Use of the all-Russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use

Math-Net.Ru

All Russian mathematical portal

G. G. Amosov, V. I. Man’ko, Evolution of probability measures associated with quantum systems, TMF , 2005, Volume 142, Number 2, 365–370

DOI: https://doi.org/10.4213/tmf1788

Use of the all-Russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use

http://www.mathnet.ru/eng/agreement Download details:

IP: 178.128.90.69

November 6, 2022, 23:01:49

”ˆ‡ˆŠ€

’®¬142,ò2

䥢ࠫì,2005

c ^{2005£. ƒ.ƒ.€¬®á®¢}

^∗

^{,‚.ˆ.Œ ­ìª®}

^†

‚Ž‹ž–ˆŸ ‚…ŽŸ’Ž‘’›• Œ…,

‘‚Ÿ‡€›• ‘ Š‚€’Ž‚›Œˆ ‘ˆ‘’…Œ€Œˆ

‚뢥¤¥­®í¢®«î樮­­®¥ãà ¢­¥­¨¥¤«ïà á¯à¥¤¥«¥­¨©¢¥à®ïâ­®á⥩¨å à ªâ¥à¨áÄ

â¨ç¥áª¨åä㭪権ª¢ ­â®¢ëå⮬®£à ¬¬,á¢ï§ ­­ëå᫨­¥©­ë¬¨¨­¥«¨­¥©­ë¬¨í¢®Ä

«î樮­­ë¬¨ãà ¢­¥­¨ï¬¨¤«ïª¢ ­â®¢ëåá®áâ®ï­¨©.

Š«î祢ë¥á«®¢ :ª¢ ­â®¢ë¥â®¬®£à ¬¬ë,í¢®«î樮­­ë¥ãà ¢­¥­¨ï.

1.‚‚…„…ˆ…

¥¤ ¢­®¡ë« ¤ ­ ä®à¬ã«¨à®¢ª ª¢ ­â®¢®©¬¥å ­¨ª¨(¢¥à®ïâ­®áâ­®¥¯à¥¤áâ ¢«¥Ä

­¨¥),¢ª®â®à®©ª¢ ­â®¢ë¥á®áâ®ï­¨ïá¢ï§ë¢ îâáïá®áâ ­¤ àâ­ë¬¨¯«®â­®áâﬨà áÄ

¯à¥¤¥«¥­¨©¢¥à®ïâ­®á⥩(â ª­ §ë¢ ¥¬ë¬¨â®¬®£à ä¨ç¥áª¨¬¨¯«®â­®áâﬨ,¨«¨á¨¬Ä

¯«¥ªâ¨ç¥áª¨¬¨â®¬®£à ¬¬ ¬¨).â®âä®à¬ «¨§¬,ᮤ¥à¦ é¨©â ªã¨­ä®à¬ æ¨î,

ª ª¢®«­®¢ ïäã­ªæ¨ï¨¬ âà¨æ ¯«®â­®áâ¨,®á­®¢ ­­ â®¬®£à ä¨ç¥áª®¬¯®¤å®¤¥ª¨§Ä

¬¥à¥­¨îª¢ ­â®¢ëåá®áâ®ï­¨©[1],[2].‚à ¡®â å[1]¡ë«®­ ©¤¥­®,çâ®â®¬®£à ä¨ç¥áª®¥

à á¯à¥¤¥«¥­¨¥¢¥à®ïâ­®á⥩á¢ï§ ­®áä㭪樥©‚¨£­¥à [3].”ã­ªæ¨ï‚¨£­¥à W(q;p)

