Foundations of Physics

, Volume 42, Issue 12, pp 1586–1608 | Cite as

Majorana Solutions to the Two-Electron Problem

Article

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 Then we discuss several, related, unpublished results obtained around the same years by Ettore Majorana, which remained unknown till recent times.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.

Keywords

Ettore Majorana Two-electron problem Helium atom Variational method 

References

  1. 1.
    Mehra, J., Rechenberg, H.: The Historical Development of Quantum Theory, vol. 1, Part 2. Springer, New York (1982) CrossRefGoogle Scholar
  2. 2.
    Rutherford, E.: Philos. Mag. 21, 669 (1911) MATHCrossRefGoogle Scholar
  3. 3.
    Bohr, N.: Philos. Mag. 26, 1, 476, 857 (1913) Google Scholar
  4. 4.
    Nagaoka, H.: Nature 69, 392 (1904) ADSCrossRefGoogle Scholar
  5. 5.
    Nagaoka, H.: Philos. Mag. 7, 445 (1904) MATHCrossRefGoogle Scholar
  6. 6.
    Sommerfeld, A.: Ann. Phys. (Berlin) 51, 1 (1916) ADSCrossRefGoogle Scholar
  7. 7.
    Wilson, W.: Philos. Mag. 29, 795 (1915) CrossRefGoogle Scholar
  8. 8.
    Briggs, J.S.: Aust. J. Phys. 52, 341 (1999) ADSGoogle Scholar
  9. 9.
    Sommerfled, A.: J. Opt. Soc. Am. 7, 509 (1923) ADSCrossRefGoogle Scholar
  10. 10.
    Born, M., Heisenberg, W.: Z. Phys. 16, 229 (1923) ADSCrossRefGoogle Scholar
  11. 11.
    van Vleck, J.H.: Philos. Mag. 44, 842 (1922) CrossRefGoogle Scholar
  12. 12.
    Bethe, H.A.: Quantenmechanik der Ein- und Zwei-Elektronen probleme. In: Handbuch der Physik, vol. 24. Springer, Berlin (1933) Google Scholar
  13. 13.
    Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of One- and Two-Electron Atoms. Academic Press, New York (1957) MATHGoogle Scholar
  14. 14.
    Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of One- and Two-Electron Atoms. Plenum, New York (1977) CrossRefGoogle Scholar
  15. 15.
    Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of One- and Two-Electron Atoms. Dover, New York (2008) Google Scholar
  16. 16.
    Ursold, A.: Ann. Phys. (Berlin) 82, 355 (1927) ADSCrossRefGoogle Scholar
  17. 17.
    Esposito, S., Recami, E., van der Merwe, A., Battiston, R.: Ettore Majorana: Research Notes on Theoretical Physics. Springer, Heidelberg (2008) Google Scholar
  18. 18.
    Slater, J.C.: Proc. Natl. Acad. Sci. 13, 423 (1927) ADSCrossRefGoogle Scholar
  19. 19.
    Slater, J.C.: Phys. Rev. 31, 333 (1928) ADSMATHCrossRefGoogle Scholar
  20. 20.
    Kellner, G.W.: Z. Phys. 44, 91 (1927) ADSMATHCrossRefGoogle Scholar
  21. 21.
    Ritz, W.: J. Reine Angew. Math. 135, 1 (1908) MATHGoogle Scholar
  22. 22.
    Courant, R., Hilbert, D.: Methoden der Mathematischen Physik. Springer, Berlin (1924) MATHGoogle Scholar
  23. 23.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience, New York (1953) Google Scholar
  24. 24.
    Hartree, D.R.: Proc. Camb. Philol. Soc. 24, 89, 111 (1928) Google Scholar
  25. 25.
    Gaunt, J.A.: Proc. Camb. Philol. Soc. 24, 328 (1928) ADSMATHCrossRefGoogle Scholar
  26. 26.
    Slater, J.C.: Phys. Rev. 35, 210 (1930) ADSCrossRefGoogle Scholar
  27. 27.
    Slater, J.C.: Phys. Rev. 32, 339 (1928) ADSMATHCrossRefGoogle Scholar
  28. 28.
    Slater, J.C.: Phys. Rev. 32, 349 (1928) ADSCrossRefGoogle Scholar
  29. 29.
    Fock, V.: Z. Phys. 61, 126 (1930) ADSMATHCrossRefGoogle Scholar
  30. 30.
    Hylleraas, E.A.: Rev. Mod. Phys. 35, 421 (1963) ADSCrossRefGoogle Scholar
