Yes. Well, we had where we had arrived with the harmonic oscillator yesterday. Woods. We had established that the expectation value of x squared in the excited state was two and plus one l 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, quantum classical physics doesn't know anything about two N plus one and the quantum number, the excitation number. But it does know about the energy and we know that n plus a half bar Omega is the energy. So we can write this is two times the energy of a h bar omega and this l squared. Well, L was defined to be h bar. I think of a2m omega. I probably had better check that my memory as that is correct. Yes. So this L squared is an H bar over two omega.
So therefore this is the energy various things cancel of omega squared. The twos cancel ups. 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 to write this down now. 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 same average should be a half.
Sure. This. This, this thing should be a half of x max squared. Right. The average of cost squared is a half. So if we are writing that x is equal to x costs 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 what is that? Maximum extension has no kinetic energy. It only has potential energy. That's how much it has. Omega squared is root omega is root K over m.
So this is a half. So so k it's some. Omega is the square root of k of rhythm for a harmonic oscillator, that being the spring constant. So if I want to get rid 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 two E over m squared omega squared, which is, but this thing is equal to two of x squared bar.
So this leads to the conclusion that x squared bar is equal to E of m squared. Omega squared in perfect agreement with the quantum mechanical result. Oh, do we need to put this up, don't we? And I've got a drift by name, so let me get rid of that stupid screen. I can do one thing at a time. Two stupid lights and blinds. Uh, screens up right screen, left screen. Left screen. No. Which. So who. Who are we talking about? You or me? Okay. Right. So there was a complaint. What went wrong?
What went wrong was the squares of m. Where did I. Where did I goof on that? That was because E was a half. Uh, no, no, that was correct. Same number of square. Yes. Because it was inside the square. Root, root sign. Yes, exactly. So this shouldn't have been there. So this shouldn't shouldn't have been there. And then everybody's happy. Thank you. Okay, so doing this check, right?
Of what? What have we done? We have. We've we've checked the classical sorry, the quantum mechanical result agrees with the classical result. Now, actually, amazingly, we've been able to do this independent of end, right? In other words, classical physics or quantum mechanics has recovered classical physics for all end. But we we we believe that we have to recover from quantum mechanics, classical physics, only in the limits of logic.
And because our classical experiments are all ones where we're moving macroscopic bodies around, where the expectation energy will be large in some natural units. So we the exercise that of QM goes to classical physics. For a large and large quantum number. Here's our first example of a quantum number, a relevant quantum number. This is the correspondence. Principle of correspondence.
But. And this is a an early example in some senses, perhaps not a brilliant example because we get perfect agreement for all end right. But what we're always requiring is agreement for login. But we really must have agreement for login because classical physics is is, is about, you know, it's been validated by experiments conducted at login. So in the same spirit. So let's talk about now the dynamics of oscillators.
So what? 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 that anti this the the state with energy and plus a half h bar omega at the time t is equal to that state at time t equals zero times e to the minus o.
Now this is e overreach bar party. 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 and plus a half, eight and 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 has a evolves in time.
None of these stationary states evolves in time with the frequency omega not one so that the amp and moreover the the oscillators that we're familiar with in the school laboratory masses on weights and stuff will have values of n, which are like ten to the ten to the 28 or 34 or that kind of simply ginormous values of. And so the frequency here will be stupendous and nothing and nothing in the laboratory is happening at that frequency.
So this is totally this is totally fantasyland. 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 is 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, is say let's at the state of our system, we have to consider we have something that moves. We have to consider a system which does not have well-defined energy, which means that it's wave function, it's its state vector wave function, whatever is a linear combination of states, a well-defined energy. And let's suppose let's take a simple example. Let's suppose there are just two. Two states present. No, sorry.
The proposal is. Let me take this. Let's do a song. Let's do it in all generality. So we get to write this as a and E to the minus I. So this is totally safe. Any state could be written like this. This is absurd. The state of my system at time times. It's a linear combination of states, of well-defined energy. There's no question I can do. That's the general initial condition. Let's work out the expectation value of X.
