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Saturday, August 13, 2011

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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