The electron-pair density matrix ρ(k1, k2; k3, k4, t) =: ρ(1, 2; 3, 4, t) changes in time, following a quantum Liouville equation
with < 1,2 | νc | 3,4 >≔ 4 π e2k0 (Ω q2)−1δk1+k2,k3+k4δk3−k1,q in the presence of a Coulomb interaction υc, where Ω is the volume. If the virtual phonon exchange is in action, the density matrix ρ is shown to change similarly with an effective interaction υe,
by using a time-dependent perturbation theory and a Markoffian approximation. The dominant longitudinal-acoustic-(optical)-phonon-exchange attraction at 0K is shown to be q-independent (-dependent). The results are used to discuss the Cooper pair size, the origin of type II superconductivity and the formation of d-wave Cooper pairs in the cuprates.