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* Page found: Dynamik des statistischen Operators (eq math.2246.1)

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\begin{align}
  & \rho \left( t \right)=\sum\limits_{i}{{{w}_{i}}\left| {{\Psi }_{i}} \right\rangle \left\langle  {{\Psi }_{i}} \right|} \\
 & \text{S}\text{.GL:}i\hbar {{\partial }_{t}}\left| {{\Psi }_{i}} \right\rangle =H\left| {{\Psi }_{i}} \right\rangle \quad |\sum\limits_{i}{{{w}_{i}}\left\langle  {{\Psi }_{i}} \right|} \\
 & \text{h}\text{.c}:-i\hbar {{\partial }_{t}}\left\langle  {{\Psi }_{i}} \right|=\left\langle  {{\Psi }_{i}} \right|H\quad |\sum\limits_{i}{{{w}_{i}}\left\langle  {{\Psi }_{i}} \right|} \\
 & \Rightarrow i\hbar {{\partial }_{t}}\sum\limits_{i}{{{w}_{i}}\left\langle  {{\Psi }_{i}} \right|}\left| {{\Psi }_{i}} \right\rangle =\sum\limits_{i}{{{w}_{i}}\left( H\left| {{\Psi }_{i}} \right\rangle \left\langle  {{\Psi }_{i}} \right|-\left| {{\Psi }_{i}} \right\rangle \left\langle  {{\Psi }_{i}} \right|H \right)} \\
\end{align}

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ρ ⁢ ( t ) = ∑ i w i ⁢ | Ψ i ⟩ ⁢ ⟨ Ψ i | S .GL: i ℏ ∂ t | Ψ i ⟩ = H | Ψ i ⟩   | ∑ i w i ⟨ Ψ i | h .c : - i ℏ ∂ t ⟨ Ψ i | = ⟨ Ψ i | H   | ∑ i w i ⟨ Ψ i | ⇒ i ⁢ ℏ ⁢ ∂ t ⁢ ∑ i w i ⁢ ⟨ Ψ i | ⁢ | Ψ i ⟩ = ∑ i w i ⁢ ( H ⁢ | Ψ i ⟩ ⁢ ⟨ Ψ i | - | Ψ i ⟩ ⁢ ⟨ Ψ i | ⁢ H ) missing-subexpression 𝜌 𝑡 subscript 𝑖 subscript 𝑤 𝑖 ket subscript Ψ 𝑖 bra subscript Ψ 𝑖 missing-subexpression fragments fragments S .GL: i Planck-constant-over-2-pi subscript 𝑡 | subscript Ψ 𝑖 ⟩ H | subscript Ψ 𝑖 ⟩ italic-  | 𝑖 w 𝑖 ⟨ subscript Ψ 𝑖 | missing-subexpression fragments h .c : i Planck-constant-over-2-pi subscript 𝑡 fragments ⟨ subscript Ψ 𝑖 | fragments ⟨ subscript Ψ 𝑖 | H italic-  | subscript 𝑖 subscript 𝑤 𝑖 fragments ⟨ subscript Ψ 𝑖 | missing-subexpression ⇒ absent 𝑖 Planck-constant-over-2-pi subscript 𝑡 subscript 𝑖 subscript 𝑤 𝑖 bra subscript Ψ 𝑖 ket subscript Ψ 𝑖 subscript 𝑖 subscript 𝑤 𝑖 𝐻 ket subscript Ψ 𝑖 quantum-operator-product subscript Ψ 𝑖 subscript Ψ 𝑖 bra subscript Ψ 𝑖 𝐻 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\rho\left(t\right)=% \sum\limits_{i}{{{w}_{i}}\left|{{\Psi}_{i}}\right\rangle\left\langle{{\Psi}_{i% }}\right|}\\ &\displaystyle\text{S}\text{.GL:}i\hbar{{\partial}_{t}}\left|{{\Psi}_{i}}% \right\rangle=H\left|{{\Psi}_{i}}\right\rangle\quad|\sum\limits_{i}{{{w}_{i}}% \left\langle{{\Psi}_{i}}\right|}\\ &\displaystyle\text{h}\text{.c}:-i\hbar{{\partial}_{t}}\left\langle{{\Psi}_{i}% }\right|=\left\langle{{\Psi}_{i}}\right|H\quad|\sum\limits_{i}{{{w}_{i}}\left% \langle{{\Psi}_{i}}\right|}\\ &\displaystyle\Rightarrow i\hbar{{\partial}_{t}}\sum\limits_{i}{{{w}_{i}}\left% \langle{{\Psi}_{i}}\right|}\left|{{\Psi}_{i}}\right\rangle=\sum\limits_{i}{{{w% }_{i}}\left(H\left|{{\Psi}_{i}}\right\rangle\left\langle{{\Psi}_{i}}\right|-% \left|{{\Psi}_{i}}\right\rangle\left\langle{{\Psi}_{i}}\right|H\right)}\\ \end{aligned}}}

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ρ(t)=∑iwi|Ψi⟩⟨Ψi|S.GL:iℏ∂t|Ψi⟩=H|Ψi⟩|∑iwi⟨Ψi|h.c:−iℏ∂t⟨Ψi|=⟨Ψi|H|∑iwi⟨Ψi|⇒iℏ∂t∑iwi⟨Ψi||Ψi⟩=∑iwi(H|Ψi⟩⟨Ψi|−|Ψi⟩⟨Ψi|H)

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