¤ ¥â®¯¨á ­¨¥ª¢ ­â®¢®© ¬¥å ­¨ª¨, ­ ¨¡®«¥¥á室­®¥áª« áá¨ç¥áª®©áâ â¨áâ¨ç¥áª®©

¬¥å ­¨ª®©£ ¬¨«ìâ®­®¢ëåá¨á⥬. DZà¨í⮬¢ë¯®«­ï¥âáïãá«®¢¨¥­®à¬¨à®¢ª¨

R

dq

∧

dpW(q;p)=1. ’¥¬­¥¬¥­¥¥W(q;p)­¥ï¢«ï¥âáﯮ«®¦¨â¥«ì­®®¯à¥¤¥«¥­­®©. ’ ª¨¬

®¡à §®¬,®­ ®¯à¥¤¥«ï¥âª¢ §¨¢¥à®ïâ­®áâ­®¥à á¯à¥¤¥«¥­¨¥.‘¤à㣮©áâ®à®­ë,¢ª¢ ­Ä

⮢®©â¥®à¨¨¨­ä®à¬ æ¨¨¯à®¨§¢®«ì­®©íନ⮢®©­ ¡«î¤ ¥¬®©¬®¦¥â¡ëâìᮯ®áâ ¢Ä

«¥­®à á¯à¥¤¥«¥­¨¥¢¥à®ïâ­®á⥩[4]. ‚­ è¨å­¥¤ ¢­¨åà ¡®â å[5],[6]¡ë« ãáâ ­®¢Ä

«¥­ á¢ï§ì⮬®£à ä¨ç¥áª®©¯«®â­®áâ¨à á¯à¥¤¥«¥­¨ïᬥன,¨á¯®«ì§ã¥¬®©¢ª¢ ­â®Ä

¢®©â¥®à¨¨¨­ä®à¬ æ¨¨. ë«®¯®ª § ­®,ç⮯ந§¢®¤­ ïâ ª®©¬¥à뮯।¥«ï¥ââ®Ä

¬®£à ä¨ç¥áªã®â­®áâìà á¯à¥¤¥«¥­¨ï.–¥«ì¯à¥¤« £ ¥¬®©à ¡®âë{¨§ãç¨â죫ã¡Ä

¦¥á¢®©á⢠¢¥à®ïâ­®áâ­®£®¯à¥¤áâ ¢«¥­¨ïª¢ ­â®¢ëåá®áâ®ï­¨©¨¢ë¢¥á⨭¥«¨­¥©Ä

­®¥í¢®«î樮­­®¥ãà ¢­¥­¨¥¤«ïª¢ ­â®¢®-¢¥à®ïâ­®áâ­ëåà á¯à¥¤¥«¥­¨©,á¢ï§ ­­ëåá

∗

Œ®áª®¢áª¨©ä¨§¨ª®-â¥å­¨ç¥áª¨©¨­áâ¨âãâ,£.„®«£®¯à㤭ë©,Œ®áª®¢áª ï®¡«.,®áá¨ï.E-mail:

