oons j.lbb@phscs.o.c.uk
To s summ Fmes of efeence Invnce une nsfomons oon of wve funcon: -funcons Eule s ngles Emple: e e - - Angul momenum s oon geneo Genec nslons n Noehe s heoem
Fmes of efeence Conse fme of efeence O n whch genec se s escbe.e. he wve funcon of n e - n he p se. If O s ffeen fme of efeence connece o O b whee G s goup of nsfomons.e. nslons oons o Loen nsfomons of he coone ssem The wve funcon n O wll n genel be wh whee s n ohonoml bss G g g D whee γ γ γ g D
Emple: nfne D well Fo he obseve O he goun se s gven b B cos π Now nsle D. In O he goun se s no bu B cos B cos π In nlogous fshon n O whee he nslon s D/ π B cos B cos B π / π Bsn π π [ cos sn ] π B sn π π 4-0 -/ 3/ The phscs s nvn
Some omc phscs We wll conse he om n one of s ece ses The egenfuncons e: u whee nlm n s he eneg egenvlue m s he egenvlue of We efne new efeence fme O whch s oe bou he s of he ognl fme b n ngle E E l l s he egenvlue of ngul momenum L n he componen of n L m l l The mlonn n L e unchnge s e he especve egenvlues n n l. In ohe wos he commue wh. oweve he econ hs chnge so m s no he sme s now pojece on new -s p wvefuncon
oon wvefuncon The new wve funcon u nlm cn be epesse b supeposon of wvefuncons wh he sme n n l bu wh ffeen m l u u u whch cn be compe o γ γ nlm nlm m m nlm m γ Now we wll use he p se nlm0 s n emple: u 0 u ϑ The ϑ ϕ whee Y cosϑ Y coeffcens ssf : ϑ Y ϑ ϕ u The l componen oes no conbue heefoe conse onl epenence on ϑ n ϕ u Y Y nlm 0 0 m 0 3 4π 0 Y m lm m 0 m m m 0 lm ϑ ϕ e he sphecl hmoncs. Fo he 3 8π m snϑ e ϕ n Y ϑ ϕ 3 8π snϑ e p cse hese e ϕ
Then subsung cosϑ cosϑ cosϑ cos sn one ges he elon cos sn cos sn mples cos sn connue The fnl pece of nfomon we eque s escpon of he oon : n nong cos cosϑ sn snϑ snϕ whch gven he nvnce of he moulus une oon gves : cosϑ cos cosϑ sn snϑ snϕ We hen fn epessons fo cosϑ n snϑ cosϕ n ems of he sphecl hmoncs π ϑ n snϑ snϕ [ Y ϑ ϕ Y ϑ ] 4 π 3 Y ϕ 0 3
connue We use he pevous esuls o epess Y 0 n ems he n sphecl hmoncs n he oe fme Y 0 ϑ 3 4π 3 4π cos cosϑ Compng o he genel epesson Y we ge he coeffcens In sml fshon ll l mm cos cosϑ sn snϑ snϕ Y ϑ sn [ Y ϑ ϕ Y ϑ ϕ ] 0 ϑ Y ϑ Y ϑ ϕ ϑ ϕ 0 00 0 0 0Y cos sn n 00 0 0 coeffcens cn be clcule sn
Tbulons of funcons fom he PDG hp://pg.lbl.gov/
Emple: e e - - Spns n he elvsc lm Onl phoon echnge n elvsc lm M <<CoM eneg<<m Z0 e e - Inl se J Fnl se J - funcons cn be use n wo ws o fn he ngul sbuon ϑ cos cosϑ O he oon fom J o J se cos ϑ Ehe w he ffeenl coss secon s σ Ω cosϑ
Angul momenum opeos s geneos of oons [ ] J J becuse Tlo epnng 0 fo sn cos sn cos he sme coone n elonshp beween We wn o fn sn cos sn cos n nvese sn cos sn cos ' s : oon bou he conse Agn we wll J s he geneo of oons bou he s. Sml esuls fo J n J
Angul momenum opeos s geneos of oons If s soluon of he Schoenge equon s? [ ] J [ ] 0 s soluon f Theefoe Opee fom lef wh he Schoenge equon s Fo eques [J ]0 n opeo
Angul momenum opeos s geneos of oons We wll now conse me von of m elemens of genec nmcl opeo h oes no epen on me.e. J The m elemen s nvn wh me n he egenvlues of e consn In ou cse hs mples he pojecon of ngul momenum s conseve wvefuncon nvn une oons ϕ ϕ ϕ ϕ ϕ ϕ Ω Ω Ω Ω Ω Ω Ω 0 ] me n [ s nepenen of If n Whee we hve use * Ω Ω ϕ ϕ
Fne oons m j J m j m j m j J J J J J j m m m j m m n n n n ep ep ecllng ep Theefoe ep ep lm oons nfnsml successve pplcons of cn be genee b ngle A fne oon of 0 δ δ δ δ δ δ δ δ J ˆ ep α. α Genel cse whee s veco n he econ of he oon s wh mgnue equl o he ngle of oon
Eule ngles Genec oons escbe b Eule ngles Defne hee successve oons: ngle bou he Z s ngle bou he new Y s ngle bou he new s The oons cn be epesene b D mces sng ngul momenum opeos s geneos of he oons n ecsng he oons s fs bou ognl bou he ognl n bou he ognl D j m m αγ j m ep αj ep αm γm ep J j m m ep 0 γj 0 0 j m
Tnslons Sml nlss cn be pple o nslons: Assume s nfnesml: ST esul s : We cn we he genel ecllng h he momenum opeo s : p p ST ST
Tnslons So fo fne nslons: ST Invnce of he wve equon une nslons consevon of momenum Smll fo me f mlonn s me nepenen: TT A τ ep A p ep τ Invnce of he wvefuncon wh espec o me nslons consevon of eneg
Smme Pncples An nvnce o smme pncple ess fo phscl ssem S n nsfomon g G f he phscl lws epesse fo S b he obseve O n hs coone ssem lso hol goo fo he sme ssem S n he coone ssem of he obseve O D g g O O whee g s he nuce un nsfomon. If he equons of moon e he sme hen he mlonns e he sme : Noehe s heoem: O Smme pncple Invnce of heo Consevon lw O O O
Summ Fmes of efeence n nvnce une nsfomons funcons o escbe oons Angul momenum s oon geneo Eule s ngles Genec nslons n Noehe s heoem