 Yes, where we arrived with the harmonic oscillator yesterday was we had established that the expectation value of x squared in the nth excited state was 2n plus 1l squared. Yes, of course, obviously on dimensional grounds. Yes, that's correct. So, and I said the next item on the agenda would be to connect that back to classical physics. Always a valuable exercise because it tells you something about quantum mechanics. It checks your results. So, of course, classical physics doesn't know anything about 2n plus 1 in the quantum number, the excitation number. But it does know about the energy and we know that n plus a half h bar omega is the energy. So, we can write this as 2 times the energy over h bar omega. And this l squared, well, l was defined to be h bar, I think, over 2m omega. I probably had better check that my memory is correct, yes. So, this l squared is an h bar over 2m omega. So, therefore, this is the energy, various things cancel over omega squared. The two's cancel. Oops, we need an m. That survives in the algebra, is that correct? So, let's ask ourselves what do we expect, what do we expect classically? Classically, we have that, what do we expect? We expect that the time average, well, I better write this down now, it's right. So, the time average of x squared, which we'll call x squared bar. Since x is a simple harmonic function of time, the time average should be a half, sorry, this thing should be a half of x max squared, right? The average of cos squared is a half, so if we are writing that x is equal to x cos omega t, it follows that x squared bar is a half of x squared, the maximum perturbation. And what's the energy? The energy classically is a half k x squared, because when it's at maximum extension it has no kinetic energy, it only has potential energies, that's how much it has. Omega squared is root k over m, so this is a half, so k omega is the square root of k over m for a harmonic oscillator, that being the spring constant. So, if I want to get root of k I have to declare it to be omega squared m squared, omega squared x squared, right? So, that leads to the conclusion that I'm expecting that x squared is equal to 2e over m squared omega squared, which is, but this thing is equal to 2 of x squared bar, so this leads to the conclusion that x squared bar is equal to e over m squared omega squared, in perfect agreement with the quantum mechanical result, oh dear, we need to put this up, don't we? I've gone on a drift by an Emsgood, let me get rid of that stupid screen, I can only do one thing at a time, too stupid, lights and blinds, screens up, right screen, left screen, left screen, no, who are we talking about, you or me? Okay, right, so there was a complaint, what went wrong, was the squares of m, where did I goof on that, that was because e was a half, no that was correct, yes because it was inside the square rooty sign, yes exactly, so this shouldn't have been there, so this shouldn't have been there and then everybody's happy, thank you. Okay, so doing this check, right, of what have we done, we've checked that a quantum mechanical result agrees with the classical result, now actually, amazingly, we've been able to do this independent of n, right, in other words our classical physics or our quantum mechanics has recovered classical physics for all n, but we believe that we have to recover from quantum mechanics classical physics only in the limit of large n because our classical experiments are all ones where we're moving macroscopic bodies around where the excitation energy will be large in some natural units, so the exercise that QM goes to classical physics for large n, large quantum number, here's our first example of a quantum number, a relevant quantum number, this is the correspondence principle, correspondence, and this is an early example, in some senses perhaps not a brilliant example because we get perfect agreement for all n, right, but all we're requiring is agreement for large n, but we really must have agreement for large n because classical physics is about, you know, it's been validated by experiments conducted at large n. So let's talk about now the dynamics of oscillators. So far we have found these stationary states and I've said several times these stationary states are highly artificial. One way you can see their artificiality is that nT, the state with energy n plus a half h bar omega at time T is equal to that state at time T equals 0 times e to the minus i. Now this is e over h bar T, but since e is n plus a half h bar omega, this is n plus a half omega T. So each and every one of these states has a phase which increments in time at a frequency n plus a half omega T, but the oscillator oscillates at a frequency omega, right? So we have to explain how it is that the oscillator oscillates at a frequency omega, but none of the states evolves in time. None of these stationary states evolves in time with a frequency omega, not one. Moreover, the oscillators that we're familiar with in the school laboratory, masses on weights and stuff, will have values of n which are like 10 to the 28 or 34 or that kind of, simply ginormous values of n, so the frequency here will be stupendous and nothing in the laboratory is happening at that frequency. So this is total fantasy land. We have to get back to reality. We get back to reality by concentrating on expectation values because it's our connection to classical physics, which is what we call, it's what we are pleased to call reality. So let's calculate