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Display information for equation id:math.1732.116 on revision:1732

* Page found: Zeitabhängige Störungsrechnung (eq math.1732.116)

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Zeitabhängige Störungsrechnung Eq: math.1732.116

Hash: 4c60a138ea2e91c6428b103e525a1b58

TeX (original user input):

\begin{align}
& {{\left| \Psi  \right\rangle }_{W}}(t)={{\left| \Psi  \right\rangle }_{W}}(t=0)-\frac{i}{\hbar }\int_{0}^{t}{{}}d\tau \left( {{{\hat{H}}}_{W}}^{1}(\tau ){{\left| \Psi  \right\rangle }_{W}}(\tau ) \right)\approx \left( 1-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{{\hat{H}}}_{W}}^{1}(\tau ) \right)\left| {{n}_{0}} \right\rangle  \\
& {{\left| \Psi  \right\rangle }_{W}}(t)\approx \left( 1-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{{\hat{H}}}_{W}}^{1}(\tau ) \right)\left| {{n}_{0}} \right\rangle =\left( 1-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\frac{i}{\hbar }{{{\hat{H}}}^{0}}\tau }}{{{\hat{H}}}_{S}}^{1}{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}\tau }} \right)\left| {{n}_{0}} \right\rangle  \\
\end{align}

TeX (checked):

{\begin{aligned}&{{\left|\Psi \right\rangle }_{W}}(t)={{\left|\Psi \right\rangle }_{W}}(t=0)-{\frac {i}{\hbar }}\int _{0}^{t}{}d\tau \left({{\hat {H}}_{W}}^{1}(\tau ){{\left|\Psi \right\rangle }_{W}}(\tau )\right)\approx \left(1-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{\hat {H}}_{W}}^{1}(\tau )\right)\left|{{n}_{0}}\right\rangle \\&{{\left|\Psi \right\rangle }_{W}}(t)\approx \left(1-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{\hat {H}}_{W}}^{1}(\tau )\right)\left|{{n}_{0}}\right\rangle =\left(1-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{e}^{{\frac {i}{\hbar }}{{\hat {H}}^{0}}\tau }}{{\hat {H}}_{S}}^{1}{{e}^{-{\frac {i}{\hbar }}{{\hat {H}}^{0}}\tau }}\right)\left|{{n}_{0}}\right\rangle \\\end{aligned}}

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\begin{aligned} {| Ψ ⟩}_{W} (t) = {| Ψ ⟩}_{W} (t = 0) - \frac{i}{ℏ} \int_{0}^{t} d τ ({\hat{H}}_{W}^{1} (τ) {| Ψ ⟩}_{W} (τ)) \approx (1 - \frac{i}{ℏ} \int_{0}^{t} d τ {\hat{H}}_{W}^{1} (τ)) | n_{0} ⟩ \\ {| Ψ ⟩}_{W} (t) \approx (1 - \frac{i}{ℏ} \int_{0}^{t} d τ {\hat{H}}_{W}^{1} (τ)) | n_{0} ⟩ = (1 - \frac{i}{ℏ} \int_{0}^{t} d τ e^{\frac{i}{ℏ} {\hat{H}}^{0} τ} {\hat{H}}_{S}^{1} e^{- \frac{i}{ℏ} {\hat{H}}^{0} τ}) | n_{0} ⟩ \end{aligned}

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Identifiers

$Ψ_{W}$

$t$

$Ψ_{W}$

$t$

$i$

$t$

$τ$

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

$τ$

$Ψ_{W}$

$τ$

$i$

$t$

$τ$

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

$τ$

$n_{0}$

$Ψ_{W}$

$t$

$i$

$t$

$τ$

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

$τ$

$n_{0}$

$i$

$t$

$τ$

$e$

$i$

$\hat{H}$

$τ$

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

$e$

$i$

$\hat{H}$

$τ$

$n_{0}$

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