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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A. Einstein: Verh. Dt. Phys. Ges. 19, 82 (1917)
M.C. Gutzwiller: Chaos in Classical and Quantum Mechanics ( Springer, New York, 1990 )
M.C. Gutzwiller: J. Math. Phys. 12, 343 (1971)
P. Cvitanovic: Classical and Quantum Chaos. Cyclist Treatise www.nbi.dk/ChaosBook/
M. Brack, R.K. Bhaduri: Semiclassical Physics ( Addison-Wesley, Reading, Mass., 1997 )
L.S. Schulman: Techniques and Applications of Path Integration ( Wiley, New York, 1981 )
G. Junker, H. Leschke: Physica 56D, 135 (1992)
M. Tabor: Physica 6D, 195 (1983)
J.P. Keating: Nonlinearity 4, 309 (1991) (Addison-Wesley, Reading, Mass., 1950; 2nd ed. 1980 )
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 )
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)
M. Morse: Variational Analysis ( Wiley, New York, 1973 )
E.T. Copson: Asymptotic Expansions ( Cambridge University Press, Cambridge, 1965 )
H.P. Baltes, E.R. Hilf: Spectra of Finite Systems ( BI Wissenschaftsverlag, Mannheim, 1976 )
R. Balian, F. Bloch: Ann. Phys. (USA) 63, 592 (1971)
M.V. Berry, C.J. Howls: Proc. Roy. Soc. Lond. A447, 527 (1994)
K. Stewartson, R.T. Waechter: Proc. Camb. Phil. Soc. 69, 353 (1971)
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)
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)
T. Prosen: Phys. Lett. A223, 323 (1997)
N.L. Balazs, A. Voros: Phys. Rep. 143, 109 (1986)
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
M.V. Berry, J.P. Keating: J. Phys. A23, 4839 (1990)
J.P. Keating: Proc. R. Soc. Lond. A436, 99 (1992)
M.V. Berry, J.P. Keating: Proc. R. Soc. Lond. A437, 151 (1992)
B. Eckhardt, E. Aurell: Europhys. Lett. 9, 509 (1989)
E.B. Bogomolny: Nonlinearity 5, 805 (1992)
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)
E.G. Vergini: J. Phys. A33, 4709 (2000); E.G. Vergini, G.G. Carlo: J. Phys. A33, 4717 (2000)
R.G. Littlejohn: J. Math. Phys. 31, 2952 (1990); J. Stat. Phys. 68, 7 (1992)
I.C. Percival: Adv. Chem. Phys. 36, 1 (1977)
S.C. Creagh, J.M. Robbins, R.G. Littlejohn: Phys. Rev. A42, 1907 (1990)
J.N. Mather: Comm. Math. Phys. 94, 141 (1984)
J. Robbins: Nonlinearity 4, 343 (1991)
J. Moser: Ergod. Theor.Dynam. Sys. 6, 401 (1986)
P.A. Boasman, U. Smilansky: J. Phys. A27, 1373 (1994)
J.H. Hannay, A.M. Ozorio de Almeida: J. Phys. A17, 3429 (1984)
M.V. Berry: Proc. R. Soc. London A400, 229 (1985)
S. Fishman, J.P. Keating: J. Phys. A31, L313 (1998)
P. Cvitanovie, B. Eckhardt: J. Phys. A24, L237 (1991); Phys. Rev. Lett. 63, 823 (1989); Nonlinearity 6, 277 (1993)
E.B. Bogomolny, J.P. Keating: Phys. Rev. Lett. 77, 1472 (1996)
O. Agam, B.L. Altshuler, A.V. Andreev: Phys. Rev. Lett. 75, 4389 (1995)
P. Gaspard: Chaos, Scattering and Statistical Mechanics ( Cambridge University Press, Cambridge, 1998 )
E.B. Bogomolny, J.P. Keating: Private communication and to be published
E.B. Bogomolny, F. Leyvraz, C. Schmit• Commun. Math. Phys. 176, 577 (1996); W. Luo, P. Sarnac: Commun. Math. Phys. 161, 419 (1994)
E. Bogomolny, C. Schmit: Nonlinearity 6, 523 (1993)
J.P. Keating, M. Sieber: Proc. R. Soc. London A447, 413 (1994)
R. Aurich, C. Matthies, M. Sieber, F. Steiner: Phys. Rev. Lett. 68, 1629 (1992)
A.V. Andreev, B.L. Altshuler: Phys. Rev. Lett. 75, 902 (1995)
M. Abramowitz, I. Stegun: Handbook of Mathematical Functions ( Dover, New York 1965 )
A.J. Lichtenberg, M.A. Liebermann: Regular and Chaotic Dynamics, Applied Mathematical Sciences, Vol. 38, 2nd ed. ( Springer, New York, 1992 )
M.V. Berry, M. Robnik: J. Phys. A17, 2413 (1984)
K.R. Meyer: Trans. Am. Math. Soc. 149, 95 (1970)
H. Schomerus: J. Phys. A31, 4167 (1998)
A.M. Ozorio de Almeida, J.H. Hannay: J. Phys. A20, 5873 (1987)
S. Tomsovic, M. Grinberg, D. Ullmo: Phys. Rev. Lett. 75, 4346 (1995)
D. Ullmo, M. Grinberg, S. Tomsovic: Phys. Rev. E54, 136 (1996)
M. Sieber: J. Phys. A29, 4715 (1996)
H. Schomerus, M. Sieber: J. Phys. A30, 4537 (1997)
H. Schomerus: Europhys. Lett. 38, 423 (1997)
M. Sieber, H. Schomerus: J. Phys. A31, 165 (1998)
T. Poston, I.N. Stewart: Catastrophe Theory and its Applications ( Pitman, London, 1978 )
M. Kus, F. Haake, D. Delande: Phys. Rev. Lett. 71, 2167 (1993)
H. Schomerus, F. Haake: Phys. Rev. Lett. 79, 1022 (1997)
M.V. Berry, J.P. Keating, S.D. Prado: J. Phys. A31, L243 (1998)
C. Manderfeld, H. Schomerus: chao-dyn/9911024
M.V. Berry, J.P. Keating, H. Schomerus: submitted to Proc. R. Soc. Lond. (Jan. 2000)
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 )
M. Khodas, S. Fishman: Phys. Rev. Lett. 84 2837 (2000), chaodyn 9910040
J. Weber, F. Haake, P. Seba: Phys. Rev. Lett. 85 3620 (2000), and chaodyn 0001013
C.W. Gardiner: Quanntum Noise ( Springer, Berlin, 1991 )
C. Manderfeld, J. Weber, F. Haake, P.A. Braun: to be published
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)
R. Fleischmann, T. Geisel, R. Ketzmerick: Phys. Rev. Lett. 68, 1367 (1992)
G. Hackenbroich, F. von Oppen: Europhys. Lett. 29, 151 (1995); Z. Physik B97, 157 (1995); K. Richter: Europhys. Lett. 29, 7 (1995)
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)
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)
M. Brack, S.M. Reimann, M. Sieber: Phys. Rev. Lett. 79, 1817 (1997)
J. Main, G. Wunner: Phys. Rev. A55, 1743 (1997)
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)
Author information
Authors and Affiliations
Rights 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