 It's been a long fall break. Lots of good stuff to talk about. All right, so we're going to have homework due today. So what we're talking about right now is weight mechanics. Weight mechanics is really what we're looking at is quantum mechanics in the position for momentum representations. And the physical context we're looking at is thinking about a particle of some mass and moving a single particle at the moment, moving under the influence of some conservative potential. And the Hamiltonian that is a familiar form for mechanics, kinetic and potential energy. And the Schrodinger equation then takes a form of weight equations. So the time-dependent Schrodinger equation, as we've derived it, takes a form of a time-dependent weight equation. And the time-independent Schrodinger equation, which tells us about the stationary states, i.e., the energy i in the system, takes a form of a time-independent weight equation of this form. And if we think about, as we discussed, the case where the particle, the free particle, where there's no force, no external potential, then the Hamiltonian is just kinetic energy. The time-independent Schrodinger equation takes this form, which is a familiar weight equation. It's the Hamiltonian's equation. And we can instantly write down the eigenstates of the Hamiltonian, the stationary states, because we see that this Hamiltonian clearly commutes with all components of momentum. And thus, the eigenstates of the momentum operator are also eigenstates of the Hamiltonian. So that's a way we can write down instantly the free particle stationary states. So they are eigenstates of the Hamiltonian with this eigenvalue. And the form of those, the wave function is, of course, the position representation of the momentum eigenvectors are plane weights. So from this, we see that if we have a free particle, a free particle, we can talk about free particle eigenstates as plane waves with definite momentum. And those plane waves, of course, we remember from our study of waves, have this form where this parameter is the wave vector. And thus, we relate from this the De Roy relationship, which relates the momentum to the wavelength or the momentum to the wave vector, the wave vector to the wave vector. And the relationship between the wavelength of the particle and the wave number is here. And that's the De Roy relationship. And from this, we see for a free particle, we have redrawed the dispersion relation that tells us the relationship between the frequency and wavelength or frequency and wave number. And that is really so the wave particle duality expression tells us that the energy relationship between energy momentum translates in the wave language into a relationship between the frequency and the wave or the frequency and the wave. And for a free particle, we have this kind of quadratic relationship, which tells us that a free particle for a non-relativistic particle of mass m has a quadratic dispersion relation, which means that the waves disperse, different wavelengths move at different phase velocities. And because of that, if you have a wave packet, which is has some spread in momentum, it will disperse. And you're finishing up your homework on that. You haven't already done that. Any questions? So I want to continue here today to get some deeper intuition and understanding of the nature of the wave function and the relationship of wave mechanics, quantum mechanics, and classical mechanics. So let me, of course, seen some of the most fundamental thing, the way we think about the wave function. Of course, if we have a wave function, in this case, a particle moving in 3D, and I look at the square wave function, well, this we have interpreted as the probability density. That is to say, if I look at some small volume, that this is the probability to find the particle in a projective measurement in the region within this. Of course, we can think about this if we like, this is the representation of the state in that position representation. Or, more generally, if I have a density operator, meaning I have a mix of a, what is the probability density to find the particle at position x if I had it right? So we know from our studies that to find the probability, it's the diagonal matrix elements of the density operator tells us the probability to be in that state. In this case, it's the probability density. We can also, though, talk about the octagonal elements in the position representation. Good. How would you interpret that? Yeah, they're the coherences. And what do the coherences mean? It's something about interference. It's the possibility of seeing interference between these two positions in space. So that's to say, if I set the way through two slits, one that adds to one of the x prime, I would want to know whether there was coherence between those two points, and whether I would see on a screen interference. So the octagonal elements tell me, I mean, if this were a pure state, well, this is, for pure state, this, which is the product of psi nx prime and psi star and a different x. So this is telling me about the coherences. And if I have generally a mixed state, it could be the case that the coherence length, the distance over which the wave function exhibits coherence, need not be the same thing as the uncertainty in the position. I can have a situation where I have a beam that has a big uncertainty where the particle is, but it may not exhibit coherence