Variational method with scattering boundary conditions imposed by the Wigner distribution

A functional form for electrons scattering off localized potentials with boundary conditions imposed through the Wigner distribution function is formulated in a manner appropriate for the study of tunneling in nanoscale junctions. Variation of the functional with respect to the electronic density matrix leads to stationary or critical points respecting open-system boundary conditions. Examples are presented for single-electron and many- (noninteracting) electron scattering and, in both instances, the usual outcome of a one-electron density matrix comprised of scattering wave functions results. The examples highlight how the degrees of freedom associated with imposing scattering boundary conditions can be constrained and removed from a variational determination of the electronic wave functions on a scattering region. The form for the scattering functional is motivated in terms of the maximum entropy principle.