Orbital Angular Momentum and Spherical Harmonics

In Notes 13, we worked out the general theory of the representations of the angular momentum operators J and the corresponding rotation operators. That theory did not make any assumptions about the kind of system upon which the angular momentum and rotation operators act, for example, whether it is a spin system or has spatial degrees of freedom, whether single particle or multiparticle, whether nonrelativistic or relativistic, etc., because most of the theory of rotations is independent of those details. We did work through the specific case of spin1 2 systems in detail in Notes 12. In this set of notes we consider another important example of the theory of rotations, namely, a system consisting of a single particle moving in three-dimensional space. We assume either that the particle is spinless, or that the spin degrees of freedom can be ignored. For most of these notes we make no assumptions about the Hamiltonian governing the single particle, but the theory of rotations is most useful in the case that the particle is moving in a central force field, that is, one for which the potential V = V (r) is a function of the radius r only. We will discuss central force problems later. In these notes we follow the usual custom in the physics literature of denoting the orbital angular momentum by L instead of the general notation J used earlier. Similarly, we use the quantum number l instead of j.