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7/3/2019 QED and Particle Physics.
http://slidepdf.com/reader/full/qed-and-particle-physics-5584618d78ab1 1/426
WC\R 678K
Wkyulb`f Wc|rlbr L
• Vdefyrukde ucf ruyvbuvyf oh ucf Rukdekye Moef`
• Bk`bv`kuf wc|rlbk` oirfyxki`fr vrldj Hf|dmkd elkjykmr
• R|mmfuylfr ld wkyulb`f wc|rlbr
• Boddfbulod ifuzffd ucfoy| kde fswfylmfdu
Jykeldj>
• 67& Krrljdmfdur
• 67& Hldk` fskm
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;
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Ucf Rukdekye Moef`• 2 hvdekmfduk` hoybfr #ruyodj zfka f`fbuyomkjdfulb, ld ufymr oh [vkduvm
Hlf`e Ucfoy|
Ilj Rmk``
R`oz B`krrlbk` Mfbckdlbr [vkduvm Mfbckdlbr
Hkru Yf`kulxlrulb Mfbckdlbr [vkduvm Hlf`e Ucfoy|
• Mkuufy lr mkef vw oh hfymlodr #rwld ;9 29 $$$ , {vkyar kde `fwuodr
• [vkyar>
7G v
e
;K7G b
r
;K7G u
i
;K
• @fwuodr>
7G f
ν f
;K7G ¶
ν ¶
;K7G υ
ν υ
;K
Wc|rlbr 678K ; - Lduyoevbulod Wkjf 9
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Mkuufy L
• Ucf {vkyar kde `fwuodr kyf oyefyfe ld jfdfykulodr$
Bodrf{vfdbf oh ovy oirfyxkulodr kde fsw`kldfe i| ucf Rukdekye Moef`$
• Uykdrlulodr ifuzffd jfdfykulodr kyf zfka oy xfy| ykyf$
Hoy fskmw`f zf ckxf dou oirfyxfe Z ∙ ↙ f∙ν ¶$
•
Ucf {vkya rfbuoy lr moyf bomw`lbkufe$Zf oirfyxf mkrr ykucfy uckd zfka fljfdrukufr kde ucf| kyf mlsuvyfr oh ucf
zfka fljfdrukufr$ Zf bkd rff Z ∙ ↙ vr efbk|r$
• Yfbfdu oirfyxkulodr oh dfvuyldo orbl``kulodr rvjjfrur k rlml`ky wkuufyd hoy
`fwuodr$
Ucf Z ∙ ↙ f∙ν ¶ lr k``ozfe ivu ucf mlsldj lr mvbc `frr kde uclr efbk| lr
wyfelbufe uo if ro rmk`` lu zl`` `laf`| dfxfy if oirfyxfe$
Wc|rlbr 678K ; - Lduyoevbulod Wkjf 2
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Mkuufy LL
Ldufyfruldj ucldjr uo douf
• Ucfyf kyf od`| 2 `ljcu dfvuyldo rwfblfr$
• Dfvuyldor ckxf mkrr ivu zf od`| adoz ucf mkrr elhhfyfdbf ifuzffd ucf 2
rwfblfr ν f mkrr `lmlu lr mfkrvyfe uo if kyovde 6 fx
• K`` {vkyar bkd hoym mfrodr #9-{vkya rukufr, oy ckeyodr #2-{vkya rukufr,$
Ucf uow efbk|r uo {vlba`| ifhoyf lu bkd ilde zluc kdoucfy {vkya
• [vkyar dou oirfyxfe rldj`| ld dkuvyf ivu ld 9 kde 2 {vkya rukufr$
Ckeyodr 2 {vkya r|rufmr #{;{9{2, f$j$ dfvuyod wyouodMfrodr 9 {vkya r|rufmr #{{, f$j$ χ A
Wc|rlbr 678K ; - Lduyoevbulod Wkjf 5
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Hoybfr
• Hoybfr kyf mfelkufe i| iorodr #rwld 7;9 $$$,
• Ucf f`fbuyomkjdfulb hoybf lr mfelkufe i| ucf wcouod #β ,
• Ucf zfka hoybf lr mfelkufe i| ucf Z » kde ^ 7 iorodr
M Z = 37$5 JfX kde M = <;$9 JfX
• Ucf ruyodj hoybf lr mfelkufe i| j`vodr #j,
• Hldk``| ucf Cljjr iorod lr yfrwodrli`f hoy wyoxleldj ucf Z kde ^ zluc mkrr$
M C 4 ;;5 JfX
Wc|rlbr 678K ; - Lduyoevbulod Wkjf 6
7/3/2019 QED and Particle Physics.
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Ucf F`fbuyomkjdfulb Hoybf
• Wcouodr bovw`f uo kd| wkyulb`f zluc f`fbuylb bckyjf
f
f
β
• Wfyuvyikulod ucfoy| bkd if vrfe uo bk`bv`kuf ucf ldufykbulodr ykufr lh ucf
ldufykbulod ruyfdjuc lr dou ruyodj
λ ≃ ;;2:
Wc|rlbr 678K ; - Lduyoevbulod Wkjf 8
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Fskmw`f> f!f∙ rbkuufyldj #Ickick,
β
f∙
f!
β
f∙
f!
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Ucf Ruyodj Hoybf
• J`vodr bovw`f uo kd| wkyulb`f zluc bo`oy bckyjf
• [vkyar bkyy| bo`oy bckyjf$$$
{
{
j
• Ruyodj ldufykbulod λR ≃;:
Wc|rlbr 678K ; - Lduyoevbulod Wkjf 3
7/3/2019 QED and Particle Physics.
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• $$$ kde ro eo j`vodr+
j
j
j
j
j
j
j
Wc|rlbr 678K ; - Lduyoevbulod Wkjf <
7/3/2019 QED and Particle Physics.
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Ucf Zfka Hoybf
•
Ucf Z kde ^ iorodr bovw`f uo fxfy|ucldj #`fwuodr kde {vkyar,• Ucf ^ 7 #k$a$k$ dfvuyk` bvyyfdu, kbur k `ou `laf ucf wcouod
h
h
^
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;7
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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 ;;
7/3/2019 QED and Particle Physics.
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• 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
{v
{e
Z
• Douf ucku `fwuod kxoy dvmify lr k`zk|r bodrfyxfe ivu ucf Z mlsfr {vkyar
hyom elhhfyfdu jfdfykulodr #kxovy bckdjf,
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;9
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• 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 ∙,
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;2
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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
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;5
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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 ivl`u 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 bo`oy
• @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,$
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;6
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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$
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;8
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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
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Bomworlulod oh ucf vdlxfyrf
Zf eo dou vdefyrukde ovy vdlxfyrf$
RVR\ mk| |lf`e ucf ekya mkuufy bkdelekuf$
Wc|rlbr 678K ; - Lduyoevbulod Wkjf ;3
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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 `odf 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 ;<
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Fskmw`f
• Mvod `lhfulmf ld dkuvyk` vdlur>υ ¶ =
;9#3χ,2M 5Z
j5Z m6¶
• Rufw ;> Mv`ulw`| mkrrfr i| b9 uo bodxfyu ucfm uo fdfyjlfr⇙ ;/b9
• Rufw 9> Zluc υ ¶ doz ld MfX∙; mv`ulw`| 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∙;
• Mv`ulw`| i| #b, ≂ ;<: MfX hm ⇙ υ ¶ = 8$8Ú ;7;: hm
• Elxlef i| b ≂ 2Ú ;792 hm/r ⇙ υ ¶ = 9$9Ú ;7∙8 r
Wc|rlbr 678K ; - Lduyoevbulod Wkjf 97
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Aldfmkulbr• @oyfdu~ Uykdrhoymkulodr
• Hovy-Xfbuoyr
• Fdfyj| Momfduvm kde Mkrr
• Bo``lrlodr
• Fskmw`fr
Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;
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@oyfdu~ Uykdrhoymkulodr• Yf`kuf booyeldkufr ld> R
x∙↙ R ‹
• Efylxfe hyom ucf woruv`kufr oh yf`kulxlu|
• Hoy moulod k`odj 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 b`obar yvd r`oz> U = βU ‹
Wc|rlbr 678K 9 Aldfmkulbr Wkjf 9
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Kww`lbkulod> Bormlb Yk| Mvodr
Zluc υ ¶ = 9$9 ¶r k mvod wyoevbfe ld ucf vwwfykumorwcfyf bov`e 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``ozr k u|wlbk` mvod uo
uykxf` 8$8 am ifhoyf efbk|ldj$
Efbk| `fdjuc ld `kioykuoy| hykmf lr βbυ
Wc|rlbr 678K 9 Aldfmkulbr Wkjf 2
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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$j m9 = w º w = w¶ w¶ = # w7,9 ∙ # w;,9 ∙ # w9,9 ∙ # w2,9
Wc|rlbr 678K 9 Aldfmkulbr Wkjf 5
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Fdfyj|, Momfduvm, kde Mkrr
• Yf`kulxlrulb fdfyj| lr F = βm
• Yf`kulxlrulb momfduvm lr w = βmx
• Efdf ucf hovy-momfduvm i| w¶ = #F,w,$
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$
Wc|rlbr 678K 9 Aldfmkulbr Wkjf 6
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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$
Wc|rlbr 678K 9 Aldfmkulbr Wkjf 8
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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 :
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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
Wc|rlbr 678K 9 Aldfmkulbr Wkjf 3
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• 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$2 JfX
Wc|rlbr 678K 9 Aldfmkulbr Wkjf <
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Hlsfe Ukyjfur xr$ 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| zov`e 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
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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 ;;
7/3/2019 QED and Particle Physics.
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Kr|mmfuylb ifkmr hoy I wc|rlbrUcf kr|mmfuylb ifkm fdfyj| yfrv`ur ld ucf II mfrod r|rufm ifldj ioorufe hyom
ucf bfduyf oh mkrr$
Uclr k``ozr odf uo mfkrvyf ucf elhhfyfdbf ld ulmf ifuzffd ucf efbk| oh fkbc Imfrod$
Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;9
7/3/2019 QED and Particle Physics.
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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
Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;2
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• 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
Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;5
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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$
Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;6
7/3/2019 QED and Particle Physics.
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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 ;$
Wc|rlbr 678K 9 Aldfmkulbr Wkjf ;8
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Bckddf`r
• Hoy K ! I ∙↙ K ! I rbkuufyldj ucf Mkdef`rukm ldxkylkdur r u kde v kyf
yf`kufe uo 2 elruldbu uowo`ojlbk` 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 ;:
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R|mmfuylfr
• Bodrfyxkulod @kzr
• Ikrlb Jyovw Ucfoy|
• Kdjv`ky Momfduvm
• H`kxoy R|mmfuylfr
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;
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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|`oy 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
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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 yfrv`u 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
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf 2
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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,
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf 5
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Ikrlb Jyovw Ucfoy|
• K jyovw J lr k rfu oh f`fmfdur zluc k ildky| bomworlulod `kz #l$f$ k
‐mv`ulw`lbkulod”, rvbc ucku>
;$ B`orvyf> ∇ 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 mv`ulw`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`$
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf 6
7/3/2019 QED and Particle Physics.
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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 rlmv`ukdfovr 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 mv`ulw`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,
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf 8
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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 :
7/3/2019 QED and Particle Physics.
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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,
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf 3
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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 <
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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
7/3/2019 QED and Particle Physics.
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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 ;;
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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
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;9
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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
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;2
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BJ Fskmw`fr LL
ˑˑˑˑ;
9
;
9 = y ;
2
p;; ˑˑˑˑ;
9
∙;
9!y 9
2
p;7 ˑˑˑˑ;
9
;
9
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;5
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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 @
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;6
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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$
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;8
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Lrorwld Krrljdmfdur
• Hoy kd| ckeyodr mkef vw oh v kde e {vkyar bodruyvbu lrorwld mv`ulw`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 ;:
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χw Byorr Rfbulodr
;5 57$ W `our 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$`i`$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
Wc|rlbr 678K 2 - R|mdmfuylfr Wkjf ;3
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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 ;
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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
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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 mv`ulw`lbkulxf {vkduvm dvmify$
Uclr lr uyvf hoy k`` elrbyfuf r|mmfuylfr$
Boduldvovr r|mmfuylfr ckxf keelulxf {vkduvm dvmifyr$
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 2
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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 `oz-mkrr mfrodr$
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 5
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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
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 6
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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 zov`e 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
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 8
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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``ozfe 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 :
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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``ozfe #kde oirfyxfe,
χ7 ↙ 9β lr dou k``ozfe #kde dou oirfyxfe 0 2$; Ú ;7∙3,
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 3
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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
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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 rcov`e 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
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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 Hlf`e 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``ozfe ld dkuvyf
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;;
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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$
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;9
BW Xlo`kulod ld ucf Akod Rfbuoy LL
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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 yfrv`u 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 = ∙;,$
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;2
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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
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;5
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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
• Zlucld ucf `kru hfz |fkyr ucf IkIky kde If``f fswfylmfdur ckxf mfkrvyfe BW
xlo`kulod ld ucf I-mfrod rfbuoy$
• BW xlo`kulod rcov`e k`ro if oirfyxki`f ld ucf E-mfrod #bckym, rfbuoy ucovjcuclr zl`` if k rmk`` fhhfbu ucku zl`` if xfy| elhbv`u uo mfkrvyf$
• BW xlo`kulod oirfyxfe ld ucf A kde I mfrodr lr dou fdovjc uo fsw`kld ucf
eomldkulod oh mkuufy ld ucf vdlxfyrf
• Zluc ucf oirfyxkulod ucku dfvuyldo ckr mkrr lu lr fswfbufe ucku zf zl``oirfyxf BW xlo`kulod ld ucf dfvuyldo rfbuoy
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;6
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Zc| ruve| BW xlo`kulod?• Rkakyox woldufe ovu ucku lu lr worrli`f uo rukyu hyom k mkuufy-kdulmkuufy
r|mmfuylb vdlxfyrf kde fde vw ld odf ucku lr kr|mmfuylb
