Lecture 1. Probability in Quantum Mechanics

Transcription

1 Polt pltus ctu Polt Qutu Mccs T ctl pol of qutu ccs stt s follows: If t c occu o t o pt pocss o pt ow o w t t totl polt su o ll pts? I ot wos wt t lws of polt o toc scl? Clsscll P P Fo c gu fg ullts t sc wt ol t so posto t polt of tctg ullt st of wt posto t locto s g p / N w s t u of ullts cugt t st N s t totl u of ullts f t gu T s ul ppls fo ullts pssg toug sco ol t cugt tcl st If ot ols op t totl polt st s g p tot p p Now suppos w o t s pt wt potos o lctos W wll plc t polt st wt t tst I Now w f tt W s ts? T tst s f lctc fl foc/ut cg V/ c sp of lgt I tot I I I cos I o potos/c /s ½ c l t cos l t λ τ l t cos l t λ τ T qutts l t t s fo ot slts u occus t t sc w ps tfo tf costuctl Tt s

2 c I tot ε T ctl sp occus cus t two pts fo soscls tgl W os t u of fst fg occu? Cll t fg spcg t spto tw t slts stc fo t slts to t sc l Fo l >> w c sow wt lttl got tt l λ wc s wll-ow sult fo ffcto optcs T cospog lw fo cog polts fo ptcls s p tot f f w p f f p f f t gll copl polt pltus fo pts T cocluso tt p tot p p pls tt t ptcl os ot spl pss toug t o ol o t ot; f t o coul spl tult t outco fo c ol t polts I ot wos t ptcl ust w-l o tt cuss tfc But wt s to stop us fo spl osg wc ol c lcto psss toug? I ot wos suppos w tg t lctos costuct two stct polt st stutos p p ptcl ptcl T toul s tt w w o ts w sto t tfc; w w tg t lctos w gt p clus p p f f p o osc ws f w o t tg t ptcls w gt fft sult p f f p p f f f f t fg p osc T so ts pps s tt loog qus us to s lgt o t ptcls t sctt lgt cus t ptcls to col lu t fgs Rucg t tst spl ucs t u of potos t g t ptcls to two clsss tgg utgg so tt P tot p tgg p utgg p o-osc p osc

3 l p I tot p osc l I l g p tot p oosc Suppos w w to uc g of ul potos so s to uc t out of scttg Ts s csg l c/u c/ w w us Plc s quto fo t potolctc ffct But t tus out tt o to pt lug of t osclltos w qu l > / wc pts us fo owg wc ol t lctos pss toug Suppos w t tl fft sttg Ist of tggg t ptcls w su t col of t plt tt ouss t slts Fo t cto of t col w c uc wc slt t lcto pss toug wtout ctull loog t t lcto t t ffc tw otu cgs fo slt slt p Wt lttl got ou c sow tt δp p l But w ow fo t Bogl pt tt t polt pltu of f ptcl oscllts pl spc wt wlgt tt s g lp Iog t ffcto quto t follows tt l λ λ δ p p p / l But g t tus out tt ttpt to costuct fo t otu stutos wc ol t lctos pss toug wps out t tfc ptt Tt s w to ow ot ol t otu of t plt ut lso t posto of t ols wl t

4 plt s og Wt w f s tt t uctt t ltt s > / It follows tt t st w c o s δ p δ > wc s t wll ow uctt pcpl / Cocluso: W t st two o o tfg ltts w ust t polt pltus fo c pt t t t squ of t su to gt t totl polt Itfg ltts f s utull clus pts fo wc o ptl ttpt s to stgus tw t Supposto stts Cos of lgt popgtg t -cto ll pol t pl T s ct o pol wc s fct so tt t tsts % of t lgt pol plll to ct s % pol ppcul to tt s W pfo t pts: W t pol s s lg t gl wt spct to t -s % of t lgt s tstt W t pol s lg log t -s fcto cos s tstt W t pol s lg log t -s fcto s s tstt Ts ostos l ustoo usg t w scpto of lgt W t tt t lctc cto of t lgt s pltu s cos j ol slct copot of t w c gt toug t pol ow o w pl ts ostos fo of ptcls potos? Fo ts pspct t pol s l flt wc psss ol fcto of t ptcls ll of t ptcls tt t tsts pol log t s of t flt W t followg uctos fo t t pts: ll of t ptcls pol t gl wt spct to t -s ll of t tct fcto cos of t potos s tstt t flt ll of t tstt ptcls pol log t -s fcto s of t potos s tstt t flt ll of t tstt ptcls pol log t -s pt sows tt t potos pu polto stt wt gl But pts sow tt t potos supposto of ppcul plll polto stts wt polt pltus popotol to cos s ow o w cocl ts two sgl 4

