Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hbar.phys.msu.ru/hbar/QOS.pdf Äàòà èçìåíåíèÿ: Tue Feb 17 00:00:00 2004 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:43:04 2012 Êîäèðîâêà:

..
1

0.2 14 2004 .

1

farid@hbar.phys.msu.ru


2




1 1.1 . . . . . . 1.1.1 . . . . . . . . . . . . . . 1.1.2 . . . . . . . . . . . . . . . . 1.1.3 - . . . . . . . . . . . . . 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 . . . . . . . . . 1.2.2 . . . . . . . . . . . . . . . 1.2.3 1.3 . . . . . . . . . . . . . . . . . . . 1.3.1 . . . . . . . . . . . . . . . . . 1.3.2 . . . . . . . . . . . . . 1.3.3 . . . . . . . . . . . . . 1.4 . . . . . 1.4.1 . . 1.4.2 . . . . . . . . . . . . . . . 1.4.3 . . . . . . . . . . . . . 1.5 . . . . . . . . . . . 1.5.1 ? . . . . . . . 1.5.2 . . . . . 1.5.3 . . . . . . . . . . . . . ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 4 5 6 8 8 9 11 13 13 14 15 18 18 19 21 24 24 24 26 28 28 28 30 31 32 34 34 36 38 38 39 40 42 44 44 46 48 48 48 49

2 2.1 . . . . . . . . . . . . 2.1.1 . . . . . . . . . . . . . . . . . . . . 2.1.2 . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 . . . 2.1.4 . . . . . . . . 2.2 . . . . . . . . . . . . . . . . . 2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 . . . . . . . . . . . . . . . . . . 2.3 . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 . . . . . . . . 2.3.4 . . . . . . . . . . 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 . . . . . . . . 2.4.2 . . . . . . . . . . . . . . . . . . . . . . .

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

3 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 . . . . . . . . . . . . . . . 3.1.2 . . . . . . . . . . . . . . . . . . . .


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 . . . . . . . . . . . . . . . . . . . 3.1.5 . . . . . . . . . . . . . . . . . . . 3.1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 . . . . . . . . . . . . . . . . . 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 . . . . . . . . . . . . . . . . . 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3

3

3.2 3.3 3.4 3.5

51 53 54 55 57 60 60 61 65 65 66 68 69 69 69 71 73 73 74 75 76 79 79 80 82 85 85 86 87

4 4.1 . . . . . . 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 . . . . . . . . . . . . 4.2 . . . . . . . . . . . . . . . . . . . . . . 4.2.1 . . . . . . . . . . . . . . . 4.2.2 . . . . . . . . . . . . . . . . . 4.2.3 . . . . . . . . . . . . . . . 4.2.4 . . . . . . . . . . 4.3 . . . 4.3.1 . . . . 4.3.2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 . . . . . . . 4.4 - . . . . . . . . . . . . . . . . . 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 . . . . . . . . . . . . . . . . . . . 4.4.3 . . . . . . . . . . . . . . . . . . 5 5.1 . . . . . . 5.1.1 5.1.2 5.2 . . . . 5.2.1 . . . . . . . . . . . . . . . 5.2.2 . . . . . . . . . 5.2.3 . . . . . . . . . . . 5.2.4 . . . . 5.2.5 : ...... ...... ...... ...... ...... ...... ...... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 . 89 . 89 . 91 . 94 . 94 . 96 . 98 . 99 . 101


4

1

1
1.1
1.1.1





. x p. , wx (x) wp (p), 1 . , : wx (x) = (x - x0 ), wp (p) = (p - p0 ). (1.1.1)

. , x: x=


xwx (x) dx

(1.1.2)

" ": x (x) = wx (x) e
i(x)

-

(1.1.3)

x (x) - , . (1.1.2) " ": x=


(x)xx (x)dx. x

(1.1.4)

wx wp "". :


-

wx (x) dx =

- , , . (. . 1). "" , , "" x. wapr (x). , x , ; . , - ( , (1)), , "" , :
x p , , , , - -
1

-

-



wp (p) dp = 1,

(1.1.5)


1.1

5

apr ........................ ......................... ..... ...... ...... ..... .... ..... ..... .... x .... ... .... ... .... ... .... .... .. ... ... ... ... .... ... .... .... ... .... ... ... ... ... ... ... . .. ... .... .... .... .... .... .... .... .... .... . . ... .... ... .... ... . ... .. .... ... ... ..... .. .. ..... ...... ..... ..... ....... .... ........ .... ......... ........ ..... ........ .... ....

w

(x)

-

x

(0)

.. ...... ..... .... ... ... ... ... ... ... ... . .. .. .. .. .. .. .. .. .. .. . . .. .. .. .. .. .. .. .. .. ..

(1) ?

(2)
(1)

(3)

wx (x)



x -

. 1: w

apr x

(x)

w (x) = x

(1) x

w

1

, x x , x x ,

(1.1.6)

0,

, , w1 =
x

w

apr x

(x) dx

(1.1.7)

. w1 , , , . . 1.1.2

x (x), - (, ) x . wx (x) = 2 (x) x (1.1.8)

, :




-

2 (x)dx E. x :

(1.1.9)


6

1

k - . 1/ 2 ,


1 k (k) = 2

-



x (x)e

-ikx

dx,

(1.1.10)

(x)dx =

2 x

, , , , , . , x (x) k (k) . , - " ": 1 . (1.1.12) 2 , , "" x xk E =
x

-

-



|k (k)|2 dk

(1.1.11)

|2 (x)| dx , x

(1.1.13)

. 1.1.3 -

, . , w x (x) wp (p). , "", - . , , . : dx, (1.1.14) 2¯ - h , p = ¯ k. h , . ( , ). : . , (, x0 ), x (x) (x - x0 ). (1.1.14) , . , p (p) = x (x)e
-ipx/¯ h

1



xp

¯ h . 2

(1.1.15)


1.1


7



1: , 1. . - , . , , .


8

1

1.2
1.2.1




"" , , (). , , . , |x| < L/2, 1 (x) = L 1 |x| < L/2 0 |x| L/2 (1.2.1)

f(p). (1.1.14) :
L/2 pL 1 ¯2 h h sin , e-ipx/¯ dx = 2L p 2¯ h 2¯ L -L/2 h :

p (p) =


(1.2.2)

f(p) =

f(p)|p (p)|2 dp =

. , , ( , , , ; ). : x , . f(p) = p:


-

¯ h 2L

-



f(p)

4 pL sin2 dp. 2 p 2¯ h

(1.2.3)

p= = 1 2¯ h

(p)pp (p)dp = p (x)x (x )e x
-

-

-

1 2¯ h

ip(x-x ) ¯ h

p dpdxdx =

=



-



(x - x ) ¯ h dxdx x (x)x (x ) - i - x ¯ x (x) h (x) (x)px (x) dx , (1.2.4) ^ dx = x x i x - p= ^ ¯ h i x (1.2.5)

-



-



(x)e x

ipx ¯ h

dx p

-



x (x)e

-

ipx ¯ h

dx dp

- . p: p
n

=

-



(x)pn x (x) dx , ^ x

(1.2.6)


1.2

9

pn = ^

¯ h i

n

n , xn

(1.2.7)

f(p), : f(p) = ^


f

n

n=0

¯ h i

n

n . xn

(1.2.8)

, , , . ¯ x (x) h (1.2.9) i x p-. , , xx (x) = xx (x), ^ px (x) = ^ xp (p) = i¯ ^ h p (p) , p pp (p) = pp (p). ^ (1.2.10)

, . . , : |p| ^ 1.2.2




-

(p)pp (p) dp p

-



(x) x

¯ h x (x) dx . i x

(1.2.11)

( ). : x (x), , , . , . (1.2.11) :


(x) x

, , x (x)


-

¯ x (x) h dx i x



=

-



x (x)

¯ (x) h x dx . -i x
±

(1.2.12)

= 0: (1.2.13)

x (x)

. , , x. . .

-

¯ x (x) h dx = - -i x

-

¯ x (x) h x (x) dx -i x


10

1

, , , , . ^ Q. | , | . ^ |Q|
^ x (x)Qx x (x) dx .

(1.2.14)

^ ^ Qx Q, . , kx2 ^ p2 ^ ^ + , Q= 2m 2 (1.2.15)

-



¯ 2 d2 h kx2 ^ Qx = - + . 2m dx2 2 (1.2.14):


(1.2.16)

^ (x)Qx x (x) dx x



=

(1.2.17) ^ , Qx . : ^ |Q|


-

-



^x x x (x)Q (x) dx

-



^x x Q (x) x (x) dx .

^ = |Q | .

(1.2.18)

[. (1.2.17)], (1.2.18) . , , ^ ^ . Q ( ) , Qx -, -, "", . , ^ ^ |Q | = |Q| , , ,


(1.2.19)

^x x Q (x) x (x) dx =

( , ). . , . . , a, ^ .

-

-



^ (x)Qx x (x) dx . x

(1.2.20)


1.2 1.2.3

11

. , , : ^ Q|q = q|q . (1.2.21)

q |q , , q |q j (j = 1, 2, ..., N), N - . : p(x) = ^ : ¯ (x) h = p(x) (x) = exp i x ^ H| = E| ipx ¯ h (1.2.22)

(1.2.23)

. . 1. . : ^ q|Q|q = q| q|q = q q|q = q. (1.2.24)

: ^ ^ q|Q |q = q|Q|q = q. , q = q.


(1.2.25)

2. . |q , |q q, q . ^ Q|q = q|q ^ q |Q = q|q

(1.2.26)

q |, |q , : 0 = (q - q ) q |q (1.2.27)

, q = q , q |q = 0. , . ^ , a | .


12

1

. , , . , . , {q (x)} , : (x) =
q

q q (x) ,

(1.2.28)

, , | =
q

q |q ,

(1.2.29)

q - . , , , (, ). , , . : | =
q

q| |q =
q

|q q| | ,

(1.2.30)

q| =
q

(1.2.31)

. , | {|q }, |q q| = 1 ,
q

(1.2.32)

. , . , - .


1.3

13

1.3
1.3.1




, 30- , , , . , ( , ). , , . 1. | , ^ Q Q, q | q| |2 . ( !), , - , -. , - . 2. , , , , |q . , . : (1) , , | ; (2) , , |q . , . , . ( ). , | q| |2 =
q q

|q q| = |
q

|q q| | = 1.

(1.3.1)

[ (1.2.32)].


14

1

apr ........................ ......................... ..... ...... ...... ..... .... ..... ..... .... x .... ... .... ... .... ... .... .... .. ... ... ... ... .... ... .... .... ... .... ... ... ... ... ... ... . .. ... .... .... .... .... .... .... .... .... .... . . ... .... ... .... ... . ... .. .... ... ... ..... .. .. ..... ...... ..... ..... ....... .... ........ .... ......... ........ ..... ........ .... ....



(x)

-

x

^ E

0

.. ...... ..... .... ... ... ... ... ... ... ... . .. .. .. .. .. .. .. .. .. .. . . .. .. .. .. .. .. .. .. .. ..

^ E

1

?

^ E

2

^ E (x)

3



ap ost x



x -

. 2: 1.3.2

, . "" (. . 2). , x , x. , , . , apr x (x) , x x , ap ost w1 (1.3.2) x (x) = 0, x x , w1 =
x

|

apr x

(x)|2 dx

(1.3.3)

, , . . : |
1

=

^ E1 |apr , w1

(1.3.4)


1.3 w1 =
apr

15

^^ |E1 · E1 | ^ E1 =

apr



apr

^ |E1 |

apr

,

(1.3.5)

|x x| dx
x

(1.3.6)

. ^^ ^ (1.3.5) , E1 · E1 E1 . , , ^ |apr , Ej , j . . {q} {q}1 , {q}2 , {q}3 .., . ^ Ej =
q{q}
j

|q q| .

(1.3.7)

, | , : 1. j ^ wj = |Ej | . (1.3.8) 2. , , j, ^ Ej | |j = wj 1.3.3 (1.3.9)

(1.3.7) , {q} ^ E
{q}

=
q{q}

|q q|

(1.3.10)

| , . : {q1 } {q2 } , ^ E
{q1 }

, (1.3.7), , :

^ ·E

{q2 }

= 0,

^ E

{q1 }

^ ·E

{q1 }

^ =E

{q1 }

,

^ E

{q2 }

^ ·E

{q2 }

^ =E

{q2 }

.

(1.3.11)


16

1

Ej =
j q

|q q| = 1 .

