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Display information for equation id:math.1732.82 on revision:1732

* Page found: Zeitabhängige Störungsrechnung (eq math.1732.82)

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Zeitabhängige Störungsrechnung Eq: math.1732.82

Hash: 79b02656cb096b0d8ab6ad4c8b3ad829

TeX (original user input):

\begin{align}
& {{g}_{n}}(t)=-\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}-\hbar \omega  \right)\tau }{\hbar } \right)}}\left\langle  n \right|\hat{F}\left| {{n}_{0}} \right\rangle -\frac{i}{\hbar }\int_{0}^{t}{d\tau }{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}+\hbar \omega  \right)\tau }{\hbar } \right)}}\left\langle  n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle  \\
& \Rightarrow {{g}_{n}}(t)=-\left\langle  n \right|\hat{F}\left| {{n}_{0}} \right\rangle \left\{ \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}-\hbar \omega  \right)t}{\hbar } \right)-1}}}{{{E}_{n}}-{{E}_{n0}}-\hbar \omega } \right\}-\left\langle  n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle \left\{ \frac{{{e}^{\left( i\frac{\left( {{E}_{n}}-{{E}_{n0}}+\hbar \omega  \right)t}{\hbar } \right)-1}}}{{{E}_{n}}-{{E}_{n0}}+\hbar \omega } \right\} \\
\end{align}

TeX (checked):

{\begin{aligned}&{{g}_{n}}(t)=-{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{e}^{\left(i{\frac {\left({{E}_{n}}-{{E}_{n0}}-\hbar \omega \right)\tau }{\hbar }}\right)}}\left\langle n\right|{\hat {F}}\left|{{n}_{0}}\right\rangle -{\frac {i}{\hbar }}\int _{0}^{t}{d\tau }{{e}^{\left(i{\frac {\left({{E}_{n}}-{{E}_{n0}}+\hbar \omega \right)\tau }{\hbar }}\right)}}\left\langle n\right|{{\hat {F}}^{+}}\left|{{n}_{0}}\right\rangle \\&\Rightarrow {{g}_{n}}(t)=-\left\langle n\right|{\hat {F}}\left|{{n}_{0}}\right\rangle \left\{{\frac {{e}^{\left(i{\frac {\left({{E}_{n}}-{{E}_{n0}}-\hbar \omega \right)t}{\hbar }}\right)-1}}{{{E}_{n}}-{{E}_{n0}}-\hbar \omega }}\right\}-\left\langle n\right|{{\hat {F}}^{+}}\left|{{n}_{0}}\right\rangle \left\{{\frac {{e}^{\left(i{\frac {\left({{E}_{n}}-{{E}_{n0}}+\hbar \omega \right)t}{\hbar }}\right)-1}}{{{E}_{n}}-{{E}_{n0}}+\hbar \omega }}\right\}\\\end{aligned}}

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\begin{aligned} g_{n} (t) = - \frac{i}{ℏ} \int_{0}^{t} d τ e^{(i \frac{(E_{n} - E_{n 0} - ℏ ω) τ}{ℏ})} ⟨ n | \hat{F} | n_{0} ⟩ - \frac{i}{ℏ} \int_{0}^{t} d τ e^{(i \frac{(E_{n} - E_{n 0} + ℏ ω) τ}{ℏ})} ⟨ n | {\hat{F}}^{+} | n_{0} ⟩ \\ \Rightarrow g_{n} (t) = - ⟨ n | \hat{F} | n_{0} ⟩ {\frac{e^{(i \frac{(E_{n} - E_{n 0} - ℏ ω) t}{ℏ}) - 1}}{E_{n} - E_{n 0} - ℏ ω}} - ⟨ n | {\hat{F}}^{+} | n_{0} ⟩ {\frac{e^{(i \frac{(E_{n} - E_{n 0} + ℏ ω) t}{ℏ}) - 1}}{E_{n} - E_{n 0} + ℏ ω}} \end{aligned}

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Identifiers

$g_{n}$

$t$

$i$

$t$

$τ$

$e$

$i$

$E_{n}$

$E_{n 0}$

$ω$

$τ$

$n$

$\hat{F}$

$n_{0}$

$i$

$t$

$τ$

$e$

$i$

$E_{n}$

$E_{n 0}$

$ω$

$τ$

$n$

$\hat{F}$

$n_{0}$

$g_{n}$

$t$

$n$

$\hat{F}$

$n_{0}$

$e$

$i$

$E_{n}$

$E_{n 0}$

$ω$

$t$

$E_{n}$

$E_{n 0}$

$ω$

$n$

$\hat{F}$

$n_{0}$

$e$

$i$

$E_{n}$

$E_{n 0}$

$ω$

$t$

$E_{n}$

$E_{n 0}$

$ω$

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