The Hydrogen Atom

Abstract

We have here a two-body problem, of an electron of charge -e and mass m, and a proton of charge +e and mass M. By using CM and relative coordinates and working in the CM frame, we can reduce the problem to the dynamics of a single particle whose mass µ = mM /(m + M) is the reduced mass and whose coordinate r is the relative coordinate of the two particles. However, since m/M ≃ 1/2000, as a result of which the relative coordinate is essentially the electron's coordinate and the reduced mass is essentially m, let us first solve the problem in the limit M→∞. In this case we have just the electron moving in the field of the immobile proton. At a later stage, when we compare the theory with experiment, we will see how we can easily take into account the finiteness of the proton mass.