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Thomas-Fermi Theory

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Part of the Texts and Monographs in Physics book series (TMP)

Abstract

In this chapter we wish to study those problems from Thomas-Fermi theory whose solutions depend almost exclusively on the methods of the calculus of variations. Questions of validity and physical interpretation are discussed by Thirring [10.1] and Lieb and Simon [10.2] and in references quoted therein. Justification for statements which we do not prove below can be found in [10.1, 3].

Keywords

Minimisation Problem Particle Number Fermi Theory Frechet Derivative Weakly Convergent Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 10.1
    Thirring, W.: A course in mathematical physics, vol. 4. Springer, Berlin Heidelberg 1986CrossRefGoogle Scholar
  2. 10.2
    Lieb, E. H., and Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. in Math.23(1977) 22–116;CrossRefMathSciNetGoogle Scholar
  3. Thomas-Fermi theory revisited. Phys. Rev. Lett.31(1973) 681–683CrossRefGoogle Scholar
  4. 10.3
    Lieb, E. H.: The stability of matter. Rev. Mod. Phys.48(1970) 553–569CrossRefADSMathSciNetGoogle Scholar
  5. 10.4
    Lieb, E. H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys.53(1981) 603–641;CrossRefzbMATHADSMathSciNetGoogle Scholar
  6. Lieb, E. H.: Errata, ibid54(1982) 311ADSMathSciNetGoogle Scholar
  7. 10.5
    Reed, M., Simon, B.: Methods of modern mathematical physics II. Fourier analysis, self adjointness. Academic Press, New York 1975zbMATHGoogle Scholar
  8. 10.6
    Brezis, H.: Nonlinear problems related to the Thomas-Fermi equation. In: Contemporary developments in continuum mechanics and partial differential equations, G. M. de la Penha and L.A. Medreiros (eds.). North-Holland, Amsterdam 1978.Google Scholar
  9. Some variational problems of the Thomas-Fermi type. In: Variational inequalities and complementary problems: theory and applications, R. W. Cottle, F. Gianessi and J. L. Lions (eds.). Wiley, New York 1980, pp. 55–73Google Scholar
  10. 10.7
    Thomas, L. H.: The calculation of atomic fields. Proc. Camb. Phil. Soc.23(1927) 542–548CrossRefzbMATHADSGoogle Scholar
  11. 10.8
    Fermi, E.: Un metodo statistico per la determinazione di alcune prioretà dell’atomo. Rend. Acad. Naz. Lincei6(1927) 602–607Google Scholar
  12. 10.9
    Messer, J.: Temperature dependent Thomas-Fermi theory. Lecture Notes in Physics147. Springer, Berlin Heidelberg 1981Google Scholar
  13. 10.10
    Baumgartner, B., Narnhofer, H., Thirring, W.: Thomas-Fermi limit of Bosejellium. Ann. Phys.150(1983) 373–391CrossRefzbMATHADSMathSciNetGoogle Scholar
  14. 10.11
    Benguria, R., Brezis, H., Lieb, E.: The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Commun. Math. Phys.79(1981) 167–180CrossRefzbMATHADSMathSciNetGoogle Scholar
  15. 10.12
    Daubechies, I., Lieb, E.: One electron relativistic molecules with Coulomb interaction. Commun. Math. Phys.90(1983) 497–510CrossRefzbMATHADSMathSciNetGoogle Scholar
  16. 10.13
    Lieb, E., Thirring, W.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys.155(1984) 494–512CrossRefADSMathSciNetGoogle Scholar
  17. 10.14
    Fröhlich, J., Lieb, E., Loss, M.: Stability of Coulomb systems with magnetic fields I. The one electron atom. Commun. Math. Phys.104(1986) 251–270CrossRefzbMATHADSGoogle Scholar
  18. 10.15
    Lieb, E., Loss, M.: Stability of Coulomb systems with magnetic fields II. The many electron atom and the one electron molecule. Commun. Math. Phys.104(1986) 271–282CrossRefzbMATHADSMathSciNetGoogle Scholar
  19. 10.16
    Loss, M., Yau, H.T.: Stability of Coulomb systems with magnetic fields III. Zero energy bound states of the Pauli operator. Commun. Math. Phys.104(1986) 283–290CrossRefzbMATHADSMathSciNetGoogle Scholar
  20. 10.17
    Lieb, E. H.: Bounds for the kinetic energy of fermions which prove the stability of matter. Phys. Rev. Lett.35(1976) 687–689;CrossRefADSMathSciNetGoogle Scholar
  21. Lieb, E. H.: Errata, ibid35(1976) 1116CrossRefADSGoogle Scholar
  22. 10.18
    Lieb, E. H.: The stability of matter: from atoms to stars. American Mathematical Society Josiah Willard Gibbs Lecture, Phoenix, AZ 1989. Bulletin of the AMS, vol. 22 no. 1, Jan. 1990, pp. 1–49CrossRefzbMATHMathSciNetGoogle Scholar

Further Reading

  1. Thirring, W.: A lower bound with the best possible constant for Coulomb Hamiltonians. Commun. Math. Phys.79(1981) 1–7CrossRefADSMathSciNetGoogle Scholar
  2. Dyson, F. J.: In: Brandeis University Summer Institute in Theoretical Physics 1966, vol. 1.Google Scholar
  3. M. Chretien, E. P. Gross and S. Deser (eds.). Gordon and Breach, New York 1978Google Scholar
  4. Lenard, A.: In: Statistical mechanics and mathematical problems. Lecture Notes in Physics20. Springer, Berlin Heidelberg 1973CrossRefGoogle Scholar
  5. Hille, E.: On the Thomas-Fermi equation. Proc. Nat. Acad. Sci. (USA)62(1969) 7–10CrossRefzbMATHADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Theoretische Physik, Fakultät für PhysikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa

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