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 2 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.
↑ 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.
↑ 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.
↑ Breit, G. (1932). "Quantum Theory of Dispersion". Rev. Mod. Phys. 4 (3): 504–576. Bibcode:1932RvMP....4..504B. doi:10.1103/RevModPhys.4.504.
↑ J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley (1967), page 56.

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