Density Matrix in Irreducible Tensor Form

This notebook calculates density matrix elements in the irreducible tensor notation and also gives the inverse transform

Download Clebsch- Gordan subroutine from where you store it or use Mathematica version 
   ClebschGordan[{(Subscript[j, 1]),(Subscript[m, 1])},{(Subscript[j, 2]),(Subscript[m, 2])},{j,m}]

Program

<<c:\pberman\mathematica\5.2\clebsh.m

ρ[k_,q_,j_,j1_]:=Sum[(-1)^(j1-m1)CG[{j,q+m1},{j1,-m1},{k,q}]ρ[q+m1,m1],{m1,-j1,j1}]

**Examples***

Table[ρ[1,-q,1/2,1/2],{q,-1,1}]

Table[ρ[1,q,1,1],{q,1,-1,-1}]

Table[ρ[2,q,1,1],{q,2,-2,-1}]

Inverse Transform

ρm[j_,m_,j1_,m1_]:=Sum[(-1)^(j1-m1)CG[{j,m},{j1,-m1},{k,q}]ρk[k,m1+m],{k,-Abs[j-j1],j+j1},{q,-k,k}]

**Examples**

ρm[1,0,1,0]

ρm[1,1,1,-1]