Advertisement

Semi-Classical Theory — Path Integrals

  • L. E. Reichl
Part of the Institute for Nonlinear Science book series (INLS)

Abstract

The “Old Quantum Theory” which is based on the Bohr-Sommerfeld quantization condition, provided a means of quantizing a classical mechanical system by quantizing the action variables associated with invariant tori. (For a historical discussion see [Born 1960].) However, it was recognized by Einstein, as early as 1917 [Einstein 1917], that this method could only be used for systems in which trajectories lie on invariant tori. The Bohr-Sommerfeld quantization condition could not be used to quantize chaotic systems and until recently no method existed by which to connect classically chaotic systems with their quantum counterpart.

Keywords

Periodic Orbit Chaotic System Path Integral Trace Formula Conjugate Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balazs, N.L. and Voros, A. (1986): Phys. Rept. 143 109.MathSciNetADSCrossRefGoogle Scholar
  2. Balazs, N.L., Schmidt, C., and Voros, A. (1987): J. Stat. Phys. 46 1067.ADSMATHCrossRefGoogle Scholar
  3. Balazs, N.L. and Voros, A. (1989): Ann. Phys. 1901.Google Scholar
  4. Balazs, N.L. and Jennings, B.K. (1984): Phys. Rep. 104 347.MathSciNetADSCrossRefGoogle Scholar
  5. Balian, R. and Block, C. (1972): Ann. Phys. 69 76.ADSMATHCrossRefGoogle Scholar
  6. Balian R. and Block, C. (1974): Ann. Phys. 85 514.ADSMATHCrossRefGoogle Scholar
  7. Berry, M.V. (1985): Proc. Roy. Soc. Lond. A400 229.ADSMATHCrossRefGoogle Scholar
  8. Berry, M.V. (1989): Proc. Roy. Soc. Lond. A423 219.ADSCrossRefGoogle Scholar
  9. Berry, M.V. and Mount, K.E. (1972): Rept. Prog. Phys. 35 315.ADSCrossRefGoogle Scholar
  10. Berry, M.V. and Tabor, M. (1977): J. Phys. A: Math. Gen. 10 371.ADSCrossRefGoogle Scholar
  11. Blumel, R. and Smilansky, U. (1988): Phys. Rev. Lett. 60 477.ADSCrossRefGoogle Scholar
  12. Bleher, S., Ott, E., and Grebogi, C. (1989): Phys. Rev. Lett. 63 919.ADSCrossRefGoogle Scholar
  13. Bogomolny, E.B. (1988): Physica D31 169.MathSciNetMATHGoogle Scholar
  14. Born, M. (1960): The Mechanics of the Atom (Frederick Ungar Pub. Co., New York)Google Scholar
  15. Choquard, Ph. (1955): Helv. Phys. Acta 28 89.MathSciNetMATHGoogle Scholar
  16. Delande, D. and Gay, J.C. (1986a): Phys. Rev. Lett. 57 2006.ADSCrossRefGoogle Scholar
  17. Delande, D. and Gay, J.C. (1986a): J. Phys. B19 L173.ADSGoogle Scholar
  18. Devaney, R.L. (1978a): J. Diff. Equa. 29 253.MathSciNetMATHCrossRefGoogle Scholar
  19. Devaney, R.L. (1978b): Inventiones Math 45 221.MathSciNetADSMATHCrossRefGoogle Scholar
  20. Du, M.L. and Delos, J.B. (1987): Phys. Rev. Lett. 58 1731.ADSCrossRefGoogle Scholar
  21. Du, M.L. and Delos, J.B. (1988a): Phys. Rev. A38 1896.ADSCrossRefGoogle Scholar
  22. Du, M.L. and Delos, J.B. (1988b): Phys. Rev. A38 1913.ADSCrossRefGoogle Scholar
  23. Eckhardt, B. and Wintgen, D. (1990): J. Phys. B23 355.ADSGoogle Scholar
  24. Eckhardt, B. (1987): J. Phys. A20 5971.MathSciNetADSGoogle Scholar
  25. Einstein, A. (1917): Verh. Dtsch. Phys. Ges. 19 82.Google Scholar
  26. Faulkner, R.A. (1969): Phys. Rev. 184 713.ADSCrossRefGoogle Scholar
  27. Feingold, M., Littlejohn, R.G., Solina, S.B., Pehling, J.S., and Piro, O. (1990): Phys. Lett. A146 199.MathSciNetCrossRefGoogle Scholar
  28. Feynman, R.P. and Hibbs, A.R. (1965): Quantum Mechanics and Path Integrals (McGraw-Hill Book Co., New York)MATHGoogle Scholar
  29. Friedrich, H. and Wintgen, D. (1989): Phys. Rept. 183 37.MathSciNetADSCrossRefGoogle Scholar
