£¤¥ { ¥ª®â®à®¥ ¢à¥¬ï ५ ªá 樨. ®à५ïâ®à ¯à¨ ᮢ¯ ¤ îé¨å ¢à¥¬¥ å å®- ¤¨âáï í«¥¬¥â à®:
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< [J ; P ] >0=< [ |
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= e2 |
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ih e2 N |
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[pi ; xi ] = |
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(10.57) |
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£¤¥ N { ç¨á«® ç áâ¨æ, ¨ ¬ë ¢®á¯®«ì§®¢ «¨áì áâ ¤ àâë¬ ª®¬¬ãâ æ¨®ë¬ á®®â- |
®è¥¨¥¬ [xi ; pi ] = ih . १ã«ìâ ⥠¯®«ãç ¥¬: |
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dte("+1=)t = |
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¨«¨, ¢ à áç¥â¥ ¥¤¨¨æã ®¡ê¥¬ : |
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ne2 |
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(10.59) |
çâ® ¯à¥¤áâ ¢«ï¥â ᮡ®© ®¡ëçãî ä®à¬ã«ã à㤥. ®¤ç¥àª¥¬, çâ® \ áâ®ï饩" § ¤ 祩 ¬¨ªà®â¥®à¨¨ ï¥âáï, ª®¥ç®, ¢ë¢®¤ ¯®¢¥¤¥¨ï ⨯ (10.56) ¨§ ⮩ ¨«¨ ¨®© ¬®¤¥«¨ ¨ à áç¥â ᮮ⢥âáâ¢ãîé¨å § ¢¨á¨¬®á⥩ ®â ⥬¯¥à âãàë (¨«¨ ª®- æ¥âà 樨 ¯à¨¬¥á¥©) ¤«ï à §«¨çëå ¬¥å ¨§¬®¢ à áá¥ï¨ï. ¬¥® §¤¥áì ¢®§¨- ª ¥â ¥®¡å®¤¨¬®áâì ¨á¯®«ì§®¢ ¨ï ᮢ६¥ëå ¬¥â®¤®¢ ⥮ਨ á¨á⥬ ¬®£¨å ç - áâ¨æ, â ª¨å, ª ª ¬¥â®¤ äãªæ¨© ਠ.
¯¥ªâà «ìë¥ ¯à¥¤áâ ¢«¥¨ï ¢à¥¬¥ëå ª®à५ïâ®à®¢ ¨ ¤¢ãå¢à¥¬¥ë¥ äãªæ¨¨à¨ .
áᬮâਬ ¥ª®â®àë¥ ®¡é¨¥ ᢮©á⢠¢à¥¬¥ëå ª®à५ï樮ëå äãªæ¨©. ¢¥- ¤¥¬, ¯® ®¯à¥¤¥«¥¨î:
FAB(t ; t0) =< A(t)B(t0) > FBA(t0 ; t) =< B(t0)A(t) > |
(10.60) |
ãáâì ¨ E { ᮡáâ¢¥ë¥ äãªæ¨¨ ¨ ᮡáâ¢¥ë¥ § ç¥¨ï £ ¬¨«ì⮨ à á- ᬠâਢ ¥¬®© á¨á⥬ë:
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H = E |
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®£¤ ¢ ¬ ¢¨¤¥ ¬®¦® ¯¨á âì: |
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< B(t0)A(t) >= Z;1 |
( ?B(t0)A(t) |
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)e; T |
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ᯮ«ì§ãï ᢮©á⢮ ¯®«®âë 1 = j |
)( ? j, ¯¥à¥¯¨è¥¬ (10.62) ª ª: |
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< B(t0)A(t) >= Z;P |
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( B(t0) )( |
A(t) )e; T |
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= Z;1 |
( ?B(0) )( ?A(0) )e; T |
exp i |
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t0) |
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200
£¤¥ ã竨, çâ® |
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e;iHt=h = e;iE t=h |
?eiHt=h = ?eiE t=h |
(10.64) |
«®£¨çë¬ ®¡à §®¬ ¬®¦® ¯®«ãç¨âì:
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< A(t)B(t0) >= Z;1 ( |
?A(0) )( |
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(E ;E )(t0 ;t)g (10.65) |
