Consider a system whose state is given in terms of a complete and orthonormal set of five vectors ∣ϕ1⟩, ∣ϕ2⟩, ∣ϕ3⟩, ∣ϕ4⟩, ∣ϕ5⟩ as follows:
∣ψ⟩=191∣ϕ1⟩+192∣ϕ2⟩+192∣ϕ3⟩+193∣ϕ4⟩+195∣ϕ5⟩
where ∣ϕn⟩ are eigenstates to the system's Hamiltonian, H^∣ϕn⟩=nε0∣ϕn⟩ with n=1,2,3,4,5, and where ε0 has the dimensions of energy.
(a) If the energy is measured on a large number of identical systems that are all initially in the same state ∣ψ⟩, what values would one obtain and with what probabilities?
(b) Find the average energy of one such system.