Drehimpulse in der Quantenmechanik by Albrecht Lindner

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By Albrecht Lindner

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Auf 1(iI(M3)i23)fm) = E 1((M2)i12f3)jm) (((iti2)f12i3)fml(iI(M3)i23)fm) , ft'J so folgt nämlich wegen der Orthonormierung solange die Richtungsquantenzahl mit i verträglich ist - sonst wird der fortgelassene Faktor + I} - mIm ± I} null und die Gleichung falsch. 3 Umkopplung von drei Drehimpulsen: 6i-Symbole Drei Drehimpulse 11 , J 2 und J 3 können verschieden zu J gekoppelt werden: Entweder diagonalisiert man neben J2, Jo, J 1 2, J 2 2 und J 8 2 noch J 12 2 oder J23 2 oder J 31 2. Die entsprechenden Darstellungen sind über unitäre Transformationen miteinander verknüpft.

13. , 1. , 1. 51'i" JA} =--:2 . , Ja . , ~ ) J. , J. je ~} . Die Summe über das Zweierprodukt läßt sich also so umformen, daß die doppelt vorkommenden Elemente (vom laufenden abgesehen) nur noch je einmal vorkommen. +I.. und die Gleichung von ARIMA, HORIE & TANABE 1. jB 1 r ia 14 la 123 112 134 1 ~2 l l 1{lI ~: ~:l· = 72 14 19 h ja 19 EARP hat Summen mit noch mehr Faktoren behandelt. 12 Summen über 31-, 6i- und gi-Symbole Aus den Gleichungen der drei letzten Abschnitte lassen sich weitere Beziehungen mit Hilfe der Unitarität herleiten.

N23 - nu + n)! nu! (nS2 - nu + n)! Abgesehen davon, daß wir nun die Summationsgrenzen bestimmt haben, sind jetzt alle Summanden ganze Zahlen, die wir rekursiv aus und 8" = - (nu + 1- n)(n22 + 1- n)(n33 + 1- n) n (n2S - nll + n) (nS2 - nu + n ) 8,,-1 berechnen können. Insbesondere gilt bei n=0 =0 =1, = 1 = n2SnS2 - n22Tt33 , = 2 = n2S(n2S -1)nS2(nS2 -I) - 2TtZSnS2n22n33 + n22(n22 -I)nss(nu -1). 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.

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