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Display information for equation id:math.1824.18 on revision:1824

* Page found: Das Wasserstoffatom (relativistsich) (eq math.1824.18)

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Das Wasserstoffatom (relativistsich) Eq: math.1826.18

Hash: d90de7f0ac9bbb963b89da35c6d047c7

TeX (original user input):

\begin{align}

& {{\left( {{\alpha }_{r}} \right)}^{2}}=\frac{1}{{{r}^{2}}}\left( \bar{\alpha }\bar{r} \right)\left( \bar{\alpha }\bar{r} \right)=\frac{1}{{{r}^{2}}}{{\alpha }^{\mu }}{{\alpha }^{\nu }}{{x}^{\mu }}{{x}^{\nu }}=\frac{1}{2{{r}^{2}}}\left( {{\alpha }^{\mu }}{{\alpha }^{\nu }}+{{\alpha }^{\nu }}{{\alpha }^{\mu }} \right){{x}^{\mu }}{{x}^{\nu }} \\

& \left( {{\alpha }^{\mu }}{{\alpha }^{\nu }}+{{\alpha }^{\nu }}{{\alpha }^{\mu }} \right)=2{{\delta }^{\mu \nu }} \\

& \frac{1}{2{{r}^{2}}}2{{x}^{\mu }}{{x}^{\mu }}=\frac{{{r}^{2}}}{{{r}^{2}}}=1 \\

& {{\alpha }_{r}}\beta +\beta {{\alpha }_{r}}=\frac{1}{r}\left( \bar{\alpha }\beta +\beta \bar{\alpha } \right)\bar{r} \\

& \left( \bar{\alpha }\beta +\beta \bar{\alpha } \right)=0\Rightarrow \frac{1}{r}\left( \bar{\alpha }\beta +\beta \bar{\alpha } \right)\bar{r}=0 \\

\end{align}

TeX (checked):

{\begin{aligned}&{{\left({{\alpha }_{r}}\right)}^{2}}={\frac {1}{{r}^{2}}}\left({\bar {\alpha }}{\bar {r}}\right)\left({\bar {\alpha }}{\bar {r}}\right)={\frac {1}{{r}^{2}}}{{\alpha }^{\mu }}{{\alpha }^{\nu }}{{x}^{\mu }}{{x}^{\nu }}={\frac {1}{2{{r}^{2}}}}\left({{\alpha }^{\mu }}{{\alpha }^{\nu }}+{{\alpha }^{\nu }}{{\alpha }^{\mu }}\right){{x}^{\mu }}{{x}^{\nu }}\\&\left({{\alpha }^{\mu }}{{\alpha }^{\nu }}+{{\alpha }^{\nu }}{{\alpha }^{\mu }}\right)=2{{\delta }^{\mu \nu }}\\&{\frac {1}{2{{r}^{2}}}}2{{x}^{\mu }}{{x}^{\mu }}={\frac {{r}^{2}}{{r}^{2}}}=1\\&{{\alpha }_{r}}\beta +\beta {{\alpha }_{r}}={\frac {1}{r}}\left({\bar {\alpha }}\beta +\beta {\bar {\alpha }}\right){\bar {r}}\\&\left({\bar {\alpha }}\beta +\beta {\bar {\alpha }}\right)=0\Rightarrow {\frac {1}{r}}\left({\bar {\alpha }}\beta +\beta {\bar {\alpha }}\right){\bar {r}}=0\\\end{aligned}}

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\begin{aligned} {(α_{r})}^{2} = \frac{1}{r^{2}} (\bar{α} \bar{r}) (\bar{α} \bar{r}) = \frac{1}{r^{2}} α^{μ} α^{ν} x^{μ} x^{ν} = \frac{1}{2 r^{2}} (α^{μ} α^{ν} + α^{ν} α^{μ}) x^{μ} x^{ν} \\ (α^{μ} α^{ν} + α^{ν} α^{μ}) = 2 δ^{μ ν} \\ \frac{1}{2 r^{2}} 2 x^{μ} x^{μ} = \frac{r^{2}}{r^{2}} = 1 \\ α_{r} β + β α_{r} = \frac{1}{r} (\bar{α} β + β \bar{α}) \bar{r} \\ (\bar{α} β + β \bar{α}) = 0 \Rightarrow \frac{1}{r} (\bar{α} β + β \bar{α}) \bar{r} = 0 \end{aligned}

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Identifiers

$α_{r}$

$r$

$\bar{α}$

$\bar{r}$

$\bar{α}$

$\bar{r}$

$r$

$α$

$μ$

$α$

$ν$

$x$

$μ$

$x$

$ν$

$r$

$α$

$μ$

$α$

$ν$

$α$

$ν$

$α$

$μ$

$x$

$μ$

$x$

$ν$

$α$

$μ$

$α$

$ν$

$α$

$ν$

$α$

$μ$

$δ$

$μ$

$ν$

$r$

$x$

$μ$

$x$

$μ$

$r$

$r$

$α_{r}$

$β$

$β$

$α_{r}$

$r$

$\bar{α}$

$β$

$β$

$\bar{α}$

$\bar{r}$

$\bar{α}$

$β$

$β$

$\bar{α}$

$r$

$\bar{α}$

$β$

$β$

$\bar{α}$

$\bar{r}$

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