  For the first time a rigorous quantum treatment of the
Landau-Pomeranchuk-Migdal effect in QED and QCD is given. The rate of photon
(gluon) radiation by an electron (quark) in medium is expressed through the
Green's function of a two-dimensional Schr\"odinger equation with an imaginary
potential. In QED this potential is proportional to the dipole cross section
for scattering of $e^{+}e^{-}$ pair off an atom, in QCD it is proportional to
the cross section of interaction with color centre of the color singlet
quark-antiquark-gluon system.
