Dirac delta function matrix elements

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... 

For further derivations we will need the matrix element of the Dirac delta function, 8(r, , ). Using the following representation of the delta function,... 

The matrix elements of the total correlation function, h, are related to all pairs of atoms. The intramolecular correlation function, to, introduced here represents the shape of the molecule. 8(r) in the diagonal element is the Dirac delta function and represents the position of an atom. The function appearing in the off-diagonal element is given by,... 

Here we are taking into account the symmetry of the functions using the Fourier reducing property of the Dirac delta functions and are calculating the matrix elements using zero-order perturbation theory. 

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... 

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