amosov@zteh.ru

†

”¨§¨ç¥áª¨©¨­áâ¨âã⨬.DZ..‹¥¡¥¤¥¢ €,Œ®áª¢ ,®áá¨ï.E-mail:manko@sci.leb edev.ru

366

ª¢ ­â®¢ë¬¨á®áâ®ï­¨ï¬¨,¨á¯®«ì§ã﨧¢¥áâ­®¥í¢®«î樮­­®¥ãà ¢­¥­¨¥¤«ï⮬®£à Ä

ä¨ç¥áª®©¢¥à®ïâ­®áâ¨. ‡ ¬¥â¨¬,çâ®í¢®«î樮­­ë¥ãà ¢­¥­¨ï¤«ïà á¯à¥¤¥«¥­¨©¢¥Ä

à®ïâ­®á⥩, ª®â®à륬ë¢ë¢®¤¨¬,¯à¨­ ¤«¥¦ â ªâ¨¯ã, ­¥¢áâà¥ç ¢è¥¬ãáï¢ â¥®à¨¨

ª« áá¨ç¥áª¨åá«ãç ©­ëå¯à®æ¥áᮢ. ’¥¬á ¬ë¬¬ë­¥¬®¦¥¬®¯à¥¤¥«¨âì­¥¬¥¤«¥­­®

â ª¨¥ ᯥæ¨ä¨ç¥áª¨¥á¢®©á⢠¢¢¥¤¥­­ëå­ ¬¨á«ãç ©­ëå¯à®æ¥áᮢ, ª ªáâ æ¨®­ àÄ

­®áâì,¬ àª®¢®áâì¨â.¯. ’¥¬­¥¬¥­¥¥®á­®¢­®¥¤®á⮨­á⢮­ è¥£®¯®¤å®¤ § ª«îç Ä

¥âá®§¬®¦­®á⨨ᯮ«ì§®¢ ­¨ïâ¥å­¨ª¨ª« áá¨ç¥áª®©â¥®à¨¨¢¥à®ïâ­®á⥩¤«ï¨§ãÄ

祭¨ïª¢ ­â®¢ëåíä䥪⮢¯à¨¯®¬®é¨¢¢¥¤¥­­ëå­ ¬¨ª« áá¨ç¥áª¨åá«ãç ©­ëå¯à®Ä

æ¥áᮢ. „«ïª¢ ­â®¢®©í¢®«î樨¢«¨­¥©­®¬á«ã砥⠪®¥ãà ¢­¥­¨¥¡ë«®­ ©¤¥­®¢

à ¡®â å[5],[6].„«ï⮬®£à ä¨ç¥áª¨å¯«®â­®á⥩à á¯à¥¤¥«¥­¨©¢¥à®ïâ­®á⥩¢à ¡®Ä

â å[7]¡ë«¨¯®«ã祭ëí¢®«î樮­­ë¥ãà ¢­¥­¨ï¤«ïª®­¤¥­á â ®§¥{©­è⥩­ ¨¤«ï

᮫¨â®­®¢,¯®ï¢«ïîé¨åá¥«¨­¥©­®¬ãà ¢­¥­¨¨˜à¥¤¨­£¥à . ‚­ áâ®ï饩ࠡ®â¥

¬ë®¯¨à ¥¬áï­ å à ªâ¥à¨áâ¨ç¥áª¨¥ä㭪樨,®¯à¥¤¥«ïî騥à á¯à¥¤¥«¥­¨¥¢¥à®ïâÄ

­®á⥩,¯® ­ «®£¨¨á¯®¤å®¤®¬,¯à¥¤«®¦¥­­ë¬¢à ¡®â¥[8]. Œë¢ë¢®¤¨¬í¢®«î樮­Ä

­®¥ãà ¢­¥­¨¥¨¬¥­­®¤«ïå à ªâ¥à¨áâ¨ç¥áª¨åä㭪権,®â¢¥ç îé¨åᨬ¯«¥ªâ¨ç¥áª¨¬

⮬®£à ¬¬ ¬.

2. ‚…ŽŸ’Ž‘’›… Œ…›, ‘‚Ÿ‡€›…

‘ Š‚€’Ž‚›Œˆ ‘ˆ‘’…Œ€Œˆ

DZãáâì

H

^{¥áâ쪮¬¯«¥ªá­®¥}£¨«ì¡¥à⮢®¯à®áâà ­á⢮. Ž¡®§­ ç¨¬ç¥à¥§(

H

^)¬­®Ä

¦¥á⢮¢á¥åá®áâ®ï­¨©(®¯¥à â®à®¢¯«®â­®áâ¨)¢

H

^{.«¥¬¥­â ¬¨(}

H

^{)ïîâáﯮ«®Ä}

¦¨â¥«ì­ë¥®¯¥à â®àëᥤ¨­¨ç­ë¬á«¥¤®¬¢

H

^.„«ï䨪á¨à®¢ ­­®£®^

∈

⁽

H

^{)äã­ªæ¨ï}

âà¥å¤¥©á⢨⥫ì­ë寠ࠬ¥â஢X,,,®¯à¥¤¥«¥­­ ïä®à¬ã«®©

!(X;;)=!

