 Shall we begin? Let's. All right. So, yeah, it's not today. It's Tuesday. Tuesday. We have the next slide. All right. So, we are talking about the simple harmonic oscillator. And as we discussed, it's one of the most important problems in physics, and so it's quite important that we focus on it in quite a bit here. And so the first thing that we did the last time and I wanted to review and emphasize the important points is just to look at the classical mechanics of the simple harmonic oscillator written in this particular location with particular emphasis on some of the particular features. So, the first thing we did is that we wrote everything in terms of dimensionless variables. That's always what you should do. So you should always get rid of the units by looking at whatever the characteristic scales are and whatever the problem is. It has some physical constants. In this case, there's two constants, the mass and the spring constant or the mass and the resonance frequency. And that's it. So everything is kind of defined in terms of fact. And at least that relates whatever characteristic units we have will be defined in terms of those physical constants. So there's some characteristic energy, there's a characteristic height scale, there's a characteristic momentum. And by this, we have, you know, they're loosely related in this way. I mean, the Oprah, that we put a factor of two in there or not, that doesn't matter. That's because, of course, everything would be rescaled in terms of that. And I keep oscillating my simple harmonically. I'm oscillating. It's inside whether I want to put that factor of two in there or not because there's nothing solution. I think I just used a factor of two somewhere. Anyway, with that, the Hamiltonian looks like this. It's symmetric. It's quadratic in position momentum. It would be symmetric. And I would say whether this factor of two is absorbed here or put there, your choice. But we'll use this. So in dimensionless units, then, the Hamilton's equation of motion for the space-space variables have this kind of symmetric form. And each one of those, the canonical variables, the position momentum satisfy the simple harmonic oscillator differential equation whose solution I've written here in terms of initial conditions, the initial condition for the position in the momentum, or I can think about it as the initial amplitude and the initial phase of the oscillator. So one important point here that I want to emphasize is that any simple harmonic motion, any simple harmonic motion that's oscillating at a fixed frequency omega can always be thought of as some amount of cosine and some amount of sine. Always. And those are sometimes called the quadratures of the oscillation because they are 90 degrees at a phase one another, so they are said to be in quadrature. So let's say I'd like to go engineering and radio jargon, radio, whatever, but that's what we talk about. We still talk about the quadratures of the oscillator. Or I could equivalently think about any harmonic motion always can be thought about as having some amplitude and some phase relative to some phase that we call the quadrature. So either I talk about the quadratures, which in this case are the x and p, the initial x and p variables, or I talk about the amplitude and phase. And of course they're interrelated, good point, right? So the amplitude is the, you know, the bagger and theorem length of that vector. And the phase is the argument of this, the imaginary part of the ratio in x. That is to say, any harmonic motion can always be expressed compactly in terms of a complex amplitude. So the complex amplitude encodes the amplitude and phase and the quadratures. Either I talk about it in terms of real and imaginary parts or I talk about it in terms of polar representation of that phaser, right? So if I have a complex number, which is, here's my complex plane, it has a real and imaginary part and those real and imaginary parts are the x and p variables, or equivalently that complex number has an amplitude and a phase. I want you to keep those things in mind because that aspect of the simple harmonic motion, that classical aspect has quantum analogies and quantum features and they're often forgotten. We learn about raising or lowering operators but we have no idea what that means relative to classical oscillators. They mean something, okay? So what this says is that written in terms of the complex amplitude, simple harmonic motion is just the rotation of this phaser on a circle. The amplitude is constant given some initial energy that energy is concerned and that energy is just given by whatever the characteristic energy of the system is times the square of the amplitude. And the trajectory is thus a trajectory on a circle in these dimensionless units, in these scale units that rotates at an angular velocity and will make that. That's what that means is that the position of the particle as I project on this is oscillating back and forth and the momentum of the particle is oscillating back and forth and they're oscillating back and forth 90 degrees at a phase with one another. So when the position is at its maximum value the kinetic energy is zero. When the position is at its minimum value the kinetic energy is its backbone value. You know that from basic simple harmonic motion. So that's the classical problem and I want you to think about it. I want you to think about the complex amplitudes. I want you to think about what the real and imaginary parts mean what the amplitude and phases mean. All right. So then what we did is we said okay now we're going to