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Display information for equation id:math.1672.13 on revision:1672

* Page found: Ortsdarstellung des Bahndrehimpulses (eq math.1672.13)

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\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}

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⟨ r ¯ | L ^ ± | l , m ⟩ = ℏ i ⁢ ( x ^ 2 ⁢ ∂ 3 - x ^ 3 ⁢ ∂ 2 ± i ⁢ x ^ 3 ⁢ ∂ 1 ∓ i ⁢ x ^ 1 ⁢ ∂ 3 ) ⁢ Ψ l ⁢ m ⁢ ( r ¯ ) = ℏ ⁢ e ± i ⁢ ϕ ⁢ ( ± ∂ ∂ ⁡ ϑ + i ⁢ cot ⁡ ϑ ⁢ ∂ ∂ ⁡ ϕ ) ⁢ Ψ l ⁢ m ⁢ ( r , ϑ , ϕ ) ℏ ⁢ e ± i ⁢ ϕ ⁢ ( ± ∂ ∂ ⁡ ϑ + i ⁢ cot ⁡ ϑ ⁢ ∂ ∂ ⁡ ϕ ) ⁢ Ψ l ⁢ m ⁢ ( r , ϑ , ϕ ) = ℏ ⁢ e i ⁢ ( m ± 1 ) ⁢ ϕ ⁢ ( ± ∂ ∂ ⁡ ϑ - m ⁢ cot ⁡ ϑ ) ⁢ f l ⁢ m ⁢ ( r , ϑ ) absent quantum-operator-product ¯ 𝑟 subscript ^ 𝐿 plus-or-minus 𝑙 𝑚 Planck-constant-over-2-pi 𝑖 minus-or-plus plus-or-minus subscript ^ 𝑥 2 subscript 3 subscript ^ 𝑥 3 subscript 2 𝑖 subscript ^ 𝑥 3 subscript 1 𝑖 subscript ^ 𝑥 1 subscript 3 subscript Ψ 𝑙 𝑚 ¯ 𝑟 Planck-constant-over-2-pi superscript 𝑒 plus-or-minus 𝑖 italic-ϕ plus-or-minus italic-ϑ 𝑖 italic-ϑ italic-ϕ subscript Ψ 𝑙 𝑚 𝑟 italic-ϑ italic-ϕ absent Planck-constant-over-2-pi superscript 𝑒 plus-or-minus 𝑖 italic-ϕ plus-or-minus italic-ϑ 𝑖 italic-ϑ italic-ϕ subscript Ψ 𝑙 𝑚 𝑟 italic-ϑ italic-ϕ Planck-constant-over-2-pi superscript 𝑒 𝑖 plus-or-minus 𝑚 1 italic-ϕ plus-or-minus italic-ϑ 𝑚 italic-ϑ subscript 𝑓 𝑙 𝑚 𝑟 italic-ϑ {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\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)\\ \par&\displaystyle\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)\\ \par\end{aligned}}}

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⟨r¯|L^±|l,m⟩=ℏi(x^2∂3−x^3∂2±ix^3∂1∓ix^1∂3)Ψlm(r¯)=ℏe±iϕ(±∂∂ϑ+icot⁡ϑ∂∂ϕ)Ψlm(r,ϑ,ϕ)ℏe±iϕ(±∂∂ϑ+icot⁡ϑ∂∂ϕ)Ψlm(r,ϑ,ϕ)=ℏei(m±1)ϕ(±∂∂ϑ−mcot⁡ϑ)flm(r,ϑ)

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