lördag 22 mars 2014

The True Meaning of Planck's Constant as Measure of Wavelength of Maximal Radiance and Small-Wavelength Cut-off.

The modern physics of quantum mechanics was born in 1900 when Max Planck after many unsuccessful attempts in an "act of despair" introduced a universal smallest quantum of action  $h= 6.626\times 10^{-34}\, Js = 4.12\times 10^{-15}\, eVs$ named Planck's constant in a theoretical justification of the spectrum of radiating bodies observed in experiments, based on statistics of packets of energy of size $h\nu$ with $\nu$ frequency.

Planck describes this monumental moment in the history of science in his 1918 Nobel Lecture as follows:
  • For many years, such an aim for me was to find the solution to the problem of the distribution of energy in the normal spectrum of radiating heat.
  • Nevertheless, the result meant no more than a preparatory step towards the initial onslaught on the particular problem which now towered with all its fearsome height even steeper before me. The first attempt upon it went wrong…
  • So there was nothing left for me but to tackle the problem from the opposite side, that of thermodynamics, in which field I felt, moreover, more confident. 
  • Since the whole problem concerned a universal law of Nature, and since at that time, as still today, I held the unshakeable opinion that the simpler the presentation of a particular law of Nature, the more general it is… 
  • For this reason, I busied myself, from then on, that is, from the day of its establishment, with the task of elucidating a true physical character for the formula, and this problem led me automatically to a consideration of the connection between entropy and probability, that is, Boltzmann's trend of ideas; until after some weeks of the most strenuous work of my life, light came into the darkness, and a new undreamed-of perspective opened up before me.
Planck thus finally succeeded to prove Planck's radiation law as a modification of Rayleigh-Jeans law with a high-frequency cut-off factor eliminating "the ultraviolet catastrophe" which had paralyzed physics shortly after the introduction of Maxwell's wave equations for electromagnetics as the culmination of classical physics.

Planck's constant $h$ enters Planck's law
  • $I(\nu ,T)=\gamma \theta (\nu , T)\nu^2 T$, where $\gamma =\frac{2k}{c^2}$,
where $I(\nu ,T)$ is normalized radiance, as a parameter in the multiplicative factor
  • $\theta (\nu ,T)=\frac{\alpha}{e^{\alpha} -1}$, 
  • $\alpha=\frac{h\nu}{kT}$,
where $\nu$ is frequency, $T$ temperature in Kelvin $K$ and $k = 1.38\times 10^{-23}\, J/K = 8.62\times 10^{-5}\, eV/K$ is Boltzmann's constant and $c\, m/s$ the speed of light.

We see that $\theta (\nu ,T)\approx 1$ for small $\alpha$ and enforces a high-frequency small-wavelength cut-off for $\alpha > 10$, that is, for   
  • $\nu > \nu_{max}\approx \frac{10T}{\hat h}$ where $\hat h =\frac{h}{k}=4.8\times 10^{-11}\, Ks$,
  • $\lambda < \lambda_{min}\approx \frac{c}{10T}\hat h$ where $\nu\lambda =c$,
with maximal radiance occuring for $\alpha = 2.821$ in accordance with Wien's displacement law.  With $T = 1000\, K$ the cut-off is in the visible range for $\nu\approx 2\times 10^{14}$ and $\lambda\approx 10^{-6}\, m$. We see that the relation 
  • $\frac{c}{10T}\hat h =\lambda_{min}$,
gives $\hat h$ a physical meaning as measure of wave-length of maximal radiance and small-wavelength cut-off of atomic size scaling with $\frac{c}{T}$. 

Modern physicsts are trained to believe that Planck's constant $h$ as the universal quantum of action represents a smallest unit of a "quantized" world with a corresponding Planck length $l_p= 1.62\times 10^{-35}$ as a smallest unit of length, about 20 orders of magnitude smaller than the proton diameter.

We have seen that Planck's constant enters in Planck's radiation law in the form $\hat h =\frac{h}{k}$, and not as $h$, and that $\hat h$ has the role of setting a small-wavelength cut-off scaling with $\frac{c}{T}$.

Small-wavelength cut-off in the radiation from a body is possible to envision in wave mechanics as an expression of finite precision analog computation. In this perspective Planck's universal quantum of action emerges as unnecessary fiction about exceedingly small quantities beyond reason and reality.

Inga kommentarer:

Skicka en kommentar