# 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

It represents the probability of the emission of photons of energy ${\displaystyle \hbar \omega _{k}^{\prime }}$ in the solid angle ${\displaystyle d\Omega _{k^{\prime }}}$ (centred in the ${\displaystyle k^{\prime }}$ direction), after the excitation of the system with photons of energy ${\displaystyle \hbar \omega _{k}}$. ${\displaystyle |i\rangle ,|n\rangle ,|f\rangle }$ are the initial, intermediate and ﬁnal states of the system with energy ${\displaystyle E_{i},E_{n},E_{f}}$ respectively; the delta function ensures the energy conservation during the whole process. ${\displaystyle T}$ is the relevant transition operator. ${\displaystyle \Gamma _{n}}$ 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.