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* Page found: Klein- Gordon- Gleichung (eq math.1816.59)

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Klein- Gordon- Gleichung Eq: math.1816.59

Hash: 9cdbeb930ff426b33f8ca70324851f4c

TeX (original user input):

\begin{align}
& -{{\partial }^{i}}{{\partial }_{i}}\Psi ={{\left( \frac{{{m}_{0}}c}{\hbar } \right)}^{2}}\Psi =i{{k}^{j}}{{\partial }^{i}}{{\Psi }_{0}}\exp \left\{ -i{{k}_{j}}{{x}^{j}} \right\}={{k}^{j}}{{k}_{j}}\Psi ={{\left( \frac{{{m}_{0}}c}{\hbar } \right)}^{2}}\Psi  \\
& \Rightarrow {{k}^{j}}{{k}_{j}}\equiv {{\left( \frac{\omega }{c} \right)}^{2}}-{{{\bar{k}}}^{2}}={{\left( \frac{{{m}_{0}}c}{\hbar } \right)}^{2}} \\
& \Rightarrow {{\omega }^{2}}={{c}^{2}}\left[{{\left( \frac{{{m}_{0}}c}{\hbar } \right)}^{2}}+{{{\bar{k}}}^{2}} \right] \\
\end{align}

TeX (checked):

{\begin{aligned}&-{{\partial }^{i}}{{\partial }_{i}}\Psi ={{\left({\frac {{{m}_{0}}c}{\hbar }}\right)}^{2}}\Psi =i{{k}^{j}}{{\partial }^{i}}{{\Psi }_{0}}\exp \left\{-i{{k}_{j}}{{x}^{j}}\right\}={{k}^{j}}{{k}_{j}}\Psi ={{\left({\frac {{{m}_{0}}c}{\hbar }}\right)}^{2}}\Psi \\&\Rightarrow {{k}^{j}}{{k}_{j}}\equiv {{\left({\frac {\omega }{c}}\right)}^{2}}-{{\bar {k}}^{2}}={{\left({\frac {{{m}_{0}}c}{\hbar }}\right)}^{2}}\\&\Rightarrow {{\omega }^{2}}={{c}^{2}}\left[{{\left({\frac {{{m}_{0}}c}{\hbar }}\right)}^{2}}+{{\bar {k}}^{2}}\right]\\\end{aligned}}

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\begin{aligned} - \partial^{i} \partial_{i} Ψ = {(\frac{m_{0} c}{ℏ})}^{2} Ψ = i k^{j} \partial^{i} Ψ_{0} \exp {- i k_{j} x^{j}} = k^{j} k_{j} Ψ = {(\frac{m_{0} c}{ℏ})}^{2} Ψ \\ \Rightarrow k^{j} k_{j} \equiv {(\frac{ω}{c})}^{2} - {\bar{k}}^{2} = {(\frac{m_{0} c}{ℏ})}^{2} \\ \Rightarrow ω^{2} = c^{2} [{(\frac{m_{0} c}{ℏ})}^{2} + {\bar{k}}^{2}] \end{aligned}

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Identifiers

$i$

$i$

$Ψ$

$m_{0}$

$c$

$Ψ$

$i$

$k$

$j$

$i$

$Ψ_{0}$

$i$

$k_{j}$

$x$

$j$

$k$

$j$

$k_{j}$

$Ψ$

$m_{0}$

$c$

$Ψ$

$k$

$j$

$k_{j}$

$ω$

$c$

$\bar{k}$

$m_{0}$

$c$

$ω$

$c$

$m_{0}$

$c$

$\bar{k}$

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