This repository contains programmes to calculate the non-zero Clebsch--Gordan coefficients of a system with given J, j1, j2 value using recursion relations.
The coupled states can be expended in the following format:
The coefficient only has a non-zero value if
$$\hat{J}{\pm} = \hat{J}^{(1)}{\pm} \otimes \hat{I}^{(2)} + \hat{I}^{(1)} \otimes \hat{J}^{(2)}_{\pm}$$
The operator's effect on a state is
$$\hat{J}{\pm}\ket{j,m} = \hbar C{\pm}(j,m)\ket{j,(m+1)}$$
where the ladder coefficient is given by
The get the recursion relations the total angular momentum raising and lowering operators have to be applied to left hand / right hand side.
To get the recursion relation the indexes have to be changed to be able to extract the
Taking the upper sign and the condition that
In the Condon–Shortley phase convention, one adds the constraint that:
Using this recursion relation and the constraint starting from the