(1.3.12)

, , : wj = 1 .
j

(1.3.13)

. , .. , .. . , , , , , ( , , , ). . , , : , , , ( ) . 60- 70- , . . , . . | - : ax | ay , (1.3.14) az -
x

| a

a

y

a

z

,

(1.3.15)

- , . 3 â 3. : , ( ): 100 000 000 ^ ^ ^ Ex = 0 0 0 , Ey = 0 1 0 , Ez = 0 0 0 , (1.3.16) 000 000 001 , ( ):


1.3 100 = 0 1 0 , 000 100 = 0 0 0 , 001 000 = 0 1 0 . 001

17

^ E

xy

^ E

xz

^ E

yz

(1.3.17)


18

1

1.4
1.4.1




, : d|(t) ^ = H|(t) , (1.4.1) dt ^ H - . , , : i¯ h ^ |(t) = U(t)|(0) , (1.4.2)

^ U(t) . , . , d|(t) d (t)| d (t)|(t) = (t)| + |(t) . (1.4.3) dt dt dt d| /dt , : ^ ^ d (t)|(t) H|(t) (t)|H = (t)| + |(t) dt i¯ h -i¯ h 1 ^ ^ (t)|H|(t) - (t)|H|(t) = i¯ h , ^ ^ d d dU (t)U(t) ^ ^ (t)|(t) = (0)|U (t)U(t)|(0) = (0)| |(0) . dt dt dt , ^ ^ dU (t)U(t) = 0. dt ^ , U(0) = 1, , ^ ^ ^ ^ U (t)U(t) = U (0)U(0) = 1 . ^ ^ U (t) = U-1 (t) . (1.4.8) (1.4.6) (1.4.5)

0 . (1.4.4)

(1.4.7)

, , . , ( , ). ,


1.4

19

^ ^ U = e i ,

(1.4.9)

^ . , i¯ h ^ dU(t) ^^ = HU(t) . dt (1.4.10)

^ , , H = const, ^ Ht ^ U(t) = exp i¯ h (1.4.11)

, , , : ^ U(t) =


n=0

^ 1 Ht n! i¯ h

n

.

(1.4.12)

, : ^ U(t) = exp 1 i¯ h
t 0

^ H(t ) dt

(!) .

(1.4.13)

, . (1.4.10) . , . - , . . ^ Q - , Q - . ^ ^^ Q(t) = (0)|U (t)QU(t)|(0) . (1.4.14)

, : ^ ^^ ^ (0)|U (t) Q U(t)|(0) = (t)|Q|(t) , (1.4.15)

^ |(t) = U(t)|(0) , . 1.4.2

: ^ ^ ^^ (0)| U (t)QU(t) |(0) = (0)|Q (t)|(0) , (1.4.16)


20

1

^ ^ ^^ Q (t) = U (t)QU(t)

(1.4.17)

^ Q . , . ^ Q (t). , ^ Q ( ) ( ): ^ ^ ^ ^ dQ (t) dU (t) ^ ^ Q ^ ^ ^ ^ dU(t) . U(t) + U (t) Q Q U(t) + U (t) = dt dt t dt : (1.4.18)

^ ^ ^^ ^ ^ U (t)H ^ ^ Q ^ dQ (t) ^ ^ ^ HU(t) , = Q U(t) + U (t) (1.4.19) U(t) + U (t) Q(t) dt -i¯ h t i¯ h , , : ^ ^^ ^ ^ ^ ^ ^ U (t)HU(t) ^ ^ ^ dQ (t) Q ^ ^ ^ ^ ^ U (t)HU(t) , = U (t)QU(t) + U (t) U(t) + U (t)Q(t)U(t) dt -i¯ h t i¯ h (1.4.20) i¯ h , , ^ ^ Q (t) dQ (t) ^ ^ = i¯ h + Q (t), H(t) , dt t (1.4.21)

^ ^ Q ^ Q (t) ^ = U (t) U(t) . (1.4.22) t t (1.4.21) . , ^^ ^ . P Q , K : ^^ ^ P, Q = K . : ^^ P , Q ^ ^^ ^ ^ ^ ^ ^ ^ ^ = U P U · U QU - U QU · U ^ ^^^ = U P QU - ^^ PU ^ ^^ ^ ^ ^^^ ^ ^ ^^ U QP U = U P , Q U = U KU = K . (1.4.24) (1.4.23)



. , K ( ), . , [x , p ] = i¯ . h , m, F(t).


1.4 p2 ^ ^ H= - F(t)x , ^ 2m x p , . : x , p ^^ dx ^ = x , H = ^^ dt 2m : i¯ h
2

21

(1.4.25)

.

(1.4.26)

x p2 - p 2 x = x p2 - p x p + p x p - p 2 x ^ ^ ^^ ^ ^ ^ ^ ^ ^^^ ^^ = (x p - p x ) p + p (x p - p x ) = i¯ p + p i¯ = 2i¯ p . (1.4.27) ^^ ^^ ^ ^ ^^ ^^ h^ ^h h^ , dx ^ p ^ = . (1.4.28) dt m , : dp ^ = F(t) . (1.4.29) dt , , , , -, . , , . , , : x = x + ^ ^ 1 ^ pt + m m
t

F(t )(t - t ) dt .
0

(1.4.30)

, , x , . 1.4.3

.. . (, ): ^ ^ ^ H = H0 + H1 , (1.4.31)

^ ^ U0 , H0 , i¯ h ^ dU0 (t) ^^ = H0 U0 (t) . dt (1.4.32)


22

1

^ U ^ ^ ^ U(t) = U0 (t)U1 (t) , : d^ ^ ^^ ^ U0 (t)U1 (t) = HU0 (t)U1 (t) . (1.4.34) dt , , : i¯ h ^ ^ dU0 (t) ^ dU1 (t) ^^ ^ ^^ ^ = H0 U0 (t)U1 (t) + H1 U0 (t)U1 (t) . U1 (t) + i¯ U0 (t) h^ dt dt (1.4.33)

i¯ h

(1.4.35)

(1.4.32), : i¯ h ^ dU1 (t) ^ =H dt
1

^ (t)U1 (t) ,

(1.4.36)

(1.4.37) ^ ^ H1 " " , U0 , ^ 0 H. ^ H . , :
1

^ H

^ ^^ (t) = U (t)H1 U0 (t) 0

^ ^ ^ ^^ (0)|U (t)QU(t)|(0) = (0)| U0 (t)U1 (t)

^ ^^ Q U0 (t)U1 (t) |(0)

^ ^ ^ ^^ ^ = (0)|U (t) · U (t)QU0 (t) · U1 (t)|(0) = (t)|Q(t)| (t) , (1.4.38) 0 1 ^ Q (t) , ^ ^ , H0 H: ^ ^ Q (t) dQ (t) ^ ^ = i¯ h + Q (t), H0 , (1.4.39) dt t | (t) ^ 1 : H d| (t) ^ = H1 (t)| (t) (1.4.40) dt , : , . . , ^ 0 , H i¯ h


1.4

23

^ , H1 . : 1 i¯ h
t 0

| (t) = |(0) +

^ dt1 H +

1

(t1 )|(0)
2 t t
1

1 i¯ h

dt
0

1 0

^ dt2 H

1

^ (t1 )H

1

(t2 )|(0) + . . . (1.4.41)

(, , ). ^ H0 ^ 1 , H . " ".


24

1

1.5
1.5.1


?

. , , , . , , . , p , , , ^ sin ap = sin a ^ (a ¯d h i dx ? (1.5.1)

, ). , : , . , p= ^
n

¯ h i

n

dn . dxn

(1.5.2)

, : x , f(x) = ^ Q ^ f(Q) =


fn x n ,

(1.5.3)

n=0

^ f n Qn ,

(1.5.4)

n=0

, , sin ap = ^


n=0

(-1)n a (2n + 1)!

2n+1

¯ h i

2n+1

d2n+1 . dx2n+1

(1.5.5)

, , (, ). , . 1.5.2 , ^^ ^ Q, P , Q = 0 ^^ ^ Q, P , P = 0 . (1.5.6)

^^ Q P

. . , . , , ,


1.5

25

, . , , , . . 3. n , ^^ Q, P
n

^ = nP

n-1

^^ Q, P .

(1.5.7)

. : ^^ ^^ = QP n - P n Q ( ^^^ ^^^ = Q, P P n-1 + P Q, P
n

^^ Q, P

^^ ^^^ ^^^ ^ ^^ ^ ^ = QP n - P QP n-1 + P QP n-1 - · · · + P n-1 QP - P n Q n - 1 ) ^ ^ ^^^ ^ ^^ P n-2 + · · · + P n-2 Q, P P + P n-1 Q, P ( n ) ^^ ^ = nP n-1 Q, P . (1.5.8)

^^ ^ ( Q, P P ). 4. f(x) , ^ df(P ) ^ ^ ^^ Q, f(P) = Q, P . ^ dP (1.5.9)

^ . f(P ) :


^^ ^ Q, f(P) = Q,

^ fn P

n

=

f

n

^^ Q, P

n

=

^ fn nP

n-1

n=0

n=0

n=0

^ df(P ) ^ ^ ^^ Q, P . Q, P = ^ dP (1.5.10)

5.

,
^^ eP Qe ^ -P

^^ ^ = Q + P, Q .

(1.5.11)

.
^^ eP Qe ^ -P

=e

^ P

^ Qe

^ -P

-e

^ -P

^ Q+e

^ -P

^ Q =e

^ P

^ Q, e

^ -P

^ +Q

( 4) =e
^ P

de ^ dP

^ -P

^ ^^ ^ ^ Q, P + Q = Q - eP e

^ -P

^^ ^^ ^ ^^ ^ Q, P = Q - Q, P = Q + P , Q . (1.5.12)

6. e
^^ Q+P

^^1 = e P eQ e 2 [

^^ Q,P

^^ ] = e Q eP e

-

1 2

[

^^ Q,P

]

(1.5.13)


26

1

, . , . e
^^ Q+P

= eQ e

^P ^

(!)

(1.5.14)

. , , :


n=0

^^ (Q + P ) n = n!

m=0 n=0





^^ Qm P n m! n! ).

(1.5.15)

^ ^ ( Q P , 1.5.3 , ^ ^^ ^ ^^ P -1 Qn P = P -1 QP
n

7. n

.

(1.5.16)

. : ^ ^ ^ ^ ^^ ^ ^ ^ ^ ^^ ^ ^^ ^^ ^^ P -1 Qn P = P -1 QQ . . . QP = P -1 Q(P P -1 )Q(P P -1 ) . . . (P P -1 )QP ^^ ( n - 1 P P -1 = 1) ^ ^^ ^ ^^ ^ ^^ ^ ^^ n = (P -1 QP )(P -1 QP ) . . . (P -1 QP ) = P -1 QP . ( ) (1.5.17) 8. f(x) , ^ ^^ ^ ^^ P -1 f(Q)P = f(P -1 QP ) . (1.5.18)

. , 4, ^ f(Q) : ^ ^^ ^ P -1 f(Q)P = P
-1

^^ f n Qn P =

n=0

n=0



f

n

^ ^^ P -1 QP

n

^ ^^ = f(P -1 QP ) .

(1.5.19)

^ , P ^ ^ U. f(Q) ( ) ^ ^ ^^ ^^ ^ U(t) f(Q)U(t) = f U(t) QU(t) = f Q (t) . 9.
^ ^ eP Qe ^ -P

(1.5.20)

, + ···+ xn ^ ^^ P , . . . , P , Q . . . + . . . (1.5.21) n!
n

2 ^ ^ ^ ^ ^^ = Q + P, Q + P, P, Q 2


1.5
^^ . eP Qe ^ -P

27 :
^ -P =0

^ ^ eP Qe

^ -P

^ ^ = eP Qe

^ -P =0

+

^^ deP Qe d

^ -P =0

+· · ·+

^^ n dn eP Qe n! dn

+ . . . (1.5.22)

:
^^ ^ deP Qe-P ^ ^ ^ ^ ^ ^ ^^ ^^ ^^ = eP P Qe-P - eP QP e-P = eP P , Q e-P , d ^^ ^ d2 eP Qe-P d ^^ ^^ ^ ^^^ ^ eP P , Q e-P = eP P , P , Q e-P , = 2 d d

(1.5.23) (1.5.24)

. , ,
^^ dn eP Qe dn ^ -P

=e

^ P

^ ^^ P, . . . , P, Q . . . e
n

^ -P

,

(1.5.25)

. ^ ^ , P Q , , 5.


28

2

2
2.1




, , 20-30 . , . , ( ), . 2.1.1

, . kq2 ^ p2 ^ ^ + , (2.1.1) H0 = 2m 2 q p , m , k . m ^^ k , : 0 = k/m = km. : 0 p 2 ^ ^ H0 = (2.1.2) + q2 . ^ 2 . LC- , , . ( ) , , : dq (t) ^ 0 = p (t) , ^ (2.1.3a) dt dp (t) ^ = -0 q (t) . ^ (2.1.3b) dt q p ip , (2.1.4) q p : X=q+ dX(t) = -io X(t) . (2.1.5) dt , , , , .


