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«®£¨çë¬ ®¡à §®¬ |
(rt) ¨ A (rt) ª®¬¬ãâ¨àãîâ: |
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[ (rt); A (r0t)] = 0 |
(7.3) |
à §«¨çë¥ ¬®¬¥âë ¢à¥¬¥¨ ¢á¥ íâ® ®âî¤ì ¥ â ª! |
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à ¢¥¨¥ ¤¢¨¦¥¨¥ ¤«ï £¥©§¥¡¥à£®¢áª®£® -®¯¥à â®à |
¨¬¥¥â ¢¨¤: |
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; i@@t = H (x) ; (x)H [H; (x)] |
(7.4) |
«ï á ¬®£® £ ¬¨«ì⮨ è।¨£¥à®¢áª®¥ ¨ £¥©§¥¡¥à£®¢áª®¥ ¯à¥¤áâ ¢«¥¨ï á®- ¢¯ ¤ îâ ¨ £ ¬¨«ì⮨ ¢ëà ¦ ¥âáï ®¤¨ ª®¢ë¬ ®¡à §®¬ ç¥à¥§ ¯®«¥¢ë¥ ®¯¥à -
â®àë ¢ ®¡®¨å íâ¨å ¯à¥¤áâ ¢«¥¨ïå.
а¨ ¢лз¨б«¥¨¨ ¯а ¢®© з бв¨ (7.4) ¬®¦® ®¯гбв¨вм ¢ £ ¬¨«мв®¨ ¥ з бвм, § ¢¨бпйго в®«мª®
®â ®¯¥à â®à A (x) (£ ¬¨«ì⮨ ᢮¡®¤®£® í«¥ªâ஬ £¨â®£® ¯®«ï), ¯®áª®«ìªã ® |
ª®¬¬ãâ¨- |
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àã¥â á . ®£¤ : |
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H = Z d3r (rt)( p + m) (rt) + e Z d3r (rt) A (rt) (rt) = |
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= Z d3r (rt)[ p + m + e A (rt)] (rt) |
(7.5) |
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ëç¨á«ïï ª®¬¬ãâ â®à [H; (x)] á ¯®¬®éìî (7.2) ¨ ãáâà ïï -äãªæ¨î ¨â¥£à¨à®¢ ¨¥¬ ¯® d3r, |
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¯®«ã稬 ãà ¢¥¨¥ ¤¢¨¦¥¨ï ¤«ï ®¯¥à â®à |
¢ ¬ ¢¨¤¥: |
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( p ; e A ; m) |
(rt) = 0 |
(7.6) |
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ª®â®à®¥, ¥áâ¥á⢥®, ᮢ¯ ¤ ¥â á ãà ¢¥¨¥¬ ¨à ª |
¢ í«¥ªâ஬ £¨â®¬ ¯®«¥. |
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à ¢¥¨ï ¤¢¨¦¥¨ï ¤«ï í«¥ªâ஬ £¨â®£® ¯®â¥æ¨ « A (rt) § à ¥¥ ®ç¥¢¨¤ë ¨§ ᮮ⢥â- á⢨ï á ª« áᨪ®© (¡®«ì訥 ç¨á« § ¯®«¥¨ï), ª®£¤ ®¯¥à â®à®¥ ãà ¢¥¨¥ ¤®«¦® ¯¥à¥©â¨ ¢ ®¡ëçë¥ ãà ¢¥¨ï ªá¢¥«« ¤«ï ¯®â¥æ¨ «®¢, â ª çâ® ¢ ¯à®¨§¢®«ì®© ª «¨¡à®¢ª¥ ¨¬¥¥¬:
@ @ A (x) ; @ @ A (x) = 4 ej (x) |
(7.7) |
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£¤¥ j (x) = (x) (x) { ®¯¥à â®à ⮪ , 㤮¢«¥â¢®àïî騩 ãà ¢¥¨î ¥¯à¥à뢮áâ¨: |
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@ j = 0 |
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(7.8) |
¨á⥬ ãà ¢¥¨© (7.6) ¨ (7.7) ¨¢ ਠâ |
®â®á¨â¥«ì® ª «¨¡à®¢®çëå ¯à¥®¡à §®¢ ¨©: |
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A ! A (x) ; @ (x) |
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(x) ! (x)e |
ie(x) |
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;ie(x) |
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(x) ! e |
(x) |
(7.9) |
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£¤¥ (x) { ¯à®¨§¢®«ìë© íନ⮢ ®¯¥à â®à, ª®¬¬ãâ¨àãî騩 (¢ ®¤¨ ¨ â®â ¦¥ ¬®¬¥â ¢à¥¬¥¨) á
. ¤¥áì à¥çì ¨¤¥â ¨¬¥® ® £¥©§¥¡¥à£®¢áª¨å ®¯¥à â®à å. ¯à¥¤áâ ¢«¥¨¨ ¢§ ¨¬®¤¥©áâ¢¨ï ª - «¨¡à®¢®ç®¥ ¯à¥®¡à §®¢ ¨¥ í«¥ªâ஬ £¨â®£® ¯®â¥æ¨ « ¢®®¡é¥ ¥ § âà £¨¢ ¥â ®¯¥à â®àë!
