QED and Particle Physics

7/3/2019 QED and Particle Physics.

http://slidepdf.com/reader/full/qed-and-particle-physics-5584618d78ab1 1/426

WC\R 678K

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Hoybfr

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M Z = 37$5 JfX kde M = <;$9 JfX

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M C 4 ;;5 JfX




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ldufykbulod ruyfdjuc lr dou ruyodj

λ ≃ ;;2:




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β

f∙

f!

β

f∙

f!

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{

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j

• Ruyodj ldufykbulod λR ≃;:




• $$$ kde ro eo j`vodr+

j

j

j

j

j

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j

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Ucf Zfka Hoybf

•

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h

h

^

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Fskmw`f> f!f∙ rbkuufyldj #Ickick,

β

f∙

f!

β

f∙

f!

^

f∙ f!

^

f∙ f!

Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;;



• Ucf Z bkd ldufykbu zluc k bckyjfe `fwuod kde lur dfvuyldo kr zf`` kr zluc kd

vw-u|wf {vkya kde k eozd-u|wf {vkya

ℒ∙

ν ℒ

Z

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Z

• Douf ucku `fwuod kxoy dvmify lr k`zk|r bodrfyxfe ivu ucf Z mlsfr {vkyar

hyom elhhfyfdu jfdfykulodr #kxovy bckdjf,




• Ucfyf kyf k`ro ldufykbulodr ifuzffd ucf zfka hoybf bkyylfyr

Z ∙

Z !

^ 7, β

Z !

Z ∙

S

\

• #S, \ , bkd if #β, β , #β, ^ 7, #^ 7, ^ 7, oy #Z !, Z ∙,




Efbk| `lhfulmfr

• Wyoikil`lu| oh k wkyulb`f efbk| lr wyowoyulodk` uo ucf ruyfdjuc oh ucf ldufykbulod

ruyodj ∈#;928, ↙ D χ ;7∙92 r

f`fbuyomkjdfulb χ7↙ ββ ;7∙;3 r

zfka Α7↙ wχ∙ ;7∙;7 r




Bodrfyxkulod @kzr

• Ld moru bkrfr zfyf ldhfyyfe hyom fswfylmfdur

• Ucfrf `kzr #kde ld romf bkrfr ucfly kirfdbf, ohufd ldrwlyfe ucf ucfoyfulbk`

efrbylwulodr

• Mkd| bodrfyxkulod `kzr kyf ivlù lduo ucf Rukdekye Moef` xlk ucf ruyvbuvyf oh

ucf xfyulbfr

Zcku lr Bodrfyxfe ld ucf Rukdekye Moef`?

•

Fdfyj| kde momfduvm ivu dou mkrr• Kdjv`ky momfduvm

• Bckyjf kde boòy

• @fwuod kxoy dvmify #kde ucfyfhoyf `fwuod dvmify,

• [vkya dvmify k$a$k$ #2Ú, iky|od dvmify #ivu dou {vkya kxoy dvmify,

• Bckyjf bodnvjkulod #B , wkylu| #W , kde ulmf yfxfyrk` #U , kyf bodrfyxfe i|

ucf ruyodj kde f`fbuyomkjdfulb ldufykbulodr ivu dou i| ucf zfka ldufykbulod

#ucovjc BW U lr bodrfyxfe,$




Dfz Wc|rlbr

• Zf yfhfy uo kd| wc|rlbr if|ode ucf Rukdekye Moef` kr Dfz Wc|rlbr$

• Ucf moru wyomlrldj bkdelekuf hoy Dfz Wc|rlbr lr rvwfyr|mmfuy| #k$a$k$ RVR\,$

• Zluc rvwfyr|mmfuy| fxfy| Rukdekye Moef` wkyulb`f kb{vlyfr k wkyudfy rwkyulb`f

oh ucf rkmf mkrr ivu owworluf rukulrulbr$

Rukdekye Moef` Rwld RVR\ Rwld

F`fbuyod f ;/9 Rf`fbuyod fː 7

Uow [vkya u ;/9 Ruow uː 7

Wcouod β ; Wcouldo βː ;/9

Zfka Iorod Z l ; Zldo Zː l ;/9

Cljjr C 7 Cljjrldo Cː ;/9

• Rlml`ky rwkyulb`fr mls ro ucku zf rwfka oh ucldjr `laf r`fwuodr r{vkyar

bckyjldor kde dfvuyk`ldor$




Zc| RVR\?

• Rvwfyr|mmfuy| wyoxlefr kd fsbf``fdu ekya mkuufy bkdelekuf$

• Ucf ikrlb lefk lr ucku RVR\ moef`r ohufd ldboywoykuf Y-wkylu| efdfe ld ufymroh iky|od dvmify #I, `fwuod dvmify #@, kde rwld #R , i|

Yw = #∙;,2#I∙@,!9R =

30

>

!; Hoy RM wky ulb`fr

∙; Hoy RVR\ wkyulb`fr

• Y-wkylu| bodrfyxkulod lmw`lfr ucku RVR\ wkyulb`fr mvru if byfkufe kde efruy

o|fe ld wklyr$ Uclr mfkdr ucku ucf `ljcufru rvwfyr|mmfuylb wkyulb`f #@RW, lr ruki`f$

• Ucf rlmw`fru zk| uo kee RVR\ uo ucf RM lr zluc ucf Mldlmk` Rvwfyr|mmfuylb

Rukdekye Moef` #MRRM,$

• Ucf MRRM `fker uo 6 Cljjr iorodr> c7 C 7 K7 C ! kde C ∙$

Ucf c7 lr mvbc `laf ucf RM Cljjr

Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;:



Bomworlulod oh ucf vdlxfyrf

Zf eo dou vdefyrukde ovy vdlxfyrf$

RVR\ mk| |lfè ucf ekya mkuufy bkdelekuf$




Dkuvyk` Vdlur> = b = ;

• Doucldj yf{vlyfr vr uo mkaf fxfy|ucldj elmfdrlod`frr

• Wkyulb`f wc|rlbr vrfr od`| = b = ; `fkxldj odf vdlu zluc elmfdrlodr

•

Zf ukaf uclr òdf elmfdrlodk` vdlu uo if fdfyj| kde zf fswyfrr lu ld MfX• Bodxfyu fdfyj| uo `fdjuc vrldj b ≂ ;<: MfX hm

• Bodxfyu `fdjuc uo ulmf vrldj b = 2Ú ;792hm/r

• Douf ucku byorr rfbulodr kyf mfkrvyfe ld ikydr> ;i ≫ ;7∙93 m9≫ ;77 hm9

Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;<



Fskmw`f

• Mvod `lhfulmf ld dkuvyk` vdlur>υ ¶ =

;9#3χ,2M 5Z

j5Z m6¶

• Rufw ;> Mvùlw`| mkrrfr i| b9 uo bodxfyu ucfm uo fdfyjlfr⇙ ;/b9

• Rufw 9> Zluc υ ¶ doz ld MfX∙; mvùlw`| i| #b, uo bodxfyu uo `fdjuc

⇙ #b,/b9 #ro hky,

• Rufw 2> Zluc υ ¶ doz ld hm elxlef i| b uo bodxfyu uo ulmf⇙ /b9

• Ekuk> jZ = 7$86 M Z = 37$5 JfX m¶ = ;78 MfX

• Rviruluvuf ekuk⇙ υ ¶ = 2$2Ú ;7;6 MfX∙;

• Mvùlw`| i| #b, ≂ ;<: MfX hm ⇙ υ ¶ = 8$8Ú ;7;: hm

• Elxlef i| b ≂ 2Ú ;792 hm/r ⇙ υ ¶ = 9$9Ú ;7∙8 r




Aldfmkulbr• @oyfdu~ Uykdrhoymkulodr

• Hovy-Xfbuoyr

• Fdfyj| Momfduvm kde Mkrr

• Bo``lrlodr

• Fskmw`fr

Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;



@oyfdu~ Uykdrhoymkulodr• Yf`kuf booyeldkufr ld> R

x∙↙ R ‹

• Efylxfe hyom ucf woruv`kufr oh yf`kulxlu|

• Hoy moulod kòdj ucf s-kslr>

s‹ = β #s ∙ xu,

|‹ = |

~‹ = ~

u‹ = β ‐

u ∙x

b9s”

zcfyf

β =;

w ; ∙ x

9

/b9

• @fdjuc Boduykbulod> moxldj oinfbur kyf rcoyufdfe> @ = @‹/β

• Ulmf El`kulod> moxldj bòbar yvd ròz> U = βU ‹

Wc|rlbr 678K 9 Aldfmkulbr Wkjf 9



Kww`lbkulod> Bormlb Yk| Mvodr

Zluc υ ¶ = 9$9 ¶r k mvod wyoevbfe ld ucf vwwfykumorwcfyf bovè uykxf` 887 m ifhoyf efbk|ldj

Ucf mvod `lhfulmf lr fdckdbfe ucyovjc ulmf el-

`kulod i| k hkbuoy oh β = F/m

#ι =w

; ∙ ;/β 9,

Rvwworldj β ∰ ;7 uclr k`òzr k u|wlbk` mvod uo

uykxf` 8$8 am ifhoyf efbk|ldj$

Efbk| `fdjuc ld `kioykuoy| hykmf lr βbυ




Hovy-Xfbuoyr kde Ufdroyr

• Zyluf s¶ = #s7, s;, s9, s2, = #bu,s,|,~, ro ucku

#s‹,¶ = Α¶νsν zluc Α =

7IIIIIG

β ∙βι 7 7∙βι β 7 7

7 7 ; 7

7 7 7 ;

;BBBBBK

• K hovy-xfbuoy lr k hovy-bomwodfdu oinfbu zclbc ifckxfr `laf s¶ vdefy @oyfdu~

uykdrhoymkulodr$

• s¶ lr ucf boduykxkylkdu kde s¶ lr ucf boxkylkdu hovy-xfbuoy

s¶ = j¶νsν zcfyf j¶ν = elkj#;,∙;,∙;,∙;,

• Ucf ldxkylkdu L = s¶s¶

f$j m9 = w º w = w¶ w¶ = # w7,9 ∙ # w;,9 ∙ # w9,9 ∙ # w2,9




Fdfyj|, Momfduvm, kde Mkrr

• Yf`kulxlrulb fdfyj| lr F = βm

• Yf`kulxlrulb momfduvm lr w = βmx

• Efdf ucf hovy-momfduvm i| w¶ = #F,w,$

Ucfd w9 = m9 lr k yf`kulxlrulb ldxkylkdu$

• Hoy mkrr`frr wkyulb`fr F = pwp = cν

• B`krrlbk``| zf k`zk|r bodrfyxf 2-momfduvm #w, k`zk|r bodrfyxf mkrr

romfulmfr bodrfyxf aldfulb fdfyj| kde k`zk|r bodrfyxf uouk` fdfyj| fxfd lh

zf eod“u affw uykba oh lu k``$

• Yf`kulxlrulbk``| zf k`zk|r bodrfyxf 2-momfduvm #w, romfulmfr bodrfyxfmkrr romfulmfr bodrfyxf aldfulb fdfyj| kde k`zk|r bodrfyxf uouk` fdfyj|$

Moyf rvbbldbu`| hovy-momfduvm lr bodrfyxfe$




Bodrfyxfe xr$ Ldxkylkdu {vkdululfr

• K bodrfyxfe {vkdulu| yfmkldr ucf rkmf ld k wkyulbv`ky hykmf ifhoyf kde khufy

kd fxfdu$

• Kd ldxkylkdu {vkdulu| lr ucf rkmf ld k`` ldfyulk` yfhfyfdbf hykmfr$

‒ Fdfyj| lr bodrfyxfe ivu dou ldxkylkdu$

‒ Mkrr lr ldxkylkdu ivu dou bodrfyxfe$




Fskmw`fr

• IkIky fswfylmfdu > Cfyf k < JfX f∙ ifkm bo``lefr zluc k 2$; JfX f! ifkm$

• Zcku kyf ucf rwffer oh ucf bo``leldj wkyulb`fr?

• Zcku kyf ucf fdfyjlfr oh ucf wkyulb`fr ld ucf bfdufy oh momfduvm #BM, hykmf?

Wc|rlbr 678K 9 Aldfmkulbr Wkjf :



Rwffer

• Vrf F = βm kde m = 7$6;; MfX uo efufymldf ucku β ∙ = ;:877 hoy ucf

f`fbuyodr kde β ! = 87:7 hoy ucf worluyodr$

• Ucfd zluc β = ;/w

;∙ ι 9 zf ro`xf hoy ι = x/b

⇙ ι =

r ;∙

;

β 9≂ ;∙

;

9β 9

• Vrldj β ∙ kde β ! zf ckxf

xf∙ = #; ∙ ;7∙<,b

xf! = #; ∙ ;7∙3,b




• Ld ucf BM hykmf wf∙ = #F BM ,wBM , kde wf! = #F BM ,∙wBM , ro ucku ucf

#ldxkylkdu, r{vkyf oh ucf uouk` hovy-momfduvm lr>

# wf∙ ! wf!,9 = #9F BM ,7,9

= 5F 9BM

• Ld ucf `ki hykmf wf∙ = #F ∙,w∙, kde wf! = #F !,w!, ro ucku

# wf∙ ! wf!,9 = w9f∙ ! w9f! ! 9 wf∙ º wf!

= m9 ! m9 ! 9#F ∙F ! ∙ w∙ º w!,

≂ 9#F ∙F ! ! pw∙ppw!p,

≂ 5F ∙F !

• F{vkuldj ucf BM kde `ki fswyfrrlodr hoy ucf ldxkylkdu zf ckxfF BM =

w F ∙F ! =

w #< JfX,#2$; JfX, = 6$2 JfX

Wc|rlbr 678K 9 Aldfmkulbr Wkjf <



Hlsfe Ukyjfur xr$ Bo``leldj Ifkmr

•Ld IkIky

#< ! 2$;, = ;9$; JfXoh ifkm fdfyj| `fker uo

#9Ú 6$2, = ;7$8 JfX

oh BM fdfyj| ucku bkd if vrfe uo mkaf dfz wkyulb`fr #ld uclr bkrf ucf Φ#5R ,,$

• Coz mvbc ifkm fdfyj| zovè lu ukaf uo wyoevbf uclr BM fdfyj| lh ucf ukyjfu

zfyf sfe?

• Vrf ucf uouk` hovy-momfduvm kr kd ldxkylkdu$ Ucf ldelxlevk` hovy-momfdukzl`` if #m,7, kde #F,w, kde ucfyfhoyf

# wf∙ ! wf!,9 = m9 ! m9 ! 9 Q#m,7, º #F,w,]

≂ 9Fm

Zluc 9Fm = 5F 9BM zf de ucku F = ;76 JfX+

Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;7



Hlsfe Ukyjfur xr$ Bo``leldj Ifkmr LLL• Zc| lr ucf fdfyj| oh ucf f`fbuyodr kde worluyodr ku R@KB elhhfyfdu?

• Zcku eofr uclr mfkd hoy efrljd oh ucf IkIky efufbuoy?

Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;;



Kr|mmfuylb ifkmr hoy I wc|rlbrUcf kr|mmfuylb ifkm fdfyj| yfrvùr ld ucf II mfrod r|rufm ifldj ioorufe hyom

ucf bfduyf oh mkrr$

Uclr k`òzr odf uo mfkrvyf ucf elhhfyfdbf ld ulmf ifuzffd ucf efbk| oh fkbc Imfrod$




Uzo-Ioe| Efbk|rBodrlefy ucf efbk| χ ↙ ¶ ! ν

• Ld ucf BM hykmf ucf dk`-rukuf fdfyjlfr kyf vdl{vf

Ucf uzo wkyulb`fr mvru fmfyjf ikba uo ikba #uo bodrfyxf momfduvm,$

• Hovy-xfbuoyr> wχ = #mχ ,7, w¶ = #F ¶,w, kde wν = #F ν ,∙w,$

• Bodrfyxkulod oh hovy-momfduvm> wχ = w¶ ! wν `fker uo

w9¶ = # wχ ∙ wν,9

m9¶ = w

9χ ! w

9ν ∙ 9 wχ º wν = m

9χ ∙ 9mχF ν

⇙ F ν =m9χ ∙ m9

¶

9mχ




• Rlml`ky`| wν = wχ ∙ w¶ `fker uo w9ν = # wχ ∙ w¶,9

7 = w9χ ! w9¶ ∙ 9 wχ º w¶

7 = m9χ ! m9

¶ ∙ 9mχF ¶

⇙ F ¶ =m9χ ! m9

¶

9mχ

• Doulbf ucku

F ν ! F ¶ =m9χ ∙m9

¶

9mχ

!m9χ ! m9

¶

9mχ

= mχ




Ucyff-Ioe| Efbk|r• Bodrlefy efbk|r rvbc kr d ↙ w ! f ! ν f kde ¶∙ ↙ f∙ ! ν f ! ν ¶

• Ld ucf BM hykmf ucf dk`-rukuf fdfyjlfr kyf dou vdl{vf$

• Ucf oirfyxkulod ucku ucfyf zkr k ykdjf oh f`fbuyod fdfyjlfr ld ucf uzo efbk|rkioxf w`k|fe k `kyjf yo`f ld ucf woruv`kuf oh ucf fslrufdbf oh dfvuyldor$




Mkdef`rukm Ldxkylkdur

• Hoy k rbkuufyldj wyobfrr ; ! 9 ∙↙ 2 ! 5 efdf ucf Mkdef`rukm ldxkylkdur i|

r = # w; ! w9,9

u = # w; ∙ w2,9

v = # w; ∙ w5,9

• Zf efdf k rbkuufyldj kdj`f γ ld ufymr oh ucf elyfbulod oh 2 zluc yfrwfbu uo ;$




Bckddf`r

• Hoy K ! I ∙↙ K ! I rbkuufyldj ucf Mkdef`rukm ldxkylkdur r u kde v kyf

yf`kufe uo 2 elruldbu uowoòjlbk` bckddf`r>

K I

K I

K I

K I

K I

K I

r-bckddf` u-bckddf` v-bckddf`

Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;:



R|mmfuylfr

• Bodrfyxkulod @kzr

• Ikrlb Jyovw Ucfoy|

• Kdjv`ky Momfduvm

• H`kxoy R|mmfuylfr

Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;



Zcku lr k R|mmfuy|?

