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* 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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MathML (92.314 KB / 8.717 KB) :

∫ R 3 d 3 ⁢ x ⁢ Ψ ⁢ ( r ¯ ) ⁢ e - i ⁢ k ¯ ⁢ ´ ⁢ r ¯ = 1 ( 2 ⁢ π ) 3 2 ⁢ ∫ R 3 d 3 ⁢ k ⁢ Φ ⁢ ( k ¯ ) ⁢ ∫ d 3 ⁢ r ⁢ e i ⁢ ( k ¯ - k ¯ ⁢ ´ ) ⁢ r ¯ ∫ R 3 d 3 ⁢ r ⁢ e i ⁢ ( k ¯ - k ¯ ⁢ ´ ) ⁢ r ¯ = ( 2 ⁢ π ) 3 ⁢ δ ⁢ ( k ¯ - k ¯ ⁢ ´ ) ⇒ 1 ( 2 ⁢ π ) 3 2 ⁢ ∫ R 3 d 3 ⁢ k ⁢ Φ ⁢ ( k ¯ ) ⁢ ∫ d 3 ⁢ r ⁢ e i ⁢ ( k ¯ - k ¯ ⁢ ´ ) ⁢ r ¯ = 1 ( 2 ⁢ π ) 3 2 ⁢ ∫ R 3 d 3 ⁢ k ⁢ Φ ⁢ ( k ¯ ) ⁢ ( 2 ⁢ π ) 3 ⁢ δ ⁢ ( k ¯ - k ¯ ⁢ ´ ) = ( 2 ⁢ π ) 3 2 ⁢ Φ ⁢ ( k ¯ ⁢ ´ ) ⇒ Φ ⁢ ( k ¯ ) = 1 ( 2 ⁢ π ) 3 2 ⁢ ∫ R 3 d 3 ⁢ x ⁢ Ψ ⁢ ( r ¯ ) ⁢ e - i ⁢ k ¯ ⁢ r ¯ absent subscript superscript 𝑅 3 superscript 𝑑 3 𝑥 Ψ ¯ 𝑟 superscript 𝑒 𝑖 ¯ 𝑘 ´ absent ¯ 𝑟 1 superscript 2 𝜋 3 2 subscript superscript 𝑅 3 superscript 𝑑 3 𝑘 Φ ¯ 𝑘 superscript 𝑑 3 𝑟 superscript 𝑒 𝑖 ¯ 𝑘 ¯ 𝑘 ´ absent ¯ 𝑟 absent subscript superscript 𝑅 3 superscript 𝑑 3 𝑟 superscript 𝑒 𝑖 ¯ 𝑘 ¯ 𝑘 ´ absent ¯ 𝑟 superscript 2 𝜋 3 𝛿 ¯ 𝑘 ¯ 𝑘 ´ absent absent ⇒ absent 1 superscript 2 𝜋 3 2 subscript superscript 𝑅 3 superscript 𝑑 3 𝑘 Φ ¯ 𝑘 superscript 𝑑 3 𝑟 superscript 𝑒 𝑖 ¯ 𝑘 ¯ 𝑘 ´ absent ¯ 𝑟 1 superscript 2 𝜋 3 2 subscript superscript 𝑅 3 superscript 𝑑 3 𝑘 Φ ¯ 𝑘 superscript 2 𝜋 3 𝛿 ¯ 𝑘 ¯ 𝑘 ´ absent superscript 2 𝜋 3 2 Φ ¯ 𝑘 ´ absent absent ⇒ absent Φ ¯ 𝑘 1 superscript 2 𝜋 3 2 subscript superscript 𝑅 3 superscript 𝑑 3 𝑥 Ψ ¯ 𝑟 superscript 𝑒 𝑖 ¯ 𝑘 ¯ 𝑟 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\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}}}}\\ \par&\displaystyle\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{\ })\\ \par&\displaystyle\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{\ })\\ \par&\displaystyle\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}% }}\\ \par\end{aligned}}}

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∫R3d3xΨ(r¯)e−ik¯´r¯=1(2π)32∫R3d3kΦ(k¯)∫d3rei(k¯−k¯´)r¯∫R3d3rei(k¯−k¯´)r¯=(2π)3δ(k¯−k¯´)⇒1(2π)32∫R3d3kΦ(k¯)∫d3rei(k¯−k¯´)r¯=1(2π)32∫R3d3kΦ(k¯)(2π)3δ(k¯−k¯´)=(2π)32Φ(k¯´)⇒Φ(k¯)=1(2π)32∫R3d3xΨ(r¯)e−ik¯r¯

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