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Additional notes for axially symmetric particles 

Former-commit-id: 989527ac17b0daea7abb3b15a71ae66d9ae224bf
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Marek Nečada 2019-08-16 12:57:40 +03:00
parent 3e6f004605
commit 3e427f35ae
1 changed files with 47 additions and 0 deletions
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notes/cylinderT.lyx
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@ -354,6 +354,53 @@ r & =\frac{h}{2\cos\left(\theta-\pi\right)}=-\frac{h}{2\cos\theta},\\
\end_inset
\end_layout
\begin_layout Standard
Let's write VSWFs in terms of the power-normalised
\begin_inset Formula $p,\pi,\tau$
\end_inset
funs:
\begin_inset Formula
\begin{align*}
\vsh_{1lm} & =\left(\uvec{\theta}\pi_{lm}-\uvec{\phi}\tau_{lm}\right)e^{im\phi}\\
\vsh_{2lm} & =\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)e^{im\phi}\\
\vsh_{3lm} & =\sqrt{l\left(l+1\right)}p_{lm}e^{im\theta}
\end{align*}
\end_inset
\begin_inset Formula
\begin{align*}
\vect y_{\kappa1lm} & =\underbrace{h_{l}^{\kappa}e^{im\phi}}_{c_{\kappa lm}^{1}}\left(\uvec{\theta}\pi_{lm}-\uvec{\phi}\tau_{lm}\right)\\
\vect y_{\kappa2lm} & =\frac{1}{kr}e^{im\phi}\left(\frac{\ud\left(krh_{l}^{\kappa}\right)}{\ud\left(kr\right)}\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)+h_{l}^{\kappa}l\left(l+1\right)\uvec rp_{lm}\right)\\
& =c_{\kappa lm}^{2}\left(\uvec{\theta}\tau_{lm}+\uvec{\phi}\pi_{lm}\right)+c_{\kappa lm}^{3}\uvec rp_{lm}
\end{align*}
\end_inset
The triple products than are (reminder:
\begin_inset Formula $\uvec{\nu}\left(\theta\right)=\uvec r\cos\beta\left(\theta\right)+\uvec{\theta}\sin\beta\left(\theta\right))$
\end_inset
:
\begin_inset Formula
\begin{align*}
\left(\vect y_{\kappa1lm}\times\vect v_{1l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{1}\left(-\pi_{lm}\tau_{l'm'}+\tau_{lm}\pi_{l'm'}\right)\\
\left(\vect y_{\kappa1lm}\times\vect v_{2l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{2}\left(\pi_{lm}\pi_{l'm'}+\tau_{lm}\tau_{l'm'}\right)\\
& +\sin\beta c_{\kappa lm}^{1}c_{\mathrm{R}l'm'}^{3}\left(-\tau_{lm}p_{l'm'}\right)\\
\left(\vect y_{\kappa2lm}\times\vect v_{1l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{1}\left(-\pi_{lm}\pi_{l'm'}-\tau_{lm}\tau_{l'm'}\right)\\
& +\sin\beta c_{\kappa lm}^{3}c_{\mathrm{R}l'm'}^{1}\left(p_{lm}\tau_{l'm'}\right)\\
\left(\vect y_{\kappa2lm}\times\vect v_{2l'm'}\right)\cdot\uvec{\nu} & =\cos\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{2}\left(\tau_{lm}\pi_{l'm'}-\pi_{lm}\tau_{l'm'}\right)\\
& -\sin\beta c_{\kappa lm}^{3}c_{\mathrm{R}l'm'}^{2}p_{lm}\pi_{l'm'}\\
& +\sin\beta c_{\kappa lm}^{2}c_{\mathrm{R}l'm'}^{3}\pi_{lm}p_{l'm'}
\end{align*}
\end_inset
\end_layout
\begin_layout Standard