Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://chronos.msu.ru/old/RREPORTS/kupervasser_osnovnie_paradoksi_stat.pdf Äàòà èçìåíåíèÿ: Sat Dec 14 13:13:36 2013 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 20:44:26 2014 Êîäèðîâêà:

.
, E-mail: olegkup@yahoo.com 2009 .
- . . , , , . , . . , . . , . . [9], [35]. , 1. , . . . , . ( ), , ( ) , , . ,
1

[7], [46] , «». [44], [45] , [10] , . . , , [4] [5]. ( ) . [59]. . .

1


. ( ) .

.
. .......................................................................................................................................4 1. ............................................................................................................................................5 1.1 . [1],[2]. 5 1.2 .[1],[2] .................................................6 1.3 . [34], [35], [9] .............................................................7 1.4 . ................................................................................8 1.5 . [1], [2], [29] .......................................................................................................8 1.6 . .......................10 1.7 . ............................ 12 1.8 . .......................................................................................................12 2. . ......................................................................14 2.1 ­ , , , , .[3], [36] .............................................................................14 2.2 . . [3], [15], [29] ...................15 2.3 . [29]................................................................................................................16 2.4 .[3], [29] ( J, O, P) .............16 2.5 «» : .[3] .......................................................17 2.6 . ............20 2


2.7 [51] .[3] ....................23 2.8 « , ». ..........25 2.9 . ........................................................................................................25 3. . . .................................................................................................................................26 3.1 . [37], [38],[3]. ............................................................. 26 3.2 . ............................................................................28 3.3 .[3], ( S, , U) .................29 4. . ................................ 29 5. . ............................................................ 30 6. . ....................................................32 7. () (). ( N) .......................................................................................35 8. /. ............................................................... 37 9. / . ...................................................................................................................39 10. . ..........................................................................42 10.1 . ? ....................................................................................................42 10.2 ? ............................................................................................................................... 45 10.3 .(Master equations).........................47 10.4 . ­ , . ............................................................................................................................... 47 10.5 «» («»): . ............................................................................................................................. 48 11. . ............................................................................................. 50 12. . ...................................................................................................................54 12.1 ­ . ..........................................54 12.2 . , ( ) . ...................................................................56 12.3 - () . ......................................................................................................................................57 12.4 . ............................................................................................................................ 61 13. . ..........................................................................................................................64 A . [1], [2], [9] ...................................................66 B . [1], [2], [9], [29] .....................................................66 . . [2].............................. 67 D . . [56] ..........................................................................................67 E. . . [2], [9] .........................................................................68 F . «». [1], [2] .............68 G. . [36], [3], [15] ........................70 I. . . [15], [29] ............72 K. . [33], [2] .............................................................................................................................. 75 3


L. « » . [9], [12] ................................................... M. . ......................................... N. . . ............ O. . [29], [3] ............................................. P. .[3], [30] ....................................................................................................................... R. . , . [3] .............................................................................................. S. -- -- . [3] ............................. T. .[3] .................................................................................. U. - -- . [3] ......................................... V. , . [58] ........................... . .................................................................................................................................

76 76 77 78 79 80 82 83 86 87 90

.
. 1) , . , , , , . , , , . 2) . (, , ..) , . , . 3) - , . , , ( , , ), . , , , , ( ). . , , . . , . [44] , [45] , [58], [46], [85] , «» . .

4


, , . 4) ( ), . 5) . . ( !), , . , , «» «» . (, ) . ( , ) .

1. .
1.1 . [1],[2].
. .

2. . , , .

, . . . , . . . 5


. , . . , , .

1.2 .[1],[2]
. . . . . .

Figure 3. . p, q. , p0, q0. (. [17])

, . , . («») , , . , . ( A ) [1], [2]

6


Figure 4. . , . . (. [17]) ( ) ( ). , . . , . , . . . , , . ( E.) [1], [2]

1.3 . [34], [35], [9]
[9], [35]: , . ( ). . . . , . ( ) [9], [35]: 7


, , , , [34], [9].

Figure 5. , , . t=t0 . (. [12]) : 1 , 4 , 8 .. . , , .

1.4 .
. , . , , . , . . ( .) [2] . , . .

1.5 . [1], [2], [29]
. 16 . 8


, «» «», , , ? 15 . . , 8 . , ? , 4 . . , . . . [1], [2], [29] ( B ) , , . , . , . . , , , . «» ? ! . , . . , , . ! , . : . , , ( F .) [2]. . , , . , .

