Angular Momentum in Spherical Symmetry
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- Linette Shaw
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1 Angu Moentu n Sphec Set Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen
2 Angu Moentu n Sphec Set The concept of ngu oentu ps cuc oe n the theedenson 3D Schödnge we equton. The ethod of septon w be used to dee the popetes of the ngu oentu opetos nd of the sphec honcs. 6 Quntu Mechncs Pof. Y. F. Chen
3 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen The opeto tht epesents ngu oentu n the pesentton s gen b 9. The thee coponents of the ngu oentu opeto e epessed s 9. B dect coputton the foowng coutton ue cn be obtned 9.3 Angu Moentu Opetos n Sphec Coodntes p h h h ; ; ] [ h h h
4 Angu Moentu n Sphec Set Angu Moentu Opetos n Sphec Coodntes Wth ccc peutton the foowng coutton ues cn be deed [ ] h ; [ ] h 9.4 A coutton ues cn be sued sboc s foows: h 9.5 Snce two dffeent coponents of the ngu oentu do not coute t s gene not possbe to specf nd esue oe thn one coponent. The sque of the ngu oentu s defned b the eton Quntu Mechncs Pof. Y. F. Chen
5 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen It cn be shown tht coutes wth. Consde to be n epe: 9.7 B set t cn be concuded tht coutes wth. In othe wods t s possbe to specf sutneous wth n coponent of. Angu Moentu Opetos n Sphec Coodntes nd ] [ ] [ h h h h nd nd
6 Angu Moentu n Sphec Set Angu Moentu Opetos n Sphec Coodntes The 3D we equton fo ptce ong wth ss M n cent-foce fed s gen b h M V ψ Eψ 9.8 It s conenent to epess the cent-foce 3D we equton n tes of the sphec po coodntes nd. The etons of the ectngu nd sphec po coodntes e gen b sn cos ; sn sn ; cos 9.9 nd ; cos ; tn 9. 6 Quntu Mechncs Pof. Y. F. Chen
7 Angu Moentu n Sphec Set Angu Moentu Opetos n Sphec Coodntes The unt ectos n the sphec po coodntes cn be epessed n tes of the ectngu unt ectos: sn cos sn sn cos cos cos cos sn sn sn cos 9. The nese etons e epessed s sn cos cos cos sn sn sn cos sn cos sn cos 9. 6 Quntu Mechncs Pof. Y. F. Chen
8 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen The gdent opeto n the sphec coodntes s gen b 9.3 In the sphec coodntes the ngu oentu opeto cn be epessed s 9.4 Angu Moentu Opetos n Sphec Coodntes sn sn sn sn ] [ p h h h h
9 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen Substtutng Eq. 9. nto Eq. 9.4 cn be wtten n the sphec coodntes s 9.5 Angu Moentu Opetos n Sphec Coodntes nd h h h cot sn cos cot cos sn
10 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen Substtutng Eq. 9.5 nto Eq. 9.6 the epesson of n the sphec coodntes s gen b 9.6 Angu Moentu Opetos n Sphec Coodntes sn sn sn cot sn cos cot sn cos cot cos sn cot cos sn h h
11 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen Wth Eq. 9. the foowng epessons fo the detes of the unt ectos cn be obtned: 9.7 Angu Moentu Opetos n Sphec Coodntes cos sn ; cos ; ;
12 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen Usng Eq. 9.3 nd 9.7 the pcn opeto cn be septed nto d pt nd n ngu pt s foows: 9.8 Usng Eq. 9.8 the 3D we equton 9.8 n the sphec coodntes cn be wtten s 9.9 Angu Moentu Opetos n Sphec Coodntes sn sn sn sn sn h V E M h h ψ ψ
13 Angu Moentu n Sphec Set Angu Moentu Opetos n Sphec Coodntes Snce Eq. 9.9 s su of d pt nd n ngu pt the egenfuncton ψ cn be epessed s the poduct of d pt nd n ngu pt ψ R Υ Substtutng 9. nto 9.9 ddng though b R Υ 9. utpng b M nd engng tes we cn obtn h R Y R M V E Y 9. 6 Quntu Mechncs Pof. Y. F. Chen
