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Progress of Theoretical Physics

Description of Thermal Effects in su(1, 1)-Algebraic Model Derived from Nuclear su(2)-Model

Y. Tsue, A. Kuriyama, Masatoshi Yamamura

Published Jun 1, 1995• 0 citations
Publisher

Abstract

A time-dependent variational approach with a mixed-mode coherent state is applied to su(1, 1)-algebraic models derived from nuclear su(2)-models, which are the single orbit shell model with pairing correlation and the Lipkin model. The equations of motion are derived in a canonical form similar to that in the time-dependent Hartree-Fock theory, and the solutions are derived by the use of conserved action variables together with an appropriate canonical transformation. It is shown that thermal effects appear in the form of the bose distribution function. Also, it is found that the energy concentrated in a pure state at initial time dissipates to the other states in a manner similar to the case of the damped-and-amplified harmonic oscillator model investigated in our previous work.

DOI
10.1143/ptp.93.1037

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A time-dependent variational approach with a mixed-mode coherent state is applied to su(1, 1)-algebraic models derived from nuclear su(2)-models, which are the single orbit shell model with pairing correlation and the Lipkin model. The equations of motion are derived in a canonical form similar to that in the time-dependent Hartree-Fock theory, and the solutions are derived by the use of conserved action variables together with an appropriate canonical transformation. It is shown that thermal effects appear in the form of the bose distribution function. Also, it is found that the energy concentrated in a pure state at initial time dissipates to the other states in a manner similar to the case of the damped-and-amplified harmonic oscillator model investigated in our previous work.
A time-dependent variational approach with a mixed-mode coherent state is applied to su(1, 1)-algebraic models derived from nuclear su(2)-models, which are the single orbit shell model with pairing correlation and the Lipkin model. The equations of motion are derived in a canonical form similar to that in the time-dependent Hartree-Fock theory, and the solutions are derived by the use of conserved action variables together with an appropriate canonical transformation. It is shown that thermal effects appear in the form of the bose distribution function. Also, it is found that the energy concentrated in a pure state at initial time dissipates to the other states in a manner similar to the case of the damped-and-amplified harmonic oscillator model investigated in our previous work.
The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.
The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.
The su(2)-algebraic model interacting with an environment is investigated from the viewpoint of treating a dissipative system. By using a time-dependent variational approach with a coherent state and with the help of the canonicity condition, the time-evolution of this quantum many-body system is described in terms of the canonical equations of motion in the classical mechanics. Then, it is shown that the su(1, 1)-algebra plays an essential role in treating this model. An exact solution with appropriate initial conditions is obtained in terms of Jacobi's elliptic function. The implications for the dissipative process are discussed.
In order to develop a previously proposed method to investigate thermal effects in quantum many-particle systems, basic ideas are formulated in more general systems obeying su(2)-algebra. It is shown that Hamiltonians of a single orbit shell model with pairing correlation, the Lipkin model, a symmetric rigid rotor model and an asymmetric rigid rotor model have almost the same structure, that is algebraic structure of su(1, 1), by the use of the Holstein-Primakoff boson representation for su(2)-algebra and so-called phase-space doubling introduced to study thermal phenomena. It is also discussed that the interaction term between intrinsic system and external environment approximately represents the interesting physical process in each system at finite temperature.
Basic idea presented in Parts (I)-(III) for the deformed boson scheme is applied to the case of the su(2)- and su(1,1)-algebras for describing many-body systems consisting of four kinds of boson operators. A possible form of the coherent state given by the present authors is generalized and the su(2)_q- and the su(1,1)_q-algebras are obtained in the form expressed in terms of four kinds of boson operators. As an illustrative example of the application, the framework for describing thermal effects observed in two-level shell model under the pairing correlation is proposed.