Maximum Number of Electrons in an Atomic Bound State

Abstract

Let $\ensuremath{\psi}$ be any normalizable energy eigenfunction for $N$ electrons in an atom having $Z$ protons in a fixed pointlike nucleus. It is shown that, for the state $\ensuremath{\psi}$, $N$ and $Z$ are rigorously related by $N<FZ+1$ with $F=4$. This is independent of any statistics. Evidence is presented that the result can be improved in the sense that $F=F(N)\ensuremath{\rightarrow}2$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$. A comparison is made with a recent result of Lieb.