Previous knowlegde required. Spherical harmonics and some of their properties. Angular momentum. References. Angular momentum operators

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1 // vious owg ui phica haoics a so o thi poptis Goup thoy Quatu chaics pctoscopy H. Haga 8 phica haoics Rcs Bia. iv «Iucib Tso thos A Itouctio o chists» Acaic ss D.A. c Quai.D. io «hii hysiu Appoch oécuai» Duo R.cWy «Quatu chaics: thos a basic appicatios» gao ss Agua otu Rotatioa spctoscopy Hyog ato pi NR ER tc pusio pi obit coupig ysta i phica haoics phica haoics Agua otu opatos ( ) z x i y x i y i i [ x y ] iz [ y z ] ix [ z x ] iy ( ) ( ) ( ) ( ) is abitay ovtioa phas choic (oo a hoty) :. osuc : phica haoics Agua otu opatos x y z si cot cosϕ i z y i ϕ y z x cos cot siϕ i x z i ϕ x x y i y x i ϕ x y z si si si ϕ si si si phica haoics z

2 // phica haoics phica haoics ( ) Θ( ) Φ( ) i ( ) A Φ Φ i ( ) Φ is oaiz Φ ( ) Φ( ) A A A phica haoics Th uctio o ca b xpss as a g poyoia Θ ( ) ( x) x cos x ( x) phica haoics 8 ( x) x ( x) ( x ) ( x) ( x x) phica haoics phica haoics Obita agua otu : a a itgs Noaisatio a othogoaity si opx cougat ( ) i ( cos ) ( ) phica haoics phica haoics phica haoics yty poptis Th stats > o ix spa a iucib pstatio D o th iiit otatio goup R. This ipis that i o appis a abitay otatio D(βγ) to th stat > o obtais a ia cobiatio o th copt st o stats > with th sa. phica haoics phica haoics [ ] R x y R z i [ ] yz R [ ] xz R i [ ] xy R phica haoics

3 // oupig o two agua ota Exap : ato with ctos o i a p obita o i a p obita Not : o spi-obit coupig p obita : p obita : - possib pouct stats o th o : tc phica syty (R ) wavuctios a iguctios o ( ( ) ) a z z z phica haoics oupig o two agua ota yty popty: p p taso as D () pouct stat tasos as D () D (). Ga u: D () D () D () D (-) D ( - ) D () D () D () D () D () () othooa uctios which a ia cobiatios phica haoics ob: oupig o two agua ota cacuat th coicits (vcto coupig o bsch-goa coicits) ( ) ( ) ( ) ( ) ( ) ( ) phica haoics oupig o two agua ota I th cas o uivat p ctos w ot that th putatio o th abs a av th stats o a uchag whi thy chag sig o. o a w a sigt spi uctio (atisytic) a o a tipt spi uctio (sytic): ts A D o th coiguatio p. phica haoics oupig o two agua ota phica haoics oupig o two agua ota Vcto coupig coicits phica haoics 8 Th agua otu iguctios o a othooa st. utipyig both sis o th uatio abov by > a itgatig o obtais : scaa pouct (a vau)

4 // phica haoics oupig o two agua ota optis: uss a - (tiag u) Ra scaa pouct ipis that <a b> <b a> phica haoics oupig o two agua ota Th othogoaity o th vcto-coupig atix ipis phica haoics - sybo Th - sybos ca b cacuat xacty. I th past tabs o ths sybos hav b pubish toay o is - sybo cacuatos o th wb g o i athatica. Not th chag o sig o btw th vcto coupig coicit a th sybo. phica haoics - sybo phica haoics - sybo phica haoics - sybo acuatio o so o th vcto coupig ts i th atix abov

5 // phica haoics - sybo o syty poptis o th - sybos: Ev putatio sa sig O putatio: i i ust b v i phica haoics - sybo si Itga o sphica haoics Not: o copx cougat i this xpssio phica haoics Th Wig-Ecat Tho I spctoscopy w us th oowig syty popty: Th atix t b i a O ψ ψ uss i Γ Γ Γ Howv this tho os ot xpoit a syty poptis. Th Wig-Ecat Tho ats atix ts to coupig coicits K T K is a costat ipt o i I th vcto coupig coicit is pac by a - sybo o obtais T T phica haoics 8 Th Wig-Ecat Tho T T Th ast t is ca a uc atix t. psts ay aitioa uatu ub cssay to spciy th stat. phica haoics Th Wig-Ecat Tho I th opato is a sphica haoic opatig o stats > th uc atix ts ca b cacuat as oows: Usig th ga itgatio o sphica haoics Usig th Wig Ecat tho phica haoics Th Wig-Ecat Tho I o uss Racah s oais sphica haoics:.8..

6 // phica haoics atix ts si si A tab o ths vaus ca b ou i:.ugao.taab a H.Kaiua utipts o Tasitio-ta Ios i ystas Acaic ss Nw o a oo p.. phica haoics - sybos Itouc to scib th coupig o agua ota. It ca b xpss i ts o - sybos: ) ( a Th Wig -sybos a tu by th athatica uctio ixybo[ ]. aso: phica haoics Ecto-cto pusio atix ts o itctoic pusio: Expasio o / usig g poyoias: cos cos ϕ ϕ ω ω > < phica haoics Ecto-cto pusio t us i th opatos: / cos ω This cospos to a scaa pouct o th vctos (i). > < This tatt aows to spaat th aia cotibutio o th agua pat xpss by th. phica haoics Ecto-cto pusio : xap p s s R R p p > < Th itgas a ca at itgas. phica haoics ouob spittig I this cas t. W ca cacuat th ativ gy o th ts () () t D () D E E E Th gou stat t is giv by Hu s u: -axiu spi utipicity (a N spis a paa i N< i p N ) -ach poctio o th obita otu is th agst aow by th xcusio u.n (p ) : (-) ½ ½ Etat oata

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