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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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\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}

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d t ⁢ 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 , ρ ~ ] - 𝔦 ⁢ U 0 † ⁢ [ H , ρ ] ⁢ U 0 = 𝔦 ⁢ [ H 0 , ρ ~ ] - 𝔦 ⁢ U 0 † ⁢ [ H 0 + H I , ρ ] ⁢ U 0 = 𝔦 ⁢ [ H 0 , ρ ~ ] - 𝔦 ⁢ [ H 0 , ρ ~ ] - 𝔦 ⁢ U 0 † ⁢ [ H I , ρ ] ⁢ U 0 = - 𝔦 ⁢ [ H ~ I , ρ ~ ] missing-subexpression subscript 𝑑 𝑡 ~ 𝜌 subscript 𝑑 𝑡 superscript subscript 𝑈 0 † 𝜌 subscript 𝑈 0 missing-subexpression absent 𝔦 subscript 𝐻 0 superscript subscript 𝑈 0 † 𝜌 subscript 𝑈 0 𝑖 superscript subscript 𝑈 0 † 𝜌 subscript 𝐻 0 subscript 𝑈 0 superscript subscript 𝑈 0 † subscript 𝑑 𝑡 𝜌 subscript 𝑈 0 missing-subexpression absent 𝔦 subscript 𝐻 0 ~ 𝜌 𝔦 superscript subscript 𝑈 0 † 𝐻 𝜌 subscript 𝑈 0 missing-subexpression absent 𝔦 subscript 𝐻 0 ~ 𝜌 𝔦 superscript subscript 𝑈 0 † subscript 𝐻 0 subscript 𝐻 𝐼 𝜌 subscript 𝑈 0 missing-subexpression absent 𝔦 subscript 𝐻 0 ~ 𝜌 𝔦 subscript 𝐻 0 ~ 𝜌 𝔦 superscript subscript 𝑈 0 † subscript 𝐻 𝐼 𝜌 subscript 𝑈 0 missing-subexpression absent 𝔦 subscript ~ 𝐻 𝐼 ~ 𝜌 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{d}_{t}}\tilde{\rho% }={{d}_{t}}\left(U_{0}^{\dagger}\rho{{U}_{0}}\right)\\ &\displaystyle=\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}}\\ &\displaystyle=\mathfrak{i}\left[{{H}_{0}},\tilde{\rho}\right]-\mathfrak{i}U_{% 0}^{\dagger}\left[H,\rho\right]{{U}_{0}}\\ &\displaystyle=\mathfrak{i}\left[{{H}_{0}},\tilde{\rho}\right]-\mathfrak{i}U_{% 0}^{\dagger}\left[{{H}_{0}}+{{H}_{I}},\rho\right]{{U}_{0}}\\ &\displaystyle=\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}}\\ &\displaystyle=-\mathfrak{i}\left[{{{\tilde{H}}}_{I}},\tilde{\rho}\right]\\ \end{aligned}}}

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dtρ~=dt(U0†ρU0)=iH0U0†ρU0−iU0†ρH0U0+U0†dt(ρ)U0=i[H0,ρ~]−iU0†[H,ρ]U0=i[H0,ρ~]−iU0†[H0+HI,ρ]U0=i[H0,ρ~]−i[H0,ρ~]−iU0†[HI,ρ]U0=−i[H~I,ρ~]

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