Mystery of Planck’s Constant Uncovered

Planck introduced Planck’s constant $h$ in his proof of Planck’s law using statistics based on an ad hoc assumption that the “smallest packet of energy” or “quantum of action” of a wave of frequency $\nu$ is equal to $h\nu$ . Later $h$ appeared in Schrödinger’s wave equation as a measure of the diameter of an atom. Planck’s constant thus has two meanings and the connection has remained a mystery, well supported by a statistical interpretation of the wave function as solution to the wave equation and with the photon as a mysterious particle of energy or $h\nu$ .

To seek to resolve the mystery let us consider wave mechanics without reference to statistics presented in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation, which offers a New View of blackbody radiation based on a generic wave model of a blackbody (as a web of atoms) with the following radiation spectrum of the principal form of Planck’s law:

$R_\nu (T) =\gamma T\nu^2$ for $\nu\le \frac{T}{h}$ ,
$R_\nu (T) = 0$ for $\nu > \frac{T}{h}$ ,

where $T$ is temperature in Kelvin, $\nu$ is frequency and $\gamma$ and $h$ are two parameters with $\nu > \frac{T}{h}$ representing a high-frequency cut-off. Radiative equilibrium between two such blackbodies defined by parameters with the same high frequency cut-off, shows that $\gamma h=C$ where $C$ is a universal constant. The generic wave model thus effectively depends on one parameter, which we take to be $h$ as a representation of Planck’s constant. We here assume that Boltzmann’s constant $k$ as a measure of energy per degree Kelvin is normalized to 1.

We observe that the cut-off condition has the form

$h\nu > T$ .

which gives the parameter $h$ in the a wave model in computational form the meaning of a mesh parameter which can be connected to the diameter of the atoms forming the web supporting the wave model. We thus have only one meaning of Planck’s constant $h$ as a mesh size parameter.

In the New View based on wave mechanics there is no need to give $h\nu$ a different meaning as a mysterious smallest “packet of energy or action” of a wave of frequency $\nu$ and there is no need to speak about a particle named photon. According to Ockham’s razor this should be a step forward.

To view $h$ as a measure of the size of atoms is rational and thus not mysterious. To view $h\nu$ as a measure of “smallest packet of energy” of a wave of frequency $\nu$ is irrational and thus mysterious.

Observing the radiation spectrum from a peephole of a cavity with graphite walls, as a physical wave model of a backbody, allows $\gamma$ and $h$ to be determined experimentally.