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Display information for equation id:math.1804.58 on revision:1804

* Page found: Dirac- Gleichung für Elektronen (eq math.1804.58)

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Dirac- Gleichung für Elektronen Eq: math.1807.58

Hash: e73ad2140491879bd47b4ae7ac79f72c

TeX (original user input):

\begin{align}

& i\hbar \left( {{\Psi }^{+}}\dot{\Psi }+{{{\dot{\Psi }}}^{+}}\Psi  \right)=-i\hbar c\left( {{\Psi }^{+}}{{\alpha }^{\mu }}\left( {{\partial }_{\mu }}\Psi  \right)+\left( {{\partial }_{\mu }}{{\Psi }^{+}} \right){{\alpha }^{\mu }}\Psi  \right) \\

& \left( {{\Psi }^{+}}\dot{\Psi }+{{{\dot{\Psi }}}^{+}}\Psi  \right)=\frac{\partial }{\partial t}\left( {{\Psi }^{+}}\Psi  \right) \\

& \left( {{\Psi }^{+}}{{\alpha }^{\mu }}\left( {{\partial }_{\mu }}\Psi  \right)+\left( {{\partial }_{\mu }}{{\Psi }^{+}} \right){{\alpha }^{\mu }}\Psi  \right)={{\partial }_{\mu }}\left( {{\Psi }^{+}}{{\alpha }^{\mu }}\Psi  \right) \\

& \Rightarrow i\hbar \frac{\partial }{\partial t}\left( {{\Psi }^{+}}\Psi  \right)+c{{\partial }_{\mu }}\left( {{\Psi }^{+}}{{\alpha }^{\mu }}\Psi  \right)=0 \\

& \Rightarrow \left( {{\Psi }^{+}}\Psi  \right)=\rho  \\

& \Rightarrow \left( {{\Psi }^{+}}{{\alpha }^{\mu }}\Psi  \right)=\frac{{{j}^{\mu }}}{c} \\

\end{align}

TeX (checked):

{\begin{aligned}&i\hbar \left({{\Psi }^{+}}{\dot {\Psi }}+{{\dot {\Psi }}^{+}}\Psi \right)=-i\hbar c\left({{\Psi }^{+}}{{\alpha }^{\mu }}\left({{\partial }_{\mu }}\Psi \right)+\left({{\partial }_{\mu }}{{\Psi }^{+}}\right){{\alpha }^{\mu }}\Psi \right)\\&\left({{\Psi }^{+}}{\dot {\Psi }}+{{\dot {\Psi }}^{+}}\Psi \right)={\frac {\partial }{\partial t}}\left({{\Psi }^{+}}\Psi \right)\\&\left({{\Psi }^{+}}{{\alpha }^{\mu }}\left({{\partial }_{\mu }}\Psi \right)+\left({{\partial }_{\mu }}{{\Psi }^{+}}\right){{\alpha }^{\mu }}\Psi \right)={{\partial }_{\mu }}\left({{\Psi }^{+}}{{\alpha }^{\mu }}\Psi \right)\\&\Rightarrow i\hbar {\frac {\partial }{\partial t}}\left({{\Psi }^{+}}\Psi \right)+c{{\partial }_{\mu }}\left({{\Psi }^{+}}{{\alpha }^{\mu }}\Psi \right)=0\\&\Rightarrow \left({{\Psi }^{+}}\Psi \right)=\rho \\&\Rightarrow \left({{\Psi }^{+}}{{\alpha }^{\mu }}\Psi \right)={\frac {{j}^{\mu }}{c}}\\\end{aligned}}

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\begin{aligned} i ℏ (Ψ^{+} \dot{Ψ} + {\dot{Ψ}}^{+} Ψ) = - i ℏ c (Ψ^{+} α^{μ} (\partial_{μ} Ψ) + (\partial_{μ} Ψ^{+}) α^{μ} Ψ) \\ (Ψ^{+} \dot{Ψ} + {\dot{Ψ}}^{+} Ψ) = \frac{\partial}{\partial t} (Ψ^{+} Ψ) \\ (Ψ^{+} α^{μ} (\partial_{μ} Ψ) + (\partial_{μ} Ψ^{+}) α^{μ} Ψ) = \partial_{μ} (Ψ^{+} α^{μ} Ψ) \\ \Rightarrow i ℏ \frac{\partial}{\partial t} (Ψ^{+} Ψ) + c \partial_{μ} (Ψ^{+} α^{μ} Ψ) = 0 \\ \Rightarrow (Ψ^{+} Ψ) = ρ \\ \Rightarrow (Ψ^{+} α^{μ} Ψ) = \frac{j^{μ}}{c} \end{aligned}

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Identifiers

$i$

$Ψ$

$\dot{Ψ}$

$\dot{Ψ}$

$Ψ$

$i$

$c$

$Ψ$

$α$

$μ$

$μ$

$Ψ$

$μ$

$Ψ$

$α$

$μ$

$Ψ$

$Ψ$

$\dot{Ψ}$

$\dot{Ψ}$

$Ψ$

$t$

$Ψ$

$Ψ$

$Ψ$

$α$

$μ$

$μ$

$Ψ$

$μ$

$Ψ$

$α$

$μ$

$Ψ$

$μ$

$Ψ$

$α$

$μ$

$Ψ$

$i$

$t$

$Ψ$

$Ψ$

$c$

$μ$

$Ψ$

$α$

$μ$

$Ψ$

$Ψ$

$Ψ$

$ρ$

$Ψ$

$α$

$μ$

$Ψ$

$j$

$μ$

$c$

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