Orbital Angular Momentum
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1 Obta Anua Moentu In cassca echancs consevaton o anua oentu s soetes teated b an eectve epusve potenta Soon we w sove the 3D Sch. Eqn. The R equaton w have an anua oentu te whch ases o the Theta equaton s sepaaton constant eenvaues and eenunctons o ths can be ound b sovn the deenta equaton usn sees soutons but aso can be soved aebaca. Ths stats b assun s conseved tue V d dt H ] P46 - anua oentu
2 P46 - anua oentu Obta Anua Moentu ook at the quantu echanca anua oentu opeato cassca ths causes a otaton about a ven as ook at 3 coponents opeatos do not necessa coute p p cos sn sn cos p p p p p p ] ]
3 Sde note Poa Coodnates Wte down anua oentu coponents n poa coodnates Supp 7-B on webe&r App M sn and wth soe t anpuatons sn sn ] sn but sae equatons w be seen when sovn anua pat o S.E. and so Y Y Y and know eenvaues o and wth spheca haoncs ben eenunctons cos cot cos cot sn sn sn sn Y ] Y P46 - anua oentu 3
4 P46 - anua oentu 4 Coutaton Reatonshps ook at a coutaton eatonshps snce the do not coute on one coponent o can be an eenuncton be daonaed at an ven te deent a sae ndces an tenso o jk k jk j ] ] ] ] ] ] ]
5 P46 - anua oentu 5 Coutaton Reatonshps but thee s anothe opeato that can be sutaneous daonaed Cas opeato u : sn ]
6 Goup Aeba The coutaton eatons and the econton that thee ae two opeatos that can both be daonaed aows the eenvaues o anua oentu to be detened aebaca sa to what was done o haonc oscato an eape o a oup theo appcaton. Aso shows how anua oentu tes ae cobned the oup theo esuts have appcatons beond obta anua oentu. Aso app to patce spn whch can have / ntee vaues Concepts ate apped to patce theo: SU SU3 U SO sus stns..usua contnuous..and to sod state phscs oten dscete Soetes oup popetes pont to new phscs SU-spn SU3-uons. But soetes not natue doesn t have an patces wth that oup s popetes P46 - anua oentu 6
7 Sdenote:Goup Theo A ve sped ntoducton A set o objects o a oup a cobnn pocess can be dened such that. I AB ae oup ebes so s AB. The oup contans the dentt AI=IA=A 3. Thee s an nvese n the oup A - A=I 4. Goup s assocatve ABC=ABC oup not necessa coutatve Abean non-abean AB AB BA BA Can oten epesent a oup n an was. A tabe a at a denton o utpcaton. The ae then soophc o hooophc P46 - anua oentu 7
8 P46 - anua oentu 8 Spe eape Dscete oup. Popetes o oup ts athetc contaned n Tabe Can epesent each te b a nube and oup cobnaton s noa utpcaton o can epesent b atces and use noa at utpcaton b a c c a c b b c b a a c b a c b a c b b a a a c b a
9 P46 - anua oentu 9 Contnuous e Goup:Rotatons Consde the otaton o a vecto R s an othoona at enth o vecto doesn t chane. A 33 ea othoona atces o a oup O3. Has 3 paaetes.e. Eue anes O3 s non-abean assue ane chane s sa ' dentt nea sae enth R ' ' ' anes sa R R cos sn sn cos R R R R
10 P46 - anua oentu Rotatons Aso need a Unta Tansoaton doesn t chane enth o how a uncton s chaned to a new uncton b the otaton U s the unta opeato. Do a Tao epanson the anua oentu opeato s the eneato o the nntesa otaton unta U R o R to chanes R U p p R
11 P46 - anua oentu Fo the Rotaton oup O3 b nspecton as: one ets a epesentaton o anua oentu notce none s daona; w daonae ate satses Goup Aeba U R R k jk j ]
12 P46 - anua oentu Goup Aeba Anothe oup SU aso satses sae Aeba. Unta tansoatons atces wth det= ves S=speca. SUn has n - paaetes and so 3 paaetes Usua use Pau spn atces to epesent. Note O3 ves ntee soutons SU ha-ntee and ntee k jk j ] U U
13 Eenvaues Goup Theo Use the oup aeba to detene the eenvaues o the two daonaed opeatos and Aead know the answe Have constants o eoet. eenvaues o ae postvedente. the enth o the -coponent can t be eate than the tota and snce s abta evese aso tue The X and Y coponents aen t ecept = but can t be daonaed and so ~ndetenate wth a ane o possbe vaues P46 - anua oentu 3
14 P46 - anua oentu 4 Eenvaues Goup Theo Dene asn and owen opeatos noe Pank s constant o now. Rase -eenvaue eenvaue whe keepn - eanvaue ed atces SU o
15 P46 - anua oentu 5 Eenvaues Goup Theo opeates on a vecto van asn o owen t s s s s
16 P46 - anua oentu 6 Can aso ook at at epesentaton o 33 othoona ea atces Choose Z coponent to be daona ves choce o atces
17 P46 - anua oentu 7 Can aso ook at at epesentaton o 33 othoona ea atces can wte down +- need sqt to noae and then wok out X and Y coponents
18 P46 - anua oentu 8 Can aso ook at at epesentaton o 33 othoona ea atces. Wok out X and Y coponents
19 P46 - anua oentu 9 Can aso ook at at epesentaton o 33 othoona ea atces. Wok out *] Identt T
20 P46 - anua oentu Eenvaues Done n deent was GasoGthsSch Stat wth two daonaed opeatos and. whee and ae not et known Dene asn and owen opeatos n and eas to wok out soe eatons ] ] ] Z
21 P46 - anua oentu Eenvaues Assue s eenuncton o and. + s aso an eenuncton new eenvaues and see ases and owes vaue hba coute opeatos o
22 P46 - anua oentu Eenvaues Thee ust be a hhest and owest vaue as can t have the - coponent be eate than the tota Fo hhest state et be the au eenvaue can eas show : n H H H H de e H H H
23 Eenvaues Thee ust be a hhest and owest vaue as can t have the - coponent be eate than the tota epeat o the owest state H equate eenvaues o o o - to n ntee steps N steps N nt ee o ha nt ee... SU on 3... tes P46 - anua oentu 3
24 P46 - anua oentu 4 Rasn and owen Opeatos can aso see GasoSch detene eenvaues b ookn at and show note vaues when = and =- ve useu when addn toethe anua oentus and budn up eenunctons. Gves Cebsch-Godon coecents C C C C
25 P46 - anua oentu 5 Eenunctons n spheca coodnates =ntee can detene eenunctons known the os o the opeatos n spheca coodnates sove st and nset ths nto the second o the hhest state = Y Y e Y Y Y Y cot e F Y cot cot cot cot F e F e e e F e Y e
26 Eenunctons n spheca coodnates sovn e cot F ves F sn Y Ae sn then et othe vaues o ebes o the utpet b usn the owen opeato w obtan Y eenunctons spheca haoncs aso b sovn the assocated eende equaton note powe o : = w have sn ; Y cos e sn cot ; cos Y P46 - anua oentu 6
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