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Display information for equation id:math.1762.28 on revision:1762

* Page found: Homöopolare chemische Bindung des Wasserstoffmoleküls (eq math.1762.28)

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Homöopolare chemische Bindung des Wasserstoffmoleküls Eq: math.1762.28

Hash: 04fbf751974a535caa1d359d211c1fad

TeX (original user input):

\begin{align}
& {{{\hat{H}}}^{(1)}}_{\alpha \beta }=\left\langle  {{\Psi }_{\alpha }} \right|{{{\hat{H}}}^{(1)}}\left| {{\Psi }_{\beta }} \right\rangle {{=}_{1}}{{\left\langle  a \right|}_{2}}\left\langle  b \right|{{{\hat{H}}}^{(1)}}{{\left| a \right\rangle }_{2}}{{\left| b \right\rangle }_{1}} \\
& \Rightarrow {{{\hat{H}}}^{(1)}}_{\alpha \beta }=\int_{{}}^{{}}{{{d}^{3}}{{r}_{1}}\int_{{}}^{{}}{{}}{{d}^{3}}{{r}_{2}}{{\Psi }_{a}}{{({{{\bar{r}}}_{1}})}^{*}}{{\Psi }_{b}}({{{\bar{r}}}_{1}}){{\Psi }_{b}}{{({{{\bar{r}}}_{2}})}^{*}}{{\Psi }_{a}}({{{\bar{r}}}_{2}}){{{\hat{H}}}^{(1)}}=}{{{\hat{H}}}^{(1)}}_{\beta \alpha }=:A \\
\end{align}

TeX (checked):

{\begin{aligned}&{{\hat {H}}^{(1)}}_{\alpha \beta }=\left\langle {{\Psi }_{\alpha }}\right|{{\hat {H}}^{(1)}}\left|{{\Psi }_{\beta }}\right\rangle {{=}_{1}}{{\left\langle a\right|}_{2}}\left\langle b\right|{{\hat {H}}^{(1)}}{{\left|a\right\rangle }_{2}}{{\left|b\right\rangle }_{1}}\\&\Rightarrow {{\hat {H}}^{(1)}}_{\alpha \beta }=\int _{}^{}{{{d}^{3}}{{r}_{1}}\int _{}^{}{}{{d}^{3}}{{r}_{2}}{{\Psi }_{a}}{{({{\bar {r}}_{1}})}^{*}}{{\Psi }_{b}}({{\bar {r}}_{1}}){{\Psi }_{b}}{{({{\bar {r}}_{2}})}^{*}}{{\Psi }_{a}}({{\bar {r}}_{2}}){{\hat {H}}^{(1)}}=}{{\hat {H}}^{(1)}}_{\beta \alpha }=:A\\\end{aligned}}

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\begin{matrix} {\hat{H}}^{(1)}_{α β} = ⟨ Ψ_{α} | {\hat{H}}^{(1)} | Ψ_{β} ⟩ =_{1} {⟨ a |}_{2} ⟨ b | {\hat{H}}^{(1)} {| a ⟩}_{2} {| b ⟩}_{1} \\ \Rightarrow {\hat{H}}^{(1)}_{α β} = \int_{}^{} d^{3} r_{1} \int_{}^{} d^{3} r_{2} Ψ_{a} {({\bar{r}}_{1})}^{*} Ψ_{b} ({\bar{r}}_{1}) Ψ_{b} {({\bar{r}}_{2})}^{*} Ψ_{a} ({\bar{r}}_{2}) {\hat{H}}^{(1)} = {\hat{H}}^{(1)}_{β α} = : A \end{matrix}

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data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>b</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msub><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="ORD"><mrow data-mjx-texclass="ORD"><mi>α</mi><mi>β</mi></mrow></mrow></msub><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><mrow data-mjx-texclass="ORD"><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"></mrow><mrow data-mjx-texclass="ORD"></mrow></msubsup><msup><mi>d</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mi>Ψ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><msup><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><msub><mover><mi>r</mi><mo>¯</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">∗</mo></mrow></msup><msub><mi>Ψ</mi><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><msub><mover><mi>r</mi><mo>¯</mo></mover><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mi>Ψ</mi><mrow data-mjx-texclass="ORD"><mi>b</mi></mrow></msub><msup><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><msub><mover><mi>r</mi><mo>¯</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow data-mjx-texclass="ORD"><mo stretchy="false" lspace="0" rspace="0">∗</mo></mrow></msup><msub><mi>Ψ</mi><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo stretchy="false">(</mo><msub><mover><mi>r</mi><mo>¯</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><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><mo stretchy="false">=</mo></mrow><msub><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="ORD"><mrow data-mjx-texclass="ORD"><mi>β</mi><mi>α</mi></mrow></mrow></msub><mo stretchy="false">=</mo><mo stretchy="false">:</mo><mi>A</mi></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Identifiers

${\hat{H}}_{α β}$

$Ψ_{α}$

$\hat{H}$

$Ψ_{β}$

$a_{2}$

$b$

$\hat{H}$

$a_{2}$

$b_{1}$

${\hat{H}}_{α β}$

$r_{1}$

$r_{2}$

$Ψ_{a}$

${\bar{r}}_{1}$

$Ψ_{b}$

${\bar{r}}_{1}$

$Ψ_{b}$

${\bar{r}}_{2}$

$Ψ_{a}$

${\bar{r}}_{2}$

$\hat{H}$

${\hat{H}}_{β α}$

$A$

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