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1988

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July

  

Reviews of topical problems


The quantum chaos problem

This review describes the present state of the quantum chaos problem: The problem of determining the specific properties of autonomous and nonautonomous quantum systems with few degrees of freedom whose classical analogs have an unstable (stochastic) motion and also that of determining the relationships between these properties and the characteristics of the classical stochastic situation. The criteria for quantum chaos which have been established to date are examined and compared for autonomous systems. These criteria make use of properties of the energy spectrum, the wave functions in various representations, the matrices of operators other than the Hamiltonian, and particular features of the evolution of time-varying states (wave packets) in such systems. For nonautonomous systems, the conditions for the applicability of classical dynamics for describing observables and certain specific quantum-mechanical effects (tunneling through a separatrix and a global quantum resonance) are analyzed.

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Fulltext is also available at DOI: 10.1070/PU1988v031n07ABEH003572
PACS: 05.45.Mt, 03.65.Fd, 03.65.Sq, 03.65.Xp (all)
DOI: 10.1070/PU1988v031n07ABEH003572
URL: https://ufn.ru/en/articles/1988/7/b/
Citation: Elyutin P V "The quantum chaos problem" Sov. Phys. Usp. 31 597–622 (1988)
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Оригинал: Елютин П В «Проблема квантового хаоса» УФН 155 397–442 (1988); DOI: 10.3367/UFNr.0155.198807b.0397

References (235) ↓ Cited by (58) Similar articles (20)

  1. Wigner E P Phys. Rev. 40 749 (1932)
  2. Bocchieri P, Loinger A Phys. Rev. 107 337 (1957)
  3. Anderson P W Phys. Rev. 109 1492 (1958)
  4. Mott N F, Twose M D Adv. Phys. 10 107 (1961)
  5. Percival I C J. Math. Phys. 2 235 (1961)
  6. Thiele, Wilson D J J. Chem. Phys. 35 1256 (1961)
  7. Sinai Ya G Dokl. Akad. Nauk SSSR 153 1261 (1963); Sinai Ya G Sov. Phys. Dokl. 8 943 (1963)
  8. Henon M, Heiles C Astron. J. 69 73 (1964)
  9. Landau L D, Lifshits E M Mekhanika (M.: Nauka, 1965)
  10. Prais D Nauka o Nauke (Per. s angl. pod red. V N Stoletova) (M.: Progress, 1966) p. 236
  11. Lakatos I Dokazatel’stva i Oproverzheniya (M.: Nauka, 1967)
  12. Mehta M L Random Matrices And The Statistical Theory Of Energy Levels (New York: Academic Press, 1967)
  13. Lloyd P J. Phys. C 2 1717 (1969)
  14. Sinai Ya G Usp. Mat. Nauk 25 141 (1970); Sinai Ya G Russ. Math. Surv. 25 (2) 137 (1970)
  15. Bell R J, Dean P Disc. Faraday Soc. 50 55 (1970)
  16. Baz’ A I, Zel’dovich Ya B, Perelomov A M Rasseyanie, Reaktsii i Raspady v Nerelyativistskoi Kvantovoi Mekhanike (M.: Nauka, 1971), Gl. VI
