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