Chapter 1: Review of Quantum Mechanics
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- Amice Newman
- 5 years ago
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1 Quum Os for Phoos Oolros Frh R Corll rsy Chr : Rw of Quum Mhs. Posuls of Quum Mhs.. Posuls O Two Thr Thr r srl osuls of quum mhs. Ths r osuls s hy o r from som ohr r hory. Thy r ow oly from rms. Th frs fw r: Th s of hysl sysm m s sr y or or o y h logs o lr s. Th s or urs ll formo h s owl ou h hysl sysm. Ery msurl quy l oso or momum of rl s sr y oror  h s h lr s. 3 Th oly ossl ou om of msurm of h quy s o of h glus of h oror Â. lr Ss lr s s us fy m for lr or s wh r rors. Vors u w log o lr s f oly f: For som oro o y h or u logs o f u log o. u u. 3 u w u w. Thr s zro or suh h. 5 For y hr ss or suh h. 6 For oml umr f. 7 For oml umr u u. 8 For oml umrs. 9 Th r rou of wo ors u o y u * u u. wh quly f oly f. s ll h mgu of h or. hs h followg rors: oror Ô g h lr s hs h rory h for y or logg o Ô s lso som or logg o. Emls: Thr r my mls of lr ss ss h lr s of quum ss. For ml h s s y -msol olum ors form lr s whh h ors
2 Quum Os for Phoos Oolros Frh R Corll rsy r olum ors u whr... r oml umrs. Th r rou f u s * *. Th orors r mrs. g h Prors of Orors lr S Egors Eglus: or s gor of oror Ô f oly f O h oml umr s ll h glu orrsog o h gor. I grl oror Ô h my glus 3... h orrsog gors r O. o Orors: Th o oror Ô orrsog o Ô s f y h followg rlo * u O O u L w O. Th lf h s s h u w. u y rory 9 of lr ss * u w w u. Comrg wh rgh h s of w g w O. W s h rors of r w orrsog o w f w u h w * * u. f w O h w O. rm Orors: oror Ô s rm or slf-o f oly f O O. I quum mhs orors orrsog o osrls r lwys rm s rm orors h rl glus. ss Vors y s of ors h log o s.. whr y or wr s sum of ors logg o hs s s ll ss s. Th mmum umr of ors ss s s ll h moly of. For ml f 3... s -msol lr s h y ohr or w wr s w. sully ss ss r hos suh h ll ors r muully orhogol. For ml for h lr s of -msol olum ors ss s s. Orhoorml ss: I h so w h so offs rm y mullyg oh ss wh h r
3 Quum Os for Phoos Oolros Frh R Corll rsy 3 w w S for orhogol ss s ulss w w If h ss s s orhoorml h hs s w. Coml ss: Th omlss of orhoorml ss s.. h f h h ors h s s h r lr s s usully rss s o h omo of h form w u s ully oror. To s hs o h f w u s o y or o os w w u whh s ohr or hs s rory of oror rory of lr s. Th oror s h y oror wh h rory h for y or.. os ohg. To s why for oml ss quls ly o y or w s h rsul w w w u s fou rlr w s h so off wh w s rss rms of h orhoorml ss s w w Thus w w. Thrfor. Th gors of rm oror hos o ll orhogol hy lso form oml s.
