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Display information for equation id:math.1796.15 on revision:1796

* Page found: Drehimpulsdarstellung und Streuphasen (eq math.1796.15)

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Drehimpulsdarstellung und Streuphasen Eq: math.1796.15

Hash: 3384f8dcee46dda0bbf8b27c9b788554

TeX (original user input):

\begin{align}

& \begin{matrix}

\lim   \\

r\to \infty   \\

\end{matrix}\frac{1}{r}{{u}_{l}}(r)=\frac{2l+1}{2}\frac{1}{ikr}\left\{ {{e}^{ikr}}-{{(-1)}^{l}}{{e}^{-ikr}} \right\}=\frac{2l+1}{2}\frac{1}{ikr}{{i}^{l}}\left\{ {{e}^{i\left( kr-l\frac{\pi }{2} \right)}}-{{e}^{-i\left( kr-l\frac{\pi }{2} \right)}} \right\} \\

& \Rightarrow \begin{matrix}

\lim   \\

r\to \infty   \\

\end{matrix}\frac{1}{r}{{u}_{l}}(r)=\left( 2l+1 \right)\frac{{{i}^{l}}}{kr}\sin \left( kr-l\frac{\pi }{2} \right) \\

\end{align}

TeX (checked):

{\begin{aligned}&{\begin{matrix}\lim \\r\to \infty \\\end{matrix}}{\frac {1}{r}}{{u}_{l}}(r)={\frac {2l+1}{2}}{\frac {1}{ikr}}\left\{{{e}^{ikr}}-{{(-1)}^{l}}{{e}^{-ikr}}\right\}={\frac {2l+1}{2}}{\frac {1}{ikr}}{{i}^{l}}\left\{{{e}^{i\left(kr-l{\frac {\pi }{2}}\right)}}-{{e}^{-i\left(kr-l{\frac {\pi }{2}}\right)}}\right\}\\&\Rightarrow {\begin{matrix}\lim \\r\to \infty \\\end{matrix}}{\frac {1}{r}}{{u}_{l}}(r)=\left(2l+1\right){\frac {{i}^{l}}{kr}}\sin \left(kr-l{\frac {\pi }{2}}\right)\\\end{aligned}}

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MathML (4.681 KB / 598 B) :

\begin{aligned} \begin{matrix} \lim \\ r \to \infty \end{matrix} \frac{1}{r} u_{l} (r) = \frac{2 l + 1}{2} \frac{1}{i k r} {e^{i k r} - {(- 1)}^{l} e^{- i k r}} = \frac{2 l + 1}{2} \frac{1}{i k r} i^{l} {e^{i (k r - l \frac{π}{2})} - e^{- i (k r - l \frac{π}{2})}} \\ \Rightarrow \begin{matrix} \lim \\ r \to \infty \end{matrix} \frac{1}{r} u_{l} (r) = (2 l + 1) \frac{i^{l}}{k r} \sin (k r - l \frac{π}{2}) \end{aligned}

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Identifiers

$r$

$r$

$u_{l}$

$r$

$l$

$i$

$k$

$r$

$e$

$i$

$k$

$r$

$l$

$e$

$i$

$k$

$r$

$l$

$i$

$k$

$r$

$i$

$l$

$e$

$i$

$k$

$r$

$l$

$π$

$e$

$i$

$k$

$r$

$l$

$π$

$r$

$r$

$u_{l}$

$r$

$l$

$i$

$l$

$k$

$r$

$k$

$r$

$l$

$π$

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