On the Coherent and the Squeezed State in the su(2)-Boson Model

Abstract

719With the use of the coherent and the squeezed state combined with canonicity condition in the time· dependent Hartree-Fock theory, quantal fluctuation appearing in the su(2)·boson model, which is well known as Schwinger boson representation, is investigated. Especially, the attention is focussed on the least quantal effect: 1) the minimal uncertainty and 2) the quantal fluctuation energy. The Lipkin model, an example obeying the su(2)-algebra, is adopted for demonstrating the basic idea. § 1. IntroductionIt is well known that the time-dependent Hartree-Fock (TDHF) theory has been recognized as a quite powerful method for investigating dynamics of many-fermion system. This method contains two characteristic aspects. First is related with obtaining a classical counterpart of many-fermion system. With the aid of Slater determinant parametrized by classical canonical variables, we can get a classical counterpart of the original quantal system. Then, under an appropriate requantization procedure, the counterpart comes back to the original system in a disguised form. Second is closely related with a problem discussed in this paper. The TDHF theory gives us a method for approximate description of time evolution of many-fermion system. This method starts from a chosen Slater determinant lsi> as a trial wave packet for the time-dependent variational procedure. In the canonical form, the wave packet contains certain parameters which obey canonicity condition and, in this sense, we can regard them as canonical variables. Under the variational procedure, we can prove that their time-dependence is determined by solving the Hamilton equation of motion for these variables. )This statement is quite interesting, because the time evolution of the quantal system is reduced to that in the classical mechanics. The Hamiltonian (H)SI, which prescribes the time evolution, can be given as the expectation value of the original Hamiltonian Ii for the wave packet, lsi>, i.e., (H)SI =<slililsl>. In the TDHF theory, anyone-body fermion operator Oi corresponds to the expectation value of Oi for the wave packet, (Oi)81( =<sll Oilsl», which is a function of the canonical variables. Therefore, we can regard (Oi)81 as the classical counterpart of Oi, i.e., the classical image of the quantal system under investigation is given. However, there exists one problem. Generally, for the wave packet 1st>,