The Atomic Density on the Thomas--Fermi Length Scale for the Chandrasekhar Hamiltonian

TL;DR

This paper demonstrates that for large neutral atoms modeled by a pseudo-relativistic Hamiltonian, the rescaled electron density converges to the non-relativistic Thomas--Fermi minimizer, indicating relativistic effects are negligible at this scale.

Contribution

It establishes the convergence of the rescaled electron density to the Thomas--Fermi minimizer in a pseudo-relativistic setting, extending non-relativistic results to relativistic models.

Findings

Density converges to Thomas--Fermi minimizer as Z→∞

Relativistic effects are negligible on the Thomas--Fermi scale

Convergence holds when Z/c is fixed up to 2/π

Abstract

We consider a large neutral atom of atomic number $Z$, modeled by a pseudo-relativistic Hamiltonian of Chandrasekhar. We study its suitably rescaled one-particle ground state density on the Thomas--Fermi length scale $Z^{-1/3}$. Using an observation by Fefferman and Seco (1989), we find that the density on this scale converges to the minimizer of the Thomas--Fermi functional of hydrogen as $Z\to\infty$ when $Z/c$ is fixed to a value not exceeding $2/\pi$. This shows that the electron density on the Thomas--Fermi length scale does not exhibit any relativistic effects.