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Display information for equation id:math.1649.62 on revision:1649
* Page found: Dynamik im Schrödinger- Heisenberg- und Wechselwirkungsbild (eq math.1649.62)
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Hash: d4717b69a27c8c4579d14cbb4abb8eaf
TeX (original user input):
\begin{align}
& \Rightarrow \frac{d}{dt}{{\left| \Psi \right\rangle }_{W}}=\frac{i}{\hbar }{{{\hat{H}}}^{0}}{{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{\left| \Psi \right\rangle }_{t}}+{{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}\frac{\partial }{\partial t}{{\left| \Psi \right\rangle }_{t}} \\
& \frac{\partial }{\partial t}{{\left| \Psi \right\rangle }_{t}}=\frac{1}{i\hbar }{{{\hat{H}}}_{S}}{{\left| \Psi \right\rangle }_{t}}=\frac{1}{i\hbar }{{{\hat{H}}}_{S}}{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{\left| \Psi \right\rangle }_{W}} \\
& \Rightarrow \frac{d}{dt}{{\left| \Psi \right\rangle }_{W}}=\frac{1}{i\hbar }\left( -{{{\hat{H}}}^{0}}{{\left| \Psi \right\rangle }_{W}}+{{{\hat{H}}}_{W}}{{\left| \Psi \right\rangle }_{W}} \right) \\
& wegen \\
& {{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{{\hat{H}}}_{S}}{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}={{{\hat{H}}}_{W}} \\
& {{e}^{+\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}{{\left| \Psi \right\rangle }_{t}}={{\left| \Psi \right\rangle }_{W}} \\
\end{align}
TeX (checked):
{\begin{aligned}&\Rightarrow {\frac {d}{dt}}{{\left|\Psi \right\rangle }_{W}}={\frac {i}{\hbar }}{{\hat {H}}^{0}}{{e}^{+{\frac {i}{\hbar }}{{\hat {H}}^{0}}t}}{{\left|\Psi \right\rangle }_{t}}+{{e}^{+{\frac {i}{\hbar }}{{\hat {H}}^{0}}t}}{\frac {\partial }{\partial t}}{{\left|\Psi \right\rangle }_{t}}\\&{\frac {\partial }{\partial t}}{{\left|\Psi \right\rangle }_{t}}={\frac {1}{i\hbar }}{{\hat {H}}_{S}}{{\left|\Psi \right\rangle }_{t}}={\frac {1}{i\hbar }}{{\hat {H}}_{S}}{{e}^{-{\frac {i}{\hbar }}{{\hat {H}}^{0}}t}}{{\left|\Psi \right\rangle }_{W}}\\&\Rightarrow {\frac {d}{dt}}{{\left|\Psi \right\rangle }_{W}}={\frac {1}{i\hbar }}\left(-{{\hat {H}}^{0}}{{\left|\Psi \right\rangle }_{W}}+{{\hat {H}}_{W}}{{\left|\Psi \right\rangle }_{W}}\right)\\&wegen\\&{{e}^{+{\frac {i}{\hbar }}{{\hat {H}}^{0}}t}}{{\hat {H}}_{S}}{{e}^{-{\frac {i}{\hbar }}{{\hat {H}}^{0}}t}}={{\hat {H}}_{W}}\\&{{e}^{+{\frac {i}{\hbar }}{{\hat {H}}^{0}}t}}{{\left|\Psi \right\rangle }_{t}}={{\left|\Psi \right\rangle }_{W}}\\\end{aligned}}
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