 So, let's get to show the road here. So, we do have the next exam coming up, as I discussed, where we'll be taking an exam. And we'll be distributed both after the problem session next Friday. So, this Friday we'll have just a discussion section of posting all of the solutions. There's only one more solution to post problem set nine. And so, a couple of questions. All right. We'll discuss that. I will put it on the web immediately after this. If you're not there at the problem session, you can download it or email it to you. You can probably submit it. And as I said, we'll discuss a little bit more the rules of the game as time approaches. All right. So, last time we were talking about an important sort of class ingredient of the formula. Maybe there's one more, you might say, postulate upon mechanics that we didn't really discuss at the beginning. It's how do you deal with composite systems? And what that means, classically, we talk about the notion of a degree of freedom. So, in classical mechanics, to each degree of freedom, we assign a set of a coordinate and a conjugate momentum, right? And if I have multiple degrees of freedom, for example, I have a particle that's moving in two dimensions or I have two particles moving on a line. Those are two examples of systems with two degrees of freedom, okay? And we assign some canonical coordinates to each degree of freedom. I'm calling them generically A and B. And in classical mechanics, what we said is that the joint phase space for the system is given by that formally what we call the Cartesian product, okay? So the coordinates on phase space are B's, okay? And that's what we call the Cartesian product. Now, as we, I mentioned the division into the degrees of freedom is not unique necessarily. For example, as I mentioned last time, if I talk about a particle moving in a plane, I could define as my, cannot, I can have one degree of freedom be, say, motion along the x-axis and the other one be motion along the y-axis and then there's a conjugate px and py. And that's one way to define the degrees of freedom. Or alternatively, I might choose to have xad, the radial, we call it rho, and the other conjugate, I mean, xbd5, right? And then there's a conjugate pthi and prl. So the division into the degrees of freedom is not unique, right? And similarly, a particle, two particles in a line, I can talk about the coordinate of each particle. I can talk about their relative coordinate and their center of mass. And we'll come back to that. It's going to be important. So that's classical physics. In quantum physics, what we said was that if I had different degrees of freedom, what is true is that observables associated with those different degrees of freedom commute. And that means we can define sort of separate Hilbert spaces for degree of freedom one, for A and degree of freedom B with observables that act on each of those Hilbert spaces. And then the joint Hilbert space that describes states of the system that describes both degrees of freedom is given by the tensor product, OK? And the tensor product is such that we can define joint states of the system that are product states. And we can grab this tensor product in between sets. And what that physically means, because of course the state, in this case I'm talking about pure states, are uncorrelated. That is to say, if I have a product state, the joint probability distribution for degree of freedom A and B is a probability for A times a probability of B, right? And that means that there's no correlations between A and B, because the probability for both things to happen is just the product of one times the product of the other. Does that make sense to everybody? Why uncorrelated probabilities are products? But why, if that's true, why is it a product? I mean that's sort of a tautology. Yeah? So what aspect of probability? What do we say? Let me say it's at the cross terms. When you do the variance, you get just the square of the thing and you also get the cross. That's one way of looking at it? What we were talking about today is like, if I want to look at the probability of flipping two heads, is the chance of flipping one head times the chance of flipping the other head? That's exactly what I meant. So if you wanted to calculate the joint probability of two heads, why did you just multiply them? Because it seems intuitive to do that, right? That's like, so if you're doing tests, you want to use just the normal average, but if you're doing something like stocks, if you want to take an average, you use the other kind of average, like the geometric average, because they depend on each other. The way I look at it is the following. If something's uncorrelated, what it means is the probability of flipping this first point has nothing to do with what happens in the second point. So what that would mean is that the marginal probability distribution, which is the probability for A to happen, I can write without knowing anything about B. That's to say, if I want to know what the probability, you know, Z, you can have 1.9, I want to flip mine, I can calculate mine independently of yours. And what that says, we know from probability theory, if I have a joint probability for two things, A and B, then the marginal probability distribution on A is given by the sum of all possible V values, whatever they are. That's the marginal distribution. Now, if this is equal to a product state of probabilities, well, then it's obvious. This is just the probability of A. It's nothing. It's just given by that. So that's the key idea from logic. Two things that are uncorrelated for this and that to happen is just a product of their happening. There's no correlation between them. And in quantum mechanics, a state like that we call separable. And in this case, what