Effective One Particle Quantum Dynamics of Electrons: A Numerical Study of the Schrodinger-Poisson-X_alpha Model

Abstract

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.

(Semi)Classical Limit of the Hartree Equation with Harmonic Potential

“…Such equations arise in the modeling of effective one particle Schrödinger equations where "exchange terms" like in the Hartree-Fock equation are simplified to functionals of the local densities, i.e. time dependent density functional theory, with the Schrödinger-Poisson-Xα equation as the simplest of such models (see [25] and [1] for a heuristic derivation and numerical simulations). Note that the additional "local" term has the opposite sign than the Hartree term (corresponding to the physical fact that the "exchange-correlation hole" weakens the direct Coulomb interaction).…”

Dimension Reduction of the Schrödinger Equation with Coulomb and Anisotropic Confining Potentials

“…In fact, the spatial confinement is an essential feature of many "nanoscale devices" and has gained much attention from both experimental and mathematical studies [3,31,40,42]. Although the SPS (1.3)-(1.4) in 2D or 1D has been used in some of the literature [1,3,12,25,27,42,43,48,50] to simulate low-dimensional quantum systems of fermions such as 2D "electron sheets" or 1D "quantum wires," it is highly debated or mathematically mysterious whether the above SPS is an appropriate model for these confining low-dimensional quantum systems. In fact, intuitively point particles confined to a 2D manifold still interact with the Coulomb interaction potential at O 1 |x| in 2D; thus it seems that the SPS (1.3)-(1.4) in 2D is not an appropriate model.…”

“…Thus the system of (1.1)-(1.2) is usually called the Schrödinger-Poisson system (SPS) in the literature [12,50]. In fact, the corresponding rigorous derivation of this kind of "Hartree equations" was started from a Hartree ansatz for the many-body (e.g., N -body) wave-function by using a "weak coupling scaling" (i.e., a factor 1/N in front of the Coulomb interaction potential) and passing to the limit N → ∞ in the BBGKY hierarchy [13,14,29].…”

“…If ϕ(x, t n ) is approximated in this way, we usually use homogeneous Dirichlet boundary condition to (3.5) and (3.1) and discretize them by the time-splitting sine pseudospectral (TSSP) method [5] and backward Euler sine pseudospectral (BESP) method [9,10,11,12], respectively. Remark 3.1.…”

TDDFT–Generalized kick perturbations and monitoring observables for calculation of excitation energies

“…Mathematically speaking TDDFT is a special, rather complicated case of a tdependent Nonlinear Schrödinger (NLS)-system [8], [9], [10] in the following called the TD-KS-NLS system.…”