So what is it? It's going to be the sum and star e to the eye and plus a half omega t times and times x times m. Times a m. Not start. It is a minus. I m plus a half omega t. So we can clean this this stuff up to. This is a sum of and an m of course it's going to be the sum and a in star. A m e to the. When we put these two together, we're going to have an E to the eye and minus M omega T times and x m and yesterday we already saw what the stylish way is to handle this expectation.
Value here is 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 two and omega characteristic length. So 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 and a m plus and a dagger.
All right. And this remember a on M produces M plus one in an amount. The square root of emphasise one. So this is going to be l root M plus one and m plus the square root. Excuse me. This A produces m minus one looks at minus one zero. I'm not concentrating at all right. But it's the square root of the bigger number, the biggest number that occurs. So this is a squared of m sorry. A on m produces m minus one.
How much? The square root of the biggest integer that's involved. That's the square root of M this is going to produce M plus one. And with this normalisation, which is the square root of the biggest number involved, which is M plus one, so and M plus one. What we want is we have a here are some 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 and minus one and a delta and plus one.
So we're going to have the 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 and star a how much? In order for this to be to be not zero, m has to be one bigger than N. So this is going to be the square root of n plus one. And everywhere where I everywhere where I see an m, I'm going to have to write an N plus one.
E to the eye. And now in this case, we've agreed that M is one bigger than ends. This is going to be easier than minus omega T, so that's what we're going to get from this term and then try from this term when that goes in there. And now we have to put this in there and now M is going to be one to get a zero contribution. M is going to N plus one is going to have to be N and so M is going to be n minus one. So we're going to get an A and store and minus one times the square root.
Of MN. E. And now in this case, M is going to be smaller than N is going to be easy. Plus, I mean, Kitty. And I've lost my l somewhere along the line that may be reinstated. Right. There's this L here. I hope we have everything. Slightly scared that we haven't. Let me just check them. Now seems to be okay. So. And we're still swimming. I've lost some sign we are still swimming over end. 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. And primed is equal to and is equal to and primed minus one. In this sum here. And then some of end primed, and then I can relabel the end, primed end. 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 rn primed l a primed primed a and primed minus one a and that start that stored and primed times the square root of and primed e to the minus I make a t and the other sum is still over. And and that's a star. And A and minus one e to the I make a t, but these n and primed are the same other. I mean, they're just dummy indices we just summing over them. So this sum is in fact the complex conjugative that some.
These two things are complex conjugates. So what we we. If we. If we write a and a and minus one is is equal to say X and E to the I phi which we can do. This is real. And this, of course, is real, too. If I'm writing a complex number. Sorry, that needs a star in it. Then I'm going to be taking x e to the minus i omega t minus ify. Plus this stuff is the omega t plus sci fi.
And we're going to be able to combine the two exponentials and discover that x is equal is equal to L times the sum of x and costs omega t plus five. Sorry. We need to fly in on that. Excuse me. We need to find him. So this obviously needs an end. And that is an end because each of these complex numbers has, as its own matter, has its own fate. So what do we discovered? We discovered that, lo and behold, the position, the expectation, value the position does oscillate with periodically.
This is now this. We have indeed sinusoidal. Oscillation. But period. Two Pi of omega. We have, in fact, recovered classical physics to the classical motion. So the motion at this frequency occurs because of interference, quantum interference between states. So these these these terms, we've got quantum interference between states of different energy. Right? It was. Why do we have states of different energy involved? It was because X that. Oh. It it was because.
So when we went to the expectation value of X, we've got this huge long sum which involved cross terms between states of it it inclu, it included the term and x and that was also involved in here in which the same the state of a given energy was was present on 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 letter operators, of these annihilation 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 as a result, all arose from interference between states which differ in energy by one by one excitation. And it's a peculiar. So that's a very general phenomenon and a peculiar feature of of of this problem is that those differences in energy are all the same. They're all H Bar Omega and 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, that the period is independent of the amplitude of the excitation, the amplitude of the excitation is controlled by which of these ends are significantly large.
