In terms of theory, nothing connotes discontinuity more than revolution. But even as scientific revolution can radically reorganize a theory's conceptual framework, examination of a revolution's fine structure reveals continuity in techniques and approaches that cut across paradigm shifts. This juxtaposition of continuity and discontinuity during theory change can be seen in the quantum revolution.
In the old quantum theory, techniques from perturbation theory in celestial mechanics were used to analyze electron orbits in atoms classically as a prelude to the translation of the results into quantum formulae. This is the procedure that John Van Vleck followed in his research and made central to his exposition of the old quantum theory in the 1926 Bulletin.(1) It inspired the closely related perturbation techniques in matrix mechanics developed in the famous Dreimaennerarbeit of Max Born, Werner Heisenberg, and Pascual Jordan.(2)
The continuity in these mathematical techniques explains things that would otherwise remain puzzling. It makes it understandable that Van Vleck could still use his 1926 Bulletin in his courses on quantum mechanics in the early 1930s. It also makes it understandable how Van Vleck could make such rapid progress once he hit upon the problem of susceptibilities not long after completion of the Bulletin. The analysis of perturbation theory in the old quantum theory and its adaptation to matrix mechanics demonstrates that even as concepts are in flux, techniques and approaches can provide continuity that spans the shift.
1. John Hasbrouck Van Vleck, Quantum Principles and Line Spectra (Washington, D.C.: National Research Council (Bulletin of the National Research Council 10, Part 4), 1926.
2. Max Born, Werner Heisenberg, and Pascual Jordan, "Zur Quantenmechanik II," Zeitschrift fur Physik 35 (1926): 557-615.