Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics

Jean Bellissard; Michel Vittot

Annales de l'I.H.P. Physique théorique (1990)

  • Volume: 52, Issue: 3, page 175-235
  • ISSN: 0246-0211

How to cite

top

Bellissard, Jean, and Vittot, Michel. "Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics." Annales de l'I.H.P. Physique théorique 52.3 (1990): 175-235. <http://eudml.org/doc/76483>.

@article{Bellissard1990,
author = {Bellissard, Jean, Vittot, Michel},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {semiclassical limit in phase space; tangent groupoid; noncommutative geometry; quantized action-angle variables; comparison between perturbation expansions in classical and in quantum mechanics through a Lie formalism using Liouville operators; Birkhoff expansion},
language = {eng},
number = {3},
pages = {175-235},
publisher = {Gauthier-Villars},
title = {Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics},
url = {http://eudml.org/doc/76483},
volume = {52},
year = {1990},
}

TY - JOUR
AU - Bellissard, Jean
AU - Vittot, Michel
TI - Heisenberg's picture and non commutative geometry of the semi classical limit in quantum mechanics
JO - Annales de l'I.H.P. Physique théorique
PY - 1990
PB - Gauthier-Villars
VL - 52
IS - 3
SP - 175
EP - 235
LA - eng
KW - semiclassical limit in phase space; tangent groupoid; noncommutative geometry; quantized action-angle variables; comparison between perturbation expansions in classical and in quantum mechanics through a Lie formalism using Liouville operators; Birkhoff expansion
UR - http://eudml.org/doc/76483
ER -

