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Progress of Theoretical Physics

A Schematic Model of Large Amplitude Collective Motions with an Exact Classical Solution. II: Classical Analysis of Two-Level Pairing Hamiltonian

S. Iida

Published Feb 1, 1985• 3 citations
Publisher

Abstract

It is investigated how classical pictures of the whole phase space are connected with the original quantal system by using the two-level pairing model.Discontinuous shape changes of classical trajectories are shown to be related with the sudden changes of patterns in the transition strength between energy eigenstates.By introducing the collective and intrinsic coordinates representing pairing rotations and vibrations, respectively, these discontinuities can be interpreted as the breakdown of the excited pairing •rotational band scheme.

DOI
10.1143/ptp.73.374

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Progress of Theoretical Physics•Apr 1, 1982•0 citations
It is investigated how classical pictures of the whole phase space are connected with the original quantal system by using the two-level pairing model.Discontinuous shape changes of classical trajectories are shown to be related with the sudden changes of patterns in the transition strength between energy eigenstates.By introducing the collective and intrinsic coordinates representing pairing rotations and vibrations, respectively, these discontinuities can be interpreted as the breakdown of the excited pairing •rotational band scheme.
A model Hamiltonian describing a two-level system with a crossing plus a pairing force is investigated using technique of large-amplitude collective motion. The collective path, which is determined by the decoupling conditions, is found to be almost identical to the one in the Born-Oppenheimer approximation for the case of a strong pairing force. For the weak pairing case, the obtained path describes a diabatic dynamics of the system. PACS number(s): 21.60.-n, 36.40.Sx Nuclei and metallic clusters are finite fermionic many-body systems which support many kinds of collective motion. While properties of high-frequency vibrations (giant resonances) may be well reproduced by the random-phase approximation (RPA), some low-frequency vibrations exhibit an anharmonic nature that the RPA cannot describe. Shape coexistence and fission phenomena in nuclei and clusters also have such a large amplitude nature. In order to investigate these kinds of large amplitude collective motion, one would like to reduce the number of degrees of freedom to a few judiciously chosen collective coordinates. In order to determine the collective coordinates self-consistently, we use the theory of adiabatic
It is the purpose of the present paper to illustrate the quantum theory of dynamical collective subspace, which has been proposed in a previous paper to describe the large-amplitude collective motion, by using a simple four-level model with the pairing interactions. The quantum theory provides us a prescription to dynamically specify a set of "global" collective operators as well as the "optimum" dynamical collective subspace spanned by them. This prescription is applied to the simple model, and the results thus obtained are compared with exact numerical solutions. It is shown that the results by our theory are in excellent agreement with the exact solutions, not only in the low-lying collective states in the vicinity of the "vacuum" with the stable mean-field but also in the higher excited collective states far from it. Through the investigation, it is demonstrated that our theory is quite powerful in disclosing microscopic structure of transfiguration of collectivity near the critical point of the "phase transition". It is the essential statement derived by our theory that the conventional (local) collective operators based on the Tamm-Dancoff approximation or the random phase approximation should be replaced by relevant (global) collective operators with changeable microscopic structure depending on the amplitude of collective motion, in order to properly specify a set of collective excited states.
A microscopic theory is proposed to describe not only the pairing rotation, the pairing vibrations and the independent-particle motions, but also their mutual couplings in a consistent manner. With the aid of a BCS-type wave packet and a quasi-particle coherent state, the quantal system is translated into a corresponding classical one. First, we develop a canonical form in a classical image. This classical system obeys certain constraints, by which the double counting in the degrees of freedom is avoided. Then, the quantization is performed with the use of the Dirac theory of a canonical system with constraints. The original fermion operators themselves are expressed in terms of three types of degrees of freedom, which correspond to the pairing rotation, the pairing vibrations and the independent-quasi-particle motions, respectively.