 And we'll have a problem session starting at 10.15 in the morning. Continue. And then we'll have the afterwards at 10.15. Right. So last time we were discussing, hold on one second. Right. So we were discussing some of the really foundational understanding of weight mechanics. How we think about the weight function. So fundamentally the main interpretation of the weight function of course is the weight function related to the ability of the weights and the square of the weight function. At a particular point telling us about the probability density to find the particle at a particular point. Space at a density obviously at some given time. And that probability density in a mechanical system will generally flow. And that flow equation, that kind of equivalent of fluid flow that we think about is described by the continuity equation. The continuity equation is familiar mostly to most of you I'm sure in the context of electricity. Magnetism where rho is the charge density and j is the current density of charge. But it's a general statement of flow of fluid, whatever that fluid might be. And the physical meaning of the continuity equation of course is conservation of whatever stuff is flowing. In this case it's thought about as a conservation of probability. Which really is to say that what we're saying in this situation is that in a system where we're described by a potential where the potential is a real function the particle is neither created nor destroyed it just moves. And that movement is described by the flow of the probability density. And that is described by the probability current which written in this form we see as whatever the local density is times a sort of effective velocity of flow and that effective velocity of flow is determined by the gradient of the local wave front to the system. So we think about the probability flowing along rays which are the normals of the wave fronts. So more over what we, of course one of the foundational ideas which made at the heart of the whole development of the quantum theory was the understanding of wave particle duality. I mean wave particle duality is where the whole idea of the probability amplitude arose and all the craziness that we've been talking about relative to quantum mechanics came from trying to understand the nature of the world and the foundations of physics Newton come from really two things mechanics and optics and trying to understand the nature of the motion of the heavenly bodies and the nature of light is really the foundation of physics. And there were sort of, of course, within that, particularly in optics there were the competing theories of optics, the particle theory, the corpuscular theory as Newton called it and the wave theory which was Huygens' idea of what the nature of light was. And so in the sort of particle picture we have the idea of a ray of light that moves along a particular trajectory. That was Newton thought as visible that light was made up of tiny little particles that were streaming at super high speeds and there were forces exerted upon them by material which caused them to refract and reflect. In modern theory we know that the ray theory of light is a limit of the wave theory of light in the short wavelength approximation and the description of rays is unified with the description of the motion of particles and the mechanics of Lagrange and Hamilton. So it's not a very nicely written chart I can try to write this out in my notes more clearly of the complete analogies between these things. So in the ray theory we have, we could leave the equation of motion of the rays follows from an action principle. That is stationary, the trajectories of the rays are the stationary functions, the solutions which make the action stationary under perturbations around those trajectories. And that's codified in Fermat's principle. Fermat said in the 17th century that we can understand the trajectory of a light ray in a medium in terms of minimizing the time it took for the light to get from one point to another. But in modern parlance what's that saying is that there's an optical pathway and we're minimizing the optical pathway. And the optical pathway is determined by the local index of refractions. And the solution to this is basically a Rayleigh-Lagrange equation that we know from this side where the motion of the particle in a potential is determined by the stationary action and the trajectories which minimize the action where the action is unchanged with small perturbations around those trajectories or the Lagrange equation, which is Newton's law, f equals ma. So that's the particle picture. Competing with that is the wave picture. In the wave picture, we have a wave equation. The equation of the wave, say for the electric field moving which has some frequency omega moving in this index of refractions is this. And the Huygens principle basically says that the rays are the normals to those wave fronts. And if we have the iconal approximation, which is to say when the wavelength is small compared to whatever the length scale for change of the same index, we have a solution which is the same as this. And the iconal approximation, under the iconal approximation, we have a wave, this is the wave solution, whose rays follow Fermat's principle. This is an approximation under that case. Now, on this side of the equation, there's kind of an interesting history. So this equation, which I have no room in my horrible plan, this is the Hamilton Jacobi equation. The Hamilton Jacobi equation is follows from classical mechanics. It's not a wave, it's a mechanical theorem. But what we have seen is that this equation