The Radiating Atom 4: Absorption vs Emission

To seek the relation between atomic absorption and emission of radiation, let us consider a near-resonantly forced harmonic oscillator with small damping as the basic model underlying the analysis presented at Computational Black Body Radiation:

• $\ddot u(t)+\nu^2u(t)+\gamma\dot u(t) = f(t)$, 

which we in mechanical terms, with the dot representing differentiation with respect to time, expresses force balance between a mass-spring oscillator with internal inertial force $\ddot u(t)$ and spring force $\nu^2u(t)$ with $u(t)$ displacement and resonance frequency $\nu$, coupled in parallel with a friction force $\gamma\dot u(t)$, which are balancing an exterior force $f(t)$. Here $\gamma >0$ is a small damping coefficient and we consider the two basic cases of 

1. Non-resonant forcing with frequency of $f(t)$ not near $\nu$: 

• $\gamma\dot u(t) \approx f(t)$ and $\ddot u(t)+\nu^2u(t)\approx 0$.  

2. Near-resonant forcing (see Computational Black Body Radiation) with frequency of $f(t)$ near $\nu$:

• $\gamma\dot u(t) \approx 0$ and $\ddot u(t)+\nu^2u(t)\approx f(t)$.
• More precisely: $\gamma \vert\dot u\vert\approx\sqrt{\gamma}\vert f\vert <<\vert f\vert$.

In case 1. the exterior force is balanced by the friction force and in case 2. by an out-of-balance harmonic oscillator. If we view $r(t) = f(t) - \gamma\dot u(t)$ as an observable net residual force, we

then have that 

1. Non-resonant forcing gives $r(t)\approx 0$: Nothing can be observed.
2. Resonant forcing $r(t) \approx f(t)$: Something can be observed.   

This gives substance to the experience that absorption and emission, as in absorption/emission spectroscopy, are related: 

• A system which can absorb radiation can also emit radiation of the same frequency, and vice versa.
• A non-resonant system does not absorb anything and nothing observable is emitted.
• Resonant absorption can be observed by some form of emission.  This does not require emission to be equal to absorption, but they come together.
• In absorption spectroscopy a cold gas is absorbing incoming radiation, which is observable as a dip in the spectrum observed after passage through the gas resulting from heating the gas.
• In emission spectroscopy of a hot gas, emission is observable but not absorption. 
• The resonance frequency connects to the difference in energy level between two electronic states since emission results from charge oscillation (connected to the Abraham-Lorentz force) of such frequencies. Hence also absorption of such these frequencies can be observable as a result of force oscillation.  

Note that both absorption and emission is a resonance phenomenon driven by forces and as such is a wave phenomenon in the spirit of Schrödinger and not a "corpuscular phenomenon", whatever that may be, as is the current wisdom rooted in Einstein's "explanation" of the photoelectric effect based on "light particles" or "photons" of energy $h\nu$ jumping stochastically back and forth seemingly without being subject to forces.

But physics is all about forces and physics without forces is non-physics.