Modèle Thomas-Fermi

Résumé

Le modèle Thomas–Fermi (TF), nommé d'après Llewellyn Thomas et Enrico Fermi, est une théorie de la mécanique quantique pour l'étude des systèmes à N corps, développée de manière semi-classique peu après l'introduction de l'équation de Schrödinger. Elle est se distingue de la théorie de la fonction d'onde, car elle est formulée uniquement en termes de densité électronique ; en tant que telle, elle est considérée comme un précurseur de la théorie fonctionnelle de la densité moderne. L'utilisation de ce modèle pour des systèmes réels donne des prédictions quantitatives très approximatives, car il ne reproduit pas certaines caractéristiques quantiques de la densité, telles que la structure en couches dans les atomes et les oscillations de Friedel dans les solides. Le modèle de Thomas-Fermi a cependant trouvé des applications modernes dans de nombreux domaines, grâce à sa capacité à extraire de manière analytique les tendances qualitatives et à la facilité avec laquelle il peut être résolu. L'expression de l'énergie cinétique dans la théorie de Thomas-Fermi est également utilisée dans la théorie fonctionnelle moderne de la densité sans orbite, comme élément de base dans une approximation plus sophistiquée. Travaillant indépendamment, Thomas et Fermi ont utilisé ce modèle statistique en 1927 pour calculer approximativement la distribution des électrons dans un atome. Bien que les électrons soient distribués de manière non uniforme, on suppose que dans chaque petit élément de volume ΔV, les électrons sont distribués uniformément avec une densité électronique , qui est locale car elle peut varier d'un élément de volume à l'autre. Le modèle développé pour l'étude des atomes a été étendu à l'étude des propriétés d'autres systèmes de fermions, comme les noyaux atomiques, les molécules ou les solides. On considère un petit élément de volume ΔV d'un atome (ou tout système de fermions) dans son état fondamental ; tous les états permis pour la quantité de mouvement (ou impulsion) sont remplis jusqu'au niveau de Fermi, soit la quantité de mouvement de Fermi p F ; le volume sphérique V F des états occupés dans l'espace des impulsions est donc, où est le vecteur position d'un point dans ΔV.

Source officielle

https://fr.wikipedia.org/wiki/Modèle_Thomas-Fermi

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Publications associées (3)

Cours associés (1)

Séances de cours associées (1)

Quantum phase transitions in low-dimensional weakly interacting disordered Bose gases

Joseph Saliba

In this thesis, we study the quantum phase transition triggered by an external random po- tential in ultra-cold low-dimensional weakly-interacting Bose gases at zero temperature. In one-dimensional systems, the quantum phases are characterized by the decay at long-range of the one-body density matrix. This decay exhibits an algebraic behaviour in the superfluid quasi-condensed phase while in the insulating Bose glass phase this it becomes exponen- tial, signaling the complete loss of coherence in the system. In two-dimensional systems, a characterization based on the long-range behaviour of the one-body density matrix appears to be highly demanding from a computational point of view. We therefore characterized the superfluid-insulator transition in two-dimensional systems by the low-energy behaviour of the cumulative density of states of the Bogoliubov excitations. While in the superfluid condensed phase this quantity grows as the square of the excitation energy in agreement with the existence of a finite velocity of sound, in the insulating Bose glass phase this power law is less than quadratic. This study is performed in the framework of an extended Bogoliubov approach properly adapted to treat low-dimensional Bose gases. Using a systematic numerical study, we draw the interaction-disorder phase diagrams of the superfluid-insulator transition in 1D and 2D. The phase boundary follows two different power- laws depending on the length scales characterizing the spatial correlations of the disorder and the strength of interactions. The power-law exponents were found to be in agreement with the ones predicted by scaling arguments, both in the white noise and the Thomas-Fermi regimes. In the two-dimensional system, the classical percolation threshold in the Thomas-Fermi limit was found to overestimate the critical disorder below which the onset of superfluidity should be observed for a given interaction strength.

EPFL2016

Atoms and quantum dots with a large number of electrons: The ground-state energy

Hervé Kunz, Rico Rueedi

We compute the ground-state energy of atoms and quantum dots with a large number N of electrons. Both systems are described by a nonrelativistic Hamiltonian of electrons in a d-dimensional space. The electrons interact via the Coulomb potential. In the case of atoms (d = 3), the electrons are attracted by the nucleus via the Coulomb potential. In the case of quantum dots (d = 2), the electrons are confined by an external potential, whose shape can be varied. We show that the dominant terms of the ground-state energy are those given by a semiclassical Hartree-exchange energy, whose N -> infinity limit corresponds to Thomas-Fermi theory. This semiclassical Hartree-exchange theory creates oscillations in the ground-state energy as a function of N. These oscillations reflect the dynamics of a classical particle moving in the presence of the Thomas-Fermi potential. The dynamics is regular for atoms and some dots, but in general in the case of dots, the motion contains a chaotic component. We compute the correlation effects. They appear at the order N ln N for atoms, in agreement with available data. For dots, they appear at the order N.

2010

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.

EPFL2009

Quantum phase transitions in low-dimensional weakly interacting disordered Bose gases

Joseph Saliba

EPFL2016