5 cotcto scptos of t ptcls? W sst s o stt t c lws cos to ptl two o o ot stts c s Itptto of t w fucto: T squ w fucto of sst gs t polt of fg t sst ptcul locto spc It s coct to t of t s gg t u st of ptcls sll olu tt soow tf wt c ot T so w t ltt tptto s fls s tt t tfc ptt pssts f ol o ptcl s su t t wt t st so low tt two ptcls pst t pptus sultousl Dsct ls ctu Polt Dstutos Cos popt osl of sst tt c ol N sct lus N pls: Nu of popl fl; u of tol qut oscllto W suts os of tos suts suts tc suc tt N T polt of osg s g wt olto P / P N P T st {P } s cll polt stuto If fo so s P s t ll ot P s t sst s s to ft o pu stt s of t osl T polt of osg l s P l P l T ost pol lu of s p w P p {P } pl: T ost pol lu of p of c s 7 T g lu lso ow s t lu o pctto lu of s g 5

6 N P wc s lso cll t fst ot of {P } T sco ot s g T oot squ lu of s g N P s T c o s g V σ s fo pu stt of T g o pctto lu fo fucto of s g Cotuous ls f N P f ctu Polt Dstutos Pt Cos cotuous l f o so o pls: posto loct lft T polt of osg lu of t l tw s p w p s polt st fucto wt possl lus uts of - Nolto: p p Totl polt o so tl of : P p T o of ps o t tu of Fo pl Sp: c Vloct: - 6

7 Rl stc: Pol gl: q p utl gl: f p Popts of t stuto fucto: p p stsfs p f f p pl scuss clss: t D Bolt stuto fucto: p p / w s t ost pol sp f T W c splf t fo of t stuto fucto fg uc sp / suc tt p Dstutos o o t o cotuous l: p w w ssu o fo - to fo ot W coos to g o o of t ls o to ot stuto o t g l: p p p p It tu out tt p s fctol wc cs t ul stutos ull ol: w p fg 7

8 f g pl wo out clss: D D Bolt stutos Cg of ls: Coot tsfoto: ˆ tϕ ˆj / Df uc l: / p F p{ / } F F / w F F Nolto: F F 8

9 T fcto of o s Jco tt ss fo t tsfoto It t clu t stuto fucto s t s o t st sptl to t olto tgl s t cs of t w fucto pl of t og w fucto ctu 4 lt Spcs t s stt wt t fl pl of Cts ctos: ˆ ˆ j ˆ ˆ ˆ ˆ Ts cll ss ctos o ss st t cto wtt s l coto of t ss ctos ˆ ˆ T coffcts gl copl Otoolt of t ss ctos: ˆj R I Dc otto fo cto ts ul: T ul ss st s wtt s Otoolt Dc otto: ˆ δ o R I δ t I / R g ˆ ˆ ˆ ˆj tc fo fo 9

10 δ gl t colu cto wtt s Its ul s ow cto tt s wtt s Sll ot ow cto wtt s u T ot o s t scl lu u u δ Dot pouct of t colu cto wt ss ow cto: It follows tt pojcto of oto pojcto opto Nucl pl 5 Clcult

11 Dot pouct of cto wt tslf: j j j Nol ctos: ˆ ˆ ˆ Two ctos otogol f u Two ctos otool f u u u Bss ctos otool coplt st of ss ctos s to sp t cto spc wc t f cus t o ot st ctos tt otogol to ll of t ss ctos Mt otto: u u Tsfotos of ctos ~ ss

12 pl: clocws otto cosθ sθ ˆ sθ cosθ cosθ sθ sθ cosθ Not: O w to t tsfoto s usg t Spcl oc to To wt cos cos cosθ s sθ w θ s t gl tw two -pl ctos wc gls wt spct to t -s ctu 5 Otool Fuctos Two gltos: T N sos N ft T ss ctos t ot poucts fuctos of so l f o tl [] Mo gll w s wgtg fucto Ts s ot t fftl olu pl of coplt st of otool fuctos: s w o s o s s δ s Cos so t ut wll- fucto f f o []