2.1

29

, ^ ^ X. X a= ^ q0 = ¯ / h (2.1.7) ^ X 2q
0

=

q + ip/ ^ ^ , 2q0

(2.1.6)

.. ( ). a ^ a+ = ^ q - ip/ ^ ^ . 2q0 (2.1.8)

a a+ : ^^ [a, a+ ] = 1 . ^^ (2.1.9)

: q0 q = (a + a+ ) , ^ ^^ 2 q0 p = (a - a+ ) . ^ ^^ i2 (2.1.10)

(2.1.1), , ¯ 0 h ^ (aa+ + a+ a) ¯ ^^ ^^ h H= 2
0

n+ ^

1 2

,

(2.1.11)

n = a+ a . ^ ^^ a a+ : ^^ [a, n] = aa+ a - a+ aa = (aa+ - a+ a)a = [a, a+ ]a = a . ^^ ^^ ^ ^ ^^ ^^ ^ ^^ ^^ ^ ^ , [a+ , n] = -a+ . ^^ ^ , : (2.1.14a) (2.1.14b) (2.1.13) (2.1.12)

na = a(n - 1) , ^^ ^ ^ na+ = a+ (n + 1) , ^^ ^^

an = (n + 1)a , ^^ ^ ^ + a n = (n - 1)a+ . ^^ ^ ^


30 2.1.2

2

, . q = 0, p = 0. , , , , . , . 0 2 p2 + q 0 2 p2 + q 0 2 2 p +

E=

2

=

2

+

2 q

.

(2.1.15)

, , , q = 0, p = 0. , p ¯ h . 2q (2.1.16)

, , : q (q) =
4

1 2
2 q

exp -

q2 42 q

.

(2.1.17)

, , E= 0 2 ¯2 h + 42 q
2 q

.

(2.1.18)

q0 q = , 2 ¯ 0 h (2.1.20) 2 , , , 2 . q0 . , , E0 = q 0 (q) = q|0 = 1
4

(2.1.19)

q

2 0

exp -

q2 2q2 0

.

(2.1.21)


2.1

31

, , ¯ 0 /2, h , , , , : ¯0 h ^ H|0 = |0 , 2 2.1.3 (2.1.22)

n|0 = 0 , ^

a|0 = 0 . ^



|n , , e , n : n|n = n|n . ^ (2.1.23)

a|n a+ |n . ^ ^ (2.1.14) : n a|n = a(n - 1)|n = a(n - 1)|n = (n - 1) a|n , ^^ ^^ ^ ^ + + + n a |n = a (n + 1)|n = a (n + 1)|n = (n + 1) a |n . ^^ ^^ ^ ^
+

(2.1.24a) (2.1.24b)

^ ^ , a|n a+ |n , , , n - 1 n + 1, : a|n = Cn |n - 1 , ^ a+ |n = Dn |n + 1 . ^ : n|a+ · a|n = n|Cn · Cn |n = C2 , ^^ n (2.1.26a) (2.1.26b) (2.1.25a) (2.1.25b)

2 n|a · a+ |n = n|Dn · Dn |n = Dn . ^^

, n|a+ · a|n = n|n|n = n , ^^ ^ + n|a · a |n = n|(n + 1)|n = n + 1 . ^ ^ , Cn = n, Dn = n+1. (2.1.28) (2.1.27a) (2.1.27b)


32

2

(2.1.25) , , a2 |n , a3 |n , . . . (a+ )2 |n , (a+ )3 |n , . . . ^ ^ ^ ^ n - 2, n - 3, . . . n + 2, n + 3, . . . (, : , , ). , ¯ 0 /2, , , . , h . , n, n - 1, n - 2, n - 3, . . . , -. , n , a|0 = 0. ^ , . : 1 |2 = (a+ )2 |0 , ^ 2 1 |n = (a+ )n |0 , ^ n!

|1 = a+ |0 , ^

...

...

(2.1.29)

, |n : n (q) = 2.1.4 1 2n n! q d q - q0 q0 dq
n

exp -

0

q2 2q2 0

.

(2.1.30)



, , ^ U0 (t) = e
^ H0 t i¯ h

=e

-io tn -io t/2 ^

e

.

(2.1.31)

, , , ^ ^ U0 (t) = R(0 t) , ^ R() = e
-in ^

(2.1.32)

.

(2.1.33)

, , , , ^ a. U (t)aU0 (t) , ^ ^^ 0 :


2.1

33

1 da (t) ^ ^ = a (t), H ^ dt i¯ h , a (t) = ae ^ ^ , ,

0

= -io a (t) . ^

(2.1.34)

-io t

,

(2.1.35)

^ R ()aR() = ae ^^ ^ :

-i

.

(2.1.36a)

^ R ()a R() = a ei . ^^ ^

(2.1.36b)

, : q0 p ^ ^ R ()xR() = ae-i + a ei = q cos + sin , ^^ ^ ^ ^ 2 q0 ^ R ()pR() = ae-i - a ei = -p sin + p cos . ^^ ^ ^ ^ ^ i2 (2.1.37a) (2.1.37b)

: {x, p/} = o t .


34

2

2.2



. , . , , . , . , . 2.2.1 -

, , , , F(t). : o ^ H= 2 p2 ^ + o q ^ o
2

- F(t)q . ^

(2.2.1)

(1.4.21), , : dq (t) ^ 0 = p (t) , ^ dt dp (t) ^ = -0 q (t) + F(t) . ^ dt ( ): 1 p ^ sin o t +
t

(2.2.2a) (2.2.2b)

q (t) = q cos o t + ^ ^

F(t ) sin o (t - t ) dt ,
0 t

(2.2.3a) (2.2.3b)

p (t) = -q sin o t + p cos o t + ^ ^ ^
0

F(t ) cos o (t - t ) dt .

, , . . (2.2.3): 1 p sin o t + q(t) = q cos o t + p(t) = - q sin o t + p cos o t +
0 t

F(t ) sin o (t - t ) dt ,
0 t

(2.2.4a) (2.2.4b)

F(t ) cos o (t - t ) dt .

, , , .


2.2

35

, (2.2.3), (2.2.4). 1 qp + pq - q p , ^^ ^^ (2.2.5) 2 (2.2.3) , (2.2.4). :
qp

=

2 qp p (t) = cos o t + 2 sin2 o t + sin 2o t
2 q 2 q 2

=

1 2

2 + q

2 p 2

+

1 2

2 - q

2 p 2

+

qp sin 2o t ,

(2.2.6a)

2 (t) = 2 2 sin2 o t + 2 cos2 o t - qp sin 2o t p q p 1 22 1 22 qp = sin 2o t , q + 2 - q - 2 - p p 2 2 1 qp (t) = 2 2 p -
2 q

(2.2.6b) (2.2.6c)

sin 2o t +

qp

cos 2o t .

. 1. F(t). , ( , ) . 2. 2o ,
2 q max



2 q min

=



2 p max


4

2 p min



=

2 2 - qp 2

2 qp

¯2 h . 42

(2.2.7)

3.

2 = q

2 p , 2



qp

= 0,

(2.2.8)

. , , , ( ).


36

2 4. , (2.2.7), , , ¯ h . 2 , q p = q = (2.2.9)

p q0 (2.2.10) = . 2 , |0 . , . . 2.2.2

( ) , qC qS . : qC (t) = q (t) cos o t - ^ ^ p (t) ^ sin o t , p (t) ^ qS (t) = q (t) sin o t + ^ ^ cos o t . (2.2.11a) (2.2.11b)

: ^ ^ ^ qC (t) = R(0 t) q (t) R (0 t) , ^ p (t) ^ ^ ^ qS (t) = R(0 t) ^ R (0 t) . (2.2.12a) (2.2.12b)

"" ( , ) , , , . : (2.2.11) . (2.2.11) (2.2.3) , , : 1 qC (t) = q - ^ ^ p1 ^ qS (t) = + ^
t

F(t ) sin o t dt ,
0 t

(2.2.13a) (2.2.13b)

F(t ) cos o t dt .
0


2.2

37

, (2.2.4, 2.2.6) : 1 qC (t) = q - p 1 qS (t) = +
t

F(t ) sin o t dt ,
0 t

(2.2.14a) (2.2.14b)

F(t ) cos o t dt ,
0

2 2 C (t) = q , q

(2.2.15a) (2.2.15b) (2.2.15c)

2 S (t) = q
qC p
S

2 p 2

,

(t) =

qp .

, , ( , , ) .


38

2

2.3
2.3.1


-

, , ? , . , , , , , , . , . (2.2.1) , F(t) (2.1.1) , : ^ H1 = -F(t)q . ^ (2.3.2) ^ ^ ^ U0 , H0 , U1 , . , : i¯ h ^ dU1 (t) ^ = -F(t)q0 (t)U1 (t) , ^ dt (2.3.3) ^ ^ ^ U(t) = U0 (t)U1 (t) , (2.3.1)

p ^ sin o t . (2.3.4) , 0 t , , , (2.3.3), . , , : q0 (t) = q cos o t + ^ ^ ^ ^ ^ U1 () U1 (0) - F(0)q0 (0)U1 (0) exp ^ i¯ h ^ , U1 (0) = 1. 2 : ^ U1 (2) exp i ^ F()q0 () U1 () exp ^ ¯ h i F()q0 () exp ^ ¯ h i F(0)q0 (0) ^ ¯ h i F(0)q0 (0) ¯ h . (2.3.5)

.

(2.3.6) , . , , 6, :


2.3

39

^ U1 (2) exp ijk =

i F(0)q0 (0) + F()q0 () ^ ^ ¯ h

exp(i10 ) ,

(2.3.7)

i i F(j)q0 (j), F(k)q0 (k), ^ ^ ¯ h ¯ h

(2.3.8)

(j, k ). , , jk . , 3, 4, . . . , : ^ U1 (N) exp i ¯ h
N-1

F(n)q0 (n) exp(i) , ^
n=0

(2.3.9)

, . , . , ( ^ ^ ) U U+ , , exp(i) exp(-i), . , , ^ H1 (t) , (2.3.3) , . , 0, N N t. , : ^ U1 (t) = exp i ¯ h
t

F(t )q0 (t ) dt ^
0

.

(2.3.10)

, , , (2.3.10) t [. (2.3.2)]. 2.3.2

q0 (t) : ^ q0 q0 (t) = ae ^ ^ 2
-io t

+ a e ^

io t

.

(2.3.11)

(2.3.10), : ^ ^ U1 (t) = D() = e iq0 F (o ) , = ¯2 h (2.3.13)
a - a ^ ^

,

(2.3.12)


40

2

t

F() =
0

F(t)e

-it

dt

(2.3.14)

F(t), 0 t. D() . , . 5, , ^ D ()aD() = a + [a, a - a] = a + [a, a ] = a + . ^^ ^ ^^ ^ ^ ^^ ^ (2.1.10), , 1
t 0 t

(2.3.15)

^ D ()qD() = q + ^^ ^ ^ D ()pD() = p + ^^ ^

2q

0

=q- ^

F(t ) sin o t dt ,
0

(2.3.16a) (2.3.16b)

2q0 = p + ^

F(t ) cos o t dt

[ (2.2.13)]. , D() , , , . , , . 6 : ^ D() = e
a ^


e

- a -||2 /2 ^

e

.

(2.3.17)

, a a , ^ ^ . . : ^ | = D()|0 (2.3.18)

, , , : q= 2q0 , p = 2q0 . (2.3.19a) (2.3.19b)

. 2.3.3

(2.1.21), :


2.3

41



q

(q) = q| =

1
4

q

2 0

exp -

(q -



2q 2q2 0

0

)

2

exp



2iq ¯ h

o



q

.

(2.3.20)

, (2.3.17): | = e


-||2 /2 a ^

e



e

a ^

|0 .

(2.3.21)

^ , e- a |0 . , : ^ a

e

|0 =

n=0

(- )nan ^ |0 . n!

(2.3.22)

, n = 0, , a|0 = 0. , ^ e
- a ^

|0 = |0 .

(2.3.23)

( a ), ^ |n -: | = e
-||2 /2

n=0

n (a )n ^ |0 = n!

n=0



e

-||2 /2





n

n!

|n .

(2.3.24)

, : ||2n . n! | n| |2 = e n = |n| = ||2 , ^
2 n 2 -||
2

(2.3.25)

(2.3.26a)
2

= |n | - |n| ^ ^

= || .

2

(2.3.26b)

, . ,


a| = ^

e

-||2 /2

(2.3.27) a , ^ . a , ^ :

n=0





n

n!

a|n = ^

n=1



e

-||2 /2



n!

n n|n - 1 =

n=0



e

-||2 /2



n!

n+1

|n = | .


