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][4]

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 discovery of the Raman effect.^[5]

Equation

The Kramers–Heisenberg (KH) formula for second order processes is^[1][6]
${\displaystyle {\frac {d^{2}\sigma }{d\Omega _{k^{\prime }}d(\hbar \omega _{k}^{\prime })}}={\frac {\omega _{k}^{\prime }}{\omega _{k}}}\sum _{|f\rangle }\left|\sum _{|n\rangle }{\frac {\langle f|T^{\dagger }|n\rangle \langle n|T|i\rangle }{E_{i}-E_{n}+\hbar \omega _{k}+i{\frac {\Gamma _{n}}{2}}}}\right|^{2}\delta (E_{i}-E_{f}+\hbar \omega _{k}-\hbar \omega _{k}^{\prime })}$

It represents the probability of the emission of photons of energy ${\displaystyle \hbar \omega _{k}^{\prime }}$ in the solid angle ${\displaystyle d\Omega _{k^{\prime }}}$ (centered 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 intrinsic linewidth of the intermediate state.

References