h is classical

Abstract

Regardless of number, standing wave modes are by definition noninteracting, and therefore cannot thermalize by themselves. Doppler shifts due to thermal motions of cavity walls provide necessary mixing, but also preserve the amplitudes and phases. The lambda/2 intervals of the modes thus preserved must have equal energy expectations, say <E>, in the resulting equilibrium. By definition again, they can be exchanged between modes only in whole numbers and hence only between harmonics. Each family of harmonic modes is thus self-contained and is disjoint from other families in such exchanges, and further, can have no more than one mode excited at any instant. The second property identifies harmonic families of standing wave modes as the harmonic oscillators of Planck&#39;s theory, since a family can only bear energy equal to exactly one of E, 2E, 3E, etc These two properties further imply that the energy expectation gets averaged only over an individual family, as the equilibrium energy <E(f)> steadily available at a given mode would have contributions from its entire harmonic family. Planck&#39;s equations reemerge, and radiation quantization arises as a classical rule <E(f)> = <E>f, as a mode must contain a whole number of exchangeable lambda/2 intervals, but it only concerns equilibrium states. The result makes Planck constant h an analogue of Boltzmann&#39;s constant k for the frequency domain, and points to postulate-free explanations of all aspects of quantum and kinetic theories.