14 Angu Moentu n Sphec Set Angu Moentu Opetos n Sphec Coodntes Equton 9. cn be stsfed on f ech of the two tes on the eft- hnd sde s septe constnt. Consequent we he obtned two dffeent equtons: nd Y λ h Y 9. h λ h V M M R E R 9. Eq. 9.3 s dentc n fo wth the D Schödnge we equton ecept fo the ddton of the te λ h M to the potent eneg. Ths te s soetes ced the centfug potent becuse t epesents the potent whose negte gdent s the centfug foce. 6 Quntu Mechncs Pof. Y. F. Chen
15 Angu Moentu n Sphec Set Egenfunctons nd Egenues of Obt Angu Moentus Appng Eq. 9.6 to Eq. 9. eds to dffeent equton fo the obt ngu oentu: sn sn sn Y λ Y 9.4 Snce Eq. 9.4 s sepbe fo the bes of nd the egenfuncton Y cn be wtten s the poduct Υ Θ Φ 9.5 Substtutng 9.5 nto 9.4 ddng though b nd engng tes we cn get d dθ sn sn Θ d d λ sn Φ d Φ d Quntu Mechncs Pof. Y. F. Chen
16 Angu Moentu n Sphec Set Egenfunctons nd Egenues of Obt Angu Moentus Equton 9.6 cn be stsfed on f ech of the two tes on the eft- hnd sde s septe constnt. As consequence we he obtned two dffeent equtons: Φ nd d Φ d 9.7 d dθ sn sn d d λ sn Θ 9.8 The egenfuncton of Eq. 9.7 s gen b Φ ce Quntu Mechncs Pof. Y. F. Chen
17 Angu Moentu n Sphec Set Egenfunctons nd Egenues of Obt Angu Moentus Note tht the functon Φ s so the egenfuncton of : Φ h Φ h Φ 9.3 To ensue tht the we functon hs snge ue t ech pont n spce the we functon needs to stsf Φ π Φ Appng Eq. 9.3 to the we functon 9.9 ges e π 9.3 so tht the egenues of ± ± ± 3 e estcted to the dscete set Quntu Mechncs Pof. Y. F. Chen
18 Angu Moentu n Sphec Set Egenfunctons nd Egenues of Obt Angu Moentus On the othe hnd Eq. 9.8 s known s the egende dffeent equton. Its egenue λ s gen b nd ts soutons cn be epessed n tes of the ssocted egende functons : Θ C P cos 9.33 whee P / d P d 9.34 nd P s the th egende pono whch s defned b the Rodgues fou: d P! d Quntu Mechncs Pof. Y. F. Chen
19 Angu Moentu n Sphec Set Egenfunctons nd Egenues of Obt Angu Moentus In othe wods the ssocted egende functons e gen b / d P! d 9.36 Fo Eqs nd 9.33 the copete egenfuncton of nd cn be epessed s Y C e P cos 9.37 whee the functon Y e ced sphec honcs nd s noton constnt such tht s noed Y wth espect to n ntegton oe the ente sod nge. C 6 Quntu Mechncs Pof. Y. F. Chen
20 Angu Moentu n Sphec Set Egenfunctons nd Egenues of Obt Angu Moentus Wth the estbshed phse conenton the noed sphec honcs fo e epessed s Y 4π 4π! e!! e! P cos sn d dcos cos 9.38 The sphec honcs fo negte e defned b Y Y ] [ Quntu Mechncs Pof. Y. F. Chen
21 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu A geneed ngu oentu opeto coponents J J nd J s defned b ts thee whch e ssued to stsf the se coutton etons s the thee Ctesn coponents of obt ngu oentu noed to h : J [ J J ] J ; [ J J ] J ; [ J J ] J 9.4 Note tht the geneed ngu oentu opetos e not necess estcted to the tpet 9.. nd defned fo the cssc eton 6 Quntu Mechncs Pof. Y. F. Chen
22 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu S the sque of the geneed ngu oentu s defned s J J J J 9.4 Theefoe s shown n 9.7 Ĵ coutes wth J J nd J. Snce the coponents J J nd J do not utu coute on one of the cn be chosen to she the se egenfunctons wth Ĵ. It s conenent to choose the egensttes to be sutneous egensttes of Ĵ nd Ĵ. 6 Quntu Mechncs Pof. Y. F. Chen
23 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Denotng the ont egensttes b λ β nd the egenues of Ĵ nd b λ nd especte the eted equtons cn be wtten s Ĵ J λ β λ λ β λ β β λ β J As n the cse of the honc oscto sng nd oweng opetos cn be defned n the cse on ngu oentu: J J J J J J Quntu Mechncs Pof. Y. F. Chen
24 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Ĵ nd Ĵ e not Hetn nd e n fct Hetn conugtes of ech othe: J J Φ Φ Φ Φ 9.46 Wth the coutton etons 9.4 the foowng coutton etons cn be obtned: [ J [ J J ] J J ] J [ J J ] J J J J ± J ± J b 9.47c 9.47d 6 Quntu Mechncs Pof. Y. F. Chen
