Mystery of Planck's Constant Revealed

This is a clarification of this post on the physical meaning of Planck's constant $h$ and so of Quantum Mechanics QM as a whole. The basic message is that the numerical value of $h=6.62607015\times 10^{-34}$ Jouleseconds is chosen to make Planck's Law fit with observation and that this value is then inserted into Schrödinger's equation to preserve the linear relation between energy and frequency established in Planck's Law. 

Quantum Mechanics is based on a mysterious smallest quantum of energy/action $h$ named Planck's constant, which was introduced by Planck in 1900 as a "mathematical trick" to make Planck's Law of blackbody radiation fit with observations of radiation energy from glowing bodies of different temperatures. 

The mysterious Planck's constant  $h$ appears in Planck's Law in the combination $\frac{h\nu}{kT}$ where $\nu$ is frequency, $k$ is Boltzmann's constant and $T$ temperature with $kT$ a measure of energy (per degree of freedom) from thermodynamics. In particular  

•  $\nu_{max}=2.821\frac{kT}{h}$                   (*)

shows the frequency of maximal radiation intensity referred to as Wien's Displacement Law, which also serves as a cut-off frequency with quick decay of radiation intensity for frequencies $\nu >\nu_{max}$.  

If we translate (*) to wave length we get a corresponding smallest wave length 

• $\lambda_{min}= 0.2015\frac{hc}{kT}=\frac{0.0029}{T}$ meter
• $\lambda_{min} \approx 10^{-5}$m for $T=300$ K 
• $\lambda_{min} \approx 5\times 10^{-7}$m for $T=5778$ K (Sun)

We see that smallest wave length is orders of magnitude bigger that atomic size of $10^{-10}$ m, which tells that blackbody radiation is a collective wave phenomenon involving many atoms per radiated wave length.

Summary: 

• Planck's constant $h$ serves the role of setting a peak frequency scaling with temperature $T$ with corresponding smallest wave length scaling with $\frac{1}{T}$.
• The smallest wave length is many orders of magnitude bigger than atomic size showing blackbody radiation to be a collective wave phenomenon involving coordinated motion of many atoms. 
• Planck's constant $h$ thus has a physical meaning of setting a smallest spatial resolution size scaling with $\frac{1}{T}$ required for coordinated collective wave motion supporting radiation. 
• Higher temperature means more active atomic motion allowing smaller coordination length. 
• The standard interpretation of $h$ as smallest quanta of energy lacks physical representation.
• Connecting $h$ to coordination length is natural and gives $h$ a physical meaning without mystery. 
• Formally h = energy x time = momentum x length representing Heisenbergs Uncertainty Relation with h connecting to spatial resolution. Formally $E=h\nu=pc$ and so $h=p\lambda$.   

PS Recall that Schrödinger's equation for atoms and Maxwell's equations for light covers a very wide range of phenomena in what is referred to as a semi-classical model as half-quantum + half classical. In this model light is not quantised and there are no photons to worry about. The above meaning of $h$ from Planck's Law is understandable. The mystery is restored in Quantum Electro Dynamics QED where Maxwell's equations are replaced by the relativistic Dirac's equations and particles/photons appear as quantised excitations of fields. QED is way too complicated to be used for the wide range covered by QM and so is reserved for very special geometrically simplified situations.