WKB and the Periodic Table

  • Charles L. Fefferman
Part of the Trends in Mathematics book series (TM)

Abstract

In this article, I’d like to motivate some work that Luis Seco and I have done on the ground-state energy of an atom of atomic number Z ≫ 1. If one ignores relativistic effects, then the mathematical problem is as follows. Fix a nucleus of charge +Z at the origin. If N electrons are located at x 1, x 2,…, x N ∈ ℝ3, then their potential energy is given by
$$ V_{Coulomb}^{NZ} (x_{1,...,} x_N ) = - \sum\limits_{k = 1}^N {\frac{z} {{|x_k |}}} + \sum\limits_{1 \leqslant j < k \leqslant N} {\frac{1} {{|x_j - x_k |}}.} $$
(1)
We regard the electrons as quantized, so that the state of the system is given by a wave function
$$ \psi \left( {{x_1}, \ldots, {x_N}} \right) \in {L^2}\left( {{\mathbb{R}^{{3N}}}} \right) $$
For simplicity, we neglect spin here. (If we had taken spin into account, then ψ would take values in the ψ-fold tensor power of ℂ2. This changes no ideas, but introduces factors of 2 into some key formulas).

Keywords

Argon Helium Seco Neon 

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Copyright information

© Birkhäuser Boston 1999

Authors and Affiliations

  • Charles L. Fefferman
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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