Clebsch-Gordan coefficient
Masatsugu Sei Suzuki and Itsuko S. Suzuki
Department of Physics, SUNY at Binghamton
(Date: March 16, 2023)

Clear["Global`*"];CCGG[{j1_,m1_},{j2_,m2_},{j_,m_}]:=Module[{s1},s1=If[Abs[m1]⩽j1&&Abs[m2]⩽j2&&Abs[m]⩽j,ClebschGordan[{j1,m1},{j2,m2},{j,m}],0]];CG[{j_,m_},j1_,j2_]:= Sum[CCGG[{j1,m1},{j2,m-m1},{j,m}] a[j1 ,m1]  b[j2, m-m1],{m1,-j1,j1}]

Clebsch-Gordan co-efficients
J1=1/2, J2=1/2
D1/2 x D1/2 = D1 +D0
j=1; m = 1, 0, -1 (triplet)
j = 0, m = 0(singlet)

j1=1/2;j2=1/2;

CG[{1,1},j1,j2]

CG[{1,0},j1,j2]

CG[{1,-1},j1,j2]

CG[{0,0},j1,j2]

Clebsch-Gordan co-efficients
J1=1, J2=1/2
D1 x D1/2 = D3/2 +D1/2
j=3/2; m = 3/2, 1/2, -1/2, and -3/2
j=1/2; m =  1/2, and -1/2

j1=1;j2=1/2;

CG[{3/2,3/2},j1,j2]

CG[{3/2,1/2},j1,j2]

CG[{3/2,-1/2},j1,j2]

CG[{3/2,-3/2},j1,j2]

CG[{1/2,1/2},j1,j2]

CG[{1/2,-1/2},j1,j2]

Clebsch-Gordan co-efficients
J1=1, J2=1
D1 x D1 = D2 +D1+D0
j=2; m = 2, 1, -0, -1, and  -2
j=1, m = 1, 0, -1
j=0, m = 0

j1=1;j2=1;

CG[{2,2},j1,j2]

CG[{2,1},j1,j2]

CG[{2,0},j1,j2]

CG[{2,-1},j1,j2]

CG[{2,-2},j1,j2]

CG[{1,1},j1,j2]

CG[{1,0},j1,j2]

CG[{1,-1},j1,j2]

CG[{0,0},j1,j2]

Clebsch - Gordan co - efficients
J1 = (3/2), J2 = (1/2)
D3/2 x D1/2 = D2 + D1 
j = 2; m  = 2, 1,  0, -1, and  -2
j = 1, m = 1,  0, -1

j1=3/2;j2=1/2;

CG[{2,2},j1,j2]

CG[{2,1},j1,j2]

CG[{2,0},j1,j2]

CG[{2,-1},j1,j2]

CG[{2,-2},j1,j2]

CG[{1,1},j1,j2]

CG[{1,0},j1,j2]

CG[{1,-1},j1,j2]

Clebsch - Gordan co - efficients
J1 = 2, J2 = 1
D2 x D1 = D3 + D2 +D1
j = 3; m  = 3, 2, 1, 0, -1,  -2, -3
j = 2; m  = ,2, 1, 0, -1,   -2, 
j = 1, m = 1,  0, -1

j1=2;j2=1;

CG[{3,3},j1,j2]

CG[{3,2},j1,j2]

CG[{3,1},j1,j2]

CG[{3,0},j1,j2]

CG[{3,-1},j1,j2]

CG[{3,-2},j1,j2]

CG[{3,-3},j1,j2]

CG[{2,2},j1,j2]

CG[{2,1},j1,j2]

CG[{2,0},j1,j2]

CG[{2,-1},j1,j2]

CG[{2,-2},j1,j2]

CG[{1,1},j1,j2]

CG[{1,0},j1,j2]

CG[{1,-1},j1,j2]