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Display information for equation id:math.1776.28 on revision:1776

* Page found: Lippmann- Schwinger- Gleichung (eq math.1776.28)

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Lippmann- Schwinger- Gleichung Eq: math.1777.28

Hash: 7248767ef5f6886f74697874b4654e6a

TeX (original user input):

\begin{align}

& \Rightarrow \left\langle  {\bar{q}} \right|\frac{1}{E-{{{\hat{H}}}_{0}}+i\varepsilon }\left| \bar{q}\acute{\ } \right\rangle =\frac{2m}{{{\hbar }^{2}}}\frac{\delta \left( \bar{q}-\bar{q}\acute{\ } \right)}{{{{\bar{k}}}^{2}}-{{{\bar{q}}}^{2}}+i\eta }=:\frac{2m}{{{\hbar }^{2}}}{{{\tilde{G}}}_{+}}(\bar{q})\delta \left( \bar{q}-\bar{q}\acute{\ } \right) \\

& \eta =\frac{2m}{{{\hbar }^{2}}}\varepsilon  \\

& \left\langle  {\bar{r}}  |  {\bar{q}} \right\rangle =\frac{1}{{{\left( 2\pi  \right)}^{\frac{3}{2}}}}{{e}^{i\bar{q}\bar{r}}} \\

\end{align}

TeX (checked):

{\begin{aligned}&\Rightarrow \left\langle {\bar {q}}\right|{\frac {1}{E-{{\hat {H}}_{0}}+i\varepsilon }}\left|{\bar {q}}{\acute {\ }}\right\rangle ={\frac {2m}{{\hbar }^{2}}}{\frac {\delta \left({\bar {q}}-{\bar {q}}{\acute {\ }}\right)}{{{\bar {k}}^{2}}-{{\bar {q}}^{2}}+i\eta }}=:{\frac {2m}{{\hbar }^{2}}}{{\tilde {G}}_{+}}({\bar {q}})\delta \left({\bar {q}}-{\bar {q}}{\acute {\ }}\right)\\&\eta ={\frac {2m}{{\hbar }^{2}}}\varepsilon \\&\left\langle {\bar {r}}|{\bar {q}}\right\rangle ={\frac {1}{{\left(2\pi \right)}^{\frac {3}{2}}}}{{e}^{i{\bar {q}}{\bar {r}}}}\\\end{aligned}}

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\begin{matrix} \Rightarrow ⟨ \bar{q} | \frac{1}{E - {\hat{H}}_{0} + i ε} | \bar{q} \overset{´}{} ⟩ = \frac{2 m}{ℏ^{2}} \frac{δ (\bar{q} - \bar{q} \overset{´}{})}{{\bar{k}}^{2} - {\bar{q}}^{2} + i η} = : \frac{2 m}{ℏ^{2}} {\tilde{G}}_{+} (\bar{q}) δ (\bar{q} - \bar{q} \overset{´}{}) \\ η = \frac{2 m}{ℏ^{2}} ε \\ ⟨ \bar{r} | \bar{q} ⟩ = \frac{1}{{(2 π)}^{\frac{3}{2}}} e^{i \bar{q} \bar{r}} \end{matrix}

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Identifiers

$\bar{q}$

$E$

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

$i$

$ε$

$\bar{q}$

$\overset{´}{}$

$m$

$δ$

$\bar{q}$

$\bar{q}$

$\overset{´}{}$

$\bar{k}$

$\bar{q}$

$i$

$η$

$m$

${\tilde{G}}_{+}$

$\bar{q}$

$δ$

$\bar{q}$

$\bar{q}$

$\overset{´}{}$

$η$

$m$

$ε$

$\bar{r}$

$\bar{q}$

$π$

$e$

$i$

$\bar{q}$

$\bar{r}$

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