• Uykdr`kuf ucf zkxfhvdbulod σ#s, i| kd kmovdu k> σ#s, ∙↙ σ#s ! k,

• Doz fswkde σ#s ! k, kr k Uk|òy rfylfr kiovu σ#s,

σ#s ! k, = σ#s, ! k∉σ

∉sˑˑˑˑ

s!

k9

9+

∉ 9σ

∉s9ˑˑˑˑ

s! $ $ $

=

∔S

d=7

kd

d+

∉ d

∉sd

+σ#s, = V #k,σ#s,

zcfyf V #k, = fsw±

k ∉ ∉s

‒• Lh ovy wc|rlbk` r|rufm lr ldeffe ldxkylkdu vdefy uykdr`kulodr ucfd

σ#s,pσ#s, = σ#s ! k,pσ#s ! k,

= V #k,σ#s,pV #k,σ#s,

=E

σ#s,pV ‧#k,V #k,σ#s,F

ucfdV ‧V = ; oy ld oucfy zoyer V ‧ = V ∙; #vdluky| mkuyls,$

Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf 9



Cfymlulkd Owfykuoyr

• Yfbk`` ucku ld [vkduvm Mfbckdlbr fxfy| wc|rlbk` oirfyxki`f lr yfwyfrfdufe

i| k Cfymlulkd owfykuoy #C ‧ = C ,$

• Fxfy| Cfymlulkd owfykuoy ckr ucf hoym V = flC $

• Hkbuoyldj ovu k bovw`f oh bodrukdur ro ucku V #k, = fsw QlCk/]

bomwkyldj uo ovy wyfxlovr yfrvù oh V #k, = fsw Qk ∉/∉s]

zf de w =

l

∉

∉s

lr k Cfymlulkd owfykuoy zclbc jfdfykufr rwkulk` uykdr`kulodr$

• Ldxkylkdbf vdefy rwkulk` uykdrohoymkulodr jlxfr momfduvm bodrfyxkulod

• Ldxkylkdbf vdefy youkulod uykdrohoymkulodr jlxfr kdjv`ky momfduvm

bodrfyxkulod




Dofucfy“r Ucfoyfm• Fxfy| r|mmfuy| lr krroblkufe zluc k bodrfyxkulod `kz$

R|mmfuy| Bodrfyxkulod @kz

Rwkbf uykdr`kulod @ldfky momfduvm

Ulmf uykdr`kulod Fdfyj|

Youkulod Kdjv`ky momfduvm

Jkvjf uykdrhoymkulod F`fbuylb bckyjf

• Romfulmfr zf ckxf bodrfyxkulod `kzr zclbc eo dou boyyfrwode uo k

wyfxlovr`| adozd r|mmfuy|$

#f$j$ @fwuod kxovy bodrfyxkulod,

• Lh ucfyf lr do adozd r|mmfuy| ucfd lu lr worrli`f ucf bodrfyxkulod `kz zl`` hkl`ld ucf hvuvyf zluc k moyf efukl`fe fswfylmfdu$

#f$j$ oirfyxkulod oh dfvuyldo orbl``kulodr mfkdr ucku @fwuod kxovy

bodrfyxkulod lr dou fskbu,




Ikrlb Jyovw Ucfoy|

• K jyovw J lr k rfu oh f`fmfdur zluc k ildky| bomworlulod `kz #l$f$ k

‐mvùlw`lbkulod”, rvbc ucku>

;$ Bòrvyf> ∇ k, i ∃ J > ki = b ∃ J9$ Lefdulu|> ∂ f ∃ J p ∇ k ∃ J > kf = fk = k

2$ Ldxfyrf> ∇ k ∃ J ∂ k∙; ∃ J p kk∙; = k∙;k = f

5$ Krroblkulxlu|> ∇ k,i,b ∃ J > #ki,b = k#ib,

• J lr kd Kif`lkd jyovw lh ucf jyovw mvùlw`lbkulod lr bommvukulxf

ki = ik ∇ k, i ∃ J$

• Ld ucf RM zf vrf RV #2,B ⊚ RV #9,@ ⊚ V #;,\ zcfyf

R ⇙ Rwfblk` ⇙ efufymldkdu ;

O ⇙ Oyucojodk` ⇙ M U M = ;

V ⇙ Vdluky| ⇙ M ‧M = ;

ucf ucyff jyovwr boyyfrwode uo ucf ucyff hoybfr ld ucf Rukdekye Moef`$




Kdjv`ky Momfduvm

• Ld [vkduvm Mfbckdlbr zf bkddou adoz fxfy|ucldj kiovu ucf kdjv`ky

momfduvm N oh k wkyulb`f ku k jlxfd ulmf$

• Ucf ifru zf bkd eo lr ucf rlmvùkdfovr adoz`fejf oh N 9 kde N ~ zluc

fljfdxk`vfr>

N 9σ = ˈ n# n ! ;,9ːσ

N ~ σ = #mn, σ

N ~ fljfdxk`vfr mvru if rwkbfe ld mvùlw`fr oh $

n lr flucfy kd ldufjfy oy k ck`h-ldufjfy$

• Uclr hoymk`lrm kww`lfr uo oyiluk` kdjv`ky momfduvm #@, kr uo lduyldrlbkdjv`ky momfduvm oy rwld #R,




Rwld ;/9 - Rwldoy Doukulod

• Mkd| oh ucf wkyulb`fr ckxf rwld-;/9 #f$j$ `fwuodr {vkyar $$$,

• Rukufr bkd if efrbylife kr k 9-bomwodfdu rwldoy>

ϊ =7G λ

ι

;KHoy fskmw`f ucf rwldoyr hoy k rwld vw kde eozd f`fbuyod kyf

ϊ =

7G ;

7

;K ϊ =

7G 7

;

;K

• Wkv`l mkuylbfr kyf

ς; =

7G 7 ;

; 7

;K ς9 =

7G 7 ∙l

l 7

;K ς2 =

7G ; 7

7 ∙;

;K

Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf :



Hfymlodr xr$ Iorodr

• Wkyulb`fr zluc ldufjfy rwld kyf iorodr kde oif| Iorf-Fldrufld rukulrulbr #l$f$

r|mmfuylb z$y$u$ fsbckdjf oh lefdulbk` wkyulb`fr,$

• Wkyulb`fr zluc ck`h-ldufjfy rwld kyf hfymlodr kde oif| Hfyml-Elykb rukulrulbr

#l$f$ kdulr|mmfuylb z$y$u$ fsbckdjf oh lefdulbk` wkyulb`fr,$

• Ldufybckdjldj uzo wkyulb`fr lr f{vlxk`fdu uo k 9χ yf`kulxf youkulod

Ucf vdluky| uykdrhoymkulod zclbc fhhfbur youkulodr lr

V #γ, = fsw

±l

N º γ

‒

• wkyulb`fr zluc ldufjfy rwld V #9χ, = fsw#9dχl, = ; ⇙ iorodr$

• wkyulb`fr zluc ck`h-ldufjfy rwld V #9χ, = fswQ9#d ! ;/9,χl] = ∙; ⇙ hfymlodr$

• Hoy k 9-wkyulb`f r|rufm σ#;9, = »σ#;9,




Uzo wkyulb`f r|rufmr L

;

p n; m; 4

99

p n9 m9 4

;

p n m 4

Bodrfyxkulod oh kdjv`ky momfduvm yf{vlyfr>

N = N; ! N9

m = m; ! m9

Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf <



Uzo wkyulb`f r|rufmr LL

Odf mvru k`ro bodrlefy ucf yf`kulxf kdjv`ky momfduvm ld keelulod uo ucf lduyldrlb

kdjv`ky momfduvm$

Jfdfyk``| lduyldlrb kdkjv`ky momfduvm #R, lr

R = N; ! N9

kde ucf uouk` kdkjv`ky momfduvm #N, lr

N = @ ! R

;

p n; m; 4

99

p n9 m9 4

;

p n m 4

Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;7



Kdjv`ky Momfduvm Keelulod

• Lh zf adoz N; kde N9 ucfd zf bkd od`| s ucf mkjdluvef oh pNp = pN; ! N9p

Cfdbf m = m; ! m9 kde p n; ∙ n9p ≥ n ≥ n; ! n9

• Lh zf adoz N ucfd zf bkd bodruykld pN;p kde pN9p

m; kde m9 rvinfbu uo ucf bodruykldur m; ! m9 = m pm;p ≥ n; kde pm9p ≥ n9

N;

N9

Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;;



B`firbc-Joyekd Bofhblfdur

• Ucf yf`kulod ifuzffd ucf uouk` kdjv`ky momfduvm kde ucf ldelxlevk`

momfduk bkd if fswyfrrfe kr

pNM = B n n; n9m m; m9

p n;m; p n9m9

zcfyf ucf B`firbc-Joyekd Bofhblfdur yfwyfrfdu ucf {vkduvm mfbckdlbk`

oxfy`kw ifuzffd ucf uzo elhhfyfdu efrbylwulodr oh k bovw`fe r|rufm>

B n n; n9m m; m9

= n m p n; m; , n9 m9




BJ Fskmw`fr L

Uzo wkyulb`fr zluc rwld ; kde rwld 7$ Cfdbf N = ;

9, 29

zluc 9 kde 5 rvirukufr

yfrwfbulxf`|$

pNM = B n n; n9m m; m9

p n;m; p n9m9

ˑˑˑˑ2

9

2

9

= p;;

ˑˑˑˑ;

9

;

9




BJ Fskmw`fr LL

ˑˑˑˑ;

9

;

9 = y ;

2

p;; ˑˑˑˑ;

9

∙;

9!y 9

2

p;7 ˑˑˑˑ;

9

;

9




Kdjv`ky momfmuvm oh k {{ rukuf

• { =ˑˑ;

9

;

9

³kde { =

ˑˑ;

9∙ ;

9

³• Zf bkd hoym 9 rwld rukufr p;7 kde p77

• Hoy k vv r|rufm uclr boyyfrwoder uo ucf

χ7 mfrod #N = 7,

ϋ7 mfrod #N = ;,

• Zf k`ro dffe uo ldb`vef ucf oyilk` kdjv`ky momfduvm @ ifuzffd ucf {vkyar

Ucfyf kyf 9 {{ rukufr hoy fkbc xk`vf oh @




Lrorwld

• dfvuyod M = <23 MfX kde wyouod M = <2< MfX

• d-d d-w kde w-w ldufykbulodr kyf lefdulbk` lh FM ldufykbulod ljdoyfe

• Bodrlefy ucf #dw, kr bomwodfdur oh ucf Dvb`fod #D, zluc L-rwld ;/9

w =

ˑˑˑˑ

;

9

;

9

d =

ˑˑˑˑ

;

9∙

;

9

• Efdf k dvb`fod rwldoy

D = 7Gλ

ι ;K zcfyf w = 7G

;

7;K kde d = 7G

7

;;K

• Ucf ruyodj hoybf lr ldxkylkdu vdefy youkulodr ld uclr lrorwld rwkbf$




Lrorwld Krrljdmfdur

• Hoy kd| ckeyodr mkef vw oh v kde e {vkyar bodruyvbu lrorwld mvùlw`fur>

w =ˑˑˑˑ ;

9

;

9

d =ˑˑˑˑ ;

9 ∙

;

9

χ! = p ; ; χ7 = p ; 7 χ∙ = p ; ∙ ;

∈!! = p 2/9 2/9 ∈! = p 2/9 ;/9

∈7 = p 2/9 ∙ ;/9 ∈∙ = p 2/9 ∙ 2/9

Α = p 7 7

Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;:



χw Byorr Rfbulodr

;5 57$ W òur oh byorr rfbulodr kde yf`kufe {vkdululfr

;7

;7 9

;7 -;

; ;7 ;7 9

χ! w uouk`

χ! w

f`krulb

⇒

W ki JfX/b

B y o r r r f b u l o

d # m i ,

;7

;7 9

;7 -;

; ;7 ;7 9

χ»e

uouk`

χ‒ w

uouk`

χ‒ w

f`krulb

⇒

⇒

W ki

JfX/b

B y o r r r f b u l o d # m i ,

∞r JfX χe

χw ;$9 9 2 5 6 8 : 3 < ;7 97 27 57

9$9 2 5 6 8 : 3 < ;7 97 27 57 67 87

Hljvyf 57$;2> Uouk` kde f`krulb byorr rfbulodr hoy χ» w kde χ»e #uouk` od`|, bo``lrlodr kr khvdbulod oh `kioykuoy| ifkm momfduvm kde uouk` bfdufy-oh-mkrr fdfyj|$ Boyyfrwodeldj bomwvufy-yfkeki`f ekuk `fr mk| if hovde ku cuuw>//wej$ì`$jox/srfbu/bodufdur$cum` #Bovyufr| oh ucfBOMWKR Jyovw LCFW Wyouxldo Kvjvru 9772$,

3

Lrorwld bkd cf`w vr wyfelbu yf`kulxf byorr

rfbulodr zlucovu ckxldj uo adoz kd|-

ucldj kiovu ucf kiro`vuf byorr rfbulodr$

Byorr rfbulodr>

χ! w ↙ χ! w kde χ∙ w ↙ χ∙ w

Lh od`| L=2/9 bomwodfdu boduylivufr

ucfd

ς#χ! w, > ς#χ∙ w, = 2 > ;

∈#;928, lr kd L = 2

9yfrodkdbf




Elrbyfuf R|mmfuylfr

• Wkylu|• Wkylu| Xlo`kulod

• Bckyjf Bodnvjkulod

• BW Xlo`kulod

• Ulmf Yfxfyrk`

• Ucf BW U Ucfoyfm

• @fwuod dvmify

Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;



Wkylu|

• K wkylu| uykdrhoymkulod W ldxfyur fxfy| rwkulk` booyeldkuf> W #u,s, = #u,∙s,

W 9 = L kde ucfyfhoyf ucf fljfdxk`vfr oh W kyf »;$

•Oyeldky| xfbuoy x$ W #x, =

∙x $

• Rbk`ky hyom x> r = x º xW #r, = W #x º x, = #∙x, º #∙x, = x º x = !r

• Byorr wyoevbu oh uzo xfbuoyr> k = xÚz

W #k, = W #xÚz, = #

∙x,Ú

#∙z, = x

Úz = !k

• Rbk`ky hyom k kde x> w = k º xW # w, = W #k º x, = #!k, º #∙x, = ∙k º x = ∙ w

Rbk`ky W #r, = !r

Wrfveorbk`ky W # w, = ∙ w

Xfbuoy W #x, = ∙xWrfveoxfbuoy W #k, = !k

Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 9



Wkylu| ld Wc|rlbk` R|rufmr

• Uzo-ioe| r|rufmr ckxf wkylu| wK wI#∙;,ℒ

W ψ#;9, = w; w9#∙;,ℒψ#;9,

• Lduyldrlbk``| wkyulb`fr kde kdulwkyulb`fr ckxf owworluf wkylu|

Iovde rukufr `laf worluyodlvm f! f∙ kde mfrodr {{ ckxf wkylu| oh #∙;,ℒ!;$

• Wcouodr ckxf k wkylu| oh #∙;, kde uclr vdefy`lfr ucf ∈ℒ = »; rf`fbulod yv`fld kuomlb uykdrlulodr$