  31. 31.
    Hylleraas, E.A.: Quantum Chemistry: Classical Scientific Papers, vol. 48, p. 469. World Scientific, Singapore (1928). English translation in, Hettema, H., 2000 Google Scholar
  32. 32.
    Lyman, T.: Astrophys. J. Lett. 60, 1 (1924) ADSCrossRefGoogle Scholar
  33. 33.
    Hylleraas, E.A.: Quantum Chemistry: Classical Scientific Papers, vol. 54, p. 347. World Scientific, Singapore (1929). English translation in Hettema, H., 2000 Google Scholar
  34. 34.
    Bethe, H.A.: Z. Phys. 55, 431 (1929) ADSCrossRefGoogle Scholar
  35. 35.
    Hylleraas, E.A.: Quantum Chemistry: Classical Scientific Papers, vol. 17, p. 982. World Scientific, Singapore (1929). English translation in Hettema, H., 2000 Google Scholar
  36. 36.
    Hylleraas, E.A.: Z. Phys. 65, 209 (1930) ADSMATHCrossRefGoogle Scholar
  37. 37.
    Majorana, E.: Nuovo Cimento 8, 22 (1931). English translation in Hettema, H.: Quantum Chemistry: Classical Scientific Papers. World Scientific, Singapore (2000) CrossRefGoogle Scholar
  38. 38.
    Majorana, E.: Nuovo Cimento 8, 78 (1931) CrossRefGoogle Scholar
  39. 39.
    Bassani, G.F. (ed.): Ettore Majorana: Scientific Papers, SIF, Bologna. Springer, Berlin (2006) MATHGoogle Scholar
  40. 40.
    Esposito, S.: See arXiv:physics/0511222
  41. 41.
    Heitler, W., London, F.: Z. Phys. 44, 455 (1927) ADSMATHCrossRefGoogle Scholar
  42. 42.
    Pucci, R., Angilella, G.G.N.: Found. Phys. 36, 1554 (2006) MathSciNetADSCrossRefGoogle Scholar
  43. 43.
    Esposito, S.: Ann. Phys. (Leipz.) 17, 302 (2008) ADSMATHCrossRefGoogle Scholar
  44. 44.
    Recami, E.: Il caso Majorana—Epistolario Documenti, Testimonianze. Mondadori, Milan 1987 and 1991. The last editions have been published by Di Renzo, Rome (2000–2011). arXiv:physics/9810023v5
  45. 45.
    Esposito, S.: La cattedra vacante—Ettore Majorana, ingegno e misteri. Liguori, Naples (2009) and references therein Google Scholar
  46. 46.
    Fender, F.G., Vinti, J.P.: Phys. Rev. 46, 77 (1934) ADSCrossRefGoogle Scholar
  47. 47.
    Wu, T.-Y.: Phys. Rev. 46, 239 (1934) ADSCrossRefGoogle Scholar
  48. 48.
    Wilson, W.S.: Phys. Rev. 48, 536 (1935) ADSCrossRefGoogle Scholar
  49. 49.
    Di Grezia, E., Esposito, S.: Found. Phys. 38, 228 (2008) ADSMATHCrossRefGoogle Scholar
  50. 50.
    Fock, V.: Z. Phys. 63, 855 (1930) ADSMATHCrossRefGoogle Scholar
  51. 51.
    Esposito, S., Majorana, E. Jr., van der Merwe, A., Recami, E.: Ettore Majorana—Notes on Theoretical Physics. Kluwer, Dordrecht (2003) Google Scholar
  52. 52.
    pp. 130–131 See Ref. [17] Google Scholar
  53. 53.
    Volumetto III, Sect. 15. Google Scholar
  54. 54.
    See Ref. [51], p. 267ff Google Scholar
  55. 55.
    See Ref. [17], pp. 133–141 Google Scholar
  56. 56.
    See Ref. [17], pp. 131–133 Google Scholar
  57. 57.
    Drake, G.W.F., Cassar, M.M., Nistor, R.A.: Phys. Rev. A 65, 054501 (2002) ADSCrossRefGoogle Scholar
  58. 58.
    Brodeur, M., et al.: Phys. Rev. Lett. 108, 052504 (2012) ADSCrossRefGoogle Scholar
  59. 59.
    Cancio Pastor, P., et al.: Phys. Rev. Lett. 108, 143001 (2012) ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.I.N.F.N. Sezione di NapoliComplesso Universitario di M. S. AngeloNaplesItaly
  2. 2.Dipartimento di Fisica “E.R. Caianiello”Università di Salerno, and CNISM Unità di Ricerca di SalernoFisciano (Salerno)Italy

Personalised recommendations