Homogeneous charge density
For the special case of a homogeneous charge density, that is one that is independent of position, equal to ρq,0 the equation simplifies to:
The proof of this is simple. Start with the definition of the charge of any volume:
Then, by definition of homogeneity, is a constant that we will denote ρq,0 to differentiate between the constant and non-constant forms, and thus by the properties of an integral can be pulled outside of the integral resulting in:
The equivalent proofs for linear charge density and surface charge density follow the same arguments as above.
where is the test position， is the charge of the ith charge carrier, whose position is .
If all charge carriers have the same charge q (for electrons q = − e) the charge density can be expressed through the charge carrier density : Again, the equivalent equations for the linear and surface charge densities follow directly from the above relations.
Quantum charge densitY
when the wavefunction is normalized as