6. A quantum system is in a state u). Suppose, in this state a physical quantity which corresponds to a Hermitian operator 2 has a definite value w. Also suppose that you perform a measurement of another physical quantity which corresponds to a Hermitian operator A. Do you expect that this measurement will always result in the same outcome (that is, A has a definite value) in case a) [N, A] = 0. b) [N, A] # 0. Explain your reasoning.

18. In differential form, what is the kinetic energy operator equal to? Does this depend on the problem that you are solving in quantum mechanics? What is the eigenvalue spectrum for this operator?

A qubit is in state |) = o|0) +₁|1) at time t = 0. It then evolves according to the Schrödinger equation with the Hamiltonian Ĥ defined by its action on the basis vectors: Ĥ0) = 0|0) and Ĥ|1) = E|1), where E is a constant with units of energy. a) Solve for the state of the qubit at time t. b) Find the probability to observe the qubit in state 0 at time t. Explain the result by referring to the way that the time-evolution transforms the Bloch sphere.

4. Consider a quantum 3-it initially in the state |v) e Cº. Suppose that the 3-it undergoes a reversible evolution associated with the unitary matrix A, immediately after which a measurement of the observable B is performed. What is (B) for this measurement if 0 0 -i A = (0 1 1 0 0 0 -i 0 0 |v) and B = |? (Express your result as an explicit real number.)