3. Consider a system described by the Hamiltonian Ĥ = €(−i|0)(1| + i]1)(0]), where {[0), [1)} form an orthonormal basis of the considered Hilbert space and e is a real- valued constant with the dimension of energy. The eigenergies of Ĥ are ±¤ and the corresponding eigenstates are le) = = |0) +i|1) √2 |-€): = |0) - i|1) √2 (a) What are the probabilities to measure & and - if the system is in the state [0)? What is the value of (H)? (b) Find the state (t)) at an arbitrary time t, when the system is initially in the state ly(t = 0)) = [0). What is the probability to find the system in the state [0) as a function of time? How does (H) change with time? Hint: you need to solve the Schrödinger equation.

3. Consider a system described by the Hamiltonian Ĥ = €(−i|0)(1| + i]1)(0]), where {[0), [1)} form an orthonormal basis of the considered Hilbert space and e is a real- valued constant with the dimension of energy. The eigenergies of Ĥ are ±¤ and the corresponding eigenstates are le) = = |0) +i|1) √2 |-€): = |0) - i|1) √2 (a) What are the probabilities to measure & and - if the system is in the state [0)? What is the value of (H)? (b) Find the state (t)) at an arbitrary time t, when the system is initially in the state ly(t = 0)) = [0). What is the probability to find the system in the state [0) as a function of time? How does (H) change with time? Hint: you need to solve the Schrödinger equation.