over that. OK, so that's one part of the interpretation, the probability of density and the coherences. Let's take a look, though. So let me call this probability density here, just for this one. I'll call it rho of x and z, that's the diagonal matrix outcome. Let's look at this, whether it doesn't really matter where it's a pure mixed state, it's easiest to deal with this for a pure state. We would just have to average over the ensemble of possible pure states if we had a general mixed state. So I'll just focus my attention for the remainder on the pure state case. And let's consider how the probability density changes as a function of time, OK? So that's equal to our state, and I'm going to say if we had a mixed state, we would just average over the ensemble of states. So it's going to change anything. And now we know how these guys evolve according to the time dependent and shorter equation, OK? So this is equal to minus i over h bar, h psi star, plus psi star minus i over h bar and a half on psi, all in the position of representation. Now, this will be equal to a lot. Well, the term that has to do with the potential will cancel, right? Because the star of this gives me a plus i. And here I have a minus i, and the potential term will cancel out. So all I'm left with here, then, is h bar plus times minus h bar squared over 2m Liposkin minus psi star bar squared minus a plus 2m Liposkin. Determining the potential cancels out. All right, so now I'm going to use the following vector calculus identity that the gradient, I'm sorry, the divergence of the vector field that is some vector times the scalar is the scalar times the gradient, I'm sorry, the divergence of the vector plus the dotted into the gradient vector calc. Again, it needs to be quickly derived with a couple of them. So I'll have to do a case or something like that. In this case, it's just a line. So using that, the following course. It's the continuity equation. Thank you. This is the continuity equation. It's familiar from fluid dynamics, but probably more familiar to you. Here's the magnetism, where j is the current of charge and rho is the charge density. So this equation represents the continuity equation is a picture in which I think about some kind of fluid. What the continuity equation tells me is that the total amount of fluid is conserved. The total amount of the stuff that makes up the fluid is conserved. And what it says, if I look at this, if I integrate this over some region of volume of space, so if I look at the integral of this over some volume, well, that by the divergence is the flow of the fluid or the flux of the current out of the surface with a minus sign, which is to say, if the stuff inside the volume is decreasing, the only way that can happen is if it's leaving. Stuff is neither created nor destroyed. It just moves around. That's what the continuity equation tells us. It tells us that the stuff is neither created nor destroyed. It just moves from one place to another. That's what the continuity equation tells us. So how do we then interpret that in the context of what we're talking about here? Rho here is y. It's the density of y, not the density of the charge. What is it? It's the probability density. And the j here is defined as the probability current. That is to say, this is, in some sense, the continuity equation in this context is saying to me about the conservation of probability, probability to find a particle somewhere. It's saying that a particle is neither created nor destroyed. It just moves around. And there's a flux of probability if the particle is moving. So let's say that probability density can change with time because a particle can move. Now, of course, we know in relativistic context, particles are created in this way. We have to think harder about that problem what we're talking about when we have the equivalent of the continuity equation in the quantum field theory. But for the moment, in relativistic theory, particles are neither created nor destroyed. And you notice that this came from the fact in the mathematics that the potential was real. If the potential had an imaginary part in it, there was somehow a complex potential, then it wouldn't have canceled in this equation. So we got from here to here because v is real. So one way to mock up absorption of a particle is to put an imaginary part in the potential, kind of like you put an imaginary part in the index of a fraction when you want to deal with absorption. So probability, current, and conservation. So to further get some insight into the nature of wave mechanics, we're going to make a connection to geometric optics through what's known as the iconophore. So what's the iconophore? So I'm going to express the wave function as it's a complex number. It has an amplitude and a phase, a magnitude and a phase. So I'm going to write it as a real amplitude and a phase. Now that phase I'm going to write in units of h bar for reasons that become apparent later. You don't have to do that, but I'm just going to write it so that it has the units. So this is the amplitude phase, the both real functions. All right, that's just an ansatz. And the assumed form, we've got no approximation. We're just going to plug it in. So let's plug this into the, well, before I do that, we could say just a couple of things quickly. Firstly, note that the probability density is just the square of the amplitude. And what we would say is that