Figure 6. . (. [14])

9


Figure 7. a) b) . (. [13])

Figure 8. : a) b) c) . (. [14])

1.6 .
, , , , . , , , . . , . , . , . , . . ( ) : 10


, , . , , 2. . . , . ( ) ( , ) . , , . - ( F .) , , . , , . «» «». «» . «» . «» «» , «» . «» [9], [35]. «» . «» «», . , . , , . . , . " " [56]. , , , . , . , . , . , ( « ») . « » . - , «». . «» . «» , . , .

2

, , . , , .

11


Figure 9. Direct process with macroscopic entropy increasing and its inverse process. Directions of shrinkage are denoted.

1.7 .
. : , . [1], [2]. . , , . . , , , , ! . , .

1.8 .
( ). . . ( D.) , . . , , . , , . , , 12


, . , . , , . , . [33], , . . , . , , ( .). () , . -, , . , . , , () , . , . ( , ) , . . ( ) , , , «» / . . . ( , ). . , , . "" [33].

13


Figure 10. . . (. [17]) , , . . . "" , . , .

2. .
2.1 ­ , , , , .[3], [36]
, . , . , " ". , . [36]. , .. . , , , . . (, ) , ( ). . ( ) . . , . 14


­ . , , « » [36]. , . , . - . , - . . [36], , . . . [36] ( G.).

2.2 . . [3], [15], [29]
. , , . . ( )? , ( « »!). . . , , . , , , . ( ­ ) , . . [15] . . . , . . , . , . , - . () . , . ( I. )

15


2.3 . [29]
, () .. 1) . 2) . , ( ). . , . 3) . ­ , . . , . - , , . . , . 4) . . . 5) , . , . , . . 6) , , . , , . . .

2.4 .[3], [29] ( J, O, P)
, . , , : . , . - , . . . . , 16


, , .. . . . , . , , . , . , . . . , . . , . . , ­ . , , , .. , . . , . , . . . , . . , . , "" "" . , - , . . , , , , . "", "", "" . , . , , . "" "" .

2.5 «» : .[3]
, . . ( U): . 17


, . , ( ) . . , , . , , : 1) ( ) . . 2) . , , . ( ), , , . , . , . . . 3) [3] . ­ , . , , , , , . , , ! . , . , . "", . , . «» -- ().[3], ( R) , «» . , " ", .. , . -, , . ( ) , . ­ , . , 18


(1/2), : (+1/2 -1/2). , . . , . - . , , . . ( Z) +1/2. . , ( ) -1/2. . Z, +1/2, 1/2. Z. , «». , , «», . , . , ? , () ? ? ( T) . (.. ) , , ( ) . .. .

11. . , , . . , ( ), . , , , "" (.. 19


) . , . , , . . , . , , . , ( ). , . , , +1/2 -1/2 .

2.6 .
- "" , . , « - -- , , 1801 ». . , S, S1 S2. , , , S1 S2, . -- , . , «» «» , . , , . , , , , . . . , , . . , .

20


Figure 12. . (. [96]) , . , ( ). , . , , . : , , , . , , , , , , , . , , .

Figure 13. . (. [97])

21


. 14 , , -- «» . , . , , , . . , , , , , . , . ­ , . , , . . . , , , . , , , .. . ­ . . , , , , . . , , . . . -, , . , , . , . , . . : 1) , "" . , , . . ­ . 10-20 . ( ). 22


2) . (D) (n) (L). : D>>(Ln), - . 3) ( , , , ) ( L), . , , , . , , . , . , , . . . (.. ) 3. , , . - ( cohaerentio - , ) [31], [39], [16], [16], [47], [48] , ( P). . (.. ) , . : . , .

2.7 [51] .[3]
, (.. ) . , . , 4.
3

, , , , . , , , , . , ( L) .
4

, , , , . . , . . , , . , .

23


, , , , . 5. , , . . , , . ­ « ». [51]( ,1935) , . . , , ( ), 50- . , . ( ), : «» «». 6.( , , . ) , . . . , . , , . , . , , . !7 . , , , . « »[3], [6], [7].

5

, - , , .
6

«» . [8], [58], ( V). 7 , [8 ], [58], ( V).. .

