Rotations of spins and of wavefunctions

[wondering12]

This is a question regarding the intrinsic angular momentum S of a particle of spin 1.
Assuming S = s(s+1)I = 2I and I is the identity operator. In our case s = 1.
Let |z> be a ket of norm 1 such that Sz |z> = 0, and let |x> and |y> be the ket vectors
obtained from it by rotations of + 1/2 Pi about the y-axis and − 1/2 Pi about the x-axis
respectively. Prove the following relations, as well as those resulting from circular
permutation of x, y, and z:
Sx |x> = 0 , Sx |y> = i|z> , Sxˆ2 |y> = |y> , Sx |z> = −i|y> , Sx2|z> = |z>
Use these to show that |x>, |y>, |y> form an orthonormal basis and that the matrices
representing Sx , Sy and Sz in that basis are:
Sx = \begin{bmatrix}
0 & 0 & 0 \\
0& 0 & -i\\
0&i &0
\end{bmatrix}
Sy = \begin{bmatrix}
0 & 0 & i \\
0& 0 & 0\\
-i&0 &0
\end{bmatrix}
Sz = \begin{bmatrix}
0 & -i & 0 \\
i& 0 & 0\\
0&0 &0
\end{bmatrix}
Show that the eigenvalues of Sz are the same as those of Jz , where Jz is the matrix
representation of the z component of the angular momentum in the |j m> basis, for
j = 1. Then find the most general unitary matrix that transforms
Sz into Jz : U Sz U† = Jz .Choose the arbitrary parameters in U so that it also
transforms Sx into Jx and Sy into Jy .
This problem can be found in Quantum Mechanics by Albert Messiah volume 1 chapter 8 problem 10 part ii and iii.
see attachment.
My humble attempt at the problem,
Used a rotation 3-d matrix for x,y,z or:
Rx = \begin{bmatrix}
1 & 0 & 0 \\
0& cos(a) & -sin(a)\\
0&sin(a) &cos(a)
\end{bmatrix}
Ry = \begin{bmatrix}
cos(a) & 0 & sin(a) \\
0& 1 & 0\\
-sin(a)&0 &cos(a)
\end{bmatrix}
Rz = \begin{bmatrix}
cos(a) & -sin(a & 0 \\
sin(a) &cos(a) & 0\\
0&0 &1
\end{bmatrix}
After that substituting angle for x and y does not give me any meaning to go further. Probably I am starting in a wrong way. Help appreciated.

 

[vela]

Those rotation matrices aren't the correct ones for this problem. You need the ones that work on the angular momentum space.

 

[DrClaude]

Those rotation matrices aren't the correct ones for this problem.

Why not? I seem to get the right answer by the unitary transform $S_x = R_y S_z R_y^{\dagger}$.

 

[blue_leaf77]

I think one could also do this problem matrix-less and hence saving lots of space, instead only works with ket notations. For example for the first task to prove $S_x|x\rangle = 0$, one can use the substitution $|x\rangle = \exp(-iS_y \pi/2) |z\rangle$. Have tried following this path, unfortunately stuck with the involved algebra. By the way the OP seems to be long absent anyway.