Right. Because and it's the amplitude to have energy and plus a half h bar omega. So a highly excited oscillator has uh, has the non-zero values of N are all clustered around a large value of N and a very gently excited oscillator has the A ends around around zero or small values of and 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 to a frequency omega regardless of which terms in this sum dominating. And that's this property that that the period does not depend on the amplitude. So let's let's be more realistic and invest. See how much more of classical physics we can we can get out of this by talking about an and harmonic and harmonic oscillator.
So I, I introduced harmonic oscillators by saying that they, they're widespread because if you have a point of equilibrium, if you plot against displacement from 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. That's right. Really should be negative, shouldn't it, actually? And I come to think of it. Sorry, I should draw the the graph this way around sun in order to get a stable force.
So 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 that's a stable equilibrium. And if a harmonic oscillator arises, if we replace if we approximate the curve, 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 is so any, 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 harmonic oscillator. And typically. And typically the force versus.
So for a harmonic oscillator, the force versus distance is a is a straight line that goes all the way to infinity, which means that in order to pull your 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 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 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 is infinity.
Sorry. And I should. I need to draw it so it becomes tangent to this and then disappears off like this. So this is a this is a more realistic curve. And the mark oscillator is is a good model if the parabola is tangent to the the realistic curve over a decent range. That's the main idea. So what we should do is investigate. Let's. Let's 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 x 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.
We have a perfectly well-defined Hamiltonian p squared over to him, 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 you can't do it. But it's pretty straightforward to solve this problem. Eight G equals E numerically.
We do it in the position representation we browse through by x and have that x squared over to m e plus x. The E is equal to e x e, which turns into this by the rules we've already discussed.
This turns into an ordinary differential equation minus h squared over to m d to you by the x squared plus v of x u is equal to EU where you of course is equal to x. Okay, so this is an ordinary differential equation, second order, etc. and it's linear and it is, it's pretty straightforward to solve numerically. If you look in the book as a footnote, that explains how to do that. Now, I should have I'm afraid I meant to bring my laptop with the with the official figures.
But when you do this. So, so. By discrete izing this differential equation, we turn it into a into a, an exercise in linear algebra, which your computer solves. So you, you write this basically as a matrix and uh, on, on you, which becomes a column vector. The value that you takes of the different of the different positions in X is equal to E. You. So you turn it into a matrix equation. And computers are very good at matrix equations.
When you do that, you discover what the values of are, and you can also discover what these what these way functions look like. And the crucial thing is that you find that if you plot the possible energies, you you get a distribution that looks like this. It starts off looking like an equally spaced ladder for the harmonic oscillator. There are steps, each one of which is separated by a bar omega for an alpha.
So we start off like that with the spacing given by the harmonic oscillator that the tangent to the bottom of the curve. But as we go up, the spacing gets gets less and less and less and less and less and less. And what essentially the the algebra is doing is giving you an infinite number of allowed energies or already in a finite range because this is v nought. So so that potential allows that potential as X goes to infinity, goes to a finite value, not mine, where it goes to zero.
So this is zero and I guess this is minus v nought. So that the lowest energy is somewhere down here. So with only a finite range in energy, you pack in an infinite number of allowed energies with a harmonic oscillator. You have to 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. Okay. So that's the this is 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 let's say our initial condition is this that it consists simply of two terms and 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 self some space and plus one E to the minus e and plus one T on h bar and plus one. So that's that's not a completely general initial condition now because I'm assuming that there are only two non vanishing amplitudes.
So my, my state, uh. It happens to be such. There are only two possible values of the energy that I can measure. There's an amplitude. And to measure the energy and and there's an aptitude a and plus one to measure the next highest energy. Okay. So this is 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 and star e to the i e in t h bar.
