This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts. For instance the Heisenberg equations for the coordinate x and the canonical momentum p = m ẋ + ⁠ e A / c ⁠ of the oscillator are: x ˙ ˙ = ( i ℏ ℏ ) − − 1 [ x . H ] = 1 m ( p − − e c A ) p ˙ ˙ = ( i ℏ ℏ ) − − 1 [ p . H ] = 1 2 ∇ ∇ ( p − − e c A ) 2 − − m ω ω 0 2 x ˙ ˙ = − − 1 m [ ( p − − e c A ) ⋅ ⋅ ∇ ∇ ] [ − − e c A ] − − 1 m ( p − − e c A ) × × ∇ ∇ × × [ − − e c A ] − − m ω ω 0 2 x ˙ ˙ = e c ( x ˙ ˙ ⋅ ⋅ ∇ ∇ ) A + e c x ˙ ˙ × × B − − m ω ω 0 2 x ˙ ˙ or: m x ¨ ¨ = p ˙ ˙ − − e c A ˙ ˙ = − − e c [ A ˙ ˙ − − ( x ˙ ˙ ⋅ ⋅ ∇ ∇ ) A ] + e c x ˙ ˙ × × B − − m ω ω 0 2 x = e E + e c x ˙ ˙ × × B − − m ω ω 0 2 x since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative A ˙ ˙ = ∂ ∂ A ∂ ∂ t + ( x ˙ ˙ ⋅ ⋅ ∇ ∇ ) A 3 .