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Display information for equation id:math.348.7 on revision:348

* Page found: Liouville-von-Neumann-Gleichung (eq math.348.7)

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Liouville-von-Neumann-Gleichung Eq: math.349.7

Hash: d4c8b71610e20c13b71edb8f92f19d75

TeX (original user input):

\begin{align}
  & \dot{\rho }={{\partial }_{t}}\left( \left| \Psi \left( t \right) \right\rangle \left\langle  \Psi \left( t \right) \right| \right) \\ 
 & =\left( {{\partial }_{t}}\left| \Psi \left( t \right) \right\rangle  \right)\left\langle  \Psi \left( t \right) \right|+\left| \Psi \left( t \right) \right\rangle \left( {{\partial }_{t}}\left\langle  \Psi \left( t \right) \right| \right) \\ 
 & =-\mathfrak{i}\hat{H}\left| \Psi \left( t \right) \right\rangle \left\langle  \Psi \left( t \right) \right|+\left| \Psi \left( t \right) \right\rangle \left\langle  \Psi \left( t \right) \right|\mathfrak{i}\hat{H} \\ 
 & =-\mathfrak{i}\left( \hat{H}\rho -\rho \hat{H} \right)\equiv -\mathfrak{i}\left[ \hat{H},\rho  \right]=\mathfrak{i}\left[ \rho ,\hat{H} \right]  
\end{align}

TeX (checked):

{\begin{aligned}&{\dot {\rho }}={{\partial }_{t}}\left(\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|\right)\\&=\left({{\partial }_{t}}\left|\Psi \left(t\right)\right\rangle \right)\left\langle \Psi \left(t\right)\right|+\left|\Psi \left(t\right)\right\rangle \left({{\partial }_{t}}\left\langle \Psi \left(t\right)\right|\right)\\&=-{\mathfrak {i}}{\hat {H}}\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|+\left|\Psi \left(t\right)\right\rangle \left\langle \Psi \left(t\right)\right|{\mathfrak {i}}{\hat {H}}\\&=-{\mathfrak {i}}\left({\hat {H}}\rho -\rho {\hat {H}}\right)\equiv -{\mathfrak {i}}\left[{\hat {H}},\rho \right]={\mathfrak {i}}\left[\rho ,{\hat {H}}\right]\end{aligned}}

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\begin{aligned} \dot{ρ} = \partial_{t} (| Ψ (t) ⟩ ⟨ Ψ (t) |) \\ = (\partial_{t} | Ψ (t) ⟩) ⟨ Ψ (t) | + | Ψ (t) ⟩ (\partial_{t} ⟨ Ψ (t) |) \\ = - i \hat{H} | Ψ (t) ⟩ ⟨ Ψ (t) | + | Ψ (t) ⟩ ⟨ Ψ (t) | i \hat{H} \\ = - i (\hat{H} ρ - ρ \hat{H}) \equiv - i [\hat{H}, ρ] = i [ρ, \hat{H}] \end{aligned}

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Identifiers

$\dot{ρ}$

$t$

$Ψ$

$t$

$Ψ$

$t$

$t$

$Ψ$

$t$

$Ψ$

$t$

$Ψ$

$t$

$t$

$Ψ$

$t$

$i$

$\hat{H}$

$Ψ$

$t$

$Ψ$

$t$

$Ψ$

$t$

$Ψ$

$t$

$i$

$\hat{H}$

$i$

$\hat{H}$

$ρ$

$ρ$

$\hat{H}$

$i$

$\hat{H}$

$ρ$

$i$

$ρ$

$\hat{H}$

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