3 CP Quantum Mechanics

Transcription

1 Quatu Mas. Bas Assutos by ut at May 6 toug ovb 7 Modlg of s basd o t followg bas assutos: () ota sbls of ato ul dsly ld u a log ad vy aow al. () T dstas btw t ul a so sall tat all ltos boud to ts ul a dloald alog t al. I ot wods: v t lto goud stat do t osst of dvdual atos. at fo a quas-o-dsoal lasa (ts ould also b s as a tal). Fgu Bas odl of a. T slaly xtds to t lft ad to t gt of ts tu. T yldal Modl of T sa ad quatu aal stat of a b vy olatd. I od to obta a sl quatu aal dsto of t followg slfatos a usd w wll subsqutly b alld t yldal odl of : () T s ftly stagt ad yldally syt.. t s ot bt to gs ls t. T s otd aalll to ad td o t -axs of t odlg yldal oodat syst. (4) T as t lgt ad otas a total ula ag Q ts o o (xlad blow). (5) T lto wav futos of t a ofd t tval <. At ts wav futo a otuously xtdd to t valu ad gadt at as f t w gs. Howv ts s at to dsb oly t ula bouday odto of t wav futos at ot t sa of t. (6) o xtal fld s ald to t. (7) A llu odl s usd fo t satal dstbuto of t ula ag. Ts as t ostv ags of t ul a odld as a ufo "ostv lly" bakgoud at ta ot ags wt dstas btw. T ul ag dsty s assud to b ostat axal ad autal dto but t dds o t adal dsta. (8) T llu s odld dfftly t o o ad t alo o. T aow o o s wt t a of t lto wav futos. T dffus alo o s outsd t a of t lto wav futos. ut at May 6 toug ovb T yss of odsd lasods ad

2 (9) T ul ag dstbuto of t o o s odld by as of a two-dsoal oal dstbuto adal dto. T fato of ul sdg t o o as wll as t stadad dvato a to b dtd by vaato su tat t total gy of t s d. Altatvly t ul ag dstbuto of t o o s odld by xotal day adal dto. T fato of ul sdg t o o as wll as t day ostat a to b dtd by vaato su tat t total gy of t s d. () It s assud tat t alo o ossts of os a avg o ostv ltay ag. T o ag dstbuto of t alo o s odld su tat t ostos of t os a qulbu wt t lt ottal of t. () T s assud to sd a vauu. Itato of t wt suoudg att s tus gltd. () Oly statoay stats a odld as t goal s to dsb t goud stat of. osqutly t odl assus t s o lto sattg.. t s o otu tasf btw ltos ad t ul. () Fo outg t ulso gy aog t ul sot-ag otos to t llu odl av to b ad w aout fo t gaulaty of t ula ags. I as t otas a xtu of dfft sots of ato ul oly t a ula ag s tak to aout fo t otos t o o at ta t dvdual ula ags. (4) T t-ddt Kl-Godo quato s usd fo odlg t lto wav futos tby gltg t agt ots of lto ss. T Kl-Godo quato s takg a of t lag latvst ffts oug.g. t ass dft stg fo t vy g bdg gy of t ltos. (laly t Da quato would b o adquat fo odlg. Howv t volvd olxts of su aoa a avodd.) (5) T agt fld of t autal lto obts s gltd. (6) Magt fld fo ula ss s gltd. (7) T lto wav futos a odld a tal fa of f w o agt fld s atd by ay olla ovts of t ul. Ts slfato aouts to a aoxato ass w t ul vlots a osto ddt. (8) T ult-lto syst s aoxatd by outg a ollto of o-lto obtals wby a lto obtal s subtd to t a lt ottal ad agt vto ottal atd by t total ag dsty ad total ut dsty of all ot oud obtals ad t ul. T aul xluso l s usd fo dtg obtal ouatos of t goud stat. xag ad olato gs a gltd. (9) Quatu fld toy s ot gagd. atl out s osvd. gstats a xludd fo ouato w t osodg total gy gvalu (ludg t lto s st gy) of t lto s gatv. () Oly boud gstats of t ltos a osdd.. t total gy of a gstat as to b lss ta t lto st gy (.. t su of t ottal gy ad t kt gy as to b gatv).. T Kl-Godo quato of a Ital alulatos of a wt t Södg quato av sow tat t sultg bdg gy of t ltos would ottally xd t st gy of t ltos by ods of agtud. Ts sults w absud lgt of sal latvty baus t ass dft boud lto sould v xd tw ts st ass. Tfo a ot-ovaat quatu aal quato s absolutly qud to odl. Gally t Da quato s gadd as t ot ot-ovaat quato fo odlg fos sally w t ffts sultg fo t atl s s s of o. Ufotuatly t Da quato volvs 4- oot wav futos ad t soluto of fou ould dfftal quatos sultg sabl atatal ad outatoal ffots. Assug tat t lto ss av oly o ffts o t bdg gy ag dsty ut dsty ad ot obsvabls t Kl-Godo quato ovds a ot-ovaat altatv to t Da quato fo odlg t ltos of. At t o-latvst lt t Kl-Godo quato s quvalt to t Södg quato wl bot quatos sa t dfy of ot odlg t s. ut at May 6 toug ovb T yss of odsd lasods ad

