Ground state properties of large atoms and quantum dots
We investigate the ground state properties of large atoms and quantum dots described by a d-dimensional N-body Hamiltonian of confinement ZV. In atoms, d = 3 and V is the Coulomb interaction; in dots, d = 2 and V is phenomenologically determined. We express the grand-canonical partition function in a path integral approach, and evaluate its expansion in Z-1. The problem can be seen as that of field theory possessing a saddle point. This saddle point results in a mean-field contribution to the energy, while the fluctuations result in the correlation energy. The mean-field contribution to the energy is self-consistently determined by the Hartree potential and contains an exchange term. Its smooth contribution is evaluated by a semiclassical method, with ε = Z-1/d in the role of ℏ, while its oscillating contribution can be related to the periodic orbits in the corresponding classical Hamiltonian. In the case of atoms, the leading order in ε of the correlation energy contains a term in Z ln Z1/3, which is essential in reproducing the behaviour shown by reference values, and a term in Z. While we have evaluated the contribution to the Z-term provided by the leading fluctuation order, the numerical evaluation of the contributions provided by higher order fluctuations remains an open problem. The self-consistent contribution to the energy corresponds to the statistical atom, composed of Thomas-Fermi and its corrections, comprehensively analysed, including oscillations, by Schwinger and Englert. In the case of dots, the leading order in ε of the correlation energy is a universal contribution of order Z, which we obtain in closed form. We then determine the expansion in ε of the smooth contributions down to this correlation order. We apply the approach to dots of quadratic and quartic confinement, including the oscillating contribution in the case of a chaotic quartic confinement.
Keywords: atom ; quantum dot ; ground state energy ; correlation energy ; mean-field theory ; self-consistent potential ; semiclassical expansion ; quantum fluctuations ; Hartree ; Hartree-Fock ; Thomas-Fermi ; Gutzwiller trace formula ; periodic orbits ; atome ; boîte quantique ; énergie fondamentale ; énergie de corrélation ; théorie de champ moyen ; potentiel autoconsistant ; développement semiclassique ; fluctuations quantiques ; Hartree ; Hartree-Fock ; Thomas-Fermi ; formule de trace de Gutzwiller ; orbites périodiquesThèse École polytechnique fédérale de Lausanne EPFL, n° 4406 (2009)
Programme doctoral Physique
Faculté des sciences de base
Institut de théories des phénomènes physiques
Groupe de chaos et désordre
Record created on 2009-04-06, modified on 2016-08-08