The equation of motion for a harmonically bound classical electron interacting with an electric field

is given by the Drude–Lorentz equation

, where

is the natural frequency of the oscillator and

is the damping constant. For an incident electromagnetic field of frequency

,

at the point

can be conveniently represented by a complex exponential

. The steady-state solution, in complex form, is given by

. The electric dipole moment of the electron,

, corresponds to the macroscopic relation for the polarizability

, where

is the complex electric susceptibility

,

being the number of polarizable electrons per unit volume. It is sufficient to approximate

, with transition frequencies lying in the optical region

rad/s. Also,

. The real part of the susceptibility

gives the frequency dependence (dispersion) of the dielectric constant

and index of refraction

. The imaginary part

represents the absorption coefficient. This function has the form of a Lorentzian.

In the more accurate quantum theory of dispersion, the frequency

is replaced by a sum over several atomic transition frequencies and the damping parameters

are determined by excited-state lifetimes.

The real and imaginary parts of the susceptibility are connected by the Kramers–Kronig relations:

and

, where

signifies the Cauchy principal value of the integral.

[1] L. Rosenfeld,

*Theory of Electrons*, New York: Dover Publications, 1965, pp. 68 ff.