Thomas-Reiche-Kuhn sum rule

The Thomas-Reiche-Kuhn sum rule (after Willy Thomas , Fritz Reiche and Werner Kuhn ) is a mathematical aid in quantum mechanics .

It states that the following applies to the radiation transitions of a particle of mass between a certain state and all other states : ${\displaystyle m_{0}}$ ${\displaystyle |m\rangle }$ ${\displaystyle |n\rangle }$

${\displaystyle \sum _{n}(E_{n}-E_{m})\left|\left\langle n|{\hat {x}}|m\right\rangle \right|^{2}={\frac {\hbar ^{2}}{2m_{0}}}}$

${\displaystyle \hbar }$... the reduced Planck quantum of action ... the energy of the state
${\displaystyle E_{n}}$ ${\displaystyle |n\rangle }$

${\displaystyle \left\langle n|{\hat {x}}|m\right\rangle =x_{nm}}$... the matrix element of the position operator , which is directly linked to the electrical dipole moment of the transition

The Thomas-Reiche-Kuhn sum rule only applies to exclusively location-dependent potentials and can therefore be used in most cases.

proof

${\displaystyle {\begin{aligned}\sum _{n}(E_{n}-E_{m})\left|\left\langle n|{\hat {x}}|m\right\rangle \right|^{2}&=\sum _{n}(E_{n}-E_{m})\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle \\&={\frac {1}{2}}\sum _{n}\left(\left\langle m\right|{\hat {x}}{\hat {H}}-{\hat {H}}{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle +\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|{\hat {H}}{\hat {x}}-{\hat {x}}{\hat {H}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\sum _{n}\left(\left\langle m\right|{\hat {x}}\left|n\right\rangle \left\langle n\right|[{\hat {H}},{\hat {x}}]\left|m\right\rangle -\left\langle m\right|[{\hat {H}},{\hat {x}}]\left|n\right\rangle \left\langle n\right|{\hat {x}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\left(\left\langle m\right|{\hat {x}}[{\hat {H}},{\hat {x}}]\left|m\right\rangle -\left\langle m\right|[{\hat {H}},{\hat {x}}]{\hat {x}}\left|m\right\rangle \right)\\&={\frac {1}{2}}\left(\left\langle m\right|[{\hat {x}},[{\hat {H}},{\hat {x}}]]\left|m\right\rangle \right)\\&=-{\frac {i\hbar }{2m_{0}}}\left\langle m\right|[{\hat {x}},{\hat {p}}]\left|m\right\rangle \\&={\frac {\hbar ^{2}}{2m_{0}}}\end{aligned}}}$

The following relationships were used:

${\displaystyle [{\hat {H}},{\hat {x}}]=-{\frac {i\hbar }{m_{0}}}{\hat {p}}}$

${\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar }$

literature

1. ^ Jeremiah A. Cronin, David F. Greenberg, Valentine L. Telegdi: University of Chicago Graduate Problems in Physics with Solutions . University Of Chicago Press, 1979, ISBN 978-0226121093 .