4 Quum Os for Phoos Oolros Frh R Corll rsy.. Posul Four Msurm of Physl Qus Quum Mhs Suos  s oror orrsog o hysl quy of hysl sysm. suos h quum s or of h hysl sysm s somhow ow. Th quso h s f s msur rmlly wh woul h rsul? Posul 3 lls us h h rsul oly o of h glus of h oror Â. Suos ll h glus of  r ow h orrsog gors r lso ow hy ssfy whr Th quso h s whh glu of s gog o o uo msurm. Th swr g y quum mhs s h o o ow h rsul of msurm for mg h msurm u wh msurm s m h roly for og h rsul s g y osul of quum mhs. S h gors form oml s o my s whr. Ths s lso Jus for h msurm s lr suroso of h gors of Â. Wh msurm s m of h quy h rsul s o wh roly. Th rols of ll ossl msurm ouoms mus u o uy hs s sy o show. Sr from..3 Posul 5 Colls of h Quum S uo Msurm Cosr hysl osrl h orrsog oror  whh hs gors glus g y. Th quum s rms of h gors of  s ssum o. Suos msurm of s m h rsul s o. Th quso s wh s h quum s us fr h msurm? Th swr s o rl. Th quum s of o os ll h formo h o ou h o y mg y of msurm. Wh som formo hs o y mg msurm h quum s fr h msurm mus rfl hs ro of formo. If h glu ws msur h h quum s us fr h msurm mus.. h gor orrsog o. If so msurm of s m us fr h frs msurm h h rsul wll o wh roly o hs s rly shlly lsg. Ths su olls of h quum s
5 Quum Os for Phoos Oolros Frh R Corll rsy from o uo msurm s ll olls of h quum s s osul of quum mhs. Cs of Dgr Eglus: Suos h frs wo glus of  r l... Th quum s for h msurm of s. Wh s h quum s fr h msurm f h rsul s o? I hs s s h msurm rsul o sgush w h quum s fr h msurm mus l h g-sus orrsog o h glu s g s I mor hl lgug h msurm ros h quum s o h g-sus orrsog o h msurm rsul. M Vlus of Orors Th m lu of oror  wh rs o quum s s f s h m lu of h osrl o y mg msurms of o my l os of h quum s. If  hs gors glus g y rss rms of s wh rs o h s s h h roly of og s. Thrfor h m lu of. Ths wr mor grlly s  or us Â. Som Commo Osrls Poso of Prl: Th oror orrsos o h oso of rl mso. Egors of r wh orrsog glus... Orhogoly rlo: Comlss rlo: Momum of Prl: Th oror orrsos o h momum of rl mso. Egors of r wh orrsog glus.. Orhogoly rlo: Comlss rlo: Wfuo: o h o wr h quum s of rl s Th mlu s usully o y s ll h wfuo of h rl. 5
6 Quum Os for Phoos Oolros Frh R Corll rsy 6 Smlrly Quso: Wh s? W o swr hs ulss w ow somhg mor ou h rors of h orors. Th lssl sro of rl s quz y mosg s osul ommuo rlo. For o-rls s-lss rl hs ommuo rlo s... O o h lu of h r rou from hs ommuo rlo s show low. X Th soluo of h o grl quo fou s Ths s qu rsg; f h follows h Smlrly I follows from h ommuo rlo h h offs of momum oso ss so h Fourr rsform rlosh!! I s lso usful o ow h o of h orors o s or.. wh r h ss. W lry ow h. W o f. : * * *
7 Quum Os for Phoos Oolros Frh R Corll rsy 7 : ll mor oml Thrfor s l ffrl oror h oso rrso! 3 : Pro s for s l ffrl oror h momum rrso. : * * * Sr Do of Osrls W sw rlr h o lu or h m lu of oror  s Â. W wll wr hs m lu s Â. Wh ou h sr o? Df w oror  s
