By Albrecht Lindner
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Extra info for Drehimpulse in der Quantenmechanik
Example text
Wie günstig dieser Weg ist, läßt sich daran erkennen, daß alle in Abschn. 8 genannten Clebsch-Gordan-Koefllzienten - außer dem letzten - zum Regge-Symbol mit nu = 0 gehören. B. bei nu = 1 und nZ2nSS = nZSnS2. 12 Rekursionsformeln für 3i-Symbole Aus den in Abschn. 7 genannten Rekursionsformeln für Clebsch-GordanKoeffizienten folgen die Gleichungen ia ) _ 0 mg ±1 bzw. v'n2dnSI + 1) [n" [ n" n21 --1 n22 u] nn23 + v'n 22(n32 +-1) n21 n31 +1 n32 n33 n12 + v'n23(nS3 Hieraus ergibt sich mit E = n11 + n12 v'n22nSs(n23 +Jn2snS2(n22 + {nutE + n13 + 1)(ns2 + 1) + 1)(ns3 + 1) + 1) n,,] n12 n23 ~2-1 + 1 nS3 nSl n32 n21 n22 n23 -1 nSl n32 n33 r" n..
Is. ((Ms )113 (M. )12. ((Ms)113(M. )i) , = r wobei über i12s zu summieren ist. +2111 2: '/ {~11 j (_ )2;',+21,. 1 i22 J ls1 ~33 121 } ~13 } 131 ~23 } JIl . (In der ersten Summe treten 113, 122 und laI nur einmal auf, in der zweiten 111, i23 und iS2 und in der letzten i12' i21 und lss). Dagegen folgt aus ((((iria )itsh )i1231. ((iri2)i12 (iai. )i3. )i! (i12(M. )iu)l) = I: (( (ini2 )imi. )i! ((ital. )i) (über i1s. , 11" . , . J 12 i2" 1r i~'} . J 54 Tritt in diesen Gleichungen eine Null auf, so folgen die Summenformeln L: (_)1+1'+i" '/ {~1 13 i h j.
1+J' '. , 111 bjjll bj'jm {iI i2 J4 ja . B. aus der ersten Summe in Abschn. 10 hergeleitet werden: Zwei der Faktoren sind in Abschn. 14 genannt. Die restliche Summe läßt sich mit der Rekursionsformel für 6j-Symbole vereinfachen. Daneben wird häufig auch noch (vgl. Abschn. 8) 63 gebraucht - die in Abschn. 10 genannte Summe enthält hier nur ein Glied, das mit Abschn. 14 noch vereinCacht werden kann. Auch die beiden 6j-Symbole sind dort noch angegeben. Zusammen mit Abschn. 13 folgt deshalb If, (' o I' 0 n) 1; ;: :) ~ 0 l 2 k 2 (±)I'+i'+!