3 I latvst ltodyas wt so-alld al oulg t Haltoa (total gy) of a atl wt ag q ovg t s of a stat (xtal) ltoagt ottal s: ˆ () H ( qa) qφ By dfg agt vto ottal ass ad ( ) Hˆ qφ w s t sd of lgt Φ s t lt ottal A s t γ v qa s t lto s aoal otu γ s t loal ot fato s t lto st Hˆ as bg t su of t kt gy ad t ottal gy ad by usg q as t ag of a lto () s ladg to t followg quato fo a lto a stat ltoagt ottal: () ( γ ) Φ ( A) Φ Tfo: Φ 4 () ( ) ( ) ( ) γ A w s t ltay ag All foulas a wtt SI uts ulss otws otd. Tougout ts dout gy sybols wt a ba o to (.g. ) dot tat t gy s asud oul. gy sybols wtout a ba o to dot tat t gy s asud uts of t Hat gy (98).. t gy s a dsolss quatty t stv foula. kws ot sybols wt a ba (.g. A ) a SI uts wl ts outats wtout t ba a atual uts (.. dsolss). By quatg t otu va t dl oato ad alyg bot sds to a lto wav futo Ψ quato () tasfos to t statoay Kl-Godo quato of a lto a stat ltoagt ottal: 4 [ ] Ψ (4) ( Φ ) Ψ ( γ ) Ψ ( A) s t dud lak ostat ad w Du to slfato (9) Ψ s alld a wav futo at ta a quatu fld. T t sts t total gy of t lto.. t su of ts st gy ottal gy ad kt gy. Usually t Kl-Godo quato s wtt su tat t total gy s sougt as t gvalu of ts dfftal quato. Howv ts dout dvats fo t ustoay aoa. Istad t quatty s sougt as t gvalu (bot aoas a quvalt t sults).... I quatu as a ult-lto syst s otly dsbd by a sgl wav futo Ψ ( ) ddg o t ostos of t ltos. T ult-lto wav futo s usually fod by a Slat dtat (o a la obato of sval Slat dtats) to su at-syty ad t aul xluso l. Howv t ub of ltos a a xd w ds a Slat dtat tly atal to out baus a oga aot adl quatos wt.g. ostos ad out dtats of ts s. Aodg to slfato (8) a goously sl aoa s usd fo odlg qug oly odat out ow: So stad of usg a ult-lto Kl-Godo quato dsbg t a-ws tato btw ltos t yldal odl uss sgl-lto Kl-Godo quatos wt wav futos Ψ ( ) dsbg a sgl lto t a ottal of all ot ltos ad t ul. a Of ous ts s ly a aoxato. Fo xal t aoa dos t aout fo t xag gy ad t olato gy usually dd otat quatu sty. ut at May 6 toug ovb T yss of odsd lasods ad

4 At fst gla ts looks stll allgg to out baus t a Kl-Godo quatos to b solvd. Fotuatly lag ubs of ts quatos a b outd gous baus ty odu aly t sa ag dsty dstbutos ad ut dsty dstbutos. xadg t gt sd of (4) ad usg (ot gaug t stat as) ylds: A 4 (5) ( Φ ) Ψ ( ) Ψ ( A A A )Ψ γ Aodg to slfato (4) ad (5) t agt fld of t lto ss ad of t autal ovt of t ltos s gltd. Tus t oly sou of t agt fld s t ut ad by t ltos ovg - dto. Tfo t vto ottal s vyw otd -dto: A A (6) T ala oato xads yldal oodats as followg: (7) s t oodat of t -axs w s t adal dsta fo t -axs φ s t aut ad Istg (6) ad (7) to quato (5) ad dvdg bot sds by s sultg t statoay Kl- Godo quato of a lto t a ottal of a s all ot ltos ad t ul: (8) A A Φ Ψ Wt slfato (6) t lt ottal Φ s ddg solly o t lto ag dsty ( ) ad t ula ag dsty ( ). T agt vto ottal A s solly ddg o t lt ut dsty ( ). T lto ag dsty ad t lt ut dsty a dvd fo t odulus squa of t ot lto s wav futos. Ts aoa as slats wt t dsty futoal toy (DFT) usd quatu sty odlg xt tat t xag ad olato gs a ot aoutd fo. Howv t flu of t agt fld o t o-la lto-lto tato s atad ad t aul xluso l wll b obyd dug obtal ouato..4 Bouday odtos fo Solutos of t Kl-Godo quato a ust b tak aodg to slfato (9) tat t total gy ostv tfo: of a gstat s always (9) > qut (9) a b fulflld by xludg gstats wt a gatv total gy dug obtal ouato. Aodg to slfato () oly boud gstats a osdd. Tfo t wav futo altud ust dsaa at ft adal dstas: () l Ψ( ) ut at May 6 toug ovb T yss of odsd lasods ad

