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, based on the correspondence principle applied to the classical dispersion formula for light. The quantum mechanical derivation was given by Paul Dirac in 1927.

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.

Equation

The Kramers–Heisenberg (KH) formula for second order processes is
d 2 σ d Ω k ′ d ( ℏ ω k ′ ) = ω k ′ ω k ∑ | f ⟩ | ∑ | n ⟩ ⟨ f | T † | n ⟩ ⟨ n | T | i ⟩ E i − E n + ℏ ω k + i Γ n 2 | 2 δ ( E i − E f + ℏ ω k − ℏ ω k ′ ) {\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 })} ${\frac {d^{2}\sigma }{d\Omega _{k^{\prime }}d(\hbar \omega _{k}^{\prime })}}={\frac {\omega _{k}^{\prime }}{\omega _{k}}}\sum _{|f\rangle }\left|\sum _{|n\rangle }{\frac {\langle f|T^{\dagger }|n\rangle \langle n|T|i\rangle }{E_{i}-E_{n}+\hbar \omega _{k}+i{\frac {\Gamma _{n}}{2}}}}\right|^{2}\delta (E_{i}-E_{f}+\hbar \omega _{k}-\hbar \omega _{k}^{\prime })$

It represents the probability of the emission of photons of energy ℏ ω k ′ {\displaystyle \hbar \omega _{k}^{\prime }} $\hbar \omega _{k}^{\prime }$ in the solid angle d Ω k ′ {\displaystyle d\Omega _{k^{\prime }}} $d\Omega _{k^{\prime }}$ (centered in the k ′ {\displaystyle k^{\prime }} $k^{\prime }$ direction), after the excitation of the system with photons of energy ℏ ω k {\displaystyle \hbar \omega _{k}} $\hbar \omega _{k}$ . | i ⟩ , | n ⟩ , | f ⟩ {\displaystyle |i\rangle ,|n\rangle ,|f\rangle } $|i\rangle ,|n\rangle ,|f\rangle$ are the initial, intermediate and final states of the system with energy E i , E n , E f {\displaystyle E_{i},E_{n},E_{f}} $E_{i},E_{n},E_{f}$ respectively; the delta function ensures the energy conservation during the whole process. T {\displaystyle T} $T$ is the relevant transition operator. Γ n {\displaystyle \Gamma _{n}} $\Gamma _{n}$ is the intrinsic linewidth of the intermediate state.

References

^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ Sakurai, J. J. (1967). Advanced Quantum Mechanics. Reading, Mass.: Addison-Wesley. p. 56. ISBN 978-0201067101. OCLC 869733.

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