Skip to main content

Semiclassical Roles for Classical Orbits

  • Chapter
  • 423 Accesses

Part of the book series: Springer Series in Synergetics ((SSSYN,volume 54))

Abstract

The precursor of quantum mechanics due to Bohr and Sommerfeld was already seen by Einstein [235] as applicable only to classically integrable dynamics. It has since been developed into a systematic semiclassical approximation, as sketched in Chap. 5. No prepotent guesses proved possible as to what quantum mechanics has to say about chaotic motion; the pertinent semiclassical approximations came into view with a delay of nearly fifty years in Gutzwiller’s pioneering work.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Einstein: Verh. Dt. Phys. Ges. 19, 82 (1917)

    Google Scholar 

  2. M.C. Gutzwiller: Chaos in Classical and Quantum Mechanics ( Springer, New York, 1990 )

    MATH  Google Scholar 

  3. M.C. Gutzwiller: J. Math. Phys. 12, 343 (1971)

    Article  ADS  Google Scholar 

  4. P. Cvitanovic: Classical and Quantum Chaos. Cyclist Treatise www.nbi.dk/ChaosBook/

    Google Scholar 

  5. M. Brack, R.K. Bhaduri: Semiclassical Physics ( Addison-Wesley, Reading, Mass., 1997 )

    Google Scholar 

  6. L.S. Schulman: Techniques and Applications of Path Integration ( Wiley, New York, 1981 )

    MATH  Google Scholar 

  7. G. Junker, H. Leschke: Physica 56D, 135 (1992)

    MathSciNet  MATH  Google Scholar 

  8. M. Tabor: Physica 6D, 195 (1983)

    MathSciNet  Google Scholar 

  9. J.P. Keating: Nonlinearity 4, 309 (1991) (Addison-Wesley, Reading, Mass., 1950; 2nd ed. 1980 )

    Google Scholar 

  10. R. Courant, D. Hilbert: Methoden der Mathematischen Physik (Springer, Berlin; 3. Aufl. 1968 ); R. Bellmann: Introduction to Matrix Analysis, 2nd. ed. ( McGraw-Hill, New York, 1970 )

    Google Scholar 

  11. S. Levit, K. Möhring, U. Smilansky, T. Dreyfus: Ann. Phys. 114, 223 (1978); S. Levit, U. Smilansky: Ann. Phys. 103, 198 (1977); ibid. 108, 165 (1977); K. Möhring, S. Levit, U. Smilansky: Ann. Phys. 127, 198 (1980)

    Google Scholar 

  12. M. Morse: Variational Analysis ( Wiley, New York, 1973 )

    MATH  Google Scholar 

  13. E.T. Copson: Asymptotic Expansions ( Cambridge University Press, Cambridge, 1965 )

    Book  MATH  Google Scholar 

  14. H.P. Baltes, E.R. Hilf: Spectra of Finite Systems ( BI Wissenschaftsverlag, Mannheim, 1976 )

    MATH  Google Scholar 

  15. R. Balian, F. Bloch: Ann. Phys. (USA) 63, 592 (1971)

    Article  ADS  Google Scholar 

  16. M.V. Berry, C.J. Howls: Proc. Roy. Soc. Lond. A447, 527 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. K. Stewartson, R.T. Waechter: Proc. Camb. Phil. Soc. 69, 353 (1971)

    Article  ADS  MATH  Google Scholar 

  18. H. Primack, U. Smilansky: J. Phys. A31 6253 (1998); H. Primack, U. Smilansky: Phys. Rev. Lett. 74,4831 (1995); H. Primack: Ph.D. Thesis, The Weizman Institute of Science, Rehovot (19997)

    Google Scholar 

  19. P. Gaspard: Prog. Th. Phys. Supp. 116, 59 (1994); P. Gaspard, D. Alonso: Phys. Rev. A47, R3468 (1993); D. Alonso, P. Gaspard: Chaos 3, 601 (1993)

    MathSciNet  Google Scholar 

  20. T. Prosen: Phys. Lett. A223, 323 (1997)

    MathSciNet  Google Scholar 

  21. N.L. Balazs, A. Voros: Phys. Rep. 143, 109 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  22. T. Kottos, U. Smilansky: Ann. Phys. 274, 76 (1999); T. Kottos, U. Smilansky:chao-dyn/9906008; H. Schanz, U. Smilansky: Proceedings of the Australian Summer School on Quantum Chaos and Mesoscopics, Canberra Australia, January 1999

    Google Scholar 

  23. M.V. Berry, J.P. Keating: J. Phys. A23, 4839 (1990)

    MathSciNet  ADS  MATH  Google Scholar 

  24. J.P. Keating: Proc. R. Soc. Lond. A436, 99 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. M.V. Berry, J.P. Keating: Proc. R. Soc. Lond. A437, 151 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. B. Eckhardt, E. Aurell: Europhys. Lett. 9, 509 (1989)

