A complete orthogonal expansion for the nuclear three-body problem: Part I. Rotational functions

Abstract

In the nuclear three-body problem, relativistic effects and noncentral forces mix states of different L and S but the same J. All sixteen of the states for J = 1 2 are exhibited in an eight-component vector notation which displays all structural details, and in a more compact spin-operator formulation. In addition, an abbreviated notation is introduced through which any state is characterized fully and clearly, showing its quartet or doublet character, its value of J and M J , the particular vector, dyadic, or polyadic used in its construction if it is not an S state, and thereby its parity and L value. With this notation many simple operators (spin, derivative, permutation) act directly on the pertinent labels. More complicated operators (tensor, spin-orbit) can all be reduced to the simple operators and a set of twelve “primary scalar operators.” The effect of each of the twelve primary operators on each of the sixteen rotational functions for J = 1 2 has been worked out and is given as a set of twelve 16-by-16 matrices which can be combined by matrix multiplication to give the effect of the more complicated secondary operators. The procedure to be followed for J = 3 2 and higher angular momenta is pointed out. Isospin functions are introduced, and the group-theoretical properties of the combined spin and isospin functions are examined.