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Bilder Eq: math.916.18

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TeX (original user input):

\begin{align}
  & \left\langle {{{\hat{A}}}_{S}} \right\rangle ={}_{t}{{\left\langle \underbrace{\psi |{{{\hat{U}}}_{0}}}_{\left\langle  {{\psi }_{W}} \right|}\underbrace{\hat{U}_{0}^{+}{{{\hat{A}}}_{S}}{{{\hat{U}}}_{0}}}_{{{{\hat{A}}}_{W}}}\hat{U}_{0}^{+}|\psi  \right\rangle }_{t}}={}_{W}{{\left\langle \psi \left| {{{\hat{A}}}_{W}} \right|\psi  \right\rangle }_{W}} \\ 
 & {{d}_{t}}{{\left| \psi  \right\rangle }_{W}}=\frac{i}{\hbar }{{{\hat{H}}}_{0,S}}\hat{U}_{0}^{+}{{\left| \psi  \right\rangle }_{t}}+\hat{U}_{0}^{+}{{\partial }_{t}}{{\left| \psi  \right\rangle }_{t}} \\ 
 & {{\partial }_{t}}{{\left| \psi  \right\rangle }_{t}}=\frac{1}{i\hbar }{{{\hat{H}}}_{0,S}}{{\left| \psi  \right\rangle }_{t}}=\frac{1}{i\hbar }{{{\hat{H}}}_{0,S}}{{{\hat{U}}}_{0}}{{\left| \psi  \right\rangle }_{W}} \\ 
 & \Rightarrow {{d}_{t}}{{\left| \psi  \right\rangle }_{W}}=\frac{1}{\hbar i}\left( -{{{\hat{H}}}_{0,S}}+\underbrace{\hat{U}_{0}^{+}{{{\hat{H}}}_{0,S}}{{{\hat{U}}}_{0}}}_{{{{\hat{H}}}_{W}}={{H}_{0,S}}+{{H}_{1,S}}} \right){{\left| \psi  \right\rangle }_{W}}=\frac{1}{i\hbar }\left( {{{\hat{H}}}_{W}} \right){{\left| \psi  \right\rangle }_{W}} \\ 
 & \Rightarrow i\hbar {{d}_{t}}{{\left| \psi  \right\rangle }_{W}}={{{\hat{H}}}_{1,S}}{{\left| \psi  \right\rangle }_{W}} \\ 
\end{align}

TeX (checked):

{\begin{aligned}&\left\langle {{\hat {A}}_{S}}\right\rangle ={}_{t}{{\left\langle \underbrace {\psi |{{\hat {U}}_{0}}} _{\left\langle {{\psi }_{W}}\right|}\underbrace {{\hat {U}}_{0}^{+}{{\hat {A}}_{S}}{{\hat {U}}_{0}}} _{{\hat {A}}_{W}}{\hat {U}}_{0}^{+}|\psi \right\rangle }_{t}}={}_{W}{{\left\langle \psi \left|{{\hat {A}}_{W}}\right|\psi \right\rangle }_{W}}\\&{{d}_{t}}{{\left|\psi \right\rangle }_{W}}={\frac {i}{\hbar }}{{\hat {H}}_{0,S}}{\hat {U}}_{0}^{+}{{\left|\psi \right\rangle }_{t}}+{\hat {U}}_{0}^{+}{{\partial }_{t}}{{\left|\psi \right\rangle }_{t}}\\&{{\partial }_{t}}{{\left|\psi \right\rangle }_{t}}={\frac {1}{i\hbar }}{{\hat {H}}_{0,S}}{{\left|\psi \right\rangle }_{t}}={\frac {1}{i\hbar }}{{\hat {H}}_{0,S}}{{\hat {U}}_{0}}{{\left|\psi \right\rangle }_{W}}\\&\Rightarrow {{d}_{t}}{{\left|\psi \right\rangle }_{W}}={\frac {1}{\hbar i}}\left(-{{\hat {H}}_{0,S}}+\underbrace {{\hat {U}}_{0}^{+}{{\hat {H}}_{0,S}}{{\hat {U}}_{0}}} _{{{\hat {H}}_{W}}={{H}_{0,S}}+{{H}_{1,S}}}\right){{\left|\psi \right\rangle }_{W}}={\frac {1}{i\hbar }}\left({{\hat {H}}_{W}}\right){{\left|\psi \right\rangle }_{W}}\\&\Rightarrow i\hbar {{d}_{t}}{{\left|\psi \right\rangle }_{W}}={{\hat {H}}_{1,S}}{{\left|\psi \right\rangle }_{W}}\\\end{aligned}}

