MAJORANA SOLUTIONS TO THE TWO-ELECTRON PROBLEM

 S. ESPOSITO AND A. NADDEO

 

Abstract. 

A review of the known different methods and results devised tostudy the two-electron atom problem, appeared in the early years of quantummechanics, is given, with particular reference to the calculations of the groundstate energy of helium. This is supplemented by several, unpublished resultsobtained around the same years by Ettore Majorana, which results did not con-vey in his published papers on the argument, and thus remained unknown untilnow. Particularly interesting, even for current research in atomic and nuclearphysics, is a general variant of the variational method, developed by Majoranain order to take directly into account, already in the trial wavefunction, theaction of the full Hamiltonian operator of a given quantum system. Moreover,notable calculations specialized to the study of the two-electron problem showthe introduction of the remarkable concept of an effective nuclear chargedif-ferentfor the two electrons (thus generalizing previous known results), andan application of the perturbative method, where the atomicnumberZwastreated effectively as acontinuousvariable, contributions to the ground stateenergy of an atom with givenZcoming also from any otherZ. Instead, con-tributions relevant mainly for pedagogical reasons count simple broad rangeestimates of the helium ionization potential, obtained by suitable choices forthe wavefunction, as well as a simple alternative to Hylleraas’ method, whichled Majorana to first order calculations comparable in accuracy with well-known order 11 results derived, in turn, by Hylleraas.

 

 https://arxiv.org/pdf/1207.1666.pdf