42

2

| =

n.n =0



e

-(||2 +||2 )/2



( )n

n

n!n !

n|n =

n

e

-(||2 -||2 )/2

( )

n

n! ||2 - ||2 + 2 2 , (2.3.28) (2.3.29)

= exp | | |2 = e

2

-|-|

.

, : | | d =
2

d

2

e

-||

n.n =0

n ( ) n!n !

2

n

|n n |

(2.3.30)

, = ||ei d2 = ||d||d:
2

| | d = = 2


n.n =0 0

|| d||
2

d
2 nn

e

-||

2

||n+n ei(n n!n !


-n )

|n n |


n.n =0 0

|| d||

e

-||

||n+n n!n !

|n n | = 2

|n n| =

e

-||

n=0

0

|| n!

2

2n

|| d||

|n n| = . (2.3.31)

n=0

, | | d2 = 1. (2.3.32)

, , , ( ). , |n = 2.3.4 e
-||2 /2



( )n d2 . | n!

(2.3.33)



, a| = | , ^ , .. a a . ^^ F , : ^ F = F (a, a ) . ^^ (2.3.34)


2.3

43

, , a a . ^^ (2.1.9), , a a : ^ ^ ^ F=


F

nn

(a )nan . ^^

(2.3.35)

n,n

. , h ¯o (aa + a a) , ^^ ^^ 2 , ¯E = h E = ¯ o (a a + 1/2) h ^^ (2.3.36)

(2.3.37)

. : ^ |F| =


F

nn

|(a ) a | = ^^

n n

n,n

n,n



F

nn

( )n n .

(2.3.38)

a , a . ^ ^ . |. . 1. : n2 = nn = a aa a . ^ ^ ^ ^ ^^ ^ 2. : a aa a = a (a a + 1)a = (a )2 a2 + a a . ^ ^^ ^ ^ ^ ^ ^ ^^ ^^ 3. a , a ^ ^ .


(2.3.39)

(2.3.40)

, n = ||4 + ||2 .


44

2

2.4
2.4.1




, , : , - (, ) . , , . : d2 q (t) ^ + 2 (1 + k sin 2o t)q (t) = 0 . ^ (2.4.1) o 2 dt , , : (). q (t) = qC (t) cos o t + qS (t) sin o t , ^ ^ ^ dq (t) ^ = o -qC (t) sin o t + qS (t) cos o t , ^ ^ dt (2.4.2a) (2.4.2b)

qS (t), qC(t) () . , , (2.2.11). , , , . , dqS (t) ^ dqC (t) ^ cos o t + sin o t = 0 , dt dt (2.4.1) : (2.4.3a)

dqC (t) ^ dqS (t) ^ sin o t + cos o t + o k sin 2o t qC (t) cos o t + qS (t) sin o t = 0 . ^ ^ dt dt (2.4.3b) (2.4.3), : - dqC (t) ^ - o k sin 2o t sin o t qC (t) cos o t + qS (t) sin o t = 0 , ^ ^ dt dqS (t) ^ + o k sin 2o t cos o t qC (t) cos o t + qS (t) sin o t = 0 . ^ ^ dt

(2.4.4a) (2.4.4b)

. , qS (t), qC (t) o (


2.4

45

, k ). . .. : dqC (t) ^ - qC (t) = 0 , ^ dt dqS (t) ^ + qS (t) = 0 , ^ dt = ko /4. : qC (t) = qC (0)et , ^ ^ qS (t) = qS (0)e-t . ^ ^ , : qC (t) = qC (0) et , qS (t) = qS (0) e-t . : 2 C (t) = 2 C e q q S (t) = S e
qC p
S

(2.4.5a) (2.4.5b)

(2.4.6a) (2.4.6b)

(2.4.7a) (2.4.7b)

2t

, ,

(2.4.8a) (2.4.8b) (2.4.8c)

2 q

2 q

-2t

(t) =

qC p

S

(0) .

, ( > 0 qC , < 0 qS ) "" ( , ), "". . . t 2 C (t) = q ¯ h e 2 ¯ h e 2 S (t) = q 2
2t

, .

(2.4.9a) (2.4.9b)

-2t

, ¯ 2 /42 , , h . . , , , . (2.4.9), , , . .


46 2.4.2

2

, ^ D(), ^ S(z) = e
1 2

[z(a )2 -z a2 ] ^ ^

.

(2.4.10)

z , ( ). , R:
R ^ ^ S(R) = e 2 [(a )2

-a ^

2

].

(2.4.11)

, ^ a, S (R)aS(R). 9, = R ^ ^^ ^ P = a2 - (a )2 /2, , ^ ^ ^ ^ Rn a2 - (a )2 a2 - (a )2 ^ ^ a2 - (a )2 ^ ^ ^ , a + · · ·+ ^ ,..., ,a ... +... ^ S (R)aS(R) = a + R ^^ ^ 2 n! 2 2 (2.4.12) a2 - (a )2 ^ ^ , a = a , ^ ^ 2 a2 - (a )2 ^ ^ ,a ^ 2
n



= a. ^

(2.4.13) a, ^

, a , ^ , , R2 R3 ^ S (R)aS(R) = a + Ra + a + a + · · · = a ^^ ^ ^ ^ ^ ^ 2! 3!


k=0

R2k +a ^ (2k)!



k=0



R2k+1 (2k + 1)!

(2.4.14)

, , ^ S (R)aS(R) = a ch R + a sh R . ^^ ^ ^ (2.4.15a)

: ^ S (R)a S(R) = a ch R + a sh R . ^^ ^ ^ (2.4.15b)

, :


2.4

47

q0 ^ ^ S (R)xS(R) = S (R)(a + a )S(R) = ^^ ^ ^^ 2 q0 ^ ^ ^ ^^ S (R)pS(R) = S (R)(a - a )S(R) = ^^ i2

q0 (a + a )(ch R + sh R) = xeR , ^^ ^ (2.4.16a) 2 q0 (a - a )(ch R - sh R) = pe-R . (2.4.16b) ^^ ^ i2

(2.4.6), , , R t. , : ^ ^ ^ ^ R ()S(R)R() = S(Re
2i

),

(2.4.17)

8 (2.1.36). , z = Re2i (2.4.17). , ^ ^ ^ ^ ^ ^ ^ ^ ^ S (z)aS(z) = R ()S (R)R() a R ()S(R)R() = R ()S (R) aei S(R)R() ^^ ^^ ^^ ^ ^ = R () a ch R + a sh R R()ei = ae-i ch R + a ei sh R ei ^ ^ ^ ^ = a ch R + a e ^ ^ , , ^ ^ ^ S (z)a S(z) = a ch R + ae ^^
-2i 2i

sh R , (2.4.18a)

sh R .

(2.4.18b)


48

3

3
3.1
3.1.1





, , , , . : I(x, t) U(x, t) =- , t x I(x, t) U(x, t) =- , C t x L

(3.1.1a) (3.1.1b)

U(x, t) , I(x, t) , L , C . , L C = L C (3.1.2)

1 v= . (3.1.3) LC . , . , t U(X, t) = U(0, t) , I(X, t) = I(0, t) , (3.1.4a) (3.1.4b)

X . :


U(x, t) = I(x, t) =

Uc (t) cos kn x + Us (t) sin kn x , n n -Is (t) cos kn x + Ic (t) sin kn x , n n

(3.1.5a) (3.1.5b)

n=1

n=1

kn =

n 2n = , X v

(3.1.6)


3.1

49

n , Uc (t), Us (t), Is , Ic (t) n n nn . (3.1.1), Uc,s , Ic,s : n n dUc (t) n dt dIc (t) n dt s dUn (t) dt dIs (t) n dt d2 Uc (t) n + 2 Uc (t) = 0 , nn dt2 d2 Us (t) n + 2 Us (t) = 0 . nn dt2 . 3.1.2 = -n Ic (t) , n = n c U (t) , n (3.1.7a) (3.1.7b) (3.1.7c) (3.1.7d)

= -n Is (t) , n = n s U (t) , n

(3.1.8a) (3.1.8b)

, , : 1. L, q1 , q2 , . . . . 2. , L pj = . q j 3. , H = j pj qj - L. 4. , [qj , pk ] = i¯ jk . ^^ h ^ H, , .. . : E= 1 2
X

LI(x, t)2 + CU(x, t)
0

2

dx .

(3.1.9)


50

3

(3.1.5) x. ,
X X

cos kn x cos km x dx =
0 X 0

sin kn x sin km x dx =

X 2

nm

,

(3.1.10a) (3.1.10b)

cos kn x sin km x dx = 0 ,
0

, : E = Ec + Es , E
c,s

(3.1.11)

CX = 4

n=1



(Uc,s )2 n + (Uc,s ) n 2 n

2

(3.1.12)

"" "" . , , , . Cn CX c,s Un . (3.1.13) 2 Cn . , , , , LC-, , . : q
c,s n

=

L = L c + Ls , L
2 n c,s

(3.1.14)

=

n=1



n Ln (qc,s )2 (qc,s ) -n 2 2Cn

2

,

(3.1.15)

Ln = 1/Cn . :
c,s n

=

:

Lc,s = Ln q n qc,s

c,s n

,

(3.1.16)

H = Hc + Hs , H
c,s

(3.1.17)

=

n=1





c,s c,s n qn

-L

c,s

=

n=1



(c,s )2 (qc,s ) n +n 2Ln 2Cn

2

.

(3.1.18)


3.1

51

, [qc , c ] = [qs , s ] = i¯ nm , ^n ^ m ^n ^ m h [qc , s ] = [qs , c ] = 0 , ^n ^ m ^n ^ m : ^ ^ ^ H = Hc + Hs , ^ H =


(3.1.19a) (3.1.19b)

(3.1.20)

c,s

n=1

^n (c,s )2 (qc,s ) ^ +n 2Ln 2Cn

2

.

(3.1.21)

3.1.3



, , . ( ) , (. 2.1.1):

q ^

c,s n

=

^ ^ H
c,s

c,s n

¯ n Cn h (an ^ 2 an ^ ¯ h = 2n Cn ¯ h
n

c,s c,s

+a ^

+ n c,s + n c,s

), , .

(3.1.22a) (3.1.22b) (3.1.22c)

-a ^ i

=

a ^

^+ n c,s an c,s

+a ^

+ ^ n c,s an c,s

n=1

( ) da ^
n c,s

(t)

dt

= -in a ^

n c,s

(t)

(3.1.8). , , , - : [an c , a+ c ] = [am s , a+ s ] = nm , ^ ^n ^ ^n + [an c , an s ] = [am s , a+ c ] = 0 . ^ ^ ^ ^n (3.1.24a) (3.1.24b)



a ^

n c,s

(t) = a ^

n c,s

e

-in t

,

(3.1.23)

, , :


52

3

^ U(x, t) =

n=1



¯ n h CX

an c e ^

-in t

+ a+ c e ^n

in t

cos kn x
-in t

+ an s e ^ ^(x, t) = - 1 I i


+ a+ s e ^n

in t

sin kn x , (3.1.25a)

n=1

¯ n h CX

an c e ^

-in t

- an c e ^+

in t

sin kn x - a+ s e ^n
in t

+ an s e ^

-in t

cos kn x . (3.1.25b)

(, Cn ). ( ) . , ( R L) : a ^ a ^ = = = = a a a a
nc + nc

nR + nR

a ^ a ^

nc + nc

nL + nL

- ia n 2 + ia+ n 2 + ian 2 - ia+ n 2

s

, , , .

(3.1.26a) (3.1.26b) (3.1.26c) (3.1.26d)

s

s

s

(3.1.24) , [an R , a+ R ] = [am L , a+ L ] = nm , ^ ^n ^ ^n + ^ ^n [an R , an R ] = [am L , a+ L ] = 0 . ^ ^ a ^ an R , an L , a+ R , a+ L , : ^ ^ ^n ^n ^ ^ U(x, t) = UR (x - vt) + UL (x + vt) , ^ ^(x, t) = 1 UR (x - vt) - UL (x + vt) , I ^ UR (x) = ^ UL (x) =
nc

(3.1.27a) (3.1.27b) , an s , a+ c , a ^ ^n ^
+ ns

(3.1.28a) (3.1.28b)

n=1

¯n h a 2CX

nR

e e

ikn x

+a

-ikn x + n Re

, .

(3.1.29a) (3.1.29b)

n=1

¯ n h a 2CX

nL

-ikn x

+a

nL

e

ikn x

, , .


3.1 3.1.4

53

: X . , , , R. , . , fn ( ):




n=1

fn

0



f()

d X = v

0



f()

d , 2

(3.1.30)

2v X , f() = f(n ) = fn ,

(3.1.31)

(3.1.32)

. . :


n=1

J() , , a() ^ . (3.1.27), ( a+ ()) ^ ^ [a(), a+( )] = 2( - ) (3.1.34)

fn a n ^

0



f() a()J() d , ^

(3.1.33)

2 ( a(), a+ () ). ^ ^ , 2: [a(), a+( )] = ( - ) . ^ ^ (3.1.34). S= ,


(3.1.35)

^ fn a n ,

n=1

n=1



^ gm a

+ m

.

