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]
$\frac{d^2 \sigma}{d\Omega_{k^\prime}d(\hbar \omega_k^\prime)}=\frac{\omega_k^\prime}{\omega_k}\sum_{|f\rangle}\left | \sum_{|n\rangle} \frac{\langle f | T^\dagger | n \rangle \langle n | T | i \rangle}{E_i - E_n + \hbar \omega_k + i \frac{\Gamma_n}{2}}\right |^2 \delta (E_i - E_f + \hbar \omega_k - \hbar \omega_k^\prime)$

It represents the probability of the emission of photons of energy $\hbar \omega_k^\prime$ in the solid angle $d\Omega_{k^\prime}$ (centred in the $k^\prime$ direction), after the excitation of the system with photons of energy $\hbar \omega_k$. $|i\rangle, |n\rangle, |f\rangle$ are the initial, intermediate and final states of the system with energy $E_i , E_n , E_f$ respectively; the delta function ensures the energy conservation during the whole process. $T$ is the relevant transition operator. $\Gamma_n$ is the instrinsic linewidth of the intermediate state.

References

1. Kramers, H. A.; Heisenberg, W. (Feb 1925). "Über die Streuung von Strahlung durch Atome". Z. Phys. 31 (1): 681–708. Bibcode:1925ZPhy...31..681K. doi:10.1007/BF02980624.
2. Dirac., P. A. M. (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proc. Roy. Soc. Lond. A 114 (769): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039.
3. Dirac., P. A. M. (1927). "The Quantum Theory of Dispersion". Proc. Roy. Soc. Lond. A 114 (769): 710–728. Bibcode:1927RSPSA.114..710D. doi:10.1098/rspa.1927.0071.
4. Breit, G. (1932). "Quantum Theory of Dispersion". Rev. Mod. Phys. 4 (3): 504–576. Bibcode:1932RvMP....4..504B. doi:10.1103/RevModPhys.4.504.
5. J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley (1967), page 56.