The external fields of Problem 13.2 are caused by a charge ze passing the origin in a straight-line path at speed v and impact parameter b. The fields are given by (11.152).
(a) Evaluate the Fourier transforms for the perpendicular and parallel components of the electric field at the origin and show that
E⊥(ω)=bvze(π2)1/2ξK1(ξ);E∥(ω)=−iγbvze(π2)1/2ξK0(ξ)
where ξ=ωb/γv, and Kν(ξ) is the modified Bessel function of the second kind and order ν. [See references to tables of Fourier transforms in Section 13.3.]
(b) Using the result of Problem 13.2, write down the energy transfer ΔE to a harmonically bound charged particle. From the limiting forms of the modified Bessel functions for small and large argument, show that your result agrees with the appropriate limit of T(b) in Problem 13.1 on the one hand and the arguments at the end of Section 13.1 on the adiabatic behavior for b≫γv/ω0 on the other.