quantize this and the way we just quantize it is to just say now my position and momentum, canonical coordinates become the position and momentum operators which no longer commute. They have the canonical commutator and now we have one more constant than the problem, HR. Before we just had two, mass and frequency. Now we have the third one, the characteristic unit to action. So it's natural to define the characteristic position momentum variables to be related to one another such that they are equal to the characteristic unit of action and quantum mechanics, H1. And now we've got three constants and three characteristic units and so we can define that. We get the natural characteristic energy scale of the harmonic oscillator HR omega and these are the characteristic scales of position and momentum. Okay. And so express in terms of those characteristic units we now have a dimensionless position and momentum operator. And moreover we have a quantum analog of the complex amplitude. So what was the complex amplitude, the complex number in the classical mechanics now is a non-hermitian operator in the quantum world whose permission part is the position and whose anti-hermitian part is the momentum. And you won't keep that in mind. What is different of course in quantum mechanics is that where it's alpha in alpha star where it's numbers now A and a dagger are operators and they don't commute. Okay. And their commutator is the canonical commutator. It's another way of writing this commutator. So express in terms of those dimensionless variables are Hamiltonian. Here's our characteristic energy scale X and P or in terms of the complex amplitudes looks like this. The point is that because A and a dagger don't commute we have a symmetrized form of what was here. Okay. So there's a new operator that appears to character the amplitude squared the square of the complex amplitude classically is the thing we call the number operator quantumly. I think quantumly should be an adder. And that thing, the number operator for reasons we know that its eigenvalues are the numbers the natural numbers has these commutator relations with A and a dagger. And we use them to solve the time independent Schrodinger equation that is to say to solve for the energy eigenvalues of the Hamiltonian. Okay. Because the energy eigenvalues are the eigenvalues of the number operator multiplied by A for omega that's a half, right? And what we showed in lightning speed at the end of last lecture is that those, that number operator its eigenvalues are the natural numbers the non-negative integer 0, 1, 2, 3 all the way up to infinity. Okay. And so the energy eigenvalues of the simple harmonic oscillator are A for omega times n plus a half. So the energy eigenvalues are, yes? Were there any other potentials that have the same space in the same way? No. Absolutely. That's a good question in the answers. So this, as it was noting, of course what is unique about or what you see about the harmonic oscillator as in fact is unique to it is that the energy eigen, the energy levels are equally spaced. They're all spaced from one another A for omega and that is unique to the simple harmonic oscillator that is a feature of the simple harmonic oscillator. If you have that, you have a harmonic potential. Of course, the lowest energy state is the ground state and the ground state is the state with 10 to the power of 0 and it has an energy that's a half A for omega above the volume potential that's called the 0 point. Okay. And moreover we showed the algebraic properties of the A and A dagger which we call the raising and lowering operators or lowering and raising operators. That's to say those operators take one of the number eigenstates and either lowers it by one or raises it by one. Okay. And of course there is a ground state that's how we derive this fact and that ground state has the property that it is annihilated by this. This is why sometimes this is also called another name and it becomes more important in the context of field theory when these ends are quanta of excitation of the field of our particles then this is known as the annihilation operator A dagger is the creation operator and with this, these two properties what it says is that I can obtain any of the energy eigenstates the number states by applying the creation operator or the raising operator to the same thing end times to lowest energy level the ground state and then we have to re-normalize it because this is not a unitary it doesn't preserve the norm of this and we re-normalize it and that norm and factorial it's easy to check So just come back to the front board here for a moment what it's saying is that quantically whereas classically sort of any, remember we said the radius of this circle represented the energy of the system and the system starts there and just rotates that the quantum state says in some sense there's quantized possible trajectories in phase space all separated by the department now of course it's not exactly true because I don't know it's supposedly a circle but you get the picture because of course I can't be at a point in phase space I mean there's a kind of uncertainty above over here and there's some kind of we're going to talk about this in much more detail think about this uncertainty here X and P in these guys but loosely that's a good picture of what's going on and the reason you can think about the existence of the zero point energy is as we discussed when we thought about the particle in the box it's from the uncertainty principle it's impossible to vocalize at position animal matrix if the ground