 I'll be available. So last time we were putting mechanics back in quantum mechanics and talking about the quantum mechanical description of the dynamics of particles, which our players now are positioned in momentum. We understand our connection that unifies the classical and quantum description is that momentum is the generator of spatial translations. It's the thing that's conserved if we have a spatially-translational invariant system. And so from that we can say that because this is the generator, we wrote down a near-identity form of the translation operator for a differential or very small translation as the identity and then a little bit generated by momentum. And so for finite translation, that tells me that the translation operator is this exponential of that anti-translational operator. And that translation operator acts as a unitary transformation that unitary to enact this translation position. Equivalently what we said that this generator implies that it must be the case that position momentum don't commute. And that commutator that follows is the canonical commutation. Now I should mention that we can kind of turn this around and look at it. We just look at the commutator in the other order. Now what this tells me is that minus position is the generator of translations in momentum so that exponential here with the position operator in the exponent acts as a translation on momentum. The position's the generator or minus position is the generator of translation momentum, momentum is the generator of translation in position. Alright, given those operators, those submission operators, we can write down representations by looking at bases. So the set of eigenvectors of position momentum are now represented by eigenvalues that are continuous variables. It can take on any real value. And so in that case instead of a sum over the screen possible values of vector, we can have integrals over those eigenvalues that form the resolutions of the identity. And we can express then the representations of vectors in Hilbert space as now functions, complex functions over the real line. So we now have instead of a column vector in the representation, we have a function. And the function is a representation of the vector. And we have in this case two different representations, the position representation and the momentum representation. They are representations of the same state in two different bases, the position basis and the momentum basis. And since we have this continuum here, we don't, if we want to talk about a projection into some finite region, then this object represents a projection operator that projects the system into a differential region of the real line between x and x plus dx. So that's the projector in that slice of space. And so by the Born rule, what we know is that the probability to find the particle in that region is the expected value of that projection. And that is given then here. So that then allows us to interpret psi star psi or the magnitude squared of that wave function, that probability amplitude as the probability density to find the particle at x. Okay. I should say we also saw here that the action of the translation operator on x is to translate the ket. And I can normally say the translation on the momentum on the conductor is to translate the momentum. Right. So the states thus can are generally in a Hilbert space and we want those states to be normalizable and so this inner product should be a finite number. We typically set it to be one. And that we can express in the position representation or the momentum representation. And so this thus defines the set of allowed functions that define the Hilbert space. The allowed functions are the ones that when I integrate them, the square of them over the real line, they must be finite. And that finite number we just said is set to one. So the space, the Hilbert space that we are discussing is the, the spined by the set of complex functions over the real line, which are square multiple lines. And that set forms a Hilbert space. That's really what Hilbert space is about. Hilbert space isn't about, in some sense, you know, two-by-two matrices. It's about this uncountably infinite set of functions and how they form an effective vector space with an effective dot product or inner product. Right. And that has a technical name we call it L2 over R, it's a square normal line. The, we can look at, for example, now, so we have these representation, these wave functions. We can look at the, say, the representations and say position momentum space of these eigenvectors. Okay. So, for example, the position eigenvector, let's call it size x naught. The position eigenvector at x naught. If I want to know what is the wave function in position space associated with this can, it's the inner product of this with x. Right. But that is this, which is zero unless x equals x naught. And it's infinity, actually, if x equals x naught. It's a Dirac delta function. All or I could look at this guy in momentum space, the momentum space representation of a position eigenvector. We calculated this last time, that's given by this function, which is, of course, a plane wave. So, if I were to look at, say, the magnitude of this, of these representations, then a position eigenvector is a delta function localized at x naught. And so in momentum space, that is completely delocalized everywhere in momentum. And this, of course, is not, this state is not the Hilbert space. You can't ever prepare a system. It would have an infinite spread of momentum. Of course, this is a reflection of the uncertainty principle. Of course, the opposite is true in momentum space. If I add a momentum eigenvector, then a momentum eigenvector, if I ask what is its wave function, what is that? That's a plane wave. So if I'm in an eigenstate of momentum at the plus sign there, as a function of x, that's a plane wave. And we have this