References

top
  1. [1] N. Ashcroft and D. Mermin, Solid state physics, Saunders, Philadelphia, Tokyo, 1976. Zbl1107.82300
  2. [2] J. Avron and R. Seiler, Quantization of the Hall Conductance for General Multiparticule Schrödinger Hamiltonians, Phys. Rev. Lett., Vol. 54, 1985, pp. 259-262. MR773033
  3. [3] R. Balian and C. Bloch, Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: I-Three-Dimensional Problem with Smooth Boundary Surface, Ann. Phys., Vol. 60, 1970, pp. 401-447. Zbl0207.40202MR270008
  4. [4] R. Balian and C. Bloch, Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: II-Electromagnetic Field, Riemannian Spaces, Ann. Phys., Vol. 64, 1971, pp. 271-307. Zbl0218.35071MR284729
  5. [5] R. Balian and C. Bloch, Distribution of Eigenfrequencies for the Wave Equation in a Finite Domain: III-Eigenfrequency Density Oscillations, Ann. Phys., Vol. 69, 1972, pp. 76-160. Zbl0226.35070MR289962
  6. [6] R. Balian and C. Bloch, Solution of the Schrödinger Equation in Terms of Classical Paths, Ann. Phys., Vol. 85, 1974, pp. 514-545. Zbl0281.35029MR438937
  7. [7] J. Bayfield and P. Koch, Multiphotonic Ionization of Highly Excited Hydrogen Atoms, Phys. Rev., Vol. 33, 1974, p. 258. 
  8. [8] J. Bayfield, Experiment and Theory for the Classically Chaotic Motion of the Driven Bound Electron, in Non linear evolution and chaotic phenomena, G. GALLAVOTTI and P. F. ZWEIFEL Eds., Plenum, New York, 1988. MR1109986
  9. [9] J. Bayfield and D.W. Sokol, Excited Atoms in Strong Microwaves: Classical Resonances and Localization in Experimental Final States Distributions, Phys. Rev. Lett., Vol. 61, 1988, p. 2007. 
  10. [10] J. Bayfield, G. Casati, I. Guarneri and D.W. Sokol, Localization of Classically Chaotic Diffusion for Hydrogen Atoms in Microwave Fields, submitted toPhys. Rev. Lett. 
  11. [11] J. Bellissard, Stability and Instability in Quantum Mechanics, in Trends and developments in the eighties, S. ALBEVERIO and P. BLANCHARD Eds., World Scientific, Singapore, 1985. Zbl0584.35024MR853743
  12. [12] J. Bellissard, K-Theory of C*-Algebras in Solid State Physics, in Statistical mechanics and field theory, mathematical aspects, T. C. DORLAS, M. N. HUGENHOLTZ and M. WINNINK Eds., Lect. Notes Phys., Vol. 257, Springer, Berlin, 1986. Zbl0612.46066MR862832
  13. [13] J. Bellissard, Ordinary Quantum Hall Effect and Non Commutative Cohomology, in Bad Schandau conference on localization, W. WELLER and P. ZIESCHE Eds., Teubner, Leipzig, 1988. MR965981
  14. [14] J. Bellissard, C*-Algebras in Solid State Physics: 2D Electrons in a Uniform Magnetic Field, in Operators Algebras and Applications, Vol. II, E. V. EVANS and M. TAKESAKI Eds., Cambridge University Press, Cambridge, 1988. Zbl0677.46055MR996451
  15. [15] J. Bellissard, Almost Periodicity in Solid State Physics and C*-Algebras, in The Harald Bohr Centennary, C. BERG and F. FLUGEDE Eds., Royal Danish Acad. Science, Copenhagen, 1989. Zbl0678.42007MR1031737
  16. [16] M.V. Berry and M. Tabor, Level Clustering in the Regular Spectrum, Proc. R. Soc. London, Vol. A356, 1977, pp. 375-394. Zbl1119.81395
  17. [17] M.V. Berry, Semiclassical Mechanics of Regular and Irregular Motion, in Chaotic Behavior of Deterministic Systems, G. Iooss, R. H. G. HELLEMAN and R. STORA Eds., North-Holland, Amsterdam, 1983. Zbl0571.70018MR724465
  18. [18] M.V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proc. R. Soc.London, Vol. A392, 1984, pp. 45-57. [19] M.V. Berry, Semiclassical Theory of Spectral Rigidity, Proc. R. Soc. London, Vol. A400, 1985, pp. 229-251. Zbl0875.35061MR738926
