation value of Q squared is equal to Q R squared,
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which implies that the variance of.
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Q you know, so far that implies that the variance of Q is as ever defined to be the expectation
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value of Q squared or minus the expectation of Q itself squared vanishes.
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So this variance expresses, of course, the fact that there's no uncertainty in the value that we get.
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But this, since this is a constant of the motion and this is the constant of the motion,
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if this variance vanishes that T equals nought, it vanishes at all times.
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And the only way that that can happen is if your system stays in the state that it was in originally.
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So if it was in a state of well-defined Q at the beginning of a T was nought, it'll be in the state.
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It will define Q at all subsequent times. So that's why that's the meaning of this good.
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That's why people talk about good quantum number.
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If I know at some particular time that the angular momentum of this isolated body is h bar or whatever,
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then I know that at all later times it's also well, it's a quantum number worth knowing because it's always a valid information.
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So. So we're very interested in quantities in observables that compete with the Hamiltonian.
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Let's move over here for the next point. Yeah, obviously.
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Okay. So we're interested in things that commute with the Hamiltonian. And a trivial observation is that h0h common h equals nought.
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This is a bad place, isn't it? Hmm. And I'm not sure that anything we can do about it.
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So the Hamiltonian commutes with itself, which means that the expectation value of H is a constant if the h by d t equals nought.
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So now we need to come back to this point that we I said usually it's going to be the case that the partial derivative of Q with respect to time,
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I've lost it. It's up there somewhere. The partial derivative of Q with respect to time vanishes.
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Now, the Hamiltonian is an interesting case where that isn't necessarily the case.
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There are very important circumstances in which the Hamiltonian does explicitly depend on time.
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The expression for the energy depends on time. For example, if you.
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If you if you put a particle in a time varying magnetic field, the expression for the Hamiltonian,
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which the Hamiltonian is going to depend on, the magnetic field depends on time.
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And in those circumstances, the energy of the particle is not going to be constant.
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And the reason is you're working on that time dependent. The Hamiltonian reflects the work that you're doing on the particle.
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But if you if the Hamiltonian is independent of time, so that will reflect you're not doing any work.
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And then the expectation value of H equals constant is conservation of energy.
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So this condition, it will turn out, is intimately connected to whether or not you're working on the particle.
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Now let's have a look at the the rate of change of the expectation value of any observable when we are in a state,
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when a stop sign happens to be a state of well-defined energy.
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Right. So these states have well-defined energy. We've explained that they're they're the key to solving the the central equation of the theory.
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So let's ask ourselves a little bit about those states. So the so the amplitude.
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Let's, let's have a look at cu e right.
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This is the amplitude to determine CU, you know, given that we're in the state.
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E Well, this is an, this is an eight let's give it this, give it and cu n so this is so this is the amplitude to find the value.
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Q And if you would measure with the observable.
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Q Given that you were in a state with well-defined energy, let's work out the time derivative of this h bar dba d t of this quantity is equal to this.
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This is so it's a very specific d cu in.
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I wonder if we should turn this off where we can turn it down.
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Three was. It looks like he's.
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Okay. So we do the same thing. This has to be a think thing we get from the commissioner, joy to the bottom of the equation.
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And we get the time to penetrating the equation. So we get A minus Q and H, so that is that.
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And then we see arms, which this is going to be the New Amsterdam by.
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Well, that's including the age increases. H e.
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And this is nothing very much. Right.
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Because H works on E to produce E, because that's the name of the h e times the 10th E focus this produces minus E,
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times Q and E and this H produces ne until we get to plus e e to these two terms cancel.
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So we've discovered that the rate of change of the amplitude to have the value q n is constant red change vanishes this amplitude.
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Q and e. For any excuse to ignore the intended result.
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We didn't make any restriction, any restriction whatsoever on what the possible queue was.
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So so the remarkable fact is that in these states,
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a well-defined integer give you a system has well-defined energy, all its property, the expectation value.
