For an electron with initial velocity v 0 x and initial position x 0 x, acted upon

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For an electron with initial velocity v0x̂ and initial position x0x̂, acted upon by an electric field E = x̂E cos( kx – ωt), show that its velocity is given by

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Using a perturbation approach, in which to lowest order E = 0, show that

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Notice that the velocity perturbation will be large only when v0 is close to the phase velocity ω/k.

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