it says is the probability of outcomes of observables of degree of freedom one, whatever that degree of freedom is, are not in any way correlated with the outcomes, they're independent random variables if the probability of distribution is of that form. Let me look at it this way as well. If we look at this in wave function land, so suppose that this was two particles moving on a line. So the wave function for this thing is given by this, where these are the position eigenvectors of particle A and particle B. If the state was a product state like this, this is the wave function. The wave function in this case is just the product of wave function A and wave function B. That means the joint probability distribution A is a function of random elements and probability distribution B, which is the square of each of these guys. Yeah? Can you try a particle's position and like stay in the same one half? Sure, that's fine too. So for every degree of freedom, if I have, you know, I can have as many degrees of freedom as I like. So if I have a tensor product, and we're going to talk about this more, we're going to get back to spin. We haven't forgotten it. We just put it on how I ate it. So every degree of freedom is in a tensor product. So for example, if I wanted to say I have one particle, I want to keep track of its position and its spin, so the system is L2, if it's moving say on a line, tensor two-dimensional complex numbers. That is the Hilbert space associated with particle having a certain position and a certain spin. That's how we perform the right thing. And we'll talk about it. They're called spinners. Yeah? So the parts we have, you know, psi AB is equal to psi of... If that's true. Yeah, if it's a product state. So that's not spanning a new space. It is not. That's one member of that new space. Correct. Indeed. And in fact, this is the definition of a tensor product space. A tensor product space is the linear span of all product states. And there are states in the tensor product space which are not product states. Okay? For example, I mean, I can write one down in terms of wave functions. Let's say, just as an example, let's say I had two particles in a harmonic well. They don't have to be in a harmonic well. If you look at the harmonic out of certain wave functions, I like to call them particle one and particle two. The joint wave function might be this. Alpha u0 x1 u1 x2 plus beta u0 or u1 x1 u0 x2. What is this? Well, this is the particle where one of them is in the ground state and the other is in the excited state. But I don't know which one it is. So there's some probability amplitude alpha that particle one is in the ground state and particle two is in the first excited state. And there's some probability amplitude here that particle one is in the first excited state and particle two is in the ground state. This is not a product state. There's no way. There is no way to write this as some psi x1, psi x2, or phi x2. This is not a product state. But it is in the tensor product space because it is spanned by the problem states. In the same way that I could say, you know, a particle or a vector in three-dimensional spaces, the span of vectors along x, y, and z. Every vector is along x, y, and z. It's the span. This kind of state is not a product state. It's not acceptable. It has another special name called the Nintendo state. Everyone knows about Nintendo. It sounds the coolest thing since Puff Weed. I don't know why Puff Weed is cool, but that's what people say. And we'll talk more about that at our semester. Okay. So, continuing on. So, formally speaking, as I said, the tensor product space is the span of all product states. And that is the space that describes the composite Hilbert space for multiple degrees of freedom. Okay? And we can always write a basis for the tensor product space as looking at the individual bases for, say, in this case, if this were just two degrees of freedom, if I have a basis for Hilbert space A, which has some vectors, and a basis for Hilbert space B, which has some vectors, then a basis for the tensor product space, there exists always a product basis. Again, not every basis needs to be a product basis, but I can always find a product basis by this construction. Okay? Question. But if the dimension of Hilbert space A is d sub A and the dimension of Hilbert space B is d sub B, what is the dimension of the joint Hilbert space? d sub A times d sub B. Okay? So the dimension of the joint Hilbert space is just in the product of the dimensions. Right? Because that means the dimension is growing if I have multiple degrees of freedom exponentially. If I have n degrees of freedom and each one of those degrees of freedom was associated with the Hilbert space of dimension d, what is the joint dimension of n in such things? In the end. Okay? What do you mean? If the dimensions of all of this space saves or is one, you understand it with one dimension? It's true, but I think you have to be careful about what you're thinking about here. This is the dimension of Hilbert space. It's not the dimension of physical space along which the particles move it. If a particle is moving along a line, what is the dimension of Hilbert space that describes all states of motion of the particle? Does it continue in? It's infinity. Okay? So if you deal with wave functions, you never see this exponentiation of the dimension. Because it started in infinity and the dimension is obvious, it's just infinity. It keeps infinity times infinity times infinity. It's only when you deal with finite dimension of Hilbert spaces that you ever see the exponentiation of the dimension of Hilbert space. Okay? Continue anyway. So another piece of the form that we talked about is, of course, we talked about the tensor product of