QPMS
/
qpms
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

Expressions for translation operators (to be checked) 

Former-commit-id: f19217f0f3f66a79d724840f633731f44bfd755f
This commit is contained in:
Marek Nečada 2019-07-31 11:43:24 +03:00
parent dba0a26877
commit 76abecce48
2 changed files with 43 additions and 16 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

2
lepaper/arrayscat.lyx
View File

@ -713,7 +713,7 @@ Abstract.
\end_layout \end_layout
\begin_layout Itemize \begin_layout Itemize
Translation operators: explicit expression, also in sph. Translation operators: rewrite in sph.
harm. harm.
convention independent form. convention independent form.
\end_layout \end_layout

57
lepaper/finite.lyx
View File

@ -290,8 +290,8 @@ outgoing
, respectively, defined as follows: , respectively, defined as follows:
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\vswfrtlm1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh1lm\left(\uvec r\right),\\ \vswfrtlm 1lm\left(k\vect r\right) & =j_{l}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
\vswfrtlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right), \vswfrtlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krj_{l}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{j_{l}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),
\end{align*} \end{align*}
\end_inset \end_inset
@ -299,8 +299,8 @@ outgoing
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\vswfouttlm1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh1lm\left(\uvec r\right),\\ \vswfouttlm 1lm\left(k\vect r\right) & =h_{l}^{\left(1\right)}\left(kr\right)\vsh 1lm\left(\uvec r\right),\\
\vswfouttlm2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh3lm\left(\uvec r\right),\\ \vswfouttlm 2lm\left(k\vect r\right) & =\frac{1}{kr}\frac{\ud\left(krh_{l}^{\left(1\right)}\left(kr\right)\right)}{\ud\left(kr\right)}\vsh 2lm\left(\uvec r\right)+\sqrt{l\left(l+1\right)}\frac{h_{l}^{\left(1\right)}\left(kr\right)}{kr}\vsh 3lm\left(\uvec r\right),\\
& \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l, & \tau=1,2;\quad l=1,2,3,\dots;\quad m=-l,-l+1,\dots,+l,
\end{align*} \end{align*}
@ -326,9 +326,9 @@ vector spherical harmonics
\begin_inset Formula \begin_inset Formula
\begin{align*} \begin{align*}
\vsh1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\\ \vsh 1lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\times\left(\vect r\ush lm\left(\uvec r\right)\right)=\frac{1}{\sqrt{l\left(l+1\right)}}\nabla\ush lm\left(\uvec r\right)\times\vect r,\\
\vsh2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\ \vsh 2lm\left(\uvec r\right) & =\frac{1}{\sqrt{l\left(l+1\right)}}r\nabla\ush lm\left(\uvec r\right),\\
\vsh3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right). \vsh 3lm\left(\uvec r\right) & =\uvec r\ush lm\left(\uvec r\right).
\end{align*} \end{align*}
\end_inset \end_inset
@ -452,7 +452,7 @@ noprefix "false"
\end_inset \end_inset
inside a ball inside a ball
\begin_inset Formula $\openball0{R^{>}}$ \begin_inset Formula $\openball 0{R^{>}}$
\end_inset \end_inset
with radius with radius
@ -470,7 +470,7 @@ noprefix "false"
\end_inset \end_inset
to have a complete basis of the solutions in the volume to have a complete basis of the solutions in the volume
\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ \begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
\end_inset \end_inset
. .
@ -496,7 +496,7 @@ The single-particle scattering problem at frequency
\end_inset \end_inset
and let the whole volume and let the whole volume
\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ \begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
\end_inset \end_inset
be filled with a homogeneous isotropic medium with wave number be filled with a homogeneous isotropic medium with wave number
@ -532,7 +532,7 @@ If there was no scatterer and
\end_inset \end_inset
due to sources outside due to sources outside
\begin_inset Formula $\openball0R$ \begin_inset Formula $\openball 0R$
\end_inset \end_inset
would remain. would remain.
@ -793,7 +793,7 @@ literal "true"
. .
Let the field in Let the field in
\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ \begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
\end_inset \end_inset
have expansion as in have expansion as in
@ -812,7 +812,7 @@ noprefix "false"
\end_inset \end_inset
to to
\begin_inset Formula $\openball0{R^{>}}\backslash B_{0}\left(R\right)$ \begin_inset Formula $\openball 0{R^{>}}\backslash B_{0}\left(R\right)$
\end_inset \end_inset
via by electromagnetic radiation is via by electromagnetic radiation is
@ -1535,10 +1535,37 @@ outside.
\begin_layout Standard \begin_layout Standard
In our convention, the regular translation operator can be expressed explicitly 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_inset Formula
\begin{multline} \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{multline}
\end_inset \end_inset