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Majorana Solutions to the Two-Electron Problem

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Abstract

The two-electron atom is the simplest nontrivial quantum system not amenable to exact solutions. Today, its relevance in the development of quantum mechanics and its pedagogical value within the realm of atomic physics are widely recognized. In this work, an historical review of the known different methods and results devised to study such a problem is presented, with an emphasis to the calculations of the ground state energy of helium. Then we discuss several, related, unpublished results obtained around the same years by Ettore Majorana, which remained unknown till recent times. Among them a general variant of the variational method appears to be particularly interesting, even for current research in atomic and nuclear physics: it takes directly into account, already in the trial wavefunction, the action of the full Hamiltonian operator of a given quantum system. Further relevant contributions, specialized to the two-electron problem, include 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. Finally a survey of results, relevant mainly for pedagogical reasons, is given; in particular we focus on simple broad range estimates of the helium ionization potential, obtained by suitable choices for the wavefunction, as well as on 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.

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Notes

  1. Or, more precisely, according to Majorana, \(W_{I}^{\mathrm{He}}=20.31\mathrm{\ eV}\) and \(W_{I}^{\mathrm{Li}} = 71.08\mathrm{\ eV}\).

  2. Note, however, that Slater applied his conclusions to any atom.

  3. The entire set of nine papers published by Majorana has been recently re-published, with translations and commentaries, in Ref. [39]. However, the interested reader should take with some caution the additional material presented in the volume, due to several inaccuracies (about Majorana’s contributions) in some commentaries and, especially, the inclusion of an apocryphal “paper no.1b”, manifestly not authored by Majorana (see [40]).

  4. For possible relation to Fano quasi-stationary states, see the discussion in [48].

  5. Majorana refers simply to the Hylleraas method (see Sect. 2.4); the paper by Fock quoted here deals with just a general subject, i.e. the virial theorem in the framework of quantum mechanics, where the Hylleraas method is mentioned as an example.

  6. Note that in Ref. [38], Majorana applied this formula to the ground state as well as to other levels (with different values of a).

  7. Recall that such contribution dates around 1928–1929.

  8. Apart from the references in [37] and [38], we do not know precisely which papers were read and studied by Majorana, but from the arguments and, especially, from the symbols he used in his notes [17, 51], we can safely infer that he was well aware of a large part of the literature quoted above.

  9. In what follows, we use consecutive labels for the equations in the quotations from the original Majorana’s manuscript; we also conform, when necessary, the original notation to the present one.

  10. Note that the predicted electron affinity of the hydrogen atom, i.e. the difference between the ground state energies of the neutral atom and the once-ionized atom, is more than twice the actual value.

  11. For technical details, see [55].

  12. In particular, Majorana deduced the numerical values for the coefficients by requiring that ψ and its first derivative have a node at the same position when the two-electron system collapses into a one-electron one, i.e. r 1=0 (or r 2=0) and r 12=0.

  13. −W/Rh=5.779 instead of −W/Rh=729/128=5.695.

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Authors and Affiliations

  1. I.N.F.N. Sezione di Napoli, Complesso Universitario di M. S. Angelo, Via Cinthia, 80126, Naples, Italy

    S. Esposito

  2. Dipartimento di Fisica “E.R. Caianiello”, Università di Salerno, and CNISM Unità di Ricerca di Salerno, Via Ponte don Melillo, 84084, Fisciano (Salerno), Italy

    A. Naddeo

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Esposito, S., Naddeo, A. Majorana Solutions to the Two-Electron Problem. Found Phys 42, 1586–1608 (2012). https://doi.org/10.1007/s10701-012-9685-1

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