(X;;)=Tr(Æ(X^

−

^{^x}

−

^{p));^}

­ §ë¢ ¥âá甆 ­â®¢®© ⮬®£à ¬¬®©. ‡¤¥áì¬ë¨á¯®«ì§®¢ «¨®¯¥à â®à­ãîÆ-äã­ª-

æ¨î,®¯à¥¤¥«¥­­ãî¯à¥®¡à §®¢ ­¨¥¬”ãà쥪 ª

Æ( ^a)= 1

2

Z

R

e ik^a

dk:

„«ï«î¡ëå䨪á¨à®¢ ­­ëå¨â®¬®£à ¬¬ !(X;;)¯à¥¤áâ ¢«ï¥âᮡ®©¯«®â­®áâì

à á¯à¥¤¥«¥­¨ï¢¥à®ïâ­®á⥩,â ªçâ®

!(X;;)

>

^0;

Z

R

!(X;;)dX=1:

”ã­ªæ¨ïà á¯à¥¤¥«¥­¨ï

M

^(X;;)=

M

^{(X;;),®â¢¥ç îé ï¯«®â­®áâ¨!(X;;),}

¬®¦¥â¡ëâ쯮«ã祭 á«¥¤ãî騬®¡à §®¬.  áᬮâਬᯥªâà «ì­®¥à §«®¦¥­¨¥­ Ä

¡«î¤ ¥¬®©^x+p:^

^x+p^=

Z

R

Xd^M((

c −∞

^;X])=

Z

R

XÆ(X

−

^{^x}

−

^{p)^ dX;}

£¤¥^M

c

^¥áâì®à⮣®­ «ì­®¥à §«®¦¥­¨¥¥¤¨­¨æë,á¢ï§ ­­®¥á®¯¥à â®à®¬^x+p.^ ’®£¤ 

ä®à¬ã« 

M

^{(X;;)=Tr(^M((}

c −∞

^;X]))

®¯à¥¤¥«ï¥âäã­ªæ¨îà á¯à¥¤¥«¥­¨ï¤«ï¯«®â­®áâ¨!(X;;),â ªçâ®

!(X;;)=

d

M

^(X;;)

dX :

”¨§¨ç¥áª¨©á¬ëá«à á¯à¥¤¥«¥­¨ï¢¥à®ïâ­®á⥩, ®¯à¥¤¥«ï¥¬®£®ä㭪樥©

M

^(X;;)

¤ î⯮ª § ­¨ïª« áá¨ç¥áª®£®¯à¨¡®à ,¨§¬¥àïî饣®­ ¡«î¤ ¥¬ãîx^(ª¢ ¤à âãàã)¢

á¨á⥬¥ª®®à¤¨­ âä §®¢®£®¯à®áâà ­á⢠,¢ª®â®à®©¡ë« ¯à®¨§¢¥¤¥­ § ¬¥­ ¬ áèâ Ä

¡ ¨á®¢¥à襭¯®¢®à®â(ª¢ ¤à âãà £®¬®¤¨­ ). ˆ§¢¥áâ­®,çâ®äã­ªæ¨ïà á¯à¥¤¥«¥­¨ï

ï¥âá鶴¢ à¨ ­â®¬¯à¥®¡à §®¢ ­¨ï¬ áèâ ¡ ,â ªçâ®

M

^(X;;)=

M

^(X;;).

DZãáâ쬭®¦¥á⢮å à ªâ¥à¨áâ¨ç¥áª¨åä㭪権à á¯à¥¤¥«¥­¨©!(X;;)®¯à¥¤¥«ïÄ

¥âáïä®à¬ã«®©

F(k;;)=F

(k;;)=

Z

R

e ik X

!

(X;;)dX=Tr(^e

ik (^x+p )^

): (1)

‚¢¥¤¥­­ ïäã­ªæ¨ï®¡« ¤ ¥â᢮©á⢮¬

F

k

;;

=F(k;;): (1

0

)

‘¢®©á⢮(1

0

)®â¢¥ç ¥â᢮©áâ¢ã®¤­®à®¤­®áâ¨â®¬®£à ¬¬.‡ ¬¥â¨¬,ç⮬®¦­®¢®ááâ Ä

­®¢¨âì!(X;;)¯®F(k;;)á«¥¤ãî騬®¡à §®¬:

!

(X;;)= 1

2

Z

R

e

−

^{ik X}

F(k;;)dk:

…᫨á®áâ®ï­¨¥=(t)㤮¢«¥â¢®àï¥âª ª®¬ã-«¨¡®í¢®«î樮­­®¬ããà ¢­¥­¨î,â®íâ®

¯à¨¢®¤¨â ª í¢®«î樮­­ë¬ ãà ¢­¥­¨ï¬ ¤«ï !(t;X;;) = !

