diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 711f2fd..b19a5ee 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -1541,56 +1541,234 @@ outside. \end_layout \begin_layout Standard -In our convention, the regular translation operator can be expressed explicitly - as (TODO CHECK CAREFULLY FOR POSSIBLE -\begin_inset Formula $(-1)^{m'}$ +In our convention, the regular translation operator elements can be expressed + explicitly as +\begin_inset Formula +\begin{align} +\tropr_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\nonumber \\ +\tropr_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)j_{\lambda}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator} +\end{align} + \end_inset - AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS) +and analogously the elements of the singular operator +\begin_inset Formula $\trops$ +\end_inset + +, having spherical Hankel functions ( +\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$ +\end_inset + +) in the radial part instead of the regular bessel functions, +\begin_inset Formula +\begin{align} +\trops_{\tau lm;\tau l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|}^{l+l'}C_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\nonumber \\ +\trops_{\tau lm;\tau'l'm'}\left(\vect d\right) & =\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}D_{lm;l'm'}^{\lambda}\ush{\lambda}{m-m'}\left(\uvec d\right)h_{\lambda}^{(1)}\left(d\right),\quad\tau'\ne\tau,\label{eq:translation operator singular} +\end{align} + +\end_inset + +where the constant factors in our convention read (TODO CHECK ONCE AGAIN + CAREFULLY FOR POSSIBLE PHASE FACTORS FACTORS) \begin_inset Note Note -status open +status collapsed \begin_layout Plain Layout -Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase -\end_layout - +Original Kristensson's +\begin_inset Formula $C,D's$ \end_inset - +from F.7: \begin_inset Formula -\begin{multline} -\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\\ +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sqrt{\frac{\varepsilon_{m}\varepsilon_{m'}}{4}}\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\\ +D_{ml,m'l'}\left(\vect d\right)=\frac{\left(-1\right)^{m+m'}}{2}\sqrt{\frac{\varepsilon_{m}\varepsilon_{m'}}{4}}\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}},\qquad\tau\ne\tau'.\label{eq:translation operator} -\end{multline} +\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}},\qquad\tau\ne\tau'. +\end{multline*} \end_inset -The singular operator -\begin_inset Formula $\trops$ +where I have found a +\begin_inset Formula $-i$ \end_inset - for re-expanding outgoing waves into regular ones has the same form except - the regular spherical Bessel functions -\begin_inset Formula $j_{l}$ + factor in the +\begin_inset Formula $\tau\ne\tau'$ \end_inset - in are replaced with spherical Hankel functions -\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$ + coefficients, so I force it here: +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\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}},\\ +D_{ml,m'l'}\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}},\qquad\tau\ne\tau'. +\end{multline*} + \end_inset -. + TODO check influence of the +\begin_inset Formula $\varepsilon_{m}$ +\end_inset + +s, whether they can be just removed as above. + If we take our definition of spherical harmonics, +\begin_inset Formula +\[ +\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right) +\] + +\end_inset + +so +\begin_inset Formula +\[ +\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}=\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right) +\] + +\end_inset + +and taking into account that we use the CS phase +\begin_inset Formula $\dlmfFer lm\left(\cos\theta\right)=\left(-1\right)^{m}P_{l}^{m}$ +\end_inset + +, and that +\begin_inset Formula $\left(-1\right)^{m+m'}=\left(-1\right)^{m-m'}$ +\end_inset + + we have +\end_layout + +\begin_layout Plain Layout +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{1}{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)\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\\ +D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{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)\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}},\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + + +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{1}{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)\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right),\\ +D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{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)\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + + +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\frac{1}{2}\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\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)\ush{\lambda}{m-m'}\left(\uvec d\right),\\ +D_{ml,m'l'}\left(\vect d\right)=-i\frac{1}{2}\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\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)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + +and finally +\begin_inset Formula +\begin{multline*} +C_{ml,m'l'}\left(\vect d\right)=\sum_{\lambda=\left|l-l'\right|}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\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)\ush{\lambda}{m-m'}\left(\uvec d\right),\\ +D_{ml,m'l'}\left(\vect d\right)=-i\sum_{\lambda=\left|l-l'\right|+1}^{l+l'}\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\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)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'. +\end{multline*} + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Formula +\begin{multline*} +C_{lm;l'm'}^{\lambda}=\left(-1\right)^{\frac{l'-l+\lambda}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\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),\\ +D_{lm;l'm'}^{\lambda}=-i\left(-1\right)^{\frac{l'-l+\lambda+1}{2}}\sqrt{\frac{4\pi\left(2\lambda+1\right)\left(2l+1\right)\left(2l'+1\right)}{l\left(l+1\right)l'\left(l'+1\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}}. +\end{multline*} + +\end_inset + + \end_layout \begin_layout Standard @@ -1611,12 +1789,11 @@ todo different notation for the complex conjugation without transposition??? or in the per-particle matrix notation, \begin_inset Formula \begin{equation} -\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}\label{eq:regular translation unitarity} +\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}.\label{eq:regular translation unitarity} \end{equation} \end_inset -. Note that truncation at finite multipole degree breaks the unitarity, \begin_inset Formula $\truncated{\troprp qp}L^{-1}\ne\truncated{\troprp pq}L=\truncated{\troprp qp^{\dagger}}L$ \end_inset