|
|
|
сделать стартовой | добавить в избранное |
 |
|
Is the nature of quantum chaos classical? |
Забавная пачка денег "100 долларов". 
Чашка "Неваляшка". 
Забавная пачка "5000 дублей". 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.
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)
9. The Impact of the Afghan War on soviet soldiers
11. The Economy of Great Britain
13. The profile of an effective manager
15. The history of Old English and its development
Чайный набор 2 предмета "Вавилон", 210 мл. 
Этажерка для обуви, узкая, 4 полки. 
Каталка-трактор "Митя" №2. 18. The Influence of English Mass Culture on Estonia
19. The role of art in our life
20. Raskolnikov and Svidrigailov: on the brink of suicide. Ф.М. Достоевский, Преступление и наказание
21. The History of Alaska (история Аляски)
25. The profile of an effective manager
28. The teaching of Hugo Gratius
29. The Problem of Holmelessness in Canada
31. London - the capital of Great Britain
32. Climate and nature of Great Britain
Цветные счетные палочки Кюизенера. 
Ростомер говорящий "Теремок". 
Подставка для ножей, 11x22 см, лавандовый. 33. The 8th of March
34. The protection of the environment
36. The commonwealth of Australia
37. The Commonwealth of Australia
41. The constitution of Ukraine
43. Bosch, Hieronymus: The Temptation of St Anthony
44. The development of computers in ukraine and the former USSR
47. The declaration of independence
48. The history of smart-cards and their place in modern Russia
Набор "Грибочки". 
Набор первоклассника, для мальчиков, 16 предметов. 
Коврик массажный "Микс ежики" от 5 лет. 50. The Feather of Finist the Falcon
51. For the Beauty of the Earth
53. The Concept of Youth Subcultures
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
63. Formation and development of political parties in the Republic of Belarus
64. Oscar Wilde "The picture of Dorian Gray"
Бумага чертежная "Mega Engineer", А1, 5 листов, 200 г/м2. 
Гель "Meine Liebe" для стирки цветных тканей, 800 миллилитров. 
Качели детские "Классик". 66. The emergence of the first Polish socialist parties
68. The history of grammar theory
73. The philosophy should meet the challenge of the new millennium
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
Подарок «Вкусный Новый год». 
Набор детской складной мебели "Познайка". 
Подставка для канцелярских принадлежностей "Attache", 7 секции, металлическая сетка, 110x165x175 мм, цвет. 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.
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)
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
Доска пробковая, деревянная рама, 80x60 см. 
Жаровня "Loraine", 1,5 л. 
Детский велосипед Jaguar трехколесный (цвет: синий). 97. Evaluating the GPRS Radio Interface for Different Quality of Service Profiles
98. Финансовые инструменты ("Financial Instruments. Teaching materials of the course")