observables. So I can define, if I have an observable, if I say O is observable, OA acts on Hilbert space, A maps vectors in this to vectors in that. And we have some other observable that acts on Hilbert space B. Then I can define an observable that acts on the composite Hilbert space, map vectors on vectors as the following that this acting on a product state equal to the tensor product of the individual action. That's quite definition. Now, what would you do if this tensor product acted on some general state and you don't know whether it's maybe it's not a product state? Well, it can always decompose it in a basis of product states. Okay? We can insert in here a resolution of the identity. Now, a resolution of the identity is, I'm going to write this, as I said, we have lots of different notations and sometimes when I write something like this, this is a shorthand for EI FJ algorithm with EI tensor FJ. Or I could think about this as the operator of the projector A tensor that's also a criminal. This is an operator that acts on Hilbert space A and this is an operator that acts on Hilbert space B. That's perfectly, they're all the same, they're just written to really in place. So this would be equal O A O B acting on, I'll call it alpha IJ EI tensor FJ where alpha IJ is a probability amplitude associated with that projection. And then I know how to act this. It's a linear operator so it comes inside the sum this guy acts on this basis vector and this guy acts on this basis vector. Let's talk about eigenvectors. Suppose I had a particular operator on a joint space which was of the following form. It was some operator A and it did nothing to B or it's kind of sum. A acts on Hilbert space A and D acts on Hilbert space B. It's a sum of two operators that act on this. What is an example of an eigenvector of this operator? Exactly. So if there are, let's suppose A has eigenvectors real A and B has eigenvectors little ket B then A B which is A tensor B written is an eigenvector with eigenvalue A plus B. Okay? So this gets confusing and this is sometimes I mean come back to that in a certain language here that I often confuse it but I'll come back to it. Alright. So this tells us something about if we're looking in particular if we're looking at let's go back to the time independent disorder equation. Okay? So we seek the eigenvectors so now there are a special class of cases where the Hamiltonian is this form. So suppose H is the Hamiltonian for multiple degrees of freedom. Particles, spatial dimensions as much as we like. Okay? So H acts on our composite open space. For the moment let's just focus on two degrees of freedom. Generally. Okay? So if, let's say EG2 if the Hamiltonian is of the form a Hamiltonian that acts on the degree of freedom A tensor one on D plus nothing, because nothing to A then we say the Hamiltonian is several. Now it's a little bit confusing that with respect to states a separable state was a product of a state on A and a state on B. Okay? But in Hamiltonian land, or in operator land this is a different notion of the term separable. Because this is not Hamiltonian 8 times Hamiltonian B. It's the identity of the other one, right? It's got this sum structure. But that, those words are used. If the Hamiltonian is separable that means that it's eigenstates our product states from the reason we just said it. Okay? So that's why it's called separable. Because you can separate the eigen vector as the product of eigenvector for Hamiltonian A and the Hamiltonian for Hamilton blah, blah, blah, blah, blah. Right? So that we can write energy B, A, B, B as some these are the eigenvectors. eigenvalues are just the sum of the eigenvalues associated with the two degrees of freedom. Now if we have a separable Hamiltonian the Hamiltonian can be separated as a Hamiltonian acting on a degree of freedom A and a Hamiltonian acting on a degree of freedom B then what about the unitary evolution, the time evolution? So let's look at that. But let's say we have the time evolution operator we call this HAB it's the Hamiltonian that acts. This is nothing over space, it's the Hamiltonian. It's got the hat on it, it's not the scripting age. So now I got my time evolution operator to say it's the time independent Hamiltonian and generally speaking we know right? But let's suppose H is separable. Then I can write this as a Hamiltonian acting on a degree of freedom A and I'm just going to write it out in its full pedantic glory T plus exactly. We can write that as the product of the exponentials because this commutes with this obviously it does and as I emphasize the whole point is that observables associated with different degrees of freedom commute and because they can use that means I can write this as a Hamiltonian and I'll be about the identity part and that product now is of this form. So this is the unitary operator acting on Hilbert space A generated by Hamiltonian HA and this is the unitary evolution UB generated by Hamiltonian UB and the joint unitary is the product of the unitary so this is where it gets confused. The Hamiltonian is a sum but the unitary is a product and we call this a separable unitary. Why is this interesting? Well, if this is true then what it says is that if I start with a product state, so if at time T equals 0 the joint state is the system state is a product state which means there's no correlation in space A and K in space B then what is what can we say about the state of the later time? Well, we applied that yet so I get UA acting on psi A tensor with UB tensor acting on phi B and this is just this tensor this it's always a product state so if my Hamiltonian is separable then if I start in a product state for all times I stay in a product state which means if there was no correlation between degree of