kind of pseudo-normalization. It's not really normalizable, but to keep it so that it has the delta function normalization, we have that factor there. So this is a plane wave. By the way, what is the wavelength of this plane wave? We usually rewrite a plane wave as e to the i k x. And the wavelength is, how is the wavelength related to k? 2 pi over k. 2 pi over k. Thank you very much. And in this case, k is p not over h bar. Writing for this, that's to say p not is h bar k. And so the wavelength here is 2 pi h bar over p not, or the plane is constant over p not. This, of course, is the devaluation that a particle with a certain momentum has an effective wavelength given by a plane's constant a little bit. Okay. Now, we were talking about last time the change of basis. So if I wanted to know, if I had the position representation, I wanted to get momentum representation. Well, what I can do is a unitary transformation by inserting a resolution of the identity. Right? And this we just said was e to the minus i dx over h bar. Similarly, if we wanted to go the other way around, I can insert a resolution of the identity with respect to the momentum basis. And this, of course, is nothing more than a Fourier transform. And as we discussed last time, we should think about the Fourier transform as effectively a change of basis. We're going from one basis, which is the position representation, to another basis, which is the momentum representation, or vice versa. And the change of basis matrix is the representation of one basis vector in terms of the other. Does anybody have any questions so far so good? All right. So what we're talking about here, thus, is wave mechanics. So wave mechanics is the way in which we talk about our quantum theory as described by wave function. And we can think about those wave functions. We're showing you're developing this in the context of understanding what we know about classical wave theory at first. But then we later understood this to be just a part of a more abstract theory of Hilbert space in which wave functions are just one representation of an arbitrary ket. So let's see, what are our players here? So we have the position momentum representation. Now, we're in, if we were to talk, yeah. Why is a basic changer of the ego to have a non-subjective as well? Because that's a... So if we talk about, for example, the inner product between two kets, then that has an equivalent in wave function language. How do you project one wave function onto another? Well, we know how to do that if we wanted to write it out by just inserting the resolution of the identity in x. That is, I take the complex conjugate of the function and I multiply it onto x and that's equivalent to projecting phi onto psi. And that's what's going on here. If I wanted to find the momentum representation, what I do is I project my state onto the momentum eigenvector and this is the momentum eigenvector represented in position space. And it's got the minus sign because it's the conjugate and that's how you remember it because the plane wave is e to the i k x and it's, when you project, you take the complex conjugate. Now, of course, if you have frequency in time, it's the opposite of opening up after a moment, but the minus sign would be the t instead of the e to the n. So that's that. Let me get my notes here to make sure I remember everything I wanted to say. Formalization. Now, matrix elements. So suppose I have some observable a matrix element. Well, again, I can write this in the position representation by putting in resolutions of the identity. This is observables in position representation and momentum representation. We haven't talked about that. Let's do that. Let's talk about the position operator and momentum in position representation. That's to say on wave functions. So if I look at the position operator, whether it's matrix element with respect to x and x prime, well, x on x prime is an eigenvector. So this is, or the other way around, matrix element. Well, what that means is that when x operator acts on a state and I want to know what is the new wave function after I've done this operator. Well, that's this. This is the new wave function, which is x. This is an eigenvector. It's a termition. It doesn't matter if I go to the left or the right. So it's this multiplication by x in the position representation. The x operator acts like a position operator is represented by x multiplied by x. What about momentum operator? Well, that's true. If I look at the momentum operator, what is its representation between these two cats? Well, we don't know how momentum acts on this. We know how it acts on momentum eigenvectors. So the way to deal with this is to insert a resolution of negative and just fit that. It's an eigenvector of a momentum operator. This is e to the plus i xp. This is e to the minus i x prime. So this together is e to the i p x minus x prime. What the heck is that? If there was no p here, that'd be a delta function, right? Direct delta function. But this messes it up. However, what I can say is that if I take this over here, it's equal to 2 pi h bar over i, the derivative with respect to say x of e to the i p over h bar. If I take the derivative of this with respect to x, it brings down an i h bar. Oh, sorry. Now the i h bar cancels that. I've got to keep it. Yeah. Is that h bar? Yes, it is. So what do I see here? Well, this tells me that this is equal to h bar over i times 1 over 2, or I'll factor out the 1 over 2 pi h bar. I've got h bar over i. I can factor the d by dx outside the integral because this is an integral over p e to the i p over h bar x minus x prime. In fact, let me bring the h bar over here. This integral, if I change p as h bar k, this is an integral dk e to the i k x minus x prime, which is equal to... Is it directly to the x minus x prime? And it's going to 2 pi. So putting that all together, the 2 pi's cancel. And