13 Rcll tt Sll N f f pl: f s Itgt pts: u u cos s s s Sttg X / w f tt ctu 6 T Stu-oull pol Fo cotol ctos Fo o gl fuctos f T fuctos coplt ss st fo Sll t fuctos fft coplt ss st fo W o ts fuctos co fo? sw: T solutos to fftl quto wt ppopt ou qutos T pl of t s fuctos solg t followg fftl quto

14 λ sujct to t ou cotos soluto of ts fftl quto s s ϕ T ou coto qus tt ϕ t ou coto qus tt w s tg so tt s T ou cotos lso pos costts o l: λ Nolto: s s Otogolt: δ s s Cgg t ou cotos fo t s fftl qutos gs fft st of otogol fuctos: ' ' cos Glto: T Stu-oull quto 4

15 5 [ ] q p λ p q l fuctos T coul costts If t ll ut w gt t s fuctos scuss pousl l s costt cll glu s wgtg fucto It lup to s Possl ou cotos: 4 p o p plc of t o Popts of t solutos: f s l soluto ssoct wt l o-gt glu l T st of fuctos fo coplt ss st Nolto: [ ] Otogolt: δ 4 Copltss: f f o gl st of fuctos s t soluto of t followg fftl quto ] [ λ It cot to t st Stu-oull fo s of t followg tfctos: p p p q p

16 You f ts ct susttuto to t Stu-oull quto Not tt p p ft susttutg w gt p λ Multplg toug /p cos t s quto p p p ctu 7 Bou Vlu Pols Qutu Mccs T popts of qutu ccl ojcts lctos tos olculs cstls tc sc ttcl ojcts cll w fuctos T popts cll osls posto otu g gul otu tc T lus of ts osls lut optg o t w fuctos wt optos ssoct wt t osl of tst Mttcll ts wtt s ˆ χ λχ w ˆ s t opto ssoct wt osl χ s gfucto of ˆ λ s ts glu If s t g t ˆ s t lto Ĥ t fftl quto s Scog s quto wc cst to Stu-oull fo I ts cs t gfuctos of t Stu-oull quto lt to t gfuctos of t lto χ t quto / χ T otoolt lto cos χ χ j j δj Ot osls fft glu qutos tt ot of t Stu-oull fo T ll t pott popts of g l optos l glus coplt sts of gfuctos T quto ˆ χ λχ ss tt f fucto χ s gfucto of ˆ t optg o t wt ˆ os ot cg t pt fo ultplct costt Wt pps f w opt o fucto tt s ot gfucto of ˆ? t s cll tt fucto W c stll opt o t wt ˆ ut w wll up wt tl w fucto To s ts lt s p s ss of gfuctos of ˆ : χ 6

17 ˆ ˆ ˆ χ χ λχ popt of ˆ tt s css fo t lst stp to wo s tt t s l opto Tt s ˆ f g f ˆ g ˆ { } pl s ˆ / Wt ts gfuctos? Is t l? W c t ul coffct s tg ot-pouct: χ s ˆ s ˆ s W coul sc ts pocss sg tt t opto s fo t fucto s ˆ pojcts out copot To s t cocto wt qutu ccs cll t St-Glc pt o t poto polto pt If t flt s lg log t -s t poto s -pol t t flt tsts t poto wtout cgg t But fo ot polto t flt wll tst ol t -copot of t poto stt Sll f t flt s lg log t -s t wll tst ol t - copot T flt s l pojcto opto Ts ls us to so postults: Fo osl t sts l opto ˆ wt glus l T l sptl coot o otu T ol possl outcos of sut t glus of ˆ Fo pl f t sst pps to pu stt of sc fucto χ t t ol possl outco of t sut s l T pctto lu of s t spl ˆ λ λ If sst s so ot stt t c wtt s supposto of wt coffcts χ If t polt of otg outco l s P t t pctto lu of s P λ W gl t pctto lu of fo t stt wtg t s ˆ j j λ j j j jλ jδj ˆ λ j j j j 7

