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Mikroskopisches Modell der Polarisierbarkeit Eq: math.2156.18

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TeX (original user input):

\begin{align}
&  \ddot{\bar{r}}={{{\ddot{\bar{r}}}}_{k}}-{{{\ddot{\bar{r}}}}_{e}}=-\frac{{{Z}^{2}}{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{R}^{3}}{{m}_{K}}}\left( {{{\bar{r}}}_{k}}-{{{\bar{r}}}_{e}} \right)+\frac{Ze}{{{m}_{K}}}{{{\bar{E}}}_{a}}\left( {{{\bar{r}}}_{\acute{\ }k}},t \right)-\frac{Z{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{R}^{3}}{{m}_{e}}}\left( {{{\bar{r}}}_{k}}-{{{\bar{r}}}_{e}} \right)+\frac{e}{{{m}_{e}}}{{{\bar{E}}}_{a}}\left( {{{\bar{r}}}_{k}},t \right) \\
& =-\frac{{{Z}^{2}}{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{R}^{3}}}\left( \frac{1}{{{m}_{K}}}+\frac{1}{Z{{m}_{e}}} \right)\left( {{{\bar{r}}}_{k}}-{{{\bar{r}}}_{e}} \right)+Ze\left( \frac{1}{{{m}_{K}}}+\frac{1}{Z{{m}_{e}}} \right){{{\bar{E}}}_{a}}\left( {{{\bar{r}}}_{k}},t \right) \\
& \left( \frac{1}{{{m}_{K}}}+\frac{1}{Z{{m}_{e}}} \right)\approx \frac{1}{Z{{m}_{e}}} \\
& \left( {{{\bar{r}}}_{k}}-{{{\bar{r}}}_{e}} \right)=\bar{r} \\
& \Rightarrow \ddot{\bar{r}}=-\frac{Z{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{m}_{e}}{{R}^{3}}}\bar{r}+\frac{e}{{{m}_{e}}}{{{\bar{E}}}_{a}}\left( {{{\bar{r}}}_{k}},t \right) \\
& \frac{Z{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{m}_{e}}{{R}^{3}}}:={{\omega }_{0}}^{2} \\
& \Rightarrow \ddot{\bar{r}}+{{\omega }_{0}}^{2}\bar{r}=\frac{e}{{{m}_{e}}}{{{\bar{E}}}_{a}}\left( {{{\bar{r}}}_{k}},t \right) \\
\end{align}

TeX (checked):

{\begin{aligned}&{\ddot {\bar {r}}}={{\ddot {\bar {r}}}_{k}}-{{\ddot {\bar {r}}}_{e}}=-{\frac {{{Z}^{2}}{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{R}^{3}}{{m}_{K}}}}\left({{\bar {r}}_{k}}-{{\bar {r}}_{e}}\right)+{\frac {Ze}{{m}_{K}}}{{\bar {E}}_{a}}\left({{\bar {r}}_{{\acute {\ }}k}},t\right)-{\frac {Z{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{R}^{3}}{{m}_{e}}}}\left({{\bar {r}}_{k}}-{{\bar {r}}_{e}}\right)+{\frac {e}{{m}_{e}}}{{\bar {E}}_{a}}\left({{\bar {r}}_{k}},t\right)\\&=-{\frac {{{Z}^{2}}{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{R}^{3}}}}\left({\frac {1}{{m}_{K}}}+{\frac {1}{Z{{m}_{e}}}}\right)\left({{\bar {r}}_{k}}-{{\bar {r}}_{e}}\right)+Ze\left({\frac {1}{{m}_{K}}}+{\frac {1}{Z{{m}_{e}}}}\right){{\bar {E}}_{a}}\left({{\bar {r}}_{k}},t\right)\\&\left({\frac {1}{{m}_{K}}}+{\frac {1}{Z{{m}_{e}}}}\right)\approx {\frac {1}{Z{{m}_{e}}}}\\&\left({{\bar {r}}_{k}}-{{\bar {r}}_{e}}\right)={\bar {r}}\\&\Rightarrow {\ddot {\bar {r}}}=-{\frac {Z{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{m}_{e}}{{R}^{3}}}}{\bar {r}}+{\frac {e}{{m}_{e}}}{{\bar {E}}_{a}}\left({{\bar {r}}_{k}},t\right)\\&{\frac {Z{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{m}_{e}}{{R}^{3}}}}:={{\omega }_{0}}^{2}\\&\Rightarrow {\ddot {\bar {r}}}+{{\omega }_{0}}^{2}{\bar {r}}={\frac {e}{{m}_{e}}}{{\bar {E}}_{a}}\left({{\bar {r}}_{k}},t\right)\\\end{aligned}}