follows as the geometrical optical approximation to the Schroder equation. In which case we understand that classical dynamics of a particle, the trajectory that a particle follows in the classical limit, is the limit where the wavelength of the brain wavelength of a particle is sufficiently small compared to the length scale where the forces change which would cause diffraction of those waves. And when we don't have such diffraction, then we have rays and those rays follow the classical. That also gives us a way of writing down a solution to the Schroder equation when we're in this approximation. And that is what is the WKP approximation where we basically use the Huygens wavefront associated with the surfaces such that the rays of those wave fronts are the classical togetheries. So when the wavelength, the de Broglie wavelength of a particle is sufficiently short compared to the length scales over which potential is changing, we can write down an approximate solution. In fact the amplitude here is related to one over. We wrote down that solution last time. So of course all these things are unified together. In some sense what we see is that these two equations are the same and the unification of them is wave part of the wavelength. So in wave part of duality, particles with certain energy have certain frequencies and particles with certain momenta have certain wave numbers for wave lengths. Obviously from rational physics. And what we see here is that the unification between the wave and the particle theory is to say that we can have our general action principle here which is again always minimizing this optical path length but these are in fact the same thing. That this action from the point of view of a Hamiltonian this is the same thing as the Le Broglie here inside here. This is the Le Broglie dt which is this. And for a constant energy this was p dx minus b times the time difference. And with this relationship here this is equal to k as a function of x the x that's a times h bar minus. So it's the same this action principle is the same as this action principle because what is k? k as a function of x is omega over c times n of x in optics or what we have over here it's the momentum as a function of x over h bar which is equal in mechanics to the square root of twice the total energy minus the potential energy. That's what's going in here. So we have exactly the same unified theory. There are waves, there are rays which are limits with short wavelengths and they undergo exactly the same action principle where the effective index of refraction in mechanics is given here. So notice that though that there is this kind of weird relationship a bigger potential here corresponds to a smaller index here. Notice in particular if I look at the phase velocity that's omega over k if I look at what that is locally at some position of space in this gray picture whereas if I write in the wave part of the duality this is this. Which is it a curious thing? What it's saying is that the velocity of these wave fronts that move in order to ensure that the normals of those wave fronts are the classical trajectories move inversely with the speed of the classical particle which is why Newton got it wrong. Newton said that inside a material light moves faster. The only way you could get refraction to work. Now no one could measure the speed of light at Newton's time. So maybe Newton was right. But Huygens' principle kind of took over and once they measured the speed of light and saw the speed of light move slower then that doomed the particle theory. The particle theory of light was thermodynamic. It was totally bogus theory. Well it's hard to measure between c and half c. We couldn't measure anything. So that's an interesting fact. However, it is quite curious that Hamilton's Jacobi equation is and Hamilton knew this that he could understand the trajectories of classical particles as the rays of a wave front. We didn't call it a wave front but these surfaces that were moving. But interestingly, there's no way that he could have guessed that in fact the classical mechanics was the ray optics limit of a more fundamental wave theory. But it is and it's all together. Now, so this unification is pretty clear. There's of course many subtleties to this. Should we think about the electric field as the wave function of the photon? Kind of. The subtlety, the problem is that the photon is a relativistic quantum. And relativistic quantum mechanics has some differences with respect to the non-relativistic mechanics. The notion of localizing particles of relative. You can't do that because particle number isn't considered. And what it means for it to measure the position of a particle relativistically is a much more subtle question. As I said, position is demoted. But nonetheless, these analogies should be kept. The final thing, of course, we talked about a little bit at the end of lecture last time, the idea of the path integral. And I would encourage you to read about that some of yourself. There's a nice beginning description of it in Saccharized Texts. And Saccharized Texts is quite good. I recommend trying to get a hold of it if you don't have it. But we can understand the quantum mechanics in saying that instead of having a single positive trajectory quantumly, in some sense, the system explores all possible trajectories. And they're weighted by a probability amplitude that's given by the local action of that trajectory. And that principle that connects classical dynamics and classical action of quantum dynamics is a foundation of quantum mechanics. And we use it all the time, particularly in quantum field