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Display information for equation id:math.1628.1 on revision:1628

* Page found: Eigenwerte und Eigenzustände von hermiteschen Operatoren (eq math.1628.1)

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TeX (original user input): 

\begin{align}

& \left\langle {{\left( \Delta \hat{F} \right)}^{2}} \right\rangle =\int_{{}}^{{}}{{{d}^{3}}r\Psi *(\bar{r}){{\left( \Delta \hat{F} \right)}^{2}}\Psi (\bar{r})=}\left\langle {{\left( \hat{F}-\left\langle {\hat{F}} \right\rangle  \right)}^{2}} \right\rangle =\left\langle {{{\hat{F}}}^{2}} \right\rangle -{{\left\langle {\hat{F}} \right\rangle }^{2}} \\

& =\int_{{}}^{{}}{{{d}^{3}}r\Psi *(\bar{r}){{\left( {\hat{F}} \right)}^{2}}\Psi (\bar{r})}-{{\left( \int_{{}}^{{}}{{{d}^{3}}r\Psi *(\bar{r})\hat{F}\Psi (\bar{r})} \right)}^{2}}=0 \\

& \Leftrightarrow \left\langle  \Psi  \right|{{{\hat{F}}}^{2}}\left| \Psi  \right\rangle =\left\langle  \Psi  \right|\hat{F}{{\left| \Psi  \right\rangle }^{2}} \\

\end{align}

TeX (checked): 

{\begin{aligned}&\left\langle {{\left(\Delta {\hat {F}}\right)}^{2}}\right\rangle =\int _{}^{}{{{d}^{3}}r\Psi *({\bar {r}}){{\left(\Delta {\hat {F}}\right)}^{2}}\Psi ({\bar {r}})=}\left\langle {{\left({\hat {F}}-\left\langle {\hat {F}}\right\rangle \right)}^{2}}\right\rangle =\left\langle {{\hat {F}}^{2}}\right\rangle -{{\left\langle {\hat {F}}\right\rangle }^{2}}\\&=\int _{}^{}{{{d}^{3}}r\Psi *({\bar {r}}){{\left({\hat {F}}\right)}^{2}}\Psi ({\bar {r}})}-{{\left(\int _{}^{}{{{d}^{3}}r\Psi *({\bar {r}}){\hat {F}}\Psi ({\bar {r}})}\right)}^{2}}=0\\&\Leftrightarrow \left\langle \Psi \right|{{\hat {F}}^{2}}\left|\Psi \right\rangle =\left\langle \Psi \right|{\hat {F}}{{\left|\Psi \right\rangle }^{2}}\\\end{aligned}}

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⟨(ΔF̂)2⟩=∫d3rΨ∗(r¯)(ΔF̂)2Ψ(r¯)=⟨(F̂−⟨F̂⟩)2⟩=⟨F̂2⟩−⟨F̂⟩2=∫d3rΨ∗(r¯)(F̂)2Ψ(r¯)−(∫d3rΨ∗(r¯)F̂Ψ(r¯))2=0⇔⟨Ψ|F̂2|Ψ⟩=⟨Ψ|F̂|Ψ⟩2
<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><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Δ</mi><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><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><mi>r</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>Δ</mi><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>Ψ</mi><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">=</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mo stretchy="false">−</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msup><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">−</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><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><mi>r</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mi>Ψ</mi><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mo stretchy="false">−</mo><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</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><mi>r</mi><mi>Ψ</mi><mo stretchy="false">∗</mo><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mi>Ψ</mi><mo stretchy="false">(</mo><mover><mi>r</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">=</mo><mn>0</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>Ψ</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><msup><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mn>2</mn></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>Ψ</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><mover><mi>F</mi><mo stretchy="false">̂</mo></mover><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mi>Ψ</mi><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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Calculated based on the variables occurring on the entire Eigenwerte und Eigenzustände von hermiteschen Operatoren page

Identifiers

  • Δ
  • F̂
  • r
  • Ψ
  • r¯
  • Δ
  • F̂
  • Ψ
  • r¯
  • F̂
  • F̂
  • F̂
  • F̂
  • r
  • Ψ
  • r¯
  • F̂
  • Ψ
  • r¯
  • r
  • Ψ
  • r¯
  • F̂
  • Ψ
  • r¯
  • Ψ
  • F̂
  • Ψ
  • Ψ
  • F̂
  • Ψ

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