Anomalous Solutions to the Dirac and Schrödinger Equations for the Coulomb Potential

Abstract

Certain anomalous solutions to the Dirac and relativistic Schr\"odinger equations for the Coulomb potential, some of which have not been investigated before, are analyzed. These are solutions for orbital angular momentum $l=0$ and $l=\frac{1}{2}$ in the Schr\"odinger case and total angular momentum $j=0$ in the Dirac case that are quadratically integrable for all energies $E<0$. The purpose is to determine if there exist fundamental reasons for discarding them, or if they describe meaningful physical systems. Reasons are given why they may be discarded for real two-body systems, such as the hydrogen atom. These solutions can only be meaningful for systems which have an exact one-body, pure Coulomb relativistic Hamiltonian. They are valid for repulsive as well as attractive potentials. It is suggested that the $l=0$ relativistic Schr\"odinger solution be identified as the wave function of a nonrigid charged spherical shell. In addition, it is suggested that a particle fluctuating about its own center of mass in zitterbewegung-type motion may experience a repulsive Coulomb self-potential and have an exact one-particle-type Hamiltonian. The charge distributions obtained in this interpretation for two of the solutions have mean radii $\ensuremath{\sim}\frac{{e}^{2}}{{m}_{0}{c}^{2}}$ and rms radii $\ensuremath{\sim}\frac{\ensuremath{\hbar}}{{m}_{0}c}$.