For the time-independent operators X and P, ∂ A /∂ t = 0 so the Heisenberg equation above reduces to: i ℏ ℏ d A d t = [ A , H ] = A H − − H A , where the square brackets [, ] denote the commutator. For a Hamiltonian which is p 2 / 2 m + V ( x ) , the X and P operators satisfy: d X d t = P m , d P d t = − − ∇ ∇ V , where the first is classically the velocity, and second is classically the force, or potential gradient. These reproduce Hamilton's form of Newton's laws of motion. In the Heisenberg picture, the X and P operators satisfy the classical equations of motion. You can take the expectation value of both sides of the equation to see that, in any state | ψ ⟩: d d t ⟨ ⟨ X ⟩ ⟩ = d d t ⟨ ⟨ ψ ψ | X | ψ ψ ⟩ ⟩ = 1 m ⟨ ⟨ ψ ψ | P | ψ ψ ⟩ ⟩ = 1 m ⟨ ⟨ P ⟩ ⟩ d d t ⟨ ⟨ P ⟩ ⟩ = d d t ⟨ ⟨ ψ ψ | P | ψ ψ ⟩ ⟩ = ⟨ ⟨ ψ ψ | ( − − ∇ ∇ V ) | ψ ψ ⟩ ⟩ = − − ⟨ ⟨ ∇ ∇ V ⟩ ⟩ .