  17. Pukhov N M, Chernavskii D S Teor. Mat. Fiz. 7 219 (1971); Pukhov N M, Chernavskii D S Theor. Math. Phys. 7 487 (1971)
  18. Thouless D J J. Phys. C 5 77 (1972)
  19. Zaslavskii G M, Filonenko N N Zh. Eksp. Teor. Fiz. 65 643 (1973); Zaslavskii G M, Filonenko N N Sov. Phys. JETP 38 317 (1974)
  20. Brody T A Lett. Nuovo Cimento 7 482 (1973)
  21. Percival I C J. Phys. B 6 L229 (1973)
  22. Bunimovich L A Funkts. Analiz Ego Prilozheniya 8 (3) 73 (1974); Bunimovich L A Funct. Analys. Appl. 8 254 (1974)
  23. Eastes W, Marcus R A J. Chem. Phys. 61 4301 (1974)
  24. Nordholm K S J, Rice S A J. Chem. Phys. 61 203 (1974); Nordholm K S J, Rice S A J. Chem. Phys. 61 768 (1974)
  25. Pomphrey N J. Phys. B 7 1909 (1974)
  26. Thouless D J Phys. Rep. 13 93 (1974)
  27. Sazonov V N, Finkel’shtein V Yu Dokl. Akad. Nauk SSSR 231 78 (1976); Sazonov V N, Finkel’shtein V Yu Sov. Phys. Dokl. 21 645 (1976)
  28. Shuryak E V Zh. Eksp. Teor. Fiz. 71 2039 (1976); Shuryak E V Sov. Phys. JETP 44 1070 (1976)
  29. Benettin G, Galgani L, Strelcyn J-M Phys. Rev. A 14 2338 (1976)
  30. Akulin V M i dr. Zh. Eksp. Teor. Fiz. 72 88 (1977); Akulin V M t al. Sov. Phys. JETP 45 47 (1977)
  31. Zaslavskii G M Zh. Eksp. Teor. Fiz. 73 2089 (1977); Zaslavskii G M Sov. Phys. JETP 46 1094 (1977)
  32. Berman G P, Zaslavsky G M Phys. Lett. A 61 295 (1977)
  33. Berry M V J. Phys. A 10 2083 (1977)
  34. Berry M V Phil. Trans. Roy. Soc. A 287 237 (1977)
  35. Berry M V, Tabor M Proc. R. Soc. London A 365 375 (1977)
  36. Izrailev F M, Shepelyanskii D L Preprint №78-97 (Novosibirsk: OYaF SO AN SSSR, 1978)
  37. Sazonov V N Teor. Mat. Fiz. 35 361 (1978)
  38. Sokolov V V Preprint №78-50 (Novosibirsk: IYaF SO AN SSSR, 1978)
  39. Efros A L Usp. Fiz. Nauk 126 41 (1978); Efros A L Sov. Phys. Usp. 21 746 (1978)
  40. Berman G P, Zaslavsky G M Physica A 91 450 (1978)
  41. G. Benettin G, Strelcyn J-M Phys. Rev. A 17 773 (1978)
  42. Arnol’d V I Matematicheskie Metody Klassicheskoi Mekhaniki (M.: Nauka, 1979)
  43. Zaslavskii G M Usp. Fiz. Nauk 129 211 (1979); Zaslavskii G M Sov. Phys. Usp. 22 788 (1979)
  44. Berman G P, Zaslavsky G M Physica A 97 367 (1979)
  45. Chirikov B V Phys. Rep. 52 263 (1979)
  46. Casati G et al. Stochastic Behavior In Classical And Quantum Hamiltonian Systems (Eds G Casati, J Ford) (New York: Springer-Verlag, 1979) p. 334
  47. Green J M J. Math. Phys. 20 1183 (1979)
  48. Heller E J Chem. Phys. Lett. 60 338 (1979)
  49. Leopold J G, Percival I C J. Phys. B 12 709 (1979)
  50. McDonald S W, Kaufman A N Phys. Rev. Lett. 42 1189 (1979)
  51. RStratt R M Handy C N, Miller W H J. Chem. Phys. 71 3311 (1979)
  52. Izrailev F M, Shepelyanskii D L Teor. Mat. Fiz. 43 417 (1980); Izrailev F M, Shepelyanskii D L Theor. Math. Phys. 43 553 (1980)
  53. Edel’man V S Usp. Fiz. Nauk 130 675 (1980); Edel’man V S Sov. Phys. Usp. 23 227 (1980)