24


. 15 . . (. [98]) , . ? , ? , ? ?

2.8 « , ».
« , ». , , . . , . , , ! . . , ( ) . , . . , , . . . , "" , , . , , . , . , «» , . , .

2.9 .
25


. . , . : , . , , . , , . , . . ( ) . . , , . : , : . , . , , . , , , . , . .

3. . .
, , , , . [3] . , , . . .

3.1 . [37], [38],[3].
. , , . . , « », ( 26


) . . , . , . . . , .

. 16. . (. Max Tegmark [99])

. , . . , . , . .. ! . , , ( ) . , . , , «» . ( .) , «» . 27


, , .

3.2 .
. , , . , . , . , . . , , . , . , , .

17. (). (. [100]) , . . , , , . , ! , [37], [38], . , , . 28


3.3 .[3], ( S, , U)
, , . , , - . [3]. , , ( - ), . , , , .. . , , . . , ( , ) , . , , . ( ) . , , , - . . , .

4. .
, . , . , , : , , . : , , . , , . , , . . , , , . . . , , , . ? ­ . , . , -. 29


5. .
, , , . : , .. . - . , , , . , , . . . . , .. . , , ? ­ . , ( ) , . . , , , . . ( ) . , , . 1) . , -.[37], [38] , . , . , - . , . , , . , , , . 2) , , . , , . ­ . , , . ( ), . , . 30


( , ) . , , , .

18. - «» . - . , , . . , , , . 3) . , . , . -, , . , , , . [40], [41] . -, , . , . , , . , . ( , , ) . . , , , , , .. , ! , ­ . 31


-, , , , , .. . , , . . , . () . [43].

.

6. . .
­ , . ­ . . , , , . , , , . , , . . . , . , , . () . , , . . , . , . . ? , . , ( ). , . 32


. , : , , . , , . . , . , ­ . .

. 19 , . . . . [2], [28], [44], [45] . , , ? 8. [2], [28], [44], [45] . , . , , , . ,
8

« » , ? [44], [45] - ! , ? [60]. , «» . « ». , , .

33


. . . / «», . . , , . . , . [49], [18] , . , .

. 20 . . . . , . , . , . , . .

34


7. () (). ( N)
, , . 1) . 2) . , . 3) , , , «». : , , . . , ( ) ( ) ( ) . 4) - . - ( ) ( ) , . 5) - ( ) . , , . . ( ) . , . , , . 6) , , . , . « », (.. ). . , 35


. 7) . , , . , . , -, , . -, . . . 8) . - . , , () «», , 9. : . , , , , , .

9

« », . - [61] , : 1) . , . , . ( ) 2) . , , . , ­ . , . - . , ( « »). , , . !

36


8. / [88-94].
. , , . , ( ). , . , . , , / ( ). , . ? ? : . ( ). - , ( - , ). , . ( . . ). () . . "". / ( ). . ( ) ( ) ! (, , ) , 37


. (, , ) . (, , ) , . "" . . . , . . , ; . , , , . , , , - . - [69], [70]: ", , (, , 1 [20])" () . - ( , ) . . , .. , /. , . - : . . . , - FAPP (.. , ). , . . , , ( , ) , () . , , (" ") (" ") , , , " ". " ? -, ? . " " " " . . ( ). 38


. , " ". - , . , , . ( ). ( ). . . , . , . . , . . . , . , . , . , , . . , . . . - . , , . . , , T, , . [0, T], - [-T, 0]. - ? : ([0, T]) 0. , [-T, 0] -T. T , . T, 0. " " . , . . , [66].

9. / .

39


, . . : . . , , . , . . , /, . , (FAPP, ) , . , , . ( ), . . . , : - ( , ) /. , /. . [71] ­ , , . ­ - . ­ , . . . [95]. - ( ) . . , - ( ) . , , . - / ! , . , /. , ( ) / . ­ . , . . 5 -- 40


, AdS/CFT , . . , - ( ) 5- 4- . , . , , . , , , . , , , .. , . -- ( ), /. , ! . , ! , [72] (). , [73]. ( , , ) (wormhole traversing space into one traversing time). , . ? , . , . . , . . , . . . , , , ! , ( ) , . « » , . , . , . [74] , (closed timelike curve) . ( , 41


) , , . . , , . ( !) , [75]. , : ! , , , . , , : , . . ( ): - () ( - , ). [75], [74]. ( ) ( ) ( ) [74] . : () , , , « » . . , , (Loshmidt) . !

