The 20th century is the one where an old dream of scientists became reality. Starting from Democritos up to Boltzmann there was a vision that the laws governing the behavior of matter around us are a consequence of and can be derived from more fundamental ones which dominate its elementary constituents, the atoms. In this century, Rutherford first discovered the structure of atoms and proposed the following picture. Atoms consist of electrically negatively charged electrons which circle in a distance of about 10−8 cm around the positively charged nucleus which is very small, 10−12–10−13 cm. He also found that the Coulomb force \(\vec xe_1 e_2 /\left| {\vec x} \right|^3 \) between two charges e 1, e 2 at a relative distance \(\vec x\) is valid down to these small dimensions and there was no sign of any other force outside the nucleus. Thus there arose for the first time the chance to deduce the properties of atoms or even bigger bodies from the mechanical laws. The ones known at that time were the rules of Hamiltonian mechanics where the fundamental equations of motion are encoded in the energy H(x i , p i ) as function of the position x i and the momentum p i of particle i, i = 1,2,...,N, N = total number of particles. This energy consists of the kinetic energy which is p 2/2m for a particle of mass m and the potential energy which is for the electrostatic interaction of two charges e 1 and e 2 and relative distance \(\vec x\) equal to \(e_1 e_2 /\left| {\vec x} \right|\) where \(\left| {\vec x} \right|\) denotes the length of the vector \(\left| {\vec x} \right|\). The force is the negative of the gradient of the potential and is just the Coulomb force mentioned earlier. It is attractive for charges of opposite sign, e 1 e 2 < 0, and repulsive if e 1 e 2 > 0. It is of the same form as Newton’s gravitational force between two masses m 1 and m 2 except that the latter is always attractive. Thus it has a potential energy \( - km_1 m_2 /\left| {\vec x} \right|\) where κ is the so-called gravitational constant.


Black Hole Kinetic Energy Potential Energy Coulomb Repulsion Nuclear Charge 


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  1. 1.
    E.H. Lieb and W. Thirring, Phys. Rev. A 34, 40 (1986).CrossRefGoogle Scholar
  2. 2.
    M. Fisher and D. Ruelle, J. Math. Phys. 7, 260 (1966).CrossRefGoogle Scholar
  3. 3.
    F. J. Dyson and A. Lenard, J. Math. Phys. 8, 423 (1967).CrossRefGoogle Scholar
  4. 4.
    E. H. Lieb and W. Thirring, Phys. Rev. Lett. 35, 687 (1975).CrossRefGoogle Scholar
  5. 5.
    J. L. Lebowitz and E. H. Lieb, Phys. Rev. Lett. 22, 631 (1969).CrossRefGoogle Scholar
  6. 6.
    Stability of matter in “From Atoms to Stars”. Selecta of E. H. Lieb, Springer, Heidelberg, 1992.Google Scholar
  7. 7.
    W. Thirring, A Course in Mathematical Physics IV, Springer, New York, 1980.Google Scholar
  8. 8.
    E. H. Lieb and B. Simon, Adv. Math. 23, 22 (1977).CrossRefGoogle Scholar
  9. 9.
    E. H. Lieb and S. Oxford, Int. J. Quant. Chem. 19, 427 (1981).CrossRefGoogle Scholar
  10. 10.
    W. Thirring, Phys. Blätter 31, 582 (1975).CrossRefGoogle Scholar
  11. 11.
    J. Beckenstein, Phys. Rev. D 7, 2333 (1973).CrossRefGoogle Scholar
  12. 12.
    S. Hawking, Commun. Math. Phys. 43, 199 (1975).CrossRefGoogle Scholar
  13. 13.
    A. Compagner, C. Bruin, A. Roelso, Phys. Rev. A 39, 5989 (1989).CrossRefGoogle Scholar
  14. 14.
    H. Posch, H. Narnhofer, W. Thirring, Phys. Rev. A 42, 1880 (1990).CrossRefGoogle Scholar
  15. 15.
    H. Narnhofer, W. Thirring, Phys. Rev. Lett. 54, 1863 (1990).CrossRefGoogle Scholar

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© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Walter Thirring
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of ViennaAustria

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