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ILA]LNQBJE-FNQQ ADQFD@DER
NJI YCL
NAAL@L]NYBJE LS_NJQBDJ DH YCL \JBXL]QL
Ibfbtrb Il`bybnjjbs_cysbas Ilpnrtfljt,Nrbstdt`l \jbvlrsbty
dh Yclssn`djbkb.
lfnb`7 il`bybnjjbsOyncdd.er (Intli7 Gu`y >1, =?><)
Ul anj lxp`nbj tcl naal`lrntbje lxpnjsbdj dh tcl ujbvlrsl my gust nssufbje tcnt tcl rlst fnss dh njy pnrtba`l ilarlnsls wbtc tcl tbfl nji tcl jdj-rl`ntbvbstba fnss idfbjntls dvlr njy dtclr adjtljtdh tcl ujbvlrsl tdiny. Adjalpts, suac ns inrk ljlrey, qubjtlssljal, vnauuf ljlrey, rlpu`sbvl hdralsnji sd dj nrl adfp l̀tl`y ujjlalssnry. Ycl adsfd`deban` fdil`, wcbac bs mnsli dj tcbs biln tsplrhlat`y td tcl nstrdjdfban` intn.
B. BJY]DI\AYBDJ.
@lt‒s nssufl tcnt tcl rlst fnss dh njy pnrtba`l il-arlnsls wbtc tcl tbfl naadribje td tcl rl`ntbdj
f 6 f?h (t) (>)
f bs tcl rlst fnss nt tbfl t, f? bs tcl prlsljt rlst fnss.Hrdf jdw dj wl wb`` kjdw tcnt tcl sumsarbpt ‐?‖ bjib-antls tcl prlsljt vn`ul dh n qunjtbty. h (t) bs n ilarlns-bje hujatbdj, wcbac wb`` ml iltlrfbjli `ntlr. Qbjal h (t)acnjels vlry s`dw`y wbtc tcl tbfl, npprdxbfntl`y bt anjml wrbttlj
h (t) ≃ (> ∝ ¼t) (=)
sd tcnt
f≃
f?
(>∝
¼t)Ns wl wb`` sll `ntlr, ¼ 6 (>.82 ¾ ?.?<) µ >?∝>3sla∝>
Hrdf (>) nji (=)
f˒
f 6
h˒
h ≃ ∝¼ (1)
Idts nrl scdrtcnji hdr tbfl ilrbvntbvls. Bj tcbs ansl,Jlwtdj‒s sladji `nw mladfls
H 6 i(fό)
it 6 fα + f˒ό
nji usbje (1), nhtlr rlnrrnjebje
α 6 H
f ∝
h˒
h ό ≃
H
f + ¼ό (9)
H bs n hdral natbje dj n fnss pdbjt f, wcbac bs fdvbjewbtc vl`dabty ό nji naal`lrntbdj α .Bt‒s dmvbdus tcnt, lvlj, bh tclrl bs jd hdral natbje dj tcl
fnss pdbjt ( H 6 ?) tclrl bs naal`lrntbdj, wcbac bs prdpdr-tbdjn` td tcl vl`dabty (α ≃ ¼ό). Ycbs cnppljs mlanusldh tcl adjslrvntbdj dh fdfljtuf. Ycl fdfljtuf dh
tcl pnrtba`l bs prdpdrtbdjn` td bts fnss f nji tcl vl`da-bty ό. Qbjal tcl fnss ilarlnsls naadribje td tcl rl`ntbdjf ≃ f?(> ∝ ¼t) tcl vl`dabty scdu`i bjarlnsls naadribjetd tcl rl`ntbdj ό ≃ ό?(> + ¼t).
