Permutation-symmetric three-particle hyper-spherical harmonics based on the S 3 ? SO(3) rot ? O(2)?SO(3) rot ? U(3)?S 2 ? O(6) subgroup chain

摘要

We construct the three-body permutation symmetric hyperspherical harmonics to be used in the non-relativistic three-body Schr?dinger equation in three spatial dimensions (3D). We label the state vectors according to the S 3 ? S O ( 3 ) r o t ? O ( 2 ) ? S O ( 3 ) r o t ? U ( 3 ) ? S 2 ? O ( 6 ) subgroup chain, where S 3 is the three-body permutation group and S 2 is its two element subgroup containing transposition of first two particles, O ( 2 ) is the “democracy transformation”, or “kinematic rotation” group for three particles; S O ( 3 ) r o t is the 3D rotation group, and U ( 3 ) , O ( 6 ) are the usual Lie groups. We discuss the good quantum numbers implied by the above chain of algebras, as well as their relation to the S 3 permutation properties of the harmonics, particularly in view of the S O ( 3 ) r o t ? S U ( 3 ) degeneracy. We provide a definite, practically implementable algorithm for the calculation of harmonics with arbitrary finite integer values of the hyper angular momentum K, and show an explicit example of this construction in a specific case with degeneracy, as well as tables of K ≤ 6 harmonics. All harmonics are expressed as homogeneous polynomials in the Jacobi vectors ( λ , ρ ) with coefficients given as algebraic numbers unless the “operator method” is chosen for the lifting of the S O ( 3 ) r o t ? S U ( 3 ) multiplicity and the dimension of the degenerate subspace is greater than four – in which case one must resort to numerical diagonalization; the latter condition is not met by any K ≤ 15 harmonic, or by any L ≤ 7 harmonic with arbitrary K. We also calculate a certain type of matrix elements (the Gaunt integrals of products of three harmonics) in two ways: 1) by explicit evaluation of integrals and 2) by reduction to known S U ( 3 ) Clebsch–Gordan coefficients. In this way we complete the calculation of the ingredients sufficient for the solution to the quantum-mechanical three-body bound state problem.