the surfaces of constant s are wave fronts. Now let's plug this in time-tending. Now it's a bit of a drag to do that, because we have to take full apportion of this thing, right? And then we have to use the product rule on that. Then we have all kinds of vector identities to deal with. But I'm not going to do all that. I'm just going to write down the ansatz. And that's why I'm cheating. Let's do that out from the board right now. You can see what's going on here, right? I mean, the right-hand side, if you take the derivative with respect to time, that's the, you know, that's the, what was the left-hand side, that would take raise over there. You've got the derivative of the amplitude and the derivative of the phase and the derivative of, and then, you know, you've got to use the i thing and I cancel that on both sides. Over here, I have a little plus here, and that has all these different derivatives. You have second derivatives on the amplitude. You have second derivatives on the phase and then all the cross terms. You take the derivative of each phase. So that's where it's going on. You just have to do it all. Okay? All right, so that doesn't look helpful. However, it is. So let's, everything here, as I said, a and s are real functions, which means that separately, the real part of this equation equals the real part of this equation or this side and the imaginary part equals the imaginary part. So we have two equations. One for the real part of this equation and one for the imaginary part. So separating real and imaginary parts, we have the problem. That still is a white look, useful, or intuitive. It doesn't seem like there's a lot yet to be going from that, but there it is. And so what I'm going to do in this equation is I'm going to multiply both sides by a and then divide by 2. Possibly, you can put in all the h-parts afterwards. I'm not going to worry about the h. And the other equation is the problem. So now we'll get, what can we say about these two equations? Well, let's look at this first equation. It's got it. It's got that. But what does it look like? It looks like the continuity equation, right? In fact, it is the continuity equation, because what is a squared? A squared is rho. It's probability density. So this is equivalent to, this is d by dt of rho. It's minus the continuity equation, equation that says that the particle isn't lost or created. But what is j? What is the probability current when expressed in the icon of 4? So look at it. It's equal to rho times this, the gradient of the phase divided by the mass. Now, if I have a fluid and the fluid, let's go back to, and let's say that right here at this position, the fluid is flowing at some velocity at that position. And suppose the local density right at that position is rho. I ask you, what is the local current density at that position? If I have fluid flowing, though there are even no fluid dynamics, you should know this, rho times p. So, it's the following. Probability is flowing with a local velocity. This is the effective local velocity at that position. And what is that? It's saying that the mass times the local, this is the effective local momentum at that position of the flow. So, how do we interpret that picture? Well, what is s? Well, s was the phase. And we said that contours were surfaces of constant s. So, this is a locust of points of surface where s of x and t is constant. What is the gradient there? The gradient are the normals to this. So, the local gradient of the wave fronts represent the way in which the probability is flowing. And these, in optics, are known as the rays. So, if I have a wave front and I want to know what direction the light ray is moving, the light ray is moving perpendicular to the wave front. It's the local ray. Okay, well, moreover, if we look at, now let's look at this equation, which tells us how those wave fronts change with time. If we look at this equation, I'll call this the limit where h bar goes to zero. Let's say, let's consider that first. Let's suppose h bar is zero. Then what is this equation? That equation is this, which is, say, the Hamiltonian evaluated at x and p is the gradient of s, the s, t, t. This equation actually has a name in classical mechanics. Probably haven't studied it, and you probably never will, because you'll never take five over a year when you study it. But it's an important equation in classical mechanics and has a name that's called Hamilton-Jacoby equation. What is the Hamilton-Jacoby equation? Well, the Hamilton-Jacoby equation is another way of solving for the classical trajectories of a particle of mass m, in this case, moving under the influence of this potential. You solve this equation, and what it tells me is that if I find s then p, and solve this equation, then whatever I get for s from this equation, the local momentum at that position of time is the gradient. This function is known as Hamilton's principle function. So what is the takeaway message for you? Well, when it's saying that in some sense, no one will have to make this much more precise, in this limit that h bar goes to 0, I don't have to explain what the heck we're doing with my data, that quantum mechanics, in some way, has as its limit classical mechanics. That's to say that the particle moves with a momentum that's given exactly from what you would get from classical mechanics with that Hamiltonian. It moves along the trajectory that can always be