freedom A and degree of freedom B at the initial time whenever are correlations generated between them which is to say that when the Hamiltonian is of this form the degrees of freedom don't interact so physically to say that the system is separable between A and B is to say that the two degrees of freedom don't talk to one another there's no interactions there's no correlations of freedom yeah I was expecting when you put UA under an eigenstate that you would just be summing the two energies in the same phase they each get their own little phase that's correct okay yeah so the point here is that when we have a separable Hamiltonian separable Hamiltonian implies no interaction physical interaction between degrees of freedom yeah yep we're about to do that indeed I just want to make one nice comment about if this is not true if the Hamiltonian is not separable okay if it didn't have this form and I started in a product state and I did the time evolution then it's not going to stay in a product state in that case we call the unitary entangling it's an entangling unitary because it creates entanglement it creates correlation quantum correlation yeah it would I mean it depends on the interaction between physical interaction so there's physics here there's enam and all that stuff quantum mechanics is not the void of physics right okay so now let's come to the kind of analysis here looking at a specific case let's consider suppose I have a free particle p squared over 2f if say this is in 3b 3 degrees of freedom correct is this separable it's separable and pxp1 is easy we have to say separable with respect to what degrees of freedom okay this is something that is almost always forgotten that what whether this state is separable or not depends on what kind of product structure we are imposing let's say which degrees of freedom are we choosing to analyze the system of respect what if we use p of r p of a that's the next step so we'll come back yeah let this one get at the moment so this of course what this really means is p sub x p on y tends to be the identity on z and this is the x tends to be the identity on z and this is blah blah blah blah why does it order form you have to decide if you're writing the state I'm saying that when I write this, tensor this tensor this, tensor this I have to talk about which Hilbert space I'm talking about so this is Hilbert space a or x you just have to keep track of it and then that's different if I went Hilbert space c, tensor Hilbert space a there are isomorphic Hilbert spaces but in like the same way when you write down a tone vector and you write down the components you just have to know what you agreed on the order I could order them anyway I like but once I write the order I've got to keep saying that that's the order it's just a bookie thing okay but I'm just emphasizing that fact just to say that this in fact is of the form I wrote and this is separable so there's a Hamiltonian that acts on the x degree of freedom there's a Hamiltonian that acts on the y degree of freedom and there's a Hamiltonian that acts on the z degree of freedom okay and the joint eigenstates of the system are just the product of the eigenstates associated with those degrees of freedom so I can write an energy eigenstate as px, tensor p1 tensor pz the eigenvalue associated with these three eigenvalues is the sum of the sums associated with each degree of freedom this is what we said I'm doing this in speuciating formalism you just sort of know this we've done this a million times I'm just showing you how it all fits together whereas I should just put a square root of qn and what is the eigenvector in position space it's a momentum eigenstate e to the i they're plane waves the momentum eigenfunctions are plane waves x px is with our usual delta normalization e to the i x px and so the three-dimensional wave function associated with the this xyz is x tensor y tensor z px dy pz px dy notice this is just normal multiplication multiplication by complex numbers and this is the i over h bar pxx plus pyy plus pzz we've got the h bar to the 3 halves or even the i over h bar p dot x so the wave function is a product state one for each degree of freedom because the Hamilton we've assembled let's get to he suggested we have different ways of assigning degrees of freedom to the system I can talk about r, theta and phi those are three degrees of freedom and then I could write I could try to say that to join Hilbert space of the system instead of having a Hilbert space and x degree of freedom y and z I could principle do this is my Hamiltonian for the free particles separable in that and yet it actually is because of course the momentum squared is equal to the artwork of p if you write if you thought about it as a Lafacian there's that act on the radio coordinates by coordinates and data coordinates of course this is just from you know from classical mechanics the kind of momentum squared over r squared yeah but I mean there's also a term in it though so that doesn't mean it's dependent on r though sure that is a good point here it is dependent on r in that way that is to say you can write eigenfunctions which are in this case separable so a free particle eigenstate I would say there is some radial part that depends on r tensored into some eigenvector that depends on theta times perfectly good because I can separate that these kinds of eigenvectors are called partial weights so separation doesn't necessarily mean that it's not dependent okay right so let's look at some example now with respect to the some potential so because the system is always separable of the kinetic energy for a particle moving in some number of spatial dimensions or some number of particles moving in some number of spatial dimensions whether or not the Hamiltonian