Consider a system whose initial state at t=0 is given in terms of a complete and orthonormal set of four vectors ∣ϕ1⟩, ∣ϕ2⟩, ∣ϕ3⟩, ∣ϕ4⟩ as follows:
∣ψ(0)⟩=12A∣ϕ1⟩+61∣ϕ2⟩+122∣ϕ3⟩+21∣ϕ4⟩,
where A is a real constant.
(a) Find A so that ∣ψ(0)⟩ is normalized.
(b) If the energies corresponding to ∣ϕ1⟩, ∣ϕ2⟩, ∣ϕ3⟩, ∣ϕ4⟩ are given by E1, E2, E3, and E4, respectively, write down the state of the system ∣ψ(t)⟩ at any later time t.
(c) Determine the probability of finding the system at a time t in the state ∣ϕ2⟩.