Group theory of harmonic oscillators: (V). The harmonic oscillator shell model

Abstract

In the present series of papers, we have dealt with the explicit construction of states for a system of n particles in a harmonic oscillator common potential. These states were classified by the irreducible representations (IR) of the symmetry group of the harmonic oscillator, i.e. the unitary group of three dimensionsU 3 and its rotational subgroupR 3. Furthermore, they are also characterized by the IR of the chain of groups U n ⊃ S n where the latter groups are respectively the unitary and symmetric group associated with the n particles. In this paper, we show that the shell structure of the n-particle states in a harmonic oscillator potential can be obtained when we characterize further the states by the IR of a group K n . This group is the semi-direct product of the group of n-dimensional diagonal unitary matrices A n and of the symmetric group, i.e. K n = A n ∧ S n . Using these chains of groups, we construct explicitly shell-model states in the harmonic oscillator potential and from them obtain the fractional parentage coefficients (fpc) in terms of the Wigner coefficients of the groups in the chain. An important result is that the fpc become much simpler when the IR ofU 3 are of one and two rows than when they have the most general form of three rows. Thus, for example, for particles in the 2d-1d shell of the harmonic oscillator, the fpc corresponding to a two-row IR ofU 3 become a product of the familiar fpc in the p-shell tabulated by Jahn and a reduced Wigner coefficient ofU 3 in theU 3⊃R 3 chain, for which many programs and tables are available. The present analysis may be useful in the many cases of physical interest when one needs the fpc in a multishell state in the harmonic oscillator potential with the IR ofU 3 not exceeding two rows.