    Article  ADS  Google Scholar 

  27. E.B. Bogomolny: Nonlinearity 5, 805 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. E. Doron, U. Smilansky: Nonlinearity 5, 1055 (1995); H. Schanz, U. Smilansky: Chaos, Solitons and Fractals 5, 1289 (1995); U. Smilansky: In E. Akkermans, G. Montambaux, J.L. Pichard: Proceedings of the 1994 Summer School on “Mesoscopic Quantum Physics” (North Holland, Amsterdam, 1991); B. Dietz, U. Smilansky: Chaos 3, 581 (1993); C. Rouvinez, U. Smilansky: J. Phys. A28, 77 (1994)

    Google Scholar 

  29. E.G. Vergini: J. Phys. A33, 4709 (2000); E.G. Vergini, G.G. Carlo: J. Phys. A33, 4717 (2000)

    MathSciNet  ADS  MATH  Google Scholar 

  30. R.G. Littlejohn: J. Math. Phys. 31, 2952 (1990); J. Stat. Phys. 68, 7 (1992)

    MathSciNet  MATH  Google Scholar 

  31. I.C. Percival: Adv. Chem. Phys. 36, 1 (1977)

    Article  Google Scholar 

  32. S.C. Creagh, J.M. Robbins, R.G. Littlejohn: Phys. Rev. A42, 1907 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  33. J.N. Mather: Comm. Math. Phys. 94, 141 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. J. Robbins: Nonlinearity 4, 343 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. J. Moser: Ergod. Theor.Dynam. Sys. 6, 401 (1986)

    MATH  Google Scholar 

  36. P.A. Boasman, U. Smilansky: J. Phys. A27, 1373 (1994)

    MathSciNet  ADS  MATH  Google Scholar 

  37. J.H. Hannay, A.M. Ozorio de Almeida: J. Phys. A17, 3429 (1984)

    MathSciNet  ADS  Google Scholar 

  38. M.V. Berry: Proc. R. Soc. London A400, 229 (1985)

    Article  ADS  MATH  Google Scholar 

  39. S. Fishman, J.P. Keating: J. Phys. A31, L313 (1998)

    MathSciNet  ADS  Google Scholar 

  40. P. Cvitanovie, B. Eckhardt: J. Phys. A24, L237 (1991); Phys. Rev. Lett. 63, 823 (1989); Nonlinearity 6, 277 (1993)

    Google Scholar 

  41. E.B. Bogomolny, J.P. Keating: Phys. Rev. Lett. 77, 1472 (1996)

    Article  ADS  MATH  Google Scholar 

  42. O. Agam, B.L. Altshuler, A.V. Andreev: Phys. Rev. Lett. 75, 4389 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. P. Gaspard: Chaos, Scattering and Statistical Mechanics ( Cambridge University Press, Cambridge, 1998 )

    Google Scholar 

  44. E.B. Bogomolny, J.P. Keating: Private communication and to be published

    Google Scholar 

  45. E.B. Bogomolny, F. Leyvraz, C. Schmit• Commun. Math. Phys. 176, 577 (1996); W. Luo, P. Sarnac: Commun. Math. Phys. 161, 419 (1994)

    Google Scholar 

  46. E. Bogomolny, C. Schmit: Nonlinearity 6, 523 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. J.P. Keating, M. Sieber: Proc. R. Soc. London A447, 413 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. R. Aurich, C. Matthies, M. Sieber, F. Steiner: Phys. Rev. Lett. 68, 1629 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. A.V. Andreev, B.L. Altshuler: Phys. Rev. Lett. 75, 902 (1995)

    Article  MathSciNet  ADS  Google Scholar 

  50. M. Abramowitz, I. Stegun: Handbook of Mathematical Functions ( Dover, New York 1965 )

    Google Scholar 

  51. A.J. Lichtenberg, M.A. Liebermann: Regular and Chaotic Dynamics, Applied Mathematical Sciences, Vol. 38, 2nd ed. ( Springer, New York, 1992 )

    Google Scholar 

  52. M.V. Berry, M. Robnik: J. Phys. A17, 2413 (1984)

    MathSciNet  ADS  MATH  Google Scholar 

  53. K.R. Meyer: Trans. Am. Math. Soc. 149, 95 (1970)

    Article  MATH  Google Scholar 

  54. H. Schomerus: J. Phys. A31, 4167 (1998)

    MathSciNet  ADS  MATH  Google Scholar 

  55. A.M. Ozorio de Almeida, J.H. Hannay: J. Phys. A20, 5873 (1987)

    MathSciNet  ADS  MATH  Google Scholar 

  56. S. Tomsovic, M. Grinberg, D. Ullmo: Phys. Rev. Lett. 75, 4346 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. D. Ullmo, M. Grinberg, S. Tomsovic: Phys. Rev. E54, 136 (1996)

