The Secret of $E=h\nu$ as Smallest Quanta of Energy/Action

Quantum Mechanics was born from the smallest quantum of energy (or action) $h\nu$, with $h$ Planck's constant and $\nu$ a frequency, appearing in Planck's mathematical analysis of blackbody radiation which captured the dependence on temperature $T$ and $\nu$ of the energy transfer $E$ from a glowing body in the form of Planck's Law 

• $E(\nu, T)=\gamma T\nu^2\times C(\alpha )$ with $\gamma =\frac{2k}{c^2}$,     (1)

where $k$ is Boltzmann's constant and $c$ the speed of light, and 

• $C(\alpha )=\frac{\alpha}{\exp^\alpha-1}$ with $\alpha =\frac{h\nu}{kT}$ 

is a high-frequency cut-off  factor with activation if $h\nu >kT$.   

The analysis concerned a hypothetical blackbody as a cavity with reflecting walls filled with waves of many frequencies brought to a common temperature by the presence of a small piece of sooth.

Planck determined a value of $h=6.55\times 10^{-34}$ Jouleseconds to make his Law fit with observation of the cut-off factor $C(\frac{h\nu}{kT})$ with $\nu$, $k$ and $T$ given.  

The smallest quantum of energy/action $h\nu$ was used as a "mathematical trick" to get a the observed dependence of the high-frequency cut-off factor on $\frac{\nu}{T}$ as an expression of Wien's displacement law. No real physics. Not yet any atom.

The next step was Einstein's association of the smallest quantum of energy/action $h\nu$ to exactly one "particle of light" of frequency $\nu$ later named "photon", in his 1905 heuristic analysis of the photo-electric effect as "one electron - one photon". No real physics. Not yet any atom. Einstein formally identified the energy $E$ of one electron through $E=h\nu$, but he did not yet have any model of an electron. 

In 1913 Bohr presented a model of a Hydrogen atom as one electron revolving around a kernel in different possible orbits of with certain specific energies, with differences $E$ of these possible one-electron energies showing to fit with the observed frequencies $\nu$ of the radiation spectrum of Hydrogen if the shift/jump of one electron was assigned an energy $E=h\nu$. Success, but an ad hoc model of an electron without physics with an ad hoc assignment of exactly the energy $E=h\nu$ to the jump of one electron between the two energy levels.   

In 1926 Schrödinger presented a wave equation model of a Hydrogen atom in terms of a wave function representing electron charge density with eigenvalues coinciding with the energy levels of the Bohr model. Again with an ad hoc association of the energy $E=h\nu$ to one electron shifting between energy levels differing by $E$. Great success in the form of a classical continuum model, but the generalisation to atoms with more than one electron carried physics into the new modern world of quantum physics fundamentally different from continuum physics with Planck's constant $h$ setting a form of smallest scale requiring quantisation of physics. We sum up:

The secret of Planck's constant $h$ with $h\nu$ smallest quantum of energy/action.  

1. A value of $h$ is determined to make the high frequency cut-off depending on $\frac{h\nu}{kT}$ in Planck's Law of blackbody radiation, fit with observation.
2. The value of $h$ so determined is connected to Schrödinger's Equation SE for a Hydrogen atom with one electron in the following way. The eigenvalues of SE represent different possible energies of the electron and differences $E$ between energy levels represent shifts of energy $E$ of one electron, which in SE is assigned a frequency $\nu = \frac{E}{h}$ so that $E=h\nu$. 
3. The shift between energy levels of the one electron of the Hydrogen atom $E=h\nu$ is associated with emission/absorption of one photon of frequency $\nu$ carrying an energy of $h\nu$. 

Altogether we understand that Planck's constant $h$ is determined to make the high-frequency cut-off in Planck's Law fit with observation. This value is then used to assign the energy $E=h\nu$ to one photon corresponding to the jump of one electron of a Hydrogen atom between energy levels differing by $E$.

We understand that high-frequency cut-off is related to small-wavelength cut-off as condition on spatial resolution as discussed in detail in Computational Blackbody Radiation. 

To understand that the smallest quantum of energy/action $E=h\nu$ is a definition without physical representation, it helps to recall that $h$ in the 2019 SI specification of units is assigned the exact value $6.62607015\times 10^{-34}$, which differs from Planck's original value $6.55 \times 10^{-34}$. If $h\nu$ really carried some real physics, it would not make sense to assign it a specific value, since it does not make sense to prescribe real physics how to behave. Ok?

PS1 A basic difficulty of understanding theoretical physics is that the distinction beween definition and physical fact is often blurred. If you do not understand that the speed of light in vacuum and smallest quantum of energy are assigned certain values rather than being given by the Creator, you miss something very essential.

PS2 The central theme in all the above models is the radiating atom in radiative equilibrium with light of certain frequencies showing up in the emission or absorption spectrum of the atom, as a resonance phenomenon analysed in Computational Blackbody Radiation. The basic mathematical model takes the following form in a wave function $\phi (t)$ depending on time $t$ satisfying

• $\ddot\phi -\nu^2\phi -\gamma\dddot\phi = f$           (2)

where a dot signals differentiation with respect to a time variable, and $f$ is forcing with frequency $\tilde\nu\approx \nu$ in near resonance and $\gamma <<\frac{1}{\nu^2}$. The essence of radiative equilibrium as output = input reads (in terms of time averages)

• $\gamma\ddot\phi^2 =\gamma T \nu^2 = f^2$,          (3)
• $T = \dot\phi^2+\nu^2\phi^2$. 

Here $T$ represents temperature for a macroscopic body, while for a single atom rather $T\sim N$ with $N$ number of electrons. The 

Planck's Law connects the differences of discrete electronic energy levels $E=h\nu$ of an atom in radiative equilibrium with light of frequency $\nu$ appearing in the emission/absorption spectrum of the atom as isolated atom or as part of a macroscopic body, with $h$ acting like a constant connecting energy with frequency.