Yclrl bs adjslrvntbdj dh fdfljtuf, mut tcl `nw dh adjslrvntbdj dh ljlrey bs vbd`ntli, mlanusl, sbjal tcl rlstfnss ilarlnsls tcl lqubvn`ljt ljlrey dh tcl pnrtba`l il-arlnsls tdd. Cdwlvlr tclrl bs nj ‐ljlrey pnrnidx‖ clrl7Bj spbtl tcnt tcl tdtn` ljlrey ilarlnsls wbtc tcl tbfl tclkbjltba ljlrey bjarlnsls wbtc tcl tbfl7 Ycl tdtn` ljlreydh n pnrtba`l nt tbfl t 6 ?, wclj bt fdvls hrll`y wbtcjdj-rl`ntbvbstba vl`dabty, bs
L ? 6 f?a= + >
=f?ό=
? (<)
Ynkbje bjtd naadujt tcnt f ≃ f?(>∝¼t) nji mlanusl dh tcl adjslrvntbdj dh fdfljtuf ό ≃ ό?(> + ¼t), tcl tdtn`ljlrey nt tbfl t wb`` ml
L ? 6 f?a=(>∝ ¼t) + >=
f?ό=
?(> + ¼t) (8)
Ns ydu anj sll, tcl sladji tlrf dj tcl rbect cnji sbildh lquntbdj (8), wcbac rlprlsljts tcl kbjltba ljlrey, bj-arlnsls wbtc tcl tbfl, n`tcduec, my adfpnrbje (<) nji(8) nji kjdwbje tcnt f?a=
>
=f?ό=
? , wl rln`bzl tcnttcl tdtn` ljlrey ilarlnsls wbtc tcl tbfl (L 0 L ?).Hbjn``y, B scdu`i jdtl tcnt fnss`lss pnrtba`ls, suac nspcdtdjs, nrl jdt nlatli dh tcl acnjel dh tcl rlst fnss.Yclrlhdrl, tcl `nw dh adjslrvntbdj dh ljlrey bs jdt vbd-`ntli my tclsl pnrtba`ls.
BB. ADQFD@DEBAN@ L^\NYBDJQ.
@lt‒s adjsbilr tcl ujbvlrsl ns n ujbhdrf lxpnjibjeflibuf, wbtc fnss iljsbty ώ nji nj dmslrvlr bj bt. Jdwadjsbilr n pnrtba`l n ibstnjal r nwny wbtc fnss f. Iultd Jlwtdj‒s tcldrlf, tcbs pnrtba`l dj`y hll`s n hdral hrdftcl fntlrbn` nt sfn``lr rnibus. Ycbs fntlrbn` cns tdtn`fnss ebvlj my F 6 9Ϗώr1/1 adjtrbmutbje n hdral
H 6 ∝EFf
r= r̍? 6 ∝
9ϏEώrf
1 r̍? (5)
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=
Ycl ujbt vlatdr r̍? bs tnjeljt td tcl ibstnjal r, nji pdbjtsbj tcl ibrlatbdj dh r bjarlnsbje. E bs tcl ernvbtntbdjn`adjstnjt.Naadribje td Cumm`l‒s `nw, tcl vl`dabty dh tcl pnrtba`l bs
ό 6 C rr̍? (3)
wclrl C bs tcl Cumm`l‒s pnrnfltlr.Bjslrtbje (5) nji (3) bj (9) wl dmtnbj
α 6 ∝9ϏEώ
1 rr̍? ∝
h˒
h Crr̍? (2)
Qbjal ∝ h˒ h
C bs jlnr`y adjstnjt wl anj trlnt α ns n adj-
slrvntbvl l`i, sd tcnt wl anj slt
α 6 ∝ Ψ (>?)
wclrl Ψ bs n san`nr pdtljtbn`.Qumstbtutbje (>?) bj (2)
Ψ 6 9ώ
1 rr̍? +
h˒
h Crr̍? (>>)
=Ψ 6
9ώ1
(rr̍?) +h˒ h
C (rr̍?)
nji sbjal (rr̍?) 6 1
=Ψ ∝ 1
h˒
h C 6 9ϏEώ (>=)
Lquntbdj (>=) bs tcl _dbssdj‒s lquntbdj wbtc tcl niib-
tbdjn` tlrf ∝1 h˒ h C . Ycl njn`de lxtljsbdj dh lquntbdj
(>=) bj tcl eljlrn` tcldry dh rl`ntbvbty bs tcl lquntbdj
E¼ζ + Λ(t)e¼ζ 6 3ϏE
a9 Y ¼ζ (>1)
wclrl
Λ(t) 6 ∝1
a=h˒
h C
E¼ζ bs Lbjstlbj‒s tljsdr, e¼ζ bs tcl fltrba tljsdr nji Y ¼ζ bs tcl ljlrey-fdfljtuf tljsdr.Bj hnat, lquntbdj (>1) bs Lbjstlbj‒s lquntbdj wbtc adsfd-`deban` adjstnjt Λ, mut wbtc djl iblrljal7 Λ bs jdt adj-stnjt. Bt acnjels vlry s`dw`y wbtc tcl tbfl. Ycnt‒s wcytcl `nw dh adjslrvntbdj dh ljlrey bs vbd`ntli, ns n`rlniycns mllj fljtbdjli.
