Question

Consider the observable M that corresponds to a measurement in the sign basis where the measurement value is 3 if the outcome is + and −2 if the outcome is −.
Write M in the standard basis.
If M is the Hamiltonian acting on a qubit which is in the state |ψ(0)⟩=|0⟩ at time 0, what is the state of the qubit at time t? Assume that you are working in units such that ℏ=1. You may use e and t in your answer.
Now you perform an X=(0110) measurement on the qubit at time 0. What is the expected value of your measurement?
What if you perform the above measurement at time t=4π3 instead? What is the expected value of your measurement?