The Contrasting Roles of Planck's Constant in Classical and Quantum Theories

TL;DR

This paper explores the historical and conceptual roles of Planck's constant in classical and quantum physics, highlighting its fundamental importance in quantum theory and its optional presence in classical electrodynamics.

Contribution

It clarifies the distinct roles of Planck's constant in classical and quantum theories and discusses the implications for the foundations of these theories.

Findings

Planck's constant originally did not relate to quanta.

In quantum theory, it is essential for the theory's structure.

In classical electrodynamics, it is optional and can be zero or non-zero.

Abstract

We trace the historical appearance of Planck's constant in physics, and we note that initially the constant did not appear in connection with quanta. Furthermore, we emphasize that Planck's constant can appear in both classical and quantum theories. In both theories, Planck's constant sets the scale of atomic phenomena. However, the roles played in the foundations of the theories are sharply different. In quantum theory, Planck's constant is crucial to the structure of the theory. On the other hand, in classical electrodynamics, Planck's constant is optional, since it appears only as the scale factor for the (homogeneous) source-free contribution to the general solution of Maxwell's equations. Since classical electrodynamics can be solved while taking the homogenous source-free contribution in the solution as zero or non-zero, there are naturally two different theories of classical electrodynamics, one in which Planck's constant is taken as zero and one where it is taken as non-zero. The textbooks of classical electromagnetism present only the version in which Planck's constant is taken to vanish.