SU(2) generators in j=1, and pauli uncertainty

[Kontilera]

Main Question or Discussion Point

Hello!
I have two questions regarding QM.

First, I tried to rotate J_z into J_y for the j=1 representation by the transformation rule for matrices.. I took my rotationvector to be (1,0,0) and rotated about 3/4 pi radians which I thought would give me S_y. Instead I got S_z again!
Wolfram alpha says that this transformation always gives S_z, no matter what angle.
My idea was: (exp(-i*pauli_x*x*pi/2)^(-1)) S_z exp(-i*pauli_x*x*pi/2)
and in wolfram alpha I wrote:

(exp(-i*{{0,1,0},{1,0,1},{0,1,0}}*x*pi/2)^(-1)){{1,0,0},{0,0,0},{0,0,-1}} exp(-i*{{0,1,0},{1,0,1},{0,1,0}}*x*pi/2)

and gets {{1,0,0},{0,0,0},{0,0,-1}} with no dependence of x.
Where does my logic break down?


Second, I just want a confirmation that I understood the exlusion principle correct.
For a three (identical)particle system we get a six dimensional subspace of the hilbertspace that corresponds to the same physical system,
|x>|y>|z>, |y>|x>|z>, ... etc.

But nature does only use the subspaces , of this six dimensional subspace, which are totally antisymmetric or symmetric. This means that we have no ambiguity in choosing the state vector, if we know wether our particles are fermions or boson.

Does this seem right? :-)

 

[Bill_K]

rotated about 3/4 pi radians which I thought would give me S_y

Don't you want to rotate through 90 degrees = π/2?

My idea was: (exp(-i*pauli_x*x*pi/2)^(-1)) S_z exp(-i*pauli_x*x*pi/2)

The rotation matrix is R(θ) = exp(iθJ_x) and to get S_y you might want to calculate R(π/2)S_z. If you calculate R(π/2)S_zR^-1(π/2) instead, you have not rotated, what you have done instead is change the basis. So now you have S_y in the new basis, in which it is again diagonal!