onsdag 2 april 2014

Universal Quantum of Action: Standard Without Universality

In recent posts on we have seen that Plank's constant $h$ in physics text books being presented as a universal quantum of action as a smallest "packet of action" as a fundamental constant of fundamental significance in the "quantized" world we happen to be part of, in fact is nothing but a conversion standard between two measures of energy, in terms of frequency $\nu$ in periods per second and electronvolt (eV), determined by Einstein's law of photoelectricity
  • $h(\nu - \nu_0) = e\, U$,
where $\nu_0$ is the smallest frequency releasing electrons from a metallic surface upon exposure of light, $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as the scale factor measured in $eVs$ $electronvolts\times second$ as $energy \times time$ as action. The reference value obtained this way is 
  • $h = 4.135667516(91)\times 10^{-15}\, eVs$,
with Millikan's value from 1916 within $0.5\%$. Determining $h$ this way makes Einstein's law of photoelectricity simply into a conversion standard (that is, a definition) of energy attributing $h\nu$ electronvolts to the frequency $\nu$. Another way of finding the conversion from frequency to electronvolt is using a Josephson junction.

We now turn to Schrödinger's equation
  • $i\bar h\frac{\partial\psi}{\partial t}+H\psi=0$,
where $\bar h=\frac{h}{2\pi}$ is Planck's reduced constant as conversion from periods $\nu$ per second to angular velocity $\omega$ with $h\nu =\bar h\omega$, and $H$ is a Hamiltonian of space dependence. An eigenvalue $E$ of the Hamiltonian represents energy with $\psi_E$ a corresponding space dependent eigenfunction satisfying $H\psi_E =E\psi_E$ and $\exp(i\omega t)\psi_E$ a corresponding solution of Schrödinger's equation with 
  • $h\nu = \bar h\omega =  E$, 
expressing energy in terms of frequency. We see that the appearance of $\bar h$ with the time derivative in Schrödinger's equation accounts for the energy conversion and is completely normal and without mystery. 

Next, we consider the space dependent Hamiltonian in the basic case of the Hydrogen atom:
  • $H\psi =  \frac{\bar h^2}{2m}\Delta\psi +  \frac{e^2}{r}\psi$  
where $\psi =\psi (x)$ with $x$ a space coordinate, $r =\vert x\vert$, and $m$ is the mass of the electron. Normalising by changing scale in space $x=a\bar x$ and time $t=b\bar t$, we obtain the Hamiltonian in normalized atomic units in the form
  • $\bar H = \bar\Delta + \frac{2}{\bar r}$ with smallest eigenvalue $1$, 
  • $a=\frac{\bar h^2}{me^2}$ as $Bohr\, radius$,
  • $b=\frac{\bar h2a}{e^2}$ as $Bohr\, time$ with $\omega =\frac{1}{b}$ angular velocity
  • $E =\frac{e^2}{2a}$ as $Rydberg\, energy$.
We now observe that
  • $E\, b = \bar h$,
  • $E = \bar h\, \omega$, 
which shows that the also the space dependent part of Schrödinger's equation is calibrated to the energy conversion standard. 

Finally, Planck's constant also appears in Planck's radiation law and then in the high-frequency cut-off factor
  • $\frac{\alpha}{\exp(\alpha )-1}$
  • $\alpha = \frac{h\nu}{kT}$,
where $k$ is Boltzmann's constant and $T$ temperature. We see that again $h\nu$ appears as an atomic energy measure with a value that is not very precisely determined in its role in the cut-off factor.
The value of $h$ from photoelectricity can then serve also in Planck's law.

We conclude that Planck's constant $h$ is a conversion standard between two energy measures and as such has no meaning as a universal quantum of action or as integral multiples $nh\nu$ with $n=1,2,3,..$ of special significance other than by connection to eigenfunctions and eigenvalues.   

Ultimately, what is measured are atomic emission spectra in terms of frequencies and wave lengths which through Planck's constant can be translated to energies expressed in electronvolts (or Joule). Nothing of the internal atomic structure (in terms of $e$ and $m$) enters into this discussion.

Planck introduced $h$ in a statistical argument in 1900 long before atoms were known, Einstein picked up $h\nu$ in his 1905 article on photoelectricity, before atoms were known, and Schrödinger put $h$ into his equation in 1926 to describe atoms. This line of events supports the idea that Planck's constant $h$ is a convention without any universal significance.

Understanding the real role of Planck's constant may help to give Schrödinger's equation a physical interpretation which is free from mysteries of "quantization" and statistics. Versions of Schrödinger's equation based on an idea of smoothed particle mechanics then naturally present themselves, with $h$ acting as a smoothing parameter.

PS Notice that the fine structure constant $\alpha = \frac{e^2}{\bar hc}=\frac{1}{137}$ can be expressed as $\alpha =\frac{2}{c}\frac{a}{b}$ which shows that $\alpha$ relates $Bohr\, speed\, =\frac{a}{b}$ to the speed of light $c$. This relation is viewed to be fundamental, but why is hidden in mystery.   

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