Quantum three body Coulomb problem in two dimensions

Laurent Hilico (1, 2) , Benoît Grémaud (1) , Thibaut Jonckheere (1) , Nicolas Billy (1, 2) , Dominique Delande (1)

1 LKB (Jussieu) - Laboratoire Kastler Brossel
2 DPM - Département de Physique et Modélisation

We study the three-body Coulomb problem in two dimensions and show how to calculate very accurately its quantum properties. The useof a convenient set of coordinates makes it possible to write the Schrödinger equation only using annihilation and creation operators of four harmonic oscillators, coupled by various terms of degree up to twelve. We analyse in details the discrete symmetry properties of the eigenstates. The energy levels and eigenstates of the two-dimensional helium atom are obtained numerically, by expanding the Schrödinger equation on a convenient basis set, that gives sparse banded matrices, and thus opens up the way to accurate and efficient calculations. We give some very accurate values of the energy levels of the first bound Rydberg series. Using the complex coordinate method, we are also able to calculate energies and widths of doubly excited states, i.e. resonances above the first ionization threshold. For the two-dimensional H- ion, only one bound state is found.

Mots clés: ab initio; helium; H-; variational method; parabolic coordinates; two dimensions

Domaines: Physique Atomique [physics.atom-ph]

Laurent Hilico, Benoît Grémaud, Thibaut Jonckheere, Nicolas Billy, Dominique Delande. Quantum three body Coulomb problem in two dimensions. Physical Review A : Atomic, molecular, and optical physics [1990-2015], 2002, 66, pp.022101. ⟨hal-00120708⟩