Since the Hamiltonian commutes with the translation operator when the translation is invariant [ H ^ ^ , T ^ ^ ( x ) ] = 0 , it also commutes with the infinitesimal translation operator [ H ^ ^ , 1 − − i x ⋅ ⋅ p ^ ^ N ℏ ℏ ] = 0 ⇒ ⇒ [ H ^ ^ , p ^ ^ ] = 0 ⇒ ⇒ d d t ⟨ ⟨ p ^ ^ ⟩ ⟩ = i ℏ ℏ [ H ^ ^ , p ^ ^ ] = 0 In summary, whenever the Hamiltonian for a system remains invariant under continuous translation, then the system has conservation of momentum, meaning that the expectation value of the momentum operator remains constant. This is an example of Noether's theorem.