18 W s tt P It s to f t opto ˆ to ssg g to ts gfuctos f If t osl s t g t ssoct opto s cll t lto Ĥ t gfuctos w fuctos cospog to spcfc g stts χ t fftl quto coctg t s ˆ χ χ If t sst s ot pu g stt t t sc w fucto wc c p t g ss χ I ts cs ˆ ut t s t polt of fg t sst spcfc g stt ot pl of osl s t posto t opto ˆ s t coot T gfucto ssoct wt ˆ s sutl cocpt wc w wll postpo utl lt If t sst s ot pu stt ot t ft coot t pctto lu of s g I ts cs t s sol to tf t polt st p of fg t sst loct tw to g tslf s cll w fucto o polt pltu I t g sut t gfucto χ t w fucto lso polt pltus fo fg t sst tw W s tt t two tps of polt stutos T fst s t polt tt ojct s ptcul gstt of ˆ Tt polt s t squ of t pso coffct T sco polt s t polt st tt t ojct s loct t posto g t squ of t w fucto T s lso ot w fucto tt gs t polt st tt t ojct s ptcul otu W wll l out ts lt χ s cotol polt t ss tt f t sst s stt t χ s t polt st fo osg t sst tw coots If w us Dc otto w o t to tl out fftl qutos so osls wo t cotol w fuctos ot fuctos of sptl coot pl s t sp of lcto W wll cos t 8

19 pojcto of t sp oto t -s T opto fo ts osl s S ˆ It s two gstts two glus s ± Sˆ / / ˆ S / / I ts otto t opto wtt s t ts gfuctos s colu ctos Cos so stt s / / Wt t polts of osg S S? Wt s t pctto lu of S? Su of t o guts: If sst s pu stt of ptcul osl t sut of tt osl ls t sst ts ogl stt Ts s sc ttcll optg o t stt: ˆ λ W t t lu of t osl pojctg t o sult c oto t ogl stt: ˆ λ λ T o sults ppl f t sst s ot pu stt of t osl W c clcult t pctto lu of t osl fst pg t stt t ss st of t osl t optg o t pso fll pojctg t sult c oto t ogl stt: f ˆ f f ˆ f λ λ Pλ 9

20 4 T o guts ppl lso f t glu pps to cotuous Ts s ot Stu-oull pol ut t pcpls t s I ts cs t su o cos tgl o t glu wc w wll ot : f f ˆ 5 If pps to coot t ˆ f f f W s ts cs tt f s t polt pltu fo fg t sst tw 6 If t osl s otu t t cospog opto s pˆ W cotu to sc t sst t coot ss wc cs t pctto lu of p s p f f pf ˆ f 7 T sst cot lws pu stt of two osls sultousl If t cot f t s pu stt of o osl t t s cssl supposto of stts of t sco osl Ts s w w spo fo out two tps of polts T t polt of g ptcul pu stt of t fst osl t stuto of stts fo t sco osl T coptlt of t two osls tt s t posslt of g pu stt of ot t t s t s pl t Uctt Pcpl wc w wll s fo lt scusso ctu 8 Ptcl Bo Cos ptcl ou o-sol wll: V fo V lsw W woul l to ow t possl g lls of t ptcl To f ts w sol t glu quto ˆ w Ĥ s t g opto s ts glu T cp fo costuctg optos to tol lt s s follows:

21 coots: ot: ˆ pˆ p T g opto s Ĥ tc g pottl g V I t pst cs V t gos w t ptcl c fou T opto fo t tc g s p wt ou cotos T Stu-oull quto tt w ust sol s W l sow tt t otool gfuctos s T glus ot susttutg c to t fftl quto: It follows tt 8 w s t o pot g 8 pl of lcto oo 5 log: 9 - Kg J c - 5 V 4 64 c - wlgt of tt poto 75

22 Popts of t gstts: Most pol lus of : s l 5 ps t < st pol lus of t - os pctto lu of : 4 s 4 cos 8 s s T c of s s ; σ pctto lu of t otu: p p ctu 9 Ptcl Bo Pt Wt s t pctto lu of t g fo ptcl tt s ot gstt of t lto? T two posslts cot tu of stts cot supposto of stts Cs pl of ts cs s tl tu T popt tt s t cot tu s tt t w fuctos fo c gstt o ft ps lto wt c ot t us suppos tt t ptcls tl qulu wt c ptcl spcfc gstt wt polt g t Bolt stuto: / T P s / T s

23 st fucto t t w fucto s to sc t sl pl: Suppos t ss scl lgt suc tt V lso lt / T K so tt T 98 V T /T 545 T 64 T oto P 64/ P / P 7/748 4 P 4 /748 4 P P / Ts su os ot qul T It tus out to 85 T W? Suppos t g s cotuous I ts cs / T p T p T T Cs cot supposto s sc sgl w fucto: Fo pl δ I t pl s o coplct pl: cs Wt s t ost pol lu of /? p c s

