= Dynamic electric polarizability
{title2=$\mathbf d_\omega=\alpha(\omega)\mathbf E_\omega$}
The dynamic electric polarizability relates the induced <electric dipole moment> at a driving frequency to the applied <electric field> at that frequency in <linear response theory>. A complex coefficient can encode both a dispersive phase and absorption. For a charged one-dimensional <quantum harmonic oscillator> with charge magnitude $e$, mass $m$, and natural frequency $\omega_0$, the undamped response is
$$
\alpha(\omega)=\frac{e^2}{m(\omega_0^2-\omega^2)}.
$$
One derivation uses the oscillator <electric-dipole selection rule> and first-order <time-dependent perturbation theory>. The same result follows from the classical <Lorentz oscillator model>: $m\ddot z+m\omega_0^2z=-eE$ and $d=-ez$ imply $\ddot d+\omega_0^2d=e^2E/m$. At exact resonance there is no bounded periodic response without damping; a finite switch-on produces secular growth rather than a finite steady-state value of the pole expression.
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