8 Quum Os for Phoos Oolros Frh R Corll rsy 8 srg ry Rlos Suos w h wo orors Â. Th srg ry Prl ss h f C whr C s som rl umr h for ll ossl ss h followg rlo hols C Proof: For som rl umr osr h s whr. ow w ow h hrfor Suos  r rm h u s C C C Th o mus hol for ll lus of hs oly h f  C C Eml: W ow h hrfor Ths shoul o surrs s w lry ow h h oso momum rrsos of quum s r rl y hs Fourr rsform rlosh mls h
9 Quum Os for Phoos Oolros Frh R Corll rsy Th mlo Oror Th rgy of rl of mss m momum ol V s g lsslly s V m I quum mhs h rgy s g y h mlo oror Ĥ. For fr rl V m.. Posul 6 Tm Dlom Quum Mhs Th m oluo of quum s s g y h Shrogr quo sh osul: Ths s frs orr lr ffrl quo. Thrfor f s ow h usg hs s h oury oo rm for ll. Th forml soluo of h o quo for m mlo s Sory Ss Th gors of h rgy oror Ĥ r ll sory ss s hy o ol m ohr h qurg m hs for. Eml: Suos Ĥ hs gors wh glus... S Ĥ s rm s gors form oml s. y rrry quum s rms of. Suos. ssum lug h Shrogr quo Mully y r o oh ss o g Soluo s 9
10 Quum Os for Phoos Oolros Frh R Corll rsy Ths mls for ll ms. Th roly of g rulr rgy gs os o hg wh m. Th s why rgy gss r ll sory ss. Mr Rrso of Orors Suos ors form oml s... y oror wr s. L. Th W rrs  mr form y hoosg mg w ss ors olum ors. For ml l Th oror  s h mr.... If h ss s hos osss of gors of  h hs ss s h oror  s rrs y gol mr mlo Oror Dffr Form W ow h for rl ol V m Suos Ĥ hs gors wh glus... sg hs rgy gors w wr Ĥ ffr form
11 Quum Os for Phoos Oolros Frh R Corll rsy ow suos w h mlo o Ĥ V m o w fou h gors glus. o. ow suos ol ol s o o Ĥ so h h full mlo s ow Ĥ whr o o o V m Th frs r s `gol h ss us. Th so r s o gol.. Dyms of Two-Ll Sysm Suos o Ĥ hs oly wo mor gors; whh r gr.. o Suos smll ol s o h mlo o Physl Rlzo: oul quum wll sysm show low s wo-ll sysm. o Ĥ orrsos o h mlo wh h wo ol wlls wr ry fr r Ĥ orrsos o h ol wh h wo wlls r los. If s smll h quum ss g `oul ulg hrough h rrr. Ths ulg s sr y h ol ol.
12 Quum Os for Phoos Oolros Frh R Corll rsy Suos m h rl s l wll # so h. W o f h rl wfuo for. Soluo y Eso h Orgl Egss: ssum h followg soluo for wh h oury oo lug o h Shrogr quo o g Mully y h rs h o g wo quos Th mr form of h o wo quos s Soluo su o h l oos s s os. s. os Th quum s oslls m w. Soluo y Eso h E Egss: Sr from
13 Quum Os for Phoos Oolros Frh R Corll rsy 3 o W golz h w mlo. I mr rrso whr Ĥ s Ĥ Eglus of Ĥ r h orrsog gors r rsly whh orrso o ss rsly. L hs ss rsly. W wor ou morl yms usg whh r h gors of h full mlo. Th l s s S r h rgy gss hy r sory ss. So usg h rlr rsul h s lr m s I follows h u ow f w lu w g os. smlrly s whh r h sm rsuls o rlr ffr mho..3 Frm s Gol Rul ow osr h rolm of o ll oul wh fly my lls s show low.