state were just that would have that position and no kinetic energy that's impossible but the ground state is the minimum energy state that consistent with the uncertainty principle so far so good very good now one thing we can see from here firstly is that what we call the number states which are the eigenstates of the number operator which are also the energy eigenstates of course here n equals 0 1, 2 a form of basis we know that the eigenvectors of any permission operator form an orthonormal basis and so and of course in the context of the electromagnetic oscillator unlike say a finite square wheel all the states are bound there are no unbound states so this is the complete basis so a resolution of the identity can be written in terms of the number states and that is to say a basis for Hilbert space and what is Hilbert space in this case is the space of square normalizable functions that is the Hilbert space we have been talking about one dimensional mechanics of a single particle so this forms a basis for Hilbert space which means that any observable or any operator I should say can be expressed in this basis so for example we can write the position operator in this basis we've written position and momentum in terms of continuous variables the x eigenstates and p eigenstates but we can and we often do write them in terms of discrete variables the number states so even if I'm not talking about the simple harmonic oscillator this is a basis right it doesn't matter whether they happen to be the eigenstates of the Hamiltonian it's always a basis yes Steve, do you have a question? yeah I know physically the annihilation of any of the operators cannot hit any people at equal zero or yes but my question is it does seem like any people change if you make a negative apart from the fact that you get imaginary terms the reason that those states don't exist is the following n itself is a positive operator meaning that its eigenvalues have to be positive and we saw that last time as because if I look at the expectation value of n with respect to any state that is equal to and that is equal to the norm squared of this vector which is always greater than or equal to zero yes but if you make n like negative 2 but then n would have a negative eigenvalue and it can't so that's how we got to that very good so because of the basis let's write the matrix representation of x and p we can do that so what about x and p how do you do that well whenever you're dealing with the number of basis the thing to do is express things in terms of a a a dagger because we know how a a dagger act on this is x in terms of a a dagger thank you very much so it's the real part with the factor of 2 in there I don't know if this is correct okay and p is what the imaginary part so that's a minus a dagger over 2i so if we know that then we can express x and p in terms of this basis because we can always write a a dagger in that basis, let's do that first okay so let's look at for example the annihilation operator in the number of basis so I'm just going to insert a resolution of the identity on both sides so I'm going to have the sum over n and n prime so this is equal to the square root of n n minus 1 delta n prime n minus 1 so this then is equal to the square root of n but because I can't have n less than 0 n equals 1 to infinity n minus 1 in the case where n equals 0 doesn't exist because in fact it's 0 so that's important sorry and a dagger is the adjoint that's just taking the adjoint or if I want to just relabel things and start at 0 I can do that too let's just relabel the sum so one of these this is just expressed as a set of our products and you see what this thing does it takes n to n minus 1 with that coefficient or I take n to n plus 1 with that coefficient let's write them as matrices of course these are all matrices so we can't write down a whole matrix but we'll write down a few of the matrix elements of it so our case is elements 0, 1, 2 dot, dot, dot of course it has nothing on the diagonals column 1 column 0, column 1, column 2 and then it has something on the off diagonals and you tell me is it on the upper diagonal or the lower diagonal lower if we see here this is row column right so it's root 2, root 3, dot, dot, dot 0 is everywhere else exactly so a dagger looks like and so x is this plus that over 2, root 2 and p is this minus so we'll have things on the upper diagonal and the lower diagonal so we can write x and p as matrices these unlike the position in momentum eigenstates the number states are valid states of Hilbert space that's why we get a representation of x and p as matrices they should be because this Hilbert space it has a countably infinite basis here's an example of such a basis right good um please so you're saying that in this basis the x is just better to hate I'm just trying to you know all I'm saying is that it has the behavior that we don't expect from any operator on a Hilbert space it's a matrix the fact that we expressed it I mean let's talk about the position of momentum basis okay so let's talk about so now I'm going to these are the eigenstates of the number operator now we can look for the position in momentum representation so the way we said that we looked for states that are eigenstates of the x operator and the p operator now these are continuous variables but that's okay they allow us to write a resolution of the identity so that's okay and let's see now so whereas these operators expressed in this basis have matrix elements that are just numbers in this case the matrix elements of x and p have singularity in them right this x is up