Sean Vig's 2011 GSOC Project

Developing Wigner-3nj Symbols in SymPy

Official GSoC Start

This week marks the official start of the Google Summer of Code. While I started getting my feet wet last week after finishing the last of my finals and grading, the bulk of the work has just started turning out. I’ll quick cover what I have from this last week and what I’m looking to get working this week.

Before the start, I worked out expanding the functionality of the currently implemented x/y/z bases, which work I have here. The previous implementation only allowed for evaluating inner products between states in the same basis and representing the states in the Jz basis and then only with j=½ states. Using the Wigner D-function, implemented with the Rotation class, I implemented represent to go between the x/y/z bases for any j values. Both _eval_innerproduct and _rewrite_as were then created to take advantage of the represent function to extend the functionality of the inner product and to implement rewrite between any bases and any arbitrary j values.

This seems like this is some documentation and tests away from being pushed, but there is something buggy with the Rotation.d function, implementing the Wigner small d-matrix. I noticed when trying to do

>>> qapply(JzBra(1,1)*JzKet(1,1).rewrite('Jx')

and I wasn’t getting the right answer. As it turns out, the Rotation.d function, which uses Varshalovich 4.3.2 Eq 7, does not give the right answer for Rotation.d(1,1,0,-pi/2) or Rotation.d(1,0,1,pi/2). Namely, there is something wrong with the equation that doesn’t change the sign of the matrix element when reversing the sign of the beta Euler angle. Running all four differential representations given by Varshalovich for the small d matrix, Eq 7-10, give the wrong result, so the derivations of these will need to be checked to fix this. I have a bug report up here.

As for what I will be implementing this week, I already have the basics of the Wigner3j and CG class implemented, the work for this going up here. This includes creating the objects, some of the printing functionality and numerical evaluation of the elements using the functions. The meat of the class that I’m currently working on is the cg_simp function, which will simplify expressions of Clebsch-Gordan coefficients. I currently have one case working, that is Sum(CG(a,alpha,0,0,a,alpha),(alpha,-a,a)) == 2a+1 which is Varshalovich 8.7.1 Eq 1. There are still some things to smooth out with the implementation, but I should have that worked out a bit better, in addition to some more simplifications by the end of the week.

That’s all I have for now, watch for updates within the week as to what I’ve gotten done and what I have yet to do.