Skip to main content
SpringerLink
Log in

Classical Invariants in the Quantum Mechanics of Chaotic Systems

  • Chapter
  • First Online:
ISCS 2013: Interdisciplinary Symposium on Complex Systems

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 8))

  • 653 Accesses

Abstract

The relevance of classical invariants in the quantization and dynamics of quantum systems is discussed. Special attention is paid to the influence of periodic orbits (“scars”) and the associated homoclinic and heteroclinic orbits. As an illustration we present some results concerning the vibrational dynamics of the LiNC/LiCN isomerizing system.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Barrow-Green, J.: Poincaré and the Three Body Problem. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  2. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    Book  MATH  Google Scholar 

  3. Berry, M.V.: AIP Conf. Proc. 46, 16 (1978)

    Article  Google Scholar 

  4. Wisniacki, D.A., Saraceno, M., Arranz, F.J., Benito, R.M., Borondo, F.: Phys. Rev. E 84, 026206 (2011)

    Article  Google Scholar 

  5. Arranz, F.J., Seidel, L., Giralda, C.G., Benito, R.M., Borondo, F.: Phys. Rev. E 87, 062901 (2013)

    Article  Google Scholar 

  6. Leboeuf, P., Voros, A.: J. Phys. A 23, 1765 (1990)

    Article  MathSciNet  Google Scholar 

  7. Jammer, M.: The Conceptual Development of Quantum Mechanics. American Institute of Physics, New York (1989)

    Google Scholar 

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

    Google Scholar 

  9. Einstein, A.: Ver. Phys. Ges. 19, 82 (1917)

    Google Scholar 

  10. Maslov, V.: Théorie des Perturbations et des Méthodes Asymptotiques. Dunod, Paris (1972)

    MATH  Google Scholar 

  11. Maslov, V., Fedoriuk, M.V.: Semiclassical Approximations in Quantum Mechanics. Reidel, Boston (1972)

    Google Scholar 

  12. Landau, L.D., Lifshitz, M.E.: Quantum Mechanics: Nonrelativistic Theory. Pergamon Press, New York (1965)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  14. Heller, E.J.: Phys. Rev. Lett. 53, 1515 (1984)

    Article  MathSciNet  Google Scholar 

  15. Heller, E.J.: In: Giannoni, M.J., Voros, A., Zinn-Justin, J. (eds.) Chaos and Quantum Physics. Elsevier, Amsterdam (1991)

    Google Scholar 

  16. Kaplan, L., Heller, E.J.: Ann. Phys. 264, 171 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bogomolny, E.B.: Physica D 31, 169 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  18. Berry, M.V.: Proc. R. Soc. Lon. A 243, 219 (1989)

    Article  Google Scholar 

  19. Tomsovic, S., Heller, E.J.: Phys. Rev. Lett. 70, 1405 (1993)

    Article  Google Scholar 

  20. Tomsovic, S., Lefebvre, J.H.: Phys. Rev. Lett. 79, 3629 (1997)

    Article  Google Scholar 

  21. Wisniacki, D.A., Vergini, E., Benito, R.M., Borondo, F.: Phys. Rev. E 70, 035202(R) (2004)

    Article  Google Scholar 

  22. Wisniacki, D.A., Vergini, E., Benito, R.M., Borondo, F.: Phys. Rev. Lett. 94, 054101 (2005)

    Article  Google Scholar 

  23. Wisniacki, D.A., Vergini, E., Benito, R.M., Borondo, F.: Phys. Rev. Lett. 97, 094101 (2006)

    Article  Google Scholar 

  24. Keating, J.P., Prado, S.D.: Proc. R. Soc. Lon. A 457, 1855 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wisniacki, D., Carlo, G.G.: Phys. Rev. E 77, 045201(R) (2008)

    Article  Google Scholar 

  26. Novaes, M., Pedrosa, J.M., Wisniacki, D., Carlo, G.G., Keating, J.P.: Phys. Rev. E 80, 035202(R) (2009)

    Article  MathSciNet  Google Scholar 

  27. Xu, H., Huang, L., Lai, Y.-C., Grebogi, C.: Phys. Rev. Lett. 110, 064102 (2013)

    Article  Google Scholar 

  28. Stöckman, H.J.: Quantum Chaos: An Introduction. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  29. Sridhar, S.: Phys. Rev. Lett. 67, 785 (1991)

    Article  Google Scholar 

  30. Stein, J., Stöckmann, H.: Phys. Rev. Lett. 68, 2867 (1991)

    Article  Google Scholar 

  31. Wilkinson, P.B., et al.: Nature (London) 380, 608 (1996)

    Article  Google Scholar 

  32. Akis, R., Ferry, D.K., Bird, J.P.: Phys. Rev. Lett. 79, 123 (1997)

    Article  Google Scholar 

  33. Nöckel, J.U., Stone, A.D.: Nature (London) 385, 45 (1997)

    Article  Google Scholar 

  34. Gmachl, C., Capasso, F., Narimanov, E.E., Nöckel, J.U., Stone, A.D., Faist, J., Sivco, D.L., Cho, A.Y.: Science 280, 1556 (1998)

