Rewrite the translation operators in terms of spherical harmonics.
Former-commit-id: 74f2ab378f3ba78fbc57807277f30c6eecb32ea3
This commit is contained in:
Marek Nečada
parent
705e61053f
commit
600e1e9c55
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@ -1541,56 +1541,234 @@ outside.
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\end_layout
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\begin_layout Standard
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In our convention, the regular translation operator can be expressed explicitly
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as (TODO CHECK CAREFULLY FOR POSSIBLE
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\begin_inset Formula $(-1)^{m'}$
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In our convention, the regular translation operator elements can be expressed
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explicitly as
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\begin_inset Formula
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\begin{align}
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\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 \\
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\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}
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\end{align}
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\end_inset
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AND SIMILAR FACTORS AND REWRITE IN TERMS OF SPHERICAL HARMONICS)
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and analogously the elements of the singular operator
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\begin_inset Formula $\trops$
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\end_inset
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, having spherical Hankel functions (
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\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
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\end_inset
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) in the radial part instead of the regular bessel functions,
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\begin_inset Formula
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\begin{align}
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\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 \\
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\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}
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\end{align}
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\end_inset
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where the constant factors in our convention read (TODO CHECK ONCE AGAIN
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CAREFULLY FOR POSSIBLE PHASE FACTORS FACTORS)
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\begin_inset Note Note
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status open
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status collapsed
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\begin_layout Plain Layout
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Teďka jsou tam zkopírovány výrazy pro C a D z Kristenssona, chybějí fase
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\end_layout
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Original Kristensson's
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\begin_inset Formula $C,D's$
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\end_inset
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from F.7:
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\begin_inset Formula
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\begin{multline}
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\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\\
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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}},\\
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\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\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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}
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\end{multline}
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\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'.
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\end{multline*}
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\end_inset
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The singular operator
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\begin_inset Formula $\trops$
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where I have found a
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\begin_inset Formula $-i$
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\end_inset
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for re-expanding outgoing waves into regular ones has the same form except
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the regular spherical Bessel functions
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\begin_inset Formula $j_{l}$
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factor in the
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\begin_inset Formula $\tau\ne\tau'$
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\end_inset
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in are replaced with spherical Hankel functions
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\begin_inset Formula $h_{l}^{(1)}=j_{l}+iy_{l}$
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coefficients, so I force it here:
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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}},\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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'.
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\end{multline*}
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\end_inset
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.
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TODO check influence of the
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\begin_inset Formula $\varepsilon_{m}$
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\end_inset
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s, whether they can be just removed as above.
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If we take our definition of spherical harmonics,
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\begin_inset Formula
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\[
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\ush lm=\left(\frac{\left(l-m\right)!\left(2l+1\right)}{4\pi\left(l+m\right)!}\right)^{\frac{1}{2}}e^{im\phi}\dlmfFer lm\left(\cos\theta\right)
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\]
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\end_inset
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so
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\begin_inset Formula
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\[
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\dlmfFer{\lambda}{m-m'}\left(\cos\theta_{\uvec d}\right)e^{i\left(m-m'\right)\phi_{\uvec d}}=\sqrt{\frac{4\pi\left(\lambda+m-m'\right)!}{\left(\lambda-m+m'\right)!\left(2\lambda+1\right)}}\ush{\lambda}{m-m'}\left(\uvec d\right)
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\]
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\end_inset
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and taking into account that we use the CS phase
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\begin_inset Formula $\dlmfFer lm\left(\cos\theta\right)=\left(-1\right)^{m}P_{l}^{m}$
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\end_inset
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, and that
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\begin_inset Formula $\left(-1\right)^{m+m'}=\left(-1\right)^{m-m'}$
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\end_inset
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we have
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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}},\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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'.
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\end{multline*}
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\end_inset
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\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),\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\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'.
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\end{multline*}
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\end_inset
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'.
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\end{multline*}
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\end_inset
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and finally
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right)\times\\
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\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}\times\\
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\times j_{\lambda}\left(d\right)\ush{\lambda}{m-m'}\left(\uvec d\right),\qquad\tau\ne\tau'.
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\end{multline*}
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\end_inset
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\end_layout
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\end_inset
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\begin_inset Formula
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\begin{multline*}
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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\\
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\times\begin{pmatrix}l & l' & \lambda\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\left(l\left(l+1\right)+l'\left(l'+1\right)-\lambda\left(\lambda+1\right)\right),\\
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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\\
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\times\begin{pmatrix}l & l' & \lambda-1\\
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0 & 0 & 0
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\end{pmatrix}\begin{pmatrix}l & l' & \lambda\\
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m & -m' & m'-m
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\end{pmatrix}\sqrt{\lambda^{2}-\left(l-l'\right)^{2}}\sqrt{\left(l+l'+1\right)^{2}-\lambda^{2}}.
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\end{multline*}
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\end_inset
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\end_layout
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\begin_layout Standard
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@ -1611,12 +1789,11 @@ todo different notation for the complex conjugation without transposition???
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or in the per-particle matrix notation,
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\begin_inset Formula
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\begin{equation}
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\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}\label{eq:regular translation unitarity}
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\troprp qp^{-1}=\troprp pq=\troprp qp^{\dagger}.\label{eq:regular translation unitarity}
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\end{equation}
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\end_inset
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.
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Note that truncation at finite multipole degree breaks the unitarity,
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\begin_inset Formula $\truncated{\troprp qp}L^{-1}\ne\truncated{\troprp pq}L=\truncated{\troprp qp^{\dagger}}L$
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\end_inset
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