we have that this matrix element is equal to h bar over i, the derivative operator acting on the delta. The moment operator in weight function land is the derivative. And so we see that if I now ask the following question, suppose I have a state side and I operated on it with the p operator. So what's the new weight function? That new weight function is this. That's the weight function after I've operated with p, right? And again, if I now insert a resolution of n and p, this is equal to the integral h bar over i, the partial derivative with respect to x, x times x prime. This, of course, is derivative acting on this. So now I'm going to play a trick. First of all, this is derivative of the delta function. What the heck is derivative of the delta function? The derivative of the delta function is this integral. That's what it is. And it has, in particular, it only makes sense inside the integral in the same way that only a delta function only makes sense inside the integral. It is a temperate distribution, as we call it. Now, the way I can do this is I can make, this is an odd function if I change x to minus, if I change x to x prime, there's a minus sign here. So I can make this minus the derivative with respect to x prime. Because I just used a chain rule with the sign. This is x minus x prime. And now I do integration by parts. Do you know what integration by parts is? Integration by parts means throw the derivative on the other side. That's what integration by parts means. If the surface vanishes, that's to say if I have an integral from minus infinity to infinity of the derivative of x dx, this is equal to minus the derivative of f of x the derivative of the other guy if the integrand vanishes at the end of this. Forget about your UDV, EDU, whatever you learned in Calc I. Forget. This is integration by parts. Throw the derivative on the other side and put it on the other side. Try it. So integration by parts. So in this case, I throw the derivative on to the other side and put a minus sign. So that says that this is equal to the plus sign at all x prime. This should be the x prime part. That's my deli variable. x prime h bar over i. And now doing the integral that says x equals x. So in a very long-winded way, we derive what we know. The momentum operator in wave mechanics is the derivative of respect to x with this h bar. So what this says is that in position space the action of the momentum operator in a state is represented by the derivative. Of course, we can do it the other way around in momentum space. The momentum operator acts as multiplication by the momentum value. The position operator acts as derivative of the other side exactly the same way. So those are the representations of position momentum operators in wave function and in wave mechanics. Now the fact that the momentum operator in position representation is the derivative should not surprise us. The reason it shouldn't surprise us is that momentum is the generator of translations in position. Suppose I have some function of position and I translate it by a small amount of dx. Well, by Taylor's theorem, this is equal to the function of the x plus dx times the derivative, in this case it's one variable, but the heck, I'll just write it as partial because it matters when we variable it. That's Taylor's theorem. It's just subtracted and divided by dx. That's the definition of the derivative, right? But look at what that is. Let's, for fun, let me multiply this by h bar and then write this as, well, look at what that is. This is equal to f of x plus i over h bar times the momentum operator acting on that dx. This is exactly what we worked out. The infinitesimal generator of translation in position is the derivative. So the derivative operator is the thing that translates you in position by a little slope. So it's not a surprise that the derivative operator is a translation in that variable. And I'll leave it to you to show that this thing, which is the exponentiation of that, generates, makes the whole Taylor series. That is to say this translates you by an arbitrary amount. The first word then is dx and then you have second derivatives, third derivatives, and that's the whole Taylor series. And this is e minus i x. So we have now in weight mechanics we have weight functions, we have inner products, we have matrix elements, we have representation of operators. Let's talk now about the shorter equation. So the shorter equation, the time depended on h bar over minus i over h bar with respect to t on the side is the Hamiltonian. Now what's a Hamiltonian? If we're talking about particle mechanics, then the Hamiltonian is kinetic and potential energy. So kinetic energy is p square root of 2m and then we have the potential. So if I now write this in the, in my time depended state, this is in the shorter depiction. So I can write this formally and just shove in a ket there, shove in a ket there. This is thus a partial derivative with respect to time of the wave function, function of x and t. And then I have to have the momentum representation, the momentum operator represented in x is the derivative with respect to x. So this becomes 1 over 2m h bar over i squared of partial squared plus v of x acting outside. Yes, sir. I mean, I don't think that, you know, notice that like, there's a side of t, but there's no t operator that you can swish onto that to a 2m outside. It's already kind of like that. So this is a subtle point that I never really dealt into. What is time? What is, how is, how do we treat time quantum mechanics? In non-autovistic quantum mechanics, time is just a parameter. It's the parameter at which we ask at this time what is the probability for this at the other time. So it's a, it's the arena, it's just a, it's a parameter. There is no Hermitian observable. We can't ask what is the time of the particle. Time is about, in non-autovistic quantum mechanics is about us. At