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Display information for equation id:math.1632.1 on revision:1632

* Page found: Eigenwerte und Eigenzustände von hermiteschen Operatoren (eq math.1632.1)

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Eigenwerte und Eigenzustände von hermiteschen Operatoren Eq: math.1632.1

Hash: 4a48f5d5914837a747af162ac1254831

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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\begin{aligned} ⟨ {(Δ \hat{F})}^{2} ⟩ = \int_{}^{} d^{3} r Ψ * (\bar{r}) {(Δ \hat{F})}^{2} Ψ (\bar{r}) = ⟨ {(\hat{F} - ⟨ \hat{F} ⟩)}^{2} ⟩ = ⟨ {\hat{F}}^{2} ⟩ - {⟨ \hat{F} ⟩}^{2} \\ = \int_{}^{} d^{3} r Ψ * (\bar{r}) {(\hat{F})}^{2} Ψ (\bar{r}) - {(\int_{}^{} d^{3} r Ψ * (\bar{r}) \hat{F} Ψ (\bar{r}))}^{2} = 0 \\ \Leftrightarrow ⟨ Ψ | {\hat{F}}^{2} | Ψ ⟩ = ⟨ Ψ | \hat{F} {| Ψ ⟩}^{2} \end{aligned}

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Identifiers

$Δ$

$\hat{F}$

$r$

$Ψ$

$\bar{r}$

$Δ$

$\hat{F}$

$Ψ$

$\bar{r}$

$\hat{F}$

$\hat{F}$

$\hat{F}$

$\hat{F}$

$r$

$Ψ$

$\bar{r}$

$\hat{F}$

$Ψ$

$\bar{r}$

$r$

$Ψ$

$\bar{r}$

$\hat{F}$

$Ψ$

$\bar{r}$

$Ψ$

$\hat{F}$

$Ψ$

$Ψ$

$\hat{F}$

$Ψ$

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