A computationally tractable version of the collective model

Abstract

A computationally tractable version of the Bohr-Mottelson collective model is presented which makes it possible to diagonalize realistic collective models and obtain convergent results in relatively small appropriately chosen subspaces of the collective model Hilbert space. Special features of the proposed model is that it makes use of the beta wave functions given analytically by the softened-beta version of the Wilets-Jean model, proposed by Elliott et al., and a simple algorithm for computing SO(5) ⊃ SO(3) spherical harmonics. The latter has much in common with the methods of Chacon, Moshinsky, and Sharp but is conceptually and computationally simpler. Results are presented for collective models ranging from the sherical vibrator to the Wilets-Jean and axially symmetric rotor-vibrator models.

An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces

“…[19]. The low-energy level spectrum of the Hamiltonian (123), calculated with a large value of α [16], is shown in Fig. 8.…”

“…It turns out that, for many purposes, other matrix elements are needed and are also needed for higher-dimensional spaces. For example, the nuclear collective model is defined on a five-dimensional space [15,16,17]. And, while harmonic oscillator bases are appropriate for spherical vibrational systems, they are not the most appropriate for rotational-vibrational systems such as diatomic molecules and non-spherical nuclei.…”

Construction of spherical harmonics and Clebsch–Gordan coefficients

“…, (38) where Φ (2) 1 is the tensor with highest-weight component Φ 1 (γ, Ω), etc., and L = 2n 1 +2n 2 +3n 4 (Sec. 2.3).…”

“…For higher L, a larger number of F K terms are involved in each calculation. Furthermore, the basis monomials tend to involve higher powers of those generating functions which carry angular momentum (Φ (2) 1 , Φ (2) 2 , and Φ…”

The O(6) description of the nuclear rotation

“…The value of determining the explicit relationship between the two models is enhanced by the recent development of an algebraic version of the BM [7,8] which makes it possible to execute a full range of BM calculations in bases that range continuously from those of a spherical vibrator to a beta-vibrational Wilets-Jean limit. Our objective in pursuing the relationships between these two models is to discover what can be learned about one model from the other and the extent to which the results of one model can be reproduced by the other.…”