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Display information for equation id:math.1618.6 on revision:1618

* Page found: Zustandsvektoren im Hilbertraum (eq math.1618.6)

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Zustandsvektoren im Hilbertraum Eq: math.1619.6

Hash: bfc4c91352daad13edb22634d0c8f56d

TeX (original user input):

\begin{align}

& \int_{{{R}^{3}}}^{{}}{{{d}^{3}}x}\Psi (\bar{r}){{e}^{-i\bar{k}\acute{\ }\bar{r}}}=\frac{1}{{{\left( 2\pi  \right)}^{\tfrac{3}{2}}}}\int_{{{R}^{3}}}^{{}}{{{d}^{3}}k\Phi (\bar{k})\int_{{}}^{{}}{{{d}^{3}}r}{{e}^{i\left( \bar{k}-\bar{k}\acute{\ } \right)\bar{r}}}} \\

& \int_{{{R}^{3}}}^{{}}{{{d}^{3}}r}{{e}^{i\left( \bar{k}-\bar{k}\acute{\ } \right)\bar{r}}}={{\left( 2\pi  \right)}^{3}}\delta (\bar{k}-\bar{k}\acute{\ }) \\

& \Rightarrow \frac{1}{{{\left( 2\pi  \right)}^{\tfrac{3}{2}}}}\int_{{{R}^{3}}}^{{}}{{{d}^{3}}k\Phi (\bar{k})\int_{{}}^{{}}{{{d}^{3}}r}{{e}^{i\left( \bar{k}-\bar{k}\acute{\ } \right)\bar{r}}}}=\frac{1}{{{\left( 2\pi  \right)}^{\tfrac{3}{2}}}}\int_{{{R}^{3}}}^{{}}{{{d}^{3}}k\Phi (\bar{k})}{{\left( 2\pi  \right)}^{3}}\delta (\bar{k}-\bar{k}\acute{\ })={{\left( 2\pi  \right)}^{\tfrac{3}{2}}}\Phi (\bar{k}\acute{\ }) \\

& \Rightarrow \Phi (\bar{k})=\frac{1}{{{\left( 2\pi  \right)}^{\tfrac{3}{2}}}}\int_{{{R}^{3}}}^{{}}{{{d}^{3}}x}\Psi (\bar{r}){{e}^{-i\bar{k}\bar{r}}} \\

\end{align}

TeX (checked):

{\begin{aligned}&\int _{{R}^{3}}^{}{{{d}^{3}}x}\Psi ({\bar {r}}){{e}^{-i{\bar {k}}{\acute {\ }}{\bar {r}}}}={\frac {1}{{\left(2\pi \right)}^{\tfrac {3}{2}}}}\int _{{R}^{3}}^{}{{{d}^{3}}k\Phi ({\bar {k}})\int _{}^{}{{{d}^{3}}r}{{e}^{i\left({\bar {k}}-{\bar {k}}{\acute {\ }}\right){\bar {r}}}}}\\&\int _{{R}^{3}}^{}{{{d}^{3}}r}{{e}^{i\left({\bar {k}}-{\bar {k}}{\acute {\ }}\right){\bar {r}}}}={{\left(2\pi \right)}^{3}}\delta ({\bar {k}}-{\bar {k}}{\acute {\ }})\\&\Rightarrow {\frac {1}{{\left(2\pi \right)}^{\tfrac {3}{2}}}}\int _{{R}^{3}}^{}{{{d}^{3}}k\Phi ({\bar {k}})\int _{}^{}{{{d}^{3}}r}{{e}^{i\left({\bar {k}}-{\bar {k}}{\acute {\ }}\right){\bar {r}}}}}={\frac {1}{{\left(2\pi \right)}^{\tfrac {3}{2}}}}\int _{{R}^{3}}^{}{{{d}^{3}}k\Phi ({\bar {k}})}{{\left(2\pi \right)}^{3}}\delta ({\bar {k}}-{\bar {k}}{\acute {\ }})={{\left(2\pi \right)}^{\tfrac {3}{2}}}\Phi ({\bar {k}}{\acute {\ }})\\&\Rightarrow \Phi ({\bar {k}})={\frac {1}{{\left(2\pi \right)}^{\tfrac {3}{2}}}}\int _{{R}^{3}}^{}{{{d}^{3}}x}\Psi ({\bar {r}}){{e}^{-i{\bar {k}}{\bar {r}}}}\\\end{aligned}}

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\begin{matrix} \int_{R^{3}}^{} d^{3} x Ψ (\bar{r}) e^{- i \bar{k} \overset{´}{} \bar{r}} = \frac{1}{{(2 π)}^{\frac{3}{2}}} \int_{R^{3}}^{} d^{3} k Φ (\bar{k}) \int_{}^{} d^{3} r e^{i (\bar{k} - \bar{k} \overset{´}{}) \bar{r}} \\ \int_{R^{3}}^{} d^{3} r e^{i (\bar{k} - \bar{k} \overset{´}{}) \bar{r}} = {(2 π)}^{3} δ (\bar{k} - \bar{k} \overset{´}{}) \\ \Rightarrow \frac{1}{{(2 π)}^{\frac{3}{2}}} \int_{R^{3}}^{} d^{3} k Φ (\bar{k}) \int_{}^{} d^{3} r e^{i (\bar{k} - \bar{k} \overset{´}{}) \bar{r}} = \frac{1}{{(2 π)}^{\frac{3}{2}}} \int_{R^{3}}^{} d^{3} k Φ (\bar{k}) {(2 π)}^{3} δ (\bar{k} - \bar{k} \overset{´}{}) = {(2 π)}^{\frac{3}{2}} Φ (\bar{k} \overset{´}{}) \\ \Rightarrow Φ (\bar{k}) = \frac{1}{{(2 π)}^{\frac{3}{2}}} \int_{R^{3}}^{} d^{3} x Ψ (\bar{r}) e^{- i \bar{k} \bar{r}} \end{matrix}

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Identifiers

$R$

$x$

$Ψ$

$\bar{r}$

$e$

$i$

$\bar{k}$

$\overset{´}{}$

$\bar{r}$

$π$

$R$

$k$

$Φ$

$\bar{k}$

$r$

$e$

$i$

$\bar{k}$

$\bar{k}$

$\overset{´}{}$

$\bar{r}$

$R$

$r$

$e$

$i$

$\bar{k}$

$\bar{k}$

$\overset{´}{}$

$\bar{r}$

$π$

$δ$

$\bar{k}$

$\bar{k}$

$\overset{´}{}$

$π$

$R$

$k$

$Φ$

$\bar{k}$

$r$

$e$

$i$

$\bar{k}$

$\bar{k}$

$\overset{´}{}$

$\bar{r}$

$π$

$R$

$k$

$Φ$

$\bar{k}$

$π$

$δ$

$\bar{k}$

$\bar{k}$

$\overset{´}{}$

$π$

$Φ$

$\bar{k}$

$\overset{´}{}$

$Φ$

$\bar{k}$

$π$

$R$

$x$

$Ψ$

$\bar{r}$

$e$

$i$

$\bar{k}$

$\bar{r}$

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