24 4 [ ] cos s s c p s s Wt s t pctto lu of /? To sw ts qusto w fst to f t olto costt: s c Nt w wt to f t pctto lu of t g T two ws to o ts Mto : ˆ Mto p Fou s ss s N to f lus of : c s s ctu Ptcl D Bo Cos o of sos ptcl of ss s plc ts o wt ou cotos suc tt t w fucto ss o ll t wlls T ft pottl outs t o guts tt t w fucto s o t s wll Scog quto: t us t followg ssupto:

25 5 wt ou cotos T Scog q ow cos: " " " D ot ss : " " " Rg: " " " T S s pt of T ol w tt t RS c lso pt of s f t quls costt Cll tt costt It follows tt t RS stsfs D Scog quto: wt t soluto s 8 Rtug to t S of t D quto t c fut g to g

26 6 cost " " T RS c sol to g: s 8 Rtug o lst t to t D quto w gt " wt t soluto s 8 T totl g s g 8 Fo cu of s 8

27 Dgc of Cu Dgc g T lst pl s cs of cctl gc C ou f o pls? Dst of Stts ow stts wt g t cu? Costuct t olu of octt wt us spc wt s 4 N 8 6 w It follows tt 8 / T st of stts s g / N 6 / N ρ / 4 pl of Bos-st cost of 87 R t K wt 5 μ 7

28 ctu Popts of Ptcls Ws Ptcl popts: Mss posto otu quto of oto: Nwto s quto: F q q& q& lto s qutos: p q q t p t p q Dts C pct qt pt f g q p 4 Cotuous ls W popts: pltu ; tst ~ ; ps; fquc; wlgt W quto: c t Uctt pcpl: If s ow pcsl t s copltl uow If s ow pcsl t s uow W c qutf ts g tsfo lt puls wc s t sotst tpol puls possl fo g spctu Ts tus out to Guss sptl pofl Fo optcs to w ow tt δδt 44 fo suc puls W c lt ts to t sptl uctt cus δλ δ λν c o λ c w δλ s su of t uctt posto λ 4 Supposto: If ot solutos of t w quto t so s 8

29 Tps of ws: Tlg ws T pltu s fucto of t qutt t Tt s t f t Wt lu os t pltu t t s t lt t? Δt t Δt f t Δt f t f t I ot wos t Δt t T w cs c to pot tt s ow o fow ow fst os t w tl? Δt cδt c / T qutt c s cll t ps loct Wt os w popgtg cws loo l? Tlg oscllto ws Wt s t po of t w? t f t ± t cos t R { t } ± t ± cos t ± cos ± t t T po s τ cus t w poucs tslf tt t ν t W c lso wt ± t cos λ τ λ wlgt w u c - λ 9

30 It gts ts fo t fct tt λ s t u of ws tt ft to ut stc ν l fquc s - ν τ po I t sos } R{ cos ϕ ϕ ± t t t f t f t Ts pl ws cus s ol to t pl wc t ps s costt Ts ws stsf t w quto: cos cos t c c t t t t ± ± ± ± ± ± ± ± ± Supposto: t cos cos ϕ ϕ t t cos cos ϕ ϕ cos cos ϕ ϕ c c t c Ts two sults cll qul Not tt tt ws t tol popt tt p λ Ts s qult to p Stg ws T supposto of fow cws tlg w t t t Fo poc osclltg ws

31 t cos t cos cos t cos t Nu of os s qut Pctlt s pol ctu W Fucto of F Ptcl Suppos f ptcl ws sc l poc w: Ψ t cos t Ψ t cos t T pol s tt t t t stgusl wc s tt w cot pct t futu fo owlg of ol t w fucto W woul l to popgto popt: Ψ t U t t Ψ t w Utt s popgto o t oluto opto tt s pt of Ψ t s t t followg fucto: Ψ t cos t t δ s t t It s coo tc to pt of t sco pt soluto of fftl quto W st t Ut t Ut to splf t otto Now lt us ssu tt Tt s Ψ t U t Ψ cos t δ s t U tcos U t δ s But cos t δ s t cos cost s st δ s cost δ cos st qutg ts qutg ts cos : s : U t cos t δ s t U t s t δ cos t δ Susttutg t fst quto to t sco w gt