  19. [20] M.V. Berry and M. Robnik, Statistics of Energy Levels Without Time-Reversal Symmetry, Aharonov-Bohm Chaotic Billards, J. Phys., Vol. A19, 1986, pp. 649-668. MR838450
  20. [21] M.V. Berry, Semiclassical Formula for the Number Variance of the Riemann Zeros, Nonlinearity, Vol. 1, 1988, pp. 399-407. Zbl0664.10022MR955621
  21. [22] G.D. Birkhoff, Dynamical systems, A.M.S. Coll. Pub., Vol. 9, AMS, Providence, Rhode Island, 1927. Zbl53.0732.01JFM53.0732.01
  22. [23] O. Bohigas, R.U. Haq and A. Pandey, Fluctuation Properties of Nuclear Energy Levels and widths: Comparison of Theory with Experiment, in Nuclear data for science and technology, K. H. BÖCKHOFF Ed., ECSC, EEC, EAEC, Brussels and Luxembourg, 1983. 
  23. [24] O. Bohigas and M.J. Giannoni, Chaotic Motion and Random Matrix Theories, in Mathematical and computational methods in nuclear physics, J. S. DEHESA, J. M. G. GOMEZ and A. POLLS Eds., Lect. Notes Phys., Vol. 209, Springer, Berlin, 1984. MR769113
  24. [25] O. Bohigas, M.J. Giannoni and C. Schmit, Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett., Vol. 52, 1984, pp. 1-4. Zbl1119.81326MR730191
  25. [26] O. Bohigas, M.J. Giannoni and C. Schmit, Spectral Fluctuations of Classically Chaotic Quantum Systems, in Quantum chaos and statistical nuclear physics, T. H. SELIGMAN and H. NISHIOKA Eds., Lect. Notes Phys., Vol. 263, Springer, Berlin, 1986. Zbl1119.81326MR870168
  26. [27] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, I, II, Springer, Berlin, 1979-1981. Zbl0421.46048
  27. [28] P. Briet, J.M. Combes and P. Duclos, On the location of resonances for Schrödinger operators in the semiclassical limit: II, Comm. P.D.E., Vol. 12, 1987, pp. 201-222. Zbl0622.47047MR876987
  28. [29] P. Briet, J.M. Combes and P. Duclos, Spectral Stability Under Tunneling, Comm. Math. Phys., 1989 (to appear). Zbl0702.35189MR1027916
  29. [30] L. Brillouin, J. Phys. Radium, Vol. 7, 1926, pp. 353-368. 
  30. [31] T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey and S.S.M. Wong, Random-Matrix Physics: Spectrum and Strenght Fluctuations, Rev. Mod. Phys., Vol. 53, 1981, pp. 385-479. MR619406
  31. [32] G. Casati, B.V. Chirikov, J. Ford and F.M. Izraelev, Stochastic Behavior of a Quantum Pendulum Under a Periodic Perturbation, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, G. CASATI and J. FORD Eds., Lect. Notes Phys., Vol. 93, Springer, Berlin, 1979. Zbl0498.60100MR550908
  32. [33] G. Casati, B.V. Chirikov, I. Guarneri and D.L. Shepelyansky, Dynamical Stability of Quantum Chaotic Motion in a Hydrogen Atom, Phys. Rev. Lett., Vol. 56, 1986, p. 2437. 
  33. [34] G. Casati, B.V. Chirikov, I. Guarneri and D.L. Shepelyansky, New Photoelectric Ionization Peak in the Hydrogen Atom, Phys. Rev. Lett., Vol. 57, 1986, p. 823. 
  34. [35] G. Casati, B.V. Chirikov, I. Guarneri and D.L. Shepelyansky, Localization of Diffusive Excitation in the Two-Dimensional Hydrogen Atom in a Monochromatic Field, Phys. Rev. Lett., Vol. 59, 1987, p. 2927. 
  35. [36] A.L. Cauchy, Résumé des leçons données à l'École Royale Polytechnique sur le calcul infinitésimal, Paris (1823) (repr. in Œuvres complètes IV, Gauthier-Villars, Paris, 1899). 
  36. [37] J. Chazarain, Spectre d'un hamiltonien quantique et mécanique classique, Comm. P.D.E., Vol. 5, 1980, pp. 595-644. Zbl0437.70014MR578047
  37. [38] B.V. Chirikov, A Universal Instability of Many Dimensional Oscillator Systems, Phys. Rep., Vol. 52, 1979, pp. 263. MR536429
  38. [39] Y. Colin De Verdiere, Compos. Math., Vol. 27, 1973, p. 83, Vol. 27, p. 159. Zbl0272.53034MR348798