5 As qud by slfato (5) t wav futo as to t ula bouday odtos: () ( ) Ψ( ) () Ψ ad ( ) Ψ( ) Ψ By dfto of a boud stat t total gy of t lto s lss ta ts st gy: () < tfo < obg (9) ad () ylds fo boud stats: < < (4) Fo outg obsvabls t Kl-Godo lto wav futos aodg to [] av to b oald su tat: d (5) Ψ γ Ψ.5 Obsvabls of t Kl-Godo lto Wav Futo Gally o a out a obsvabl O ~ fo t Kl-Godo wav futos Ψ as followg: ~ (6) O γ ( ) ~ Ψ OΨd w Aodg to [] t loal ot fato outs as: (7) γ ( ) Φ( ) Ψ s t ougat olx of Ψ ad γ s t loal ot fato T fato γ (6) a b udstood by sal latvst t dlato: A lto statstally sds by a fato of γ o t aas of g vloty baus t t t lto s fa of ta lass slow ta t t t fa of ta of t obsv wo asus t obsvabl. Tfo ts aas av to b wgtd g by a fato of γ dug tgato. T loal ottal gy of a lto s solly stg fo t oulob fld: (8) ( ) Φ( ) ot T loal kt gy of t lto s wat s lft w t ottal gy s subtatd fo : (9) ( ) Φ( ) [ γ ( ) ] k Aodg to [] t volu ag dsty dstbuto of lto ub a stat ltoagt ottal outs as followg: (4) γ Ψ w γ s t loal ot fato at t osto of Sug ts u fo all ltos of t s sultg : (4) γ Ψ ut at May 6 toug ovb T yss of odsd lasods ad

6 Aodg to [] ad [] t ut dsty dstbuto of lto ub a stat ltoagt ottal outs as followg: Ψ Ψ Ψ Ψ AΨ (4) ( ) Sug (4) u fo all ltos of t ovds: (4) ( Ψ Ψ Ψ Ψ ) AΨ Usg odut asat (6) odulus squa fatoato (6) ad oodats) of t ut dsty (4) a outs as: (44) π Ψ Ψ Ψ Ψ A Ψ Ψ w (45) Ψ -soluto (67) t -oot ( yldal A k A s t -oot of t lto s kt otu W (4) (4) ad (44) wll b usd fo dtg t lt ad agt ottals t Kl-Godo quato (8) t lto ub s otly xosd also to ts ow ottal. Howv ts o s qut sall f t otas vy ay ltos. T total ut -dto ad by all ltos of t a b outd by tgatg (44) ov all adus valus ad aut valus: π (46) I ( ) dd Ψ ( ) [ k A ( ) ] Ψ ( ) d d T xtd valu of t lto gou vloty s -oot (avagd ov all ltos of t ) a b outd fo t -oot of t total ut: (47) v I T -oot of t loal gou vloty of a lto a b outd fo (45): (48) v γ k A γ Aodg to [] t xtd valu of t lto obt adus fo gstats of quato (8) s: Ψ (49) γ ( ) d.6 T ltoagt ottal of a T lt ottal of a slts as follows: (5) Φ Φ Φ w Φ s t lt ottal of t ula llu aodg to slfato (7) ad Φ s t lt ottal of t ltos ut at May 6 toug ovb T yss of odsd lasods ad

7 As a tool fo outg t ltoagt ottal t followg goty s aalyd: A sal ag at dsta fo t -axs (og) ad aut φ sall at as t ot of asut fo vto ottal A ad t lt ottals Φ ad Φ. T followg fgu llustats ts fut: Fgu 4 S fo outg t ltoagt ottal. Ts sows a ut dula to t -axs. Fo t goty of Fgu 4 t a b oludd: (5) os (5) s D os s os (5) ( ) ( ) T followg fgu sall llustat t goty -dto: Fgu 5 S fo outg t ltoagt ottal. Ts sows a ut aalll to t -axs. Fo Fgu 5 t a b oludd: (54) D D Fgu 5 sows a ftsal t l of ag xtdg fo to. Ts l s aalll to t -axs. T volu ag dsty ( ) s ostat alog t l. A ftsal ag dsty lt wt a volu of otas a ag of: d dd (55) dq ( ) d d d w (56) ( ) [ ( ) ( )].. t su of t ula ag dsty ad t lto ag dsty ut at May 6 toug ovb T yss of odsd lasods ad

