Use a completeness relation to rewrite each of the states below in the \(z\)-direction.

• \(|\psi_A\rangle = |-1\rangle_x\)
• \(|\psi_B\rangle = \frac{1}{\sqrt{2}}|+1\rangle_z - \frac{1}{\sqrt{2}}|-1\rangle_z\)
• \(|\psi_C\rangle = |-1\rangle_z\)

Suppose the Hamiltonian is given by \(\hat{H} = \omega_o \hat{S}_x\).

1. Identify the energy eigenstates and their corresponding energy eigenvalues.
2. Use them to write each state at a later instant in time.
3. Represent each state using Arms as complex numbers.

For \(|\psi_C\rangle\):

1. List the possible values of a measurement of \(S_z\) and calculate the corresponding probabilities.

2. List the possible values of a measurement of \(S_x\) and calculate the corresponding probabilities.

3. Graph any probabilities that depend on time.