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Display information for equation id:math.1732.86 on revision:1732

* Page found: Zeitabhängige Störungsrechnung (eq math.1732.86)

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Zeitabhängige Störungsrechnung Eq: math.1732.86

Hash: 493eab67641b10c9002dd297e1edd06e

TeX (original user input):

\begin{align}
& \Rightarrow {{\left| {{\left\langle  n | \Psi  \right\rangle }_{t}} \right|}^{2}}={{\left| {{g}_{n}} \right|}^{2}}=\frac{2\pi }{\hbar }{{\left| \left\langle  n \right|\hat{F}\left| {{n}_{0}} \right\rangle  \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega ) \\
& +\frac{2\pi }{\hbar }{{\left| \left\langle  n \right|{{{\hat{F}}}^{+}}\left| {{n}_{0}} \right\rangle  \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )+A{{e}^{-i\gamma }}\left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{-}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{+}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\} \\
& +A{{e}^{i\gamma }}\left\{ \frac{\left( {{e}^{\left( -i{{\Omega }^{+}}t \right)}}-1 \right)\left( {{e}^{\left( i{{\Omega }^{-}}t \right)}}-1 \right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}} \right\} \\
\end{align}

TeX (checked):

{\begin{aligned}&\Rightarrow {{\left|{{\left\langle n|\Psi \right\rangle }_{t}}\right|}^{2}}={{\left|{{g}_{n}}\right|}^{2}}={\frac {2\pi }{\hbar }}{{\left|\left\langle n\right|{\hat {F}}\left|{{n}_{0}}\right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}-\hbar \omega )\\&+{\frac {2\pi }{\hbar }}{{\left|\left\langle n\right|{{\hat {F}}^{+}}\left|{{n}_{0}}\right\rangle \right|}^{2}}t\delta ({{E}_{n}}-{{E}_{n0}}+\hbar \omega )+A{{e}^{-i\gamma }}\left\{{\frac {\left({{e}^{\left(-i{{\Omega }^{-}}t\right)}}-1\right)\left({{e}^{\left(i{{\Omega }^{+}}t\right)}}-1\right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}}}\right\}\\&+A{{e}^{i\gamma }}\left\{{\frac {\left({{e}^{\left(-i{{\Omega }^{+}}t\right)}}-1\right)\left({{e}^{\left(i{{\Omega }^{-}}t\right)}}-1\right)}{{{\hbar }^{2}}{{\Omega }^{+}}{{\Omega }^{-}}}}\right\}\\\end{aligned}}

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\begin{aligned} \Rightarrow {| {⟨ n | Ψ ⟩}_{t} |}^{2} = {| g_{n} |}^{2} = \frac{2 π}{ℏ} {| ⟨ n | \hat{F} | n_{0} ⟩ |}^{2} t δ (E_{n} - E_{n 0} - ℏ ω) \\ + \frac{2 π}{ℏ} {| ⟨ n | {\hat{F}}^{+} | n_{0} ⟩ |}^{2} t δ (E_{n} - E_{n 0} + ℏ ω) + A e^{- i γ} {\frac{(e^{(- i Ω^{-} t)} - 1) (e^{(i Ω^{+} t)} - 1)}{ℏ^{2} Ω^{+} Ω^{-}}} \\ + A e^{i γ} {\frac{(e^{(- i Ω^{+} t)} - 1) (e^{(i Ω^{-} t)} - 1)}{ℏ^{2} Ω^{+} Ω^{-}}} \end{aligned}

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Identifiers

$n$

$Ψ$

$t$

$g_{n}$

$π$

$n$

$\hat{F}$

$n_{0}$

$t$

$δ$

$E_{n}$

$E_{n 0}$

$ω$

$π$

$n$

$\hat{F}$

$n_{0}$

$t$

$δ$

$E_{n}$

$E_{n 0}$

$ω$

$A$

$e$

$i$

$γ$

$e$

$i$

$Ω$

$t$

$e$

$i$

$Ω$

$t$

$Ω$

$Ω$

$A$

$e$

$i$

$γ$

$e$

$i$

$Ω$

$t$

$e$

$i$

$Ω$

$t$

$Ω$

$Ω$

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