14 Quum Os for Phoos Oolros Frh R Corll rsy Th mlo wr s Th lls o r sr y sy of lls or sy of ss E D h hs us qul o h umr of lls r u rgy rl wr s E E D O o golz hs g mlo. Suos h o my s h quso: wh s h s m of h rl from h wll? s for l h soluo of h form lug hs soluo h Shrogr quo o g h followg quos Th l oos r 3 Suos o Ths gs us s hs rsso h quo for o g
15 Quum Os for Phoos Oolros Frh R Corll rsy 5 S s o hg m slowly omr o h ol rm w ull ou of h grl o g Ths gs Rl Rl Th o quos shows h h roly of fg h rl y lr m o h l s ys s whr h y r s If w rl h summo y h grl E E D y E h w g D E E D E E Th o rlo s ll Frm s Gol Rul. Coluso: Wh h umr of ossl fl ss s f hr r o osllos. Thr s us y of h l s o h fl ss h y r s g y h Frm s Gol Rul.. srg Shrogr Purs Quum Mhs.. Th Shrogr Pur I quum mhs o s usully rs lulg o lus of orors.g. qus l. Th lulo ros s follows: G h l s lul for usg h Shrogr quo
16 Quum Os for Phoos Oolros Frh R Corll rsy Th forml soluo s. Th oror ol s o rr s s Tylor srs so.. O s ow lul. Ths mho whh w h usg so fr s ll h Shrogr s ur. I h Shrogr ur h s ors r m h orors r m... Th srg Pur Thr s ohr qul wy o lul o lus of orors. Frs o h If o fs m- oror s h oms. I hs w form h quum s os o hg wh m u h oror ols m. Ths s ll h srg ur. I h srg ur h orors r m O ffr oh ss wh rs o m o g Th o quo s ll h srg quo. Th lulo rour h srg ur s s follows: G l s oror  lul usg h srg quo wh s h oury oo. O s ow h m lu of  m s o s follows I h srg ur h orors r m h s ors r m o us uss h l s for lulos. o h: If C h mlo oror s m-. h h qul m ommuo rlo lr m s 6
17 Quum Os for Phoos Oolros Frh R Corll rsy 7 C C Th form of h qul-m ommuo rlos o o hg wh m. Th qul-m ommuo rlos rrs fuml rors of hysl sysms hr form r m. Two-Ll Sysm h srg Pur Th wo-ll sysm suss rlr s sr y h mlo Suos. W o f usg h srg ur. W f h followg umr orors Th h sr qus wr s. So w o f. W lso f orors s Ĥ s h You rfy h h followg ommuo rlos hol sg h srg quo w g o h Df. Th quo for s Th o quo sol wh h wo oury oos Soluo s s os
18 Quum Os for Phoos Oolros Frh R Corll rsy 8 I follows h s s os s os s Th o wo oror rssos for mgh loo srg. Th swr for h srg orors hs rss rms of h Shrogr orors w lry ow h o of hs Shrogr orors o h quum ss. W ow o. os. s Ths rsuls r h sm s fou rlr usg h Shrogr quo..5 Quum Mhl Msurms.5. Commuo Rlos Quum Mhs Physl Msurms I quum mhs ommuo rlos h m oo wh hysl msurms hs oo wll lor h followg Sos. Commuo Rlos Commo Egors W ow from lr lgr h f wo mrs ommu h hy oh h h sm s of gors. I quum mhs f wo orors  ommu. h hy lso h h sm s of gors. Oul of h Proof: Suos  hs gor wh glus.. S
19 Quum Os for Phoos Oolros Frh R Corll rsy s lso gor of  wh glu. If  hs ll s glus h mus roorol o. hs ms h s lso gor of. If  hs my gors wh h sm glu h mus ls l hs gsus of  f s o roorol o. I hs s h ors hs gsus of  hos suh h hy r lso gors of roof of hs fou y o lr lgr. Commuo Rlos Smulous Msurms Cosr wo orors  h h gors ll s glus g y u u Suos h osrl s msur for s. Th -ror roly of og h rsul s. Suos ws o. Immly fr h msurm h quum s ollss o. ll susqu msurms of o fs ough so h o oluo sr y h Shrogr quo s l urg hs m wll yl h