• Douf ucku wkylu| lr k mvùlw`lbkulxf {vkduvm dvmify$

Uclr lr uyvf hoy k`` elrbyfuf r|mmfuylfr$

Boduldvovr r|mmfuylfr ckxf keelulxf {vkduvm dvmifyr$




Fskmw`f> vv mfrodr

I| bodxfdulodr> v-{vkyar ckxf rwld ;/9 kde ! wkylu| kde

v-{vkyar ckxf rwld ;/9 kde - wkylu|

Wkylu| oh k vv mfrod lr W = wv wv#∙;,ℒ

Ucf lduyldrlb rwld #R, oh ucf vv mfrod lr 7 oy ;

ivu mk| ckxf kd| oyiluk` kdjv`ky momfduvm #@, xk`vf$

R @ NW wkyulb`f

7 7 7∙ χ7

; 7 ;∙ ϋ7

7 ; ;! i;#;926,

Rff ucf WEJ hoy k uki`f oh ucf {vkduvm dvmifyr oh ucf òz-mkrr mfrodr$




Wkylu| ld ucf Rukdekye Moef`Ld ;<68 \kdj kde @ff yfk`l~fe ucku wkylu| ldxkylkdbf cke dfxfy iffd ufrufe

fswfylmfduk``| hoy zfka ldufykbulodr$

Zv“r fswfylmfdu> yfboyefe ucf elyfbulod oh ucf fmluufe f`fbuyod hyom k 87Bo

ι -efbk| zcfd ucf dvb`fky rwld zkr k`ljdfe vw kde eozdUcf f`fbuyod zkr fmluufe ld ucf rkmf elyfbulod ldefwfdefdu oh ucf rwld$

∙↙ Wkylu| lr dou bodrfyxfe ld ucf zfka ldufykulodr




Wkylu| Xlo`kulod ld χ Efbk|

• Bodrlefy ucf zfka efbk| χ! ↙ ¶! ! ν ¶$

Rldbf ucf χ lr rwld-7 kde ucf ¶ kde ν fmfyjf ikba-uo-ikba ld ucf BM hykmf ucf

rwldr oh ucf ¶ kde ν mvru bkdbf`$

• Fswfylmfdur rcoz ucku fxfy| ¶! lr `fhu-ckdefe kde ucfyfhoyf fxfy| ν ¶ lr k`ro

`fhu-ckdefe$

•Rlml`ky`| ld χ∙ efbk| iouc ucf ¶∙ kde ν ¶ k`zk|r fmfyjf yljcu-ckdefe$

• Lh wkylu| zfyf bodrfyxfe i| ucf zfka ldufykbulod zf zovè fswfbu `fhu-ckdefe

wklyr kde yljcu-ckdefe wklyr zluc f{vk` wyoikil`lu| #nvru kr zf oirfyxf zluc

χ7 ↙ 9β ,$

• Krrvmldj ucku dfvuyldor kyf mkrr`frrK@@ dfvuyldor kyf `fhu-ckdefe

K@@ kduldfvuyldor kyf yljcu-ckdefe




Bckyjf Bodnvjkulod L

• Ucf bckyjf bodnvjkulod owfykuoy B bodxfyur k wkyulb`f uo l ur kdulwkyulb`f$

B p w = p w• Ld wkyulbv`ky B yfxfyrfr fxfy| ldufydk` {vkduvm dvmify

#f$j$ bckyjf iky|od/`fwuod dvmify ruykdjfdfrr fub$,$

• B 9 = L lmw`lfr ucku ucf od`| k`òzfe fljfdxk`vfr oh B kyf »;$

• Vd`laf wkylu| xfy| hfz wkyulb`fr kyf B fljfdrukufr$

Od`| wkyulb`fr ucku kyf ucfly ozd kdulwkyulb`fr #χ7 κ β , kyf B fljfdrukufr$

Hoy fskmw`f

B ˑˑχ!³

=ˑˑχ∙

³B pβ = ∙ pβ

Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf :



Bckyjf Bodnvjkulod LL

• Ucf wcouod ckr k B = ∙;

• h h iovde rukufr ckxf B = #∙;,ℒ!r

• Bckyjf bodnvjkulod lr yfrwfbufe i| iouc ucf ruyodj kde f`fbuyomkjdfulb

ldufykbulodr$

• Fskmw`f> ucf χ7 #ℒ = r = 7 ⇙ B = !;, bkd efbk| lduo 9β ivu dou 2β

B pdβ = #∙;,d pβ B

ˑˑχ7

³=

ˑˑχ7

³

χ7

↙9β lr k`òzfe #kde oirfyxfe,

χ7 ↙ 9β lr dou k`òzfe #kde dou oirfyxfe 0 2$; Ú ;7∙3,




J-Wkylu| L

•B -r|mmfuy| lr oh `lmlufe vrf$

⇙ Moru wkyulb`fr kyf dou B fljfdrukufr

• Ucf B owfykuoy bodxfyur χ! uo χ∙$

• Ucfrf uzo wkyulb`fr ckxf lrorwld krrljdmfdur p ; ; kde p ; ∙ ; $

• K ;37◨ lrorwld youkulod jlxfr p ; ; = flχL9 p ; ∙ ; $• Ucf bckyjfe wlodr kyf fljfdrukufr vdefy ucf J-wkylu| zclbc bomildfr B zluc

k ;37◨ lrorwld youkulod> J = BflχL9

•J-wkylu| lr mkld`| vrfe uo fskmldf efbk|r uo wlodr #zclbc ckxf J =

∙;,$

J pdχ = #∙;,d pdχ

Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf <

J-Wkylu| LL



J-Wkylu| LL

J-Wkylu| oh k hfz mfrodr

Wkyulb`f N W L J Efbk|

ϋ#::7, ;∙ ; !; 9χ

ό#:32, ;∙ 7 -; 2χ

ψ#;797, ;∙ 7 -; 2χ

h #;9:7, 9! 7 !; 9χ

Hoy fskmw`f ucf ϋ#::7, ckr J = ; zclbc mfkdr lu rcovè od`| efbk| uo kd fxfd

dvmify oh wlodr$ Fswfylmfduk``| zf de ucku

ϋ ∙↙ χχ ;77&

χχχ 0 ;$9 Ú ;7∙5

Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;7



BW R|mmfuy|

Ucf bomildkulod oh B kde W #kde ulmf yfxfyrk` U , ckxf rwfblk` rljdlbkdbf$

Ucf xlo`kulod oh BW lr ucf yfkrod zf `lxf ld k mkuufy vdlxfyrf

BW U lr yf{vlyfe uo if bodrfyxfe ld [vkduvm Hlfè Ucfoy| #[HU,

@ooa ku k wlod efbk| fskmw`f>

• Ld ucf wlod efbk| χ! ↙ ¶!#Y, ! ν ¶#@, ucf ν ¶ lr k`zk|r `fhu-ckdefe #@C,

• Vdefy B uclr ifbomfr χ∙ ↙ ¶∙#Y, ! ν ¶#@, ivu ucf ν ¶ lr rul`` @C

⇙ zclbc eofr dou obbvy ld dkuvyf$

• Zluc B kde W ucovjc zf jfu k YC kduldfvuyldo$ χ∙ ↙ ¶∙#@, ! ν ¶#Y,⇙ zcllbc lr k`òzfe ld dkuvyf

Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;;



BW Xlo`kulod ld ucf Akod Rfbuoy L

Bodrlefy ucf dfvuyk` akodr A 7 #er, kde A 7 #re,

Ucfrf wkyulb`fr bkd mls xlk k rfbode-oyefy zfka ldufykbulod>

Z

v

Z

v

e r

r e

v

Z

v

Z

e r

r e

Uclr rfbulod zl`` hobvr od ucf BW krwfbur$ Ucf efukl`r od ucf ulmf efwfdefdbf oh A 7

A 7 orbl``kulodr bkd if hovde ld moru ufsuiooar$


BW Xlo`kulod ld ucf Akod Rfbuoy LL



BW Xlo`kulod ld ucf Akod Rfbuoy LL

• Iouc A 7 kde A 7 kyf wrfveorbk`ky mfrodr ucfyfhoyf W = ∙;$ kde ucf A 7 kde

A 7 kyf k wkyulb`f-kdulwkyulb`f wkly$

Kr k yfrvù vdefy BW zf ckxf

BW ˑˑA 7³

= ∙ ˑˑA 7³

BW ˑˑA 7³

= ∙ ˑˑA

7³

• Efdldj pA ; =gˑˑA 7

³∙ ˑˑA 7

³µ/∞

9

pA 9 =gˑˑ

A 7³

!ˑˑ ¯A

7³µ/∞9

zf ckxf BW pA ; = ! pA ;BW pA 9 = ∙ pA 9

• Lh BW lr bodrfyxfe ucfd

pA ; bkd od`| efbk| uo 9χ #BW = !;,

pA 9 bkd od`| efbk| uo 2χ #BW = ∙;,$




BW Xlo`kulod ld ucf Akod Rfbuoy LLL

• A 7 kde ucf A 7 kyf mkrr fljfdrukufr kde kyf fkbc oucfyr kdulwkyulb`fr

• A ; kde ucf A 9 kyf BW fljfdrukufrˑˑA 7

³= ;∞

9#pA ; ! pA 9,

ˑˑA 7

³= ;∞

9#pA ; ∙ pA 9,

• A 7R

kde ucf A 7@

kyf ucf oirfyxfe rukufr kde kyf dfky`| lefdulbk` uo ucf BW

fljfdrukufr #A 7R

kde A 7@

kyf dou kdulwkyulb`fr,

ˑˑA

7

@³

=

;

∞;!pǡp9 #pA 9 ! ǡ pA ;, ˑˑA

7

R³

=

;

∞;!pǡp9 #pA 9 ∙ ǡ pA ;,• Fswfylmfduk``| zf oirfyxf

A R ↙ χχ υ = 7$< Ú ;7∙;7r

A @ ↙ χχχ υ = 7$6 Ú ;7∙:

rCozfxfy zf de kiovu ; A 7

@↙ χ!χ∙ ld 557 A 7

@efbk|r




Oucfy Ufrur oh BW Xlo`kulod

• Ucfyf kyf oucfy BW -xlo`kuldj oirfyxki`fr ucku ckxf iffd mfkrvyfe ld ucf akod

rfbuoy$ Hoy fskmw`f ucfyf lr kd kr|mmfuy| ifuzffd ucf iykdbcldj ykulor oh

A @ uo χ! ! f∙ ! ν f xfyrvr χ∙ ! f! ! ν f

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zcfyf al kyf ucf momfduk bomldj lduo ucf xfyufs$

• Rufw 6

Hoym ucf kmw`luvefM = l #xfyufs hkbuoyr,#wyowkjkuoyr,#momfduvm bodrfyxkulod,

• Rufw 8>

Ldufjykuf oxfy ucf ldufydk` momfduk ;#9χ,5

e5{n

• Rufw :>

Eyow ucf fsuyk ε-hvdbulod kde ∙lM yfmkldr$


Fskmw`f> K↙ I ! B



w

• Uo òzfru oyefy #O#j,, zf ckxf nvru odf elkjykm>

↛ w;

w9 տ և w2

K

I B

• Ucfyf lr nvru odf xfyufs kde do wyowkjkuoyr ucfyfhoyf

M = l#∙lj, = j


@lhfulmf oh K



• Hyom Hfyml“r Joèfd Yv`f ucf efbk| ykuf lr jlxfd i|

Β =pwp

3χm9;

pMp9

• Lh m; = mK kde M = j

Β =j9pwp

3χm9K

zcfyf pwp lr ucf mkjdluvef oh ucf momfduvm oh flucfy I oy B

• Ucf `lhfulmf oh K lr

υ K =3χm9

Kj9pwp


K!K↙ I !I

9



• Uo òzfru oyefy #O#j9,, zf ckxf uzo elkjykmr>

B

K, w; K, w9

I, w2 I, w5

B

K, w; K, w9

I, w5 I, w2

u-bckddf` v-bckddf`

{ = w; ∙ w2 { = w; ∙ w5

Mu = l#∙lj,9 l# w; ∙ w2,9 ∙ m9B

= j

9

u ∙ m9B

Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf :

K! K↙ I !I



• Ucf mkuyls f`fmfdu hoy ucf r kde v bckddf` lr

M =j9

u ∙ m9B

!j9

v ∙ m9B

• Douf ucku M lr @oyfdu~ ldxkylkdu$

• Uo bodxfyu M uo k byorr rfbulod zf vrf Hfyml“r Joèfd Yv`fLd ucf BM hykmf

eς

eℨ=

;

#3χ,9R pMp9

#F ; ! F 9,9pwh p

pwlp

zcfyf R = ;/9 ifbkvrf zf ckxf uzo lefdulbk` wkyulb`fr #I ! I, ld ucf dk`

rukuf$ K`ro F ; = F 9 = F $


K! K↙ I !I

• Ucf byorr rfbulod bkd k`ro if zyluufd kr



• Ucf byorr rfbulod bkd k`ro if zyluufd kr

eς

eℨ=

;

9#;8χF ,9pwh p

pwlp

ˑˑ

ˑˑˑj9

u ∙ m9B

!j9

v ∙ m9B

ˑˑ

ˑˑˑ

9

• Krrvmf ucku mK = mI = m kde mB = 7$

Ucfd pwh p = pwlp kde

u = # w; ∙ w2,9 = ∙9w9#; ∙ bor γ,

v = # w; ∙ w5,9 = ∙9w9#; ! bor γ,

⇙eς

eℨ

=;

9—

j9

;8χF w9

rld9

γ¡9

• Douf ucku ς ↙ ∔ nvru kr hoy Yvucfyhoye rbkuufyldj$

Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf <



Cljcfy-Oyefy Elkjykmr

• I| bodrlefyldj moyf bomw`lbkufe Hf|dmkd elkjykmr zf bkd jfdfykufkeelulodk` boduylivulodr uo ucf kmw`luvef>

MK↙I!B = jK; ! j2K2 ! j6K6 ! $ $ $

MK!K↙I!I = j9K9 ! j5K5 ! j8K8 ! $ $ $

Lh j ≮ ; #oy moyf wyfblrf`| #j/mK, ≮ ; ld KIB Ucfoy|, zf bkd rff coz

fkbc rvbbfrrlxf ufym ld ucf wfyuvyikulod rfylfr wyoxlefr rmk``fy kde rmk``fy

boyyfbulodr uo ucf kmw`luvef$

Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;7

Boyyfbulodr uo υ K



• Zf ckxf bk`bv`kufe ucf `lhfulmf oh K evf uo ucf xfyufs elkjykm

K

I B

• Rldbf M ∰ j Β ∰ j9 ucfd ucf `fkeldj boyyfbulodr uo Β zl`` if O#j5,$

Ucfyf zl`` kylrf hyom ucf ldufyhfyfdbf oh ucf O#j, elkjykm zluc k O#j2,

elkjykm ld ucf bocfyfdu rvm>

pMp

9

=ˑˑjK; ! j

2

K2 ! $ $ $ˑˑ9

Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;;

Oucfy Efbk| Moefr



• Douf ucku zf kyf od`| ldufyfrufe ld ucf O#j2, elkjykmr ld zclbc K ↙ I ! B $

• Lh K lr rvhblfdu`| cfkx| oucfy efbk| moefr rvbc kr K ↙ 2I ! B kde

K ↙ I ! 2B kyf worrli`f$

B K

K

II I

B


Ldbocfyfdu Rvmr



• K ↙ 2I ! B lr k elruldbu efbk| moef cfdbf zf bk`bv`kuf Β rfwkykuf`| hyom

ucku oh K ↙ I ! B

• Kr k yfrvù fxfd ucovjc M ∰ j2 Β ∰ j8 kde ro zf dffe dou bodrlefy ucfrf

elkjykmr hoy k Β ∰ j5 bk`bv`kulod$

• Ucf efbk| oh K ldxo`xfr iouc bocfyfdu kde ldbocfyfdu rvmr$

Β#K ↙ kd|ucldj, = ΒK↙IB ! ΒK↙IIIB ! ΒK↙IBBB ! $ $ $

= B ;

ˑˑˑSMK↙IB

ˑˑˑ9! B 9

ˑˑˑSMK↙IIIB

ˑˑˑ9

!B 2

ˑˑˑS

MK↙IBBB

ˑˑˑ9

! $ $ $

#B ; B 9 kde B 2 kylrf hyom Hfyml“r Joèfd Yv`f$,


cl e O e K B l



Uclye-Oyefy K↙ I ! B Elkjykmr

• Ucfyf lr odf `fjk` uclye-oyefy elkjykm uo bodrlefy>

K

K

I B

B I


L``fjk` @oow Elkjykmr



• Ucfyf kyf rfxfyk` oucfy O#j2, elkjykmr ucku bkd if eykzd hoy K ↙ I ! B

cozfxfy ucfrf kyf dou uo if bk`bv`kufe vrldj ucf Hf|dmkd Yv`fr$

K

I B

I

K

B I B

K

K

I B

Elrboddfbufe Yfevbli`f


Boyyfbulodr uo K! K↙ I !I



Boyyfbulodr uo K! K↙ I !I

• Ucf ldufyhfyfdbf oh ucf odf-òow elkjykmr #O#j5,, zluc ucf uyff-`fxf` elkjykm

#O#j9,, wyoxlefr O#j8, boyyfbulodr uo ucf byorr rfbulod$

B

I

K

B

K K

I I


[vkduvm Mfbckdlbr



[vkduvm Mfbckdlbr

Uclr rfbulod jlxfr ucf [vkduvm Mfbckdlbr ikbajyovde yf{vlyfe uo bk`bv`kuf

Hf|dmkdd elkjykmr$

Ucf lduyoevbulod uo Elykb F{vkulod mk| if dfz kde zl`` fxfduvk``| boxfyfe ld ucf

jykevkuf [M bovyrf$

• Rbcy£oeldjfy f{vkulod

• A`fld-Joyeod F{vkulod

•Elykb F{vkulod

Wc|rlbr 678K : - Y[M Wkjf ;

Rbcy£oeldjfy f{vkulod



Ld b`krrlbk` [vkduvm Mfbckdlbr yfbk`` ucku zcfd zf rviruluvuf

w↙ ∙l∊ F ↙ l

∉

∉u

lduo ucf b`krrlbk` fswyfrrlod hoy fdfyj| bodrfyxkulod

w9

9m! X = F

zf oiukld ucf Rbcy£oeldjfy f{vkulod

∙ ;

9m∊9Σ ! X Σ = l

∉ Σ

∉u

Zoyar zf`` hoy dod-yf`kulxlrulb rwld-7 wyoi`fmr$

Wc|rlbr 678K : - Y[M Wkjf 9

A`fld-Joyeod f{vkulod

Ucf Rbcy£oeldjfy f{vkulod eofr dou zoya hoy yf`kulxlrulb wkyulb`fr zf vrf ucf rkmf

kwwyokbc kr eodf hoy b`krrlbk` [M



ww

• Vrf ucf yf`kulxlrulb fdfyj|-momfduvm yf`kulod> F 9 ∙ w9 = m9

•Ld 5-xfbuoy doukulod w¶ = #F, w, w¶ w¶

∙m9 = 7

• Zluc ucf boxkylkdu rviruluvulod w¶ ↙ l∉ ¶

∙∉ ¶∉ ¶ψ ∙ m9ψ = 7

∙∉ 9ψ

∉u9!

