Matrix elements and their selection rules from ladder operator considerations

Within the Schrodinger–Infeld–Hull factorization framework it is shown that, by introducing a parameter e in the quantization condition, that is, e(j–|m|)=integer ≥ 0, and, thus, considering “symmetrized” ladder operators, one can use the same formulas to handle both class I (e = +1) and class II (e = −1) problems. Starting from this unified point of view, after building up the associated angular momentum operators and their e-dependent eigenfunctions, one unique closed-form expression of the coupling coefficients is obtained. This expression embodies many sparse and known previous results, without being more intricate than any of them. The basic material, allowing the application of a Wignera–Eckart theorem to matrix elements of an operator on the basis of eigenfunctions of factorizable equations, and a quick determination of the associated selection rules are given. Some examples are treated as an illustration.