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Display information for equation id:math.1816.59 on revision:1816

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

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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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MathML (41.024 KB / 4.597 KB) :

- ∂ i ⁡ ∂ i ⁡ Ψ = ( m 0 ⁢ c ℏ ) 2 ⁢ Ψ = i ⁢ k j ⁢ ∂ i ⁡ Ψ 0 ⁢ exp ⁡ { - i ⁢ k j ⁢ x j } = k j ⁢ k j ⁢ Ψ = ( m 0 ⁢ c ℏ ) 2 ⁢ Ψ ⇒ k j ⁢ k j ≡ ( ω c ) 2 - k ¯ 2 = ( m 0 ⁢ c ℏ ) 2 ⇒ ω 2 = c 2 ⁢ [ ( m 0 ⁢ c ℏ ) 2 + k ¯ 2 ] missing-subexpression superscript 𝑖 subscript 𝑖 Ψ superscript subscript 𝑚 0 𝑐 Planck-constant-over-2-pi 2 Ψ 𝑖 superscript 𝑘 𝑗 superscript 𝑖 subscript Ψ 0 𝑖 subscript 𝑘 𝑗 superscript 𝑥 𝑗 superscript 𝑘 𝑗 subscript 𝑘 𝑗 Ψ superscript subscript 𝑚 0 𝑐 Planck-constant-over-2-pi 2 Ψ missing-subexpression ⇒ absent superscript 𝑘 𝑗 subscript 𝑘 𝑗 superscript 𝜔 𝑐 2 superscript ¯ 𝑘 2 superscript subscript 𝑚 0 𝑐 Planck-constant-over-2-pi 2 missing-subexpression ⇒ absent superscript 𝜔 2 superscript 𝑐 2 delimited-[] superscript subscript 𝑚 0 𝑐 Planck-constant-over-2-pi 2 superscript ¯ 𝑘 2 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle-{{\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\\ &\displaystyle\Rightarrow{{k}^{j}}{{k}_{j}}\equiv{{\left(\frac{\omega}{c}% \right)}^{2}}-{{{\bar{k}}}^{2}}={{\left(\frac{{{m}_{0}}c}{\hbar}\right)}^{2}}% \\ &\displaystyle\Rightarrow{{\omega}^{2}}={{c}^{2}}\left[{{\left(\frac{{{m}_{0}}% c}{\hbar}\right)}^{2}}+{{{\bar{k}}}^{2}}\right]\\ \end{aligned}}}

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MathML (4.45 KB / 572 B) :

−∂i∂iΨ=(m0cℏ)2Ψ=ikj∂iΨ0exp⁡{−ikjxj}=kjkjΨ=(m0cℏ)2Ψ⇒kjkj≡(ωc)2−k¯2=(m0cℏ)2⇒ω2=c2[(m0cℏ)2+k¯2]

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