theory and in statistical physics as well. So I would encourage you to take a peek at that. No key difference. All right, so now let's get into some nitty-gritty. So, given this, we're just going to go at light and speed today talking about something you've studied over and over and part of a box, all that stuff. I just want to get some of this, some of the foundations down. So we have some potential as a function of x. And say it looks like this. What we call the zero tetra-energy is irrelevant. So, classically, of course, we have different kinds of motion that can exist in such a potential. Remember, of course, we have a potential energy to conserve. So specifying the energy and the initial condition completely specifies the trajectory, classically. That's the standard trick to do. You want to throw up a ball how high it would go. You could just use conservation of energy to do it. So, for example, if a particle started at some position, if its energy was this and it started at some position, it will oscillate between these two points, their turning points. Because, of course, the total kinetic energy will always be bigger than or equal to the potential for all x. So, this is one kind of solution. If we started up here, then I'll have this one turning point. This is another possible trajectory. Come in here, bounce off, and come back. If I started over here at this energy, I'll hit this guy and bounce back. If I started over here at the same energy, then, of course, I'll hit this turning point and bounce back. This is classical. So, now it's, of course, talking about the quantum mechanics. So, classically, we have this supposed to be at the same energy. So, this, of course, in this case, this kind of solution, we call scattering. So, a particle will scatter off this potential. This is also scattering. This is bound motion. This is the bounce motion of the particle. Now, of course, you know, and it's even alluded to, if I look at this, so I look at my Schrodinger equation in 1B. This is the equation that we just, the time independent Schrodinger equation has this form, momentum over HR, which is 2. So, as we said here, classically, k squared can be negative. Classically, p is a real number, which means that it must always be the case that p squared is positive. So, with that, but of course, quantumly, k squared can be negative. There's nothing ever to be said, which means that k of x for when the, for p less than v of x, this is imaginary. No. So, that means the local solution, if I think wk be like, That'd be k squared if it looked like k, k was squared. So, if k squared is minus k, because k squared can be negative. So, the wkb solution, which gives us a hint of things, will be of the form, e to the minus kappa, at least kind of the exponential tails. Of course, what that means is that what was classically forbidden, this is this region, the particle, if it had this energy, it cannot be in this region to become classically forbidden. Now it's classically allowed. It means, of course, that we can tunnel it. And so, we have tunneling, unequally. Moreover, these two solutions now can, in some sense, become mixed. So, what that means is that when I have this kind of situation, when I can have a scattering resonance, as we will study next semester, if you come in with a certain energy, which is resonant with a wasi balance state, then there is a particular kind of scattering that occurs, usually the fact that the particle can tunnel in and rattle around at a resonant energy, and then tunnel back out. This is another kind of scattered interaction. Moreover, if I started the system here, classically whereas it would be bound, quantumly it has a finite lifetime, and it's only quasi-bound, and it can ultimately tunnel out. It means, in some sense, that the particle inside this potential has a finite lifetime. So, what can we say in general? So, firstly, if we look at this kind of potential, if it's non-singular, it doesn't blow up anywhere. At your first study in the public of your homework, if the delta function potential is quite high, if the potential is non-singular, and we say a few things, just from the solution to this kind of question. By the way, this kind of differential equation events under a certain, you know, 19th century study of ODEs. It's known as Stern-Liebel theory, and it's been well studied back then. Maybe I don't know if you guys talk about it in 466. I don't know. Probably not. Anyway, what can we say? The solution is derivative is everywhere, defined. Moreover, the solutions we seek in the solutions of those are the, that's our Hilbert space that we talked about, the square normalization of functions on the line. Now, what that means is that for bound states, because we have to satisfy the boundary conditions and because we don't want the solutions to blow up because we want to be able to look for normalizable solutions, then we have discrete possible unbound states are trickier. Generally, the unbound states, well, as we know, if we have a free particle, the energy eigenstates we can write down are plane waves. Plane waves are not normalizable. They are not in this set of square normalizable functions. Nonetheless, they form a basis of possible functions for this. And so we care about them for that reason because we can always write a wave packet or a superposition of the unbound states, which is a physical state. But that means because we don't have this kind of boundary conditions to worry about here, then the, what we, all we don't want is, so the unbound states is, they should be normalizable like a double function. In other words, they don't blow up