the expectation value of x. If we do it for n, we know we're going to get a constant, right? Because when we take the complex conjugate of this complex number, it will multiply together with a complex number over here and make one, but we already know this. We already know that a stationary state has no time evolution whatsoever. So to get time evolution, what we need to do is say the state of our system, we have to consider, to have something that moves, we have to consider a system which does not have well-defined energy, which means that its wave function, its state vector wave function, whatever, is a linear combination of states of well-defined energy. And let's suppose, let's take a simple example, let's suppose there are just two states present. No, sorry, the proposal is that we take, so let's do a sum. Let's do it in all generality, so we're going to write this as a n e to the minus i. So this is totally safe. Any state can be written like this. So this is psi, the state of my system at time t. It's a linear combination of states of well-defined energy. There's no question that I can do that. It's a general initial condition. And now let's work out the expectation value of x. So what is it? The sum a n star e to the i n plus a half omega t times n times x times m times a m not star e to the minus i m plus a half omega t. So we can clean this stuff up to, this is a sum of n and m, of course. It's going to be the sum n m a n star a m e to the, when we put these two together, we're going to have an e to the i n minus m omega t times n x m. And yesterday we already saw what the stylish way is to handle this expectation value here. It's to take advantage of an expression that we showed that the operator x can be written as l times a plus a dagger, where l is the thing we were discussing earlier on. It's a square root of h bar over 2 m omega, characteristic length. And we also saw what happened when we took an expectation value where we were doing a slightly harder problem yesterday. So this is going to be very straightforward now because it's going to be l n a m plus n a dagger. And this, remember, a on m produces m plus 1 in an amount, the square root of m plus 1. So this is going to be l root m plus 1 of n m plus the square root. Excuse me, this a produces m minus 1. Whoops, m minus 1 because I'm not concentrating at all. But it's the square root of the biggest number that occurs. So this is a square root of m. Sorry, a on m produces m minus 1. How much the square root of the biggest integer that's involved, that's the square root of m. And this is going to produce m plus 1. And with normalisation, which is the square root of the biggest number involved, which is m plus 1. So n m plus 1. What we want is we have here a sum of n and m. Let's do the sum of m first, right? Bearing in mind that that thing is the sum of a delta n m minus 1 and a delta n m plus 1. So we're going to have, for our oscillator, the expectation value of x is going to be, okay, let's take this first one. When we do this first one, sum over n, we're going to have that this is a n star, a how much. In order for this to be not 0, m has to be 1 bigger than n. So this is going to be the square root of n plus 1. And everywhere where I see an m, I'm going to have to write an n plus 1, e to the i. And now, in this case, we've agreed that m is 1 bigger than n. So this is going to be e to the minus i omega t. So that's what we're going to get from this term. And then, sorry, from this term when that goes in there. And now we have to put this in there. And now, m is going to be 1 to get a non-zero contribution. m plus 1 is going to have to be n. So m is going to be n minus 1. So we're going to get an a n star a n minus 1 times the square root of n, e. And now, in this case, m is going to be smaller than n. So this is going to be e to the plus i omega t. And I've lost my l somewhere along the line. Let me reinstate it. There's this l here. I hope we have everything. Slightly scared that we haven't. Let me just check that. No, it seems to be OK. And we're still summing. I've lost a sum sign. We are still summing over n. Why don't we declare that in this? These are two separate sums. And in this sum, I can introduce a new notation. I can say that n prime is equal to, sorry, n is equal to n prime minus 1 in this sum here. And then sum of n primed. And then I can relabel the n primed n. And this term becomes the same as that term, becomes the complex conjugate of that term. So when I do this, I'm going to have a sum now of n primed, l a primed, no, not a primed, a n primed minus 1, a n that's star, that's star a n primed times the square root of n primed e to the minus i omega t. And the other sum is still over n, and that's a star n a n minus 1 e to the i omega t. But n and n primed are the same other, I mean, they're just dummy indices, we're just summing over them. So this sum is in fact the complex conjugate of that sum. These two things are complex conjugates. So what we, if we write a n a n minus 1 is equal to say x n e to the i phi, which we can do where this is real, and this of course is real too. So I'm writing a complex number, sorry that needs a star on it. Then I'm going to be taking x e to the minus i omega t minus i phi plus this stuff e to the omega t plus i phi. And we're going to be able to combine the two exponentials and discover that x is equal to L times the sum of x n cos omega t plus phi. Sorry, we need a phi n on that. Excuse me, we need a phi n on that. So