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Display information for equation id:math.1008.25 on revision:1008

* Page found: Master Gleichung (eq math.1008.25)

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Master Gleichung Eq: math.1008.25

Hash: 8841e534c6871532fad0fdf12d2f7168

TeX (original user input):

\begin{align}
  & {{d}_{t}}\tilde{\rho }={{d}_{t}}\left( U_{0}^{\dagger }\rho {{U}_{0}} \right) \\
 & =\mathfrak{i}{{H}_{0}}U_{0}^{\dagger }\rho {{U}_{0}}-iU_{0}^{\dagger }\rho {{H}_{0}}{{U}_{0}}+U_{0}^{\dagger }{{d}_{t}}\left( \rho  \right){{U}_{0}} \\
 & =\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}U_{0}^{\dagger }\left[ H,\rho  \right]{{U}_{0}} \\
 & =\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}U_{0}^{\dagger }\left[ {{H}_{0}}+{{H}_{I}},\rho  \right]{{U}_{0}} \\
 & =\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}U_{0}^{\dagger }\left[ {{H}_{I}},\rho  \right]{{U}_{0}} \\
 & =-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},\tilde{\rho } \right] \\
\end{align}

TeX (checked):

{\begin{aligned}&{{d}_{t}}{\tilde {\rho }}={{d}_{t}}\left(U_{0}^{\dagger }\rho {{U}_{0}}\right)\\&={\mathfrak {i}}{{H}_{0}}U_{0}^{\dagger }\rho {{U}_{0}}-iU_{0}^{\dagger }\rho {{H}_{0}}{{U}_{0}}+U_{0}^{\dagger }{{d}_{t}}\left(\rho \right){{U}_{0}}\\&={\mathfrak {i}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}U_{0}^{\dagger }\left[H,\rho \right]{{U}_{0}}\\&={\mathfrak {i}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}U_{0}^{\dagger }\left[{{H}_{0}}+{{H}_{I}},\rho \right]{{U}_{0}}\\&={\mathfrak {i}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}\left[{{H}_{0}},{\tilde {\rho }}\right]-{\mathfrak {i}}U_{0}^{\dagger }\left[{{H}_{I}},\rho \right]{{U}_{0}}\\&=-{\mathfrak {i}}\left[{{\tilde {H}}_{I}},{\tilde {\rho }}\right]\\\end{aligned}}

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\begin{aligned} d_{t} \tilde{ρ} = d_{t} (U_{0}^{†} ρ U_{0}) \\ = i H_{0} U_{0}^{†} ρ U_{0} - i U_{0}^{†} ρ H_{0} U_{0} + U_{0}^{†} d_{t} (ρ) U_{0} \\ = i [H_{0}, \tilde{ρ}] - i U_{0}^{†} [H, ρ] U_{0} \\ = i [H_{0}, \tilde{ρ}] - i U_{0}^{†} [H_{0} + H_{I}, ρ] U_{0} \\ = i [H_{0}, \tilde{ρ}] - i [H_{0}, \tilde{ρ}] - i U_{0}^{†} [H_{I}, ρ] U_{0} \\ = - i [{\tilde{H}}_{I}, \tilde{ρ}] \end{aligned}

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Identifiers

$d_{t}$

$\tilde{ρ}$

$d_{t}$

$U$

$ρ$

$U_{0}$

$i$

$H_{0}$

$U$

$ρ$

$U_{0}$

$i$

$U$

$ρ$

$H_{0}$

$U_{0}$

$U$

$d_{t}$

$ρ$

$U_{0}$

$i$

$H_{0}$

$\tilde{ρ}$

$i$

$U$

$H$

$ρ$

$U_{0}$

$i$

$H_{0}$

$\tilde{ρ}$

$i$

$U$

$H_{0}$

$H_{I}$

$ρ$

$U_{0}$

$i$

$H_{0}$

$\tilde{ρ}$

$i$

$H_{0}$

$\tilde{ρ}$

$i$

$U$

$H_{I}$

$ρ$

$U_{0}$

$i$

${\tilde{H}}_{I}$

$\tilde{ρ}$

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