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Expressions for translation operators (to be checked) 

Former-commit-id: f19217f0f3f66a79d724840f633731f44bfd755f
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Marek Nečada 2019-07-31 11:43:24 +03:00
parent dba0a26877
commit 76abecce48
2 changed files with 43 additions and 16 deletions
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2
lepaper/arrayscat.lyx
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@ -713,7 +713,7 @@ Abstract.
\end_layout
\begin_layout Itemize
Translation operators: explicit expression, also in sph.
Translation operators: rewrite in sph.
harm.
convention independent form.
\end_layout

31
lepaper/finite.lyx
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@ -1535,10 +1535,37 @@ outside.
\begin_layout Standard
In our convention, the regular translation operator can be expressed explicitly
as
as (TODO CHECK CAREFULLY FOR POSSIBLE
\begin_inset Formula $(-1)^{m'}$
\end_inset
AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS)
\begin_inset Note Note
status open
\begin_layout Plain Layout
Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase
\end_layout
\end_inset
\begin_inset Formula
\begin{multline}
\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\dots.\label{eq:translation operator}
\tropr_{\tau lm;\tau l'm'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\
\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right)=-i\frac{\left(-1\right)^{m+m'}}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\left(2\lambda+1\right)\sqrt{\frac{\left(2l+1\right)\left(2l'+1\right)\left(\lambda-\left(m-m'\right)\right)!}{l\left(l+1\right)l'\left(l'+1\right)\left(\lambda+\left(m-m'\right)\right)!}}\times\\
\times\begin{pmatrix}l & l' & \lambda-1\\
0 & 0 & 0
\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
m & -m' & m'-m
\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
\times j_{\lambda}\left(d\right)P_{\lambda}^{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}.\label{eq:translation operator}
\end{multline}
\end_inset