Die Thomas-Reiche-Kuhn-Summenregel [ 1] Willy Thomas , Fritz Reiche und Werner Kuhn ) ist ein mathematisches Hilfsmittel in der Quantenmechanik .
Sie besagt, dass für die Strahlungsübergänge eines Teilchens der Masse
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{\displaystyle \sum _{n}(E_{n}-E_{m})\left|\left\langle n|{\hat {x}}|m\right\rangle \right|^{2}={\frac {\hbar ^{2}}{2m_{0}}}}
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{\displaystyle \hbar }
reduzierte plancksche Wirkungsquantum
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{\displaystyle \left\langle n|{\hat {x}}|m\right\rangle =x_{nm}}
Matrixelement des Ortsoperators , das direkt mit dem elektrischen Dipolmoment des Überganges verknüpft istDie Thomas-Reiche-Kuhn-Summenregel gilt nur für ausschließlich ortsabhängige Potentiale und kann somit in den meisten Fällen angewandt werden.
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{\displaystyle {\begin{aligned}\sum _{n}(E_{n}-E_{m})\left|\left\langle n|{\hat {x}}|m\right\rangle \right|^{2}&=\sum _{n}(E_{n}-E_{m})\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle \\&={\frac {1}{2}}\sum _{n}\left(\left\langle m\right|{\hat {x}}{\hat {H}}-{\hat {H}}{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle +\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {H}}{\hat {x}}-{\hat {x}}{\hat {H}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\sum _{n}\left(\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|[{\hat {H}},{\hat {x}}]\left|m\right\rangle -\left\langle m\right|[{\hat {H}},{\hat {x}}]\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\left(\left\langle m\right|{\hat {x}}[{\hat {H}},{\hat {x}}]\left|m\right\rangle -\left\langle m\right|[{\hat {H}},{\hat {x}}]{\hat {x}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\left(\left\langle m\right|[{\hat {x}},[{\hat {H}},{\hat {x}}]]\left|m\right\rangle \right)\\&=-{\frac {i\hbar }{2m_{0}}}\left\langle m\right|[{\hat {x}},{\hat {p}}]\left|m\right\rangle \\&={\frac {\hbar ^{2}}{2m_{0}}}\end{aligned}}}
Dabei wurden folgende Beziehungen verwendet:
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{\displaystyle [{\hat {H}},{\hat {x}}]=-{\frac {i\hbar }{m_{0}}}{\hat {p}}}
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{\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar }
↑ Jeremiah A. Cronin, David F. Greenberg, Valentine L. Telegdi: University of Chicago Graduate Problems in Physics with Solutions . University Of Chicago Press, 1979, ISBN 978-0-226-12109-3 . 
![{\displaystyle {\begin{aligned}\sum _{n}(E_{n}-E_{m})\left|\left\langle n|{\hat {x}}|m\right\rangle \right|^{2}&=\sum _{n}(E_{n}-E_{m})\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle \\&={\frac {1}{2}}\sum _{n}\left(\left\langle m\right|{\hat {x}}{\hat {H}}-{\hat {H}}{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle +\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {H}}{\hat {x}}-{\hat {x}}{\hat {H}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\sum _{n}\left(\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|[{\hat {H}},{\hat {x}}]\left|m\right\rangle -\left\langle m\right|[{\hat {H}},{\hat {x}}]\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\left(\left\langle m\right|{\hat {x}}[{\hat {H}},{\hat {x}}]\left|m\right\rangle -\left\langle m\right|[{\hat {H}},{\hat {x}}]{\hat {x}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\left(\left\langle m\right|[{\hat {x}},[{\hat {H}},{\hat {x}}]]\left|m\right\rangle \right)\\&=-{\frac {i\hbar }{2m_{0}}}\left\langle m\right|[{\hat {x}},{\hat {p}}]\left|m\right\rangle \\&={\frac {\hbar ^{2}}{2m_{0}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2cca6884cb7d26a650320fde7cec529d712e1f0)
![{\displaystyle [{\hat {H}},{\hat {x}}]=-{\frac {i\hbar }{m_{0}}}{\hat {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/263c797c67d32f86da815bb6349bbbb834a255ce)
![{\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar }](https://wikimedia.org/api/rest_v1/media/math/render/svg/42dbbd0db710385288536bcf4f4a1b7cceb75d9a)