C: Quantization of the electromagnetic field

Abstract

The cross-section for either scattering or absorption is evaluated from time-dependent perturbation theory, as outlined in Appendix A. In any perturbation problem it is of course first necessary to specify completely the non-interacting Hamiltonian, H 0 , of the system, before the effect of the perturbing Hamiltonian, H I , may be calculated. For the scattering or absorption of an X-ray this amounts to establishing a quantum mechanical description of both the electromagnetic field and the sample. The former may well be unfamiliar to many readers and here we explain briefly how this is achieved. The starting point in quantizing the electromagnetic field is the classical expression for its energy in terms of the electric and magnetic fields, both of which may be derived from the vector potential A (Appendix B). When seeking a quantum mechanical description of the electromagnetic field it would therefore seem natural to focus on A. Indeed quantizing the electromagnetic field amounts to quantizing the vector potential. It also transpires that the Hamiltonian, H I , that describes the interaction of the X-ray and the sample, is a simple function of A. As a consequence the matrix elements of H I that enter into the perturbation theory may be calculated readily, and in the last section we work through the example of the Thomson cross-section. Classical energy density of the radiation field The total energy of the electromagnetic field in free space is E rad = 1 2 V 0 E 2 + μ 0 H 2 dV = V 0 E 2 dV Here it is assumed that the field is confined to some volume V, and 0 E 2 =μ 0 H 2 , where the brackets · · · · indicate a temporal average. The E field is related to vector potential A through E = − ∂A ∂t Elements of Modern X-ray Physics, Second Edition. Jens Als-Nielsen and Des McMorrow