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13: Thomas-Fermi Analysis

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    • 13.1: Prelude to Thomas-Fermi Analysis
      This page explores the difficulties in pinpointing electron locations in atoms, highlighting the effects of their motion and the Heisenberg uncertainty principle. It introduces charge density as a statistical representation of probable electron locations, affecting chemical and electrical characteristics.
    • 13.2: Preliminary Ideas
      This page covers derivatives and integrals of scalar functions in spherical coordinates, focusing on the gradient and Laplacian, their definitions, and simplifications for uniform functions. It highlights notation, especially for voltage and position, and introduces reciprocal space related to atomic properties. Additionally, it references figures and tables to clarify variable meanings concerning energy and voltage.
    • 13.3: Derivation of the Lagrangian
      This chapter explores the voltage \(V(r)\) and charge density \(\rho_{ch}(r)\) around a spherically symmetric atom at near absolute zero. It links charge density to the quantum wave function and discusses energy interactions among electrons and the nucleus. Key concepts include kinetic energy, Fermi energy, and wave vectors, establishing relationships between charge density, kinetic energy density, and voltage.
    • 13.4: Deriving the Thomas-Fermi Equation
      This page explores classical mechanics principles concerning electrons in atoms, emphasizing the Principle of Least Action and the Euler-Lagrange equation to calculate electron motion. It elucidates the connection between voltage and charge density, introducing the Thomas-Fermi equation derived from these concepts.
    • 13.5: From Thomas-Fermi Theory to Density Functional Theory
      This page examines early electron location calculations (1926-1928) via the Thomas-Fermi method, noting critical assumptions like neglecting angular dependencies and the assumption of absolute zero. It introduces density functional theory (DFT) as a more refined method that reduces these assumptions and allows for better exploration of electron properties.
    • 13.6: Problems
      This page covers generalized momentum within Lagrangian mechanics, detailing its calculation, units, and alternative formulations of Hamiltonian and Lagrangian. It discusses generalized paths and potentials and showcases transformations between the two frameworks. Furthermore, it evaluates solutions to the Thomas Fermi equation, highlighting its nonlinear characteristics and the physical significance of these mathematical constructs.

    This page titled 13: Thomas-Fermi Analysis is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Andrea M. Mitofsky via source content that was edited to the style and standards of the LibreTexts platform.