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\begin{aligned} ⟨ {\hat{A}}_{S} ⟩ =_{t} \underset{t}{⟨ \underset{⟨ ψ_{W} |}{\underset{⏟}{ψ | {\hat{U}}_{0}}} \underset{{\hat{A}}_{W}}{\underset{⏟}{{\hat{U}}_{0}^{+} {\hat{A}}_{S} {\hat{U}}_{0}}} {\hat{U}}_{0}^{+} | ψ ⟩} =_{W} {⟨ ψ | {\hat{A}}_{W} | ψ ⟩}_{W} \\ d_{t} {| ψ ⟩}_{W} = \frac{i}{ℏ} {\hat{H}}_{0, S} {\hat{U}}_{0}^{+} {| ψ ⟩}_{t} + {\hat{U}}_{0}^{+} \partial_{t} {| ψ ⟩}_{t} \\ \partial_{t} {| ψ ⟩}_{t} = \frac{1}{i ℏ} {\hat{H}}_{0, S} {| ψ ⟩}_{t} = \frac{1}{i ℏ} {\hat{H}}_{0, S} {\hat{U}}_{0} {| ψ ⟩}_{W} \\ \Rightarrow d_{t} {| ψ ⟩}_{W} = \frac{1}{ℏ i} (- {\hat{H}}_{0, S} + \underset{{\hat{H}}_{W} = H_{0, S} + H_{1, S}}{\underset{⏟}{{\hat{U}}_{0}^{+} {\hat{H}}_{0, S} {\hat{U}}_{0}}}) {| ψ ⟩}_{W} = \frac{1}{i ℏ} ({\hat{H}}_{W}) {| ψ ⟩}_{W} \\ \Rightarrow i ℏ d_{t} {| ψ ⟩}_{W} = {\hat{H}}_{1, S} {| ψ ⟩}_{W} \end{aligned}

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Identifiers

${\hat{A}}_{S}$

$t$

$ψ$

${\hat{U}}_{0}$

$ψ_{W}$

$\hat{U}$

${\hat{A}}_{S}$

${\hat{U}}_{0}$

${\hat{A}}_{W}$

$\hat{U}$

$ψ$

$t$

$W$

$ψ$

${\hat{A}}_{W}$

$ψ$

$W$

$d_{t}$

$ψ_{W}$

$i$

${\hat{H}}_{0, S}$

$\hat{U}$

$ψ_{t}$

$\hat{U}$

$t$

$ψ_{t}$

$t$

$ψ_{t}$

$i$

${\hat{H}}_{0, S}$

$ψ_{t}$

$i$

${\hat{H}}_{0, S}$

${\hat{U}}_{0}$

$ψ_{W}$

$d_{t}$

$ψ_{W}$

$i$

${\hat{H}}_{0, S}$

$\hat{U}$

${\hat{H}}_{0, S}$

${\hat{U}}_{0}$

${\hat{H}}_{W}$

$H_{0, S}$

$H_{1, S}$

$ψ_{W}$

$i$

${\hat{H}}_{W}$

$ψ_{W}$

$i$

$d_{t}$

$ψ_{W}$

${\hat{H}}_{1, S}$

$ψ_{W}$

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