8 T lt ottal at dsta D fo t ftsal l of ag ad at axal osto outs as followg: dq D (57) dφ( D) ( ) d d 4πε 4πε ( ) D ( ) d dl 4πε D D d Istg (5) to (57) ad tgatg ov ' ad φ ylds t otbuto of t t to t lt ottal ( ot gaug stat as): d 4πε (58) Φ ( ) ( ) G( ) (59) ( ) ( ) w π os l d os G s t goty tgal lag ( ) ε wt µ ( ) ot gaug stat as): (58) ovds t -oot of t s agt vto ottal ( µ µ [ ] Ψ ( ) G d 4π 8π (6) A ( ) ( ) G( ) d A ( ) I s t total ut of t ltos -dto µ s t vauu ablty ad ( ) w -oot of t ut dsty s t ot tat A s ddg o tslf quato (6). Tfo t valus of tatvly utl slf-ossty. A ad d to b dtd Basd o t ula bouday odto (5) t lt ottal (58) ad t vto ottal (6) a ad to b ostat -dto. Ts aoxato s qud fo atag t full yldal syty of t odl. T adal ad autal (s slfato (5)) oots of t vto ottal ad t ut dsty s o vyw. Du to slfato (7) t ula llu s ot otbutg to t ut dsty. ot tat l Φ ( ) ad l A ( ) bdg gy of ltos to a wtout gagg a o-o f ottal..7 odut Asat T followg odut asat s ad to fato t lto wav futo: (6) Ψ( ) Ψ ( ) Ψ ( ) Ψ ( ) o sot: Ψ Ψ Ψ Ψ. quatos (58) ad (6) tfo a b usd fo dtg t T wav futo of a sgl lto s suosd to b oald ad t sts a statoay stat. I autal dto ad axal dto t ltoagt ottal s ostat. Tfo t odulus squa of Ψ ad s also ostat: (6) Ψ Ψ Ψ ad Ψ π Ψ Ψ H t odulus squa of t t wav futo fatos as: (6) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ π Ψ ut at May 6 toug ovb T yss of odsd lasods ad

9 T oalato ta (5) ould t b ad out as: Ψ (64) Ψ γ ( ) ( ) d.8 Saato of t Kl-Godo quato Wt odut asat (6) t atal dvatvs of t wav futo a: (65) Ψ dψ Ψ Ψ d ad Ψ dψ ΨΨ d ad Ψ Ψ Ψ d d Ψ Istg ts to t quato (8) ad dvdg bot sds by Ψ ylds: d dψ Ψ Ψ Ψ d d A d A (66) Ψ d d Ψ d Ψ d Ψ d Φ ak fo t atatal uty: T dvso by Ψ s do out of ov. It ould av b ostod to a lat st wtout afftg t d sult su tat wav futos (w a av os) v sow u t doato. T followg wav futo s solvg t -ddt at of (66): (67) k Ψ w k Du to slfato (5) t gy gvalus a quatd to a dst stu baus wav ub k as to t t followg bouday odto: (68) π k l w l Z Itg l ats as a axal quatu ub (Ts quatu ub l sould ot b ofusd wt t l θ ala s sal ao futo ( ) Y usd fo odlg t ltos of atos). T followg wav futo s solvg t φ-ddt at of (66): l (69) π Ψ w Z Itg s t autal quatu ub. Istg (45) (67) ad (69) to (66) ovds t adal Kl-Godo quato of a : d d d d (7) Ψ At t o-latvst lt t t x ( Φ) ( ) Taylo ss of ( ) x about (7) ( x ) x Φ aoas o. By usg oly t fst two ts of t x o a aoxat: ut at May 6 toug ovb T yss of odsd lasods ad

10 Wt ts aoxato quato (7) bos t adal Södg quato of a : d d d d (7) Φ Ψ T adal Södg quato (7) s basd o t o-latvst Haltoa fo a lto a ltoagt fld wt al oulg: ˆ A A ( ) ( ) (7) H Φ Φ T gstats of dfftal quato (7) o (7) ovd t adal wav futos Ψ. T gvalus of boud stats a dst.. ty a outabl by a al quatu ub t autal quatu ub ad t axal quatu ub l. T al quatu ub s dfd aalogous to t ydog ato: quals o lus t ub of od ls of Ψ Ψ tfo (I a stt ss Ψ as o od ls. Howv a stadg wav of two suosd autal wav futos dffg oly t sg of quatu ub as od ls.) al quatu ub as o xlt stato (7) o (7) o ay of t followg foulas. It s usful owv as a odg s fo outatoal sults. O as to k d tat t gvalus t gstats Ψ Ψ ad Ψ as wll as t quatu ubs ad l a gally dstt fo a lto of t. I od to as adablty t lto ub as a dx as b ottd fo ts sybols ulss t dx s dd a suato..9 T llu Modl of t ula ag Dstbuto Aodg to slfato (7) t ag of t ul s tatd as f t w a ufo "ostv lly" bakgoud at ta ot ags wt dstas btw. T ula ag dsty dstbuto ( ) of t llu as yldal syty.. t dos t dd o φ ad. It s a futo of t adal dsta. Aodg to quatos (5) (56) ad (58) t lt ottal of t ula llu s: d 4πε (74) Φ ( ) ( ) G( ) A ftsal ag dsty lt ( ) d d d (75) d ( ) Φ ( ) d d d bougt to ottal Φ as t ottal gy: Itgatg (75) ov t t sa ad dvdg t sult by two ylds t ula slf-ulso gy: G G s t gaulaty oto (84) > ad > π (76) ( ) Φ ( ) ddd π ( ) Φ ( ) G G d w T dvso by two (76) taks a of t fat tat t llu s tatg wt tslf ad t ulso gy ust ot b aoutd tw dug tgato. ut at May 6 toug ovb T yss of odsd lasods ad

