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Display information for equation id:math.1668.13 on revision:1668

* Page found: Ortsdarstellung des Bahndrehimpulses (eq math.1668.13)

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Ortsdarstellung des Bahndrehimpulses Eq: math.1672.13

Hash: aacd57de9a148c038ead707c39c2c582

TeX (original user input):

\begin{align}

& \left\langle  {\bar{r}} \right|{{{\hat{L}}}_{\pm }}\left| l,m \right\rangle =\frac{\hbar }{i}\left( {{{\hat{x}}}_{2}}{{\partial }_{3}}-{{{\hat{x}}}_{3}}{{\partial }_{2}}\pm i{{{\hat{x}}}_{3}}{{\partial }_{1}}\mp i{{{\hat{x}}}_{1}}{{\partial }_{3}} \right){{\Psi }_{lm}}(\bar{r})=\hbar {{e}^{\pm i\phi }}\left( \pm \frac{\partial }{\partial \vartheta }+i\cot \vartheta \frac{\partial }{\partial \phi } \right){{\Psi }_{lm}}(r,\vartheta ,\phi ) \\

& \hbar {{e}^{\pm i\phi }}\left( \pm \frac{\partial }{\partial \vartheta }+i\cot \vartheta \frac{\partial }{\partial \phi } \right){{\Psi }_{lm}}(r,\vartheta ,\phi )=\hbar {{e}^{i\left( m\pm 1 \right)\phi }}\left( \pm \frac{\partial }{\partial \vartheta }-m\cot \vartheta  \right){{f}_{lm}}(r,\vartheta ) \\

\end{align}

TeX (checked):

{\begin{aligned}&\left\langle {\bar {r}}\right|{{\hat {L}}_{\pm }}\left|l,m\right\rangle ={\frac {\hbar }{i}}\left({{\hat {x}}_{2}}{{\partial }_{3}}-{{\hat {x}}_{3}}{{\partial }_{2}}\pm i{{\hat {x}}_{3}}{{\partial }_{1}}\mp i{{\hat {x}}_{1}}{{\partial }_{3}}\right){{\Psi }_{lm}}({\bar {r}})=\hbar {{e}^{\pm i\phi }}\left(\pm {\frac {\partial }{\partial \vartheta }}+i\cot \vartheta {\frac {\partial }{\partial \phi }}\right){{\Psi }_{lm}}(r,\vartheta ,\phi )\\&\hbar {{e}^{\pm i\phi }}\left(\pm {\frac {\partial }{\partial \vartheta }}+i\cot \vartheta {\frac {\partial }{\partial \phi }}\right){{\Psi }_{lm}}(r,\vartheta ,\phi )=\hbar {{e}^{i\left(m\pm 1\right)\phi }}\left(\pm {\frac {\partial }{\partial \vartheta }}-m\cot \vartheta \right){{f}_{lm}}(r,\vartheta )\\\end{aligned}}

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\begin{aligned} ⟨ \bar{r} | {\hat{L}}_{\pm} | l, m ⟩ = \frac{ℏ}{i} ({\hat{x}}_{2} \partial_{3} - {\hat{x}}_{3} \partial_{2} \pm i {\hat{x}}_{3} \partial_{1} \mp i {\hat{x}}_{1} \partial_{3}) Ψ_{l m} (\bar{r}) = ℏ e^{\pm i ϕ} (\pm \frac{\partial}{\partial ϑ} + i \cot ϑ \frac{\partial}{\partial ϕ}) Ψ_{l m} (r, ϑ, ϕ) \\ ℏ e^{\pm i ϕ} (\pm \frac{\partial}{\partial ϑ} + i \cot ϑ \frac{\partial}{\partial ϕ}) Ψ_{l m} (r, ϑ, ϕ) = ℏ e^{i (m \pm 1) ϕ} (\pm \frac{\partial}{\partial ϑ} - m \cot ϑ) f_{l m} (r, ϑ) \end{aligned}

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Identifiers

$\bar{r}$

$\hat{L}$

$l$

$m$

$i$

${\hat{x}}_{2}$

${\hat{x}}_{3}$

$i$

${\hat{x}}_{3}$

$i$

${\hat{x}}_{1}$

$Ψ_{l m}$

$\bar{r}$

$e$

$i$

$ϕ$

$ϑ$

$i$

$ϑ$

$ϕ$

$Ψ_{l m}$

$r$

$ϑ$

$ϕ$

$e$

$i$

$ϕ$

$ϑ$

$i$

$ϑ$

$ϕ$

$Ψ_{l m}$

$r$

$ϑ$

$ϕ$

$e$

$i$

$m$

$ϕ$

$ϑ$

$m$

$ϑ$

$f_{l m}$

$r$

$ϑ$

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