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Display information for equation id:math.1729.119 on revision:1729

* Page found: Zeitabhängige Störungsrechnung (eq math.1729.119)

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TeX (original user input):

\begin{align}
& {{c}_{n}}(t)=\left\langle  n \right|\left| \Psi  \right\rangle =\left\langle  n \right|U(t,0)\left| {{n}_{0}} \right\rangle =\left\langle  n \right|{{e}^{-\frac{i}{\hbar }{{{\hat{H}}}^{0}}t}}\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  \\
& \Rightarrow {{c}_{n}}(t)={{e}^{-\frac{i}{\hbar }{{E}_{n}}t}}\left( {{\delta }_{n{{n}_{0}}}}-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\frac{i}{\hbar }{{E}_{n}}\tau }}\left\langle  n \right|{{{\hat{H}}}_{S}}^{1}\left| {{n}_{0}} \right\rangle {{e}^{-\frac{i}{\hbar }{{E}_{n0}}\tau }} \right) \\
& {{\delta }_{n{{n}_{0}}}}={{g}_{n}}^{(0)} \\
& -\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\frac{i}{\hbar }{{E}_{n}}\tau }}\left\langle  n \right|{{{\hat{H}}}_{S}}^{1}\left| {{n}_{0}} \right\rangle {{e}^{-\frac{i}{\hbar }{{E}_{n0}}\tau }}=\varepsilon {{g}_{n}}^{(1)} \\
& {{c}_{n}}(t)={{e}^{-\frac{i}{\hbar }{{E}_{n}}t}}\left( {{\delta }_{n{{n}_{0}}}}-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\frac{i}{\hbar }{{E}_{n}}\tau }}\left\langle  n \right|{{{\hat{H}}}_{S}}^{1}\left| {{n}_{0}} \right\rangle {{e}^{-\frac{i}{\hbar }{{E}_{n0}}\tau }} \right)={{e}^{-\frac{i}{\hbar }{{E}_{n}}t}}{{g}_{n}}(t) \\
\end{align}

TeX (checked):

{\begin{aligned}&{{c}_{n}}(t)=\left\langle n\right|\left|\Psi \right\rangle =\left\langle n\right|U(t,0)\left|{{n}_{0}}\right\rangle =\left\langle n\right|{{e}^{-{\frac {i}{\hbar }}{{\hat {H}}^{0}}t}}\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 \\&\Rightarrow {{c}_{n}}(t)={{e}^{-{\frac {i}{\hbar }}{{E}_{n}}t}}\left({{\delta }_{n{{n}_{0}}}}-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{e}^{{\frac {i}{\hbar }}{{E}_{n}}\tau }}\left\langle n\right|{{\hat {H}}_{S}}^{1}\left|{{n}_{0}}\right\rangle {{e}^{-{\frac {i}{\hbar }}{{E}_{n0}}\tau }}\right)\\&{{\delta }_{n{{n}_{0}}}}={{g}_{n}}^{(0)}\\&-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{e}^{{\frac {i}{\hbar }}{{E}_{n}}\tau }}\left\langle n\right|{{\hat {H}}_{S}}^{1}\left|{{n}_{0}}\right\rangle {{e}^{-{\frac {i}{\hbar }}{{E}_{n0}}\tau }}=\varepsilon {{g}_{n}}^{(1)}\\&{{c}_{n}}(t)={{e}^{-{\frac {i}{\hbar }}{{E}_{n}}t}}\left({{\delta }_{n{{n}_{0}}}}-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{e}^{{\frac {i}{\hbar }}{{E}_{n}}\tau }}\left\langle n\right|{{\hat {H}}_{S}}^{1}\left|{{n}_{0}}\right\rangle {{e}^{-{\frac {i}{\hbar }}{{E}_{n0}}\tau }}\right)={{e}^{-{\frac {i}{\hbar }}{{E}_{n}}t}}{{g}_{n}}(t)\\\end{aligned}}

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\begin{aligned} c_{n} (t) = ⟨ n | | Ψ ⟩ = ⟨ n | U (t, 0) | n_{0} ⟩ = ⟨ n | e^{- \frac{i}{ℏ} {\hat{H}}^{0} t} (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} ⟩ \\ \Rightarrow c_{n} (t) = e^{- \frac{i}{ℏ} E_{n} t} (δ_{n n_{0}} - \frac{i}{ℏ} \int_{0}^{t} d τ e^{\frac{i}{ℏ} E_{n} τ} ⟨ n | {\hat{H}}_{S}^{1} | n_{0} ⟩ e^{- \frac{i}{ℏ} E_{n 0} τ}) \\ δ_{n n_{0}} = {g_{n}}^{(0)} \\ - \frac{i}{ℏ} \int_{0}^{t} d τ e^{\frac{i}{ℏ} E_{n} τ} ⟨ n | {\hat{H}}_{S}^{1} | n_{0} ⟩ e^{- \frac{i}{ℏ} E_{n 0} τ} = ε {g_{n}}^{(1)} \\ c_{n} (t) = e^{- \frac{i}{ℏ} E_{n} t} (δ_{n n_{0}} - \frac{i}{ℏ} \int_{0}^{t} d τ e^{\frac{i}{ℏ} E_{n} τ} ⟨ n | {\hat{H}}_{S}^{1} | n_{0} ⟩ e^{- \frac{i}{ℏ} E_{n 0} τ}) = e^{- \frac{i}{ℏ} E_{n} t} g_{n} (t) \end{aligned}

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Identifiers

$c_{n}$

$t$

$n$

$Ψ$

$n$

$U$

$t$

$n_{0}$

$n$

$e$

$i$

$\hat{H}$

$t$

$i$

$t$

$τ$

$e$

$i$

$\hat{H}$

$τ$

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

$e$

$i$

$\hat{H}$

$τ$

$n_{0}$

$c_{n}$

$t$

$e$

$i$

$E_{n}$

$t$

$δ_{n n_{0}}$

$i$

$t$

$τ$

$e$

$i$

$E_{n}$

$τ$

$n$

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

$n_{0}$

$e$

$i$

$E_{n 0}$

$τ$

$δ_{n n_{0}}$

$g_{n}$

$i$

$t$

$τ$

$e$

$i$

$E_{n}$

$τ$

$n$

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

$n_{0}$

$e$

$i$

$E_{n 0}$

$τ$

$ε$

$g_{n}$

$c_{n}$

$t$

$e$

$i$

$E_{n}$

$t$

$δ_{n n_{0}}$

$i$

$t$

$τ$

$e$

$i$

$E_{n}$

$τ$

$n$

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

$n_{0}$

$e$

$i$

$E_{n 0}$

$τ$

$e$

$i$

$E_{n}$

$t$

$g_{n}$

$t$

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