Uclr yf{vlyfr ucku ucfyf if romf wyobfrr #oy wyobfrrfr, ucku xlo`kufr ucf BW
r|mmfuy|$
• Ucf RM eofr dou wyfelbu BW xlo`kulod #lu bkd kbbomoekuf do BWX oy BWX,$
Cozfxfy ucf RM wyoxlefr od`| odf rovybf oh BW xlo`kulod #BAM wckrf kdj`f,
zclbc lr od`| worrli`f lh ucfyf kyf moyf uckd 2 {vkya jfdfykulodr$
• Ucf bvyyfdu`| oirfyxfe RM BWX #vrldj A kde I mfrodr, lr uoo rmk`` uo
fsw`kld ucf mkuufy-eomldkufe vdlxfyrf$
• Ucfyf lr kdoucfy worrli`f rovybf oh BWX zlucld ucf RM ld ucf dfvuyldo rfbuoy
kdk`ojvovr uo ucf BAM mkuyls$• If|ode ucf RM> BW xlo`kulod lr vil{vluovr ld ucfoylfr oh Dfz Wc|rlbr
⇙ RVR\ bkd wyoxlef fdovjc BW xlo`kulod uo if oirfyxki`f ku `oz fdfyjlfr$
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;8
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Bomworlulod oh ucf vdlxfyrf
Ucf vdlxfyrf lr moyf bomw`lbkufe kde
lu lr oirfyxfe ucku od`| k rmk`` hykbulod#5&, oh ucf vdlxfyrf lr mkef oh adozd
mkuufy$
Oirfyxkulodr k`ro rcoz ucku ucfyf lr
moyf mkuufy #ekya mkuufy, hyom vd-
adozd rovybfr #9;&,$
Ld keelulod ucfyf lr kdoucfy bomwodfdu
#ekya fdfyj|, zclbc zf adoz fxfd `frrkiovu #:6&,$
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;:
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BW Xlo`kulod ld υ efbk|r
Ucf jok` lr uo rfkybc hoy elyfbu BW xlo`kulod ld ucf efbk| υ ∙ ↙ c∙A 7R
g≦ 7χ7µ
ν υ
Ucfyf lr do bckyjf kr|mmfuy| ifuzffd υ ∙ ↙ χ∙A 7ν υ kde υ ! ↙ χ!A 7ν υ
Fswfylmfduk``| zf oirfyxf ucf A 7R
kde A 7@
mfrodr zclbc kyf `ldfky bomildkulodr
oh A 7 kde A 7 rukufr$
Fky`lfy fswfylmfdur ckxf rcozd ucku ucf A 7R
kde A 7@
kyf dou fskbu BW fljfdrukufr$
Kr k yfrv`u ucf bckyjf kr|mmfuy| lr
K[ =Βg
υ ! ↙ χ!A 7R
ν υ µ∙ Β
gυ ∙ ↙ χ∙A 7
Rν υ
µ
Βg
υ ! ↙ χ!A 7R
ν υ µ
! Βg
υ ∙ ↙ χ∙A 7R
ν υ µ
Ucfoy| wyfelbur K[ = #2$2 » 7$;, Ú ;7∙2
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;3
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Ulmf Yfxfyrk` R|mmfuy|
• Ulmf yfxfyrk` r|mmfuy| kr |ov mljcu jvfrr yfxfyrfr ucf ulmf bomwodfdu>
U #u,s, = U #∙u,s,
• K`ucovjc zf fswfbu ucf zfka ldufykbulod uo xlo`kuf U elyfbu U xlo`kulod ckr
dou iffd efdlulxf`| oirfyxfe |fu$
• Fswfylmfduk``| odf uylfr uo mfkrvyf ucf ykuf oh k yfkbulod ld iouc elyfbulodr
K ! I
↙B ! E ivu uclr lr dou ro fkr|
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf ;<
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Ucf BWU Ucfoyfm
• Ucf bomildkulod BWU lr k`zk|r bodrfyxfe ld kd| `obk` {vkduvm f`e ucfoy|$
BWU xlo`kulod ld frrfdulk``| r|dod|movr zluc k xlo`kulod oh @oyfdu~ ldxkylkdbf
• BW U r|mmfuy| mkdekufr ucku wkyulb`fr kde kdulwkyulb`fr mvru ckxf bfyukld
lefdulbk` wyowfyulfr rvbc kr ucf rkmf mkrr `lhfulmf bckyjf kde mkjdfulb
momfdu
• Rff ucf WEJ rvmmky| uki`fr>
cuuw>//wej$ i`$jox/97;;/uki`fr/yww97;;-bodrfyxkulod-`kzr$weh
Fskmw`fr #hykbulodk` elhhfyfdbfr,>
M #A 7,∙ M #A 7, 0 3 Ú ;7∙;<
M #f!,
∙M #f∙, 0 3
Ú;7∙<
υ #¶!, ∙ υ #¶∙, 0 9 Ú ;7∙6
j#f!, ∙ j#f∙, 0 ;7∙;9
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 97
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@fwuod Dvmify• Ucfyf kyf 2 `fwuod dvmifyr> @f @¶ kde @υ
@f = !; hoy f∙ kde ν f
@f =
∙; hoy f! kde ν f
• Bodrfyxfe ld ucf FM kde Zfka ldufykbulodr β ↙ f!f∙ kde χ! ↙ ¶!ν ¶ kyf
k``ozfe zcfyfkr ¶! ↙ f!β lr hoyileefd
• Dfvuyldo orbl``kulodr lmw`| ucku `fwuod dvmify lr xlo`kufe #ku k xfy| rmk``
`fxf`,
• Rff ucf WEJ rvmmky| uki`fr>
cuuw>//wej$ i`$jox/97;;/uki`fr/yww97;;-bodrfyxkulod-`kzr$weh
Fskmw`fr #hykbulodk` elhhfyfdbfr,>
Β#¶∙ ↙ f∙β , 0 ;7∙;;
Β#^ 7 ↙ f∙¶!, 0 ;7∙8
Β#υ ∙ ↙ f∙f∙f!, 0 ;7∙3
Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 9;
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Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 99
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Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 92
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Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 95
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Wc|rlbr 678K 5 - Elrbyfuf R|mmfuylfr Wkjf 96
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Oirfyxki`fr
Zf zkdu uo yf`kuf fswfylmfduk` mfkrvyfmfdur uo ucfoyfulbk` wyfelbulodr
• Efbk| zleucr kde `lhfulmfr
Β = c/υ #vdlur oh fdfyj|,
• Byorr Rfbulodrς lr ucf uouk` byorr rfbulodeςeℨ
lr ucf kdjv`ky elruylivulodeςeF
lr ucf fdfyj| elruylivulod
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;
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@lhfulmf oh kd Vdruki`f Wkyulb`f
• Ucf efbk| ykuf Β yfwyfrfdur ucf wyoikil`lu| wfy vdlu ulmf oh ucf wkyulb`f
efbk|ldj>
eD = ∙ΒD eu
⇙ D #u, = D #7, f∙Βu
• Ucf efbk| ykuf efufymldfr ucf #mfkd, `lhfulmf oh ucf wkyulb`f>
υ =;
Β
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf 9
Iyflu-Zljdfy Yfrodkdbf
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j
Zkxfhvdbulod hoy k wkyulb`f zluc yfru fdfyj| F Y kde zleuc Β lr
Σ#u, = Σ#7,f∙l#FY∙lΒ/9,u
Hovylfy uykdrhoym ϊ#F , ∝ ;
#FY∙
F,∙
lΒ/9
Iyflu-Zljdfy hoymv`k > pϊ#F ,p9 ∝ ;#FY∙F,
9!Β9/5
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf 2
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Fskmw`f L
OWK@;<<9 ! ;<<2
ekuk
efbk| `fdjuc #bm,
d o $
e f b k | r / 7 $ 7
6 b m
977
577
877
377
;777
;
;7
;7 9
;72
-9 -; 7 ; 9 2 Ukv `fwuod `lhfulmf 933 hr
Efbk| `fdjuc = βbυ ≃ 9$6 mm
Ku @FW ucf ukv ckr w ≃ 67 JfX
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf 5
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Fskmw`f LL
∞[9
ς
# d i k y d r ,
O@\K92
BME99
EM;95
EM9;3
∞[9
ς # d
i k y d r ,
7
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577
877
377
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;977
;577
;877
277 577 677 877 :77 377 <77 ;777
;7-;
;
;7
;79
377 ;777 ;977 ;577 ;877 ;377 9777 9977
f!f∙ ↙ χ!χ∙ byorr rfbulod kr k hvdb-
ulod oh ucf f!f∙ bfduyf-oh-mkrr fdfyj|
ϋ#::7, yfrodkdbf
Β = ;6; MfX
υ = c/Β = 5Ú ;7∙95 rfboder
Douf ucf ϋ∙ ό ldufyhfyfdbf ld ucf vwwfy
w`ou kde ucf ϋ∙ϋ‹ ldufyfdbf ld ucf `ozfyw`ou
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf 6
Fskmw`f LLL
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,9Mkrr #JfX/b
7$53 7$6 7$69 7$65 7$68 7$63 7$8 7$89
, 9
F x f
d u r / 7 $ 7
7 9 J f X / b
97
57
87
37
;77;97
;57
;87
;37
977
,9Mkrr #JfX/b
;$;6 ;$9 ;$96 ;$2 ;$26 ;$5
, 9
F x f d u r / 7 $ 7
7 9 J f X / b
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IkIky υ efbk|r>
υ ∙ ↙ χ∙χ∙χ!κν #κ ↙ ββ ,
Ucf dkuvyk` zleuc oh ucf κ lr dkyyoz#hfz afX, ro ucf zleuc oh ucf yfrodkdbf
lr eomldkufe i| efufbuoy yfro`vulod kde
lr u zluc k Jkvrrlkd elruylivulod$
υ ∙ ↙ χ∙h ;#;936,κν #h ; ↙ χ!χ∙κ,
Ucf dkuvyk` zleuc oh ucf h ;#;936, lr
bomwkyki`f #hfz MfX, uo ucf efufbuoy
yfro`vulod$ Ucf wfka lr u zluc k Iyflu-
Zljdfy #hoy ucf h ; `ldfrckwf, bodxo`vufe
i| k Jkvrrlkd eluylivulod #hoy ucf efufb-
uoy yfro`vulod,
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf 8
l l
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@vmldorlu|
• Zf yf`kuf byorr rfbulodr uo oirfyxfe efufbulod ykufr wfy vdlu ulmf i|
eD = @ eς
zcfyf D lr ucf dvmify oh fxfdur oirfyxfe wfy vdlu ulmf
@ ckr ucf elmfdrlodr oh kd ldxfyrf byorr-rfbulod wfy vdlu ulmf$
Hoy fskmw`f>
Yvd LL ku Hfyml`ki> @ ≂ ;729 bm∙9r∙;
@CB> @ ≂ ;725 bm∙9r∙;
• Uclr ldufjykufe `vmldorlu| lr vrvk``| {vouffe ld wkwfyr kde boyyfrwoder uo ucfrl~f oh ucf ekuk rfu #mfkrvyfe ld wi∙;$,
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf :
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@vmldorlu| - IkIky/KU@KR
Ek| ld 97;;
93/79 93/72 96/75 92/76 97/78 ;3/7: ;6/73
] - ;
U o u k ` L d u f j y k u f e @ v m l d o r l u | Q h i
7
7$6
;
;$6
9
9$6
2
Ek| ld 97;;
93/79 93/72 96/75 92/76 97/78 ;3/7: ;6/73
] - ;
U o u k ` L d u f j y k u f e @ v m l d o r l u | Q h i
7
7$6
;
;$6
9
9$6
2 = : UfXr KU@KR Od`ldf @vmldorlu|
@CB Ef`lxfyfe
KU@KR Yfboyefe
-;Uouk` Ef`lxfyfe> 9$58 hi-;
Uouk` Yfboyefe> 9$25 hi
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf 3
@ l lu F ` @FW
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@vmldorlu| Fskmw`f - @FW
Ku @FW @ = ;72; bm∙
9 r∙
;
Ucf f!f∙ ↙ {{ byorr rfbulod ku ucf wfka
oh ucf ^ 7 yfrodkdbf lr 27 di$
; ikyd = ;7
∙95
bm
9
oy ; di = ;7
∙22
bm
9
Cfdbf @ = ;7∙9di∙;r∙;
kde D wfy rfbode lr = ς@ = 7$2 r∙;
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf <
Iykdbcldj Ykulor
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• Lh k wkyulb`f bkd efbk| xlk mv`ulw`f yovufr zf ckxf
Βuou =Sl
Βl υ =;
Βuou
zcfyf υ lr ucf wkyulb`f `lhfulmf kde Βl lr ucf wkyulk` efbk| zleuc hoy ucf wkyulb`fefbk|ldj uo ucf l-uc dk` rukuf
• Zf efdf ucf iykdbcldj ykulo hoy k wkyulbv`ky efbk| moef kr
Il =Βl
Βuou
Hoy fskmw`f
υ ↙ fνν ;:&
¶νν ;:&
ϋν 95&
χν ;9&
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;7
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Byorr Rfbulod
• K wkyulb`f fdbovdufyr k woufdulk`kde lr rbkuufyfe ohh ku rbkuufyldj kd-
j`f γ
• Ucf rbkuufyldj kdj`f lr k hvdbulod oh
ucf lmwkbu wkykmfufy i$
• Ucf rmk``fy ucf lmwkbu wkykmf-
ufy ucf `kyjfy ucf effbulod ivu
ucf hvdbulod hoym efwfder od ucf
wyowfyulfr oh ucf woufdulk`$
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;;
Ckye rwcfyf rbkuufyldj
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Ckye rwcfyf rbkuufyldj
• Ucf wkyulb`f zluc lmwkbu wkykmfufy if-
uzffd i kde I ! eI zl`` fmfyjf zluc rbku-
ufyldj kdj`f ifuzffd γ kde γ ! eγeς = E#γ,eℨ
• E#γ, = eς/eℨ lr ohufd yfhfyfe uo kr ucf
kdjv`ky elruylivulod$ Ucf hoym oh E#γ, ef-
wfder od ucf wyowfyulfr oh ucf woufdulk`
• Uouk` byorr rfbulod lr oiukldfe i| ldufjyku-
ldj ucf kdjv`ky elrulivulod oxfy k`` ucf ro`le
kdj`f eℨ = rldγeγeψ
ς = Y eς
• Odf bkd k`ro jlxf byorr rfbulod k hvdbulod oh
ucf fdfyj| oh odf oh ucf wkyulb`fr eς/eF
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;9
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Byorr rfbulod kde efbk| ykufr• Jylhucr ckr bckdjfe ucf rfbulod od byorr rfbulodr kde efbk| ykufr ld Xfyrlod 9$
• Oucfy kwwyokbcfr bkd if hovde ld Ck`~fd kde Mkyuld #kde [M ufsur,
• Hobvr od ucf lmwoyukdu f`fmfdur kde rvjjfru |ov zoya ucyovjc ucf efylxkulod
ld Jylhucr
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;2
Efbk| ykuf ;↙ 9 ! 2
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Uzo ioe| efbk|r ld ucf BM hykmf
#Do woldu vrldj kd| oucfy hykmf,
wh = p w9p = p w2p
Efbk| zleuc hoy k uzo ioe| efbk| #; ↙ 9 ! 2,
Β =R
29χ9m;
^ pM p9
ε5# w; ∙ w9 ∙ w2,{ w9
9 ! m99
{ w2
9 ! m92
e2w9e2w9
zcfyf
R = ;/D + lr k boyyfbulod hkbuoy lh ucfyf kyf lefdulbk` wkyulb`fr ld ucf dk` rukuf1
29χ9m; lr k doymk`l~kulod hkbuoy yf`kufe uo ucf ldlulk` rukuf1
pM p9 ucf mkuyls f`fmfdu bodukldr ucf wc|rlbr1 kde
ucf yfmkldldj wkyur kyf ucf wckrf rwkbf f`fmfdur$
Ucf wckrf rwkbf f`fmfdur kyf wyowoyulodk` uo ucf wyoikil`lu| oh ucf wkyulb`f
efbk|ldj uo ucf uzo wkyulb`f wkyulb`fr ld k rwfblb aldfmkulb rukuf$
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;5
Efbk| ykuf ;↙ 9 ! 2
• Efbk| zleuc
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• Efbk| zleuc
eΒ =;
29χ9 wh
m;pM p9 eℨ
• Ldufjykuf oxfy k`` kdj`fr
Β =wh
29χ9m9;
^ pM p9 eℨ
• Hoy k rbk`ky wkyulb`f #zluc mkrr m, ucf mkuyls f`fmfdu M kde cfdbf ucf efbk|ykuf lr ldefwfdefdu oh ucf ro`le kdj`f kde ucf ldufjyk` lr k bodrukdu$
Β = B wh
29χ9m9
zcfyf B lr k bodrukdu ucku mk| ldb`vef momfduvm oy fdfyj| efwfdefdu
ufymr$
• Ucf Cljjr Iorod lr kd fskmw`f oh k rbk`ky wkyulb`f zluc k bovw`ldj
wyowoyulodk` uo ucf mkrr oh ucf wkyulb`fr ld ucf dk` rukuf$
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;6
Rbkuufyldj Byorr Rfbulod #BM Hykmf,
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Ucf hykmf ld zclbc odf fxk`vkufr ucf
byorr rfbulod lr lmwoyukdu$
Hoy fskmw`f ucf kdj`f ku zclbc k wkyul-
b`f fmfyjfr yf`kulxf uo ucf ldblefdu wky-ulb`f zl`` if elhhfyfdu ld ucf @KI kde BM
hykmfr
w¶; = #F ;, w,
w¶9 = #F 9,∙ w,
w¶2 = #F 2, w‹,
w¶5 = #F 5,∙ w‹,
p wp = p
w‹
p = wIkIky bo``lefr f! kde f∙ ku elhhfyfdu fdfyjlfr ld oyefy uo rfwkykuf ucf ovujoldj I
mfrodr i| iooruldj ucfly momfduvm$
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;8
Rbkuufyldj Byorr Rfbulod #BM Hykmf,
• Ucf byorr rfbulod bkd if zyluufd kr
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• Ucf byorr rfbulod bkd if zyluufd kr
#rlml`ky hoym kr ucf efbk| ykuf,
ς =R
85χ9#F ;
! F 9
, pw;p
^ pM p9
ε5# w; ! w9 ∙ w2 ∙ w5,
{ w29 ! m
92{ w59 ! m
95
e2w2e2w5
R = ;/D + lr k boyyfbulod hkbuoy lh ucfyf kyf lefdulbk` wkyulb`fr ld ucf dk` rukuf1
85χ9#F ; ! F 9, pw;p lr k doymk`l~kulod hkbuoy yf`kufe uo ucf ldlulk` rukuf1
pM p9 ucf mkuyls f`fmfdu bodukldr ucf wc|rlbr1 kde
ucf yfmkldldj wkyur kyf ucf wckrf rwkbf f`fmfdur$
• Odf bkd efylxf ucf elhhfyfdulk` byorr rfbulod ld ucf BM hykmf uo if
eς
eℨ= pM p9
;
#3χ,9R
#F ; ! F 9,9p wlp
ˑˑ wh ˑˑ
zcfyf p wlp lr ucf mkjdluvef oh flucfy ldbomldj momfduvm kde p wh p lr ucf
mkjdluvef oh flucfy ovujoldj momfduvm$
pM p9 zl`` `laf`| efwfde od ucf elyfbulod oh odf oh ucf ovujoldj wkyulb`fr
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;:
9-ioe| rbkuufyldj ld ucf @KI hykmf
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9-ioe| rbkuufyldj ld ucf @KI hykmf
w¶; = #F ;, w;,
w¶9 = #m9, 7,
w¶2 = #F 2, w2,
w¶5 = #F 5,∙ w5,
eς
eℨ=
;
85χ9pM p9
R pw2p
m9 pw;p #F ; ! m9 ∙ pw;p borγ,
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;3
Fskmw`fr oh byorr rfbulodr
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ς#f!f∙ ↙ ckeyodr,
ς#Z ∙
↙ f∙
ν ,
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf ;<
KU@KR nfu byorr rfbulod
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Ldb`vrlxf nfu byorr rfbulod
QJfX]U
w ;77 977 277 577 677 877
Q w i / J f X ]
U w
/ e ς e
;
;7
9;7
2;7
5;7
6;7
8;7
p 0 9$3| p
R|rufmkulb Vdbfyukldulfr
Dod-wfyu$ boyy$ÚD@O w[BE #BUF[ 8$8,
KU@KR
=7$5Y nfuru
kdul-a
=: UfX,r #-;
eu=;: di@∡
QJfX]U
w ;77 977 277 577 677 877
E k u k / U c f o y |
7
7$6
;
;$6
9
Wc|rlbr 678K 6 - Oirfyxki`fr Wkjf 97
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KIB Ucfoy|
• Hf|dmkd Elkjykmr
• Hf|dmkd Yv`fr
• Bk`bv`kuldj Efbk| Ykufr
• Bk`bv`kuldj Byorr Rfbulodr
• Cljcfy-Oyefy Elkjykmr
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;
KIB Ucfoy|
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KIB Ucfoy|
• 2 rwld`frr wkyulb`fr K I kde B #fkbc oh zclbc lr lur ozd kdulwkyulb`f,
• Od`| odf xfyufs zclbc mvru ldb`vef k`` ucyff wkyulb`f u|wfr #KIB,
Hoy fskmw`f #KKK, lr dou k``ozfe
• Lh mK 4 mI ! mB ucfd K bkd efbk| lduo I kde B$
K
I B
B
K I
I K
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf 9
Ucf Hf|dmkd Yv`fr
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Ucf Hf|dmkd yv`fr wyoxlef ucf yfblwf hoy bodruyvbuldj kd kmw`luvef M hyom k
Hf|dmkd elkjykm$
• Rufw ;>
Eykz ucf Hf|dmkd elkjykm zluc ucf mldlmvm dvmify oh xfyulbfr$ Ucfyf
mk| if moyf uckd odf$
• Rufw 9>
@kif` ucf hovy-momfduvm oh fkbc `ldf #zluc kyyozr, fdhoybldj
hovy-momfduvm bodrfyxkulod ku fxfy| xfyufs$
B
K, w; K, w9
I, w2 I, w5
w;, w9, $ $ $ fsufydk` momfduk
{;, {9, $ $ $ ldufydk` momfduk
kyyozr ldelbkuf worlulxf elyfbulod
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf 2
• Rufw 2>
Fkbc xfyufs boduylivufr k hkbuoy oh #∙lj,
c ` ` c e h i
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Fkbc ldufydk` `ldf zluc mkrr m kde hovy-momfduvm { boduylivufr k
wyowkjkuoy oh l{9∙m9
• Rufw 5>
Bodrfyxf 5-momfduvm ku fkbc xfyufs#9χ,5ε#5,#a; ! a9 ! a2,
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$
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf 5
Fskmw`f> K↙ I ! B
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w
• Uo `ozfru 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
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf 6
@lhfulmf oh K
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• Hyom Hfyml“r Jo`efd 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
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf 8
K!K↙ I !I
9
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• Uo `ozfru 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
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• 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 Jo`efd 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 $
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf 3
K! K↙ I !I
• Ucf byorr rfbulod bkd k`ro if zyluufd kr
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• 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 <
7/3/2019 QED and Particle Physics.