And this is all very similar to the other case and plus one star e i e in plus 1th bar. X. And then the same stuff on and then. And he's the minus I ian to your natural. Now, when we multiply this stuff out, we will get. We 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 and 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 nought. So this follows from symmetry. A V of x that v of x is an even function of x so long as vivek's the potential is an even function. The same behaviour at minus x is plus x. This will. This will vanish. We will show this as we as we go along. We haven't shown it yet, but that that will be true.
So given that that so. Quite generally. My expectation value here is going to be a star. So it's going to be this on this e to the I D and minus E and plus one t on h bar times the matrix element and X and plus one. Plus excuse me and I'm needing here some a and still a and plus one. Right. So that's that. On that. And then we will have this on this a and and plus one star e to the minus I e and minus C and plus one T on both times end plus one and x and.
Hope I've done that right. So what do we have now? We have a gain that this term is the complex. Well, we have this term is a complex conjugate of this term. So we're looking at a sinusoidal function. Plus it's complex conjugate. Therefore, we're looking at something which is x and could be written as x and cos e and cos e and minus C and plus one T on h bar plus a possible phase factor. Right. Where just to be concrete x and is the model is the modulus of and and plus one.
So and AA plus one is a complex number. I have its modulus sticking out here and I stuck its phase into that. So what do we what do we observe? Again, we have harmonic motion, sinusoidal motion. But look now at the period of this sinusoidal motion, the period of the frequency of this sinusoidal motion now depends on PN because it's the again, it's the difference of two energies of adjacent energies, which counts.
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 in. So we're recovering a classical fact, which is that if you have if you have an uncommon oscillator of a typical type and you make bigger, you kick it to a bigger oscillations, it it's period will slow down. And that's true of an ordinary pendulum. An ordinary pendulum has its highest frequency.
If you you have it go to and fro with a small amplitude the clock makers make their pendulums go to. But the period goes if you as you increase the amplitude of the oscillations. So this is this is, as it were, high, high omega for a pendulum. As you boost the period to the point at which it's about to go overtop dead centre, you know, if you have if you break it swings 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 to the point at which it would go or just keep keep on going. It had enough energy to go right through top to centre. So so this slowing of the period with increasing amplitude is, is manifested in a, in a standard pendulum and we see how it emerges from the structure. So here we're learning something important, the way in which the dynamics is encoded in the spacing of the energy levels of the of the stationary.
The energies of the stationary states. This is these are just simple examples of what's totally generic. Okay. Something else that we can learn about this on harmonic oscillator is that so more generally, this was a simple example. I said let's consider in order to get something to move. I considered a state that was under it had undefined energy.
To keep it simple, I took just two non vanishing amplitudes the only two 3 to 5 non vanishing amplitudes in the expansion of the state and stationary states. Realistically, we would have if you take an ordinary pendulum like that and you and you give it a jog, you will the energy will be uncertain by zillions of values of Omega. And many, many, many, many, many of these coefficients will be non will be non vanishing.
So more generally. We're going to have the upside is equal to we will have many terms and let's just write down a few of these terms and minus one e to the minus i e n minus one t on each bar plus a and e to the I in, etc. W So there will be many, many of these coefficients, but if we know pretty much what the energy of the oscillator is, right, we've, we've lifted our bulb up to 30 degrees or something and let it go.
The energy is not completely undetermined. And what that means is that the 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 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.
Right? There'll be an N at which the at which the amplitudes peak and they'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 the generic situation. And when we come in of calculate the expectation value of X, what we now have is the same sort of thing up as up there. But it's it's somewhat more complicated. We're going to have an and still a n minus one.
Uh, some of the things that we had before e to the I n minus E and minus one T on h bar times, some matrix element and x and minus one. And then we will have to do this the other way around and. And then I need to put in a minus sign. Right. Then I will have the next one. I will have because an x and will vanish. The next one I will buy the cemetery property. I will have a and plus one store and e to the minus I e and minus E and plus one t on each bar each four times and x and plus one.
