For the dipole oscillator under consideration it can be assumed that the radiative damping rate is small compared with the natural oscillation frequency, i.e., τω 0 ≪ 1. Then the integrand above is sharply peaked at ω = ω 0 and: [ z ( t ) , p z ( t ) ] ≈ ≈ 2 i ℏ ℏ e 2 3 π π m c 3 ω ω 0 3 ∫ ∫ − − ∞ ∞ ∞ ∞ d x x 2 + τ τ 2 ω ω 0 6 = ( 2 i ℏ ℏ e 2 ω ω 0 3 3 π π m c 3 ) ( π π τ τ ω ω 0 3 ) = i ℏ ℏ the necessity of the vacuum field can also be appreciated by making the small damping approximation in x ¨ ¨ + ω ω 0 2 x − − τ τ x . . . = e m E 0 ( t ) x ¨ ¨ ≈ ≈ − − ω ω 0 2 x ( t ) x . . . ≈ ≈ − − ω ω 0 2 x ˙ ˙ and x ¨ ¨ + τ τ ω ω 0 2 x ˙ ˙ + ω ω 0 2 x ≈ ≈ e m E 0 ( t )