so it's in some sense not very well behaved right and p in that in the representation p eigenstates the momentum operator in position representation was the derivative from operator I'm sorry the position operator in momentum representation was the hr went away because this is dimensionless units this looks a lot like what we did before but in the character scale that's correct that's exactly what it is indeed let's just just emphasize that let me say one quick point recall that with the units of the problem the kets and p at units so recall with units we had these are my little variables having units which meant that this thing as units of one over the square root of length and p kets has units of one over the square root of momentum which is the square root of momentum operator so I can define the dimension full ket in terms of the dimension less ket as one over the square root of the character is the unit where x is equal to xc now this thing has units or said another way the dimensionless ket is related to the dimension full ket so by getting rid of the units then when you do the change of variables this so if I look at for example x prime in the dimension full variables they said that was h bar over i d by d x delta x minus x but if I wanted to do all that re-expressing over here this is going to be you know there's going to be a one over root x that gives me xc put it over here and then I get p, one over pc moreover of course what we have is that I can talk about position and momentum representations I mean let me just quickly go back over here before I say of course in the number of basis that I can write any the state as an expansion over the number I'm in states and the c sub n's are just the probability amplitudes in the end basis with respect to the continuous variables we have a similar thing but I start to write every side as a superposition of position I do vectors that way or momentum those are the position and momentum wave functions so psi of x is that which if I wanted to write it in dimension full form is that times that full variables gets rid of that and the momentum space in dimensionless variables is finally what about a and a dagger well what I can see is let me first say the following thing if I look at the momentum operator acting on a state and look at the wave functions we get the derivative respect to x so the momentum operator is the derivative of acting on the wave function or in momentum space the x operator acting on the state looks like a derivative with respect to p so what about the creation violation operators or the grazing and lowering operators what do they look like in position momentum space they are combinations of x and p so let's just do it so if I look at I can take the annihilation operator and I act it or lower an operator and I act it on the side what is the wave function well to do that I look at the most position representation so that's equal to the position represent a is x plus i p so this is equal to x let's write it all out 1 over 2 x on side taking the position representation of course so good what is this it's x times the wave function and what is this it's the derivative the momentum operator in position space is the derivative all these x's are the the new dimensionless everything to dimensionless sorry yes probably be quite much more careful about my notation here plus i times the derivative perspective so in the position representation the annihilation operator acting on this is x plus the derivative respect of x what about and of course a dagger is just the adjoint of that and the adjoint of the derivative is minus what about in momentum space what is the position the creation of the annihilation operator x plus i p in the momentum representation and what do we get this guy he's got some minds in there you gotta think about it 1 over minus i d by dp and then what is this guy i times p thank you very much times the momentum space so in the momentum representation this is i over root 2 p plus d by dp up to this overall phase of i the annihilation operator and the position of momentum representation look the same which you kind of expect because they're kind of symmetric in x and p except an overall fact some i's floating around there and of course a dagger if you put the minus sign there very very good alright so given how do we find the position or momentum representation of the energy eigenvectors that is to say the wave functions that are the energy levels associated with the energy levels of the ionic oscillator we could so one way we could do it is that we could express so we see the position or momentum representation of the energy eigenvectors now one way we could get at them is to solve the differential equation i over root p the wave and so you know sorry i mean what that would you know one way is to say there's a differential equation in the position representation we know that uh the wave function so this is the time this is h side equals e side and we could just try to solve that differential equation which is the way it's always done or used to be always done in introductory quantum mechanics you plug in a power series and it seems the power series eventually has to truncate because you can't satisfy the boundary conditions unless that and that's one way to get at the eigenvalue equation that we got algebraically just based on the theory of operators and positive operators and so forth but another way is to use the one thing we know which is that there is for the ground state we have a very simple equation it says for the ground state we're calculated by the lower component