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MathML (12.356 KB / 753 B) :

\begin{aligned} \ddot{\bar{r}} = {\ddot{\bar{r}}}_{k} - {\ddot{\bar{r}}}_{e} = - \frac{Z^{2} e^{2}}{4 π ε_{0} R^{3} m_{K}} ({\bar{r}}_{k} - {\bar{r}}_{e}) + \frac{Z e}{m_{K}} {\bar{E}}_{a} ({\bar{r}}_{\overset{´}{} k}, t) - \frac{Z e^{2}}{4 π ε_{0} R^{3} m_{e}} ({\bar{r}}_{k} - {\bar{r}}_{e}) + \frac{e}{m_{e}} {\bar{E}}_{a} ({\bar{r}}_{k}, t) \\ = - \frac{Z^{2} e^{2}}{4 π ε_{0} R^{3}} (\frac{1}{m_{K}} + \frac{1}{Z m_{e}}) ({\bar{r}}_{k} - {\bar{r}}_{e}) + Z e (\frac{1}{m_{K}} + \frac{1}{Z m_{e}}) {\bar{E}}_{a} ({\bar{r}}_{k}, t) \\ (\frac{1}{m_{K}} + \frac{1}{Z m_{e}}) \approx \frac{1}{Z m_{e}} \\ ({\bar{r}}_{k} - {\bar{r}}_{e}) = \bar{r} \\ \Rightarrow \ddot{\bar{r}} = - \frac{Z e^{2}}{4 π ε_{0} m_{e} R^{3}} \bar{r} + \frac{e}{m_{e}} {\bar{E}}_{a} ({\bar{r}}_{k}, t) \\ \frac{Z e^{2}}{4 π ε_{0} m_{e} R^{3}} : = {ω_{0}}^{2} \\ \Rightarrow \ddot{\bar{r}} + {ω_{0}}^{2} \bar{r} = \frac{e}{m_{e}} {\bar{E}}_{a} ({\bar{r}}_{k}, t) \end{aligned}

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Identifiers

$\ddot{\bar{r}}$

${\ddot{\bar{r}}}_{k}$

${\ddot{\bar{r}}}_{e}$

$Z$

$e$

$π$

$ε_{0}$

$R$

$m_{K}$

${\bar{r}}_{k}$

${\bar{r}}_{e}$

$Z$

$e$

$m_{K}$

${\bar{E}}_{a}$

$\bar{r}$

$\overset{´}{}$

$k$

$t$

$Z$

$e$

$π$

$ε_{0}$

$R$

$m_{e}$

${\bar{r}}_{k}$

${\bar{r}}_{e}$

$e$

$m_{e}$

${\bar{E}}_{a}$

${\bar{r}}_{k}$

$t$

$Z$

$e$

$π$

$ε_{0}$

$R$

$m_{K}$

$Z$

$m_{e}$

${\bar{r}}_{k}$

${\bar{r}}_{e}$

$Z$

$e$

$m_{K}$

$Z$

$m_{e}$

${\bar{E}}_{a}$

${\bar{r}}_{k}$

$t$

$m_{K}$

$Z$

$m_{e}$

$Z$

$m_{e}$

${\bar{r}}_{k}$

${\bar{r}}_{e}$

$\bar{r}$

$\ddot{\bar{r}}$

$Z$

$e$

$π$

$ε_{0}$

$m_{e}$

$R$

$\bar{r}$

$e$

$m_{e}$

${\bar{E}}_{a}$

${\bar{r}}_{k}$

$t$

$Z$

$e$

$π$

$ε_{0}$

$m_{e}$

$R$

$ω_{0}$

$\ddot{\bar{r}}$

$ω_{0}$

$\bar{r}$

$e$

$m_{e}$

${\bar{E}}_{a}$

${\bar{r}}_{k}$

$t$

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