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]
The Kramers–Heisenberg (KH) formula for second order processes is[1] [6]
| d2\sigma | = | |||||||||||
|
| |||||||
| \omegak |
\sum|f\rangle\left|\sum|n\rangle
| \langlef|T\dagger|n\rangle\langlen|T|i\rangle | |||||||||
|
\right|2\delta(Ei-Ef+\hbar\omegak-\hbar
| \prime) | |
| \omega | |
| k |
It represents the probability of the emission of photons of energy
\hbar
| \prime | |
| \omega | |
| k |
| d\Omega | |
| k\prime |
k\prime
\hbar\omegak
|i\rangle,|n\rangle,|f\rangle
Ei,En,Ef
T
\Gamman
. J. J. Sakurai. Advanced Quantum Mechanics. 1967. Addison-Wesley. 978-0201067101. Reading, Mass.. 869733. 56.