this obviously needs an n and that needs an n. Because each of these complex numbers has its own face. So what have we discovered? We discovered that lo and behold, the position, the expectation value of the position, does oscillate with periodically. This is now, this, what we have indeed, sinusoidal oscillation at period 2 pi of omega n. We have in fact recovered classical physics, the classical motion. So the motion at this frequency occurs because of interference, between states, so these terms we've got are quantum interference between states of different energy. Why do we have states of different energy involved? It was because x, it was because, so when we took the expectation value of x, we got this huge long sum which involved the cross terms between states of, it included the term n x n. That was also involved in here in which the same, the state of a given energy was present on both sides of x. But that made no contribution to the sum because x is a sum, can be written as a sum of these ladder operators, of these annihilation and creation operators, and if you put the same state on either side, you get nothing. You only get something if the states on either side differ in energy by one unit of excitation. So our result all arose from interference between states which differ in energy by one excitation. So that's a very general phenomenon, and a peculiar feature of this problem is that those differences in energy are all the same. They're all H bar omega. And the frequency, well, we'll see this in a moment, the frequency of these oscillations. So all these terms have the same sinusoidal, we have an infinite number of contributions still, but they all have the same sinusoidal behaviour. So we've recovered the important feature of a harmonic oscillator that the period is independent of the amplitude of the excitation. The amplitude of the excitation is controlled by which of these ANs are significantly large, right? Because AN is the amplitude to have energy, N plus a half H bar omega. So a highly excited oscillator has the non-zero values of AN are all clustered around a large value of N and a very, only gently excited oscillator has the ANs around zero or small values of N being fairly large, and therefore this sum will be, and this sum will be dominated by whatever region has the large values of A. But the result we've got is that there's harmonic motion at frequency two, at frequency omega, regardless of which terms in this sum are dominating. And that's this property that the period does not depend on the amplitude. So let's be more realistic and investigate, see how much more of classical physics we can get out of this by talking about an an harmonic oscillator. So I introduced harmonic oscillators by saying that they're widespread because if you have a point of equilibrium, if you plot against displacement from point of equilibrium you plot the force, you have some curve that looks like this and should pass through zero and should pass through zero at the point of equilibrium by definition of a point of equilibrium, but that if you displace yourself from either side of the point of equilibrium, if it's stable, the force slopes like this, it's positive. Sorry, it really should be negative shouldn't it, actually, and I come to think of it. Sorry, I should draw the graph this way around, shouldn't I, in order to get a stable force. If I displace myself positively in X, the force becomes negative and pushes me back. If I displace myself negatively, the force becomes positively and pushes me back. So that's a stable equilibrium. And if an harmonic oscillator arises, if we replace, if we approximate the curve of this force versus distance by the straight line that's tangent to it at that point. So basically what we're doing, so any force versus distance curve could be expanded as some kind of a Taylor series. And if we just take the first non-trivial term in that Taylor series, we have a harmonic oscillator, if we take subsequent terms, we will have a not harmonic oscillator and a harmonic oscillator. And typically the force versus, so for a harmonic oscillator, the force versus distance is a straight line that goes all the way to infinity, which means that in order to pull your spring apart, you have to do infinite work. Because to get X to go to infinity, you have to overcome a force that goes to infinity. So infinite work is required to pull this thing apart, but all real oscillators, all macroscopic ones, certainly you can just break them. So only a finite amount of energy is required to push X off to infinity, and that's reflected in the fact that typically the force versus distance curve slopes over like this, so that if we plot the potential, V versus X, in the harmonic case we have a parabola that looks like this and disappears off to infinity, so this is the harmonic oscillator, but in a real oscillator, the force, the potential curve starts from some finite value at infinity. Sorry, and I need to draw it so it becomes tangent to this and then disappears off like this. So this is a more realistic curve, and the harmonic oscillator is a good model if the parabola is tangent to the realistic curve over a decent range. That's the main idea. So what we should do is investigate, to see what quantum mechanics has to say about more realistic oscillators, let us take, so this is just an example, supposing we take V of X is minus some constant A squared plus X squared, and I suppose we need an A squared on top to get the dimension