For some time now I use a nice transformation for double integrals. Okay in my case they are not exactly double, but 6th order integrals, nonetheless it is the same. The problem I'm trying to solve is a double integral over the radius vectors of two particles, so it is 3rd order by the coordinates of a single particle (I have two particles). You would ask why I need to do that ... well I have to simplify otherwise very heavy calculation. This double volume integral is calculated for permutations (which are quite a lot) of the quantum numbers of the particle system. For convenience, because of the problem symmetries, I work in cylindrical coordinate system. Up until now I've managed to separate the linear part, which is pretty straightforward, and was able to factorize the double integration through the transformation of coordinates mentioned.
Here it goes the simple but elegant method:
Say we have a double volume integral over two variables x and y. And they are used as sums and differences throughout the function we are integrating. It is obvious we would want to separate them in some manner, so we would calculate two one dimensional integrals, instead of one two dimensional. It's not hard to imagine that:
would do the trick. But there is also a kick in all that. I've realized yesterday that this is nothing more than rotating the integration area ... fun huh?
This transformation being a orthogonal one (rotation at fixed angle) means it has a constant Jacobian. It even gets better. If we normalize the new variables (divide by square root of 2) the change preserves volume:
This last change of variables is called Moshinsky transformation. There is a little bit more on the subject but goes out of the scope of this post, maybe someday I'll write down the Gogny separation method as well.