more than a double function. The unbound states, though, don't have these boundary conditions, so we have a continuum of possible energy eigenstates. So what that means is that the solutions to the energy eigenstates, which is in the position representation, is this equation, written in the position representation, right, that the solutions to that form a basis, which means we can write a resolution of the identity in terms of generally the unbound states and then in addition the scatter of the states. So we have to integrate over all unbound energies. That's the resolution we're in. Note, when we talk about the scattering states, typically, because of the way we integrate over them, we typically give them units, right? The kets have to have the units of one over two over three. Just like when we had the kets in X, they have the units of one over the square root of length This is often forgotten that this has to be a resolution that it does. Now, it might be that the support of your wave function is dominated, has support on mostly on these vectors and not on the scatter of states, but if you really want to be formal and be careful, you have to think about both the bound and unbound contribution to your resolution of the identity. Now, one thing we can say about the solution. Another thing we can say about the solution is about the generous. Well, for the unbound states, the scattering states, we have uncountably infinite. We already saw that in the case of the free particle. The free particle, the energy is p squared over 2n but any plane wave moving with that momentum but in a different direction is also a unbound state. What about the bound states? Well, the answer to that is we will show in a moment that we probably have seen before. The bound states, there cannot be generous, no generous quantity. Of course, you know about higher dimensions of what we'll get to that in a little while. How do we see that? Get in here. You give me in the same way as classically, you give me energy and that determines the trajectory. In quantum mechanics, you give me the energy and that determines the wave function, the bound state wave function. That's not true in higher dimensions. If I tell you the energy of a planet moving around the sun, the energy alone is not enough to tell you the trajectory. We also need to know the momentum and the projection angle. If I have those three, then I can do the Kepler orbit and the same thing is true of quantum. Those are the three quantum numbers, the principal quantum number, the L quantum number, the M. All right, how do we prove this? We have to remember how to do it because it's not going to be here. So let's say, consider two solutions at the same energy. So I have two solutions, psi 1. So let's multiply the first equation by psi 2 star and this equation by psi 1 star. It's a trap. Actually, I don't think I want to multiply by the star. I just want to multiply by the star. So that means that d by dx of psi 2 d psi 1 dx minus psi 1 d psi 2 dx is equal to... which means that this, which is the Ronskian, is a constant. So what that means then is that the derivative of psi 1 dx over psi 1 is equal to d by... which means, if you just integrate that, that psi 1 of x is proportional to psi 2. So if you integrate that, you get the log is equal to the log and that means that 1 is just the constant of... and because you can always renormalize it, it means that's a different point because this doesn't follow. What is this constant? Because this says the constant is zero. S of this is equal to this. Otherwise, it comes here. For a balanced state, it must be the case that at infinity, so if the domain extends to... x goes to plus or minus infinity, the wave function has to go to zero for a balanced state. Right? Because otherwise it would be... even the plus x and it would blow up and that's not a allowed solution. So if that's the case, this goes to infinity, then the psi goes to zero which means the constant is zero. And then it follows. So that's why it's important to be talking about balanced states. Otherwise this just doesn't follow. So that means because a proportional normalization says that psi 1 must equal psi 2. So this is the same solution. So if we have the energy, there is a unique wave function for a balanced state corresponding to that. That's the same thing. That's true in the classical philosophy. If you give me the energy, if it's a balanced state, there's one trajectory. All right. What else can we say in general to things about this? So for balanced states, from experiment to experiment, first of all, you can prove the following thing. The lowest energy solution has no limits. Of course, what we call a balanced state. So over each subsequent higher energy state, does it kind of make sense physically that excited states and more and more excited states, states that have higher and higher energy have more and more limits. Why would that? Forgetting about Stern-Gildo theory, which is where this stuff follows, you can prove this mathematically. Why would we expect that we have anything here? That's not good. So let's have you know, would we expect this wave function to be a higher energy state than this wave? Well, I mean, what we know is something about the relationship between kinetic energy and the derivative of the wave function. So the kinetic energy, states which have bigger changes in their curvature have higher kinetic energy. So you kind of expect that. You're going to keep curving the wave function more and more, a wave function with lots of curvature has higher kinetic energy. So curvature. So what next? Well, there's one important class