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Display information for equation id:math.1728.14 on revision:1728

* Page found: Zeitabhängige Störungsrechnung (eq math.1728.14)

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Hash: ca71560ec384381e41b97ef64b58bd73

TeX (original user input):

\begin{align}

& i\hbar \sum\limits_{n}{{}}\frac{d}{dt}{{c}_{n}}(t)\left\langle  m \right|\left| n \right\rangle =\sum\limits_{n}{{}}{{c}_{n}}(t)\left\langle  m \right|\left( {{{\hat{H}}}_{0}}+{{{\hat{H}}}^{1}}(t) \right)\left| n \right\rangle =\sum\limits_{n}{{}}{{c}_{n}}(t)\left\langle  m \right|\left( {{E}_{n}}+{{{\hat{H}}}^{1}}(t) \right)\left| n \right\rangle  \\

& =\sum\limits_{n}{{}}{{c}_{n}}(t)\left( \left\langle  m \right|{{E}_{n}}\left| n \right\rangle +\left\langle  m \right|{{{\hat{H}}}^{1}}(t)\left| n \right\rangle  \right)=\sum\limits_{n}{{}}{{c}_{n}}(t){{E}_{n}}{{\delta }_{mn}}+\sum\limits_{n}{{}}{{c}_{n}}(t)\left\langle  m \right|{{{\hat{H}}}^{1}}(t)\left| n \right\rangle  \\

& \Rightarrow i\hbar \sum\limits_{n}{{}}\frac{d}{dt}{{c}_{n}}(t)\left\langle  m \right|\left| n \right\rangle ={{c}_{m}}(t){{E}_{m}}+\sum\limits_{n}{{}}{{c}_{n}}(t)\left\langle  m \right|{{{\hat{H}}}^{1}}(t)\left| n \right\rangle =i\hbar \frac{d}{dt}{{c}_{m}}(t) \\

\end{align}

TeX (checked):

{\begin{aligned}&i\hbar \sum \limits _{n}{}{\frac {d}{dt}}{{c}_{n}}(t)\left\langle m\right|\left|n\right\rangle =\sum \limits _{n}{}{{c}_{n}}(t)\left\langle m\right|\left({{\hat {H}}_{0}}+{{\hat {H}}^{1}}(t)\right)\left|n\right\rangle =\sum \limits _{n}{}{{c}_{n}}(t)\left\langle m\right|\left({{E}_{n}}+{{\hat {H}}^{1}}(t)\right)\left|n\right\rangle \\&=\sum \limits _{n}{}{{c}_{n}}(t)\left(\left\langle m\right|{{E}_{n}}\left|n\right\rangle +\left\langle m\right|{{\hat {H}}^{1}}(t)\left|n\right\rangle \right)=\sum \limits _{n}{}{{c}_{n}}(t){{E}_{n}}{{\delta }_{mn}}+\sum \limits _{n}{}{{c}_{n}}(t)\left\langle m\right|{{\hat {H}}^{1}}(t)\left|n\right\rangle \\&\Rightarrow i\hbar \sum \limits _{n}{}{\frac {d}{dt}}{{c}_{n}}(t)\left\langle m\right|\left|n\right\rangle ={{c}_{m}}(t){{E}_{m}}+\sum \limits _{n}{}{{c}_{n}}(t)\left\langle m\right|{{\hat {H}}^{1}}(t)\left|n\right\rangle =i\hbar {\frac {d}{dt}}{{c}_{m}}(t)\\\end{aligned}}

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\begin{aligned} i ℏ \sum_{n} \frac{d}{d t} c_{n} (t) ⟨ m | | n ⟩ = \sum_{n} c_{n} (t) ⟨ m | ({\hat{H}}_{0} + {\hat{H}}^{1} (t)) | n ⟩ = \sum_{n} c_{n} (t) ⟨ m | (E_{n} + {\hat{H}}^{1} (t)) | n ⟩ \\ = \sum_{n} c_{n} (t) (⟨ m | E_{n} | n ⟩ + ⟨ m | {\hat{H}}^{1} (t) | n ⟩) = \sum_{n} c_{n} (t) E_{n} δ_{m n} + \sum_{n} c_{n} (t) ⟨ m | {\hat{H}}^{1} (t) | n ⟩ \\ \Rightarrow i ℏ \sum_{n} \frac{d}{d t} c_{n} (t) ⟨ m | | n ⟩ = c_{m} (t) E_{m} + \sum_{n} c_{n} (t) ⟨ m | {\hat{H}}^{1} (t) | n ⟩ = i ℏ \frac{d}{d t} c_{m} (t) \end{aligned}

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$i$

$n$

$d$

$d$

$t$

$c_{n}$

$t$

$m$

$n$

$n$

$c_{n}$

$t$

$m$

${\hat{H}}_{0}$

$\hat{H}$

$t$

$n$

$n$

$c_{n}$

$t$

$m$

$E_{n}$

$\hat{H}$

$t$

$n$

$n$

$c_{n}$

$t$

$m$

$E_{n}$

$n$

$m$

$\hat{H}$

$t$

$n$

$n$

$c_{n}$

$t$

$E_{n}$

$δ_{m n}$

$n$

$c_{n}$

$t$

$m$

$\hat{H}$

$t$

$n$

$i$

$n$

$d$

$d$

$t$

$c_{n}$

$t$

$m$

$n$

$c_{m}$

$t$

$E_{m}$

$n$

$c_{n}$

$t$

$m$

$\hat{H}$

$t$

$n$

$i$

$d$

$d$

$t$

$c_{m}$

$t$

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