straight. So supposing we take that to be our potential curve, then we can no longer, we can no longer, we now sit down, have a perfectly well-defined Hamiltonian, P squared over 2M plus this V of X, but we can no longer solve this analytically any more than we can actually analytically integrate the equations of motion classically in this potential. So in either case, you can't do it. But it's pretty straightforward to solve this problem, HE equals EE numerically. We do it in the position representation, we bra-through by X and have that X, P squared over 2M EE plus X, V EE is equal to EE XE, which turns into, by the rules we've already discussed, this turns into an ordinary differential equation, minus H bar squared over 2M D2U by the X squared plus V of X, U is equal to EU, where U of course is equal to X, E. So this is an ordinary differential equation, second order, et cetera, and it's linear, and it's pretty straightforward to solve numerically. If you look in the book, there's a footnote that explains how to do that. I'm meant to bring my laptop with official figures, but when you do this, so by discretising this differential equation, we turn it into an exercise in linear algebra, which your computer solves, so you write this basically as a matrix, M on U, which becomes a column vector, the value that U takes at the different positions in X is equal to EU, so you turn it into a matrix equation, and computers are very good at matrix equations, so when you do that, you discover what the values of E are, and you can also discover what these wave functions look like, and the crucial thing is that you find that if you plot the possible energies, you get a distribution that looks like this. It starts off looking like an equally-splaced ladder. For the harmonic oscillator, there are steps here, each one of which is separated by H bar omega. We start off like that with the spacing given by the harmonic oscillator that's tangent to the bottom of the curve, but as we go up, the spacing gets less and less and less and less and less and what essentially the algebra is doing is giving you an infinite number of allowed energies, or already in a finite range, because this is V naught. That potential allows that potential, as X goes to infinity, goes to a finite value, well, it goes to zero, sorry, this is zero, and I guess this is minus V naught, so the lowest energy is somewhere down here. With only a finite range in energy, you pack in an infinite number of allowed energies. With a harmonic oscillator, you pack in an infinite number of these things, but in an infinite energy range, because this ladder goes on forever right up to the heavens. This is a very generic behaviour that we will encounter again in real systems. Now, what's the physical consequence of that? Suppose we have, so now let's say our initial condition is this, that it consists simply of two terms, A n of n plus A n plus one of n plus one, and so the time evolution is going to be, and this one's going in A, give me some space A n plus one, E to the minus I E n plus one, T on H bar n plus one. So that's not a completely general initial condition now, because I'm assuming that there are only two non-vanishing amplitudes, so my state, it happens to be such that there are only two possible values of the energy that I can measure, there's an amplitude A n to measure the energy E n, and there's an amplitude A n plus one to measure the next highest energy. So this is kind of a special case. If we now work out what the expectation value of x is for this special case, we find that it is A n star E to the I E n T H bar n, this is all very similar to the other case, A n plus one star E I E n plus one, T on H bar x, and then the same stuff on that, and then A n E to the minus I E n T on H bar. Now when we multiply this stuff out, we will get, we will quite generally get only two terms. We'll have this on this and this on this. The reason for that is that we will show later on that for that potential, x n, so for the harmonic oscillator this is true, but it's not only true for a harmonic oscillator that this thing vanishes here. It's going to be true for any potential well which is symmetrical around x equals naught. So this follows from symmetry of v of x, that v of x is an even function of x. So long as v of x, the potential is an even function, the same behaviour at minus x is plus x, this will vanish. We will show this as we go along. I haven't shown it yet, but that will be true. Given that that's so, quite generally, my expectation value here is going to be A n star, so it's going to be this on this, E to the I E n minus E n plus 1 T on H bar times the matrix element n x n plus 1 plus, excuse me, and I'm needing here some A n star A n plus 1, right? So that's that on that, and then we will have this on this, A n A n plus 1 star E to the minus I E n minus E n plus 1 T on H bar times n plus 1 n x n. Hope I've done that right. So what do we have now? We have again that this term is the complex, well, we have this term is the complex conjugate of this term. So we're looking at a sinusoidal function plus its complex conjugate, therefore we're looking at something which is x n, could be written as x n cos E n cos E n minus E n plus 1 T on H bar plus a possible phase factor, right? Where just to be concrete, x n is the modulus of A n A n plus 1. So A n A plus 1 is a complex number. I have its modulus sticking out here and I stuff its phase into there. So what do we observe? Again we have harmonic