  54. Brumer P, Shapiro M Chem. Phys. Lett. 72 528 (1980)
  55. Casati G, Valz-Gris F, Guarneri I Lett. Nuovo Cimento 28 279 (1980)
  56. Davis M J, Stechel E B, Heller E J Chem. Phys. Lett. 76 21 (1980)
  57. Heller E J J. Chem. Phys. 72 1337 (1980)
  58. Heller E J, Stechel E B, Davis M J J. Chem. Phys. 73 4720 (1980)
  59. Hutchinson J H, Wyatt R E Chem. Phys. Lett. 72 378 (1980)
  60. Kay K G J. Chem. Phys. 72 5955 (1980)
  61. Noid D W, Koszykowski M L, Marcus R A Chem. Phys. Lett. 73 269 (1980)
  62. Noid D W et al. J. Chem. Phys. 72 6169 (1980)
  63. Waite B A, Miller W H J. Chem. Phys. 73 3713 (1980)
  64. Yonezawa F J. Non-Cryst. Sol. 35-36 29 (1980)
  65. Zimmerman M L, Kash M M, Kleppner D Phys. Rev. Lett. 45 1092 (1980)
  66. Berman G P, Zaslavskii G M, Kolovskii A R Zh. Eksp. Teor. Fiz. 81 506 (1981); Berman G P, Zaslavskii G M, Kolovskii A R Sov. Phys. JETP 54 272 (1981)
  67. Matinyan S G, Savvidi G K, Ter-Arrutyunyan-Savvidi N G Pis’ma ZhETF 34 613 (1981)
  68. Chirikov B V, Shepelyanskii D L Pis’ma ZhETF 34 171 (1981)
  69. Shepelyanskii D L Dokl. Akad. Nauk SSSR 256 (1981); Shepelyanskii D L Sov. Phys. Dokl. 26 80 (1981)
  70. Shepelyanskii D L Teor. Mat. Fiz. 49 117 (1981); Shepelyanskii D L Theor. Math. Phys. 49 925 (1981)
  71. Berry M V Ann. Physics 131 163 (1981)
  72. Brody T A et al. Rev. Mod. Phys. 53 385 (1981)
  73. Delande D, Gay J C Phys. Lett. A 82 393 (1981)
  74. Hutchinson J S, Wyatt R E Phys. Rev. A 23 1567 (1981)
  75. R. A. Pullen R A, Edmonds A R J. Phys. A 14 L319 (1981)
  76. Pullen R A, Edmonds A R J. Phys. A 14 L477 (1981)
  77. Ramaswamy R, Marcus R A J. Chem. Phys. 74 1379 (1981)
  78. Richens P J, Berry M V Physica D 2 495 (1981)
  79. Waite B A, Miller W H J. Chem. Phys. 74 3910 (1981)
  80. Weissman Y, Jortner J Chem. Phys. Lett. 78 224 (1981)
  81. Zaslavsky G M Phys. Rep. 80 157 (1981)
  82. Apenko S M, Kirzhnits D A, Lozovik Yu E Pis’ma ZhETF 36 172 (1982)
  83. Berman G P, Zaslavsky G M Physica A 111 17 (1982)
  84. Berman G P, Zaslavsky G M, Kolovsky A R Phys. Lett. A 87 152 (1982)
  85. Bixon M, Jortner J J. Chem. Phys. 77 4175 (1982)
  86. Buch V, Gerber R B, Ratner M A J. Chem. Phys. 76 539 (1982)
  87. Edmonds A R, Pullen R A, Percival I C Chem. Phys. Lett. 91 273 (1982)
  88. Farrelly D, Reinhardt W P J. Chem. Phys. 78 606 (1983)
  89. Fishman S, Grempel D R, Prange R E Phys. Rev. Lett. 49 509 (1982)
  90. Grempel D R, Fishman S, Prange R E Phys. Rev. Lett. 49 833 (1982)
  91. Gutzwiller M C Physica D 5 183 (1982)
  92. Heller E J, Davis M J J. Phys. Chem. 86 2118 (1982)
  93. Heller E J, Stechel E B Chem. Phys. Lett. 90 484 (1982)
  94. Hietarinta J Phys. Lett. A 93 55 (1982)
  95. Hogg T, Huberman B A Phys. Rev. Lett. 48 711 (1982)
  96. Hose G, Taylor H S J. Chem. Phys. 76 5356 (1982)
