Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15797/doc.pdf Дата изменения: Tue Oct 30 15:36:50 2007 Дата индексирования: Mon Oct 1 20:47:59 2012 Кодировка:

.., .. (. , .) (), . , . . , . - . FINITE-DIMENSIONAL MODELS OF NONLINEAR QUANTUM MECHANICS Yu.M. Aponin, E.A. Aponina (Pushchino, Moscow Region) Finite-dimensional models of quantum mechanics fall in to the complex dynamical systems (CDS), which behavior is determined by ordinary differential equations in a finite-dimensional complex state space. The alternative classes of nonlinear CDS, which inherit some of the features of linear quantum theory models, are introduced. The CDS with unitary nonlinearity are studied. A description is given of CDS with closed dynamical equation for density matrix. This class of systems is more extensive than the class of CDS with unitary nonlinearity.

376


.. .-- -10, 2002, .376-377

, :
ih z = F ( z ), z= ( z1 ,..., zn )T &

n,

(1)

= dz/dt, t ­ , ­ . z. (1) , , : i = A z, A = H + iD, (2) 20- ­ . H = [H], D = [D] ­ (nвn)- H, D . « » [1] , (D = 0) (2). , (D 0), , [2, 3]. (), . [4]. :
ih z = &
=1



n

A z + z

=1



n

B z

2



, =1,...,n,

(3)

, , [5]. - , , 377


3. (II)

(. [6], . 87 ­ 88). [7­ 10]. [7] [8], [9] [10]. (1), . , (2) = z z: (2) ih = A - A* . & (1), ? , , (2) D. , , , - (3)? . , , , (1), F = (F1,..., Fn)T, n, .. 2n n. , n z , z
378


.. .-- -10, 2002, .376-379

(. [11], .12). F F z =z ,

(nвn)- F z F z
F F z = z , , = 1,...,n,
T

: n

. = ,..., * z1 z n z
, K Kn (nвn)- K. iff " , ". (1) , n : Im Tr ( F/ z) < 0. (1) , n Im Tr ( F/ z) = 0. (1) , H: n , n F = H/ z*. H (1). . , , (1) , 2n. , (1) 2n. () () 2n. , . (1) . (1) n . z 379


3. (II)

, | z | 2 , . N = N(z) = z*z, z n. . (1) , N(z) , .. . N (1)
& hN =2Im z * F ( z ) , z n.

[

]

(4)

, (1) . . 1. (1) H ­ . (1) iff H(ei z) = H(z) , z n. . (5) (6) (7)

(, z) = H(ei z), , z n.
(6), ( , z ) = 2Im[ * F ( )], = ze i .

, (1) . (4) Im [z*F(z)] = 0 z n. (8) (7) / 0. , (, z) (0, z) (5). , (5), (, z) (0, z). & , / 0. (4) (7) N =0, .. (1) . . 2. (1) iff n
380


.. .-- -10, 2002, .376-381

F F F F z = z, z =z.

*

T

(9)

. (1) H: n . F = H / z*. , (9). , F 2H 2H F = = = , , = 1,...,n, z z z z z z .. (9). . : (9). (9) , = F1d z1 +...+ Fn d zn -, .. =0. , ­ (. [12], . 42) W: n , = W , ..
F = W / z , =1,...,n.

-. ,
(10)

= Re W Q = Im W, (10) (9) : 2 Q / z z =0 ( , =1,...,n ), , Q(z) ­ . , G, Q = Re G (. 2.14.1 [11], . 130). R = Im G. i Q / z = i (G -i R )/ z = R/ z . (10) F = H / z H = P + R. . 2 - (3). , n ­ (nвn)-. 3. (3) iff A* = A, BT = B n.
381

(11)


3. (II)

H ( z ) = z * Az +

1 2

, =1



n

B z z

2

2



(12)

(3). . (3) (9) :
A - A + ( B - B
( B - B




)z z +



=1



n

( B - B ) z 0 ,

2

(13) (14)

) z z 0

­ z n , = 1, ..., n. , (11). , (11), (12), , . , H / z* = F ­ (3). . (3)
h& * N = z Dz+ 2
, =1



n

(ImB ) z z

2

2



,

(15)

D = (A ­ A*)/2i ­ . , (3) :
F Tr z = Tr( A) +




n

B =1



+

=1



n

B z

2



.

(16)

(15), (16) 4. : (I) (3) iff A* = A, (B + BT) n. (II) (3) iff Im Tr (A) = 0, Im(B + B1 + ...+ Bn) = 0 = 1, ..., n. (III) (3) iff Im Tr (A) < 0, Im(B + B1 + ... + Bn) 0 = 1, ..., n. 3 4
382


.. .-- -10, 2002, .376-383

5. - (3) . , N(z) = z*z . , (1), F = (F1, ..., Fn)T, = { z n | | z1| < r1, ..., | zn| < rn }
F ( z ) =
a ,b



F
n

,a ,b

z a z b , z , =1,...,n,

(17)

z . r1 > 0, ..., rn > 0; N = {0,1,2, ...}; z =( z1 ,...,zn )T z = (z1, ..., zn)T
a a n; z a = z1 1 ...zn n z n, a Nn. :

A=

, =1



n





A

=

[ ]

n, A = [A] Nn.

, z = z z* U n: U = {Z n | | Z | < r r , = 1, ..., n}. . , (1) () iff (1)
ih =G ( ) = z z * , z &

(18)

G: U n, G =[G ]. 6. : (I) (1) , (18)
G ( ) =
A, B



G
n

, , A, B

A B , U , , =1,...,n ,

(19)

U .
383


3. (II)

(II) (1) :
ih z = ( zz * ) z + f ( z ) z , z , &

(20)

:U n ­ f: ­ ,
f ( z) =
a ,b



f
n

a ,b

z a z b , z,

(21)

z. (I) (II)
( z z * )z + f ( z )z = F (z ), z,

(22)

( ), f ( z ), . (II) (I) : (20) (18),
G ( ) = ( ) - ** ( ), U .

(23)

(23) , ( , f ) - (20) (18) f. (20) f 0:
ih z = ( z z * )z, z. &

(24)

. , (24) z(t) = exp(­ i t), n . ( *) = , , n, (25) .
384


.. .-- -10, 2002, .376-385

(24) ­ . , "" (24): E(z) = Re[z* (zz*) z], z n. , , E(z) N(z) = z*z : E = N. , (, f)- (20) , , (,0)- (24). , z(t) (20) z(t) = (t)exp[­i (t)], (t) ­ (24),

& (t) h = f ( e ) . , (24) , .
INTAS 97 ­ 30804.

- i

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385


3. (II)

6. 7. 8.

9. 10. 11. 12.

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