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Display information for equation id:math.1777.9 on revision:1777

* Page found: Lippmann- Schwinger- Gleichung (eq math.1777.9)

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Lippmann- Schwinger- Gleichung Eq: math.1777.9

Hash: acf5e02150b795e92c7b4f01c0bb6124

TeX (original user input):

\begin{align}

& \left( E-{{{\hat{H}}}_{0}} \right)\left| \Psi  \right\rangle =\left( E-{{{\hat{H}}}_{0}} \right)\left| \Phi  \right\rangle +\left( E-{{{\hat{H}}}_{0}} \right)\frac{1}{\left( E-{{{\hat{H}}}_{0}} \right)}{{{\hat{H}}}^{(1)}}\left| \Psi  \right\rangle  \\

& \left( E-{{{\hat{H}}}_{0}} \right)\frac{1}{\left( E-{{{\hat{H}}}_{0}} \right)}:=1 \\

& \Rightarrow \left( E-{{{\hat{H}}}_{0}} \right)\left| \Psi  \right\rangle ={{{\hat{H}}}^{(1)}}\left| \Psi  \right\rangle \Leftrightarrow \left( E-{{{\hat{H}}}_{0}} \right)\left| \Phi  \right\rangle =0 \\

\end{align}

TeX (checked):

{\begin{aligned}&\left(E-{{\hat {H}}_{0}}\right)\left|\Psi \right\rangle =\left(E-{{\hat {H}}_{0}}\right)\left|\Phi \right\rangle +\left(E-{{\hat {H}}_{0}}\right){\frac {1}{\left(E-{{\hat {H}}_{0}}\right)}}{{\hat {H}}^{(1)}}\left|\Psi \right\rangle \\&\left(E-{{\hat {H}}_{0}}\right){\frac {1}{\left(E-{{\hat {H}}_{0}}\right)}}:=1\\&\Rightarrow \left(E-{{\hat {H}}_{0}}\right)\left|\Psi \right\rangle ={{\hat {H}}^{(1)}}\left|\Psi \right\rangle \Leftrightarrow \left(E-{{\hat {H}}_{0}}\right)\left|\Phi \right\rangle =0\\\end{aligned}}

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\begin{matrix} (E - {\hat{H}}_{0}) | Ψ ⟩ = (E - {\hat{H}}_{0}) | Φ ⟩ + (E - {\hat{H}}_{0}) \frac{1}{(E - {\hat{H}}_{0})} {\hat{H}}^{(1)} | Ψ ⟩ \\ (E - {\hat{H}}_{0}) \frac{1}{(E - {\hat{H}}_{0})} := 1 \\ \Rightarrow (E - {\hat{H}}_{0}) | Ψ ⟩ = {\hat{H}}^{(1)} | Ψ ⟩ \Leftrightarrow (E - {\hat{H}}_{0}) | Φ ⟩ = 0 \end{matrix}

<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Ψ</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Φ</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">+</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mfrac></mrow><msup><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Ψ</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mfrac></mrow><mo stretchy="false">:=</mo><mn>1</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Ψ</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><msup><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Ψ</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">⇔</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>E</mi><mo stretchy="false">−</mo><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Φ</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

$E$

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

$Ψ$

$E$

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

$Φ$

$E$

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

$E$

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

$\hat{H}$

$Ψ$

$E$

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

$E$

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

$E$

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

$Ψ$

$\hat{H}$

$Ψ$

$E$

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

$Φ$

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