(t)

(X;;),

M

^(t;X;;)=

M

(t)

(X;;)¨F(t;k;;)=F

(t)

(k;;). ­¥¥(á¬.[5])¬ë¯®«ãç¨Ä

«¨(¢­¥ª®â®àëåç áâ­ëåá«ãç ïå)í¢®«î樮­­ë¥ãà ¢­¥­¨ï¤«ï¯«®â­®áâ¨!(t;X;;)

¨®â¢¥ç î饩¥©ä㭪樨à á¯à¥¤¥«¥­¨ï

M

^{(t;X;;).‚í⮩ࠡ®â¥¬ë¢ë¢¥¤¥¬í¢®Ä}

«î樮­­ë¥ãà ¢­¥­¨ï¤«ïF(t;k;;)¨

M

^(t;X;;).

3.‚Ž‹ž–ˆŽŽ… “€‚…ˆ…

„‹Ÿ •€€Š’…ˆ‘’ˆ—…‘Šˆ•

”“Š–ˆ‰ ‚ ‹ˆ…‰ŽŒ ‘‹“—€…

 áᬮâਬª¢ ­â®¢ãîí¢®«îæ¨î,®¯¨á뢠¥¬ãîãà ¢­¥­¨¥¬

@

t

^

=

−

^i[H

b

^{;];^ (2)}

368

£¤¥^

∈

^(H)¨H

b

^{¯à®¨§¢®«ì­ë©£ ¬¨«ìâ®­¨ ­.‚뢥¤¥¬í¢®«î樮­­®¥ãà ¢­¥­¨¥¤«ï å à ªâ¥à¨áâ¨ç¥áª®©ä㭪樨F(t;k;;),®â¢¥ç îé¥©í¢®«î樨á®áâ®ï­¨ï(2) . „¨äÄ

ä¥à¥­æ¨àãﯮt,¬ë¯®«ãç ¥¬¤«ïä㭪樨(1)

@

t

F(t;k;;)=

−

^iTr([H

b

^{;]e^ ik (x+^ p )^ )=iTr(^;[H}

b

^{;eik (^x+p )^ ]): (3)}

DZਭ¨¬ ï¢®¢­¨¬ ­¨¥,çâ®e

ik (^x+p )^

=e ik x^

e ik p^

e k =2

=e ik p^

e ik ^x

e

−

^{k =2}

,­ å®Ä

¤¨¬

Tr(^^p s

e

ik (^x+p )^

)=

e

−

^{k =2 1}

ik

@

e

k =2

^s

Tr(e^

ik (^x+p )^

);

Tr(^ e

ik (^x+p )^

^ p s

)=

e k =2

1

ik

@

e

−

^{k =2}

^s

Tr(e^

ik (^x+p )^

);

Tr(^^x s

e

ik (^x+p )^

)=

e

−

^{k =2 1}

ik

@

e

k =2

^s

Tr(^ e

ik (^x+p )^

);

Tr(^e

ik (^x+p)^

^ x s

)=

e k =2

1

ik

@

e

−

^{k =2}

^s

Tr(^ e

ik (^x+p )^

);

s=1;2;3;:::.‘«¥¤®¢ â¥«ì­®,¥á«¨£ ¬¨«ìâ®­¨ ­¨¬¥¥â¢¨¤^H

b

^{=W(^p)+V(^x ),£¤¥W¨}

V {¯à®¨§¢®«ì­ë¥ ­ «¨â¨ç¥áª¨¥ä㭪樨,â®í¢®«î樮­­®¥ãà ¢­¥­¨¥¤«ïF(t;k;;)

¯à¨­¨¬ ¥âä®à¬ã

@

t

F(t;k;;)=i

W

e

−

^{k =21}

ik

@

e

k =2

−

^W

e k =2

1

ik

@

e

−

^{k =2}

F(t;k;;)+

+i

V

e

−

^{k =21}

ik

@

e

k =2

−

^V

e k =2

1

ik

@

e

−

^{k =2}

F(t;k;;): (4)

