If the general angular momentum quantum number j is 1, there is a triplet of |j,mj) states: {|1,1), |1,0), and |1,-1).
In this case, a matrix representation for the operators Jz, Jy, and J can be constructed if we represent the |j,mj) triplet by three-component column vectors as follows: |1,1) [1,0) (8), |1,-1) -(8) (4.1). J can then be represented by the matrix:
J = h
Construct matrix representations for the raising and lowering operators, J+ and J-, acting on the eigenstates |1,1), |1,0), and |1,-1) in the representation given in equation (4.1).
Use the relationships
J^2 = J+J- + Jz
Jy = (J- - J+)/2
to construct matrix representations for J and Jy. Show that the matrix representations of Jz, Jy, and J obey the commutation relation [Jz, Jy] = ihJ.