Kramers-Heisenberg formula

The Kramers-Heisenberg formula or Kramers-Heisenberg dispersion formula is a mathematical expression for describing the cross-section in the scattering of photons at an electron that binds to an atom is bound. Hendrik Kramers and Werner Heisenberg applied it in 1925, based on the correspondence principle , to the classic dispersion formula for light. The quantum mechanical derivation was derived by Paul Dirac in 1927, before quantum mechanics was established .

The Kramers-Heisenberg formula was a significant achievement at the time of publication because it explains, among other things, the notion of “negative absorption” ( stimulated emission ), the Thomas-Reiche-Kuhn sum rule and the inelastic scattering - in which the energy of the scattered photon is larger or smaller than that of the incident photon . This also establishes the connection to the Raman effect . ${\displaystyle \hbar \omega _{k}^{\prime }}$ ${\displaystyle \hbar \omega _{k}}$

formula

The Kramers-Heisenberg formula for second order processes is

${\displaystyle {\frac {d^{2}\sigma }{d\Omega _{k^{\prime }}d(\hbar \omega _{k}^{\prime })}}={\frac {\omega _{k}^{\prime }}{\omega _{k}}}\sum _{|f\rangle }\left|\sum _{|n\rangle }{\frac {\langle f|T^{\dagger }|n\rangle \langle n|T|i\rangle }{E_{i}-E_{n}+\hbar \omega _{k}+i{\frac {\Gamma _{n}}{2}}}}\right|^{2}\delta (E_{i}-E_{f}+\hbar \omega _{k}-\hbar \omega _{k}^{\prime })}$

It makes a statement about the probability of the emission of photons of energy in the solid angle (centered in the -direction), after the excitation of the system with photons of energy . Here is the initial state, the intermediate state and the final state of the system with the energy . As in Fermi's golden rule , the delta function ensures that energy is conserved throughout the entire spreading process. is the transition matrix or transition operator. is the intrinsic line width (natural line width) of the intermediate state. ${\displaystyle \hbar \omega _{k}^{\prime }}$ ${\displaystyle d\Omega _{k^{\prime }}}$ ${\displaystyle k^{\prime }}$ ${\displaystyle \hbar \omega _{k}}$ ${\displaystyle |i\rangle }$ ${\displaystyle |n\rangle }$ ${\displaystyle |f\rangle }$ ${\displaystyle E_{i},E_{n},E_{f}}$ ${\displaystyle \delta (E)}$ ${\displaystyle T}$ ${\displaystyle \Gamma _{n}}$

Individual evidence

1. ↑ ^{a b} H. A. Kramers , W. Heisenberg: About the scattering of radiation by atoms . In: Z. Phys. . 31, No. 1, Feb 1925, pp. 681-708. bibcode : 1925ZPhy ... 31..681K . doi : 10.1007 / BF02980624 .
2. ^ PAM Dirac. : The Quantum Theory of the Emission and Absorption of Radiation . In: Proc. Roy. Soc. Lond. A . 114, No. 769, 1927, pp. 243-265. bibcode : 1927RSPSA.114..243D . doi : 10.1098 / rspa.1927.0039 .
3. ^ PAM Dirac. : The Quantum Theory of Dispersion . In: Proc. Roy. Soc. Lond. A . 114, No. 769, 1927, pp. 710-728. bibcode : 1927RSPSA.114..710D . doi : 10.1098 / rspa.1927.0071 .
4. ↑ G. Breit: Quantum Theory of Dispersion . In: Rev. Mod. Phys. . 4, No. 3, 1932, pp. 504-576. bibcode : 1932RvMP .... 4..504B . doi : 10.1103 / RevModPhys.4.504 .
5. ↑ JJ Sakurai: Advanced Quantum Mechanics. Addison-Wesley, 1967, p. 56.