25 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Fo 9.43 nd 9.47 we he J J λ β J J J λ β β J λ β In the se w fo 9.43 nd 9.47b we obtn J J λ β J J J λ β β J λ β Eqs nd 9.49 ndctes tht J± λ β s so n egenstte of Ĵ wth the egenue β ±. Now snce Ĵ nd Ĵ coute J λ β ust so be n egenstte of Ĵ. ± 6 Quntu Mechncs Pof. Y. F. Chen
26 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Wth the coutto [ J J ± J± ] t cn be es shown tht the stte λβ s so n egenstte of Ĵ wth egenue λ : J λ β J J λ β λ J λ β ± ± J ± 9.5 Fo t cn be concude tht when the opeto Ĵ ± cts on the stte λβ t does not ffect the fst quntu nube λ but t ses o owes the second quntu nube β b one unt. Epct J λ β s popoton to λ β ± : ± J ± ± λ β λ β λ β ± ±C 9.5 whee C λβ e the constnts to be detened te. 6 Quntu Mechncs Pof. Y. F. Chen
27 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Fo the fct tht J J J J ; we cn he the nequt λ β J J λ β λ β J J λ β λ β β λ 9.5 In othe wods fo gen egenue of fo the egenue β. λ thee ests n uppe t Snce β hs n uppe t β thee ust est stte whch cnnot be sed futhe: J λ β λβ Quntu Mechncs Pof. Y. F. Chen
28 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Mutpng on the eft of 9.53 b obtned tht Ĵ nd usng 9.47d t cn be J J λ β J J J λ β λ β β 9.54 Consequent λ β β S thee s stte λ β n 9.54 whch cn be obtned b n successe ppctons of Ĵ on J λ βn λβ nd cnnot be oweed futhe: Quntu Mechncs Pof. Y. F. Chen
29 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Mutpng on the eft of 9.55 b Ĵ nd usng 9.47d t cn be obtned tht J J λ β n J J J λ β n λ β n n β 9.56 Consequent λ β β n n 9.57 Copng 9.54 nd 9.57 t cn be found tht thee e two soutons: β n β o β n β 9.58 The second souton s obous uncceptbe becuse t otes the defntons of β nd β. n 6 Quntu Mechncs Pof. Y. F. Chen
30 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Snce β cn be eched b n successe ppctons of Ĵ on n we cn he β n β n. λβ Cobnng wth the fct tht n β In shot β t foows tht 9.59 ust be ethe nonnegte ntege o hf-ntege dependng on n beng een o odd. β n β Epct the on owed ues fo β e β Quntu Mechncs Pof. Y. F. Chen
31 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Accodng to the boe nss t s pope to use the notton nd to denote β nd β especte: β ; β 9.6 Wth the new notton the egenue of λ Ĵ s then gen b 9.6 Fo gen ue of the owed egenues of Ĵ e 9.63 The tot nube s hence thee e othogon egensttes fo ee ue of. 6 Quntu Mechncs Pof. Y. F. Chen
32 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Snce cn epesent n poste ntege t s possbe to constuct ecto spce fo n denson tht s cosed unde the opetons of the geb of thee opetos etons 9.4. J J nd J whch stsf the coutton Ths s the fundent de of n educbe epesentton of the otton goup see te. In tes of the new nottons 9.6 nd 9.6 the ont egensttes λ β cn be epced wth. 6 Quntu Mechncs Pof. Y. F. Chen
33 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu As consequence 9.4 nd 9.43 cn be ewtten s J J In the se w 9.5 cn be epessed s J C ± ± ± Snce J J we cn he J J J J C Quntu Mechncs Pof. Y. F. Chen
34 Angu Moentu n Sphec Set Opeto Method fo the Angu Moentu Wth the d of 9.47d t cn be found tht C J J J J J 9.68 As usu the bt phse of C s ssued to be eo hence C 9.69 S the constnt C cn be found to be C Quntu Mechncs Pof. Y. F. Chen
35 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen As the defntons n 9.44 nd 9.45 we cn defned the sng nd oweng opetos fo the obt ngu oentu: 9.7 Fo the gene epesentton theo of ngu oentu t s known tht becuse cnnot be oweed n oe hence 9.7 Usng 9.65 we he 9.73 Deton of Sphec Honcs wth Opeto Method ± ± cot e h cot Y e Y h Y Y Y h h