Jdw `lt‒s nssufl n nt Hrblifnj-]dmlrtsdj-Un`klrujbvlrsl, bj wcbac tcl jdj-rl`ntbvbstba fnss idfbjntls
dvlr njy dtclr adjtljt dh tcl ujbvlrsl. Bj Anrtlsbnj addr-ibjntls xb 6 (x?, x>, x=, x1) 6 (t ,x,y,z) tcl ]dmlrtsdj-Un`klr `bjl l`lfljt is nji tcl tljsdrs e¼ζ , E¼ζ nji Y ¼ζ nrl ebvlj my tcl lquntbdjs ml``dw.
is= 6 a=it= ∝ ξ=(t)[ix= + iy= + iz=V
e¼ζ 6 ibne(a=,∝ξ=,∝ξ=,∝ξ=)
E¼ζ 6 ibne(1 ξ˒
ξ,∝
ξ˒= + =ξξ̈
a= ,∝
ξ˒= + =ξξ̈
a= ,∝
ξ˒= + =ξξ̈
a= )
Y ¼ζ 6 ibne(ώa9, ?, ?, ?)
¼, ζ 6 ?, >, =, 1 nji ξ(t) bs tcl san`l hnatdr dh tcl ujb-vlrsl.Bjslrtbje tcl `nst tcrll lquntbdjs bj (>1), bj tcl usun`wny, wl dmtnbj tcl rst nji sladji dh tcl hd``dwbje lqun-tbdjs
Naal`lrntbdj lquntbdj 7
ξ̈
ξ 6∝
9ϏEώ
1 ∝
h˒
h C Hrblifnj lquntbdj 7
C = 6 3ϏE
1 ώ ∝
h˒
h C
H`ubi lquntbdj 7
ώ˒
ώ + 1C 6
h˒
h
(>9)
C 6 ξ˒ξ bs tcl Cumm`l‒s pnrnfltlr.
Ycl lnsy ilrbvntbdj dh tcl ubi lquntbdj, tcl tcbri dh lquntbdjs (>9), bs ns hd`̀ dws7 Bh F ? bs tcl fnss dh tclfntlrbn` bj n spclrban` rlebdj dh rnibus r nt t 6 ?, tcljnt tbfl t bt wb`` ml F 6 F ?h dr 9
1Ϗώξ1 6 9
1Ϗώ?ξ1
?h nji
jn``y
ώξ1 6 ώ?ξ1
?h (><)
My iblrljtbntbje lquntbdj (><) wbtc rlsplat td tcl tbflwl dmtnbj tcl ubi lquntbdj.
BBB. ILYL]FBJNYBDJ DH YCL H\JAYBDJh (t).
Ul nrl stb`` jdt bj n pdsbtbdj td sd`vl tcl lquntbdjs(>9), mlanusl wl idj‒t kjdw tcl hujatbdj h (t). Hdr tcliltlrfbjntbdj dh tcl hujatbdj h (t) nji hdr juflrban` sbf-u`ntbdjs, lquntbdjs scdu`i ml put bj ibfljsbdj`lss hdrf.Ns n rst stlp td tcbs ibrlatbdj wl wb`` usl tcl `denrbtcfbatbfl vnrbnm`l
ω 6 `j ξ
ξ?
