Alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the Hamiltonian, just as the corresponding Poisson brackets in classical theory must vanish in order to generate the correct Hamilton equations. The formal solution of the field equation is: a k λ λ ( t ) = a k λ λ ( 0 ) e − − i ω ω k t + i e 2 π π ℏ ℏ ω ω k V ∫ ∫ 0 t d t ′ e k λ λ ⋅ ⋅ x ˙ ˙ ( t ′ ) e i ω ω k ( t ′ − − t ) and therefore the equation for ȧ k λ may be written: x ¨ ¨ + ω ω 0 2 x = e m E 0 ( t ) + e m E R R ( t ) where E 0 ( t ) = i ∑ ∑ k λ λ 2 π π ℏ ℏ ω ω k V [ a k λ λ ( 0 ) e − − i ω ω k t − − a k λ λ † † ( 0 ) e i ω ω k t ] e k λ λ and E R R ( t ) = − − 4 π π e V ∑ ∑ k λ λ ∫ ∫ 0 t d t ′ [ e k λ λ ⋅ ⋅ x ˙ ˙ ( t ′ ) ] cos ⁡ ⁡ ω ω k ( t ′ − − t )