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Display information for equation id:math.2403.32 on revision:2403

* Page found: Quantenmechanische Gleichgewichtsverteilungen (eq math.2403.32)

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Hash: 167cc59f8927973fd20f250692dd9f5f

TeX (original user input):

\begin{align}
& tr\hat{X}:=\sum\limits_{\alpha }^{{}}{{}}\left\langle  \alpha  \right|\hat{X}\left| \alpha  \right\rangle =\sum\limits_{\alpha ,\beta ,\beta \acute{\ }}^{{}}{{}}\left\langle  \alpha  \right|\left| \beta  \right\rangle \left\langle  \beta  \right|\hat{X}\left| \beta \acute{\ } \right\rangle \left\langle  \beta \acute{\ } \right|\left| \alpha  \right\rangle =\sum\limits_{\beta ,\beta }^{{}}{{}}\left\langle  \beta  \right|\hat{X}\left| \beta \acute{\ } \right\rangle \sum\limits_{\alpha }^{{}}{{}}\left\langle  \beta \acute{\ } \right|\left| \alpha  \right\rangle \left\langle  \alpha  \right|\left| \beta  \right\rangle  \\
& \sum\limits_{\alpha }^{{}}{{}}\left\langle  \beta \acute{\ } \right|\left| \alpha  \right\rangle \left\langle  \alpha  \right|\left| \beta  \right\rangle =\left\langle  \beta \acute{\ } \right|\left| \beta  \right\rangle ={{\delta }_{\beta \acute{\ }\beta }} \\
& tr\hat{X}=\sum\limits_{\beta }^{{}}{{}}\left\langle  \beta  \right|\hat{X}\left| \beta  \right\rangle  \\
\end{align}

TeX (checked):

{\begin{aligned}&tr{\hat {X}}:=\sum \limits _{\alpha }^{}{}\left\langle \alpha \right|{\hat {X}}\left|\alpha \right\rangle =\sum \limits _{\alpha ,\beta ,\beta {\acute {\ }}}^{}{}\left\langle \alpha \right|\left|\beta \right\rangle \left\langle \beta \right|{\hat {X}}\left|\beta {\acute {\ }}\right\rangle \left\langle \beta {\acute {\ }}\right|\left|\alpha \right\rangle =\sum \limits _{\beta ,\beta }^{}{}\left\langle \beta \right|{\hat {X}}\left|\beta {\acute {\ }}\right\rangle \sum \limits _{\alpha }^{}{}\left\langle \beta {\acute {\ }}\right|\left|\alpha \right\rangle \left\langle \alpha \right|\left|\beta \right\rangle \\&\sum \limits _{\alpha }^{}{}\left\langle \beta {\acute {\ }}\right|\left|\alpha \right\rangle \left\langle \alpha \right|\left|\beta \right\rangle =\left\langle \beta {\acute {\ }}\right|\left|\beta \right\rangle ={{\delta }_{\beta {\acute {\ }}\beta }}\\&tr{\hat {X}}=\sum \limits _{\beta }^{}{}\left\langle \beta \right|{\hat {X}}\left|\beta \right\rangle \\\end{aligned}}

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\begin{matrix} t r \hat{X} := \sum_{α}^{} ⟨ α | \hat{X} | α ⟩ = \sum_{α, β, β \overset{´}{}}^{} ⟨ α | | β ⟩ ⟨ β | \hat{X} | β \overset{´}{} ⟩ ⟨ β \overset{´}{} | | α ⟩ = \sum_{β, β}^{} ⟨ β | \hat{X} | β \overset{´}{} ⟩ \sum_{α}^{} ⟨ β \overset{´}{} | | α ⟩ ⟨ α | | β ⟩ \\ \sum_{α}^{} ⟨ β \overset{´}{} | | α ⟩ ⟨ α | | β ⟩ = ⟨ β \overset{´}{} | | β ⟩ = δ_{β \overset{´}{} β} \\ t r \hat{X} = \sum_{β}^{} ⟨ β | \hat{X} | β ⟩ \end{matrix}

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$α$

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$\hat{X}$

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$\overset{´}{}$

$α$

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$\hat{X}$

$β$

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$β$

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$α$

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$β$

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$α$

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$α$

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$α$

$α$

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$β$

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$δ$

$β$

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$t$

$r$

$\hat{X}$

$β$

$β$

$\hat{X}$

$β$

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