áâ ®¢¨¬ ⥯¥àì á¢ï§ì ¬¥¦¤ã ®¯¥à â®à ¬¨ ¢ £¥©§¥¡¥à£®¢áª®¬ ¯à¥¤áâ ¢«¥¨¨ ¨ ¯à¥¤áâ ¢«¥¨¨ ¢§ ¨¬®¤¥©á⢨ï. ᮮ⢥âá⢨¨ á ¤¨ ¡ â¨ç¥áª®© £¨¯®â¥§®© ¯à¥¤- ¯®«®¦¨¬, çâ® ¢§ ¨¬®¤¥©á⢨¥ HI(t) ¬¥¤«¥® \¢ª«îç ¥âáï" ®â ¬®¬¥â t = ;1 ª
ª®¥çë¬ ¢à¥¬¥ ¬. ®£¤ ¯à¨ t ! ;1 ®¡ ¯à¥¤áâ ¢«¥¨ï (£¥©§¥¡¥à£®¢áª®¥ ¨ ¢§ - ¨¬®¤¥©á⢨ï) ¯à®á⮠ᮢ¯ ¤ îâ. ®¢¯ ¤ îâ ¨ ᮮ⢥âáâ¢ãî騥 ¢®«®¢ë¥ äãªæ¨¨
(¢¥ªâ®à á®áâ®ï¨©) ¨ int:
int(t = ;1) = |
(7.10) |
¤à㣮© áâ®à®ë, ¢®«®¢ ï äãªæ¨ï ¢ £¥©§¥¡¥à£®¢áª®¬ ¯à¥¤áâ ¢«¥¨¨ ®â ¢à¥¬¥¨ ¢®®¡é¥ ¥ § ¢¨á¨â (¢áï ¢à¥¬¥ ï § ¢¨á¨¬®áâì ®¯¥à â®à å!), ¢ ¯à¥¤áâ ¢«¥¨¨ ¢§ ¨¬®¤¥©áâ¢¨ï § ¢¨á¨¬®áâì ¢®«®¢®© äãªæ¨¨ ®â ¢à¥¬¥¨ ¨¬¥¥â, ª ª ¬ë ¢¨¤¥«¨, ¢¨¤:
int(t) = S(t; ;1) int(;1) |
(7.11) |
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153 |
£¤¥1 |
t2 |
dt0HI(t0) |
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S(t2; t1) = T exp ;i Zt1 |
(7.12) |
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á ®ç¥¢¨¤ë¬¨ ᢮©á⢠¬¨: |
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S(t; t1)S(t1; t0) = S(t; t0) |
S;1(t; t1) = S(t1; t) |
(7.13) |
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à ¢¨¢ ï (7.11) ¨ (7.10) 室¨¬: |
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int(t) = S(t; ;1) |
(7.14) |
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çâ® ãáâ ¢«¨¢ ¥â á¢ï§ì ¢®«®¢ëå äãªæ¨© ¢ ®¡®¨å ¯à¥¤áâ ¢«¥¨ïå. ®®â¢¥âáâ¢ã- îé ï ä®à¬ã« ¯à¥®¡à §®¢ ¨ï ®¯¥à â®à®¢ ¨¬¥¥â ¢¨¤:
(rt) = S;1(t; |
;1 |
) int(rt)S(t; ) = S( |
;1 |
; t) int(rt)S(t; |
;1 |
) |
(7.15) |
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;1 |
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¨ «®£¨ç® ¤«ï ¨ A . ⨠ä®à¬ã«ë à¥è îâ ¯®áâ ¢«¥ãî § ¤ çã.