rsul. ow suos h osrl s msur for s. Suos ws o mmly fr hs msurm h quum s olls o. ow suos h osrl s msur. Th -ror roly of og o. Immly fr h msurm h quum s g msur h roly of og s u s u. Suos ws ollss o u. If ow s. Th msurm of `sur h quum s so h h so msurm of g ffr rsul h h frs. W sy h r o smulously msurl. Msurm of o quy surs h lu of h ohr quy. 3 Suos. Th  h h sm s of gors sy. I ohr wors ow suos h followg squ of s: s msur rsul s o s msur rsul o. u ow ll susqu msurms of /or wll g h rsuls rsly. W sy h f  ommu hy r smulously msurl.. msurm of o quy os o sur h lu of h ohr quy. Of ours hs mlly ssum h ll msurms r o m ro shor ough h o m oluo of h quum s ours. 9
20 Quum Os for Phoos Oolros Frh R Corll rsy.5. Quum Mhl Dohr Dohr s o of h ls ursoo s wll s h mos msursoo o quum mhs. sml ur of ohr s rs hr. Cosr h suroso s of rl. msurm s m o rm whhr h rl s or. If h rsul s h s mmly fr h msurm s... If h rsul s h us fr h msurm. I hr s h o of msurm sroys h lr suroso s g y rls y hr or g uo h msurm ouom. I ohr wors h quso of formo y llg gs sroy quum mhl lr surosos. llg g os o o m r msurm. /sh rhs us sy hoo or hoo sr off h rl o rm whhr h rl s s or s. y suh rour h gs h llg g formo ou whhr h rl s or sroys h lr suroso ollss h quum s of h rl o hr or. ow suos h s rl s rg wh s rom. If hr s wy y whh llg g rm whhr h rl s or y us osrg h rom h h lr suroso s of h rl wll sll g sroy. Lr surosos hrfor shor l sly g sroy y ro wh h rom f o llg g s ly mg msurm. quso o s hr s f hr wy o qufy hs sruo of quum mhl lr surosos. For h s osr h rous. Ths rous r of h lr suroso h s. If hs rous r zro h lr suroso os o s. Th orors gr hs rous g s Th ro of rl wh s rom m hs rous go o zro s m rogrsss.. Ths homo whh rsuls h sruo of quum mhl surosos s ll quum mhl ohr. Pro: W sr from s rh h oluso h s m rogrsss wll hr olls o or y ro wh h rom.. h lr suroso wll g sroy. Suos w ss s s rou
21 Quum Os for Phoos Oolros Frh R Corll rsy Th srg from o s u or whh o lso wr s rsly. W s lr surosos g sroy y ro wh h rom. Th ow om w u lr suroso ss fr ro wh h rom? Of ours y s wr s lr suroso s y hoosg ror ss s. Whhr or o lr surosos rulr ss rrso g sroy s o h ur of h ro w h rl h rom ly wh formo s r y h rom urg h ro. Suos h ro of h rl lly s s suh h llg osrr y loog lr h rom fgur ou whhr h rl ws or. W wr s s m rogrsss h lr suroso w wll g sroy h rl s wll u or whh r rsly..6 Th Quum Mhl Dsy Oror.6. Pur Ss Ssl Murs Cosr wo ss of quum ss: S : lrg umr of l os of h lr suroso s S : lrg umr of ss. Th umrs of ss ss r h ro :. S osss of ur ss. S s ssl mur of ss. Suos h ss r gss of oror Ô wh orrsog glus rsly. If h m lu of Ô s msur for ss h sm rsul wll o for oh ss. Is hr wy o hl h so w ur ss s ssl mur s? Th swr s ys h sy oror s h ool sg o hl ur ss ssl murs o qul foog..6. Th Dsy Oror h Dsy Mr Quum Mhs Dsy orors r usful wy o rrs quum ss. Dsy orors lso sr yms of quum ss lug ohr sml wy. Dsy orors lso rrs ssl murs o o ur quum ss. Mos grlly quum s s o rrs y s or u y sy oror. For ur s h sy oror s o s. If s h s of s