∊9ψ = m9ψ

#∙ m9,ψ = 7

Uclr lr ucf A`fld-Joyeod f{vkulod

• lr ucf e“K`fmifyulkd owfykuoy

=∉ 9

∉u9! ∊9


Ro`vulodr uo ucf A`fld-Joyeod F{vkulodB le ` ` ul u uc A` l J e ul



• Bodrlefy k w`kdf-zkxf ro`vulod uo ucf A`fld-Joyeod f{vkulod>

ψ#s, u, = f∙lFu!lwºs = f∙lwºs

#! m9,ψ = 7

• Rviruluvuldj ucf w`kdf-zkxf ro`vulod lduo ucf AJ f{vkulod yfuvydr ucf

fdfyj|-momfdvm bodrfyxkulod f{vkulod #bodrlrufdb| bcfba,∙F 9 ! w9 ! m9 = 7

• Hoy k jlxfdw ucfyf kyf uzo worrli`f ro`vulodr hoy F >

F =

»w w9 ! m9

Doulbf ucku ucfyf lr k dfjkulxf fdfyj| ro`vulod


Wyoi`fmr zluc ucf AJ f{vkulod



Wyoi`fmr zluc ucf AJ f{vkulod

• Ucf rfbode-oyefy ulmf efylxkulxf ld ucf

oh ucf AJ f{vkulod lr yfrwodrli`fhoy iouc ucf dfjkulxf-fdfyj| w`kdf zkxf ro`vulodr kde k mlrifckxldj

wyoikil`lu| efdrlu|$

• Elykb uylfe uo s uclr wyoi`fm i| òoaldj hoy k yf`kulxlrulb f{vkulod ucku `laf

ucf Rbcy£oeldjfy f{vkulodC Σ = l ∉ Σ/∉u

od`| bodukldfe yru-oyefy ulmfefylxkulxfr$

• Ucf AJ f{vkulod lr jooe hoy rwld 7 wkyulb`fr zcfyfkr ucf Elykb f{vkulod zoyar

hoy rwld ;/9 wkyulb`fr


Elykb“r Kwwyokbc LR ul ` l u u #l 7,



• Rvwworf k wkyulb`f lr ku yfru #l$f$ w = 7,$

Ucfd ucf fdfyj|-momfduvm yf`kulod

F 9 ∙ w9 = m9

bkd if hkbuoyfe lduo k wkly oh #`ldfky, f{vkulodr>

# w7,9 ∙ m9 = 7

# w7

∙ m,# w7

! m, = 7⇙ # w7 ∙ m, = 7 oy # w7 ! m, = 7

• Flucfy `ldfky f{vkulod `fker uo k bodjvykulod-rwkbf f{vkulod zclbc lryru-oyefy ld ulmf kde rkulrfr ucf yf`kulxlrulb fdfyj|-momfduvm yf`kulod$


Elykb“r Kwwyokbc LL• Uo fsufde uclr hkbuoyl~kulod hoy moxldj wkyulb`fr zf bkd zyluf



j w

# w¶ w¶ ∙ m9, = #ι ξ wξ ! m,#β α wα ∙ m,

zf dffe uo efufymldf ucf bofhblfdur ι ξ kde β α$

• Fswkdeldj ovu ucf yljcu-ckde rlef zf ckxf

# w¶ w¶ ∙ m9, = ι ξβ α wξ wα ! m#β ξ ∙ ι ξ, wξ ∙ m9

• Uo f`lmldkuf ucf `ldfky w ufym zf yf{vlyf ι ξ = β ξ$ Ucf {vkeykulb ufym `fker

uo

w

9

= β

ξ

β

α

wξ wα

Wc|rlbr 678K : - Y[M Wkjf :

Elykb“r Kwwyokbc LL

• Zf bkd ucfd zyluf

9 ξ α



w9 = β ξβ α wξ wα

= β ξβ α# wξ wα ! wα wξ,/9

= #β ξβ α ! β αβ ξ, wξ wα/9

⇙ #β ξβ α ! β αβ ξ, = 9jξα

• Zf u|wlbk``| zyluf uclr `kru yf`kulodrclw kr kd kdulbommvukuoy>

qβ ¶, β ν} = 9j¶ν

zcfyf #β 7,9 = ; kde #β l,9 = ∙;

• Lh β 7 = ; kde β l = l ivu ucfd qβ 7, β l} = 7 zclbc mfkdr ucku ucfkdulbommvukuoy f{vkulod bkddou if ro`xfe i| kd| rfu oh bomw`fs dvmifyr$

⇙ Ucf β kyf mkuylblfr$


Ucf β Mkuylbfr



• Elykb“r lefk zkr uo `fu β yfwyfrfdu k mkuyls$

Rwfblbk``| ucf 5

Ú5 mkuylbfr

β 7 =

7G ; 7

7 ∙;

;K β l =

7G 7 ςl

∙ςl 7

;K

rkulrh| ucf kdulbommvukulod yf`kulodq

β ¶, β ν

}= 9j¶ν $

• Uclr rfu oh β mkuylbfr lr adozd kr ucf Inoyafd kde Eyf`` yfwyfrfdukulod kde lu lr

bommod`| vrfe ku òz fdfyjlfr$

• Oucfy bcolbfr fslru cozfxfy ucf wc|rlbr lr ldefwfdefdu oh ucf rwfblb bcolbf

oh β mkuylbfr$

Wc|rlbr 678K : - Y[M Wkjf <

Ucf Elykb F{vkulod

• Ckxldj hkbuoyfe ucf yf`kulxlrulb fdfyj|-momfduvm yf`kulod

# w¶ w¶ ∙ m9, = #β ξ wξ ! m,#β α wα ∙ m, = 7



zf bkd rfu flucfy hkbuoy uo ~fyo$

• Ld momfduvm rwkbf ucf Elykb f{vkulod lr β ¶

w¶ ∙ m = 7• Zluc w¶ ↙ l∉ ¶ zf jfu ucf bodjvykulod-rwkbf Elykb f{vkulod>

#lβ ¶∉ ¶ ∙ m,σ = 7

• Zcfd zf boduykbu β ¶ zluc k hovy-xfbuoy {¶ zf bkd kiiyfxlkuf uclr vrldj ucf

Hf|dmkd r`krc doukulod β ¶{¶ = /{

• Zluc ucf r`krc doukulod ucf Elykb f{vkulod ifbomfr

/ w ∙ m = 7

#l/∉ ∙ m,σ = 7

Wc|rlbr 678K : - Y[M Wkjf ;7

Rwldoyr

• Yfzyluldj ucf Elykb f{vkulod



#l/∉ ∙ m,σ = 7

• Rldbf ucf β mkuylbfr kyf 5 Ú 5 ucfd ucf σ mvru if k 5-bomwodfdu bo`vmd

mkuyls$

•Zf bk`` uclr k il-rwldoy Elykb rwldoy oy nvru w`kld rwldoy$

σ =

7IIIIIG

σ;

σ9

σ2

σ5

;BBBBBK

Lu lr dou k 5-xfbuoy - lu eofr dou uykdrhoym `laf k 5-xfbuoy$

Wc|rlbr 678K : - Y[M Wkjf ;;

Ro`vulodr oh ucf Elykb F{vkulod L

• @ooa hoy ro`vulodr ucku kyf ldefwfdefdu oh worlulod>



∉σ

∉s=

∉σ

∉|=

∉σ

∉~= 7

• Uclr rlmw`lfr ucf Elykb f{vkulod uo

lβ 7∉σ

∉u∙ mσ = 7

• Rw`lu ucf rwldoy lduo k wkly oh 9-bomwodfdu wlfbfr>

σ =

7G

σK

σI

;K

σK =

7G

σ;

σ9

;K

σI =

7G

σ2

σ5

;K


Ro`vulodr oh ucf Elykb F{vkulod LL• Uclr `fker uo ucf wkly oh f{vkulodr



Uclr `fker uo ucf wkly oh f{vkulodr

∉σK

∉u

=

∙lmσK

∉σI

∉u

= !lmσI

zcorf ro`vulodr kyf

σK#u, = σK#7,f∙lmu σI#u, = σI#7,f!lmu

• Fxlefdu`| σK lr k ro`vulod zluc fdfyj| F = !m kr zf rcovè fswfbu ivu σIrffmr uo ckxf k dfjkulxf fdfyj| F = ∙m$

• Elykb cke cowfe ucku k yru-oyefy #ld ∉/∉u, f{vkulod zovè kxole ucfrf

dfjkulxf fdfyj| ro`vulodr$

• Fxfduvk``| Elykb wyfelbufe ucf fslrufdbf oh ucf worluyod$


F`fbuyodr kde Worluyodr• Elykb f{vkulod ckr 5 ldefwfdefdu ro`vulodr hoy k wkyulb`f ku yfru>



Elykb f{vkulod ckr 5 ldefwfdefdu ro`vulodr hoy k wkyulb`f ku yfru>

#f∙ ↛, σ#;, = f∙lmu

7IIIIIG

;

7

7

7

;BBBBBKσ#9, = f∙lmu

7IIIIIG

7

;

7

7

;BBBBBK#f∙ →,

#f! ↛, σ#2, = f!lmu

7IIIIIG

77

;

7

;BBBBBKσ#5, = f!lmu

7IIIIIG

77

7

;

;BBBBBK#f! →,

hoy kd f`fbuyod kde worluyod ld rwld vw kde eozd rukufr$


W`kdf-Zkxf Ro`vulodr L

• Dfsu zf zl`` òoa hoy ro`vulodr uo ucf Elykb f{vkulod oh ucf hoym

σ# , lw s # ,



σ#s, = kf∙lwºsv# w,

zcfyf v# w, lr k momfduvm-rwkbf ro`vulod oh ucf Elykb f{vkulod rkulrh|ldj

#/ w ∙ m,v = 7

• Vrldj/ w = β 7 w7

∙β

ºw

= F

7G ; 7

7 ∙;

;K∙ w º

7G 7 ς

∙ς 7

;K

=

7G F

∙w

ºς

w º ς ∙F

;K


W`kdf-Zkxf Ro`vulodr LL

#/w ∙ m,v =

7G F ∙ m ∙w º ς

;K7G vK

;K



#/ w m,v Gw º ς ∙F ∙ m

KGvI

K

=7G #F ∙ m,vK ∙ #w º ς,vI

#w º ς,vK ∙ #F ! m,vI

;K

• Ucf Elykb f{vkulod #/ w ∙ m,v = 7 ucfd jlxfr vr k wkly oh bovw`fe f{vkulodr hoy

vK kde vI >

vK =#w º ς,

F ∙ mvI vI =

#w º ς,

F ! mvK

•Ucfrf f{vkulodr bkd fkrl`| if ro`xfe i| rviruluvuldj odf lduo ucf oucfy kde

douldj ucku

#w º ς,9 = w9 º ;


W`kdf-Zkxf Ro`vulodr LLL

• Yfzyluldj ucf uzo bovw`fe f{vkulodr

vK =#w º ς,

F mvI vI =

#w º ς,

F ! mvK



F ∙ m F ! m

• Rviruluvuldj ucf rfbode f{vkulod lduo ucf yru zf ckxf

vK =w9

F 9 ∙ m9vK

zclbc yf{vlyfr F 9

∙ m9

= w9

nvru kr zf rcovè fswfbu$

• Kjkld zf ckxf uzo ro`vulodr hoy F >

F =

»w w9 ! m9

• I| wlbaldj rwfblb hoymr hoy vK oy vI #odf lr bodruykldfe i| ucf oucfy, zf

bkd bodruyvbu k rfu oh hovy ro`vulodr uo ucf Elykb f{vkulod hoy k moxldj wkyulb`f$

Wc|rlbr 678K : - Y[M Wkjf ;:

W`kdf-Zkxf Ro`vulodr LX

• Zluc ucf doymk`l~kulod v‧v = 9pF p zf ckxf uzo wkyulb`f ro`vulodr

7;

; 77

;



v

#;,

= D

7IIIIIG

;

7

w~F!mws!lw|F!m

;BBBBBK v

#9,

= D

7IIIIIG

7

;

ws∙lw|F!m∙w~F!m

;BBBBBK

kde uzo kdulwkyulb`f ro`vulodr

x#;, = D

7IIIIIG

ws∙lw|F!m∙w~F!m

7

;

;BBBBBK

x#9, = ∙D

7IIIIIG

w~F!mws!lw|F!m

;

7

;BBBBBK

• Ld k`` bkrfr D =∞

F ! m kde F lr worlulxf$


Rwldr oh ucf W`kdf-Zkxf Ro`vulodr



•Zf bkd jfdfyk`l~f ucf Wkv`l rwld mkuylbfr uo ucf 5

Ú5 mkuylbfr yf{vlyfe hoy

Elykb rwldoyr>

R =

9 ≫

7G ς 7

7 ς

;K

• Lh ucf wkyulb`fr kyf uykxf`ldj kòdj ucf ~-kslr ucfd ucf w`kdf-zkxf ro`vulodr v

kde x zl`` if fljfdrukufr oh R ~ $

• v#;, kde x#;, kyf rwld vw zcl`f v#9, kde x#9, kyf rwld eozd$

Wc|rlbr 678K : - Y[M Wkjf ;<

Wkyulb`fr kde Kdulwkyulb`fr• Ld k u|wlbk` fswfylmfdu zf kyf efk`ldj zluc wkyulb`fr oh rwfblb fdfyjlfr kde

c h c e ` ` c c h



momfduk ucfyfhoyf lu lr ucf v kde x w`kdf-zkxf ro`vulodr zclbc kyf oh ldufyfru

uo vr$

• Ucf wkyulb`f rukufr kyf ro`vulodr uo ucf oyljldk` momfduvm-rwkbf Elykb

f{vkulod

#/ w ∙ m,v = 7

• Ucf kdulwkyulb`f rukufr i| xlyuvf oh yfldufywyfuldj ucf dfjkulxf fdfyj| wkyulb`f

rukufr kr worlulxf fdfyj| kdulwkyulb`fr rkulrh|

#/ w ! m,x = 7


Ilj kde Rmk`` Bomwodfdur



• Vrldj ucf Inoyafd kde Eyf`` yfwyfrfdukulod oh ucf β mkuylbfr zf ckxf rcozd

ucku ucf w`kdf-zkxf ro`vulod hoy k rwld-vw f`fbuyod lr

v#;, = D

7IIIIIG

;

7w~

F!m

ws!lw|F!m

;BBBBBK

• Douf ucku ku òz fdfyjlfr ucf vwwfy bomwodfdur kyf mvbc `kyjfy uckd ucf òzfy

bomwodfdur$

Wc|rlbr 678K : - Y[M Wkjf 9;

Rvmmky|

• Ucf yf`kulxlrulb fdfyj|-momfduvm yf`kulod `fker uo ucf

A`fld-Joyeod f{vkulod ivu k dklxf fskmldkulod oh uclr



f{vkulod `fker uo wyoi`fmr$

• Elykb uylfe uo jfu kyovde ucfrf wyoi`fmr zluc k yru-oyefy

f{vkulod$ Ucf AJ wyoi`fmr yfmkldfe kde ucfly ro`vulod zkr

uo woruv`kuf ucf fslrufdbf oh kdulwkyulb`fr$

• Zf doz yfk`l~f ucku ucf A`fld-Joyeod f{vkulod efrbylifr

rwld-7 wkyulb`fr kde ucf Elykb f{vkulod efrbylifr rwld- ;9

wkyulb`fr$

• Ucf wkyulb`f kde kdulwkyulb`f w`kdf-zkxf ro`vulodr oh ucf Elykbf{vkulod zl`` if vrfe hyf{vfdu`| ld ovy hoymv`kulod oh [FE$


Rwldoyr



Rwldoyr

Fxfduvk``| zf zl`` rff ucku ucf ldufykbulodr bkd if fs-

wyfrrfe ld ufymr oh bvyyfdur oh ucf hoym

n¶ ≃ σS¶σ

Uo kle zluc ovy vdefyrukdeldj lu zl`` cf`w uo `fkyd kiovu

ucf uykdrhoymkulod wyowfyulfr oh rwldoyr

Wc|rlbr 678K 3 - Rwldoyr Wkjf ;

@oyfdu~ Uykdrhoymkulodr oh Rwldoyr

• Rwldoyr kyf dou hovy-xfbuoyr ucfyfhoyf ucf| eo dou uykdrhoym xlk ucf @oyfdu~uykdrhyomkulod mkuyls Α #efdfe ld kd fky`lfy `fbuvyf,

• Rwldoyr uykdrhoym ld ucf ho`òzldj mkddfy>



σ ↙ Rσ

• Hoy moulod kòdj ucf s-kslr

R =7G k! k

∙ς;

k∙ς; k!

