Bound States in Coulomb Systems - Old Problems and New Solutions

TL;DR

This paper investigates bound states in hydrogen using quantum statistical methods, analyzing formation and destruction of these states across densities, and compares different models to improve understanding of the equation of state.

Contribution

It provides a detailed quantum statistical analysis of bound states in hydrogen, avoiding chemical models, and explores the effects of density and degeneracy on bound state behavior and pressure calculations.

Findings

Bound states occur only in a specific valley in the T-p plane.

Fugacity representations show saturation of bound state populations at high density.

The combination of canonical and grand canonical ensembles yields rapidly converging equations of state.

Abstract

We analyze the quantum statistical treatment of bound states in Hydrogen considered as a system of electrons and protons. Within this physical picture we calculate isotherms of pressure for Hydrogen in a broad density region and compare to some results from the chemical picture. First we resume in detail the two transitions along isotherms : (i) the formation of bound states occurring by increasing the density from low to moderate values, (ii) the destruction of bound states in the high density region, modelled here by Pauli-Fock effects. Avoiding chemical models we will show, why bound states according to a discrete part of the spectra occur only in a valley in the T-p plane. First we study virial expansions in the canonical ensemble and then in the grand canonical ensemble. We show that in fugacity representations the population of bound states saturates at higher density and that a combination of both representations provides quickly converging equations of state. In the case of degenerate systems we calculated first the density-dependent energy levels, and find the pressure in Hartree-Fock-Wigner approximation showing the prominent role of Pauli blocking and Fock effects in the selfenergy.