Atoms and anharmonic oscillations

Apply Gauss' law
${\int \int}_{Surface} \vec{E} \cdot \mathrm{d} \vec{A} = \frac{Q_{enc}}{\epsilon_0}$

We can deduce the resonance frequency of the oscillation of the proton w.r.t. the electron cloud, which yields

${\omega_0}^{2} = \frac{k}{m} = \frac{e^2}{4\pi \epsilon_0 a^3}$

This resonance frequency compared with the driving frequency of visible light tells us that the system is heavily damped and the oscillation amplitude is stiffness controlled.

Damping in Atomic oscillators

Consider the Larmor's radiation formular in electrodynamics for a charge moving at $x_0 e^{i \omega t}$
$P_{rad} = \frac{e^2{x_0}^2 \omega^4}{12 \pi \epsilon_0 c^3}$
The mean energy of the electron oscillating is $W= \frac{m_e {\omega_0} ^2 {x_0}^2 }{2}$

$\frac{\mathrm{d}W}{\mathrm{d} t} = - P_{rad}$

Integrate the previous formular to get the time decaying coefficient of the energy.

Notice the under the illumination of visible light, the proton-electron cloud system is heavily damped with a time constant $\frac{1}{\gamma}$
Hence we estimate the values of $\omega_0, \gamma, Q = \frac{\omega_0}{\gamma}$

Linear system driven at multiple harmonic frequencies

Due to the linearity of the linear system, the resulting oscillation is the linear combination of the solutions in the cases of independently driven harmonic ocsillations.
Consider phase shifts $\alpha_1, \alpha_2$ for two displacement solutions in the two forces case.
The compound motion's amplitude is given by trigonometric manipulations
$A^2 = {A_1} ^2 + {A_2} ^2 + \cos(\alpha_1 - \alpha_2)A_1 A_2$