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And by thinking about, you know, in fact it then follows with a couple of extra steps.
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The probability distribution of measuring any observable whatsoever is completely constant.
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Nothing ever changes. So that seems to be being called stationary state.
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These states really are forever if they are completely internal, unchanging.
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They are not of this world. And in particular, you can never get the system into consciousness, that you can never get the system into a state of,
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well, because you it's going into there, we claim like something is changing.
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Who can get a check? That's kind of remarkable.
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So now we have a new a new topic, the position representation.
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To bring us much closer to the races of the world was in reaching it.
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So so far we've talked about we use abstract representation through a little bit like the
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energy representation of let's say we've assumed that our observable has discrete spectrum.
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So the spectrum is made up of discrete numbers. Think about the position operator.
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So this is the thing made up. I mean, this is business.
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The operator, which encodes of the status of well-defined position on the x axis of this has it
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has a spectrum which is usually continuous and it runs your that you can physically.
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It's not discrete, it's continuous. And this requires some some adjustment.
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So we. We used to write, we'd have been writing Let's divide board.
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We have in writing, which is equal to some and shall we say in representation.
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Now we're going to write that a side is equally integral as some over discrete set of possible and shrink values numbers,
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and the spectrum becomes an integral over the possible values of the spectrum.
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Search legislation to infinity of some of you have to be an x close.
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The states will find that they're being added.
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So this is the state of the system is in when it is at X when our value or whatever is at X and this is the amplitude to be x, right?
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And if you. B, x and of x.
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We used to have that e m e and you still have.
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And what are we going to have now? Let's roll this thing three times and we're going to have the integral, the x primes.
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Sorry, x x prime x upside of x.
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This side of this side is going to equal one side.
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Sorry, sorry, sorry. What do we want? What we want is that this?
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Well, it's obvious that this thing vanishes. Except when x prime is equal to X.
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Because if it's definitely an x, sorry, it's definitely an x, then it's certainly.
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Isn't it x prime if x prime is different for. So this thing here is nothing except when x equals x, right?
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And it must be it must be non-zero, presumably rather like that.
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But I think we do this right. And this is what is this by definition?
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This is the amplitude to be X prime. And I've already said that is the amplitude to be in its prime, right x.
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So it's clear that this thing here is a slide of x find the amplitude to be an next prime.
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So we have the amplitude of x prime. That function is equal to the depth of explained x.
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And I hope that you I'm sure you've already met this relationship or stuff like this,
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but this could also be written in equations delta primes minus x9x.
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So we have so that result of the two being generalised or is morphed into statement.
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The next prime x is equal to a direct running.
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The delta ls from x 1 to 6 prime.
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We used to have that upside side,
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which was the sum of the and squares was won by conservation of probability that this was
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the sum of the probabilities to get the value and it was that you had to get some value.
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That sum of probabilities have to be one. Now what do you have to turn into this?
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This turns into rain here, turns into upside side.
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Should we one? And how do we how do we express it like this?
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We say that this is the integral of the x of sine x x.
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Sure. Where we we're using the idea. Sorry, but we we used to have the sons D and he and I was the identity operator.
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Now we have to go back to the integral, the x, x, x and the ID brings up every all of these.
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Some are turning into integrals so of this relationship becomes this because because this is the identity operation snuck into there
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and this is what is the amplitude to be an X that we already call that the wave function so that this is the complex conjugate,
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the class level, the complex conjugate of this. So this becomes the integral, the x squared.
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So the integral of what size squared should be one.
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And in this continuous rate, these are just natural transformations, what we've already done in the discrete case to the continuous case.
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But the one more thing that we need to write down, we used to have we used to have the sign.
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This was the sum at the end.
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And by. B and E and then the complex number by side who is the sum, the end, and some of them.
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And as a result, we have had was the analogous thing here.
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You know, this thing is going to be the fi side is going to be integral into here we stick it identity operator the integral
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the x of x price so this becomes the interval the x so you stick an identity operation into that of 5xx sign.
