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Semiclassical Roles for Classical Orbits

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Quantum Signatures of Chaos

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Abstract

The precursor of quantum mechanics due to Bohr and Sommerfeld was already seen by Einstein [1] 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 50 years in Gutzwiller’s pioneering work.

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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. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay: Chaos: Classical and Quantum, ChaosBook.org (Niels Bohr Institute, Copenhagen 2005)

    Google Scholar 

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

    Google Scholar 

  6. S. Müller, S. Heusler, A. Altland, P. Braun, F. Haake: New J. Phys. 11, 103025 (2009), arXiv:0906.1960v2

    Article  Google Scholar 

  7. K. Richter, M. Sieber: Phys. Rev. Lett. 89, 206801 (2002)

    Article  ADS  Google Scholar 

  8. H. Schanz, M. Puhlmann, T. Geisel: Phys. Rev. Lett. 91, 134101 (2003)

    Article  ADS  Google Scholar 

  9. O. Zaitsev, D. Frustaglia, K. Richter: Phys. Rev. Lett. 94, 026809 (2005)

    Article  ADS  Google Scholar 

  10. R.S. Whitney, P. Jacquod: Phys. Rev. Lett. 94, 116801 (2005); ibid. 96, 206804 (2006)

    Google Scholar 

  11. S. Heusler, S. Müller, P. Braun, F. Haake: Phys. Rev. Lett 96, 066804 (2006)

    Article  ADS  Google Scholar 

  12. P. Braun, S. Heusler, S. Müller, F. Haake: J. Phys. A 39, L159 (2006)

    ADS  Google Scholar 

  13. S. Müller, S. Heusler, P. Braun, F. Haake: New J. Phys. 9 (2007) 12

    Google Scholar 

  14. P. Brouwer, S. Rahav: Phys. Rev. Lett. 96 196804 (2006)

    Article  ADS  Google Scholar 

  15. P. Brouwer, A. Altland: Phys. Rev. B78, 075304 (2008)

    ADS  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  20. 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 

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

    MATH  Google Scholar 

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

    Google Scholar 

  23. J.P. Keating: Nonlinearity 4, 309 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    Article  MATH  Google Scholar 

  29. 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 (1997)

    Google Scholar 

  30. 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)

    Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  33. 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 

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

    MathSciNet  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  39. E. Doron, U. Smilansky: Nonlinearity 5, 1055 (1995); H. Schanz, U. Smilansky: Chaos, Solitons & 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 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  47. J.B. Conrey, D.W. Farmer, M.R. Zirnbauer: arXiv:math-ph0511024v2 (9 Sep 2007)

    Google Scholar 

  48. A. Huckleberry, A. Püttmann, M.R.AZirnbauer: arXiv:07091215v1 [math-ph] (9 Sep 2007)

    Google Scholar 

  49. S. Müller, S. Heusler, P. Braun, F. Haake, A. Altland: Phys. Rev. Lett. 93, 014103 (2004).

    Article  ADS  Google Scholar 

  50. S. Müller, S. Heusler, P. Braun, F. Haake, A. Altland: Phys. Rev. E 72, 046207 (2005)

    ADS  Google Scholar 

  51. S. Heusler, S. Müller, A. Altland, P. Braun, F. Haake: Phys. Rev. Lett. 98, 044103 (2007)

    Article  ADS  Google Scholar 

  52. M. Turek, D. Spehner, S. Müller, K. Richter: Phys. Rev. E 71, 016210 (2005)

    ADS  Google Scholar 

  53. J.P. Keating, S. Müller: Proc. Roy. Soc. A463, 3241 (2007)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  55. M. Sieber, K. Richter: Phys. Scr. T90, 128 (2001); M. Sieber: J. Phys. A35, L613 (2002)

    Google Scholar 

  56. A.D. Sokal: J. Math. Phys. 21, 261 (1980)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  ADS  Google Scholar 

  58. J. Bolte, S. Keppeler: J. Phys. A32, 8863 (1999)

    MathSciNet  ADS  Google Scholar 

  59. S. Keppeler: Spinning Particles –– Semiclassics and Spectral Statistics (Springer, Berlin, 2003)

    MATH  Google Scholar 

  60. K. Saito K, T. Nagao: Phys. Lett. A 352 380 (2006)

    Article  ADS  Google Scholar 

  61. T. Nagao, P. Braun, S. Müller, K. Saito, S. Heusler S, F. Haake: J. Phys. A: Math. Theor. 40 47(2007)

    Article  MATH  ADS  Google Scholar 

  62. J. Kuipers, M. Sieber J. Phys. A: Math. Theor. 40 935 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  63. T. Nagao, K. Saito: J. Phys. A 40, 12055 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  64. K. Saito, T. Nagao, P. Braun, S. Müller S: arXiv:0906.2033[nlin.CD] 11 Jun 2009

    Google Scholar 

  65. T. Nagao, S. Müller: J. Phys. A 42, 375102 (2009)

    Article  MathSciNet  Google Scholar 

  66. E.B. Bogomolny, B. Georgeot, M.J. Giannoni, C. Schmit: Phys. Rev. Lett. 69 1477 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  67. P. Braun, F. Haake: arXiv:1001.3339v2 [nlin.CD] (2010)

    Google Scholar 

  68. P. Brouwer, S. Rahav: Phys. Rev. Lett. 96 196804 (2006)

    Article  ADS  Google Scholar 

  69. R.S. Whitney, P. Jacquod: Phys. Rev. Lett. 96 206804 –(2006)

    Article  ADS  Google Scholar 

  70. G. Berkolaiko, J.M. Harrison, M. Novaes: J. Phys. A: Math. 41 365102 (2008)

    Article  MathSciNet  Google Scholar 

  71. J. Müller, A. Altland: J. Phys. A 38, 3097 (2005); J. Müller, T. Micklitz, A. Altland: arXiv:0707.2274v1 [nlin.CD] (2007)

    Google Scholar 

  72. J. Kuipers, C. Petitjean, D. Waldner, K. Richter: arXiv:0907.2660v1[cond-mat.mes-hall]

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

  88. C. Manderfeld, H. Schomerus: Phys. Rev. E63, 066208 (2001)

    ADS  Google Scholar 

  89. M.V. Berry, J.P. Keating, H. Schomerus: Proc. R. Soc. A456, 1659 (2000)

    MathSciNet  ADS  Google Scholar 

  90. 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 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  94. C.W. Gardiner, P. Zoller: Quantum Noise (Springer, Berlin, 2004)

    MATH  Google Scholar 

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

    Google Scholar 

  96. 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)

    Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Google Scholar 

  99. 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 

  100. 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. Capasso, 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 

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

    Article  ADS  Google Scholar 

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

    ADS  Google Scholar 

  103. M. Fließer, A. Csordás, 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 

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Haake, F. (2010). Semiclassical Roles for Classical Orbits. In: Quantum Signatures of Chaos. Springer Series in Synergetics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-05428-0_10

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