(3.1.36)

S=

^^ fn gm [an , a ] =

+ m

f n gm

nm

=

n,m=1

n,m=1

n=1

,

f n gn

f()g()

0

d .

(3.1.37)


54

3

S

0



f()a()J() d, ^ =


0



g()a+ ()J() d ^

f()g( )[a(), a+( )]J()J( ) dd ^ ^


0

= 2

f()g( )( - )J()J( ) dd = 2


0

f()g()J2() d . (3.1.38)

0

, J() =


1 1 = 2 2 X v


X , v d . 2

(3.1.39)

n=1

(3.1.29): ^ UR (x) = ^ UL (x) = 3.1.5


fn a n ^

f() a() ^

(3.1.40)

0

0

0

¯ h aR ()e ^ 2 ¯ h aL ()e ^ 2

ix/v

+ a+ ()e ^R

-ix/v

d , 2 d . 2

(3.1.41a) (3.1.41b)

-ix/v

^L + a+ ()e

ix/v



, . , . ^= R,L = CU2 (x) ^ R (3.1.43) ^ I CU2 (x, t) L^2 (x, t) + 2 2 dx = R + L , ^ ^ (3.1.42)

, , . , , , . " R".


^(x) =

0

¯ h 2v

a()a+ ( )e ^ ^

i(- )x/v

+ a+ ()a( )e ^ ^

i( -)x/v

+ a()a( )e ^ ^

i(+ )x/v

+ a+ ()a+ ( )e ^ ^

-i( +)x/v

dd . (3.1.44) (2)2


3.1

55

(, ):


^(x) =

0

¯ h 2v

a+ ( )a() + 2( - ) e ^ ^
i(+ )x/v

i(- )x/v

+ a+ ()a( )e ^ ^

i( -)x/v

+ a()a( )e ^ ^

+ a+ ()a+ ( )e ^ ^

-i( +)x/v

= ^(N)


dd (2)2 ¯ d h , (3.1.45) (x) + 2v 2 0

^

(N)

(x) =

0

¯ h 2v

2a+ ()a( )e ^ ^

i( -)x/v

+ a()a( )e ^ ^

i(+ )x/v

+ a+ ()a+ ( )e ^ ^

-i( +)x/v

dd (2)2

(3.1.46)

. (3.1.45), . , . , . , , . , (). ( ), . :


^ E

(N)

=



(N)

-

¯ h (x) dx = 2a+ ()a( )( - ) ^ ^ 2 0 dd + a()a( ) + a+ ()a+ ( ) ( + ) ^ ^ ^ ^ 2


= 3.1.6

0



¯ a+ ()a() h^ ^

d . (3.1.47) 2

, , .. , . . " " , y z, z, . :


56

3

CU 2

2

E ( , A ). , E c = v , 4C U= A 4 U. Ac (3.1.49)



E2 A E2 ds = , 8 8

(3.1.48)

. , , , ,

^ ER (x) =

4 ^ UR (x) = Ac

0



2 ¯ h aR ()e ^ Ac

ix/c

+ a+ ()e ^R

-ix/c

d . 2

(3.1.50)

, , :


^ E(x) =
=1,2

n



0

2 ¯ h a ()e ^ Ac

ix/c

+ a+ ()e ^

-ix/c

d , 2

(3.1.51)

n1 , n

2

, . , n1 = ny , n2 = n
z

(3.1.52)

n1 = . ny + i nz , 2 n2 = ny - i n 2
z

(3.1.53)


3.2

57

3.2



|0 , . . . , . , ( ), , , . , . . - , , . , , . , ( ). , , , d. , , . , . . .




0



¯ d ; h

(3.2.1)

d c/d: d


¯ d . h

(3.2.2)

c/d

()
c/d



0

¯ d h

¯ h . d2

(3.2.3)

d 2 d ( !), , ,
c/d



0

¯ 2 d h

¯ h . d4

(3.2.4)

c, , A ,


58

3

C

l

R

. 3: ¯c h A. (3.2.5) d4 , , , F 2 ¯ hc F= A 1.2 · 10 240d4
-18

â

1 d

4

â

A 1

2

.

(3.2.6)

, d, . , d = 0.1 A = 1 2 120 . , . , . , ¯ 2 5 h A. (3.2.7) S() = 1204 c4 , , (- ¯ 2 /c4 ), h . , , , . . , C, d, l = v, v (. . 3). . ( ) , , . LC-, C Lline = .


3.2

59

( , ). , "-" . . , , . F ¯1 h , d (3.2.8)

1 (3.2.9) Lline C "-" . , C, , Cline = /. (3.2.9) C = C line , , 1 1/ 1 F , ¯ h , (3.2.11) d , v, , d = 10-4 l = 102 2 · 10-16 . , d2 d, 1.7d/l 1. , , "" , , , . F 0.07 ¯ h , d (3.2.10)


60

3

3.3
3.3.1




- : ? | =


|

n

,

(3.3.1)

n=1

|n . ( , ?) , . . . : |1(1 , 2 ) = 1 |1|0 + 2 |0|1 , (3.3.2)

1,2 , , . , , |1 |2 + |2 |2 = 1 . |1(1 , 2 ) = (1a+ + 2 a+ )|0 , ^1 ^2 (3.3.4) (3.3.3)

a+ ^ 1,2 . ( , , ). , (, ) : |1(1 , 2 , . . . ) =


n a+ |0 , ^n

(3.3.5)

n=1

n


|n |2 = 1 .

(3.3.6)

n=1

, ( , ). a+ n a+ ()), n ^ (), : |1[()] =


()a+ () ^

0

d |0 . 2

(3.3.7)


3.3

61

, |1 n , (). 3.3.2

, , |1 |1[()] , , , |1 (). (3.1.34) , a()|0 = 0. ,


1|1 =

()( ) 0|a()a+ ( )|0 ^ ^ =


0

dd (2)2 dd (2)2 d . (3.3.8) |()|2 2

()( ) 0| a+ ( )a() + 2( - ) |0 ^ ^ =


0

0

, " " ()


|()|

2

0

d = 1. 2

(3.3.9)

" " ( ) . . , ^ N=


a+ ()a() ^ ^

0

d . 2

(3.3.10)

:


^ N|1 =

a+ ()a()( )a+ ( ) ^ ^ ^


0

dd |0 = (2)2 dd |0 (2)2 d a+ ()() ^ |0 = 1 · |1 . (3.3.11) 2

a+ ()( ) a+ ( )a() + 2( - ) ^ ^ ^ =


0

0

, , , . , , , "" .


62

3

, , , ^ E= :


¯ a+ ()a() h ^ ^

0

d . 2

(3.3.12)

^ E|1 =

¯ a+ ()a()( )a+ ( ) h^ ^ ^


0

dd |0 = (2)2 dd |0 (2)2

¯ a+ ()( ) a+ ( )a() + 2( - ) h^ ^ ^ =


0

¯ a+ ()() h ^

0

d |0 = const ·|1 . (3.3.13) 2

! . (3.3.13) , n ^ En |1 = ,


(¯ )n ()a+ () h ^

0

d |0 . 2

(3.3.14)

^ 1|En |1 = 0|

( )a( )(¯ )n ()a+ () ^ h ^

0

dd |0 (2)2 =
0

(¯ )n |()| h

2

d . (3.3.15) 2

, |()|2 /2 - ( ) . , E =¯ =¯ h h (E)2 = ¯ 2 ()2 = ¯ 2 h h


|()|

2

0

d , 2 d . 2

(3.3.16)

0



( - )2 |()|

2

(3.3.17)

() . , - . , (. (3.1.46)):


3.3

63

1|^

(N)

(x)|1 =

0



¯ h ( )( ) c
i( -)x/v

â 0|a( )a+ ()a( )a+ ( )|0 e ^ ^ ^ ^

dd d d , (3.3.18) (2)4

c ( , v = c). , (3.1.34): 0|a( )a+ ()a( )a+ ( )|0 ^ ^ ^ ^ = 0| a+ ()a( ) + 2( - ) ^ ^ a+ ( )a( ) + 2( - ) |0 ^ ^ = ( - )( - ) . (3.3.19) "" :


1|^

(N)

(x)|1 =

0

¯ h ()( )e c

i( -)x/c

¯ h = c0 .

dd (2)2 ()e

ix/c

d 2

2

. (3.3.20)

. () (). , . , , ( ), t (3.3.20), t = -x/c (, , , , t, , x). . , - (3.1.45), 1 = ^ c


a+ ( )a()e ^ ^

i(- )x/c

0

+

1 a()a( )e ^ ^ 2

i(+ )x/c

+ a+ ()a+ ( )e ^ ^

-i( +)x/c

dd . (3.3.21) (2)2

( ). ,


64

3

, E = ¯ n. h (3.3.20) 1 c


1|(x)|1 = ^

()( )e

i( -)x/c

0

dd 1 = 2 (2) c

0



()e

ix/c

d 2

2

. (3.3.22)

, ( ) - (). (3.3.22) , t 1 . (3.3.23) 2 , E = ¯ , h E t: t Et , E = pc , x = -ct , (3.3.25a) (3.3.25b) ¯ h . 2 (3.3.24)

p x: xp ¯ h . 2 (3.3.26)


3.4

65

3.4
3.4.1




, , . , .. . . , n 1, "" "" . , , , , . , : ^ | = D()|0 , ^ D() = e
a+ - a ^ ^

(3.4.1)

(3.4.2)

. 1 , 2 : |1 , ^ D1,2 (1,2 ) = exp(1,2a ^
+ 1,2 2

^ ^ = D1 (1 )D2 (2 )|0 ,

(3.4.3)

- a1,2 ) 1,2 ^

(3.4.4)

. : |1 , ^ D(1 , 2 ) = e
1 a+ +2 a+ - a2 - a ^1 ^2 1^ 1^
2

2

^ = D(1 , 2 )|0 ,

(3.4.5)

(3.4.6)

. , , , : |1 , 2 , . . . =


^ ^ Dn (n )|0 = D(1 , 2 , . . . )|0 ,

(3.4.7)

n=1

^ D(1 , 2 , . . . ) = exp

n=1



(na+ - an ) . ^n n^

(3.4.8)


66

3

{1 , 2 , . . . }. , d/2 {1 , 2 , . . . } (): ^ | = D[]|0 , ^ D[] = exp


(3.4.9)

()a+ () - ()a() ^ ^

0

d 2

.

(3.4.10)

, (). 3.4.2

^ ^ D+ []a()D[] = a() + () , ^ ^ + + ^ ^ D []a ()D[] = a+ () + () . ^ ^ ^ ^ D[]D+ [] = 1 , , ^ ^ ^ ^ ^ ^ D+ [] a+ ()a() D[] = D+ []a+ ()D[] D+[]a()D[] ^ ^ ^ ^ = a+ () + () a() + () ^ ^ ^ ^ D+ [] a+ ()a()a+ ( )a( )D[] ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = D+ []a+ ()D[] D+[]a()D[] D+ []a+( )D[] D+[]a( )D[] ^ ^ ^ ^ = a+ () + () a() + () a+ ( ) + ( ) a( ) + ( ) ^ ^ ^ ^ d ^ ^ |0 D+ []a+ ()a()D[] ^ ^ 2 0 d d = |()|2 , (3.4.15) 0| a+ () + () a() + () |0 ^ ^ 2 2 0


(3.4.11a) (3.4.11b)

(3.4.12)

(3.4.13)

(3.4.14)

^ ^^ N = 0|D+ [] N D[]|0 = 0| =


0


3.4

67

N =

2

0



dd ^ ^ |0 D+ []a+ ()a()a+ ( )a( )D[] ^ ^ ^ ^ (2)2 0 dd 0| a+ () + () a() + () a+ ( ) + ( ) a( ) + ( ) |0 ^ ^ ^ ^ (2)2 d = ()( ) 0| a() + () a+ ( ) + ( ) |0 ^ ^ 2 0 d ()( ) 2( - ) + () ( ) = 2 0 = N 2 + N . (3.4.16) ^ ^^ = 0|D+ [] N2 D[]|0 = 0|



, (n)2 = n
2

-n

2

=n

(3.4.17)

, . : d ^ ¯ D+ []a+ ()a()D[] h^ ^ ^ |0 2 0 d d + ¯ 0|[a () + ()][a() + ()]|0 h ^ ^ ¯ |()|2 h = (3.4.18) 2 2 0


^ ^^ E = 0|D+ [] E D[]|0 = 0| =


0

: E
2

^ ^^ = 0|D+ [] E2 D[]|0 = 0|
0

dd ^ ^ |0 ¯ 2 D+ []a+ ()a()a+ ( )a( )D[] h ^ ^ ^ ^ (2)2
0

=

¯ 2 0| a+ () + () a() + () h ^ ^ dd (2)2

â a+ ( ) + ( ) a( ) + ( ) |0 ^ ^ =
0

¯ 2 ()( ) 0| a() + () a+ ( ) + ( ) |0 h ^ ^


=

0

dd (2)2 dd ¯ 2 ()( ) 2( - ) + () ( ) h (2)2 d (¯ )2 |()|2 h + E 2 . (3.4.19) = 2 0

, (E)2 = E
2

-E

2

=

0



(¯ )2 |()| h

2

d . 2

(3.4.20)


68

3 : E ¯ = = ¯N h


|()|

0

0



d 2 . d 2 |()| 2
2

(3.4.21)

: (E)2 = ¯ E + h¯


¯ ( - )2 |()|

2

0

d . 2

(3.4.22)

¯ E h , - . , , , .

