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 discovery of the Raman effect.[4]
Equation
The Kramers–Heisenberg (KH) formula for second order processes is[1][5]
\( {\displaystyle {\frac {d^{2}\sigma }{d\Omega _{k^{\prime }}d(\hbar \omega _{k}^{\prime })}}={\frac {\omega _{k}^{\prime }}{\omega _{k}}}\sum _{|f\rangle }\left|\sum _{|n\rangle }{\frac {\langle f|T^{\dagger }|n\rangle \langle n|T|i\rangle }{E_{i}-E_{n}+\hbar \omega _{k}+i{\frac {\Gamma _{n}}{2}}}}\right|^{2}\delta (E_{i}-E_{f}+\hbar \omega _{k}-\hbar \omega _{k}^{\prime })} \)
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 E i , E n , E f {\displaystyle E_{i},E_{n},E_{f}} respectively; the delta function ensures the energy conservation during the whole process. T is the relevant transition operator. \( {\displaystyle \Gamma _{n}} \) is the intrinsic linewidth of the intermediate state.
References
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.
Breit, G. (1932). "Quantum Theory of Dispersion". Rev. Mod. Phys. 4 (3): 504–576. Bibcode:1932RvMP....4..504B. doi:10.1103/RevModPhys.4.504.
Sakurai, J. J. (1967). Advanced Quantum Mechanics. Reading, Mass.: Addison-Wesley. p. 56. ISBN 978-0201067101. OCLC 869733.
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
