Transforming to the measurement basis

The coupled-spin basis vectors are transformed from the energy eigenbasis to the  basis using transVec[].

transVec[{rep_,order_,scale_}] :=
  Block[ { vec={},n=Length[rep],q,j1,m1,j2,m2,j12,m12,newrep } ,
         q = 2 order ;
         If[ order!=0 ,
             j1  = rep[[q-1]] ;
             j2  = rep[[q]]   ;
             j12 = rep[[q+1]] ;
             m12 = rep[[q+2]] ;
             For[ m1=-j1,m1<=j1,m1++,
                  m2=m12-m1 ;
                  If[ ((-j2<=m2) && (m2<=j2)),
                      newrep=Delete[rep,q+1] ;
                      newrep=Delete[newrep,q+1] ;
                      newrep=Insert[newrep,m1,q] ;
                      newrep=Insert[newrep,m2,q+2] ;
                      vec=Join[vec,
                      transVec[{newrep,
                      order-1,
                      scale*cg[j1,j2,m1,m2,j12,m12]}] ] ;
                    ] ;
                ] ; ,
            Return[{{rep,order,scale}}] ;
          ] ;
        Return[vec] ;
      ] ;

The argument of transVec is a length-three list consisting of rep, the representation of the basis vector in the basis produced by genVecs, order, the order of the representation given, usually N-1, where N is the number of spins (note that transVec is recursive, order keeps track of the depth), and scale, the value of the normalization of the vector generated. For example,

transforms the vector  to the vector

which is a list of three lists, each of these being a vector in the  basis, along with an order and scale indicator, and the whole list to be interpreted as the sum of these vectors.

Along with transVec, the routine transVecs transforms a list of many vectors, such as would be produced by the output of genVecs.

transVecs[x_] :=
  Block[ {n,vecs={},i} ,
         n=Length[x] ;
         For[ i=1,i<=n,i++,
              vecs=Join[vecs,{transVec[x[[i]]]}] ;
            ] ;
         Return[vecs] ;
       ] ;

For example

produces the output

which is a list of the four vectors in the  basis corresponding to the four vectors in the coupled spin basis produced by the genVecs command in equation 13.1.
 

Tue Mar 25 08:11:49 CST 1997