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Display information for equation id:math.1636.61 on revision:1636

* Page found: Die Quantisierung (eq math.1636.61)

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\begin{align}
& f({{\lambda }_{0}})=\alpha +\frac{{{\beta }^{2}}}{2\gamma }-\frac{{{\beta }^{2}}}{4\gamma }=\alpha +\frac{{{\beta }^{2}}}{4\gamma }\ge 0 \\
& {{\beta }^{2}}={{\left\langle \left[ \Delta \hat{F},\Delta \hat{G} \right] \right\rangle }^{2}}={{\left\langle \left[ \hat{F},\hat{G} \right] \right\rangle }^{2}}=-\left\langle \left[ \hat{G},\hat{F} \right] \right\rangle \left\langle \left[ \hat{F},\hat{G} \right] \right\rangle =-\left\langle \left[ \hat{F},\hat{G} \right] \right\rangle *\left\langle \left[ \hat{F},\hat{G} \right] \right\rangle =-{{\left| \left\langle \left[ \hat{F},\hat{G} \right] \right\rangle  \right|}^{2}} \\
& \Rightarrow \left\langle {{\left( \Delta \hat{F} \right)}^{2}} \right\rangle \left\langle {{\left( \Delta \hat{G} \right)}^{2}} \right\rangle \ge \frac{1}{4}{{\left| \left\langle \left[ \hat{F},\hat{G} \right] \right\rangle  \right|}^{2}} \\
\end{align}

TeX (checked):

{\begin{aligned}&f({{\lambda }_{0}})=\alpha +{\frac {{\beta }^{2}}{2\gamma }}-{\frac {{\beta }^{2}}{4\gamma }}=\alpha +{\frac {{\beta }^{2}}{4\gamma }}\geq 0\\&{{\beta }^{2}}={{\left\langle \left[\Delta {\hat {F}},\Delta {\hat {G}}\right]\right\rangle }^{2}}={{\left\langle \left[{\hat {F}},{\hat {G}}\right]\right\rangle }^{2}}=-\left\langle \left[{\hat {G}},{\hat {F}}\right]\right\rangle \left\langle \left[{\hat {F}},{\hat {G}}\right]\right\rangle =-\left\langle \left[{\hat {F}},{\hat {G}}\right]\right\rangle *\left\langle \left[{\hat {F}},{\hat {G}}\right]\right\rangle =-{{\left|\left\langle \left[{\hat {F}},{\hat {G}}\right]\right\rangle \right|}^{2}}\\&\Rightarrow \left\langle {{\left(\Delta {\hat {F}}\right)}^{2}}\right\rangle \left\langle {{\left(\Delta {\hat {G}}\right)}^{2}}\right\rangle \geq {\frac {1}{4}}{{\left|\left\langle \left[{\hat {F}},{\hat {G}}\right]\right\rangle \right|}^{2}}\\\end{aligned}}

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\begin{aligned} f (λ_{0}) = α + \frac{β^{2}}{2 γ} - \frac{β^{2}}{4 γ} = α + \frac{β^{2}}{4 γ} \geq 0 \\ β^{2} = {⟨ [Δ \hat{F}, Δ \hat{G}] ⟩}^{2} = {⟨ [\hat{F}, \hat{G}] ⟩}^{2} = - ⟨ [\hat{G}, \hat{F}] ⟩ ⟨ [\hat{F}, \hat{G}] ⟩ = - ⟨ [\hat{F}, \hat{G}] ⟩ * ⟨ [\hat{F}, \hat{G}] ⟩ = - {| ⟨ [\hat{F}, \hat{G}] ⟩ |}^{2} \\ \Rightarrow ⟨ {(Δ \hat{F})}^{2} ⟩ ⟨ {(Δ \hat{G})}^{2} ⟩ \geq \frac{1}{4} {| ⟨ [\hat{F}, \hat{G}] ⟩ |}^{2} \end{aligned}

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Identifiers

$f$

$λ_{0}$

$α$

$β$

$γ$

$β$

$γ$

$α$

$β$

$γ$

$β$

$Δ$

$\hat{F}$

$Δ$

$\hat{G}$

$\hat{F}$

$\hat{G}$

$\hat{G}$

$\hat{F}$

$\hat{F}$

$\hat{G}$

$\hat{F}$

$\hat{G}$

$\hat{F}$

$\hat{G}$

$\hat{F}$

$\hat{G}$

$Δ$

$\hat{F}$

$Δ$

$\hat{G}$

$\hat{F}$

$\hat{G}$

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