Comparing Blackbody Radiation Spectrum to Atomic Emission Spectrum

Planck's constant $h$ appears with different roles in a blackbody radiation spectrum and an atomic emission spectrum.  Blackbody radiation can be described as a near-resonance phenomenon in a forced harmonic oscillator with small damping in a mathematical model of the form

• $\ddot u (t) +\omega^2u(t) -\gamma\dddot u = f(t)\approx \sin(\omega t)$, 

where $u(t)$ is displacement as function of time $t$, $\omega$ is angular velocity, $\gamma$ is a small damping constant, $f(t)$ is forcing in near-resonance with $\omega$ and the dot signifies time differentiation. Here the oscillator described by $\ddot u (t) +\omega^2u(t)$ carries energy as background temperature and the dissipative term $-\gamma\dddot u$ gives off radiation balancing forcing $f(t)$.

The dynamics of near-resonance is quite subtle as explained in detail on Computational Blackbody Radiation showing that Planck's constant enters as a parameter in a high-frequency cut-off reflecting Wien's displacement law.   

Atomic emission can be described as an eigenvalue problem for Schrödinger's equation of the form

• $ih\dot\psi = E\psi$,

where $E$ is a real eigenvalue of an atomic Hamiltonian, with solution

• $\psi (t) =\exp(i\omega t) =\cos(\omega t)+i\sin(\omega t)$, 

which can be seen as a periodic exchange of two forms of energy represented by the real part $\cos(\omega t)$ and the complex part $\sin(\omega t)$ reflecting incoming-outgoing radiation. Atomic emission is thus a direct resonance phenomenon without background temperature. Planck's constant serves to convert angular velocity (angular momentum) $\omega$ to atomic energy $E$ as $\bar h\omega$ with $\bar h=\frac{h}{2\pi}$ with $E$ the sum of kinetic and potential energy. 

We conclude:

1. Blackbody radiation is a near-resonance phenomenon of molecules or collections of atoms modeled as a forced harmonic oscillator with small damping. Collections of atoms vibrate without electron configurations changing energy.   
2. Atomic radiation is a direct resonance phenomenon which can be modeled by a harmonic oscillator. Electrons oscillate between two energy levels representing eigenstates of an atom.

In both cases $h$ enters combined with frequency $\nu$ in the form $h\nu =\bar h\omega$ as quantity of energy serving in a threshold condition in blackbody radiation, and as an energy eigenvalue in atomic emission.

The value of $h$ as setting a conversion scale between light energy and electronic energy can be determined by the photoelectric effect and can then be used by definition in blackbody radiation and Schrödinger's equation.