The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion

TL;DR

This paper proves that for large numbers of bosonic electrons in an atom or ion, the ground state energy converges to a Hartree theory limit, with the wave function factorizing into identical one-body functions.

Contribution

It demonstrates the Hartree limit for Born's ensemble in bosonic atoms or ions, extending mean-field results to quantum N-body Coulomb systems with a novel variational approach.

Findings

Ground state energy scales with N^3 and converges to Hartree energy.

Wave functions factorize into identical one-body functions in the limit.

The variational principle involves Fisher information, analogous to classical free energy.

Abstract

The non-relativistic bosonic ground state is studied for quantum N-body systems with Coulomb interactions, modeling atoms or ions made of N "bosonic point electrons" bound to an atomic point nucleus of Z "electron" charges, treated in Born--Oppenheimer approximation. It is shown that the (negative) ground state energy E(Z,N) yields the monotonically growing function (E(l N,N) over N cubed). By adapting an argument of Hogreve, it is shown that its limit as N to infinity for l > l* is governed by Hartree theory, with the rescaled bosonic ground state wave function factoring into an infinite product of identical one-body wave functions determined by the Hartree equation. The proof resembles the construction of the thermodynamic mean-field limit of the classical ensembles with thermodynamically unstable interactions, except that here the ensemble is Born's, with the absolute square of the ground state wave function as ensemble probability density function, with the Fisher information functional in the variational principle for Born's ensemble playing the role of the negative of the Gibbs entropy functional in the free-energy variational principle for the classical petit-canonical configurational ensemble.