10. .
10.1 . ?
, , . , , , , , (master equations) - . . , . 42


. ? , . , (.. ), , , . ? , .. ? ! ( ) , . . , master equations ( , , , , ) . . . , . . , , . , , , , . . ? , . / , . , .

43


. , , . , , . , , , . . ( ) , ? [11]. . , . , . [56]. , , . . , . , , , , , , , ? , , , . ( ), . , , . . , . . , ? , . [42]. , . ? , , ( ). .. .10

10

, , . , , , , , , . . « » . , . , , ( )

44


. , , . : , [10], [44], [45]? , , . -, , . , , .11 , , . .. , . , - , . , .

10.2 ?
, . . , , , ( ), . , . , , . . ,
«» «» , . .[13] 11 , . , . ( ) .[43],[10]. . . , , , . , , , . (, , , [53] [54] ).

45


, , . , , . , («») . . (.. ), . , . ( ) ( ) , .. . , .

23 ) b) : , [14]? , . , . . , . [15], [16], [47], [48] .. . ? . . . (pointer states) [39], [31]. , , , . (pointer states) . . , , , . . , master equations (, , , ) . 46


«», . , , . , , , . . -, . . , . -, ( ). . , . . .

10.3 .(Master equations)
. , ( ) . (, ). [15] , . , («») , [35], [2], ( ). , ­ . , («») () . . [3] . , . , . ( ) ( ) , , . . - .

10.4 . ­ , .
47


, . ( ) . N ( ) . t0 N0 . . , . , , , . () , . (t>>/E, t ­ , ­ , E ­ ) , .( /E << t < , ­ ). ( ), - . (nt << Treturn , n ­ [] , Treturn ­ ) , .. . (). ( N=N0exp(-(tt0)/) ) , , .

10.5 «» («»): .
.[1],[2] («») ( K) . . , , , . . . « ». , . , . , (.. ). « » . [9], [12], ( L) . («»), ( , , ) . . , 48


, ( ). , ( ). («») . , « ». « », , « ». «». « », , , . . , ( « ») . « » . - , «». » . , « » . , , .

. 24 («») . . (« » .) « » . , ( ).

49


, . « » 12. . , , . , . . - ( ). ( , ) , , . , (, ) () . « » . « » .[28] , , « ». , . , , « » ! « » (, , ), . , . . , . .

11. .
( ) . . , , . , , . . , . . ( - ) . , . , . , , . . 1) . , , -
12

. , ().

50


, . , . . , .. . 2) , . , , . , - . , . (, , , , .) ­ , . , , . . 3) , , . . (.. ) . . . . , . , , ! .. . .[16], [47], [48]

. 25

, . . 51


, « ». «» , , (). , , - , , . .

26. . (. [102]) « » ( , , ) , . , , «» . , . , , [16], [47], [48] . , , . , . «» , , , , . , . , . , [86-87]. [52, 8687]. . , , , . 52


, , , . , . . . 4) , , . , . [25] ­ . , . - . , . , , . , . ( , .) ( ) . 5) . . , . 13. ­ . , .. . , , . , ( ) . , . . , , , . , - .
13

, , , .

53


. . . . , . , . , . ! , , .[44], [45] , «» , ( ). 6) . , ­ . (, -, ) . , . .

12. .
, , . , . : ? ? ? ? , , , , , , , , . , : , , .

12.1 ­ .
[11]. , .. 14, , , .
14

.. . «» . , .

54


, : , . : « » . [26]. , , . [19] . , . . , , , , , . .. ! , [20]. ( ) , . , , . , , . 15 [17] , (, , ) , . , . , . . , , . . .. , . , , . - . - , - () , . . , , () . . [18]. . , ( () , ) , , , () .
15

, . .

55


. [18]. : 1) , , . 2) , , , [32]. , , , . , , . 3) , , , [50], , . , , , , . , , [50] , . .

12.2 . , ( ) .
. . , ( ) . , . , , . , . ­ . , , . ­ , , , . , . . , , . . . , . , . , .. . 56


, , . , ( ) , . , . . , . . , - , . . , . , .. .

