Symbolic-Numeric Algorithms for Computing Orthonormal Bases of SU(3) Group for Orbital Angular Momentum

We have developed symbolic-numeric algorithms implemented in the Wolfram Mathematica to compute the orthonormal canonical Gel’fand–Tseitlin (G-T), non-canonical Bargmann-Moshinsky (B-M) and Elliott (E) bases of irreducible representations SU(3) ⊃ SO(3) ⊃ SO(2) group for a given orbital of angular momentum. The algorithms resolve the missing label problem by solving eigenvalue problem for the “labeling” B-M operator X^{(3 )}. The effective numeric algorithm for construction of the G-T basis provides a unique capability to perform large scale calculations even with 8 byte real numbers. The algorithms for the construction of B-M and E bases implemented very fast modified Gramm–Schmidt orthonormalization procedure. In B-M basis, a very effective formula for calculation of the matrix X^{(3 )} is derived by graphical method. The implemented algorithm for construction of the B-M basis makes it possible to perform large scale exact as well as arbitrary precision calculations. The algorithm for the construction of the E basis resolves the missing label problem by calculation of the matrix X^{(3 )} in an orthogonal basis from this matrix previously built in non-orthogonal basis. The implementation of this algorithm provides large scale calculations with arbitrary precision.