  97. Jensen R V Phys. Rev. Lett. 49 1365 (1982)
  98. Pechukas P Chem. Phys. Lett. 86 553 (1982)
  99. Peres A Phys. Rev. Lett. 49 1118 (1982)
  100. Shapiro M, Brumer P Chem. Phys. Lett. 90 481 (1982)
  101. Shapiro M, Child M S J. Chem. Phys. 76 6176 (1982)
  102. Weissman Y, Jortner J J. Chem. Phys. 77 1469, 1486 (1982)
  103. Delone N B, Krainov V P, Shepelyanskii D L Usp. Fiz. Nauk 140 355 (1983); Dalone N B, Krainov V P, Shepelyanskii D L Sov. Phys. Usp. 26 551 (1983)
  104. Tatarskii V I Usp. Fiz. Nauk 139 587 (1983); Tatarskii B I Sov. Phys. Usp. 26 311 (1983)
  105. Bai Y Y et al. Chem. Phys. Lett. 99 342 (1983)
  106. Berman G P, Kolovsky A R Phys. Lett. A 95 15 (1983)
  107. Berry M V, Hannay J H, Ozorio de Almeida A M Physica D 8 229 (1983)
  108. Camarda H S, Georgopulos P D, Phys. Rev. Lett. 50 492 (1983)
  109. Casati G, Guarneri I Phys. Rev. Lett. 50 640 (1983)
  110. Haller E, Koppel H, Cederbaum L S Chem. Phys. Lett. 101 215 (1983)
  111. Hogg T, Huberman B A Phys. Rev. A 28 22 (1983)
  112. Hose G, Taylor H S J. Chem. Phys. 78 5845 (1983)
  113. Hose G, Taylor H S Phys. Rev. Lett. 51 947 (1983)
  114. Inozemtsev V I Phys. Lett. A 96 447 (1983)
  115. Kay K G J. Chem. Phys. 79 3026 (1983)
  116. Korsch H J Phys. Lett. A 97 77 (1983)
  117. Moiseyev N, Peres A J. Chem. Phys. 79 5945 (1983)
  118. Mukamel S J. Chem. Phys. 78 5843 (1983)
  119. Noid D, Koszykowski M L, Marcus R A J. Chem. Phys. 79 4018 (1983)
  120. Pechukas P Phys. Rev. Lett. 51 943 (1983)
  121. Pechukas P J. Chem. Phys. 78 3999 (1983)
  122. Ramaswamy R Chem. Phys. 76 15 (1983)
  123. Robnik M J. Phys. A 16 3971 (1983)
  124. Schuster H G Phys. Rev. B 28 443 (1983)
  125. Shepelyansky D L Physica D 8 208 (1983)
  126. Taylor R D, Brumer P Farad. Disc. Chem. Soc. 75 170 (1983)
  127. Uzer T, Noid D W, Marcus R A J. Chem. Phys. 79 4412 (1983)
  128. Elyutin P V Preprint №24/1984 (M.: Fizicheskii fakul’tet MGU, 1984)
  129. Zaslavskii G M Stokhastichnost’ Dinamicheskikh Sistem (M.: Nauka, 1984)
  130. Likhtenberg A, Liberman M Regulyarnaya i Stokhasticheskaya Dinamika (M.: Mir), 1984)
  131. Benjamin I et al. Chem. Phys. Lett. 107 515 (1984)
  132. Berman G P, Kolovsky A R, Zaslavky G M Phys. Lett. A 105 483 (1984)
  133. Berry M V Physica D 10 369 (1984)
  134. Berry M V Phys. Lett. A 104 306 (1984)
  135. Berry M V, Robnik M J. Phys. A 17 2413 (1984)
  136. Berry M V, Wilkinson M Proc. R. Soc. London A 392 15 (1984)
  137. Blumel R, Smilansky U Phys. Rev. Lett. 52 137 (1984)
  138. Blumel R, Smilansky U Phys. Rev. A 30 1040 (1984)
  139. Bohigas O, Giannoni M J, Schmit C Phys. Rev. Lett. 52 1 (1984)
  140. Brickmann J, Schmidt P C Chem. Phys. Lett. 104 591 (1984)
  141. Carnegie A, Percival I C J. Phys. A 17 801 (1984)
  142. Casati G, Chirikov B V, Shepelyansky D L Phys. Rev. Lett. 53 2525 (1984)
