Hello everyone welcome back to exploring quantum physics. I'm Charles Clark, and in this lecture we're going to use the theory of Angular Momentum to understand how matter interacts with electric fields. In our previous lecture we looked at the role of angular momentum. In terms of this contribution to the kinetic energy of a free particle. So, just to recall what we showed was, that, here's the kinetic energy operator, for a particle moving in three dimensions, squared. And this turns out to break down into a part that just depends upon the distance from the coordinate origin, spherical symmetric, and then a contribution of the angular momentum operator. Which is dependent upon the angles, the directions in space that a particle makes. In fact, you're probably familiar with, an instance of this in a somewhat different context. And that is for the case of a rigid rotator in space. So in classical mechanics, the kinetic energy of a rigid rotator of mass, M, and radius, R, Is given by this, its L squared over 2Mr squared. And, so, if we look at a simple rotating molecule with two massive pieces, then we get this expression for the kinetic energy, just the square of the angular momentum divided by twice the reduced mass and the distance between these two other masses taken as points. And you can easily derive this just by looking at the kinetic energy of the two particles as they rotate but are constrained to move at a constant distance. It turns out that this is a rather good approximation for the rotational. Motions of molecules which hare very important in say thermodynamics and gases. The rotational motor of a molecule is sort of lowest energy beyond just simple translation, and the rotational states of molecules coming into great interest. In the particular context of production of altered code molecules for study of quantum gases. So we're going to look at an example that is based on one such molecule, potassium-rubidium and understand what the rotational dynamics for that system are also in the previous lecture we introduced a system of a set of special function the spherical harmonics which we generally useful for describing states of angular momentum. Here's a conventional definition, notation and you can find more information about these in the, this particular section. The digital library of mathematical functions. Now just as you saw that the role of angular momentum for a free particle is very similar to its role for a constrained system of particles. It turns out that for describing angular motion this set of functions has a sort of universal applicability. Many branches of physics describe wave motion optics electrodynamics theory of elasticity and so on. And I'm not going to dwell In depth of the properties of these functions. But you should know a little bit about the simple ones, and the general things that they make possible. So basically, you can think of these as a set of functions that's useful for describing variations of any type of function defined on the surface of a sphere. So they're defined. In terms of the polar coordinates of three dimensions, the polar angle theta and the angle phi. They're the indices index L, which represents the magnitude of angular momentum goes from 0 to infinity. And then m which is a projection of that projection of m particular chosen axis, typically the z axis or x three axis runs from line to cell to l. Now if you just open a book and look at expressions for circle harmonics They can tend to be a little bit confusing. So I'm going to give you a useful mnemonic to thinking about them. Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. That is, it consists of,products of the three coordinates, x1, x2, x3, where the net power, a plus b plus c, is equal to l, the index of the spherical harmonic. All divided by an inverse power, r to the minus l. So you see this is dimensionless says this is inverse units of length the alpha this is inverse units of length. And furthermore the spherical harmonics all obey this relation. So in other words if you take the Laplacian and apply it to the numerator here Then, it vanishes. Now you might ask how helpful this is. Well, I think you'll see quickly that this enables you to test whether something is spherical harmonic and even to be constructive in creating your own. So you see for the low order harmonics l equals zero, one, and two, you can quickly recognize them just by applying the Laplacian and you can even make your own to suit your own purposes. That is you can form a polynomials that will suit your particulars at the moment. From just more or less arbitrary hours coordinates. So I'd like to also note another important result of this definition is that, you see all the spherical mnemonics of a degree l have a net power of l In the cartesian coordinates, and so that means that under inversion of all coordinates, the circle harmonics either remain the same if L is even or change sign if L is odd. So for the next two, few parts of this lecture, we're going to look at an application to discussing the rotational motion of the molecule potassium. This is one of the great targets for Ultracold molecule research. Here's a recent paper by one of the leading troops in the field. That's accessible through science magazine or you can look on Google Scholar and find references to this work. Basically there's a big milestone in making very cold molecules which are of interest to wide range of scientific applications and just technically fascinating and challenging to work with. So the properties of potassium-rubidium are well known. The reduced of this molecule is given by this number. The equilibrium radius is not that well known at the moment, but we'll take this common value of, 0.4 nanometers. And one important aspect of this molecule is that it has a permanent electrical dipole. For a moment in it's body frame. Now what I mean by that is, if you just hold the molecule in one place there's an imbalance of charge. There's more negative charge around the potassium and more positive charge around the rubidium. And that means that this molecule acts like an electric dipole when it's exposed to an external field. We'll discuss that momentarily. But, when there's no electric field present then the Hamiltonian Conlan that describes the rotational motion of this molecule is just given by this L squared over two mu r and e squared where r is equilibrium bond length, and I've converted that so that's L times L plus one times h bar squared over two mu re squared and I've converted that to an energy In temperature equivalent units just because it's convenient for these cold molecule applications. And so you can see that this equivalent temperature which is basically half the distance between the l equals zero. And the allele equal one rotational state is equivalent to 55 milicalvin. Very low temperature. Much cooler than heat for example. Now we use the dimoment for the molecule As a convenient reference through axis so we're going to describe the orientation of the molecule with respect to some arbitrarily chosen laboratory frame the x 3 x laboratory frame. In terms of the angle what it returns we In terms of the given Cartesian frame, we describe the orientation of the molecule by the angle that its dipole moment makes with respect to the various axes. Now, in the ground state of the molecule, it's the The lowest spherical harmonic is just a constant, that's something you should always remember. The first, l equal one is just either xy or x1, x2 or x3 or some other combination of them. And this means that the dipole moment is oriented randomly in space, so there is no net dipole. Moment of the molecule when we go to make a measurement of it, it could be in any arbitrary direction. So we're now going to address the question of how we orient and control