11 quato (76) ds to b otd by G od to aout fo t gaulaty of t ula ags. Fo ts uos t followg aoxato s ad: gadg slfato () t ul a assud to av a a ag of t avag of t ula ags sot: (77) Z F Z w T volu oud by o ulus would b: Z of t ato sot wgtd by t fato Z. T a ag s dtd by F < of t stv ato < F ad s t ub of dfft ato sots of t xtu (78) Z 4 V π H t adus of a s wt volu V would b: Z (79) 4 π Assug tat t ag dsty s ostat wt V t lt ottal of t s s: (8) Φ ( ) 6ε Z ε 4πε Z 6ε 4π ( ) > T slf-ulso gy of t llu wt s V (.. fo as ) would b as followg: π π (8) ( ) Φ ( ) dq Φ ( ) dsθdθd π Φ ( ) θ V π ( ) 4 d d d π 6ε ε 4π 5ε 5 5ε 4 5 Z π 5 I alty.. t slf-ulso gy of a sgl ulus s o. Ts s baus a ulus dos t l tslf. Tfo t slf-ulso gy (76) ds to b subtatd by (8) fo a ulus t. A ftsal yldal o of a wt adus (8) dv π d T ub of ul sdg volu dv s: (8) d dv V π Z ( ) d d d as t volu: d ut at May 6 toug ovb T yss of odsd lasods ad

12 Multlyg (8) wt ad tgatg ov ylds t gaulaty oto of t o llu s slf-ulso gy: (84) G π 5ε 4 ( ) Z π d Aodg to slfato () t llu s ag dstbuto of t alo o sall b odld su tat t ostos of t atos a qulbu wt t lt ottal of t. Ts s t qulbu of two ssus sultg fo oulob fos: T ulsv ssu btw gbog os ad t attatv ssu fo t s lt ottal (58). a o as t followg adus: (85) 4 π A s wt ts adus as a sufa aa of: (86) A 4π T ulsv oulob fo btw two gbo os s: F 4πε (87) As a aoxato t s assud tat t oulob fo oot dula to t sufa aa (86) s ostat ov ts aa. Tfo t oulob fo s sultg a ssu of: ε ε π 6 (88) F A 6π A ftsal yldal o (8) otas t ag: dq d (89) π Ts ag s attatd by t s lt ottal (58) wt t followg fo: d d d ε d (9) F dq dq Φ( ) ( ) G( ) Fo (9) s dula to t sufa F (9) d ( ) d ( ) G( ) π of t yldal o (8) tus atg a tal ssu of: d d d π 4πε Itgatg (9) ov t adus ylds t ssu at adal dsta : (9) ( ) ( ) ( ) G( ) 4πε d d d d ssus ad av to b dtal od to ata a qulbu of t ulso btw gbog os ad t global attato of t llu by t lt ottal of t as s qud by slfato (). ut at May 6 toug ovb T yss of odsd lasods ad

13 Tfo: π 6 (9) 4 ( ) ( ) ( ) G( ) d d d d T o ag dsty dstbuto of t alo o as to b outd tatvly (by vaato) su tat quato (9) bos aoxatly tu fo all valus of t alo o. Dug ts outatoal tatos t lto wav futos sould b kt ostat baus t s suably vy lttl ddy btw t lto ag dstbuto ad t ula ag dstbuto of t alo. Aodg to slfato (9) t ul ag dstbuto of t o o s odld by as of a twodsoal oal dstbuto adal dto: Q x πs s (94) ( ) w s s t stadad dvato t Q s t ula ag t o o T dstbuto futo (94) s oald su tat t tgal ov all sa ( atsa oodats) ylds t total ula ag Q of t o: Q πs s x y (95) ( ) x ddydx Q w x y T fato of ul sdg t o o as wll as t stadad dvato a to b dtd tatvly (by vaato) su tat t total gy of t s d. Dug a outatoal tato t lto gstats av to b outd as ty stogly dd o t ula ag dstbuto of t o. Aodg to t altat slfato (9) t ul ag dstbuto of t o o s odld by xotal day adal dto: Q x πs s w s s t day fato t Q s t ula ag t o o (96) ( ) T dstbuto futo (96) s oald su tat t tgal ov all sa ( yldal oodats) ylds t total ula ag Q of t o: Q s x π s π (97) ( ) dd d Q T fato of ul sdg t o o as wll as t day ostat a to b dtd by vaato su tat t total gy of t s d.. Tasfoato to atual Uts I t followg txt t Hat gy wll b usd as a ut of asu fo gy. It s dfd as: a 4 πε (98) 7. V w 4πε (99) a 5.98 s t Bo adus ad () 4πε - s t f stutu ostat. a ut at May 6 toug ovb T yss of odsd lasods ad

