Qualitative properties of eigenfunctions of classically chaotic Hamiltonian systems

  • Eric J. Heller
II. Spectra and States
Part of the Lecture Notes in Physics book series (LNP, volume 263)


We begin by discussing the properties expected of eigenfunctions of a classically chaotic Hamiltonian system, using simple Correspondence Principle arguments. The properties involve nodal surfaces, coordinate and momentum space amplitude distribution, and phase space distribution. The eigenfunctions of the stadium billiard are examined, and it is found that the periodic orbits of shortest periods and smallest stability parameter profoundly affect the eigenfunctions: “scars” of higher wavefunction density surround the special periodic orbits. Finally a theory is presented for the scars, showing that they must exist, and relating them directly to the special periodic orbits. These same periodic orbits cause level density fluctuations.


Periodic Orbit Wave Packet Break Time Classical Trajectory Nodal Line 


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  1. 1a.
    M.C. GUTZWILLER, Physica 5D, 183 (1982); b.J. Math. Phys. 12, 343(1971).Google Scholar
  2. 2.
    M.V. BERRY and M. TABOR, Proc. Roy. Soc. A349,101 (1976).Google Scholar
  3. 3.
    M.V. BERRY, Proc. Roy. Soc., 229 (1985).Google Scholar
  4. 4.
    J.H. HANNAY and A.M. ORZORIO de ALMEIDA, J. Phys A 17,3429 (1984)Google Scholar
  5. 5.
    D.A. HEJHAL, Duke Math. Jour. 43, 441 (1976).Google Scholar
  6. 6a.
    M. V. BERRY in “Chaotic Behavior of Deterministic Systems”, Les Houches Summer School Lectures. 1981) North-Holland 1983, p171;b. J. Phys. A: 10 2083 (1977).Google Scholar
  7. 7.
    P. PECHUKAS, J. Chem. Phys. 51, 5777 (1972).Google Scholar
  8. 8.
    N. DeLEON and E.J. HELLER, Phys. Rev. A30,5 (1984)Google Scholar
  9. 9.
    B.V. CHIRIKOV, Phys. Rep. 52, 263 (1979).Google Scholar
  10. 10a.
    S.W. McDONALD, Ph. D. Thesis, Lawence Berkeley Laboratory Report LBL 14837; b. S.W. McDONALD and A.N. KAUFMAN, Phys. Rev. Lett. 42,1189 (1979).Google Scholar
  11. 11.
    E. J. HELLER, Phys. Rev. Lett. 53, 1515 (1984).Google Scholar
  12. 12.
    T.H. SELIGMAN and J.J.H. VERBAARSCHOT, “Long Range Stiffness of Spectral Fluctuations in Integrable Scale Invarient Systems”(pre-print 1986)Google Scholar
  13. 13.
    J. W. HELTON and M. TABOR, Physica 14D, 409 (1985).Google Scholar
  14. 14a.
    T.A. BRODY J. FLORES, J.B.FRENCH, P.A. MELLO, A. PANDEY and S.S.M. WONG, Rev. Mod. Phys. 53, 385 (1981); b. O. BOHIGAS, M. J. GIANNONI and C. SCHMIT, J. Phys.Lett. 45, L1015 (1984).Google Scholar
  15. 15.
    G.A. HAGEDORN, Comm. Math. Phys. 71, 77 (1980).Google Scholar
  16. 16.
    R. LITTLEJOHN, Physics Reports, to be published.Google Scholar
  17. 17.
    P.W. ANDERSON Phys. Rev. 103.1492 (1958)Google Scholar
  18. 18.a
    S. FISHMAN, D.R. GREMPEL, and R. E. PRANGE, Phys.. Rev. Lett. 449. 509 (1982); b. Phys. Rev. Lett. 49, 509 (1982).Google Scholar
  19. 19.
    K. ISHII, Progr. Theor. Phys. Suppl. 53, 77 (1973).Google Scholar
  20. 20.
    G. CASATI and F. VALZ-GRIS, and I. GUARNIERI, Nuovo Cimento Lett. 28, 279 (1980).Google Scholar
  21. 21.
    G.HOSE, H.S. TAYLOR, and Y.Y. BAI, J. Chem. Phys. 80,4363 (1984).Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Eric J. Heller
    • 1
  1. 1.Departments of Chemistry and PhysicsUniversity of WashingtonSeattleUSA

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