Sommerfeld’s fine structure constant approximated as a series representation in e and $\pi$

Abstract

Sommerfeld in 1916 introduced the dimensionless fine structure constant, $\alpha$, in to the context of atomic physics, in the course of working out the relativistic theory of the H atom, under the old quantum theory of Bohr. He was able to account for the fine structural detail of the atomic line spectrum of H by introducing this dimensionless constant which emerged naturally from his relativistic theory of the H atom. Since this time, the fine structure constant has emerged in several other contexts within experimental and theoretical physics. It has attained a status of being a mysterious number in physics that defies understanding as to its experimentally verified magnitude and identity. Being physically dimensionless, such a number invites a suggestion (or approximation) of its value in terms of mathematical constants in some formulation. Feynman most famously has conjectured that it might be possible to account for $\alpha$ in some type of series or product expression in “e”, the base of natural logarithms, and “$\pi$” the familiar circular constant. Here we propose an infinite series in the product $\mathrm{e} \cdot \pi$ that converges, within a few terms, to better than 9999 parts in 10,000 of the true value of $\alpha$.