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 m_{0}}
Zustand
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{\displaystyle |m\rangle }
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{\displaystyle |n\rangle }
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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 E_{n}}
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{\displaystyle |n\rangle }
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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 ist
Die Thomas-Reiche-Kuhn-Summenregel gilt nur für ausschließlich ortsabhängige Potentiale und kann somit in den meisten Fällen angewandt werden.
Beweis
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{\displaystyle \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 }
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{\displaystyle ={\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)}
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{\displaystyle ={\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)}
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{\displaystyle ={\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}}}}
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 }
Literatur
↑ 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 ={\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf2394f2bc6f9ec784f86c01531bafc21189569)
![{\displaystyle ={\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07c44ff326f2da7017742c125f38bb15197e3e67)
![{\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)