Return to Separation of Variables

Abstract

In Chapter 3 we have seen that the Hamilton-Jacobi Equation as well as the Schrödinger Equation of the 3-dimensional Kepler Problem are both separable in four different orthogonal coordinate systems. This has, however, only been verified by a direct calculation, working explicitly with those coordinates. In this chapter we show that general theorems about separable orthogonal systems, combined with the knowledge of the invariance group SO(2) x SO(4) of the Kepler Problem, enable us toderivethese four coordinate systems, and to prove that they are in fact the only ones consistent with the separability.