®çë© ä®â®ë© ¯à®¯ £ â®à.
®çë© ä®â®ë© ¯à®¯ £ â®à ®¯à¥¤¥«ï¥âáï ä®à¬ã«®©: |
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D (x ; x0) = i < 0jTA (x)A (x0)j0 > |
(7.16) |
¢ ª®â®à®© A (x) { £¥©§¥¡¥à£®¢áª¨¥ ®¯¥à â®àë ¯®«ï, ⮣¤ ª ª ¢ëè¥ ¬ë, ä ªâ¨ç¥- ᪨, à áᬠâਢ «¨:
D (x ; x0) = i < 0jTAint(x)Aint(x0)j0 > |
(7.17) |
¢ ª®â®àãî ¢å®¤¨«¨ ®¯¥à â®àë ¢ ¯à¥¤áâ ¢«¥¨¨ ¢§ ¨¬®¤¥©á⢨ï. ¥«¨ç¨ã (7.17) ®¡ëç® §ë¢ î⠯ய £ â®à®¬ ᢮¡®¤ëå ä®â®®¢ (\ã«¥¢®©" äãªæ¨¥© ਠ).
ëà §¨¬ ⥯¥àì â®çë© ¯à®¯ £ â®à |
D |
ç¥à¥§ ®¯¥à â®àë ¢ ¯à¥¤áâ ¢«¥¨¨ ¢§ - |
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int |
(7.15) ¨¬¥¥¬: |
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¨¬®¤¥©á⢨ï. ãáâì t > t0, ⮣¤ ¨á¯®«ì§ãï á¢ï§ì A ¨ A ⨯ |
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D (x ; x0) = i < 0jTA (x)A (x0)j0 >= |
(7.18) |
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= i < 0 |
j |
S( |
;1 |
; t)Aint(x)S(t; |
;1 |
)S( |
;1 |
; t0)Aint(x0)S(t0; |
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0 > |
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;1 j |
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ᯮ«ì§ãï (7.13) ¨¬¥¥¬: |
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S(t; ;1)S(;1; t0) = S(t; t0) |
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S(;1; t) = S(;1; +1)S(1; t) |
(7.19) |
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®£¤ (7.19) § ¯¨áë¢ ¥âáï ª ª: |
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D (x ; x0) = i < 0jS;1[S(1; t)Aint(x)S(t; t0)Aint(x0)S(t0; ;1)]j0 > |
(7.20) |
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£¤¥ ¤«ï ªà ⪮á⨠®¡®§ 祮: |
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S = S(+1; ;1) |
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(7.21) |
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1 ¬¥â¨¬, çâ® «®£¨çë© ®¯¥à â®à ¢ ¯à¥¤ë¤ã饩 £« ¢¥ ®¡®§ ç «áï ª ª U(t2; t1).
154 |
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¨ ¬ë ã竨, çâ® S;1(1; ;1)S(1; t) = S(;1; t). ®áª®«ìªã S(t2; t1) ᮤ¥à¦¨â ⮫쪮 ®¯¥à â®àë ¢ ¬®¬¥âë ¢à¥¬¥¨ ¬¥¦¤ã t1 ¨ t2, à ᯮ«®¦¥ë¥ ¢ åà®®«®£¨-
ç¥áª®¬ ¯®à浪¥, â® ®ç¥¢¨¤®, çâ® ¢®®¡é¥ ¢á¥ ®¯¥à â®àë¥ ¬®¦¨â¥«¨ ¢ ª¢ ¤à ⮩ ᪮¡ª¥ ¢ (7.20) à ᯮ«®¦¥ë ¢ ¯®à浪¥ ã¡ë¢ ¨ï ¢à¥¬¥ á«¥¢ ¯à ¢®. ®áâ ¢¨¢ ¯¥à¥¤ ᪮¡ª®© ᨬ¢®« T -㯮à冷票ï, ¬®¦® ¯®â®¬ ¯à®¨§¢®«ì® ¯¥à¥áâ ¢«ïâì ¯®- à冷ª ¬®¦¨â¥«¥©, ¯®áª®«ìªã T -㯮à冷票¥ ¢á¥ à ¢® à ááâ ¢¨â ¨å ¢ 㦮¬ ¯®à浪¥. ®£¤ ¬®¦® ¯¥à¥¯¨á âì íâã ᪮¡ªã ¢ ¢¨¤¥:
T[Aint(x)Aint(x0)S( |
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; t)S(t; t0)S(t0; |
;1 |
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(7.22) |
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ª¨¬ ®¡à §®¬ ¯®«ãç ¥¬:
D (x ; x0) = i < 0jS;1T Aint(x)Aint(x0)Sj0 > |
(7.23) |
®¢â®àïï ¢á¥ à áá㦤¥¨ï ¥âà㤮 ã¡¥¤¨âìáï, çâ® íâ ä®à¬ã« |
á¯à ¢¥¤«¨¢ ¨ |
¤«ï t < t0. |
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®¦® ¯®ª § âì, çâ® ¬®¦¨â¥«ì S;1 ¢ë®á¨âáï ¨§ ¯®¤ § ª |
ãá।¥¨ï ¯® |
¢ ªãã¬ã ¢ ¢¨¤¥ ¥ª®â®à®£® ä §®¢®£® ¬®¦¨â¥«ï. ¥©á⢨⥫ì®, £¥©§¥¡¥à£®¢áª ï
¢®«®¢ ï äãªæ¨ï ¢ ªã㬠int0 (ª ª ¨ ¢áïª ï ¤àã£ ï £¥©§¥¡¥à£®¢áª ï äãªæ¨ï) |
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ᮢ¯ ¤ ¥â, ᮣ« á® (7.10), á® § 票¥¬ int0 (;1) ¢®«®¢®© äãªæ¨¨ ¢ ªã㬠¢ |
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¯à¥¤áâ ¢«¥¨¨ ¢§ ¨¬®¤¥©á⢨ï. ¤à㣮© áâ®à®ë, ¨¬¥¥¬: |
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S int0 (;1) S(+1; ;1) int0 (;1) = int0 (+1) |
(7.24) |
® ¢ ªã㬠(®á®¢®¥ á®áâ®ï¨¥), ¢ ãá⮩稢®© á¨á⥬¥, ¯à¥¤áâ ¢«ï¥â ᮡ®© áâண® áâ 樮 ஥ á®áâ®ï¨¥, ¢ ¥¬ ¥¢®§¬®¦ë á ¬®¯à®¨§¢®«ìë¥ ¯à®æ¥ááë ஦¤¥¨ï ¨ ã¨ç⮦¥¨ï ç áâ¨æ. ç¥ £®¢®àï, á â¥ç¥¨¥¬ ¢à¥¬¥¨ ¢ ªã㬠®áâ ¥âáï ¢ ªãã-
¬®¬. â® ®§ ç ¥â, çâ® 0int(+1) ¬®¦¥â ®â«¨ç âìáï ®â 0int(;1) ⮫쪮 ¥ª®â®àë¬ ä §®¢ë¬ ¬®¦¨â¥«¥¬ ei . ®£¤ :
S int0 (;1) = ei int0 (;1) =< 0jSj0 > int0 (;1) |
(7.25) |
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¨«¨, ¯à®¨§¢®¤ï ª®¬¯«¥ªá®¥ ᮯà殮¨¥ ¨ ãç¨âë¢ ï ã¨â à®áâì S: |
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0 ( |
;1 |
)S;1 =< 0 S |
0 >;1 0 ( |
;1 |
) |
(7.26) |
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int |
j j |
int |
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âáî¤ ïá®, çâ® (7.23) ¬®¦¥â ¡ëâì ¯¥à¥¯¨á ® ¢ ¢¨¤¥: |
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(x |
; |
x0) = i< 0jT A (x)A (x0)j0 > |
(7.27) |
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D |
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< 0jSj0 > |
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®¤áâ ¢«ïï áî¤ ¢ ç¨á«¨â¥«ì ¨ § ¬¥ ⥫ì à §«®¦¥¨¥ S-¬ âà¨æë ¢ àï¤ â¥®- ਨ ¢®§¬ã饨©, ®¯à¥¤¥«ï¥¬®¥ ¨§ (6.55), ¨ ¯à®¢®¤ï ãá।¥¨¥ á ¯®¬®éìî ⥮६먪 , ¬®¦® ¯®«ãç¨âì à §«®¦¥¨¥ D ¯® á⥯¥ï¬ ª®áâ âë á¢ï§¨ e2.