3.5


() , v. , , v . , , .

1. " "

2. "" , (). - , , . .


69

4
4.1
4.1.1





. , . , , . , - (, ), - , (, ), . , , ¯ /2 " ". h , , , . . , . A () , , |0 , | . B () ( , ). , | . |0 0| + | | . (4.1.1) 2 - , | . , , , . "" . , , | j pj , (j = 1, 2, . . . ), = ^ = ^
j

pj |

j

j | .

(4.1.2)

, pj , p1 , , . = | ^
1

1 | .

(4.1.3)

.


70

4

. {|n } - (^ ) N. n ||n ^ ^ , N: ^ Sp = ^
n

n||n = ^
n j

pj n|

j

j |n =
j

p

j n

| n|j |2 =
j

pj = 1 . (4.1.4)

^ N , (Sp, ) . |j ; ^ Q=
j

^ pj j |Q|

j

(4.1.5)

{|n } - () . |n n| 1 , (4.1.6)

n

(4.1.5) :

^ Q=
j

^ pj j | Q
n

|n n| |j =
j

p

j n

n|

j

^ j |Q|n n|Q|n , (4.1.7) ^^
n

=
n

n|
j

pj |

j

^ j |Q |n =

^ Q = Sp(Q) ^^ (4.1.8)

|j , ( ): d(t) ^ = dt d j (t)| d|j (t) + j (t)| dt dt =
j

p
j

j

|j (t)

p

j

^ ^ j(t)|H H|j (t) j (t)| + |j (t) i¯ h -i¯ h

, (4.1.9)




4.1

71

d(t) ^ ^^ = [H, (t)] . dt : i¯ h ^^^ (t) = U(t)(0)U+ (t) , ^ ^ U(t) . 4.1.2 , (0) ^

(4.1.10)

(4.1.11)

-



, , ( ). . ( a b), , . | = ca |1 a |0
b

+ cb |0 a |1

b

,

(4.1.12)

ca,b , a b, , |ca |2 + |cb |2 = 1 . () : = | | . ^ (4.1.14) (4.1.13)

. (4.1.14) (|n b ): a = Spb = ^ ^


n| |n = |ca |2 |1

a

1|a + |cb |2 |0

a

0|a .

(4.1.15)

nb =0

, , . , , - . (4.1.15) | = ca |1
a

+ cb |1

b

,

(4.1.16)

, , (|ca |2 |cb |2 , ). : , a (4.1.15) (4.1.16). . , , . , , . . , .


72

4

. , , , ( 1023 ) . , - , . - , , . .


4.2

73

4.2
4.2.1




, . , N p1 , p2 , . . . ,
N

S = -

B n=1

pn ln pn .

(4.2.1)

.. S = - ln = -B Sp( ln ) . ^ ^^ (4.2.2)

, . , ( ). , , . N (, N = 2), 1/N, B N ln N. ( , ), . : , E . : (4.2.2) Sp( H) = E B T , ^^ Sp = 1 . ^ (4.2.3a) (4.2.3b)

T . . , ^ -B Sp( ln ) + Sp ( H) + µ Sp = Sp (-B ln + H + µ) , ^^ ^^ ^ ^ ^ (4.2.4)

, µ . ( ^ ), : ^ Sp (-B ln - B + H + µ) = 0 . ^ ^ (4.2.5)

, ^ ,


74

4

µ ^ - 1 + H . (4.2.6) B (4.2.3), ^ B ln = -B + H + µ = exp ^ ^ ^ H 1 exp - Z B T = ^ Z = Sp exp - .. . 4.2.2 ^ H B T (4.2.8) , (4.2.7)

, 1 ^ H = ¯ n+ h ^ . (4.2.9) 2 = ^ n =


n |n n| ,

(4.2.10)

n=0

¯ h 1 1 exp - n+ = (1 - e- )e-n Z B T 2 , n , a


(4.2.11)

¯ h . B T

(4.2.12)

ne

-n

= (-1)



n=0

n=0

e-n 1 = (-1) 1-e

-

(4.2.13)

( ), : n=


nn =

n=0

1 e -1


(4.2.14)

( ), (n) = n
2 2

-n

2

=

n=0



n 2 n - n

2

=n

2

+n.

(4.2.15)


4.2

75

(T 0) , n , 0 , . : = |0 0| . ^ (4.2.16)

T = 0 , , |0 . 4.2.3

, , . () Q ^ e-iQ , ( ). , , ^ () = e
^ -iQ

= Sp e ^

^ -iQ

.

(4.2.17)

^ Q . ^ Q=


|q q q| dq ,

(4.2.18)

q , |q . , e
^ -iQ

-

=

, :
-iq -iq -iq

-



|q e

-iq

q| dq ,

(4.2.19)

() =

e

Sp (|q q|) dq = ^

e

q||q dq = ^

e

w(q) dq , (4.2.20)

-

-

-

Q . ^ , - - . , , : w(q) =


w(q) q||q ^

(4.2.21)

()e

iq

-

d . 2

(4.2.22)


76

4

, , . 4.2.4

, . ( ) , ^ A = a + a+ , ^ ^ (4.2.23)

. A , . (4.2.10) () = Sp e
^ -iA

= (1 - e

-

)

n=0



e

-n

n|e

^ -iA

|n .

(4.2.24)

6, : n|e
^ -iA

|n = e

-2 ||2 /2

n|e

i a ^

+

e

ia ^

|n .

(4.2.25)

, |n : (a+ )n ^ |n = |0 , n! 4: 1 (a+ )n ^ |0 = e n! n! (4.2.26)

e

ia ^

|n = e

ia ^

ia ^

(a+ )n e ^

-ia ^

e

ia ^

|0
ia ^

1 = (+ + i)n e ^ n! , e
ia ^

|0 . (4.2.27)

|0 = |0 ,

(4.2.28)

(+ + i)n , : ^ 1 |n = n!
n

e

ia ^

k=0

n! (a+ ) ^ k!(n - k)!

n n-k

(i)k |0 =
k=0 i


0|e (4.2.25): !!!!!!

n! (i)k |n - k . k! (n - k)! (4.2.29) a+ ^ -




4.2

77

n

n|e

^ -iA

|n = e

-2 ||2 /2 k,l=1

k!l!

n! (i )l (i)k n - l|n - k (n - k)!(n - l)!
n

=e Ln . ,

-2 ||2 /2 k=1

n! =e (k!) (n - k)!
2

-2 ||2 /2

Ln (2 ||2 ) , (4.2.30)

() = (1 - e

-

)e

-2 ||2 /2

n=0



e

-n

Ln (2 ||2 ) .

(4.2.31)

,


zn Ln (x) =

e

-xz/(1-z)

n=0

1-z

,

(4.2.32)

: () = e
-2 ||2 ( n +1/2)

,

(4.2.33)

n [. (4.2.14)]. A :


w(A) =

e

-2 ||2 ( n +1/2)+iA

-

d 2 1 . (4.2.34)

=

A2 exp - 4||2 ( n + 1/2) 4||2 ( n + 1/2)

, , : 1. , ; 2. ; 3. ( n = 0) 2 n + 1 . , , , "" 2 n + 1. . (4.2.23) q


78

4

q0 = = 2

¯ h , 2

(4.2.35)

( ) . , w(q) = 1 2q2 ( n + 1/2) 0 exp - q2 2q2 ( n + 1/2) 0 . (4.2.36)

p, , 1 q0 = = i i2 w(p) = 1 22 q2 ( n + 1/2) 0 exp - p2 22 q2 ( n + 1/2) 0 . (4.2.38) ¯ h , 2 (4.2.37)

, F(t). , . : .


4.3

79

4.3
4.3.1




, . (). , , , : dq(t) d2 q(t) + 2 + 2 q(t) = 0 , (4.3.1) o 2 dt dt . , , : dq(t) ^ d2 q(t) ^ + 2 + 2 q(t) = 0 . (!) (4.3.2) o^ 2 dt dt , , , . , : o . , . (4.3.2) : q(t) = ^ q cos o t + ^ p ^ sin o t e
-t

. (!)

(4.3.3)

, , : dq(t) ^ = (-q sin o t + p cos o t) e ^ ^ dt
-t

p(t) = ^

. (!)

(4.3.4)

, t , . , , , , , , ¯ h . (!) (4.3.5) 2 , q(t) ^ p(t): ^ qp < [q(t), p(t)] = [q, p]e ^ ^ ^^
-2t

= i¯ e h

-2t

. (!)

(4.3.6)

, :


80

4

^ ^ ^ ^ [q(t), p(t)] = [U+ (t)qU(t), U+ (t)pU(t)] = U+ (t)[q, p]U(t) = U+ (t)i¯ U(t) = i¯ . ^ ^ ^^ ^^ ^^^ h^ h (4.3.7) "", , () , . , . . , . ( ) , , . (4.3.1). T = 0 , (4.3.2) . , , . 4.3.2 :

. (), . , . . : ^2 ^2 ^2 ^2 ^ = 1p1 + 1 1 q1 + 2 p2 + 2 2q2 - q1 q2 , H ^^ 21 2 22 2 (4.3.8)

1,2 1,2 , q1,2 ^ p1,2 , . ^ a+ a+ ( ), , ^ 1,2 ^ 1,2 1 2 1 2

^ H=¯ h

1

a+ a1 + ^1 ^

+¯ h

2

a+ a2 + ^2 ^

- ¯ (a1 + a+ )(a2 + a+ ) , h ^ ^1 ^ ^2

(4.3.9)

= . 2 1 2

(4.3.10)

( ):


4.3

81

da1 (t) ^ + i1 a1 (t) = i[a2 (t) + a+ (t)] , ^ ^ ^2 dt da2 (t) ^ + i2 a2 (t) = i[a1 (t) + a+ (t)] , ^ ^ ^1 dt

(4.3.11a) (4.3.11b)

. , . , ,
1,2

.

(4.3.12)

(4.3.11) . , , = 0. : a1 (t) = a1 e ^ ^ a2 (t) = a2 e ^ ^
-i1 t -i2 t

, .

(4.3.13a) (4.3.13b) (4.3.13c)

(4.3.11): da1 (t) ^ + i1a1 (t) = i[a2 e-i2 t + a+ ei2 t ] , ^ ^ ^2 dt da2 (t) ^ + 2a2 (t) = i[a1 e-i1 t + a+ ei2 t ] , ^ ^ ^1 dt (4.3.14a) (4.3.14b)

. ( ) :
-i1 t

a1 (t) = a1 e ^ ^ a2 (t) = a2 e ^ ^

+ +

-i1 t

- 1 - -i1 t e - 2 - e

-i2 t

ei2 t - e-i1 t + e-i1 t a2 + ^ a2 ^ 2 1 + 2 e-i2 t ei1 t - e-i2 t + a1 + ^ a1 ^ 1 2 + 1

, .

(4.3.15a) (4.3.15b)

, : |1 - 2 |
1,2

.

(4.3.16)

a+ (4.3.15) ^ 1,2 , a1,2 . , ^
-i1 t

a1 (t) = a1 e ^ ^ a2 (t) = a2 e ^ ^

+

-i1 t

- 1 - -i1 t e - + 2 - e

-i2 t

e-i1 t a2 , ^ 2 e-i2 t ^ a1 . 1

(4.3.17a) (4.3.17b)


82

4

(4.3.11), , : da1 (t) ^ + i1a1 (t) = ia2 (t) , ^ ^ dt da2 (t) ^ ^ + i2a2 (t) = ia1 (t) . ^ dt

(4.3.18a) (4.3.18b)

, : 1 2 1 2

^ H=¯ h

1

a+ a1 + ^1 ^

+¯ h

2

a+ a2 + ^2 ^

- ¯ (a1 a+ + a+ a2 ) . h ^ ^ 2 ^ 1 ^

(4.3.19)

, , , , : [ (4.3.12)] [ (4.3.16)]. a , a+ [. ^ ^ ++ (4.3.18)], - aj ak aj ak , j, k ^^ ^^ [. (4.3.19)]. 4.3.3

4.3.1, , . ( , , ). ( ) a(t) : ^ da(t) ^ = -io a(t) . ^ (4.3.20) dt ( ). "" ( ) : da(t) ^ = -(io + )a(t) . (!) ^ dt : a(t) = ae ^ ^
-(io +)t

(4.3.21)

. (!)