solved from the Hamilton-Jacoby equation as the local rays to wave fronts. That was Hamilton do that well before quantum mechanics. Well, not Hamilton. Some decades, I forget what Hamilton was, but it's hard to know quantum mechanics. Sometimes in the 1960s. Now, what is that limit here? What we did is that what we really did is that we ignored the term here. So let's go back to that. This term here, the term that has h bar, and that's the term that we neglected. So the Hamilton-Jacoby equation is a good approximation, for example, when this is much, much bigger than this. Let's get a feeling for what that might be. Let's suppose I had something that was kind of like a plane wave with some amplitude in the eye, something that looked like that. So this approximation is saying that h bar squared h squared over 2n is much, much bigger than h bar squared over 2n. Or just getting rid of all that, that k is much, much bigger than the square root. Now, let's say that there's some characteristic length scale over which the wave function changes the amplitude. Let me go above here. So I'm just going to say this thing is on the order of some length scale, which changes just a very rough kind of thing. Then this equation here is basically saying that k is much, much bigger than 1 over that length. Which is to say that the wave length over 2tie, which is very rough, is tiny compared to this one. So the length scale is saying something about the scale over which this amplitude is changing. So in some sense, what that length scale is telling us, that amplitude is changing because it's not really repartible to the potential. So really, and that's an important point here, that this length scale is sort of the scale over which the changes. So it's saying something about the particles moving in this potential. Potential is not uniform, although this would be no. And how does the particle move in this potential? Well, if the wavelength associated with that momentum is tiny compared to that scale, then the wave nature of the problem is negligible. And it looks more like trajectories of rays. We know this from our studies of basic optics. If I put a laser pointer through this door, it doesn't diffract. The reason it doesn't diffract is the length scale associated with this length is huge compared to the wavelength. And under that condition, I can just think about ray optics. I don't have to think about diffraction or any wave-like property. The thing that's what we're learning here is that the geometric optics limit of quantum mechanics is a classical mechanic. That is to say, so let me make this a little bit more accurate. Let's talk more about the geometric optics limit. So let's go back to the Helmholtz equation that we had before. Let's think about now. Let's talk about an electromagnetic wave moving in an inhomogeneous. So let's say I have a dielectric that's a function of position. And I want to know what is the wave equation that describes this? Yeah, you've got to scratch your head for a moment. You go through max equations, you get the problem. Let's say this is a wave with frequency omega. So let's just talk about the time in the head. That's the equation. Now, generally, you have to solve that. The boundary conditions, it's not an easy thing to solve. However, the way in which this is changing over space is much, much, much slower than omega over k. Then I have an approximate solution. There's a local wave vector. I would say if this were a constant, we know what the wave vector is. The wave vector is omega over c times the index of refraction. And the index of refraction is the square root of the dielectric constant. Now, what I'm saying is that there's a local index of refraction at every position in this inhomogeneous dielectric. And it says there's a ray that moves through there with a local wave vector that depends on the gradient of the index. And as long as the wavelength is much, much, much smaller than the length scale over which the dielectric changes, then I don't have to think about solving the wave equation. I just have a trajectory. So the upshot of all of this is that wave optics is the single particle wave mechanics as ray optics is the classical. Classical mechanics is the ray optics limit of the wave equation. Let's look at this particular, oh, I should say, that, of course, if we didn't take this limit, and we just looked at this equation, including this particular term, which is sometimes known as the quantum potential, this is exact. We just took Schrodinger's equation. We separated out the real matter of the parts. And it's tempting, and people do follow this, that to say that really what's going on is that there's a particle that is moving in a combination of the physical potential and this quantum potential. And that's known as the de Broglie-Bohm guided pilot wave theory. As I say, the particle is guided through the wave function. The wave function itself, after the wave function, provides something that guides the particle. It's sort of an effective potential. It's very hard to generalize this theory to anything more than a single particle moving in a potential. So whether this really has anything to do with the reality of quantum mechanics or not, I think it's highly questionable. But people work quite hard to try to think about, well, that's really what's going on. There's really both a particle and a wave. The wave just guides the particle. So let's look