Based on a second-quantized representation of the nonrelativistic Hamiltonian of a system of
N electrons with pairwise Coulomb interactions, we demonstrate the exact statistical reduction of the
N-body problem to a three-dimensional Schr\"odinger equation for the motion of a single active electron with all other $N\ensuremath{-}1$ electrons acting as spectators. As a by-product, three-dimensional Schr\"odinger equations for the ground and excited states of two-electron atoms and ions are derived and the dynamical role of Pauli's exclusion principle is established. The classical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ of the quantal all-electron equations is examined, and the Thomas-Fermi equation including the Amaldi correction is obtained.
A new operator approach to the solution of the non-relativistic Schrödinger equation for few-body quantum systems with pair-wise interaction is presented. The problem is solved on the basis of the sequential account of configurations (SAC) method proposed. The wave-function of a few-body system determined with allowance for the configuration interaction (CI-wavefunction) is expressed in terms of the one-configuration wavefunctions with use of the configuration mixing operator (±) introduced in the paper. The examination is carried out with an example of a few-electron atomic system. An analytical expression for the (±) -operator has been represented and an integral equation for the CI-wavefunction has been obtained. It is shown that: (I) the correlation interaction inside an isolated configuration does not destroy the independent particle approximation; (ii) in the framework of the present approach the self-action is excluded. The present approach has enabled to receive an analytical expression for the (±) -operator without applying the non-physical procedure of 'switching-on' the interconfiguration interaction.