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This is what we have been calling the wave function of CI of x. It is the amplitude b, it is the function.
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It is a function to make public, I think all eight side x and by analogy we should hold we should call x upside x five.
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Sorry, we should call the wave function by x that being.
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So this becomes the complex coming it. This becomes a years.
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So. So then both of these things are possible.
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So this is this is this is precisely a transformation of that with some wind and X and the sun up and becomes a digital print.
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This is the stuff you have to do with the spectrum of x is continuous, not discrete.
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Let's just do a little practice with this by asking ourselves how does the Operation X work on an arbitrary state side in this representation?
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So what we want to know is the thing to do is to ask ourselves what wavefunction represents.
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That is to say, what is this complex number as a function of X?
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So here's the operator. X, there is an arbitrary value of x.
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I would like to know what the amplitude v that is to this stage, but you get when you operate x world function and when you see an operator x,
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the obvious thing to do is dig into here and identify the operator made up of the eigen functions of x.
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So in order to understand what this is, what we do is we slide into one of the identity operators.
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X is busy. Some identity operator is going to have to give some of x primes a new value, some independent value of x x x x primes, x primes sun.
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So here is the identity operator along with that.
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Now, life is relatively straightforward because this is an eigen function of that operator with this eigenvalue.
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That's the definition, which is that. Yeah. So an x means this, it produces simply x time its prime the number of times the can x probed.
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So this becomes the integral that the x prime or x prime.
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There's the eigenvalue popped out when we have our x next primes.
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This we recognise to be the wave function of sine evaluation of primes.
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But this we recognise now we have seen that this is the direct build function of x minus x, right?
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So when we do the integration over x prime,
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this shows we get no contribution except for that little second when x from is equal to x oh well and we get the values,
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the integral and evaluated it is exercise turned into x, so this is equal to x5x.
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So at the end of a long story, what have we discovered? We've just got the wave function associated with with the result using the operator
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x on some stage it could be x times the wave function of the original state.
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We can express that. But the way to remember that is to say that the operator X or Y functions like all the multiplication.
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So you don't usually go to this kind of a performance.
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You know what's going to happen when you do it, but that's the logical basis for this statement.
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Let's introduce another very important operator, the mental breakdown.
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Now I'm going to make an understandable claim about what this operator, how this operator looks at the position representation.
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I don't expect you to think. A-ha, that makes sense. It doesn't make sense.
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It's a complete leap in the dark. We will understand later, considerably later, why these operations take the form that they do.
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But I hope soon to build some kind of sense of what are we going to do just right now is completely going to go up.
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I'm going to say, well, let's investigate the operators. I I'd like to know from the investigative which is defined find us and have.
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Now let's just make sure we understand positively what's happening here and
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operate fundamentally is something which turns the state into another state.
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When we're in the position representation we are. When we're in the position of representation, we are.
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We're working with functions our way out of space represented by their weight functions, which is now what you see.
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So the next operator, the ex operator has to turn away function or some other function.
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And look, exercise is another function. It depends on x is a different way from website us.
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So similarly, this momentum of this operator claims the momentum operator without any basis is going to try again.
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This momentum operator. Is.
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Is turning the wave function of PSI into its derivative and derivative.
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Is a function different from the function we first thought of? So indeed it's turning a function into a function.
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So that's kind of it means it is at least a band operating. Is it a mission?
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It's not obvious that is the mission. And if it isn't the mission, it certainly can't be the momentum operator.
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So let's let's check that out if we. So let's write down the complex number five.
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He had a sign. Let's let's evaluate this using this hocus pocus here.
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Right. So this is the integral de x. What am I going to do?
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I'm going to put an identity operator just in here.
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Made up of X is right. Why? Because I know I define P in terms of what is what happens when it has an X on the left of it.
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So this is going to be fine. X rays, p hat, everybody.
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And now we can turn this into wave function language.
