Dirac delta function energy-conserving

This is an application of Fermi s golden rule. The first term is the square of the matrix element of the perturbation, which appears in all versions of perturbation theory. In the second term 8(x) denotes the Dirac delta function. For a full treatment of this function we refer to the literature [2]. Here we note that S(x) is defined such that S(x) = 0 for x 7 0 at the origin S(x) is singular such that / ( r) dx — 1. The term 8 (Ef — Ei) ensures energy conservation since it vanishes unless... 

The transition energies are not sharp but rather have finite widths due to thermal vibrations in the solid. Thus, the wave functions and the density of states can be treated as functions of energy. The wave functions can be normalized with respect to energy and a Dirac delta function used for the density of final states to insure conservation of energy. Then Einstein s A and B coefficients can be used to relate the transition matrix elements to experimentally measurable quantities such as oscillator strengths and luminescence lifetimes. For electric dipole-dipole interaction the energy transfer rate becomes... 

The procedure for obtaining the double-differential scattering cross-section is to now sum over possible final states f) of the scatterer, subject to the condition that the energy of the combined system of radiation -i- target is conserved. The latter is accomplished by attaching the Dirac delta function, d( — ), to each term in the sum. Finally, we can introduce... 

In (10.3) i) and (f denote the initial and final states of the radiative process whiled,. . are the so-called virtual intermediate states that are not physical states that take part in the process.This is indicated by the Dirac delta function that reflects the energy-conservation law, and it covers the energies of the initial and final states between which the physical transition occurs. 

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