Construction of the Pauli potential for semiconductor quantum wells and superlattices

The usual approach to get the electron density n( overliner) for a model system of 2 N non-interacting particles is to solve N wave equations for a self-consistent potential V( overliner) and summing up N wave function moduli. A modern way to attain a direct solution of n( overliner) without recourse to N wave functions is to solve a wave equation only for the ground state, but to add to the potential V( overliner) a so-called Pauli potential Vp( overliner) which simulates the effect of the Pauli-exclusion principle. For one-dimensional problems as occurring in quantum wells and superlattices an explicit exact expression for the Pauli potential is derived and applied to illustrative examples. The extension to three-dimensional problems is discussed.