Saturation of Energy Levels of the Hydrogen Atom in Strong Magnetic Field

TL;DR

This paper investigates how the energy levels of a hydrogen atom saturate under extremely strong magnetic fields, showing that vacuum polarization effects lead to a finite limit, with the effective potential becoming nonsingular due to screening.

Contribution

It demonstrates that vacuum polarization causes saturation of hydrogen atom energy levels in strong magnetic fields within the Euler-Heisenberg approximation, revealing a different mechanism from Furry picture calculations.

Findings

Energy levels saturate at finite values in strong magnetic fields.

Vacuum polarization induces a nonsingular effective potential.

Limiting energies are computed for various magnetic quantum numbers.

Abstract

We demonstrate that the finiteness of the limiting values of the lower energy levels of a hydrogen atom under an unrestricted growth of the magnetic field, into which this atom is embedded, is achieved already when the vacuum polarization (VP) is calculated in the magnetic field within the approximation of the local action of Euler--Heisenberg. We find that the mechanism for this saturation is different from the one acting, when VP is calculated via the Feynman diagram in the Furry picture. We study the effective potential that appears when the adiabatic (diagonal) approximation is exploited for solving the Schr\"{o}dinger equation for the longitudinal degree of freedom of the electron on the lowest Landau level in the atom. We find that the (effective) potential of a point-like charge remains nonsingular thanks to the growing screening provided by VP. The regularizing length turns out to be $\sqrt{\alpha /3\pi }\lambdabar_{\mathrm{C}}$, where $\lambdabar_{\mathrm{C}}$ is the electron Compton length. The family of effective potentials, labeled by growing values of the magnetic field condenses towards a certain limiting, magnetic-field-independent potential-distance curve. The~limiting values of even ground-state energies are determined for four magnetic quantum numbers using the Karnakov--Popov method.