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 Zāā 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.
Work partially supported by U.S. National Science Foundation grant MCS 75-21684.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
BalĆ zs, N., 1967, āFormation of stable molecules within the statistical theory of atoms,ā Phys. Rev. 156, 42ā47.
Barnes, J. F., 1975, private communicatiou.
Birman, M. S., 1961, Mat. Sb. 55 (97), 125ā174 [āThe spectrum of singular boundary value problems,ā Am. Math. Soc. Transl. Ser. 2 53, 23ā80 (1966)].
Dirac, P. A. M., 1930, āNote on exchange phenomena in the Thomas atom,ā Proc. Camb, Phil. Soc. 26, 376ā385.
Dyson, F. J., 1967, āGround-state energy of a finite System of charged particles,ā J. Math. Phys. 8, 1538ā1545.
Dyson, F. J., and A. Lenard, 1967, āStability of matter. I.ā J. Math. Phys. 8, 423ā434.
Fermi, E., 1927, āUn metodo statistico per la determinazione di alcune prioretĆ dellāatome,ā Rend. Acad. Naz. Lincei 6, 602ā607.
Fock, V., 1930, āNƤherungsmethode zur Lƶsung des quanten-mechanischen Mehrkƶrperproblems,ā Z. Phys. 61, 126ā148; see also V. Fock, 1930, āSelfconsistent fieldā mit austausch fĆ¼r Natrium,ā Phys. 62, 795ā805.
GombƔs, P., 1949, Die statistischen Theorie des Atomes und. ihre Anwendungen (Springer Verlag, Berlin).
Hartree, D. R., 1927ā1928, āThe wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods,ā Proc. Camb. Phil. Soc. 24, 89ā110.
Heisenberg, W., 1927, āĆber den anschaulichen Inhalt der quanten-theoretischen Kinematik und Mechanik,ā Z. Phys. 43, 172ā198.
Jeans, J. H., 1915, The Mathematical Theory of Electricity and Magnetism (Cambridge University, Cambridge) 3rd ed., p. 168.
Kato, T., āFundamental properties of Hamiltonian operators of Schrƶdinger type,ā Trans. Am. Math. Soc. 70, 195ā211.
Kirzhnits, D. A., 1957, J. Exptl. Theoret. Phys. (USSR) 32, 115ā123 [Engl. transl. āQuantum corrections to the Thomas-Fermi equation,ā Sov. Phys. JETP 5, 64ā71 (1957)].
Kompaneets, A. S., and E. S. Pavlovskii, 1956, J. Exptl. Theoret. Phys. (USSR) 31, 427ā438 [Engl. transl. āThe self-consistent field equations in an atom,ā Sov. Phys. JETP 4, 328ā336 (1957)].
Lenard, A., and F. J. Dyson, 1968, āStability of matter. II,ā J. Math. Phys. 9, 698ā711.
Lenz, W., 1932, āĆber die Anwendbarkeit der statistischen Methode auf lonengitter,ā Z. Phys. 77, 713ā721.
Lieb, E. H., 1976, āBounds on the eigenvalues of the Laplace and Schrƶdinger operators,ā Bull. Am. Math. Soc, in press.
Lieb, E. H., and J. L. Lebowitz, 1972, āThe constitution of matter: existence of thermodynamics for Systems composed of electrons and nuclei,ā Adv. Math. 9, 316ā398. See also J. L. Lebowitz and E. H. Lieb, 1969, āExistence of thermo-dynamics for real matter with Coulomb forces,ā Phys. Rev. Lett. 22, 631ā634.
Lieb, E. H., and H. Narnhofer, 1975, āThe thermodynamic limit for jellium,ā J. Stat. Phys. 12, 291ā310; Erratum: J. Stat. Phys. 14, No. 5 (1976).
Lieb, E. H., and B. Simon, āOn solutions to the Hartree-Fock problem for atoms and molecules,ā J. Chem. Phys. 61, 735ā736. Also, a longer paper in preparation.
Lieb, E. H., and B. Simon, 1977, āThe Thomas-Fermi theory of atoms, molecules and solids,ā Adv. Math., in press. See also E. H. Lieb and B. Simon, 1973, āThomas-Fermi theory revisited,ā Phys. Rev. Lett. 33, 681ā683.
Lieb, E. H., and W. E. Thirring, 1975, āA bound for the kinetic energy of fermions which proves the stability of matter,ā Phys. Rev. Lett. 35, 687ā689; 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, āInequalities for the moments of the Eigenvalues of the Schrƶdinger Hamiltonian and their relation to Sobolev in-equalities,ā in Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, edited by E. H. Lieb, E. Simon, and A. S. Wightman (Princeton University, Princeton).
Rosen, G., 1971, āMinimum value for c in the Sobolev in-equality ā„ Ļ ā„ 3 ā¤ ā„ āĻ ā„ 3,ā SIAM J. Appl. Math. 21, 30ā32.
Schwinger, J., 1961, āOn the bound states of a given potential,ā Proc. Nat. Acad. Sci. (U.S.) 47, 122ā129.
Scott, J. M. C, 1952, āThe binding energy of the Thomas Fermi atom,ā Phil. Mag. 43, 859ā867.
Sheldon, J. W., 1955, āUse of the statistical field approximation in molecular physics,ā Phys. Rev. 99, 1291ā1301.
Slater, J. C, 1930, āThe theory of complex spectra,ā Phys. Rev. 34, 1293ā1322.
Sobolev, S. L., 1938, Mat. Sb. 46, 471. See also S. L. Sobolev, 1950, āApplications of functional analysis in mathematical physics,ā Leningrad; Am. Math. Soc. Transl. Monographs 7 (1963).
Sommerfeld, A., 1932, āAsymptotische Integration der Differential-gleichung des Thomas-Fermischen Atoms,ā Z. Phys. 78, 283ā308.
Teller, E., 1962, āOn the stability of molecules in the Thomas-Fermi theory,ā Rev. Mod. Phys. 34, 627ā631.
Thomas, L. H., 1927, āThe calculation of atomic fields,ā Proc. Camb. Phil. Soc. 23, 542ā548.
Von WeizsƤcker, C. F., 1935, āZur Theorie der Kernmassen,ā Z. Phys. 96, 431ā458.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2005 Springer-Verlag Berlin Heidelberg New York
About this chapter
Cite this chapter
Lieb, E.H. (2005). The stability of matter. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27056-6_37
Download citation
DOI: https://doi.org/10.1007/3-540-27056-6_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22212-5
Online ISBN: 978-3-540-27056-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)