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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

Atomic Number Periodic Table Lower Eigenvalue Electron Repulsion Small Irregularity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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