Abstract
A fundamental paradox of classical physics is why matter, which is held together by Coulomb forces, does not collapse. The resolution is given here in three steps. First, the stability of atom is demonstrated, in the framework of nonrelativistic quantum mechanics. Next the Pauli principle, together with some facts about Thomas-Fermi theory, is shown, to account for the stability (i.e., saturation) of bulk matter. Thomas-Fermi theory is developed in some detail because, as is also pointed out, it is the asymptotically correct picture of heavy atoms and molecules (in the Z?8 limit). Finally, a rigorous version of screening is introduced to account for thermodynamic stability.
Work partially supported by U. S. National Science Foundation grant MCS 75–21684.
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References
Balàzs, N., 1967, “Formation of stable molecules within the statistical theory of atoms,” Phys. Rev. 156, 42–47.
Barnes, J. F., 1975, private communication.
Birman, M. S., 1961, Mat. Sb. 55 (97), 125–174
Birman, M. S. [“The spectrum of singular boundary value problems,” Am. Math. Soc. Transi. Ser. 2 53, 23–80 (1966)].
Dirac, P. A. M., 1930, “Note on exchange phenomena in the Thomas atom,” Proc. Camb. Phil. Soc. 26, 376–385.
Dyson, F.J., 1967, “Ground-state energy of a finite system of charged particles,” J. Math. Phys. 8, 1538–1545.
Dyson, F. J., and A. Lenard, 1967, “Stability of matter. I.” J. Math. Phys. 8, 423–434.
Fermi, E., 1927, “Un metodo statistico per la determinazione di alcune prioretà dell’atome,” Rend. Acad. Naz. Lincei 6, 602–607.
Fock, V., 1930, “Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems,” Z. Phys. 61, 126–148;
see also V. Fock, 1930, “Selfconsistent field mit austausch fÜr Natrium,” Phys. 62, 795–805.
Gombâs, P., 1949, Die statistischen Theorie des Atomes und ihre Anwendungen (Springer Verlag, Berlin).
Hartree, D. R., 1927–1928, “The wave mechanics of an atom with a non-Coulomb central field. Parti. Theory and methods,” Proc. Camb. Phil. Soc. 24, 89–110.
Heisenberg, W., 1927, “Über den anschaulichen Inhalt der quanten-theoretischen Kinematik und Mechanik,” Z. Phys. 43, 172–198.
Jeans, J. F.I., 1915, The Mathematical Theory of Electricity and Magnetism (Cambridge University, Cambridge) 3rd ed., p. 168f.
Kato, T., “Fundamental properties of Hamiltonian operators of Schrödinger type,” Trans. Am. Math. Soc. 70, 195–211.
Kirzhnits, D. A., 1957, J. Exptl. Theoret. Phys. (USSR) 32, 115–123
Kirzhnits, D. A., [Engl, transi. “Quantum corrections to the Thomas-Fermi equation,” Sov. Phys. JETP 5, 64–71 (1957)].
Kompaneets, A. S., and E. S. Pavlovskiy 1956, J. Exptl. Theo ret. Phys. (USSR) 31, 427–438
Kompaneets, A. S., and E. S. Pavlovskiy [Engl, transi. “The self-consistent field equations in an atom,” Sov. Phys. JETP 4, 328–336 (1957)].
Lenard, A., and F.J. Dyson, 1968, “Stability of matter. II,” J. Math. Phys. 9, 698–711.
Lenz, W., 1932, “Über die Anwendbarkeit der statistischen Methode auf Ionengitter,” Z. Phys. 77, 713–721.
Lieb, E. H., 1976, “Bounds on the eigenvalues of the Laplace and Schrödinger operators,” Bull. Am. Math. Soc, in press.
Lieb, E. H., and J. L. Lebowitz, 1972, “The constitution of matter: existence of thermodynamics for systems composed of electrons and nuclei,” Adv. Math. 9, 316–398.
See also J. L. Lebowitz and E.H. Lieb, 1969, “Existence of thermodynamics for real matter with Coulomb forces,” Phys. Rev. Lett. 22, 631–634.
Lieb, E.H., and H. Narnhofer, 1975, “The thermodynamic limit for jellium,” J. Stat. Phys. 12, 291–310; Erratum: J. Stat. Phys. 14, No. 5 (1976).
Lieb.’E. H., and B. Simon, “On solutions to the Hartree-Fock problem for atoms and molecules,” J. Chem. Phys. 61, 735–736. Also, a longer paper in preparation.
Lieb, E. H., and B. Simon, 1977, “The Thomas-Fermi theory of atoms, molecules and solids,” Adv. Math., in press. See also E. H. Lieb and B. Simon, 1973, “Thomas-Fermi theory revisited,” Phys. Rev. Lett. 33, 681–683.
Lieb, E.H., and W. E. Thirring, 1975, “A bound for the kinetic energy of fermions which proves the stability of matter,” Phys. Rev. Lett. 35, 687–689; Errata: Phys. Rev. Lett. 35, 1116.
For more details on kinetic energy inequalities and their application, see also E.H. Lieb and W. E. Thirring, 1976, “Inequalities for the moments of the Eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities,” in Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, edited by E. H. Lieb, B. Simon, and A. S. Wightman (Princeton University, Princeton).
Rosen, G., 1971, “Minimum value for c in the Sobolev inequality || Ø ||3 ≤ c|| ∇Ø ||3,” SIAM J. Appl. Math. 21, 30–32.
Schwinger, J., 1961, “On the bound states of a given potential,” Proc. Nat. Acad. Sci. (U.S.) 47, 122–129.
Scott, J. M. C., 1952, “The binding energy of the Thomas Fermi atom,” Phil. Mag. 43, 859–867.
Sheldon, J. W., 1955, “Use of the statistical field approximation in molecular physics,” Phys. Rev. 99, 1291–1301.
Slater, J. C., 1930, “The theory of complex spectra,” Phys. Rev. 34, 1293–1322.
Sobolev, S. L., 1938, Mat. Sb. 46, 471.
See also S. L. Sobolev, 1950, “Applications of functional analysis in mathematical physics,” Leningrad; Am. Math. Soc. Transi. Monographs 7 (1963).
Sommerfeld, A., 1932, “Asymptotische Integration der Differ-ential-gleichung des Thomas-Fermischen Atoms,” Z. Phys. 78, 283–308.
Teller, E., 1962, “On the stability of molecules in the Thomas-Fermi theory,” Rev. Mod. Phys. 34, 627–631.
Thomas, L. H., 1927, “The calculation of atomic fields,” Proc. Camb. Phil. Soc. 23, 542–548.
Von Weizsäcker, C.F., 1935, “Zur Theorie der Kernmassen,” Z. Phys. 96, 431–458.
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Lieb, E.H. (2001). The stability of matter. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_37
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