Let |a> and |a> be eigenstates of a Hermitian operator A with eigenvalues and , respectively

Let |a’> and |a”> be eigenstates of a Hermitian operator A with eigenvalues α’ and α”, respectively (α’ ≠ α”). The Hamiltonian operator is given by H = | α’> δ < α”| + | α”>δ< α’|, where is just a real number.

a. Clearly, | α’> and | α”> are not eigenstates of the Hamiltonian. Write down the eigenstates of the Hamiltonian. What are their energy eigenvalues?

b. Suppose the system is known to be in state |a’> at t = 0. Write down the state vector in the Schrödinger picture for t > 0.

c. What is the probability for finding the system in | α”) for t > 0 if the system is known to be in state |α” > at ‘= 0?

d. Can you think of a physical situation corresponding to this problem?