The new eigenstates | ψ ψ + ⟩ ⟩ and | ψ ψ − − ⟩ ⟩ can be found by solving the eigenvalue equations H ′ | ψ ψ + ⟩ ⟩ = E + | ψ ψ + ⟩ ⟩ and H ′ | ψ ψ − − ⟩ ⟩ = E − − | ψ ψ − − ⟩ ⟩ . From simple calculations it can be shown that