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This is the complex conjugate of the wave function by.
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So this is the integral the x phi, the star of x.
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And we define what that is. It's minus h bar D by the x sign.
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So we can now integrate by part two, integrating minds and pages infinity.
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So we can integrate by pass this this partial derivative to get this part moderated by and onto the PHI.
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So what does that give me? We get we get a square bracket term.
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We have a five star registered minus.
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Let's put the minus H bar outside some fast bracket.
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All right. So we can have a square bracket term. Now we're going to have an ABC star sorry, a five star.
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ABC minus infinity. Infinity. And then we're going to have minus.
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The integral. The X, the ci.
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Sorry. A sci fi star by ex close the big bracket.
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So we're going to operate under the assumption that this thing vanishes. Now, this is a rather hairy.
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Don't don't press too hard as to whether this really does vanish.
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But the general idea is that the amplitude to find your particle of the edge of the universe is zero.
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So we dispose of this on the grounds that it is the amplitude to find the particle infinitely far away.
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We're going to say that that zero we will actually be working.
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You'll see quite soon with some way functions where that doesn't vanish.
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And this is an example where physicists are rather fast and loose. And but fortunately, this doesn't lead to any bad effects.
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So this we put in the pin and this we can see is more or less what we want.
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Let's just tidy up a bit. So what is this that survives, including this minus H bar?
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So we're going to have a minus. Well, let's leave the minus out.
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We. Yep. D x of a CI.
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And then we got to have an h bar defined by the x.
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Supposing I take the star of all that, then I think I need a minus sign because this coming, this minus will cancel on this minus.
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So we'll have an h bar times this stuff with the CI start.
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If I take that of CI that star and put it around the whole caboodle, including the I,
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I'll need an additional minus sign to cancel the minus sign that will arise when that star is evaluated on here.
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And I can say now that that is the integral de x upside star minus h bar defined by the x.
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Stahl So if I take the store, this store completely outside the whole thing,
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then I will need a store which will be cancelled by the global store and otherwise everything be okay.
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And what is this? This that we have in here, in in direct notation is absurd.
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I. P hat. Fi.
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And that still sits outside, right? What's inside the store is by definition this.
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So that we are the answer is mission. Provided we get rid of that surface term that that minus infinite the square bracket.
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One. Okay, let's calculate the commentator x hat, comma p hat.
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We're going to calculate it like this. So what we're going to do is calculate the action.
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So what we know at the moment is the action of p hat on any wave function and we know the action of X on any wave function.
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So I want to work with wave functions, which means I, I put a bra at this end here, a bra x at this end here.
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And what does this give me? So this is going to be obviously x x p ABC minus.
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X. P hat x hat sign.
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No prizes for that. Now this we've discovered X on this.
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We discovered that X on any wave function,
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on any object gives you x times the wave function you first thought of, the wave function you were operating on.
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What is the wave function that this produces? The wave function that this produces is minus bar deep upside by the X.
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So we all we really have to take that and multiply it by X and then that's what you get there, right?
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This is a complex number depending on X, if it's a complex number depending on X.
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So P on this is is a certain wave function.
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It's this and then x hat on that produces x times that wave function.
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So that's it.
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So here same stuff x on this is going to produce x abassi x of CI and then p on that is going to produce minus h bar, cbd, x of that stuff.
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And there's a minus sign floating here. I mean, this minus sign is this, that minus belongs to the, to the p operator.
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So I think you can see that the X dips liberty x terms.
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When you differentiate on this product, you'll get two terms.
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You'll get x deep side of the x, which will cancel on this because of the two minus signs.
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And you will also get an upside to the derivative of X with respect to x, which is obviously one.
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So we're going to get H bar of CI of x, which can also be written by H for x oops vici.
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So what have we learned? What we've learned is that for any state of SCI whatsoever, we never said what it was.
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The wave function associated with x CI is simply by the times the wave function of CI.
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So that means that we can now write down an operator statement that x hat comma p hat is equal to h bar.
