3. Magic wand

‘Magic wand’ refers to the ‘correspondence principle’ that Bohr devised and deployed to investigate the interface between quantum and ordinary (‘classical’) physics. The chapter covers various lines of work, some inspired by his approach and some independent of it, all of which confirmed its fertility. The mobilization of the international brotherhood of physicists for the Great War gave Bohr breathing space to develop the correspondence principle with the help of Hendrik Kramers, who had come to neutral Denmark to study with him, and in friendly competition with Arnold Sommerfeld, who made important formal extensions of the theory.

Fine-structure constant from Sommerfeld to Feynman

The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant.

Successes

The set of principles formulated in 1915-1918, and now collectively called the old quantum theory, were successfully applied to a number of problems in atomic and X-ray spectroscopy. The three most notable successes are all associated with the Munich school headed by Arnold Sommerfeld. First, there was the derivation of a relativistic fine-structure formula which predicted splittings of stationary state energies for orbits of varying eccentricity at a given principal quantum number. These splittings were empirically verified by Paschen for ionized helium, and constituted the first quantitative confirmation of the special relativistic mechanics introduced by Einstein a decade earlier. The relativistic fine-structure formula was also applied successfully to the splitting of lines in the X-ray spectra of atoms of widely varying atomic number. Finally, the principles of the old quantum theory (in particular, the use of Schwarzschild quantization in combination with Hamilton-Jacobi methods of classical mechanics) were successfully applied to explain the first order splitting spectral lines in the presence of an external electric field (Stark effect).

On social and psychological aspects of a negligible reception of Natanson’s article of 1911 in the early history of quantum statistics

Possible reasons are studied why Ladislas (Władysław) Natanson’s paper on the statistical theory of radiation, published in 1911 both in English and in the German translation, was not cited properly in the early history of quantum statistics by outstanding scientists, such as Arnold Sommerfeld, Paul Ehrenfest, Satyendra Nath Bose and Albert Einstein. The social and psychological aspects are discussed as background to many so far discussions on the academic evaluation of his theory. In order to avoid in the future such Natansonian cases of very limited reception of valuable scientific works, it is proposed to introduce a digital tag in which all the information of relevant papers published so far should be automatically accumulated and updated.