Atoms and Quantum Dots With a Large Number of Electrons: the Ground State Energy

TL;DR

This paper calculates the ground state energy of large-electron atoms and quantum dots, revealing semiclassical behaviors, oscillations, and correlation effects, with implications for understanding quantum many-body systems in different dimensions.

Contribution

It provides a detailed analysis of the ground state energy for large N electrons in atoms and quantum dots, connecting semiclassical theory with quantum correlations and classical dynamics.

Findings

Ground state energy dominated by semiclassical Hartree-exchange energy.

Oscillations in energy linked to classical particle dynamics.

Correlation effects scale as N ln N for atoms and N for dots.

Abstract

We compute the ground state energy of atoms and quantum dots with a large number N of electrons. Both systems are described by a non-relativistic 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 to 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.