Symbolic Evaluation of Expressions from Racah’s Algebra

Abstract

Based on the rotational symmetry of isolated quantum systems, Racah’s algebra plays a significant role in nuclear, atomic and molecular physics, and at several places elsewhere. For N-particle (quantum) systems, for example, this algebra helps carry out the integration over the angular coordinates analytically and, thus, to reduce them to systems with only N (radial) coordinates. However, the use of Racah’s algebra quickly leads to complex expressions, which are written in terms of generalized Clebsch–Gordan coefficients, Wigner n-j symbols, (tensor) spherical harmonics and/or rotation matrices. While the evaluation of these expressions is straightforward in principle, it often becomes laborious and prone to making errors in practice. We here expand Jac, the Jena Atomic Calculator, to facilitate the sum-rule evaluation of typical expressions from Racah’s algebra. A set of new and revised functions supports the simplification and subsequent use of such expressions in daily research work or as part of lengthy derivations. A few examples below show the recoupling of angular momenta and demonstrate how Jac can be readily applied to find compact expressions for further numerical studies. The present extension makes Jac a more flexible and powerful toolbox in order to deal with atomic and quantum many-particle systems.

Keywords

• atomic structure
• nuclear structure and reactions
• open-shell system
• Racah algebra
• recoupling coefficient
• rotation matrix