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So the computation of these two operators is a constant, small,
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constant but constant and a canonical and a commutation relation of this sort is called a canonical.
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Commutation relation.
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We will meet other relations of this type with with a commentator of two operators is equal to each bar and they will be declared canonical as well.
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The word this canonical of course comes from classical mechanics, Hamiltonian mechanics.
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And this arises because in classical mechanics, momentum is canonically conjugated quote unquote to two x.
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Right. So now now that we've done that, let's.
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Yeah. We've just got time, I think to do this. Let's apply Ehrenfest nice theorem to.
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So I'll begin. Let's work out this bar.
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Divide e t of the expectation value of x.
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What do you think this should be? The rate of change of the expectation value of X should be the speed.
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Right. Does so that the rate of change of the expectation value of X should be the should be the speed velocity, whatever.
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So we're hoping that this turns out to be i b which should be i.
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P upon m if. If. If we're doing this right, according to Aaron.
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First, what's this equal to? It's equal to ABC X comma, Hamiltonian upside.
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Right. That's Aaron Fest Theorem. Concrete example of application.
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So in order to go further, we need to say, So what's H? H is the energy operator.
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What do we know about the energy of of some.
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Of a particle that's moving in May, possibly with some potential present.
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Right. So the energy should be a half classically.
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Claire Skelly. If we're doing this classically I should replace this with an energy.
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Should be a half and v squared plus the potential energy depending on x,
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which could also be written as the momentum squared over to m plus the potential energy.
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Right. Because P classically is an.
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So let's let's just suppose that we can carry this forward into the quantum domain
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and say that the Hamiltonian operator is the momentum operator over to M plus v.
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The function V evaluated on the position operator.
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Then we're going to have that HBO DVD.
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Of the expectation value of X that is going to be.
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The expectation value of x computed with p squared over two m plus v.
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But we know that. So this but this committee is it can be broken down into a sum of two computations, the computation,
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the combination of X with P and the computational of X with V, but v is a function of x and therefore x.
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The position operator is going to compute with this. So we're going to have that x comma v equals nought because v is a function of x.
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So what we're left with is. What we're left with is the expectation value of p.
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P. Sorry. Expectation.
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Value of the commentator. Of X with.
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P. P. Sigh over to em.
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Okay, I can take the two m out of the commentator because it's just a number and I can express p squared is p.
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But we discussed we discussed probably yesterday what how we took the commentator of a product.
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We used the rule analogous to differentiation of a product.
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So this is equal to a sigh onto x comma p.
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Commentator P. Standing idly by plus p.
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Standing on the first piece. Standing idly by while excuse with a second p.
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But we've discovered that this animal is is a half. Sorry. Is is bar right?
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This is a bar and this is a bar. I was just a boring number, so it can come out front.
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So that becomes a bar over to M, and then we have P plus P, which is two P, so I can rub out that two times.
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So what have we discovered? We've discovered that we can cancel this on the right side with what we had
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on the left side and say the DVD t of the expectation value of the position.
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Is in fact equals the expectation value of the momentum.
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What I claim is the momentum anyway over time, which is exactly what we were hoping for.
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Right. So we've recovered the definition well.
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The relationship between velocity and momentum, which in classical in Hamiltonian mechanics is a rather is rather.
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Those of you who've done a seven will realise that the the connection between
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momentum and velocity is not as simple as elementary Hamiltonian mechanics.
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Right? Elementary.
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Newtonian mechanics would lead you to believe it can be quite subtle and it's determined by this, which is one of this is one of Hamilton's equations.
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What we've done is derived one of Hamilton's equations which supersede Newton's laws of motion.
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In classical physics. So we derive from quantum mechanics a classical result which was already known.
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But this is the justification for Hamilton's equations, because this is true that Hamilton's equation is true.
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And we'll leave it on that. And tomorrow morning, I'll start by driving the other of Hamilton's equation, which is analogous to F equals Emma.