Consider a system whose state is given in terms of an orthonormal set of three vectors: ∣ϕ1⟩, ∣ϕ2⟩, ∣ϕ3⟩ as
∣ψ⟩=33∣ϕ1⟩+32∣ϕ2⟩+32∣ϕ3⟩
(a) Verify that ∣ψ⟩ is normalized. Then, calculate the probability of finding the system in any one of the states ∣ϕ1⟩, ∣ϕ2⟩, and ∣ϕ3⟩. Verify that the total probability is equal to one.
(b) Consider now an ensemble of 810 identical systems, each one of them in the state ∣ψ⟩. If measurements are done on all of them, how many systems will be found in each of the states ∣ϕ1⟩, ∣ϕ2⟩, and ∣ϕ3⟩?