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Display information for equation id:math.1672.21 on revision:1672

* Page found: Ortsdarstellung des Bahndrehimpulses (eq math.1672.21)

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Ortsdarstellung des Bahndrehimpulses Eq: math.1672.21

Hash: 0e6ec2a958d8d0e7c75e574c7877b638

TeX (original user input):

\begin{align}

& {{Y}_{l}}^{m}(\vartheta ,\phi )=\frac{{{e}^{im\phi }}}{\sqrt{2\pi }}\cdot \frac{{{\left( -1 \right)}^{m}}}{{{2}^{l}}l!}\sqrt{\frac{\left( 2l+1 \right)\left( l-m \right)!}{2\left( l+m \right)!}}\frac{1}{{{\left( \sin \vartheta  \right)}^{m}}}\frac{{{d}^{l-m}}}{d{{\left( \cos \vartheta  \right)}^{l-m}}}{{\left( \sin \vartheta  \right)}^{2l}} \\

& {{Y}_{l}}^{m}(\vartheta ,\phi )=\frac{{{e}^{im\phi }}}{\sqrt{2\pi }}\cdot {{\left( -1 \right)}^{m}}\sqrt{\frac{\left( 2l+1 \right)\left( l-m \right)!}{2\left( l+m \right)!}}{{P}^{m}}_{l}(\cos \vartheta ) \\

\end{align}

TeX (checked):

{\begin{aligned}&{{Y}_{l}}^{m}(\vartheta ,\phi )={\frac {{e}^{im\phi }}{\sqrt {2\pi }}}\cdot {\frac {{\left(-1\right)}^{m}}{{{2}^{l}}l!}}{\sqrt {\frac {\left(2l+1\right)\left(l-m\right)!}{2\left(l+m\right)!}}}{\frac {1}{{\left(\sin \vartheta \right)}^{m}}}{\frac {{d}^{l-m}}{d{{\left(\cos \vartheta \right)}^{l-m}}}}{{\left(\sin \vartheta \right)}^{2l}}\\&{{Y}_{l}}^{m}(\vartheta ,\phi )={\frac {{e}^{im\phi }}{\sqrt {2\pi }}}\cdot {{\left(-1\right)}^{m}}{\sqrt {\frac {\left(2l+1\right)\left(l-m\right)!}{2\left(l+m\right)!}}}{{P}^{m}}_{l}(\cos \vartheta )\\\end{aligned}}

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MathML (5.146 KB / 603 B) :

\begin{aligned} {Y_{l}}^{m} (ϑ, ϕ) = \frac{e^{i m ϕ}}{\sqrt{2 π}} \cdot \frac{{(- 1)}^{m}}{2^{l} l!} \sqrt{\frac{(2 l + 1) (l - m)!}{2 (l + m)!}} \frac{1}{{(\sin ϑ)}^{m}} \frac{d^{l - m}}{d {(\cos ϑ)}^{l - m}} {(\sin ϑ)}^{2 l} \\ {Y_{l}}^{m} (ϑ, ϕ) = \frac{e^{i m ϕ}}{\sqrt{2 π}} \cdot {(- 1)}^{m} \sqrt{\frac{(2 l + 1) (l - m)!}{2 (l + m)!}} {P^{m}}_{l} (\cos ϑ) \end{aligned}

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Identifiers

$Y_{l}$

$m$

$ϑ$

$ϕ$

$e$

$i$

$m$

$ϕ$

$π$

$m$

$l$

$l$

$l$

$l$

$m$

$l$

$m$

$ϑ$

$m$

$d$

$l$

$m$

$d$

$ϑ$

$l$

$m$

$ϑ$

$l$

$Y_{l}$

$m$

$ϑ$

$ϕ$

$e$

$i$

$m$

$ϕ$

$π$

$m$

$l$

$l$

$m$

$l$

$m$

$P$

$m$

$l$

$ϑ$

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