22 Quum Os for Phoos Oolros Frh R Corll rsy Th sy oror l ll quum mhl orors u mr form. I mr rrso usg h ss.. h sy oror s Th gol lms of h ouo rols h off-gol lms rrs ohrs. For ssl mur s h s of s o h sy oror s I mr form Comrg o w s h h off-gol lms of h sy mr r s for ssl mur. W lso ow h ohr sroy lr suroso ru lrg ollo of ss of h form o ssl mur h hs ss wh ros. Thus w Dohr Dohr sroy off gol lms of h sy oror or h sy mr. M Vlus of Orors sg h Dsy Oror/Mr: Th m lu of osrl wh rs o h s ws. Th m lu of wh rs o h sy oror s f s Tr Suos whr form oml ss. Th Tr O h ohr h f rrss ssl mur of wh rols h Tr Tr
23 Quum Os for Phoos Oolros Frh R Corll rsy 3 Mor Coml Eml: Suos w h ssl mur of ss wh rols rsly. I hs s w h ssl mur of lr suroso ss. Th W wr h o rsso s o h h sy mr ws gol h ss u hs off-gol lms h ss. Th rg lu of h osrl s h. Tr Th o mls show h sy orors/mrs hl h mos grl ys of suos..6.3 Tm Dlom of Dsy Orors I h s of ur ss h rg lu of quy m ws show o m srg Pur h oror s m Pur h s s Shrogr Thr r lso wo wys o lul rgs wh quum ss r sr y sy orors m srg Pur h oror s Tr m Pur h s s Shrogr Tr Tr To s hr qul rll h So Tr Tr s h rsul Tr Tr o g
24 Quum Os for Phoos Oolros Frh R Corll rsy Tr Tr Tr whr fr ffrg h o quo wh rs o m w o h Shrogr quo o h srg quo for h sy oror o h h o quo s ll ffr from h srg quo for y ohr oror g of h srg ur s h w lul orrlo fuos of osrls suh s Tr I s fful o omu hs qus h Shrogr ur. I h Shrogr ur w oly sly omu qul m rgs.g. Tr Tr Two-ll Sysm h Dsy Oror Formlsm h Shrogr Pur Cosr h wo-ll sysm sr y h mlo.. Th sy oror s h Suos I h mr form wh Our gol s o f Tr s h ss w g Sr from T h mr lm of h o quo wh o g T h mr lm wh o g
25 Quum Os for Phoos Oolros Frh R Corll rsy Smlrly o h Th o quo sr h osro of roly urg m oluo. L s Th s s Comg h o wo quo gs Th oury oos r s Th soluo s os. So ρ s. os os. T r os Ths s h sm rsul s o rlr y ffr mhos.. 5
26 Quum Os for Phoos Oolros Frh R Corll rsy Dohr h Dsy Oror Formlsm W frs suss h ffs of ohr. rgs omu h sy oror shoul rr s rg ss.. h rg Tr oul m hr o of h followg: Srl l os of sysm r rr h s of h sysm s rrs y h sy oror. m msurm of quy s m o ll os of h sysm. Th rg of h rsuls o orrsos o h quy Tr. sgl sysm s rr h s of h sysm s rrs y h sy oror. m msurm of quy s m. Th sysm s h u y som ms h sm s h ross s r my ms. Th rg of ll h msurms orrsos o h quy Tr. W r ow oso o urs how ohr wll ff our rsuls. Suos w h ffr os of wo-ll sysm ll os r rr h s m. So ssumg rro h sy oror of h sysm s lr m ssumg o ohr os s os s os s m osr h s of h -h sysm os s Suos h -h sysm olls o oms os s s rsul of ohr. For os s whr. ow suos h -h sysm olls o. For os s whr. s m rogrsss mor mor sysms wll ohr olls o hr or. S h ms wh h ss of ffr sysms olls r rom h hss 6
27 Quum Os for Phoos Oolros Frh R Corll rsy 7 wll lso rom. Thrfor wh follows o sor h hs rms. Th ross of ohr usully hs sso m sl whh w wll ll. For mos sysms wll o h r s olls. For mos sysms woul h r s olls. m h sml rg of y osrl quy rg or h r sysm wll s os s os s os If for ry sysm wll o-zro wll h rom lu. For lrg o rl h sum or y rgg wh rs o ssumg h s rom rl uformly sru h rl from h o rsso oms s s s os Th o rsul s qul o h sy mr for g y sro of ohr mus hrfor g h sm rsuls s o o. Ths o y mofyg h quos for h off-gol omos of h sy mr roug y rm ow w g for h quo sg h sm oury oos s for w g h rsul