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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
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• 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
7/3/2019 QED and Particle Physics.
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• 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
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;9
Ldbocfyfdu Rvmr
7/3/2019 QED and Particle Physics.
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• K ↙ 2I ! B lr k elruldbu efbk| moef cfdbf zf bk`bv`kuf Β rfwkykuf`| hyom
ucku oh K ↙ I ! B
• Kr k yfrv`u 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 Jo`efd Yv`f$,
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;2
cl e O e K B l
7/3/2019 QED and Particle Physics.
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Uclye-Oyefy K↙ I ! B Elkjykmr
• Ucfyf lr odf `fjk` uclye-oyefy elkjykm uo bodrlefy>
K
K
I B
B I
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;5
L``fjk` @oow Elkjykmr
7/3/2019 QED and Particle Physics.
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• 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
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;6
Boyyfbulodr uo K! K↙ I !I
7/3/2019 QED and Particle Physics.
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Boyyfbulodr uo K! K↙ I !I
• Ucf ldufyhfyfdbf oh ucf odf-`oow 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
Wc|rlbr 678K 8 - KIB Ucfoy| Wkjf ;8
[vkduvm Mfbckdlbr
7/3/2019 QED and Particle Physics.
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[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
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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
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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
Wc|rlbr 678K : - Y[M Wkjf 2
Ro`vulodr uo ucf A`fld-Joyeod F{vkulodB le ` ` ul u uc A` l J e ul
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• 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
Wc|rlbr 678K : - Y[M Wkjf 5
Wyoi`fmr zluc ucf AJ f{vkulod
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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| `ooaldj 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
Wc|rlbr 678K : - Y[M Wkjf 6
Elykb“r Kwwyokbc LR ul ` l u u #l 7,
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• 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$
Wc|rlbr 678K : - Y[M Wkjf 8
Elykb“r Kwwyokbc LL• Uo fsufde uclr hkbuoyl~kulod hoy moxldj wkyulb`fr zf bkd zyluf
7/3/2019 QED and Particle Physics.
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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
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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$
Wc|rlbr 678K : - Y[M Wkjf 3
Ucf β Mkuylbfr
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• 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 `oz 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
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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
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#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>
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∉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
Wc|rlbr 678K : - Y[M Wkjf ;9
Ro`vulodr oh ucf Elykb F{vkulod LL• Uclr `fker uo ucf wkly oh f{vkulodr
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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 rcov`e fswfbu ivu σIrffmr uo ckxf k dfjkulxf fdfyj| F = ∙m$
• Elykb cke cowfe ucku k yru-oyefy #ld ∉/∉u, f{vkulod zov`e kxole ucfrf
dfjkulxf fdfyj| ro`vulodr$
• Fxfduvk``| Elykb wyfelbufe ucf fslrufdbf oh ucf worluyod$
Wc|rlbr 678K : - Y[M Wkjf ;2
F`fbuyodr kde Worluyodr• Elykb f{vkulod ckr 5 ldefwfdefdu ro`vulodr hoy k wkyulb`f ku yfru>
7/3/2019 QED and Particle Physics.
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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$
Wc|rlbr 678K : - Y[M Wkjf ;5
W`kdf-Zkxf Ro`vulodr L
• Dfsu zf zl`` `ooa hoy ro`vulodr uo ucf Elykb f{vkulod oh ucf hoym
σ# , lw s # ,
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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
Wc|rlbr 678K : - Y[M Wkjf ;6
W`kdf-Zkxf Ro`vulodr LL
#/w ∙ m,v =
7G F ∙ m ∙w º ς
;K7G vK
;K
7/3/2019 QED and Particle Physics.
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#/ 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 ;
Wc|rlbr 678K : - Y[M Wkjf ;8
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
7/3/2019 QED and Particle Physics.
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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 rcov`e 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
;
7/3/2019 QED and Particle Physics.
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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$
Wc|rlbr 678K : - Y[M Wkjf ;3
Rwldr oh ucf W`kdf-Zkxf Ro`vulodr
7/3/2019 QED and Particle Physics.
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•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 k`odj 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
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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
Wc|rlbr 678K : - Y[M Wkjf 97
Ilj kde Rmk`` Bomwodfdur
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• 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 `oz fdfyjlfr ucf vwwfy bomwodfdur kyf mvbc `kyjfy uckd ucf `ozfy
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
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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$
Wc|rlbr 678K : - Y[M Wkjf 99
Rwldoyr
7/3/2019 QED and Particle Physics.
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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``ozldj mkddfy>
7/3/2019 QED and Particle Physics.
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σ ↙ Rσ
• Hoy moulod k`odj 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
7/3/2019 QED and Particle Physics.
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• 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
Wc|rlbr 678K 3 - Rwldoyr Wkjf 2
β 6
mkuyls• Efdf kd keelulodk` β -mkuyls i| β 6 ≫ lβ 7β ;β 9β 2
7/3/2019 QED and Particle Physics.
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• 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$
Wc|rlbr 678K 3 - Rwldoyr Wkjf 5
Wkylu|• Vdefy k wkylu| uykdrhoymkulod
σ 7σ
7/3/2019 QED and Particle Physics.
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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$
Wc|rlbr 678K 3 - Rwldoyr Wkjf 6
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>
7/3/2019 QED and Particle Physics.
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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 ]
Wc|rlbr 678K 3 - Rwldoyr Wkjf 8
Rljdlbkdbf oh i`ldfky boxkylkdur?
q 6 6
7/3/2019 QED and Particle Physics.
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• 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 bov`e #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¶
7/3/2019 QED and Particle Physics.
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• 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``ozfe 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 `odj kr α = 7$
=⇙ Zf bcoorf uo rfu K7 = 7 kde zoya ld ucf Bov`omi jkvjf> ∊ º K = 7
Wc|rlbr 678K 3 - Rwldoyr Wkjf 3
Hyff Wcouodr
• Hoy k wcouod ld hyff rwkbf #N ¶ = 7, ucf woufdulk` lr jlxfd i| K¶ = 7$
7/3/2019 QED and Particle Physics.
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• Ucf w`kdf-zkxf ro`vulod lrK¶#s, = kf∙lwºsǡ¶# w,
zcfyf ǡ¶ lr ucf wo`kyl~kulod xfbuoy kde w¶ w¶ = 7
• K`ucovjc ǡ
¶
ckr 5 bomwodfdur dou k`` kyf ldefwfdefdu$Ucf @oyfdu~ bodelulod yf{vlyfr ucku w¶ǡ¶ = 7
Hvyucfymoyf ucf Bov`omi 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
7/3/2019 QED and Particle Physics.
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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
7/3/2019 QED and Particle Physics.
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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,
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• @oyfdu~ bodelulod
ǡ¶ w¶ = 7
• oyucojodk`lu|
ǡ¶#;,
ǡ¶#9, = 7
•doymk`l~kulodǡ¶√ǡ¶ = ;
• Bov`omi jkvjf
ǡ7 = 7 kde ǡ º w = 7
• Bomw`fufdfrrWr#ǡ#r,,l#ǡ√
#r,,n = εln ∙ wl wn
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 2
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$
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| 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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 5
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
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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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 6
Xfyufs Hkbuoyr
F [FE u
f
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• 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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 8
Wyowkjkuoyr
• Fkbc ldufydk` wcouod boddfbur uzo xfyulbfr oh ucf hoym ljfβ ¶ kde ljfβ ν ro zf
rcov`e fswfbu ucf wyowkjkuoy uo boduykbu ucf ldelbfr ¶ kde ν $
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• 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
7/3/2019 QED and Particle Physics.
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od ucf ovurlef$• Zoya ikbazkyer k`odj fxfy| hfymlod `ldf vrldj>
f∙ ld f∙ ovu f! ld f! ovu β ld β ovu
v v x x ǡ¶ ǡ√¶
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 3
Mkuyls f`fmfdur L
ho``oz hfymlod `ldfr ikbazkye uo jlxf v#9,ljβ ¶v#;,
7/3/2019 QED and Particle Physics.
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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 if`oz$
7/3/2019 QED and Particle Physics.
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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
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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
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• 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 ,$
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;9
Fskmw`f> Ickick Rbkuufyldj
f∙, w2 f!, w5 f∙, w2 f!, w5
7/3/2019 QED and Particle Physics.
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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;]
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;2
Fskmw`f> Bomwuod Rbkuufyldj
β, w2 f∙, w5 β, w2 f∙, w5
7/3/2019 QED and Particle Physics.
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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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;5
Wo`kyl~fe Wkyulb`fr
• K u|wlbk` [FE kmw`luvef mljcu `ooa romfucldj `laf
M ∰ Qv;Β¶x9] ǡ2¶
7/3/2019 QED and Particle Physics.
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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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;6
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
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;$ 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$
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;8
Rwld Rvmr L
• @fu“r rlmw`lh| ucldjr fxfd hvyucfy kde rvwworf ucku zf ckxf
M ∰ Qv;Βv9]
7/3/2019 QED and Particle Physics.
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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
7/3/2019 QED and Particle Physics.
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• 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;]
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf ;3
Rwld Rvmr LLL
• Ucf yljcu-ckde rlef lr nvru k dvmify ivu lh zf yfwyfrfdu ucf mkuyls
mv`ulw`lbkulod zluc rvmmkulodr oxfy ldelbfr zf bkd yfzyluf lu kr
Qv;[v;] = #v;, [ln #v;,
7/3/2019 QED and Particle Physics.
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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]
7/3/2019 QED and Particle Physics.
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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 ;$
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 97
Bkrlmly“r Uylba
• Uclr wyobfevyf oh bk`bv`kuldj rwld-kxfykjfe kmw`luvefr ld ufymr oh uykbfr lr
7/3/2019 QED and Particle Physics.
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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 `ou oh
7/3/2019 QED and Particle Physics.
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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,
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 99
Jkmmk mkuylblfr kde uykbfr
• Ucf uzo mknoy lefdululfr ucku zf zl`` dffe ld oyefy uo ivl`e moyf bomw`lbkufe
uykbf lefdululfr kyf
j¶νj¶ν = 5
¶ ν ¶ν
7/3/2019 QED and Particle Physics.
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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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 92
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
7/3/2019 QED and Particle Physics.
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• 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”
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 95
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
7/3/2019 QED and Particle Physics.
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• 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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 96
Ucf β 6 Uykbf
O ` luc 5 # , uc u l iu l u
7/3/2019 QED and Particle Physics.
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• 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
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 98
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,
7/3/2019 QED and Particle Physics.
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# w¶; wν9 ! w¶9 wν; ,$
• 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
”
$$$
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 9:
Fskmw`f ;
• Odf oh ucf uykbfr ldxo`xfe ld Ickick rbkuufyldj lr
U = Uy Qβ ¶#/ w; ! m,β ν #/ w2 ! m,]
7/3/2019 QED and Particle Physics.
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• Zf bkd fswkde uclr ovu uo byfkuf 5 ufymr ivu 9 oh ucfrf ufymr #ucf odfr `ldfky
ld m, zl`` ldxo`xf 2 β -mkuylbfr kde kyf ucfyfhoyf ~fyo$
• Ucvr
U = Uy#β ¶/ w;β ν/ w2, ! m9Uy#β ¶β ν ,
= 5g w¶; wν2 ! w¶2 wν; ∙ # w; º w2,j¶ν
µ! 5m9j¶ν
• Uclr yfrv`u zl`` if boduykbufe zluc kdoucfy uykbf ucku lr boxkylkdu #l$f$ ¶ν krowworfe uo boduykxkylkdu ¶ν , ld ¶ kde ν $
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 93
Fskmw`f 9
• Bodrlefy
K = Uy#β ¶/ w;β ν/ w9, Uy#β ¶/ w;β ν/ w9,
7/3/2019 QED and Particle Physics.
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• Fxk`vkuldj ucf uykbfr
K = 5 w¶; wν9 ! wν; w¶9 ∙ # w; º w9,j¶νː
Ú5 Q w;¶ w9ν ! w;ν w9¶ ∙ # w; º w9,j¶ν ]= ;8
ˈ9 w9
; w99 ! 9# w; º w9,9 ! 5# w; º w9,9 ∙ 5# w; º w9,9
ː= 29
ˈm9
;m99 ! # w; º w9,9
ː
Wc|rlbr 678K < - Hf|dmkd Yv`fr Wkjf 9<
F`fbuyod Rbkuufyldj
7/3/2019 QED and Particle Physics.
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• F`fbuyod-Mvod Rbkuufyldj
#Mouu Rbkuufyldj,
• F`fbuyod-worluyod rbkuufyldj
• Bomwkylrodr zluc fswfylmfduk` ekuk
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;
Fskmw`f> F`fbuyod-Mvod Rbkuufyldj L
β
f∙, w2 ¶∙, w5
7/3/2019 QED and Particle Physics.
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β
f∙, w; ¶∙, w9
Od`| odf elkjykm
M = l Qv2#ljfβ ¶,v;]
—∙lj¶ν
# w; ∙ w2,9
¡Qv5#ljfβ ν,v9]
= ∙j9f
# w; ∙ w2,9 Qv2β ¶v;] Qv5β ¶v9]
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 9
f ¶ Rbkuufyldj LL
M = ∙j9f
# w; ∙ w2,9 Qv2β ¶
v;] Qv5β ¶v9]
Kxfykjf pMp9 oxfy ldlulk` rukuf rwldr kde rvm oxfy dk` rukuf rwldr
EpMp9F =j5f
5#w; ∙ w2,5
Uy Qβ ¶#/ w; ! m,β ν #/ w2 ! m,]
U Q #/ ! M , #/ ! M ,]
7/3/2019 QED and Particle Physics.
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E F5# w w ,
Ú Uy Qβ ¶#/ w9 ! M ,β ν#/ w5 ! M ,]
=j5f
5# w; ∙ w2,5
ˈ5g w¶;
wν2 ! w¶2
wν; ! #m9 ∙ w; º w2,j¶νµː
Ú ˈ5 g w9¶ w5ν ! w5¶ w9ν ! #M 9 ∙ w9 º w5,j¶νµː=
5j5f# w; ∙ w2,5
q9# w; º w9,# w2 º w5, ! 9# w; º w5,# w9 º w2,
! 9m9# w9 º w5, ! 9M 9# w; º w2,
∙ 5# w; º w2,# w9 º w5, ! 5#m9 ∙ w; º w2,#M 9 ∙ w9 º w5,¯
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 2
f ¶ Rbkuufyldj LLL
Rlmw`lh| kde bomildf mkrr ufymr
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Rlmw`lh| kde bomildf mkrr ufymr
EpMp9
F=
3j5f# w; ∙ w2,5
q# w; º w9,# w2 º w5, ! # w; º w5,# w9 º w2,
∙ m9
# w9 º w5, ∙M 9
# w; º w2, ! 9m9
M 9¯
Uclr lr k xfy| jfdfyk` yfrv`u ucku bkd if kww`lfe uo f`fbuyod rbkuufyldj ohh oh kd|
bckyjfe wkyulb`f fsbfwu hoy kdoucfy f`fbuyod oy worluyod
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 5
Mkdef`rufm xkylki`fr
Hoy ucf yfkbulod ; ! 9 ↙ 2 ! 5 zf efdf
r = # w; ! w9,9 = # w2 ! w5,9
u = # w; ∙ w2,9 = # w9 ∙ w5,9
9 9
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v = # w; ∙ w5,9 = # w9 ∙ w2,9
r ! u ! v = m9; ! m9
9 ! m92 ! m9
5
Ld ucf BM hykmf kde dfj`fbuldj ucf mkrrfr oh ucf wkyulb`fr zf jfu
r = 9 w; º w9 = 9 w2 º w5 = 5 w9
u = ∙9 w; º w2 = 9 w9 º w5 = ∙9 w9#; ∙ borγ,
v = ∙9 w; º w5 = 9 w9 º w2 = ∙9 w9#; ! borγ,
r ! u ! v = 7
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 6
f¶ Rbkuufyldj ld BM Hykmf
Ljdoyldj ucf mkrrfr oh ucf wkyulb`fr ucf mkuyls f`fmfdu lr
pM p9 =3j9
# w; ∙ w2,9## w; º w9,# w2 º w5, ! # w; º w5,# w9 º w2,,
3j9 ;# 9 9,
7/3/2019 QED and Particle Physics.
