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Display information for equation id:math.2680.3 on revision:2680

* Page found: Weitere Eigenschaften der Dirac-Gleichung (eq math.2680.3)

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Weitere Eigenschaften der Dirac-Gleichung Eq: math.2685.3

Hash: f1d8e7895289e0189dd6c1c66d50e766

TeX (original user input):

\begin{align}

& \mathfrak{i} {{\Psi }^{+}}\dot{\Psi }\quad ={{\Psi }^{+}}\left( \underline{\alpha }.\hat{\underline{p}}+\beta m \right)\Psi  \\

& -\mathfrak{i} {{{\dot{\Psi }}}^{+}}\Psi \quad ={{\left( \underline{p}\Psi  \right)}^{+}}\underline{\alpha }\Psi +m{{\Psi }^{+}}\beta \Psi  \\

& ------------------------ \\

& \mathfrak{i} {{\partial }_{t}}\underbrace{\left( {{\Psi }^{+}}\Psi  \right)}_{:=\rho }\quad ={{\Psi }^{+}}\underline{\alpha }\left( \underline{p}\Psi  \right)-{{\left( \underline{p}\Psi  \right)}^{+}}\underline{\alpha }\Psi  \\

& \quad =-\mathfrak{i} \sum\limits_{k}{{{\Psi }^{+}}{{\alpha }_{k}}\left( {{\partial }_{k}}\Psi  \right)-{{\left( {{\partial }_{k}}\Psi  \right)}^{+}}{{\alpha }_{k}}\Psi } \\

& \quad =-\mathfrak{i} \sum\limits_{k}{{{\partial }_{k}}\underbrace{\left( {{\Psi }^{+}}{{\alpha }_{k}}\Psi  \right)}_{:={{j}_{k}}}} \\

\end{align}

TeX (checked):

{\begin{aligned}&{\mathfrak {i}}{{\Psi }^{+}}{\dot {\Psi }}\quad ={{\Psi }^{+}}\left({\underline {\alpha }}.{\hat {\underline {p}}}+\beta m\right)\Psi \\&-{\mathfrak {i}}{{\dot {\Psi }}^{+}}\Psi \quad ={{\left({\underline {p}}\Psi \right)}^{+}}{\underline {\alpha }}\Psi +m{{\Psi }^{+}}\beta \Psi \\&------------------------\\&{\mathfrak {i}}{{\partial }_{t}}\underbrace {\left({{\Psi }^{+}}\Psi \right)} _{:=\rho }\quad ={{\Psi }^{+}}{\underline {\alpha }}\left({\underline {p}}\Psi \right)-{{\left({\underline {p}}\Psi \right)}^{+}}{\underline {\alpha }}\Psi \\&\quad =-{\mathfrak {i}}\sum \limits _{k}{{{\Psi }^{+}}{{\alpha }_{k}}\left({{\partial }_{k}}\Psi \right)-{{\left({{\partial }_{k}}\Psi \right)}^{+}}{{\alpha }_{k}}\Psi }\\&\quad =-{\mathfrak {i}}\sum \limits _{k}{{{\partial }_{k}}\underbrace {\left({{\Psi }^{+}}{{\alpha }_{k}}\Psi \right)} _{:={{j}_{k}}}}\\\end{aligned}}

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\begin{aligned} i Ψ^{+} \dot{Ψ} = Ψ^{+} (\underline{α} . \hat{\underline{p}} + β m) Ψ \\ - i {\dot{Ψ}}^{+} Ψ = {(\underline{p} Ψ)}^{+} \underline{α} Ψ + m Ψ^{+} β Ψ \\ - - - - - - - - - - - - - - - - - - - - - - - - \\ i \partial_{t} \underset{: = ρ}{\underset{⏟}{(Ψ^{+} Ψ)}} = Ψ^{+} \underline{α} (\underline{p} Ψ) - {(\underline{p} Ψ)}^{+} \underline{α} Ψ \\ = - i \sum_{k} Ψ^{+} α_{k} (\partial_{k} Ψ) - {(\partial_{k} Ψ)}^{+} α_{k} Ψ \\ = - i \sum_{k} \partial_{k} \underset{: = j_{k}}{\underset{⏟}{(Ψ^{+} α_{k} Ψ)}} \end{aligned}

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Identifiers

$i$

$Ψ$

$\dot{Ψ}$

$Ψ$

$\underline{α}$

$\hat{\underline{p}}$

$β$

$m$

$Ψ$

$i$

$\dot{Ψ}$

$Ψ$

$\underline{p}$

$Ψ$

$\underline{α}$

$Ψ$

$m$

$Ψ$

$β$

$Ψ$

$i$

$t$

$Ψ$

$Ψ$

$ρ$

$Ψ$

$\underline{α}$

$\underline{p}$

$Ψ$

$\underline{p}$

$Ψ$

$\underline{α}$

$Ψ$

$i$

$k$

$Ψ$

$α_{k}$

$k$

$Ψ$

$k$

$Ψ$

$α_{k}$

$Ψ$

$i$

$k$

$k$

$Ψ$

$α_{k}$

$Ψ$

$j_{k}$

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