28 Quum Os for Phoos Oolros Frh R Corll rsy 8 s os s os s os s Thrfor Ths s us wh ws sr. Thus rouo of ohr y rms.. wors s. o h w rrrly sr hs ohr y rms h quos for. Quso rss f hr s r sro or r quo for h sy oror of h form?? whos mr lms o h ohr rms for h off-gol omos. I rms ou hr s suh quo u s oml. I s muh sr o o hs s h srg ur usg Lg quos. W shll o hs lr h ours. Two-ll Sysm h Dsy Oror Formlsm h srg Pur I h srg ur Tr whr W h s for
29 Quum Os for Phoos Oolros Frh R Corll rsy S 3 Tr Tr Tr Tr Tr Tr Tr h rg of quos - wh rs o rlr for Tr gs h quos w o worg h Shrogr ur. Th soluo of - s os s s Tr Tr os s. ow w w o rou ohr. W ry hgg h quos for s follows 5 6 Th moo for g hs y rms s h w ow h Tr Tr Tr S w h lry s h h quos for ow hy h h y rms h mof quos for sur h w g h rgh quos for. Th soluo of 5 6 wh l oos s 9
30 Quum Os for Phoos Oolros Frh R Corll rsy 3 s s os s os Flly s os Tr Tr whh s h sm rsul s for. g Prolm: W h s rlr h m oluo h srg ur rsrs h ommuo rlos w orors. Proof: If C h C C Commuo rlos r fuml grs of h quum mhl sro of y sysm r ol. u f o s r rms o h rgh h s of h srg quo h hr s o gur h h ommuo of  wh ohr orors wll rsr urg m oluo. For ml o mol ohr w hg h quo for g y o Th orgl quo rsr h ommuo rlos suh s Th mof quo os o rsr suh ommuos I molg ohr w hg h quos wy h sol h quum mhl ossy of h quos. W o hs rolm h Shrogr ur s w oo rgs of h sy oror quo
31 Quum Os for Phoos Oolros Frh R Corll rsy 3 rou ohr o o h oror quos u h quos for oml umrs.g.. Lr h ours w wll s h ohr rous os o h sysm h s som ss fuml. Ths os wll su wh srg-lg quos..7 Prou lr Ss Th lr s of wo quum sysms s o y sg oghr h lr ss of h ul sysms. For ml osr wo sysms sysm sysm wh quum ss rsly. Th s of h om sysm s wr s oror hs lrg lr s s wr s sor rou whr h oror  or s oly h lr s of sysm or sysm s follows Eml: Two Dffr Two-Ll Sysms: Th mlo for wo ffr wo-ll sysms s whh wh slgh us of oo s mor ommoly wr s r gs of h om sysm s for ml h o of h mlo o h gs s o h h oror h sor rou s o h s logg o s ow lr s. Eml: Two Dffr Ro Mos Cy Th mlo for wo ffr mos of ro y s
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33 Quum Os for Phoos Oolros Frh R Corll rsy 3 roly roly ow suos w m rgy msurm o sysm o h rsul. Rgh fr hs msurm h s of h full sysm s. If fr h msurm o sysm w m msurm o sysm h ossl ouoms h orrsog sso rols r roly roly Thrfor msurm o sysm hs o hg h msurm rsuls. ossl ouoms h orrsog rols for sysm. ow osr h gl s 3 For rgy msurm o sysm ossl ouoms h orrsog rols r 3 roly roly For rgy msurm o sysm w h roly 3 roly Suos w m rgy msurm o sysm o h rsul. Rgh fr hs msurm h s of h om sysm ollss o h s If fr h msurm o sysm w m rgy msurm o sysm h oly ossl ouom s wh roly o. Thrfor for gl ss msurm o o susysm hgs h msurm rsuls. ossl ouoms h orrsog rols for h ohr susysm..9 Dsy Orors for Jo lr Ss.9. Dsy Orors for Egl gl Ss If h quum s of sysm ossg of wo susysms s ugl s l h h sy oror for h sysm s 33
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