14 T lto st gy uts of () tfo bos: T followg quato dfs a f adus: () () (4) a w Q s t la ula ag dsty t o o atual uts Q s t ula ag t o o ad s t lgt uts of t Bo adus. a T dfto of t f adus was aftd su tat t latv adal xtt of t lto obts at t olatvst lt bos ddt of t la ula ag dsty. T latv adus s dfd as: (5) T volu ag dsty atual uts s dfd as: (6) ( ) a a T ut dsty atual uts s dfd as: a (7) Addtoally t followg quatts a dfd : (8).. t su of t ottal gy ad t kt gy of t lto w s futog as t gy gvalu of t Kl-Godo quato Φ Φ Φ (9).. t ottal gy latd to t oulob ottal of t ltos ad t ul as s by a lto (gatv sal ag) () a l π s t t axal aoal otu of t lto atual uts () A a A s t axal agt vto ottal atual uts. T latd ts A ad A a t agt lto-lto tato gy ad t so-alld daagt gy stvly () a A l a A π s t axal kt otu of t lto atual uts. T latd t s t axal kt gy atual uts ().. t ula slf ulso gy.. t gaulaty o of t ula llu (4) G G ut at May 6 toug ovb T yss of odsd lasods ad

15 ut at May 6 toug ovb T yss of odsd lasods ad T adal wav futo atual uts s dfd as: (5) Ψ (6) ( ) ( ) [ ] γ s t loal ot fato outd fo t gs atual uts Dvdg bot sds of (7) by usg t odut ul of alulus ad substtutg va () (4) (8) (9) () ad () s sultg : (7) Ψ a d d a d d a Substtutg (5) ad (5) (7) usg t otato ad fo t fst ad sod dvatv to of adal wav futo ad ultlyg bot sds of t quato by ylds t adal Kl-Godo quato atual uts: (8) T Södg quato (7) atual uts s: (9) T goty tgal (59) a b xssd atual uts as: () ( ) π os os 4 l d G Multlyg (4) wt a ad usg (6) ad (5) ylds t volu ag dsty atual uts: () ( ) ( ) ( ) γ π Multlyg (44) wt ( ) a ad usg (68) () (4) () ad (5) ylds t ut dsty atual uts: () ( ) ( ) [ ] ( ) A π By usg () (5) () () (5) quato (46) t total ut (avagd ov all ltos as) a b outd fo t quatts atual uts as followg: () ( ) [ ] ( ) d A a I Multlyg bot sds of (58) by ad substtutg va (5) (6) (9) () ad () ovds t oulob gy atual uts: (4) ( ) ( ) ( ) ( ) ( ) ( ) x d G s s d G γ π

16 Multlyg bot sds of (6) by ε µ a ovds t agt vto ottal atual uts: (5) A ( ) ( ) G( ) d A ( ) substtutg va () () (5) (7) () ad () ad usg [ ] ( ) ( ) G d π By dvdg (49) by ad usg () (5) ad (5) t xtd valu of t lto obt adus atual uts bos: (6) γ ( ) d Dvdg (76) by ad usg (84) (98) () (6) (9) ad (4) ad takg a of t fat tat t sal ags a ostv ylds t ula slf-ulso gy atual uts: π (7) ( ) ( ) G G 8 5 π 4 d ( ) Z d w > ad s t gaulaty o atual uts > Usg () (5) ad t oalato ta (64) atual uts bos: d (8) γ ( ) ( ) T stadad dvato of t ula ag dstbuto t o o atual uts s: G (9) s s a Multlyg bot sds of (94) wt ad substtutg va () () (4) (5) ad (9) ylds t ula ag dstbuto of t o o atual uts: x πs s () ( ) a Altatvly ultlyg bot sds of (96) wt ad substtutg va () () (4) (5) ad (9) ylds t ula ag dstbuto of t o o atual uts: x πs s () ( ). Aoxat Soluto of t adal Wav Futo T followg asat wll b usd fo aoxatg t adal wav futo: () ( ) f ( ) ( ζ) x w f ( ) s assud to b a olyoal ad ζ s a tuabl salg fato. T adal Kl-Godo quato (8) as a sod soluto w s la ddt of t soluto gad by asat (). T sod soluto would b std by t followg asat: () ( ) f ( ) x( ζ) w ζ ut at May 6 toug ovb T yss of odsd lasods ad