12.3 - () .
. , () . , ( ). ( ), . 1) ­ «» . . 2) , ­ , (). , . 3) . . , . . : 1) ­ [62-64]. ( , ) . , . , «» , (, !) 2) , , , . 57


«» ( + + ), . , , . . . , . , [27] [6] , . , . , . , , , . . [27] , , , , [25], . , « »[47]. , . ( .[47]) , , . «, », . , « » , . , . - (, ). . , . , , . , , .[44],[45] - , (, ). . .[16], [47], [48] , . , ­ . . . ­ , . , . , , , 58


.[10] , , , , ­ . : -> -> -> -> . « » « » - . . , . , , [10]. , . . , . () . , , , . . , , , ... . , . , . ( , ) . , . , , , , , . , , . , , . , . «» () , . «» (blow up) [62-64], . . , . «» . , , . ( ) [65]. ­ , ­ . . («») , «». , , , . «» . , «» , 59


« »[66]. , , «» , . «» . [67-68]. , , . . . , , () . - . ( ). , ­ . m . , , . . . ( , ). ( ) . . . . . , , . . , ­ . , . . , - . «-». , . , , , («») , . , «-» , «» . , ( «-»), « ». , «», () «», ( ) «». «» - «» (.. ) . 60


, . .

12.4 .
, , , , , .. . ( ) , . , .. , . . , . « «»» [55]. . , "": «... "". , - , , , - . "" " " , , - . - - , " ", ....
, , , , , "". , . " " , , , , , . - , . , . , " ", , , , - . , . . - , , . . - , . , » , "" : « , , , ""

61


. " " , . , , . , ad hoc( (.)). , , , , , , , . , , - , - . , , , . , . "", , "" . "" , ,, - , -, , . , , , , . , - - . , . , , . . , . (Cetonia aurata), - . : " ". . . - " ", , , . , , . ., , . : 1) , , (, ), , , , . 2) ( - ) , , , (, ). 3) , , . 2 3 , , , , . "", ""»

, «» , . , «» . 62


, (.. ). , , . . , « ». , , . , «». «» . «» , «» . , , . . . , , , . . . , . ! , . ? ? , . , « ». , , «». . . [21], [22]. . . [46] . 1) «» , , « » «» , . «» . «» . , , «» . «» «» . «» «» . 2) , , . , , . 63


3) . . . , , « ».[56] , « » ! ( , ) [56] .

13. .
. . . - ( ) . . ( M) , , , . , . , . , . , , , . . , - , , «» . [23] , , , . , , . [24] «» , , , . , , . , . , . , ­ , 64


. , «» . , , . . , . , . . , , , «». , . . , . , , , . : 1) , ? 2) ? 3) « » «» (, , ), ( , ), ? , : a) , , ( ) , . b) , c . , (.. ) , , . , . , . () . , . . , . , . c) . , , . . [58] ( M V) d) . [57] «» , [76], [77], [78]. , , . 65


A . [1], [2], [9]
t N r1, ..., rN p1, ..., pN N . xi=(ri,pi) (i=1, 2, ..., N) X= (x1, ..., xN)=(r1, ..., rN , p1, ..., pN) . 6N 6N- . , - . . , , , t X. dX X. t , dM M. lim dM/M=fN(X,t)dX
m

t. fN(X,t)dX=1 f i N Lf N {H , f N } t L - : H H L i i p x x p , H - .

B . [1], [2], [9], [29]
S=-k
(X)fN(X,t)

ln fN(X,t)

[29] S=-k tr ln

66


tr - . . fN . [35]

. . [2]
, g , - . . , G. , g. , g. , , , . g' , , g, . g g'T . g'T,. , g, G. , g g'T , , . . Tg'
G , g' . -->, (1) g'-->0, .

D . . [56]
. X Y, : .. - xi yi . , (xi , yi ), , . . , . , .. , . : . , , , Y , X. , . , 67


. ,

E. . . [2], [9]
-- , , E.


, . . , . , , (, a) = E = const (1)

() , . () -->0 , , - { -- }. , :
(X) =

1 {-(,)}, ( E , a )

(2)

1/(, ) -- , , . . (,)=

(X )



{E H ( X , a)}dX

(3)

(2) ,

F

(X )



F(X )

1 {E H ( X , a)}dX ( E , a)

(4)

(, ) . (, )dE , (,) = H(X,a) = E + dE..

