Please help with the QM question below regarding Commuting Hermitian Operators.

Consider two Hermitian operators (2 and A that commute: [(2, A] = 0. (3) Using Dirac braket notation, show that there exists a common basis of eigenvectors for Q and A. (b) Explain why we can label the states of hydrogen in terms of their eigenvalues under the Hamiltonian, L2, and Lz. Why do we not also label the states by their p2 eigenvalues? (0) Imagine that we make a measurement of Q and nd the system to have a value ml. We then measure A and nd the system to have have value A1 for that observable. We now make another measurement of Q. \\Vhat is the probability to nd the measured value of Q to be L01 again? (Assume that the successive measurements are done immediately after each other in quick succession so that we do not need to timeevolve the states. What property of Q and A would we need for your answer to hold for arbitrary time intervals between measurements?)