Arguments for the Anomalous Solutions of the Dirac Equations

In this paper, we look into the difficult question of electron deep levels in the hydrogen atom. An introduction shows some general considerations on these orbits as “anomalous” (and usually rejected) solutions of relativistic quantum equations. The first part of our study is devoted to a discussion of the arguments against the deep orbits and for them, as exemplified in published solutions. We examine each of the principal negative arguments found in the literature and show how it is possible to resolve the questions raised. In fact, most of the problems are related to the singularity of the Coulomb potential when considering the nucleus as a point charge, and so they can be easily resolved when considering a more realistic potential with finite value inside the nucleus. In a second part, we consider specific works on deep orbits as solutions of the relativistic Schrodinger and of the Dirac equations, named Dirac Deep Levels (DDLs). The latter presents the most complete solution and development for spin ½ particles, and includes an infinite family of DDL solutions. We examine particularities of these DDL solutions and more generally of the anomalous solutions. Next we analyze the methods for, and the properties of, the solutions that include a corrected potential inside the nucleus, and we examine the questions raised by this new element. Finally we indicate, in the conclusion, open questions such as the physical meaning of the relation between quantum numbers determining the deep levels and the fact that the angular momentum seems two orders-of-magnitude lower than the values associated with the Planck constant. As a prerequisite to a deep comprehension of the resolution methods, we recall in the appendices some essential elements of the Dirac theory