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Display information for equation id:math.1737.27 on revision:1737

* Page found: Induzierte Emission und Absorption von Lichtquanten in Atomen (eq math.1737.27)

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Induzierte Emission und Absorption von Lichtquanten in Atomen Eq: math.1737.27

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TeX (original user input):

\begin{align}
& {{\Psi }_{nlm}}(\bar{r})=\frac{{{u}_{nl}}(r)}{r}{{Y}_{l}}^{m}\left( \vartheta ,\phi  \right)\tilde{\ }{{P}_{l}}^{m}(\cos \vartheta ){{e}^{im\phi }} \\
& \left\langle  n\acute{\ }l\acute{\ }m\acute{\ } \right|\hat{\bar{\xi }}\left| nlm \right\rangle \tilde{\ }\int_{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left( \vartheta  \right){{P}_{l\acute{\ }}}^{m\acute{\ }}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta )\int_{0}^{2\pi }{d}\phi {{e}^{i\left( m-m\acute{\ }+1 \right)\phi }} \\
& \int_{0}^{2\pi }{d}\phi {{e}^{i\left( m-m\acute{\ }+1 \right)\phi }}\tilde{\ }{{\delta }_{m\acute{\ },m+1}} \\
& \Rightarrow \left\langle  n\acute{\ }l\acute{\ }m\acute{\ } \right|\hat{\bar{\xi }}\left| nlm \right\rangle \tilde{\ }\int_{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left( \vartheta  \right){{P}_{l\acute{\ }}}^{m+1}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta ) \\
& \int_{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left( \vartheta  \right){{P}_{l\acute{\ }}}^{m+1}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta )\tilde{\ }{{\delta }_{l\acute{\ },l\pm 1}} \\
& \Rightarrow \left\langle  n\acute{\ }l\acute{\ }m\acute{\ } \right|\hat{\bar{\xi }}\left| nlm \right\rangle \tilde{\ }{{\delta }_{m\acute{\ },m+1}}{{\delta }_{l\acute{\ },l\pm 1}} \\
\end{align}

TeX (checked):

{\begin{aligned}&{{\Psi }_{nlm}}({\bar {r}})={\frac {{{u}_{nl}}(r)}{r}}{{Y}_{l}}^{m}\left(\vartheta ,\phi \right){\tilde {\ }}{{P}_{l}}^{m}(\cos \vartheta ){{e}^{im\phi }}\\&\left\langle n{\acute {\ }}l{\acute {\ }}m{\acute {\ }}\right|{\hat {\bar {\xi }}}\left|nlm\right\rangle {\tilde {\ }}\int _{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left(\vartheta \right){{P}_{l{\acute {\ }}}}^{m{\acute {\ }}}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta )\int _{0}^{2\pi }{d}\phi {{e}^{i\left(m-m{\acute {\ }}+1\right)\phi }}\\&\int _{0}^{2\pi }{d}\phi {{e}^{i\left(m-m{\acute {\ }}+1\right)\phi }}{\tilde {\ }}{{\delta }_{m{\acute {\ }},m+1}}\\&\Rightarrow \left\langle n{\acute {\ }}l{\acute {\ }}m{\acute {\ }}\right|{\hat {\bar {\xi }}}\left|nlm\right\rangle {\tilde {\ }}\int _{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left(\vartheta \right){{P}_{l{\acute {\ }}}}^{m+1}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta )\\&\int _{0}^{\pi }{d}\vartheta {{\sin }^{2}}\left(\vartheta \right){{P}_{l{\acute {\ }}}}^{m+1}(\cos \vartheta ){{P}_{l}}^{m}(\cos \vartheta ){\tilde {\ }}{{\delta }_{l{\acute {\ }},l\pm 1}}\\&\Rightarrow \left\langle n{\acute {\ }}l{\acute {\ }}m{\acute {\ }}\right|{\hat {\bar {\xi }}}\left|nlm\right\rangle {\tilde {\ }}{{\delta }_{m{\acute {\ }},m+1}}{{\delta }_{l{\acute {\ }},l\pm 1}}\\\end{aligned}}

