|
|
| Line 1: |
Line 1: |
| The '''Kramers-Heisenberg dispersion formula''' is an expression for the [[cross section (physics)|cross section]] for [[scattering]] of a [[photon]] by an [[atom]]ic [[electron]]. It was derived before the advent of [[quantum mechanics]] by [[Hendrik Anthony Kramers|Hendrik Kramers]] and [[Werner Heisenberg]] in 1925,<ref name=KH>{{cite journal
| | She is recognized by the title of Myrtle Shryock. For years he's been operating as a receptionist. To gather coins is what his family and him appreciate. California is our beginning place.<br><br>Here is my website - at home std test ([http://dtekorea.com/xe/DS_030111/1073195 visit the next internet site]) |
| | last = Kramers
| |
| | first = H. A.
| |
| | authorlink = Hendrik Anthony Kramers
| |
| | last2 = Heisenberg
| |
| | first2 = W.
| |
| | author2-link = Werner Heisenberg
| |
| | title = Über die Streuung von Strahlung durch Atome
| |
| | journal = Z. Phys.
| |
| | volume = 31
| |
| | issue = 1
| |
| | pages = 681–708
| |
| | date = Feb 1925
| |
| | url = http://www.springerlink.com/content/x2x7220805540747
| |
| | doi = 10.1007/BF02980624
| |
| |bibcode = 1925ZPhy...31..681K }}</ref> based on the [[correspondence principle]] applied to the classical dispersion formula for light. The quantum mechanical derivation was given by [[Paul Dirac]] in 1927.<ref>{{cite journal
| |
| | last = Dirac.
| |
| | first = P. A. M.
| |
| | authorlink = Paul Dirac
| |
| | title = The Quantum Theory of the Emission and Absorption of Radiation
| |
| | journal = Proc. Roy. Soc. Lond. A
| |
| | volume = 114
| |
| | issue = 769
| |
| | pages = 243–265
| |
| | year = 1927
| |
| | doi = 10.1098/rspa.1927.0039
| |
| |bibcode = 1927RSPSA.114..243D }}</ref><ref>{{cite journal
| |
| | last = Dirac.
| |
| | first = P. A. M.
| |
| | authorlink = Paul Dirac
| |
| | title = The Quantum Theory of Dispersion
| |
| | journal = Proc. Roy. Soc. Lond. A
| |
| | volume = 114
| |
| | issue = 769
| |
| | pages = 710–728
| |
| | year = 1927
| |
| | doi = 10.1098/rspa.1927.0071
| |
| |bibcode = 1927RSPSA.114..710D }}</ref>
| |
| | |
| 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 [[Raman effect]].<ref>{{Cite journal
| |
| | last = Breit
| |
| | first = G.
| |
| | title = Quantum Theory of Dispersion
| |
| | journal = Rev. Mod. Phys.
| |
| | volume = 4
| |
| | issue = 3
| |
| | pages = 504–576
| |
| | year = 1932
| |
| | url = http://link.aps.org/doi/10.1103/RevModPhys.4.504
| |
| | doi = 10.1103/RevModPhys.4.504
| |
| |bibcode = 1932RvMP....4..504B }}</ref>
| |
| | |
| == Equation ==
| |
| The Kramers-Heisenberg (KH) formula for second order processes is <ref name=KH/><ref>J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley (1967), page
| |
| 56.</ref><BR>
| |
| <math> \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)</math>
| |
| | |
| It represents the probability of the emission of photons of energy <math> \hbar \omega_k^\prime </math> in the
| |
| solid angle <math>d\Omega_{k^\prime}</math> (centred in the <math>k^\prime</math> direction), after the excitation of the system with photons of energy <math> \hbar \omega_k</math>. <math>|i\rangle, |n\rangle, |f\rangle</math> are the initial, intermediate
| |
| and final states of the system with energy <math>E_i , E_n , E_f</math> respectively; the delta
| |
| function ensures the energy conservation during the whole process. <math>T</math> is the relevant
| |
| transition operator. <math>\Gamma_n </math> is the instrinsic linewidth of the intermediate state.
| |
| | |
| == References ==
| |
| <references />
| |
| | |
| {{DEFAULTSORT:Kramers-Heisenberg formula}}
| |
| [[Category:Particle physics]]
| |
| | |
| | |
| {{physics-stub}}
| |
She is recognized by the title of Myrtle Shryock. For years he's been operating as a receptionist. To gather coins is what his family and him appreciate. California is our beginning place.
Here is my website - at home std test (visit the next internet site)