Modification of Coulomb law and energy levels of the hydrogen atom in a superstrong magnetic field

TL;DR

This paper derives an analytical formula for the electric potential near a point charge in superstrong magnetic fields, showing significant deviations from Coulomb's law and analyzing the altered energy levels of the hydrogen atom.

Contribution

It provides the first analytical description of how superstrong magnetic fields modify Coulomb potential and hydrogen atom energy levels, especially in the ultra-relativistic regime.

Findings

Deviation from Coulomb's law becomes significant for magnetic fields > 6×10^{16} G.

Energy spectrum of the hydrogen atom is substantially altered in superstrong magnetic fields.

Electrons are ultra-relativistic except for those in the lowest Landau level.

Abstract

We obtain the following analytical formula which describes the dependence of the electric potential of a point-like charge on the distance away from it in the direction of an external magnetic field B: \Phi(z) = e/|z| [ 1- exp(-\sqrt{6m_e^2}|z|) + exp(-\sqrt{(2/\pi) e^3 B + 6m_e^2} |z|) ]. The deviation from Coulomb's law becomes essential for B > 3\pi B_{cr}/\alpha = 3 \pi m_e^2/e^3 \approx 6 10^{16} G. In such superstrong fields, electrons are ultra-relativistic except those which occupy the lowest Landau level (LLL) and which have the energy epsilon_0^2 = m_e^2 + p_z^2. The energy spectrum on which LLL splits in the presence of the atomic nucleus is found analytically. For B > 3 \pi B_{cr}/\alpha, it substantially differs from the one obtained without accounting for the modification of the atomic potential.