F . «». [1], [2]
xk,, yk,, z
k

(k = l, 2, . .., ).

rk . N 3 :



68


qn(x1,. . . , zN) (n=1, 2, . . . , N). :

d L L 0 . dt q q k k

( k =1 , 2, . . . , 3N). (1)

L=K--U-- , ; -- ; U-- . :

H p k H pk q k qk H


k 1

3N

k 1,2,...,3N , L p k q k L, p k q k



(2)

-- , , a (q1 , q2 ,..., q3N ; p1 , p2, ,..., p3) -- . &, : qk=Xk , pk = Xk
+3N

{k=1, 2, ..., 3).

(3)

(Xl, 2 ..., 6N) (X), dXu dX2, ..., dX6N dX. , X t , 0 t = 0. . , . , , . , , , . , . ,



div div t

(4)

= const :

x y z div x y z

(5)

, , , . .

69



k 1

6N

X X

k k

0

(6)

, , , (2)


k 1

6N

X k X k


k 1

6N

qk pk q p k k




k 1

6N

2H 2H q p p q k k kk

0

(7)

. , , .

G. . [36], [3], [15]
. , ( , ) (q), q: ||2dq , dq . . . f, . , , . , . , , , . , , . (, ), ; , . , , , ; . , f . f fn, n 1, 2, 3, .... , f fn n. n f. , | n | 2 dq = 1. (1)

, f fn. , n -

70


fn , , , . = nn, (2)

, n -- . , , , , . , , . (2) ( ) fn f. , | |2 (2) fn f . fn ; ,


n

| an |2 1

(3)

, . . k k :

A k= k k
A ­ .

(4)

- . U(x, y, z) :

H

2 U ( x, y, z ) 2m

(5)

2 2 2 2 2 2 - x y z
. , , , , . . . . .

. 71




i

2 U ( x, y, z ). t 2m

(6)

(6) 1926 . . .


x



x x , p x
2



px p

x

2

,

. -- (x), ; , x .







d x dx 0 dx
2

2

-- . ,




x 2 dx (x) 2 ,
x d * dx d d * d 2 dx x x * dx dx 1, dx dx dx 2 d d 1 1 2 2 dx * 2 dx 2 * p x dx 2 p x , dx dx
2





x
2 2

p x 2

2

0

( ) , . x px /2 (7)

/2. (7) 1927 . , (. . ), , . , ( = y = z = 0), = = = pz = . , . , , .

I. . . [15], [29]
72


, . , , , ( ) . |a>, |b> . , z ( |+1/2> |-1/2>) |a> |b> |a>=a1(a) |+1/2>+a2(a) |-1/2>, |b>=a1(b) |+1/2>+a2(b) |-1/2>. , a |a>: (a ( a1( a ) | a ( a ) |2 a 1 1 ) a2a )* (a1(a)* , a2(a)* )= ( a1* ( a ) a = ( a(a) a ) a | a2a ) |2 2 1 2 b ( a1( b ) | a1(b ) | 2 a 1(1 ) a 2b )* (b)* (b)* b = a ( b ) (a1 , a2 )= a (b )* a (b ) | a (b ) | 2 2 2 1 2 b |b>:

, . , , . , |a>:

a

=

|a> < a|

a |> :

a |> = < a|> |a>,
< a|> , : < a|>= a dV
*

Nb ,

: Na , |a> , |b> . .

. ;

73


W a ( a ) 2 W a (b ) 2 a1 b1 =Wa a + Wb b = ( a )* ( a ) ( Wa a1 a2 Wb a1(b )*a2b

)

( ( Wa a1( a ) a2a )* Wb a1(b ) a2b )* (a) 2 (b ) 2 Wa a2 Wb a2

Wa=Na/N, Wb=Nb/N, N=Na+Nb. |±1/2>, { |±1/2>}-. |a>, |b>, .

1 0 a = 0 0 1 0 b = 0 0 : 0 W =Wa a + Wb b = a 0 W b
P0, , ( , , - , , ..). , , , . 1,...,k,- k; , k, - , P0=


k

pP k ,


k

p 1,

.., 0 - , . k , P0 (P0)kl=pkl . P0. A, - P0. . 74


N L N t L - : L=H-H=[H,], H - . i

A ­ , : =trA tr ­ . J. . [6], [3] « » (1), (2), ... . , , 1, 2, ... . a. (), « » , a X () . : a X ()a() X () « », 1 (1) + 2 (2)+... . a X [()][a() X ()] , , : « » ( - - ) . - . , , , , .. a() X () ||2. * ||2 . .

