Due to the orthonormality of the eigenstates where ⟨ ⟨ ψ ψ m 0 | ψ ψ n 0 ⟩ ⟩ = δ δ n m , we can easily observe that the off-diagonal matrix elements are zero: ⟨ ⟨ ψ ψ m 0 | H ^ ^ ψ ψ 0 | ψ ψ n 0 ⟩ ⟩ = E m δ δ n m . Physically, the matrix elements here represent the transition probability amplitude for a particle in eignestate n to transition to eigenstate m due to the interaction governed by the unperturbed Hamiltonian. Since in the unperturbed Hamiltonian system these eigenstates are uncoupled, H ^ ^ ψ ψ ( 0 ) will not cause any such transitions to occur. Mathematically, since all the off-diagonal elements are 0, the matrix elements calculated by the above expectation value will always return 0.