36 Angu Moentu n Sphec Set Deton of Sphec Honcs wth Opeto Method Ths equton ndctes tht the uth dependence of gen b e hence cn be epessed s Y Y s Y f e 9.74 Substtutng 9.74 nto 9.7 we he d cot d f df f cot d 9.75 n f nsn const 6 Quntu Mechncs Pof. Y. F. Chen
37 Angu Moentu n Sphec Set Deton of Sphec Honcs wth Opeto Method Theefoe f C sn 9.76 nd Y C sn e 9.77 The bsoute ue of C cn be found b the noton: π * Y Y dω sn d d C sn π 9.78 The nteg s conenent done usng cos hence π C d Quntu Mechncs Pof. Y. F. Chen
38 Angu Moentu n Sphec Set Deton of Sphec Honcs wth Opeto Method Wtng nd futhe chnge the be to t the ght sde of 9.79 becoes π C [t t] dt π C t t dt Fo the popet of the Bet functon p q Γ p Γ q B p q t t dt Γ p q Wth we obtn π C Γ Γ Γ 4π C!! 9.8 C!! 4π 6 Quntu Mechncs Pof. Y. F. Chen
39 Angu Moentu n Sphec Set Deton of Sphec Honcs wth Opeto Method The coon used conenton s not to he n ddton phse fcto hence Y! sn! 4π e 9.83 Fo 9.66 nd 9.68 we he Eq ndctes tht Y ctng on. Y 9.84 cn be obtned b successe 6 Quntu Mechncs Pof. Y. F. Chen
40 Angu Moentu n Sphec Set Deton of Sphec Honcs wth Opeto Method Wth 9.7 nd 9.84 the sphec honc Y cn be epessed s Y h h h h he cot Y -! e cot Y!! 9.85 The dete ws ge egenues. 6 Quntu Mechncs Pof. Y. F. Chen
41 Angu Moentu n Sphec Set Deton of Sphec Honcs wth Opeto Method But ech te cts thee s fcto of e nd kes the egenue of ncese one b one. As esut 9.85 cn be epessed s! -! d Y e cot! 4 π!! d d d d cot cot cot sn d d d 9.86 Wth the popet tht d d n cot sn n d sn d n Quntu Mechncs Pof. Y. F. Chen
42 Angu Moentu n Sphec Set Deton of Sphec Honcs wth Opeto Method Eq cn be ewtten s Y -! d! 4 π! sn d e sn d d d sn sn sn sn sn d sn d sn d -! e! 4 π! d sn sn d sn / cos cos -! d e! 4 π! dcos whee cos. / -! d e! 4 π! d Quntu Mechncs Pof. Y. F. Chen
43 Angu Moentu n Sphec Set Gphc Vsuton of Sphec Honcs It s of sgnfcnt potnce to sue the dstbuton pttens of sphec honcs n pedgogc w. Befoe peentng the suton of sphec honcs n epct pono epesson fo the ssocted egende functons s essent. Fo 9.36 the epesson of the ssocted egende functons fo s gen b / d P! d Quntu Mechncs Pof. Y. F. Chen
44 Angu Moentu n Sphec Set Gphc Vsuton of Sphec Honcs To obtn n epct epesson we epnd d d d d k d k d k k d d k k 9.9 Athough the su etends fo k the te n the fst sque pentheses nshes fo k > nd the te n the second sque pentheses nshes fo k >. Theefoe the su s tken on fo k. 6 Quntu Mechncs Pof. Y. F. Chen
45 Angu Moentu n Sphec Set Gphc Vsuton of Sphec Honcs Then k k d d d k k d k k d d k! k! k k k k! k! s s! s! s s s! s! s! s! s s s s! s! Quntu Mechncs Pof. Y. F. Chen
46 Angu Moentu n Sphec Set Gphc Vsuton of Sphec Honcs s!! s! s s s! s! s! s! s!!! s! s! s s! s! s! s! s s! d d s s! s s d d! d! d Quntu Mechncs Pof. Y. F. Chen
47 Angu Moentu n Sphec Set Gphc Vsuton of Sphec Honcs Wth 9.9 the tente defnton of the ssocted egende functons s gen b P!! / d d!! / d d 9.9 Wth 9.9 the tente epesson fo the sphec honcs wth s gen b 6 Quntu Mechncs Pof. Y. F. Chen
48 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen 9.93 Gphc Vsuton of Sphec Honcs d d e d d e P e Y cos cos sn!! 4! cos cos sn!! 4! cos!! 4 π π π
49 Angu Moentu n Sphec Set Mt Repesentton of Angu Moentu Snce the egensttes coespondng to dffeent ngu oent e othogon nd snce the spect e dscete the othnot condton s gen b δ δ The copeteness condton s epessed b I The t eeents of Ĵ nd δ J δ Ĵ Ĵ ± n the bss e gen b δ J δ J ± δ δ ± 6 Quntu Mechncs Pof. Y. F. Chen
50 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen Fo 9.44 nd 9.45 the t eeents of e gen b Consde the cse the tces of e gen b Mt Repesentton of Angu Moentu J J nd [ ] J δ δ δ [ ] J δ δ δ ± Ĵ J J J nd J J J J J
