abstract-nakane – Niels Bohr Archive

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Niels Bohr Archive > Calendar > 2013 > 100 years of the Bohr atom > Programme > abstract-nakane

Michiyo Nakane, Kawasaki, Japan
Bohr’s Introduction to Action-Angle Variables in a 1918 Paper

Action-angle variables provide one of the most important mathematical techniques for quantum theory. These variables originated in Charlier’s books on celestial mechanics published in 1902 and 1907. Charlier performed a canonical transformation defined by a particular generating function and attained new canonical variables constructed by angle variables and action integrals. Noting Charlier’s argument, Schwarzschild defined action-angle variables and used them for an explanation of Stark effects in 1916. However, he did not present action variables in the form of Ii=∫pidqi, but he noted variables that have the same dimension as the action integrals that construct canonical variables together with angle variables, ωi=nit+βi, where t is time, ni is the mean motion, and βi is the initial value of the angle. In a 1918 paper entitled “On the Quantum Theory of Line-Spectra,” Bohr mentioned that Kramers showed him a way to make Ii and ωi canonical conjugates. An origin of formation of action-angle variables that we find in textbooks of mechanics today first appeared here. Then, Bohr developed his idea of a conditionally periodic system using the action variables Ii.