(4.3.22a)

, , :


4.3

83

a+ (t) = a+ e ^ ^

(io -)t

. (!)

(4.3.22b)

, , , , , t : [a(t), a+(t)] = [a, a+ ]e ^ ^ ^^
-2t

=e

-2t

. (!)

(4.3.23)

, (4.3.21) ^ f(t) + g(t) ( ): ^ da(t) ^ ^ = -(io + )a(t) + f(t) + g(t) . ^ ^ (4.3.24) dt ^ f(t) ( ) , , g(t) , ^ . , g(t): ^ da(t) ^ ^ = -(io + )a(t) + f(t) . ^ (4.3.25) dt , , : ^ + b(t)]e-io t , ^ a+ (t) = [a+ e-t + b+ (t)]eio t , ^ ^ a(t) = [ae ^ ^
-t

(4.3.26a) (4.3.26b)

^ b(t) = e
-t

^ f(t )e

(io +)t

dt .

(4.3.27)

0

^ , b(t) , . , |0 , ^ b(t)|0 0 . (4.3.28)

, a(t) a+ (t) ^ ^ , ^ ^ b(t) b+ (t): 1 = [a(t), a+(t)] = [a, a+ ]e ^ ^ ^^ , ^ ^ [b(t), b+(t)] = 1 - e
-2t -2t

^ ^ + [b(t), b+(t)] = e

-2t

^ ^ + [b(t), b+(t)] .

(4.3.29)

.

(4.3.30)

, (4.3.26), (4.3.28), (4.3.30) , , .


84

4

, . ¯ n, (n)2 . , . , . | . , , |0 . : ¯ n(t) = 0||a+ (t)a(t)||0 = 0|| a+ e ^ ^ ^ (4.3.28), : ¯ n(t) = 0||a+ e ^
-t -t

^ + b+ (t)

ae ^

-t

^ + b(t) ||0 .

(4.3.31)

ae ^

-t

¯ ||0 = ne

-2t

.

(4.3.32)

: n2 (t) = 0|| a+ (t)a(t) ||0 ^ ^ ^ = 0|| a+ e ^
-t 2

^ + b+ (t) a(t)a+ (t) ae ^^ ^

-t

^ + b(t) ||0 . (4.3.33)

(4.3.28), : n2 (t) = 0||a+ a(t)a+ (t)a||0 e ^ ^^ ^ ^
-2t -4t

= 0||a+ aa+ a||0 e ^ ^^ ^

^ + 0||a+ b(t)b+ (t)a||0 e ^^ ^

-2t

. (4.3.34)

(4.3.30): ^ ^ 0||a+ b(t)b+ (t)a||0 = 0||a ^^ ^
+

^ ^ b+ (t)b(t) + 1 - e

-2t

a||0 ^
-2t

= 0||a+ a||0 (1 - e ^^ : n2 (t) = n2 e ^ ^ , , ¯ (n(t))2 = n2 (t) - n2 = (n)2 e ^
-4t -4t

) . (4.3.35)

¯ + ne

-2t

(1 - e

-2t

).

(4.3.36)

¯ + ne

-2t

(1 - e

-2t

)

(4.3.37)

- (), . , t n0 n, n(1 - n/no ) ( ). (4.3.37).


4.4 -

85

frag replacements L C (a) R L C (b)

R

. 4:

4.4
4.4.1

-


- () . -, . : S = 2B TR . (4.4.1)

, B T ¯ , . h - . , , " " . , , , . LC- R (. . 4(a)). , R R, . (. 4(b)). : ^ d2q(t) q(t) ^ ^ + = U(0, t) , dt2 C dq(t) ^ ^ + I(0, t) = 0 . dt

L

(4.4.2a) (4.4.2b)

^ U(0, t) I(0, t) ( ), x = 0, . (3.1.28), ^ ^ ^ U(0, t) = UR (-vt) + UL (vt) , ^ ^ ^(0, t) = 1 UR (-vt) - UL (vt) , I R (4.4.3a) (4.4.3b)

(4.4.3) (4.4.2), (4.4.2) R ,


86

4

, : L d2 q(t) ^ dq(t) q(t) ^ ^ ^ +R + =U 2 dt dt C ^ U ^ (t) = 2UL (vt) .
fluct

(t) ,

(4.4.4)

fluct

(4.4.5)

, , , " " , , - ( ). , . , , , [. (3.1.41b)]: ^ UL (vt) =


0

¯ R h aL ()e ^ 2

-it

+ a+ ()e ^L

it

d . 2

(4.4.6)

, ( - LC-) ( ; ) . " " , . . 4.4.2

, T . , . ^ Ufluct (t) ( , "" ). ^ Bns (t) = U =4
0

fluct

^ ^ ^ (0)Ufluct (t) = 4 UL (0)UL (vt) ¯ R h aL () + a+ () aL ( )e ^ ^L ^ 2

-i t

+ a+ ( )e ^L

i t

dd . (2)2 (4.4.7)

, , : Bns (t) = 2¯ hR




0|aL ()a+ ( )|0 e ^ ^L

i t

0

dd . (2)2

(4.4.8)

, , (3.1.34):


4.4 -

87

0|aL ()a+ ( )|0 = 0| a+ ( )aL () + 2( - ) |0 , ^ ^L ^L ^ :


(4.4.9)

Bns (t) = 2¯ R h



2( - )e

i t

0

dd = 2¯ hR (2)2

0



e

it

d . 2

(4.4.10)

"" () : 1^ U 2 1 Bns (t) + Bns (-t) 2 d d ||eit = ¯R h . (4.4.11) 2 2 - =

B(t) =

fluct

^ (0)U

fluct

^ (t) + U


fluct

^ (t)U
it

fluct

(0)

= ¯R h

e

+e

-it

0

, , , , -: S() =


B(t)e

-it

, R , , ... .. (4.4.12). : , (4.4.12). . 4.4.3

-

d = ¯ ||R . h 2

(4.4.12)

(4.2.10). - , () , . X, ^ Ufluct (t), . (3.1.29b) ,

^ Bns (t) = U

fluct

^ (0)U


fluct

^ ^ (t) = 4 UL (0)UL (vt) a ^
kL

=4

k,k =1

¯ k k Rv h 2X

+a ^

+ kL

a ^

kL

e

-ik t

+a ^

+ kL

e

ik t

. (4.4.13)

,


88

4

a k L ak L = 0 , ^^ a+L a+ L = 0 , ^k ^k a+L ak L = nT (k )kk , ^k ^ ak L a+ L = [nT (k ) + 1]kk , ^ ^k

(4.4.14a) (4.4.14b) (4.4.14c) (4.4.14d)

(4.4.15) -1 e [. (4.2.14)]. , nT (k ) =
¯ k h B T

1

2¯ Rv h Bns (t) = X

k=1





k

[nT (k ) + 1]e

ik t

+ nT (k )e

-ik t

.

(4.4.16)

, (3.1.30): Bns (t) = 2¯ hR


[nT () + 1]e

it

+ nT ()e

-it

0

d . 2

(4.4.17)

, . : 1 Bns (t) + Bns (-t) = ¯ hR 2


B(t) =

[2nT () + 1][e = ¯R h


it

+e

-it

]

0

d 2
it

||[2nT (||) + 1]e

, -, :
-it

-

d , (4.4.18) 2

S() =

B(t)e

: 1 , cth x|x 1 , (4.4.20) x , T = 0 (4.4.19) (4.4.12), B T ¯ h (4.4.1). cth x|
x0

-

d ¯ | | h = ¯ ||R [2nT (||) + 1] = ¯ ||R cth h h . 2 2B T

(4.4.19)




89

5
5.1
5.1.1





, ( ). , , . , . , ( ) , , ( ) . , ( ) , . . , . . , , , " " (t, t ), ^ q(t) U(t): ^ ^ U(t) = U(t) +
t

(t, t )q(t ) dt ,
-

(5.1.1)

^ ^ U(t) U(t) , . , R dq(t) ^ ^ U(t) = U(t) + R , dt ^ q(t) , U(t) . , (t, t ) = (t - t ) d . dt (5.1.2) -

(5.1.3)

, ^ ^ ^ H = H0 - q(t)U , (5.1.4)

^ H0 . , ^ U(t) ( ) :


90 5

1 ^ ^ U(t) = U(t) - i¯ h +

t

^ ^ dt [U(t), U(t )]q(t ) 1 (i¯ ) h
- t t

2

dt
- -

^ ^ ^ dt [ [U(t), U(t )], U(t )] q(t )q(t ) + . . . . (5.1.5)

^ (5.1.5) (5.1.1) , U(t) : ^ ^ ^ [ [U(t), U(t )], U(t )] 0 t, t , t . (5.1.6)

(5.1.5), , . (5.1.5) (5.1.1) , ^ ^ [U(t), U(t ): i [U(t) , U(t )] , t t , ^ ^ h (t, t ) = ¯ 0, t > t ( ) . ^ ^ [U(t), U(t )] = ^ ^ [U(t), U(t )] = i¯ (t , t) - (t, t ) . h (5.1.9) h -i¯ (t, t ) , t i¯ (t , t) , h t t, t,

(5.1.7)



(5.1.8)

, . (5.1.9) 1^ ^ ^ ^ U(t)U(t ) + U(t )U(t) . (5.1.10) 2 , , - . ^ Q, B(t, t ) = ^ Q=


^ Q(t)U(t) dt ,

(5.1.11)

Q(t) . , Q : ^ Q+ =


-

^ ^ Q (t)U(t) dt = Q .

(5.1.12)

^ " " Q:

-


5.1

91

^^ Q+ Q =

(5.1.10) (5.1.9): 1^ ^ ^ ^ ^^ ^ ^ ^^ U(t)U(t ) = U(t)U(t ) + U(t )U(t) + U(t)U(t ) - U(t )U(t) 2 1^ i¯ h ^ (t , t) - (t, t ) . (5.1.14) = B(t, t ) + [U(t), U(t )] = B(t, t ) + 2 2 , (5.1.13) . , Q(t)


-



^^ Q (t)Q(t ) U(t)U(t ) dtdt .

(5.1.13)

Q (t)Q(t )B(t, t ) dtdt

. 5.1.2

-

h i¯ 2

-



Q (t)Q(t ) (t, t ) - (t , t) dtdt . (5.1.15)

(5.1.15), , , . , , . (t, t ), B(t, t ) , :
i(t-t )

(t, t ) = B(t, t ) =

()e

-

S()e

i(t-t )

^ S() U(t). (5.1.1) ^ ^ U() = U() + ()q() . , , iR ^ ^ , U() = U() + q() , () = iR . (5.1.19) (5.1.18) (5.1.17)

-

d , 2 d , 2

(5.1.16a) (5.1.16b)


92 5 (5.1.16) (5.1.15), (t , t) = :


()e

i(t-t )

-

d , 2

(5.1.20)

Q (t)e


it

dt
it

Q(t )e


-it

dt S()
-it

-

i¯ h 2

-

Q (t)e

dt

-

d 2 () - () dtdt . (5.1.21)

Q(t )e

dt



-

-

- -it

Q() =

Q(t )e

dt

(5.1.22)

Q(t) ( ), (5.1.21)


-

, Q() S() -¯ () . h (5.1.24)

-

d |Q()| S() 2
2

-¯ h

-



|Q()|2 ()

d . 2

(5.1.23)

, S() (), , S(-) = S () , (-) = () . (5.1.25a) (5.1.25b)

, B(t, t ) = B(t, t ), S() , (5.1.25), : S(-) = S() . (5.1.26a)

(), (5.1.25), : (-) = - () . , (5.1.24) -, , S(-) = S() = -¯ (-) = ¯ (-) . h h (5.1.27) (5.1.26b)

, , S() ¯ | ()| . h (5.1.28)


5.1

93

, ... R S() ¯ |||R| . h (5.1.29)

(5.1.28) - . , , . (5.1.28) , . "" , ( ), . , . , R < 0. (5.1.29) , |R|.


94 5

5.2
5.2.1




, , , "" . , . , - . , , , , . (, , ) 2 â 2 = 11 (t, t ) 12 (t, t ) 21 (t, t ) 22 (t, t ) , (5.2.1)

"" (, ): ^ ^ U1 (t) = U1 (t) + ^ ^ U2 (t) = U2 (t) +
t t

11 (t, t )q1 (t ) dt +
- t -

12 (t, t )q2 (t ) dt ,
- t -

(5.2.2a) (5.2.2b)

21 (t, t )q1 (t ) dt +

22 (t, t )q2 (t ) dt .

