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Display information for equation id:math.1826.3 on revision:1826

* Page found: Das Wasserstoffatom (relativistsich) (eq math.1826.3)

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Das Wasserstoffatom (relativistsich) Eq: math.1826.3

Hash: 4ef3d42f2ed230d08d3ef03d5e5441d8

TeX (original user input):

\begin{align}

& {{\alpha }_{r}}{{p}_{r}}+\frac{i}{r}{{\alpha }_{r}}\beta \hbar Q={{\alpha }_{r}}\left[ \frac{1}{r}\left( \bar{r}\bar{p}-i\hbar  \right)+\frac{i}{r}{{\beta }^{2}}\left( \tilde{\bar{\sigma }}\bar{L}+\hbar  \right) \right] \\

& {{\beta }^{2}}=1 \\

& =\frac{{{\alpha }_{r}}}{r}\left( \bar{r}\bar{p}+i\tilde{\bar{\sigma }}\bar{L} \right)=\frac{1}{{{r}^{2}}}\left[ \left( \bar{\alpha }\bar{r} \right)\left( \bar{r}\bar{p} \right)+i\left( \bar{\alpha }\bar{r} \right)\left( \tilde{\bar{\sigma }}\bar{L} \right) \right] \\

& i\left( \bar{\alpha }\bar{r} \right)\left( \tilde{\bar{\sigma }}\bar{L} \right)=i\left( \bar{\alpha }\bar{r} \right)\left( \bar{r}\bar{p} \right)-i{{r}^{2}}\left( \bar{\alpha }\bar{p} \right) \\

& \Rightarrow {{\alpha }_{r}}{{p}_{r}}+\frac{i}{r}{{\alpha }_{r}}\beta \hbar Q=\frac{1}{{{r}^{2}}}\left[ \left( \bar{\alpha }\bar{r} \right)\left( \bar{r}\bar{p} \right)+i\left( \bar{\alpha }\bar{r} \right)\left( \tilde{\bar{\sigma }}\bar{L} \right) \right]=\bar{\alpha }\bar{p} \\

\end{align}

TeX (checked):

{\begin{aligned}&{{\alpha }_{r}}{{p}_{r}}+{\frac {i}{r}}{{\alpha }_{r}}\beta \hbar Q={{\alpha }_{r}}\left[{\frac {1}{r}}\left({\bar {r}}{\bar {p}}-i\hbar \right)+{\frac {i}{r}}{{\beta }^{2}}\left({\tilde {\bar {\sigma }}}{\bar {L}}+\hbar \right)\right]\\&{{\beta }^{2}}=1\\&={\frac {{\alpha }_{r}}{r}}\left({\bar {r}}{\bar {p}}+i{\tilde {\bar {\sigma }}}{\bar {L}}\right)={\frac {1}{{r}^{2}}}\left[\left({\bar {\alpha }}{\bar {r}}\right)\left({\bar {r}}{\bar {p}}\right)+i\left({\bar {\alpha }}{\bar {r}}\right)\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)\right]\\&i\left({\bar {\alpha }}{\bar {r}}\right)\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)=i\left({\bar {\alpha }}{\bar {r}}\right)\left({\bar {r}}{\bar {p}}\right)-i{{r}^{2}}\left({\bar {\alpha }}{\bar {p}}\right)\\&\Rightarrow {{\alpha }_{r}}{{p}_{r}}+{\frac {i}{r}}{{\alpha }_{r}}\beta \hbar Q={\frac {1}{{r}^{2}}}\left[\left({\bar {\alpha }}{\bar {r}}\right)\left({\bar {r}}{\bar {p}}\right)+i\left({\bar {\alpha }}{\bar {r}}\right)\left({\tilde {\bar {\sigma }}}{\bar {L}}\right)\right]={\bar {\alpha }}{\bar {p}}\\\end{aligned}}

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MathML (9.441 KB / 641 B) :

\begin{aligned} α_{r} p_{r} + \frac{i}{r} α_{r} β ℏ Q = α_{r} [\frac{1}{r} (\bar{r} \bar{p} - i ℏ) + \frac{i}{r} β^{2} (\tilde{\bar{σ}} \bar{L} + ℏ)] \\ β^{2} = 1 \\ = \frac{α_{r}}{r} (\bar{r} \bar{p} + i \tilde{\bar{σ}} \bar{L}) = \frac{1}{r^{2}} [(\bar{α} \bar{r}) (\bar{r} \bar{p}) + i (\bar{α} \bar{r}) (\tilde{\bar{σ}} \bar{L})] \\ i (\bar{α} \bar{r}) (\tilde{\bar{σ}} \bar{L}) = i (\bar{α} \bar{r}) (\bar{r} \bar{p}) - i r^{2} (\bar{α} \bar{p}) \\ \Rightarrow α_{r} p_{r} + \frac{i}{r} α_{r} β ℏ Q = \frac{1}{r^{2}} [(\bar{α} \bar{r}) (\bar{r} \bar{p}) + i (\bar{α} \bar{r}) (\tilde{\bar{σ}} \bar{L})] = \bar{α} \bar{p} \end{aligned}

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Identifiers

$α_{r}$

$p_{r}$

$i$

$r$

$α_{r}$

$β$

$Q$

$α_{r}$

$r$

$\bar{r}$

$\bar{p}$

$i$

$i$

$r$

$β$

$\tilde{\bar{σ}}$

$\bar{L}$

$β$

$α_{r}$

$r$

$\bar{r}$

$\bar{p}$

$i$

$\tilde{\bar{σ}}$

$\bar{L}$

$r$

$\bar{α}$

$\bar{r}$

$\bar{r}$

$\bar{p}$

$i$

$\bar{α}$

$\bar{r}$

$\tilde{\bar{σ}}$

$\bar{L}$

$i$

$\bar{α}$

$\bar{r}$

$\tilde{\bar{σ}}$

$\bar{L}$

$i$

$\bar{α}$

$\bar{r}$

$\bar{r}$

$\bar{p}$

$i$

$r$

$\bar{α}$

$\bar{p}$

$α_{r}$

$p_{r}$

$i$

$r$

$α_{r}$

$β$

$Q$

$r$

$\bar{α}$

$\bar{r}$

$\bar{r}$

$\bar{p}$

$i$

$\bar{α}$

$\bar{r}$

$\tilde{\bar{σ}}$

$\bar{L}$

$\bar{α}$

$\bar{p}$

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