  39. [40] Y. Colin De Verdiere, Quasi-modes sur les variétés Riemanniennes, Inv. Math., 1977, Vol. 43, pp. 15-52. Zbl0449.53040MR501196
  40. [41] A. Connes, Non Commutative Differential Geometry, Pub. I.H.E.S., 1986, Vol. 62, pp. 43-144. Zbl0592.46056
  41. [42] W.J. De Haas and P.M. Van Alphen, Proc. Acad. Sci. (Amsterdam), Vol. 36, 1933, p. 262. 
  42. [43] J.B. Delos, S.K. Knudson and D.W. Noid, High Rydberg States of an Atom in a Strong Magnetic Field, Phys. Rev. Lett., Vol. 50, 1983, pp. 579-583. 
  43. [44] J. Dixmier, Les C*-algèbres et leurs représentations, Gauthiers-Villars, Paris, 1969. Zbl0174.18601MR246136
  44. [45] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verhandl. Deutsch. Phys. Ges., Vol. 19, 1917, p. 82-92. 
  45. [46] G.A. Elliot, Gaps in the Spectrum of an Almost Periodic Schrödinger Operator, C.R. Acad. Sci., Royal Soc. of Canada, Vol. IV, 1982, p. 255-259. Zbl0516.46048MR675127
  46. [47] S. Fishman, D.R. Grempel and R.E. Prange, Chaos, Quantum Recurrence and Anderson Localization, Phys. Rev. Lett., Vol. 49, 1982, pp. 509-512. MR669169
  47. [48] S. Fishman, D.R. Grempel and R.E. Prange, Quantum Dynamics of a Non Integrable System, Phys. Rev., Vol. A29, 1984, pp. 1639-1647. 
  48. [49] G. Gallavotti, Quasi-Integrable Mechanical Systems, in Critical Phenomena, Random Systems, Gauge Theories, K. OSTERWALDER and R. STORA Eds., North Holland, 1986. Zbl0662.70022MR880535
  49. [50] T. Geisel, G. Radons and J. Rubner, Kolmogorov-Arnol'd-Moser Barriers in the Quantum Dynamics of Chaotic Systems, Phys. Rev. Lett., Vol. 57, 1986, pp. 2883- 2886. 
  50. [51] S. Graffi and T. Paul, The Schrödinger Equation and Canonical Perturbation Theory, Comm. Math. Phys., Vol. 108, 1987, pp. 25-40. Zbl0622.35071MR872139
  51. [52] J.M. Greene, A Method for Determining a Stochastic Transition, J. Math. Phys., Vol. 20, 1979, pp. 1183-1201. 
  52. [53] V. Guillemin and S. Sternberg, The Metaplectic Representation, Weyl Operators and Spectral Theory, in Differential Geometric Methods in Mathematical Physics, P. L. GARCIA, A. PÉREZ-RENDÓN and J. M. SOURIAU Eds., Lect. Notes Math., Vol. 836, Springer, Berlin, 1979. Zbl0457.58018MR607713
  53. [54] M.C. Gutzwiller, Energy spectrum According to Classical Mechanics, J. Math. Phys., Vol. 11, 1970, pp. 1791-1806. 
  54. [55] M.C. Gutzwiller, Periodic Orbits and Classical Quantization Conditions, J. Math. Phys., Vol. 12, 1971, pp. 343-358. 
  55. [56] B. Helffer and J. Sjöstrand, Analyse semi-classique pour l'équation de Harper (avec application à l'étude de l'équation de Schrödinger avec champ magnétique) I, II, III, Preprint Univ. Orsay, Bull. Soc. Math. France, 1988 (submitted). Zbl0714.34130
  56. [57] E.J. Heller, Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits, Phys. Rev. Lett., Vol. 53, 1984, pp. 1515-1518. MR762412
  57. [58] E.J. Heller and R.L. Sundberg, Quantum Ergodicity and Intensity Fluctuations, in Chaotic behavior in quantum systems, G. CASATI, Ed., NATO ASI, Vol. B120, Plenum, New York, 1985. 
  58. [59] C. Jaffe and W.P. Reinhardt, Uniform Semiclassical Quantization of Regular and Chaotic Classical Dynamics on the Hénon-Heiles Surface, J. Chem. Phys., Vol. 77, 1982, pp. 5191-5203. MR681223
  59. [60] Y. Katznelson, An introduction to harmonic analysis, Wiley, New York, 1968. Zbl0169.17902MR248482
  60. [61] J.B. Keller, Corrected Bohr-Sommerfeld Quantum Conditions for Nonseparable Systems, Ann. Phys., Vol. 4, 1958, pp. 180-188. Zbl0085.43103MR99207
  61. [62] H.A. Kramers, Z. Physik, Vol. 39, 1926, pp. 828-840. JFM52.0969.04
