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

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 $Z^{-1\mathrm{/}3}$. 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 $Z\to \mathrm{\infty }$ when $Z\mathrm{/}c$ is fixed to a value not exceeding $2\mathrm{/}\pi$. This shows that the electron density on the Thomas–Fermi length scale does not exhibit any relativistic effects.