Majorana solutions to the two-electron problem

TL;DR

This paper reviews early methods for the two-electron atom problem, highlights unpublished Majorana results including a generalized variational approach, and presents new insights into effective nuclear charge and perturbative techniques.

Contribution

It uncovers Majorana's unpublished work on a generalized variational method and introduces novel concepts like effective nuclear charge for two electrons.

Findings

Majorana developed a generalized variational method incorporating the full Hamiltonian.

Introduction of an effective nuclear charge concept for each electron.

Perturbative approach treating atomic number Z as a continuous variable.

Abstract

A review of the known different methods and results devised to study the two-electron atom problem, appeared in the early years of quantum mechanics, is given, with particular reference to the calculations of the ground state energy of helium. This is supplemented by several, unpublished results obtained around the same years by Ettore Majorana, which results did not convey in his published papers on the argument, and thus remained unknown until now. Particularly interesting, even for current research in atomic and nuclear physics, is a general variant of the variational method, developed by Majorana in order to take directly into account, already in the trial wavefunction, the action of the full Hamiltonian operator of a given quantum system. Moreover, notable calculations specialized to the study of the two-electron problem show the introduction of the remarkable concept of an effective nuclear charge different for the two electrons (thus generalizing previous known results), and an application of the perturbative method, where the atomic number Z was treated effectively as a continuous variable, contributions to the ground state energy of an atom with given Z coming also from any other Z. Instead, contributions relevant mainly for pedagogical reasons count simple broad range estimates of the helium ionization potential, obtained by suitable choices for the wavefunction, as well as a simple alternative to Hylleraas' method, which led Majorana to first order calculations comparable in accuracy with well-known order 11 results derived, in turn, by Hylleraas.