, , q1,2 ^ , U1,2 (t) , ^ 1,2 (t) , " U ", . , . , ,
2

^ ^ H = H0 -
j=1

^ qj (t)Uj ,

(5.2.3)

^ H0 . , ^ U1 , 2(t) ( ) : 1 ^ ^ U1 (t) = U1 (t) - i¯ h 1 + (i¯ ) h
t 2 t - t - 2

^ dt U1 (t),
j=1

^ Uj (t )qj (t )
2 2

dt
-

dt

^ U1 (t),
j=1

^ Uj (t )qj (t ) ,
j=1

^ Uj (t )qj (t ) + . . . , (5.2.4a)


5.2 1 ^ ^ U2 (t) = U2 (t) - i¯ h 1 + (i¯ ) h
t 2 t t 2

95

^ dt U2 (t),
- j=1 2

^ Uj (t )qj (t )
2

dt
-

dt

^ U1 (t),
j=1

^ Uj (t )qj (t ) ,
j=1

^ Uj (t )qj (t ) + . . . . (5.2.4b)

^ , (5.2.4) (5.2.2) , U1,2 (t) : ^ ^ ^ [ [Uj (t), Uk (t )], Ul (t )] 0 t, t , t , j, k, l = 1, 2 . (5.2.5)

-

(5.2.2), , , i [U (t) , U (t )] , t t ^j ^k h jk (t, t ) = ¯ 0, t>t ^ ^ [Uj (t), Uk(t )] = ^ ^ [Uj (t), Uk(t )] = i¯ kj (t , t) - jk (t, t ) . h ^ Q,
2

(j, k = 1, 2) , ( ) .

(5.2.6)

(5.1.7), h -i¯ jk (t, t ) , t i¯ kj (t , t) , h t t, t, (5.2.7)

(5.2.8)

^ Q=
j=1

Q1,2 (t) . ^ " " Q:
2

-



^ Qj (t)Uj (t) dt ,

(5.2.9)

^^ Q+ Q =

: 1^ ^ ^ ^ ^ ^ ^ ^ ^ ^ Uj (t)Uk (t ) = Uj (t)Uk (t ) + Uk (t )Uj (t) + Uj (t)Uk (t ) - Uk (t )Uj (t) 2 1^ i¯ h ^ = Bjk (t, t ) + [Uj (t), Uk(t )] = Bjk (t, t ) + kj (t , t) - jk (t, t ) , (5.2.11) 2 2

j,k=1 -



^ ^ Q (t)Qk (t ) Uj (t)Uk (t ) dtdt . j

(5.2.10)


96 5

Bjk (t, t ) =

1^ ^ ^ ^ Uj (t)Uk (t ) + Uk (t )Uj (t) 2

(5.2.12)

^ ^ U1 (t), U2 (t). (5.2.10) , , Q(t)
2

Q (t)Qk(t )Bjk (t, t ) dtdt j i¯ h 2
2

jk=1 -

Q (t)Qk(t ) jk (t, t ) - kj (t , t) dtdt , (5.2.13) j

, (5.1.15), . , , , , 2N, . , , 2 N. :
N

jk=1 -

Q (t)Qk(t )Bjk (t, t ) dtdt j i¯ h 2
N

jk=1 -

Q (t)Qk(t ) jk (t, t ) - kj (t , t) dtdt , (5.2.14) j

Q1 (t), Q2 (t), . . . QN (t) 5.2.2

jk=1 -

.



, (5.2.13) , . , . , . , , , , , , ( , , ). , jk (t, t ) = jk (t)(t - t ) , Bjk (t, t ) = Sjk (t)(t - t ) , (5.2.15a) (5.2.15b)


5.2

97

jk (t) Sjk (t) , , , . , B12 (t, t ) B21 (t , t), S12 (t) = S21 (t). Sjk (t) . , , ^ , , U1,2 (q S12 ). (5.2.13), , :
2

Q (t)Qk(t)Sjk (t) dt

j

Q1,2 (t), t:
2

jk=1 -

i¯ h 2

2

jk=1 -



Q (t)Qk(t) jk (t) - kj (t) dt , j

(5.2.16)

Q (t)Qk(t)Sjk (t)
jk=1

j

i¯ h 2

2

Q (t)Qk(t) jk (t) - kj (t) j
jk=1

t.

(5.2.17)

Q

1,2

: Q
1,2

= |Q1,2 |e

i

1,2

,

(5.2.18)

: 0. (5.2.19) |Q1 |. :
22 2

|Q1 |2 S11 + 2S12 cos(1 - 2 ) - ¯ (12 - 21 )) sin(1 - 2 ) |Q1 ||Q2 | + |Q2 |2 S h

2S12 cos(1 - 2 ) - ¯ (12 - 21 )) sin(1 - 2 ) h

- 4S11 S

22

0.

(5.2.20)

1 - 2 . 1 - 2 , , h 4S2 + ¯ 2 (12 - 21 )2 - 4S11 S22 , 12 (5.2.21)

, (5.2.20) : S11 (t)S22 (t) - |S12 (t)|
2

¯2 h 12 (t) - 21 (t) 4

2

.

(5.2.22)


lacements 98 5 q ^ ^ U
in fluct out fluct

q ^ U

out

q ^

^ U
in fluct

fluct

q

out

Uin q ^ (a) (b)

K
(a)

out

Uin q ^

K
(b)

^ U

out

. 5: : (a)

; (b)



, , (5.2.13) . , "" "" 12 21 . , , 12 = 21 , S11 (t)S22 (t) - |S12 (t)|
2

0.

(5.2.23)

, ( , , , , "" ). , ( ) , . . 5.2.3

. , ( ) , K, ( ) . , K, Sjk , ( ). q1,2 (t) ( , ) Uin (t) qout (t) ( , , -). qin (t) Uout (t) (. . 5(a)). qfluct (t) Ufluct out (t), Sq SU out , SqU out ( "" out ). , q U, ( (5.2.2), :


5.2

99

(5.2.2) : qin (t) = qfluct (t) , ^ ^ ^ (t) = Uout fluct (t) + KUin (t) . (5.2.25a) (5.2.25b)

q1 (t) Uin (t) , ^ U1 (t) qin (t) , ^ ^ U1 (t) qfluct (t) , ^ S11 Sq ,

q2 (t) qout (t) , ^ ^ U2 (t) Uout (t) , ^ ^ U2 (t) Uout fluct (t) , S12 SqU out ,

(5.2.24a) (5.2.24b) S
22

S

U out

.

(5.2.24c) (5.2.24d)

^ U

out

, = 12 = 22 = 0 , 21 = K , (5.2.26a) (5.2.26b) (5.2.26c)

11

(5.2.22) : Sq S
U out

-S

2 qU out

¯2 2 h K. 4

(5.2.27)

^ , U U

out fluct

(t) U
out fluct

fluct

(t) =

(t)

K S

.

(5.2.28)

SU = q
fluct U out K2

,

(5.2.29)

(t) S
qU

. (5.2.30) K : = Sq SU - S 5.2.4
2 qU

S

qU out

¯2 h . 4

(5.2.31)



, , , (5.2.13) , , jk (t, t ), Bjk (t, t )


100 5 t t . (. (5.1.16)):


jk (t, t ) = Bjk (t, t ) =

jk ()e

i(t-t )

-

Sjk ()e

i(t-t )

(5.2.32) (5.2.13). :
2

-

d , 2 d . 2

(5.2.32a) (5.2.32b)

Q ()Qk()Sjk () j i¯ h 2

j,k=1 -

d 2
2

Q ()Qk () jk () - () j kj



j,k=1 -

d , (5.2.33) 2

Qj () =

Qj (t )e

-it

dt

(5.2.34)

Qj (t) ( ). Qj (), (5.2.33) :
2

-

Q Qk Sjk () j
j,k=1

i¯ h 2

2

Q Qk jk () - () , kj j
j,k=1

(5.2.35a)

Q1,2 . , - Sjk () jk (), ,
2

Q Qk Sjk () j
j,k=1

-

h i¯ 2

2

Q Qk jk () - () . j kj
j,k=1

(5.2.35b)

(5.2.35) 2 â 2 Sjk () i¯ h jk () - () kj 2 . (5.2.36)

S11 () S22 () ¯ | 11 ()| , h ¯ | 11 ()| , h (5.2.37a) (5.2.37b)


5.2

101

( , )

S11 () ± ¯ 11 () h

S22 () ± ¯ 22 () h S12 () i¯ h 12 () - () 21 2
2

(5.2.37c)

( ± , ). (5.2.37a) (5.2.37b) (5.1.28), . , , ( 11 22 ). (5.2.37c). , ( 12 21 ). , . , , , jk 0. (5.2.37a) (5.2.37b) , (5.2.37c) :

S11 ()S22 () - |S12 ()|

2 2

¯2 h 12 () - 21 () 4

+ ¯ S12 () 12 () - 21 () h

. (5.2.38)

, 12 21 , (5.2.22). , . , "" ( ) "" ( ). 5.2.5 :

, , , /, /. , - . - . "", . , (. . 6). S, ,


102 5 S

PSfrag replacements

R

BS

M
x

D1

D2

-

I

. 6: M, x . : ^ E
signal

^ ^ (t) = (A0 + A(t)) sin(o t + (t)) ,

(5.2.39) -

^ A0 , A(t) , SA = ^ 2¯ o h , c

(5.2.40)

, S (t) = ¯ o h , 4W (5.2.41)



cA2 0 W= (5.2.42) 8 . : (5.2.40) (5.2.41), 3.4. 1 , x, ^ E ^
out out

^ ^ (t) = (A0 + A(t)) sin(o t +

out

(t)) ,

(5.2.43)

2o x(t) cos ^ . (t) = (t) - c

(5.2.44)


5.2

103

, , "" BS R: ^ ^ Eref (t) = (B0 + B(t)) cos(o t +
LO

^ + ref (t)) ,

(5.2.45)

B0 , LO ^ (t) LO (t) ^ (.. ), B . , , B
0

A0 .

(5.2.46)

, , ^ ^ Eref (t) + E ^ E1 (t) = 2 ^ ref (t) - E ^ E ^ E2 (t) = 2
out

(t) (t)

, .

(5.2.47a) (5.2.47b)

out

. , , . e ec ^ (E1,2 (t))2 , (5.2.48) W1,2 = ¯ o h 4¯ o h W1,2 , , . I
1,2

=

ec ^ ^ Eref (t)Eout (t) . (5.2.49) 2¯ o h , ^ ^ ^ ^ out (t), ref (t), A(t) B(t): I- (t) = I- (t) = ec 4¯ h ^ ^ A0 B0 + B0 A(t) + A0B(t) cos
o LO

^ + A 0 B0

out

^ (t) - ref (t) sin

LO

.

(5.2.50) , A0 B0 , , . , (5.2.46), , : ecB0 ^ ^ A(t) cosLO +A0 out (t) sin LO . (5.2.51) 4¯ o h ( ) , (5.2.25b): I- (t) =


104 5

I- (t) = I I
fluct

fluct

(t) + Kx(t) ,

(5.2.52)

(t) =

ecB0 4¯ h

^ ^ A(t) cosLO +A0 sin
o

LO

(5.2.53)

"", K=- eA0 B0 cos sin 2¯ h
LO

(5.2.54)

" ". , , Ifluct (t) x
fluct

(t) =

I

fluct

(t) c =- K 2o cos

^ A(t) cot A0

LO

^ + (t)

,

(5.2.55)

(5.2.52) : I- (t) = K x
fluct

(t) + x(t) ,

(5.2.56)

^ (5.2.25) Uoutfluct qfluct , ^ . x, , M. , M. ^ (t) cos (Esignal (t))2 cos = . (5.2.57) c 2 (5.2.39) , , F
fluct

(t) =

2W

signal

F

fluct

(t) = F0 + F

fluct

,

(5.2.58)

2W cos (5.2.59) c , , F0 = ^ A0 A(t) cos 2 . F
fluct

(t) =

(5.2.60)


5.2

105

, "". . . (5.2.40), (5.2.41) (5.2.55) , xfluct (t) Sx = ¯ o c2 h 16W cos2 sin2 .
LO fluct

(5.2.61a) (t)

(5.2.40) (5.2.60) , F SF =

4¯ o W cos2 h . (5.2.61b) c2 , (5.2.40), (5.2.41),(5.2.55) (5.2.60) , ¯ h cot LO . (5.2.61c) 2 , S
xF

=-

¯2 h Sx SF - S = , (5.2.62) 4 ( ) (5.2.31). , . , , .
2 xF