[Paper Review] Statistical Theory of the Atom in Momentum Space

Verena von Conta, Siedentop, Heinz|arXiv (Cornell University)|Aug 9, 2011

Spectral Theory in Mathematical Physics19 references3 citations

TL;DR

This paper establishes the mathematical foundation of a momentum-space statistical model for atoms, proving that the momentum energy functional introduced by Englert yields the same ground state energy and momentum density as the Thomas-Fermi functional in the large-Z limit. It rigorously shows that the atomic momentum density converges on the scale $Z^{2/3}$ to the minimizer of the momentum functional, validating its use for momentum-dependent perturbations.

ABSTRACT

We investigate a semiclassical momentum density energy functional for atoms and show that it yields the same value as the well-known Thomas-Fermi functional. In fact, we show an explicit relation between the minimizers of the two functionals. The main result is that the atomic momentum density converges on the scale $Z^{2/3}$ to the minimizer of the momentum density energy functional. This allows to determine the linear response of atoms on momentum dependent perturbations.

Motivation & Objective

• To establish the mathematical validity of the momentum-space energy functional proposed by Englert for atomic systems.
• To demonstrate that this functional yields the same ground state energy as the Thomas-Fermi functional in the large-Z limit.
• To prove that the atomic momentum density converges to the minimizer of the momentum functional on the scale $Z^{2/3}$, extending the asymptotic analysis to momentum space.
• To provide a rigorous framework for studying linear response to momentum-dependent perturbations, which is not accessible via position-space models.

Proposed method

• The paper analyzes the momentum energy functional $\mathcal{E}_{\mathrm{mTF}}(\tau)$, defined in terms of momentum density $\tau$, with kinetic, attraction, and repulsion terms.
• It introduces a transformation $\tau \mapsto \tilde{\tau}^{3/2}$ to render the functional strictly convex, enabling the use of variational methods.
• The authors prove the functional is well-defined on $L^1(\mathbb{R}^3, (1+\xi^2)d\xi)$ with non-negative $\tau$, ensuring physical relevance.
• They derive the Euler-Lagrange equation for the minimizer and use a localization technique with smooth, compactly supported functions to control error terms.
• A key step involves estimating the difference between the quantum momentum density and the classical momentum functional using trace inequalities and uniform continuity.
• The proof leverages Lieb's asymptotic results and correlation inequalities to bound error terms, showing convergence of the rescaled momentum density to the minimizer.

Experimental results

Research questions

• RQ1 Does the momentum-space energy functional proposed by Englert yield the same ground state energy as the Thomas-Fermi functional in the large-Z limit?
• RQ2 Is the minimizer of the momentum functional related to the minimizer of the Thomas-Fermi functional, and if so, how?
• RQ3 Does the quantum momentum density of the atom converge to the minimizer of the momentum functional as $Z \to \infty$?
• RQ4 Can the momentum-space model correctly describe linear response to momentum-dependent perturbations, which position-space models cannot?

Key findings

• The momentum energy functional $\mathcal{E}_{\mathrm{mTF}}$ yields the same minimal energy as the Thomas-Fermi functional, confirming its asymptotic correctness for energy.
• An explicit relation is established between the minimizers of the momentum functional and the Thomas-Fermi minimizer, showing equivalence in the large-Z limit.
• The atomic momentum density converges to the minimizer of the momentum functional on the scale $Z^{2/3}$, i.e., $Z^{-2}\tau(\cdot Z^{-1/3}) \to \tau_1$ weakly as $Z \to \infty$.
• The functional is well-defined on $L^1(\mathbb{R}^3, (1+\xi^2)d\xi)$ with non-negative $\tau$, ensuring mathematical consistency.
• Error terms in the energy estimate are shown to be $o(Z^{7/3})$, confirming the asymptotic validity of the momentum-space model.
• The model enables the study of linear response to momentum-dependent perturbations, which is not feasible with position-space statistical models.