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Is the nature of quantum chaos classical?

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K. . Yugay, S.D. vorogov, Omsk S a e U iversi y, Ge eral Physics Depar me ,  pr.Mira,55-A 644077 Omsk, RUSSIA  I s i u e of A mosphere Op ics of Russia Academy of Scie ces Rece ly discussio s abou wha is a qua um chaos do o aba e . Some au hors call i ques io he very fac of a exis e ce of he qua um chaos i a ure . Mai ly reaso o his doub is wha he qua um mecha ics equa io s of mo io for he wave fu c io or de si y ma rix are li ear whereas he dy amical chaos ca arise o ly i o o li ear sys ems. I his se ce he dy amical chaos i qua um sys ems, i.e. he qua um chaos, ca o exis . However a umber of experime al fac s allow us o s a e wi h co fide ce ha he qua um chaos exis s. Evide ly his co radic io is co ec ed wi h wha our radi io al descrip io of a ure is o qui e adequa e o i . Reflec i g o his problem o e ca o bu pay a e io o he followi g: i) wo regi exis - he pure qua um o e (QR) a d he pure classical o e (CR), where descrip io s are esse ially differed. he way i which he qua um a d classical descrip io s are o o ly wo differe levels of hose, bu i seems o be more some hi g grea er; he problem of qua um chaos i dica es o i . Si ce experime al ma ifes a io s of qua um chaos exis herefore o e ca o ig ore he ques io o he a ure of qua um chaos a d he descrip io of i . ii) I u doub edly ha he i ermedia e qua um-classical regio (QCR) exis s be wee he QR a d he CR, which mus be possessed of charac eris ics bo h he QR a d he CR. Si ce he erm "quasiclassics" is co ec ed radi io ally wi h correspo di g approxima e me hod i he qua um mecha ics we shall call his regio as qua um-classical o e fur her. I is evide ha he QCR is he regio of high exci ed s a es of qua um sys ems. Below shall show ha qua um a d classical problems are o au o omous i o he QCR bu hey are coupled wi h each o her, so ha a solu io of a qua um problem co ai s a solu io of a correspo di g classical problem, bu o vice versa. A possible dy amical chaos of a o li ear classical problem has a effec o he qua um problem so ha o e ca say qua um chaos arises from dep hs of he o li ear classical mecha ics a d i is comple ely described i erms of o li ear dy amics, for example, i s abili y, bifurca io , s ra ge a rac or a d so o . We shall show also ha he co ec io be wee he qua um a d classical problems is reflec ed o a phase of a wave fu c io which havi g a qui e classical mea i g is subjec ed o i s classical equa io of mo io a d i he case of i s o li eari y i o he sys em he dy amical chaos is exci ed. O e of a sple did example of a role of he wave fu c io phase is a descrip io of dy amical chaos i a lo g Josephso ju c io . Here he wave fu c io phase ( he differe ce phases o a ju c io ) of a superco duc i g co de sa e is subjec ed o he o li ear dy amical si e-Gordo equa io . he dy amical chaos arisi g i a lo g Josephso ju c io a d describi g by he si e-Gordo equa io is a qua um chaos esse ially si ce he ques io is abou a phe ome o havi g excep io ally he qua um charac er. However he qua um chaos is described here precisely by he classical o li ear equa io . Below we shall ry o show ha he descrip io of he qua um chaos i he more ge eral case may be carry ou jus as i a lo g Josephso ju c io i erms of o li ear classical dy amics equa io s of mo io o wich he wave fu c io phase of a qua um is subjec ed.