  62. [63] H. Kunz, The Quantum Hall Effect for Electron in a Random Potential, Comm. Math. Phys., Vol. 112, 1987, pp. 121-145. Zbl1108.81314MR904141
  63. [64] R.A. Marcus, Aspects of Intramolecular Dynamics in Chemistry, in Chaotic behavior in quantum systems, G. CASATI Ed., NATO ASI, Vol. B120, Plenum, New York, 1985. 
  64. [65] H.P. Mckean, Selberg's Trace Formula as Applied to a Compact Riemann Surface, Comm. Pure Appl. Math., Vol. 25, 1972, pp. 225-246. MR473166
  65. [66] N.N. Nekhoroshev, The Behavior of Hamiltonian Systems that Are Close to Integrable Ones, Funct. Anal. Appl., Vol. 5, 1971, pp. 338-339. Zbl0254.70015MR294813
  66. [67] N.N. Nekhoroshev, Exponential Estimates of the Time of Stability for Nearly Integrable Hamiltonians, Russ. Math. Surveys, Vol. 32, 1977, pp. 1-63. Zbl0389.70028
  67. [68] L. Onsager, Interpretation of the Haas-van Alphen Effect, Phil. Mag., Vol. 43, 1952, pp. 1006-1008. 
  68. [69] G. Pedersen, C*-Algebras and their Automorphisms Groups, Academic, New York, 1979. Zbl0416.46043MR548006
  69. [70] I.C. Percival, Regular and Irregular Spectra, J. Phys., Vol. B6L, 1973, pp. 229-232. 
  70. [71] I.C. Percival, Regular and Irregular Spectra in Molecules, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, G. CASATI and J. FORD Eds., Lect. Notes Phys., Vol. 93, Springer, Berlin, 1979. MR550901
  71. [72] N. Pomphrey, Numerical Identification of Regular and Irregular Spectra, J. Phys., Vol. B7, 1974, pp. 1909-1915. 
  72. [73] R. Rammal, Landau Level Spectrum of Bloch Electron in a Honeycomb Lattice, J. Phys. France, Vol. 46, 1985, pp. 1345-1354. 
  73. [73b] Y.Y. Wang, PANNETIER and R. Rammal, Quasi Classical Approximations for Almost-Mathieu Equation, J. Phys. France, Vol. 48, 1987, pp. 2067-2079. 
  74. [74] J. Renault, A Groupoid Approach to C*-Algebras; Lect. Notes Math., Vol. 793, Springer, Berlin, 1980. Zbl0433.46049MR584266
  75. [75] M. Robnik and M.V. Berry, False Time-Reversal Violation and Energy Level Statistics : the Role of Anti-Unitary Symmetry, J. Phys., Vol. A19, 1986, pp. 669-682. MR838451
  76. [76] A. Selberg, Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces with Applications to Dirichlet Series, J. Indian. Math. Soc., Vol. 20, 1956, pp. 47-87. Zbl0072.08201MR88511
  77. [77] R.B. Shirts and W.P. Reinhardt, Approximate Constants of Motion for Classically Chaotic Vibrational Dynamics: Vague Tori, Semiclassical Quantization, and Classical Intramolecular Energy Flow, J. Chem. Phys., Vol. 77, 1982, pp. 5204-5217. MR681224
  78. [78] J.B. Taylor, Unpublished (1968) (quoted in [52]). 
  79. [79] D.J. Thouless, M. Kohmoto, M.P. Nightingale and M. Den Nijs, Quantized Hall Conductance in a Two Dimensional Periodic Potential, Phys. Rev. Lett., Vol. 49, 1982, pp. 405-408. 
  80. [80] M. Vittot, A Simple and Compact Presentation of Birkhoff Series, in Non Linear Evolution and Chaotic Phenomena, G. GALLAVOTTI and P. F. ZWEIFEL Eds., Plenum, New York, 1987. Zbl0707.70015MR1109977
  81. [81] A. Voros, Asymptotic h-Expansions of Stationnary Quantum States, Ann. I.H.P., Vol. A26, 1977, pp. 343-403. MR456138
  82. [82] M. Wilkinson, Critical Properties of Electrons Eigenstates in Incommensurate Systems, Proc. R. Soc. London, Vol. A391, 1984, pp. 305-350. MR739684
  83. [83] M. Wilkinson, Von Neumann Lattices of Wannier Functions for Bloch Electrons in a Magnetic Field, Proc. R. Soc. London, Vol. A403, 1986, pp. 135-166. MR828687
  84. [84] M. Wilkinson, An Example of Phase Holonomy and WKB Theory, J. Phys., Vol. A17, 1984, pp. 3459-3476. MR772333

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.