at the Hamilton-Jacobi equation in the case that I have for a stationary state. So now my wave function, the time dependence factor factor out. OK? And so this thing is a function of position. And then there's a phase that is typically called w. So that tells me that this Hamilton's principal function separates into a spatial part and a temporal part of this form to bind these two together. This is known as Hamilton's characteristic function. All right, let's plug that into the Hamilton-Jacobi equation, which is the equation that comes from Schrodinger's equation when we've taken the limit of geometric optics. And what we get then is the following. The gradient of w squared plus v of x is equal to, that's to say that the gradient of w is the square root of the energy minus the potential energy times 2x. That is to say, this is the classical momentum at that position. Subtract the potential energy. You've got kinetic energy. This is the momentum. So that tells me integrating this, that wx is the integral of this over some path from some initial point to some final point p classical. So that gives us in this geometric optics to make the same approximation to the wave function. In the short wavelength limit, wave function or the stationary state wave function, the function of position is some amplitude, which we still have to calculate, e to the i, the classical value. The momentum of the position is the square root of 2np, just like we wrote down over here. So this is approximate and this limit. It's a solution to the wave equation in that limit. What about the amplitude? What is the amplitude of the wave function, as to say the magnitude is what we want to say? We have the continuity equation. This is a stationary state, right? So I'm looking at here. What can you tell me about the probability density as a function of time? It's not changing, it's not a function of time. So the rho dt is 0. But the rho dt is minus the divergence of the probability current, right? And what's the probability current? Well, the probability current is the local probability density times the gradient of the phase. That's what we said. So this is equal to whatever that classical momentum is at that position over here. And this, obviously, doesn't matter. OK, let's look at this, for example, in one spatial dimension. So that would just have everything moves in 1d. Well, we have a solution to this equation. The solution, it says the local probability density at that position is some constant divided by this. Does that make sense in a kind of classical limit? This is saying that in this limit, the probability density to find the particle moving at this position is 1 over its velocity. If it's moving very slowly, it's very likely to be at that position. It's moving very fast, then at any time, it's going to be very unlikely to find it there. So this is just what you would expect from just classical particle moving. And so putting that all together, we get an approximate form of the wave function. It says that the wave function across the stationary state of the wave function is approximately to some normalization constant, the square root of n over, let's say this is square root of that constant, whatever, this is called some normalization constant, classical at that x e to the i. And this form, this approximate solution for the wave function in this limit of short wavelengths compared to the particle is known as the WKV approximation. And I forget who these are. I know this is Rihwan. This is probably Kramers, but I forget who W is. The WKV approximation is an extremely simple way to solve for the energy eigenstates. You've done it. You just know what the classical momentum is, and that's the geometric optics limit. Now, I want to get just one piece deeper into here. And I want to talk more deeply about particle versus waves, trajectories, and make this connection deeper between, in what sense is the classical dynamics like the quantum dynamics in what sense it was different? To do that, I want to first quickly review in two minutes classical mechanics. So what do we know classically? So let's say we know a particle moves along the trajectory. So there's some trajectory as a function of time. There's some position at the initial time, and there's some function for transition at some later time. And we know that this trajectory can be obtained as the stationary solution to the action. So we have the action, which is a function of the trajectory. So the action, what we're calling it, s, depends on the trajectory. And given that trajectory, what I do is I integrate the Lagrangian, which I substitute in that trajectory, and it's time derivative for that particular trajectory integrated over time from the initial. The Lagrangian, for the simple case of a particle moving in a potential. And we know the solutions to one of the Lagrangian equations. We also know that we can write this in terms of Hamiltonian mechanics by the canonical momentum. And we define the Hamiltonian as a function of position of momentum as the canonical momentum dotted into x dot minus the Lagrangian evaluated x dot. So with that, we have another expression for the action expressed in terms of position and momentum trajectory rather than position. Solving this, this is that integral. That's why I did it loosely like that. And let's put all of that in. So that's equal to the integral p dot x dot dt minus the integral of the Hamiltonian, the