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Physical Review C

Concept of Transition - Operator Boson and Its Application to an Exactly Soluble Model

R. M. Dreizler, Abraham Klein

Published Feb 1, 1973• 10 citations
Publisher

Abstract

In phenomenological descriptions of nuclear vibrations a close connection is implied or expressed between the most-collective transition operator and the harmonic-oscillator (boson) variables which are introduced to describe the collective motion. With the help of a widely used example - the monopole vibration model of Meshkov, Lipkin, and Glick - it is shown how this relationship suggests, for the microscopic case, a new and flexible class of boson representations. As shown, these are capable of yielding accurate descriptions of the system not only in the extremes of weak and strong coupling, but in the usually more inaccessible intermediate regions as well.

DOI
10.1103/physrevc.7.512

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In phenomenological descriptions of nuclear vibrations a close connection is implied or expressed between the most-collective transition operator and the harmonic-oscillator (boson) variables which are introduced to describe the collective motion. With the help of a widely used example - the monopole vibration model of Meshkov, Lipkin, and Glick - it is shown how this relationship suggests, for the microscopic case, a new and flexible class of boson representations. As shown, these are capable of yielding accurate descriptions of the system not only in the extremes of weak and strong coupling, but in the usually more inaccessible intermediate regions as well.
In a previous paper the concept of an ideal collective coordinate was introduced as the basis for a phenomenological theory of nuclear collective motion. The application of this idea to the microscopic domain is illustrated by the study of a simple two-level model with monopole-monopole interactions, previously considered by Lipkin and collaborators. The complete scheme of calculation which is presented is seen as an alternative to the boson expansion methods currently in fashion for the study of anharmonic vibrations. The new scheme should, however, be applicable to a more extended range of problems such as the adiabatic limit for vibrations and the general problem of rotation-vibration interactions.
In a generic quantum mechanical description of a molecule interacting with its thermal environment, the molecule is represented as a few level system (in the simplest description just two, for example, ground and excited states) and the environment is often modeled as a bath of harmonic oscillators. The resulting theoretical framework is known as the spin–boson model, a term that seems to have emerged in the Kondo problem literature (which deals with the behavior of magnetic impurities in metals) during the 1960s, but is now used in a much broader context. Indeed, it has become one of the central models of theoretical physics, with applications in physics, chemistry, and biology that range far beyond the subject of this book. Transitions between molecular electronic states coupled to nuclear vibrations, environmental phonons, and photon modes of the radiation field fall within this class of problems. The present chapter discusses this model and some of its mathematical implications. The reader may note that some of the subjects discussed in Chapter 9 are reiterated here in this more general framework. In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field and provides good starting points for describing nuclear motions in molecules and in solid environments. It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach. In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models: Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property! We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9.
The boson expansion method was developed with the purpose of continuously following even nuclear systems near their ground states, from the region of slightly anharmonic vibrations through the transitional region formed by a nonsharp phase transition to a region of permanent distortion characterized by the occurrence of rotational spectra. However, as the boson quantum was defined in the extreme vibrational limit, it was not possible to obtain quantitatively correct spectra in the transitional or deformed region. Believing that the major cause for the lack of success was the disregard of the self- consistent change of the basic degree of freedom, a simple variational method to be used with the boson expansion technique is presented. Its ability to cover the vibration-to-rotation transition is demonstrated by comparison with an exact solution for a four-level model with pairing force. Further the coverage of a quadrupole phase transition is shown for a system interacting via pairing-Plus- quadrupole forces. A successful calculation of static quadrupole moments of Cd and Te isotopes is presented. Finally the approach in general situations with more complicated forces is discussed. (auth)
It is shown that nuclear matter described by quasispin or pseudospin operators may exhibit irreversible behavior, in the sense of modern kinetic theory. With the help of the group-contraction technique, a harmonic oscillator can be associated to the many-body system and it is seen that in the standard Lipkin-Meshkov-Glick model the ground-state phase transition gives rise to a significant change in the spectral properties of the characteristic operators. This fact makes room for the introduction of the microscopic operators of irreversible dynamics. Their meaning and the consequences of their existence are examined and it is seen that the time evolution of the quasispin arrangement closely resembles the approach to equilibrium of a macroscopic system.NUCLEAR STRUCTURE Operators of irreversible Hamiltonian dynamics; quasispin nuclear matter; group contraction, Lipkin-Meshkov-Glick model; ground-state phase transition; entropy; irreversible approach to equilibrium.
The Beliaev-Zelivinsky boson expansion and particle-vibration coupling methods are studied in comparison with an exact treatment of the monopole model. Two of the three boson expansions considered were valid for all interaction strengths. A third expansion and the particle-vibration coupling method were found to be equivalent to perturbation theory. The difficulty associated with the overcompleteness of the basis in the particle-vibration coupling method was resolved both in the computation of the energies and the matrix element of the monopole specific operator.
The three-dimensional harmonic oscillator plays a central role in nuclear physics. It provides the underlying structure of the independent-particle shell model and gives rise to the dynamical group structures on which models of nuclear collective motion are based. It is shown that the three-dimensional harmonic oscillator features a rich variety of coherent states, including vibrations of the monopole, dipole, and quadrupole types, and rotations of the rigid flow, vortex flow, and irrotational flow types. Nuclear collective states exhibit all of these flows. It is also shown that the coherent state representations, which have their origins in applications to the dynamical groups of the simple harmonic oscillator, can be extended to vector coherent state representations with a much wider range of applicability. As a result, coherent state theory and vector coherent state theory become powerful tools in the application of algebraic methods in physics.