(>8)
\sbje lquntbdj (>8) wl dmtnbj tcl hd``dwbje lquntbdjs7
iω
it 6 C, ώ˒ 6 ώC, h˒ 6 h C, ξ˒ 6 ξC
ξ̈ 6 ξ
C =
+ ξ
C
C, ξ 6 ξ
6 ξ
(>5)
_rbfls nrl scdrtcnji hdr `denrbtcfba tbfl ilrbvntbvlsξ 6 iξ
iω
. Ul n`sd bjtrdiual tcl ibfljsbdj`lss pnrnf-
ltlrs
ώ̄ 6 ώ
ώa?, C̄ 6
C
C ?(>3)
wclrl ώa? 6 1C =?
/3ϏE bs tcl prlsljt arbtban` iljsbty hdrnt spnal. \sbje lquntbdjs (>5) nji (>3), tcl ibfljsbdj-
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1
`lss hdrf dh lquntbdjs (>9) nrl
Naal`lrntbdj lquntbdj 7
= C̄ C̄ 6 ∝1ώ̄
Hrblifnj lquntbdj 7
C̄ = 6 ώ̄
> + h
h
H`ubi lquntbdj 7ώ̄
ώ̄ + 1 6
h
h
(>2)
Hrdf lquntbdjs (>2) wl dmtnbj
h
h 6
(ώ̄? ∝ >) l9ω
ώ̄? ∝ (ώ̄? ∝ >) l9ω (=?)
Hbjn``y, tcl sd`utbdj dh lquntbdj (=?) bs
h (ω ) 6
ώ̄? ∝ (ώ̄? ∝ >) l9ω ∝>/9
(=>)
Qll tcl ilrbvntbdj dh lquntbdjs (=?) nji (=>) clrl [>V.B scdu`i jdtl tcnt n`` tcl nmdvl lquntbdjs nrl vn`bi hdrω : ∝9 wclj iljsbty bj fnttlr idfbjntls dvlr iljsbtybj rnibntbdj.
Hdr sfn`` ω , l9ω ≃ > + 9ω nji (=>) mladfls
h (ω ) ≃ > ∝ (>∝ ώ̄?)ω
Ul acnjel tcl vnrbnm`l ω td t usbje ω ≃ C ?t
h (t) ≃ > ∝ (>∝ ώ̄?)C ?t (==)
Adfpnrbje lquntbdjs (=) nji (==)
¼ 6 (> ∝ ώ̄?)C ? (=1)
Ul wb`` iltlrfbjl ώ̄? nji C ? `ntlr. Ul wb`` n`sd usl (=1)td iltlrfbjl tcl vn`ul dh ¼.
BX. YCL ILA]LNQBJE-FNQQ FDIL@.
Ycl ibfljsbdj`lss hdrf dh lquntbdj (><) bs
ώ̄ 6 ώ̄?l∝1ω h (ω ) (=9)
Bjslrtbje lquntbdjs (=?), (=>) nji (=9) bj tcl ibfljsbdj-`lss Hrblifnj lquntbdj wl dmtnbj
C̄ = 6 [ώ̄?l∝9ω + (> ∝ ώ̄?)V1/9 (=<)
Jdw `lt‒s an`` tcl adsfd`deban` fdil` wcbac dmlys tcllquntbdj (=<) ‐ilarlnsbje-fnss fdil`‖ nji sll bh btnerlls wbtc tcl nstrdjdfban` intn.
Ycl nstrdjdfban` intn B cnvl bj fy ibspdsn` nrl tclflnsurlfljts dh tcl ibstnjal fdiu`us nji tcl rliscbht dh 83> suplrjdvn-Bn. Rdu anj sll tclsl intn clrl [=V. Yclsl83> suplrjdvn cnvl mllj acdslj dh n `bst dh 5>< suplrjdvnhrdf wcbac cnvl mllj lxa`uili tcdsl wbtc rliscbht `lss
tcnj ?.?>. B ibi tcbs td nvdbi lrrdrs iul td tclbr lxtrnvl`dabty, dtclr tcnj ‐Cumm`l vl`dabty‖.
