lördag 15 juni 2019

Demystifying the New SI Base Units.

In the previous post we observed that Planck's constant $h$ appears as a conversion factor connecting light of frequency $\nu$ with attributed energy $h\nu$ (in eV or Joule) through the photoelectric effect with the release of an electron from a surface exposed to light (of sufficient high frequency). The inner mechanics of the atoms delivering the electrons upon excitation by exposure to light does not enter into the discussion and so Planck's constant can be given a meaning in macroscopic physics, thus without quantum mechanics, as a trade between light and electron energy and then further to mechanical energy. Its role in quantum mechanics then appears as an after construction.

Let us now turn to Boltzmann's constant $k$ to see its connection to Planck's constant and macroscopic physics. Boltzmann's constant appears in Planck's universal law of blackbody radiation law of the form
  • $E(\nu ,T) = W(a)\, kT\nu^2$,
  • $W(a) = \frac{a}{\exp(a )-1}$ with $a = \frac{h\nu}{kT}$,
where $E(\nu ,T)$ is the (suitably normalised) intensity of radiation of frequency $\nu$ from a blackbody of temperature $T$ and $W(a)$ is a cut-off factor with $W(a)=1 $ for small $a$ and
$W(a)$ small for medium to large $a$, expressing Wien's displacement law stating cut-off of high frequencies. We see that Planck’s constant only appears in the cut-off factor.

Experimental observation of $E(\nu ,T)$ makes it possible to determine $W(a)$ and thus $kT$ in terms of $h\nu$, from which Boltzmann's constant $k$ can be determined with respect to a chosen scale for temperatur $T$, or the other way around as in the new SI units by specifying by definition 
  • $k=1.380650\times 10^{-23}\, J/K$,  
thus setting a new standard for Kelvin $K$ as measure of temperature. The connection between the energy measures $h\nu$ and $kT$ then shows to be
  • $h\nu_{max} \approx 2.8214391\times kT$,
where $\nu = \nu_{max}$ gives maximum of the spectrum $E(\nu ,T)$.

Again, this can be done without having to invoke quantum mechanics in its standard form with $h$ as a "smallest quantum of action" as exposed in detail on Computational Blackbody Radiationwhich presents a derivation of Planck's law using deterministic continuum physics instead of as usual statistics of discrete quanta. In particular, the new derivation captures the universality of blackbody radiation beyond specific inner atomic mechanics.

The universality of Planck's law is expressed by the fact that an ideal blackbody can take the form of a set of oscillators without very specific inner structure. In particular different blackbodies with different inner structure can share the same temperature scale.

To sum up, both Planck's constant and Boltzmann's constant are specified by definition in the new SI units, from which the new units kilogram and Kelvin can be determined by macroscopic experiments without resort to quantum mechanics in its standard form.

Hopefully this helps to demystify both Planck's and Boltzmann's constant, and the new SI units.

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