Mystery of Planck’s Constant Uncovered

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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.

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