  143. Casati G, Guarneri I Comm. Math. Phys. 95 121 (1984)
  144. Davis M J, Heller E J J. Chem. Phys. 80 5036 (1984)
  145. De Leon N, Davis M J, Heller E J J. Chem. Phys. 80 794 (1984)
  146. B. Dorizzi B, Grammaticos B, Pomeau Y J. Stat. Phys. 37 93 (1984)
  147. Farrelly D Phys. Let. A 104 63 (1984)
  148. Feingold M, Moiseyev N, Peres A Phys. Rev. A 30 509 (1984)
  149. Graham R, Höhnerbach M Phys. Lett. A 101 61 (1984)
  150. Grempel D R, Fishman S, Prange R E Phys. Rev. Lett. 53 1212 (1984)
  151. Grempel D R, Prange R E, Fishman S Phys. Rev. A 29 1639 (1984)
  152. Guarneri I Lett. Nuovo Cimento 40 171 (1984)
  153. Haller E, Köppel H, Cederbaum L S Phys. Rev. Lett. 52 1665 (1984)
  154. Heller E J Pys. Rev. Lett. 53 1515 (1984)
  155. Hanson J D, Ott E, Antonsen T M (Jr.) Phys. Rev. A 29 819 (1984)
  156. Hirooka H et al. Phys. Lett. A 101 115 (1984)
  157. Hose G, Taylor H S, Bai Y Y J. Chem. Phys. 80 4363 (1984)
  158. Hose G, Taylor H S, Tip A J. Phys. A 17 1209 (1984)
  159. Jensen R V Phys. Rev. A 30 386 (1984)
  160. Matsushita T, Terasaka T Chem. Phys. Lett. 105 511 (1984)
  161. Meyer H-D et al. J. Phys. A 17 L831 (1984)
  162. Ott E, Antonsen T M (Jr.), Hanson J D Phys. Rev. Lett. 53 2187 (1984)
  163. Pechukas P J. Phys. Chem. 88 4823 (1984)
  164. Peres A Phys. Rev. Lett. 53 1711 (1984)
  165. Peres A Phys. Rev. A 30 504 (1984)
  166. Peres A Phys. Rev. A 30 1610 (1984)
  167. Robnik M J. Phys. A 17 1049 (1984)
  168. Seligman T H, Verbaarschot J J M, Zirnbauer M R Phys. Rev. Lett. 53 215 (1984)
  169. Shapiro M, Goelman G Phys. Rev. Lett. 53 1714 (1984)
  170. Shapiro M, Taylor R D, Brumer P Chem. Phys. Lett. 106 325 (1984)
  171. Thiele E, Stone J J. Chem. Phys. 80 5187 (1984)
  172. Andrianov A N i dr. Vychislitel’nye Protsessy i Sistemy. Vyp. 2 (Pod red. G I Marchuka) (M,: Nauka, 1985) p. 13
  173. Berman G P, Zaslavskii G M, Kolovskii A R Zh. Eksp. Teor. Fiz. 88 1551 (1985)
  174. Bogomol’nyi E B Pis’ma ZhETF 41 55 (1985)
  175. Bai Y Y et al. Phys. Rev. A 31 2821 (1985)
  176. Bardsley J N, Sundaram B Phys. Rev. A 32 689 (1985)
  177. Bayfield J E, Pinnaduwage L A Phys. Rev. Lett. 54 313 (1985)
  178. Berman G P, Kovolsky A R Physica D 17 183 (1985)
  179. Brickmann J, Levine R D Chem. Phys. Lett. 120 252 (1985)
  180. Casati G, Chirikov B V, Guarneri I Phys. Rev. Lett. 54 1350 (1985)
  181. Chang S-J, Shi K-J Phys. Rev. Lett. 55 269 (1985)
  182. Chirikov B V Phys. Lett. A 108 68 (1985)
  183. Christoffel K M, Brumer P Phys. Rev. A 31 3466 (1985)
  184. Deng Z, Hioe F T Phys. Rev. Lett. 55 1539 (1985)
  185. Farantos S C, Tennyson J J. Chem. Phys. 82 800 (1985)
  186. Feingold M Phys. Rev. Lett. 55 2626 (1985)
  187. Feingold M et al. Phys. Rev. B 31 6852 (1985)
  188. Feingold M, Peres A Phys.Rev. A 31 2472 (1985)
  189. Frahm H, Mikeska H J Z. Phys. B 60 117 (1985)
