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* Page found: Dynamik des statistischen Operators (eq math.2246.10)

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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 \right|\left| j \right\rangle } \\
 & =\sum\limits_{j}{\underbrace{\left\langle  j \right|\left| j \right\rangle }_{1}\sum\limits_{i}{\left\langle  {{\Psi }_{i}} \right|\left| n \right\rangle }\left\langle  n \right|{{w}_{i}}\left| {{\Psi }_{i}} \right\rangle } \\
 & =\sum\limits_{i}{{{w}_{i}}\left\langle  n \right|\left| {{\Psi }_{i}} \right\rangle \left\langle  {{\Psi }_{i}} \right|\left| n \right\rangle }=\left\langle  n \right|\rho \left| n \right\rangle  
\end{align}

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p n = Tr ⁡ ( ρ ⁢ | n ⟩ ⁢ ⟨ n | ) = ∑ j ⟨ j | ∑ i w i | Ψ i ⟩ ⁢ ⟨ Ψ i | ⁢ | n ⟩ ⁢ ⟨ n | ⁢ | j ⟩ = ∑ j ⟨ j | ⁢ | j ⟩ ⏟ 1 ⁢ ∑ i ⟨ Ψ i | ⁢ | n ⟩ ⁢ ⟨ n | w i | Ψ i ⟩ = ∑ i w i ⁢ ⟨ n | ⁢ | Ψ i ⟩ ⁢ ⟨ Ψ i | ⁢ | n ⟩ = ⟨ n | ρ | n ⟩ missing-subexpression subscript 𝑝 𝑛 Tr 𝜌 ket 𝑛 bra 𝑛 missing-subexpression absent subscript 𝑗 quantum-operator-product 𝑗 subscript 𝑖 subscript 𝑤 𝑖 subscript Ψ 𝑖 bra subscript Ψ 𝑖 ket 𝑛 bra 𝑛 ket 𝑗 missing-subexpression absent subscript 𝑗 subscript ⏟ bra 𝑗 ket 𝑗 1 subscript 𝑖 bra subscript Ψ 𝑖 ket 𝑛 quantum-operator-product 𝑛 subscript 𝑤 𝑖 subscript Ψ 𝑖 missing-subexpression absent subscript 𝑖 subscript 𝑤 𝑖 bra 𝑛 ket subscript Ψ 𝑖 bra subscript Ψ 𝑖 ket 𝑛 quantum-operator-product 𝑛 𝜌 𝑛 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{p}_{n}}=% \operatorname{Tr}\left(\rho\left|n\right\rangle\left\langle n\right|\right)\\ &\displaystyle=\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\right|\left|j\right\rangle}\\ &\displaystyle=\sum\limits_{j}{\underbrace{\left\langle j\right|\left|j\right% \rangle}_{1}\sum\limits_{i}{\left\langle{{\Psi}_{i}}\right|\left|n\right% \rangle}\left\langle n\right|{{w}_{i}}\left|{{\Psi}_{i}}\right\rangle}\\ &\displaystyle=\sum\limits_{i}{{{w}_{i}}\left\langle n\right|\left|{{\Psi}_{i}% }\right\rangle\left\langle{{\Psi}_{i}}\right|\left|n\right\rangle}=\left% \langle n\right|\rho\left|n\right\rangle\end{aligned}}}

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pn=Tr⁡(ρ|n⟩⟨n|)=∑j⟨j|∑iwi|Ψi⟩⟨Ψi||n⟩⟨n||j⟩=∑j⟨j||j⟩⏟1∑i⟨Ψi||n⟩⟨n|wi|Ψi⟩=∑iwi⟨n||Ψi⟩⟨Ψi||n⟩=⟨n|ρ|n⟩

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