motion, sinusoidal motion, but look now at the period of this sinusoidal motion. The frequency of this sinusoidal motion now depends on n because again it's the difference of two energies, of adjacent energies which count and as we increase n, according to my bad sketch up there, the difference between adjacent energies gets smaller. So the frequency becomes smaller with increasing n. So we're recovering a classical fact which is that if you have an anharmonic oscillator of a typical type and you make bigger, you kick it to a bigger oscillations, its period will slow down and that's true of an ordinary pendulum, has its highest frequency if you have it go to and fro with a small amplitude that clockmakers make their pendulums go to, but the period goes to, as you increase the amplitude of the oscillations, so this is as it were, high omega for a pendulum, as you boost the period to the point at which it's about to go over top dead center, if you'll make it swing so it goes like this and then like this, the period goes to infinity formally, well it does go to infinity, it's hard to do experimentally as you increase the amplitude at the point at which it would just keep on going. It had enough energy to go right through top dead center. So this slowing of the period with increasing amplitude is manifested in a standard pendulum and we see how it emerges from the structure. So here we're learning something important, in which the dynamics is encoded in the spacing of the energy levels, of the energies of the stationary states. These are just simple examples of what's totally generic. Okay, something else that we can learn about this anharmonic oscillator is that more generally, this was a simple example. I said let's consider, in order to get something to move, a state that had undefined energy, to keep it simple, I took just two non-vanishing amplitudes. There were only two non-vanishing amplitudes in the expansion of the stationary states. Realistically we would have, if you take an ordinary pendulum like that and you give it a jog, the energy will be uncertain by zillions of values of h bar omega and many, many, many, many, many of these coefficients will be non-vanishing. So more generally, we're going to have that if psi is equal to, it will have many terms, and let's just write down a few of these terms, a n minus one e to the minus i en minus one t on h bar plus a n e to the i en etc. So there will be many, many of these coefficients, but if we know pretty much what the energy of the oscillator is, we've lifted our bob up to 30 degrees or something and let it go, the energy is not completely undetermined, and what that means is that many of these will be non-zero, but they will all be clustered around some particular value of n. So that if you look at the value of one of these amplitudes, the modulus of it, as a function of n, you'll find that you'll get a pattern sort of like this somehow. There'll be an n at which the amplitudes peak and there'll be small values here because we're pretty certain the energy isn't that small and small values here because we're pretty certain the energy isn't that large. So that's the generic situation. And when we come and calculate the expectation value of x, what we now have is the same sort of thing up as up there, but it's somewhat more complicated. We're going to have an a n star a n minus one sort of the things that we had before, e to the i e n minus e n minus one t on h bar times some matrix element n x n minus one. And then we will have, actually I want to do this the other way around, and then I need to put in a minus sign there. Then I will have the next one because n x n will vanish, the next one I will, by the symmetry property, I will have a n plus one star a n e to the minus i e n minus e n plus one t on h bar times n x n plus one. And then I will have, not plus, it's not equals but plus a n plus three star a n, this will be the next term e to the i e n minus e n plus three t over h bar n x n plus three plus dot dot dot dot. This is a specimen of a disgusting expression which would give us the expectation value of x. This combination of terms we've already seen, this is nothing really new. The harmonic oscillator had just this kind of thing. In the case of a harmonic oscillator, this energy difference was exactly minus this energy difference, making this exponential, this exponential complex conjugates. They will not now be exactly, this will not be exactly this because this is the difference between n and n minus one and this is the one step in the ladder and this is the size of the step one above it in the ladder which will be slightly smaller. Thank you. I changed my mind how I was going to do this. Thank you very much. This should be n plus one and that should be n and this should be n plus three. It's good to know that there's understanding in the room. That's one thing that's going to happen because this is a more realistic oscillator is these frequencies will be changing and crucially this frequency here will be present. Where it wasn't present in the harmonic oscillator case, in the harmonic oscillator case, this number here vanished. In principle this would have been here but this matrix element vanishes for the harmonic oscillator. But it's not generally going to vanish. It's going to stick around. Now that has an important consequence because this frequency, so en plus three minus en, is going to be on the order of three times en minus en minus one. Because it's the difference, it's