Resolution of the hyperfine puzzle and its significance for two fermion Dirac atoms

TL;DR

This paper uses a variational Dirac equation approach to explain why hydrogen and positronium atoms remain stable despite the hyperfine interaction's strong dependence on atomic size, revealing a natural cutoff at small scales.

Contribution

It introduces a minimax variational method based on the Dirac equation to analyze hyperfine interactions and stability in two-fermion Coulombic systems, including hydrogen and positronium.

Findings

Hyperfine interaction is bounded and does not cause collapse.

Electron magnetic moment effectively softens at small atomic sizes.

Framework for relativistic treatment of diquarks in QCD.

Abstract

The hyperfine interaction in the ground state of a hydrogen atom of assumed radius $R$ is proportional to $-1/R^3$, raising the question of why the hyperfine interaction does not lead to collapse of hydrogen, or positronium. We approach the problem in terms of a minimax variational calculation based on the exact Gordon solution of the Dirac equation for the hydrogen atom ground state. The full Dirac treatment leads to the result that in an assumed variational state of size $R$, when $R$ minimizes the total energy the magnetic moment of the electron assumes its usual value, $e\hbar/2mc$, but when $R<\hbar/mc$, the effective electron magnetic moment becomes essentially $eR/2$, softening the hyperfine interaction and eliminating an energy minumum at small $R$. The magnetic moment of the proton is similarly suppressed, and the hyperfine interaction of a small size atom becomes bounded by the kinetic energy, thus assuring stability. We extend the Dirac variational calculation to positronium where we find simple results for the ground state energy and hyperfiine interaction, and then extend this variational calculation to Coulombic atoms of two fermions of arbitrary masses. This paper also lays out a framework for treating diquarks as relativistic Coulombic systems, in the presence of color electric and magnetic interactions.