K. . [33], [2]
. fN*(X,t)= g(X-Y)fN(Y,t) 75

(y)


g(X)=1/ D(X/) D(x)= 1 for |X|<1 D(x)= 0 for |X|1 - « » f (x1,x2,t) -> f(x1,t)f(x2,t)

L. « » . [9], [12]
. , [9, 12]

~ 1
-1 1=1 ~

-1 . ~ , : ~~ ~ ~ tr tr ln d/dt0
~ ~ t = -1 L - -1(L)=+(-L) -1 . . P, . , P: ~ ~ ~ t

M. . , . 76


- , . , . , . , , . , , , , .

. 27. . (. [101])

N. . .



,

,

77


. . [3] (q, ) | > (q, p) , dqdp=l = - tr = l =|> <| ( 1)

H , t



i

[H , ] t

(q, )



dq dp

tr ()

O. . [29], [3]
, |0>, -- |> = ci|i>, |i>-- . 0=|> |0><0| <|. (5.40) , , , trA(0)=n |n> -- - . , trA(0)= |> <0|n><|=|><|, (5.41)

78


|n> . , , |>. , , , |>=cieii|i>|0>|. (5.42) 0=|> <|=cicj*ei(i-j)|i>|i>. (5.43) trA()=n |= =(ij)cicj* ei(i-j) |i> ); , trA()=|ci|2|i>. . 5.5 ( ). S , |i> S |i> A, trA() (S ) , , S, |>=ici|i>. , , , , .

P. .[3], [30]







, . , (, ) , . , |0> |0,s>; (5.40), 0=sps|> |0,s><0,s| <|. (5.46)

|0,s> |i> , |i,s>, , , i; iH / (|> |0,s>)=ii,s |> |i,s>. (5.47) , s. |0,s> , i,s(mod 2) 0 2. 79


(5.46) (5.47) , |>=ici|i> : =(s,i,j) pscicj* ei(i,
s-j,s)

|i> |i,s>
(5.48)

(5.48) (5.45), , (5.48) , . , («» , «» ). , , . . |i>|i,s> |j>|j,s> r s. tr (A) =(s,i,j) pscicj* ei(i,s-j,s) |i,s>= =(i,j) cicj* ai,js psei(i,s-j,s) (5.49)

i,s , s ij; , tr (A) = |ci|2aii= tr ('A). (5.50) '= |ci|2aii ps|i> |i,s>, , |i,s>, , |i,s>. , . 5.6. . , . -- , , ' -- , . , '. , , 5.6, , . -- -- [30].

R. . , . [3]
: -- , 0 1. , 0 = t0, t1, ..., tN = [0, T] . -- , tn 0. , , , max(pn+1 -- n) 0, 80


pN--p0 0

(5.52)

( , t = 0, t = T). . 0 , 0, P1= 1 -- 0 1. -- , tn. , ' = 0 0 + P1 P1 (5.53)

, , tn+1,
+1

= e-i

Hn

' ei

Hn

(5.54)

-- , = tn+1-- tn. , k |><|, +1 2k ; =|0><0|, , . (5.54),
+1

= ' ­ in[H, ' ]+O(n2).

(5.55)

02 =0, 0 P1 = 0, 0
+1

0 = 0 0 ­ in[0 H 0, 0 0]+O(n2).

(5.56)

, , tn 0, pn+1= tr ( +1 0) = tr(0 +1 0) = =tr(0 0) ­ in tr[0 H 0, 0 0]+O(n2). (5.57) , 02 = 0 , 00 |><| X tr (X |><|) = <||>= tr (|> <| ). (5.58) , (5.57) , pn+1= pn+O(n2). (5.59) ( = max n); k, pn+1- pn kn 2 kn (5.60)

p N p0


n 0

N 1

(p

n 1

p n ) k


n 0

N 1

n

kT 0

0. 81


S. -- -- . [3]
, , - : , , , , . , 1935 . , 0. 1/2, . . 1 (5.62) | | | | | 2 |> |> -- sz +1/2 --1/2 , (5.62) . , (5.62) , . , 1 | | | | (5.63) | 2 |> |> -- sx. , , , z - . , sz(e--) ; : +1/2, |>|>. , |> z sz(e+) --1/2. , , , , - . , , ( sz(e+)= --1/2) , . , z-, x- . (5.63) , |>|>, |>|>, x- sx(e+). . , sz(e+), sx(e+). , : - , . , , - «/», . 82


, , , , , () . , , |>|>, sz(e-) +1/2, |>|>, sx(e-) +1/2. , , |>, |> , , . , , , . , - , . (5.62) (5.63), ( )
tr | | 1 2 | | | |
e


(5.64)

e



1

| | | | 2

(5.65)

. . , 1/2, . , , . sz, |>, |> , (5.64). sx, |>, |> , (5.65), . . , . ( , sz sx) - , , . , , . . .

