Effective One Particle Quantum Dynamics of Electrons: a Numerical Study of the Schrödinger-poisson-xα Model∗

The Schrödinger-Poisson-Xα (S-P-Xα) model is a “local one particle approximation” of the time dependent Hartree-Fock equations. It describes the time evolution of electrons in a quantum model respecting the Pauli principle in an approximate fashion which yields an effective potential that is the difference of the nonlocal Coulomb potential and the third root of the local density. We sketch the formal derivation, existence and uniqueness analysis of the S-P-Xα model with/without an external potential. In this paper we deal with numerical simulations based on a time-splitting spectral method, which was used and studied recently for the nonlinear Schrödinger (NLS) equation in the semiclassical regime and shows much better spatial and temporal resolution than finite difference methods. Extensive numerical results of position density and Wigner measures in 1d, 2d and 3d for the S-P-Xα model with/without an external potential are presented. These results give an insight to understand the interplay between the nonlocal (“weak”) and the local (“strong”) nonlinearity.