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Progress of Theoretical Physics

On the Classical Interpretation of Schwinger Boson Representation for the Quantized Rotator

H. Ichihashi, M. Yamamura

Published Sep 1, 1978• 4 citations
Publisher

Abstract

A classical correspondence of Schwinger boson representation for the quantized rotator is investigated by the use of a coherent state. As one of the applications, the asymmetric rotator in the high spin limit is described, and the results are consistent with those given by Bohr and Mottelson.
DOI
10.1143/ptp.60.753

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With the aid of the correspondence between commutator and Poisson bracket, a set of canonical variables is derived from the Schwinger boson representation for the quantized rotator. By the use of these variables, Euler's equations of motion are given through Hamilton's equations of motion in the case of the rotator. Further, wobbling motion is described in the classical sense. Finally, for the purpose of applying the ATDHB theory to a microscopic description of collective roation, a basic idea is given.
Various definitions of the coherent states of the angular momentum are shown to be special cases of the oscillator-like coherent states of the groups SU(2) and SO(3) obtained by Mikhailov on the basis of a generalized Schwinger boson realization of the angular momentum algebra. This realization is then generalized to that of the angular momentum algebra of an asymmetric top by means of a transformation from the Euler angles to the Cayley–Klein parameters. The oscillator-like coherent states of an asymmetric top, analogous to those of Mikhailov, are then constructed. It is, then, shown that Janssen's and Mostowski's definitions of the coherent states of a top are special cases of these.
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.
Using the bosonic coherent state representation and the Schwinger bosonic operator realization of angular momentum we find the formula for the quantum Hamiltonian H = ia † i ij U † jl al for SU(2) rotation U , in this way we further specify the angular velocity , where is the Pauli matrix. Though the spin as a quantum observable has no classical correspondence, we may still mimic it as a rigid body rotation characterized by 3 Euler angles, and calculate its Pseudo-classical rotational partition function of spin one-half.