Problem Set 6

Excercise on transforming from Ls to jj coupling.

Prob 1

Write down the 2 by 2 matrix that transforms the jj coupled states |(1(Subscript[s, 1/2]) 2(Subscript[p, 1/2])) 1> 
and  |(1(Subscript[s, 1/2]) 2(Subscript[p, 3/2])) 1>  into LS coupled states (1s2p) (^1)(Subscript[P, 1]) and  (1s2p)  (^3)(Subscript[P, 1]).

As a preliminary step define the 9j symbol  nineJ and the LS-jj transformation matrix t[LS,jj].

nineJ[a_, b_, c_, d_, e_, f_, g_, h_, 
        j_]  := 
Sum[(-1)^(2 x) (2 x  +  1) SixJSymbol[{a, b, c}, {f, j, x}] 
          SixJSymbol[{d, e, f}, {b, x, h}] SixJSymbol[{g, h, j}, {x, a, 
              d}], {x, Max[Abs[a - j], Abs[b - f], Abs[d - h]], 
          Min[a + j, b + f, d + h]}]

t[l1_, s1_, j1_, l2_, s2_, j2_, l_, s_, 
        j_]  := 
Sqrt[(2  l + 1) (2  s + 
                1) (2  j1 + 1) (2  j2 + 1)] nineJ[l1, s1, j1, l2, 
          s2, j2, l, s, j]

ϕLS[0]   =  
      Sum[t[0, 1/2, 1/2, 1, 1/2, j2, 1, 0, 1] ψjj[j2], {j2, 1/2, 
          3/2}]

ϕLS[1]   =  
      Sum[t[0, 1/2, 1/2, 1, 1/2, j2, 1, 1, 1] ψjj[j2], {j2, 1/2, 
          3/2}]

mat1  =  {{ϕLS[0][[1, 1]], ϕLS[0][[2, 
                1]]}, {ϕLS[1][[1, 
                1]], ϕLS[1][[2, 1]] ϕLS[1][[2, 
                  2]]}} // 
        MatrixForm

Prob 2

Write down the 3 by 3 matrix that transforms the jj coupled states 
|(2(Subscript[p, 1/2]) 3(Subscript[d, 3/2])) 1>,  (2(Subscript[p, 3/2]) 3(Subscript[d, 3/2])) 1>, and |(2(Subscript[p, 3/2]) 3(Subscript[d, 5/2])) 1 > 
 into LS coupled states 
(2p3d) (^1)(Subscript[P, 1]),   (2p3d) (^3)(Subscript[P, 1]) ,  and  (2p3d) (^3)(Subscript[D, 1]).

mat2 = 
      Simplify[{{t[1, 1/2, 1/2, 2, 1/2, 3/2, 1, 0, 1], 
            t[1, 1/2, 3/2, 2, 1/2, 3/2, 1, 0, 1], 
t[1, 
              1/2, 3/2, 2, 1/2, 5/2, 1, 0, 1]}, {t[1, 1/2, 1/2, 2, 1/2, 3/2, 
              1, 1, 1], 
            t[1, 1/2, 3/2, 2, 1/2, 3/2, 1, 1, 1], 
t[1, 
              1/2, 3/2, 2, 1/2, 5/2, 1, 1, 1]}, {t[1, 1/2, 1/2, 2, 1/2, 3/2, 
              2, 1, 1], 
            t[1, 1/2, 3/2, 2, 1/2, 3/2, 2, 1, 1], 
t[1, 
              1/2, 3/2, 2, 1/2, 5/2, 2, 1, 1]}}]

MatrixForm[mat2]

MatrixForm[Inverse[mat2]]

Note that the inverse of this matrix is its transpose, as it should be for an orthogonal matrix.

Some Other Examples

Look at the case  |(1(Subscript[s, 1/2]) 2(Subscript[p, 1/2])) 0>  coupled to give  (^3)(Subscript[P, 0]). In this case the matrix is  1 x 1.  We find

t[0, 1/2, 1/2, 1, 1/2, 1/2, 1, 1, 0]

Here is another 1 x 1 case:  |(1(Subscript[s, 1/2]) 2(Subscript[p, 3/2])) 2>  coupled to give  (^3)(Subscript[P, 2]).  We find

t[0, 1/2, 1/2, 1, 1/2, 3/2, 1, 1, 2]

In these cases of only one intremediate state,  the transformation is just the identity or a phase!