this particular incident of time we do, we do a measurement, what is the position, what is the momentum, what is the state. So time is a parameter that parameterizes states in the Schrodinger picture or parameterizes observables in the Heisenberg picture. But it's not an observable. Now, there's something weird about that because of course in relativistic theory, space and time are interrelated. So what the heck? How can time and space have different roles? What happens in relativistic quantum mechanics? Does time suddenly become an observable? The answer is no, position gets demoted. Positions no longer talk about the position of a relativistic particle. Now we'll come back, I mean, in some sense we measure time, we're poorly in this stupid lecture hall, but, you know, so we'll come back to that question. What does it mean to measure time? But from the point of view of the mathematics, your time is not an eigenvalue of an observable, it's just the parameter. Very subtle. Okay, so this is a dependent Schrodinger equation in wave mechanics. Let me just do a quick little aside. I talked about the Schrodinger equation in this case, or we've talked about representation just in one spatial dimension. Let's talk just quickly. We can write down the results in 3D. Okay, in 3D we have now position, which I'll call x to the vector. So this is x hat, y hat, z hat, and the momentum operator has the x operator of y momentum operator. These are called vector operators. And we can have position eigenvectors which have vector numbers, eigenvalues. There are three eigenvalues. x, y, and z. And these satisfy the same kind of orthogonality relation now with a three-dimensional delta function. x prime delta y minus y prime delta z minus z. We can have resolutions of the identity integrated over x, y, and z. And we can have wave functions as a function of three-dimensional space which are representations. And we can have normalizations. So now this is L2 and R3. The action of the position operator on a vector in position space is multiplying by that position on the wave function. And what is the momentum operator in this case? How does it function in position representation? The gradient, right? This is h bar over i the gradient. So in 3D, one-stage dimension and three-stage dimensions this becomes h bar minus i d by dt. The wave function in three-stage dimensions is minus h bar over 2m the reflection of psi of z of x. Finally, next, of course we can talk about the time independent order equation which is the energy eigenvalue equation. Independent is just another name for the eigenvalue equation for energy. And in position representation of the Hamiltonian square over 2m is minus h bar squared del squared over 2m plus the potential on psi is equal to e. Let me call this u just to emphasize that this is not a general state but it's an energy eigenstate with some eigenvalue. The energy eigenfunctions. So that's the time sugar equation as a wave equation. Of course, we know in a simple case we can solve instantly what you know which is the free particle. That means no forces. No external forces. That means the potential energy is zero. In that case, the Hamiltonian is just e squared what you had. And the sugar equation if I rewrite it looks like this. So I set the potential equal to zero and just remove some stuff. What are the solutions to that equation? Well, there are lots of different solutions. By the way, this equation has a name. Do you know what this Pb is called? It's the Helmholtz equation. It has lots of different kinds of solutions. But there's one solution we can write down instantly which are plane wave solutions. How do we know that plane waves are our solution? Well, their solution is to just plug them in where if I plug that in because it's lost in on this guy is minus k squared. So that means that k squared is equal to 2ne over h bar squared or the energy of our squared k squared 2ne. We knew that from the start because of course for a free particle that Hamiltonian I'm going to write a free particle that Hamiltonian commutes with the momentum operator which means that eigenstates of p are eigenstates of h or at least there exists as we know we can always construct eigencommon sets of eigenvectors if they can use it. There exists common eigenvectors Let me write it that way because it's a subtle point that I'm going to emphasize in a moment. There exists common sets of eigenvectors of the momentum operator so the momentum eigenvectors are and they are eigenvectors of the Hamiltonian too with eigenvalue p squared over 2n which is equivalent to saying that the wave vector h bar k is the eigenvalue or I could say that there is degeneracy degeneracy what set of different eigenvectors have the same energy eigenvalue that's what it means to be degenerate different vectors that have the same eigenvalue the same magnitude of momentum exactly so the direction is relevant it's only the magnitude so any plane wave which has the same magnitude of the wave vector so the energy eigenvalue so what that tells me is just specifying the energy doesn't specify the eigenvector because could be that plane wave so how do we know in our first weeks of class we discuss that in order to fully specify an eigenvector if there is degeneracy we need to specify a complete set of commuting operators so in 3D we need three mutually commuting operators to fully specify the state what are they what are the ones that specify suggestions could be that do we use the special or XYZ that sounds good what XY and Z position X position Y position Z that doesn't commute those are vector close PX PY and PZ they all commute one another and they all commute with the Hamiltonian so that is if we specify that then the unique eigenvector which is simultaneously an eigenstate has the three eigenvalues so the unique eigenvector which is simultaneously eigenvector those three operators that state as these