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Display information for equation id:math.1619.6 on revision:1619
* Page found: Zustandsvektoren im Hilbertraum (eq math.1619.6)
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\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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data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow></mstyle></mrow></mrow></msup></mrow></mfrac></mrow><mstyle displaystyle="true" scriptlevel="0"><munderover><mo texclass="OP">∫</mo><mrow data-mjx-texclass="ORD"><msup><mi>R</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover></mstyle><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><mi>x</mi></mrow><mi mathvariant="normal">Ψ</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo stretchy="false">)</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mi>i</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>k</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow 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