söndag 23 mars 2014

Why the Same Universal Quantum of Action $h$ in Radiation, Photoelectricity and Quantum Mechanics?

Planck's constant $h$ as The Universal Quantum of Action was introduced by Planck in 1900 as a mathematical statistical trick to supply the classical Rayleigh-Jeans radiation law $I(\nu ,T)=\gamma T\nu^2$ with a high-frequency cut-off factor $\theta (\nu ,T)$ to make it fit with observations including Wien's displacement law, where
  • $\theta (\nu ,T) =\frac{\alpha}{\exp(\alpha )-1}$,
  • $\alpha =\frac{h\nu}{kT}$, 
$\nu$ is frequency, $T$ temperature in Kelvin $K$ and $k =1.38066\times 10^{-23}\, J/K$ is Boltzmann's constant and $\gamma =\frac{2k}{c}$ with $c\, m/s$ the speed of light in vaccum. Planck then determined $h$ from experimental radiation spectra to have a value of $6.55\times 10^{-34} Js$, as well as Boltzmann's constant to be $1.346\times 10^{-23}\, J/K$ with $\frac{h}{k}= 4.87\times 10^{-11}\, Ks$ as the effective parameter in the cut-off.  

Planck viewed $h$ as a fictional mathematical quantity without real physical meaning, with $h\nu$ a fictional mathematical quantity as a smallest packet of energy of a wave of frequency $\nu$, but in 1905 the young ambitious Einstein suggested an energy balance for photoelectricity of the form 
  • $h\nu = W + E$,
with $W$ the energy required to release one electron from a metallic surface and E the energy of a released electron with $h\nu$ interpreted as the energy of a light photon of frequency $\nu$ as a discrete lump of energy. Since the left hand side $h\nu$ in this law of photoelectricity was determined by the value of $h$ in Planck's radiation law, a new energy measure for electrons of electronvolt was defined by the relation $W + E =h\nu$. As if by magic the same Universal Quantum of Action $h$ then appeared to serve a fundamental role in both radiation and photoelectricity.

What a wonderful magical coincidence that the energy of a light photon of frequency $\nu$ showed to be exactly $h\nu \, Joule$! In one shot Planck's fictional smallest quanta of energy $h\nu$ in the hands of the young ambitious Einstein had been turned into a reality as the energy of a light photon of frequency $h\nu$, and of course because a photon carries a definite packet of energy a photon must be real. Voila!

In 1926 Planck's constant $h$ showed up again in a new context, now in Schrödinger's equation
  • $-\frac{\bar h^2}{2m}\Delta\psi = E\psi$
 with the formal connection   
  • $p = -i\bar h \nabla$ with $\bar h =\frac{h}{2\pi}$,
  • $\frac{\vert p\vert^2}{2m} = E$, 
as a formal analog of the classical expression of kinetic energy $\frac{\vert p\vert ^2}{2m}$ with $p=mv$ momentum, $m$ mass and $v$ velocity.

Planck's constant $h$ originally determined to make theory fit with observations of radiation spectra and then by Planck in 1900 canonized as The Universal Quantum of Action, thus in 1905 served to attribute the energy $h\nu$ to the new fictional formal quantity of a photon of frequency $\nu$ . In 1926 a similar formal connection was made in the formulation of Schrödinger's wave equation.  

The result is that the same Universal Quantum of Action $h$ by all modern physicists is claimed to play a fundamental role in both (i) radiation, (ii) photolelectricity and (iii) quantum mechanics of the atom. This is taken as an expression of a deep mystical one-ness of physics which only physicists can grasp,  while it in fact it is a play with definitions without mystery, where $h$ appears as a parameter in a high-frequency cut-off factor in Planck's Law, or rather in the combination $\hat h =\frac{h}{k}$,  and then is transferred into (ii) and (iii) by definition.  Universality can this way be created by human hands by definition. The power of thinking has no limitations, or cut-off.

No wonder that Schrödinger had lifelong interest in the Vedanta philosophy of Hinduism "played out on one universal consciousness".

But Einstein's invention of the photon as light quanta in 1905 haunted him through life and approaching the end in 1954, he confessed:
  • All these fifty years of conscious brooding have brought me no nearer to the answer to the question, "What are light quanta?" Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken. 
Real physics always shows up to be more interesting than fictional physics, cf. Dr Faustus ofd Modern Physics.

PS Planck's constant $h$ is usually measured by (ii) and is then transferred to (i) and (iii) by ad hoc definition.

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