Dynamic Case

The obvious generalization of the static model for the polarization is to assume a damped linear response to a harmonic (plane wave) driving electric field. That is, every molecule will be viewed as a collection of damped, driven (charged) harmonic oscillators. Magnetic and non-linear effects will be neglected. This is valid for a variety of materials subjected to ``weak'' harmonic EM fields^9.6 which in practice (with optical frequencies) means nearly everything but laser light.

The equation of motion^9.7 for a single damped, driven harmonically bound charged electron is:

$$m\left [ \ddot{\mbox{\boldmath$x$}} + \gamma \dot{\mbox{\bo... ...^2 \mbox{\boldmath$x$} \right ] = -e E(\mbox{\boldmath$x$},t)$$ (9.103)

where $\gamma$ is the damping constant (so $-m \gamma \dot{\mbox{\boldmath$x$}}$ is the velocity dependent damping force). If we assume that the electric field $\mbox{\boldmath$E$}$ and $\mbox{\boldmath$x$}$ are harmonic in time at frequency $\omega$ (or fourier transform the equation and find its solution for a single fourier component) and neglect the transients we get:

$$\mbox{\boldmath$p$} = -e\mbox{\boldmath$x$} = \frac{e^2}{m}... ...math$E$}_\omega }{ (\omega_0^2 - \omega^2 - i \omega \gamma)}$$ (9.104)

for each electron^9.8.

Actually, we have $N$ molecules/unit volume each with $Z$ electrons where $f_i$ of them have frequencies and damping constants $\omega_i$ and $\gamma_i$, respectively (whew!) then (since we will stick in the definitions $\mbox{\boldmath$P$}_\omega = \epsilon_0 \chi_e \mbox{\boldmath$E$}_\omega$ and $\epsilon = 1 + \chi_e$)

$$\epsilon(\omega) = \epsilon_0\left (1 + \frac{N e^2}{m} \su... ...frac{f_i}{(\omega_i^2 - \omega^2 - i \omega \gamma_i)}\right)$$ (9.105)

where the oscillator strengths satisfy the sum rule:

$$\sum_i f_i = Z .$$ (9.106)

These equations (within suitable approximations) are valid for quantum theories, and indeed, since quantum oscillators have certain discrete frequencies, they seem to ``naturally'' be quantum mechanical.