# Kramers–Heisenberg formula

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