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Zeitabhängige Störungsrechnung Eq: math.1732.119

Hash: 887aed10e3ea7d66afff41a587e2b02e

TeX (original user input):

\begin{align}
& {{c}_{n}}(t)=\left\langle  n  |  \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|\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{matrix} 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{matrix}

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data-mjx-texclass="ORD"><mi>n</mi></mrow></msub><mi>τ</mi></mrow></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>n</mi><mo data-mjx-texclass="CLOSE">|</mo></mrow><msup><msub><mover><mi>H</mi><mo stretchy="false">̂</mo></mover><mrow data-mjx-texclass="ORD"><mi>S</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"><mi alternate="1">ℏ</mi></mrow></mfrac></mrow><msub><mi>E</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>n</mi><mn>0</mn></mrow></mrow></msub><mi>τ</mi></mrow></mrow></msup><mo 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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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