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=j
u9 5#r9 ! v9,
Cfdbf ucf elhhfyfdulk` byorr rfbulod lr
eς
eℨ =
;
85χ9 pM p
9 wh
wlr
=λ9
9r
v9 ! r9
u9
=λ9
9r—
#; ! borγ,9 ! 5
#; ∙ borγ,9 ¡
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 8
f¶ Rbkuufyldj ld @KI Hykmf
f¶ rbkuufyldj lr lmwoyukdu zcfd zf elrbvrr f-wyouod rbkuufyldj
kde ucf fxlefdbf hoy {vkyar$
Ucf hovy-momfduk kyf
w; = #F,w;,
7/3/2019 QED and Particle Physics.
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# ,
w9 = #M,7,
w2 ≂ #F,w2,
w5 ≂ #M,7,
Ucf momfduvm uykdrhfy lr
{ = # w; ∙ w2,9
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf :
f¶ Rbkuufyldj Kmw`luvefUcf rwld-kxfykjfe kmw`luvef hoy f¶ rbkuufyldj lr
EpMp9F =3j5f
# w; ∙ w2,5q# w; º w9,# w2 º w5, ! # w; º w5,# w9 º w2,
9# , M9# , 9 9M9¯
7/3/2019 QED and Particle Physics.
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E F∙ m9# w9 º w5, ∙M 9# w; º w2, ! 9m9M 9
¯
Zf bkd eyow ucf ufymr ld yfe kr m lr krrvmfe uo if ~fyo
Khufy romf mkdlwv`kulod
EpMp9
F=
;8j5fM 9F ;F 2
{5
±bor9
γ
9∙
{9
9M 9rld9
γ
9
‒
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 3
f¶ Rbkuufyldj Byorr Rfbulod
Yfbk`` ucf hoym hoy ucf elhhfyfdulk` byorr rfbulod ld ucf `ki hykmf
eς
eℨ=
;
85χ9pM p9
w2
m9 w; ‐F ; ! m9 ∙w;F2borγ
w2”
F9
7/3/2019 QED and Particle Physics.
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‐ ”=
F 285χ9M 9F 9
;
pM p9
Ldrfyuldj ucf rbkuufyldj kmw`luvef lduo ucf kioxf hoymv`k
eς
eℨ=
λ9
5F 9;
rld5 γ9
F 2
F ;
±bor9
γ
9∙
{9
9M 9rld9
γ
9
‒
Lh wyouodr zfyf woldu-`laf oinfbur ucfd ucf f-wyouod rbkuufyldj byorr rfbulod zov`e
if lefdulbk` uo uclr rbkuufyldj
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf <
Ickick Rbkuufyldj L
Kd lmwoyukdu byorr rfbulod mfkrvyfe i| mkd| fswfylmfdur
f∙, w2 f!, w5 f∙, w2 f!, w5
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β
f∙
, w; f!
, w9
β
f∙
, w; f!
, w9
Ucf mkuyls f`fmfdu lr
M = ∙j9
# w; ∙ w2,9Qv2β ¶v;] Qx5β ¶x9]∙ ∙j9
# w; ! w9,9Qv2β ¶x5] Qx9β ¶v;]
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;7
Ickick Rbkuufyldj LL
Ucf hoym oh ucf uouk` mkuyls f`fmfdu lr
M 9 = M 9; ! 9M 9ldu ! M 99
zcfyf
M 9; = 3j5
,5Q# w; º w9,# w2 º w5, ! # w; º w5,# w9 º w2,]
7/3/2019 QED and Particle Physics.
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; w; ∙ w2,5Q#w; w9,#w2 w5, #w; w5,#w9 w2,]
Ld ucf BM hykmf
M 9; = 9j5r9 ! v9
u9
Rlml`ky`|
M 9; = 9j5v9 ! u9
r9
kde
9M 9ldu = 9j59v9
ru
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;;
Ickick Rbkuufyldj LLLUcf elhhfyfdulk` byorr rfbulod ld ucf BM hykmf lr
eς
eℨ
=λ9
9r±
r9 ! v9
u9
!v9 ! u9
r9
!9v9
ru‒
Ucf yru ufym lr ucf fsbckdjf ufym oy u bckddf` wyobfrr
7/3/2019 QED and Particle Physics.
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± ‒Ucf yru ufym lr ucf fsbckdjf ufym oy u-bckddf` wyobfrr$
Ucf rfbode ufym lr ucf kddlcl`kulod ufym oy r-bckddf` wyobfrr
Ld hv`` hoym
eς
eℨ=
λ9
9r
); ! bor5 γ
9
rld5 γ9
!; ! bor9 γ
9
9∙
9bor5 γ9
rld9 γ9
"
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;9
f!
f∙
Kbbf`fykuoyrBFRY #Boydf``, 3!3 i-{vkya kde υ wc|rlbr
UYLRUKD #AFA, 27!27 [FE ufrur
R@B #R@KB, 67!67 #wo`kyl~fe, ^ 7 wc|rlbr
@FW #BFYD, 67 67 # c L, ^7 c l
7/3/2019 QED and Particle Physics.
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@FW #BFYD, 67!67 #wckrf L, ^ 7 wc|rlbr
;77!;77 #wckrf LL, Z Z wc|rlbr
WFWLL #IkIky, 3!2 i-{vkya kde υ wc|rlbr
If``f 3!2 i-{vkya kde υ wc|rlbr
RvwfyI 3!2 i-{vkya kde υ wc|rlbr
L@B ?? Ruve| oh Cljjr iorod kde RVR\
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;2
R@KB
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;5
R@KB
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;6
f!
f∙ efufbuoyr
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;8
Fswfylmfduk` yfrv`ur L
Kdjv`ky elruylivulod hyom o`e
fswfylmfdur
7/3/2019 QED and Particle Physics.
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Douf ucku byorr rfbulod lr ldef-
wfdefdu od r od`|$
reς
eℨ= H #γ,
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;:
Fswfylmfduk` yfrv`ur LL
U u ` ul
7/3/2019 QED and Particle Physics.
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Uouk` byorr rfbulod
ς ∝;
r
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;3
Fswfylmfduk` yfrv`ur LLL
Uouk` byorr rfbulod
7/3/2019 QED and Particle Physics.
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Ku cljcfy bfduyf-oh-
mkrr fdfyjlfr ucf zfkaldufykbulod ifbomfr
lmwoyukdu$
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf ;<
@vmldorlu| mfkrvyfmfdurMkd| fswfylmfdur mfkrvyf ucf `vmldorlu| vrldj ucf f!f∙ ↙ f!f∙ yfkbulod ku
xfy| rmk`` kdj`fr$ Yfbk`` ucku
eς
eℨ=
λ9
9
); ! bor5 γ
9
5 γ!
; ! bor9 γ9
9∙
9bor5 γ9
9 γ
"
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eℨ 9r
)rld5 γ
99 rld9 γ
9
"
Ku γ 7$; ykelkdr #: efjyffr, rld γ
9
≃ γ
9
kde bor γ
9
≃ 7 ro
eς
eℨ=
λ9
9r
)9
# γ9
,5! ; ∙
9
# γ9
,9
"
ucf yru ufym eomldkufr ucf byorr rfbulod ku rmk`` kdj`fr$
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 97
Fxlefdbf hoy ucf zfka ldufykbulod
Zf zl`` `ooa ku ucf f!f∙ ↙ h h yfkbulod ld ucf dfsu `fbuvyf #h lr k hfymlod,$
Ld ucf BM hykmf kde ljdoyldj ucf hfymlod mkrrfr ucf byorr rfbulod lr
eς
eℨ =λ9
9r
v9 ! u9
r9 =λ9
9r #; ! bor9
γ,
7/3/2019 QED and Particle Physics.
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Odf efdfr ucf hoyzkye-ikbazkye kr|mmfuy| kr
KHI =
Y ;7
eςeℨ
eℨ ∙Y 7∙;
eςeℨ
eℨY ;
∙;eςeℨ
eℨ
Hoy ucf FM ldufykbulod KHI = 7 vrldj ucf kioxf byorr rfbulodr cozfxfy uclr lr
dou uyvf hoy ucf zfka ldufykbulod$
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 9;
Hoyzkye-Ikbazkye Kr|mmfuy| L
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Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 99
Cljcfy-Oyefy Elkjykmr ld [FE• Ucf moru hkmovr cljcfy-oyefy wyobfrr ld [FE lr ucf kdomk`ovr mkjdfulb
momfdu oh ucf f`fbuyod #oy mvod, kylrldj hyom ucf elkjykm
β f∙ f∙
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β #I,
Ld ;<53 Rbczldjfy rcozfe ucku uclr moelfr ucf f`fbuyod j-hkbuoy hyom 9 uo
#9 ! λ/χ,$ Lu lr bvyyfdu`| adozd uo λ5 boyyfrwodeldj uo kd vdbfyukldu| ld jf
oh kiovu ;7∙;9$
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 92
IkIky efufbuoy
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Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 95
f!
f∙ Byorr rfbulod dfky ucf Φ yfrodkdbfr
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Ucf `ozfy Φ yfrodkdbfr kyf if`oz ucf ucyfrco`e hoy wyoevbldj I-mfrodr$ IkIky
kde If``f fswfylmfdur ukaf ekuk ku BM fdfyjlfr oh ucf Φ#5R ,
Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 96
IkIky Fxfdur
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Wc|rlbr 678K ;7 - f`fbuyod rbkuufyldj Wkjf 98
F`fbuyod-wyouod rbkuufyldj
• Wyoevbulod oh ckeyodr hyom f! f∙
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• F`krulb f w rbkuufyldj
• Ldf`krulb f w rbkuufyldj
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;
f! f∙↙ h h
β
f∙, w; f!, w9
h, w2 h , w5
O ` el
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• Od`| odf elkjykm
M = l ˈv2#∙l[h jfβ ¶,x5ː—∙lj¶ν
# w; ! w9,9¡ Qx9#ljfβ ν,v;]
=[h j
9f
# w; ! w9,9Qv2β ¶x5] Qx9β ¶v;]
• h h β xfyufs hkbuoy lr ∙l[h jfβ ¶
• M #f! f∙ ↙ h h , = [h M #f! f∙ ↙ ¶! ¶∙,
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf 9
Ucf Rwld-Kxfykjfe Kmw`luvef
M =[h j
9f
# w; ! w9,9Qv2β ¶x5] Qx9β ¶v;]
• Bk`bv`kuldj ucf rwld-kxfykjfe kmw`luvef
EpMp9
F=
[9j5f5#w; ! w9,5
Uy Qβ ¶#/ w5 ∙M ,β ν#/ w2 ! M ,]
7/3/2019 QED and Particle Physics.
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5# w; ! w9,
Ú Uy Qβ ¶#/ w; ! m,β ν #/ w9 ∙m,]
=
5[9j5f# w; ! w9,5 w
¶
2 wν
5 ! w¶
5 wν
2 ∙g w2 º w5 ! M
9µj¶νː
Ú w;¶ w9ν ! w9¶ w;ν ∙g w; º w9 ! m9
µj¶ν
ː
=5[9j5f
# w; ! w9,5Q9# w; º w2,# w9 º w5, ! 9# w; º w5,# w9 º w2,
! 9m9# w2 º w5, ! 9M 9# w; º w9, ! 5m9M 9ː
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf 2
f! f∙ ↙ h h Byorr Rfbulod
Ucf byorr rfbulodr hoy f!
f∙
↙ h ¯
h lr
eς
eℨ=
[9h
λ9
5r#; ! bor9γ, ς =
5χλ9
2r[9h
Hoy f! f∙ ↙ { { zcfyf { lr k rwfblb {vkya kxovy
ς =5χλ9
2r[9{
7/3/2019 QED and Particle Physics.
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Cozfxfy lh zf eo dou elruldjvlrc ifuzffd {vkya kxovyr ucfd
ς#f! f∙
↙ { {, =5χλ9
2r 2Sl
[9l
zcfyf zf ukaf lduo kbbovdu ucf 2 bo`ovyr oh fkbc {vkya kxovy$
Kde ucf ykulo
Y = ς#f!
f∙
↙ { {,ς#f! f∙ ↙ ¶! ¶∙,
= 2Sl
[9l
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf 5
Ykulo oh f!f∙ Byorr Rfbulodr
Y#F , =ςg
f! f∙ ↙ ckeyodrµ
ς #f! f∙ ↙ ¶! ¶∙,= 2
Sh
[9h
H`kxoy Mkrr #MfX, [ Y
9 5
7/3/2019 QED and Particle Physics.
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v ;$6-5 ! 92
52
e 5-3 ∙ ;2
62
r 37-;27 ∙ ;2 9
b ;;67-;267 ! 92
;72
i 5;77-5577 ∙ ;2
;;2
u ;:3777 ! 92
6
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf 6
Fswfylmfduk` Mfkrvyfmfdu oh Y
;
;7
;7 9
;7 2
ϋ
ό
ψ
ϋ
N/ σ σ #9R,^Y
7/3/2019 QED and Particle Physics.
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;7 -;
;
; ;7 ;7 9
ϋ
R JfX
Fswfbu> 9;7
2
;;
26
Mkd| fswfylmfdur kde kbbf`fykuoyr ckxf boduylivufe uo ucf ekuk ld uclr w`ou$
f!f∙ mkbcldfr ckxf `lmlufe fdfyj| ykdjf
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf 8
Cljc fdfyj| f!f∙ rbkuufyldj
OWK@ zkr odf oh 5 fswfylmfdur
ku @FW #BFYD,
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f! f∙ ↙ f f f! f∙ ↙ υ υ f! f∙ ↙ { {
Yvd>fxfdu 57<2> ;;67 Ekuf <2769: Ulmf 97:6;
F ifkm 56 $ 863 F x l r <5 $ 5 Fm l r r -2 $ ; X u s # -7 $ 76 7 $ 73 7 $ 28 ,
I~=5$267 Ucyvr u=7$<<:< Kw`kd=7$7777 Oi`ku=7$ 772< Rwcfy=7$777;
Buya #D= 9 Rvmw= <9$5, Fbk`#D= < RvmF= <7$6, Cbk`#D= 7 RvmF= 7 $7,
Mvod#D= 7, Rfb Xus#D= 7, Hefu #D= ; RvmF= 7$7,
\
S^
977$ bm$
Bfd u y f o h r b y ffd l r # 7 $ 7777 7 $ 7777 7 $ 7777 ,
67 JfX97;76
\
S^
977$ bm$
Bfd u y f o h r b y ffd l r # 7 $ 7777 7 $ 7777 7 $ 7777,
67 JfX97;76
Yvd>fxfdu 5279> :68:9 Ekuf <27:;: Ulmf 996725
F ifkm 56 $ 8;7 Fx l r ;9; $ < Fm l r r -27 $ : X u s # -7 $ 75 7 $ 75 7 $ 9< ,
I~=5$267 Ucyvr u=7$<<<2 Kw`kd=7$777; Oi`ku=7$ 778; Rwcfy=7$7778
Buya #D= 5 Rvmw= :9$;, Fbk` #D= ;5 RvmF= 92$:, Cbk `#D= < RvmF= 58$5,
Mvod#D= ;, Rfb Xus#D= 7, Hefu #D= 7 RvmF= 7$7,
\
S^
977$ bm$
Bfd u y f o h r b y ffd l r # 7 $ 7777 7 $ 7777 7 $ 7777,
67 JfX97;76
Yvd>fxfdu 57<2> ;777 Ekuf <2769: U lmf 97:;8
F i f km 56 $8 6 3 F x l r < < $< F m l r r - 3 $8 X us # - 7 $7 : 7 $7 8 - 7 $3 7 ,
I ~ = 5 $2 6 7 U c y v r u= 7 $< 3 : 2 K w ` k d = 7 $7 7 ; : O i ` k u= 7 $7 9 5 3 R w c f y = 7 $7 7 :2
Buya #D= 2< Rvmw= :2$2, Fbk` #D= 96 RvmF= 29$8, Cbk`# D=99 RvmF= 99$8,
M v o d #D = 7 , R f b X us # D = 2 , H e f u # D = 7 R v m F= 7 $7 ,
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf :
Y-Mfkrvyfmfdu vrldj LRY L
Doxf` kwwyokbc uo fswkde ucf ykdjf oh ucf Y-Mfkrvyfmfdu ku k rldj`f kbbf`fykuoy
Doymk``| odf ckr uo s ucf bfduyf-oh mkrr #BM, f!f∙ fdfyj| oh ucf kbbf`fykuoy kde
mfkrvyf ucf byorr rfbulod$ Bckdjldj ucf ifkm fdfyj| lr dou k`zk|r fkr|$
LRY lr Ldlulk` Rukuf Ykelkulod
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Ucf f`fbuyod oy worluyod ckr ykelkufe kd
fdfyjfulb wcouod
Vrf ucf fdfyj| oh ucf wcouod uo yfbk`bv-
`kuf ucf BM fdfyj|$
IkIky kde If``f kyf vrldj fxfdur zluc ldlulk` rukuf ykelkulod #LRY, uo mfkrvyf ucff! f∙ ↙ { { byorr rfbulod
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf 3
Y-Mfkrvyfmfdu vrldj LRY LL
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Wc|rlbr 678K ;; fw rbkuufyldj Wkjf <
F`fbuyod-wyouod Rbkuufyldj L
K ifkm oh f`fbuyodr zluc fdfyj| F kyf rbkuufyfe hyom k rukulodky| dvb`fvr zluc
mkrr M D
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Xlyuvk` wcouod {9 = ν 9 ∙ { = 7
F`krulb rbkuufyldj> ovujoldj dvb`fvr yfmkldr ldukbu
Ldf`krulb rbkuufyldj> ovujoldj dvb`fvr lr ld kd fsblufe rukuf #elhhfyfdu mkrr, oy lr
iyoafd kwkyu
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;7
F`fbuyod-wyouod Rbkuufyldj LL
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Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;;
Romf efdlulodr
Ldlulk` rukuf> wD = #M D , 7,
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wD # D , ,
Hldk` rukuf> w‹D
= #M D ! ν, {,
ν = F ∙F = [9/9M D zcfyf ∙[9 = {9
ν lr ucf fdfyj| `oru i| ucf f`fbuyod
[9 lr ucf r{vkyf oh ucf 5-momfduvm uykdrhfyfe ifuzffd ucf f`fbuyod kde dvb`fvr
i| ucf wcouod
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;9
F`krulb F`fbuyod-Wyouod Rbkuufyldj
• F`krulb rbkuufyldj mfkdr ucf wyouod yfmkldr ldukbu
• Lh ucf wyouod zfyf ruyvbuvyf`frr zf bov`e vrfy f`fbuyod-mvod
rbkuufyldj>EpMp9
F=
j5f5# w; ∙ w2,5
ˈ5g w¶; wν2 ! w¶2 wν; ! #m9 ∙ w; º w2,j¶ν
µː
7/3/2019 QED and Particle Physics.
