If we could shrink the ground-state wave function of a hydrogen atom to a size $R$, at fixed electron mass and charge, the singlet hyperfine attraction in a spin singlet state would scale as $1/R^3$—faster than the increase in kinetic energy—raising the question of why the hyperfine interaction does not lead to collapse of hydrogen or similarly, positronium. We resolve this issue within the framework of the full Dirac equation for the hydrogen atom ground state, employing a minimax variational calculation. The result is that in a variational state of size $R$, the magnetic moment of the electron assumes its usual value, $e\hslash /2mc$, when $R$ minimizes the total energy. However, for $R<\hslash /mc$ there is no energy minimum, and the effective electron magnetic moment becomes essentially $eR/2$, softening the hyperfine interaction compared with the zero-point energy. When the Dirac variational calculation is extended to Coulombic atoms with two oppositely charge pointlike fermions of arbitrary masses, e.g., positronium or muonium, the effective magnetic moment of both particles are similarly suppressed in small atomic configurations, leading to stability against collapse, while in hydrogen, the finite size of the proton is enough to ensure stability. This same framework is, in addition, applicable for treating diquarks as relativistic Coulombic systems in the presence of color electric and magnetic interactions.

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• Relativistic & quantum electrodynamic effects in atoms, molecules,& ions