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

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. R. 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. R. Soc. Lond. A. 114 (769): 710–728. Bibcode:1927RSPSA.114..710D. doi:10.1098/rspa.1927.0071.
4. Forbes, Kayn A.; Salam, A. (2019-11-21). "Kramers-Heisenberg dispersion formula for scattering of twisted light". Physical Review A. 100 (5): 053413. doi:10.1103/PhysRevA.100.053413. S2CID 214221551.
5. Breit, G. (1932). "Quantum Theory of Dispersion". Rev. Mod. Phys. 4 (3): 504–576. Bibcode:1932RvMP....4..504B. doi:10.1103/RevModPhys.4.504. S2CID 4133208.
6. Sakurai, J. J. (1967). Advanced Quantum Mechanics. Reading, Mass.: Addison-Wesley. p. 56. ISBN 978-0201067101. OCLC 869733.