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#w w2,
Ú
ˈ5
g w9¶ w5ν ! w5¶ w9ν ! #M 9 ∙ w9 º w5,j¶ν
µː= j5f
{5@¶νf`fbuyod
@¶ν mvod
zluc { = w; ∙ w2 kde @¶νf`fbuyod
= 9 g w9¶ w5ν ! w5¶ w9ν ! #m9 ∙ w9 º w5,j¶νµ
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;2
Ivu ucf Wyouod Lrd“u Ruyvbuvyf`frr$$$
• Lh ucf wyouod zfyf k uyvf woldu wkyulb`f ucfd
EpMp9
F=
j5f{5
@¶ν
f`fbuyod@¶ν wyouod
• Efdf A ¶ν zclbc zl`` yfwyfrfdu ucf wyouod ruyvbuvyfEpMp9
F=
j5f5
@¶νf`fbuyod
A ¶ν wyouod
7/3/2019 QED and Particle Physics.
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Ep p
F{5 f`fbuyod ¶ w ouo
β
f∙, w; w, w9
f∙, w2 w, w5
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;5
Coz Eo Zf Bk`bv`kuf A ¶ν wyouod?
• Zf adoz ucku A ¶ν
wyouod lr k rfbode-ykda ufdroy$
• Zf bkd bodruyvbu ufdroyr hyom ucf hovy-xfbuoyr j¶ν w9 w5 kde
{$
• Rldbf { = w5 ∙ w9 od`| 9 oh ucfrf hovy-xfbuoyr kyf ldefwfdefduhyom zclbc zf bcoorf { kde w9 = w$ Ucvr ovy bcolbfr kyf
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j¶ν w¶ wν {¶{ν # w¶{ν ! wν{¶, # w¶{ν ∙ wν{¶,
• Hoy f`fbuyomkjdfulb ldufykbulodr @¶ν
f`fbuyod lr r|mmfuylb ld ¶
kde ν ucfyfhoyf zf dffe dou ldb`vef # w¶{ν ∙ wν{¶,$
#⇙ uclr ufym lr yf{vlyfe hoy ucf zfka ldufykbulod,
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;6
Hoym Hkbuoyr
• Vrldj ucf hovy r|mmfuylb rfbode-ykda ufdroyr zf zyluf
A ¶νwyouod = ∙A ;j¶ν !A 9
M 9 w¶ wν !
A 5
M 9{¶{ν !
A 6
M 9# w¶{ν ! wν{¶,
zcfyf A ; A 9 A 5 kde A 6 kyf vdadozd hvdbulodr zclbc zf yfhfy uo kr hoym
hkbuoyr$
• Ucf hoym hkbuoyr bkd efwfde od {9 ucf od`| rbk`ky xkylki`f kxkl`ki`f uo vr
rldbf w9 = M9 kde w º { = ∙{9/9$
7/3/2019 QED and Particle Physics.
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rldbf w M kde w { { /9$
• Vrldj ucf Zkye lefdulu| {¶A ¶νwyouod = 7
zf de ucku ucfyf kyf od`| 9 ldefwfdefdu hoym hkbuoyr>
A ¶νwyouod = A ;
—∙j¶ν !
{¶{ν
{9
¡!
A 9
M 9
— w¶ !
{¶
9
¡— wν !
{ν
9
¡
Ucf jok` lr ucfd uo mfkrvyf ucfrf hoym hkbuoyr fswfylmfduk``| kde uo uy| uo
bk`bv`kuf ucfm ucfoyfulbk``|$
Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;8
f w Byorr Rfbulod
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Wc|rlbr 678K ;; fw rbkuufyldj Wkjf ;:
Wkyuod Moef`
• Ucf Wkyuod Moef`
• Inoyafd Rbk`ldj
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• Inoyafd Rbk`ldj
•
Wkyuod Elruylivulod Hvdbulodr
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;
Fsufdeldj ucf Yvucfyhoye Fswfylmfdu
Yfbk`` ucku ikrfe od k rvywylrldj`| cljc dvmify oh `kyjf-kdj`f fxfdur ld f`krulb
λ Kv rbkuufyldj Yvucfyhoye efevbfe kuomlb rviruyvbuvyf #l$f$ ucf dvb`fvr,
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Ld k rlml`ky hkrclod odf bkd ldxfruljkuf ucf ruyvbuvyf oh ucf wyouod ld f w rbkuufyldj
wkyulbv`ky`| ld ucf effw ldf`krulb rbkuufyldj yfjlmf zcfyf {9 lr `kyjf$
Ucf wyouod zkr hovde uo ckxf rviruyvbuvyf #R@KB `kuf ;<87r,$ Ucfrf bodruluvfdur
bkmf uo if adozd kr wkyuodr zclbc kyf doz yfbojdl~fe kr {vkyar kde j`vodr$
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 9
F`fbuyod-wyouod rbkuufyldj ifckxlovy
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 2
Romf efdlulodr
Ldlulk` rukuf> wD = #M D , 7,
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Hldk` rukuf> w‹
D = #M D ! ν, {,
ν = F ∙F = [9/9M D zcfyf ∙[9 = {9
ν lr ucf fdfyj| `oru i| ucf f`fbuyod
[9 lr ucf r{vkyf oh ucf 5-momfduvm uykdrhfyfe ifuzffd ucf f`fbuyod kde dvb`fvr
i| ucf wcouod
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 5
#ν,[9
, w`kdf
F`fbuyod `ki fdfyj| lr ;7 JfX
#R@KB fdfyj|,
ν = F ∙ F ‹ lr ucf fdfyj|
h e h c `
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uykdrhfyfe hyom ucf f`fbuyod uo
ucf wyouod
[9 lr wcouod fdfyj| r{vkyfe
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 6
Byorr rfbulod
7/3/2019 QED and Particle Physics.
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Douf ucf zfka efwfdefdbf od {9 hoy ucf ELR rbkuufyldj zcfyf
Z 9 = M 9 ! 9M #F ∙F ‹,∙ 5FF ‹rld9#γ/9,
hoy f`krulb rbkuufyldj Z = M
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 8
Wkyuod rbkuufyldj
• Ucf byorr rfbulod f ! w ↙ f ! S rcov`e yfevbf uo f ! { ↙ f ! { #zclbc lr
lefdulbk` uo f ! ¶ ↙ f ! ¶,
eς
eℨeF 2#f¶ ↙ f¶, =
5λ9F 92
{5
±bor9
γ
9∙
{9
9mrld9
γ
9
‒ε#ν !
{9
9m,
eς
eℨeF 2 #fw ↙ fS, =
5λ9F 92
{5±
Z 9bor9
γ
9 ! 9Z ;rld9
γ
9‒
zcfyf ν = F ; ∙F 2 #ldlulk` kde dk` fdfyjlfr,
• Hoy bodxfdlfdbf efdf [9 = ∙{9 #[9 lr k dfjkulxf {vkdulu|,
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• Hoy bodxfdlfdbf efdf [ = ∙{ #[ lr k dfjkulxf {vkdulu|,$
• Yf`kuldj f¶ ↙ f¶ kde fw ↙ fS byorr rfbulodr jlxfr
9Z woldu;
=[9
9mε#ν ∙
[9
9m, Z woldu
9= ε#ν ∙
[9
9m,
• Ku `kyjf [9 ldf`krulb fw rbkuufyldj lr xlfzfe kr f`krulb f-{vkya rbkuufyldj ohh k
hyff {vkya ldrlef ucf wyouod
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf :
Rbk`ldjUcf Z ; ruyvbuvyf hvdbulod lr
oirfyxfe uo if ldefwfdefdu oh
[
9
Ucf rbk`ldj ifckxlovy zkr fs-
w`kldfe i| Hf|dmkd zco rkle
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ucku ucf wyouod zkr mkef vw oh
woldu-`laf wkyuodr$Ruyvbuvyf hvdbulodr Z ↙ H #s,
kyf jlxf ucf elruylivulod oh wky-
uodr zlucld ucf wyouod
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 3
Wkyuod Elruylivulod Hvdbulodr
• Lh {vkyar kyf uyv`| hyff zlucld ucf dvb`fvr hoy rvhblfdu`| cljc-fdfyj| wyoifr
ucf WEHr zl`` if ε-hvdbulodr$ Hoy ucf wyouod ucfd
H w9
#s, = s
#9
—9
2
¡9
ε‐
s∙mv
M
”!
—∙
;
2
¡9
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s∙me
M
”,
• Moyf jfdfyk``| zf bkd jfdfyk`l~ldj ucf WEHr>
H w9
#s, = s
#—9
2
¡9
v#s, !
—;
2
¡9
e#s,
,
7/3/2019 QED and Particle Physics.
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#—2
¡ —2
¡ ,
• Ucf wyfblrf efufymldkulod oh v#s, kde e#s, lr mfkrvyfe i| fswfylmfdu ivu
ucfrf hvdbulodr mvru rkulrh| bfyukld rvm yv`fr>
^ ;
7
sv#s, es = 9^ ;
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se#s, es
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf <
Bodruykldur od WEHr
• Fswfylmfduk` rvywylrf> iouc rlefr oh ucf kioxf f{vkulod kyf mfkrvyfe uo if
7$28 mfkdldj ucku od`| 65& oh ucf wyouod“r momfduvm lr kbbovdufe hoy$
Zcku ckwwfdfe uo ucf oucfy 58&?
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;7
J`vodr
• Rldbf j`vodr kyf f`fbuylbk``| dfvuyk` ucf| eo dou boduylivuf uo f w rbkuufyldj
ivu ucf| kyf fxlefdu`| cokyeldj kzk| romf oh ucf wyouod momfduvm #kde
rwld uoo,$
• Uclr `fker uo k `odj `lru oh WEHr ucku zl`` if yf{vlyfe uo efrbylif ucf wyouod
kbbvykuf`|>
7/3/2019 QED and Particle Physics.
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v#s, e#s, r#s, $ $ $ v#s, e#s, r#s, $ $ $ j#s,
• Zcfyf zf odbf cke nvru odf vdadozd hvdbulod H 9#s, zf doz ckxf ;2+
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;;
Yf`kuldj ucf WEHr
• I| elruldjvlrcldj ifuzffd xk`fdbf kde rfk {vkyar zf bkd b`fky vw moru oh ucf
b`vuufy$ Rldbf ucf rfk {vkyar kyf k`` wyoevbfe i| ucf rkmf j`vod-rw`luuldj
mfbckdlrm
v#s,≂
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r#s,≂
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• Hoy v#s, kde e#s, zf rfwkykuf ucf xk`fdbf kde rfk boduylivulodr ro ucku
7/3/2019 QED and Particle Physics.
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v#s, = vx#s, ! r#s, e#s, = ex#s, ! r#s,
• Ucf dfvuyod WEHr kyf yf`kufe uo ucf wyouod WEHr i| lrorwld #l$f$
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;9
Wyouod kde Dfvuyk` WEHr L
Wyouod ruyvbuvyf hvdbulod #vve,
H w9
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5
<Qvw ! vw] !
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cew ! e
wl
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H d9
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s =
5
< Qvd
! vd
] !
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ed
! edl
!
;
< Qrd
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]
Vrldj lrorwld ldxkylkdbf
vw = ed = v#s,
7/3/2019 QED and Particle Physics.
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ew = vd = e#s,
rw = rd = r#s,
Zf jfu
H w9
#s,
s=
5
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s=
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<
ce ! e
l!
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;2
Wyouod kde Dfvuyk` WEHr LL
Ucf wyouod bodrlrur oh 2 xk`fdbf {vkyar #vx , vx, ex, kbbomwkdlfe i| mkd|
{vkya-kdul{vkya wklyr
v = vx ! vr e = ex ! er
vr = vr = er = er = rr = rr = r
Ucf {vkya elruylivulodr mvru jlxf ucf boyyfbu {vkduvm dvmifyr^ Qv∙ v] es = 9
e∙ e es = ;
7/3/2019 QED and Particle Physics.
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^ ce el es = ;^ Qr∙ r] es = 7
Ro uckuH w9
#s,
s=
;
<Q5vx ! ex ] !
5
2r
H d9 #s,
s=
;
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5
2r
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Bomwkylrod zluc fswfylmfdu L
H w9
#s,
s=
;
<Q5vx ! ex ] !
5
2r
H d
9 #s,s
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r
7/3/2019 QED and Particle Physics.
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Ucf ykulo H d
H wufder uo
; lh r eomldkufr
5 lh ex eomldkufr;
5lh vx eomldkufr
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;6
Bomwkylrod zluc fswfylmfdu LL
Elhhfyfdbf ifuzffd wyouod kde dfvuyod
hvdbulodr
H w9
#s,∙ H d9 #s, =s
2Qvx ∙ ex ]
7/3/2019 QED and Particle Physics.
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2
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;8
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;:
Wkyuod Elruylivulod Hvdbulodr
7$2
7$5
7$6
7$8
7$:
s h
# s ,
7/3/2019 QED and Particle Physics.
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7
7$;
7$9
7 7$; 7$9 7$2 7$5 7$6 7$8 7$: 7$3 7$< ;
s
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;3
Iyfkaeozd oh rbk`ldj
Ku cljc momfduvm #cljc [9, ucf
wkyuodr kyf hyff ldelbkuldj ucku ucf
rbkuufyldj oh ucf wkyuod eofr dou khhfbu
ucf oucfy wkyuodr$
7/3/2019 QED and Particle Physics.
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Ku rmk``fy xk`vfr oh s ucfyf lr k ruyodjxlo`kulod oh rbk`ldj$
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf ;<
F`fbuyod-wyouod rbkuufyldj ku CFYK
Ucf EFR\ @kioykuoy| ld Ckmivyj owfykufr ucf CFYK f-wyouodbo``lefy$
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 97
CFYK
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 9;
^fvr kde C; fswfylmfdur
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 99
Effw ldf`krulb rbkuufyldj ELR
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 92
F`fbuyod wyouod byorr rfbulod
Ucf elhhfyfdulk` byorr rfbulod hoy ucf dfvuyk` bvyyfdu wyobfrrf D ↙ f S bkd if zyluufd kr
e9ς
es es=
5χλ9
[5#r∙M 9, ;∙ | ∙
M 9s|
r M9H 9 ! s|9 H ;
7/3/2019 QED and Particle Physics.
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es es [5 ±— r∙M 9¡ ‒
Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 95
Dfvuyk` bvyyfdu ELR
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 96
Bckyjfe bvyyfdu ELR
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 98
Aldfmkulb yfkbc oh CFYK
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 9:
H 9 ruyvbuvyf hvdbulod
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 93
Wkyuod efdrlu| hvdbulodr #WEH,
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 9<
Zc| lr uclr lmwoyukdu?
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Wc|rlbr 678K ;9 - Wkyuod Moef` Wkjf 27
[BE
• Efrbylildj Bo`oy• Ucf Hf|dmkd Yv`fr
• Romf Rlmw`f Fskmw`fr
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• Ckeyod bo``lefy wc|rlbr
Wc|rlbr 678K ;2 - [BE Wkjf ;
[vkya Bo`oy
• Yfbk`` ucku zf ckxf ldhfyyfe ucf fslrufdbf oh 2 {vkya bo`oyr hyom ucf ckeyod
wyoevbulod ykuf
Y =ς#f! f∙ ↙ ckeyodr,
ς#f! f∙
↙¶! ¶∙,
• Efdouldj ucf ucyff bo`oyr kr ––yfe” ‐i`vf” kde ‐jyffd” zf zl`` dffe uo
kwwfde k bo`oy xfbuoy b uo fxfy| fsufydk` {vkya zkxfhvdbulod #v oy x,>
7/3/2019 QED and Particle Physics.
