Field equations for the coupled system of Schrödinger and Maxwell fields are derived from a Lagrange density including external sources. To use Coulomb gauge, the concept of transverse fields is taken. Green functions are defined via functional derivatives. Vertex functions are introduced for the longitudinal Coulomb interaction and for the transverse interaction. The polarization tensor is discussed in the one-loop or RPA-approximation. The static limit is derived for the longitudinal and the transverse components of the polarization tensor. For the longitudinal part we get the square of the inverse Debye radius, as usual. The transverse part vanishes in first order approximation. The dielectric tensor is introduced, the long wavelength limit is discussed. The classical limit of our results is identical with the results of Klimontovich.