Effective One Particle Quantum Dynamics of Electrons: A Numerical Study of the Schrodinger-Poisson-X alpha 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.

[1]  J. C. Slater A Simplification of the Hartree-Fock Method , 1951 .

[2]  W. Kohn,et al.  Self-Consistent Equations Including Exchange and Correlation Effects , 1965 .

[3]  Norbert J. Mauser,et al.  THE CLASSICAL LIMIT OF A SELF-CONSISTENT QUANTUM-VLASOV EQUATION IN 3D , 1993 .

[4]  P. Markowich,et al.  Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation , 2003, cond-mat/0303239.

[5]  Olivier Bokanowski,et al.  LOCAL APPROXIMATION FOR THE HARTREE–FOCK EXCHANGE POTENTIAL: A DEFORMATION APPROACH , 1999 .

[6]  T. Paul,et al.  Sur les mesures de Wigner , 1993 .

[7]  P. Markowich,et al.  On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime , 2002 .

[8]  P. Markowich,et al.  Homogenization limits and Wigner transforms , 1997 .

[9]  François Golse,et al.  Derivation of the Schrödinger–Poisson equation from the quantum N-body problem , 2002 .

[10]  V. Bach Accuracy of mean field approximations for atoms and molecules , 1993 .

[11]  P. Dirac Note on Exchange Phenomena in the Thomas Atom , 1930, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  N. SIAMJ. AN EXPLICIT UNCONDITIONALLY STABLE NUMERICAL METHOD FOR SOLVING DAMPED NONLINEAR SCHRÖDINGER EQUATIONS WITH A FOCUSING NONLINEARITY , 2003 .

[13]  P. Miller,et al.  On the semiclassical limit of the focusing nonlinear Schrödinger equation , 1998 .

[14]  R. Gáspár,et al.  Über eine Approximation des Hartree-Fockschen Potentials Durch eine Universelle Potentialfunktion , 1954 .

[15]  Esben Skovsen,et al.  Quantum state tomography of dissociating molecules. , 2003, Physical review letters.

[16]  B. Herbst,et al.  Split-step methods for the solution of the nonlinear Schro¨dinger equation , 1986 .

[17]  Norbert J. Mauser,et al.  Mean field dynamics of fermions and the time-dependent Hartree-Fock equation , 2002 .

[18]  R. Parr Density-functional theory of atoms and molecules , 1989 .

[19]  François Golse,et al.  Weak Copling Limit of the N-Particle Schrödinger Equation , 2000 .

[20]  Rémi Carles,et al.  Remarques sur les mesures de Wigner , 2001 .

[21]  Olivier Bokanowski,et al.  Local density approximations for the energy of a periodic Coulomb model , 2003 .

[22]  Ping Zhang,et al.  The limit from the Schrödinger‐Poisson to the Vlasov‐Poisson equations with general data in one dimension , 2002 .

[23]  H. Steinrück The one-dimensional Wigner-Poisson problem and its relation to the Schro¨dinger-Poisson problem , 1991 .

[24]  Pierre-Louis Lions,et al.  Solutions of Hartree-Fock equations for Coulomb systems , 1987 .

[25]  Norbert J. Mauser,et al.  The Schrödinger-Poisson-X equation , 2001, Appl. Math. Lett..

[26]  Hans Peter Stimming THE IVP FOR THE SCHRÖDINGER–POISSON-Xα EQUATION IN ONE DIMENSION , 2005 .

[27]  Peter A. Markowich,et al.  A Wigner-Measure Analysis of the Dufort-Frankel Scheme for the Schrödinger Equation , 2002, SIAM J. Numer. Anal..