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by =
7IIG;7
7
;BBK bi =
7IIG7;
7
;BBK bj =
7IIG77
;
;BBK
Wc|rlbr 678K ;2 - [BE Wkjf 9
3 J`vod Bo`oyr
• Zcfd k j`vod ldufykbur zluc k {vkya ucf {vkya bo`oy mljcu bckdjf$ Uclr
mfkdr ucku ucf j`vod bkyylfr odf vdlu oh bo`oy kde odf vdlu oh kdulbo`oy
• [BE lr ikrfe od kd RV #2, bo`oy r|mmfuy| ro ucfyf kyf 3 j`vodr
• Fsw`lblu`|2⊚ 2
=3⊖ ;
mfkdr ucku zf ckxf k bo`oy obufu
p; = #yi ! iy,/∞
9 p6 = ∙l#yj ∙ jy,/∞
9
p9 = ∙l#yi∙ iy,/∞
9 p8 = #ij ! ji,/∞
9
2 = #yy ii,/∞
9 : = l#ij ji,/∞
9
7/3/2019 QED and Particle Physics.
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p ∙ p ∙ ∙p5 = #yj ! jy,/∞9 p3 = #yy ! ii∙ 9jj,/∞8
kde k bo`ovy rldj`fu p< = #yy ! ii ! jj,/∞
2
Wc|rlbr 678K ;2 - [BE Wkjf 2
Ucf Jf``-Mkdd Mkuylbfr
• Ucf 3 j`vod rukufr bkd if yfjkyefe kr 2Ú 2 mkuylbfr ld ‐bo`oy-rwkbf”$ Ucfrfkyf ucf Jf``-Mkdd mkuylbfr>
α; =
7IIG
7 ; 7
; 7 7
7 7 7
;BBK α9 =
7IIG
7 ∙l 7
l 7 7
7 7 7
;BBK α2 =
7IIG
; 7 7
7 ∙; 7
7 7 7
;BBK
α5 =
7IIG
7 7 ;
7 7 7
; 7 7
;BBK α6 =
7IIG
7 7 ∙l
7 7 7
l 7 7
;BBK α8 =
7IIG
7 7 7
7 7 ;
7 ; 7
;BBK
α: =
7I 7 7 7
7 7 ∙l
;B
α3 =;∞
2
7I ; 7 7
7 ; 7
;B
7/3/2019 QED and Particle Physics.
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IG7 l 7
BK IG7 7 ∙9
BK
• Ucf Jf``-Mkdd mkuylbfr zl`` kwwfky ld ucf {vkya-j`vod xfyufs hkbuoy hoy [BE$
• Ucf Jf``-Mkdd mkuylbfr αλ kyf ucf RV #2, bovdufywkyur oh ucf Wkv`l mkuylbfr
ςl hoy RV #9,$
Wc|rlbr 678K ;2 - [BE Wkjf 5
J`vod Zkxfhvdbulodr
p; ⇙ k =
7
IIIIIIIIII
;
7
7
7
7
7
;
BBBBBBBBBBUcf j`vod zkxfhvdbulodr zl`` bodrlru
oh k wo`kyl~kulod xfbuoy #ǡ¶, kde kd 3-
bomwodfdu bo`oy xfbuoy kλ>
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IIIIIIG 7
7
BBBBBBK
Wc|rlbr 678K ;2 - [BE Wkjf 6
Xfyufs Hkbuoyr
• Ucf ikrlb [BE xfyufs
{
{
j ,λ,¶
boduylivufr k hkbuoy oh
∙ ljr
9αλβ ¶
7/3/2019 QED and Particle Physics.
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• jr lr k elmfdrlod`frr bovw`ldj bodrukdu kde lr yf`kufe uo ucf [BE xfyrlod oh
ucf df-ruyvbuvyf bodrukdu i|
λr =j9r
5χ
Wc|rlbr 678K ;2 - [BE Wkjf 8
J`vod Rf`h-Bovw`ldjr
• Nvru kr wcouodr bovw`f uo wkyulb`fr zluc dod-~fyo f`fbuylb bckyjf j`vodr bovw`f
uo wkyulb`fr zluc bo`oy$
• Wcouodr kyf f`fbuylbk``| dfvuyk` ro ucf| eo dou bovw`f #elyfbu`|, uo oucfy
wcouodr$ J`vodr ckxf bo`oy ucfyfhoyf ucf| bkd bovw`f elyfbu`| uo fkbc oucfy >
7/3/2019 QED and Particle Physics.
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O#jr, O#j9r,
Wc|rlbr 678K ;2 - [BE Wkjf :
Wyowkjkuoyr
• Fkbc ldufydk` j`vod boddfbur uzo xfyulbfr“r oh ucf hoym αλβ ¶ kde αιβ ν ro zf
rcov`e fswfbu ucf j`vod wyowkjkuoy uo boduykbu ucf ldelbfr λ zluc ι kde ¶
zluc ν $
J`vod wyowkjkuoy> ∙lj¶ν
ελι
{9
• Ldufydk` {vkyar ckxf ucf hkml`lky hfymlod wyowkjkuoy
[ a u l#/ ! ,
7/3/2019 QED and Particle Physics.
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[vkya wyowkjkuoy> l#/{ ! m,{9 ∙m9
Ucf rljd oh { mkuufyr cfyf ― zf ukaf lu uo if ld ucf rkmf elyfbulod kr ucf
hfymlod kyyoz$
Wc|rlbr 678K ;2 - [BE Wkjf 3
Fsufydk` @ldfr• Kr ld [FE ucf fsufydk` `ldf hkbuoyr mvru rlu od ucf ovurlef ld oyefy uo mkaf k
dvmify ovu oh ucf kmw`luvef$
• Zoya ikbazkyer k`odj fxfy| {vkya `ldf vrldj>
7/3/2019 QED and Particle Physics.
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{ ld { ovu { ld { ovu j ld j ovu
v b v b‧ xb‧ x b ǡ¶ kλ ǡ√¶ kλ√
Wc|rlbr 678K ;2 - [BE Wkjf <
Fskmw`f> Ucf [vkya-Kdul{vkya Ldufykbulod
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; 9
2 5
• Hyom ucf Hf|dmkd yv`fr zf ckxf
7/3/2019 QED and Particle Physics.
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M = lc
v2b‧2
l—∙ ljr
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Wc|rlbr 678K ;2 - [BE Wkjf ;7
Ucf Oyljld oh Bo`oy Hkbuoyr
M = ∙ j9r h
{9Qv2β ¶v;] Qx9β ¶x5]
h =;
5
‐b‧2
αλb;”‐
b‧9
αλb5”
• Uclr [BE kmw`luvef `ooar nvru `laf ucf [FE kmw`luvef hoy f∙ ¶! rbkuufyldj
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• Uclr mfkdr ucku ldrohky ucku λr lr rvhblfdu`| rmk`` uo nvrulh| wfyuvyikulxf
[BE zf zl`` ckxf k Bov`omi `laf woufdulk`
7/3/2019 QED and Particle Physics.
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[BE zf zl`` ckxf k Bov`omi-`laf woufdulk`
X {{#y, = ∙ h λr
y
Wc|rlbr 678K ;2 - [BE Wkjf ;;
Bo`ovy hkbuoyr
• Obufu Bodjvykulod
h = ;5
‐b‧2αλb;
”‐b‧9αλb5
”= ∙;
8
• Rldj`fu Bodjvykulod
h = ;5
‐b‧2αλb;
”‐b‧9αλb5
”= 5
2
• Zluc ucf bo`oy hkbuoyr zf ckxf bk`bv`kufe ucf {{ woufdulk`r kyf>
X # ,30 ∙5λr/2y #bo`oy rldj`fu,
7/3/2019 QED and Particle Physics.
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X {{#y, =30> ∙5λr/2y #bo`oy rldj`fu,
!λr/8y #bo`oy obufu,
• Uclr mfkdr ucku ucf {vkya kde kdul{vkya kyf kuuykbufe uo fkbc oucfy ld ucf
rldj`fu bodjvykulod ivu yfwf``fe ld ucf obufu bodjvykulod$⇙ Uclr fsw`kldr zc| mfrodr kyf bo`oy rldj`fur$
Wc|rlbr 678K ;2 - [BE Wkjf ;9
Hf|dmkd elkjykmr
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf ;
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7/3/2019 QED and Particle Physics.
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xlk ucf wyobfrrfr
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf 5
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• Ucf wyowojkuoy lr
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7/3/2019 QED and Particle Physics.
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M = l
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∙! ν f
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M =j9z
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• Douf ucf `fkeldj hkbuoy oh ;9
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7/3/2019 QED and Particle Physics.
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Qw9¶w5ν ! w9νw5¶ #w9 w5,j¶ν ¶νξυ w9w5 ]
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7/3/2019 QED and Particle Physics.
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± ‒• Uclr jlxfr
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M 5
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7/3/2019 QED and Particle Physics.
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eℨ=
9
5χM 9Z
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mfkdr hoy ufruldj ucf Rukdekye Moef` kde rfkybc hoy fxlefdbf oh Dfz Wc|rlbr
Z
; > ¶
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M =j9z
3M9
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7/3/2019 QED and Particle Physics.
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3M 9Z
ˈ ː ˈ¶
ː
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9
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Mvod Efbk| LL
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EpMp9F =9j5z
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• Rldbf ucf aldfmkulbr oh ¶ efbk| kyf elhhfyfdu hyom ucorf oh f ν ¶ rbkuufyldj zf
zl`` dffe uo rukyu ovy zoya cfyf$
• Ld ucf mvod yfru hykmf # w; º w9, = m¶F 9 kde
# w2 º w5, =ˈ
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• Ucf rwld-kxfykjfe r{vkyfe mkuyls f`fmfdu rlmw`lfr uo
EpMp9F
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9
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• Ykucfy uckd `lru ucf efylxkulod ld ucf `fbuvyfr ucf yfkefy lr fdbovykjfe uo
7/3/2019 QED and Particle Physics.
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yfxlfz lu ld Jylhucr
Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf ;2
Mvod Efbk| LX
• Ucf efylxkulod oh ucf mkuyls f`fmfdu |lf`er xk`vki`f aldfmkulb ldhoymkulod
mksqF 9, F 5} 0 ;
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• Ucf rvm oh ucf momfduk hoy ucf 2 dk`-rukuf wkyulb`fr mvru if ~fyo ucfyfhoyf
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• Ucf rvm oh ucf momfduk hoy ucf 2 dk` rukuf wkyulb`fr mvru if ~fyo ucfyfhoyf
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf ;5
F`fbuyod fdfyj| rwfbuyvm ld mvod efbk|
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Ohufd odf mfkrvyfr ucf momfduvm rwfbuyvm oh ucf ovujoldj f`fbuyod
• Ucf ho``ozldj f{vkulod efrbylifr ucf f`fbuyod-fdfyj| rwfbuyvm oh mvod efbk|$
e¶
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9#5χ,2 —;∙ 5F
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf ;6
F`fbuyod fdfyj| rwfbuyvm ekuk
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf ;8
Mvod Efbk| Ykuf
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3M 9
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7/3/2019 QED and Particle Physics.
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υ ¶ =;<9χ2
J9
H m6¶
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J ; ;882:#;, ;7∙6 J X∙9
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JH = ;$;882:#;,Ú ;7∙6 JfX∙9
Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf ;3
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3M 9Z
= ;$;88Ú ;7∙6 JfX∙9
• Zf bkd vrf ucf Z mkrr mfkrvyfmfdu M Z = 37$5 JfX uo efufymldf jz$
jz = 7$86 ⇙ λz =j9z5χ
=;
9<
• Ucf zfka ldufykbulod lr ldcfyfdu`| ruyodjfy uckd ucf FM ldufykbulod+
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Lu lr od`| ucf rvwwyfrrlod hkbuoy F 9/M 9Z zclbc mkafr ucf zfka hoybf rffm ro
hffi`f$
Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf ;<
Uzlru fswfylmfdu ku UYLVMH L
• Ucf f`fbuyod fdfyj| rwfbuyvm lr mfkrvyfe i| ucf #kbulxf, UZLRU fswfylmfdu
ku UYLVMH$• Rldbf uclr fswfylmfdu vrfr wo`kyl~fe mvodr ucf elyfbulod oh ucf f`fbuyod lr do
`odjfy kyiluyky| kde k elruylivulod ld F kde γ lr byfkufe$
• lh ucf r|rufmkulb vdbfyukldulfr kyf mkef rvhblfdu`| rmk`` odf bkd `ooa hoy
fxlefdbf oh wc|rlbr if|ode ucf Rukdekye Moef`#f$j$ yljcu-ckdefe bckyjfe-bvyyfdu ldufykbulodr xlk Z Y iorodr,$
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf 97
Mvod wyoevbulod ku UYLVMH
Mvodr kyf elyfbufe lduo ucf efufbuoy zcfyf ucf f`fbuyod momfduvm lr mfkrvyfe$
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf 9;
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Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf 99
Mlbcf` wkykmfufyr
• Fswfylmfdur rvbc kr Uzlru mfkrvyf ucf kdjv`ky kde momfduvm efwfdefdbfoh ucf ovujoldj f`fbuyod
• Ykucfy uckd krrvmf ucf domldk` hoym oh ucf zfka ldufykbulod X ∙ K zf bkd
ufru hoy rbk`ky wrfveorbk`ky kde oucfy ufdroy bomwodfdur
• Mlbcf` rwfbuyvm #rff ¶ rfbulod ld WEJ,
eΒ
eseborγ≃ s9
2#;∙ s, !
9ϋ
2#5s∙ 2, ! 2κ s7
;∙ s
s
» s9
W ¶ δ borγ±
;∙ s !
9ε
2 #5s∙ 2,‒
zcfyf s = F ¶/Z s7 = mf/Z kde Z = #m9¶ ! m9
f,/9m¶
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¶
• Ld ucf Rukdekye Moef` ϋ = δε = 7$:6 δ = ; kde κ = 7
Wc|rlbr 678K ;5 - Zfka ldufykbulodr Wkjf 92
Bckyjfe Zfka Ldufykbulodr
• Dfvuyod Efbk|
• Wlod Efbk|
• Ucf BAM Mkuyls
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;
Dfvuyod Efbk|
• Ljdoyldj dfvuyod rviruyvbuvyf zf bov`e moef` dfvuyod efbk| kr k zfka
ldufykbulod wyobfrr mvbc `laf mvod efbk|>
Z
; > d
2 > w9 > ν f
5 > f
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• Ld mvod efbk| k`` 2 dk`-rukuf wkyulb`fr #ν ¶ ν f kde f, kyf frrfdulk``| mkrr`frr$
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 9
Aldfmkulbr oh Dfvuyod Efbk|
• Ld dfvuyod efbk| ucf wyouod mkrr lr oixlovr`| {vluf `kyjf$
• Ld keelulod ucf mkrr oh ucf f`fbuyod #7$6 MfX, lr k rljdlbkdu hykbulod oh ucf
dfvuyod-wyouod mkrr elhhfyfdbf #;$2 MfX, ro zf bkddou ljdoyf mf$
•Kr k yfrv`u ucf wckrf-rwkbf bk`bv`kulod hoy dfvuyod efbk| lr moyf elhbv`u uckd
ucku oh mvod efbk|$ Bodrv`u Jylhucr lh |ov zov`e `laf uo rff ucf efukl`r$
• Vrldj k wvyf X ∙ K xfyufs hkbuoy zf oiukld k dfvuyod `lhfulmf oh
υ d = ;2;8 r
• Ucf fswfylmfduk``| mfkrvyfe xk`vf lr 336$: » 7$8 r #kiovu ;6 mld,
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• Wyouod `lhfulmf 4 ;79< |fkyr
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 2
Wlod Efbk| L
• Zcl`f ucf χ7 #vv ∙ ee, efbk|r uo β ! β xlk kd f`fbuyomkjdfulb ldufykbulod ucf
bckyjfe wlodr #ve kde ev, efbk| uo k `fwuod-dfvuyldo wkly ucyovjc ucf zfka
ldufykbulod$
•Ld romf yfrwfbur χ
∙efbk| bkd if yfjkyefe kr k rbkuufyldj wyobfrr>
Z
e v
ν ℒ ℒ
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 5
Wlod efbk| LL
• Lh zf“yf joldj uo ckxf romf vdadozd hkbuoy kwwfkyldj ld ovy yfrv`ur zf
rcov`e mkaf ucf yfru oh ucf bk`bv`kulod kr rlmw`f kr worrli`f$
• @fu“r moef` χ∙ efbk| i|>
Z
χ∙
ν ℒ ℒ
zcfyf ucf χ Z ldufykbulod ku ucf i`oi lr efrbylife i| ucf xfyufs hkbuoy
∙ljz∞ hχw¶χ
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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,
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 6
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
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efufymldfe hyom χ! ↙ ¶!ν efbk|r #WEJ 977:,
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 8
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 $9
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⇙ EpMp9
F = j9zh χ
3M
9
Z +9
w;¶ w;ν 3 ˈ w¶9
wν2 ! wν9 w¶2
∙# w9
º w2,j¶νː
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf :
Ucf Wlod Efbk| Kmw`luvef LLL
EpMp9F =
;
3 j9zh χ
M 9Z +9
ˈ9# w; º w9,# w; º w2, ∙ m
9
χ# w; º w9,ː• Zf bkd fxk`vkuf ucf xkylovr eou wyoevbur i| vrldj w; = w9 ! w2>
w9;
= # w9 ! w2,9
m9χ = m9
ℒ ! 9# w9 º w2,
⇙ # w9 º w2, = #m9χ ∙ m9
ℒ ,/9
Rlml`ky`| # w; º w9, = #m9χ ∙ m9
ℒ ,/9
# w; º w2, = #m9χ ! m9ℒ ,/9
9
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⇙ EpMp9F =;
;8 j9zh χ
M 9Z +
9
m9ℒ #m9
χ ∙ m9ℒ ,
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 3
Ucf Wlod Efbk| Ykuf L
•Yfbk`` hyom kd fky`lfy `fbuvyf
Β =pwh ppMp9
3χm9χ
• Zluc pwh p f{vk``ldj ucf dfvuyldo fdfyj|
pwh p = F 9
= # w; º w9,/mχ
= #m9χ
∙m9ℒ ,/9mχ
⇙ Β =h 9χ
2
—jz
¡5
m9ℒ #m9
χ ∙ m9ℒ ,9
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⇙χm2
χ
—5M Z
¡ℒ # χ ℒ ,
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf <
Ucf Wlod Efbk| Ykuf LL
Β =h 9χ
m2χ
—jz
5M Z
¡5m9ℒ #m9
χ ∙ m9ℒ ,9
• h χ bkd if fsuykbufe hyom uclr fswyfrrlod$• @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 `
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• 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
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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
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;9
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 $
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;2
χ∙ kde A
∙ Efbk|r L
Ucf ykulo oh ucf efbk| ykufr oh χ∙ ↙ `∙ν ¶ uo A ∙ ↙ `∙ν ¶ rcov`e 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
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Β#A ∙
↙`∙ν ¶, =
h 9A
χm2A — jz
5M Z ¡5
m9` #m9
A
∙m9` , bor9γb
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;5
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
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e‹ = 7$<:e ! 7$92r
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;6
Wyoi`fmr zluc ucf Bkilio moef`
• Ucf Bkiliio kdj`f ro`xfe mkd| wyoi`fmr ivu dfz odfr kwwfkyfe$
• Ucf iykdbcldj ykulo hoy A 7 ↙ ¶! ¶∙ lr <$; Ú ;7∙<
Ucf wyfelbufe efbk| ykuf lr wyowoyulodk` uo rldγbborγb zclbc jlxfr k
rljldbkdu`| `kyjfy yfrv`u uckd ucf fswfylmfduk` yfrv`u$
A 7 #er, rukuf
v
Z
ν ¶
Z
¶ ¶
M ∰ !rld γB bor γB
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e r
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;8
Ucf JLM Mfbckdlrm
• J`krcoz L`lowov`or kde Mklkdl #JLM, wyoworfe k ro`vulod ld ;<:7 uo fsw`kld
ucf `oz 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
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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
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf ;3
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
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#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
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w | |
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 97
Vdlukylu| Uylkdj`fr L
• Ucf vdlukylu| uylkdj`fr kyf vrfe uo xlrvk`l~f ucf yfrv`ur 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
7/3/2019 QED and Particle Physics.
