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Progress of Theoretical Physics

Minimal Uncertainty in Quantal System Obeying the su(2)-Algebra

M. Yamamura, A. Kuriyama

Published Oct 1, 1992• 1 citation
Publisher

Abstract

A viewpoint for describing time development of wave packet with the least quantal effect is proposed in the case of system obeying the su(2)-algebra, i.e., the quasi-spin. The basic idea is a possible extension of a method developed by the present authors, together with Tsue and Fujiwara, for one-dimensional system. Concerning the least quantal effect, the uncertainty relations coming from the non-commutability among the components of the quasi-spin are investigated and a certain condition for the least quantal effect in the quasi-spin is presented.

DOI
10.1143/ptp/88.4.711

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A viewpoint for describing time development of wave packet with the least quantal effect is proposed in the case of system obeying the su(2)-algebra, i.e., the quasi-spin. The basic idea is a possible extension of a method developed by the present authors, together with Tsue and Fujiwara, for one-dimensional system. Concerning the least quantal effect, the uncertainty relations coming from the non-commutability among the components of the quasi-spin are investigated and a certain condition for the least quantal effect in the quasi-spin is presented.
A viewpoint for describing time development of wave packet with the least quantal effect is proposed in the case of system obeying the su(2)-algebra, i.e., the quasi-spin. The basic idea is a possible extension of a method developed by the present authors, together with Tsue and Fujiwara, for one-dimensional system. Concerning the least quantal effect, the uncertainty relations coming from the non-commutability among the components of the quasi-spin are investigated and a certain condition for the least quantal effect in the quasi-spin is presented.
Minimal quantal effect in many-nucleon system which is correlated through pairing interaction is investigated in the frame of the quasi-spin coherent and the quasi-spin squeezed state. Basic idea can be found in the case of the su(2)-boson model reported by the present authors. For simplicity, the single-j shell model is adopted. The quasi-spin coherent state, which is well known as the BCS ground-state, shows the minimal uncertainty for quasi-spin operators under a rather restricted condition. In order to relax this restriction, the quasi-spin squeezed state is introduced and it is shown that, with the help of a certain phase angle parameter, the minimal uncertainty for quasi-spin operators can be controlled.
In terms of the boson coherent state and a possible extension, three forms of wave packets are presented in the Schwinger boson representation. Each wave packet gives classical counterpart for a certain set of operators, for example, three components of the su(2)-spin. Furthermore, uncertainty relations which come from the boson coherent state and its extension are examined. The content discussed in this paper is, in a certain sense, an extension of that discussed in a paper presented recently by the present authors.
In quantum mechanics with minimal length uncertainty relations the Heisenberg-Weyl algebra of the one-dimensional harmonic oscillator is a deformed SU(1,1) algebra. The eigenvalues and eigenstates are constructed algebraically and they form the infinite-dimensional representation of the deformed SU(1,1) algebra. Our construction is independent of prior knowledge of the exact solution of the Schrödinger equation of the model. The approach can be generalized to the DDD-dimensional oscillator with non-commuting coordinates.