j2 m2 j m j m k j1 j2 j k j1 m1

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Unformatted text preview: + 1) 2 m= J max m− m =1 J min −1 m m =1 (2 j + 1) = J max ( J max + 1) − J min ( J min + 1) + J max − J min + 1 = = (2 j1 + 1)(2 j2 + 1) J min = j1 − j2 j1 − j2 ≤ J ≤ j1 + j2 Clebsch-Gordon coefficients: Clebsch general formula > , j1m1 j2 m2 jm = δ m1 + m2 ,m -? . - ?/ . ( j1 + j2 − j )!( j + j1 − j2 )!( j + j2 − j1 )!(2 j + 1) ( j + j1 + j2 + 1)! ( j1 + m1 )!( j1 − m1 )!( j2 + m2 )!( j2 − m2 )!( j + m )!( j − m )! k !( j1 + j2 − j − k ) !( j1 − m1 − k )!( j2 + m2 − k ) !( j − j2 + m1 + k )!( j − j1 − m2 + k )! (−1) k × k ; Clebsch-Gordon coefficients: symmetry relations ) ) @@ j1 − m1 j2 − m2 j − m = (−1) j1 + j2 − j j1m1 j2 m2 jm j2 m2 j1m1 jm = (−1) j1 + j2 − j j1m1 j2 m2 jm j − mj2 m2 j1 − m1 = (−1) j2 + m2 2 j1 + 1 j1m1 j2 m2 jm 2J + 1 Three-j symbols j1 j2 j3 m1 m2 m3 = (−1) j1 − j2 − m3 j1m1 j2 m2 j3 − m3 2 j3 + 1 m1 + m2 + m3 = 0 ? Three-j symbols: symmetry relations j3 m3 j1 m1 j2 j =2 m2 m2 j3 m3 j1 j =1 m1 m1 j2 m2 j3 m3 312 − 231 − 123 : no change with even permutations j2 m2 j1 m1 j3 j = (−1) j1 + j2 + j3 1 m3 m1 j2 m2 j3 m3 213 -132 - 321: phase change with odd permutations j1 j2 j3 −m1 −m2 −m 3 = (−1) j1 + j2 + j3 j1 j2 j3 m1 m2 m3 Three-j symbols: Three orthogonality relations j1 m 1m 2 j2 j3 ' j1 j2 j3 m1 m2 m3' m1 m2 m3 j1 j2 j3 j1 j2 m1 m2 m3 (2 j3 + 1) j 3m 3 = 1 δ j ' j δ m 'm 2 j3 + 1 3 3 3 3 j3 m1 ' m2 ' m 3 = δ m1 ' m1δ m 2 ' m2 & Irreducible tensor operators Irreducible ! 2k+1 ! q=-k,-k+1, …, k J z , Tqk = qTqk J ± , Tqk = (k ± q + 1)(k q )Tqk±1 Wigner-Eckart theorem Wigner + j1m1 Tqk j2 m2 = (−1) j1 − m1 j1m1 Tqk j2 m2 ≠ 0 if q = m1 − m2 j1 − j2 ≤ k ≤ j1 + j2 j1 − m1 k j2 q m2 j1 T k j2...
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This document was uploaded on 04/05/2014.

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