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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
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oh ucf kdj`fr
Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 99
Vdlukylu| Uylkdj`fr LLL
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 92
Vdlukylu| Uylkdj`fr LX
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J`oik` ur uo ucf ekuk |lf`e
α = 7$99:9
K = 7$3;3
ϋ = 7$99;
κ = 7$257
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 95
BW Xlo`kulod zluc `fwuodr
• BW xlo`kulod ld ucf vdlxfyrf bkddou fsw`kldfe i| BWX zf rff zluc {vkyar
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• RvwfyA kde RDO ckxf rcozd ucku dfvuyldo mls ν f ⇗⇙ ν ¶ kde ν f ⇗⇙ ν υ
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• Ucfyf mk| if k BAM-`laf mkuyls hoy dfvuyldor7IIG
ν f
ν ¶
ν υ
;BBK =
7IIG
X f; X f9 X f2
X v; X v9 X v2
X u; X u9 X u2
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ν ;
ν 9
ν 2
;BBK
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;6 - Bckyjfe Zfka Ldufykbulod Wkjf 96
Dfvuyk` Zfka Ldufykbulodr
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• Hf|dmkd Yv`fr
• Ucf Zfka Mlsldj Kdj`f
• Yfrodkdbf ld f!
f∙ Rbkuufyldj
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;
Hfyml“r Ucfoy|
• Yfbk`` ucku Hfyml moef`fe ucf zfka ldufykbulod kr k hovy hfymlod bodukbu
ldufykbulod zluc k bovw`ldj oh JH >
h ; h 9
h 2 h 5
M =JH ∞
9
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• Hyom {vkduvm rbkuufyldj ucfoy| odf ckr k wyoi`fm lh pMp∰ ;
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• Hoy ucf Hfyml moef` zf bkd rukuf ucku lu zl`` dou zoya if|ode F ∰ 277 JfX$
c ` e i i # , c c
7/3/2019 QED and Particle Physics.
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Uclr lr yfro`xfe i| ldboywoykuldj k Z -iorod #M Z
∰37 JfX, lduo ucf ucfoy|
oh zfka ldufykbulodr
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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 !
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 2
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
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 5
Elrboxfy| oh ucf ^ 7
• Ucf ^ 7 zkr ldelyfbu`| oirfyxfe ld ;<:2 ld ucf Jkyjkmf``f fswfylmfdu ku BFYD
xlk ucf wyobfrrfr
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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 = <;$;3:8#9;, JfX
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 6
Ucf Zfka Mlsldj Kdj`f
• Mkd| oh ucf wkykmfufyr oh ucf f`fbuyozfka ldufykbulod kyf yf`kufe uo fkbc oucfy$
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
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Uclr lr zc| ucf zfka hoybf lr ldcfyfdu`| ruyodjfy uckd ucf f`fbuyomkjdfulb
hoybf$
7/3/2019 QED and Particle Physics.
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• Fswfylmfduk``| rld9 γz#M , = 7$92;97#;6,
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• Ucf ^ 7 wyowkjkuoy `ooar nvru `laf ucku oh ucf Z >
∙l—
j¶ν ∙ {¶{ν
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∙lj~9
β ¶#bX ∙ bKβ 6,
h
h
^ 7
7/3/2019 QED and Particle Physics.
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h
Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf :
Hfymlod Bovw`ldjr uo ucf ^ 7
• Ucf xfbuoy kde kslk` bovw`ldjr bX kde bK kyf rwfblfe i| ucf JZR moef`>
h bX bK
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Jkvjf Iorod Rf`h-Bovw`ldjr
• Nvru `laf [BE ucf f`fbuyozfka iorodr bkd ldufykbu zluc fkbc oucfy>
Z ∙
Z !
^ 7, β
Z !
Z ∙
\
S
zcfyf #S, \ , bkd if #β, β , #β, ^ 7, #^ 7, ^ 7, oy #Z !, Z ∙,$
Bodrv`u Kwwfdels E oh Jylhucr hoy xfyufs hkbuoyr$
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf <
Bomwkyf β kde ^ 7 elkjykmr
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β ↙ h h ^ ↙ h h
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ifbkvrf ku `oz 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
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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
7/3/2019 QED and Particle Physics.
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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
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^
2::6
Ú±
x9
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9β ν#bfX ∙ bfKβ 6,
¡v;
‒
• Ku `oz fdfyjlfr {9 ≮ M 9^ ucf ^ 7-mfelkufe elkjykm lr rlml`ky ld ruyvbuvyf uo
ucf [FE elkjykm$ Ucf β kmw`luvef zov`e eomldkuf kr {∙9 ≡ M ∙9^
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;9
Rbkuufyldj kmw`luvef ku {9 =M 9^
> L
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Affwldj ucf hv`` wyowkjkuoy
M = ∙ j9~5#{9 ∙M 9
^,
cv5β ¶#bh
X ∙ bh
Kβ 6,x2
lj¶ν ∙ {¶{ν
M 9^
+
Ú ˈx9β ν#bfX ∙ bfKβ
6
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• Lh zf bkd dfj`fbu k`` hfymlod mkrrfr ucf{¶{ν
M 9^
wkyu oh ucf wyowkjkuoy zl``
boduylivuf doucldj rldbf zf bkd zyluf { kr flucfy w; ! w9 oy w2 ! w5$• Zf bkd zyluf { kr flucfy w; ! w9 oy w2 ! w5
ucfd ucf /{ hkbuoyr `fke uo bomildkulodr `laf v5/ w5 kde / w2x2
zclbc i| ucf Elykb f{vkulod kyf v5m5 kde∙m2x2$
7/3/2019 QED and Particle Physics.
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Rbkuufyldj kmw`luvef ku {9 =M 9^
> LL
• Ucf rbkuufyldj kmw`luvef bkd if zyluufd kr
M = ∙ j9~
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X ∙ bh
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#bX ∙ bKβ 6,/ w2β ν#bX ∙ bKβ 6, = #bX ∙ bKβ 6,9/ w2β ν
= #b9X ! b9K,/ w2β ν ∙ 9bX bKβ 6/ w2β ν
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;5
Rbkuufyldj kmw`luvef ku {9 ≃M 9^
> LLL
• Ucf byorr rfbulod hoy ^ 7-mfelkufe f! f∙ ↙ h h lr
ς =;
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9 ! #bfK,9]
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7/3/2019 QED and Particle Physics.
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Wyowojkuoy hoy vdruki`f wkyulb`fr
• Ucf rovybf oh ucf wyoi`fm lr ucku ucf aldfmkulbr kyf rvbc ucku f! f∙ ↙ ^ 7 lr k
wc|rlbk``| k``ozki`f wyobfrr fxfd zlucovu k rvirf{vfdu efbk| uo h h $
• Kr k yfrv`u zf dffe uo moelh| ucf ^ 7-wyowkjkuoy ld oyefy uo kbbovdu hoy ucf
ldrukil`lu| oh ucf ^ 7$ Cfyf“r zcku zf eo>
;$ Zf yfbk`` ucf hkml`lky bodjvykulod-rwkbf zkxfhvdbulod oh k ruki`f
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
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W #u, =
^ pΣp9 e2y = ;
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;8
2$ Lh ucf wkyulb`f lr vdruki`f zf fswfbu ucf wyoikil lu| oh deldj ucf wkyulb`f uohk`` ohh zluc ulmf kbboyeldj uo ucf efbk| ykuf Β
W #u, =
^ pΣp9 e2y = f∙Βu
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
wkyulb`f kde krrvmf ucku Β lr rvhblfdu`| rmk`` ucku zf bkd dfj`fbu ucf Β9 ufym>
;
{9 ∙M 9↙ ;
{9 ∙ #M ∙ lΒ/9,9
≂ ;
{9 ∙M 9 ! lM Β
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;:
Byorr Rfbulod
• Zluc ucf moelbkulod uo ucf ^ 7 wyowkjkuoy
;
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ς ∰ ;
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Ucf cfljcu kde zleuc oh ucf yfrodkdbf wfka kyf efufymldfe i| ucf efbk|
zleuc Β^ $
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;3
Mfkrvyfmfdu oh ucf ^ 7 Wfka > L
Ucf Iyflu-Zljdfy rckwf lr moelfe i| cljcfy-oyefy #ykelkulxf, fhhfbur
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf ;<
Mfkrvyfmfdu oh ucf ^ 7 Wfka > LL
f! f∙ ↙ h h
[FE eomldkufr ku `oz fdfyjlfr
ς^
ςβ≂ 9
—F
M ¡5
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dkufr dfky ucf yfrodkdbf wfka
ς^
ςβ≂ ;
3
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Β^
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 97
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 9;
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7
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2$ Boyyf`kulodr zluc ucf UJX
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 99
Zkufy `fxf`r ld dfkyi| @kaf Jfdfxk
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 92
Ulek` elruoyulodr oh ucf yldj
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 95
Boyyf`kulodr zluc ucf UJX
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 96
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 98
@FW-LL ZZ kde ^^ yfrv`ur>
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 9:
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 93
KU@KR Z kde ^ yfrv`ur> Z ∙
↙ f∙
ν f
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 9<
KU@KR Z kde ^ yfrv`ur Z ∙ ↙ υ ∙ν υ
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 27
KU@KR Z kde ^ yfrv`ur ^ 7↙ f
!
f∙
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 2;
KU@KR Z kde ^ yfrv`ur ^ 7 ↙ υ !υ ∙
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 29
KU@KR Z kde ^ yfrv`ur
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 22
KU@KR Z kde ^ yfrv`ur
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Wc|rlbr 678K ;8 - Dfvuyk` bvyyfdu ldufykbulodr Wkjf 25
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 bov`e 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
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;
Cleldj ucf X ∙K
•F`fbuyomkjdfulrm kde ucf zfka hoybfr kyf xfy| rlml`ky
;$ Ucf β lr mkrr`frr ucf zfka iorodr Z » ^ 7 kyf mkrrlxf
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 » `ooa `laf k wvyf xfbuoyldufykbulod lh zf krroblkuf wkyu oh ucf ldufykbulod zluc ucf hfymlod
zkxfhvdbulod>
∙ljz∞9
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9
¡v
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9Qvβ ¶v@]
zcfyfv@ ≫
#;∙ β 6,
9v
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 9
β 6> Bclyk`lu| xr$ Cf`lblu|
β 6v =
7G 7 ;
; 7
;K7G vK
vI
;K
= 7G vI
vK;K
=
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;
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7
7 #wºς,F∙m
;K7G vK
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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
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 2
Bclyk` Hfymlodr• Rldbf β 6 kbur nvru `laf ucf cf`lblu| owfykuoy º i w hoy mkrr`frr hfymlodr
v@ ≫ ;9
#;∙ β 6,v = 30> 7 lh v ckr cf`lblu| !;
v lh v ckr cf`lblu| ∙;
• Rlml`ky`| zf bkd wyonfbu ovu ucf yljcu-ckdefe wkyu oh k rwldoy>
vY ≫ ;9
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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,
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 5
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#;
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• Zf bkd uclda oh ucf bckyjfe zfka ldufykbulod kr k wvyf xfbuoy ldufykbulod
ifuzffd `fhu-ckdefe hfymlodr$
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 6
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,
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9
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9
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v@β ¶vY = v ;9
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#; ! 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
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 8
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
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf :
Zfka Eovi`fur
• Rldbf ucf Z bovw`fr `fhu-ckdefe `fwuodr kde ucfly dfvuyldor uojfucfy
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ϊ@ =
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f
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• 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 ∙ ≫
7G 7 7
; 7
;K
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 3
K Dfvuyk` Bvyyfdu• Rvwworf zf efdf k uclye υ mkuyls ld oyefy uo bomw`fuf ucf r|mmfuy|>
υ 2
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf <
C|wfybckyjf
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;9
ν @β ¶ν @ ∙ ;9
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;7
Zfka lrorwld kde c|wfybckyjf bvyyfdur
• Zfka lrorwld bvyyfdu n¶ = ;9
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wkyulb`fr Z ! Z ∙ ^ 7 kde β $
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;;
Z ! kde Z ∙
• Odf bkd rcoz ucku n¶ ºZ¶ = n;
¶Z ¶; ! n9¶Z ¶9 ! n2
¶Z ¶2
= ;9
g n;¶ ! ln9
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9 g n;¶
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n!¶ Z ¶! ! ;∞
9 n∙¶ Z ¶∙ ! n2
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Z »¶ ≫ ;∞ 9gZ ;¶ − lZ 9¶µ
zclbc boyyfrwoder uo ucf zkxf hvdbulodr yfwyfrfduldj ucf Z » wkyulb`fr
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;9
Z ! kde Z ∙ Xfyufs Hkbuoyr
• Hyom ucf jfdfyk` ldufykbulod
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9 n\
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zf rff ucku ucf bovw`ldj ldxo`xldj ucf Z ∙ lr
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n∙¶ Z ¶∙
• Zluc n∙¶ = ν @β ¶f@
= 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,
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;2
Zfka mlsldj
• Ucf uzo dfvuyk` rukufr Z
2
kde I mls ld ucf JRZ ucfoy|7G K¶
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• Zluc ucf f`fbuyozfka mlsldj ucf ldufykbulod ufymr hoy ucf dfvuyk` wkyulb`fr kyf
∙l
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9bor γz n\ ¶
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∙l ±jz bor γz n2¶ ∙ j‹
9rld γz n\ ¶
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7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;5
FM bovw`ldj bodrukdur
• Zf zl`` rviruluvuf n\ ¶ = 9 nfm¶ ∙ 9 n2¶$
Lh K¶ lr uo yfwyfrfdu ucf f`fbuyomkjdfulb f`e ucfd
nfm
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z n2
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z n\
¶ ‒=
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¶ ! j‹ bor γzg nfm¶ ∙ n2
¶
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⇙jf = j‹ bor γz = jz rld γz
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;6
^ 7 bovw`ldj bodrukdu
• Vrldj jf = j‹ bor γz = jz rld γz ucf ^ 7 ldufykbulod ufym lr
∙l
±jz bor γz n2
¶ ∙j‹
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rld γz
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rld γz bor γz n2¶ ∙ rld9 γz nfm¶
ː^ ¶
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j~ =jf
rld γz bor γz
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;8
bX kde bK
• @fu“r `ooa ku ucf vw {vkya$ Zluc
∙lj~ n2¶ ∙ rld9 γz nfm¶ ː^ ¶
=∙lj~
9^ ¶
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ucf hoym ∙lj~9
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Wc l 678K ;: F ` a l l W ;:
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;:
Bovw`ldjr uo ucf ^ 7
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h bX bK
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Wc l 678K ;: F ` u a l ul W ;3
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;3
Mfkrvyldj bx kde bK
-7$75;
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Wc|rlbr 678K ;: Ff`fbuyozfka vdlbkulod Wkjf ;<
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf ;<
Mfkrvyldj rld9γZ
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 97
7/3/2019 QED and Particle Physics.
