Probability of Measuring Spin-1 Particle in State |1,-1> at Time t?

[cragar]

Homework Statement

A spin-1 particle is placed in a constatn external B field
with $B_0$ in the x direction. the intial spin of the particle
is spin up in the z direction.
Take the spin Hamiltonian to be $H=\omega_0 S_x$
determine the probability that the particle is in the state |1,-1> at time t.

The Attempt at a Solution

Would I start with using the $S_x$ matrix in the z basis and then
set this equal to spin 1 and then find the eigenvector for this equation and that
will give me the amplitudes for spin-1, spin-0 and spin minus 1 and these will be amplitudes
in the x basis then I will just time evovle that intial state.
So I will have $S_xQ=1Q$
where Q is a generic column vector and 1 is the eigenvalue because we are
spin up in the z direction. Or do I need everything in the x basis.
But it seems like I first have to work in the z basis because we know it is spin up in the z direction.

 

[mfb]

If you can express "spin in z-direction" in the basis of spin in x-direction, you don't have to work with a basis of z-spin.
The basis is just a mathematical tool - you can choose any basis you like (even y-direction or weird linear combinations of those, but that would be impractical).

 

[cragar]

so i just need to write spin up in the z in the x basis.
I am not really sure how to do that with spin-1.
Do I use a rotation matrix.