By Brain Judd (Auth.)

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8 Reduced Matrix Elements vector. The sign of i can have no significance in quantum mechanics: one merely has to be consistent. However, if we accept Eq. 8) and its cyclic permutations as defining an angular momentum vector, a normal angular momentum vector Ρ satisfies anomalous commutation relations. 10) and now the components of Ρ satisfy commutation relations of the type ( £ 8 ) . ) are to be used, in which Μ ρ is the eigenvalue of Pf. The change of sign of an angular momentum vector corresponds t o time reversal.

7. So E q . 31) becomes 2 C w- i ( c o s x ) C n ' - i ( c o s x ) = Σ f m {2τ η"/nn ) f(nn'n") n" X A(n - 1, n' - 1, n " - l ) C n - - i ( c o s X) . 7, we m u s t have /(nn'n") = (ηη'727rV) 1 / 2 . This can now be fed b a c k into E q . 31). 27 can be used for t h e C G coefficient. T h e result is YnlmW Yn'l'm' (Ω) = , Σ 2 1 2 {(2i + 1) (2V + l ) n n V / 2 x } / ( Z m , Vm' \ I V ) n"i"m" i(n X \(η' - i(n" - 1) 1) 1) J(n - 1) I 1(η' - 1) V ) Ι^ιι"ί"ιι>"(Ω). 9 Addition Theorems 39 T h e 9-j symbol in this equation automatically vanishes when Δ ( η — 1, , ,f η' — 1, η " — 1) = 0 , so our derivation of f(nn n ) holds good in t h e general case.

Consider, then, the effect of Ae, as given in Eqs. 16), on F n i m ( G ) . 3). m(V), namely, ((kk)l + 1, m\ A9\ R(4) of (kk)lm). W e first note t h a t Az = K\z — Kit = 2Ku ~ h, a n d since lz is diagonal with respect t o Z, t h e last t e r m can be dropped. Next, we observe t h a t Ku acts on t h e first p a r t of t h e coupled system, so we use E q . 66) after first applying t h e W E t h e o r e m : ((kk)l+ 1, m\Az\ (kk)lm) «2(-i)«*-( 1 Z + 1 \ —m m = 2(-l)^ {(2l+ l )«kk)l m/ 0 + l | | K i | | (fcfc)Z) 1 2 l)(2l + 3) j ' Λ + 1 1 Z \ f* 1 k \ —m 0 m/ k I + 1J [l ) ,, ,, T o complete t h e calculation, we p u t 1 2 (* || K i l l * ) = {fc(fc+ l ) ( 2 f c + l ) } ' , which follows from E q .