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* Page found: Dynamik des statistischen Operators (eq math.2249.10)

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Dynamik des statistischen Operators Eq: math.2250.10

Hash: d5c830aa58a7bca6ed2173fef9904708

TeX (original user input):

\begin{align}
  & {{p}_{n}}=\operatorname{Tr}\left( \rho \left| n \right\rangle \left\langle  n \right| \right) \\
 & =\sum\limits_{j}{\left\langle  j \right|\sum\limits_{i}{{{w}_{i}}\left| {{\Psi }_{i}} \right\rangle \left\langle  {{\Psi }_{i}} \right|}\left| n \right\rangle \left\langle  n  |  j \right\rangle } \\
 & =\sum\limits_{j}{\underbrace{\left\langle  j  |  j \right\rangle }_{1}\sum\limits_{i}{\left\langle  {{\Psi }_{i}}  |  n \right\rangle }\left\langle  n \right|{{w}_{i}}\left| {{\Psi }_{i}} \right\rangle } \\
 & =\sum\limits_{i}{{{w}_{i}}\left\langle  n  |  {{\Psi }_{i}} \right\rangle \left\langle  {{\Psi }_{i}}  |  n \right\rangle }=\left\langle  n \right|\rho \left| n \right\rangle  
\end{align}

TeX (checked):

{\begin{aligned}&{{p}_{n}}=\operatorname {Tr} \left(\rho \left|n\right\rangle \left\langle n\right|\right)\\&=\sum \limits _{j}{\left\langle j\right|\sum \limits _{i}{{{w}_{i}}\left|{{\Psi }_{i}}\right\rangle \left\langle {{\Psi }_{i}}\right|}\left|n\right\rangle \left\langle n|j\right\rangle }\\&=\sum \limits _{j}{\underbrace {\left\langle j|j\right\rangle } _{1}\sum \limits _{i}{\left\langle {{\Psi }_{i}}|n\right\rangle }\left\langle n\right|{{w}_{i}}\left|{{\Psi }_{i}}\right\rangle }\\&=\sum \limits _{i}{{{w}_{i}}\left\langle n|{{\Psi }_{i}}\right\rangle \left\langle {{\Psi }_{i}}|n\right\rangle }=\left\langle n\right|\rho \left|n\right\rangle \end{aligned}}

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MathML (4.438 KB / 537 B) :

\begin{matrix} p_{n} = Tr (ρ | n ⟩ ⟨ n |) \\ = \sum_{j} ⟨ j | \sum_{i} w_{i} | Ψ_{i} ⟩ ⟨ Ψ_{i} | | n ⟩ ⟨ n | j ⟩ \\ = \sum_{j} \underset{1}{\underset{⏟}{⟨ j | j ⟩}} \sum_{i} ⟨ Ψ_{i} | n ⟩ ⟨ n | w_{i} | Ψ_{i} ⟩ \\ = \sum_{i} w_{i} ⟨ n | Ψ_{i} ⟩ ⟨ Ψ_{i} | n ⟩ = ⟨ n | ρ | n ⟩ \end{matrix}

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Identifiers

$p_{n}$

$ρ$

$n$

$n$

$j$

$j$

$i$

$w_{i}$

$Ψ_{i}$

$Ψ_{i}$

$n$

$n$

$j$

$j$

$j$

$j$

$i$

$Ψ_{i}$

$n$

$n$

$w_{i}$

$Ψ_{i}$

$i$

$w_{i}$

$n$

$Ψ_{i}$

$Ψ_{i}$

$n$

$n$

$ρ$

$n$

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