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¬¥ïï §¤¥áì ¨¤¥ªáë á㬬¨à®¢ ¨ï ) ¨ áà ¢¨¢ ï á (10.63), ¢¨¤¨¬, çâ® ®¡ |
ª®à५ïâ®à ¬®¦® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥: |
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< B(t0)A(t) >= |
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Z;1 JBA(!)ei!(t;t |
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< A(t)B(t0) >= |
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Z;1 JBA(!)e T |
ei!(t |
;t) |
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(10.66) |
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£¤¥ ¢¢¥«¨: |
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( ?B(0) )( |
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JBA(!) = 2 Z;1 |
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)e; T (E |
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(10.67) |
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®®â®è¥¨ï (10.66) §ë¢ îâáï ᯥªâà «ì묨 ¯à¥¤áâ ¢«¥¨ï¬¨, |
¢¥«¨ç¨ |
JBA(!) ¯à¥¤áâ ¢«ï¥â ᮡ®© â ª §ë¢ ¥¬ãî ᯥªâà «ìãî ¯«®â®áâì ª®à५ï樮- ®© äãªæ¨¨ < B(t0)A(t) >. § áà ¢¥¨ï ®¡®¨å ¢ëà ¦¥¨© ¢ (10.66) ¢¨¤¨¬ ¢ ¦®¥
᢮©á⢮: |
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JAB(;!) = JBA(!)e T |
(10.68) |
«ï ¢á¥å ॠ«ìëå á¨á⥬ ¨¬¥¥â ¬¥áâ® § âãå ¨¥ ª®à५ï権 ¢® ¢à¥¬¥¨, â ª çâ® |
lim |
< A(t)B(t0) >=< A >< B > |
(10.69) |
jt;t0j!1 |
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£¤¥ ¯®¤а §г¬¥¢ ¥вбп, зв® ¢¥«¨з¨л (®¯¥а в®ал) A ¨ B ¥ п¢«повбп ¨в¥£а « ¬¨ |
¤¢¨¦¥¨ï3. ᫨ < A >= 0 ¨ < B >= 0, â® |
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®¡é¥¬ á«ãç ¥ ¬®¦® ®¯à¥¤¥«¨âì ®¢ë¥ ®¯¥à â®àë A(t); < A > ¨ B(t); < B >, ¤«ï ª®â®àëå ¢á¥£¤ ¢ë¯®«ï¥âáï ãá«®¢¨¥ (10.70).
।¥¥ ¯® ¢à¥¬¥¨ ®â ª®à५ï樮®© äãªæ¨¨ à ¢® ã«î:
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1 d!JBA(!)e;i!t = |
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1!112T Z;T |
2 Z;1 |
JBA(0) |
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T |
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= lim |
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d! (!)JBA(!) = lim |
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(10.71) |
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¥á«¨ ᯥªâà «ì ï ¯«®â®áâì ª®¥ç |
¯à¨ ! = 0, çâ® å à ªâ¥à® ¤«ï ᯥªâà |
í࣮- |
¤¨ç¥áª®£® á«ãç ©®£® ¯à®æ¥áá . ¤ «ì¥©è¥¬ í⮠᢮©á⢮ ¯à¥¤¯®« £ ¥âáï4. ਠí⮬ ¢¥§¤¥ ¯®¤à §ã¬¥¢ ¥âáï â¥à¬®¤¨ ¬¨ç¥áª¨© ¯à¥¤¥« V ! 1 (V=N ! const).
3 ᫨ A ¨ B ¨â¥£à «ë ¤¢¨¦¥¨ï, â® ª®à५ï樮 ï äãªæ¨ï, ®ç¥¢¨¤®, ¢®®¡é¥ ¥ § ¢¨á¨â ®â ¢à¥¬¥¨.
4 ç áâ®áâ¨, íâ® ¨áª«îç ¥â ¨§ à áᬮâà¥¨ï ®á®¡¥®á⨠ᯥªâà «ì®© ¯«®â®á⨠⨯ (!), å à ªâ¥àë¥ ¤«ï á¨á⥬ á ¥í࣮¤¨ç¥áª¨¬ ¯®¢¥¤¥¨¥¬ [4].