three steps on the ladder and that's only one step on the ladder. If we think of the size of the steps in the ladder becoming small only gradually as we go up the ladder, which is a good picture to use, then this term is going to be essentially three times the frequency associated with the other two which we can regard as about the same. So when we assemble all this stuff we're going to find that the expectation value of x looks like some number times cos of en minus en plus one. Let's declare this to be three omega n. We're defining omega n to be this quantity here and we're going to find that we have a cos, some term like cos omega nT and we're going to have some other term with some other coefficient times cos three omega nT and we're going to have some other term with some other coefficient times cos five omega nT. This number will contain products of stuff like an, an plus three star and an plus three, an plus five star, et cetera. This will contain things like an, an plus five. We will have these other frequencies present and this is what leads to, so this, this series implies periodic motion but anharmonic motion. So in a musical instrument you, the, the, the note you, the motion of the, of the string in a piano or the vibrations in an organ tube or a flute tube or whatever has a, it's, it has a, it has a well-defined frequency which sets its pitch but the, but the particular tone of the instrument is determined by the characteristic numbers of, of higher harmonics which are present because it's an anharmonic oscillation typically. But there's, there's more that we can do here which is connecting to classical physics which is to make the points that, if we arrange this stuff, right, so this expectation value of x, we take out the leading term, we say that this is e to the minus i omega n t and then we're going to have some sum, no, let's, let's, let's, sorry, it's better, it gets very complicated if you think about the expectation value of x. It's, it's easier if we think about upside itself as a function of time. We can take an e to the minus i e n t on h bar out and we can say that this is dot dot dot a n n plus e to the i e n plus one e n plus one t on h bar, n plus one times a n, sorry, sorry, this needs to be, yep, let's need a minus sign there, I need a t on h bar there. So I've taken a common factor out, so this one doesn't have any exponential. This one should have its proper exponential minus the thing that I've taken out. The next one should have, and this should be a n plus one, this should be plus a n plus two e to the minus i e n plus two minus e n t over h bar n plus two and so on and so forth. So to a lowest order approximation, these differences here are all multiples of a common frequency and after a period, so when e n t over h bar is equal to two pi, these things, sorry, n plus one minus e n, after the time it takes for this to come around to two pi, this will come around to four pi almost and so on and so forth and so the wave function will look, all this sum will be the same as it was at t equals naught because all of these exponentials would have come around to one again. That's in the case that these things are all multiples of the same frequency but as we've seen, they're not quite multiples of the same frequency. This is slightly smaller than twice this and so in the time it takes for this one to come around to two pi, this one isn't quite round to two pi and even more so further down the line and therefore the wave function isn't quite back to where it was when at t equals naught and as time goes on and we count more periods so this becomes two pi n, these discrepancies become more and more and more important and these terms down here, when this one has come around to two n pi, this one will be significantly off two n pi and this one even more so and that means that the wave function is not returning to its original value and we're looking at motion which is not periodic and whereas initially, because we'd released our particle from some particular point in the potential well, these wave functions all constructively interfered here at a particular value of x, after a certain number of basic periods the interference, the constructive interference here and the destructive interference everywhere else will become less and less exact and the distribution of our particle will become more and more vague until after a very long period the phases of these will be essentially random and will have no knowledge of where it is and this is precisely mirrored in classical physics in classical physics, the small uncertainty in energy that was associated with having more than one a n in this series was associated with a small uncertainty in period if the energy was very high the period would be very long and after a long time the particle would have gone around and around a million times a million and a bit times and here it would be but if it was slightly different a slightly lower energy it would have a slightly higher frequency and it would have done a million and one oscillations and it would be over here so that you can see that this small uncertainty in energy is going to lead after a long time through the small uncertainty in period to a large uncertainty in phase and a total scrambling of our prediction of where it is so, again, quantum mechanics is returning in a rather complicated way and through quantum interference a result that we're very familiar with if we think about the classical situation it's time to stop