# Cleaning Up Simplification and Moving Into Coupled States

So, as I stated in my last post, the first thing I dealt with was fixing up the _cg_simp_add method by implementing pattern matching and move the logic for determining if the simplification can be performed and performing the simplification out of the _cg_simp_add method and developing a system that can easily be expanded to include additional simplifications. To do this, I created another method, _check_cg_simp, which takes various expression to determine if the sum can be simplified. Using Wild variables, the method takes an expression which is matched to the terms of the sum. The method uses a list to store the terms in the sum which can be simplified, so additional expressions are used to determine the length of the list and the index of the items that are matched. There are also additional parameters to handle the leading terms and the sign of the terms. There are still some issues with this method, as when there is more than 1 Clebsch-Gordan coefficient in the sum, then the leading term cannot be matched on the term.

In addition to the finishing of this component of the Clebsch-Gordan coefficient simplification, I have started into working on the coupled spin states and the methods to rewrite them in coupled and uncoupled bases. Coupled spin states are set by passing j1 and j2 parameters when creating the state, for example:

These states can be given in the uncoupled basis using the rewrite method and passing coupled=False, so:

This can also be done with a normal state and passing j1 and j2 parameters to the rewrite method, as:

How the coupled states will be handled by rewrite still needs to be addressed, but that will need some thinking and with another GSoC project doing a lot of changes to the represent function, it may take some coordination to get this and the TensorProducts of states and operators working.

Note that in the python expressions above, the states are given as uncoupled states written as tensor products. Uncoupled states will be written as TensorProduct’s of states, which will be extended later to spin operators, being written in the uncoupled basis as a TensorProduct. I’ve just started playing with the uncoupled states and the various methods that will be used to go from uncoupled to coupled states and I’ve been putting them in a separate TensorProductState class, which subclasses TensorProduct, which keeps all the spin logic separate from the main TensorProduct class, tho this will have to be expanded to include operators. Developing the logic for these uncoupled spin states will be the primary focus of this next week of coding.