¯¥ªâà «ì ï ¯«®â®áâì JA+A(!) ¢à¥¬¥®£® ª®à५ïâ®à , ®¡à §®¢ ®£® ¨§ á®- ¯à殮ëå ®¯¥à â®à®¢ A ¨ A+ ¯®«®¦¨â¥«ì :
JA+A(!) = 2 Z;1 |
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E ; E |
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! > 0 (10.72) |
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¯®áª®«ìªã ¢á¥ ç«¥ë ¯®¤ § ª®¬ áã¬¬ë ¯®«®¦¨â¥«ìë. á®, çâ® ¨ JAA+ (!) > 0.ãáâì ãà ¢¥¨ï ¤¢¨¦¥¨ï ¤«ï A ¨ B ¨¢ ਠâë ®â®á¨â¥«ì® ®âà ¦¥¨ï ¢à¥¬¥¨, ¯à¨ ª®â®à®¬ A ! "AA; B ! "BB, £¤¥ "A; "B = 1, ¢ § ¢¨á¨¬®á⨠®â ç¥â- ®á⨠®¯¥à â®à®¢ ¯à¨ ®¡à 饨¨ ᪮à®á⥩. áᬮâਬ ᯥªâà «ì®¥ à §«®¦¥¨¥:
< A(t)B(t) >= 21 Z 1 d!ei!(t;t0)JAB(!)
;1
ª¢ ⮢®© ¬¥å ¨ª¥ ®¯¥à æ¨ï ®¡à é¥¨ï ¢à¥¬¥¨ ᢮¤¨âáï ª §
;t0; i ! ;i, â ª çâ® «¥¢ ï ç áâì í⮣® à ¢¥á⢠㬮¦ ¥âáï ç á⨠JAB(!) ¯¥à¥å®¤¨â ¢ JAB? (!) (¢á«¥¤á⢨¥ § ¬¥ë i ! ;i). à áᬠâਢ ¥¬®¬ á«ãç ¥:
(10.73)
¬¥¥ t ! ;t; t0 ! "A"B, ¢ ¯à ¢®©«¥¤®¢ ⥫ì®, ¢
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(10.74) |
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â ª ç⮠ᯥªâà «ì ï ¯«®â®áâì ¤«ï ®¯¥à â®à®¢ ®¤¨ ª®¢®© ç¥â®á⨠¤¥©á⢨- ⥫ì .
à ¢¨¢ ï (10.73) ¨ ᮯà殮®¥ ¥¬ã á®®â®è¥¨¥: |
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< B+(t0)A+(t) >= |
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Z;1 d!e;i!(t;t |
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(10.75) |
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£¤¥ ¬ë ã竨 ¢¥é¥á⢥®áâì ᯥªâà «ì®© ¯«®â®áâ¨, ã¡¥¦¤ ¥¬áï, çâ® ¤«ï ®¯¥à - â®à®¢ ®¤¨ ª®¢®© ç¥â®áâ¨:
< A(t)B(t0) >=< B+(t)A+(t0) > |
(10.76) |
᫨ á¨á⥬ 室¨âáï ¢® ¢¥è¥¬ ¬ £¨â®¬ ¯®«¥, ᯥªâà «ì ï ¯«®â®áâì 㦥 ¥ ¡ã¤¥â ¢¥é¥á⢥ . ®áª®«ìªã ãà ¢¥¨ï ¤¢¨¦¥¨ï ¨¢ ਠâë ®â®á¨- â¥«ì® t ! ;t á ®¤®¢à¥¬¥®© § ¬¥®© H ! ;H, ⮠ᯥªâà «ì ï ¯«®â®áâì 㤮¢«¥â¢®àï¥â ᢮©áâ¢ã ᨬ¬¥âਨ:
J |
(!) = J? |
(!)"A"B |
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AB;H |
AB;;H |
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â ª çâ® |
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< B+(t)A+(t0) >H=< A(t)B(t0) > |
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(10.78) |
®бª®«мªг ¤¢ге¢а¥¬¥л¥ дгªж¨¨ а¨ (10.22) ®¯а¥¤¥«повбп з¥а¥§ ¢а¥¬¥- л¥ ª®аа¥«пв®ал, ®¨ «¥£ª® ¢ла ¦ овбп з¥а¥§ б¯¥ªва «мго ¯«®в®бвм [4, 29].®®в¢¥вбв¢¥®, ¤«п ¨е ¨ ¨е дгам¥ { ®¡а §®¢ ¯® а §®бв¨ ¢а¥¬¥ ¯®«гз овбп
«®£¨çë¥ á®®â®è¥¨ï ᨬ¬¥âਨ:
<<A(t)B(t0) >>=<< B+(t)A+(t0) >>
<< AjB >>!=<< B+jA+ >>! |
(10.79) |
1 d!e;i!tAi(!)
