Addition of Angular Momentum
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1 Addition of Angula Moentu We ve leaned tat angula oentu i ipotant in quantu ecanic Obital angula oentu L Spin angula oentu S Fo ultielecton ato, we need to lean to add angula oentu Multiple electon, eac wit l i and i Spin-obit inteaction couple L and S to fo a total angula oentu J
2 Addition of Angula Moentu Wat i te total pin of two electon? Again we ut take into account tat electon ae identical paticle Syetic pin wave function Syetic unde intecange of,, ( + ) paticle Antiyetic pin wave function Antiyetic unde intecange of ( ) paticle
3 Addition of Angula Moentu Coent Te yetic pin tate i called S te tiplet tate coepond to ( + ) coepond to coepond to Te antiyetic pin tate i called S ( ) coepond to te inglet tate
4 Addition of Angula Moentu Coent Te yety popetie unde paticle intecange of te wave function fo feion and boon efe to te TOTAL wave function ψ(pace)χ(pin) Fo feion, ince te oveall wave function ut be antiyetic unde intecange of paticle and, we can ave eite ψ ψ ( yetic) χ( antiyetic) ( antiyetic) χ( yetic) o
5 Addition of Angula Moenta Te coplete wave function ae Ψa ψ x ψ n x + ψ x ψ n x Ψa ψ x ψ n x ψ x ψ n x ( ( ) ( ) ( ) ( )) ( ) ( ( ) ( ) ( ) ( )) ( + ) Te yetic pin function could be any of te S tiplet tate Note tat fo te S tiplet tate te patial wave function a low pobability fo x ~x Paallel pin epel Note tat fo te S inglet tate te patial wave function a ig pobability fo x ~x Oppoite pin attact Ti pin paiing aie fo te excange foce we talked about ealie 5
6 Addition of Angula Moentu Coent Look at te He ato again ( ) Wat i te pin of te gound tate of He? Wat i te pin of te fit excited tate of He? Etiate te gound tate enegy of He Etiate fo te fit excited tate of He 6
7 Addition of Angula Moentu Coent Baed on ti exaple, te ule fo addition of (pin) angula oentu ae S S + Eigenvalue of Eigenvalue of un fo un fo + + S,..., -S S S z,..., ae ae ( S + ) Te ae ule old fo te addition of any two (o oe) angula oenta S 7
8 Spin-Obit Inteaction Magnetic field poduced by obiting poton (nuclea dipole oent) 8
9 Spin Obit Inteaction Quick eview of obital and pin agnetic oent Obital agnetic oent g μ l B μl L wee gl gl μb μl Lz gl μb z l ΔE μ B l Spin agnetic oent g μ B μ S wee g gμb μ Sz gμb z ΔE μ B 9
10 Spin-Obit Inteaction Ti intenal agnetic field give ie to a pinobit inteaction te in te Hailtonian ΔE μ B μi B (Biot - Savat law) Ze π I and L v T T Ze B L πε c g μb e μ S S Ze ΔE S L πε c and c μ ε
11 Spin-Obit Inteaction Actually ti in t quite igt ince te electon i in a non-inetial fae Te coect eult diffe by a facto of ½ and i know a te Toa peceion Ze Δ E 8πε c S L
12 Spin-Obit Inteaction Aide ( ) ( ) ( ) ev E ev MeV E c a c E c a c a E a c e E a c e E, n,l L S c Ze E fo 8 Etiate Δ Δ Δ Δ Δ Δ Δ πε πε πε
13 Spin-Obit Inteaction Aide, Etiate B ΔE μ B B B ΔE μ B ev ev / T.T
14 Spin-Obit Inteaction We left te pin-obit inteaction out of te Hailtonian te fit tie we did te ydogen ato In pat, becaue it a all coection Te full Scodinge equation fo te ydogen ato i (neglecting a elativitic coection) ψ + Ze πε + Ze 8πε e c S L ψ Eψ
15 Spin-Obit Inteaction Ou oiginal quantu nube fo te ydogen ato wee n, l,,, l And [ ] [ H, L H, S ] [ H, L ] [ H, S ] Z Z But wit te pin-obit te in te Hailtonian, H no longe coute wit oe of tee opeato Befoe you anwe note tat Fo J L + S J L + S L + S + L ( ) S 5
16 Total Angula Moentu Te pin-obit inteaction couple te obital (L) and pin (S) angula oentu to fo te total angula oentu (J) Te intenal agnetic field i deteined by L and ti act on te pin agnetic dipole of te electon deteined by S o te two angula oenta ae not independent J L + S Te new good quantu nube of te ydogen ato ae n, l,, j, j 6
17 Total Angula Moentu Coupling of L and S to fo J L and S pece aound J J pecee aound te z axi 7
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