Kramers–Heisenberg formula

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The Kramers-Heisenberg dispersion formula is an expression for the cross section for scattering of a photon by an atomic electron. It was derived before the advent of quantum mechanics by Hendrik Kramers and Werner Heisenberg in 1925,[1] based on the correspondence principle applied to the classical dispersion formula for light. The quantum mechanical derivation was given by Paul Dirac in 1927.[2][3]

The Kramers–Heisenberg formula was an important achievement when it was published, explaining the notion of "negative absorption" (stimulated emission), the Thomas-Reiche-Kuhn sum rule, and inelastic scattering - where the energy of the scattered photon may be larger or smaller than that of the incident photon - thereby anticipating the Raman effect.[4]

Equation

The Kramers-Heisenberg (KH) formula for second order processes is [1][5]

It represents the probability of the emission of photons of energy in the solid angle (centred in the direction), after the excitation of the system with photons of energy . are the initial, intermediate and final states of the system with energy respectively; the delta function ensures the energy conservation during the whole process. is the relevant transition operator. is the instrinsic linewidth of the intermediate state.

References

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  5. J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley (1967), page 56.


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