32 δ t δ st st δ cost cos Sol fo δ : δ δ ± B coto coos t sg T w fucto fo f ptcl s t t s t Ψ t cos T popgto s U t cost st U t t tt t If Ψ t t t tt t t Ψ t T w fucto fo ptcl og t gt cto c ot t t plcg Tt s Ψ t U stll s t s fo: tt t t U t t Ψ t Gl lu of t t oluto opto: Rplc tt / potl Tlo ss: U t t Wt s t w quto fo ts fuctos? / p t Ψ t Ψ t Ψ t But w ow tt ν Tfo Ψ t Ψ

33 Sll Ψ Ψ Ψ p Ψ Ψ p Ψ Ψ Ψ w us t Bogl lto wc ls to p / Ts lst sult s ow s t stto stt o t-pt Scog quto qutg t two sults Ψ t Ψ Not tt t quto s ol fst o t cus w o t t tt of Ψ t to pct ts futu Ts s o s to of t Scog quto; t s ol ostto fo t f ptcl T ost gl cs clus t pottl g Ψ t Ψ V Ψ T t-pt quto ppls ol f t w fucto c fcto to sptl tpol pts: Ψ t T t I ts cs w c us t spto of ls tcqu: T V cost T fo wc t st stt Scog quto ops out T fctoto ssupto s qult to g stg w wc s t popt of ou stt Solg t S gs T t t

34 ctu Scttg Pottl B pottl g tt s lu of fo V fo > Cos ptcl wt tc g >V fo t cs t Fo B Fo > t tc g s -V C D V Cs : D fo ptcl ct fo t lft Cs : fo ptcl ct fo t gt W ott t tpol ps cus t s o ffct o t qutts w ws to clcult W wll cos t fst cs T sco s ssg fo owo Ict w: I Rflct w: R B Tstt w: T C Cotut of t gs B C Cotut of t gs B C Susttutg t fst quto to t sco gs B B B C T osl tt pls t ol of polt scttg pol s t ptcl flu j wc s t u of ptcls sctt p ut p ut t It s lso cll t cut st 4

35 T flu of ptcls toug cotol sufc s g t st ts t loct j ρ T flu stsfs quto of cotut wc s tt gt t flu sults gowt o plto of ptcl st p ut t: Dto of t quto of cotut: Rt t out t of tl gowt t t Mss s t tst olu ρ Δ out ρ t ρ t ρ ρ Gowt t s t olu Δ ρ ρ Δ Cotut: t tout t ρ ρ t ρ Δ ρδ ρ t t costt : ρ ρ t o ρ gt lu o t RS s tt ptcls ccuultg s t olu I t sos ρ ρ t lctc cut: ρ J t w J s cut/ ρ s t totl cg tgt o t olu 5

36 6 T qutu ccl qult o so s t j j ρ ρ Dto of t flu psso: V t V t Multpl t fst quto t sco quto sutct t sco quto fo t fst T sult s t W c wt t sco t o usful fo otg tt Sutctg t sco fo t fst gs Puttg ll t pcs togt gs t Copg ts sult wt t quto of cotut gs t flu: j

37 7 wt cospog quto t so Flu of ptcl wt l-lu w fucto s o T flu of f ptcl s: ρ p j Rtug to t scttg pol: C j B j j T R I Rflct fcto: j j R I R Tstt fcto: j j T I T 4 4 RT Fo V >> T R Cos ow t cs of < V T w fucto fo < s t s s fo ltoug wt fft lu fo B/ Fo > V κ V T soluto ts cs s l: D C κ κ Fo ftl w D ust o

38 pplg t ou cotos t B C B Cκ Solg fo t tos of coffcts w gt B κ κ C κ Fo > s l plg tt t flu T tt go o T flctc s B κ κ R κ κ It s lso stuct to wt t coffcts potl fo: B κ κ κ κ θ cost κ κ κ It s s to sow tt t costt t ps gl s g κ tθ κ W s tt t ffct of t s to flct t w lstcll wl toucg ps sft θ < w t ps cos fo t us sg ctu 4 Clsscl oc Oscllto Ptcls sss loct t postos T focs t t o c ot : F F Ts focs t popts: F F F fo F < fo > V Fo t fto of t pottl g F w gt 8

39 9 V W f t tsfoto: R t R p p & & wc s t fto of t ct-of-ss W sol ts qutos fo t s tsfoto: M R M R w M I ts coots t pottl s g V W lso pss t tc g s MR T & & & & μ w M μ Not tt & & Nwto s qutos g: && μ oto of t oscllto