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\begin{aligned} Ψ_{n l m} (\bar{r}) = \frac{u_{n l} (r)}{r} {Y_{l}}^{m} (ϑ, ϕ) \tilde{} {P_{l}}^{m} (\cos ϑ) e^{i m ϕ} \\ ⟨ n \overset{´}{} l \overset{´}{} m \overset{´}{} | \hat{\bar{ξ}} | n l m ⟩ \tilde{} \int_{0}^{π} d ϑ \sin^{2} (ϑ) {P_{l \overset{´}{}}}^{m \overset{´}{}} (\cos ϑ) {P_{l}}^{m} (\cos ϑ) \int_{0}^{2 π} d ϕ e^{i (m - m \overset{´}{} + 1) ϕ} \\ \int_{0}^{2 π} d ϕ e^{i (m - m \overset{´}{} + 1) ϕ} \tilde{} δ_{m \overset{´}{}, m + 1} \\ \Rightarrow ⟨ n \overset{´}{} l \overset{´}{} m \overset{´}{} | \hat{\bar{ξ}} | n l m ⟩ \tilde{} \int_{0}^{π} d ϑ \sin^{2} (ϑ) {P_{l \overset{´}{}}}^{m + 1} (\cos ϑ) {P_{l}}^{m} (\cos ϑ) \\ \int_{0}^{π} d ϑ \sin^{2} (ϑ) {P_{l \overset{´}{}}}^{m + 1} (\cos ϑ) {P_{l}}^{m} (\cos ϑ) \tilde{} δ_{l \overset{´}{}, l \pm 1} \\ \Rightarrow ⟨ n \overset{´}{} l \overset{´}{} m \overset{´}{} | \hat{\bar{ξ}} | n l m ⟩ \tilde{} δ_{m \overset{´}{}, m + 1} δ_{l \overset{´}{}, l \pm 1} \end{aligned}

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In Mathematica:

Identifiers

$Ψ_{n l m}$

$\bar{r}$

$u_{n l}$

$r$

$r$

$Y_{l}$

$m$

$ϑ$

$ϕ$

$\tilde{}$

$P_{l}$

$m$

$ϑ$

$e$

$i$

$m$

$ϕ$

$n$

$\overset{´}{}$

$l$

$\overset{´}{}$

$m$

$\overset{´}{}$

$\hat{\bar{ξ}}$

$n$

$l$

$m$

$\tilde{}$

$π$

$ϑ$

$ϑ$

$P$

$l$

$\overset{´}{}$

$m$

$\overset{´}{}$

$ϑ$

$P_{l}$

$m$

$ϑ$

$π$

$ϕ$

$e$

$i$

$m$

$m$

$\overset{´}{}$

$ϕ$

$π$

$ϕ$

$e$

$i$

$m$

$m$

$\overset{´}{}$

$ϕ$

$\tilde{}$

$δ$

$m$

$\overset{´}{}$

$m$

$n$

$\overset{´}{}$

$l$

$\overset{´}{}$

$m$

$\overset{´}{}$

$\hat{\bar{ξ}}$

$n$

$l$

$m$

$\tilde{}$

$π$

$ϑ$

$ϑ$

$P$

$l$

$\overset{´}{}$

$m$

$ϑ$

$P_{l}$

$m$

$ϑ$

$π$

$ϑ$

$ϑ$

$P$

$l$

$\overset{´}{}$

$m$

$ϑ$

$P_{l}$

$m$

$ϑ$

$\tilde{}$

$δ$

$l$

$\overset{´}{}$

$l$

$n$

$\overset{´}{}$

$l$

$\overset{´}{}$

$m$

$\overset{´}{}$

$\hat{\bar{ξ}}$

$n$

$l$

$m$

$\tilde{}$

$δ$

$m$

$\overset{´}{}$

$m$

$δ$

$l$

$\overset{´}{}$

$l$

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