ç¨á«¨â¥«¥ (7.27) ãáà¥¤ï¥¬ë¥ ¢ëà ¦¥¨ï ®â«¨ç îâáï ®â ¬ âà¨çëå í«¥- ¬¥â®¢ ¬ âà¨æë à áá¥ï¨ï, à áᬮâà¥ëå ¢ ¯à¥¤ë¤ã饩 £« ¢¥, «¨èì ⥬, çâ®
¢¬¥áâ® \¢¥è¨å" ®¯¥à â®à®¢ ஦¤¥¨ï ¨«¨ ã¨ç⮦¥¨ï ä®â®®¢ ¢ ¨å áâ®ïâ ®¯¥à â®àë Aint(x) ¨ Aint(x0). ®áª®«ìªã ¢á¥ ¬®¦¨â¥«¨ ¢ ãá।塞ëå ¯à®¨§¢¥-
¤¥¨ïå áâ®ïâ ¯®¤ § ª®¬ T -¯à®¨§¢¥¤¥¨ï, â® ¯®¯ àë¥ á¢¥à⪨ íâ¨å ®¯¥à â®à®¢ á \¢ãâ२¬¨" ®¯¥à â®à ¬¨ Aint(x1); Aint(x2) ¡ã¤ãâ ¤ ¢ âì ä®â®ë¥ ¯à®¯ £ â®àë
D . ª¨¬ ®¡à §®¬, १ã«ìâ âë ãá।¥¨ï ¢ëà §ïâáï ᮢ®ªã¯®áâﬨ ¤¨ £à ¬¬
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155 |
¨á. 7-1
¨á. 7-2
á ¤¢ã¬ï ¢¥è¨¬¨ ª®æ ¬¨, á®áâ ¢«¥ë¬¨ ¯® ¯à¨¢¥¤¥ë¬ ¢ ¯à¥¤ë¤ã饩 £« ¢¥ ¯à ¢¨« ¬, á ⮩ à §¨æ¥©, çâ® ¢¥è¨¬ (ª ª ¨ ¢ãâ२¬) ä®â®ë¬ «¨¨ï¬ ¤¨ - £à ¬¬ë ®â¢¥ç îâ ⥯¥àì ¯à®¯ £ â®àë D , ¢¬¥áâ® ¬¯«¨â㤠ॠ«ìëå ä®â®®¢.
ã«¥¢®¬ ¯à¨¡«¨¦¥¨¨, ª®£¤ |
S = 1, ç¨á«¨â¥«ì (7.27) ¯à®á⮠ᮢ¯ ¤ ¥â á D (x x0). |
«¥¤ãî騥 ®â«¨çë¥ ®â ã«ï ç«¥ë ¨¬¥îâ ¯®à冷ª e2. ¨ ¨§®¡à ¦ îâáï;¤¨ - |
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£à ¬¬ ¬¨, ᮤ¥à¦ 騬¨ ¤¢ |
¢¥è¨å ª®æ ¨ ¤¢¥ ¢¥àè¨ë, ¯®ª § 묨 ¨á.7- |
1. â®à ï ¨§ íâ¨å ¤¨ £à ¬¬ á®á⮨⠨§ ¤¢ãå ¥ á¢ï§ ëå ¬¥¦¤ã ᮡ®© ç á⥩: ¯ãªâ¨à®© «¨¨¨ (ª®â®à®© ®â¢¥ç ¥â ;iD ) ¨ § ¬ªã⮩ ¯¥â«¨. â® ®§ ç ¥â, ç⮠ᮮ⢥âáâ¢ãî饥 ¤ ®© ¤¨ £à ¬¬¥ «¨â¨ç¥áª®¥ ¢ëà ¦¥¨¥ à ᯠ¤ ¥âáï ¤¢ ¥§ ¢¨á¨¬ëå ¬®¦¨â¥«ï. ਡ ¢¨¢ ª ¤¨ £à ¬¬ ¬ ¨á.7-1 ¯ãªâ¨àãî «¨¨î ã- «¥¢®£® ¯à¨¡«¨¦¥¨ï ¨ \¢ë¥áï ¥¥ § ᪮¡ªã" ¯®«ã稬, çâ® á â®ç®áâìî ¤® ç«¥®¢e2 ç¨á«¨â¥«ì (7.27) ¨§®¡à ¦ ¥âáï ¤¨ £à ¬¬ ¬¨ ¨á.7-2. ëà ¦¥¨¥ < 0jSj0 > ¢ § ¬¥ ⥫¥ (7.27) ¯à¥¤áâ ¢«ï¥â ᮡ®© ¬¯«¨âã¤ã \¯¥à¥å®¤ " ¢ ªã㬠{ ¢ ªãã¬. £® à §«®¦¥¨¥ ¢ àï¤ â¥®à¨¨ ¢®§¬ã饨© ᮤ¥à¦¨â ¯®í⮬㠫¨èì ¤¨ £à ¬¬ë ¡¥§ ¢¥è-