  30. Gaspard, P. and Rice, S.A. (1989): J. Chem. Phys. 90 2225, 2242, 2255.MathSciNetADSCrossRefGoogle Scholar
  31. Gelfand, I.M. and Yaglom, A.M. (1960): J. Math. Phys. 1 48.ADSCrossRefGoogle Scholar
  32. Gutzwiller, M.C. (1967): J. Math. Phys. 8 1979.ADSCrossRefGoogle Scholar
  33. Gutzwiller, M.C. (1970): J. Math. Phys. 11 1791.ADSCrossRefGoogle Scholar
  34. Gutzwiller, M.C. (1971): J. Math. Phys. 12 343.ADSCrossRefGoogle Scholar
  35. Gutzwiller, M.C. (1973): J. Math. Phys. 14 139.MathSciNetADSCrossRefGoogle Scholar
  36. Gutzwiller, M.C. (1977): J. Math. Phys. 18 806.MathSciNetADSCrossRefGoogle Scholar
  37. Gutzwiller, M.C. (1980): Phys. Rev. Lett. 45 150.ADSCrossRefGoogle Scholar
  38. Gutzwiller, M.C. (1982): Physica 5D 183.MathSciNetMATHGoogle Scholar
  39. Gutzwiller, M.C. (1985): Physica Scripta T9 184.MathSciNetCrossRefGoogle Scholar
  40. Gutzwiller, M.C. (1990): Chaos in Classical and Quantum Mechanics (Springer-Verlag, Berlin).MATHGoogle Scholar
  41. Hannay, J.H. and Ozorio de Almeida, A.M. (1984): J. Phys. A: Math. Gen. 17 3429.ADSCrossRefGoogle Scholar
  42. Hasagawa, H., Robnik, M., and Wunner, G. (1989): Prog. Theor. Phys. Suppl. 98 198.ADSCrossRefGoogle Scholar
  43. Hao, B. (1989): Elementary Symbolic Dynamics (World Scientific, Singapore).MATHGoogle Scholar
  44. Heller, E.J. (1984): Phys. Rev. Lett. 53 1515.MathSciNetADSCrossRefGoogle Scholar
  45. Heller, E.J. (1986): in Quantum Chaos and Statistical Nuclear Physics (Lecture Notes in Physics 263) edited by T.H. Seligman and H. Nishioka (Springer-Verlag, Berlin).Google Scholar
  46. Husimi, K. (1940): Proc. Phys. Math. Soc. Japan. 22 264.Google Scholar
  47. Jung, C. and Scholz, H-J. (1987): J. Phys. A20 3607.MathSciNetADSMATHGoogle Scholar
  48. Littlejohn, R. (1987): Unpublished Lecture Notes.Google Scholar
  49. Littlejohn, R. (1991): in Quantum Chaos, edited by H.A. Cerdeira, M.C. Gutzwiller, R. Ramaswamy, G. Casati. (World Scientific Pub. Co., Singapore).Google Scholar
  50. Louisell, W.H. (1973): Quantum Statistical Properties of Radiation (Wiley-Interscience, New York).Google Scholar
  51. McDonald, S.W. (1983): Lawrence Berkley Lab. Report LBL-14837.Google Scholar
  52. McKean, H.P. (1972): Comm. Pure Appl. Math 25 225.MathSciNetCrossRefGoogle Scholar
  53. Merzbacher, E. (1970): Quantum Mechanics (John Wiley and Sons, New York).Google Scholar
  54. Montroll, E.W. (1952): Comm. Pure and Appl. Math. 5 415.MathSciNetMATHCrossRefGoogle Scholar
  55. Morette, C. (1951): Phys. Rev. 81 848.MathSciNetADSMATHCrossRefGoogle Scholar
  56. Papadopoulos, G.J. (1975): Phys. Rev. D11 2870.ADSGoogle Scholar
  57. Richens, P.J., and Berry, M.V. (1981): Physica D2 495.MathSciNetMATHGoogle Scholar
  58. Saraceno, M. (1990): Ann. Phys. 199 37.MathSciNetADSMATHCrossRefGoogle Scholar
  59. Schulman, L.S. (1981): Techniques and Applications of Path Integration (Wiley-Interscience, New York)MATHGoogle Scholar
  60. Selberg, A. (1956): J. Indian Math. Soc. 20 47.MathSciNetMATHGoogle Scholar
  61. Taylor, R.D. and Brumer, P. (1983): Faraday Discuss.Chem.Soc. 75 170.CrossRefGoogle Scholar
  62. Van Vleck, J.H. (1928): Proc. Natl. Acad. Sci. (USA) 14 178.ADSMATHCrossRefGoogle Scholar
  63. Wunner, G. and Ruder, H. (1987): Physica Scripta 36 291.ADSCrossRefGoogle Scholar
  64. Wintgen, D. and Friedrich, H. (1986): J. Phys. B: At. Mol. Phys. 19 991.ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • L. E. Reichl
    • 1
  1. 1.Center for Statistical Mechanics and Complex Systems, Department of PhysicsUniversity of Texas at AustinAustinUSA

Personalised recommendations