Il Nuovo Cimento A (1965-1970)

, Volume 18, Issue 3, pp 551–573 | Cite as

Electron wave functions in over-critical electrostatic potentials

  • B. Müller
  • J. Rafelski
  • W. Greiner


The mathematical properties of the solutions of the Dirac equation in over-critical external potentials are investigated. The 1/r Coulomb potential is treated as the limitR → 0 of cut-off Coulomb potentials (R is the cut-off parameter). The results are interpreted in terms of quantum electrodynamics. A number of new physical phenomena occur in quantum electrodynamics of strong fields. One of the most interesting ones is the autoionization of positrons (energyless creation of electron-positron pairs) in over-critical external fields.

Электронные волновые функции в сверхкритических злектростатических потенциалах


Исследуются математические свойства уравнения Дирака в сверхкритических внещних потенциалах. Кулоновский потенциал 1/r рассматривается как предел обреэанных кулоновских потенциалов приR→0 (гдеR есть параметр обреэания). Полученные реэультаты интерпретируются в терминах квантовой злектродинамики. Ряд новых фиэических явлений появляется в квантовой злектродинамике сильных полей. Одно иэ наиболее интересных явлений представляет автоиониэация поэитронов (рождение злектрон-поэитронных пар) в сверхкритических внещних полях.


Si studiano le proprietà matematiche delle soluzioni dell’equazione di Dirac per potenziali esterni ipercritici. Si tratta potenziale Coulombiano del tipo 1/r come il limite perR → 0 di potenziali Coulombiani di taglio (R è il parametro di taglio). Si interpretano i risultati in termini di elettrodinamica quantistica. Nell’elettrodinamica quantistica dei campi forti intervengono parecchi fenomeni fisici nuovi. Uno dei più interessanti è l’autoionizzazione di positoni (creazione senza energia di coppie elettronepositone) in campi esterni ipercritici.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    W. Gordon:Zeits. Phys.,48, 11 (1928).CrossRefADSGoogle Scholar
  2. (2).
    I. Pomeranchuk andJ. Smorodinsky:Journ. Phys. USSR,9, 97 (1945).Google Scholar
  3. (3).
    A. I. Achiezer andV. B. Berestetzky:Quantum Electrodynamics (New York, 1965).Google Scholar
  4. (4).
    W. Pieper andW. Greiner:Zeits. Phys.,218, 327 (1969).CrossRefADSzbMATHGoogle Scholar
  5. (5) a).
    V. S. Popov:Yad. Fiz.,12, 429 (1970);Google Scholar
  6. (5) b).
    V. S. Popov:Yad. Fiz.,14, 458 (1971);Google Scholar
  7. (5) c).
    V. S. Popov:Žurn. Ėksp. Teor. Fiz.,60, 1228 (1971).Google Scholar
  8. (6).
    B. Müller, J. Rafelski andW. Greiner:Zeits. Phys.,257, 183 (1972).CrossRefADSGoogle Scholar
  9. (7).
    B. Müller, H. Peitz, J. Rafelski andW. Greiner:Phys. Rev. Lett.,28, 1235 (1972).CrossRefADSGoogle Scholar
  10. (8).
    K. M. Case:Phys. Rev.,80, 797 (1950).MathSciNetCrossRefADSGoogle Scholar
  11. (9).
    General properties of the Dirac equation in central fields and solutions for < 1 can be found comprehensively inE. M. Rose:Relativistic Electron Theory (New York, 1961).Google Scholar
  12. (10).
    D. Rein: Diploma thesis, Frankfurt am Main (1964).Google Scholar
  13. (11).
    L. J. Slater:Confluent Hypergeometric Functions (Cambridge, 1960).Google Scholar
  14. (*).
    See also ref. (5c) in this connection.zbMATHGoogle Scholar
  15. (12).
    M. Abramowitz andJ. Stegun:Handbook of Mathematical Functions (New York, 1965).Google Scholar
  16. (13).
    U. Fano:Phys. Rev.,124, 1866 (1961).CrossRefADSGoogle Scholar
  17. (*).
    The localization of the 1s bound state at the boundary of the lower continuum has also been discussed in ref. (4–7).Google Scholar
  18. (14).
    V. F. Weisskopf:Kgl. Danske Videnskab. Selskab.,14, 1 (1936).Google Scholar
  19. (15).
    E. A. Uehling:Phys. Rev.,48, 55 (1935).CrossRefADSGoogle Scholar
  20. (16).
    Y. B. Zeldovich andV. S. Popov:Sov. Phys. Usp.,14, 673 (1972).CrossRefADSGoogle Scholar
  21. (**).
    Bawin andLavine (17) maintain that the bound state ϕ cannot penetrate into the negative-energy continuum. This statement is erroneous since the bound-state energy is lowered by the matrix element ΔE (9.6), which involves no interaction with the continuum, and not byΓ E (9.7), which vanishes forE=−m.Google Scholar
  22. (17).
    M. Bawin andJ. P. Lavine:On the existence of a critical charge for superheavy nuclei, preprint, University of Liège, Belgium (1972).Google Scholar
  23. (18).
    This idea was first suggested by the authors during various GSI seminars, 1969–1971. See alsoL. P. Fulcher, J. Rafelski andW. Greiner:Phys. Rev. Lett.,27, 958 (1971).CrossRefADSGoogle Scholar

Copyright information

© Società Italiana di Fisica 1973

Authors and Affiliations

  • B. Müller
    • 1
  • J. Rafelski
    • 1
  • W. Greiner
    • 1
  1. 1.Institut für Theoretische Physik der Universität FrankfurtFrankfurt/Main

Personalised recommendations