torsdag 6 mars 2014

Mystery of Planck's Constant h 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 (spatial) 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 $h\nu$.

To seek to resolve the mystery let us consider the 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 vibrating 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, $\nu$ 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 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 finite precision 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. This makes it possible to determine the size of atomistic microscopics in a macroscopic experiment.

From the standard non-normalized values $h = 6.626\times 10^{-34}$ and $k = 1.381\times 10^{-23}$, we have with normalization to $k=1$ that $h = 4.8\times 10^{-11}$ which is of atomistic size. With $T = 480\, K$ we thus find a cut-off frequency of $10^{13}$ in the infrared visible range. 

2 kommentarer:

  1. Could you clarify what you mean with h=4.8*10^-11 being of atomistic scale?

    It looks like you are dividing h with k, but since they both have units you end up with

    h/k = 4.8*10^-11 Js/(J/K) = 4.8*10^-11 sK

    In what way is this on atomistic scale?

  2. h/k has the dimension of T/f with f frequency connected to wave length as highest frequency which can be carried by a web of oscillating atoms at temp T.