DOC: show how to collect Wigner D-functions

[redeboer]

Hi @mmikhasenko, here's an example of how to 'hack' the dynamics builder so that it produces a symbol instead of an expression for the dynamics. As you said, this can then be done to collect the angular functions or one can use the same expression to insert dynamics lineshapes later on.

It's just a sketch, but you can see what it looks like in the collect-wigners.ipynb notebook.

[mmikhasenko]

Nice! I think it might make a very popular application of the ampform-dpd. So, cool to have this example.
The expressions can be simplify-ed and trigsimp-ed.
Where are the non-digonal terms?

[redeboer]

The expressions can be simplify-ed and trigsimp-ed.

Cool O.o

$$ \begin{eqnarray} X_{0}^{0,1/2;0,0}\left(\sigma_{1}\right) &:& 1 \\ X_{1}^{0,1/2;1,0}\left(\sigma_{1}\right) &:& \frac{1}{3} \\ X_{2}^{1,3/2;2,0}\left(\sigma_{1}\right) &:& \frac{2}{5} - \frac{3 \sin^{2}{\left(\theta_{23} \right)}}{10} \\ X_{1/2}^{0,1/2;0,1/2}\left(\sigma_{2}\right) &:& 9 \\ X_{1/2}^{0,1/2;1,1/2}\left(\sigma_{2}\right) &:& 1 \\ X_{3/2}^{1,3/2;2,1/2}\left(\sigma_{2}\right) &:& 4 - 3 \sin^{2}{\left(\theta_{31} \right)} \\ X_{3/2}^{1,3/2;1,1/2}\left(\sigma_{3}\right) &:& 4 - 3 \sin^{2}{\left(\theta_{12} \right)} \\ X_{3/2}^{1,3/2;2,1/2}\left(\sigma_{3}\right) &:& 1 - \frac{3 \sin^{2}{\left(\theta_{12} \right)}}{4} \\ \end{eqnarray} $$

[mmikhasenko]

Often, it is needed to give not just diagonal terms, i.e. coefficients in front of |X|^2, but also interference terms,
in front of

X_i X_j*

It matches the Hessian with a factor 1/2. So, autodiff gives a general approach to obtain the matrix.
I have been very happy discovering this simple trick for other project, recently

https://github.com/mmikhasenko/PhotoAmbiguities.jl/blob/master/src/experiments.jl#L10-L19