we can express this equation as a differential equation and just solve that and once we have the ground state we have all other states because all other states are obtained by just raising that n times so what we're going to do is do that much easier and let's just do that but I agree this is perfectly correct alright so how are we going to solve this equation from the wave function what we think is the wave function u0 ground state wave function n would be 0 so we don't want to solve this equation we want to solve that equation so how are we going to use this equation to find u0 if I did that I'm going to get 0 equals 0 say again yeah I want to express this in the position representation that's what I want to do I want to link that this equation because that's going to be a differential equation and we have that over there somewhere the annihilation operator in the position representation is equal to 1 over root 2 x plus the derivative with respect to x on the wave function which is what we're calling u0 alright, good enough ground state satisfies the following differential equation x d by dx ground state wave function is 0 so that's a simple differential equation we can solve but we can solve it of course this is really in 1d without time dependence it's a simple PDE not a PDE so how do you solve that differential equation separate variables thank you very much this says that the derivative with respect to x is equal to minus x which says that the integral x squared and this is the natural log of u0 divided by some constant a so we have a solution the ground state wave function we just call it some constant what is that function? it's like a homework it's a Gaussian how do we find this constant c? normalize it very much excellent alright, so the norm of vector squared we can do as an integral in position representation so that's minus square equal to of pi I know if it was over 2 it would be yeah I think it's this do you do like square root of pi over whatever in front of you? yeah, that's right that's right I only know when it's normal I just move c and you move c okay, it's good alright, and we want this to be equal to 1 normalize it alright, so where would you c be? now of course there's an overall phase that we can always put in front here but by convention we choose this to be we do that okay, so that says in dimensionless units u0 is 1 over the fourth root of pi e to the minus a half x squared in dimensionful units I have to put in a 1 over the square root of that putting in what xc was the characteristic remember xc was the square root of hr over m omega this is equal to hr over m omega and pi to the 1 over the power e to the minus a half and x squared yeah, so I mean I guess to classically think about a harmonic oscillator I would expect it to be most likely to be found where its speed was approaching zero at the ends and not in the middle where its speed is the highest indeed, that is an excellent piece of optimization and that is something which is saying that this kind of motion is extremely non-physical now, I was going to say this first next section but let's talk about this now so let me hint at something so azic was let's think about a pendulum a pendulum that's a sublimonic oscillator spends a lot of time at the turning point and then is speeding through the center so you kind of expect the highest probability for it to be is at its turning points and the smallest chance of finding the particle to be in the center this is the opposite if I look at the wave function is it because here when it has a tip of the swing it's just very narrow but so nothing must go from very large no, you cannot talk about the problem is you can't talk about the position and momentum simultaneously classically so this is this wave function supposed to be a bell shape this is the square of the wave function so classically we can talk about the position at a particular I mean the momentum at a particular position we can do that classically because we can talk about position and momentum simultaneously and what is that given a certain amount of energy well it's whatever the kinetic energy is at that point of the momentum so that's the square root of 2 and v-8 right so where the energy at the turning point the momentum is 0 right and you expect it to stay there all the time now what we talked about very briefly but we did talk about this is the WKB approximation is a way of writing down the wave function when the momentum is large compared to the characteristic scale of change of the potential so when the the wave plane was tiny compared to something having to do with the rate of change of the momentum in this case this is the spring constant so for very very short wave lengths which in this case means high energy we expect the following to be true that this is equal to 1 over some constant normalization constant divided by the square root of the classical momentum either the I or the classical momentum remember this so this is a semi classical approximation so we expect this to be true which is to say that for large energies where the the wave length is very small so here's my harmonic we have the ground state the ground state wave function looks like that we'll talk about the intermediate cases in a moment but if I'm at large n well then this should be an approximate wave function which means the wave function should have a lot of its probability amplitude where the classical momentum goes to zero and very very little a little bit so up here I mean classically the probability goes to infinity small number here very high this is the classical