    Article  Google Scholar 

  35. Lee, S.-B., Lee, J.-H., Chang, J.-S., Moon, H.-J., Kim, S.W., An, K.: Phys. Rev. Lett. 88, 033903 (2002)

    Article  Google Scholar 

  36. Harayama, T., Fukushima, T., Davis, P., Vaccaro, P.O., Miyasaka, T., Nishimura, T., Aida, T.: Phys. Rev. E 67, 015207(R) (2003)

    Article  Google Scholar 

  37. Song, Q.H., Ge, L., Stone, A.D., Cao, H., Wiersig, J., Shim, J.-B., Unterhinninghofen, J., Fang, W., Solomon, G.S.: Phys. Rev. Lett. 105, 103902 (2010)

    Article  Google Scholar 

  38. Wiersig, J., Eberspächer, A., Shim, J.-B., Ryu, J.-W., Shinohara, S., Hentschel, M., Schomerus, H.: Phys. Rev. A 84, 023845 (2011)

    Article  Google Scholar 

  39. Doya, V., Legrand, O., Mortessagne, F., Miniatura, C.: Phys. Rev. Lett. 88, 014102 (2002)

    Article  Google Scholar 

  40. Michel, C., Doya, V., Legrand, O., Mortessagne, F.: Phys. Rev. Lett. 99, 224101 (2007)

    Article  Google Scholar 

  41. Huang, L., Lai, Y.-C., Ferry, D.K., Goodnick, S.M., Akis, R.: Phys. Rev. Lett. 103, 054101 (2009)

    Article  Google Scholar 

  42. Revuelta, F., Benito, R.M., Borondo, F., Vergini, E.G.: Phys. Rev. E 87, 042921 (2013)

    Article  Google Scholar 

  43. Borondo, F., Benito, R.M.: In: Yurtsever, E. (ed.) Frontiers of Chemical Physics. NATO ASI Series C. Kluwer, Dordrecht (1995) (references therein)

    Google Scholar 

  44. Esser, R., Tennyson, J., Wormer, P.E.S.: Chem. Phys. Lett. 89, 223 (1982)

    Article  Google Scholar 

  45. Benito, R.M., Borondo, F., Kim, J.-H., Sumpter, B.G., Ezra, G.S.: Chem. Phys. Lett. 161, 60 (1989)

    Article  Google Scholar 

  46. Losada, J.C., Estebaranz, J.M., Benito, R.M., Borondo, F.: J. Chem. Phys. 108, 63 (1998)

    Article  Google Scholar 

  47. Borondo, F., Losada, J.C., Benito, R.M.: Found. Phys. 31, 147 (2001)

    Article  Google Scholar 

  48. Losada, J.C., Benito, R.M., Arranz, F.J., Borondo, F.: Int. J. Quantum Chem. 86, 167 (2002)

    Article  Google Scholar 

  49. Bačić, Z., Light, J.C.: Annu. Rev. Phys. Chem. 40, 469 (1989)

    Article  Google Scholar 

  50. Arranz, F.J., Borondo, F., Benito, R.M.: J. Chem. Phys. 104, 6401 (1996)

    Article  Google Scholar 

  51. Arranz, F.J., Borondo, F., Benito, R.M.: Phys. Rev. Lett. 80, 944 (1998)

    Article  Google Scholar 

  52. Arranz, F.J., Borondo, F., Benito, R.M.: J. Chem. Phys. 107, 2395 (1997)

    Article  Google Scholar 

  53. de Polavieja, G.G., Borondo, F., Benito, R.M.: Phys. Rev. Lett. 73, 1613 (1994)

    Article  Google Scholar 

  54. Sibert III, E.L., Vergini, E., Benito, R.M., Borondo, F.: New J. Phys. 10, 053016 (2008)

    Article  Google Scholar 

  55. Revuelta, F., Vergini, E.G., Benito, R.M., Borondo, F.: Phys. Rev. E 85, 026214 (1012)

    Article  Google Scholar 

  56. Borondo, F., Zembekov, A.A., Benito, R.M.: Chem. Phys. Lett. 246, 421 (1995)

    Article  Google Scholar 

  57. Borondo, F., Zembekov, A.A., Benito, R.M.: J. Chem. Phys. 106, 5068 (1996)

    Article  Google Scholar 

  58. Borondo, F., Zembekov, A.A., Benito, R.M.: J. Chem. Phys. 107, 7934 (1997)

    Article  Google Scholar 

  59. Vergini, E.G., Sibert III, E.L., Revuelta, F., Benito, R.M., Borondo, F.: Europhys. Lett. 89, 40013 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Química and Instituto de Ciencias Matemáticas (ICMAT), Universidad Autónoma de Madrid, CANTOBLANCO, 28049, Madrid, Spain

    F. Borondo

Authors
  1. F. Borondo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Borondo .

Editor information

Editors and Affiliations

  1. University of Tübingen, Tübingen, Germany

    Ali Sanayei

  2. Department of Computer Science, VŠB-Technical University of Ostrava, Ostrava, Czech Republic

    Ivan Zelinka

  3. Institute of Physical and Theoretical Chemistry, University of Tübingen, Tübingen, Germany

    Otto E. Rössler

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Borondo, F. (2014). Classical Invariants in the Quantum Mechanics of Chaotic Systems. In: Sanayei, A., Zelinka, I., Rössler, O. (eds) ISCS 2013: Interdisciplinary Symposium on Complex Systems. Emergence, Complexity and Computation, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45438-7_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-45438-7_12

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45437-0

  • Online ISBN: 978-3-642-45438-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation