Introduction: systems with different symmetries

It is obvious that the Cartesian coordinate system which we have been using up to this point is ideally suited for those situations in which one is trying to solve problems with a rectangular, or “box”-type geometry. Determining the electric potential inside of a cube, for instance, would be an ideal problem to use Cartesian coordinates, for one can choose coordinates such that faces of the cube each correspond to a coordinate being constant. However, it is also clear that there exist problems in which Cartesian coordinates are poorly suited. Two such situations are illustrated in Fig. 3.1. A monochromatic laser beam has a preferred direction of propagation z, but is often rotationally symmetric about this axis. Instead of using (x, y) to describe the transverse characteristics of the beam, it is natural to instead use polar coordinates (ρ,ϕ), which indicate the distance from the axis and the angle with respect to the x-axis, respectively. A point in three-dimensional space can be represented in such cylindrical coordinates by (ρ,ϕ,z). In the scattering of light from a localized scattering object, the field behaves far from the scatterer as a distorted spherical wave whose amplitude decays as 1/r, r being the distance from the origin of the scatterer.