Kramers–Heisenberg formula

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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]


The Kramers-Heisenberg (KH) formula for second order processes is [1][5]

It represents the probability of the emission of photons of energy in the solid angle (centred in the direction), after the excitation of the system with photons of energy . are the initial, intermediate and final states of the system with energy respectively; the delta function ensures the energy conservation during the whole process. is the relevant transition operator. is the instrinsic linewidth of the intermediate state.


  1. ^ a b 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.