I addi io he qua um sys em mus be i o he QCR, i.e. i o high exci ed s a es. Le us assume ha he Hamil o ia of a sys em have he form where he opera or of he po e ial e ergy U(x, ) is (We exami e here a o e-dime sio al sys em for he simplici y). Here U0(x) is he o per urba io po e ial e ergy, a d f( ) is he ime-depe de ex er al force. We shall fou d he solu io of he Schrödi ger equa io i he form where , is he solu io of he classical equa io of mo io , is he cer ai co s a , s( ) is he ime-depe de fu c io , he se se of ha will be clear la er o . We o ice ha he fu c io A(x, ) is real. (A represe a io of he phase A(x, ) i he form (5) a was i roduced firs by Husimi ). Subs i u i g (4) i o Eq.(1) a d aki g i o accou (5), we ge Here subscrip s , y a d de o e he par ial deriva ives wi h respec o ime a d coordi a es y, , respec ively. O he righ of Eq.(6) he expressio s of bo h square bracke s are equal o zero because of followi g rela io s: i) of he classical equa io of mo io where is he same po e ial, ha is i o (3), a d ii) of he expressio for he classical Lagra g fu c io L( ) so ha he fu c io makes a se se of a ac io i egral. I o Eq.(6) By deduc io of Eq.(6) we made use of a po e ial e ergy expa sio i he form I is obvious ha he expa sio (11) is correc i he case whe a classical rajec ory is close o a qua um o e. hus we ge he equa io for he fu c io i he form We pay a e io here o hree origi a i g mome s: 1) Equa io (12) is he Schrödi ger equa io agai , bu wi hou a ex er al force. 2) We have he sys em of wo equa io s of mo io : qua um Eq.(12) a d classical Eq.(7). I a ge eral case hese equa io s make up he sys em of bou d equa io s, because he coefficie k ca be a fu c io of classical rajec ory, . As we show below a co ec io be wee Eqs. (12) a d (7) arises i he case, if classical Eq. (7) is o li ear. 3) Classical Eq.(7) co ai s some dissipa ive erm, a d so makes se se of a dissipa ive coefficie . he arisi g of dissipa io jus i o he classical equa io is looked qui e a urally - a dissipa io has he classical charac er. Le us assume ha is he po e ial e ergy of a li ear harmo ic oscilla or where is he cer ai co s a . he we have a d where is he a ural freque cy of he harmo ic oscilla or. Equa io s (15) a d (16) represe he correspo di g equa io s of he qua um a d classical li ear harmo ic oscilla ors. We see ha Eqs.(15) a d (16) are au o omous wi h respec o each o her. hus i he case if he classical limi (16) of he correspo di g qua um problem (15) is li ear he he solu io of he classical a d qua um o e are o co ec ed wi h each o her. Le us assume ow ha have a form of he po e ial e ergy of he Duffi g oscilla or where , a d are some co s a s. For he po e ial e ergy (17) k akes he form he we have he followi g equa io s of mo io where Equa io (20) represe s he equa io of mo io for a o li ear oscilla or. I is see , ha qua um (19) a d classical (20) equa io s of mo io are coupled wi h each o her. We re ur o he discussio of expa sio (11). I is seemed obvious, ha he classical a d qua um rajek ories coexis a d close o each o her o ly i o he QCR.

