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 \(\alpha \).

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References

  1. 1.

    A. Sommerfeld, Atomic Structure, Spectral Lines, 1st edn. (Methuen Press, London, 1923)

    Google Scholar 

  2. 2.

    M. Born, Atomic Physics, 8th edn. (Dover Books, Mineola, 1989)

    Google Scholar 

  3. 3.

    P.J. Mohr, B.N. Taylor, D.B. Newell, Fine structure constant, in CODATA Internationally recommended 2014 values of the fundamental physical constants, National Institute of Standards and Technology (2015)

  4. 4.

    R.P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University Press, Princeton, 1985), p. 129

    Google Scholar 

  5. 5.

    M.E. Tobar, An alternative view of the fine structure constant and its variation: bringing the flux quanta into the definition of the electron, in Proceedings of the 10th Marcel Grossmann Meeting, vols. I–III, Rio de Janeiro, Brazil, World Scientific (2006), pp. 2073–2075

  6. 6.

    A.S. Eddington, The constants of nature, in The World of Mathematics, vol. II, ed. by J.R. Newman (Simon and Schuster, New York, 1956), pp. 1074–1093

    Google Scholar 

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Correspondence to Michael J. Bucknum.

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Bucknum, M.J., Castro, E.A. Sommerfeld’s fine structure constant approximated as a series representation in e and \(\pi \) . J Math Chem 56, 651–655 (2018). https://doi.org/10.1007/s10910-017-0835-8

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Keywords

  • Fine structure constant \(\alpha \)
  • Sommerfeld
  • Infinite series
  • e, \(\pi \)