Bound electrons, with resonant frequency being forced by a sinusoidal electric field

will obey the equation of motion

where is the charge on the electron and m is its mass. The solution is

Given a dipole number density of N, the polarization is

From the definition of polarization,

where is the permittivity of free space. Using the Maxwell relation for index of refraction,

This result can be derived rigorously from quantum mechanics in a slightly different form

where are the oscillator strengths (or transition probabilities), and

Including a damping term, the equation becomes

In a dense medium, the polarization adds an electric field

leading to

If n increases with ,it is called normal dispersion. If it decrease with increasing ,it is called anomalous dispersion.