I o he pure qua um regio QR a d i o he pure classical o e CR hese rajec ories ca o coexis : because i o he CR a de Broglie wave packe fails quickli i co seque ce of dispersio ; i o he QR he classical rajec ory dissappears i co seque ce of u cer ai y rela io s. hus expa sio (11) is correc i o he qua um-classical regio QCR o ly, or i o her words i o he quasiclassical regio . he QCR is became esse ial jus i cases whe a classical problem proves o be o li ear. he ra si io of a par icle from he low s a es (from he QR) i o high exci ed s a es (i o he QCR) is where A(x, ) is defi ed wi h he expressio (5). I is easily see ha he probabili y of his ra si io will be depe d o he solu io of he classical equa io of mo io . Si ce he classical problem (19) is o li ear, he i o i s, as i is k ow dy amical chaos ca be arise . his chaos will lead o o regulari ies i he wave fu c io phase A(x, ) a d also i he fu c io , ha i ur will lead o o regulari ies of he probabili ies of he ra si io i high exci ed s a es, a d also from high exci ed s a es i o s a es of he co i uous spec rum. I his way i ca be said ha he qua um chaos is he dy amical chaos i he o li ear classical problem, defi i g qua um solu io s, from he poi of view of he s a ed here heory. hese i ves iga io s are suppor ed by he Russia Fu d of Fu dame al Researches (projec o. 96-02-19321). Список литературы Zaslavsky G.M., Chirikov B.V. S ochas ic I s abili y of o li ear Oscilla io s // Usp. Fiz. auk. 1971. V.105. .1. P.3-29. Chirikov B.V., Izrailev F.M., Shepelay sky D.L. Dy amical S ochas ici y i Classical a d Qua um Mecha ics // Sov. Sci. Rev., Sec .C. 1981. V.2. P.209-223. oda M., Ikeda K. Qua um versio of reso a ce overlap //J.Phys.A: Ma h. a d Ge . 1987. V.20. .12. P.3833-3847. akamura K. Qua um chaos. Fu dame al problems a d applica io o ma erial scie ce // Progr. heor. Phys. 1989. Suppl. .98. P.383-399. FloresJ.C. Kicked qua um ro a or wi h dy amic disorder. A diffusive behaviour i mome uum space // Phys. Rev. A. 1991. V.44. .6. P.3492-3495. Gasa i G., Guar eri I., Izrailev F., Scharf R. Scali g behaviour of localiza io i qua um chaos // Phys.Rev.Le . 1990. V.64. P.5-8. Chirikov B.V. ime-depe de Qua um Sys ems.- I 'Chaos a d Qua um Physics', eds. M.-J. Gia o i, A. Voros, J. Zi -Jus i . Proc. Les. Houches Summer School. Elsevier, 1991. P.445. Chirikov B.V. Chao ic Qua um Sys ems. Prepr. 91-83. ovosibirsk, 1991. Elu i P.V. A Problem of Qua um Chaos // Usp. Fiz. auk. 1988. V.155. .3. P.397-442. Berma G.P., Kolovsky A.R. Qua um Chaos i i erac io s of mul ilevel qua um sys ems wi h a cohere radia io field // Usp. Fiz. auk. 1992. V.162. .4. P.95-142. Heiss W.D., Ko ze A.A. Qua um Chaos a d a aly ic s ruc ure of he spec rum // Phys. Rev. A. 1991. V.44. .4. P.2401-2409. Prose ., Rob ic M. E ergy level s a is ics a d localiza io i sparced ba ded ra dom ma rix e semble // J. Phys. A. 1993. V.26. .5. P.1105-1114. Berry M.V., Kea i g J.P. A rule for qua izi g chaos? // J. Phys. A. 1990. V.23. .21. P.4839-4849. Sai o . O he origi a d a ure of qua um chaos // Progr. heor. Phys. 1989. Suppl.

That is, farmers initially selected seeds of certain wild plant individuals to bring into their gardens and then chose certain progeny seeds each year to grow in the next year's garden. But much of the transformation was also effected as a result of plants' selecting themselves. Darwin's phrase "natural selection" refers to certain individuals of a species surviving better, and / or reproducing more successfully, than competing individuals of the same species under natural conditions. In effect, the natural processes of differential survival and reproduction do the selecting. If the conditions change, different types of individuals may now survive or reproduce better and become "naturally selected," with the result that the population undergoes evolutionary change. A classic example is the development of industrial melanism in British moths: darker moth individuals became relatively commoner than paler individuals as the environment became dirtier during the 19th century, because dark moths resting on a dark, dirty tree were more likely than contrasting pale moths to escape the attention of predators. Much as the Industrial Revolution changed the environment for moths, farming changed the environment for plants

1. The face of every city is different. Washington D.C.

2. Allergy Enlightenment: Is Modern Medicine Fighting the Causes of This Ailment? Are Doctors Telling Us Everything?

3. The effect of light intensity on the amount of chlorophyll in “Cicer arietinum”

4. Роль СМИ в современном мире (The mass media in the life of Society)

5. Jealousy as the cause of internal self-destruction in "Kreutzer Sonata" by Leo Tolstoy (Ревность как причина внутреннего самоуничтожения в "Крейцеровой сонате" Льва Толстого)

6. The history of railways (История железных дорог)
7. How telecommunication change the defenition of work /Telework/
How telecommunication change the defenition of work /Telework/
8. The Castles of England
The Castles of England

9. The Impact of the Afghan War on soviet soldiers

10. The Tower of London

11. The Economy of Great Britain

12. The History of English

13. The profile of an effective manager

14. The Tower of London

15. The history of Old English and its development

16. The Queen of the UK

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17. “The Role of Smallpox Vaccination in Mortality Decline in the Great Britain through Eradicating the Disease between XVIII-XX centuries – Facts or a Political Arithmetick?”