40 Clsscl soluto fo t oscllto: M R & oto of t ct of ss It follows tt t cos μ&& μ t cos t t μ Not tt ν s t ccul fquc s/s T s so cofuso t otto us fft utos T t w u s t u of ws p c o / λ wt uts c - T qutt / λ soul cll t ccul w u o t w cto two o o sos Ts soul ot cofus wt t us to ot t foc costt g uss t otto ν / λ I s otto ν c / λ s t fquc ν cν c W wll us t o coo otto of ν wll f to / λ s spl t w u Wt s t lu of? It s t u lu of wc occus t T t t tug pot V Tfo / Wt s t clsscl polt st? Df w l θ t t t cosθ t Wl s o-ll fo to θ s ll t fo to T polt of fg t oscllto coot spcfc g os ot p o t coot sst: p p θ θ θ pθ θ pθ θ p sθ θ pθ θ cos θ p θ θ But p θ cost 4

41 4 p p p θ θ θ T clsscl oscllto sps ost of ts t t t tug pots ctu 5 Qutu Mccl oc Oscllto μ μ W wll tt ts s Stu-oull pol But fst lt s cst t solss fo W f two scl pts: μ μ / 4 / c μ μ β c - T Scog quto ow cos 4 β W t f solss stc: / o

42 T solss Scog quto cos: β β W woul l to cst ts quto Stu-oull fo Spl pplg t tsfoto g ctu 6 os t wo To gt ou ts pol w f t uow fucto loog t ts popts t lg : ± / Sow tt ts s ppot soluto fo lg W jct t sg s upscl W ow opt tl fo fo t ct soluto: w ϕ ϕ cos ou uow fucto Ist ts fucto to t solss Scog quto sow tt t cos β Ts quto s t s fo s t s quto: wt β f f f Ts s Stu-oull quto wt t wgtg fucto T solutos to ts quto t t polols T otogolt fo ts polols coto s fo T olto coto s 4

43 4! T gfuctos of t lto ccogl / / / /!! pls: / / 4 / / / 4 / Otoolt: δ Qutto coto: β Pscl g of : T clsscl tug pot s g Tulg polt fo t gou stt: / 57 f P

44 Wg: s s fft lu If w cll s pt t s lt to t pst lu / ctu 6 Popts of t oc Oscllto Gtg Fucto: Rcuso ltos: W c po fo t Tlo ss soluto tt But fftto of t gtg fucto gs qutg t two sults gs o pplcto to pol sopto of sgl poto:!! / / /!! But ulss ± 44

45 45 t s cos t cs of Notg tt t follows tt!!!! You soul wo out t cs of s cs Nt wt s t pctto lu of? Mto lut t tgl Ts s ffcult Mto Us t cuso lto twc 4 tctg t t t follows tt T lst stp follows fo t fct tt /!!

46 46 Mto Us t cuso lto oc It follows tt / Tfo o V Ts s pl of t l to Fo cost foc F F V T Fo ctl foc V T V V V F V pls: oc oscllto: V V V T V T / 4! 4!! 4 /!!!

47 Coulo foc V / T V T V V T Ts to ppls ol to stto stts It s ot tu fo pl fo w ± Fst sow tt / ν P / P / w P P T lst t lt s o-o fo ± Nt sow tt V / ν P / P / ow o l olculs? T ot oc W wll loo t g s tls N t u of t pottl R o Usg Tlo ss pso R V V V V V 6 T t l s cssl o T g o ts tt s ptutos sultg g c c c c 4 Not t us sg t sco t wc s tt t lls gt clos togt wt lg I g s otto ν ~ / c ν wt uts of c T g spcg s g c c cus / / / / I us t t s t qutu u T o t o tol g of fo Suppos w c wt t clsscl lto ts of ol coots Q Q Q N t cojugt ot P P P N : 47

48 P PN Q μ μ N N Q N It s ot to sow usg t spto of ls to tt t tol w fucto s fctol wt totl tol g g Q Q Q Q Q Q N N N / / / N N If t olcul s tos t s gs of fo of wc tsltol oto of t ct of ss o ottol fo l o ol olculs It follows tt t u of tol gs of fo s N-5 fo l olcul N-6 fo o-l olcul Fo CO N4 T gs fo stc sttc stc sttc oul gt : Q Q Q s Q Q Q θ T pottl g s g V s Qs Q θ 48