An observable quantity Q will be a constant of motion if it commutes with the Hamiltonian, H, and it does not itself depend explicitly on time. This is because d d t ⟨ ⟨ ψ ψ | Q | ψ ψ ⟩ ⟩ = − − 1 i ℏ ℏ ⟨ ψ ψ | [ H , Q ] | ψ ψ ⟩ + ⟨ ψ ψ | d Q d t | ψ ψ ⟩ where [ H , Q ] = H Q − − Q H is the commutator relation.