Two particles are under the influence of an interaction with a Hamiltonian that is proportional to \(\hat{S}_{z}\):

\[\hat{H} = \omega_0 \hat{S}_{z}\]

At \(t=0\):

1. one is in the state \(\vert + \rangle _{x}\).
2. the other particle is in the state \(\vert + \rangle\)

For each particle:

1. What values of energy could you measure?
2. What are the energy eigenstates?

3. What state is each particle in at a later time t?

   
   

4. What are the probabilities for each energy measurement?

5. What is the probability that you would measure \(S_x = {\hbar \over 2}\) state at time \(t\)? Does this probability change with time?

   
   

6. What is the probability that you would measure \(S_z = {\hbar \over 2}\) at time \(t\)? Does this probability change with time?

   
   

7. Given a Hamiltonian, how would you determine which states are stationary states (states where no probabilities change with time)? Under what circumstances do measurement probabilities change with time?