three eigenvalues which we lump together in a vector and these are plane waves that is to say these vectors are plane waves and typically we normalize them with delta normalization we will see next semester another set of three commuting operators if I have if H is only a function of the magnet rotationally symmetric potential spherically symmetric potential oh sorry we're talking about a free particle never mind in a free particle I have another example because that is rotationally symmetric a Hamiltonian the magnitude squared of the angular momentum and the Z component we will see that this is all commuting with one another you showed that in your first homework assignment or second homework assignment that LZ commuted with p squared you know squared so these three commute for free particles but they're not they don't commute with momentum so these are three different sets and these things are instead of plane waves these are spherical waves the Helmholtz equation but they're not plane waves and they're not eigenstates of momentum they're eigenstates of this stuff and these things have a name also they're also called partial waves and they're very important in the theory of scattering we're going to talk about that next semester so just because it's a free particle doesn't mean it's in a plane wave if it's an eigenstake of energy it could be a superposition of different plane waves all of which have the same length of the vector but different directions any superposition of that sort is also an energy eigenstate because you have degeneracy and a superposition of degenerate vectors has the same energy eigenvalue if I look at the solution to the time dependent equation for free particle so if I initially start in an energy eigenstate a momentum plane wave only know generally solutions are stationary states they have that frequency the energy itself is a function of the momentum though so my energy is a function of momentum is p squared or a free particle in a plane wave eigenstate this is just the frequency of the wave this is h bar omega that's the frequency in time so this is e to the i p over h bar minus times minus v to the right of this x wave vector k dot x k so that's a free particle this is a plane wave notice the relationship here between frequency and wavelength is known as the dispersion relation so e equals the energy relation using the wave particle duality omega that's a function of k is h bar k that's the dispersion relation for a free particle and given the dispersion relation we can talk about phase and group velocities so the phase velocity how is that possible omega over k wave theory and so that's equal to h bar k over h should the k of that dispersion relation yeah that's thank you sorry that's why that's screwed up that's right and the group velocity is what now what the fact that this is not a linear relation means that the phase velocity depends on the wavelength which means that different waves of different wavelengths move at different phase velocities which means the wave disperses that's why it's called a dispersion relation okay now I ask you a question because this always confuses people which people forget if I have an electromagnetic wave in free space in the vacuum does it disperse in the vacuum because it's a speed of light all it's only when you're in a medium where as an electron if I thought about it as a wave an electromagnetic wave in the vacuum it disperses because it has mass that's the difference between dispersion this is the non-autonomistic for a photon that's a linear relationship between omega and k that's why this doesn't disperse what this does so next suppose that I look at the this general case for the time-dependent record equation in free space so I'm given this which is to say I'm given a wave function at time t and I see the wave function at some later time this is not necessarily an energy ion state some arbitrary part and I want to know what is the solution how do I solve the time-dependent record equation well a generic procedure is the following take the initial state decompose it into energy ion vectors you know how every energy ion vector evolves and then resum them right so my initial state at time t equals zero can be expanded in terms of energy ion in this case the energy ion values are continuum of energies energy goes from zero to infinity where the energy is given by that all this thing c sub t and so my state at any later time is that clear so in this case e and p are just related so I can write this as the integral over momentum remember the energy is p squared over 2m so t is just the square root of 2m e so I can equivalently write this as the integral over momentum the magnitude of the momentum e to the minus i t squared over 2m hr times time momentum so in terms of functions what this says is that the weight function at arbitrary later time is the integral over all the momentum p well c of p is this which is the projection onto for a transform it's the minus sign what have we done here what we say here is that and I can write this is really just another name for the momentum space weight function at time equals zero if I know at the momentum space weight function at time equals zero then I know the weight function is at any time because every one of these guys just gets a face if time equals zero these of course are equal this is just an inverse 4a transform but at a later time well each momentum component picks up the space factor and this will generally lead to this version of a weight packet which is the spreading so if I start with a localized weight packet in position because it has a spreading momentum and because each momentum has a different phase velocity though different momentum to get out of step with one another and the weight function let's do this in homework alright so you're back to the future we're back to weighted mechanics without you and so I'll connect