of problems that we often solve. And that is the reflection symmetric. That is to say, if a molecule has reflection symmetry, whatever that is, then there is important properties that follow to the solution to the Schrodinger equation. And that means the following. So the reflection symmetry is known as parity. Parity is a... And we've talked about briefly of continuous symmetries like space translations, rotations. Parity, which is reflection through the origin. So in 3B, that means x goes to minus x. So if I draw this, here's my position vector x, and I reflect it through the origin. In terms of minus x, minus x minus y, y minus y is even minus z. So that's reflection through the origin. In 1B, that's just along the x-axis going from x. So the parity operator is a unitary operator because we know that all symmetry operations are... What's the word I'm looking for in there? They can be performed by unitary transformations. So the parity operator, I'll call p, or pi, 3, does the following. It also turns a momentum operator to minus the momentum operator. And we see clearly from this that this is unitary because if I do it by, in the inverse of it, well, it's a unitary operator. We see here under this definition the parity operator is actually... its adjoint is actually equal to itself because the inverse operation is again the same operation. So this says that when you do it twice, you get the origin, identity. So what are the eigenvalues of this operation? Plus or minus 1. We know they have to be unit magnitude complex numbers. They're e to the i0 and e to the i pi. Because every unitary operator is eigenvalues or phases. All right. So, how does the parity operator act on eigenket? Well, it turns the eigenket to the position eigenket at minus x. So, if I look at the wave function, so I have a state psi and it's wave function where it's the position representation of the state. So what is the position representation of the parity operator acting on site? Well, if I look at the position representation of that vector, it's that. And this is the equivalent of the dagger of phi dagger acting on that. But this is equal to its own adjoint. So that is equal to minus x. So that's this. So the parity operator flips the state to what its value is on the minus x. All right. So, what can we say about this? Well, let's suppose we have a potential is reflection symmetric. Written as an operator, so my Hamiltonian is p squared over 2 m plus v max. If the potential is reflection symmetric, what it says is that if I view the unitary transformation on this potential operator, that evaluates to this, right? And if this is reflection symmetric, this is this. The kinetic energy is reflection symmetric. Because p goes to minus p, but this is p squared. Which means that the Hamiltonian, if the potential is reflection symmetric, the kinetic energy is always reflection symmetric. Therefore, the Hamiltonian is invariant under that is to say if I do this transformation and what that means is that I'm multiplied by a pi on the left or equivalently h can use. Commutes. So this is an important result for reflection symmetric potentials the Hamiltonian commutes to the periodic operator. Now, if two operators commute, what can we say about their eigenvectors? That at least they can be the same. There exists common set of eigenvectors of both the Hamiltonian and the periodic operator. But to this, there exists common eigenstates. We can say more because the only way they can be different is if there's degeneracy. Right? Because when we can have degenerate the degeneracies of the eigenvalues then there can be different eigenfunctional positions of which might be eigenvectors of one operator but not the other. As we discussed with the block diagonal matrices and all that at the beginning of the semester. However, if we're talking about bound states that extend to infinity which means for these cases energy eigenstates are eigenstates of parity. So that means that if I have a solution that is the nth bound state that this solution is an eigenstate of parity which means that this has to be plus or minus that. Which means in weight function language if I multiply the cat here that u n of minus x that's x on that is plus or minus which means that the solutions the bound state solutions in one day are for as reflections metrics potential are either symmetric or anti-symmetric. Correct? Then so let's look at some simple solutions. Of course, the solution we all know off the top of our head is the particle in a box. Before I do that I guess we should say the problem thing of course. The solutions that are easy to write down are easy but quickly if you write down are solutions with piecewise constant. What I mean by that is that my potential a constant be 1, 2, 3, 4, etc. So we can write down those solutions instantly because in that case in any region of space where the potential is constant then my Schrodinger equation takes the form of effectively a simple harmonic oscillator where k0 is the square root of 2 and e minus v0 over h1. So we have two kinds of solutions and the classically allowed k0 is real. In the classically forbidden regions k0 is imaginary. In this region the solutions are of the form e to the plus or minus i k0 x or sine and cosine and in this region the solutions are either the plus or minus kappa x or the cosh and cinch. So depending upon the energy here in this region is classically forbidden here is classically allowed all these regions are classically forbidden this region is classically allowed this region is classically forbidden and so what you do and you've done this is you put these solutions