Quantization Conditions, 1900–1927: From Ambiguity to Uncertainty

I give a brief overview of the development of quantization conditions from Planck to Heisenberg. It is widely accepted by now that Planck did not quantize the energy of his resonators when he introduced the constant named after him. Planck’s work, however, did sow the seeds for Sommerfeld's later phase-integral approach to generalizing Bohr's model of the hydrogen atom. Schwarzschild connected this approach to some powerful techniques from celestial mechanics. The resulting theory, now remembered as the old quantum theory, was less than a decade old when it ran into insurmountable difficulties. In his famous "Umdeutung" [= reinterpretation] paper, Heisenberg showed that the way out of the impasse was to replace single-component quantities occurring in the classical laws by many-component ones, which Born and Jordan soon recognized to be matrices. Following this "Umdeutung" procedure, Heisenberg, Born and Jordan rewrote the basic quantization condition of the old quantum theory in the form of the now familiar commutation relations for position and momentum. In 1927, drawing on Jordan's unification of matrix mechanics and Schrödinger's wave mechanics, Heisenberg showed that these commutation relations express what we now know as the uncertainty principle.