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Display information for equation id:math.1762.25 on revision:1762

* Page found: Homöopolare chemische Bindung des Wasserstoffmoleküls (eq math.1762.25)

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Hash: ba3ceef49506d4d723846f82e0de7f2f

TeX (original user input): 

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

TeX (checked): 

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

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MathML (38.814 KB / 4.322 KB) :

H ^ ( 1 ) α , α = ⟨ Ψ α | H ^ ( 1 ) | Ψ α ⟩ = 1 ⟨ a | 2 ⁢ ⟨ b | H ^ ( 1 ) | b ⟩ 2 ⁢ | a ⟩ 1 ⇒ H ^ ( 1 ) α , α = ∫ d 3 ⁢ r 1 ⁢ ∫ d 3 ⁢ r 2 ⁢ | Ψ a ⁢ ( r ¯ 1 ) | 2 ⁢ | Ψ b ⁢ ( r ¯ 2 ) | 2 ⁢ H ^ ( 1 ) = H ^ ( 1 ) β ⁢ β missing-subexpression subscript superscript ^ 𝐻 1 𝛼 𝛼 quantum-operator-product subscript Ψ 𝛼 superscript ^ 𝐻 1 subscript Ψ 𝛼 subscript 1 subscript bra 𝑎 2 subscript quantum-operator-product 𝑏 superscript ^ 𝐻 1 𝑏 2 subscript ket 𝑎 1 missing-subexpression ⇒ absent subscript superscript ^ 𝐻 1 𝛼 𝛼 superscript 𝑑 3 subscript 𝑟 1 superscript 𝑑 3 subscript 𝑟 2 superscript subscript Ψ 𝑎 subscript ¯ 𝑟 1 2 superscript subscript Ψ 𝑏 subscript ¯ 𝑟 2 2 superscript ^ 𝐻 1 subscript superscript ^ 𝐻 1 𝛽 𝛽 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{{\hat{H}}}^{(1)}}_% {\alpha,\alpha}=\left\langle{{\Psi}_{\alpha}}\right|{{{\hat{H}}}^{(1)}}\left|{% {\Psi}_{\alpha}}\right\rangle{{=}_{1}}{{\left\langle a\right|}_{2}}\left% \langle b\right|{{{\hat{H}}}^{(1)}}{{\left|b\right\rangle}_{2}}{{\left|a\right% \rangle}_{1}}\\ &\displaystyle\Rightarrow{{{\hat{H}}}^{(1)}}_{\alpha,\alpha}=\int{{{d}^{3}}{{r% }_{1}}\int{{}}{{d}^{3}}{{r}_{2}}{{\left|{{\Psi}_{a}}({{{\bar{r}}}_{1}})\right|% }^{2}}{{\left|{{\Psi}_{b}}({{{\bar{r}}}_{2}})\right|}^{2}}{{{\hat{H}}}^{(1)}}=% }{{{\hat{H}}}^{(1)}}_{\beta\beta}\\ \end{aligned}}}

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MathML (5.224 KB / 613 B) :

H^(1)α,α=⟨Ψα|H^(1)|Ψα⟩=1⟨a|2⟨b|H^(1)|b⟩2|a⟩1⇒H^(1)α,α=∫d3r1∫d3r2|Ψa(r¯1)|2|Ψb(r¯2)|2H^(1)=H^(1)ββ

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Calculated based on the variables occurring on the entire Homöopolare chemische Bindung des Wasserstoffmoleküls page

Identifiers

  • H^α,α
  • Ψα
  • H^
  • Ψα
  • a2
  • b
  • H^
  • b2
  • a1
  • H^α,α
  • r1
  • r2
  • Ψa
  • r¯1
  • Ψb
  • r¯2
  • H^
  • H^ββ

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