¨å ª®æ®¢. ã«¥¢®¬ ¯à¨¡«¨¦¥¨¨ < 0jSj0 >= 1, á â®ç®áâìî ¤® ç«¥®¢ e2 íâ ¬¯«¨â㤠¢ëà ¦ ¥âáï £à ä¨ç¥áª¨, ª ª íâ® ¯®ª § ® ¨á.7-3. §¤¥«¨¢ á ⮩ ¦¥ â®ç®áâìî e2 ç¨á«¨â¥«ì (7.27) § ¬¥ â¥«ì ¯®«ã稬 ¤¨ £à ¬¬ë, ¯®ª § ë¥ ¨á.7-4. â ª çâ® ¢ª« ¤ \¢ ªãã¬ëå" ç«¥®¢ (¢ë¤¥«¥ëå à¨á㪠å 䨣ãன ᪮¡ª®©) ¯®«®áâìî ᮪à é ¥âáï. ª¨¬ ®¡à §®¬ ¥á¢ï§ ï ¤¨ £à ¬¬ ¨á.7-1(¡) ¢ë¯ ¤ ¥â ¨§ ®â¢¥â . â®â १ã«ìâ â ¨¬¥¥â, á ¬®¬ ¤¥«¥, ®¡é¨© å à ªâ¥à. ®- à §¡¨à ï ¯®¤à®¡¥¥ ᯮᮡ ¯®áâ஥¨ï ¤¨ £à ¬¬, ᮯ®áâ ¢«ï¥¬ëå ç¨á«¨â¥«î ¨ § ¬¥ â¥«î ¢ (7.27), ¬®¦® ¯®ïâì, çâ® à®«ì § ¬¥ ⥫ï < 0jSj0 > ᢮¤¨âáï ª ⮬ã, çâ® ¢ «î¡®¬ ¯®à浪¥ ⥮ਨ ¢®§¬ã饨© â®çë© ¯à®¯ £ â®à D ¨§®¡à - ¦ ¥âáï ⮫쪮 ¤¨ £à ¬¬ ¬¨, ¥ ᮤ¥à¦ 騬¨ ®â¤¥«¥ëå ¤à㣠®â ¤à㣠ç á⥩, ¨«¨, ª ª ¯à¨ïâ® £®¢®à¨âì, ⮫쪮 á¢ï§ë¬¨ ¤¨ £à ¬¬ ¬¨.
¬¥â¨¬, çâ® ¤¨ £à ¬¬ë ¡¥§ ¢¥è¨å ª®æ®¢ (§ ¬ªãâë¥ ¯¥â«¨) ¢®®¡é¥ ¥ ¨¬¥îâ 䨧¨ç¥áª®£® á¬ëá« , ¯®áª®«ìªã â ª¨¥ ¯¥â«¨ ¯à¥¤áâ ¢«ïîâ ᮡ®© à ¤¨ æ¨- ®ë¥ ¯®¯à ¢ª¨ ª ¤¨ £® «ì®¬ã í«¥¬¥âã S-¬ âà¨æë, ®¯¨áë¢ î饬㠯¥à¥å®¤ë
¨á. 7-3
156 |
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¨á. 7-4
¨á. 7-5
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D (k) = D (k) + ie2D (k) Z |
d4p |
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Sp G(p + k) G(p)D (k) |
(7.28) |
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(2 )4 |
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«¥ë á«¥¤ãîé¨å ¯®à浪®¢ áâà®ïâáï |
«®£¨ç® ¨ ¨§®¡à ¦ îâáï ¤¨ £à ¬¬ ¬¨ á |
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¨á. 7-6