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Display information for equation id:math.1777.55 on revision:1777

* Page found: Lippmann- Schwinger- Gleichung (eq math.1777.55)

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Lippmann- Schwinger- Gleichung Eq: math.1777.55

Hash: 78e6434e5ebc14510be4bad4df1820b7

TeX (original user input):

\begin{align}
 \left\langle  {\bar{r}}  |  {{\Psi }^{(+)}} \right\rangle &=\left\langle  {\bar{r}}  |  \Phi  \right\rangle +\int_{{}}^{{}}{{{d}^{3}}r\acute{\ }\left\langle  {\bar{r}} \right|\frac{1}{\left( E-{{{\hat{H}}}_{0}}+i\varepsilon  \right)}\left| \bar{r}\acute{\ } \right\rangle }\left\langle  \bar{r}\acute{\ } \right|{{{\hat{H}}}^{1}}\left| {{\Psi }^{(+)}} \right\rangle  \\
& =\left\langle  {\bar{r}}  |  \Phi  \right\rangle +\frac{2m}{{{\hbar }^{2}}}\int_{{}}^{{}}{{{d}^{3}}r\acute{\ }{{G}_{+}}\left( \bar{r}-\bar{r}\acute{\ } \right)}\left\langle  \bar{r}\acute{\ } \right|{{{\hat{H}}}^{1}}\left| {{\Psi }^{(+)}} \right\rangle ={{e}^{i\bar{k}\bar{r}}}+\frac{2m}{{{\hbar }^{2}}}\int_{{}}^{{}}{{{d}^{3}}r\acute{\ }\frac{-{{e}^{ik\left| \left( \bar{r}-\bar{r}\acute{\ } \right) \right|}}}{4\pi \left| \left( \bar{r}-\bar{r}\acute{\ } \right) \right|}}\left\langle  \bar{r}\acute{\ } \right|{{{\hat{H}}}^{1}}\left| {{\Psi }^{(+)}} \right\rangle  \\
\end{align}

TeX (checked):

{\begin{aligned}\left\langle {\bar {r}}|{{\Psi }^{(+)}}\right\rangle &=\left\langle {\bar {r}}|\Phi \right\rangle +\int _{}^{}{{{d}^{3}}r{\acute {\ }}\left\langle {\bar {r}}\right|{\frac {1}{\left(E-{{\hat {H}}_{0}}+i\varepsilon \right)}}\left|{\bar {r}}{\acute {\ }}\right\rangle }\left\langle {\bar {r}}{\acute {\ }}\right|{{\hat {H}}^{1}}\left|{{\Psi }^{(+)}}\right\rangle \\&=\left\langle {\bar {r}}|\Phi \right\rangle +{\frac {2m}{{\hbar }^{2}}}\int _{}^{}{{{d}^{3}}r{\acute {\ }}{{G}_{+}}\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)}\left\langle {\bar {r}}{\acute {\ }}\right|{{\hat {H}}^{1}}\left|{{\Psi }^{(+)}}\right\rangle ={{e}^{i{\bar {k}}{\bar {r}}}}+{\frac {2m}{{\hbar }^{2}}}\int _{}^{}{{{d}^{3}}r{\acute {\ }}{\frac {-{{e}^{ik\left|\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\right|}}}{4\pi \left|\left({\bar {r}}-{\bar {r}}{\acute {\ }}\right)\right|}}}\left\langle {\bar {r}}{\acute {\ }}\right|{{\hat {H}}^{1}}\left|{{\Psi }^{(+)}}\right\rangle \\\end{aligned}}

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\begin{aligned} ⟨ \bar{r} | Ψ^{(+)} ⟩ & = ⟨ \bar{r} | Φ ⟩ + \int_{}^{} d^{3} r \overset{´}{} ⟨ \bar{r} | \frac{1}{(E - {\hat{H}}_{0} + i ε)} | \bar{r} \overset{´}{} ⟩ ⟨ \bar{r} \overset{´}{} | {\hat{H}}^{1} | Ψ^{(+)} ⟩ \\ = ⟨ \bar{r} | Φ ⟩ + \frac{2 m}{ℏ^{2}} \int_{}^{} d^{3} r \overset{´}{} G_{+} (\bar{r} - \bar{r} \overset{´}{}) ⟨ \bar{r} \overset{´}{} | {\hat{H}}^{1} | Ψ^{(+)} ⟩ = e^{i \bar{k} \bar{r}} + \frac{2 m}{ℏ^{2}} \int_{}^{} d^{3} r \overset{´}{} \frac{- e^{i k | (\bar{r} - \bar{r} \overset{´}{}) |}}{4 π | (\bar{r} - \bar{r} \overset{´}{}) |} ⟨ \bar{r} \overset{´}{} | {\hat{H}}^{1} | Ψ^{(+)} ⟩ \end{aligned}

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Identifiers

$\bar{r}$

$Ψ$

$\bar{r}$

$Φ$

$r$

$\overset{´}{}$

$\bar{r}$

$E$

${\hat{H}}_{0}$

$i$

$ε$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$\hat{H}$

$Ψ$

$\bar{r}$

$Φ$

$m$

$r$

$\overset{´}{}$

$G_{+}$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$\hat{H}$

$Ψ$

$e$

$i$

$\bar{k}$

$\bar{r}$

$m$

$r$

$\overset{´}{}$

$e$

$i$

$k$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$π$

$\bar{r}$

$\bar{r}$

$\overset{´}{}$

$\bar{r}$

$\overset{´}{}$

$\hat{H}$

$Ψ$

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