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Display information for equation id:math.1777.54 on revision:1777

* Page found: Lippmann- Schwinger- Gleichung (eq math.1777.54)

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TeX (original user input): 

\begin{align}
& \delta (\bar{r}-\bar{r}\acute{\ })=\left\langle  {\bar{r}} | \bar{r}\acute{\ } \right\rangle ={{G}_{+}}(\bar{r},\bar{r}\acute{\ })=\left\langle  {\bar{r}} \right|\left( E-{{{\hat{H}}}_{0}}+i\varepsilon  \right)\frac{1}{\left( E-{{{\hat{H}}}_{0}}+i\varepsilon  \right)}\left| \bar{r}\acute{\ } \right\rangle  \\
& \cong \left\langle  {\bar{r}} \right|\left( \frac{{{\hbar }^{2}}{{k}^{2}}}{2m}-\frac{{{{\hat{p}}}^{2}}}{2m} \right)\frac{1}{\left( E-{{{\hat{H}}}_{0}}+i\varepsilon  \right)}\left| \bar{r}\acute{\ } \right\rangle =\frac{{{\hbar }^{2}}}{2m}\left( {{k}^{2}}+\Delta  \right)\left\langle  {\bar{r}} \right|\frac{1}{\left( E-{{{\hat{H}}}_{0}}+i\varepsilon  \right)}\left| \bar{r}\acute{\ } \right\rangle  \\
\end{align}

TeX (checked): 

{\begin{aligned}&\delta ({\bar {r}}-{\bar {r}}{\acute {\ }})=\left\langle {\bar {r}}|{\bar {r}}{\acute {\ }}\right\rangle ={{G}_{+}}({\bar {r}},{\bar {r}}{\acute {\ }})=\left\langle {\bar {r}}\right|\left(E-{{\hat {H}}_{0}}+i\varepsilon \right){\frac {1}{\left(E-{{\hat {H}}_{0}}+i\varepsilon \right)}}\left|{\bar {r}}{\acute {\ }}\right\rangle \\&\cong \left\langle {\bar {r}}\right|\left({\frac {{{\hbar }^{2}}{{k}^{2}}}{2m}}-{\frac {{\hat {p}}^{2}}{2m}}\right){\frac {1}{\left(E-{{\hat {H}}_{0}}+i\varepsilon \right)}}\left|{\bar {r}}{\acute {\ }}\right\rangle ={\frac {{\hbar }^{2}}{2m}}\left({{k}^{2}}+\Delta \right)\left\langle {\bar {r}}\right|{\frac {1}{\left(E-{{\hat {H}}_{0}}+i\varepsilon \right)}}\left|{\bar {r}}{\acute {\ }}\right\rangle \\\end{aligned}}

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δ(r¯−r¯´)=⟨r¯|r¯´⟩=G+(r¯,r¯´)=⟨r¯|(E−H^0+iε)1(E−H^0+iε)|r¯´⟩≅⟨r¯|(ℏ2k22m−p^22m)1(E−H^0+iε)|r¯´⟩=ℏ22m(k2+Δ)⟨r¯|1(E−H^0+iε)|r¯´⟩
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Identifiers

  • δ
  • r¯
  • r¯
  • ´
  • r¯
  • r¯
  • ´
  • G+
  • r¯
  • r¯
  • ´
  • r¯
  • E
  • H^0
  • i
  • ε
  • E
  • H^0
  • i
  • ε
  • r¯
  • ´
  • r¯
  • k
  • m
  • p^
  • m
  • E
  • H^0
  • i
  • ε
  • r¯
  • ´
  • m
  • k
  • Δ
  • r¯
  • E
  • H^0
  • i
  • ε
  • r¯
  • ´

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