Chapter 6 Perturbation theory

Transcription

1 Ct 6 Ptutio to 6. Ti-iddt odgt tutio to i o tutio sst is giv to fid solutios of λ ' ; : iltoi of si stt : igvlus of : otool igfutios of ; δ ii Rlig-Södig tutio to ' λ..6. ; : gl iltoi ': tutio λ : sll u gl solutio of..6. d λ λ λ λ L L i i : it-od otios to ; sustitut qutio ito 6. λ ' λ λ λ L λ L λ λ L λ λ ' λ λ ' L L Qutu is ltu 69

2 M ' ' kow fistl Fist-od to i fist-od otio to t g Fo qutio 6.5 ' tk i odut wit ' is iti : ' ' t fist-od otio to t g l fo t ifiit squ wll si tu t sst ostt tutio ' ' 7 Qutu is ltu

3 l If t tutio tds lfw oss t wll / ' / si d d ii fist-od otio to t wv futio / fo ' '..6.7 T otool igfutios ssud to fo olt st. Tus dd i ts of t st w f solutios of qutio 6.7 d dod fo..6.8 sustitut qutio 6.8 ito qutio 6.7 ' ' ' Qutu is ltu 7

4 tk i odut wit l δ l δ l l ' If l δ ' l If l ol w l t lft sid of qutio 6.9 is ol o t : d δ l ' ' so '..6. t fist-od otio to t wv futio Nodgttio oditio : T ov tutio tod l ol to o-dgt igvlu of t utud sst. Nl difft utud stts osodig to difft gis. i gl Sod-od solutio of fo qutio 6.6 tk i odut wit ' ' 7 Qutu is ltu

5 ' ' ' ' ' 6. Ti-iddt dgt tutio to If so distit stts s t s g qutio 6. fils Twofold dg Suos s t s g Is is otd tt α β is lso igstt of Wil t tutio will k t dg : s is λ t oo utud g slits ito two. ` λ ' d λ λ LL λ λ LL Qutu is ltu 7

6 sustitut ito λ ' L λ L t od λ : ' i tk i odut wit ' d ' ' α β α β α ' β ' α lt W ij i ' j α W βw α..6. ii siill tk i odut wit α W βw β..6. fo qutio W W W W α β α β [ W W ± W W W ] is t fudtl sult of dgt tutio to ± 7 Qutu is ltu

7 if W W t W ' ' W t s s w otid usig t Nodgt tutio to To Lt iti oto tt outs wit If igfutios of wit distit igvlus ' µ ν ; µ ν t W d good stts to us i tutio to oof : [ '] [ '] ' ' µ ' ' ν ' ' ν µ ν ' µ ν W W Fo -fold dg ig od dg: w look fo t igvlu of t W-ti W ij i ' j Qutu is ltu 75

8 l / / t-disiol ifiit uil wll < < < < < < otwis T tutio : < < < < otwis / / ' fo lt s d d isid t wll d d si siill si si si si si / t osodig llowd gis : 76 Qutu is ltu

9 i t goud stt is odgt / si si si tus t fist-od otio to t goud stt g is ' / si d / si d si d ii t fist itd stt is til dgt lt v t s g tus t fist-od otio to t fist itd stt ust fid W-ti t ti lts : W / / si d si d si d siill W W d W W W W W W 6 9 W k w k lt igvlus of W is t t tisti qutio is : k 8 Qutu is ltu 77

10 k k k T igvlus t otio to gis Tus t fist-od otio fo t itd stt to g is k k λ λ λ λ λ λ λ T fi stutu of dog fi stutu du to otio oulig oit si otio ltivisti fi stutu ostt α i t ltivisti otio ltivisti foul fo kiti g : / v T ltivisti foul fo otu : / v v / / ] / [ T v v v v Istd of vloit t ltivisti d is T 78 Qutu is ltu

11 LL LL 8 8 T lowst od ltivisti ' 8 ' 8 8 Now t Södig qutio to t utud stt is ] [ fo dog : / / d l l / / * l l / / * d / l w /!] [! φ θ l l l l l Y L l l is t Bo dius / l sustitut ito Qutu is ltu 79

12 l / 5 t ltivisti otio is sll t ii Si-oit oulig otio lto i gti fild siig gd til ostituts gti diol. gti diol ot : μ γs γ is gogti tio w gti diol ld i gti fild B t g -μ B - γb S gti diol ot of t lto fo lssil ltodis t gti diol ot of t ig is q µ T is t iod T µ S t gul otu is L T ig q μ q L fo siig gd til q t lto s gti diol ot isμ g S g S w g is fto g so µ S fo t lto oit ositiv g sts u gti fild B wi ts toqu o t lto tdig to lig its gti ot log t ditio of B 8 Qutu is ltu

13 fo t Biot-Svt lw t gti fild of t oto is µ I B µ T is t iod of t oit T d L v T µ µ B L T B L µ L L - w µ T si-oit ittio fo µ B S L S L..fo itil sst fo t lto oits oud t ulus ' SO 8 S L..is t si-oit ittio d fid t S L fo t si-oit oulig t iltoi o log outs wit L d S ' SO out wit L S totl gul otu J L S osvd J L S L S L S L S L S J L S Qutu is ltu 8

14 L S J L S [ j j l l s s ] ' SO SO 8 S L S L 8 * w S L [ j j l l s s ] d [ j j l l s s ] l l / l so SO / [ j j l l s s ] 8 l l / l w fs SO g lvls fo t fi stutu of dog 8 Qutu is ltu

15 6. T vitol iil To T goud-stt g g fo sst dsid t iltoi g oof : fo olid wv futio t ukow igfutios of fo olt st ss s li oitio of t wit * * t goud-stt g g is t sllst igvlu g g l fo t o-disiol oi osillto : d d us til wv futio : oli : Qutu is ltu 8

16 d / d d d d d d 8 iii wit st to 8 d d so i I tis s it is t is fo of t tul goud stt i g 6. T WKB oitio WKB tod is tiqu fo otiig oit solutios to t ti-iddt Södig qutio i o disio It is tiull usful i lultig oud-stt gis d tulig ts toug ottil is. T oss of WKB tod : Fo d d d d d d..6.5 w ] [ 8 Qutu is ltu

17 i ssu tt > I gl is so ol futio ss iφ fid d d d dφ i d d d d d i d d d d iφ dφ d φ dφ i d d d iφ sustitut ito qutio 6.6 d i d d d dφ d φ dφ i d d d d dφ d d d dφ d φ d d d fo qutio 6.7 d dφ d d d dφ d d dφ C d C d dφ d i qutio 6.6 w ssu vis slowl so tt d d is gligil d d is u lss t dφ d / dφ d dφ ± d so φ ± d d i C ± d Qutu is ltu 85

18 l Fo t ottil wll wit two vtil wlls so sifid otwis futio < < oud oditio Outsid t wll Isid t wll ssuig > iφ iφ [ C C ] [ C siφ C osφ ] w φ ' d' φ [ C siφ C osφ ] C C siφ φ L so d tis qutitio oditio is i sult it dti t oit llowd gis ii ssu tt < t s s vious oss C ± d 86 Qutu is ltu