‚ç áâ­®¬á«ãç ¥W(^p)=p^ 2

=2,¯®«ãç ¥¬

@

t

F(t;k;;)=@

F(t;k;;)+i

V

e

−

^{k =21}

ik

@

e

k =2

−

−

^V

e k =2

1

ik

@

e

−

^{k =2}

F(t;k;;):

4.…‹ˆ…‰Ž… ‚Ž‹ž–ˆŽŽ… “€‚…ˆ…

 áᬮâਬ¢ª ç¥á⢥¯à¨¬¥à ­¥«¨­¥©­®¥í¢®«î樮­­®¥ãà ¢­¥­¨¥¤«ï¢®«­®¢®©

ä㭪樨

∈ H

^{á«¥¤ãî饣®¢¨¤ :}

i

t

=

^ p 2

+V(

| |

^{2) :}

¥«¨­¥©­®áâì ®¡ãá«®¢«¥­ § ¢¨á¨¬®áâìî ¯®â¥­æ¨ «ì­®© í­¥à£¨¨®â ¢®«­®¢®© äã­ªÄ

樨. ˆ¬¥¥âá﨧¢¥áâ­ë©¯à¨¬¥à­¥«¨­¥©­®£®ãà ¢­¥­¨ï˜à¥¤¨­£¥à áªã¡¨ç¥áª®©­¥Ä

«¨­¥©­®áâìî[9].‚¢¥¤¥¬ãà ¢­¥­¨¥¤«ïª®¬¯«¥ªá­®-ᮯà殮­­®©ä㭪樨

∗

:

−

ⁱ

^∗

t

=

^ p 2

2

∗

+V(

| |

²⁾

^∗

^:

’®£¤ ¤«ïá®áâ®ï­¨ï(t)^ =

|

^(t)

ih

^(t)

|

^{¯®«ãç ¥âáï}í¢®«î樮­­®¥ãà ¢­¥­¨¥‹¨ã¢¨«Ä

«ï{ä®­¥©¬ ­ 

^

t

=

−

ⁱ

^ p 2

2

;^

−

^i[V

b

⁽

| |

^{2);];^ (5)}

£¤¥®¯¥à â®à^V

b

⁽

| |

^{2)¤¥©áâ¢ã¥â¯®ä®à¬ã«¥(V}

b

⁽

| |

^2))(X)=V(

|

^(X)

|

^2)(X),

∈ H

^.

“¬­®¦¨¢®¡¥ç áâ¨(5)­ e

ik (^x+p )^

¨¢§ï¢á«¥¤,­ å®¤¨¬

@

t

F(t;k;;)=@

F(t;k;;)+iTr(^;[^V

b

⁽

| |

^{2);eik (^x+p^]):}

Ž¡®§­ ç¨¬^V

e

^(X)=V(

|

^(X)

|

^{2).’®£¤ ¨§(4)á«¥¤ã¥â,çâ®}

@

t

F(t;k;;)=@

F(t;k;;)+i

e

V

e

−

^{k =21}

ik

@

e

k =2

−

−

^V

e

e k =2

1

ik

@

e

−

^{k =2}

F(t;k;;):

‡ ¬¥â¨¬,çâ®

|

^(X)

|

^{2=!(X;1;0)= 1}

2

Z

R

e

−

^{ik X}

F(t;k;1;0)dk:

’ ª¨¬®¡à §®¬,¯®«ãç ¥âáïí¢®«î樮­­®¥ãà ¢­¥­¨¥¢¨¤ 

@

t

F(t;k;;)=@

F(t;k;;)+

+i

V

1

2

Z

R

exp

−

^ike~

⁻

^{k =21}

ik

@

e

k =2

F(t;

~

k;1;0)d

~

k

−

−

^V

1

2

Z

R

exp

−

^{i~kek =21}

ik

@

e

−

^{k =2}

F(t;

~

k;1;0)d

~

k

F(t;k;;): (6)

