Students do calculations for time evolution for spin-1.
Spin-1 Time Evolution
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 \(\hat{H} = \omega_o \hat{S}_x\).
- Identify the energy eigenstates and their corresponding energy eigenvalues.
- Use them to write each state at a later instant in time.
- Represent each state using Arms as complex numbers.
For \(|\psi_C\rangle\):
List the possible values of a measurement of \(S_z\) and calculate the corresponding probabilities.
List the possible values of a measurement of \(S_x\) and calculate the corresponding probabilities.
- Graph any probabilities that depend on time.