is separable depends on the potential because we can always separate the kinetic energy we can do it in spherical coordinates we can do it in cartesian coordinates but the question is about the potential okay so when we have a Hamiltonian which is p squared over 2m plus the potential of the function position then separability depends alright so let's consider for example a particle in a box so what I mean here let's say a 2D box so let's just talk about two spatial dimensions both my potentials are the form that it's 0 if it's inside of the square and it's infinity outside so I have a box where the potential is 0 in here and everywhere out here it's infinite behind is this separable meaning can I write this as a sum of a potential on the x-coordinate and a potential on the y-coordinate as I can it is this is equal to v of x is 0 if x is between 0 outside and the same thing with so this is a separable potential okay what if I had a finite potential well suppose that this were true suppose I had something like that where everywhere outside here the height is v0 is this separable I mean what if I wrote this is that the product is this plus this that potential no it's not right that's the one that would be you know it would be v0 v0 2v0 2v0 I think right so that's not this potential is not the one I wanted the one where it's v0 and everywhere is not separable okay this is not separable now let's go back to the infinite case what are the eigenfunctions and eigenvalues of the Hamiltonian exactly so for each one of these guys we have now my energy eigenfunctions they're specified by two quantum numbers for each one of these guys it depends where I put this guy typically I write in terms of even an odd this is cosine and i xi over l I mean actually it's important I'm going to make this not a square let's say it's a rectangle let's start with that case so let's put that and then I have this with the sine one or two mean x or y for even parity in the odd parity the energy let's go to another photo energy is just the sum of the energies associated with each of those for the energy E and x E and y hr square over 2m k x of nx squared for k to meet the boundary conditions for the i coordinate equal to that's how it meets the boundary conditions as we know so this energy is hr square over 2m pi squared I'm going to factor out it lx squared nx squared plus ny squared lx squared so now one thing we see here is that whether or not there are degeneracies in the system that is to say if there are two different eigen functions that have the same eigenvalue depends on something about the ratio of these guys so for example if the ratio of lx and ly is rational then we have degeneracies ok so for example suppose that lx is y's ly suppose that were the case I had that kind of reaction right then e 1 1 here is the same thing as what well this is 2 squared right so if this is 1 and this is 2 that's the same thing if this is 1 right those two are the same they're both equal to 5 times that mass hr squared over 2m pi squared over lx squared because it's 1 plus this is 1 get that backwards well whenever nx is equal to twice ny and this is and it's it's hard to do that this is the case where I should look at my notes no I have to go the other way around so the other one is the same thing the way I've written it is this like ly is 2lx hold on a second you just do 2 2 on the left side and then it should be I think 2 1 2 1 which way do we want to do this ly is we want so let's say that ly is 2lx it doesn't matter which way I do this right then then we have that the eigenvalue squares here are proportional to and it's a constant times this divided by 4 right so if nx equals 1 and ny equals 2 then the energy in this case is this constant times 2 right now is there another case where that's true the other way around yeah n equals 2 nx equals 2 and y equals 1 why am I screwing this up so good I think a higher I'm just going to switch over algebra this looks better your journals can't submit anything you know I could see this here I'm going to come back because I can't see the algebra right now and obviously you can't do that so I'm going to just let this go for a moment and just say the following which is are degeneracies in this system I'm just screwing it up so now but the point is here that degeneracies are called accidental degeneracies although I can't seem to find an accident meaning that you know if it just happens to be that this were a rational number there can be an alignment of these two things now there's a very simple case where we can automatically see degeneracies and that is suppose that lx equals del y please 2, 2 and 4, 1 so nx equals 2 and ny equals 1 where ny equals 2 also and then the other one is 4 and 1 4 and 1 so in this case we have for this case 4 plus 4, yep that was thank you yay but that's an accident and I couldn't find it is that why they call it that? yeah because it's not in any way obvious it just happens to be that it just happens to line up but if this were true well then it's obvious if this were true then this is what we call if this is the case but this is a indication of is symmetry the reason for this kind of degeneracy is the fact that if I have a square potential then I have reflection symmetry that is to say that v of x, y is equal to v of y, x that is to say if I flip these coordinates the Hamiltonian is the same so if I have a symmetry operator let me just call it the reflection operator and it's I'll call it pi since x into y and y into x and px into p y and p y into p x the Hamiltonian is unchanged because the potential is reflection symmetry so if I have a reflection symmetric potential then the Hamiltonian is invariant under the reflection symmetry operation which means another way of saying that is the