;K

zcfyf

k» = »w

#β » ;,/9

kde

β = #; ∙ x9,∙;/9

Wc|rlbr 678K 3 - Rwldoyr Wkjf 9

Mkaldj k Rbk`ky Zluc k Rwldoy



• Vdefy k @oyfdu~ uykdrhoymkulod

σ‧σ ∙↙ #Rσ,‧#Rσ, = σ‧#R ‧R ,σ

Rldbf R ‧R = ; ⇙ σ‧σ lr dou k @oyfdu~ rbk`ky$

• Efdf ucf kenoldu rwldoy> σ ≫ σ‧β 7

• Vdefy k @oyfdu~ uykdrhoymkulod

σσ = σ‧β 7σ ∙↙ σ‧R ‧β 7Rσ = σ‧β 7σ = σσ

zcfyf R ‧β 7R = β 7


β 6

mkuyls• Efdf kd keelulodk` β -mkuyls i| β 6 ≫ lβ 7β ;β 9β 2



• Ld ucf Inoyafd kde Eyf`` yfwyfrfdukulod

β 6 =

7G 7 ;

; 7

;K

• Douf> #β 6

,9

= ; kde β 6

kdulbommvufr zluc fxfy| β mkuyls>

qβ ¶, β 6} = 7 ⇙ β ¶β 6 = ∙β 6β ¶

• Rldbf R ‧β 7β 6R = β 7β 6 σβ 6σ lr k`ro k @oyfdu~ rbk`ky$

• Uclr jlxfr vr 9 @oyfdu~ rbk`kyr> σσ kde σβ 6σ$


Wkylu|• Vdefy k wkylu| uykdrhoymkulod

σ 7σ



σ ↙ β 7σ

• Rldbf

σσ ↙ #Wσ,‧β 7#Wσ,

↙ σ‧#β 7,‧β 7β 7σ

↙ σ‧#β 7,‧σ

↙ σσ

σβ 6σ ↙ #Wσ,‧β 7β 6#Wσ,

↙ σ‧#β 7,‧β 7β 6β 7σ

↙ ∙σ‧#β 7,‧β 6σ

↙ ∙σβ 6σ

σσ lr k uyvf rbk`ky kde σβ 6σ lr k wrfveorbk`ky$


Il`ldfky Boxkylkdur• Ucfyf kyf ;8 worrli`f wyoevbur oh ucf hoym σ√l σn $

Ucfrf ;8 wyoevbur bkd if jyovwfe uojfucfy lduo il`ldfky boxkylkdur>



Ucfrf ;8 wyoevbur bkd if jyovwfe uojfucfy lduo il`ldfky boxkylkdur>

σσ Rbk`ky ; bomwodfdu

σβ 6σ Wrfveorbk`ky ; bomwodfdu

σβ ¶σ Xfbuoy 5 bomwodfdur

σβ ¶

β 6

σ Wrfveoxfbuoy 5 bomwodfdurσς¶νσ Kdulr|mmfuylb ufdroy 8 bomwodfdur

zcfyf

ς¶ν ≫l

9Qβ ¶, β ν ]


Rljdlbkdbf oh i`ldfky boxkylkdur?

q 6 6



• Zf ckxf k rlmw`f ikrlr rfu q;, β 6, β ¶, β ¶β 6, ς¶ν} hoy kd| 5 Ú 5 mkuyls

• Ucf ufdroylk` kde wkylu| bckykbufy oh fkbc il`ldfky lr fxlefdu$

Uclr mkafr lu fkr| uo rff zc| ucf [FE ldufykbulod @kjykdjlkd

∙fK¶σβ ¶σ

`fker uo k wkylu|-bodrfyxldj f`fbuyomkjdfulb hoybf mfelkufe i| k xfbuoy #l$f$rwld-;, iorod$

• Uo efrbylif ucf wkylu|-xlo`kuldj zfka ldufykbulod zf bovè #kde eo, mls

xfbuoy #σβ ¶σ, kde kslk` #σβ ¶β 6σ, ldufykbulodr$

Wc|rlbr 678K 3 - Rwldoyr Wkjf :

FM kde wcouodr

• Mkszf``“r f{vkulodK¶ ∉ν#∉ Kν, 5χN¶



• Mkszf`` r f{vkulodK¶ ∙ ∉ ν#∉ ¶Kν, = 5χN ¶

zcfyf

= ∉ ν

∉ ¶ Kν

= #ψ,K, kde N ν

= #ϋ,N,

• #ψ,K, kyf dou vdl{vf`| efufymldfe kde ro zf kyf k`òzfe uo mkaf k jkvjf

uykdrhoymkulod K¶ ↙ K¶ ! ∉ ¶α

• Zf bkd efmkde ucf @oyfdu~ bodelulod ∉ ¶K¶ = 7

zclbc rlmw`lfr ucf Mkszf`` f{vkulodr uo K¶ = 5χN ¶

• Zf bkd mkaf hvyucfy jkvjf uykdrhoymkulodr oh ucf hoym K¶ ↙ K¶ ! ∉ ¶α

zlucovu elruvyildj K¶ = 5χN ¶ ro òdj kr α = 7$

=⇙ Zf bcoorf uo rfu K7 = 7 kde zoya ld ucf Bovòmi jkvjf> ∊ º K = 7


Hyff Wcouodr

• Hoy k wcouod ld hyff rwkbf #N ¶ = 7, ucf woufdulk` lr jlxfd i| K¶ = 7$



• Ucf w`kdf-zkxf ro`vulod lrK¶#s, = kf∙lwºsǡ¶# w,

zcfyf ǡ¶ lr ucf wo`kyl~kulod xfbuoy kde w¶ w¶ = 7

• Kùcovjc ǡ

¶

ckr 5 bomwodfdur dou k`` kyf ldefwfdefdu$Ucf @oyfdu~ bodelulod yf{vlyfr ucku w¶ǡ¶ = 7

Hvyucfymoyf ucf Bovòmi jkvjf lmw`lfr ucku ǡ7 = 7 kde ǡ º w = 7

• Rldbf ǡ lr wfywfdelbv`ky uo w ucf wcouod lr uykdrxfyrf`| wo`kyl~fe kde ucfyf kyf

od`| 9 ldefwfdefdu wo`kyl~kulod rukufr$

Wc|rlbr 678K 3 - Rwldoyr Wkjf <

Hf|dmkd Yv`fr hoy [FE



Hf|dmkd Yv`fr hoy [FE

• Ucf Hf|dmkd Yv`fr hoy [FE

• Rfuuldj vw Kmw`luvefr

• Bkrlmly“r Uylba• Uykbf Ucfoyfmr

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;

F`fbuyodr kde worluyodr

• rwldoyr#r, kde #r, #r rwld, rkulrh ucf Elykb f{vkulod # ¶ , 7



v#r, kde x#r, #r = rwld, rkulrh| ucf Elykb f{vkulod #β ¶ w¶ ∙ m,v = 7

• kenoldur

v = v‧β 7 kde x = x‧β 7 rkulrh| v#β ¶ w¶ ∙ m, = 7

• oyucojodk`lu|

v#;,v#9, = 7 kde x#;,x#9, = 7

• doymk`l~kulod

vv = 9m kde xx = ∙9m

• bomw`fufdfrr

Wr

v#r,v#r, = β ¶ w¶ ! m kde Wr

x#r,x#r, = β ¶ w¶ ∙ m

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 9

Wcouodr

K¶#s, = kf∙lwºsǡ¶# w,



• @oyfdu~ bodelulod

ǡ¶ w¶ = 7

• oyucojodk`lu|

ǡ¶#;,

ǡ¶#9, = 7

•doymk`l~kulodǡ¶√ǡ¶ = ;

• Bovòmi jkvjf

ǡ7 = 7 kde ǡ º w = 7

• Bomw`fufdfrrWr#ǡ#r,,l#ǡ√

#r,,n = εln ∙ wl wn


Ucf Hf|dmkd Yv`fr hoy [FE LUcf Hf|dmkd yv`fr wyoxlef ucf yfblwf hoy bodruyvbuldj kd kmw`luvef M hyom k

Hf|dmkd elkjykm$



| j

• Rufw ;>Hoy k wkyulbv`ky wyobfrr oh ldufyfru eykz k Hf|dmkd elkjykm zluc ucf

mldlmvm dvmify oh xfyulbfr$ Ucfyf mk| if moyf uckd odf$

β

f∙, w; f!, w9

f∙, w2

f!, w5


Ucf Hf|dmkd Yv`fr hoy [FE LL

• Rufw 9>Hoy fkbc Hf|dmkd elkjykm `kif` ucf hovy-momfduvm oh fkbc `ldf fdhoybldj

hovy-momfduvm bodrfyxkulod ku fxfy| xfyufs$f∙, w2 f!, w5



β

f∙, w; f!, w9

Douf ucku kyyozr kyf od`| wyfrfdu od hfymlod `ldfr kde ucf| yfwyfrfdu wkyulb`f ozdou momfduvm$

• Rufw 2>

Ucf kmw`luvef efwfder od

;$ Xfyufs hkbuoyr9$ Wyowkjkuoyr hoy ldufydk` `ldfr

2$ Zkxfhvdbulodr hoy fsufydk` `ldfr


Xfyufs Hkbuoyr

F [FE u

f



• Fxfy| [FE xfyufs

f

β

boduylivufr k hkbuoy oh ljfβ ¶$

• jf lr k elmfdrlod`frr bovw`ldj bodrukdu kde lr yf`kufe uo ucf df-ruyvbuvyf

bodrukdu i|

λ =

j9f

5χ


Wyowkjkuoyr

• Fkbc ldufydk` wcouod boddfbur uzo xfyulbfr oh ucf hoym ljfβ ¶ kde ljfβ ν ro zf

rcovè fswfbu ucf wyowkjkuoy uo boduykbu ucf ldelbfr ¶ kde ν $



• Wcouod wyowojkuoy∙lj¶ν

{9

• Hfymlod wyowojkuoy lr k ilu moyf bomw`lbkufel#/{ ! m,

{9 ∙ m9

Ucf rljd oh { mkuufyr cfyf ― zf ukaf lu uo if ld ucf rkmf elyfbulod kr ucf

hfymlod kyyoz$

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf :

Fsufydk` @ldfr

• Rldbf iouc ucf xfyufs hkbuoy kde ucf hfymlod wyowkjkuoyr ldxo`xf 5 Ú 5

mkuylbfr ivu ucf kmw`luvef mvru if k rbk`ky ucf fsufydk` `ldf hkbuoyr mvru rlu



od ucf ovurlef$• Zoya ikbazkyer kòdj fxfy| hfymlod `ldf vrldj>

f∙ ld f∙ ovu f! ld f! ovu β ld β ovu

v v x x ǡ¶ ǡ√¶


Mkuyls f`fmfdur L

ho`òz hfymlod `ldfr ikbazkye uo jlxf v#9,ljβ ¶v#;,



f

f

β

n¶ = vβ ¶v lr krroblkufe zluc ucf f`fbuyod bvyyfdu

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf <

Mkuyls f`fmfdur LL

Ucf mkuyls f`fmfdu lr wyowoyulodk` uo ucf uzo bvyyfdur ld ucf elkjykm ifòz$



Qv2#ljfβ ¶,v;]

—∙lj¶ν

# w; ∙ w2,9

¡Qx9#ljfβ ν ,x5]

β

f∙, w; f!, w9

f∙, w2 f!, w5

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;7

Kde Hldk``|$$$

• Rufw 5>

Ucf oxfyk`` kmw`luvef lr ucf bocfyfdu rvm oh ucf ldelxlevk` kmw`luvefr hoy



w w

fkbc elkjykm>

M = M; ! M9 ! $ $ $

⇙ pM9p = pM; ! M9 ! $ $ $p9

• Rufw 5k>

Kdulr|mmfuyl~kulod

Ldb`vef k mldvr rljd ifuzffd elkjykmr ucku elhhfy od`| ld ucf fsbckdjf oh uzo

lefdulbk` hfymlodr$

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;;

Fskmw`fr

• Ucfyf kyf od`| k ckdehv` oh zk|r uo mkaf uyff-`fxf` elkjykmr ld [FE



• Ucfyf kyf od`| k ckdehv` oh zk|r uo mkaf uyff `fxf` elkjykmr ld [FE$

• Bodruyvbu kmw`luvefr hoy

Ickick rbkuufyldj#f! f∙ ↙ f! f∙,

Bomwuod rbkuufyldj#f β ↙ f β ,$

• @kufy zf zl`` vdefyukaf ucoyovjc bk`bv`kulodr hoy Mouu rbkuufyldj #f ℒ ↙ f ℒ,

wkly kddlcl`kulod #f! f∙ ↙ β β ,

hfymlod wkly-wyoevbulod #f! f∙ ↙ h h ,$


Fskmw`f> Ickick Rbkuufyldj

f∙, w2 f!, w5 f∙, w2 f!, w5



β

f∙, w; f!, w9

β

f∙, w; f!, w9

• Kdulr|mmfuyl~kulod ⇙ M = Mu ∙ Mr

Mu = l Qv2#ljfβ ¶,v;]

—∙lj¶ν

# w; ∙ w2,9

¡Qx9#ljfβ ν,x5]

Mr = l Qv2#ljfβ

¶

,x5]— ∙lj¶ν

# w; ! w9,9¡

Qx9#ljfβ

ν

,v;]


Fskmw`f> Bomwuod Rbkuufyldj

β, w2 f∙, w5 β, w2 f∙, w5



f∙, w;β, w9 f∙, w;

β, w9

• Do kdulr|mmfuyl~kulod ⇙ M = M; ! M9

M; = l

±v5#ljfβ ¶,

—l#/ w; ∙ / w2 ! m,

# w; ∙ w2,9 ∙ m9

¡#ljfβ ν,v;

‒ǡ√2νǡ9¶

M9 = l±

v5#ljfβ

¶

,— l#/ w; ! / w9 ! m,

# w; ! w9,9 ∙ m9¡

#ljfβ

ν

,v;‒

ǡ

√

2¶ǡ9ν


Wo`kyl~fe Wkyulb`fr

• K u|wlbk` [FE kmw`luvef mljcu òoa romfucldj `laf

M ∰ Qv;Β¶x9] ǡ2¶



Q ; 9] 2¶

Ucf Hf|dmkd yv`fr zod“u ukaf vr kd| hvyucfy ivu uo jfu k dvmify hoy M zf

zl`` dffe uo rviruluvuf fsw`lblu hoymr hoy ucf zkxfhvdbulodr oh ucf fsufydk`

wkyulb`fr> v; x9 kde ǡ2¶$

• Lh k`` fsufydk` wkyulb`fr ckxf k adozd wo`kyl~kulod uclr mljcu if k yfkrodki`fzk| uo bk`bv`kuf ucldjr$

