8 - One-Electron Atoms

Summary

In this chapter we are going to make a very detailed study of the one-electron atom using relativistic quantum mechanics. The one-electron atom is the simplest bound system that occurs in nature, and plays a central role in both classical and quantum theory. There are two reasons for this. Firstly, it is a model that can be solved exactly at many levels of theory — the classical Bohr atom, the non-relativistic quantum mechanical one-electron atom, and the relativistic one-electron atom with spin zero and spin 1/2. Secondly, most of our understanding of multielectron atoms, molecules and solids is based on this model.

Of course, a good description of the hydrogen atom is provided by non-relativistic Schrödinger theory. However, once the relativistic quantum theory had been discovered it was of great importance to validate the theory by describing hydrogen at least as well as Schrödinger theory. As we shall see, the Dirac theory was exceedingly successful in this, and furthermore, at several points in the theory the non-relativistic limit can be taken and the effect of relativity in one-electron atoms can be seen explicitly. This connection with the simpler theory will be made here wherever insight may be gained from doing so.

One of the key advances brought about by the advent of quantum theory was that it defined eigenfunctions as well as the eigenvalues predicted by the Bohr theory.