integral p dot dx. Suppose we were moving along the trajectory that had a constant energy, then h to the constant. And this is just the time to the time difference. This action, classical action, is exactly what we had over there. The phase factor that sits in our wave function is the action. However, we know that there is only one possible trajectory that takes us from here to here, the one that is the stationary action. That's the one that leads. However, quantumly, there's no reason that I have just the one that has the stationary phase. In principle, all possible paths can take me from here to here. And I weight them by this probability amplitude. The one that was the classical solution is the one where slight perturbations around that don't change anything. So what it's saying is that when I'm in the geometric optics limit, the path that has the dominant contribution is the classical path. But when I'm not in the geometric optics limit, I have all possible paths. And that leads us to the notion of the path integral, which unfortunately, we don't have time to discuss. Maybe we'll have a whole other problem. I was going to try to debate lower at a time. And everyone wants to know about the path integral, because it's cool. But I'll just state what the result is. What's the result? So if I want to treat this problem quantum mechanically, what am I doing? I'm saying I'm looking at the problem where, suppose at t0, I'm at a position. I can never really do that, but I can analyze it mathematically. So I'm at a position, I'm at a state. And I want to know the probability amplitude, so that the state at a later time is that. How do I calculate that? What's the mathematical expression? I told you at that time t0, this is the wave function. And I want to know what is the wave, how would I calculate it? Exactly. So initially, this is my state. I evolve for a time that takes me from t0 to t1. And then I look at this. This is known as the propagator. This is the position matrix element of the kind of ocean operator. It's sometimes called k or kernel. So this we can plug in. I'll have notes on this. What we can show is that this is equal to the following. This was part of Feynman's PhD thesis at Rainey Bridge. What is this service? This is every possible path that can connect me from those two points. I weight that with a phase factor that depends on the classical path. This is a integral over all paths. In the limit that h bar is a tiny, tiny thing, generally, all these different paths are going to kind of cancel each other out. Because they don't get the same phase. They have one of the different phases. So I have waves up like this, and then waves up like that. And they all destructively interfere. Because if I slightly change the path, and if s changes slightly, but h bar is tiny, then that phase changes by a lot. And because the phase changes by a lot, it goes from this phase to all of a sudden, that phase. And so on average, it washes out. It's only the ones that are near the case where the phase doesn't change a lot for a small change in the trajectory. That is to say the case of the stationary solutions, which will contribute the most to the central one when this is big compared to h bar. But this is not just about the classical limit. This is about, in general, this is a way of writing the kind of equation that connects the classical picture with the quantum picture. The connection is classically, the trajectory is the one that gives us stationary action. And that's this path. But quantumly, all possible paths contribute. So we're going from here to here. And they are way they interfere with one another. When I have the geometric optics limit, there is a dominant ray. But in general, all contribute. Yes? So quantum mechanically, if they go past themselves, they don't have to be close to the, I guess, the classical. We have to have higher amplitude or higher probability. That is exactly what I'm talking about. So for example, suppose I have a potential well. And I send in a part of classically what's going to happen. Well, here it has an energy. Now it's going to speed up. And then it's going to slow down. Quantum mechanically, it can reflect. Classically, that will never happen. That's a path very far. Now, of course, that's only going to be true if the wavelength of the particle is on the order of this well. If the wavelength is tiny compared to the size of this well, well, then it'll just be like geometric optics. There'll be no reflection. I want to make one last comment in that question, because although we describe this in terms of pure states and we talked about the classical limit coming from being at short wavelengths, that's not exactly the full story. What's important is the coherence length. The question of whether there is interference. I can have a situation where the wavelength is small, but somehow I've allowed the coherence length to get big and still see wave effects of something, something microstomping. For example, there have been recent experiments that have seen diffraction of a virus. If you can cool down the position of the virus and know to keep its coherence big enough, even though it's a somewhat macroscopic thing, it can diffract and end up on the screen with a probability of the pen's wave mechanics. So it's not just that it's macroscopic. It's about its coherence. Let us quit for today. And we'll talk about that part in about six minutes.