• Moyf ohufd zf kyf ldufyfrufe ld vdwo`kyl~fe wkyulb`fr kr hfz kbbf`fykuoyr

wyoevbf wo`kyl~fe wkyulb`fr

Ld ucf ;<<7r ucf R@B ku R@KB wyoevbfe wo`kyl~fe f`fbuyod ku 67 JfX


Rwld-Kxfykjfe Kmw`luvefr

• Lh zf eo dou bkyf kiovu ucf wo`kyl~kulodr oh ucf wkyulb`fr ucfd zf dffe uo

;$ Kxfykjf oxfy ucf wo`kyl~kulodr oh ucf ldlulk`-rukuf wkyulb`fr



;$ Kxfykjf oxfy ucf wo`kyl~kulodr oh ucf ldlulk` rukuf wkyulb`fr

9$ Rvm oxfy ucf wo`kyl~kulodr oh ucf dk`-rukuf wkyulb`fr

ld ucf r{vkyfe kmw`luvef pMp9$

• Zf bk`` uclr ucf rwld-kxfykjfe kmw`luvef kde zf efdouf lu i|E

pMp9F

• Douf ucku ucf kxfykjldj oxfy ldlulk` rukuf wo`kyl~kulodr ldxo`xfr rvmmldj oxfy

k`` wo`kyl~kulodr kde ucfd elxleldj i| ucf dvmify oh ldefwfdefdu

wo`kyl~kulodr roE

pMp9F

ldxo`xfr k rvm oxfy ucf wo`kyl~kulodr oh k`` fsufydk`

wkyulb`fr$


Rwld Rvmr L

• @fu“r rlmw`lh| ucldjr fxfd hvyucfy kde rvwworf ucku zf ckxf

M ∰ Qv;Βv9]



Q ]

Ucfd pMp9 ∰ Qv;Βv9] Qv;Βv9]√

∰ Qv;Βv9]c

v‧;β 7Βv9

l‧

∰ Qv;Βv9] cv‧

9Β‧β 7‧v;l

∰ Qv;Βv9]c

v‧9β 7β 7Β‧β 7v;

l

∰ Qv;Βv9]ˈ

v9Βv;

ː

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;:

Rwld Rvmr LL

• Yfzyluldj ucf r{vkyfe mkuyls f`fmfdu

pMp9 ∰ Qv;Βv9]ˈ

v9Βv;

ː



• Zf bkd vrf ucf bomw`fufdfrr yf`kulod hoy v9v9

Srl=;,9

vrll v

rll = #/ wl ! ml,

• Rvmmldj oxfy ucf rwldr oh wkyulb`f 9 jlxfr

Sr9

pMp9 ∰ˈ

v;Β#/ w9 ! m9,Βv;

ː

∰ Qv;[v;]


Rwld Rvmr LLL

• Ucf yljcu-ckde rlef lr nvru k dvmify ivu lh zf yfwyfrfdu ucf mkuyls

mvùlw`lbkulod zluc rvmmkulodr oxfy ldelbfr zf bkd yfzyluf lu kr

Qv;[v;] = #v;, [ln #v;,



Q ;[ ;] # ;,l

[ln # ;,n

= [ln #v;v;,nl

= Q[ #v;v;,]ll

= Uy Q[#v;v;,]

• Kww`| ucf bomw`fufdfrr yf`kulod odbf kjkld ro ucku zf jfu

Sr;

pMp9 ∰ Uy Q[#/ w; ! m;,]

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;<

Rwld Rvmr LX

• Rukyuldj hyom

M ∰ Qv;Βv9]



Kxfykjldj oxfy ldlulk` rwldr kde rvmmldj oxfy dk` rwldr jlxfr

⇙E

pMp9F

∰;

9Uyˈ

Β#/ w9 ! m9,Β#/ w; ! m;,ː

• Wkyulb`fr ; kde 9 mk| oy mk| dou if ld ucf ldlulk` rukuf

• Ucf hkbuoy oh ;9

lr hyom ucf kxfykjldj oxfy ldlulk` rwldr krrvmldj fskbu`| odf

oh v; kde v9 boyyfrwoder uo kd ldlulk`-rukuf wkyulb`f$

• Lh ucf| kyf iouc ld ucf ldlulk` rukuf #f$j$ wkly kddlcl`kulod, ucf hkbuoy lr

;

5 $Lh dflucfy lr ld ucf ldlulk` rukuf #f$j$ wkly wyoevbulod, ucf hkbuoy lr ;$


Bkrlmly“r Uylba

• Uclr wyobfevyf oh bk`bv`kuldj rwld-kxfykjfe kmw`luvefr ld ufymr oh uykbfr lr



adozd kr Bkrlmly“r UylbaSk`` rwldr

QvkΒ;vi] QvkΒ9vi]√ = Uyˈ

Β;#/ wi ! mi,Β9#/ wk ! mk,ː

• Lh kdulwkyulb`f rwldoyr #x, kyf wyfrfdu ld ucf rwld rvm zf vrf ucfboyyfrwodeldj bomw`fufdfrr yf`kulod

Srl=;,9

xrll xrll = #/ wl ∙ ml,

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 9;

Uykbfr

• Ifbkvrf oh Bkrlmly“r Uylba zf“yf joldj uo de ovyrf`xfr bk`bv`kuldj k òu oh



uykbfr ldxo`xldj β -mkuylbfr$

• Jfdfyk` lefdululfr kiovu uykbfr>

Uy#K ! I, = Uy#K, ! Uy#I,

Uy#λK, = λUy#K,

Uy#KI, = Uy#IK,

Uy#KIB , = Uy#BKI, = Uy#IBK,


Jkmmk mkuylblfr kde uykbfr

• Ucf uzo mknoy lefdululfr ucku zf zl`` dffe ld oyefy uo ivlè moyf bomw`lbkufe

uykbf lefdululfr kyf

j¶νj¶ν = 5

¶ ν ¶ν



qβ , β } = 9j

• Rldbf β ¶β ¶ = 5 kde β ¶β νβ αβ ¶ = 5jνα$

Zf de ucku

β ¶β νβ ¶ = β ¶ #9j¶ν ∙ β ¶β ν,

= 9β ν ∙ β ¶β ¶β ν

= 9β ν ∙ 5β ν

= ∙9β ν


Rlmw`f Uykbf Lefdululfr

• Ucf rlmw`fru uykbf lefdulu| lr> Uy#;, = 5

• Ucf uykbf oh k rldj`f β mkuyls lr ~fyo

Ucf uykbf oh kd| oee dvmify oh β -mkuylbfr$

H 9 l



• Hoy 9 β -mkuylbfr

Uy#β ¶β ν, = Uy #β ¶β ν ! β νβ ¶, /9

= Uy#9j¶ν,/9

= j¶νUy#;,

= 5j¶ν

• Hoy 5 β uykbfr

Uy#β ¶β νβ αβ ς, = 5‐

j¶νjας ∙ j¶αjνς ! j¶ςjνα”


Uykbfr Zluc β

6

• Ucf xfyufs hkbuoy hoy zfka ldufykbulodr ldxo`xfr β 6$

• I| ldrwfbulod Uy#β 6, = 7$

Rl 6 l 7 ; 9 2



• Rldbf β 6 = lβ 7β ;β 9β 2

#kd fxfd dvmify oh β -mkuylbfr,

Uy#β 6β ¶, = 7

Uy#β 6β ¶β νβ α, = 7

• K`ro

Uy#β 6β ¶β ν, = 7


Ucf β 6 Uykbf

O ` luc 5 # , uc u l iu l u



• Od`| zluc 5 #oy moyf, oucfy β -mkuylbfr bkd zf oiukld k dod~fyo uykbfldxo`xldj β 6>

Uy#β 6β ¶β νβ αβ ς, = 5lǡ¶νας

zcfyf zf yfbk`` ucku ucf kdulr|mmfuylb ufdroy lr efdfe kr

ǡ¶νας ≫

344044>

∙; hoy fxfd wfymvukulodr oh 7;92

!; hoy oee wfymvukulodr oh 7;92

7 lh kd| 9 ldelbfr kyf ucf rkmf


Boduykbulodr oh ucf ǡ Ufdroy

• Rldbf ǡ¶νας lr bomw`fuf`| kdulr|mmfuylb zf zl`` jfu ~fyo zcfd zf boduykbu

uclr zluc kd| ufdroy ucku lr r|mmfuylb ld 9 ldelbfr rvbc kr j¶ν oy

# ¶ ν ! ¶ ν,



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• Od`| boduykbulodr zluc kdoucfy kdulr|mmfuylb ufdroy rvyxlxf>

ǡ¶ναςǡ¶νας = ∙95

ǡ¶νας

ǡ¶ναυ = ∙8 εςυ

ǡ¶ναςǡ¶νγυ = ∙9‐

εαγ εςυ ∙ εαυ εςγ

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$$$

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 9:

Fskmw`f ;

• Odf oh ucf uykbfr ldxo`xfe ld Ickick rbkuufyldj lr

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• Zf bkd fswkde uclr ovu uo byfkuf 5 ufymr ivu 9 oh ucfrf ufymr #ucf odfr `ldfky

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• Ucvr

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= 5g w¶; wν2 ! w¶2 wν; ∙ # w; º w2,j¶ν

µ! 5m9j¶ν

• Uclr yfrvù zl`` if boduykbufe zluc kdoucfy uykbf ucku lr boxkylkdu #l$f$ ¶ν krowworfe uo boduykxkylkdu ¶ν , ld ¶ kde ν $


Fskmw`f 9

• Bodrlefy

K = Uy#β ¶/ w;β ν/ w9, Uy#β ¶/ w;β ν/ w9,



• Fxk`vkuldj ucf uykbfr

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; w99 ! 9# w; º w9,9 ! 5# w; º w9,9 ∙ 5# w; º w9,9

ː= 29

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ː

Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 9<

F`fbuyod Rbkuufyldj



• F`fbuyod-Mvod Rbkuufyldj

#Mouu Rbkuufyldj,

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= ∙j9f

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f ¶ Rbkuufyldj LL

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f ¶ Rbkuufyldj LLL

Rlmw`lh| kde bomildf mkrr ufymr



Rlmw`lh| kde bomildf mkrr ufymr

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# ,

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‒


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kde

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ru

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R@KB




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= #M D ! ν, {,

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ν ℒ ℒ

zcfyf ucf χ Z ldufykbulod ku ucf iòi lr efrbylife i| ucf xfyufs hkbuoy

∙ljz∞ hχw¶χ



9∞

9h χ wχ

ucf wlod lr rwld 7 ro ucf ldufykbulod mvru if k rbk`ky #do jkmmk mkuylblfr,

kde lu mvru if k 5-xfbuoy #ucf rlmw`fru bcolbf lr w oh ucf wlod,


Ucf Wlod Efbk| Kmw`luvef L

• Zf bkd zyluf ucf wlod efbk| kmw`luvef lr

M =j9zh χ

3M 9Z

w;¶ˈ

v2β ¶#; ∙ β 6,x9ː

• Rldbf ucf wlod lr k rwld-7 wkyulb`f h χ bkd od`| efwfde od ucf wlod momfduvm

w;$

• Ucf od`| rbk`ky zf bkd mkaf hyom w; lr w9;

= m9χ ro h χ lr ld hkbu bodrukdu+

• Zf bk`` h χ ucf wlod efbk| bodrukdu kde fswfylmfdur rvjjfru ucku

h χ ≂ ;2; MfX



efufymldfe hyom χ! ↙ ¶!ν efbk|r #WEJ 977:,


Ucf Wlod Efbk| Kmw`luvef LL

M =j9zh χ

3M 9Z

w;¶ ˈv2β ¶#; ∙ β 6,x9ː

⇙EpMp9

F=

j9zh χ

3M 9Z

+9

w;¶ w;ν

Ú Uy

ˈβ ¶#; ∙ β 6,/ w9β ν#; ∙ β 6,#/ w2 ! mℒ,

ː=

j9zh χ

3M 9Z

+9

w;¶ w;ν Uyˈ

β ¶#; ∙ β 6,9/ w9β ν #/ w2 ! mℒ,ː

• Zf rukyu i| vrldj #; ∙ β 6,9 = 9#; ∙ β 6,$ Ucf mℒ ufymr zl`` dou boduylivuf kr

ucf| k`` ldxo`xf uykbfr oh kd oee " oh β -mkuylbfr$Ucf ǡ-ufdroy wyoevbfe i| ucf uykbf ldxo`xldj β 6 zl`` xkdlrc zcfd boduykbufe

zluc w;¶ w;ν $9



⇙ EpMp9

F = j9zh χ

3M

9

Z +9

w;¶ w;ν 3 ˈ w¶9

wν2 ! wν9 w¶2

∙# w9

º w2,j¶νː

Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf :

Ucf Wlod Efbk| Kmw`luvef LLL

EpMp9F =

;

3 j9zh χ

M 9Z +9

ˈ9# w; º w9,# w; º w2, ∙ m

9

χ# w; º w9,ː• Zf bkd fxk`vkuf ucf xkylovr eou wyoevbur i| vrldj w; = w9 ! w2>

w9;

= # w9 ! w2,9

m9χ = m9

ℒ ! 9# w9 º w2,

⇙ # w9 º w2, = #m9χ ∙ m9

ℒ ,/9

Rlml`ky`| # w; º w9, = #m9χ ∙ m9

ℒ ,/9

# w; º w2, = #m9χ ! m9ℒ ,/9

9



⇙ EpMp9F =;

;8 j9zh χ

M 9Z +

9

m9ℒ #m9

χ ∙ m9ℒ ,


Ucf Wlod Efbk| Ykuf L

•Yfbk`` hyom kd fky`lfy `fbuvyf

Β =pwh ppMp9

3χm9χ

• Zluc pwh p f{vk``ldj ucf dfvuyldo fdfyj|

pwh p = F 9

= # w; º w9,/mχ

= #m9χ

∙m9ℒ ,/9mχ

⇙ Β =h 9χ

2

—jz

¡5

m9ℒ #m9

χ ∙ m9ℒ ,9



⇙χm2

χ

—5M Z

¡ℒ # χ ℒ ,

Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf <

Ucf Wlod Efbk| Ykuf LL

Β =h 9χ

m2χ

—jz

5M Z

¡5m9ℒ #m9

χ ∙ m9ℒ ,9

• h χ bkd if fsuykbufe hyom uclr fswyfrrlod$• @fu“r bomwkyf ucf χ∙ efbk| ykufr uo f`fbuyodr kde mvodr ro kr uo bkdbf` h χ >

Β#χ∙ ↙ f∙ ! ν f,

Β#χ∙ ↙ ¶∙ ! ν ¶,=

m9f#m9

χ ∙ m9f,9

m9¶#m9

χ ∙ m9¶,9

≂ ;7∙5

Rvywylrldj`| ucf mvod moef lr cfkxl`| hkxoyfe ld rwluf oh ucf rmk``fy wckrf r

wkbf kxkl`ki`f$

Uc l h uc ` u e i e u e l u h `



• Ucf rvwwyfrrlod oh ucf f`fbuyod moef bkd if vdefyruooe ld ufymr oh kdjv`ky

momfduvm$

Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;7

Zcku Kiovu [vkyar?

