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Communications in Theoretical Physics

An SU(1,1) Boson Realization Involved in Solving the Three Coupled Harmonic Oscillators' Problem

Hong-Yi Fan

Published Sep 1, 1991• 1 citation
Publisher

Abstract

We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. The unitary operator U, which is found to be able to transform the Fock space of a three–dimensional isotropic harmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum–mechanical rotation in three–mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.

DOI
10.1088/0253-6102/16/2/245

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We find that a three-mode boson realization of the SU(1,1) Lie algebra is involved in solving the three coupled harmonic oscillators' problem. The unitary operator U, which is found to be able to transform the Fock space of a three–dimensional isotropic harmonic oscillator into the space in which the Hamiltonian of three coupled oscillators is diagonized, is further decomposed as a quantum–mechanical rotation in three–mode Hilbert space followed by an SU(1,1) squeezing transformation. The coordinate representation of this SU(1,1) unitary operator is obtained.
We have constructed the multimode bosonic realization of the su(1,1) Lie algebra, on the basis of which the su(1,1) generalized coherent state in the multimode Fock space is derived. It is shown that the multimode coherent state is, in fact, the generalized multimode squeezed vacuum state discussed by Lo and Sollie [Phys. Rev. A 47, 733 (1993)], and that the unitary displacement operator can be identified as the generalized multimode squeezing operator. Hence, similar to the single- and two-mode cases, our multimode bosonic realization of the su(1,1) Lie algebra has immediate relevance to the squeezing properties of boson fields.
In this paper we have constructed the multiparticle ( q, p )-realization of the su (1, 1) Lie algebra, and studied its relation to the multimode bosonic realization and squeezing properties. We believe that this multiparticle realization should be very useful in the study of a system of coupled general time-dependent quantum oscillators whose Hamiltonian can be written as a linear combination of the three su (1, 1) generators; in particular, it is more capable to give a vivid picture of the system in the configuration space.
Based on the rotation transformation in the phase space and the technique of integration within an ordered product of operators,the coherent state representations of the rotation operator in quantum phase space,the Fourier transform operator and the parity operator are derived. Then the three-mode rotation operator is introduced.By the three-mode rotation operator,the Hamilton of the three-dimensional coordinate-momentum coupled harmonic oscillators is simply decoupled. Then its exact energy spectrums and eigenstates are obtained.
Two new closely related boson realizations of the Heisenberg–Weyl algebra are derived that are directly equivalent to the introduction of spherical coordinates for the isotropic three-dimensional harmonic oscillator. This provides another way to obtain matrix elements of all physical operators connecting angular-momentum eigenstates, which are represented here by simple products of boson operators acting on the vacuum. Two of these bosons carry the angular-momentum quantum numbers while the third is a novel ‘‘radial boson.’’ In this approach, one begins by finding the nonunitary Dyson and then the unitarized Holstein–Primakoff realizations for the spectrum-generating SU(1,1)×SO(3) algebra of the oscillator. The corresponding mappings of the Heisenberg–Weyl generators are then obtained from their tensor properties under SU(1,1)×SO(3). The Dyson map of the Heisenberg–Weyl generators is not uniquely determined in this way, but the Holstein–Primakoff map is nevertheless unique. The possibility exists for generalization to isotropic oscillators of higher dimension, which are of interest in connection with nuclear collective motion.