Hamiltonian commutes with reflection which means that the eigenfunctions there exist simultaneous eigenfunctions of the reflection operator and the Hamiltonian so um if nx and y is an eigenfunctor this way with the same eigenvalue that must be true yes just a tensor product so as I say often we just leave out the o times here it's just too annoying to write it all the time if I just write two kets to one another that's the tensor product is it kind of like if you were like to put in a projection into x you just have the two functions correct it's exactly what it is that is to say the joint eigenfunction of the system is equal to this eigenfunction and that's the eigenfunction that's what it means and so what is this eigenfunction see this acts on this which is x and y so that's u and x evaluated at y and u and y evaluated at x that's the reflective one now I claim that this these two functions are eigenfunctors simultaneously eigenfunctions of the Hamiltonian the same eigenvalue they are degenerate you see that because so now I'm just going to simplify notation even further just specify them by their eigenvalues the Hamiltonian acting on this with some energy what is acting on the reflected ket same energy but the right it's exactly the same energy because these commute because they commute I can move the Hamiltonian to the other side this then is the same energy eigenvalue so this is an important lesson the point is that if we have a symmetry in this case we have reflection symmetry if there is a symmetry of the Hamiltonian then there will be degeneracies symmetry and degeneracy are interrelated to one another because there is a symmetry operator that this commutes with there are a set of different eigen vectors which share the same energy eigenvalue that's sometimes instead of called an accidental degeneracy this kind of the degeneracy is called an essential degeneracy let's do another example before our time is up here let's talk about a particle in a simple harmonic oscillation well in 2D do you mean that it has a one half kx squared or a one half ky squared with a potential or is it a derivative? exactly so let me explain that so for example here is a simple which is a classical picture let's say I have two kinds of spring I have a particle that's attached by springs to a wall and there's a spring constant kx and the spring constant ky so I can pull it in either direction and there is a resonance frequency associated with that spring constant and a resonance frequency associated with that spring constant okay? what is the Hamiltonian for this? is this a separable Hamiltonian in x and y? sure is what are the energy eigenvalues and eigenfunctions of the system so the energy from omega x times n plus a half so there are two quantum numbers in n and y okay? again there will be there can be accidental degeneracies in this system but there's a special case that I just want to conclude the isotropic say in 2D let's say that the two frequencies were the same the springs were the same so it doesn't matter how I pull it it has a restoring force in that case which is just proportional to the distance from the origin it doesn't depend whether I'm pulling it along x or I'm pulling it along the y it just says as long as I pull it some distance r from the origin it feels the same force it's isotropic okay? in that case we have something special well first of all the energy in that case let's call that omega we see this guy has degeneracy right? so I have energies let's just make a little table here nx and ny so nx is 0 and ny is 0 the energy is I'll just call it 1 okay? when this is 0 this is equal to 2 etc when this is 1 and that's 0 f2 3 now when this is 1 and this is 1 then I have 3 when this is 1 this is 3 I'm sorry 4 5 etc so we see the degeneracies here there are many different combinations of nx and ny which in fact we could write this I mean they're always integers so the energy here is hr omega times sum n plus 1 n can be 0 1 2 3 and what is that degeneracy? n plus 1 so there's a degeneracy in this system which means that there's really another point on the m but this guy is completely independent of that point it only depends on n and not n what kind of degeneracy is this? is this an essential degeneracy? or an accidental degeneracy? it's essential it's associated with what kind of symmetry? it's true it has reflection symmetry but in fact this guy had even more symmetry than that let's look at the Hamiltonian again p squared over 2n plus 1 half n omega squared plus y squared this is my so let me coordinate in row this is p rho squared plus p phi squared right? what kind of symmetry does this have? remember this is just the rate this is just x squared plus y squared it's rotational symmetry the potential is independent it's out of a potential here it's independent of phi it's admiratively symmetric it's rotationally symmetric this has rotation symmetry it doesn't matter where how much I pull it from the origin at what angle it sees the same force okay? so this has rotational symmetry there's a rotation operator for the function of phi and my Hamiltonian is in varying respect to rotations around the z axis we see that classically because the potential is independent of phi the conjugate momentum is conserved right? remember that from Lagrangian mechanics so there's a conserved quantity what is that conserved quantity? it's this which is of course angular momentum around the z axis the angular momentum around the z axis is conserved because we have no torques because the potential is independent of phi which means that there is a symmetry conserved quantities and symmetries are one and the same and quantum mechanics are also related to degeneracies we will pick all that up and talk about angular momentum for many weeks to come