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¢ ®вбгвбв¢¨¥ ¬ £¨в®£® ¯®«п ¨ ¤«п ®¯¥а в®а®¢ ®¤¨ ª®¢®© з¥в®бв¨. ¯а¨бгвбв¢¨¥ |
¬ £¨â®£® ¯®«ï ¨¬¥¥¬: |
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<< B+(t)A+(t0) >>H=<< A(t)B(t0) >> |
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H "A"B |
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<< B+jA+ >>!;H=<< AjB >>!;;H "A"B |
(10.80) |
⨠᢮©á⢠|
ᨬ¬¥âਨ ®ª §ë¢ îâáï ¢ ¦ë¬¨ ¯à¨ ¢ë¢®¤¥ ¯à¨æ¨¯ |
ᨬ¬¥âਨ |
á £¥à ¤«ï ®¡®¡é¥®© ¢®á¯à¨¨¬ç¨¢®áâ¨.
¨á¯¥àá¨®ë¥ á®®â®è¥¨ï à ¬¥àá {ந£ ¨ ¯à¨æ¨¯ ᨬ¬¥âਨ á £¥à .
ãáâì á¨á⥬㠤¥©áâ¢ã¥â § ¢¨áï饥 ®â ¢à¥¬¥¨ ¢®§¬ã饨¥ ¬¥å ¨ç¥áª®£® ⨯ , ¢ª«îç ¥¬®¥ ¤¨ ¡ â¨ç¥áª¨ ¨ ®¯¨áë¢ ¥¬®¥ ¤®¡ ¢ª®© ª £ ¬¨«ì⮨ ã ¢¨¤ :
n
Ht1 = ; XFj(t)Bj (10.81)
j=1
£¤¥ Fj(t) e"t ¯à¨ t ! ;1; " ! +0, Bj { ¥ª®â®àë¥ ¤¨ ¬¨ç¥áª¨¥ ¯¥à¥¬¥ë¥ (®¯¥à â®àë), Fj(t) { c { ç¨á«®¢ë¥ \ᨫë", á ª®â®à묨 ¢¥è¨¥ ¯®«ï ¤¥©áâ¢ãîâ
¯¥à¥¬¥ë¥ Bj. «ï ¯à®áâ®âë ¯®« £ ¥¬, çâ® ¢ á®áâ®ï¨¨ áâ â¨áâ¨ç¥áª®£® à ¢- ®¢¥á¨ï (¯à¨ Fj = 0) ¨¬¥¥¬ < Aj >0= 0, â ª ç⮠ॠªæ¨î á¨áâ¥¬ë ¢®§¬ã饨¥ (10.81) ¬®¦® ᮣ« á® (10.21) § ¯¨á âì ¢ ¢¨¤¥:
t |
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< Ai >= Z;1 dt0 ij(t ; t0)Fj(t0) |
(10.82) |
£¤¥ |
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ij(t ; t0) = ; << Ai(t)Bj(t0) >> |
(10.83) |
{ ®¡®¡é¥ ï ¬ âà¨æ ॠªæ¨¨ (®âª«¨ª ). ®áª®«ìªã § ¯ §¤ë¢ îé ï äãªæ¨ï |
ਠ®â«¨ç ®â ã«ï «¨èì ¯à¨ ¯®«®¦¨â¥«ìëå à£ã¬¥â å, â® |
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ij(t ; t0) = 0 ¯à¨ t < t0 |
(10.84) |
ç⮠ï¥âáï ¢ëà ¦¥¨¥¬ ¯à¨æ¨¯ ¯à¨ç¨®áâ¨: ॠªæ¨ï á¨áâ¥¬ë ¥ ¬®¦¥â ¯à¥¤- è¥á⢮¢ âì ¢® ¢à¥¬¥¨ ⮬㠢®§¬ã饨î, ª®â®à®¥ ¥¥ ¢ë§ë¢ ¥â.
§«®¦¨¬ Fj(t) ¨ < Ai > ¢ ¨â¥£à «ë ãàì¥:
< Ai >= 1 Z
2 ;1
Fj(t) = 1 Z 1 d!e;i!tFj(!)
£¤¥ äãàì¥ { ª®¬¯®¥âë:
2 ;1
Ai(!) = Z 1 ei!t < Ai(t) >
;1