Excercise on distinguishable particles interacting with Hamiltonian

[venetiano77]

Please, help me with this problem!

Two distinguishable particles of spin 1/2 interact with Hamiltonian
H=A*S1,z*S2,x
with A a positive constant. S1,z and S2,x are the operators related to the z-component of the spin of the first particle and to the x-component of the spin of the second particle, respectively.
If at t=0 the system is in its simultaneous eigenket of S1,z for the eigenvalue h/4pi (ħ/2) and of S2,x for the eigenvalue h/4pi (ħ/2), FIND THE STATE OF THE SYSTEM AT THE TIME t

Attempt of solution:
Firstly, one should use the formulas for temporal evolution. Please, help me!

 

[TSny]

If at t=0 the system is in its simultaneous eigenket of S1,z for the eigenvalue h/4pi (ħ/2) and of S2,x for the eigenvalue h/4pi (ħ/2), FIND THE STATE OF THE SYSTEM AT THE TIME t

Note that the problem in the attachment states that the initial state is an eigenket for S2,z rather than S2,x.