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Wc|rlbr 678K ;: - Ff`fbuyozfka vdlbkulod Wkjf 97
@lmlu od Cljjr mkrr
37$2
37$5
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Wc|rlbr 678K ;: Ff`fbuyozfka vdlbkulod Wkjf 9;
Ucf Rukdekye Moef` Cljjr
• Cljjr bovw`fr uo fxfy|
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| jj j
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Ucf Z !Z ∙ ↙Z !Z
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| jj j
Cljjr kr k ro`vulod uo ucf vdlukylu| wyoi`fm
Zf bkd yfro`xf ucf Z !Z ∙ ↙Z !Z ∙ byorr rfbulod elxfyjfdbf zluc uzokeelulodk` elkjykmr
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Cljjr wyowfyulfr> efbk| zleuc
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Cljjr wyowfyulfr> iykdbcldj ykulor
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Z iorod mkrr efwfdefdbf od Cljjr mkrrUcf mkrr oh ucf Z iorod lr rfdrlulxf uo cljcfy-oyefy boyyfbulodr ucku efwfde od ucf
Cljjr mkrr
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Uow {vkya, Z-iorod, Cljjr efwfdefdbf
Ucf mfkrvyfmfdu oh mZ kde muow bkd if vrfe uo w`kbf `lmlur od ucf k``ozfe
xk`vfr oh mC
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Uow {vkya, Z-iorod mkrr mfkrvyfmfdur
Uow-[vkya Mkrr QJfX]
mu QJfX]
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@FW;/R@E/mu 37$286 » 7$797
Nv`| 97;7
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RM wyfelbulod hoy Cljjr mkrr
Ucf |f``oz yfjlodr kyf fsb`vefe i| elyfbu fswfylmfduk` rfkybcfr$
Ucf i`vef ikde lr ucf ucfoyfulbk` wyfelbulod vrldj ucf uow {vkya kde Z yfrv`ur$
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Nv`| 97;7 m@lmlu
= ;63 JfX
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Elyfbu mfkrvyfmfdu ku @FW
Wc|rlbr 678K ;3 - Cljjr iorod Wkjf ;7
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Byorr rfbulod ku ckeyod bo``lefy
Wc|rlbr 678K ;3 - Cljjr iorod Wkjf ;;
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Cljjr wyoevbulod
Wc|rlbr 678K ;3 - Cljjr iorod Wkjf ;9
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Elyfbu mfkrvyfmfdu ku Ufxkuyod
Wc|rlbr 678K ;3 - Cljjr iorod Wkjf ;2
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Elrboxfy| woufdulk` ku ucf @CB
Wc|rlbr 678K ;3 - Cljjr iorod Wkjf ;5
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Lmwkbu oh @CB yvddldj ku : UfX
Wc|rlbr 678K ;3 - Cljjr iorod Wkjf ;6
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KU@KR 97;7 Cljjr `lmlur
Ucf Cljjr iorod mkrr ykdjfr
hyom ;58 JfX uo 927 JfX 968JfX uo 939 JfX kde 9<8 JfX uo
56< JfX kyf fsb`vefe ku ucf <6&
B@
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Dfvuyldor• rwld ;
9wkulb`f zluc do f`fbuylb bckyjf kde zfka`| ldufykbuldj
• ν f elrboxfyfe ld ;<62 ν ¶ ld ;<89 kde ν υ ld ucf ;<<7r
• dfvuyldo wc|rlbr lr xfy| uowlbk`
ro`ky dfvuyldo wyoi`fm dfvuyldo mlsldj dfvuyldo mkrrfr
U9A wyonfbu
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ν f elrboxfy|
• ν f elrboxfyfe i| Yfldfr kdeBozkd ld ;<62 ku Ckdhoye #Doi`f
Wyl~f ld ;<<6,
• Ucf Zkufy ! BeB` efufbuoy zkr rlu-
vkufe dfky k dvb`fky yfkbuoy
• Dfvuyldo wyoevbfr dfvuyod hyom
ldufykbulod zluc wyouod
• Dfvuyod zkdefyr kyovde kde lr
bkwuvyfe i| Bkemlvm #Be√,
• Be√ efbk|r uo jyovde rukuf i| wco-
uod fmlrrlod
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ν ¶ elrboxfy|
• @fefymkd Rzkyu~ Rufldifjfy elr-boxfyfe ν ¶ ku ID@ KJR #Doif`
Wyl~f ;<33,
• ;6 JfX wyouodr wyoevbu wlod
ifkmr zclbc ucfd efbk| ¶↙ fνν
• F`fbuyodr kyf ykdjfe ovu `fkxldj k
dfvuyldo ifkm
• Dfvuyldor kyf rbkuufyfe ohh efufb-
uoy lh f`fbuyod fmfyjfr uckd dfvuyl-
dor lr ν f kde mvod fmfyjfr ucfd lu
lr ν ¶
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ν υ elrboxfy|
Elrboxfy| oh υ `fwuod ld f!f∙ ↙ f!¶∙S yf{vlyfr k ν υ oucfyzlrf `fwuod dvmify lr
xlo`kufe
Wc|rlbr 678K ;< - Dfvuyldor Wkjf 5
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Od`| 2 `ljcu dfvuyldor
Ucf ^ 7 ↙ hh lr k``ozfe hoy kd| hfymlod zluc mh 0;9m^ $ Ucf efbk| ykuf oh ucf
^ 7 lr efwfdefdu od ucf dvmify oh hfymlodr #ldb`veldj dfvuyldor,$
Rfkybcfr hoy k hovyuc-jfdfykulod #mkrrlxf, `fwuod yfrv`u ld `lmlur oh 4 ;77 JfX$
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Eo Dfvuyldor Ckxf Mkrr?
• Kr hky kr elyfbu mfkrvyfmfdur jo>
ν H`kxoy Mkrr @lmlu Wyobfrr
ν f mν 0 2 fX 2C ↙2 Cf ! f∙ ! ν f
ν ¶ mν 0 ;<7 afX χ ↙ ¶ ! ν ¶
ν υ mν 0 ;3$9 MfX υ ↙ 2χ ! ν υ
• Ucfoyfulbk``| ucf Rukdekye Moef` krrvmfr ucku dfvuyldor kyf wyfblrf`|
mkrr`frr$
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ν f mkrr mfkrvyfmfdu
ν f mkrr lr mfkrvyfe ku ucf fdewoldu oh ucf f`fbuyod fdfyj| rwfbuyvm
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ν ¶ mkrr mfkrvyfmfdu
• Ucf ν ¶ mkrr lr mfkrvyfe hyom wlod efbk| ekuk χ! ↙ ¶!ν ¶
• Hoy k wlod ku yfru
m9ν¶ = m9χ ! m9¶ ∙ 9mχ{ wp9¶ ! m9¶
• Zf vrf ucf adoz wlod kde mvod mkrrfr
mχ = ;2<$68<<6 » 7$777726 MfX
m¶ = ;76$86323<» 7$7777725 MfX
• K`odj zluc ucf wlod momfduvm
w¶ = 9<$:<97:» 7$777;9 MfX
• Uclr jlxfr kd vwwfy `lmlu oh mν¶ 0 7$;8 MfX
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ν υ mkrr mfkrvyfmfdu@FW fswfylmfdur mfkrvyfe ucf `lmlu od ucf ν υ mkrr vrldj υ ↙ 2χ∙9χ!ν υ efbk|r$
M #ν υ , 0 ;3 MfX
m 6χ
QJfX ]
F 6 χ Q
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Ó Ó Ò Ù Á ¾ × Û Ù Ó × ¾ µ Ã Ø Ó × Ó Ø Ò Ü Õ ¶ ³ Ã Ý Ú ¾ Å Ó Ó ¾ ´ ³ ½ µ ´ ° ° ¶ Ù ´ ° ° Õ Ø Ó × Ü Õ Ó Ò Ø Ó Ò Ù ´ ° ° µ Í Î Å Ó Ó Õ ± ± Õ ¾ × × ¾ Ò Þ ¶
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Mknoykdk Dfvuyldor• Odf oh ucf moru lmwoyukdu owfd {vfrulodr ld dfvuyldo wc|rlbr lr ucf {vfrulod
oh zcfucfy dfvuyldor kyf Mknoykdk oy Elykb wkyulb`fr$
Elykb dfvuyldor> wkyulb`fr kyf elruldbu hyom ucfly kdul-wkyulb`fr$Mknoykdk dfvuyldor> wkyulb`fr kyf lefdulbk` uo ucfly kdul-wkyulb`fr
• Kuufmwur uo efufbu ucf Mknoykdk dkuvyf oh dfvuyldor hobvr kyovde ucf eovi`f
ifuk efbk| wyobfrr #K,^ , ↙ #K,^ ! 9, ! 9f∙ ! 9ν f
• Lh dfvuyldor kyf Mknoykdk wkyulb`fr ucf kdul-dfvuyldo fmluufe i| odf oh ucfdfvuyodr bkd if kiroyife kr k dfvuyldo i| ucf oucfy$ Ucf yfrv`uldj wyobfrr ld
zclbc do dfvuyldor kyf fmluufe lr dfvuyldo`frr eovi`f ifuk efbk|
#K,^ , ↙ #K,^ ! 9, ! 9f∙
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Eovi`f ι Efbk|
Ucf f`fbuyod fdfyj| rwfb-uyvm hyom ιινν lr boduldvovr
zcfyfkr ucf 7νιι rwfbuyvm
`ooar `laf k 9-ioe| efbk|
Wc|rlbr 678K ;< - Dfvuyldor Wkjf ;9
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RDO! Fswfylmfdu• Ucf oyljldk` RDO fswfylmfdu lr dou ifldj moelfe hoy eovi`f ifuk efbk|
fswfylmfdur
• Ucf cfkx| zkufy lr ifldj yfw`kbfe i| `l{vle rbldul``kuoy w`vr k ykelokulxfrovybf ;67De zclbc ckr iffd oirfyxfe uo efbk| xlk ιινν zluc k `lhfulmf oh
<$: Ú ;7;3 |fkyr
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Ro`ky Dfvuyldo Wyoi`fm• Ld ;<8: Yk| Ekxlr wvu ;77777 jk`-
`odr oh ey|-b`fkdldj vle ld k ukda
k ml`f vdefyjyovde uo uy| uo mfk-
rvyf ucf ro`ky dfvuyldo vs$
• Ucf mfkrvyfe vs #ldhfyyfe hyom ;
B` uo Ky bodxfyrlod fxfy| 9 ek|r,
zkr kiovu k hkbuoy oh 2 if`oz ucf
ucfoyfulbk` fswfbukulodr hyom ucfRukdekye Ro`ky Moef`$
• Rlml`ky dfvuyldo efblur zfyf `kufy
oirfyxfe hoy ucf kumorwcfylb dfv-
uyldor jfdfykufe i| bormlb yk|r$
• Iuc fswfylmfdu kde ucfoy| uvydfe
ovu uo if yljcu kde ucfrf fhhfbur kyf
doz vdefyruooe kr dfvuyldo orbl``k-
ulod fhhfbur$
Wc|rlbr 678K ;< - Dfvuyldor Wkjf ;5
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Wc|rlbr 678K ;< - Dfvuyldor Wkjf ;6
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Wc|rlbr 678K ;< - Dfvuyldor Wkjf ;8
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Efufbuldj dfvuyldor
Fswfylmfdur dffe uo if
`kyjf kr ucf ykufr kyf `oz
effw uo yfevbf ucf bormlb yk| ikbajyovde
b`fkd uo f`lmldkuf ucf ykelokbulxf ikbajyovder
Zkufy Bfyfdaox ef-ufbuoyr kyf k bommod
bcolbf #fj$ Rvwfy-A,
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Rvwfy-Akmloakdef
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Wc|rlbr 678K ;< - Dfvuyldor Wkjf 97
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Dfvuyldo Mlsldj
• Lh dfvuyldor ckxf mkrr ucfd zf mvru doz k``oz k mlsldj ifuzffd ucf zfka
fljfdrukufr kde ucf mkrr fljfdrukufr$
#Nvru kr zf ckxf eodf hoy ucf {vkyar ,
• Ucf dfvuyldo kdk`ojvf oh ucf BAM mkuyls lr ucf Mkal-Dkakjkzk-Rkakuk
#MDR, mkuyls$
• @laf ucf BAM mkuyls ucf MDR mkuyls bkd if wkykmfuyl~fe ld ufymr oh 2
mlsldj kdj`fr kde ; BW-xlo`kuldj bomw`fs wckrf$
• Zf `kif` ucf dfvuyldo mkrr fljfdrukufr kr ν ; ν 9 kde ν 2$
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Dfvuyldo Orbl``kulodr> 9 H`kxoy Moef`
• Dfvuyldor kyf k`zk|r wyoevbfe kr zfka fljfdrukufr$Rvwworf ucku ku u = 7 zf wyoevbf kd f`fbuyod dfvuyldo>
pν #7, = pν f
• Dfvuyldor wyowkjkuf kr mkrr fljfdrukufr$
Ld k 9-dfvuyldo moef` ucf zfka kde ucf mkrr fljfdrukufr kyf yf`kufe i|
7G
ν f
ν ¶
;K =
7G
bor γ rld γ
∙ rld γ bor γ
;K7G
ν ;
ν 9
;K
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Dfvuyldo Orbl``kulodr
• Ld ufymr oh ucf mkrr fljfdrukufr ovy oyljldk` ν f lr
pν #7, = bor γ pν ; ! rld γ pν 9
• Ucf uzo mkrr fljfdrukufr wlba vw elhhfyfdu wckrfr kr ucf| wyowkjkuf ro ucku
pν #u, = f∙lF;u bor γ pν ;! f∙lF9u rld γ pν 9
• Joldj ikba uo kxoy fljfdrukufr uclr lr
pν #u, = f∙lF;u bor γ #bor γ pν f ∙ rld γ pν ¶,
! f∙lF9u rld γ #rld γ pν f ! bor γ pν ¶,
• Ucf wyoikil`lu| oh kd orbl``kulod hyom ν f uo ν ¶ lr ucfd
W orb#u, = pν ¶pν #u,p9 =
ˑˑ
ˑrld γ bor γ
‐∙f∙lF;u ! f∙lF9u
”ˑˑ
ˑ9
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Dfvuyldo Orbl``kulodr> W orb
W orb#u, =ˑˑ
ˑrld γ bor γ
‐∙f∙lF;u ! f∙lF9u
”ˑˑ
ˑ9
= ;5 rld9 9γc
9 ∙‐fl#F9
∙
F;,u ! f∙
l#F9∙
F;,u”l
= ;9
rld9 9γ Q; ∙ bor#F 9 ∙F ;,u]
•Zluc
∈F = ∈m9/9 wkde
F ≂ wb ( @ ≂ ub ⇙u
w≂
@
F
• Zf jfu
W orb#u, = ;9
rld9 9γˈ
; ∙ borg
∈m9@/9F µː
= rld9 9γ rld9—
;$9: ∈m9 #fX9, @ #am,
F #JfX,
¡
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Dfvuyldo Orbl``kulodr
W orb#u, = rld9 9γ rld9—
;$9: ∈m9 #fX9, ‹@ #am,
F #JfX,
¡
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Oirfyxldj Dfvuyldo Orbl``kulodr
Ucf orbl``kulod wyoikil`lulfr efwfde od γ ∈m9
@ kde F $
Zf bkd flucfy `ooa hoy ucf kwwfkykdbf oh k elhhfyfdu dfvuyldo kxoy #vrvk``| `lmlufe
i| ikbajyovde, oy zf bkd mfkrvyf ucf elrkwwfkykdbf oh ucf fswfbufe kxoy #`lmlufe
i| bk`liykulod oh rovybf kde ukyjfu,$
Rovybf ν U|wfr Moef Kexkdukjf
Ro`ky ν f Elrkwwfkykdbf Jyfku elrukdbf
Kumorwcfylb ν f ν f ν ¶ ν ¶ Elrkwwfkykdbf Xkylki`f elrukdbf
Yfkbuoy ν f Elrkwwfkykdbf @oz fdfyj|
Kbbf`fykuoy ν ¶ ν ¶ Flucfy Boduyo` F kde @
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Dfvuyldo orbl``kulod fswfylmfdur
Dkuvyk` dfvuyldo rovybfr
• Ro`ky rovybfr> Comfrukaf RKJF!DO RvwfyA RDO Ioyfsldo
• 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,
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Dfvuyldo mlsldj mkuyls
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Ro`ky dfvuyldo yfrv`ur
rld9γ;9 = 7$275!7$799∙7$7;8 ∈m9
9; = :$86!7$92∙7$97 Ú ;7∙
6fX 9
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Kumorwcfylb dfvuyldo yfrv`ur
rld9γ92 = 7$67!7$7:∙7$78 ∈m9
2; = 9$57!7$;9∙7$;; Ú ;7∙2fX 9
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Iovde odγ;2
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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@,
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U9a Fswfylmfdu
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