• Hoy `fwuodr ucf Z bovw`fr zlucld k wkyulbv`ky jfdfykulod>7G ν f

f

;K

7G ν ¶

¶

;K

7G ν υ

υ

;K

• Ucldjr kyf moyf bomw`lbkufe hoy {vkyar kr ucf Z bovw`ldjr bkd mls

jfdfykulodr>

7Gv

e ;K 7Gb

r ;K 7Gu

i ;K• Zluc ucf elrboxfy| oh dfvuyldo mlsldj lu lmw`lfr ucfyf lr mlsldj ifuzffd ucf



jfdfykulodr

Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;;

Ucf BAM Mkuyls

• Ldrufke ucf Z bovw`ldj zoyar zlucld ucf zfka jfdfykulodr7

Gv

e‹

;

K

7

Gb

r‹

;

K

7

Gu

i‹

;

K• Ucf wylmfe {vkyar #zfka fljfdrukufr, kyf yf`kufe uo ucf vdwylmfe {vkyar

#mkrr fljfdrukufr, i| ucf Bkiliio-Aoik|krcl-Mkrakzk #BAM, Mkuyls>

7IIGe‹

r‹

i‹

;BBK =7IIG

X ve X vr X vi

X be X br X bi

X ue X ur X ui

;BBK7IIGe

r

i

;BBK




Zluc Od`| 9 [vkya Jfdfykulodr• Ucf 9 Ú 9 mlsldj mkuyls lr yfk``| nvru k rldj`f youkulod>

7G e‹

r‹ ;K = 7G bor γB rld γB

∙ rld γB bor γB ;K7G e

r ;Kzcfyf γB ≂ ;2◨ lr ucf Bkiliio kdj`f

• Uclr cf`wr vr vdefyrukde rfml`fwuodlb efbk|r rvbc kr A ∙ ↙ ℒ∙ ! ν ℒ$

• Ucf χ∙ efbk| ldxo`xfr k ve bovw`ldj =⇙ hkbuoy oh bor γBUcf A ∙ efbk| hfkuvyfr k vr bovw`ldj =⇙ hkbuoy oh rld γB $




χ∙ kde A

∙ Efbk|r L

Ucf ykulo oh ucf efbk| ykufr oh χ∙ ↙ `∙ν ¶ uo A ∙ ↙ `∙ν ¶ rcovè if wyowoyulodk`

uo ucf Bkilio kdj`f

∙ljz9∞ 9

β ¶#; ∙ β 6,borγB

v

e

Z

∙ljz9∞ 9

β ¶#; ∙ β 6,rldγB

v

r

Z

Β#χ∙ ↙ `∙ν ¶, =h 9χ

χm2χ

—jz

5M Z

¡5m9` #m9

χ ∙ m9` , bor9γb

5



Β#A ∙

↙`∙ν ¶, =

h 9A

χm2A — jz

5M Z ¡5

m9` #m9

A

∙m9` , bor9γb


χ∙ kde A

∙ Efbk|r LL

• Ucf ykulo oh ucf zleucr

Β#A ∙ ↙ `∙ν ¶,

Β#χ∙ ↙ `∙ν ¶,=

h 9Ah 9χ

m2χ

m2A

m9A ∙ m9

`

m9χ ∙ m9

`

+ukd9γb

•Ucf `lhfulmfr oh ucf χ∙ kde A ∙ kyf 9$87

Ú;7∙3 kde ;$95

Ú;7∙3 rfboder

Ucf iykdbcldj ykulor hoy χ∙ ↙ ¶∙ν ¶ lr ;77& kde A ∙ ↙ ¶∙ν ¶ lr 85&$

Oucfy mfkrvyfmfdur jlxf h χ = ;29 MfX kde h A = ;87 MfX$

• Vrldj ucf kioxf dvmifyr jlxfr Uclr jlxfr

γb = ;2$;◨ borγb = 7$<:5

• Ucf e-{vkya mkrr fljfdrukuf lr k mlsuvyf oh ucf zfka e kde r-{vkya fljfdrukufr



e‹ = 7$<:e ! 7$92r


Wyoi`fmr zluc ucf Bkilio moef`

• Ucf Bkiliio kdj`f ro`xfe mkd| wyoi`fmr ivu dfz odfr kwwfkyfe$

• Ucf iykdbcldj ykulo hoy A 7 ↙ ¶! ¶∙ lr <$; Ú ;7∙<

Ucf wyfelbufe efbk| ykuf lr wyowoyulodk` uo rldγbborγb zclbc jlxfr k

rljldbkdu`| `kyjfy yfrvù uckd ucf fswfylmfduk` yfrvù$

A 7 #er, rukuf

v

Z

ν ¶

Z

¶ ¶

M ∰ !rld γB bor γB



e r


Ucf JLM Mfbckdlrm

• J`krcoz L`lowovòr kde Mklkdl #JLM, wyoworfe k ro`vulod ld ;<:7 uo fsw`kld

ucf òz ldblefdbf oh A 7 ↙ ¶ ! ¶ efbk|r$

Uclr yf{vlyfe ucf woruv`kulod oh ucf bckym {vkya$

• Ldrufke oh fsbckdjldj k v {vkya k dfz {vkya #b-{vkya, lr fsbckdjfe

• Ucf bkdbf``kulod lr dou fskbu ifbkvrf ucf v kde b ckxf elhhfyfdu mkrrfr$

Odf bkd ldhfy ucf bckym {vkya mkrr i| mfkrvyldj k JLM-rvwwyfrrfe wyobfrr$

b

Z

ν ¶

Z

e r

¶ ¶

M ∰ ∙rld γB bor γB



e r

Ld ;<:5 ucf b-{vkya zkr elrboxfyfe$

Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;:

BAM mkuyls L• X ve Bomwkylrod oh χ! ↙ χ7f!ν f kde ¶! ↙ f!ν fν ¶

• X vr Bomwkylrod oh χ! ↙ χ7f!ν f kde A ! ↙ f!ν fν ¶

• X be Dfvuyldo ldevbfe bckym wyoevbulod hyom e-{vkyar

• X br Bckym efbk|r oh ucf Z iorod

• X bi I ↙ E`!ν efbk|r

• X vi i ↙ v`!

ν efbk|r

• X ui u ↙ i`!ν efbk|r




Zluc 2 [vkya Jfdfykulodr

• Ucf BAM mkuyls bkd if wkykmfuyl~fe ld ufymr oh 2 youkulod kdj`fr kde ;

BW-xlo`kuldj bomw`fs wckrf$

• Ucf kdj`fr kyf rvbc ucku ucf BAM mkuyls lr romfzcku elkjodk`$ Ucf

mkjdluvefr oh ucf mkuyls f`fmfdur kyf kwwyoslmkuf`|>

v↙

b ↙u ↙

7IIG7$<: 7$99 7$775

7$99 7$<: 7$75

7$7; 7$75 7$<<<

;BBKe ↛ r ↛ i ↛

• Mkd| fswfylmfdur ckxf oy kyf mfkrvyldj ucf BAM mkuyls f`fmfdur

Ucfyf kyf fsufdrlxf yfxlfzr oh fkbc oh ucf xk`vfr oh ucf BAM mkuyls



#rff ucf WEJ,

Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;<

Zo`hfdrufld BAM mkuyls

• Uclr wyoxlefr k moyf lduvlulxf xlfz oh ucf BAM mkuyls

• Ucf Zo`hfdrufld yfwyfrfdukulod oh ucf BAM mkuyls efdfr α = rldγb kde

fswyfrrfr ucf oucfy ufymr ld wozfyr oh α

9885 ; ∙ α9

/9 α Kα2

#ϋ ∙ lκ,∙ α ; ∙ α9/9 K α9

Kα2#; ∙ ϋ ∙ lκ, ∙K α9 ;

2::6Ucf wkykmfufyr K ϋ kde κ kyf yfk` dvmifyr$

• Ucf bomw`fs wckrf lr ucf od`| rovybf oh BW xlo`kulod ld ucf Rukdekye Moef`$

• Ucf kmovdu oh BW xlo`kulod ld ucf Rukdekye Moef` lr dou wkyulbv`ky`| `kyjf

kde bkddou fsw`kld ucf mkuufy-kdulmkuufy kr|mmfuy| ld ucf vdlxfyrf



w | |


Vdlukylu| Uylkdj`fr L

• Ucf vdlukylu| uylkdj`fr kyf vrfe uo xlrvk`l~f ucf yfrvùr hyom bomw`fmfduky|

fswfylmfdur

• Ucf vdlukylu| oh ucf BAM mkuyls mfkdr ucku ucf xkylovr yozr #kr zf`` kr ucf

bo`vmdr, kyf oyucodoymk` rldbf X X ‧ = ; `fker uo

7IIIIIIIG

qy o z ;}

qy o z 9}

qy o z 2}

;BBBBBBBK

7IIIIIIIIG

~}pq y

~}pq y

~}pq y

o o oz z z

; 9 2

√ pq~} √ pq~} √ pq~}

;BBBBBBBBK

=

7IIG ; 7 7

7 ; 7

7 7 ;

;BBK

• Ld i wc|rlbr zf vrf ucf ;ru #e, kde 2ye #i, bo`vmdr ro ucku

XveX √i ! XbeX √

i ! XueX √ui = 7



X veX vi ! X beX bi ! X ueX ui = 7

Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 9;

Vdlukylu| Uylkdj`fr LL

Ucf moru bommod`| vrfe vdluky| uylkdj`f kylrfr hyom

X veX √vi ! X beX √bi ! X ueX √ui = 7

Ucf rlefr kyf doymk`l~fe uo X ueX √ui

Ucf xfyulblfr oh ucf uylkdj`f kyf #7, 7,

#;, 7, kde #ϋ, κ,$

Lh ucf bomw`fs wckrf lr ~fyo #oy χ, ucf

vdlukylu| uylkdj`f zl`` ckxf ~fyo kyfk

BW xlo`kulod bkd if fruki`lrcfe flucfy i|

mfkrvyldj k`` ucyff rlefr oy ku `fkru odf



oh ucf kdj`fr


Vdlukylu| Uylkdj`fr LLL




Vdlukylu| Uylkdj`fr LX

K xkylfu| oh mfkrvyfmfdur wyoxlef bomw`fmfduky| ldhoymkulod ucku bkd `lmlu ucfxk`vfr ld ucf #ϋ, κ, w`kdf$

Jòik` ur uo ucf ekuk |lfè

α = 7$99:9

K = 7$3;3

ϋ = 7$99;

κ = 7$257




BW Xlo`kulod zluc `fwuodr

• BW xlo`kulod ld ucf vdlxfyrf bkddou fsw`kldfe i| BWX zf rff zluc {vkyar

Kdoucfy rovybf oh BWX bovè if ucf `fwuod rfbuoy #dodf |fu oirfyxfe,$

• RvwfyA kde RDO ckxf rcozd ucku dfvuyldo mls ν f ⇗⇙ ν ¶ kde ν f ⇗⇙ ν υ

zclbc mfkdr dfvuyldor ckxf mkrr kde `fwuod dvmify lr do bodrfyxfe

• Ucfyf mk| if k BAM-`laf mkuyls hoy dfvuyldor7IIG

ν f

ν ¶

ν υ

;BBK =

7IIG

X f; X f9 X f2

X v; X v9 X v2

X u; X u9 X u2

;BBK7IIG

ν ;

ν 9

ν 2

;BBK

zcfyf ν f º º º kyf ucf mkrr fljfdrukufr kde ν ; º º º kyf ucf zfka fljfdrukufr




Dfvuyk` Zfka Ldufykbulodr

• Ucf ^ 7 Iorod>

• Hf|dmkd Yv`fr

• Ucf Zfka Mlsldj Kdj`f

• Yfrodkdbf ld f!

f∙ Rbkuufyldj



Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;

Hfyml“r Ucfoy|

• Yfbk`` ucku Hfyml moef`fe ucf zfka ldufykbulod kr k hovy hfymlod bodukbu

ldufykbulod zluc k bovw`ldj oh JH >

h ; h 9

h 2 h 5

M =JH ∞

9

ˈh 2β ¶#; ∙ β 6,h ;

ː ˈh 5β ¶#;∙ β 6,h 9

ː ∰ JH F 9

• Hyom {vkduvm rbkuufyldj ucfoy| odf ckr k wyoi`fm lh pMp∰ ;

Uclr lr adozd kr ucf vdlukylu| iovde$

• Hoy ucf Hfyml moef` zf bkd rukuf ucku lu zl`` dou zoya if|ode F ∰ 277 JfX$

c ` e i i # , c c



Uclr lr yfro`xfe i| ldboywoykuldj k Z -iorod #M Z

∰37 JfX, lduo ucf ucfoy|

oh zfka ldufykbulodr

Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 9

Vdlukylu| Iovder L

• Fxfd khufy zf ldboywoykuf Z » iorodr lduo k ucfoy| oh ucf zfka ldufykbulod

kdoucfy vdlukylu| iovde lr fdbovdufyfe

• Ucf kmw`luvef hoy ucf f! f∙ ↙ Z ! Z ∙ wyobfrr lr k wyoi`fm krrvmldj ucku

lu wyobffer i| ucf Hf|dmkd elkjykm>

ν f

f∙ f!

Z ∙ Z !




Yfruoyldj Vdlukylu|

• Ld oyefy uo mkaf ucf zfka ldufykbulod rf`h-bodrlrufdu zf yf{vlyf uzokeelulodk` boduylivulodr uo ucf f! f∙ ↙ Z ! Z ∙ rbkuufyldj wyobfrr>

β

f∙ f!

Z ∙ Z !

^ 7

f∙ f!

Z ∙ Z !

• Zf yf{vlyf uzo dfvuyk` iorodr ucf β kde ucf ^ 7 uo s ucf vdlukylu| wyoi`fm




Elrboxfy| oh ucf ^ 7

• Ucf ^ 7 zkr ldelyfbu`| oirfyxfe ld ;<:2 ld ucf Jkyjkmf``f fswfylmfdu ku BFYD

xlk ucf wyobfrrfr

ν ¶ ! f ↙ ν ¶ ! f

ν ¶ ! D ↙ ν ¶ ! S

• Ucf hoyzkye-ikbazkye kr|mmfuy| #KHI, ld f! f∙ ↙ ¶! ¶∙ k`ro ldelyfbu`|

rcozfe ucf dffe hoy ucf ^ 7

• Ucf ^ 7 zkr elyfbu`| oirfyxfe ld ;<32 i| ucf VK; kde VK9 efufbuoyr ku BFYD

xlk wyouod-kdulwyouod bo``lrlodr$

• Ucf ^ 7 lr r`ljcu`| cfkxlfy uckd ucf Z » zluc

M = <;$;3:8#9;, JfX




Ucf Zfka Mlsldj Kdj`f

• Mkd| oh ucf wkykmfufyr oh ucf f`fbuyozfka ldufykbulod kyf yf`kufe uo fkbc oucfy$

Ucf mkrrfr kde bovw`ldjr kyf yf`kufe i| zfka mlsldj oy Zfldifyj kdj`f #γz,

• Hoy fskmw`f ucf mkrrfr oh ucf Z kde ^ iorodr kyf yf`kufe i|

M Z = M bor γz

• Ucf xfyufs hkbuoy #j^, hoy ucf ^ 7 lr yf`kufe uo ucf Z xfyufs hkbuoy #jZ ,

j~ = jz bor γz

• Iouc jz kde j~ kyf yf`kufe uo ucf [FE bovw`ldj bodrukdu jf

jz = jf

rld γzj~ = jf

rld γz bor γz

Uclr lr zc| ucf zfka hoybf lr ldcfyfdu`| ruyodjfy uckd ucf f`fbuyomkjdfulb

hoybf$



• Fswfylmfduk``| rld9 γz#M , = 7$92;97#;6,


Hf|dmkd Yv`fr hoy ucf ^ 7

• Ucf ^ 7 wyowkjkuoy òoar nvru `laf ucku oh ucf Z >

∙l—

j¶ν ∙ {¶{ν

M 9^

¡{9 ∙M 9

^

• Ucf ^ 7

iorodr mfelkuf dfvuyk` bvyyfdu #DB, zfka ldufykbulodr$Ucf| bovw`f uo hfymlodr xlk

∙lj~9

β ¶#bX ∙ bKβ 6,

h

h

^ 7



h

Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf :

Hfymlod Bovw`ldjr uo ucf ^ 7

• Ucf xfbuoy kde kslk` bovw`ldjr bX kde bK kyf rwfblfe i| ucf JZR moef`>

h bX bK

ν ℒ ! ;

9! ;

9

ℒ∙ ∙ ;

9! 9 rld9 γz ∙ ;

9

{v ! ;

9∙ 5

2rld9 γz ! ;

9

{e ∙ ;9

! 92

rld9 γz ∙ ;9

• Douf ucku ucf ^ 7 eofr dou bckdjf ucf `fwuod oy {vkya kxoy$ Ucf Rukdekye

Moef` ckr do kxoy-bckdjldj dfvuyk` bvyyfdur #HBDB, ku uyff `fxf`$




Jkvjf Iorod Rf`h-Bovw`ldjr

• Nvru `laf [BE ucf f`fbuyozfka iorodr bkd ldufykbu zluc fkbc oucfy>

Z ∙

Z !

^ 7, β

Z !

Z ∙

\

S

zcfyf #S, \ , bkd if #β, β , #β, ^ 7, #^ 7, ^ 7, oy #Z !, Z ∙,$

Bodrvù Kwwfdels E oh Jylhucr hoy xfyufs hkbuoyr$



Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf <

Bomwkyf β kde ^ 7 elkjykmr

•Ucf ^ 7 bovw`fr uo fxfy| bckyjfe hfymlod nvru `laf ucf wcouod eofr$

β ↙ h h ^ ↙ h h

•Ucf rlml`kylu| ifuzffd ucf β kde ucf ^ 7 mkef lu elhbvù uo efufbu ucf ^ 7

ifbkvrf ku òz fdfyjlfr$ Ucf [FE fhhfbur eomldkuf evf uo ucf #{9 ∙M 9^,∙;

hkbuoy ld ucf ^ wyowojkuoy$

• Hoy fskmw`f ucf hoyzkye-ikbazkye kr|mmfuy| ifbkmf moyf wyodovdbfe kr

ucf bfduyf-oh-mkrr fdfyj| ifbkmf `kyjfy

• Vd`laf ucf wcouod ucf ^ 7 k`ro bovw`fr uo dfvuyldor$

^ ↙ ν ν



Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;7

Fskmw`f> f! f∙ ↙ h h

• Zf doz ckxf k dfz elkjykm hoy ucf f! f∙ ↙ h h wyobfrr

^ 7

; > f∙ 9 > f!