in and you match them at the boundary conditions with the condition that the wave functions doesn't blow up so here the wave function has to grow like e to the plus kappa 1x and here it goes like e to the minus kappa of all this then inside here you have to match that the wave function is continuous and the wave function is derivative is continuous because let's move this even though the potential is not continuous the wave function is now if a couple v not goes to infinity then kappa goes to infinity which means the wave function does not into the city so if I have a potential that's a square well and this goes up to infinity then we know that the solutions can in fact we can sketch solutions they have to be eigenstates of parity so they're cosine sine cosine etc so we have cosine in this case we can write that solutions automatically because the solutions are eigenstates of parity the eigenstates of the Hamiltonian are also eigenstates of parity and because every wave function has one more node we go from an even to a not to an even to a not okay if this was a finite potential again the same thing would be true so if I had a potential where this is now v not to infinity well now I can penetrate in a little bit and I have solutions and there's a finite number of down states again we can alternate them from even parity odd parity even parity and know from the fact that they must go from even to odd to even how this guy must be either the plus or the minus half of x or even guy whereas on the other side over here we have the other kind of solution and this is minus because it has to be odd so you should use parity when you have it now I want to conclude with one last thing about this and again I'm not going to go through these details or in the notes before many times a part of my box is kind of what we do in mechanics we're not going to do it here you'll have some homework to review those things but I'm not going to go through that in detail but I do want to say one last thing in our laboratory suppose that I have a double well doesn't look so perfect but it's supposed to be reflection symmetric reflection symmetric of this guy so we know in this situation that the eigenstates are the eigenstates of parity and they alternate from even to odd to even to odd and the ground state has no notes so we can sketch that the ground state solution the part of the molecule tunnels in that's the ground state what's that? we're going to quit in two minutes but you can that's it the first excited state has one note and where is that note it has to be in the middle because it's anti-symmetric so the first excited state with some energy over here looks quite similar it looks like that the next excited state will get a much higher energy of course it's got to be an even parity solution and then the next one is not a parity solution which looks quite similar but has the opposite parity so these solutions come in pairs of symmetric and symmetric combinations of the particle localized in one or the other well what would happen if that barrier between this went up to infinity well they're not necessarily they're both even in odd solutions particle can tunnel into from one side but it can't it can tunnel into this but it can't get into that region this region is impenetrable it's a hard wall it's another word for infinite barrier it's a hard wall which means that in that case the particle really has those notes there's a little bit different we have those opinions so there is a solution that looks like this and it's a perfectly good solution it satisfies all the boundary conditions the wave function is just localized there another solution where the particle is localized here and then goes to zero and then it's never anywhere else those solutions are degenerate they don't violate this theorem that we proved because we said something about the particle being able to extend from plus and minus infinity when you have this case where you have an infinite barrier you can have degeneracies and this kind of solution a solution which doesn't respect the symmetry of the Hamiltonian does it say psi is not an eigenstate of a parity even though the Hamiltonian is an invariant it's known as a spontaneously symmetry broken state in terms of spontaneous development here I can just call this a symmetry broken state that's only allowed if I have degeneracies if I don't have degeneracies it's impossible because you have to have simultaneous eigenstates of the symmetry and the Hamiltonian but when you have degeneracies then I can take this symmetry respecting eigenstates well they're symmetric an anisometric combination so there is another solution which is perfectly degenerate which is this one this solution is even it's a linear combination of the two of you there's another solution that guy is the degenerate the degenerate solution this is the anisometric combination so as we said when two operatives commute there always exists commonizing space but if there's degeneracy I can add these two they all have the same energy and they would be localized on this side or I could subtract these two and it would be localized on that side so final comment given this and given this degeneracy what can you tell me about this energy splitting between the symmetric and anisometric states for a finite barrier well they have something to do with how much the particle can penetrate into the barrier because as I raise this barrier these these doublets come degenerate so there's something about the mixing of these two states that breaks the degeneracy and that's an important problem I will stress in more detail degenerate perturbation