  190. Helton J W, Tabor M Physica D 14 409 (1985)
  191. Ishikawa T, Yukawa T Phys. Rev. Lett. 54 1617 (1985)
  192. Jensen R V Phys. Rev. Lett. 54 2057 (1985)
  193. Kus M Phys. Rev. Lett. 54 1343 (1985)
  194. Lanza E G, Kwong N H, Ibarra R H Phys. Lett. A 107 149 (1985)
  195. Mayer I L et al. J. Phys. B 18 3835 (1985)
  196. Nakamura K, Nakahara Y, Bishop A R Phys. Rev. Lett. 54 861 (1985)
  197. Petrosky T Y, Schieve W C Phys. Rev. A 31 3907 (1985)
  198. Razavy M Phys. Lett. A 113 297 (1985)
  199. Reinhardt W P Chaotic Behavior In Quantum Systems. Theory And Applications (Ed. G Casati) (New York: Plenum, 1985) p. 235
  200. Seligman T H, Verbaarschot J J M J. Phys. A 18 2227 (1985)
  201. Seligman T H, Verbaarschot J J M Phys. Lett. A 108 183 (1985)
  202. Seligman T H, Verbaarschot J J M Zirnbauer M R J. Phys. A 18 2751 (1985)
  203. Steeb W-H, Villet C M, Kunick A J. Phys. A 18 3269 (1985)
  204. Steeb W-H, Villet C M, Kunick A Phys. Rev. A 32 1232 (1985)
  205. Terasaka T, Matsushita T Phys. Rev. A 32 538 (1985)
  206. Toda M, Ikeda K Phys. Lett. A 110 235 (1985)
  207. Van Leeuwen K A H et al. Phys. Rev. Lett. 55 2231 (1985)
  208. Yukawa T Phys. Rev. Lett. 54 1883 (1985)
  209. Bardsley J N et al. Phys. Rev. Lett. 56 1007 (1986)
  210. Berry M V, Robnik M J. Phys. A 19 649 (1986)
  211. Berry M V, Robnik M J. Phys. A 19 1365 (1986)
  212. Bhattacharya K, Mukherjee D J. Chem. Phys. 84 3212 (1986)
  213. Casati G et al. Phys. Rev. Lett. 56 2437 (1986)
  214. Christoffel K M, Brumer P Phys. Rev. A 33 1309 (1986)
  215. Furusawa T Prog. Theor. Phys. 75 59 (1986)
  216. Hioe F T, Deng Z Phys. Rev. A 34 3539 (1986)
  217. Izrailev F M Phys. Rev. Lett. 56 541 (1986)
  218. Jose J V, Cordery R Phys. Rev. Lett. 565 290 (1986)
  219. Lozada A, Torres P L J. Phys. A 19 L237 (1986)
  220. Meyer H-D J. Chem. Phys. 84 3147 (1986)
  221. Nakamura K, Bishop A R Phys. Rev. B 33 1963 (1986)
  222. Pomeau Y, Dorizzi B, Grammaticos B Phys. Rev. Lett. 56 681 (1986)
  223. Reichl L E, Lin W A Phys. Rev. A 33 3598 (1986)
  224. Robnik M, Berry M V J. Phys. A 19 669 (1986)
  225. Samuelides M et al. Europhys. Lett. 1 203 (1986)
  226. Seligman T H, Verbaarschot J J Phys. Rev. Lett. 56 2767 (1986)
  227. Shepelyansky D L Phys. Rev. Lett. 56 677 (1986)
  228. Steeb W-H, Louw J A, Villet C M Phys. Rev. A 34 3489 (1986)
  229. Takahashi K J. Phys. Soc. Jpn. 55 1443 (1986)
  230. Wintgen D, Friedrich H J. Phys. B 19 1261 (1986)
  231. Yukawa T Phys. Lett. 116 227 (1986)
  232. Zimmerman Th et al. Phys. Rev. A 33 4334 (1986)
  233. Mol’ A, Fuks V, Kassler M Iskusstvo i EVM (M.: Mir, 1975)
  234. Feinman R, Leiton R, Sends M Feinmanovskie Lektsii Po Fizike (M.: Mir, 1966) p. 25, Vyp. 5
  235. Delande D, Gay J C Phys. Rev. Lett. 57 2006 (1986)

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