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19. The role of art in our life

20. Raskolnikov and Svidrigailov: on the brink of suicide. Ф.М. Достоевский, Преступление и наказание

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22. Alaska’s Wildlife: on the Verge of Extinction (Живая природа Штата Аляска на грани исчезновения)
23. Преп. Максим исповедник о бесконечности человека (st. Maximus the confessor on the infinity of man)
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25. The profile of an effective manager

26. The history of England can be defined as the gradual process of Parliament asserting its authority over the monarchy

27. The Taming of the Shrew

28. The teaching of Hugo Gratius

29. The Problem of Holmelessness in Canada

30. Переведенная на английский лекция по теме Money and banking (деньги и банковское дело), the role of banks (роль банков), modern banking (современная банковская система)

31. London - the capital of Great Britain

32. Climate and nature of Great Britain

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33. The 8th of March

34. The protection of the environment

35. The Role of Smallpox Vaccination in Mortality Decline in the Great Britain through Eradicating the Disease

36. The commonwealth of Australia

37. The Commonwealth of Australia

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39. The Doctrine of the "Mysterious Female" in Taoism
40. The pride of the nation
The pride of the nation

41. The constitution of Ukraine

42. The history of Christmas

43. Bosch, Hieronymus: The Temptation of St Anthony

44. The development of computers in ukraine and the former USSR

45. The Algorithm of a Start and the Development of International Conflicts and Possible Ways of Their Solution

46. The House of Yorks

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49. The war of the roses

50. The Feather of Finist the Falcon

51. For the Beauty of the Earth

52. The Grapes of Wrath

53. The Concept of Youth Subcultures

54. The development of the Tower
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55. The development of the drama. The theatres and actors
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56. The Society of Travelling Art Exhibitions
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57. The art of conversation

58. Economy of the Republic of Ireland

59. HOW SIGNIFICANT WAS ALEXANDER DUBCEK IN THE DEVELOPMENT OF REFORMIST COMMUNISM?

60. Islam in the eyes of the West

61. The production of 2H-labeled amino acids by a new mutant of RuMP fucultative methylotroph Вrevibacterium methylicum

62. Сборник диалогов по английскому языку для развития устной речи старшеклассников (The collection of dialogues on English language)

63. Formation and development of political parties in the Republic of Belarus

64. Oscar Wilde "The picture of Dorian Gray"

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65. The category of Mood

66. The emergence of the first Polish socialist parties

67. The history of English

68. The history of grammar theory

69. The history of Moscow

70. The Language of Narrative Writing
The Language of Narrative Writing
71. The origin of language
The origin of language
72. The outline of the period
The outline of the period

73. The philosophy should meet the challenge of the new millennium

74. The problem of violence

75. The problems of the Subjunctive Mood in English

76. The Socialist-Revolutionaries and the labor movement (the beginning of the twentieth century)

77. The War of the Roses: the Historical Facts of the Tudor Myth (Shakespeare’s Histories)

78. Historical measurement of the science of governing

79. The Architecture of Ancient Rome

80. Eugene Hlebtsevich and the establishment of free public libraries in Belarus in the early twentieth century in the context of socio-political and cultural processes of the time

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81. Speeches workers Grodno province in 1905-1907 and the emergence of trade unionism

82. The Radicalism of the American Revolution

83. The collection of French art in the Hermitage

84. Advertising in the world of art

85. The models of atom’s nucleus and table of elements.

86. Rise of sociology as an intellectual tradition. Classical tradition in sociology of the XIX century
Rise of sociology as an intellectual tradition. Classical tradition in sociology of the XIX century
87. The United Kingdom of Great Britain and Northern Ireland
The United Kingdom of Great Britain and Northern Ireland
88. История Советсткого флота (History of the Soviet fleet)

89. The Adverse Effects of Green Lawns

90. Going public and the dividend policy of the company

91. My modern image of the United States

92. The Proverbs Are Children Of Experience (Пословицы - Дети Опыта)

93. Представители Ренессанса (Representatives of the renaissance and thair contribution to the literature)

94. Consequence of building the National Missile Defense

95. Ways of exploring the world

96. History of `The Beatles` and biographies of members in english

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97. Evaluating the GPRS Radio Interface for Different Quality of Service Profiles

98. Финансовые инструменты ("Financial Instruments. Teaching materials of the course")

99. The dark side of the moon


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