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 −.

Question

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?

Answer 1

We have:

|+> = 1/sqrt(2)[|0> + |1>]

|-> = 1/sqrt(2)[|0> - |1>]

M = 3|+><+| - 2 |-><-|

Insert the above expressions and expand, or just evaluate the matrix elements. E.g.:

<0|M|0> =

3<0|+><+|0> - 2 <0|-><-|0> =

3/2 - 1 = 1/2

Answer 2

I was able to do part 1 and part 2. I am stuck on
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?