17 Howv ts sod soluto ad all la obatos wt t w oatbl wt bouday odto (). Tfo ts sod soluto asat wll ot b usd. T fst dvatv of t adal wav futos () ads: (4) ( f ζf ) x ( ζ) T sod dvatv of t adal wav futos s: (5) ( f ζf ζ f ) x( ζ) T valu of ζ a b dtd by aalyg t asytot bavo of t wav futo at : T ltoagt ottal (ad tfo t ts Also t ts ootoal to to: ad (6) ( ) Istg () ad (5) to (6) lads to: ad dsaa at (7) ( f ζf ζ f ) ( ) f A ) bo o w t adus aoas fty.. T Kl-Godo quato (8) t slfs Assug futo f a b aoxatd by a olyoal of ft dg t futo doats ov ts dvatvs at ad tfo t xotal salg fato s: (8) ( ) ζ w < T as of dos ot aly baus of slfato (). Oly t ostv valu of t squa oot s vald baus of asat (). Solvg (8) fo t gy ovds: ζ (9) Oly t ostv valu of t squa oot s vald baus of bouday odto (4). quato (9) outo wt bouday odto (4) as tstg osqus: (4) < ζ < at t low d ad by as of t st gy at t g d At t o-latvst lt t xotal salg fato outs as: (4) ( ) ζ tus ( ζ ).. t salg fato ζ s ltd by as of t axal aoal otu Istg () (4) (5) ad (9) to adal Kl-Godo quato (8) s ladg to: ζ ζ ζ ζ f (4) f f ut at May 6 toug ovb T yss of odsd lasods ad

18 Fo t o-latvst lt t Södg quato (9) s ladg to: f ζ ζ f A (4) A Solutos to dfftal quato (4) o (4) osst of gvalus of ζ ad gstats of olyoal f. Ts solutos a t b usd to out t gvalus of ad gstats of of t adal Kl-Godo quato (8) o Södg quato (9). Futo f() a b aoxatd by a olyoal of as followg: f (44) f ( ) fo ß ad T suato us ov a ub of ts ddg o t dsd auay of t aoxato ( at ds to b about wt 8-bt floatg ot ubs fo asoabl auay). T (gally abtay) as of t (gally olx) wav futo s os su tat t offts bo al ubs. Gally ostats ad ζ a ddg o quatu ubs ad l. Fo slty asos ts ddy s ot fltd t stv ds of ts ostats. T fst dvatv of (44) ads: (45) f ( ) ( ) T sod dvatv of (44) s: (46) f ( ) ( )( ) I quato (4) a ub of ts a b aoxatd by a olyoal of dg w s dvdd by : ad b (47) ζ b w At t o-latvst lt (47) slfs to: A (48) A b b ζ w ad b Aoxatos (47) ad (48) obably av a ltd ovg adus o att ow lag s ad ad ow t offts a os. Howv fo a gv losd tval of adus valus t aoxatos a b ad abtaly s by oosg ad t offts aoatly. A sutabl aoxato a b foud by fst dtg t ag b of lvat adus valus fltg t adal xtt of t lto s wav futo. Fo xal o a oos ad su a way tat t lto sds wt 99.9% obablty btw ts ad ad at t sa t t ag s ad as sall as ossbl. Basd o ts ag addtoal ods toug d to b dtd btw ad. T ods sould b os su tat t aoxato o s d (.g. va bysv ods). Ts ods a t b usd.g. by wto olyoals fo tolato. ut at May 6 toug ovb T yss of odsd lasods ad

19 ut at May 6 toug ovb T yss of odsd lasods ad Istg (44) (45) (46) ad (47) to (4) ad ultlyg bot sds wt ylds: (49) ( )( ) ( ) ( ) b ζ ζ By gltg t ts wt ots of g ta t sult a b wtt as: (5) ( ) [ ] ( ) ut b b ε ζ w fo < ad ut ε s t ut-off o odud by gltg ots of g ta T ut-off o outs as: (5) ( ) ut b b ζ ε T lft ad sd of quato (5) quals o fo all valus of. Ts a oly b tu f t offts of fulfll t followg quato: (5) ( ) [ ] ( ) b b ζ Aalyg t as gvs: (5) tfo Istg (5) to (5) ylds t tatv foula fo outg t offts fo t valu of : (54) ( ) ( ) b b ζ w fo < ot tat t offts a all ootoal to a ot. Foula (54) stays t sa at t o-latvst lt. quato (5) uts addtoal quts o t offts - toug w otadt t quts of quato (54). Tfo t olyoal aoxato of t adal wav futo wt ft aot b ad s. T aoxato o bos al w t last offt s o w s t as oly fo t gvalus of ξ. Tfo ts dfs a tod fo dtg t gvalus. Altatvly o ould dt t gvalus by usg t ogal Kl-Godo quato (8) outo wt (9) as a asu of o: (55) ( ) ζ δ ζ b ζ