ˆá¯®«ì§ãﮡà â­®¥¯à¥®¡à §®¢ ­¨¥”ãàì¥,¨§ãà ¢­¥­¨ï(6)¬®¦­®¢ë¢¥áâ¨í¢®«îæ¨Ä

®­­®¥ãà ¢­¥­¨¥¤«ïà á¯à¥¤¥«¥­¨ï¢¥à®ïâ­®á⥩

M

^(t;X;;):

@

t

M

^(t;X;;)=@

M

^(t;X;;)+i

V

1

2

Z

R

exp

−

^@

@

~

X e

−

(i=2)(@=@X)

×

×

@

@X

−

¹

@

e

(i=2)(@=@X)

d

M

^(t;X

e

^;1;0)

−

−

^V

1

2

Z

R

exp

−

^@

@^X

e

^e

(i=2)(@=@X)

×

×

@

@X

−

¹

@

e

−

(i=2)(@=@X)

d

M

^(t;X;

e

^1;0)

M

^(t;X;;):

‡¤¥áì¬ë®¡®§­ ç¨«¨ç¥à¥§(@=@X)

−

¹

®¯¥à â®à,®¯à¥¤¥«¥­­ë©ä®à¬ã«®©

@

@X

−

¹

f

(x)= 1

2

Z

R

dk

Z

R

dye

ik (y

−

^{x) 1}

ik f(y):

370

5.‡€Š‹ž—…ˆ…

Œë¢ë¢¥«¨í¢®«î樮­­ë¥ãà ¢­¥­¨ï¤«ïà á¯à¥¤¥«¥­¨©¢¥à®ïâ­®á⥩¨¨åå à ªÄ

â¥à¨áâ¨ç¥áª¨å ä㭪権, á¢ï§ ­­ëå᪢ ­â®¢®©í¢®«î樥©, ¢ ¤¢ãåá«ãç ïå: ¤«ï «¨Ä

­¥©­®£®í¢®«î樮­­®£®ãà ¢­¥­¨ïᣠ¬¨«ìâ®­¨ ­®¬¢¨¤ ^H

b

^{=W(^p)+V(^x )(4)¨¤«ï}

í¢®«î樨,®¯à¥¤¥«ï¥¬®©­¥«¨­¥©­ë¬ãà ¢­¥­¨¥¬˜à¥¤¨­£¥à (6).

‘¯¨á®ª«¨â¥à âãàë

[1] J.Bertrand, P.Bertrand. Found.Phys.1987. V. 17. P.397; K.Vogel,H.Risken. Phys.

Rev.A.1989.V.40.P.2847.

[2] D.T.Smithey,M.Beck,M.G.Raymer,A.Faridani. Phys.Rev.Lett.1993.V.70.P.1244.

[3] E.P.Wigner. Phys.Rev.1932.V.40.P.749.

[4] A.S.Holevo.Statisticalstructureofquantumtheory.Lect.NotesPhys.V.67.Berlin:Springer,

2001.

[5] G.G.Amosov,V.I.Man'ko. Phys.Lett.A.2003.V.318.P.287.

[6] G.G.Amosov,V.I.Man'ko. J.Russ.LaserRes.2003. V.25.P.253;Quantumtomograms

asdensitiesofvonNeumannprobabilitydiistributions. In:Squeezed StatesandUncertainty

Relations.Pro c.ofthe8thInt.Conf.onSqueezedStatesandUncertaintyRelations(Puebla,

Mexico,June9{13,2003).Eds.H.Moya-Cessaetal.princeton:Rinton,2003.P.7.

[7] S.DeNicola,R. Fedele,M.A.Man'ko, V.I.Man'ko. Eur.Phys.J.B.2003. V.36.P.385;

J.Russ.LaserRes.2004. V.25.P.1.

[8] K.E.Cahill,R.J.Glauber. Phys.Rev.1969.V.177.P.1882.

[9] ‹.€. ’ åâ ¤¦ï­, ‹.„. ” ¤¤¥¥¢. ƒ ¬¨«ìâ®­®¢¯®¤å®¤¢â¥®à¨¨á®«¨â®­®¢. Œ.:  ãª ,

1986.