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

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

2012-04-11

arxiv.org/abs/1204.2511v1

Physics

Mathematical Physics

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Scientific paper

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 minimum amount of nuclear charge Z*(N) required to bind N "bosonic electrons" grows over-proportional to N, viz. Z*(N)/N grows with N, and thus converges to some l* not bigger than 1 when N goes to infinity. It is also shown that the (negative) ground state energy E(Z,N) yields the monotonically growing function (E(l N,N) over N cubed). 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.

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