51 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen Fo the ngu oentu opeto coesponds to ptce wth spn /. The t epesentton s usu n tes of the Pu tces whch e eted to the spn ecto s foows Usng ths eton the Pu tces cn be found to be Mt Repesentton of Angu Moentu / σ h S σ σ σ
52 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen The Pu tces stsf the foowng two popetes whee the subscpts nd k efe to s the unt t nd s the e-ct tenso. These etons cn be condensed nto Mt Repesentton of Angu Moentu I σ k k k σ σ σ σ k k σ ε σ σ ] [ Î k ε k k k σ ε δ σ σ
53 Angu Moentu n Sphec Set Mt Repesentton of Angu Moentu Wth ths equton t cn be efed tht f A nd B e two ectos tht coute wth σ then σ A σ B A B I σ A B Appng ths dentt to the powe sees epnson of n eponent t cn be shown tht α U ep α β n σ e Icos β n σ sn β whch s geneed de Moe fou. It cn be efed tht n unt t cn be epessed n ths fo. 6 Quntu Mechncs Pof. Y. F. Chen
54 Angu Moentu n Sphec Set Mt Repesentton of Angu Moentu Fo γ we he det U nd the t U s unodu. The set of unt unodu tces consttutes the goup SU. In the D spn fos the otton t tkes the fo R n ϕ ϕ ep n σ ϕ Icos n ϕ σ sn The foowng conenton s dopted fo the geneed otton: R denoted R β α w be the poduct of otton b n nge β bout O foowed b one b n nge bout O: R β α R α R β 6 Quntu Mechncs Pof. Y. F. Chen
55 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen t cn be found tht Mt Repesentton of Angu Moentu / cos / sn / sn / cos / cos / sn / sn / cos / sn / cos / sn / cos / / / / / / β β β β β α α β β β β β β σ β β α σ α α α α α α α α e e e e R R R I R e e I R
56 Angu Moentu n Sphec Set Addton of Angu Moent The tot gnetc oent of n eecton s the su of ts ntnsc nd obt gnetc oents. The nube of owed oenttons of ths tot gnetc oentu s detened b the ecto su of the spn h S nd the obt ngu oentu h nd s ced the tot ngu oentu h J : J S. In dscussng toc stuctue t s often necess to fnd the esutnt obt ngu oentu of see eectons o the esutnt spn o to fnd the tot ngu oentu ethe of snge eecton o of the to s whoe. 6 Quntu Mechncs Pof. Y. F. Chen
57 Angu Moentu n Sphec Set Addton of Angu Moent The gene pobe s to fnd the egenues of the esutnt of two ngu oentu opetos of n tpe J J whee nd J J J efe to two dstnct ptces o to two dffeent popetes of the se ptce. The densons of the spces to whch J nd J beong e gen b nd especte. The fou opetos J J fo copete set of coutng opetos. J J 6 Quntu Mechncs Pof. Y. F. Chen
58 Angu Moentu n Sphec Set 6 Quntu Mechncs Pof. Y. F. Chen Snce the coodntes of nd e ndependent the ont egensttes cn be wtten s dect poducts of nd : wth the foowng popetes: Addton of Angu Moent J J J J J J
59 Angu Moentu n Sphec Set Addton of Angu Moent The kets fo copete nd othono bss of coutng opetos { J J }. We sh constuct nothe bss n whch the opetos { J J J Ĵ } e dgon. The egensttes of the opetos { J J J Ĵ } e denoted b tht stsf the foowng popetes: J J J J J J 6 Quntu Mechncs Pof. Y. F. Chen
60 Angu Moentu n Sphec Set Addton of Angu Moent Snce nd e usu fed t s conenent to use the shothnd notton to bbete. An egenstte egensttes cn be epessed s ne cobnton of the b ens of unt tnsfoton. The eeents of the unt t tht pefos ths tnsfoton e ced the Cebsch-Godn CG coeffcents: C ; whee the Cebsch-Godn CG coeffcents e gen b C ; 6 Quntu Mechncs Pof. Y. F. Chen
61 Angu Moentu n Sphec Set Addton of Angu Moent Snce J J we he. J On the othe hnd the owed ues of e octed wthn the nge Theefoe the CG coeffcents cn be noneo on f. nd These e known s the seecton ues fo the CG coeffcents. 6 Quntu Mechncs Pof. Y. F. Chen
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