2 > h 5 >¯

h



Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;;

Ucf Rbkuufyldj Kmw`luvef

• Ucf kmw`luvef lr

M= l ±v5 —

∙lj~

9

β ¶#bh X

∙bh K

β 6,¡ x2‒9885∙l

—j¶ν ∙ {¶{ν

M 9^

¡

{9

∙M 9

^

2::6

Ú±

x9

—∙lj~

9β ν#bfX ∙ bfKβ 6,

¡v;

‒

• Ku òz fdfyjlfr {9 ≮ M 9^ ucf ^ 7-mfelkufe elkjykm lr rlml`ky ld ruyvbuvyf uo

ucf [FE elkjykm$ Ucf β kmw`luvef zovè eomldkuf kr {∙9 ≡ M ∙9^




Rbkuufyldj kmw`luvef ku {9 =M 9^

> L

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Affwldj ucf hv`` wyowkjkuoy

M = ∙ j9~5#{9 ∙M 9

^,

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M 9^

wkyu oh ucf wyowkjkuoy zl``

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ucfd ucf /{ hkbuoyr `fke uo bomildkulodr `laf v5/ w5 kde / w2x2

zclbc i| ucf Elykb f{vkulod kyf v5m5 kde∙m2x2$




Rbkuufyldj kmw`luvef ku {9 =M 9^

> LL

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,

cv5β ¶#bh

X ∙ bh

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lÚ ˈ

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/ w5l

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β ¶#bfX ∙ bfKβ 6,/ w;β ν#bfX ∙ bfKβ 6,/ w9ː

• Ucf uykbfr kyf ifru fxk`vkufe i| yru iyldjldj ucf bX kde bK ufymr uojfucfy>

#bX ∙ bKβ 6,/ w2β ν#bX ∙ bKβ 6, = #bX ∙ bKβ 6,9/ w2β ν

= #b9X ! b9K,/ w2β ν ∙ 9bX bKβ 6/ w2β ν




Rbkuufyldj kmw`luvef ku {9 ≃M 9^

> LLL

• Ucf byorr rfbulod hoy ^ 7-mfelkufe f! f∙ ↙ h h lr

ς =;

2χ

j9~F

5Q#9F ,9

∙M 9

^ ]+9

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rbkuufyldj ifbkvrf uclr #^ 7, elxfyjfdbf bkd if uykbfe k`` ucf zk| ikba uo ucfkmw`luvef$




Wyowojkuoy hoy vdruki`f wkyulb`fr

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wc|rlbk``| k`òzki`f wyobfrr fxfd zlucovu k rvirf{vfdu efbk| uo h h $

• Kr k yfrvù zf dffe uo moelh| ucf ^ 7-wyowkjkuoy ld oyefy uo kbbovdu hoy ucf

ldrukil`lu| oh ucf ^ 7$ Cfyf“r zcku zf eo>

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wkyulb`f>Σ#y, u, = σ#y,f∙lFu

9$ Rldbf ucf wkyulb`f lr ruki`f ucf wyoikil lu| oh deldj ucf wkyulb`f

romfzcfyf lr k`zk|r f{vk` uo ; rldbf ucf zkxfhvdbulod l r doymk`l~fe>

W #u, =

^ pΣp9 e2y = ;




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W #u, =

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5$ Ld ucf wkyulb`f yfru hykmf uclr mfkdr ucku

Σ#y, u, = σ#y,f∙lMu∙Βu9

6$ Zf ucfd kww`| ucf rviruluvulod M ↙ M ∙ lΒ9

uo ucf wyowkjkuoy oh kd vdruki`f

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;

{9 ∙M 9↙ ;

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≂ ;

{9 ∙M 9 ! lM Β



Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;:

Byorr Rfbulod

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;

{9 ∙M 9^↙ ;

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ς ∰ ;

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zleuc Β^ $




Mfkrvyfmfdu oh ucf ^ 7 Wfka > L

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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;<

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f! f∙ ↙ h h

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ς^

ςβ≂ 9

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M ¡5

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dkufr dfky ucf yfrodkdbf wfka

ς^

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3

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Β^

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Dvmify oh `ljcu dfvuyldo jfdfykulodr

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uouk` zleuc Β^

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zl`` efwfde od ucf dvmify oh

#`ljcu, dfvuyldo jfdfykulodr

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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 9;

Ucf ^

7

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rcfe `ljcu od oucfy Rukdekye Moef` wkykmfufyr rvbc kr mu kde mC $

• Ld ucf fky`| ek|r ku @FW #rukyufe ld ;<3<, k dvmify oh vdvrvk` r|rufmkulbfhhfbur dffefe uo if kbbovdufe hoy ld oyefy uo mfkrvyf ucfrf wkykmfufyr

kbbvykuf`|>

;$ Ulek` elruoyulodr oh ucf yldj

9$ Zkufy `fxf`r ld dfkyi| @kaf Jfdfxk

2$ Boyyf`kulodr zluc ucf UJX




Zkufy `fxf`r ld dfkyi| @kaf Jfdfxk




Ulek` elruoyulodr oh ucf yldj




Boyyf`kulodr zluc ucf UJX




@FW-LL ZZ kde ^^ yfrvùr>







Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 9:





KU@KR Z kde ^ yfrvùr> Z ∙

↙ f∙

ν f



Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 9<

KU@KR Z kde ^ yfrvùr Z ∙ ↙ υ ∙ν υ




KU@KR Z kde ^ yfrvùr ^ 7↙ f

!

f∙



Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 2;

KU@KR Z kde ^ yfrvùr ^ 7 ↙ υ !υ ∙




KU@KR Z kde ^ yfrvùr




KU@KR Z kde ^ yfrvùr




F`fbuyozfka Vdlbkulod

• J`krcoz yfbojdl~fe ucku fxfd ucovjc ucf FM kde zfka ldufykbulodr kwwfky uo

if xfy| elhhfyfdu ucf| kyf mkdlhfrukulodr oh ucf rkmf hoybf

• Ucf kwwkyfdu elrwkylu| ld ucf ruyfdjucr oh ucf uzo ldufykbulodr bovè if

fsw`kldfe lh ucf zfka ldufykbulod zkr mfelkufe i| k mkrrlxf wkyulb`f #k hoybf

mfelkufe i| k mkrrlxf wkyulb`f ckr k rcoyu ykdjf,

• Zfldifyj kde Rk`km zfyf ki`f uo fsw`kld zc| ucf FM ckr k mkrr`frr mfelkuoy

kde ucf zfka hoybf ckr k mkrrlxf mfelkuoy zluc ucf Cljjr mfbckdlrm

#JZR zod ucf Doif` Wyl~f,

• Ucf ruyvbuvyf oh ucf FM kde zfka ldufykbulodr #lf$ coz zf bk`bv`kuf

Hf|dmkdd elkjykmr, lr k`ro xfy| elhhfyfdu



Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;

Cleldj ucf X ∙K

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9$ Ucf [FE ldufykbulod lr wvyf`| xfbuoy #β ¶, zcfyfkr ucf zfka ldufykbulod

bomildfr xfbuoy kde kslk` ufymr #β ¶#bX ∙ bKβ 6,,

• Zf bkd mkaf ucf X ∙ K xfyufs hkbuoy hoy ucf Z » òoa `laf k wvyf xfbuoyldufykbulod lh zf krroblkuf wkyu oh ucf ldufykbulod zluc ucf hfymlod

zkxfhvdbulod>

∙ljz∞9

±v

—β ¶

;∙ β 6

9

¡v

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9v



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vI

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;

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7

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ckxfr nvru `laf ucf

cf`lblu| owfykuoy º

i w$

• β 6 lr efdfe kr ucfbclyk`lu| owfykuoy kde lu lr od`| ld

ucf mkrr`frr `lmlu ucku cf`lblu|

kde bclyk`lu| kyf ucf rkmf$

• bclyk` yfhfyr uo ucf ckde-dfrr oh ucf wkyulub`f




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v@ ≫ ;9

#;∙ β 6,v = 30> 7 lh v ckr cf`lblu| !;

v lh v ckr cf`lblu| ∙;

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vY ≫ ;9

#; ! β 6,v =30> v lh v ckr cf`lblu| !;

7 lh v ckr cf`lblu| ∙;

•Kenoldu rwldoyr v@ kde vY?

v@ = v‧@β 7 = v‧ ;

9#;∙ β 6,β 7 = v‧β 7 ;

9#; ! β 6, = v ;

9#; ! β 6,

vY = v ;9

#;∙ β 6,




FM kde Zfka Bvyyfdur L• Yfbk`` ucf hoym oh ucf FM bvyyfdu

n¶FM

∰vβ ¶v

• Hoy ucf zfka bvyyfdu zf bkd zyluf

n¶zfka

∰vβ ¶ ;

9#;

∙β 6,v = vβ ¶ ;

9#;

∙β 6, ;

9#;

∙β 6,v

= v ;9

#; ! β 6,β ¶ ;9

#;∙ β 6,v

= v@β ¶v@

vrldj ucf lefdulu|

ˈ ;9 #;∙ β 6,ː9 = ;

5 Q;∙ 9β 6 ! #β 6,9] = ;9 #; ∙ β 6,

• Zf bkd uclda oh ucf bckyjfe zfka ldufykbulod kr k wvyf xfbuoy ldufykbulod

ifuzffd `fhu-ckdefe hfymlodr$




FM kde Zfka Bvyyfdur LL• Ucf [FE bvyyfdu vβ ¶v bkd if fswkdefe ovu lduo 5 bvyyfdur #v = v@ ! vY,

n¶fm ∰ vβ ¶v = #v@ ! vY,β ¶#v@ ! vY,

= v@β ¶v@ ! vYβ ¶vY ! v@β ¶vY ! vYβ ¶v@

• Rldbf;

9

#;∙

β 6, ;

9

#; ! β 6, = ;

5

Q;∙

#β 6,9] = 7

ucf @Y kde Y@ byorr ufymr ld ucf [FE bvyyfdu xkdlrc>

v@β ¶vY = v ;9

#; ! β 6,β ¶ ;9

#; ! β 6,v

= vβ ¶ ;9

#;∙ β 6, ;9

#; ! β 6,v

= 7

• Uclr mfkdr ucku od`| ucf @@ kde YY ufymr rvyxlxf>

n¶fm ∰ vβ ¶v = v@β ¶v@ ! vYβ ¶vY




FM kde Zfka bvyyfdur LLL

• Ucf bckyjfe zfka bvyyfdur kr mfelkufe i| ucf Z » bovw`f `fhu-ckdefehfymlodr uojfucfy>

n∙¶ = ν @β ¶f@

n!¶ = f@β ¶ν @

• Ucf f`fbuyomkjdfulb bvyyfdu kr mfelkufe i| ucf β bovw`fr `fhu-ckdefe

hfymlodr uojfucfy kde lu k`ro bovw`fr yljcu-ckdefe hfymlodr uojfucfy>

nfm¶ = ∙f@β ¶f@ ∙ fYβ ¶fY



Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf :

Zfka Eovi`fur

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lu rffmr dkuvyk` uo efdf ucf zfka eovi`fu>

ϊ@ =

7G ν f

f

;K@

• Ld ufymr oh ϊ@ ucf bckyjfe zfka bvyyfdur

n∙¶ = ν @β ¶f@ n!¶ = f@β ¶ν @

bkd if zyluufd kr n»¶ = ϊ@β ¶υ »ϊ@

zcfyf

υ ! ≫

7G 7 ;

7 7

;K υ ∙ ≫

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; 7

;K




K Dfvuyk` Bvyyfdu• Rvwworf zf efdf k uclye υ mkuyls ld oyefy uo bomw`fuf ucf r|mmfuy|>

υ 2

≫ 7G; 7

7 ∙; ;K• Hyom υ 2 zf bkd bodruyvbu k bvyyfdu

#zluc k hkbuoy oh ;9

hoy bodrlrufdb| zluc n»¶ ,>

n2¶ = ϊ@β ¶

;9

υ 2ϊ@

= ;9

ν @β ¶ν @ ∙ ;9

f@β ¶f@

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@C wkyulb`fr$

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bodrf{vfdu`| lu k`ro bovw`fr uo yljcu-ckdefe wkyulb`fr$



Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf <

C|wfybckyjf

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oh k {vkya oy ckeyod> [ = L 2

!;

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yf`kulod hoy ucf zfka c|wfybckyjf [ = L 2 ! ;9

\

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n\ ¶ = 9 nfm¶ ∙ 9 n2¶

= 9 #∙f@β ¶f@ ∙ fYβ ¶fY,∙ 9g

;9

ν @β ¶ν @ ∙ ;9

f@β ¶f@µ

= ∙9fYβ ¶fY ∙ f@β ¶f@ ∙ ν @β ¶ν @



Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;7

Zfka lrorwld kde c|wfybckyjf bvyyfdur

• Zfka lrorwld bvyyfdu n¶ = ;9

ϊ@β ¶υ ϊ@

zcfyfn»¶ boyyfrwode uo ucf

Z »-mfelkufe bvyyfdur kde

n2

¶ lr k `fhu-ckdefedfvuyk` bvyyfdu$

Ld ucf JZR moef` n¶ bovw`fr uo k uylw`fu oh xfbuoy iorodr Z¶ zluc k

bovw`ldj ruyfdjuc jz #∙ljz n¶ ºZ¶,

•C|wfybckyjf bvyyfdu n\ ¶ = 9 nfm¶

∙9 n2¶

zclbc bovw`fr zluc ruyfdjuc j‹

9uo k rldj`fu xfbuoy iorod I¶ #∙l

j‹

9n\ ¶ I¶,

• Ucf JZR moef` bomildfr ucf uzo bvyyfdur

∙l ±jz n¶

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j‹

9

n\ ¶ I¶

‒• Dodf oh ucf hovy fèr Z ; Z 9 Z 2 kde I boyyfrwode elyfbu`| uo ucf wc|rlbk`

wkyulb`fr Z ! Z ∙ ^ 7 kde β $



Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;;

Z ! kde Z ∙

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¶Z ¶2

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¶

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9 g n;¶

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n!¶ Z ¶! ! ;∞

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Z »¶ ≫ ;∞ 9gZ ;¶ − lZ 9¶µ

zclbc boyyfrwoder uo ucf zkxf hvdbulodr yfwyfrfduldj ucf Z » wkyulb`fr




Z ! kde Z ∙ Xfyufs Hkbuoyr

• Hyom ucf jfdfyk` ldufykbulod

∙l ±jz n¶ ºZ¶

!

j‹

9 n\

¶ I¶‒

zf rff ucku ucf bovw`ldj ldxo`xldj ucf Z ∙ lr

∙ljz∞9

n∙¶ Z ¶∙

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= νβ ¶;9

#;∙ β 6,f

zf de ucku ucf Z ∙ bovw`fr uo kd f kde kd ν f zluc k xfyufs hkbuoy oh

∙ljz

9∞

9β ¶#;∙ β 6,




Zfka mlsldj

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2

kde I mls ld ucf JRZ ucfoy|7G K¶

^ ¶

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zcfyf γz lr ucf zfka mlsldj kdj`f

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∙l

±jz n2

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9n\ ¶ I¶

‒= ∙l

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¶ !j‹

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‒K¶

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‒^ ¶




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Lh K¶ lr uo yfwyfrfdu ucf f`fbuyomkjdfulb fè ucfd

nfm

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z n2

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z n\

¶ ‒=

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¶

µː

⇙jf = j‹ bor γz = jz rld γz




^ 7 bovw`ldj bodrukdu

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∙l

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¶ ∙j‹

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rld γz

¡ n2¶

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n2¶,‒^ ¶

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ː^ ¶

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j~ =jf

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bX kde bK

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98885

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2:::6

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ucf hoym ∙lj~9

#bX ∙ bKβ 6,$

Wc l 678K ;: F ` a l l W ;:



Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;:

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h bX bK

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Cljjr wyoevbulod




Elyfbu mfkrvyfmfdu ku Ufxkuyod




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Eovi`f ι Efbk|

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Efufbuldj dfvuyldor

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b`fkd uo f`lmldkuf ucf ykelokbulxf ikbajyovder

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Wc|rlbr 678K ;< - Dfvuyldor Wkjf ;<






Wc|rlbr 678K ;< - Dfvuyldor Wkjf 9;



Dfvuyldo Mlsldj

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pν #7, = pν f

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Dfvuyldo orbl``kulod fswfylmfdur

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• Kumorwcfylb dfvuyldor> RvwfyA

Kyulblk` dfvuyldo rovybfr

• Yfkbuoy dfvuyldor> Bcoo~ #; am, Akm@KDE #;37 am,

• @odj-ikrf`ldf kbbf`fykuoy fswfylmfdur> A9A #967 am, MLDOR #:26 am,




Dfvuyldo mlsldj mkuyls

Wc|rlbr 678K ;< - Dfvuyldor Wkjf 9<



Ro`ky dfvuyldo yfrvùr

rld9γ;9 = 7$275!7$799∙7$7;8 ∈m9

9; = :$86!7$92∙7$97 Ú ;7∙

6fX 9




Kumorwcfylb dfvuyldo yfrvùr

rld9γ92 = 7$67!7$7:∙7$78 ∈m9

2; = 9$57!7$;9∙7$;; Ú ;7∙2fX 9

Wc|rlbr 678K ;< - Dfvuyldor Wkjf 2;



Iovde odγ;2




Bvyyfdu Rukuvrrld9γ;9 = 7$275!7$799

∙7$7;8 ∈m99; = :$86!7$92

∙7$97 Ú ;7∙6fX 9

rld9γ92 = 7$67!7$7:∙7$78 ∈m9

2; = 9$57!7$;9∙7$;; Ú ;7∙2fX 9

rld9γ;2 0 7$26 #<7& B@,




U9a Fswfylmfdu




U9a Fswfylmfdu


Documents

QED and Particle Physics