Physics: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][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]
d2σdΩkd(ωk)=ωkωk|f||nf|T|nn|T|iEiEn+ωk+iΓn2|2δ(EiEf+ωkωk)

It represents the probability of the emission of photons of energy ωk in the solid angle dΩk (centered in the k direction), after the excitation of the system with photons of energy ωk. |i,|n,|f are the initial, intermediate and final states of the system with energy Ei,En,Ef respectively; the delta function ensures the energy conservation during the whole process. T is the relevant transition operator. Γn is the intrinsic linewidth of the intermediate state.

References

  1. Jump up to: 1.0 1.1 Kramers, H. A.; Heisenberg, W. (Feb 1925). "Über die Streuung von Strahlung durch Atome". Z. Phys. 31 (1): 681–708. doi:10.1007/BF02980624. Bibcode1925ZPhy...31..681K. 
  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. doi:10.1098/rspa.1927.0039. Bibcode1927RSPSA.114..243D. 
  3. Dirac, P. A. M. (1927). "The Quantum Theory of Dispersion". Proc. R. Soc. Lond. A 114 (769): 710–728. doi:10.1098/rspa.1927.0071. Bibcode1927RSPSA.114..710D. 
  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. https://link.aps.org/doi/10.1103/PhysRevA.100.053413. 
  5. Breit, G. (1932). "Quantum Theory of Dispersion". Rev. Mod. Phys. 4 (3): 504–576. doi:10.1103/RevModPhys.4.504. Bibcode1932RvMP....4..504B. 
  6. Sakurai, J. J. (1967). Advanced Quantum Mechanics. Reading, Mass.: Addison-Wesley. p. 56. ISBN 978-0201067101. OCLC 869733.