(Solved): Let la) and la) be eigenstates of an observable with different eigenvalues. Consider the ...

Let la?) and la?) be eigenstates of an observable  with different eigenvalues. Consider the Hamiltonian ? = 6(la?)(a? + a?)(??1), (3) where 8 is a real number. (a) Are |a?) (i = 1,2) eigenstates of this Hamiltonian? If so, provide proof. If not, write down the eigenstates of the Hamiltonian and their corresponding energy values. (b) Suppose the system is in state |a?) at time t = 0. Write down the state vector in the Schrödinger picture for t > 0. (c) What is the probability for finding the system in state |??) at some time t > 0 if the system is known to be in state |??) at t = 0? (d) Can you think of a physical situation corresponding to this problem?