Yd nsslss tcl eddijlss dh t dh tcl ‐ilarlnsbje-fnssfdil`‖ td tcl nstrdjdfban` intn B usli tcl fltcdi dh rlsbiun`s. ]lni nmdut tcbs fltcdi clrl [1V. B npp`blilquntbdj (=<) nt n sbfu`ntbdj prdernf nji B envl ns bjputintn tcl vn`uls ώ̄?6?.=??, ?.=?>, ..., ?.1<? nji C ?682.?,82.>, ..., 5>.? Kf/sla/Fpa. Ycl fdil` ts plrhlat̀ y td
tcl nstrdjdfban` intn bh C ? 6 5?¾ ?.1 kf/sla/Fpa
ώ̄? 6 ?.=8¾ ?.?=
q ? 6 ∝?.8= ¾ ?.?1
ω t 6 ∝?.99 ¾ ?.?1
zt 6 ?.<8¾ ?.?9
¼ 6 (>.82 ¾ ?.?<) µ >?∝>3 sla∝>
(=8)
Qll tcl ilrbvntbdj dh tcl rlsu`ts nmdvl clrl [9V.C ? bs tcl prlsljt Cumm`l‒s pnrnfltlr.ώ̄? bs tcl rntbd dh tcl prlsljt iljsbty bj fnttlr td tclprlsljt arbtban` iljsbty.q ? bs tcl prlsljt ilal`lrntbdj pnrnfltlr.ω t bs tcl trnjsbtbdj tbfl. Ycnt bs, tcl ω wclj tcl ujbvlrslfnil n trnjsbtbdj hrdf ilal`lrntbdj td naal`lrntbdj.zt bs tcl trnjsbtbdj rliscbht. Bs adfputli tcrduec lqun-tbdj ω t 6 ∝`j(> + zt).¼ bs ebvlj my lquntbdj (1) nji adfputli vbn lquntbdj(=1).
Bj adfpnrbsdj td tclsl rlsu`ts `lt fl scdw ydu tclrlsu`ts B‒vl edt usbje tcl ΛAIF fdil`, wcbac dmlys tcllquntbdj C̄ = 6 ώ̄?l∝1ω + (> ∝ ώ̄?).
C ? 6 82.3 ¾ ?.1 kf/sla/Fpa
ώ̄? 6 ?.1? ¾ ?.?=
q ? 6 ∝?.<< ¾ ?.?1
ω t 6 ∝?.<= ¾ ?.?9
zt 6 ?.83 ¾ ?.?8
(=5)
Qll tcl ilrbvntbdj dh tcl rlsu`ts nmdvl clrl [<V.Jdw `lt‒s adfpnrl nji adjtrnst tcl prlibatbdjs dh tcl
twd fdil`s, hdr ώ̄?, q ? nji zt. Ycl dmslrvntbdjn` vn`ulsnrl
ώ̄? 6 ?.=2 ¾ ?.?8
q ? 6 ∝?.88 ¾ ?.>?
zt 6 ?.98 ¾ ?.>9
Ns ydu sll tcl prlibatbdjs dh tcl ‖ilarlnsbje-fnssfdil`‖ hdr ώ̄?,q ? nji zt nrl bj nerllfljt wbtc tcl dm-slrvntbdjn` vn`uls, wclrlns tcl prlibatbdj dh tcl ΛAIFfdil` hdr zt hnb`s nji tcl prlibatbdj hdr q ?, jlnr`y hnb`s.
X. _@NJLYN]R D]MBYQ.
Bh tcl biln dh ilarlnsbje dh fnss bs rbect, tclj wl scdu`ilxplat td dmslrvl sdfl llats td tcl p`njltnry drmbts.
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Qplaban``y, lnac p`njlt bs erniun``y rlalibje hrdf tclsuj bjtd n cbeclr drmbt nji tcl drmbtn` rdtntbdj bs s`dwbjes`bect̀ y wbtc tcl tbfl. Nssufbje, hdr sbfp b̀abty, tcnttcl drmbt dh tcl p`njlt bs abrau`nr nji r? bs tcl prlsljtibstnjal dh tcl p`njlt hrdf tcl suj, tclj, wl anj prdvltcnt tcl bjarlfljt dh tcbs ibstnjal nhtlr tbfl t wb`` ml
∈r 6 r?1¼t (=3)
nji bh Y ? bs tcl sbilrln` plrbdi dh tcl `nst rlvd`utbdj dh tclp`njlt nt tbfl t 6 ?, tclj tcl bjarlfljt dh tcl sbilrln`plrbdi dh tcl `nst rlvd`utbdj nt tbfl t wb`` ml
∈Y 6 Y ?<¼t (=2)
\sbje (=3) nji (=2) bj tcl ansl dh tcl Quj-Lnrtc systlf,wl dmtnbj tcl hd``dwbje rlsu`ts7>) Ycl bjarlfljt dh tcl nvlrnel ibstnjal Lnrtc-Quj bs∈r 6 (=1.29 ¾ ?.5>)fltlrs/ylnr=) Ycl bjarlfljt dh tcl sbilrln` plrbdi dh Lnrtc bs∈Y 6 (3.9= ¾ ?.=9)fb``bsladjis/ylnrQll tcl ilrbvntbdj dh tcl rlsu`ts nmdvl clrl [8V.
