Quantum Mechanics: Solving for j(square) Matrix with Clebsch Gordan Coefficients

[TURK]

Please urgent help.[quantum]

I have a homework I have tried a lot of thinks but I couldnot manage to solve.

problem: for two particle j1 = 2 ; j2 = 1 and the condition is m=m1 + m2 = 2.
by using Clebsch Gordan coefficients for the possible states !22> and !32>, create a matrix which will be a 2x2 matix lest say U and create j square for the system.
find the matrix product U(transpose).j(square).U.
I know that this pruduct will be the identity matris I if you take the h(bar) as 1
I have created the matrix U that is -squareroot(1/3) squareroot(2/3)

squareroot(2/3) squareroot(1/3)

but I have problem with the j(square)

I know the j(square)= j1(square) + j2(square) + [(j1-.j2+) + (j1-.j2+)] + 2j1z.j2z

here j1- and j2- are lowering and j1+ and j2+ are rising optrs.


please someone help me with the building of j(square) matris. I really have done whatever I can do.

 

[TURK]

for example for j1(square) matrix elements are j1(j1+1) that is 6 in the diagonal of 2x2 matris and off diagonal elements are zero.

 

[TURK]

please someone help me. I really worked on this problem.