probability based on this amplitude and what we get is something that looks like this supposed to look symmetric I'm sorry so there's a limit of large quantum numbers that's to say a high energy we have a limitation that you intuited and that you expected the probability amplitude should be very high at the classical turning point and very small at the place where it's speeding by the fastest the classical dynamics is informing in some way what the wave function is okay and we saw that when we looked at how the amplitude I mean at phase of the wave function obeyed equations that were the classical Hamilton Jacobi equation okay so this is sometimes known as the correspondence principle now I wonder we're going to talk about this a little bit more and try to understand that somehow at the limit as n gets large I should just say whatever the potential is n is large to get to the classical limit now that's what we mean by classical is a very slippery term in the sense in which this state is supposed to be symmetric is a classical state well that can you think about the semantics of the word classical but what is certainly true is that this is true as to say the wave function is well approximated such that its magnitude is guided by the classical trajectory but for short I mean for long wavelengths for low quantum numbers the wave nature of the motion dominates not the ray trajectory in which case it has nothing to do with the classical trajectory nothing excellent, excellent optimization so we have the ground state what about the higher excited states well firstly what can you tell me just from what we know about the nature of the potential and the solutions to wave mechanics at 1D about what you kind of expect the ray function to look like yet it's got another node so you kind of expect it to look something like that right and then what about the next excited state mean it's supposed to be there kind of like something like this and you know that this is symmetric and isometric symmetric and isometric okay it could be one two why am I saying that the eigenfunctions of the Hamiltonian are necessarily symmetric and isometric functions of X potential is symmetric potential is symmetric potential is reflection symmetric and thus we know that the eigenfunctions are eigenfunctions of parity alright very good okay so algebraically we have this which means that the eigenfunction is given by and so this is the nth power of a dagger which is over there a dagger in the X representation so this is 1 over the square root of n factorial 1 over the square root of 2 to the power of n X minus d by dX n times acting on so we just have to do that but the answer is that this every time I take this do this derivative well I'm going to get the Gaussian back because you take the derivative you get the Gaussian and then you're going to get some derivatives of the things in the exponents which are going to be polynomial in X and so this thing is equal to some polynomial the nth polynomial X times the Gaussian and these polynomials have a name they're called the Hermite the Hermite polynomials I have to double check the fact because the Hermite polynomials have a certain factor yeah I forgot that stuff okay so this is equal to this I'm just going to write my name so those are called the Hermite polynomials and the Hermite polynomial so by definition I could write a Hermite polynomial as what I did when I take this derivative and times on the Gaussian and then get rid of the Gaussian part so here are some of the Hermite polynomials H0 is 1 H1 is linear H2 is quadratic notice they alternate from even to odd functions so H and use the reflection symmetry I either get plus or minus and this is what they look like you take those polynomials you multiply them by Gaussians and you get these are the odd functions now I'll leave it as a simple exercise for you to show the following that in the momentum space representation so and then before I do that let me just quickly say in the position representation is equal to some normalization constant a sub n times the Hermite polynomial either the minus x squared over 2 and if I write out all the factors here that in the dimension full version if I had to talk whatsoever just in a drawer putting back all the units here to 1 over xc xc either the minus x or 2 what about in the momentum space well firstly we can see right away what is this well again we start with the equation that we annihilate the ground state and in momentum space this is an annihilation operator times i over root 2 and this I can cancel so what can you tell me about the momentum space wave function of the ground state look at that equation for position space same it's exactly the same in these characteristic units so in the characteristic units this guy is exactly the same it's exactly the same Gaussian it's also Gaussian in dimensions units with exactly the same width therefore what are the excited states they're exactly the same except there's some factor of i by convention so this thing is again whatever normalization constant I have her mean polynomial up to that phase convention they're exactly the same and this again is unique to the simple harmonic oscillator and the reason is that in the simple harmonic oscillator x and t are the same the simple harmonic oscillator in the Hamiltonian was x squared plus e squared in some units this is also true for p so the position wave functions look just like the momentum wave functions they're the same it's the symmetry in x and p that makes the simple harmonic oscillator special alright we shall continue this discussion next time