Bj tcl ansl dh tcl Lnrtc-Fddj systlf, mut n`sd tnkbjebjtd naadujt tcl tbin` llats tcl Fddj cns dj Lnrtcsrdtntbdj, wl dmtnbj tcl hd``dwbje rlsu`ts7>) Ycl bjarlfljt dh tcl nvlrnel ibstnjal Fddj-Lnrtc bs∈r 6 (>?.3< ¾ ?.=>) af/ylnr=) Ycl bjarlfljt dh tcl sbilrln` plrbdi dh tcl Fddj bs∈Y 6 (>.?8 ¾ ?.?=) fb``bsladjis/ylnrQll tcl ilrbvntbdj dh tcl rlsu`ts nmdvl clrl [5V.
XB. LS_L]BFLJYN@ YLQY.
Ul anj tlst tcl vn`bibty dh tcl biln tcnt tcl rlst fnss dh
njy pnrtba`l ilarlnsls wbtc tcl tbfl my lxnfbjbje tcl rd-
tntbdj dh n sntl``btl. Ycl njeu`nr fdfljtuf dh tcl sntl`-`btl bs prdpdrtbdjn` td bts fnss f nji tcl njeu`nr vl`dabtyς. Mlanusl dh tcl adjslrvntbdj dh tcl njeu`nr fdflj-tuf, sbjal tcl fnss ilarlnsls naadribje td tcl rl`ntbdjf ≃ f?(>∝¼t) tcl njeu`nr vl`dabty scdu`i bjarlnsls na-adribje td tcl rl`ntbdj ς ≃ ς?(> + ¼t). Bj dtclr wdris, bh Y ? bs tcl plrbdi dh tcl rdtntbdj tdiny, tclj tcl ilarlfljtdh tcl plrbdi nt tbfl t wb`` ml ∈Y 6 Y ?¼t. @lt‒s sny hdr
bjstnjal tcnt Y ? 6 >sla. Nhtlr djl ylnr tcl plrbdi wb`` mlscdrtlr my ∈Y 6 Y ?¼t 6 (>sla)(>.82 µ >?∝>3)(>ylnr) 6<1.1 pbadsladjis.
XBB. ADJA@\QBDJ.
Ul anj lxp`nbj tcl naal`lrntbje lxpnjsbdj dh tcl ujb-vlrsl my gust nssufbje tcnt tcl rlst fnss dh njy pnrtba`lilarlnsls wbtc tcl tbfl. Ycl fdil`, wcbac bs mnsli djtcbs biln (ilarlnsbje-fnss fdil`) ts plrhlat`y td tclnstrdjdfban` intn. Bh tcl biln dh ilarlnsbje dh fnss bsrbect, tclj wl scdu`i lxplat td dmslrvl sdfl llats tdtcl p`njltnry drmbts7 >) Lnac p`njlt bs erniun``y rlalibjehrdf tcl suj bjtd n cbeclr drmbt. =) Ycl drmbtn` rdtntbdjbs s`dwbje s`bect`y wbtc tcl tbfl. Ul anj tlst tcl vn`bibtydh tcl biln tcnt tcl rlst fnss dh njy pnrtba`l ilarlnslswbtc tcl tbfl my flnsurbje tcl bjarlfljt dh tcl rdtntbdj
dh n sntl``btl.
[>V cttps7//www.sarbmi.adf/ida/=53>=?=3</Iltlrfbjntbdj-dh-tcl-Hujatbdj-h-t
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