The magic numbers of the quantum

''It was in the course of similar investigations that Liveing and Dewar discovered in the spectra of sodium and potasium the existence of two series of lines, one of sharp and one of diffuse lines, and established the existence of homologous series, that is, series of lines of the same type in chemically analogous elements. [...].
Three years later, such an empirical formula, the first to comprise correctly all lines of a spectral series, was published by Johan Jakob Balmer, a schoolteacher in Basel. [...]. He therefore expressed the wavelengths of these lines by the formula [in millimeters]

lambda=h m^2/(m^2-2^2) mm

where h=3645.6 x 10^-7 and m=3, 4, 5, 6. Balmer also predicted the existence of a fifth line with wavelength 3969.65 x 10^-7 mm. Informed by von Hagenbach that this line and other additional lines had indeed been discovered by Huggins, Balmer showed in a second paper that this formula applied to all twelve then known hydrogen lines. He also predicted correctly that in the series which subsequently carried his name, no lines of wavelength longer that 6562 x 10^-7 mm would ever be discovered and that the series could converge at 3645.6 x 10^-7 mm. The agreement between the calculated and the observed values of the wavelength was extremely close in the visible region, but for shorter wavelengths slight systematic discrepancies were evident. In view of these discrepancies Balmer expressed some doubts whether the fault lay with the formula or with the data.''

In Jammer, The Conceptual Development of Quantum Mechanics, pp. 65-66.
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Planck and the Quantum

''What did Planck do in 1900? Conventional wisdom holds that he quantized harmonic oscillators (or resonators, to use his term) in equilibrium with electromagnetic radiation in a cavity, at some fixed temperature. But physicists interested in the history of their discipline have learned to beware of such conventional wisdom, with good reason. If we turn to the historians for guidance, however, we find little consensus. To be sure, most would agree that the unqualified statement that ‘Planck quantized the oscillators’ can be misleading; on any interpretation, Planck’s understanding of this phrase would have been quite different from our own. Nevertheless, many historians, following the lead of Martin J. Klein, do assert that, however tentatively and uncertainly, Planck – with his finite ‘energy elements’ – did introduce something very new into physics in 1900, and almost certainly knew that he had done so. This situation changed in 1978 with the publication of Thomas S. Kuhn’s Black‐Body Theory and the Quantum Discontinuity. Kuhn gave a highly detailed account of Planck’s work and the initial reaction to it. But in the process, he argued that Planck could not possibly have intended such a far‐reaching step. In Kuhn’s words: My point is not that Planck doubted the reality of quantization or that he regarded it as a formality to be eliminated during the further development of his theory. Rather, I am claiming that the concept of restricted resonator energy played no role in his thought .

[…]

In Kuhn's scenario, the prospect of discontinuous energies did not appear until 1905 and 1906, in the work of the young and little‐known physicists Paul Ehrenfest and Albert Einstein. And even then, according to Kuhn, Planck did not take the idea seriously until 1908. Another historian of physics, Olivier Darrigol, has reached similar conclusions, though in part for different reasons, on the basis of his own detailed analysis of Planck’s work. Others have maintained the older view. And from the standpoint of Allan Needell, who also has written extensively on Planck, even to put the issue in these terms is to ask the wrong question: Our goal should be to understand Planck on his own terms, rather than focus too exclusively on a question that would have had little meaning in 1900.

All of these works have led to a much more detailed understanding of Planck, and even to a considerable measure of agreement. Nevertheless, on the central question – how did Planck think about his derivation in 1900? – no consensus has emerged.''

In Gearhart, “Planck, the Quantum, and the Historians,” pp. 171-173.
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