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* Page found: Dynamik im Schrödinger- Heisenberg- und Wechselwirkungsbild (eq math.1642.62)

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\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}

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⇒ d d ⁢ t ⁢ | Ψ ⟩ W = i ℏ ⁢ H ^ 0 ⁢ e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ | Ψ ⟩ t + e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ ∂ ∂ ⁡ t ⁢ | Ψ ⟩ t ∂ ∂ ⁡ t ⁢ | Ψ ⟩ t = 1 i ⁢ ℏ ⁢ H ^ S ⁢ | Ψ ⟩ t = 1 i ⁢ ℏ ⁢ H ^ S ⁢ e - i ℏ ⁢ H ^ 0 ⁢ t ⁢ | Ψ ⟩ W ⇒ d d ⁢ t ⁢ | Ψ ⟩ W = 1 i ⁢ ℏ ⁢ ( - H ^ 0 ⁢ | Ψ ⟩ W + H ^ W ⁢ | Ψ ⟩ W ) w ⁢ e ⁢ g ⁢ e ⁢ n e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ H ^ S ⁢ e - i ℏ ⁢ H ^ 0 ⁢ t = H ^ W e + i ℏ ⁢ H ^ 0 ⁢ t ⁢ | Ψ ⟩ t = | Ψ ⟩ W missing-subexpression ⇒ absent 𝑑 𝑑 𝑡 subscript ket Ψ 𝑊 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ket Ψ 𝑡 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 𝑡 subscript ket Ψ 𝑡 missing-subexpression 𝑡 subscript ket Ψ 𝑡 1 𝑖 Planck-constant-over-2-pi subscript ^ 𝐻 𝑆 subscript ket Ψ 𝑡 1 𝑖 Planck-constant-over-2-pi subscript ^ 𝐻 𝑆 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ket Ψ 𝑊 missing-subexpression ⇒ absent 𝑑 𝑑 𝑡 subscript ket Ψ 𝑊 1 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 subscript ket Ψ 𝑊 subscript ^ 𝐻 𝑊 subscript ket Ψ 𝑊 missing-subexpression 𝑤 𝑒 𝑔 𝑒 𝑛 missing-subexpression superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ^ 𝐻 𝑆 superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ^ 𝐻 𝑊 missing-subexpression superscript 𝑒 𝑖 Planck-constant-over-2-pi superscript ^ 𝐻 0 𝑡 subscript ket Ψ 𝑡 subscript ket Ψ 𝑊 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\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}}\\ &\displaystyle\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}}\\ &\displaystyle\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)\\ &\displaystyle wegen\\ &\displaystyle{{e}^{+\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}{{{\hat{H}}}_{S}}{{e}^% {-\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}={{{\hat{H}}}_{W}}\\ &\displaystyle{{e}^{+\frac{i}{\hbar}{{{\hat{H}}}^{0}}t}}{{\left|\Psi\right% \rangle}_{t}}={{\left|\Psi\right\rangle}_{W}}\\ \end{aligned}}}

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⇒ddt|Ψ⟩W=iℏH^0e+iℏH^0t|Ψ⟩t+e+iℏH^0t∂∂t|Ψ⟩t∂∂t|Ψ⟩t=1iℏH^S|Ψ⟩t=1iℏH^Se−iℏH^0t|Ψ⟩W⇒ddt|Ψ⟩W=1iℏ(−H^0|Ψ⟩W+H^W|Ψ⟩W)wegene+iℏH^0tH^Se−iℏH^0t=H^We+iℏH^0t|Ψ⟩t=|Ψ⟩W

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