20 At t o-latvst lt o would us t Södg quato (9) outo wt (4) as a asu of o: δ ζ (56) ( ) A b A ζ ζ T futo δ ( ) s aoag o fo all valus of oly at t gy gvalus o ζ. T valu of offt a b dtd fo ζ by oalato of t wav futo. obg () (44) ad (5) lads to: (57) x( ζ ) T oalato odto (8) qus: d (58) γ ( ) ( ) d γ ( ) x( ζ) Tat as o as to sal all. Total Bdg gy O ould avly assu tat t total bdg gy ula slf-ulso gy: (59) B ootoally su tat (58) ylds t valu. B of a s t su of t gs of all ltos lus t w s t gy gvalu of lto ub Ufotuatly ts aoa would out t lto-lto tato gs ltos a tatg wt tslvs. ad M tw baus t Istad t followg tod wll b usd fo outg t total bdg gy: I t fst st t gy gvalus wll b doosd va t adal Kl-Godo quato (8) to t xtd valus of a gy t t. Sodly t xtd valus otag t tato gs a o-quadat fo wll b dvdd by two ad t xtd valus otag ts tato gs a quadat fo wll b dvdd by fou. Fally wt ts adustt t otd gs ~ wll b assbld va t sa Kl-Godo quato. T total bdg gy of a ( uts of ) t outs as: ~ (6) B w ~ s t otd gy of lto ub ut at May 6 toug ovb T yss of odsd lasods ad

21 ut at May 6 toug ovb T yss of odsd lasods ad Multlyg (8) by γ usg (9) ad xadg t squad atss ovds: (6) ( ) γ γ Itgatg (6) ov all adus valus ad usg (8) tasfos t to a quato of xtd valus tby doosg : (6) sqt w (6) ( ) ( ) d γ s t adal kt gy (64) s t squa of t gy gvalu (65) ( ) d γ s t autal kt gy (66) ( ) ( ) d A l A γ π s t axal kt gy (67) ( ) d γ s t squa of t oulob gy fo lto-ulus tato (68) ( ) d γ s t squa of t oulob gy fo lto-lto tato ad so o. Dvdg t xtd valus (6) otag t tato gy a o-quadat fo by two ad dvdg t xtd valus otag ts tato gy a quadat fo by fou ylds t otd gs: (69) 4 ~ sqt At t o-latvst lt t outato of t total bdg gy s sl. Multlyg (9) by ad xadg t squad atss ovds: (7) ( )

22 Itgatg (7) ov all adus valus tasfos t to a quato of xtd valus tby doosg (7) H t otd gs at t o-latvst lt bo: ~ (7). Goug Obtal Ouato Slf-osstt Fld Itatos T lto ofguato of a ossts of ay obtals w a aatd by t quatu ubs ad l. Aodg to t aul xluso l a obtal a oly b oud by a axu of two ltos (o wt s u ad o wt s dow). T a too ay ltos a fo outg all oud obtals dvdually. Istad ags of obtals wt otguous valus fo l a goud togt. Wt a gou all obtals av t sa quatu ubs ad. Ts obtals of su gous dff quatu ub l. T att a of t quatu ubs l sts t gou dug outato. T ost sl aoa s to lt a gou ota t sa ub of obtals. O o ad t gous sould b sall oug to av a f sag t lto gs (fo auay). O t ot ad t gous d to b oas oug su tat outato t bos affodabl. quatos () ad () a outd by lttg t suato u ov t oud ub of gous. a suad s ultld by t ub of ltos t sts. Fo goud stat outatos t ouato sould stat wt t lowst gy. It sould ogss to gous wt sussvly g gy utl t tagtd ub of ltos foud t obtal. quatos () () ad (54) as wll as t ouato oss a ddg o a ot a ula a. Tus ty a b outd oly tatvly utl ag slf-ossty btw gstats ottal ad ouato. Aodg to slfato (9) obtals wt gvalu gs blow : a fobdd to ouy. a ust b tak ad aoat ual dag ust b ald su tat flutuatos of obtals btw fobdd ad allowd a ot dg ovg of t SF algot. Wt a of ts SF-tatos (slf-osstt fld tatos) t s a d fo sub-tatos: Aodg to (6) () ad (4) t loal ot fato γ t lto ag dsty ad t ltolto oulob gy ad a utually ddg o a ot. T sub-tatos a qud fo akg γ slf-osstt wl lavg t gstats uagd. Also aodg to () ad (5) t axal ut dsty ad t vto ottal A a utually ddg o a ot. Sub-tatos a qud fo akg ts quatts osstt wt a ot wl lavg t gstats uagd. ut at May 6 toug ovb T yss of odsd lasods ad