T. .[3]
, , , . , , , , , , . . , , , . 83


. , . . , , ( , , , , ). p(|) , . , F, , . , (5.75) pE F | = p ( | ) p F ( | ). , 1/2 , . , F . . , , , F ; ,
pE
F

( ) pE

F

( ) 0 (5.76)

() -- , , ; (5.76)

p

EF

( )



pE (| ) pF (| ) d



p

EF

( ) d

(5.77)

, , , . , () = 0, pE(|) = 0, F (|)=0. (5.78) , () = 0, pE(|) = 0, F (|)=0. (5.79) , (5.80) pE (| ) 0 pE (| ) 1 (5.78) -- (5.80) , () 0, 0, 1. , , , . , , , , , . , , , , ( «» «»). 1 , 1: , + --. b1, c1, a2, b2, 2. , ; 1=--a2. 84


, v. , . . b1=--b2 c1=--c2. , , b . ( =1, b =1) , b. (b = 1, = --1) = ( =1, b = 1, = --1) + ( = --1,b = 1, = --1) ( =1, b = 1) + ( = --1, = --1). (5.81) , , b , (b1 = 1, 2= 1) (1=1, b2=--1) + (1=--1, 2= 1). (5.82) , ; , , , . (5.82) , , . , c 1/2 , 0, ; , . , B, , , B , . P(b1 = 1, 2 = 1), (5.82); , 1 2 , , +1/2. 1 z; 1 1/2, 1 | >, 2 -- |>. , 2, |> |> ( x); , , + 1/2, | e iJ x | [cos 1 2iJ x sin 1 ] | 2 2 (5.83) cos 1 | i sin 1 | 2 2 , P(b1 = 1, 2 = 1) =1/2 |<+() | >|2=1/2 sin2( 1/2 ) (5.84) ( +1/2 1 1/2). (5.82)









(a1 = 1, b2 = --1) = 1/2 cos2 (1/2 ) (a1 = --1, 2 = 1) = 1/2s2[1/2( + )]. , (5.82) sin2 (1/2) cos2(1/2) + cos2 [1/2 ( + )], cos + cos + cos ( + ) -- 1, (5.85) = = 3/4. . 85


5.8 (). , , , . , , , (5.82). 1/2, , 0.

U. - -- . [3]
( , ) . , , ( ) [29], . , . , V(r). , t (r, t), q(t), 2 (5.69) i U ( x, y, z ). t 2m q

dq j (q, t ) dt (q, t )
j -- : j Im , = ||2. (5.71) m , t = 0 , (r, 0), q. , dV, q, (q, 0)dV; t (q, t)dV. , q , u = j/p (5.70).

u 0 t
..

(5.72)

j (5.73) t (r, t), (r, 0), j(r, t) p(r, t). = , (5.73) (3.42), , , (5.69). , q t = 0 , .
86


, , , (5.69), q , , ; |(q)|2dV dV q. , , , |(q)|2dV dV. q , (5.69), (5.70), , - . , . , , . q1 q2, (r1, r2). dq1 j1 dq2 j2 , , dt dt

j

Im 1 , j Im 2 , = ||2. m m

j1, , dq1/dt q2: . , , , V(r1, r2) . , , . , .

V. , . [58]

87


Figure 28. . . ( 1956) "" , , " (. 142). , , , , , , , , ; - -- , -- , , , , , -- !? , , . , (. 143):

88


Figure 29. « » . . . (""); (""); ("") , , . ( 144):

Figure 30. . : , . . 144 , " ": . , . " ":

Figure 31. 29. 89


. " ", ? . 146.

Figure 32. 29. , , "". , " ", " "? , . , . , , , -- , "" " ". , , , , , . -- -- . , . , , , , " " , . . .

.
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