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34 3j,6j,9j SymbolsProperties

§34.5 Basic Properties: 6j Symbol

Contents

§34.5(i) Special Cases

In the following equations it is assumed that the triangle inequalities are satisfied and that J is again defined by (34.3.4).

If any lower argument in a 6j symbol is 0, 12, or 1, then the 6j symbol has a simple algebraic form. Examples are provided by:

34.5.1 {j1j2j30j3j2} =(-1)J((2j2+1)(2j3+1))12,
34.5.2 {j1j2j312j3-12j2+12} =(-1)J((j1+j3-j2)(j1+j2-j3+1)(2j2+1)(2j2+2)2j3(2j3+1))12,
34.5.3 {j1j2j312j3-12j2-12} =(-1)J((j2+j3-j1)(j1+j2+j3+1)2j2(2j2+1)2j3(2j3+1))12,
34.5.4 {j1j2j31j3-1j2-1} =(-1)J(J(J+1)(J-2j1)(J-2j1-1)(2j2-1)2j2(2j2+1)(2j3-1)2j3(2j3+1))12,
34.5.5 {j1j2j31j3-1j2} =(-1)J(2(J+1)(J-2j1)(J-2j2)(J-2j3+1)2j2(2j2+1)(2j2+2)(2j3-1)2j3(2j3+1))12,
34.5.6 {j1j2j31j3-1j2+1} =(-1)J((J-2j2-1)(J-2j2)(J-2j3+1)(J-2j3+2)(2j2+1)(2j2+2)(2j2+3)(2j3-1)2j3(2j3+1))12,
34.5.7 {j1j2j31j3j2} =(-1)J+12(j2(j2+1)+j3(j3+1)-j1(j1+1))(2j2(2j2+1)(2j2+2)2j3(2j3+1)(2j3+2))12.

§34.5(ii) Symmetry

The 6j symbol is invariant under interchange of any two columns and also under interchange of the upper and lower arguments in each of any two columns, for example,

34.5.8 {j1j2j3l1l2l3}={j2j1j3l2l1l3}={j1l2l3l1j2j3}.

Next,

34.5.9 {j1j2j3l1l2l3} ={j112(j2+l2+j3-l3)12(j2-l2+j3+l3)l112(j2+l2-j3+l3)12(-j2+l2+j3+l3)},
34.5.10 {j1j2j3l1l2l3} ={12(j2+l2+j3-l3)12(j1-l1+j3+l3)12(j1+l1+j2-l2)12(j2+l2-j3+l3)12(-j1+l1+j3+l3)12(j1+l1-j2+l2)}.

Equations (34.5.9) and (34.5.10) are called Regge symmetries. Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). See Srinivasa Rao and Rajeswari (1993, pp. 102–103) and references given there.

§34.5(iii) Recursion Relations

In the following equation it is assumed that the triangle conditions are satisfied.

34.5.11 (2j1+1)((J3+J2-J1)(L3+L2-J1)-2(J3L3+J2L2-J1L1)){j1j2j3l1l2l3}=j1E(j1+1){j1+1j2j3l1l2l3}+(j1+1)E(j1){j1-1j2j3l1l2l3},

where

34.5.12 Jr =jr(jr+1),
Lr =lr(lr+1),
34.5.13 E(j)=((j2-(j2-j3)2)((j2+j3+1)2-j2)(j2-(l2-l3)2)((l2+l3+1)2-j2))12.

For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99).

§34.5(iv) Orthogonality

34.5.14 j3(2j3+1)(2l3+1){j1j2j3l1l2l3}{j1j2j3l1l2l3}=δl3,l3.

§34.5(v) Generating Functions

For generating functions for the 6j symbol see Biedenharn and van Dam (1965, p. 255, eq. (4.18)).

§34.5(vi) Sums

34.5.15 j(-1)j+j+j′′(2j+1){j1j2jj3j4j}{j1j2jj4j3j′′}={j1j4jj2j3j′′},
34.5.16 (-1)j1+j2+j3+j1+j2+l1+l2{j1j2j3l1l2l3}{j1j2j3l1l2l3}=j(-1)l3+l3+j(2j+1){j1j1jj2j2j3}{l3l3jj1j1l2}{l3l3jj2j2l1}.

Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the 6j symbol.

34.5.17 j(2j+1){j1j2jj1j2j} =(-1)2(j1+j2),
34.5.18 j(-1)j1+j2+j(2j+1){j1j2jj2j1j} =(2j1+1)(2j2+1)δj,0,
34.5.19 l{j1j2lj2j1j} =0,
2μ-j odd, μ=min(j1,j2),
34.5.20 l(-1)l+j{j1j2lj1j2j} =(-1)2μ2j+1,
μ=min(j1,j2),
34.5.21 l(-1)l+j+j1+j2{j1j2lj2j1j} =12j+1((2j1-j)!(2j2+j+1)!(2j2-j)!(2j1+j+1)!)12,
j2j1,
34.5.22 l(-1)l+j+j1+j21l(l+1){j1j2lj2j1j} =1j1(j1+1)-j2(j2+1)((2j1-j)!(2j2+j+1)!(2j2-j)!(2j1+j+1)!)12,
j2<j1.
34.5.23 (j1j2j3m1m2m3){j1j2j3l1l2l3}=m1m2m3(-1)l1+l2+l3+m1+m2+m3(j1l2l3m1m2-m3)(l1j2l3-m1m2m3)(l1l2j3m1-m2m3).

Equation (34.5.23) can be regarded as an alternative definition of the 6j symbol.

For other sums see Ginocchio (1991).