The Physical Meaning of Planck's Constant $h$

Planck's constant $h$ appears in several different contexts in quantum physics including 

1. Planck's Law of blackbody radiation.
2. Schrödinger's Equation SE describing the spectrum of the Hydrogen atom. 

Here 1. involves collective behaviour of atoms, while 2. concerns one atom. The standard view is that the double appearance of $h$ in two fundamentally different contexts is an expression of a deep connection in a mysterious quantum world, which cannot be understood.  

Let us see if we can uncover some the mystery. We recall from this post  that $h$ was introduced by Planck in 1900 to make Planck's Law fit with observation, specifically in a high frequency cut-off factor $C(\alpha )$ with $\alpha =\frac{h\nu}{kT}$ where $\nu$ is frequency, and $k$ as Boltzmann's constant combined with temperature $T$ into $kT$ serves as an energy per degree of freedom in thermodynamics.  For $h\nu << kT$ then $C(\alpha ) =1$ and for $h\nu >kT$, $C(\alpha )$ decays to zero. The quantity $h\nu$ is thus compared to the elementary energy $kT$ and so $h\nu$ is referred to as a basic quantum of energy $E=h\nu$ connected to the frequency $\nu$. This is a linear relation and $h$ is the constant of proportionality between energy and frequency. 

Energy $E$ is thus connected to frequency $\nu$ by $E=h\nu$ with physical meaning of setting the high-frequency cut-off in Planck's Law in relation to $kT$.  This explains the function of $h$ in 1. as serving in high frequency cut-off. High-frequency connects to small-wavelength which connects to spatial resolution. Wien's displacement law takes the form $\nu \approx \frac{k}{h}T$ with linear scaling between frequency and temperature as measure of energy.  We see that $h$ appears in a high-frequency threshold condition scaling with $T$ which gives $h$ a physical meaning but not as measure of physical quantity representing some smallest quantum of energy $h\nu$. Threshold condition and not physical quantity.  

We now turn to 2. recalling Schrödinger's Equation SE with $H$ a Hamiltonian:

• $ih\frac{\partial\psi}{\partial t}+H\psi =0$

with solution $\exp (i\frac{E}{h}t)\Psi$ where $\Psi$ is an eigenfunction satisfying $H\Psi =E\Psi$ with $E$ an eigenvalue as energy. We see frequency $\nu =\frac{E}{h}$ with thus $E=h\nu$. We thus see that SE is constructed to carry a linear relation between energy and frequency, more specifically the relation $E=h\nu$. Notice that temperature $T$ does not appear in SE, only in the context of blackbody radiation as collective behaviour.

We understand the value of $h=6.62607015\times 10^{-34}$ Jouleseconds is chosen to make Planck's Law fit with observation as concerns high frequency cut-off based on $h\nu$ scaling linearly with frequency $\nu$, and with $T$ in Wien's displacement law.. 

Once the value of $h$ was determined to fit 1. with linear scaling of energy vs frequency as $E=h\nu$, that value of $h$ was inserted into SE with the effect of preserving the linear scaling of energy vs frequency as $E=h\nu$. The appearance of $h$ is SE was not due to a miraculous intervention of Nature, but simply the effect of putting it in by the design of SE. 

Summary: 

1. The value of $h$ as conversion factor between energy and frequency as $E=h\nu$ in a linear relation, is set to make Planck's Law fit with observation. 
2. That value of $h$ is then inserted by design into SE to keep the $E=h\nu$ linear relation between energy and frequency. 
3. The design is elaborated by connecting a one electron energy jump $E$ to emission/absorption of one photon of frequency $\nu$ determined by $\nu =\frac{E}{h}$ as the mechanism behind the spectrum of Hydrogen.
4. In short: The appearance of  $h$ is SE is not deep mystery, but simply the design of SE.  
5. Is the fact that SE describes the observed spectrum of Hydrogen a mystery? Not with charge density representation of the wave function solution to SE, in which case SE has classical continuum mechanical form of a vibrating elastic body and as such not a mystery. 
6. Extension to atoms with more than one electron presents new questions to be answered differently by Standard QM and RealQM.