And then I will have not plus equals, but plus a and plus three store. And this will be the next term. Each of the i e in minus e and plus three t over h bar and X and plus three plus two. Not at all. Right. This is this is a specimen of a disgusting expression which would give us the expectation value of X to the. 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 in this extra amount complex conjugates. They will not now be. Exactly. This will not be exactly this, because this is the difference between an and N minus one. And this is the. So one step in the latter. And this is the the size of the step one above it, in the ladder, which we slightly smaller. And I think you have your problem. Oh, thank you.
I have. That's right. Because I changed my mind how I was going to do this. And I thank you very much. So this should be in my plus one and that should be in and this should be plus three. Okay. It's good to know that. There's understanding in the room. So. So that's one thing that's going to happen because this is a more realistic oscillator as these frequencies will be changing. And this 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 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 e n plus three minus e n is going to be on the order of three times E and minus E and minus one.
Right. Because it's the difference. It's it's three steps on the ladder and that's only one step on the ladder. And if we think of of that, the size of the steps in the ladder becoming small only gradually as we go up the ladder, which will be is, is, is a good picture to use. Then then this term is going to be essentially three times the frequency associated with the other two, which we can regard is about about the same.
So when we, when we assemble all this stuff, we're going to find that x the expectation value of X looks like some number times costs of e n minus the N plus one. Now let's, let's, let's declare this to be three omega n, right?
So we defining omega n to be this quantity here and we're going to find that we have a cos some sum term like cos Omega and T and we're going to have some other term with some other coefficient times cos three Omega and T and we're going to have some other term with x five some other coefficient times cos five omega n t. So the, this number will depend on, uh, will contain products of stuff like a and a and plus three star and a and plus three A and plus five star, etc.
Right. And this will contain things like a and a and, uh. Plus. Plus five. But we will have these other frequencies present. And this is what leads to. So this. This series implies periodic motion. But on harmonic motion. So in a musical instrument. You know, the the. The. The note. The motion of the of the string in a piano or the vibrations in an organ tube or flute tube or whatever has its it has a it has a well-defined frequency which that 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 and harmonic oscillation. Uh. Typically. But there's there's more that we can do here, which is connecting to classical physics, which is to make the point that if we arrange this stuff, right, this expectation value of X, we take out the leading term. We say that this is E to the minus omega and T, and then we're going to have some some.
Well. No. Let's, let's. Let's try it. It's better. It gets very complicated using about the expectation, but it makes it's easier if we think about website itself as a function of time. If we look at upside cells as a function of time, we can take an E to the minus I, e and T on each bar out. And we can say that this is dot, dot, dot a and and plus e to the i e in plus one e and plus one t on each bar. And plus one times a and sorry. Right. This needs to be. Yeah.
That's you need a minus sign there or I need a T on all that. So I've taken the 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 and plus one. This should be plus and plus to e to the minus i e n plus two minus e n t over h for and plus two and so on and so forth.
So you have to a lowest order approximation. These differences here are all multiples of a common frequency. And after a, after a period. So when, uh, e and t over h bar is equal to two pi, these things. Uh, sorry. And. Plus one minus C in. After after the time it takes for this to come around to to pay, this will come round two for 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 nought, because all of these exponentials would have come round to one. Again, that's in the case. That's the 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 PI, this one has come round isn't quite round to pi and even more so further down the line. And therefore the wave function isn't quite back to where it was when a t equals nought. And as we as as this, as time goes on and we count more periods so this becomes two pi end, these discrepancies become more and more and more important in these terms down here.
When this one is come round to two to end pi, this one will be significantly off to end pi and this one even more so. And that means that the wave function is not returning to its original value. And we're have we're looking at motion which is not periodic.
And whereas initially because we 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 disruption 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 we'll 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 and in this series was associated with a small uncertainty in period. If the periods, if the 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 and a million and a bit times. And here it would be.
But if it was slightly different, slightly lower energy would have a slightly higher frequency and it would have done it 1,000,001 oscillations, and it would be over here so that you can see that the small uncertainty in energy is going to lead off for a long time through the small uncertainty in period to a large uncertainty and 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.