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* Page found: Dynamik des statistischen Operators (eq math.2250.1)

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Dynamik des statistischen Operators Eq: math.2250.1

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TeX (original user input):

\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}

TeX (checked):

{\begin{aligned}&\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{aligned}}

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\begin{aligned} ρ (t) = \sum_{i} w_{i} | Ψ_{i} ⟩ ⟨ Ψ_{i} | \\ S .GL: i ℏ \partial_{t} | Ψ_{i} ⟩ = H | Ψ_{i} ⟩ | \sum_{i} w_{i} ⟨ Ψ_{i} | \\ h .c : - i ℏ \partial_{t} ⟨ Ψ_{i} | = ⟨ Ψ_{i} | H | \sum_{i} w_{i} ⟨ Ψ_{i} | \\ \Rightarrow i ℏ \partial_{t} \sum_{i} w_{i} ⟨ Ψ_{i} | | Ψ_{i} ⟩ = \sum_{i} w_{i} (H | Ψ_{i} ⟩ ⟨ Ψ_{i} | - | Ψ_{i} ⟩ ⟨ Ψ_{i} | H) \end{aligned}

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Identifiers

$ρ$

$t$

$i$

$w_{i}$

$Ψ_{i}$

$Ψ_{i}$

$i$

$t$

$Ψ_{i}$

$H$

$Ψ_{i}$

$i$

$w_{i}$

$Ψ_{i}$

$i$

$t$

$Ψ_{i}$

$Ψ_{i}$

$H$

$i$

$w_{i}$

$Ψ_{i}$

$i$

$t$

$i$

$w_{i}$

$Ψ_{i}$

$Ψ_{i}$

$i$

$w_{i}$

$H$

$Ψ_{i}$

$Ψ_{i}$

$Ψ_{i}$

$Ψ_{i}$

$H$

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