    ADS  Google Scholar 

  58. M. Sieber: J. Phys. A29, 4715 (1996)

    MathSciNet  ADS  MATH  Google Scholar 

  59. H. Schomerus, M. Sieber: J. Phys. A30, 4537 (1997)

    MathSciNet  ADS  MATH  Google Scholar 

  60. H. Schomerus: Europhys. Lett. 38, 423 (1997)

    Article  ADS  Google Scholar 

  61. M. Sieber, H. Schomerus: J. Phys. A31, 165 (1998)

    ADS  MATH  Google Scholar 

  62. T. Poston, I.N. Stewart: Catastrophe Theory and its Applications ( Pitman, London, 1978 )

    MATH  Google Scholar 

  63. M. Kus, F. Haake, D. Delande: Phys. Rev. Lett. 71, 2167 (1993)

    Article  ADS  Google Scholar 

  64. H. Schomerus, F. Haake: Phys. Rev. Lett. 79, 1022 (1997)

    Article  ADS  Google Scholar 

  65. M.V. Berry, J.P. Keating, S.D. Prado: J. Phys. A31, L243 (1998)

    MathSciNet  Google Scholar 

  66. C. Manderfeld, H. Schomerus: chao-dyn/9911024

    Google Scholar 

  67. M.V. Berry, J.P. Keating, H. Schomerus: submitted to Proc. R. Soc. Lond. (Jan. 2000)

    Google Scholar 

  68. M.V. Berry: In: G. Casati, I. Guarneri and U. Smilansky (eds): Proceedings of the International School of Physics “Enrico Fermi” Course CXLIII G. Casati, I. Guarneri and U. Smilansky (Ios Press, Amsterdam, 2000 )

    Google Scholar 

  69. M. Khodas, S. Fishman: Phys. Rev. Lett. 84 2837 (2000), chaodyn 9910040

    Google Scholar 

  70. J. Weber, F. Haake, P. Seba: Phys. Rev. Lett. 85 3620 (2000), and chaodyn 0001013

    Google Scholar 

  71. C.W. Gardiner: Quanntum Noise ( Springer, Berlin, 1991 )

    Google Scholar 

  72. C. Manderfeld, J. Weber, F. Haake, P.A. Braun: to be published

    Google Scholar 

  73. D. Weiss, M.L. Roukes, A. Menschig, P. Grambow, K. von Klitzing, G. Weimann: Phys. Rev. Lett. 66, 2790 (1991); D. Weiss, K. Richter, A. Menschig, R. Bergmann, H. Schweizer, K. von Klitzing, G. Weimann: Phys. Rev. Lett. 70, 4118 (1993)

    Article  ADS  Google Scholar 

  74. R. Fleischmann, T. Geisel, R. Ketzmerick: Phys. Rev. Lett. 68, 1367 (1992)

    Article  ADS  Google Scholar 

  75. G. Hackenbroich, F. von Oppen: Europhys. Lett. 29, 151 (1995); Z. Physik B97, 157 (1995); K. Richter: Europhys. Lett. 29, 7 (1995)

    Google Scholar 

  76. T.M. Fromhold, L. Eaves, F.W. Sheard, M.L. Leadbeater, T.J. Foster, P.C. Main: Phys. Rev. Lett. 72, 2608 (1994); G. Mueller, G.S. Boebinger, H. Mathur, L.N. Pfeiffer, K.W. West: Phys. Rev. Lett. 75, 2875 (1995); E.E. Narimanov, A.D. Stone: Phys. Rev. B57, 9807 (1998); E.E. Narimanov, A.D. Stone: Physica D131, 221 (1999); E.E. Narimanov, A.D. Stone, G.S. Boebinger: Phys. Rev. Lett. 80, 4024 (1998)

    Google Scholar 

  77. A. Mekis, J.U. Nöckel, G. Chen, A.D. Stone, R.K. Chang: Phys. Rev. Lett. 75, 2682 (1995); J.U. Nöckel, A.D. Stone: Nature 385,47 (1997); C. Gmachl, F. Ca-passo, E.E. Narimanov, J.U. Nöckel, D.A. Stone, J. Faist, D.L. Sivco, A.Y. Cho: Science 280, 1493 (1998); E.E. Narimanov, G. Hackenbroich, P. Jacquod, A.D. Stone: Phys. Rev. Lett. 83, 4991 (1999)

    Google Scholar 

  78. M. Brack, S.M. Reimann, M. Sieber: Phys. Rev. Lett. 79, 1817 (1997)

    Article  ADS  Google Scholar 

  79. J. Main, G. Wunner: Phys. Rev. A55, 1743 (1997)

    Article  ADS  Google Scholar 

  80. M. Fließer, A. Csordâs, P. Szépfalusy, R. Graham: Phys. Rev. A56, R2533 (1997); ibid. A56, 4879 (1997); M. Fließer, R. Graham: Phys. Rev. A59, R27 (1999); Physica D131, 141 ( 1999 ); M. Fließer: Dissertation, Essen (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Haake, F. (2001). Semiclassical Roles for Classical Orbits. In: Quantum Signatures of Chaos. Springer Series in Synergetics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04506-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-04506-0_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08722-6

  • Online ISBN: 978-3-662-04506-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics