Variation and Series Approach to the Thomas-Fermi Equation

TL;DR

This paper introduces a new variational method for solving the Thomas-Fermi equation, providing more accurate asymptotic behavior and initial slope, surpassing existing series solutions in precision.

Contribution

It presents a novel variational approach that improves the accuracy of the numerical solution and asymptotic behavior of the Thomas-Fermi equation.

Findings

More precise asymptotic behavior at large distances.

Exact initial slope value obtained.

Enhanced series solution accuracy.

Abstract

The Thomas - Fermi equation describing the screening of the Coulomb potential inside heavy neutral atoms is reconsidered. An accurate representation for its numerical solution was obtained by means of the variational principle. The proposed new solution has more precise asymptotic behaviour at large distances from the origin and allows us to obtain the exact value of the initial slope. The obtained new variational solution can also be developed in power series similar to the Baker's ones but more precise even than some series solutions that have been recently obtained within the homotopy analysis method and a modified variational method.